Recent Advances in Experimental Mechanics. In Honor of Isaac M. Daniel

Recent Advances in Experimental Mechanics

Table of Contents Editor’s Preface

v

Biography of Isaac M. Daniel

xv

List of Publications by Isaac M. Daniel

xxi

List of Contributors

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1. Mechanical Behavior of Materials Hopkinson Techniques for Dynamic Triaxial Compression Tests J. Rome, J. Isaacs and S. Nemat-Nasser

3

Asymptotic Scaling of Gradient Theory of Micro-Scale Plasticity of Metals

13

The Role of Pressure in the Behavior of Time-Dependent Materials T. Prodan and I. Emri

19

High Strain Rate Testing of Sandwich Core Materials M. Vural and G. Ravichandran

31

Development of a Shear Test for Low Modulus Foam Materials A. K. Roy and J. D. Camping

43

Indentation of a PVC Cellular Foam E. E. Gdoutos and J. L. A bot

55

Nanomanipulation and Characterization of Individual Carbon Nanotubes R. S. Ruoff, M.-F. Yu and H. Rohrs

65

Quasi-Static and Dynamic Torsion Testing of Ceramic Micro and Nano-Structured Coatings Using Speckle Photography F. Barthelat, K. Malukhin and H. Espinosa

75

2. Composite Materials Measured Response: State Variables for Composite Materials K. Reifsnider and M. Pastor

87

On the Modeling of the Mechanical Properties of Composite Materials at High Strain Rates J. R. Vinson and S. Xiao

99

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Damage Quantification in Metal Matrix Composites G. Z. Voyiadjis, A. R. Venson, and R. K. Abu-Alrub

109

Study of Damage in Particulate Composites C. A. Sciammarella and F. M. Sciammarella

121

Hygric Characterization of Composite Laminates S.-C. Wooh and C.-L Tsai

133

Pneumatic Behavior of Composite Materials C.-L. Tsai and Y.-S. Tsai

145

The Effect of Specimen Size on the Compressive Strength of Carbon Fibre-Epoxy Laminates C. Soutis and J. Lee

153

Interfacial Strength and Toughness Characterization Using a Novel Test Specimen G. P. Tandon, R. Y. Kim and V. T. Bechel

163

A Model for the Accurate Prediction of the Residual Strength after Damage Due to Impact and Erosion of FRPs G. C. Papanicolaou, G. Samoilis, S. Giannis, N.-M. Barkoula and J. Karger-Kocsis

175

3. Fracture and Fatigue The Origin and Inception of Fatigue in Steel – A Probabilistic Model S. A. Guralnick and J. Mohammadi

187

Fatigue Damage Tolerant Analysis Using the Fatigue Damage Map C. A. Rodopoulos and J. R. Yates

197

Crack Growth Behavior and SIF’s as Observed by Optical Methods C. W. Smith

209

A Model for Failure Initiation in Ductile Materials J. Zuo, M. A. Sutton and X. Deng

217

Crack Paths in Adhesive Bonds L. Banks-Sills and J. Schwartz

225

Experimental Determination of Fracture Parameters for Predicting Crack Growth in Viscoelastic Polymers D. H. Allen and J. J. Williams

235

Failure of Spot Weld: A Competition Between Crack Mechanics and Plastic Collapse Y. J. Chao

245

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Investigating the Effects of Specimen Thickness and Pressure on the Crack Growth Behavior of a Particulate Composite Material C. T. Liu

257

Dynamic Fracture Experiments Using Point Impact D. Rittel

267

Experimental and Numerical Investigation of Shear-Dominated Intersonic Crack Growth and Friction in Unidirectional Composites A. J. Rosakis, C. Yu, M. Ortiz, D. Coker and A. Pandolfi

275

4. Optical Methods Moiré Interferometry - Past, Present and Future D. Post

291

Optical Fibre Bragg Grating Sensors in Experimental Mechanics of Composite Laminated Plates J. Botsis, F. Bosia, M. Facchini, Th. Gmür

303

Optoelectronic Displacement Measurement Method for Rotating Disks C. E. Bakis, B. J. Haldeman and R. P. Emerson

315

Deformation Measurement of Sheet Metal Forming Using Photogrammetry K. Andresen

325

Fracture Processes of Quasi-Brittle Materials Studied with Digital Image Correlation J. S. Lawler and S. P. Shah

335

On the Use of Different Wavelengths to Digitally Determine the Isochromatic Fringe Order T. Y. Chen, Y. C. Chou, H. L. Lee and S. H. Tsao

345

The Application of Speckle Metrology to Heart Mechanics G. R. Gaudette, E. U. Azeloglu, J. Todaro, L. Keene, I. B. Krukenkamp and F. P. Chiang

353

5. Non-Destructive Evaluation Line Focus Acoustic Microscopy for Thin-Film Measurements Z. Guo and J. D. Achenbach

367

Recent Advances in Acoustography-Based NDE J. S. Sandhu and H. Wang

381

Experimental Limitations to Guided Wave Generation in Elastic Materials D. A. Sotiropoulos and E. Babatsouli

389

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Nondestructive Testing Using Shearography M.Y.Y. Hung

397

Theoretical and Experimental Study of Laser-Ultrasonic Signal Characteristics Enhanced by Wetting the Surface S.-C. Wooh

409

Defect Detection by the Scattering Analysis of Flexural Waves P. Fromme and M. B. Sayir

421

Evaluation of Fiber Waviness in Thick Composites by Ultrasonic Test H.-J. Chun

433

Development of a Dry-Contact Ultrasonic Technique and its Application to NDE of IC Packages H. Tohmyoh and M. Saka

443

6. Neutron Diffraction and Synchrotron Radiation Methods High-Resolution Neutron Diffraction Techniques for Strain Scanning P. Mikula, M. Vrana, P. Lukas and V. Wagner

457

Draft Standard for the Measurement of Residual Stresses by Neutron Diffraction G. A. Webster, A. G. Youtsos, C. Ohms and R. C. Wimpory

467

Microstresses Determined by Neutron Diffraction and Self-Consistent Model A. Baczmanski, C. Braham, R. Levy-Tubiana, A. Lodini and K. Wierzbanowski

477

Residual Stress Measurements at the Metal/Ceramic Interface Using Modelling of Neutron Diffraction Spectrometer A. Carrado, J.-M. Sprauel, L. Barrallier and A. Lodini

487

Elastoplastic Deformation of Two Phase Steels Studied by Neutron Diffraction and Self-Consistent Modelling M. R. Daymond, H. G. Priesmeyer and A. M. Korsunsky

495

Residual Stresses and Elastic Constants in Thermal Deposits T. Gnäupel-Herold, H. J. Prask and F. S. Biancaniello

507

Neutron Diffraction Assisted Residual Stress Analysis in Welded Structures C. Ohms and A. G. Youtsos

515

TABLE OF CONTENTS

Synchrotron Radiation In-Situ Analyses of AA 6061 + During Tensile Deformation at Ambient and Elevated Temperature A. Pyzalla, B. Reetz, A. Jacques, J.-P. Feiereisen, O. Ferry and T. Buslaps

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527

7. Hybrid Methods Reflections on the Importance of Experimental Results to all Mechanicists, Especially Theoreticians R. M. Jones

537

Mixed Numerical-Experimental Techniques : Past, Present and Future A. H. Cardon, H. Sol, W. P. De Wilde, J. De Visscher, K. Hoes and D. Dinescu

551

A Moiré-FE Method for Internal CTOA Determination J. H. Jackson and A. S. Kobayashi

561

Patterns of Modern Experimental Mechanics J. T. Pindera

571

Inverse Methods in Experimental Mechanics J. F. Doyle

585

Complex Stiffness Identification by Inverse Methods H. Sol and W. P. De Wilde

595

Considerations of a Flutter Prediction Methodology Using a Combined Analytical-Experimental Procedure P. Marzocca, L. Librescu and W. A. Silva

609

Displacement-Based Smoothing Hybrid Finite-Element Representation for Stress Analyzing Perforated Composites K. Y. He and R. E. Rowlands

619

8. Composite Structures Future Experimental Methods Needed to Verify Composite Life-Cycle Simulations C. C. Chamis and L. Minnetyan

631

Experimental Observations on the Delamination Behavior in Composite Structures G. A. Kardomateas, V. La Saponara and G. J. Simitses

645

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Direct Identification of Elastic Properties of Composite Structures A Wave-Controlled Impact Approach S. Abrate

661

Static Behaviour of Pre-Stressed Polymer Composite Sandwich Beams R. A. W. Mines, Q. M. Li, R. S. Birch, R. Rigby, M. Al-Khalil and A. Tanner

671

On Debond Failure of Foam Core Sandwich L. A. Carlsson

683

Core Crush Mechanisms and Solutions in the Manufacturing of Sandwich Structures H. M. Hsiao, S. M. Lee and R. A. Buyny

689

Displacement Fields Around a Circular Hole in Composite Laminates S. M. Chern and M. E. Tuttle

701

9. Structural Testing and Analysis Recent Advances in Long-Term Monitoring of Bridges J. R. Casas

715

An Experimental Mechanics Approach to Structural Health Monitoring for Civil Aircraft E. W. O'Brien

727

Smart Structures Application to Airworthiness and Repair R. Jones, I. H. McKenzie, S. Galea and S. Pitt

737

Experimental Investigation of Shrinkage Strains in Multilayered Stereolithography Parts D. E. Karalekas

749

Suppression of Dimpling In Sheet Metal Parts Formed on Discrete Tooling R. C. Schwarz, J. Nardiello and J. M. Papazian

757

Effect of Loading Rate and Geometry Variation on the Dynamic Shear Strength of Adhesive Lap Joints V. Srivastava, V. Parameswaran, A. Shukla and D. Morgan

769

Author Index

781

Subject Index

797

Editor’s Preface

This book contains 71 papers presented at the symposium on “Recent Advances in Experimental Mechanics” which was organized in honor of Professor Isaac M. Daniel. The symposium took place at Virginia Polytechnic Institute and State University on June 23-28, 2002, in conjunction with the 14th US National Congress of Applied Mechanics. The book is a tribute to Isaac Daniel, a pioneer of experimental mechanics and composite materials, in recognition of his continuous, original, diversified and outstanding contributions for half a century. The book consists of invited papers written by leading experts in the field. It contains original contributions concerning the latest developments in experimental mechanics. It covers a wide range of subjects, including optical methods of stress analysis (photoelasticity, moiré, etc.), composite materials, sandwich construction, fracture mechanics, fatigue and damage, nondestructive evaluation, dynamic problems, fiber optic sensors, speckle metrology, digital image processing, nanotechnology, neutron diffraction and synchrotron radiation methods. The papers are arranged in the following nine sections: Mechanical characterization of material behavior, composite materials, fracture and fatigue, optical methods, nondestructive evaluation, neutron diffraction and synchrotron radiation methods, hybrid methods, composite structures, and structural testing and analysis. The first section on mechanical characterization and material behavior contains eight papers dealing with dynamic tests using Hopkinson bar techniques, gradient theory of micro-scale plasticity, time-dependent materials, foam materials, carbon nanotubes and nanostructured coatings. The second section on composite materials contains nine papers dealing with state variables, properties at high strain rates, metal matrix composites, particulate composites, hygric characterization, pneumatic behavior, compressive strength, interfacial strength and toughness, and residual strength after damage due to impact and erosion. The third section on fracture and fatigue contains ten papers dealing with a probabilistic model of fatigue, damage tolerance analysis, crack growth, failure initiation, crack paths in adhesive bonds, failure of spot welds, and dynamic and intersonic crack growth. The fourth section on optical methods contains seven papers dealing with moiré interferometry, optical fiber sensors, displacement measurements based on optoelectronics and photogrammetry, fracture processes using digital image correlation, determination of fringe order using different wavelengths, and heart mechanics problems by means of speckle metrology. The fifth section on nondestructive evaluation contains eight papers dealing with acoustic microscopy, acoustography, waves in elastic materials, shearography, laser-ultrasonics, scattering of flexural waves, and ultrasonic applications. The sixth section on neutron diffraction

Biography of Isaac M. Daniel Isaac M. Daniel was born on October 7, 1933 in Thessaloniki (Salonica), Greece, second of four children of Mordochai Daniel and Bella (Modiano) Daniel. Thessaloniki, the second largest city in Greece, is an ancient city named by Alexander the Great for his sister. It is also near Aristotle’s birthplace. His family, for the most part, traces its origins to Spain from where they were expelled in 1492 for religious reasons, although it also has roots in a community that lived in the area for over 2000 years. His father was born and raised in the nearby town of Veria (Veroia) also an ancient city. His mother’s family came from Livorno, Italy and is distantly related to the painter Amedeo Modigliani. Thessaloniki, originally the center of Alexander’s Macedonia became successively part of the Roman, Byzantine and Ottoman empires until 1912 when it was incorporated in Greece. Sometimes when talking about his family’s background, Isaac says that his mother was born in Turkey, raised in Greece as an Italian citizen, attended an Italian school where she learned French, but at home she spoke Ladino (a medieval Spanish dialect with admixtures of Hebrew, Portuguese, Italian, Greek, French and Turkish). Before the Second World War, Isaac’s family moved to Veria (about 45 miles west of Thessaloniki) which at the time had a population of around 15,000. He entered the local Jewish school and attended the Synagogue that the apostle Paul visited some 2000 years ago. Isaac’s father was a cabinet maker and occasionally worked as a grain dealer. His peaceful life came to a halt with the German invasion and its consequences. He still remembers all the horrible details of that period. The roundup of the Jewish community came suddenly, but his family miraculously escaped and hid in their basement. They lived in that basement, just like Ann Frank, for about a month until they were found out, arrested and put in the Baron Hirsch transit camp in Thessaloniki. They were scheduled to be shipped within days to the death camps of Auschwitz, when another miracle occurred. They were put on somebody’s list, the list of Guelfo Zamboni, the Italian consul in Thessaloniki. Through his intervention, 280 inmates, with direct or remote connection to Italian citizenship, out of more than 55,000 who passed through that camp, were released and taken to Athens which was then under Italian control. After Italy surrendered to the allies, life in Athens during the remaining war years became turbulent and full of dangerous adventures and more miraculous and narrow escapes. After the liberation on October 12, 1944, the troubles were not completely over, as they were caught in the middle of a brief but bloody civil war. Finally, they managed to return to their home in Veria and tried to start rebuilding their broken lives. They had lost over fifty of their relatives and most of their friends. In 1945 Isaac Daniel enrolled in one of the public schools in Veria and then entered, after an entrance examination, the Gymnasium (High School). He has very

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good memories of his high school years. He was very popular and had many friends. The school followed a classical curriculum which included among others, six years of ancient Greek, and four years of Latin and French. In the senior year the students put on a play and they raised enough funds for a class trip to southern Greece and the islands. It was Isaac’s first and only attempt as a playright. He had no favorite or less favorite subjects. He was equally at ease with Math and Science as with history and literature. He wrote poetry and had one poem published. He was selected to give the valedictory address at his graduation. He was told that a couple of his teachers had tears in their eyes during the speech. Graduation from high school was the end of a care-free period and the beginning of a new phase of career planning and preparation. The high school principal who counseled all the graduating students individually told Isaac that he had no special advice to give him, that he would do well in whatever career he followed. It was a choice among Literature, Medicine and Engineering. He chose the latter because it seemed to be the most challenging at the time. The entrance exams for the National Technical University in Athens (the only technical school in the country at that time) were the toughest of all. The best graduating students from all high schools would take a year off and enter a preparatory school before attempting the entrance exams. Upon graduation from high school Isaac Daniel attended a preparatory school for less than a month and he spent the rest of the summer studying by himself. He took the exams in the fall and it was one of the greatest days of his life when he heard that he was successful. He entered the School of Civil Engineering at the National Technical University (Metsovion Polytechnion). They had classes from 5 to 8 hours a day, six days a week. He finished two years and had enrolled for the third year when he immigrated with his family to the United States. The family settled in Chicago in 1955 and Isaac transferred to the Illinois Institute of Technology. It was a bit of a cultural shock. He was more advanced in technical and math subjects, but he had to take English literature and other liberal arts courses. He was a straight A student and graduated with distinction, first in his class as well as the entire Engineering Division of IIT in 1957. He received a Fellowship from the Chicago Bridge and Iron Co. and entered graduate school at IIT. IIT had an illustrious tradition in Mechanics with names like Enrico Volterra, Bill Ramberg, Eli Sternberg, Daniel Drucker and others associated with it before Isaac came. He had great teachers like August Durelli, Max Frocht, Lloyd Donnell and Phil Hodge. Durelli was his first mentor who introduced him to the field of Experimental Mechanics. Durelli was a scientist, philosopher and humanist. Frocht taught him precision and quality of presentation, especially in the case of photoelastic fringe patterns. Donnell impressed him by demonstrating the thought processes and approximations involved in developing complex relations. Hodge was clear and methodical in his lectures on continuum mechanics and plasticity. In addition to the technical knowledge, he learned one wise life guideline: "what is not worth doing is not worth doing well.” As he was finishing his Master’s degree studies, Durelli offered Isaac a job at the Armour Research Foundation (later IIT Research Institute) the research affiliate of IIT, where he became deeply involved in Experimental Stress Analysis. At that time, the

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late 50’s and early 60’s, the IIT campus was a hotbed of activity in experimental stress analysis. Albert Kobayashi and Pericles Theocaris had both worked in Durelli’s laboratory. He was fortunate to have supervisors and colleagues like Jim Dally, Bill Riley, Cesar Sciammarella, Vince Parks, Bob Rowlands and Ted Liber. The biggest compliment that Durelli paid Isaac, when he ran the first dynamic photoelasticity test and showed him the results, was a quotation from El Cid: “leurs coups d’essai veulent des coups de maître” (their trial strokes are master strokes). His first technical presentation, based on his Master’s thesis, was at the ASME meeting in New Orleans in 1960. The work at Armour (IITRI) was exciting. Isaac Daniel worked on stress analysis problems using photoelastic, moiré and strain gage methods. He worked on fracture of brittle materials and published work on fracture probabilities with N. A. Weil. He worked on creep of rubberlike materials and was impressed to verify experimentally that the equilibrium modulus of these materials is proportional to the absolute temperature as predicted. He became interested in viscoelasticity and this led him to write a proposal to the Air Force for the development of dynamic photoviscoelasticity and application to the study of stress waves in earth media. This led him to his Ph.D. research which he did while working full-time at IITRI. The research was reviewed by a committee including K. H. Chu, Jim Dally, Phil Hodge and Frank Essenburg of IIT and IITRI. In the course of this research he received valuable advice from Y. H. Pao of Cornell who was visiting at IITRI for one summer. He developed a theory of photothermoviscoelasticity and applied it to wave propagation problems. In 1964, Isaac received his Ph.D. He continued research in the areas of wave propagation and rock mechanics and started applying experimental stress analysis methods to the emerging technology of composite materials. Initially, he studied the micromechanics of composites by means of photoelasticity. In the middle 60’s there was a government thrust to develop advanced composites. Texaco had developed a new boron prepreg tape, which was 1/8 in. wide. Isaac Daniel and his co-workers made their first advanced composite sample from this tape, the size of a postage stamp. The government announced a large program and encouraged the collaboration of universities, research institutes and industry. Isaac’s director at IITRI wrapped the postage stamp sized specimen in his wallet and they both flew to Fort Worth to meet with General Dynamics. At a high level technical meeting they asked Isaac how much experience he had with advanced composites. Before he had a chance to say “not much” his boss kicked him under the table and said that they had expertise in fabricating composites and showed them the specimen. The General Dynamics people were impressed. It was proposed then that the first application be a flat part, namely the horizontal stabilizer of the F-111. Isaac Daniel was appointed manager of the Experimental Stress Analysis Section at IITRI, a position previously held by Durelli and Riley. It was a heady time for research in composites and the funding was lavish. There were no established testing procedures for composites and the common practice was to adopt the same methods used for metals. For example, tensile testing was done by ultrasonically machining a narrowed section in a boron/epoxy coupon. This process, besides being very expensive and time consuming, did not work. It was then that Isaac tried a straight-sided coupon tabbed with glass/epoxy tapered tabs, a method that has become an ASTM standard. In the

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following years his work with composites expanded in all directions including processing, development of test methods, micromechanics, material characterization, special test methods (torsion, impact, creep/relaxation, fracture toughness, hygrothermal effects, high rate testing, multi-axial testing, fatigue), stress and failure analysis (stress concentrations, joints, warpage), damage mechanics and nondestructive evaluation. In the early 70’s he worked with Bob Rowlands and Jim Whiteside of Grumman on the effects of stress concentrations in composite panels. They applied birefringent coatings and moiré techniques to study the deformation and failure around holes. This work was extended later to investigations of size effects and biaxial loading. He built a new biaxial testing system for flat plates. He had a chance to work on speckle interferometry with Mike Hung during his brief sojourn in his group at IITRI. In the 70’s he also worked with Ted Liber on developing an embedded strain gage technique for determination of curing residual stresses, biaxial testing using tubular specimens, high strain rate testing and nondestructive evaluation of composites. He developed and built a fully controlled test system for multiaxial testing of thin wall tubular specimens. He also developed a new method for testing composites at very high strain rates using thin ring specimens subjected to an internal pressure pulse produced by an explosive inside a liquid container. This method did not have the limitations of other methods using test coupons such as the Hopkinson bar method, where stress wave propagation over the length of the specimen produces a nonuniform stress distribution. In the case of the thin ring specimen under internal pressure there are no end effects and the relevant dimension is the thickness of the ring. He also found out that they could determine dynamic compressive properties by loading thin wall cylinders under external pressure without producing buckling, if the pressure were applied at a sufficiently high rate. Isaac Daniel received two best paper awards from the Society for Experimental Stress Analysis (SESA and later SEM). He became Fellow of SESA in 1981 and was invited to be the keynote speaker at the 7th International Conference on Experimental Mechanics in 1982 in Haifa. In the same year he coauthored with Jim Whitney and Byron Pipes a monograph on Experimental Mechanics of Composites. In January 1982 he was invited to join IIT as Professor and Director of the Experimental Stress Analysis Laboratory (later Mechanics of Materials Laboratory). The first grant he received at IIT was from ONR (Yapa Rajapakse) which has continued its support to date. During this period he worked on damage mechanics of composites, including damage characterization, evolution and accumulation, accelerated testing and life prediction; dynamic fracture toughness, and nondestructive evaluation of composites. In 1984 he received the B. J. Lazan Award for outstanding original contributions in Experimental Mechanics. In the same year he spent the summer in Poitiers, France, as a visiting Professor at the invitation of Professor Alexis Lagarde, and gave several seminars on composites. In early 1986 he was invited to give a seminar at Northwestern University. During lunch, Toshio Mura, in his inimitable fashion, said to him: “After you join us in the fall, we will write a proposal together on solitons.” That was the first hint he got of the offer from Northwestern that came later and he couldn’t refuse. He joined Northwestern and he wrote the proposal with Toshio Mura to NSF. It received two “excellent” and one

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“very good” rating, but it was not funded. He continued the work in damage mechanics and damage evolution in composites. This included development of models for stiffness and strength degradation and closed form expressions for constitutive behavior of cracked composite laminates with students Jae-Won Lee and Cho-Liang Tsai, now at Samsung Company in Korea and Yun Lin University in Taiwan, respectively. With Tsai he developed a unique method for determining all three shear moduli of an orthotropic composite. He developed advanced ultrasonic and acoustic emission techniques for flaw and damage detection and characterization with his student ShiChang Wooh, now Professor at MIT. This work included ultrasonic methods for threedimensional imaging of internal damage, detecting porosity and for determination of elastic properties of composites. He worked with Tao-Ming Wang (now at Illinois Tool Works) on control of residual stresses and warpage in circuit boards, a problem of great interest in electronic packaging. One result of this work was a method for measuring the chemical cure shrinkage in composite laminates. A multi-year program, started in the late 80’s, was the investigation of constitutive behavior of metal matrix composites sponsored by NASA-Lewis (Chris Chamis). He worked with his students Dimitri Karalekas and Heoung-Jae Chun, now Professors at the University of Piraeus, Greece and Yonsei University in Korea, respectively. They did a very extensive characterization of elastic and viscoelastic behavior of metal matrix composites at ambient and elevated temperatures. Starting in the late 80’s and continuing into the 90’s he embarked on the study of micromechanics of failure and constitutive behavior of ceramic matrix composites with his students Jae-Won Lee, George Anastasopoulos and Jyi-Jiin Luo. Based on real time observations of microscopic failure modes and acoustic emission output, they proposed analytical constitutive models describing material response up to failure. The investigation was extended to elevated temperatures and determination of residual stresses. He worked with his former student and now research associate Jyi-Jiin Luo on the development of an analytical model and a general theory for characterizing deformation of damaged composites. One of the most satisfying activities for Isaac Daniel in the early 90’s was writing the textbook “Engineering Mechanics of Composite Materials,” with Ori Ishai of the Technion in Israel. The book has been very well received by the community and is widely used as a textbook throughout the world. He is currently working with Ishai on the second edition. In the early 90’s the composites thrust of ONR shifted to the compressive behavior of thick composites. With his student Hao-Ming Hsiao he developed methods for manufacturing thick composites of uniform quality and a new method for compressive testing of such composites using a specially designed fixture (NU fixture). They studied and recorded the failure mechanisms of kink band formation and its propagation. They investigated experimentally and modeled the behavior, both linear and nonlinear, of thick composites with fiber waviness. The above studies were followed by extensive investigations of the effects of high strain rate on the stress-strain behavior and properties of composites under tension, compression and shear. This work was followed by research in the area of composite sandwich structures with the help of Emmanuel Gdoutos, visiting Professor from Democritus University of Thrace, Greece, and former student and now research associate Jandro Abot. They

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developed methods for fabrication and testing of sandwich structures statically and under impact and studied experimentally and analytically the various failure modes and criteria for their occurrence. A major expansion of his activity in composites occurred at the end of 1997 when Professor Daniel became Director of the new Center for Intelligent Processing of Composites. The objective of this center is to develop a comprehensive, commercially viable approach for fabrication of affordable, functional and reliable composite structures. This is an interdisciplinary program conducted by a team including University faculty, industrial firms and the government. Contributions of Daniel’s group in this Center include the design and fabrication of an innovative computer controlled system for resin transfer molding (RTM); new methods for characterizing reinforcement preforms; flow and cure models; and a new noninvasive approach for online quality control of reinforcement quality, a method of great commercial potential. Isaac Daniel received several recognitions and awards in the 90’s. He became Fellow of the American Academy of Mechanics (1994); was invited to be a keynote speaker at the International Conference on Composites Engineering in New Orleans (1995); received the Distinguished Research Award of the American Society for Composites (1996); was elected Chairman of the Theoretical and Applied Mechanics program at Northwestern University (1996); received the William M. Murray Medal of the Society for Experimental Mechanics and gave the associated prestigious lecture to the society (1998); he was appointed the Walter P. Murphy Professor of Civil and Mechanical Engineering at Northwestern University (1998); became Fellow of the American Society of Mechanical Engineers (1999); received the Professional Achievement Award from his alma mater, the Illinois Institute of Technology (1999). Besides all the research and professional activities, Isaac leads a busy and happy family life. His wife Elaine Krule Daniel of Chicago shares many interests with him. She studied at the Sorbonne and the University of Illinois. Elaine is a polyglot with degrees in French and teaching English as a second language and has taught at the University of Illinois in Champaign-Urbana and the University of Louisville in Kentucky. She is also a gourmet cook and a quilter. They have three children, Belinda, 13, Rebecca, 11, and Max 9. Isaac spends a lot of time with his family taking them along on trips to summer conferences in the United States and abroad. He attends and participates in the children’s activities, such as concerts and Boy Scout projects. He is active in community affairs, reads, travels and is interested in photography. He has a sense of humor which Elaine appreciates by laughing at his jokes even when she hears them for the fifth time. He shares a special interest with Elaine in linguistics and their dream is to write a Ladino-English dictionary. Isaac was very close to his father and mother who passed away in 1993 at the ages of 94 and 86. His older brother, Aaron, is an architect living in California. When he lived in Chicago he contributed to the design of the Sears Tower and the Civic Center plaza. His sister, Sarah Spector, has a degree in nutrition and lives in Chicago. His younger brother Sam, works for Motorola on artificial intelligence and lives in Arizona. Isaac Daniel has no immediate plans for retirement. People say that his children keep him young and active. He hopes to continue working and help his children with their education and, maybe, have them in his class.

1. Mechanical Behavior of Materials

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HOPKINSON TECHNIQUES FOR DYNAMIC TRIAXIAL COMPRESSION TESTS

JACOB ROME, JON ISAACS, SIA NEMAT-NASSER Center of Excellence for Advanced Materials Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, CA 92093-0416

Abstract

A modified Hopkinson technique is developed to simultaneously load a sample in both the axial and radial directions. This is accomplished by using a composite incident-bar and a composite transmission-bar system. The incident-bar system consists of a largediameter bar, followed by a small-diameter bar that fits inside a tube whose outside diameter is equal to that of the large bar. The transmission-bar system consists of a tube and a bar that fits inside the tube. The sample is placed inside a Teflon sleeve, and the Teflon sleeve is placed inside an aluminum sleeve. The Teflon sleeve is loaded during the test by the stress pulse traveling in the incident tube, producing a radial stress on the sample. The stress pulse traveling in the incident bar loads the sample that is sandwiched between the small-diameter incident and transmission bars. The radial stress is calculated from the hoop strain measured by a strain gauge placed on the outer surface of the aluminum sleeve. Mortar samples have been tested under a range of confining stresses and strain rates, and some of the results of these experiments are presented. 1.

Introduction

Compressive properties and failure modes of many materials, particularly brittle materials, are dramatically affected by the stress triaxiality. This has been demonstrated through various laboratory experiments, since the early work of Bridgman [1] who demonstrates several failure modes peculiar to high pressures, leading to seemingly paradoxical results. The common feature of these paradoxes is that failure always occurs by the formation of tension cracks in specimens subjected to pure compression. While all these paradoxes have been fully understood and resolved (Scholtz et al., [2]; Nemat-Nasser & Horii [3]; Horii & Nemat-Nasser [4]; Horii and Nemat-Nasser [5]) they do emphasize the importance of controlled triaxial experiments in material characterization. Indeed, in axial compression, brittle materials such as rocks or ceramics fail by axial splitting, faulting, or plastic deformation (barreling), depending 3 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 3–12. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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on the relative magnitude of the confining pressure which may accompany the axial compression (Nemat-Nasser & Horii [3]; Horii & Nemat-Nasser [4]; Horii and NematNasser [5]). Model studies show that similar results emerge at high strain rate (NematNasser and Deng [6]; Nemat-Nasser and Hori [7]). The Split Hopkinson pressure bar (SHPB) was invented by Kolsky [8], following the early work of John Hopkinson [9] and his son, Bertram Hopkinson [10,11], and Davies [12], on elastic stress pulse traveling in a single elastic bar. Several investigators (Harding et al. [13]; Lindholm [14]; Lindholm & Yeakley [15]; Follansbee [16]; Hartman et al. [17]; Nicholas and Bless [18]) have since developed new methods for Hopkinson bar testing, including recovery experiments (Nemat-Nasser et al. [19]). A Hopkinson bar can be modified to subject a sample simultaneously to axial and radial compressions (Nemat-Nasser et al. [20]). In the present paper, this technique is described in detail, and the results are illustrated using mortar samples. 2.

Test Apparatus and Procedure

2.1. CLASSICAL SYSTEM A classical split Hopkinson bar (the Kolsky bar) is shown in Figure 1, in its simplest configuration. It consists of a gas gun with a perforated barrel (not shown), a striker bar, an incident bar, and a transmission bar with common cross-sectional area, and Young's Modulus, The sample of length and cross-sectional area is sandwiched in between the incident and transmission bars. Strain gauges are placed on the incident and transmission bars, equidistant from the sample.

A test begins by firing the gas gun that propels the striker bar through the barrel at a prescribed velocity, impacting with the incident bar and producing in it an incident elastic pulse, which travels through the incident bar at the bar's elastic wave speed The corresponding elastic strain in the bar, is measured as a function of time by the strain gauge. Once the incident pulse reaches the sample, part of it is reflected from the sample and travels back through the incident bar, while the remaining part is transmitted through the sample into the transmission bar, as it loads the sample. The reflected elastic strain in the incident bar, and the transmitted one, are measured by the corresponding strain gauge.

DYNAMIC TRIAXIAL COMPRESSION TESTS

5

In a well-designed test, the magnitude of the reflected pulse is proportional to the strain rate in the sample, The sample stress is proportional to the strain in the transmission bar. The sample strain is obtained by integrating the corresponding strain rate with respect to time. In this manner, the stress-strain response of a sample can be measured. The sample strain rate and sample stress are calculated based upon onedimensional elastic waves in the bar,

2.2. TRIAXIAL SYSTEM In order to test a sample in triaxial compression at a high strain rate, several modifications are made to the Hopkinson bar. The modified Hopkinson-bar system now consists of a gas gun with a perforated barrel, a striker bar, incident bar 1, incident bar 2, incident tube, transmission bar and transmission tube (Figure 2).

The sample, which has the same diameter as the transmission bar and incident bar 2, is placed inside of a Teflon sleeve. The Teflon sleeve has roughly the same cross section as the incident and transmission tubes; there is a slight interference fit (0.03mm) between the sample and the Teflon. The Teflon, with the sample inside of it, is placed inside an aluminum sleeve with an inside diameter, and an outside diameter, that can be adjusted for each test. The sample is placed between incident bar 2 and the transmission bar. The Teflon, slightly longer than the sample, fits around incident bar 2 and the transmission bar, and is sandwiched between the incident and transmission tubes. The aluminum sleeve, slightly longer than the Teflon sleeve, fits around the incident and transmission tubes. We have used Teflon and aluminum sleeves in our tests to show the basic approach, but other material may be used as appropriate.

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In the current study, a 19.1-mm Hopkinson bar was used. The system configuration where sample is to be sandwiched between the incident and transmission bars, is shown in Figure 3. The striker bar and the first incident bar are each 27.1mm in diameter. The second incident bar and the transmission bar have a diameter of 19.1mm. The incident and transmission tubes both have an inside diameter of 19.1mm and an outside diameter of 27.1mm.

2.3. TEST PROCEDURE Before the test, the entire Hopkinson bar apparatus is pressed together to ensure that there are no gaps between the contacting bars, the sample or the Teflon sleeve. This is vital to ensure that the radial load is applied at the same time as the axial load, and to ensure that the data collected during the experiment is an accurate measure of the sample response. Incident bar 1 must be in direct contact with both incident bar 2 and the incident tube. The incident tube must be in a tight contact with the Teflon sleeve, which in turn must be in contact with the transmission tube. In our setup, incident bar 2 is slightly longer than the incident tube. Due to the radial expansion accompanying the sample deformation, it is necessary to use a pre-split Teflon sleeve to facilitate sample recovery without any damage (Figure 4). Pushing the Teflon sleeve through the aluminum sleeve easily separates the sample, the pre-split Teflon sleeve and the aluminum sleeve. Since the fit is very important, the Teflon is sliced with a knife instead of being machined. The residual stresses in the Teflon then produce expansion, leading to a tight fit with the sample and the aluminum sleeve. Experiments that follow this technique will simultaneously load the sample in the axial and radial directions, depending on the mechanical properties of the sleeves and the geometry of the setup. Refer to Figures 5a,b.

DYNAMIC TRIAXIAL COMPRESSION TESTS

7

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J. ROME, et al.

The gas gun propels the striker bar (A) through the barrel at a prescribed velocity. The striker bar (A) impacts the first incident bar (B), generating the incident pulse. The incident pulse travels through the first incident bar (B) and the strain it causes in the bar, is measured at the strain gauge. When the incident pulse reaches the end of the first incident bar (B), it continues into the second incident bar (C) and the incident tube (F). The resulting strain in the second incident bar (C) and the incident tube (F) is still When the incident pulse reaches the end of the second incident bar (C), the sample (D) is loaded in the axial direction. Part of the incident pulse, proportional to the sample strain rate, is reflected back into the second incident bar (C); the reflected pulse (being tensile) does not travel into the first incident bar (B) and therefore can not be measured directly. The rest of the incident pulse travels through the sample (D) and generates the transmission pulse that travels through the transmission bar; the strain it generates, proportional to the sample stress, is measured at the strain gauge. Simultaneous with the incident pulse loading the sample, the incident pulse also reaches the end of the incident tube (F) and loads the Teflon (G) as in much the same way as a normal incident bar loads a sample. As the Teflon (G) is loaded between the incident tube (F) and the transmission tube (I), it is restrained outwardly by the aluminum sleeve (H), and inwardly by the sample (G). As a result, a hydrostatic stress is produced in the Teflon (G), which loads the sample (D) in the radial direction. The hoop strain in the aluminum sleeve (H) is measured, and the pressure in the Teflon (G) (equal to radial confining stress on the sample) is calculated. The radial stress can be controlled independently from the axial stress and strain to a limited extent, by, e.g., altering the thickness of the aluminum sleeve to control if and when the sleeve yields, or using a different material for the outer sleeve. The Teflon sleeve could also be replaced by another appropriate material to produce different loading conditions. The simultaneous loading in the radial and axial directions is assured by the design of the bar. The elastic waves in the incident bar and incident tube are generated at the same time. The bar and the tube are made of the same material and they have nearly the same length. Thus, the elastic wave in the incident bar reaches the sample (loading it axially) at the same time as the stress wave in the incident tube reaches the Teflon (loading the sample in the radial direction). 3. Data Analysis With this design, the reflected pulse can not be measured directly. Instead it is calculated as the difference between the incident and reflected pulses. To do this, the incident and reflected pulses must be time-shifted very precisely to ensure reliable results. The constructed reflected pulse is:

The axial strain rate and axial stress in the sample are then calculated according to equations (1).

DYNAMIC TRIAXIAL COMPRESSION TESTS

9

In our tests, 7075 aluminum (Young's modulus, Poisson’s ratio is used as the sleeve to confine the Teflon, with outside diameters of 28.6mm, 31.8mm and 38.1mm, depending on the test requirement. This material is chosen for its high strength (yield modulus, and low degree of hardening. The relation between the hydrostatic pressure in the Teflon (which is equal to the radial confining stress on the sample) and the (measured on the outside surface of the aluminum sleeve) hoop strain, falls into one of three distinct regions. The first is before the aluminum sleeve has yielded. In this region, the stress is found using basic elastic theory for a thick-walled cylinder. The radial stress at the inner surface (the hydrostatic pressure in the Teflon), is:

The second region is when the aluminum sleeve is elastic-plastic. The third and final region is when the entire material has undergone plastic deformation. When plastic deformation occurs in the metal sleeve, additional information on the elasticplastic properties of the material is required in order to calculate the radial stress on the sample; see, for example, Mendelson [21]. 4.

Results and Discussion

Several samples have been tested using this method. The material being studied is a concrete mortar. This material has been chosen because it is known to have a much different response under triaxial loading than it does under uniaxial loading. Under uniaxial compression tests, the elastic modulus has been measured as 32GPa, and the failure strength is 50MPa. If the aluminum sleeve was suspected to have undergone plastic deformation, it was replaced and a new sleeve was used in its place. New Teflon sleeves were used for each test. The samples were tested at average strain rates of about 200/s, 400/s or 600/s; the strain rate during each test was not constant due to the increasing stress in the sample during the experiment. Figure 6 shows a comparison between the response of an unconfined sample and a confined sample. The confined sample carried a much greater load and underwent significantly more strain after the unconfined sample had failed. Figure 7 compares three samples that had the same nominal strain rate (400/s), but each had a different radial stress. It is shown that the greater the confining stress, the greater the axial stress at a given strain. Figure 8 demonstrates that mortar has rate sensitivity at high strain-rates under triaxial compression. In this case, the radial stress is about the same, but the axial stress is higher for the sample with the higher strain rate. Even in these graphs, the effect of the confinement can be seen. As is seen, at about 2.5% strain, the radial stress for sample tested at 200/s increases; at the same time, the axial stress rises relative to the axial stress in the higher strain rate sample.

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DYNAMIC TRIAXIAL COMPRESSION TESTS

5.

11

Concluding Comments

This new method to test materials under triaxial compression at high strain rates is important for a number of reasons. First, the radial stress is applied at the same time as the sample is loaded axially. Since the radial stress is generated by the incident tube compressing the Teflon sleeve, this method does not wholly depend on the dilatancy of samples to create the radial stress, rather, the magnitude of the radial stress can be controlled by proper design. 6.

Acknowledgement

This work has been supported by an ARO MURI DAAH04-96-1-0376 grant to the University of California, San Diego. 7. References 1. 2.

3. 4. 5.

Bridgman, P.W. (1931) The Physics of High Pressure, Bell, London. Scholtz, C.H., G. Boitnott, and S. Nemat-Nasser The Bridgman ring paradox revisited, PAGEOPH, 124, (1986), 587-599. Nemat-Nasser, S. and H. Horii Compression-induced nonplanar crack extension with application to splitting, exfoliation, and rockburst, J. Geophys. Res., 87, (1982), 6805-6821. Horii, H. and S. Nemat-Nasser Compression-induced microcrack growth in brittle solids: Axial splitting and shear failure, J. Geophys. Res., 90, (1985), B4, 3105-3125. Horii, H. and S. Nemat-Nasser Brittle failure in compression: splitting, faulting, and brittle-ductile transition, Phil. Trans. Roy. Soc. Lond., 319, (1986), 1549, 337-374.

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6.

7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

18. 19.

20. 21.

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Nemat-Nasser, S. and H. Deng Strain-rate effect on brittle failure in compression, Acta Melallurgica et Materialia, 42, (1994), 3, 1013-1024. Nemat-Nasser, S. and M. Hori (1999) Micromechanics: Overall properties of heterogeneous materials, Elsevier Science Publishers, 2nd edition. Kolsky, H. An investigation of the mechanical properties of materials at very high rates of loading, Proc. R. Soc. Lond. B, 62, (1949), 676-700. Hopkinson, J. Original Papers by J. Hopkinson, 2, B. Hopkinson Ed., Cambridge at University Press, (1901), 316-324. Hopkinson, B., The effects of momentary stresses in metals, Proc. R. Soc. Lond. B, 74, (1905), 498. Hopkinson, B., A method of measuring the pressure produced in the detonation of high explosives or by the impact of bullets, Phil. Trans. R. Soc. Lond. A, 213, (1914), 437-456. Davies, R.M., A critical study of the Hopkinson pressure bar, Phil. Trans. R. Soc. Lond. A, 240, (1948), 375-457. Harding, J., E.O. Wood, and J.D. Campbell, Tensile testing of materials at impact rates of strain, J. Mech. Eng. Sci., 2, (1960), 88-96. Lindholm, U.S., Some experiments with the split Hopkinson pressure bar, J. Mech. Phys. Solids, 12, (1964), 317-335. Lindholm, U.S. and L.M. Yeakley, High strain-rate testing: tension and compression, Exp. Mech., 8, (1968), 1-9. Follansbee, P.S., Metals Handbook, 8, American Society of Metals, (1985), 198-203. Hartman, K.H., H.D. Kunze and L.W. Meyer, Metallurgical effects in impact loaded materials, Shock Waves and High-Strain-Rate Phenomena in Metals, Editors: M.A. Meyers and L.E. Murr) (1986), p. 325, New York: Plenum. Nicholas, T. and S.J. Bless, Metals Handbook, 8, American Society of Metals, (1985), 208-214. Nemat-Nasser, S. J.B. Isaacs, and J.E. Starrett, Hopkinson techniques for dynamic recovery experiments, Proc. Royal Soc. London, 435, (1991), 371-391. Nemat-Nasser, S., J. Isaacs and J. Rome, Triaxial Hopkinson techniques, ASM Mechanical Testing and Evaluation Handbook, 8, (2000), 516-518. Mendelson, A., Plasticity: Theory and Application, Chp. 8, A. Mendelson Ed., Macmillan Company, New York, (1968), 156-163.

ASYMPTOTIC SCALING OF GRADIENT THEORY OF MICRO-SCALE PLASTICITY OF METALS

Departments of Civil Engineering and Materials Science Northwestern University Evanston, IL 60208

Abstract The paper deals with the question of asymptotic behavior of Gao, Huang, Hutchinson and Nix’s dislocation-based theory of strain-gradient plasticity on the micrometer scale, which is currently of keen interest for micro-mechanics. Certain peculiar features of the nano-scale extension of the existing theory are identified and possible remedies pointed out. 1. Introduction The recent interest in small-scale extensions of continuum models capturing the transition to atomic lattice models for metallic crystals and polycrystals brings about interesting problems of scaling. In these problems, one can benefit from a formal analogy with the previous studies of scaling in quasibrittle heterogeneous materials such as concrete (e.g., and Planas [1], and Chen [2], [3-5], and Novák [6]). Exploiting such analogy, the small-scale asymptotic scaling properties of the recent theory of Gao, Huang, Nix and Hutchinson [7,8], which is based on dislocations that are geometrically necessary for producing a curvature or twist of the atomic lattice (Fig. 1 top) are examined in the lecture briefly summarized in this paper. A full presentation of the present scaling analysis is given in [9]. 2.

Constitutive Model with Strain Gradients

The constitutive relation proposed by Gao et al. [7,8] and Huang et al. [10] reads:

where 13 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 13–18. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

14

SCALING OF MICRO-SCALE GRADIENT PLASTICITY

Here K = elastic bulk modulus; strains;

15

deviatoric strains, 2nd

and 3rd order tensors of components

displacement curvature (or twist), reflecting the effect of geometrically

necessary

dislocations

(Fig.

1

top)

[the

strain

gradient

volumetric (hydrostatic) part of order stresses work-conjugate to

third-

While Gao et al. (7,8) characterize the plastic

constitutive properties by the semi-empirical formula interesting to consider a slightly more general formula

with positive exponents p and q;

is

yield stress,

it is

stress and strain intensities;

effective strain gradient proportional to the density of geometrically stored dislocations (i.e., to lattice curvature or twist);

classical plastic hardening

function, which reflects the effect of statistically stored dislocations and is an increasing function of monotonically decreasing slope, size of the so-called ‘mesoscale cell’ or the spacing of geometrically necessary dislocations (which is the material length characterizing the transition from standard to gradient plasticity and is interpreted by Gao et al. [7,8] as the minimum volume on which the macroscopic deformation contributions of the geometrically necessary dislocations may be smoothed by a continuum); magnitude of Burgers vector of edge or screw dislocation (e.g., 0.255 nm for copper); positive dimensionless material characteristics expressed in terms of Taylor factor, Nye factor and the ratio of elastic shear modulus to the dislocation reference stress. The principle of virtual work yields the following field equations of equilibrium [7,8]:

The following dimensionless variables (labeled by an overbar) may be introduced:

While the derivatives with respect to are denoted by subscript preceded by a comma, the derivatives with respect to dimensionless coordinates are denoted as

16

To avoid struggling with the formulation of the boundary conditions, one may consider that the applied surface tractions and applied couple stresses vanish at all parts of the boundary where the displacements are not fixed as 0, and that all the loading characterized by nominal stress is applied as body forces whose distributions are assumed to be geometrically similar; is considered as a parameter of these forces having the dimension of stress. As for the loading by applied surface tractions and applied couple stresses, one may consider them replaced by equivalent body forces (proportional to ) that act within a surface layer of a very small thickness proportional to D. 3. Small-Size Asymptotic Scaling

Now the problem of scaling and size effect. For one must recover the standard theory of plasticity, and this is indeed the case. But it is also of interest to check the opposite asymptotic behavior for It can be shown that for the asymptotic form of the field equations is:

where

From this it follows that the asymptotic scaling law is:

where constant, and the exponent According to Gao et al.’s theory, and so (Fig. 1 middle) This asymptotic size effect is curiously strong. It is much stronger than the size effect of linear elastic fracture mechanics (LEFM) for similar sharp cracks on the macroscale, which is It is also possible to determine the asymptotic load-deflection curve for the special case in which the displacement distribution (or relative displacement profile) remains constant during the loading process. This is for example typical of the micro-bending and micro-torsion tests. It can be generally shown that

SCALING OF MICRO-SCALE GRADIENT PLASTICITY

17

In particular, for Gao et al.’s theory, the exponent is 3/2 (see Fig. 2 bottom). Now it is hard to escape noticing that for metals such a behavior is queer. The tangential stiffness of the structure for an infinitely small load is infinitely small, which is physically hard to accept, and then it increases with increasing deflection. It must be emphasized, however, that the small-size asymptotic behavior is approached only for much smaller sizes than those of the available experiments, and at dimensions much smaller than the theory was intended for. Nevertheless, a gradual transition to such odd behavior is doubtless the reason that the comparisons with test data reveal this theory to be too soft for small loads and too stiff for high loads. 4. Closing Comments

Thus it seems that some more fundamental modification of Gao et al.’s (1999) theory might be desirable in order to eliminate the strangely strong small-size asymptotic size effect and the queer small-size load-deflections curve. Some ways to do that are being studied at Northwestern University by Z. Guo. The lecture will also present the scaling properties of the original theory of Fleck and Hutchinson [11,12] and of a new modified theory of Gao and Huang [13]. As found by and Guo, the small-size asymptotic scalings of these theories are as follows:

and

respectively, where is the hardening exponent of the power-law stress-strain curve on the large scale. The size effect in the former is still curiously strong. The size effect in the latter, which is the limiting case of the scaling according to [7,8,13], is of the same type as in the case of similar cracks in linear elastic fracture mechanics. 5.

Acknowledgement

Partial support under U.S. National Science Foundation Grant CMS-9732791 to Northwestern University is gratefully acknowledged. 6.

References

1.

Z. P. and J. Planas, Fracture and Scaling of Concrete and Other Quasibrittle Materials, CRC Press, Bocca Raton, Florida, and London, 1998. Z. P. and E.-P. Chen, “Scaling of structural failure,” Applied Mechanics Reviews, ASME 50 (1997) (No. 10), 593-627. Z. P. Scaling of Structural Strength. Hermes Scientific Publications, Oxford and Paris 2002.

2. 3.

18

4.

5. 6.

7.

8. 9. 10. 11. 12. 13.

Z. P. “Scaling of quasibrittle fracture: Asymptotic analysis,” Int. J. of Fracture, 83 (1997) (No. 1), 19-40. Z. P. “Size effect on structural strength: a review,” Archives of Applied Mechanics (Ingeniur-Archiv, Springer Verlag) 69 (1999), 703-725. Z. P. and D. Novák, D., “Probablilistic nonlocal theory for quasibrittle fracture initiation and size effect. I. Theory. II. Application,” J. of Engrg. Mech., ASCE 126 (2000) (No. 2), 166174, 175-185. H. Gao, Y. Huang, W. D. Nix and J. W. Hutchinson, “Mechanism-based strain gradient plasticity –I. Theory,” J. of the Mechanics and Physics of solids 47 (1999), 1239-1263. H. Gao, Y. Huang and W. D. Nix, “Modeling plasticity at the micrometer scale,” Naturwissenschaften , 86 (1999), 507-515. Z. P. “Scaling of dislocation-based strain-gradient plasticity,” J. of the Mechanics and Physics of Solids – in press. Y. Huang, H. Gao, W. D. Nix and J. W. Hutchinson, “Mechanism-based strain gradient plasticityII. Analysis,” J. of the Mechanics and Physics of Solids, 47, (2000), 99-128. N. A. Fleck and J. W. Hutchinson, “A phenomenological theory for strain gradient effects in plasticity,” J. of Mechanics and Physics of Solids, 41 (1993), 1825-1857. N. A. Fleck and J. W. Hutchinson, “Strain gradient plasticity,” In Advances in Applied Mechanics, vol. 33, J. W. Hutchinson and T. Y. Wu, eds., Academic Press, New York, pp. 295361, 1997. H. Gao and Y. Huang, “Taylor-based nonlocal theory of plasticity,” Int. J. of Solids and Structures, in press.

THE ROLE OF PRESSURE IN THE BEHAVIOR OF TIME-DEPENDENT MATERIALS TED PRODAN, IGOR EMRI Center for Experimental Mechanics University of Ljubljana Ljubljana, Slovenia 1000

Abstract We improved and upgraded the apparatus that Fillers, Moonan, and Tschoegl used several years ago to investigate the influence of pressure and temperature on the mechanical properties of time dependent polymeric materials. The new apparatus can measure the volume and the shear relaxation moduli of solid polymer specimens, subjected to a combination of temperatures from –60°C to +110°C, and pressures from atmospheric to 360 MPa. The paper demonstrates the capabilities of the new apparatus from the scientific as well as engineering stand point. Shear relaxation measurements on poly(vinyl acetate) (PVAc) and styrene-butadiene rubber (SBR) are also presented. For PVAc we present also the bulk creep compliance, coefficient of thermal expansion and the bulk modulus.

1. Introduction Although high isotropic pressure is commonly employed in many demanding structural applications and during several manufacturing processes, there is little information available about its effect on the mechanical properties of polymeric materials. Few instruments exist that can measure the effect of pressure on polymers. This paper describes an apparatus, built at the Center of Experimental Mechanics (CEM) of the University of Ljubljana, for researching the effect of pressure on the time-dependent properties of solid polymers. An extended measurement description is published in another paper [4]. For liquid polymers, Bair built a rheometer that can determine rheological properties of liquid polymer lubricants under high hydrostatic pressure [1]. A detailed account of the history and development of various attempts to describe mathematically the time-dependence of the bulk properties is presented elsewhere [7]. The CEM apparatus is an improved and upgraded version of the apparatus developed originally by Fillers, Moonan, and Tschoegl. The first version [3] measured stress relaxation in uniaxial tension. This was later modified extensively [6] to allow the determination of stress relaxation in shear. The new version, described here, improves the measurement technique in many details and greatly extends the usefulness of the original device. The apparatus now also performs various types of volumetric measurements. One of the more noteworthy features of the new CEM apparatus is that it can perform measurements of shear relaxation either isothermally or isobarically on the same specimen. Thus, it allows the determination of both the temperature shift factors, and the pressure shift factors, required to predict the effect of the 19 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 19–30. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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temperature, or of the pressure, on the mechanical properties of polymeric materials. This is becoming more and more important because of the rather high pressures utilized in the state-of-the-art molding operations. In addition, measurement of the time-dependent properties, not only isothermally but isobarically as well, lead to the removal of the ambiguity in the way the temperature shift factors, are currently determined. The background to this is presented in Section 6 of this paper. This paper describes the CEM apparatus, along with results on two polymeric materials: poly(vinyl acetate) (PVAc) and styrene-butadiene rubber (SBR). The temperature and pressure shift factors were determined for both polymers.

2. The CEM pressure apparatus The assembly is shown schematically in Fig. 1. The pressure is generated by the pressurizing system using silicone fluid. The pressure vessel is contained within the thermal bath, where another silicone oil circulates from the stirred circulator. The apparatus utilizes two separate measuring devices that can be inserted into the pressure vessel: a relaxometer, shown in Fig. 2a, and a dilatometer, shown in Fig. 2b. Signals from the measuring inserts pass through the carrier amplifier prior to being collected in digital format by the data acquisition system (DAQ). The magnet and motor charger supplies current to the electric magnet, which activates the measurements. The same charger also supplies current to the electric motor of the relaxometer, shown in Fig. 2a, which pre-loads the spring that then applies the desired deformation to the specimen. Specimens can be subjected to pressures of up to 360 MPa with a precision of ±0.1 MPa, and to temperatures from –60°C to +110°C with a precision of ±0.01°C. The pressure limit is currently being extended to 850 MPa.

2.1 THE RELAXOMETER The relaxometer insert, shown in Fig. 2a, measures the shear relaxation modulus by applying a constant torsional strain to a cylindrical specimen, and by monitoring the induced moment as a function of time. The two main parts of the insert are the loading device, and the load cell. The loading device, applies the torsional strain by twisting the specimen for a few degrees in less than 0.01 seconds. To effect the deformation of the specimen, the electric motor is used to first pre-load an

APPARATUS FOR PRESSURE AND TEMPERATURE

21

Archimedes spring build into the triggering mechanism. Once twisted, the spring is kept in its pre-loaded position by a rack-and-pawl mechanism. Activation of the electric magnet, mounted outside the pressure vessel (see Fig. 1), pulls the pawl out of the rack, and the energy of the spring deforms the specimen to a pre-determined angle. The angle can range from 1° to 15°. The induced moment is then measured by the load cell, which is attached to the slider mechanism to compensate for possible changes in the length of the specimen resulting from various temperature and pressure conditions. After the shear relaxation measurement is complete, the electric motor brings the specimen to its original undeformed state, while maintaining the pressure vessel fully pressurized. The specimen diameter can range from 2.5 mm to 12 mm, while its length may vary from 50 mm to 58 mm. The relaxometer can measure shear moduli ranging from 0.01 MPa to 10 000 MPa, with an accuracy within ±5% of the modulus value.

2.2 THE DILATOMETER The dilatometer insert is used to measure bulk properties such as the bulk creep compliance, the equilibrium bulk modulus, and the coefficient of thermal expansion. A schematic of the insert is shown in Fig. 2b. The measurements are performed by monitoring the volume change of the specimen, which results from the imposed changes in pressure and temperature, by measuring the change in length of the specimen with the aid of a built-in Linear Variable Differential Transformer (LVDT). The volume estimate can be considered accurate if the change in volume is small (up to a few percent) and the material is isotropic. Specimens for the dilatometer can be up to 16 mm in diameter and 50 mm in length. Measurements are accurate within 0.1% of the specimen’s specific volume, which corresponds to resolutions better than 0.0001 Length dilatometry has several advantages over the mercury confinement volume dilatometry. One of the most important is its high accuracy, which, combined with easily automated measurement procedure, allows tracking of any possible transient volume changes. However, there is also an important limitation that one needs to consider when working at temperatures above Above specimens creep downward due to their own weight. Given the arrangement of the LVDT rod (see Fig. 2b), there will be an additional creep downward caused by the

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T. PRODAN, I. EMRI

weight of the rod. If the combined creep downward is significant, the measurement will over time considerably underestimate the true volume of the specimen.1

3. Materials and specimen dimensions We used two materials: poly(viny1 acetate) (PVAc) and styrene-butadiene rubber (SBR). Pertinent data are assembled in Table 1. The specimen used for shear relaxation measurements had a diameter of 4.3 mm and a length of 58 mm. The specimen used for volume measurements had a diameter of 12.1 mm and a length of 52.2 mm. Both specimens were cast from the same material batch and were both thermally annealed prior to testing.

4.

Experimental methods

We determined the shear relaxation modulus, G(t), by measuring the decaying moment M(t), after applying a sudden angular displacement, to the cylindrical specimen with diameter at time The modulus is calculated from

On the poly(vinyl acetate) (PVAc) sample we additionally measured the timedependent bulk compliance, B(t), the equilibrium bulk modulus as function of temperature, K(T), and its pressure dependence at the reference temperature using the Murnaghan equation,

where the asterisk indicates that the modulus is referred to zero pressure. The measurements started at the lowest temperature and were continued in increasing temperature order. The specimen was held at each temperature for 2.5 hours before G(t) was measured. The bulk creep compliance, B(t), is the time-dependent relative cubical contraction, v(t), measured in the dilatometer in response to an instantly applied hydrostatic pressure, P, while the temperature is maintained constant. It is obtained from

where

and V(t) is the volume at time t, while

is the original undeformed volume.

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5.

23

Experimental results

We first present results on the shear relaxation moduli measured on both materials at different temperatures (i.e., isothermally) and different pressures (i.e., isobarically). The left half of each figure shows either the isothermal or the isobaric measurement segments, while the right half of each figure the mastercurves obtained by shifting the segments into superposition at the chosen reference temperature or pressure. The shift factors, and are shown in the inserts on the right side of each figure.

5.1 DETERMINATION OF THE SHEAR RELAXATION MODULI, G(t) The two materials were subjected to shear relaxation measurements in the CEM relaxometer.

5.1.1. PVAc We begin by presenting the results obtained on PVAc. The left plot in Fig. 3 shows G(t) for PVAc at various temperatures and constant atmospheric pressure. 40°C was chosen as the reference temperature. This temperature is roughly 10°C above the glass transition temperature of PVAc, No vertical correction factors were applied.

A second series of measurements of G(t) were performed at various pressures while maintaining the specimen at the reference temperature. The measurements shown in Fig. 4 started at atmospheric pressure and continued in ascending pressure order, as indicated in the plot on the left side. The specimen was held at each pressure for 2.5 hours before G(t) was measured. The reference pressure was 0.1 MPa.

5.1.2.

SBR

Figures 5 and 6 show the same information for the SBR rubber, In Fig. 5 the reference pressure was 200 MPa. At atmospheric pressure the material was in the

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T. PRODAN, I. EMRI

rubbery region even at the lowest temperature used. The specimen was first pressurized at room temperature. Then it was cooled down to the lowest experimental temperature where it was equilibrated over night. Afterwards, the measurements were performed at the temperatures in the order indicated in the figure. Between each measurement the specimen was equilibrated for 2.5 hours.

The isobaric experiments shown in Fig. 6 were performed at the reference temperature, The specimen was first cooled to the reference temperature at atmospheric pressure, and then subjected to the pressures in increasing order as indicated in the figure. The glassy region could not be reached because of our 360 MPa pressure limit.

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25

5.2. DETERMINATION OF THE BULK COMPLIANCE, B(t) The bulk creep compliance, B(t), is determined by monitoring the volume change of the specimen over time, after a pressure drop is applied, while the temperature is maintained constant. The PVAc sample was first heated to the reference temperature pressurized to the (absolute) pressure ( P = 10.1 MPa ), and then equilibrated in this state overnight. A pressure step function of was then applied by rapidly releasing the pressure to atmospheric, and the change in the volume of the specimen was monitored as function of time. The measurement is shown in Fig. 7.

5.3. DETERMINATION OF THE EQUILIBRIUM BULK MODULUS, K(P) The volume change as function of the pressure was measured in the dilatometer at the reference temperature for pressures ranging from atmospheric (i.e., 0.1 MPa) to 100.1 MPa. The specimen was first heated to the reference temperature at atmospheric pressure, which was then gradually increased in steps of

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T. PRODAN, I. EMRI

MPa. At each pressure the sample was equilibrated for 2.5 hours before taking the measurements. The results are shown in Fig. 8.

We fitted a quadratic equation to these data in order to calculate the equilibrium bulk modulus as a function of pressure, using the equation,

The result is presented in Fig. 9.

Fitting the Murnaghan equation, Eq. (2), to this data we obtained the parameters MPa and which define the pressure dependence of the equilibrium bulk modulus of PVAc at zero pressure and at the reference temperature

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27

5.4. MEASUREMENTS OF THE THERMAL EXPANSION COEFFICIENT OF PVAc The volumetric thermal expansion coefficient was measured in the dilatometer at the two different pressures, 0.1 MPa and 50 MPa. The specimen was first pressurized to the experimental pressure, and then heated to the experimental temperatures shown in the figure. After being exposed to these conditions for 2.5 hours, the measurements were taken. The results are shown in Fig. 10. The thermal expansion coefficient was then calculated from

6.

Background to the WLF and FMT equations

We present here the theoretical background that is required to understand the advantage, or more appropriately, the need to carry out time-dependent measurement on a linear viscoelastic material, both isothermally and isobarically, if its mechanical behavior is to be fully characterized. The temperature dependence of the mechanical properties of a polymeric material is commonly described by the well-known WLF (Williams, Landel, Ferry) equation

where an are material constants. This equation is derived from the Doolittle equation, which may be written in the form [2]

where is the fractional free volume, is the fractional free volume in the reference state, and B is a proportionality factor. The free volume is a function of temperature. Letting

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T. PRODAN, I. EMRl

where is the expansion coefficient of the fractional free volume and chosen reference temperature, leads to the equation,

is the

Introducing the constants

and

leads to Eq. (6). The parameters and are obtained experimentally by shifting isothermal segments of any linear viscoelastic response curve into superposition with the reference segment at Thus we have only the two experimentally determined parameters, and to represent the three parameters of the Doolittle equation, B, and Thus, the WLF equation is underdetermined unless either B or is known. To deal with this predicament, it is customary to set the bothersome proportionality factor B to unity. One then obtains

and

However, as had been shown already by Moonan and Tschoegl (1985), and is confirmed in this paper, B is generally not equal to unity. These authors have also shown that by carrying out isobaric as well as isothermal measurements as function of time the ambiguity can be removed. Measurements of isobaric as well as isothermal time dependence lead to the FMT (Fillers, Moonan, Tschoegl) equation

where

and

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29

The first superscripts on the c’s refer to the reference temperature, The second superscripts refer to the reference pressure, and are the Murnaghan parameters in the equilibrium (rubbery) state. The subscript identifies the same parameters for the occupied volume. An asterisk indicates that the bulk modulus data are referred to zero pressure. and are determined in the CEM apparatus using the dilatometer and therefore the constant is known. For the determination of the remaining c’s see [5]. The FMT parameters and are obtained by solving the Eqs. (14-16) and (18-19). At the reference pressure vanishes and we regain the WLF equation. However, since we now have information on B, and may be found from the equations

and

7.

Discussion

The primary purpose of this paper is to demonstrate the capabilities of the CEM apparatus. As could be gathered from Section 5, in its present form, the apparatus is able to determine G(T, P, t), B(T, P, t), K(P,T), and . It is a noteworthy feature that measurements can be made under isothermal as well as isobaric conditions. Such simultaneous measurements prove to be indispensable for the unambiguous determination of the semi-molecular parameters as well as of and These parameters are needed as input into computer codes for numerical analysis of technological processes and the stress analysis of structural elements. As discussed in Section 6, isothermal measurements often lead to incorrect values for these parameters. For the four materials we investigated, this is shown in Table 2. For each material there are two columns. The first, denoted as T, indicates isothermal measurements only, while the second, denoted as T&P, indicates measurements performed isothermally as well as isobarically. Table 2 clearly shows that the assumption that leads to the wrong estimation of and For PVAc the values of the two parameters calculated on the assumption that are underestimated, while for SBR they are overestimated. In both cases the difference is definitely not acceptable for any engineering application.

30

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T. PRODAN, I. EMRI

Conclusions

Finally, we conclude that the CEM apparatus is a highly useful tool for acquiring the necessary materials input data for the numerical optimization of processing technologies, as well as for the long-term prediction of the geometric stability and durability of structural elements composed of polymers and composites.

9. Acknowledgement We wish to point out that our endeavors are a continuation of the work initiated by Professor N. W. Tschoegl of the California Institute of Technology. We are grateful for his abiding interest and guidance.

10. References 1. 2. 3. 4. 5.

6. 7.

Bair S. (1996) Normal stress difference in liquid lubricants shared under high pressure, Rheol. Acta, 35, 13-23. Ferry, J.D. (1983) Viscoelastic Properties of Polymers, 3rd ed., John Wiley and Sons, 285-286. Fillers, R.W. and Tschoegl, N.W. (1977) The Effect of Pressure on the Mechanical Properties of Polymers, Trans. Soc. Rheol., 21, 51-100. Kralj, A., Prodan, T., Emri, I (2001) An Apparatus for Measuring the Effect of Pressure on the Time-Dependent Properties of Polymers, J. of Rheology, 45/4, 929-943. Moonan, W.K. and Tschoegl, N.W. (1983) The Effect of Pressure on the Mechanical Properties of Polymers 2. Expansivity and Compressibility Measurements, Macromolecules, 16, 55-59. Moonan, W.K., and Tschoegl, N.W. (1985) The Effect of Pressure on the Mechanical Properties of Polymers. 4. Measurements in Torsion, Polymer Sci., Phys. Ed., 23, 623-651. Tschoegl, N.W., Knauss, W.G., Emri, I. (2002) The Effect of Temperature and Pressure on the Mechanical Properties of Thermo- and/or Piezorheologically Simple Polymeric Materials in Thermodynamic Equilibrium – A Critical Review, Mech. Time-Dep. Mater, 5/1 (in print).

HIGH STRAIN RATE TESTING OF SANDWICH CORE MATERIALS M. VURAL Graduate Aeronautical Laboratories, California Institute of Technology Pasadena, CA 91125 Department of Aeronautical Engineering Istanbul Technical University Maslak, Istanbul, 80626, TURKEY G. RAVICHANDRAN Graduate Aeronautical Laboratories, California Institute of Technology Pasadena, CA 91125

Abstract

Mechanical response of a cellular sandwich core material, balsa wood, is investigated over its entire density spectrum from 55 to Specimens were compression loaded along the grain direction in both quasi-static and dynamic strain rates from to Results show that while the initial failure stress is very sensitive to the rate of loading, plateau (crushing) stress remains unaffected by the strain rate. Kinematics of deformation and micro inertial effects are suggested and discussed to explain this different behavior. Specific energy dissipation capacity of balsa wood was measured and determined to be comparable with those of fiber-reinforced polymers. As in quasistatic loading, buckling and kink band formation were identified to be two major failure modes in dynamic loading as well. 1. Introduction

Sandwich structures, which are commonly used to increase structural stiffness, are also used in critical applications as energy dissipators. In these applications, energy is transmitted to the system mostly by the impact of a foreign object and dissipated through the compressive inelastic deformation of the core material used in the sandwich structure. Flow (crushing) stress of the core material determines the load transmitted to the system during dissipation process and the deceleration of impactor as well. Thus it is desired to be kept at an optimum level, which is not too high to damage the container and its contents and not too low to dimmish the amount of energy dissipated. Once the flow stress is established the stroke of deformation (or the maximum strain before stress elevation) determines the amount of energy that can be dissipated safely. Therefore, the constitutive response of an ideal energy dissipator is that of a rigid-perfectly plastic 31 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 31–42. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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material capable of tolerating as large strains as possible. Within this frame balsa wood, as a naturally occurring porous biocomposite, offers salient mechanical and physical properties. As its cellular/porous microstructure allows the application of a long deformation stroke, fine composite nano-architecture of wood cell material increases its specific strength and stiffness, giving rise to a high specific energy dissipation capacity. Moreover the fact that balsa wood can be found in a wide range of densities from 40 to 380 , depending on the average size and the wall thickness of cells, provides the flexibility in design since the strength is a monotonic function of its density. As shown in Fig.1, the cellular microstructure of balsa (Ochroma) is mainly composed of long prismatic cells (fibers, grains) of nearly honeycomb shape and the blocks of these cells are separated by narrower rays in which the cells are smaller and of a different shape. Balsa wood has three orthogonal axes in the longitudinal (L, along the grain), radial (R, across the grain and along the rays) and tangential (T, across the grain and transverse to rays) directions forming a heterogeneous porous composite. This composite is highly anisotropic with a high ratio of longitudinal to transverse properties, the latter having also difference in itself due to so-called ray cells oriented in radial direction. For applications where balsa wood is considered as a potent material to dissipate energy, its dynamic constitutive response in longitudinal direction as well as corresponding failure mechanisms are of particular interest.

Numerous uniaxial quasi-static tests have been performed to determine the mechanical properties and deformation mechanisms of balsa wood in longitudinal and/or transverse directions by Knoell [1], Soden and McLeish [2], Easterling et al. [3] and Vural and Ravichandran [4]. Knoell [1] investigated the effects of environmental and physical variables (temperature, moisture content and ambient pressure) on the mechanical response of balsa wood. Soden and McLeish [2] carried out extensive tests and mainly concentrated on the variation of tensile strength with fiber alignment. They

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also reported compressive strength data. Easterling et al. [3] paid particular attention to the micromechanics of deformation in their experiments, during which they performed in-situ SEM observations and defined the end-cap collapse of grains as the dominant compressive failure mechanism in longitudinal direction. Vural and Ravichandran [4] documented the compressive strength, plateau stress and densification strain of balsa wood in its entire density range, identified the variations in failure mechanisms with density and described simple analytical models to represent experimental strength data. On the other hand, dynamic compression behavior of balsa wood (and in general wood) has received limited attention. Daigle and Lonborg [5] performed dynamic compression tests over balsa wood and several other crushable materials using drop towers and compared their specific energy absorption capacities. However, a systematic investigation on the dynamic response of balsa wood for different strain rates and specimen densities that covers its full range is still lacking. In the present study, the quasi-static and dynamic response of balsa wood is investigated at strain rates from to with a particular emphasis on the energy dissipation characteristics and their dependence on the density and strain rate. The dominant deformation modes at micro-scale during the process of energy dissipation (crushing) and their dependence on density are discussed and documented through the examinations performed on the sectioned surfaces of recovered specimens by scanning electron microscope (SEM). 2. Experimental Procedure 2.1. SPECIMENS

Experiments were performed on balsa wood specimens over a wide range of densities between 55 and _ Cylindrical specimens were machined from big blocks of balsa wood at five different nominal densities provided by BALTEK Corporation (Northvale, NJ) and polished using 320 grit sand papers. Densities were measured in a consistent way, paying considerable attention particularly to the moisture content since it has significant effect on both density and properties [6]. Due to the variation of density within each block of a certain nominal density, specimens were prepared with densities varying continuously between the ranges specified above. In order to find moisture contents of balsa specimens a series of specimens were heat-treated in furnace at 110 °C and intermittent weight measurements were performed until they reached a constant weight. Measured moisture content ranged from 8 to 11 %, which is the typical value for well-seasoned dry woods. 2.2. QUASI-STATIC TESTS

The specimens used in quasi-static tests were typically 19.05 ± 0.12 mm in diameter and 25.4 ± 0.12 mm in length, corresponding to a length to diameter ratio of 1.33. A compression fixture was used in the quasi-static compression tests. It ensured that two loading rods were perfectly aligned with each other so that any unwanted shear forces on the specimen are minimized. The specimen was sandwiched in between the loading rods and the compression was applied by a screw-driven materials testing system

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(Instron, model 4204). The quasi-static tests were performed at an overhead displacement rate of 2 mm/min, which corresponds to nominal strain rate of The use of bonded resistance strain gages on highly compliant balsa specimens was not considered satisfactory because the reinforcing effect of bonding these gages with adhesive might lead to large errors in measurement of strains. Therefore, deformation data of specimens was obtained from the crosshead displacement transducer, which was calibrated to eliminate the machine and fixture compliance. 2.3. DYNAMIC TESTS For dynamic tests, cylindrical specimens of a smaller dimension, 9.5 ± 0.12 mm in diameter and 5.08 ± 0.12 mm in length, were loaded using a modified Kolsky (split Hopkinson) pressure bar as shown in Fig. 2. Kolsky pressure bar, originally developed by Kolsky [7], has been widely used and modified to measure the dynamic compressive behavior of engineering materials. The present modification through the use of a quartz single crystal, first introduced by Chen et al. [8], enables stress measurements three orders of magnitude more sensitive than in a conventional split Hopkinson pressure bar. Hence, low stress profiles experienced by balsa specimens could be measured with high precision by adopting this modification.

The dimensions of the bars in Kolsky pressure bar setup used in this study are 1219, 584 and 610 mm in length for the incident and transmission bars respectively, with a common diameter of 12.7 mm. The striker bars of 12.7 mm diameter were used with varying lengths to achieve desired pulse duration. All the bars are made of 7075T651 precision ground aluminum alloys. Depending on the striker velocity, either a thin, annealed copper (C11000) disc of 4.8 mm in diameter and 0.81 mm in thickness or a layer of tissue paper was typically used as pulse shaper to control the rise time of the incident pulse and to ensure stress equilibration within the specimen [9]. A high resistance strain gage (Micro-measurements, WK-06-250BF-10C) with excitation voltage of 30 volts was used to measure the surface strain on incident bar. The raw strain gage signal was recorded using a high-speed 12-bit digital oscilloscope

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(Nicolet 440). A Valpey-Fisher X-cut quartz single crystal of 12.7 ± 0.025 mm in diameter and 0.254 ± 0.025 mm in thickness, coated with a 30 nm thick chrome/gold layer on both of the flat surfaces, was glued to the end of the specimen-contact half of the transmission bar using a conductive epoxy (CW2400, Chemtronics). Other surface remained in contact with the other half of the transmission bar without bonding, allowing the compressive transmitted pulse to pass the quartz without reflection. Before each experiment, a thin layer of conductive grease (CW7100, Chemtronics) was applied at the contact surface to ensure electrical conductivity. Thin, silver coated copper wires were wrapped and glued (using the same conductive epoxy) to the bar surface near the quartz crystal. The wires were connected to a charge amplifier (Kistler 5010 B1) whose output is recorded using the same oscilloscope for data processing. Using the modified Kolsky pressure bar system described above dynamic compression tests were performed at nominal strain rates between and 3.

Results and Discussion

3.1 MECHANICAL RESPONSE

In all experiments, specimens of balsa wood were loaded in longitudinal (L) direction (i.e., along the grain). Characteristic features of the typical engineering stress-strain curves for balsa wood obtained from these tests in both quasi-static and dynamic range are as shown in Fig. 3. The stress-strain curve is almost linear up to a maximum stress (failure strength) beyond which, as the deformation is increased, the stress level either remains nearly constant or experience a sudden drop depending mainly on the density of specimen. After this sudden drop, if there is, the specimen continues to deform at a lower level of stress (plateau stress). Eventually, at high strains (after the wood cells collapse sufficiently so that opposing cell walls touch (or their broken fragments pack together) and further deformation compresses the cell wall material itself. This gives the final, steeply rising portion of the stress-strain curve called densification regime.

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Figure 4 illustrates the dependence of compressive failure strength and plateau stress, respectively, on the density and strain rate for balsa wood. Solid lines in Fig. 4a represent the best linear fits to the experimental data. It is obvious that both the density and the strain rate have considerable effects on the failure strength of balsa wood. Higher density means increased ratio of cell wall thickness to cell diameter, and thus less porosity, therefore its effect towards fortifying the structure is simply expected and experimentally observed in both failure strength and plateau stress. However the effect of strain rate seems to be more complicated due to the fact that the plateau stress is practically insensitive to the strain rate whereas the failure strength is significantly affected by the increase in strain rate. In other words, experimental evidence indicates that as the critical stress for failure initiation has a strong dependence on the strain rate driving force for progressive deformation remains unaffected by the dynamics of deformation. This different response to increasing strain rate can simply be explained in terms of the differences in the kinematics of deformation, associated inertia and triggering mechanism at micro scale prior to initial failure and during progressive deformation. Initial failure at the peak stress level is preceded by elastic loading. Micromechanics of deformation in this regime is characterized by the uniform axial compression of wood cells in longitudinal direction. Initial failure is mostly triggered by an elastic instability in the form of local buckling or kink band formation depending mainly on the density of balsa wood, as will be discussed in the next section. In dynamic loading, both kind of instability is associated with the lateral inertia in deforming elements, which in turn serve as increasing the critical stress for failure initiation in longitudinal direction. Simple calculations based on the kinematics of buckling/bending geometries show that even though the initial effect of this inertial hardening is very strong it rapidly falls down and becomes negligible during the evolution of deformation in a localized zone. However, progressive deformation phase is also characterized, in addition to the continuation of deformation in initially localized regions, by the creation of new localization zones. Following the initial failure/localization, the state of stress experienced by neighboring cells is perturbed and far from being uniaxial due to the kinematics of deformation at micro scale. This new local situation (state of stress) makes the following progressive deformation (localizations) easier and at a lower nominal stress level by taking and exaggerating, in a sense, the role of local geometric and material imperfections that trigger the initial failure. Within this frame, regarding the formation of new strain localizations, it appears that the softening effect due to the perturbations in stress field created by preceding localization, suppress the effect of inertial hardening. As conclusion the effect of strain rate, and hence micro inertia, turns out to be restricted to the initiation phase of first instability due to the kinematics of failure (buckling/kink banding). During the following progressive deformation phase the effect of inertia is suppressed either by the kinematical considerations, as in the evolution of deformation within localized region, or by the perturbations in stress field as in the creation of new localization sites. Figure 4b also shows an increased scatter in plateau stress when the specimen density exceeds 170-190 interval. This is attributed to the transition in deformation mode from buckling to kink band formation and resulting increased susceptibility of plateau stress to the random nature of increased perturbations in stress field. As the length scale of the deformation geometry increases from a fraction of cell

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diameter in buckling mode to as much as 10-15 times the cell diameter in kink band formation mode, an increased level of perturbation is induced in the stress field and it generates the increased scatter in plateau stress for denser specimens.

Another aspect of stress-strain response is the significant decrease observed in densification strain with increasing density and strain rate (see Fig. 5), which is defined in this study as the strain at the last local minimum before the stress starts rising steeply. For quasi-static loading rates, densification strain decreases as a function of

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specimen density and varies between 0.68 and 0.86. As expected, the denser is the balsa wood the lower is the densification strain due to its close correlation with the level of initial porosity. In dynamic loading regime, the same declining linear trend in densification strain is preserved. However, it is translated downwards and the bounds decrease to the range between 0.58 and 0.74, suggesting that cellular packing efficiency at microstructural level degrades by the increasing rate of strain. This could be attributed to the micro-inertial effect associated with the cell wall collapse.

Plateau (crushing) stress and densification strain are two important quantities that directly determine the energy dissipation capacity of the material under consideration. Specific energy dissipation capacity of balsa wood is shown in Fig. 6 for both unit volume and unit mass. Data of Fig. 6 were obtained by integrating the area under stressstrain curves, rather than simply multiplying plateau stress and densification strain, up to the strain schematically denoted by in Fig. 2. Therefore, the data presented represent the maximum amount of energy that can be dissipated safely, i.e., by limiting maximum transmitted force to the one associated with the failure strength of balsa wood. The effect of transition in progressive deformation mode from buckling to kink band formation is apparent in Fig. 6 either in the form of plateau region (volume based data) or in the form of a temporary decrease (mass based data). When considered together with the decrease in densification strain with density (Fig. 5), slight leveling of plateau stress (see Fig. 4) during this transition results in the obvious leveling of the energy dissipated per unit volume (Fig. 6). The effect of increasing density in this

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transitive plateau region can be seen as a transient drop in the energy dissipated per unit mass. As a whole the energy dissipation capacity of balsa wood lies between 30-90 kj/kg, which is comparable or better than the most fiber reinforced composite tubes. Mamalis et al. [10] report in their review for the crashworthy capability of composite material structures that specific energy dissipation/absorption capacities of fiber reinforced hallow tubes made of Carbon/Epoxy, Aramid/Epoxy, Glass/Epoxy and Carbon/PEEK lie between 50-99, 9-60, 30-53, 127-180 kj/kg respectively, variations arising mostly from the fiber orientation angle and stacking sequence. Actually balsa wood owes its remarkable energy dissipation capacity to its naturally tailored microand nano-structural features such as the tubular honeycomb geometry of its cells and the fine nano-architecture and composite nature of its cell walls.

Figure 6 also shows the effect of elevated strain rate on the energy dissipation capacity. Even though there appears a slight increase in dynamic data for low densities it is highly scattered and intermixed with the quasi-static data for higher densities. Once again this phenomenon is attributed to the transition in deformation mode and, thus, to

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the increased effect of random stress perturbations on the plateau stress level for high densities. For low-density balsa the cells deform by buckling. Since the length scale of local deformation is limited to the length of buckles, which is much smaller than that in kink banding mode, stress perturbations are also small and therefore plateau stress benefits from micro inertial effects. However, with the transition in deformation mode, the length scale involved becomes higher and, therefore, induces larger stress perturbations which, in turn, both suppress the inertial effects and result in a wider scatter. 3.2. FAILURE MODES In order to reveal microstructural failure modes some of the balsa specimens are first compressed to predetermined strains and then carefully sectioned to perform SEM examination on the failure surfaces. The SEM observations on recovered specimens from both quasi-static and dynamic tests clearly show that the failure mode undergoes transition from elastic/plastic buckling of cell walls to kink band formation as the density of the specimen tested increases. The micrographs of two specimens showing the typical dynamic deformation mode in low- and high-density balsa wood are presented in Figure 7. The rate of loading does not change the deformation mode. Similar micrographs for quasi-statically loaded specimens can be found in Ref. [4]. The major mechanism during both initial failure and progressive deformation is buckling for low-density balsa wood. When the specimen density goes beyond 170-190 range, thickness to diameter ratio of tubular cells increases and the critical stress for buckling becomes too high so that another mechanism (kink band formation) starts operating. Initiation of this new deformation mode is very sensitive to and triggered by initial fiber (cell, grain) misalignments which are naturally present in the microstructure. Therefore, the appearance of kink band formation in denser specimens is attributed to the increase in fiber misalignments caused by the thicker ray cells penetrating the entire microstructure in radial direction. Ref. [4] reports the experimental evidence of larger fiber misalignments in denser balsa wood as well as analytic models that explain the transition in failure modes. 4. Conclusions The mechanical behavior of balsa wood over its entire range of densities has been investigated under quasi-static and dynamic loading conditions. In both cases, the compressive failure strength and plateau stress of balsa wood increase as a function of its density while the densification strain decreases. Even though the application of high loading rates elevates the initial failure stress as much as 1.5 to 2 times, plateau stress remains unaffected by the strain rate. This different response is attributed to and explained by the kinematics of deformation and associated micro inertial effect. Specific energy dissipation capacity (per unit mass) of balsa wood, as a naturally occurring material, was found to be comparable with advanced polymeric composites. The post-test SEM examinations on dynamically loaded specimens reveal that failure mode undergoes transition from elastic/plastic buckling to kink band formation with increasing density, as also observed in quasi-statically loaded specimens.

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5.

41

Acknowledgements

This research was supported by the Office of Naval Research (Dr. Y. D. S. Rajapakse, Scientific Officer) and is gratefully acknowledged. MV gratefully acknowledges the support provided by (The Scientific and Technical Research Council of Turkey) through the NATO Advanced Science Fellowship.

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6. References 1. 2.

3. 4. 5. 6.

7. 8. 9. 10.

Knoell, A.C.: Environmental and physical effects of the response of balsa wood as an energy dissipator, JPL Technical Report No. 32-944, California Institute of Technology, Pasadena, CA, 1966. Soden, P.D., McLeish, R.D.: Variables affecting the strength of balsa wood, J. Strain. Anal. 11(4), (1976), 225-234. Easterling, K.E., Harrysson, R., Gibson, L.J., Ashby, M.F.: On the mechanics of balsa and other woods, Proc. R. Soc. Lond. A 383, (1982), 31-41. Vural, M. and Ravichandran, G.: Microstructural aspects and modeling of failure in naturally occurring porous composites, Mech. Mater., (2001), submitted for publication. Daigle, D.L., and Lonborg, J.O.: Evaluation of certain crushable materials, JPL Technical Report No. 32-120, California Institute of Technology, Pasadena, CA, 1961. Dinwoodie, J.M.: Timber – a review of the structure-mechanical property relationship, Journal of Microscopy 4(1), (1975), 3-32. Kolsky, H.: An investigation of mechanical properties of materials at very high rates of loading, Proc. Phys. Soc. Lond. Ser. B 62, (1949), 676-700. Chen, W., Lu, F., and Zhou, B.: A quartz-crystal-embedded split Hopkinson pressure bar for soft materials, Experimental Mechanics 40(1), (2000), 1-6. Ravichandran, G., and Subhash, G.: Critical appraisal of limiting strain rates for compression testing of ceramics in split Hopkinson pressure bar, J. of the American Ceramic Society 77(1), (1994), 263-267. Mamalis, A.G., Robinson, M., Manolakos, D.E., Demosthenous, G.A, Ioannidis, M.B. and Carruthers, J.: Review: Crashworthy capability of composite material structures, Composite Structures 37, (1997), 109-134.

DEVELOPMENT OF A SHEAR TEST FOR LOW MODULUS FOAM MATERIALS AJIT K. ROY Air Force Research Laboratory ARFL/MLBC, Wright-Patterson AFB OH 4543 3-7 750 JOHN D. CAMPING University of Dayton Research Institute Dayton, OH 45469-0168 Abstract The shear properties of carbon (graphitic) foam, being brittle and highly porous, are not reliably measured with most of the standard test methods, such as, single rail, double rail, Iosipescu shear, etc. A new testing device is developed to accurately measure the shear stiffness and strength of carbon foam or other porous materials. Specimens of cylindrical cross-section are used to reduce high stress concentration that normally occurs in the vicinity of the grip section. Since strain gages could not be installed on the specimen surface (due to porosity), the shear strain is determined from the specimen end rotation. A high accuracy in the rotational measurement is achieved by using a stepper motor with multiple gear reduction. In view of testing low modulus material, the load cell of fixture was mounted on an axial roller to relieve axial constraint while twisting the specimens. The accuracy of the measurement and calibration of the test fixture was demonstrated by measuring shear modulus of PVC plastic.

1. Introduction The intrinsic superior mechanical and thermal properties of carbon material (carbon fiber, in particular) than metals make carbon (graphitic) foam an attractive ultralightweight material for space and thermal management applications. Its process tailorability makes it work as an insulator and a radiator as well. Its macroscopic near isotropic property makes it an attractive core material for sandwich structures over traditional honeycomb. The material also a good candidate for net shape fabrication of structural components providing three-dimensional reinforcements through its ligaments. The micrograph of a graphitic (open-cell) foam microstructure is shown in Figure 1. Based on the minimum surface energy during the foaming process (i.e., the bubble nucleation process), the microstructure of an open-cell foam usually possesses a tetrahedral structure of the foam ligaments oriented approximately 109° with each other. Due to the tetrahedral cell microstructure, the macroscopic properties, such as foam modulus and strength, are critically influenced by the deformation characteristics of the cell ligaments. Thus, to develop a basic understanding of the performance of open-cell foam materials, as a first step, there 43 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 43–54. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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is need to measure its bulk (macroscopic) properties and then correlate the bulk properties with the foam ligament microstructure.

There is an abundance of literature on modeling foam properties, but very limited on measuring foam properties in the contrary. Among the modeling efforts, the most widely used reference is that of Gibson and Ashby [1]. Christensen [2, 3] developed a few closed form relations for bulk properties of foam assuming membrane behavior of foam ligaments. Warren and Kraynik [5] incorporated foam ligament bending in their model. Zhu, et at [6, 7] analyzed the elastic properties and compression characteristics of open cell foam with tetrakaidecahedral cell geometry. Although some effort has been undertaken to measure foam properties in the past, such as the work of Kau et al. [8, 9] and Bilkhu [10], a comprehensive effort of correlating foam ligament structure with foam bulk properties appears to be lacking to assess or validate the existing foam models. Christensen [4] also reiterates the need for comprehensive experimental efforts for measuring foam properties in his recent review paper. The foam material considered here contains porosity of around 80 percent or even higher, with pore size of in diameter (Figure 1). In general, foam being a highly porous and low modulus (about one order of magnitude lower than that of standard PVC plastic) material, traditional strain measuring devices (such as, surface mount strain gages) fail to work reliably in measuring its strains. Surface mount strain gages are not compliant enough to measure the true strain of foam. Strain gages tend to add stiffness to this type of material; hence its apparent measured strain is lower than the strain in the material. Clip extensometers, another strain measuring device used in many engineering materials, do not work with foam due to open material porosity and lack of fixed extensometer clipping point on the material surface. Further, measuring shear strain even gets more difficult in porous materials like foam. The objective of this study is to develop a test method to

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measure the shear modulus and strength of foam. In developing the test method a careful attention is given on the low load requirement and the shear strain measurement accuracy, for measuring shear modulus and strength of foam (porous) materials.

2. Test method considerations for shear properties of carbon foam There are several standard shear tests, such as Single Rail, Double Rail (ASTM Standard D 4255), and Iosipescu, (ASTM Standard D 5379) which are used in measuring shear properties of engineering materials [11]. The Single and Double Rail shear tests are not suitable for measuring low strength, brittle and porous materials, such as foam, because the adhesive that is used to bond foam to the rails tend to ingress through the foam pores and induce an artifact in the test data. The present ASTM standard for measuring shear properties of sandwich core materials (ASTM C 273-94), using the single rail configuration suffers from the above deficiency [11]. In addition, in order to eliminate the effect of stress concentration due to bonding of the rails with the specimen, specimen dimension (width and length) in these two test configurations need to be sufficiently large. Such large specimen configurations, in general, are difficult to meet in testing new materials like carbon foam, when these materials in its developmental stage are produced in small quantities.

It is known that Iosipescu shear test configuration requires a relatively small specimen compared to that of single or double rail shear. Thus, Iosipescu shear test was initially performed to measure the shear strength, although the test configuration is not suitable for measuring foam shear modulus. The standard Iosipescu shear test specimens were machined with v-notches at the specimen midlength (see ASTM Standard D 5379, [11]). In order to prevent a premature failure under the loading pins the loading faces of the specimens were reinforced by bonding tab materials, as shown in Figure 2. Even with reinforced loading faces specimens mostly failed near the face-tab ends and under the loading pins, as shown in Figure 2. This undesirable failure mode may be attributed to high porosity and the brittle nature of carbon foam. Thus, Iosipescu shear test appeared not to be suitable for highly porous and brittle carbon foam materials.

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Miniature torsion fixture design and calibration

The difficulty in generating a desirable failure mode in Iosipescu shear specimens may be attributed to excessive stress concentrations of the foam ligaments in the vicinity of loading rollers. Thus, a cylindrical specimen configuration (for example, circular rods) subjected to an axial torsional load is considered to be a preferred test configuration for this material to measure its shear properties. The schematic configuration of the specimen is shown in Figure 3, as well as, the foam specimen with slotted end tab is shown in Figure 4.

The cylindrical specimen of applying torque at the specimen ends, in contrast to Iosipescu, Double rail and Single rail shear tests, significantly reduces the stress concentration at the loading locations, i.e., the specimen ends. A special miniature torsion fixture was designed to measure the shear stiffness and strength of the material, as shown in Figure 5 [12]. Due to low bulk modulus of carbon foam (generally, one order of magnitude lower than that of PVC plastic), a careful attention was given to the low load requirement in designing the test fixture. Since strain gages could not be installed on the specimen surface, shear strain was determined from specimen end rotation with a special design to obtain an angular rotational resolution of 0.005 degrees. Further, to relieve the axial constraint while twisting (applying torque) the specimen, the specimen-gripping end attached to the load cell is mounted on a linear roller. In addition to testing foam, this miniature torsion tester is also designed to allow the convenient and accurate torsional testing

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of small samples of various low stiffness materials. The fixture consists of several components described separately as follows.

General Configuration: The test machine consists of a torsion drive section and a reaction section that are both mounted on an aluminum base. The torsion drive section uses a computer-controlled stepper motor to apply a torsional displacement to the material under test. The reaction section contains a torque load cell that measures the applied torque (see Figures 6(a), 6(b)).

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Support Base: The base is made of a 76.2x44.5x6.35 mm (3" x 1.75" x 0.25") aluminum channel with 12.7 mm (0.5") thick aluminum end plates. The overall size of the base is 127 x 343 x 76.2 mm (5" x 13.5" x 3"). The main consideration in constructing the base is that it be able to resist the torque produced by the machine without significant deflection. The top surface of the base is machined flat to provide a stable platform for the drive and reaction sections. Drive Section: The drive section is constructed on a 89 x 165 x 12.7 mm (3.5" x 6.5" x 0.5") aluminum plate, which is bolted to one end of the base. The drive itself consists of a Frame 23-stepper motor with a 30-oz capacity geared to the output shaft through two sets of gear reduction. The first gear reduction is a 3:1 timing belt drive that drives a 60:1 worm gear reducer to the output shaft for a total reduction of 180:1. All elements of the torsion drive are mounted in precision ball bearings to reduce friction in the system as much as possible. The worm drive consists of a hardened and ground steel worm driving a brass worm gear. The output shaft is a 6.35-mm (0.25") diameter steel shaft aligned parallel to the long axis of the base. All of the mechanical parts of the drive section are protected by an extruded aluminum cover. Angular Measurements: Provisions have been made in the drive section for either a potentiometer or a shaft encoder to be added to the drive unit to measure the angular deflection applied to the specimen. The current version accomplishes this measurement in the control software by simply counting the number of steps applied to the motor. The capacity of the motor coupled with the high torque multiplication due to the gear reduction ensures that the motor will never "stall" so the software calculation of the applied angle will always be accurate to within one step. The stepper motor is a standard 200-step-per-revolution motor so the final resolution of the drive is: Steps/Degree (or 0.01 Degree/Step) Since the control software operates the motor in "half-step" mode, this resolution doubles to 200 Steps/Degree or 0.005 Degree/Step step Radian/Step). This resolution was deemed satisfactory for the preliminary testing. Reaction Section: The reaction section consists of a reaction base, which contains two commercial ball slide assemblies aligned, parallel to the test axis of the machine (the long axis of the base) and the mounting plate for the reaction torque cell. The purpose of the ball slides is to allow the torque cell the freedom to move along the torque axis of the machine while resisting the applied torque. This ensures that no axial loads are applied to the specimen. The total range of motion of the torque cell is approximately 1 inch (25.4 mm). This allows the operator sufficient room to insert each specimen into the gripping fixtures of the machine. The location of the reaction section on the base is also adjustable over about a 4inch (101-mm) range. Once the reaction section has been located in a suitable position for the specimen being tested, it can be locked in place using two locking screws. This allows specimens to be tested in a range from less than 0.5 inch (12.7 mm) to about 3.5 inch (76 mm) in length. Torque Measurement: The torque cell is a commercial unit with a 100-lb capacity (Transducer Techniques, Inc., TRT-100). The signal from the torque cell is amplified by a commercial strain gage signal conditioning system and fed to a 16-bit resolution Data Translation data acquisition card in an IBM PC-compatible computer. The torque cell is calibrated before each set of tests using a 10.00-inch (254-mm) lever arm and a set of standard laboratory gram weights.

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Specimen Attachment: The carbon foam specimens for this program were limited in size to approximately 0.3-inch (7.6 mm) diameter by 2 to 3 inch (50 to 75 mm) long due to the small size of the foam preforms. The foam specimens were fitted with slotted end ring or solid tabs (Figure 4 shown with slotted ring tab) for easy gripping to the torsion tester. Due to this limitation, the torsion tester was fitted with grooved collar adapters to grip the specimens through the slotted end tabs. These adapters can be easily modified or replaced to accommodate specimens of different diameters. Control Software: The control software was written in Microsoft Quick Basic, Version 4.5. The program allows the operator to enter specific information about the specimen and designate a file name for the data. The output data file contains the torque and angular displacement values in engineering units for each step of the motor (every 0.005°). This data is in ASCII format and may be imported into a spreadsheet program, such as Excel, for further analysis or plotted using any of the commercially available plotting programs. Calibration of the Torsion Fixture: The tester was calibrated in two stages. First, the response of the load cell was calibrated to known applied torque and then the whole torsion fixture by measuring shear modulus of PVC plastic. The PVC plastic was chosen as the material for calibrating the fixture because of its modulus being closest to that of foam that we could obtain. To assess the response of the load cell, the real time response (dynamic response) of the load cell due to applied quasi-static torque was recorded and is shown in Figure 7. The quasi-static torque to the load cell was generated by hanging a known mass at the end of a torsion arm attached to the fixture perpendicular to the torsional axis. A known amount of torque was applied to the tester for a few minutes and then changed to another value several times to assess the stability of the load cell response. The real time response of the load cell shown in Figure 7 indicated the reliability and stability of the load cell response

The objective of developing this test fixture was to measure shear properties of brittle porous (carbon foam) materials. Since strain gages could not be installed on porous specimen surface, the emphasis for this test method was to determine the shear strain in the specimen from the specimen end rotation rotation between

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the specimen gage length. To calibrate the torsional response of the fixture, a solid PVC tubular specimen (Figure 8) with slotted ring aluminum tabs was tested in torsion to calibrate the torsional response of the fixture. The modulus of aluminum tabs is generally over 10 times of that of PVC and over 100 times of that of the foam considered here for testing. Thus the portion of the PVC or foam specimen that was bonded into the aluminum ring tabs, assuming a perfect bond between the specimen surface and the tabs, was considered to

behave rigid compared to the rest of the PVC or foam specimen. Thus, the specimen length excluding the end tab portion is considered the gage length (G.L.) of the specimen. The specimen was also instrumented with a strain gage to compare the strain gage strain data with that of specimen end rotation. The torque versus the specimen end rotation of the specimen is shown in Figure 9. The expression for calculating shear modulus from the torque (T) versus the Specimen End Rotation plot is given below.

where, G is the shear modulus of the material, T is the applied torque, L is the specimen gage length, is the specimen end rotation, J is the torsional rigidity of the specimen. For circular cross-section the torsional rigidity (J) is where and are the outer and inner diameter of the specimen, respectively. In Equation (1), is the slope of the torque (T) versus Specimen End Rotation curve, as shown in Figure 9. The shear strain on the specimen surface of circular cross section can be calculated from the Specimen End Rotation by using the following relation

The values of the specimen surface shear strain associated with the specimen end rotation of the PVC specimen tested is shown at the top abscissa in Figure 9. The initial shear modulus of the PVC material obtained from the data shown in Figure 9 was found to be 1.112 GPa (0.161 Msi). In order to assess the accuracy of the torsional measurement, a flat specimen of dimension 101 x 12.7 x

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1.016 mm (4”x0.5”x0.040”) made out of the same PVC material (Figure 10), was tested in tension to measure its Young's modulus and Poisson’s ratio. In view of low modulus of PVC, two extensometers were used to measure its strain in longitudinal and transverse directions. In our experience, adhesion of strain gages to the specimen surface tends to locally stiffen PVC by as much as by 19 percent; thus strain gages were not used measuring strain in PVC. The tensile stress-strain curves of the PVC flat specimen are shown in Figure 11. Using the stress-strain data in Figure 11, the measured Young's modulus (E) and Poisson’s ratio (v) of the PVC specimen were found to be 2.987 GPa (0.433 Msi) and 0.3683, respectively. Assuming material isotropy, the initial shear modulus of PVC was independently calculated from its Young's modulus (E) and Poisson’s ratio (v) using the following expression

The initial shear modulus of the specimen calculated from the shear test (Equation (1) and Figure 14) was within 1.8 percent of the shear modulus calculated from the tensile test (using Eq. (3) and Figure 11) that illustrated the accuracy of the test method.

4. Measured Shear Properties of Foam Materials The shear modulus and strength of graphitized foam of two different densities were measured using the miniature torsion fixture described above. The measured shear stress-strain curves of the two materials are shown in Figures 12 and 13, respectively. The densities of the two carbon foams were 0.16 and 0.56 g/cc, respectively. The shear modulus and strength of the foam, as expected increased with its density. All the specimens failed in the gage section in shear, as revealed from the failure surface shown in Figure 14. The test data in these to figures

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indicate that both the ring and solid tabs are equally efficient for gripping the foam specimens to the test fixture.

The failure surface of the specimens in Figure 14 revealed that shear failure occurred at approximately 45° with respect to the specimen longitudinal axis, which is the usual failure mode observed in specimens subjected to pure shear. It is known from the principle of strain transformation that the shear strain when transformed to a 45° rotation, becomes orthogonal tensile and compressive strains (see Figure 15), yielding Thus a comparison was made between the tensile failure strain (measured from the tensile tests) and the tensile strain calculated from transformed shear failure strain (measured from the shear test). The failure shear strain of the foam of density 0.16 g/cc (data shown in Figure 12) was compared with its tensile failure strain measured by Roy, et al. [13]. The average tensile failure strain of the same foam reported in Figure

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7(b) in [13], was (with standard deviation of . The failure shear strain of the 0.16 g/cc foam found in this study to be , which yielded a tensile strain of value transformed at an angle of 45° with respect to the specimen longitudinal axis. The value of this tensile strain, obtained after transforming the shear failure strain, was within the scatter of the failure tensile strain data obtained in reference [13], which inferred that the shear failure was in fact a tensile failure at an angle of 45° with respect to the shear plane.

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5. Conclusions The shear properties of carbon (graphitic) foam, being brittle and highly porous, are not reliably measured with most of the standard test methods, such as, single rail, double rail, Iosipescu shear, etc. A new testing fixture is developed to accurately measure the shear stiffness and strength of carbon foam or other porous materials. A digital step motor was used in the fixture to exert torsion to specimens of cylindrical cross section. Since strain gages could not be installed on the specimen surface (due to porosity), the shear strain is determined from the specimen end rotation. To ensure a high accuracy in the specimen end rotation, the test fixture was designed to a rotational resolution of 0.005 Degree/Step (or Radian/Step). In view of testing low modulus material, the load cell of fixture was mounted on an axial roller to relieve axial constraint while twisting the specimens. The accuracy of the measurement with the test fixture was demonstrated by measuring shear modulus of PVC plastic.

6. 1.

References

Gibson, L. J. and Ashby, M. F., 1988, Cellular Solids – Structures and Properties, Pergamon Press, Chapters 5 and 6. 2. Christensen, R. M., 1986, Mechanics of Low Density Materials, J. Mech. Phys. Solids, Vol. 34, No. 6, pp. 563-578. 3. Christensen, R. M., 1996, On the Relationship of Minimal Conditions to Low Density Material Microstructures, J. Mech. Phys. Solids, Vol. 44, No. 12, pp. 2113-2123. 4. Christensen, R. M., 2000, Mechanics of Cellular and Other Low-density Materials, Intl J. Solids and Structures, Vol. 37, pp. 93-104. 5. Warren, W. E. and Kraynik, A. M., 1988, The Linear Elastic Properties of Open-Cell Foams, J. Applied Mechanics, Vol. 55, pp. 341-346. 6. Zhu, H. X., Knott, J. F., and Mills, N. J., 1997, Analysis of the Elastic Properties of Open-cell Foams with Tetrakaidecahedral Cells, J. Mech, Phys. Solids, Vol. 45, No. 3, pp. 319–343. 7. Zhu, H. X., Mills, N. J., and Knott, J. F , 1997, Analysis of the High Strain Compression of Open-cell Foam, J. Mech. Phys. Solids, Vol. 45, No. 11/12, pp. 1875-1904. 8. Kau, C., Huber, L., Hiltner, A., and Baer, E., 1992, Hard Elastic Behavior in Polyurethane Foams, J. Applied Polymer Science, Vol. 44, pp. 2081-2093. 9. Kau, C., Huber, L., Hiltner, A., and Baer, E., 1992, Hard Elastic Behavior in Polyurethane Foams, J. Applied Polymer Science, Vol. 44, pp. 2069-2079. 10. Bilkhu, S. S., Founas, M., and Nusholtz, G. S., 1994, Determination of Cellular Foam Material Properties for use in Finite Elements Analysis, Experimental Techniques, pp. 23-25. 11. Annual Book of ASTM Standards, Vol. 15.03, Standards C 273-94, D 4255, D 5379, ASTM, 1916 Race Street, Philadelphia, PA 19103, USA. 12. Roy, A. K. and Camping J. D., A Novel Miniature Test Fixture to Measure Shear Stiffness and Strength of Foam (Highly Porous Materials), Invention Disclosure, Submitted, August 1999, U. S. Air Force. 13. Roy, A. K., Pullman, D., and Kearns, K. M., Experimental Methods for Measuring Tensile and Shear Stiffness and Strength of Graphitic Foam, Proc. Intl. SAMPE Conference and Exhibition, Anaheim, CA, 31 May – 4 June, 1998

INDENTATION OF A PVC CELLULAR FOAM

E. E. GDOUTOS and J. L. ABOT Department of Civil Engineering Northwestern University Evanston, IL 60208 Abstract

An investigation of the indentation response of a PVC closed-cell cellular foam (Divinycell H80) by a cylindrical indenter was undertaken. The material has a density of 80 and shows a linear elastic almost perfectly plastic behavior in simple compression up to a critical strain at which densification takes place. The plastic Poisson’s ratio of the material approaches zero. The load-indentation depth relation was monitored for a progressively increasing applied load. It consists of an initial linear part followed by a softening behavior. A simple model based on a rigid plastic behavior of the foam in compression and a maximum stress failure criterion was used to predict the load-indentation depth curve. Results for the indentation hardness indicate that it takes values close to one for small indentation depths, while it increases as the indentation depth increases. 1. Introduction

The compressive stress-strain behavior of many cellular foams consists of an initial relatively small and stiff elastic regime, an extended stress plateau regime and a final regime in which densification of the material takes place [1]. In the stress plateau regime the cells of the foam collapse, while the average stress remains almost constant during the instability spread through the material. Axial compression produces little lateral spreading resulting to almost zero Poisson’s ratio. When all of the cells collapse the material is densified and its stiffness increases. As a consequence of such behavior foams change their volume during plastic compression. This is contrary to dense solids which are incompressible during plastic deformation. Because of this, the indentation behavior of foams is quite different than that of dense solids. The first study on the plastic behavior and indentation hardness of cellular materials was conducted by Shaw and Sata [2]. They observed that during the plastic deformation Luder’s-like bands and upper and lower yield points appear, as in mild steel. Furthermore, they found that indentation hardness was close to the yield stress in compression and they attributed this behavior to the very low Poisson’s ratio. Patel and Finnie [3] presented a model for the prediction of properties of foams assuming a unit cell in the form of a pentagonal dodecahedron. An analysis of the indentation of compressible foams by cylindrical and spherical indenters was conducted by Wilsea et al.[4]. They presented a simplified model for the prediction of the indentation hardness based upon idealization of the stress-strain behavior. Cousins [5] presented a simplified theory to describe the impact behavior of padding materials with rate dependent deformation when struck by flat or spherical indenters. 55 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 55–64. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Following the above early investigations, a number of works on the indentation behavior of honeycombs have appeared. Klintworth and Stronge [6] studied the stress and deformation fields in an anisotropic half-space made of a ductile honeycomb loaded by a flat punch. They found that after a small regime of elastic deformations the honeycomb crushes in localized bands. They obtained a lower and upper estimate of the indentation force. Papka and Kyriakides [7,8] studied the in-plane crushing of hexagonal aluminum honeycombs between parallel rigid surfaces. The forcedisplacement behavior is initially stiff and it is then followed by an instability behavior under constant load. Localized crushing occurs and is spread through the material under constant load. When the whole specimen is crushed the load-deformation response stiffens again. A finite element analysis was performed to simulate the crushing process. The crushing response of a polycarbonate circular cell honeycomb subjected to in-plane biaxial loading was studied by Chung and Waas [9]. Fortes et al. [10] based on the micromechanical model of Gibson and Ashby [1] analyzed the contact behavior of a cellular solid for both open cell and closed cell materials. They derived a simple equation for the prediction of the indentation behavior. A thorough investigation of the mechanical behavior of a PVC closed cell foam under uniaxial and multiaxial stress conditions was conducted by Gdoutos et al. [11,12]. Two types of Divinycell, H100 and H250, with densities 100 and 250 , respectively, were investigated. The uniaxial tensile, compressive and shear stressstrain curves along the in-plane and the through-the-thickness directions of both materials were obtained. A series of biaxial tests using strip, ring and tube specimens subjected to uniaxial compressive or tensile loads and combined tension, torsion and internal pressure were conducted. The failure envelopes of the materials were constructed and satisfactorily modeled by the Tsai-Wu failure criterion. An experimental and analytical study of the indentation behavior of composites facesheets supported by a foam foundation and subjected to a concentrated load was conducted by Abot et al. [13]. The load-deflection curve includes a linear range followed by a nonlinear portion. In the linear range the load-indentation behavior was predicted by the Hetenyi model based on Winkler’s solution of an elastic beam resting on an elastic foundation. In the nonlinear range the load-indentation relationship was obtained by fitting experiments. A new method was described for direct measurement of the foundation modulus using cylindrical and rectangular section indenters on foam blocks. In the present work the indentation behavior of Divinycell H80 by a cyclindrical indenter was studied. The load-indentation depth and indentation hardness curves were obtained experimentally and were modeled by a simplified plastic analysis. 2.

Characterization of foam material

2.1 MATERIAL

The material studied in this work was a fully cross-linked PVC closed-cell cellular foam (Divinycell H80) with density of 80 This is a commonly used core material in sandwich construction by marine structures. The material is produced by introducing gas bubbles into the hot PVC polymer and allowing the bubbles to grow and stabilize. The material was produced by the manufacturer in the form of 25.4 mm (1 in.) thick plates with the rise direction along the thickness direction.

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For modeling the indentation response of the material its compressive stress-strain behavior is needed. 2.2 COMPRESSION TEST Divinycell H80 is relatively isotropic. Its mechanical properties along the in-plane and the through-the-thickness directions are almost equal. To determine the in-plane stressstrain behavior of the material in compression, prismatic specimens of dimensions 25.4 x 25.4 x 76.2 (1 x 1 x 3 in.) were tested quasi-statically in an Instron servohydraulic testing system. Both longitudinal and transverse strains were measured with extensometers. The longitudinal strains were monitored on opposite sides of the specimen to insure that there was no bending effect during loading. The tests were terminated after the load dropped and remained almost constant following a peak value. The above specimens during loading in the stress plateau region of the compressive stress-strain behavior failed due to buckling. Thus, they cannot be used to obtain the complete stress-strain curve. For this reason short prismatic specimens of dimensions 25.4 x 25.4 x 25.4 (1 x l x 1 in.) were tested. The longitudinal strains were determined from the stroke of the grips of the testing machine. Figure 1 shows the in-plane longitudinal and transverse stress-strain curves of Divinycell H80. The longitudinal stress-strain curve shows an initial elastic portion followed by a yield plateau during which the applied stress remains nearly constant under increasing strain. The elastic portion of the curve shows an initial linear part followed by a nonlinear portion with a decreasing tangent modulus. At the end of the nonlinear elastic part the stress drops abruptly from an upper to a lower yield point. The complete stress-strain curve of the material after densification takes place is shown in Fig. 2. Note that the yield portion of the curve extends up to a strain of 50 percent. Following the end of the yield region, densification takes place and the stress rises abruptly. From a micromechanical pint of view this corresponds to collapse of the cells of the foam. Opposing cell walls come into contact and the stress-strain curve corresponds to that of the cell material. This behavior is similar to that of aluminum honeycomb in which localized yielding or crushing occurs after peak load. The transverse stress-strain curve (Fig. 1) shows similar characteristics as the longitudinal stress-strain curve up to the stress drop to the lower yield point. After that point, the transverse strain does not increase any more, that is, longitudinal compression does not produce any lateral deformation. In other words, Poisson’s ratio in the yield region is almost zero. This behavior can be explained from the collapse of the foam cells as the foam is compressed. Thus, foams, unlike dense solids, cannot be considered to be incompressible when deformed in the plastic region. This behavior of foams has been noticed by previous investigators [1,2]. The in-plane longitudinal and transverse stress-strain curve of Divinycell H80 are almost the same as the through-the-thickness curves. 3. Indentation tests A series of panel specimens of width 254 mm (10 in.) and thickness 24.1 mm (0.95 in.) were cut from Divinycell H80 plates. The height of the specimens took the values 25.4, 76.2, 152.4, 254 and 381 mm (1,3,6,10 and 15 in.).

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The panels were supported on a rigid base and loaded by a cylindrical roller of diameter 25.4 and 50.8 mm (1 and 2 in.) in an Instron servo-hydraulic testing machine. The indentation of the cylindrical roller into the specimens was measured from the stroke of the testing machine. The load was applied at a rate 0.0085 mm/sec (0.02 in./min). The load and the stroke of the testing machine were monitored during the test. Figure 3 shows a picture of the experimental setup for a panel specimen of width 254 mm (10 in.) and height 12.7 mm (5 in.). The deformation of a series of straight lines parallel to the surface of the panel near the indenter is shown in Fig. 4. Note that two

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inclined cracks on both sides of the indenter starting from the surface of the panel have initiated outside the indentation zone beneath the indenter.

4. Theoretical modeling For the prediction of the indentation behavior of the foam a simplified model developed by Wilsea et al.[4] will be used. The model is based on an idealized stress-strain behavior of the foam under compression, as shown in Fig. 5. The material presents a small linear elastic region which is omitted. When the stress reaches a critical value

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the material yields under constant strain up to a critical value at which densification occurs. For strains higher than the material hardens and the stress increases. Failure of a material under a multiaxial state of stress occurs when the maximum principal stress becomes equal to and is uninfluenced by the other two principal stresses. This is a simplified idealization of the material behavior. Gdoutos et al. [11,12] have shown that the failure behavior of Divinycell is influenced by all three principal stresses. They showed that the failure surfaces of Divinycell under combined normal and shear stresses along the in-plane and through-the-thickness directions obtained from a host of biaxial tests are predicted well by the Tsai-Wu failure criterion. Under such conditions, the plastic zone has the shape of a lune under the indenter. The indentation depth d corresponding to an indentation load P is obtained from the following equations

where 2a is the width of indent, R is the radius of the indenterand L is the indenter length. The angle defines the plastic lune under the indenter. For the predicaiotn of the load versus indentation depth curve based on Eqs. (1) to (3) the following procedure is followed. For a given value of indentation depth d the value 2a of the width of indent is calculated from Eq. (1) and then the value of angle from Eq. (2) with a material constant determined from the uniaxial compression stress-strain curve of the material. It was obtained Having the values of a and the load P is calculated from Eq. 3 in which is the yield stress of the material in uniaxial compression. Following the above procedure the load versus indentation depth curve can be predicted when the values and defined from the stress-strain curve of the material in compression, the radius, R, and the length, L, of the indenter are known. 5.

Results and Discussion

Figure 6 shows the load-indentation depth curves for five Divinycell H80 panels of width 25.4 mm (10 in), thickness 25.4 mm (1 in.) and heights 76.1, 152.4, 254 and 381 mm (1,3,6,10 and 15 in.) for indentation depth up to 20 mm (0.79 in.). Note that for height higher than 76.1 mm (3 in.) all curves coincide. This means that the panel height does not affect the indentation behavior. Thus, for panel heights higher than 76.1 mm (3 in.) the indentation curve depends on the foam and the diameter of the indenter. It can be considered as a material property for a given indenter diameter. The initial part of the indentation curves for an indentation length up to 5 mm (2 in.) are shown in Fig. 7. Finally, Fig. 8 shows the indentation curves for a panel of height 150 mm (5.9 in.) indented by a cylindrical indenter of diameter 25.4 and 50.8 mm (1and 2 in.).

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In Fig. 8 the predicted indentation curves based on the simplified rigid-perfectly plastic model proposed by Wilsea et al. [4] are shown. Note that for the case of indenter of diameter 25.4 mm (1 in.) the experimental results coincide with the theoretical predictions for an indentation depth up to 10 mm (0.4 in.), while for an indenter of diameter 50 mm (2 in.) up to 5 mm (0.2 in.). After these indentation depths there is a small deviation of the experimental results from the theoretical predictions.

A characteristic feature of the indentation behavior is the indentation hardness H defined by

H is equal to the indentation pressure assumed uniformly distributed along the width of indent. The relationship between the indentation hardness H and the width 2a of the indenter can be predicted from Eq.s (2) to (4). For a given value of a the angle is calculated from Eq. (2), then the load P from Eq. (3) and finally the indentation hardness from Eq. (4).

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Figure 9 shows the indentation hardness H normalized to the yield stress plotted versus the width of indent 2a divided by the diameter d of the indenter In the same figure the theoretical curves based n the model of Wilsea et al. [4] for different values of are shown. Note that the indentation hardness H for small indentation widths is about equal to the yield stress. H increases with 2a/D up to a value of for . These results are in agreement with those of references 2 and 4.

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Conclusions

The indentation behavior of a PVC closed-cell cellular foam was investigated. Panel specimens of various different dimensions were indented by a cylindrical indenter of two different diameters. The load versus indentation depth response was monitored during the test. From the results of this study the following conclusions may be drawn: 1.

The indentation behavior of the foam material studied in this work consists of a nonlinear load versus indentation depth curve with decreasing stiffness as the load is increased.

2.

The load versus indentation depth response becomes independent of the height of the panel for the two indenter diameters used for heights higher than 75 mm (3 in.). Under such conditions the indentation curve is a material property and depends only on the diameter of the indenter.

3.

The indentation hardness is almost equal to the yield stress of the foam material for small indentation depths. As the indentation depth increases the hardness increases up to a limiting value of for indentation depth versus indenter diameter ratio equal to 1.5. This behavior is contrary to incompressible materials, like metals, where the indentation hardness is equal to three tunes the yield stress. This is because the foams are compressible materials with Poisson’s ratio almost equal to zero in the plastic region. This behavior has been reported in the literature.

4.

The indentation behavior can well be predicted by a simple model based on a rigid plastic behavior of the foam in compression and a maximum stress failure criterion.

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7. Acknowledgements The authors wish to thank Professor I. M. Daniel for his advice, guidance and support during the course of this work. This research was sponsored by the Office of Naval Research (ONR). We are grateful to Dr. Y. D. S. Rajapakse of ONR for his encouragement and cooperation and to Mrs. Yolande Mallian for typing the manuscript. 7. References 1. 2. 3. 4. 5. 6.

7. 8. 9.

10. 11. 12.

13.

L. J. Gibson and M. F. Ashby, Cellular Solids, Cambridge University Press, Cambridge, 1997. M. C. Shaw and T. Sata, The plastic behavior of cellular materials, Int. J. Mechanical Sciences, 8 (1966) 469-478. M. R. Patel and I. Finnie, Structural features and mechanical properties of rigid cellular plastics, Journal of Materials, JMLSA, 5 (1970) 909-932. M. Wilsea, K. L. Johnson and M. F. Ashby, Indentation of foamed plastics, Int. J. Mechanical Sciences, 17 (1975) 457-460. R. R. Cousins, A theory for the impact behavior of rate-dependent padding materials, J. Applied Polymer Science, 20 (1976) 2893-2903. J. W. Klintworth and W. J. Stronge, Plane punch indentation of a ductile honeycomb, Int. J. Mechanical Sciences, 31 (1989) 359-378. S. D. Papka and S. Kyriakides, In-plane compressive response and crushing of honeycombs, J. Mechanics and Physics of Solids, 42 (1994) 1499-1532. S, D. Papka and S. Kyriakides, Experiments and Full-scale numerical simulations of in-plane crushing of a honeycomb, Ada Metallurgies, 46 (1998) 2765-2776. J. Chung and A M. Waas, In-plane biaxial crush response of polycarbonate honeycombs, J. Engienering Mechanics-ASCE, 127 (2001) 180-193. M. A. Fortes, R. Colaço and M. Fátima Vaz, The contact mechanics of cellular solids, Wear, 230 (1999) 1-10. E. E. Gdoutos, I. M. Daniel and K.-A. Wang, Failure of cellular foams under multiaxial loading, Composites: Part A 33 (2002) 163-176.. E. E. Gdoutos, I. M. Daniel and K.-A. Wang, Multiaxial characterization and modeling of a PVC cellular foam, J. Thermoplastic Composite Materials, to appear. J. L. Abot, I. M. Daniel and E. E. Gdoutos, Contact law for composite sandwich beams, J. Sandwich Structures and Materials, to appear.

NANOMANIPULATION AND CHARACTERIZATION OF INDIVIDUAL CARBON NANOTUBES RODNEY S. RUOFF Department of Mechanical Engineering Northwestern University 2145 Sheridan Road, Evanston, IL 60208 MIN-FENG YU Advanced Technologies Group Zyvex Corporation 1321 North Plano Road, Richardson, TX 75081 HENRY ROHRS Mass Sensors, Inc. 1350 Baur Blvd. St. Louis, MO 63132-1904 Abstract We describe the implementation of new tools for the measurement of the mechanics of nanostructures. These tools have been used to measured the stiffness, fracture strength, and various tribological properties, of carbon nanotubes.

1. Introduction The rapid advancement in miniaturization in the last decade has seen the discovery and generation of numerous types of nanomaterials whose size is in the nanometer scale. In particular, carbon nanotubes (CNT) have received significant attention. Multi-walled carbon nanotubes (MWCNT) were discovered in 1991[1] and single wall carbon nanotubes (SWCNT) in 1993 [2,3], and since then over 4000 scientific publications on CNTs have appeared. The electrical and mechanical properties of CNTs have been the subject of numerous theoretical/modeling and experimental studies. CNTs can be considered to be a seamless cylinder (SWCNT), or a set of nested cylinders (MWCNT), of graphene sheet(s), and can have open or capped ends. CNTs made to date typically have a diameter of a few tens of nanometers (MWCNT), and about 1 nm (SWCNT), and lengths of a few microns. Various novel discoveries related to the natural properties of a graphene sheet, of quantum confinement in small dimensions, and the detailed geometry including in particular the chirality of the shell structure of CNTs have been made, and potential applications are likely in materials reinforcement, field emission panel display, electronic nanowires and devices, chemical sensors, gas storage, as tips for scanning microscopy, for use in batteries, and so on. It has been theoretically predicted that CNTs should possess mechanical properties far superior to commercially available carbon fibers, due to their expected structural perfection. High Young’s modulus (~ 1 TPa) similar to the inplane modulus of high quality graphite and high tensile strength (~ 50 to 100 GPa, 65 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 65–74. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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thus much greater than any other known material) are predicted, and for the modulus experimentally verified, for CNT [4-7]. For comparison, the highest strength carbon fiber in industry has a strength of ~ 7 GPa [8]. The small dimensions of CNTs imposes a significant challenge for experimental study of their mechanical properties. The general requirements for such study include: (i) the challenge of CNT placement in an appropriate configuration for testing, such as of pick up followed by release and placement; (ii) in certain cases the fabrication of appropriate clamps and boundary conditions for loading; (iii) successful application of the desired loading; (iv) characterizing and measuring the mechanical deformation at the nanometer or perhaps atomic length scale. Various types of high-resolution microscopes allow the characterization of nanomaterials, and recent developments in the new area of “nanomanipulation,” based on inserting or adapting new tools to such high-resolution microscopes, have enhanced our ability to mechanically test such nanoscale objects. In this paper, we briefly summarize our effort in the development of new tools for nanoscale characterization and the study of various mechanical properties of CNTs. We close by speculating about future directions worth pursuing for mechanics studies on individual nanostructures.

2.

A brief review of related instruments for nanoscale material characterization

Scanning probe microscopy and electron microscopy have been the most widely used methods for resolving and characterizing nanoscale objects. We and others have primarily used SPM and EM instruments to study nanotube mechanics. For this reason, we give a brief review of the methods of operation of these types of microscopes. Electron microscopes (EM) use high-energy electron beams (several keV up to several hundred keV) as a source for scattering and diffraction from a sample, which results in high resolving power down to sub-nanometer resolution because of the extremely short wavelength (a fraction of a nanometer) of high kinetic energy electrons. In TEM, an accelerated electron beam from a thermal or a field emitter is used to interrogate samples. The beam transmits through the sample and passes several stages of electromagnetic lenses, and projects the image of the studied sample region to a phosphor screen or other image recording media. A thin (normally several tens of nanometer or less in thickness) and small (no more than sample is a typical requirement since only a small sample chamber is available and a dedicated holder is typically used in these expensive instruments (which are thus typically time-shared among many users) for sample transfer. TEM is limited by such factors as the spread in the electron beam energy and the quality of the ion optics. It normally has a resolution on the order of 0.2 nm. A good textbook on TEM is reference [9]. Fig. 1C and D shows typical high-resolution images of CNTs [10]. Fringes corresponding to the projection of the shell(s) in each CNT are easily resolved. In SEM, a focused electron beam (nanometers in spot size) is used to raster across the sample surface and the amplified image of the sample surface is formed by recording the secondary electron signal or the back scattering signal generated from the sample. There is no strict limit on sample size in principle, and normally a large sample chamber is available such that samples can be surveyed over large

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areas. SEM is limited by the scattering volume of the electrons interacting with sample material, and high end instruments commercially available are capable of achieving a resolution of a few nanometers. Typical SEM images of a SWCNT sample, and of a purified MWCNT sample, are shown in Fig. 1E and F, respectively. Individual MWCNTs can be seen sticking out from the bundle, while only bundles of SWCNTs having diameters ~10 nm can be seen in the SWCNT sample. Scanning probe microscopes (SPM) use extremely sharp probes (that can have 10 nm or smaller radius of curvature at the tip) controlled by sensitive sensing and actuation feedback electronics for obtaining nanoscale and even atomic scale information. The Atomic force microscope (AFM) has been particularly useful for the mechanical studies of CNTs. Since its invention in 1986 [11], it has been rapidly accepted as a useful tool for many applications related to surface characterization. High-resolution (nanometer to atomic resolution) mapping of surface topology on either conductive or nonconductive material can be achieved with an AFM. The principle of operation of the microscope is relatively simple. A probe with a sharp tip having a force-sensitive cantilever is used as a sensor to scan, in close proximity to, the sample surface. The probe is driven by a piezoelectric tube capable of nanometer resolution translation in the x, y and z directions, and the tip normally has a radius of curvature of ~10 nm. The force interactions between tip and sample results in deflection of the cantilever. The deflection of the cantilever, during scanning of the sample surface in the x and y directions, is constantly monitored by a simple optical method, or other approaches. A feedback electronic circuit that monitors the deflection signal and controls the piezoelectric tube maintains a constant force between the tip and the sample surface; a surface profile of the sample can thus be obtained. Fig. 1A shows a typical image acquired by AFM of MWCNTs on a substrate. For in-depth introduction of AFM see references [11-13].

The detection mechanism of the scanning tunneling microscope (STM) is based on the tunneling current (on the order of nA and less) between a sharp conductive tip and a sample when positioned in close proximity and under a constant bias. A non-insulating sample is thus required for STM imaging. At appropriate imaging

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conditions on many crystalline samples atomic resolution can be obtained, due to the strong dependence of the tunneling current on the distance between the conductive tip and the sample surface (there is an exponential dependence of the current on this distance). Fig. 1B shows an atomically resolved image of a SWCNT by Odom et al. [14]. The lattice structure and thus the chirality of the SWCNT can be determined from such images, as well as the electronic structure from additional scanning tunneling spectroscopy measurements. Similar atomically resolved STM images of SWCNTs were also obtained at the same time by Wildoer et al. [15]. A detailed explanation on the principle of STM can be found in ref. [12].

3. New tools for nanoscale mechanical measurement The high resolution and the large sample chamber space have made SEM a good candidate for inclusion of three dimensional nanomanipulation capabilities. A multiple degree-of-freedom manipulation device inside SEM was developed by Yu et al. to allow handling, characterizing, and even building with CNTs and other nanomaterials [16]. The device possesses the capability and precision to probe a collection of CNTs, and then to isolate an individual CNT and extract it from the ensemble for study. The manipulator can be used as a testing stage once a CNT has been isolated from the group, to allow measurement of the mechanics and transport properties while the CNT is freely suspended in vacuum and thus free from contact with a surface. The device fits well inside the vacuum chamber of an SEM, and can be operated without disturbing the electron microscope's imaging quality or interfering with the microscope's components. The design of the manipulator is depicted in Fig. 2 Only the major components are shown. The manipulator was designed with small size, low-cost, wide translation range, reasonable precision, and rapid assembly in mind. To avoid interference with the SEM electron optics, the x-y and z-theta motions are grouped into two low-profile, opposing stage sub-modules anchored symmetrically on the SEM platform around the axis of the electron beam column. The SEM stage manipulator occupies roughly

As observed with the SEM, the linear stage has a "dynamic" step resolution of 25 nm. The rotational actuator gives angular step sizes of <0.02 degrees. The piezotube atop the z-stage is necessary for continuous motions after the linear stages are brought to within a micron or less of the final desired positions of a manipulation operation. Although no minimum step resolution for the piezotube could be directly observed in the SEM, the spatial resolution can be estimated, conservatively, to be less than 1 nm.

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Commercially-available AFM tips with rigid cantilevers and soft cantilevers were used (Fig. 3), along with electrochemically etched tungsten and mechanically sharpened gold and platinum-iridium tips.

Attaching an individual CNT or CNT bundle to an AFM tip is done by a rather simple process. A visible quantity of purified CNT material is loaded onto a Tungsten or PtIr tip and placed into the tip holder on the x-y stage along with three to six other AFM tips that are ready for CNT attachment. The AFM tip on the piezotube can then be brought close to the raw material. When the tip is brought close enough to a protruding CNT, it is held in place through the van der Waals attraction. A stronger attachment, thus clamp, can be made by using the electron beam to perform localized electron impact-induced deposition of carbonaceous materials due to the presence of the gases in the chamber, a process which we have referred to as "nano-welding." (Figs. 3C and D.) Attaching CNTs to AFM tips has been previously done by Dai et al. [17] with light microscope observation and a compliant adhesive. Their work involved the first example of a CNT as a tip for AFM, and the first detailed studies of the mechanical performance of such a CNT AFM Tip. Our technique, attachment with simultaneous viewing in the SEM, allows the CNT tip to be inspected at much higher spatial resolution, and immediately altered if desired. Of course, this general approach is not limited to CNTs, and we expect that a large number of different types of nanostructures, such as metal nanowires, inorganic nanowhiskers, and nanoplatelets of various layered materials, will be “nanoclamped” in various types of testing stages configured to be in SEMs, TEMs, or SPMs, or adapted to various spectroscopic probes, such as Raman. The method of making clamps, and the types of clamps, will likely be sample dependent. “Pull-

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out” experiments of relevance to the understanding of nanocomposites, could be done with such an approach. Owing to the high resolving power of TEM, recent effort has also been devoted to the development of manipulation tools for TEM. Typical manipulation tools for TEM incorporate a single piezotube for fine motion in x, y and z in the range of several microns and an external mechanical drive for coarse adjustment along the axial direction of the TEM holder. The limited maneuverability is mostly constrained by the small sample space available in TEM and the design requirement imposed by a standard-sized TEM holder that can fit into the general commercial TEM’s that are typically used for standard imaging/analysis experiments, and not nanomanipulation studies. A TEM tensile loading tool has recently been developed that uses microelectromechanical system (MEMS) technology to include the direct force load and sensing capabilities essential for a detailed mechanical property study of CNT. Shown in Fig. 4 is a schematic of this “TEM nano-stressing stage.” The stage is designed as a separate unit that can fit into a TEM holder but is detachable. The central part in the stage is a MEMS-like chip fabricated in a silicon wafer by deep reactive ion etching process, which is composed of two main functional parts: a thin cantilever on a rigid fixture, and a linear pusher and a thick cantilever on a soft fixture. The pusher is in contact with a piezostack, which provides the fine linear translation as well as the load. The device can be placed inside a modified TEM holder (EM-SHH4, JEOL) and in operation was inserted into a TEM (Jeol FX2000). The piezostack is connected to a power supply to control the moving stage position. Voltage supplied to the piezostack is regulated with an accuracy of 10mV, which allows the control of the movement of the pusher with sub-nanometer resolution.

In operation, the unit is detached from the TEM holder, and the gap between two cantilevers is adjusted to within and fixed in place by locking the fixed end of the piezostack. The preload is done using a linear piezo inchworm actuator under the monitoring of a high magnification optical microscope. For deposition of nanowires such as CNT, a liquid suspension of CNTs was drawn into a glass micropipette, which was then positioned above the gap between the cantilevers. The position of the micropipette tip was controlled by an x-y-z stage and monitored with an optical microscope. The micropipette was then lowered into contact with the cantilevers so that capillary action can draw a very small volume (10-100 femtoliter) of CNT suspension into the gap. The capillary force in the gap holds the liquid drop in between the cantilevers and typically makes the free end portion of the thin cantilever stick to the thick cantilever. When the droplet dries, the thin cantilever typically smoothly releases and returns to its original position, and a small number of CNTs can be found being placed directly

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across the gap between the cantilevers. For the case of SWCNT suspension, such operation tends to gently draw the SWCNT nanorope or individual SWCNTs across the gap, leaving them mechanically loaded by the deposition. The SWCNTs are at times held in place by a residue left after the solvent evaporates. Separate AFM imaging indeed showed the presence of small amounts of residual organic material. SWCNTs so deposited are often under an initial tension, and the preload can be calculated from the deflection of the thin cantilever if the force constant of the cantilever is known. The force constant can be calculated using the cantilever dimensions measured by SEM, and the standard cantilever beam-bending model. Or, AFM can be used to calibrate the cantilever force constant (by cleaving them to expose the cantilever for such an AFM calibration) using a calibrated AFM probe. The two methods agree within 10%. After the deposition, the device was inserted into the TEM. Load is applied by extending the piezostack to push the thick cantilever. The deflection of the thin cantilever was then measured during the loading process, which gives the applied load, and the length of CNTs can be obtained using a formula which includes the calibrated piezostack extension under different applied voltage and the initial gap distance and the deflection of the cantilever, or can be measured from the recorded TEM images in some cases. Such a device was used to break SWCNT ropes having only several SWCNTs as shown in Fig. 5A [18], and also was used to directly observe the fracture process of CNTs under tensile load [19] as shown in Fig. 5B by Lourie et al.

3. Experimental research on the mechanics of individual carbon nanotubes The response to axial tensile loading of individual MWCNTs was realized by Yu et al. using a testing stage based on a nanomanipulation tool operating inside an SEM introduced above [20]. The nanomanipulation stage allowed for the threedimensional manipulation -- picking, positioning, and clamping -- of individual MWCNTs as demonstrated previously [16]. The individual MWCNTs were attached to AFM probes having sharp tips by localized electron beam induced deposition (EBID) of carbonaceous material inside the SEM. A MWCNT, so clamped between two AFM probes, was then tensile loaded by displacement of the rigid AFM probe, and the applied force was measured at the other end by the cantilever deflection of the other, compliant, AFM probe. The measured force vs elongation were converted, by SEM measurement of the

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MWCNT geometry, to stress versus strain and the breaking strength of each MWCNT was obtained by measuring the tensile loading force at break (Fig. 6). The experiment also clearly resolved that MWCNT normally breaks in a sword-insheath breaking mechanism, where the MWCNT so attached under tensile load breaks at its outmost layer followed by the sliding out of the inner shells during the continuous pulling. Such tensile loading of MWCNTs yielded a Young’s modulus from 270 to 950 GPa, and tensile strength ranging from 11 to 63 GPa.

Yu et al. applied a similar approach for the tensile strength measurement of nanoropes of SWCNTs [21]. The entangled and web-like agglomeration of SWCNTs in raw samples made it difficult to find an individual SWCNT and resolve it by SEM or to pick out individual SWCNT nanoropes, so a modified approach was used for the experiment. SWCNT ropes having a strong attachment at one end to the sample surface were selected as candidates for the measurement. The free end of such a SWCNT nanorope was then approached and attached to an AFM tip by the same EBID method outlined above. The AFM tip was used to stretch the SWCNT rope up to the breaking point and the same AFM tip also served as the force sensor to measure the applied force. Results of 15 single-walled carbon nanotubes (SWCNT) ropes yielded Young's modulus values in the range from 320 to 1470 GPa (mean: 1002 GPa). The average SWCNT tensile strength ranged from 13 to 52 GPa, calculated by assuming the load being applied primarily on the SWCNTs at the perimeter of each nanorope. The shear strength between the shells of a MWCNT is also an interesting subject for experimental study. Yu et al. were able to directly measure the friction force between the neighboring layers while pulling the inner shells out of the outer shells of a MWCNT using the same apparatus for measuring the tensile strength of individual MWCNTs as shown in Fig. 7 [22]. The realization of such a measurement was based on the discovery that a tensile-loaded MWCNT normally broke with a sword-in-sheath mechanism. The separated outer shell can still be in contact with the underlying inner shell in certain cases (in other cases, the “snap back” of the loading and force-sensing cantilevers leads to two separated

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fragments). A model was developed to include forces (as shown in Fig. 7) such as (i) the applied force from the deflection of the soft AFM cantilever; (ii) the static shear interaction force between shells present during the “stick” event; (iii) the dynamic shear interaction force between shells in the “slip” event; (iv) the solid-solid surface tension interface force that is due to the creation of a new shell surface area in the pullout event (surface tension effect); (v) other forces, for example, a force perhaps present and due to the interaction of the dangling bonds on the edge of the fractured MWCNT cylinder with the internal shell surface. Shear strength was related to the shear interaction force. The continuous measurement of force and “contact length” (the overlap length between the outer shell and its neighbor) in the pullout process provided the necessary data for obtaining the dynamic (0.08 MPa) and static shear strength (0.30 MPa in one case and 0.08 MPa in another case) between the shells. The surface energy of graphite could also be estimated..

4. Conclusions The new developments in the area of nanoscale manipulation and measurement as reflected in the studies presented in the last section have certainly helped our understanding of CNT mechanics. Since CNTs possess unique structures that maintain their conformation while being manipulated they represent a “nano-tinker toy” for manipulation on the nanoscale. Therefore, such types of approaches also provide a window on current capabilities for exploring and exploiting the “nanoworld,” and provide an avenue for future advancement in methods and tools useful in nanotechnology. But what has not yet achieved? We have not yet measured the tensile loading response of an individual SWCNT, nor have we applied a known torque or a controlled, and reversible, twist along a CNT. The influence of environment on NT mechanics has not yet been explored in any detail—such as effects of temperature, chemical environment, loading rate, defect density—nor do we have a clear and detailed picture of the nucleation, propagation, and ultimate failure resulting from, defects. From the experimental perspective, such advances will come with new approaches and tools generated by innovative thinking. It is clear that focused effort in developing new measurement tools that can be integrated into high spatial resolution imaging instruments is necessary for further advances in nanostructure mechanics.

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5. Acknowledgements. For the work summarized here, RSR appreciates support from NSF, NASA, ONR, and Zyvex as well as the efforts of former Ruoff group members such as Hui Huang, Oleg Lourie, Kevin Ausman, and the coauthors Rohrs and Yu; in addition we acknowledge the efforts of our collaborators, particularly Tomek Kowalewski, Wing Kam Liu, and Dong Qian.

6. References 1. 2. 3.

4.

5. 6.

7. 8. 9.

10. 11. 12. 13. 14.

15.

16.

17. 18. 19. 20.

21. 22.

Iijima, S. (1991) Helical microtubules of graphitic carbon. Nature (London) 354, pp.56-58. Iijima, S., Ichihashi, T. (1993) Single-shell carbon nanotubes of 1-nm diameter. Nature (London) 363, pp.603-605. Bethune, D. S., Kiang, C. H., de Vries, M. S.,Gorman, G., Savoy, R., Vazquez, J., Beyers, R. (1993) Cobalt-catalyzed growth of carbon nanotubes with single-atomic-layer walls. Nature (London) 363, pp.605-607. Overney, G., Zhong, W., Tomanek, D. (1993) Structure Rigidity and Low-Frequency Vibrational-Modes of Long Carbon Tubules. Zeitschrift Fur Physik D-Atoms Molecules and Clusters 27(1), pp.93-96. Ruoff, R. S., Lorents, D. C. (1995) Mechanical and thermal properties of carbon nanotubes. Carbon 33, pp. 925-930. Yakobson, B. I. (1997) Dynamic topology and yield strength of carbon nanotubes. In: Ruoff RS, Kadish KM (eds.) Proceedings of the Symposium on Fullerenes, Electrochem. Soc. ECS, Pennington, pp 549-560. Lu, J. P. (1997) Elastic Properties of Carbon Nanotubes and Nanoropes. Phys. Rev. Lett. 79, pp.1297-1300. Dresselhaus, M. S., Dresselhaus, G., Sugihara, K., Spain, I. L., Goldberg, H. A. (1988) Graphite Fibers and Filaments. Springer-Verlag, New York. Williams, D. B., Carter, C. B. (1996) Transmission electron microscopy. Plenum Press, New York. M., A. P., W., E. T. (1997) Nanometer-size tubes of carbon. Reports on Progress in Physics 60, pp. 1025. Binnig, G., Quate, C. F. (1986) Atomic force microscope. Phys. Rev. Lett. 56, pp.930-933 Wiesendanger, R. (1994) Scanning probe microscope: methods and applications. Cambridge University Press, Oxford. See: http://www.thermomicro.com/Spm guide/contents.htm. Odom, T. W., Huang, J.-L., Kim, P., Lieber, C. M. (1998) Atomic structure and electronic properties of single-walled carbon nanotubes. Nature (London) 391, pp. 62-64. Wildoer, J. W. G., Venema, L. C., Rinzier, A. G., Smalley, R. E., Dekker, C. (1998) Electronic structure of atomically resolved carbon nanotubes. Nature (London) 391, pp.5962. Yu, M.-F., Dyer, M. J., Skidmore, G. D., Rhors, H. W., Lu, X. K., Ausman, K. D., Ehr, J. R V., Ruoff, R. S. (1999) 3-dimensional manipulation of carbon nanotubes under a scanning electron microscope. Nanotechnology 10, pp.244. Dai, H., Hafner, J. H., Rinzler, A. G., Colbert, D. T., Smalley, R. E. (1996) Nanotubes as nanoprobes in scanning probe microscopy. Nature (London) 384, pp.147-150. Lourie, O., Rohrs, H., Huang, H., Ausman, K., Piner, R., Yu, M.-F., Dyer, M., Gibbons, P., Ruoff, R (2001) Mechanics of single walled carbon nanotubes. unpublished result. Lourie, O., Rohrs, H., Huang, H., Ausman, K., Piner, R, Yu, M.-F., Dyer, M., Gibbons, P., Ruoff, R. (2001) TEM/SEM nanostressing stage, unpublished result. Yu, M.-F., Lourie, O., Dyer, M. J., Moloni, K., Kelly, T. F., Ruoff, R. S. (2000) Strength and breaking mechanism of multiwalled carbon nanotubes under tensile load. Science (Washington, D. C.) 287, pp.637-640. Yu, M.-F., Files, B. S., Arepalli, S., Ruoff, R. S. (2000) Tensile loading of ropes of single wall carbon nanotubes and their mechanical properties. Phys. Rev. Lett. 84, pp.5552-5555. Yu, M.-F., Yakobson, B. I., Ruoff, R. S. (2000) Controlled Sliding and Pullout of Nested Shells in Individual Multiwalled Carbon Nanotubes. J. Phys. Chem. Bpp. 8764-8767.

QUASI-STATIC AND DYNAMIC TORSION TESTING OF CERAMIC MICRO AND NANO-STRUCTURED COATINGS USING SPECKLE PHOTOGRAPHY FRANCOIS BARTHELAT, KOSTYANTIN MALUKHIN, HORACIO ESPINOSA Northwestern University 2145 Sheridan Road Evanston, IL 60208-3111

Abstract

An experimental methodology for the quasi-static and dynamic testing of and WC/Co coatings on aluminum substrates is presented. A stored energy Kolsky bar apparatus was used for both quasi-static and dynamic loading. Acquisition of speckle images at different loading stages allowed the determination of the shear strains in the gage region of the specimen and the identification of deformation mechanisms as a function of coating grain size. The interaction between coating and substrate is investigated by correlating damage to measured stress-strain curves. It was identified that as cracks develop in the coating, they are bridged by the underlying ductile substrate. Optical and SEM interrogation of the coating fracture surface also show a low percentage of porosity and severe damage created by residual stresses. These residual stresses arise during the coating manufacturing process. The pre-existing damage is believed to be the cause of the low shear moduli and strengths exhibited by the coatings. A comparison between quasi-static and dynamic response is carried out, to assess strainrate effects, as well as micro versus nano grain size coatings of the same composition. 1. Introduction

Ceramic coatings are employed mainly as wear resistant materials in a variety of applications, including mining, grinding and metal cutting. Recent developments towards nano-structured cermets showed that reduction in grain size leads to enhanced mechanical properties [1],[2],[3],[4],[5]. The objective of this work was to determine mechanical properties for nanostructured and micro-structured and WC/Co coatings, and more specifically shear modulus and strength from static and dynamic torsion tests. The coatings were sprayed on an aluminum 6061-T6 substrate. The experimental procedure combined different elements: The Kolsky bar [6], which was used for both static and dynamic tests, high speed photography and speckle correlation technique for the determination of strain fields as well as localized deformations. To get a full picture of the mechanisms of deformation and failure, optical microscopy and SEM were used to investigate the 75 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 75–84. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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microstructures of the coatings in an attempt to relate them to the observed mechanical behavior. 2. Specimens

The specimens were composed of an aluminum 6061-T6 substrate on which a coating was deposited by spraying (Figure 1).

The spraying procedure was performed by A&A (A&A Company, South Plainfield, NJ). The substrates were blasted with aluminum oxide grit before the deposition process to ensure better adhesion. coatings were made using a plasma deposition technique; WC/Co coatings were made by High Velocity Oxide Fuel (HVOF). For all materials, both nano and micro-structured specimens were obtained. At the end of the coating deposition, the specimens were cooled down in air without any post-treatment procedures. 3. Testing methodology

The torsion tests were performed on a Kolsky bar apparatus (Figure 2). The specimen was glued between input and output bars. A torque was applied on the bar and was transmitted to the specimen. For the quasi-static tests, a lever arm and a platen attached to it were installed on the output bars, and the input bar was clamped. Weights were added on the platen to increase the torque by increments. The transmission of the torque to the specimen was verified using the strain gage station #1 (Figure 2). For the dynamic test, the torque was stored between the torque actuator and the clamp, and suddenly released by fracture of an aluminum pin. Upon release of the stored torque, a torsional stress wave traveled down the input bar to the specimen. For both quasi-static and dynamic tests, pictures of the surface were taken at different loading stages. For the quasi-static test these pictures were triggered manually (at each load increment), and for the dynamic case the strain gage station #1, upon arrival of the wave, triggered the highspeed camera. Adequate timing allowed taking eight high-speed pictures at different stages of the rising pulse of the wave. Dark and bright features on the specimen surface were used to correlate the pictures with each other, allowing the determination of the strain in the specimen.

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Speckle correlation technique The principle of the speckle technique is to correlate an "undeformed" image with a "deformed" image, using features present in the two images [7],[8]. These features constitute the speckle, a random distribution of dark and light features. They can be naturally present on the surface of the specimen (roughness, etched surface, etc.) or they can be created artificially (toner powder on white paint or lithography). The correlation procedure uses a reference picture of the speckle, and a "deformed" picture where the speckle has been distorted by deformations. The following numerical scheme is used [7],[8]: consider a square subset in the undeformed, reference image, and a point P, of coordinates (x,y), located inside this subset. After deformation, the subset may have moved to a new location and P moved to of coordinates viz.,

where u and v are the horizontal and vertical components of the displacement of the center of the subset, and and are the distances from the point (x,y) to the subset center. Using the dark and light features of the subset, in other words the gray level values F(x,y) in the undeformed subset and in the deformed subset, the goal is to find the values of and that satisfies the best match between undeformed and deformed subsets. A way to do so is to minimize the correlation function S:

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A Newton-Raphson scheme is then used to optimize S. Because of the discrete nature of a digital image, the resolution is limited by the size of the pixel. In order to increase this resolution, the gray values of the subset are fitted with a function, typically a cubic B-spline. Grey values between pixels can therefore be interpolated, and the search is performed using this function. This procedure allows the determination of subpixel displacements between two images. Once the displacements and deformations of the first subset are found, another subset is taken in the undeformed image and the same procedure is repeated. This allows the determination of a full field displacement between the undeformed and deformed images. The correlations were performed using the software VIC 2D (Video Image Correlation Package for 2-dimensional problem. © Correlated Solutions, 1998). Validation tests on aluminum substrates Aluminum substrates (without coatings) were tested in the first place using this method. Because of the smooth, shiny aspect of the surface of the specimen, the speckle pattern was created artificially, spraying toner power on a thin layer of white acrylic paint applied on the surface of the gage. Both quasi static and dynamic tests were performed. An example of displacement field given by the correlation method is shown in Figure 3. The pictures were taken using the high-speed camera during a dynamic torsion test on aluminum 6061-T6. The subset size used was 25x25 pixels, and correlations were performed every 5 pixels. The observed displacements are due to deformations of the specimen, but also to mechanical drift in the acquisition system (rigid body motion). Only the relative displacements (strains) are of interest in this test, and the results show an almost perfect state of pure shear in the specimen. From the displacements the shear strains were determined. Figure 4 shows a plot of the vertical displacement V as a function of the horizontal position x, along different horizontal lines. From this plot, the shear strain is simply given by the slope of these curves. The shear stress in the gage was computed from the torque using a thin wall approximation (constant stress throughout the thickness of the gage), namely,

where T is the applied torque, D is the diameter of the specimen through the centerline, and t is the wall thickness. For the quasi-static tests the torque is well known and related to the weights on the platen of the lever arm. For the dynamic test, the torque is measured by gage station #1. Accurate synchronizations between load pulse and images allow correlation between torque and each high-speed picture, as shown on Figure 5. The resulting quasi-static and dynamic stress-strain curves are shown in Figure 6. They exhibit properties are consistent with literature values for aluminum 6061-T6; shear modulus G=25 GPa and yield shear stress MPa. The yield strength for the dynamic case is slightly higher, which is also expected. Figure 6 also shows the dynamic stress-strain curve obtained by the integration of the reflected pulse [6]. The shear stresses are computed in the same way as above, but the shear strains are obtained by analysis of the incident, reflected and transmitted

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torques recorded during the dynamic test. Details of this approach can be found elsewhere [6].

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These tests provided accurate stress-strain curves for the aluminum substrate. They also showed that the speckle correlation is very accurate, even when acquiring pictures through high-speed photography. 4. Results for

and WC/Co coatings

The coated specimens were tested using the same procedure. The natural roughness of the surface of the coating provided the speckle pattern. Because the measured torque is due to both substrate and coating, it is necessary to subtract the torque in the substrate to obtain the torque in the coating. This approach relies on the assumption that the strain in the coating and substrate are identical. Figure 7 illustrates this operation, for a quasistatic test on a coating. The strains were determined by the speckle correlation method. From the total torque, the torque produced by the aluminum substrate (obtained in the previous test) is subtracted to obtain the torque of the coating alone.

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Using Eq. (3) the shear stress in the coating is computed from the torque. Figure 8 shows the resulting stress-strain curve for the coating. The curve exhibits a linear behavior up to 50 MPa, and then shows a strain hardening like behavior. Further investigation on the displacement fields reveals that this behavior cannot be attributed to the coating alone. Figure 9 shows a sequence of pictures and displacement fields at different loading stages. Figure 9a shows the displacement field in the linear elastic region. The displacements reveal inhomogeneities in the coating. Furthermore, in this article, it is shown that they are likely related to initial defects in the specimen. Figure 9b shows the displacement fields in the plateau region of the stress-strain curve. Although no crack or damage is visible in the picture, the displacement fields, measured by interrogating the speckle, reveal at least two cracks running at about 45°. Either these cracks are underneath the surface or the crack opening is so small that they cannot be seen in the pictures. The underlying aluminum substrate bridges these cracks, which prevents the specimen from catastrophic failure. Figure 9c confirms the locations of the cracks. The figure shows the last frame taken before total failure of the specimen. The image clearly shows the two cracks with a significant opening. At this point the assumption of having the same strain in both coating and substrate does not hold and, therefore, the strain hardening like part of the stress-strain curve is computed under inhomogeneous deformations.

All of the coatings, micro and nano grain size, exhibited the same type of behavior in both quasi-static and dynamic loading. Figure 10 shows stress-strain curves obtained from the tested specimens, in both quasi-static and dynamic mode. The shear moduli were in the 20-25 GPa range for and 30 GPa for WC/Co. It should be noted that these moduli are well below the theoretical values as a result of the initial microcraking present in the coatings, which is later shown by means of microscopy. The shear strengths did not exceed 60

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MPa. These values are below the reference values for these materials. This must be attributed to porosity in the coatings, as well as internal damage created by residual stresses during the coating process. Figure 11 shows micrographs of a microspecimen, which was not tested. The specimen was saw cut in the middle of the gage and the surfaces were polished. These pictures reveal porosity in the coating, as well as severe internal damage resulting from residual stress build-up during the spray deposition process. The coating is debonded from the substrate and cracks are present in both circumferential and radial directions. The circumferential cracks, running parallel to the coating-aluminum interface, are much less detrimental to the coating torsional strength than the radial cracks.

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Figure 12 shows an optical micrograph and an SEM picture of a nano-WC/Co specimen. The same kind of defects and damage is observed. The degree of initial damage varied somewhat from specimen to specimen. We tested those that exhibited the least amount of initial damage and no apparent radial cracks.

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5. Conclusions A methodology for quasi-static and dynamic torsion testing of micro and nanostructured coatings was presented. The Kolsky bar apparatus was used, as well as highspeed photography and the speckle correlation method. Experiments conducted on aluminum 6061-T6 demonstrated the accuracy of the method. Results on nano and micro-structured and WC/Co coatings were obtained. The displacement field given by the speckle correlation technique allowed the identification of the deformation mechanism, i.e., early crack propagation and crack bridging by the ductile aluminum substrate. No significant difference in modulus and strength was found between the nano and micro-structured materials, but an eventual difference between them is probably masked by the initial damage in the coatings, as observed in optical and SEM pictures. Manufacturing coatings free of initial microcracks remains a significant challenge. Research on optimization of the spray deposition parameters should be pursued to produce high quality nanostructured coatings that can fully exploit the benefits of nanosize grains. 6. Acknowledgements Horacio Espinosa and his students would like to express a special thank to Dr. Larry Kabacoff for providing the research support, through ONR-YIP award No. N0014-971-0550, and continuous encouragement through this research effort. Likewise, the support from Dr. Robert Ringel, president of A&A Co., in the manufacturing of the coated specimens is highly appreciated. 7.

References

1.

K. Jia, T.E. Fischer, B. Gallois, Microstructure, Hardness and Toughness of Nanostructured and Conventional WC/Co Composites, NanoStructured Materials, Vol. 10, No.5, 1998, pp.875-891. H. Conrad and J. Narayan, On the grain size softening in nanocrystalline materials, Scripta materilia, 42, 2000, pp. 1025-1030. B.H. Kear and W.E. Mayo. Thermal Sprayed Nanostructured Hard Coatings, Nanostructured Films and Coatings, edited by Gan-Moog Chow, Ilya A. Ovid'ko, Thomas Tsakalakos, Kluwer Academic Publishers, 2000, pp.113-129. J. He, M. Ice, and E.J. Lavernia, Synthesis and Characterization of Nanocomposite Coatings, Nanostructured Films and Coatings, edited by Gan-Moog Chow, Ilya A. Ovid’ko, Thomas Tsakalakos, Kluwer Academic Publishers, 2000, pp.131-148. H.D. Espinosa, Z. Wu and B. Prorok, “Failure Mechanisms of Nano-WC/Co Coatings Subjected to Dynamic Torsion,” submitted to Acta Materiala, 2000. ASM Handbook Vol.8: Mechanical Testing and Evaluation. ASM international, 2000; H.D. Espinosa, A. Patanella, and M. Fischer, A Novel Dynamic Friction Experiment Using a Modified Kolsky Bar Apparatus, Experimental Mechanics, Vol. 40, No. 2, 2000, pp. 138-153. H.A. Bruck, S.R. McNeill, M.A. Sutton and W.H. Peters, Digital Image Correlation Using Newton-Raphson Method of Partial Differential Correction, Experimental Mechanics, Vol. 29, 1989, pp.261-267. T.C. Chu, W.F. Ranson, M.A. Sutton and W.H. Peters. Application of Digital-Image-Correlation Techniques to Experimental Mechanics, Experimental Mechanics, Vol. 25, 1985, pp.232-244.

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2. Composite Materials

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MEASURED RESPONSE: STATE VARIABLES FOR COMPOSITE MATERIALS

KEN REIFSNIDER and MICHAEL PASTOR Virginia Polytechnic Institute and State University Blacksburg, VA 24061

Abstract

The present paper examines the concept of state variables for composite materials, and advances the premise that for irreversible processes in composite materials, the state variables are uniquely defined only for specific damage and failure modes, and that the popular "internal variables" typically used in the context of free energy concepts for the creation of constitutive equations can be replaced by a measurable engineering failure function. The paper will begin with a general discussion of the thermodynamic foundations that support the constructs of damage mechanics and continue with an outline of the manner in which strength can be cast as a state variable. Implications of the specific forms that result for evolution equations and rate processes will be discussed. Specific consequences for the case of temperature dependent material properties will be defined and compared to experimental data. The focus of the paper will be observables, i.e., physical quantities that are clearly defined and measurable. 1.

State Variable Concepts

It can be said that the state of a (material) system is defined when all the information necessary for a complete characterization of the system is in hand. The measurable quantities (observables) required for such a "complete characterization" are called state variables. Their definition depends on the purpose at hand, i.e., there is an implied function of the system when such variables are defined. Our focus here will be the function of composite material systems under the influence of mechanical, thermal, and chemical applied conditions. The literature provides ample evidence that state variables are well defined for reversible functions, especially for familiar material systems such as metals and polymers. The corresponding literature for composite materials is remarkably vacant. The need for a definition of state variables (and state functions) for irreversible processes in composite materials has spawned the field of "damage mechanics," still in it's infancy. However, the literature associated with the sub-discipline of damage mechanics is extensive. Many concepts are ‘generally accepted’ and common to much of that corpus of thought. Physical interpretations and identifications of the material parameters that typically appear in such rigor are less well established. This is a fundamental problem, 87 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 87–98. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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since one would prefer that state variables be observables, like strain and temperature, which can be measured and are known to be independent characterizations of the state of the material. In the present paper, we introduce the concept of strength as a state variable, on physical grounds, and present some related ideas for examination by the community of scholars that work in this field. A suitable framework for this exercise is provided by the thermodynamical formulations commonly used for such discussions. With the notation of Abdel-Tawab and Weitsman[1], we consider a Helmholtz free energy of the form

where represents the point-wise strain field, the damage tensor, an internal state variable representing the internal degrees of freedom of molecular motion of a polymer, when viscoelastic behavior is considered, and T represents the temperature. With the functional form of equation (1), and the entropy production inequality of Coleman and Gurtin [2] in the form

where S is entropy per unit volume, are components of the heat flux vector, and an over-dot represents differentiation with respect to time, the usual thermodynamic forces conjugate to the state variables can be written as

for the stress, entropy, molecular degrees of freedom, and internal thermodynamic force respectively. There is the question of the physical interpretation and measurement of the damage parameters, We will make the assertion that a possible interpretation of these parameters can be made in terms of anisotropic strength, defined in terms of an engineering failure function that depends on the operative failure mode associated with the engineering function of the material for a given thermodynamical situation. 2.

Strength of a Composite Material: "Critical Element" Concepts

Defining strength in fibrous composite materials cannot generally be done by simply identifying a single “stress level” that causes failure. Such composites are generally anisotropic and inhomogeneous, so that the stress state in the material is nearly always complex, even when only one stress is applied at the global level. In addition, the material strength at the local level is also anisotropic and spatially nonuniform. One must select, then, specific stress values and compare them to the correct corresponding strength values to construct a proper “strength” concept at the global level. The most common engineering method for doing this is to define a “failure function.” A failure

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function is a scalar generally written as where lies between 0 and 1. defines failure of the material. A number of different expressions based on this form have been proposed over the years. The more “natural” choice is to compare the stresses in the composite directions (1: fiber direction, 2: perpendicular or matrix direction) to their respective strengths (X: tensile strength, Y: transverse strength, S: shear strength). Many continuous fiber composites are made up of plies or laminae that are bonded together to form a laminate. The discussion of strength for each of those laminae is essentially the same as the details mentioned in the previous section, with the addition of the effect on ply stresses of the accommodation each ply must make to deform with the same strain as it’s neighbors. For the laminate, each ply may fail, in various ways, and the failure of the laminate can be defined by “last ply failure,” in some sense. The most typical approach to the definition of the strength of laminates is to address the failure of each ply with increasing load level or time. In that sense, one can define a “first ply failure,” and subsequent ply failures as the internal stress state changes due to internal relaxation. There is an accepted practice in the composite engineering field for calculating the progressive internal stress state, by reducing (or ‘discounting’) the stiffness in damage-affected directions in those plies that fail. This “ply discount” method assumes that stress in the matrix of those plies is redistributed, with the effect of increasing various stress components in the other plies. If one calculates the strength of the last ply to fail as a function of this redistribution process, then it is clear that this “failure function” will be increasing with time, even for a constant applied global loading. If failure is defined by fracture, then the last ply to fail can be called a “critical element,” in the sense that global failure is defined by the local failure of that ply. Then one might consider the failure function in that critical element as a canonical parameter for the definition of strength, or remaining strength in the process of progressive failure of the plies in the laminate. This is, in fact, the fundamental foundation of the “critical element theory,” that we have developed (Reifsnider and Stinchcomb [3], Reifsnider [4, 5, 6, 7, 8, 9]). The concept of a local failure function in the critical element changing with loading history, or more precisely, with changes in internal stress state (the numerator of the failure function) or material state (the denominator of the failure function) is the foundation of the remaining strength philosophy that we have constructed over a period of about 15 years, and the measurable that we wish to discuss. 3.

Strength Evolution

The preceding concepts and methods can be extended to the most general case of the calculation of remaining strength and life of composite materials and structures under mechanical, thermal, and environmental applied conditions that produce combinations of fatigue, creep, and stress rupture (time-dependent failure). The first step in the philosophy is to carefully identify the failure mode that is induced by the applied conditions, using experimental methods. Then, using the precepts described above, a failure function form is selected to describe the final failure event (e.g., fiber failure in a critical element, etc.). Then all of the processes that cause changes in the stress state or

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material state in that critical element are characterized by rates as a function of the applied conditions and (generalized) time. Failure modes can change due to applied environments such as temperature, chemical agents, and time or cycles. For the present philosophy, the failure mode must be determined for the conditions to be modeled, preferably by experimental characterization. If that is not possible, experience available in the literature can often be used to anticipate that failure mode. Another essential feature of this philosophy is that the initial stress state and material state of the material are greatly altered by the history of loading. In fact, for the low stress levels applied for practical engineering applications, the amount of damage that occurs before failure is very great, much larger than the damage that occurs in laboratory tests for shorter times at larger stress levels. The central theme of the present approach, then, is to track the changes in the stress state and material state in the critical element as functions of the duration and history of the applied conditions, as well as the constituent properties and the rates of the degradations processes that act to change properties. Our problem is to combine those elements in a self-consistent way, such that the interactions and collective effects of, say, fatigue, creep, stress rupture, and other phenomena are retained, in the spirit of the critical element philosophy. Elements of this approach have been described in our earlier publications (noted above). Neither thermodynamics or statistical mechanics are applicable to systems that are not in equilibrium. Many non-equilibrium processes are, however, of great importance to engineering problems. Classical examples include chemical reactions and solidification of super-cooled liquids. In the present context, these processes include micro (or sub-micro) defect formation and accumulation, which is generally dissipative and spontaneous. 4. Damage Evolution

We make the claim that damage is developing under applied conditions that are constant in time with respect to each other, i.e., while applied stress or strain components may vary proportionately in magnitude, their ratios remain constant with respect to each other and with respect to any anisotropy of the subject body. Under those conditions, if the magnitude of those applied conditions reach a sufficient level, failure will be induced with a specific and unique failure mode. Hence, since we will discuss strength, we will discuss strength individually for each such failure mode, and the constitutive equations will be cast individually as well, for a constant failure mode. This is a significant departure from “classical” formulations. We make the claim that our resulting constitutive forms (for damage evolution) apply to a specific damage mode. For such conditions, the form of the scalar failure function, is constant. For the moment, we also assume that we will not consider viscoelastic behavior, so we ignore the dependence of the Gibbs free energy function on the molecular degrees of freedom and postulate the form

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For such a form, the damage parameter is an observable that does not have units (like strain and temperature). Also, it can be shown that the corresponding conjugate force for this state variable for damage is proportional to the elastic stored strain energy, an argument quite similar to the formulation of Kachanov (10). Then we ask, what are the consequences of this postulate, and how does it relate to engineering behavior? We follow classical arguments for the case of a scalar state variable for damage, and introduce a damage rate with the form

where is the (scalar) damage variable such that , is the corresponding continuity of the materials such that , and G is the scalar damage potential. For our physical interpretation, we make the claim that For the moment, we only justify this assertion by the observation that the limits of these two functions are identical, and that they have the same physical sense in that fracture is defined (for a quasi-static test) by the condition of for which must have a null value. Then for slowly varying values of we postulate,

so that

where j is a material parameter and is the characteristic time for the damage evolution process that controls strength. From this rate equation, an expression for the remaining strength of our material system has been developed by Reifsnider (7).

Before using this form to solve engineering problems, we consider a further physical interpretation of the forms and parameters introduced in the above formulation, assisted by the concepts generally associated with kinetic theory. In an earlier publication [Reifsnider and Pastor, (11), in press], the authors used kinetic theory to discuss the probability of occurrence of discrete events such as micro-crack formation, debonding, constituent-boundary separation, and constituent fracture in inhomogeneous systems. We were able to show that the probability had the form

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where 'e' is the rate of applied energy (per unit time), A is a material constant, and is the characteristic energy of all active damage events. Interestingly, this probability has the form of a Boltzman distribution. On physical grounds, the probability function in (9) has the limit of 0 when there is no energy applied to the system, and 1 when the energy (rate) is so large that "all" of the damage events are active and the material fails - but that is exactly the physical interpretation of in equations (5-7). If we set and make the approximation that on the same physical grounds just discussed, then equation (9) gives the same form as the postulate in equation (7) for the special case of j and when the energy rate is proportional to time and collects all of the other material constants. This is a remarkably simple and direct physical interpretation of the damage parameters, and leads directly to the remaining strength equation, (8). This basic form of the strength evolution equation has been used by us for a dozen years, with many industrial partners, to solve various applied problems in which time, temperature, and cyclic loading are explicit influences [c.f. Reifsnider and Stinchcomb (3), Reifsnider (4, 5, 6)]. The nature of the inputs to equation (9), are determined by two requirements. First, it must be possible to measure all necessary inputs to any practical model using straight forward engineering tests, like fatigue, creep, and stress rupture characterizations. Second, the measurements must produce independent input data, not coupled to other behavior. Hence, fatigue data for input is usually developed at temperatures and under conditions that eliminate any time-dependent behavior. Creep is carefully measured in such a way to avoid cracking and fatigue. The effects of matrix cracking (on stiffness) are measured under conditions that avoid creep or stiffness changes induced by other phenomena, and so on. These independent characterizations are recombined in equation (8) by their collective effect on the arguments in If creep changes stiffness, for example, and matrix cracking also change stiffness (in the same increment of time), the changes are added together to alter the state of stress in the numerator of If corrosion processes change material strengths and thermodynamical microstructural changes also alter material strengths, then their respective incremental changes are added to alter the denominator of in that time increment. So the method of combining effects is piecewise linear. It should also be noted that the coupling of effects is accounted for in this process. If creep reduces the local stress that is driving crack initiation, for example, the reduced stress level is used in the calculation of the next incremental matrix cracking rate. The incremental evaluations of the integral in equation (8) bring all coupled effects together by updating the independent variables that enter the rate equations for all degradation processes with each incremental evaluation of the integral. The resulting remaining strength, , and life is a highly nonlinear, sequencedependent function of all applied conditions. Numerous predictions of remaining strength and life for numerous combinations of fatigue, creep, and stress rupture effects have been made and checked against experimental data for applications from jet engine parts to off-shore flexible pipe. Figure 1 shows some examples of some of these comparisons.

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In Fig. 1, predicted remaining strength is compared to observed values for an enhanced material (an early DuPont Lanxide developmental version of that material) at Literally hundreds of these comparisons have been made with comparable results. These predictions are somewhat unique; very few remaining strength predictions have appeared in the literature. However, they are essential for man-rated structures since damage tolerance is the certification procedure required by the FAA, for example. These and many other comparisons were done in connection with a variety of industrial partners including the General Electric Engine Division, Pratt & Whitney, United Technologies, Westinghouse, Allied Signal Composites, Martin Marietta, Babcock and Wilcox (now McDermott Technologies, Inc.), DuPont, and Solar Turbines. The physical data and general guidance provided by those partners was critical to the construction of a robust approach to the present problem. 5. Temperature Temperature is another observable state variable. It is known to effect stiffness and strength, especially of polymer materials. However, our recent results provide the surprising fact that the tensile strength in the fiber direction of a polymer composite coupon, for example, can change by 15-34 percent when the matrix properties or the fiber-matrix coupling changes due to temperature or local constituent variations, even though the fibers are unaffected by those changes (Mahieux and Reifsnider [12], Mahieux, et. al [13]). The present kinetic approach has been extended to the description of the explicit variation of polymer stiffness with temperature. The model uses Weibull statistics to discuss the relaxation of the constraints of secondary bonds with increasing temperature. For polymers which display a primary and secondary transition with temperature, the resulting expression for Youngs modulus across those transitions has the form

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where and are the instantaneous moduli at the beginning of the glassy and rubbery plateaus, respectively, is the glass transition temperature, is the onset temperature for flow, and and are the Weibull parameters associated with the glass and flow transitions. All quantities except in this expression can be found by independent measurements. Typical values of are about 20 for many common polymers, but is dependent on the degree of crystallinity in thermoplastics, or amount of filler in something like rubber. Mahieux, et. al [13] applied this methodology to amorphous and highly crystallized PPS, with excellent fidelity to measured results. Mahieux and Reifsnider [12] have postulated that the stiffness of a unidirectional composite, in the fiber direction can be given in the form

where are the fiber and matrix modulii, are the volume fractions of the constituents, and is an “efficiency factor,” given by

Combining equations (11) and (12), we obtain the prediction of the explicit dependence of fiber-direction stiffness on temperature for unidirectional composites shown below. If the independent data for the polymer PPS are substituted into equation (12) and the results compared to experimental data for the secondary transition, the comparison shown in Fig. 2 are obtained.

Mahieux, et. al, [12] were also successful in using this approach to estimate the changes in unidirectional, uniaxial strength of the same composite with temperature, by

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introducing the same efficiency factor into the shear transfer stress between the fiber and matrix.

6.

Conclusions

We have made the assertion that the internal state variable that appears in the free energy expression for damage evolution can be interpreted in terms of the failure function that is appropriate for the failure mode that ultimately controls failure. We have also shown how this failure function can be related to the measurable quantity, remaining strength. The generality of these assertions has not been established. Many of the implications are pertinent to engineering applications. In that context, we have presented an outline of an approach to the prediction of remaining strength and life of brittle material systems in which damage accumulation is responsible for the degradation of properties and performance. The approach, based on the critical element method developed earlier, and founded on the concept of a generalized failure function and thermodynamical damage evolution, follows accepted engineering practice for the interpretation of strength in composite materials and structures. The practicality, utility, and validity of the approach has been established by applications in association with more than two dozen industrial organizations and groups over 15 years. In Fig. 3, predicted life values (developed from equation (9)) are compared to observed data for a variety of high-temperature ceramic composite tests, including mission loading, hold times, different shaped loading ramp combinations, and spike and hold sequences. Many of the predictions combine the effects of matrix cracking (which can cause large stiffness changes), creep rupture (largely due to oxidation), and cyclic fatigue loading.

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Frontiers and opportunities associated with this approach greatly outnumber current achievements. Although we have not had space for the development here, the entire philosophy can be cast in terms of micromechanics, of stiffness and strength. But micromechanical strength models are still incompletely developed for many damage and failure modes. Some of our efforts are found in Gao and Reifsnider [14], Reifsnider and Gao [15], Xu and Reifsnider [16], and Subramanian, et. al [17]. Viscoelastic representations currently used are essentially linear; robust nonlinear models based on measurable quantities are only now being introduced. Stiffness change due to damage development is the subject of countless papers and a large body of literature. But even crack density as a function of time or cycles is still difficult to estimate from first principles. Kinetics is a well developed analytical concept, but there are only a few first-principle simulations of the kinetics of stress rupture, and those are for failure modes and models that are incompletely verified. And the application of such models to structures has another set of challenges. This approach can be (and is being) applied on an element by element basis for FEM analysis of high-temperature structures, such as combustor liners. However, these are highly nonlinear problems, so that the sequence of internal stepping and adjustments of stiffness and material strength variables is significant – but largely unexplored.

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Despite these and many other needs, the present approach offers the field a general method that is soundly based on mechanics and materials fundamentals, sensitive to mechanistic degradation mechanisms, and easily supported by well established engineering characterization methods that provide independent data. The method can support advances by other investigators in nearly all areas of mechanics and materials, and, therefore, offers a method of bringing other fundamental work to the applications community. Finally, the approach can be used as a guide to the development of systematic experimental data for the development of material and structural systems, and for the life cycle cost estimation of engineering systems. 7. Acknowledgements

The authors gratefully acknowledge the support of the Air Force Office of Scientific Research under grant no. F49620-95-1-0217 and NASA Langley under contract no. NAS1-19610 for the work on high temperature polymer composites, and the support of the National Science Foundation under grant no. DMR9120004 for the micromechanical modeling. Thanks also go to McDermott Technologies Inc. under contract no. 90189/CER1286 and the Virginia Center for Innovative Technologies under grant no. MAT-94-015 for support of the ceramic composites research. 8. 1.

References

Abdel-Tawab, K., and Weitsman, Y.J. (1997) “A Strain-Based Formulation for the Coupled Viscoelastic/Damage Behavior,” Contractors report MAES97-3.0-CM, The University of Tennessee, Knoxville, TN, July. 2. Coleman, B.D. and Gurtin, M. (1967) “Thermodynamics with Internal Variables,” J. Chem. Phys., 47, 597-613. 3. Reifsnider, K.L. and Stinchcomb, W.W. (1986) "A Critical Element Model of the Residual Strength and Life of Fatigue-loaded Composite Coupons,” Composite Materials: Fatigue and Fracture, ASTM STP 907, H.T. Hahn, Ed., Philadelphia, PA: American Soc. for Testing and Materials, 298-313. 4. Reifsnider, K.L. (1991) Editor, Fatigue of Composite Materials, London: Elsevier Science Publishers. 5. Reifsnider, K.L. (1992) "Use of Mechanistic Life Prediction Methods for the Design of Damage Tolerant Composite Material Systems," ASTM STP 1157, M.R. Mitchell & O. Buck Eds., Philadelphia, PA: American Society for Testing and Materials, 205-223. 6. Reifsnider, K.L. (1995) “Evolution Concepts for Microstructure-Property Interactions in Composite Systems,” Proc. IUTAM Symp. On Microstructure-Property Interactions in Composite Materials, Aalborg, Denmark, R. Pyrz, Ed., New York, NY: Kluwer, 327-348. 7. Reifsnider, K.L. (1996) "A Micro-Kinetic Approach to Durability Analysis: The Critical Element Method,” Progress in Durability of Composite Systems, A.H. Cardon, K.L. Reifsnider, & H. Fukuda, Eds., Balkema, Rotterdam, 3-11. 8. Reifsnider, K.L. (1996) “Recent Advances in Composite Damage Mechanics,” Proc. Conf. On Materials and Mechanical Testing, European Space Agency, Noordwijk, Netherlands, 27-29 March 1996, SP-386, European Space Agency, 483-490. 9. Reifsnider, K.L. (1997) "Durability and Damage Tolerance: Testing, Simulation, and Other Virtual Realities," in Composite Materials: Testing and Design, Thirteenth Volume, ASTM STP 1242, S.J. Hooper, Ed., American Society for Testing and Materials, 45-59. 10. Kachonov, L.M. (1986) "Introduction to Continuum Damage Mechanics," Martinus Nijhoff Pub., Boston.

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11. Reifsnider, K.L. and Pastor, M. (in press), "Relatioinship of Continuum Damage Mechanics to 12. 13.

14. 15. 16. 17.

Strength and Performance Evolution," Proc. Recent Advances in Continuum Damage Mechanics for Composites, 20-22 September, 2000, LMT-Cachan, France. Mahieux, C.A. and Reifsnider, K.L. (2001) “Property Modeling across Transition Temperatures in Polymers; A Robust Stiffness-Temperature Model,” Polymer, Vol. 42, 3281-3291. Mahieux, C.A., Reifsnider, K.L, and Case, S.W. (2001) “Property Modeling across Transition Temperatures in PMC’s”, Applied Composite Materials, Vol. 8, Part 1. Tensile Properties, pp. 217-234; Part II. Stress Rupture in End-Loaded Bending, pp. 235-248; Part III. Bending Fatigue, 249-261. Gao, Z. and Reifsnider, K.L. (1993) "Micromechanics of Tensile Strength in Composite Systems," Proc. Fourth Symp. on Composite Materials: Fatigue and Fracture, Indianapolis, IN: ASTM 1156, W.W. Stinchcomb and N.E. Ashbaugh, Eds., Philadelphia, PA: ASTM, 453-470. Reifsnider, K.L. and Gao, Z. (1991) "Micromechanical Concepts for the Estimation of Property Evolution and Remaining Life," Proc. Intl. Conf. on Spacecraft Structures and Mechanical Testing, Noordwijk, The Netherlands, 24-26 April 1991: ESA SP-321, Oct. 1991, 653-657. Xu, Y.L. and Reifsnider, K.L. (1993) “Micromechanical Modeling of Composite Compression Strength," Journal of Composite Materials, 27 (6), 572-588. Subramanian, S., Reifsnider, K.L. and Stinchcomb, W.W. (1995) “Tensile Strength of Unidirectional Composites: The Role of Efficiency and Strength of Fiber-Matrix Interface,” Journal of Composites Technology and Research, JCTRER, 17 (4) (October), 289-300.

ON THE MODELING OF THE MECHANICAL PROPERTIES OF COMPOSITE MATERIALS AT HIGH STRAIN RATES

JACK R. VINSON and SHENGKUAN XIAO Department of Mechanical Engineering University of Delaware Newark, DE 19716, U.S.A.

Abstract High strain rate compressive and tensile ultimate strength properties for a wide variety of composite materials as function of strain rate are modeled, using the Weeks – Sun equation. These include unidirectional, quasi-isotropic and woven composites, made of glass, graphite and aramid fibers in various thermoset and thermoplastic polymeric matrices. Through modeling, using the Weeks – Sun equation, dynamic composite material properties can be easily used in the design and analysis of composite material structures rather than static material properties. By doing so, excess weight or unexpected failure may be avoided. 1.

Introduction

Almost all structures are subjected to dynamic loads. The U.S. Navy, for example, has stated that under some circumstances, naval vehicles can encounter strain rates up to 1200/second. Because the dynamic properties of composites may vary widely with strain rate, it is important to use dynamic properties when the loading conditions on a structure require it. Vinson and his colleagues have found that through testing over thirty varied composite material over a range of strain rate values up to 1600/sec, that in comparing the high strain rate values to static values, yield stresses can increase by a factor up to 3.6, yield strains can change by factor 3.1, strains to failure can change by factors up to 4.7, moduli of elasticity can change by factors up to 2.4, elastic strain energy densities can vary by factors up to 6.0, while strain energy densities to failure can change by factors up to 8.1. In addition, as strain rates increase, some composites can change from a ductile to a brittle behavior. Thus the use of static material properties to analyze and design structures subjected to impact, explosions, crashes or other dynamic loads should be carefully reviewed. All too few materials have been characterized at high strain rates. Still less effort has been spent in trying to model the high strain rate properties to develop a predictive capability. It has been hoped that earlier modeling for metals, such as Johnson and Cook [1], and Zerilli and Armstrong [2] might be used for composite materials. The Johnson – Cook model was modified by Weeks and Sun [3] for composite materials. Other recent modeling research has been performed by Theruppukuzki and Sun [4], Hsiao, 99 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 99–108. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Daniel and Cordes [5] and Tsai and Sun [6]. Woldesenbet and Vinson [7] have characterized the high strain rate and fiber orientation effects on one typical graphite/epoxy composite. Most of these characterizations model ultimate strengths only. Vinson and Xiao [8] have modeled the ultimate strength and moduli of elasticity of numerous composite materials over a wide variety of strain rates, using polynomial expressions. Over the last several years a program has been conducted to experimentally determine the dynamic compressive and tensile material properties of various composite materials that are of interest to industry and to various government agencies. A Split Hopkinson Pressure Bar was used for all of the compression experiments. The tensile tests were carried out in an Instron high strain rate machine. In all cases at least three replicate specimens were tested, and subsequently the data was analyzed to determine both mean values and standard deviations. The Johnson – Cook equation for depicting high strain rate effects on ultimate strength is written as

where is the ultimate compressive strength is the strain

is the strain rate

and

n and are empirically determined constants. The Weeks – Sun equation is written as

where all of the symbols have been defined previously, except which is the static ultimate strength. It is seem that in the Weeks – Sun analysis, has replaced the first term in the Johnson – Cook equation, but in each equation the terms involving strain rate effects are identical. In the Weeks – Sun equation it is seen that

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The first and third terms in the above are both constants, so (3) can be written simply as

In this equation A is a constant and B is a parameter indicating the magnitude of the effect of high strain rate. 2.

Analysis

In the following, (4) is used to provide the equation for the ultimate strength as a function of strain rate using the data obtained and referenced. The data fall into two categories. Early on, in the experiments for each material, data was collected from the Split Hopkinson Pressure Bar at a variety if strain rates, but no static data was obtained for the following reason. If the static material properties are wanted there are two options: (1) use standard ASTM test procedures but in that case the size and shape of the test pieces differ greatly from those used in the SHPB high strain rate tests, or (2) use the same test piece configuration as used in the high-strain rate tests to obtain static properties using an Instron or other testing machine, but then the data is not ASTM data. After some testing without static property determination it was decided to use alternative (2) above to obtain static properties. It was wise to do so because, with the static test date used in the characterization, all values of A in equation (4) above are positive and are the static values for the material tested under non-ASTM methods, but with the same configuration as the SHPB test pieces. Without the static data some values of A in (4) are negative and although the equation gives accurate values of the dynamic strength within the range of strain rates tested, the equation cannot be used to determine strength values below or above the strain rates of the data obtained. 3.

Results

All ultimate compressive strengths reported below are in units of pounds per square inch (psi). All data for Materials 1 through 12 are in compression, while the data for Material 13 and 14 are tensile data. Material 1 and 2 are polymer matrix composites with glass fibers; Materials 3 through 8 involves graphite fibers, while Material 9 involves aramid fibers; Material 10 is a metal matrix composite; Materials 11 and 12 are graphite/epoxy composites which show the effects of through – thickness stitching, as stated before, and Materials 13 and 14 involve contact molded woven fabric composites tested in tension. In all of the papers referenced it was found that the standard deviations of the high strain rate data obtained were small, even with three replicate testes, hence the coefficients of variation are small. In each of the following, the data from the lowest strain rate test and the highest strain rate test are used to form two equations from which the constants A and B can be solved. A typical calculation will be performed for Material 1.

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GLASS-FIBER REINFORCED COMPOSITE MATERIALS

Material 1: Cycom 5920/1583, parallel lay up (Reference [9])

Using equation (4) above, From From

So

Material 2: Unidirectional E-glass/epoxy (Scotchply 1003, [10], Tables 3 and 4)

1 (fiber) direction

3 direction

HIGH STRAIN RATE PROPERTIES

4.2

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Material 3: IM7/8551-7 Unidirectional composite ([11]) 1 (fiber) direction

2 direction

3 direction

Note: Because none of these data have static test values and have differing upper limits, one can draw no comparison for static values, nor can one extrapolate the equations beyond the limits specified, e.g. one cannot conclude that the static value in the 2 directions is larger than the static value in the 1 direction. Material 4: IM7/E7T1-2 Composite ([12], Tables 4 and 5) 1 (fiber) direction

3 (thickness) direction

Material 5: IM7/K3B Quasi-isotropic Composite ([13], Tables 3 and 4)

1 and 2 (in-plane) directions

3 (thickness) direction

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Material 6: AS4-shell 9470/9405 Quasi-isotropic Composite ([14], Tables 5 and 6)

1 and 2 (in-plane) directions

3 (thickness) direction

Material 7: AS4/K3B Quasi-isotropic Composite ([14], Tables 7 and 8)

1 and 2 (in-plane) directions

3 (thickness) direction

Material 8: AS4/PEKK Quasi-isotropic Composite ([15], Tables 2 and 3)

1 and 2 (in-plane) directions

3 (thickness) direction

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4.3 ARAMID FIBER POLYMER MATRIX COMPOSITE Material 9: Kevlar 49/3501-6 Composites ([16], Tables 2, 3, 5 and 6) 1 direction

3 direction

4.4 SILICON CARBIDE METAL MATRIX COMPOSITES Material 10: SiC /Al Composites ([17])

4.5

EFFECTS OF STITCHING Material 11: AS4 Unidirectional Graphite Fabric/3501-6 Epoxy Composites ([18], Tables 3a, 4a and 5)

2 (in-plane) direction, without stitch

3 (thickness) direction, without stitch

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Material 12: AS4 Unidirectional Graphite Fabric/3501-6 Epoxy Composite, same as Material 11, with a stitch ([18], Tables 3b, 4b and 5)

2 direction, with stitch

3 (thickness) direction, with stitch

4.6 WOVEN COMPOSITE MATERIALS IN TENSION Material 13: 510A/WR Contact Molded Composite ([19], Tables 5 and 6)

Warp direction

Fill direction

Material 14: 8084/WR Composites ([19], Tables 7 and 8)

Wasp direction

Fill direction

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4. Discussion and Conclusions

From the above equations several conclusions can be drawn, 1. In several graphite fiber composites the high strain rate effects are deleterious, i.e. the strength decreases with increased strain rate, among these materials are: AS4 Graphite K3B (thermoplastic) composites in the 1 and 2 directions (in-plane directions). AS4 PEKK (thermoplastic) composite in the 1 and 2 direction (in-plane directions). AS4 Uniweave Graphite Fabric/3501-6 Epoxy with and without a stitch. IM7/E7T1-2 in the 3 (thickness) direction. Note that this occurs in both thermosetting and thermoplastic matrix composites with graphite fibers. 2. The highest strain rate sensitivity (i.e., highest value of B in equation (2)) by far occurs with the quasi – isotropic K3B (thermoplastic) matrix composites in the thickness direction with both AS4 and IM7 graphite fibers. 3. Below those materials listed in 2 above, the next highest sensitivity occurs in the SiC / 2080 Aluminum MMC. 4. In the woven composite materials in tension (Material 13 and 14 above), the strain rate effects are significant, even though the highest strain rate is only about 4 This adds evidence to the belief that the most significant changes in properties due to dynamic effects occur at low strain rates. 5. Concerning the effects due to stitching, at static loads the stitch has little effect in the 2 direction, but the stitch reduces properties in the thickness direction at static loading, and reduces properties in both directions under dynamic loads. 6. In the woven composites in tension, the 510 composite is superior to the 8084 composite in the warp direction, but the two materials have the same behavior in the fill direction. 5. Acknowledgements

Appreciation is expressed to the Office of Naval Research, which supported all of this research through Grant N000 14-93-1-1014, and to Dr. Yapa D.S. Rajapakse, the Project Manager. 6. References 1. 2. 3.

Johnson, G.R. and Cook, W.H. (1985) "Fracture Characteristics of Three Metals Subjected to Various Strains, Strain Rates, Temperatures and Pressures," Engineering Fracture Mechanics, Vol. 21, 31-48. Zerilli, F.J. and Armstrong, R.W. (1987) "Dislocation Mechanics – Based Constitutive Relations for Material Dynamic Calculations," Journal of Applied Physics, Vol.6, 1816-1825. Weeks, C.R. and Sun, C.T. (1995) "Nonlinear Rate Dependent Response of Thick-Section Composite Laminates," ASME Bound Volume AD-Vol. 48 "High Strain Rate Effects on Polymer, Metal and Ceramic Matrix Composites and other Advanced Materials," co edited by Y.D.S Rajapakse and J.R. Vinson, 109-114.

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7. 8. 9. 10.

11. 12. 13.

14.

15.

16. 17. 18.

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Thorukpukuzki, S.V. and Sun, C.T. (1998) "Testing and Modeling High Strain Rate Behavior of Polymeric Composites," Composites, Part B, 29B, 535-546. Hsiao, H.M., Daniel, I.M. and Cordes, R.D. (1999) "Strain Rate Effects on the Transverse Compressive and Shear Behavior of Unidirectional Composites," Journal of Composite Materials, Vol. 33, No. 17, 1620-1642. Tsai, J. and Sun, C.T. (2000) "Nonlinear Constitutive Model for High Strain Rate Response in Polymeric Composites," Proceedings of the Annual Technical Conference of the American Society for Composites. Vinson, J.R. and Woldensenbet, E. (2001) "Fiber Orientation Effects on High Strain Rate Properties of Graphite/Epoxy Composites," Journal of Composite Materials, Vol. 35, No. 6, 509522. Vinson, J.R. and Xiao, S. (2001) "On Predicting the Dynamic Failure in Composite Materials under Compressive and Tensile Loads," ASME-IMECE, New York, N. Y., November 2001. Powers, B.M., Vinson, J.R., I.W. Hall and Hubbard, R.F. (1995) "High Strain Rate Mechanical Properties of Cycom 5920/1583," Proceedings of the AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 2386-2392. Dee, A.T. and Vinson, J.R. (1997) "The Effect of High Strain Rate on the Compressive Mechanical Properties of Unidirectional Glass/ Epoxy (3M Scotchply 1003)," Proceedings of the American Society for Composites Annual Technical Conference, 399-408. Powers, B.M. and Vinson, J.R. (1995) "High Strain Rate Mechanical Properties of IM7/8551-7 Graphite Epoxy," Proceedings of the American Society for Composites Annual Technical Conference, 227-238. Dee, A.T., Vinson, J.R. and Leon, G. (1997) "High Strain Rate Mechanical Properties of a Torospherical Shell Composed of IM7/E7T1-2 Graphite Epoxy Composite," International Conference on Composite Materials, Australia. Powers, B.M., Vinson, J.R., Wardle, M. and Scott, B. (1996) "High Strain Rate Effects on Two Graphite Fiber K3B Polyimide Matrix Composites," Proceedings of the AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 30-38. Powers, B.M., Vinson, J.R., Wardle, M. and Scott, B. (1995) "High Strain Rate Effects on Two AS4 Graphite Fiber Polymer Matrix Composites," ASME Bound Volume AD-Vol. 48, "High Strain Rate Effects on Polymer, Metal and Ceramic Matrix Composites and other Advanced Materials," co edited by Y.D.S. Rajapakse and J.R. Vinson, 179-189. Powers, B.M., Vinson, J.R., Wardle, M.D. and Scott, B. (1996) "High Strain Rate Effects on AS4/PEKK Graphite Fiber Thermoplastic Matrix Composites," Proceedings of the American Society for Composites Annual Technical Conference, 486-494. Preissner, R.C., Woldesenbet, E. and Vinson, J.R. (1997) "High Strain Rate Compression Testing of K49/3501-6 Kevlar/Epoxy Composites," Proceedings of the AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 935-944. Powers, B.M., Vinson, J.R., Hall, I.W. and Nardone, V. (1995) "High Strain Rate Mechanical Properties of Silicon Carbide Reinforced 2080 Aluminum Metal Matrix Composites," Proceedings of the International Conference on Composite Materials, Vol. II, 317-322. Dee, A.T., Vinson, J.R. and Sankar, B.V. (1997) "Effects of High Strain Rate Compression on the Mechanical Properties of a Uniweave A54/3501-6 Composite Laminate with Through-Thickness Stitching," Proceedings of the AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 945-955. Greenfield, R., Vinson, J.R. and Telegadas, H. (1998) "High Strain Rate Tensile Properties of Woven Glass Fabric Composites," Proceedings of the AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 1362-1371.

DAMAGE QUANTIFICATION IN METAL MATRIX COMPOSITES

G. Z. VOYIADJIS, A. R. VENSON, and R. K. ABU-ALRUB Department of Civil and Environmental Engineering Louisiana State University Baton Rouge, LA 70803

Abstract

An experimental procedure is presented to quantify damage in terms of microcrack density. This is accomplished by experimentally evaluating the components of a second-order damage tensor for a metal matrix composite material. The procedure involves the use of a scanning electron microscope and image analyzing software to quantify physical damage features found on a representative volume element. These features are quantified in terms of crack density, which is used in developing the second-order damage tensor. This procedure is applied to a titanium aluminide SiCreinforced laminate. Laminates of the following staking sequences, and are tested under uniaxial tensile loadings. Damage evolution is obtained by loading the specimens over a range of load intensities from rupture load down to 70% rupture load. A proposed formulation for a coupled anisotropic damage model for the inelastic response of composite materials is presented in this work. 1.

Introduction

Various theoretical models exist for investigating the effects of damage on metal matrix composites. Limited experimental investigations have been performed to relate physical damage to a corresponding theoretical definition. These investigations are primarily confined to damage as a result of fatigue and or fracture [1], with little correspondence between physical and theoretical damage. Additionally, these investigations do not present the damage evolution in terms of a function of the measured physical damage over the load history. In a recent publication, [2] presented a thorough examination and explanation of the microstructural evolution of damage. The work from their investigation still needs to be incorporated into a constitutive theory for the quantification and evolution of physical damage. Other more recent experimental procedures have also been introduced to quantify damage due to microcracks and microvoids through X-ray diffraction, tomography, and so forth [3]. However, these procedures are in need of refinement that will allow the ease of differentiating different types of damage such as voids and cracks (radial, debonding, and z-type). 109 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 109–120. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Although some steps have been taken to correlate physical and damage and its theoretical definition, additional experiments are needed to quantify damage parameters and evaluate the corresponding damage theory. The work that has been done uses a continuum definition with various schemes to measure damage. These schemes measure damage as a ratio between an effective quantity and its respective damage value. [4] listed several such schemes that defined the damage parameter ratio based on the area of resistance, material density, or elastoplastic modulus, among others. The ratio defined by the elastoplastic modulus is commonly used to obtain the damage parameter due to the ease in evaluating the damaged and undamaged elastoplastic moduli. The material presented in this work will outline a technique to experimentally evaluate different types of damage in a metal matrix composite material. It will also show how the resulting damage parameters can be incorporated into a micromechanical damage theory. The focus of the technique presented is in support of an elastoplastic micromechanical damage model developed in [5] and [6]. However, the results of this work can be used with any damage theory based on a tensorial damage parameter. 2.

Design and Manufacture of Specimens

The material investigated is a titanium aluminide composite reinforced with continuous SiC (SCS-6) fibers. Typical properties of the SiC fibers are shown in Table 1. It is reported that these fibers have good wettability characteristics for metals, which minimizes the chance of voids being induced during the manufacturing process. These fibers are also coated with a carbon rich coating that assists in protecting the inner SiC from damage during handling. Both of these features are important in reducing the opportunity for inducing damage during manufacturing.

The matrix material originates as a titanium aluminide foil in an phase, with typical properties as shown in Table 2. The titanium aluminide foil is made from rolled ingot material. Niobium is added to the matrix to improve overall composite ductility [7]. Material properties of a composite lamina for 0° and 90° orientations were obtained from experimental tests. These values are as reported in Table 3.

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Two 304.80 × 304.80 mm plates with layups of and are fabricated using hand layup techniques from SCS – 6 SiC fiber mats and Ti – 14A1 – 21Mb foils. Each of the plates contained four plies with a 0° or +45° as the top and bottom ply for the respective layups. Fibers are included in each ply in the form of a fiber mat, were fibers are held together with molybdenum wires. This mat aide in keeping the fibers aligned and equally spaced during the consolidation process. Consolidation was accomplished by hot-isostatic-pressing (HIP) in a steel vacuum bag at 1010 °C ±25° under 103.00 MPa pressure for 2 hrs. The most severe warpage, resulting from the differences in coefficients of thermal expansion for the fiber and matrix, was confined to the edges of each plate, with a maximum relative elevation difference of 2.24 cm for the plate and 1.30 cm for the plate. Care is taken during specimen preparation to ensure that this warpage does not induce damage. Each of the laminates was machined using diamond tooling to produce six tensile test specimens with shape of dogbone type specimen. The shape of the dogbone type specimen was selected based on a successful use by previous researchers [8]. This shape will ensure specimen failure within the gage section and not the grips. Aluminum tabs are arc-welded onto the ends of each test specimen in order to prevent the mechanical grips from damaging the specimen.

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Tensile Testing of Specimens

The experimental model used was designed to collect quantitative and qualitative data. Recommendations for specimen preparation suggested in [9] were used in determining the method and procedure needed to collect quantitative data. It was decided to use foilresistance strain gages to collect the quantitative strain data. Placement of strain gages follows the recommendations suggested in [10]. Each of the dogbone type specimens contained a transverse and longitudinal strain gage pair mounted on the top and bottom surfaces. Mechanical testing is done utilizing a computer controlled load frame with hydraulic grips. The material tested failed at a maximum load of 5.23 kN and 8977 Since these maximum were at such low levels, specimens are loaded at a crosshead rate of 4.23 mm/hr to allow enough time to collect sufficient data during the test. A personal computer was used to control the load frame as well as the data acquisition system. This allowed data to be sampled continuously without interrupting the test. Calibration factors were obtained for all specimen strain gages before testing and used later during data reduction. Immediate feedback of load vs strain was obtained by attaching an extensometer to the specimen with results being plotted on an oscilloscope while the test was being run. Results from the extensometer matched within ±3% the longitudinal results of the strain gages. One dogbone type test specimen from each laminate layup is loaded to rupture. The remaining five specimens are loaded to 90, 85, 80, 75 and 70% of rupture load. By loading specimens to different load levels a measure of the evolution of damage through the progression of loading can be obtained. The actual quantification of damage is obtained by measuring damage features on a representative cross section. Load vs longitudinal %strain curves for selected dogbone shaped specimens of orientations and are shown in Figures 1 and 2, respectively.

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SEM Analysis and Damage Quantification of Specimens

Following mechanical testing, all specimens were sectioned into small samples for further investigation using image analysis. The purpose of the image analysis is to identify and quantify all damage features that may exist on the cross section of the sample. Results of information collected are then compiled for characterization of damage. Details of the complete procedure and results are given in the following paragraphs. A scanning electron microscope was used to perform an SEM analysis of a representative cross section of all specimens to obtain qualitative information. Photographs are also taken during this analysis so that the visible damage features can be measured later using image processing software. Longitudinal and transverse sections are taken from all samples in the vicinity of the strain gages. The transverse cross-sections contained a portion of the free edge. In order to eliminate any possible free edge effects, information within two fiber diameters of the specimen free edge is disregarded. Since the longitudinal cross sections are carefully taken from the middle of the specimen, free edge effects did not have to be considered. All SEM samples were cut from the original test specimen using a low speed diamond saw. The low speed diamond saw eliminates the possibility of introducing damage on the cross section during sectioning. In addition, the cut surfaces were ground and polished to eliminate any surface defects that can be introduced by the cutting operation. Careful observance of this procedure increases the probability that defects observed during the SEM analysis reflect damage as a result of the loading only. Although, the cross section could contain radial cracks as a result of the fabrication cool down process, it is assumed that a well-controlled manufacturing process was used such that the number of these cracks is low and can be neglected. Therefore, all measured cracks are attributed to loading.

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The entire cross-sectional areas of the longitudinal and transverse sections are scanned at low magnification (<1000X). Based on the observations of this analysis, photographs are taken on an area of the cross section that is 1% of the total area and contains an average representation of damage features for the complete cross-section. The aforementioned area is taken as one of the three mutually perpendicular planes used in defining the representative volume element (RVE). Similar areas were considered on the remaining mutually perpendicular sample sections to complete the planes of the RVE.

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Results of the SEM analysis showed a higher occurrence of damage in the fibers than in the matrix. The type of damage was mainly confined to cracks in the fiber or in the matrix material adjacent to the fiber/matrix interface. The prevalence of the linear crack damage feature is somewhat expected, since the matrix is ductile and the fibers are very brittle. Because of the low strains obtained and the ductile nature of the matrix, damage features such as matrix voids are not expected and are not observed on any section. A sample of the damage features found on the longitudinal face of specimens investigated is shown in Figures 3 and 4. Similar damage features were found on transverse sections. Crack length quantification was obtained through use of SEM equipment and software. The SEM photographs collected during the image analysis were scanned at a resolution of 600 dots per inch (dpi). A semi-automatic technique is used to measure crack lengths in that the cracks are digitized by hand before being processed with the image analyzing software. This software automatically computed the crack lengths with respect to the photograph scale. Information collected during the SEM and image analysis process must be related to a damage parameter that can be incorporated into a constitutive model. [5] and [6] developed a damage model that defines a second order tensorial damage parameter, . One of the difficulties in using this model is being able to evaluate components of the damage tensor. In this work it is postulated that the area reduction due to damage in the effective configuration is essentially due to the development of micro cracks during the process of loading. The measure of micro crack damage, is postulated to be , in terms of the crack density, Damage for an elasto-plastic response is evaluated by defining the damage tensor, as a function of crack density, ([7]; [11]; and [12]). For off axis laminates, the general matrix representation of in terms of is given as

Crack density, is normalized in equation (1) to smooth out the slight variability in evaluating crack density. The components of (i.e. where xparallel to the load axis, y -transverse to the load axis and z -normal to the xy plane) represent the normalized crack density on a cross section whose normal is along the i axis. The off diagonal terms in equation (1) capture damage due to the interaction of cracks on the three mutually perpendicular planes of the RVE. These terms also imply that shearing stresses impose this interactive damage. Information collected during the image analysis process is used in evaluating the normalized crack density defined as,

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Total crack length on the cross-section is represented by the quantity in equation (2) and is the corresponding cross sectional area, represents a normalization factor chosen so that the values of the damage variable, falls within the range < 1. Normally the magnitude of this quantity will be 1 or close to land the term is an expression selected such that it will yield a dimeusiouless average value for the One possible form for is given by

where represents the maximum crack density of all load levels. In the case of uniaxial tension, this maximum value corresponds to the crack density evaluated at the maximum load before macrocrack initiation. is a dimensionless quantity while , has the dimensions of Crack densities, are tabulated at the five load levels below rupture for each of the specimen orientations used in this investigation ([11]; [12]). Results are given in Tables 4 and 5.

These values are the crack densities computed with equation (3). Densities on the z-section are not measured and assumed to be one half the magnitude of those on the respective y -section. This assumption is made for convenience and in future damage investigations the z -plane damage should be measured. Using equation (2) and the tabulated information, the components of the damage tensor given by equation (1) for each laminate layup at 90% of the rupture load are computed as

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similarly the damage tensor, , for each laminate layup at different %load of the rupture can be computed. Damage parameter curves for each component of the damage tensor as a function of the strain have the same general shape as those presented in [11]; and [12]. Each of the curves shown has a concave shape with a vertical asymptote as a critical damage value is approached. The shape and form of these curves are consistent with the theoretically generated curves of [5]. 5.

Damage Tensor as an Internal State Variable

The anisotropic phenomenon of the microcracks distribution in the material can be interpreted using the second-order damage tensor, , as an internal state variable of thermodynamics of irreversible processes. It is expressed as [6]:

where are the principal values of the damage tensor and are the principal directions. The linear elastic constitutive equation for the damaged material is written according to the principal of strain energy equivalence between the virgin material and damaged material. That is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the Cauchy stress tensor, , is replaced by the effective stress tensor, , and the strain tensor, , is replaced by the effective strain tensor, [6]:

where

where

is a fourth order damage effect tensor defined as:

is the Kronecker delta.

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The free energy here is expressed as follows:

where and p are the flux related to the kinematic hardening and the cumulative plastic strain related to the isotropic hardening in plasticity. Assuming small elastic strains in equation (9), the average strain tensor, is decomposed into two parts: the elastic part, and the plastic part, Using the Clausus-Duhem inequality [8] one obtains:

where

is the material density and

conjugate to

and R are the thermodynamic forces

and p, respectively.

Assuming the decoupling between the plastic and damage dissipation processes, one can define the dissipation process as the sum of the product of the associated variable with the respective conjugate force as follows:

The rate of the internal stat variables associated with plastic and damage deformations are obtained by utilizing the calculus of function of several variables with the Lagrange multipliers and , respectively, as follows:

Extremizing the function

where

we obtain:

and are determined using the consistency conditions and respectively. Once a material is damaged, further loading can only affect the undamaged material. Thus, the damage potential function is defined in terms of the effective stresses and strains. By combining plasticity with damage, it seems natural that plasticity can only affect the undamaged material skeleton. Thus the plastic potential is originally defined in terms of the effective stresses and strains and subsequently converted to the damaged space. The plastic potential and the damage potential can have several forms [6]:

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where is the effective backstress tensor and is the effective isotropic hardening. When considering the damage and plastic deformations of metal matrix composites, we can use the combined damage and plastic formulations for each of the constituents. The information obtained from the individual properties of the different materials at the local level can be linked to the overall properties by using a certain homogenization technique. For further details on the subject consult reference [13], [14], and [15]. 6. Conclusions

Described within this work is an experimental procedure by which damage can be quantified in terms of microcrack density. Damage quantification is accomplished through use of a scanning electron microscope and image analyzing software. This equipment is used to measure physical damage features (microcracks) on a representative volume element (RVE) taken from test specimens. The primary focus of this investigation is to convert physical damage into a mathematically defined damage tensor. Given that physical damage can occur in different forms (e.g. fiber and/or matrix cracks, matrix voids, fiber-matrix debonding, etc.) an expression must be selected to quantify damage in terms of a total quantity irrespective of individual forms or in terms of each individual form. Within this work, damage is quantified in terms of crack density as defined by equation (2). The method of data collection is such that crack densities can be obtained as a total value for the laminate RVE or individual values for the fiber and matrix. Although equation (2) only considers damage resulting from features in the form of cracks, they can be replaced by expressions for other damage features and combined as given by equation (1). Results of this investigation show that physical damage can be quantified into a tensorial quantity suitable for any damage constitutive model. Additional investigations were conducted [6] using the experimental data presented here in order to verify the formulation for a coupled anisotropic damage model for the inelastic response of composite materials. A physical interpretation of the second order damage tensor, , is presented in that work. 7. References 1. 2.

3. 4.

Allix, O., Ladeveze, P., Gilletta, D. and Ohayon, R. (1989) A damage prediction method for composite structures, Int. J. Num. Meth. Eng. 27, 271–283. Majumdar, B.S., Newaz, G.M., and Ellis, J.R. (1993) Evolution of damage and plasticity in titanium-based, fiber-reinforced composites, Metall. Trans. Ser. A 24A, 1597–1610. Breuing, T.M., Stock, S.R., Kinney, J.H., Guvenilir, A., and Nichols, M.C. (1991) Impact of x-ray tomographic microscopy on deformation studies of a SiC/Al MMC, in Material Research Society Symposium Proceedings, Material Research Society, pp. 135–141. Lemaitre, J. and Dufailly, J. (1987) Damage measurements, Eng. Frac. Mech. 28, 643–661.

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10. 11. 12.

13. 14. 15.

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Voyiadjis, G.Z. and Kattan, P.I. (1993) Damage of fiber-reinforced composite materials with micromechanical characterization, Int. J. Solids Struc 30, 2757–2778. Voyiadjis, G.Z. and Deliktas, B. (2000) A coupled anisotropic damage model for the inelastic response of composite materials, Computer Methods in Applied Mechanics and Engineering 183, 159–199. Voyiadjis, G.Z., Kattan, P.I., and Venson, A.R. (1993) Evolution of a damage tensor for metal matrix composites, in MECAMAT 93 International seminar on micromechanics of materials, Moret-sur-Loing, France, 84, pp. 406-417. Lemaitre, J. and Chaboche, J.-L. (1990) Mechanics of Solid Materials, Cambridge University Press, London. Carlsson, L.A. and Pipes, R.B. (1987) Experimental Characterization of Advanced Composite Materials, Prentice-Hall, Inc. Turtle, M.E. and Brinson, H.F. (1984) Resistance-foil strain-gage technology as applied to composite materials, Exp. Mech. 24, 54–65. Voyiadjis, G.Z. and Venson, A.R. (1995) Experimental damage investigation of a SiC-Ti aluminide metal matrix composite, Int. J. Damage Mech. 4, 338–361. Venson, A.R. and Voyiadjis, G.Z. (2001) Damage quantification in metal matrix composites, J. Engrg. Mech. 127, 291-298. Voyiadjis, G.Z. and Kattan, P.I. (1999) Advances in Damage Mechanics:Metals and Metals Matrix Composites , Elsvier, Oxford. Park, T. and Voyiadjis, G.Z. (1997) Kinematic description of damage, Journal of Applied Mechanics 65, 93-98. Voyiadjis, G.Z., and Park, T. (1997) Anisotropic damage effect tensors for the symmetrization of the effective stress tensor, Journal of Applied Mechanics 64, 106-110.

STUDY OF DAMAGE IN PARTICULATE COMPOSITES

C. A. SCIAMMARELLA and F. M. SCIAMMARELLA Illinois Institute of Technology Mechanical Materials and Aerospace Engineering Department Chicago IL, 60616 U.S.A

Abstract The problem of damage in particulate composites with a soft matrix and rigid inclusions is experimentally studied. Two different approaches are used. In one approach the damage is evaluated though the increment of the material porosity. In the other approach a microscopic measurement of adherence between particle and matrix is used. The last approach provides a more realistic picture of the damage process and leads to simple models for numerical simulation. 1. Introduction Solid rocket propellants are used for the propulsion of space launch vehicles. The rocket propellant is a mixture of crystalline and metallic particles with a matrix and binders that are basically rubbers with different chemical compositions. When designing rocket motors it is necessary to take into consideration the mechanical properties of the propellant. If damage occurs in the propellant due to incipient cracking, the damage can alter the burning process of the propellant and cause a malfunction in the ballistic trajectory of the rocket. Since the particles in the propellant are much stiffer and stronger than the matrix, particle/matrix interface debonding called dewetting becomes the major damage model. To address this problem a number of models have been introduced in the literature. These models vary greatly in scope and complexity. The basis of the model can be a michromecanics unit cell, for example Schapery [1]. Or it can be a more complex model with a unit cell (micromechanics) interacting with a structure (macromechanics) and utilizing elements of the theory of damage of materials. Other models are purely phenomenological [2]. The aim of the models is to establish procedures to relate basic mechanical parameters to the stiffness and strength of the propellant. This paper is devoted to the experimental study of the propellant behavior using information gathered by experimentally measuring deformations on the surface of a tensile specimen subjected to creep loading applied by a dead weight [3], [4]. The load is applied in steps and the measurements are carried out at equal intervals of time providing an average constant strain rate. The Holographic-Moiré technique is used to measure displacements and strains. The Holo-Moiré Strain Analyzer is used for data 121 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 121–132. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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gathering and processing. The Holo-Moiré Strain Analyzer is a system for the analysis of optical signals that contains hardware and software that gives as an output displacement and strain information. To get the strains of the specimen in the axial and in the transversal direction a region of 3.5×3.5 mm is captured by a CCD camera attached to the Holo-Moiré Strain Analyzer. In this case the output of the system is the average strains of the specimen. To study local fields a region of about 1000x1000 microns is observed with a microscope. The output of the system is a map of the principal strains. In both cases since the material deformation is time dependent the observation of the two displacement fields, the axial field in the direction of the load and transversal field, perpendicular to the load, is done simultaneously. Two different approaches have been followed to study damage. In one approach the effect of the matrix and particle separation was analyzed from the point of view of void formation. In the second approach, damage was characterized by measuring the particle and matrix separation through the loss of stiffness. The aim of this study is to provide the basis for models that can be used to predict the behavior of the propellant. Some preliminary results of numerical analysis are presented. 2.

Damage Analysis Through the Process of Void Formation

2.1 TESTING PROCEDURES Tensile loads up to 90 % of the failure load were applied in six consecutive steps. Displacement and strain information was gathered for each step. The same procedure was followed for unloading. Around 72 hours after the specimen unloading, residual deformations were determined. A total of six cycles were applied to the specimen. Figure 1 shows the first cycle of loading and Figure 2 shows the last loading cycle. In Figures land 2 the thin lines represent the recovered delayed elasticity and have been obtained trough interpolation.

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2.2. ANALYSIS OF THE CYCLIC LOADING The obtained hysteresis loops differ from the ones that are often shown in the literature. The difference arises from the way that the load is applied. The tensile test results are the consequence of the interaction specimen-machine. A “hard” testing machine is a machine that imposes displacement conditions to the specimen. A “soft” machine imposes a mix of load and displacement condition. The testing procedure used in the present tests is equivalent to an “infinitely soft” machine since measurements are taken after one minute of applying or removing load. Figure 3 represents the permanent strains added to the specimen after each applied cycle after the 72 hours of rest from the time of unloading. Figure 4 gives the hysteresis loop energies corresponding to the total loop, the recoverable part of the loop energy and the energy lost in function of the applied cycles of load. The results of the cyclic loading indicates that most of the damage done to the specimen takes place in the initial cycles and then is asymptotically reduced in the later cycles. The hysteresis loop energy of the later cycles consists mostly of the delayed elastic energy. There is an internal process of accommodation similar to the “shake down” in plastic deformations. After certain amount of damage is done to the interfaces a new configuration is established and the composite becomes stable.

2.3. QUANTIFICATION OF THE DAMAGE TROUGH POROSITY The separation between matrix and particles produces voids. Void formation can be used to quantify damage experienced by the propellant in each of the loading cycles. The following equation [5] can be used to obtain the porosity at each loading level,

In equation (1) In , which is the volumetric deformation, is the stretch in the axial direction of the specimen, and is the Eulerian deformation, is

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Poisson’s ratio, determined by the equation, x is the transversal direction of the specimen

The values of Poisson’s ratio are determined from the differentiation of the plots of versus Figures 5-6 show the porosity versus the stress for the first and six cycles.

In order to gain an understanding of the total damage producing the fracture of the specimen a specimen was tested to fracture. The room temperature for the test was 21 and the average rate of loading was 0.0068 1/min. Figure 7 shows the corresponding results. In this plot one can see the extrapolated unloading path and the extrapolated recovery path obtained from data of the first cycle of the loading up to 90 % of the failures stress.

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The Poisson’s ratio to failure was also obtained from the Poisson’s ratio of the first load cycle. This value of Poison’s ratio was used to compute the porosity at failure. The results are shown in Figure 8. Figure 9 shows the porosity as a function of the irrecoverable energy invested in damage. The energy balance is as follows, or 18.6 % of the total energy corresponds to damage, is delayed elasticity energy and is elastic energy recovered upon unloading. To clarify the meaning of parameters defining the fracture, the following analysis was performed. The values of the parameter defined by Shapery [6] were computed for different propellants whose data are given in [7]. These propellants have properties similar to the propellant tested. Figure 10 shows the porosity data for the tested specimen as a function of the plotted simultaneously with the parameter It is possible to see that both measures of damage are close in the order of magnitude. It is possible to see that porosity as defined by (1) can be used as a measure of the damage done to the specimen. However the value of porosity at fracture for the tested specimen 8.23 % is a quantity that depends on the rate of loading and temperature as can be seen in Figure 10.

3.

Measure of the Damage by the Loss of Stiffness

3.1. MICROSCOPIC MEASURE OF ADHERENCE An alternative way to define damage is to evaluate the loss of stiffness of the composite as the load is being applied. To have a direct understanding of the influence of the interfaces in this process a direct measure of adherence was introduced. In order to evaluate the process of separation between particle and matrix an integral definition was introduced.

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The integral in the numerator represents the elongation of a closed circuit taken about 10 microns from the boundary of a particle. is the component of the strain tensor along the tangent direction to the closed circuit. The integral in the denominator is the elongation of the same circuit computed using the modulus of elasticity and Poisson’s ratio of the composite under tensile load. If we divide the numerator and denominator by the length of the circuit we obtain the following equation,

where is the average strain in the chosen circuit obtained from the actual strain distribution and is the average strain of the same circuit corresponding to the uniaxial state of stresses for the same load. If it means that the matrix sticks to the particle since the modulus of elasticity of the particles is three orders of magnitude larger than the matrix, the elongation of the contour will be negligible. In this case the adherence will be equal to one. If on the contrary the adherence will be equal to zero. If is larger than the adherence becomes negative meaning that the contour deformation is larger than the one corresponding to the tensile specimen contour.. The definition of adhesion is related to the quantity damage in the following way,

when is equal to one the damage will be zero and when the damage will have the maximum value equal to one. The previously introduced definition of adherence is based on a contour integral performed on the surface of the specimen based on the motion of the continuum. It can be shown mathematically that the experimentally determined contour integral provides a measure of the detachment of the particle from the matrix. As a result, the surface values can be taken as a representative measure of the process not only on the surface but also internally. 3.2 DAMAGE RELATIONSHIP OBTAINED FROM THE STRESS STRAIN CURVE OF MATRIX AND COMPOSIITE In the literature of composite materials it is usual to measure damage from the change of the elastic modulus of the stress strain curve. To apply this definition to the particulate composite presents the following problem. The stress strain curve of the composite has a non-linear behavior due to the non-linearity of the matrix material. Consequently there are two effects that produce the non-linear behavior. The first effect is the non-linearity of the matrix, and the second is the separation between the particles from the matrix. Consequently to measure damage or adherence it is necessary to separate these two effects. In order to separate these two effects the following definition of adherence is introduced.

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where is the measured composite tangent modulus for the corresponding stress level, is the matrix modulus for the same stress and is the value of the composite modulus assuming that the adherence between the particle and the matrix is perfect. We need to know the value of Figure 11 gives an important clue towards the answer of the problem of evaluation of damage. Figure 11 shows the ratio of the modulus of elasticity of the composite and the matrix for zero stress as a function of the volumetric matrix content, 1 indicating pure matrix.

The data comes from references [5], [8] and [9]. In [5] and [8] the values were evaluated form the stress strain curves of the composite and the matrix material, in [9] these values are provided in tabulated form. The trend line of the experimental points is to a high degree of accuracy of the form

where and represent the volumetric contents of the matrix and the particles respectively. In (7) the contribution of the second term is negligible because is a very large value, and therefore only the first term counts. Equation (7) provides the traverse modulus of a unidirectional composite and also is a limit form that the Halpin-Tsai equation takes for short fiber reinforced composites. If we make the hypothesis that this equation can be applied for the tangent modulus of the composite as the matrix deforms, we can obtain the composite tangent moduli for the corresponding stresses assuming that there is perfect adherence between the matrix and the particles. Actually the whole process can be expressed in dimensionless form if we divide the numerator and denominator of (6) by and the corresponding stress are expressed as the ratio of the actual stresses S divided by the composite rupture stress Sr. Equation (6) becomes,

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We have two measures of adherence obtained from two separate procedures; the analysis of the experimental data will provide a way to compare the results coming from these two different definitions. 3.3. EXPERIMENTAL RESULTS

Measurements were performed on five separate specimens. In each specimen several particles were analyzed. The measurements performed need to be interpreted in the statistical sense. The adherence changes with the shape, dimensions and orientation of the particles in the strain field. Most of the particles that we have analyzed have irregular contour and therefore a large number of orientations are represented in the obtained integrals. Since we can measure only a limited number of samples of a very large population, the question to be answered is whether the obtained sample represents a meaningful trend for the whole population. This is not an uncommon situation with these kinds of problems but fortunately we have found and independent way to evaluate adherence that reflects the overall behavior of the composite. Values of adherence measured in five specimens were used to obtain the trend line shown in Figure 12, together with the 95 % level of confidence interval that indicates that values within this envelope belong to the population that has the trend line indicated in Figure 12. The obtained adhesion curve goes practically to one for zero stress and to zero for a stress very close to the actual fracture stress of the composite. The trend line can be represented by the cubic equation shown in Figure 12.

To apply equation (6) for our composite material, the tangent modulus of the matrix material and the composite were determined and plotted in dimensionless form in Figure 13.

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The agreement between the two trend lines is excellent particularly if we consider that the experimental values have been obtained up to The agreement between the experimentally measured values of adhesion with the values of adhesion obtained from the stress-strain curves indicates the soundness of the model that has been used to predict adhesion from the stress-strain curves. 3.4. OBSERVATIONS CONCERNING THE LOSS OF STIFFNESS The introduction of the definition of adhesion as an experimentally measurable quantity through a contour integral provides a tool to understand the behavior of the interface between particle and matrix. It can be shown theoretically that the surface measurements provide information of the events that take place in 3-D and not only on the surface. The sampling of the population of interfaces through this tool has proved to be very fruitful because it has been feasible to ascertain from a reduced number of measurements population means. These experimentally determined values lend credibility to the model of damage that has been introduced in this section. It has been evident to us for a long time that the changes of moduli of the composite with increasing load were not only caused by gradual damage of the interfaces as many authors have modeled it but also were greatly influenced by the matrix behavior. A link was missing between these two facts and this link was provided by the experimentally obtained data that relate volumetric content of matrix of the composite and the tangent modulus at zero stress. This relationship became clear when a dimensionless plot was introduced, thus removing the dispersion of the same data when the absolute modulus was plotted vs. the volumetric content of the matrix. It is very interesting to observe that the modulus prediction follows the well-known relationship used in composites to obtain transverse moduli of uniaxially fiber reinforced composites and short fiber composites. The difference between the predicted modulus of the composite and the actual modulus can be attributed to the process of damage and leads to the adhesion definition given in (3). The fact that this definition of adhesion agrees with the definition of adhesion given in (6) indicates two important things. First the integral definition of adhesion is successful in measuring the degree of efficiency of the particle-matrix interface. Second a reduced number of samples provides enough information to

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ascertain the trend of the population. It is also interesting to point out that measurements have been carried out in large particles. The large particles seem to be the ones that determine the behavior of the composite in the process of damage. The gradient theory supports this conclusion. The adhesion between particle and matrix is affected by particle size, orientation in the field of stresses, geometry of the particle and therefore the observed quantity is statistical in nature. The measurement has to be performed by selecting at random a large enough number of samples so that a good estimate of the population mean can be obtained. It became clear to us, that the faces most unfavorably oriented with respect to the applied tension can separate while the other faces subjected to compressive forces can remain adherent to the particle up to very high stresses. We can conclude that to understand the evolution of the defined quantity adhesion we have two processes to analyze, separation of the matrix in the direction of tensile stress components perpendicular to the interface, and gradual sliding where compressive stress act perpendicular to the interface. The other component of adhesion, the relative sliding of the matrix and particles, seems to be a reversible process since cyclic loading deformation after a while settles to a stable value. The initial drop of the adhesion curve vs. stress is very likely due to mostly matrix separation. Though the techniques introduced in this paper it is possible to quantify these effects by analyzing the cyclic stress-strain curves. 4.

Numerical Simulation

To verify the behavior of the interfaces a simple numerical model has been analyzed. Figure14 shows the model. The model is two-dimensional and a square particle is assumed. The volume ratio of particle/matrix is assumed to be 25 %, which is the ratio of the large particles to the matrix in the studied propellant. In this model it is assumed that the damage starts in the interface of the large particles and the process is controlled by the large particles. The surrounding matrix is assumed homogeneous and the modulus of this region is increased to take into consideration the effect of the smaller particles. A uniaxial displacement is applied and a plain strain problem is considered. The finite element software of ABAQUS was used to perform the study. The particle is assumed to be rigid. The interface between the rigid particle and the matrix may be bound, may slide or may separate according to position and stress level.

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This simple model can represent the behavior of the composite quite well. Three conditions are considered: a) the particle is totally bonded to the matrix, b) the particle is partially bonded the matrix, c) the particle is completely separated from the matrix. Figure 15 illustrates the results of the finite element analysis for the different interface bound conditions along with the experimental data obtained. The experimental data falls between the fully bonded and the fully free. By combining the three states it is possible to duplicate the experimental stress strain curve seen in Figure 16. In the partially bounded state the crack propagates on the horizontal face while the lateral phase slides with respect to the matrix. 5.

Conclusions

Two different approaches have been introduced to measure damage in particulate composites. These two models can be used as a measure of the damage experienced by the composite upon loading. The first model combined with the cyclic loadings provides a connection with the energy invested in permanent deformations. This model puts in evidence a mechanism of recovery revealed by the stabilization of the specimens under cyclic loadings, the porosity increases are partially recovered upon unloading. The second measure adherence provides a glimpse of the damage process and shows that the effect of the orientation of the particles faces with respect the applied tensile stress is important. In the first approach the shape of the particles is immaterial, thus the spherical particle approach of Shapery works well with this approach. The second approach provides a better glimpse of the physical process of dewetting and leads to a simple numerical model that can simulate quite well the damage process. The frictional properties of rubbers play a very important role in this last model since rubbers can slide over a surface while still showing an adhesion with the surface. 6.

Acknowlegement.

The authors would like to thank Dr. T.C. Liu and the United States Air Force for supporting this research through successive research grants.

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7.

References

1.

Schapery, R.A. A Micromechanical Model for Non-Linear Viscoelastic Behavior of Particle Reinforced Rubber with Distributed Damage, Engineering Fracture Mechanics, 25(5/6), (1986), 845-867. De Souza Neto, Peric D., Owen Dr.J, A Phenomenological Three-Dimensional Independent Continuum Damage Model for Highly Filled Polymers: Formulation and Computational Aspects, 142(10), (1994), 1533-1550. Sciammarella C.A., Sciammarella F.M., A Lagarde (ed.), Advanced Optical Methods and Applications in Solid Mechanics, Kluwer Academic Publishers, Dordecht, 2000. Sciammarella C.A., Sciammarella F.M., Interfacial Deformation Between Particles and Matrix in Particle Reinforced Composites, Proceedings of the SEM IX International Congress on Experimental Mechanics, SEM, Bethel CT., USA, 788-791, 2000. Kugler, H.P., Stacer R.J, and C.Steimle, Direct Measurement of Poisson’s Ratio in Elastometers, Rubber Chemistry and Technology, 63(4), 473-487, October 1990. Schapery, R.A., Models for Damage Growth and Fracture in Non-Linear Viscoelastic Particulate Composites, Y.H. Pao (ed.), Proceedings of the Ninth USA National Congress of Applied Mechanics, ASME,N.Y., 1982. Bencher C.D., Dauskard R.H., Richie R.O., Microstructural Damage and Fracture Process in Composite Solid Rocket Propellant, Journal of Spacecraft and Rockets, 32(2), 1995. Sciammarella, C.A, and F.M. Sciammarella, Damage Mechanisms in solid propellants, Fourth Report, Air force Subcontract 97-517, May 1998. Smith, T.L., Volume Changes and Dewetting in Glass Bead-Polyvinyl Chloride Elastomeric Composites under Large Deformations, Transactions of the Society of Rheology, III, 113-136, 1959.

2.

3. 4. 5. 6.

7. 8. 9.

HYGRIC CHARACTERIZATION OF COMPOSITE LAMINATES SHI-CHANG WOOH Department of Civil and Environmental Engineering Massachusetts Institute of Technology Cambridge, Massachusetts CHO-LIANG TSAI Department of Construction Engineering National Yunlin Univsersity of Science and Technology Yunlin, Taiwan

Abstract A novel experimental method based on Fick's diffusion law is developed to determine the distribution of moisture concentration as well as the coefficients of moisture expansion of a composite laminate immersed in water. The technique is based on measuring the curvature of a antisymmetric crossply laminate introduced by the unbalanced in-plane interlaminar resultant forces, which can be related to the hygric expansion. The technique is verified experimentally for and graphite/epoxy specimens. The results show the enhanced measurement sensitivity of the new technique.

1 Concept The behavior of polymeric composite materials is relatively well understood [1]. Hygric deformation due to moisture absorption is determined by the coefficients of moisture expansion (CMEs) in longitudinal and transverse directions and [2], which are typically determined by measuring the moisture concentration and hygric expansions in the corresponding directions. High precision balances is used to measure the average moisture concentration of the laminate. However, it is very difficult, although not impossible, to measure the variation of moisture concentration through the thickness. According to the lamination theory, the hygric expansion is proportional only to the average moisture concentration, if the distribution of moisture is symmetric around its midplane and if the specimen is subjected to no other mechanical or thermal loadings. The CMEs can thus be determined directly by measuring the hygric expansions using such devices such as micrometers, calipers or strain gages [3–6]. These devices work well for measuring large expansions in the transverse direction but usually they do not provide high sensitivity required to measure the longitudinal expansion [2]. 133 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 133–144. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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In order to improve the sensitivity of measurement, a novel technique is proposed in this work, which is based on measuring the curvatures and weight gain of an unbalanced laminate immersed in water. If an unbalanced laminate is subjected to hygric loading, the mismatches of CMEs between the layers induce unbalanced hygric strains and stresses in the thickness direction, resulting in warpage of the laminate. In order to measure the CME, the curvatures and weight of a specimen are measured during its life in aqueous environment. The moisture absorption of the material is computed from the weight gain ratio, and the distribution of moisture concentration and the CMEs are determined from the one-dimensional Fick’s law. In this work, crossply composites of the stacking sequence of are considered for calculating the CMEs. From this perspective, the technique may be referred to as Antisymmetric Crossply Curvature Technique (ACCT).

2 Theory of Moisture Diffusion One-dimensional Fick’s diffusion law may be used to describe the diffusion of moisture in a laminate through the thickness as

where is the weight gain ratio (weigh of absorbed moisture/weigh of dry material), D is the effective diffusivity of the material, is time and is the out-of-plane axis with its origin at the midplane (Fig. 1). For an immersed laminate, the boundary conditions are given as

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where is the thickness of the laminate and M is the weight gain ratio of the material in its saturated state. As approaches M at any point in the laminate. Assuming that the specimen is perfectly dry before it is immersed in water, the initial condition is simply written as

To describe the symmetric distribution of moisture along the dition is expanded into a Fourier cosine series as

the initial con-

Assuming that the solution is of the form

and using the separation of variables, the following relationship is obtained:

The values

and

for the

term are solved as:

where

for

Then, the solution becomes

The average weight gain ratio through the thickness can be evaluated as

which is a function of M , D , and The values for M and D can be calculated from a set of data measured at different immersion times.

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3 Theoretical Background The classical lamination theory [2] allows us to write the lamina strains in hygrothermomechanical terms under the state of plane stresses as:

where are the midplane strains (at are the curvatures, is the difference between the temperature of the specimen and the stress-free temperature of the laminate. is the change in moisture concentration, are the coefficients of thermal expansion (CTE), and are the coefficients of moisture expansion (CME). With no mechanical loading, the hygrothermal forces and moments can be expressed as:

and

where is the reduced stiffness matrix under the condition of plane stresses, and and matrices respectively denote the extensional, coupling and bending stiffness of the laminate. For crossply laminates, Eqs. (14)–( 16) can be simplified as:

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and

where is the longitudinal modulus. is the transverse modulus, is the in-plane, and and are the major and minor Poisson’s ratios of the lamina, respectively, and

Since our focus is on studying the hygric behavior, it is assumed that the temperature change remains constant throughout the test, i.e.,

where is the room temperature and is the temperature free of thermal stresses. If the specimen is initially dry. then and we also have the condition that the shear stress. and all the external loadings be zero. Equations (17), (21) and (22) become:

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and

where

so that

It should be noted that this expression is valid only if the changes of material properties are negligibly small or independent of the moisture concentration. This assumption has been experimentally verified although not reported in this paper. The moisture concentration representing the relative water volume ratio in the material is related to the weight gain ratio as [2]:

where

and

is the specific weight of the material. At

let us define

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Equation (29) can be simplified as:

Then, the curvature at time

can be written as:

Defining

and from Eqs. (33)–(36), we obtain

and

where

There are three unknowns in this equation: and The curvature can be measured experimentally as a function of immersion time by fitting the data. It is also possible to find the stress-free temperature from the computed as

4 Experimental Procedure For convenience, we consider only laminates. Equation (38) implies that a thinner plate will result in bigger curvatures and higher sensitivity. From that point, a

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[0/90] laminate is expected to be the best configuration. However, such a specimen is so thin and fragile that it is highly probable that the specimen may produce initial cracks that may jeopardize the test. Thus, we have chosen specimens for our experiments. In addition, specimens were also tested for comparison. Table 1 shows the measured mechanical properties of the material (AS-4 3501-6 carbon epoxy). Three specimens of each configuration were cut into 1 cm wide and 30 cm long coupons. The measured thicknesses of and samples are 0.59 mm and 1.48 mm. respectively. Figure 2 shows the fixture designed to measure the curvatures of the strip shaped specimens. A specimen of thickness is rigidly clamped at one end and the other end is left free. The specimen should be placed in such a way that it is bent upward from the base of the holding fixture. The curvature is expected to reduce due to higher hygric expansion in the 90° layer. At a horizontal distance H (fixed at 25 cm) from the clamped end, a ruler is used to measure the distance the displacement of the

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point on the top surface at with respect to the reference top surface at With the aid of an optical telescope, the measurements could be made accurately. From the simple geometry, the inner radius of the specimen, can be written as a function of

d: and the radius of the midplane is:

so that the curvature of the specimen can be expressed as

Before running tests, each specimen was dried and weighed using an analytical balance with an accuracy of 0.01 mg. The specimens were loaded in the fixture for taking the initial measurements at a dry state. They were removed from the fixture and immersed in a water tank to start living in aqueous environment. At an interval of two or three days, the specimens were removed to measure and record their curvatures and weight. After a full set of data was obtained, the CMEs and WT were computed using a least square curve fitting.

5 Experimental Results Figure 3 shows the measured average weight gain ratio (through the thickness) of the laminate plotted as a function of immersion time, which is a symptotically increasing. The values M and the effective diffusivity D were evaluated by fitting these data; yielding and Figure 4 shows the measured displacements, d. as a function of exposure time t, which illustrates the superb sensitivity of the proposed technique. The d value of a specimen changed from 7 cm to 2 cm in 25 days of aqueous life. The required measurement accuracy is a fraction of centimeters. As expected, the specimen is less sensitive. It may be more difficult to notice the change for thicker laminates The computed curvatures are shown in Fig. 5. It can be observed from these results that the initially larger curvature (displacement) is relaxed asymptotically as the moisture absorption in the specimen increases. The curvatures measures in Fig. 5 are plugged into Eqs. (38)–(40) to determine the CMEs and Table 2 shows the calculated values and their standard deviations. The measured values are very close to those given by Daniel and Ishai [2].

6 Summary and Conclusions A novel method, ACCT, has been developed to measure the moisture concentration and coefficients of moisture expansion using an antisymmetric crossply laminate. This technique relates the curvature of an antisymmetric crossply laminate immersed in water to its distribution of moisture concentration. The relationship involves material properties,

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specimen geometry and immersion time. Diffusion related properties of the material, M and D, can be characterized, CMEs of the material and stress-free temperature of the laminate, and are calculated. The ACCT may provide excellent accuracy which cannot be achieved by traditional techniques relying on strain measurements, such as the ones using calipers, micrometers or strain gages. This is especially true for obtaining the properties in the fiber direction, where the hygric strains are extremely small. The technique may also have the merit that the M and D, can be characterized at the same time using the same specimens. Experiments revealed that the measurement accuracy required for this technique is just a fraction of a centimeter. No complicated or expensive devices are required. It is also recommended to use thin plates in order to achieve high sensitivity and accuracy. The measured hygric properties agree well with the ones reported elsewhere. The tech-

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nique may have future potentials in characterizing other properties such as coefficients of thermal expansion or chemical shrinkage.

7 Acknowledgments The second author would like to acknowledge the support and encouragement from the National Science Foundation under the Contract Number CMS-9733157.

8

References

1. Shen, C. W. and Springer, G. S. (1976) Moisture Absorption and Desorption of Composite Materials, Journal of Composite Materials, Vol. 10, 1976, pp. 2–20. 2. Daniel, I.M. and Ishai, O. (1994) Engineering Mechanics of Composite Materials, Oxford University Press Inc., New York. 3. Gazit, S. and Ishai, O. (1977) Hygroelastic Behavior of Glass-Reinforced Plastics Exposed to Different Relative Humidity Levels, Proceedings of Conference on Environmental Degradation of Engineering Materials, Virginia Polytechnic Inst., Blacksburg, VA, pp. 383-392. 4. Hahn, H. T. and Kim, R. Y. (1978) Swelling of Composite Laminates”, Environmental Effects on Composite Materials, ASTM STP 658, J. R. Vinson, Ed., American Society for Testing and Materials, Philadelphia, pp. 98–120. 5. Whitney, J. M., Daniel, I. M., and Pipes, B. R. (1985) Experimental Mechanics of Fiber Reinforced Composite Materials, Monograph No.4, Society for Experimental Mechanics, Bethel, CT, Prentice-Hall, Englewood Cliffs, NJ. 6. Yaniv, G., Peimanidis, G., and Daniel, I. M. (1987) Method for Hygrothermal Characterization of Graphite/Epoxy Composite, J. Composites Technol. Res., Vol. 9, pp. 21– 25. 7. Tsai, C.-L. and Wooh, S. C. (2001) Hygric Characterization of Woven Glass/Epoxy Composites, Experimental Mechanics, 41(1), pp. 70–76.

PNEUMATIC BEHAVIOR OF COMPOSITE MATERIALS

CHO-LIANG TSAI Dept. of Construction Engineering National Yunlin University of Sci. & Tech. Yunlin 640, Taiwan YI-SHIUN TSAI General Education Center Hsiuping Institute of Technology Dali 412, Taiwan Abstract

The dimensional change of some composite materials induced by ambient air pressure change was discovered and dubbed as pneumatic strain in 2000. This pneumatic behavior is similar to the hygric behavior. The pneumatic strain is proportional to the ambient air pressure change by the coefficients of pneumatic expansion. In this work, a technique termed suspending method was employed for characterizing pneumatic behaviors of different types of materials including fibers, plates and thin film specimens. Specimens including Kevlar 49 fiber, aluminum plate, aluminum foil, paper, celluloid sheet, pure epoxy plate, unidirectional T700 carbon/epoxy composite laminates were characterized. Results showed that Kevlar 49 fiber and aluminum have no pneumatic behavior. Paper, celluloid film, epoxy plate, and T700 carbon/epoxy composite laminate have pneumatic behavior. 1.

Introduction

Since the first fiberglass boat was made in 1942, the application of composites becomes more and more popular. Composites have unique advantages over conventional materials, but also have complicated mechanic behaviors, like anisotropy and hygric behavior. A novel behavior of composites was discovered by Tsai and Du in 2000 [1, 2]. They verified that carbon/epoxy and woven glass/epoxy expand when the ambient air pressure increases and shrink when the air pressure decreases. They dubbed this phenomenon as pneumatic behavior and the induced strain as pneumatic strain. The pneumatic strain of composites is usually small. Resolution of conventional tools like micrometer and caliper is sometimes not fine enough for the measurement. Beside, during the experiment the specimen must be isolated inside a vacuum chamber. Operating a micrometer or caliper inside the vacuum chamber is not convenient. Several methods can be employed for characterizing pneumatic behavior of materials. 145 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 145–152. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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For not woven continuous fiber composite laminates, crossply method can be employed [1]. In the crossply method, the specimen is a strip shape laminate. The specimen always bends into an arch with some curvature due to the residual strain from manufacturing process. When ambient air pressure change, the induced pneumatic strain of the 90° layer is normally much bigger than that of the 0° layer. The curvature of the specimen will change due to the mismatch between the dimensional changes of 0° and 90° layers. By analyzing the curvature changes of two specimens with different thickness, the pneumatic strains of the 0° and 90° layers can be obtained. Usually, one and one laminates were chosen for each characterization. The pneumatic strains of 90° and 0° layers of AS4 carbon/epoxy (Hercules, Inc., USA) induced by 1 atm air pressure change measured by this method were 0.0015 and 0.00008 [1]. By the same method, those of T700 (Toray, Inc., Japan) carbon/epoxy were 0.0008±0.0005 and 0.0000410.00002 based on 30 experiments. It is a very sensitive method. The theoretical resolution can be as good as that of strain gage. But the deviation of the data tended to be not small. The biggest problem of this technique was that when the specimens were cut by diamond saw a lot of cracks were generated randomly around the edges. It was very difficult if not impossible to prevent those cracks and they were too random to be quantified. Among all the T700 carbon/epoxy specimens, those specimens with less cracks tended to have much smaller pneumatic strains. Mechanically, this method was effective. The problem was that the cutting damage of the specimens couldn't be prevented and resulted in much higher pneumatic strain. If the specimen is plate shape with high modulus, embedded strain gage can be used [2], This method is easy and has high resolution. In this method specimens don't have to be cut into strip shape. Cutting damage can be prevented completely. It is suitable for characterizing woven or 0° unidirectional composite laminates. However, embedding strain gage into the specimen does alert the geometry of the laminate slightly. Unless the specimen is so thick or strong that the effects can be neglected, or the measurement must has error. For conductive material like carbon/epoxy, the strain gage, soldering tin and lead wires must be protected carefully from touching the material. Data from five embedded strain gages in and T700 carbon/epoxy laminates showed that the pneumatic strains of the 90° and 0° layers induced by 1 atm air pressure change were 0.0001 ±0.00002 and 0.00003±0.00001. Those of and were 0.00012±0.00002 and 0.00003±0.00001. The two values of the and laminates were same and close to that from crossply method but with smaller deviations. Because there was no cutting damage in the specimen, the values of the and the were much smaller than that from crossply method. The problem here is that the value of the was smaller than that of the It is because that the modulus of 90° layer of carbon/epoxy composite is usually not high in comparison with that of the strain gage and when strain gage is embedded in the weaker or thinner laminate, it tends to reinforce the laminate and reduces the pneumatic strain. So, the pneumatic strain in the 90° layer of thicker laminate tended to be bigger than those from the thinner one. This method is not suitable for weak specimens such as 90° composite laminate.

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A technique termed suspending method is presented herein [3,4]. In this method, a piece of flexible single fiber was suspended horizontally, with both ends glued tightly, on the surface of a vertical plate and a small weight was hung at its center to straighten it (Figure 1). When the ambient air pressure changed the deflection at the center of the fiber should change if the fiber or the plate has dimensional change. The deflection can be measured from pictures taken by a camera with a close up lens. Its resolution is not so good as that of strain gage. But, it could be the only method so far that can effectively characterize the thermal, hygric and pneumatic behaviors of single fibers, soft plates, and thin films.

First of all, Kevlar 49 fiber and 2mm thick glass plate were chosen as the fiber and vertical plate of the method. Because the glass has no pneumatic behavior [2], the deflection at the center of the fiber must be caused by the pneumatic behavior of the fiber when there is air pressure change. From simple geometry analysis of the deflection, the pneumatic strain of the fiber can easily be obtained. Experimental results showed that when ambient air pressure changed, the deflection at the center of the fiber didn't change at all. The result indicated that Kevlar 49 fiber has no pneumatic behavior. The suspending method can also be employed to characterize plate shape and semi-rigid film specimens. Hold the specimen flatly, loosely and vertically on a piece of glass and stick a piece of Kevlar 49 fiber on it as prescribed in the last section (Fig.2). Since, the Kevlar 49 provides no pneumatic strain, the deflection at the center of the fiber must be caused by the pneumatic strain of the specimen when the ambient air pressure changes. From simple geometry analysis of the deflection, the pneumatic strain of the specimen can easily be obtained.

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Theoretical Background

In suspending method, a piece of single fiber was suspended horizontally with both ends glued tightly on the surface of a vertical flat base plate (sheet, film or foil) and a small weight was hung at its center to straighten it (Figs.1, 2). The glued spots should be as small as possible and the weight was roughly 1mg. Denote ambient air pressure, the total length of the fiber, the horizontal distance between both glued ends, and the deflection at the center of the fiber as and The relationship between them is

i is the stage index.

and

stand for those values at the initial stage. The

pressure change equals The can be measured by caliper easily at the very beginning of every experiment. The deflection can be measured from pictures taken by a camera with a close up lens fixed in front of the vacuum chamber window. The accuracy is about 0.001cm. If the plate has no pneumatic strain, the should remain a constant, The change of the deflection should be induced by the pneumatic strain of the fiber only. The pneumatic strain of the fiber is

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On the contrary, if the fiber has no pneumatic strain, the should remain a constant, The change of the deflection should be induced by the pneumatic strain of the base plate only. The pneumatic strain of the plate is

In the experiment, the and of with respect to is

From Eq.(3), the gradient of

were about 16cm and 0.7cm. From Eq.(2), the gradient

with respect to

is

If the varies from 0.7cm to 0.75cm, the and calculated from Eqs.(2-5) are shown in Figs.(3-6). It can be calculated easily from Eqs.(4,5) that the each 0.001cm change of the deflection indicates 0.00001 pneumatic strain change of the specimen, fiber or plate, when the and are 16cm and 0.7cm. This implies that if the accuracy of the is 0.001cm, the accuracy of the calculated pneumatic strain should be 0.00001. The was measured from the picture taken by a camera fixed in front of the vacuum chamber window. The camera had a close up lens to magnify the image. The accuracy of the measured in this way is 0.001cm if not better and the accuracy of calculated pneumatic strain should be 0.00001cm which was good enough for the characterization of pneumatic strain of materials.

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Experimental Process

First of all, Kevlar 49 fiber and glass plate were chosen for the experiment. The glass plate was verified to have no pneumatic strain by using embedded strain gage [2]. Kevlar 49 fiber is flexible but very tough. A piece of single Kevlar 49 fiber was glued on the vertical glass plate by both ends (Fig.1). A 1mg weight was hung at the center of the fiber and a ruler was placed between the fiber and the plate as a reference for measuring the The was measured by a micrometer. Then, the glass plate with the fiber was placed in a vacuum chamber and a camera with a close up lens was fixed

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tightly in front of the chamber window for taking pictures to measure the deflections. The environmental temperature was controlled to be A vacuum pump was used to pump out the air until the air pressure inside the chamber was almost zero (less than 0.1torr), The vacuum was maintained for at least two weeks and then a picture was taken to record After the was recorded, air was released to the chamber to recover 1 atm air pressure, The outlet was then sealed to maintain the 1atm pressure. All the air passed through a moisture trapper to enter the chamber in order to keep the air inside the chamber as dry as possible, less than 0.1%. Another picture was taken after another two weeks for measuring From Eq.(2) the pneumatic strain of the Kevlar 49 fiber induced by 1atm air pressure change can be calculated. Results of five experiments verified that Kevlar 49 fiber has no measurable pneumatic strain. Since the Kevlar 49 fiber has no pneumatic strain, it can be used for characterizing plate (sheet, film or foil) specimens. Hold the plate specimen flatly, loosely and vertically on a piece of glass by rubber bands and glud a piece of Kevlar 49 fiber on it as prescribed (Fig.2). Since, the Kevlar 49 provides no pneumatic strain, Eq.(3) can be used to calculated the pneumatic strain of the plate specimen. Same experimental procedures as prescribed can be taken. Specimens of 0.02mm aluminum foil, 1mm aluminum plate, 0.1mm paper (normally used for printer), 0.1mm celluloid sheet (transparency), 1mm pure epoxy plate (RTM resin, Golden Point Chemical Co., Ltd., Taiwan, cured under 200ºC for 1 hour), and T700 carbon/epoxy laminates were tested. Results and their standard deviations from five experiments each were listed in Table 1.

4.

Conclusions and Discussions

The suspending method is suitable for a wide range of materials with different geometry. It has almost no interruption to the specimens. Kevlar 49 fiber, aluminum foil, aluminum plate, paper, celluloid sheet, pure epoxy plate and T700 carbon/epoxy composite laminate were tested. Experimental results showed that except for Kevlar 49 fiber and aluminum, all the others specimens have pneumatic behavior.

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More materials than expected have pneumatic behavior including the material used so frequently and supposed to be fully understood, such as paper. Composite material also has pneumatic behavior. The pneumatic strain is small, but has never been taken into consideration in designing the structure serving in the environment of varying air pressure. For critical structures, such as the body of aircraft, pneumatic behavior could be as important as hygric behavior [5-10]. Several different methods were developed for characterizing pneumatic behavior of materials. Crossply method is a sensitive one. But, the specimens are always damaged in the process of preparation and result in much higher pneumatic strain. Embedded strain gage is good for materials of higher modulus. For weaker material, the gage reinforces the specimen and gives lower value of strain measurement. Among those methods, suspending method is an easy, effective, and accurate one. The basic idea is that the deflection at the center of the fiber suspended on the surface of the plate is so sensitive that even a very small dimensional change of the fiber or the plate induced by air pressure change may cause visible change of the deflection. An important contribution of this work is that a wide range of materials such as fiber, plate, foil, and sheet specimens can be characterized by this method. No tedious work and complex theory are required and the process induces almost no interruption or damage to the specimens. The deflection used for calculating pneumatic strain was measured optically. No contact was required and the accuracy is good. It is the only method for effectively characterizing the pneumatic behavior of some materials. Different techniques should be developed to verify the accuracy of the suspending method. 5. 1.

References

Tsai, C.-L. and Du, Y, “Characterization for Pneumatic Behavior of Carbon/Epoxy Composites”, Composites Science and Technology 61 (2001), 551-555. 2. Tsai, C.-L., Du, Y, Tsai, Y.-S. and Hou, C.-K., “Pneumatic behavior of woven glass/epoxy composite laminates”, Accepted by Composites Science and Technology (2001), CST-AA01-363. 3. Tsai, C.-L. and Chiang, C.-H., “Characterization of Hygric Behavior of Single Fiber”, Composite Science and Technology 60 (2000), 2725-2729. 4. Tsai, C.-L. and Daniel, I.M., “Method for Thermomechanical Characterization of Single Fibers”, Composite Science and Technology 50 (1994), 7-12. 5. Daniel, I. M. and Ishai, O., Engineering Mechanics of Composite Materials, Oxford University Press Inc., New York, 1994. 6. Yaniv, G., Peimanidis, G. and Daniel, I. M., “Method for Hygrothermal Characterization of Graphite/Epoxy Composite”, J. Composites Technol. Res., Vol. 9, 1987, 21-25. 7. Tsai, C.-L. and Wooh, S.-C., “Hygric Characterization of Woven Glass/Epoxy Composite”, Experimental Mechanic, Vol.41, No. 1, March 2001, 70-76. 8. Shen, C.W. and Springer, G.S., “Moisture Absorption and Desorption of Composite Materials”, Journal of Composite Materials, Vol. 10, 1976,2-20. 9. Gazit, S. and Ishai, O., “Hygroelastic Behavior of Glass-Reinforced Plastics Exposed to Different Relative Humidity Levels”, Proceedings of Conference on Environmental Degradation of Engineering Materials, Virginia Polytechnic Inst., Blacksburg, VA, Oct. 1977, 383-392. 10. Hahn, H. T. and Kim, R. Y, “Swelling of Composite Laminates”, Environmental Effects on Composite Materials, ASTM STP 658, J. R. Vinson, Ed., American Society for Testing and Materials, Philadelphia, 1978, 98-120.

THE EFFECT OF SPECIMEN SIZE ON THE COMPRESSIVE STRENGTH OF CARBON FIBRE-EPOXY LAMINATES C. SOUTIS and J. LEE Department of Aeronautics

Imperial College of Science, Technology and Medicine, Prince Consort Road, London SW7 2BY UK.

ABSTRACT The effect of specimen gauge section (length x width) was investigated on the compressive behaviour of a T300/924C carbon fibre-epoxy laminate. A modified Imperial College compression test fixture was used together with an anti-buckling device to test 3 mm thick specimens with a 30x30, 50x50, 70x70, and 90mm x 90 mm gauge length by width section. In all cases failure was sudden and occurred mainly within the gauge length. Post-failure examination suggests that 0° fibre microbuckling is the critical damage mechanism that causes final failure. This is a matrix dominated failure mode and its triggering depends very much on initial fibre waviness. It is suggested that manufacturing plays a significant role in determining the compressive strength and may be more important as the section thickness of the composite increases. Additionally, compressive tests on specimens with an open hole are performed. The local stress concentration arising from the hole dominates the strength of the laminate rather than the stresses in the bulk of the material. It is observed that the remote failure stress decreases with increasing hole size and specimen width but is generally well above the value one might predict from the elastic stress concentration factor. This suggests that the material is not ideally brittle and some stress relief occurs around the hole. X-ray radiography reveals that damage in the form of fibre microbuckling and delamination initiates at the edge of the hole at approximately 80% of the failure load and extends stably under increasing load before becoming unstable at a critical length of 2-3 mm (depends on specimen geometry). This damage growth and failure are analysed by a linear cohesive zone model. Using the independently measured laminate parameters of compressive unnotched strength and in-plane fracture toughness the model predicts successfully the notched strength as a function of hole size and width.

1.

INTRODUCTION

It has long been known that the strength of brittle materials like ceramics depends on the volume of stressed material and the nature of stress distribution. Both of these effects arise because brittle materials are flaw sensitive and flaw severity and distribution are generally statistical in nature. As the probability of 153 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 153–162. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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finding a serious defect increases with increasing material volume, large brittle bodies tend to fail at lower stress levels than smaller ones when both are subjected to the same kind of uniform stress field, such as uniaxial tension. Also, it is widely recognised that the flexural strength is much higher than the uniaxial tensile strength of a specimen having the same dimensions, because the region of high tensile stress in flexural specimen is much smaller. This is known as the ‘size effect’. The influence of size on the strength of metallic structures is rarely, if ever, considered, because plastic yielding in metals tends to reduce stress concentrations arising from defects, and these materials display much less strength scatter and size effect than ceramics. Continuous fibre reinforced plastics have a number of characteristics that are typical to brittle materials. They lack plasticity to reduce the influence of stress concentrations arising from defects, have more or less linear stress-strain curves when loaded in tension or compression along the fibre direction and display significantly more scatter in strength than metallic materials [1]. In light of these facts, it is natural to ask whether the strength of composites depends on material volume. This question has been around since the 1960s and there is a significant, but inconclusive, amount of evidence that there is a size effect in composites. For instance, the tensile strengths of glass, carbon and aramid filaments, decrease with increasing length [2, 3]; the flexural strengths of unidirectional specimens can exceed tensile strengths by as much as 44% and compressive strengths by as much as 56% [4,5]; the flexural strength of 100-ply unidirectional specimens is 15% lower than for 25-ply specimens and apparent size effects occur in both tension and compression [6]; the reduction in laminate strength caused by open holes increases with increasing hole size, possibly because the volume of material subjected to a stress concentration increases with increasing hole diameter [7]. The stress gradients and associated through-thickness effects can explain the difference between flexural and unidirectional strengths. A recent study by Lavoie et al [8] indicates that strength-scaling in 0° fibre-dominated laminates should be regarded as an artefact of the test procedure and failure mode. When tensile tests of composite laminates containing unidirectional plies fractured consistently in the specimen gauge section, then the 0° ply stress and strain at failure was the same as the unidirectional beams tested in 3-point bending. This implies that the static strength of the 0° plies was insensitive to the damage state of neighbouring plies. The study [8] also found that the tensile strength of continuous carbon fibre-epoxy composites did not have a strength versus volume dependency of a magnitude sufficient to be distinguishable from the data scatter. A strength scaling effect in connection to length was not ruled out, only in connection to volume. The aim of the present paper is to investigate the existence of size effect on the compressive strength of a T300/924C, carbon fibre-epoxy laminate. Unnotched specimens with three different gauge sections are tested statically in uniaxial compression; the hole size effect is also examined experimentally and analytically. A cohesive zone model is employed to estimate the notched strength and the results are compared to experimental measurements.

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2. Experimental Procedure 2.1 MATERIAL The material used was Toray T300 carbon fibre in a Ciba-Geigy 924C epoxy resin (T300/924C). The pre-impregnated tapes (pre-preg) were laid up by hand into a quasi-isotropic lay-up and cured according to the manufacturer’s recommended procedure. The quality of the moulded laminates was examined by using ultrasonic C-scanning. The lamina elastic in-plane properties of the T300/924C composite system are presented in TABLE 1.

2.2 SPECIMEN GEOMETRY Several rectangular specimens were cut from the 3 mm thick panels and glass fibre-epoxy reinforcement tabs (30 mm long) were bonded giving gauge sections of 30 x 30, 50 x 50, 70 x 70 and 90 mm x 90 mm. The geometry of the 30 mm long by 30 mm wide specimen is based on the Airbus Industry test method (AITM1.008) [9]. Circular holes of diameter 1.5-36 mm (diameter/width ratio, a/W= 0.05 - 0.5) were drilled at the centre of the specimens using a tungsten carbide bit to minimise fibre damage and delamination at the hole boundary. At least five specimens for each configuration were tested.

2.3 COMPRESSIVE TESTING Static compressive tests were carried out on a screw-driven Zwick 1488 universial testing machine with a load capacity of 200kN; a crosshead displacement rate of 1 mm/min was used. Load introduction to the specimen was mainly by end loading using a modified ICSTM fixture [10]. For all 50mm x 50mm, 70mm x 70mm and 90mm x 90mm specimens an anti-buckling device similar to that used by Soutis [11] was employed to prevent column buckling. It contains a window at the device centre, allowing damage around the hole to occur but restraining the specimen from general bending. Frictional effects were minimised by lining the inner faces of the fixture with Teflon tape; the clearance between the anti-buckling device and the specimen face was less than The column length of the device is slightly shorter than the specimen gauge length to allow for endshortening of the specimen. Back-to-back strain gauges were attached on the specimens to monitor out-of-plane bending and of course record the failure strain. Several of the tests were interrupted before final failure in order to examine damage growth. Examination was by dye-enhanced X-ray radiography. Zinc iodide solution has been shown to be an effective penetrant for highlighting damage regions in composite laminates. The location and nature of damage in individual plies was obtained by using the de-ply technique and scanning electron microscopy (SEM).

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3. Test Strength Results The test results obtained include stress-strain curves to failure and post-failure examination of fracture patterns. The stress-strain response, axial stiffness, strength and final failure patterns were studied as a function of specimen gauge section; thickness of all specimens was approximately 3 mm.

3.1 UNNOTCHED SPECIMENS Failure of the unnotched specimens was sudden and occurred mainly within the specimen gauge length. Post-failure examination suggests that in-plane fibre microbuckling in the 0° plies is the critical damage mechanism, which causes the catastrophic fracture. Longitudinal splits, fibre/matrix de-bonding and delamination between neighbouring plies do not occur gradually but take place suddenly and concurrently with the final failure. This is supported by failure strain measurements. The average failure strain measured by the two back-to-back strain gauges at the point of failure was in the region of 1%, which similar to the strain of the 0° unidirectional material. In general, test results for all sizes were good and reproducible. The scatter in axial stiffness and strength for all specimen configurations was less than 5%, see TABLE 2; the results quoted in TABLE 2 are based on the average of five specimens tested for each different size. The scatter in strength is probably due to imperfections introduced during manufacturing of the laminates resulting in fabrication defects and non-uniform laminate thickness. Imperfections in specimen geometry can produce misalignment of the specimen in the testing fixture that causes bending and reduction in the measured compressive strength. The average elastic modulus (measured at 0.25% applied strain) for all specimens was 56 GPa, which is in a good agreement with that estimated by the laminate plate theory (58 GPa). However, the average strength of the specimen with the 30 mm x 30 mm gauge section was at least 20% lower than that of the bigger size specimens. For this configuration no anti-buckling device was used and failure occurred prematurely due to global Euler buckling. A typical stress-strain curve of such a specimen is shown in Figure 1; the strain readings of the two backto-back strain gauges are almost the same up to 0.6% applied strain and start to deviate at higher applied loads indicating out-of-plane deflection that leads to premature failure of the laminate and lower overall strength.

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From the limited strength data presented in TABLE 2 there is no evidence of a size effect, which is not the case for specimens loaded in tension or flexure.Under compressive load, such a size effect does not seem to exist probably because of the different failure mechanism. The mechanism of failure under compressive loading is fibre microbuckling that initiates in regions of maximum fibre misalignment or waviness introduced during the fabrication process [12]. The fibre waviness is sensitive to thickness change of the specimen rather than the two-dimensional area change introduced in this study. Therefore, an investigation on thickness effects should be performed.

3.2 OPEN HOLE SPECIMENS All specimens failed from the hole in a direction almost perpendicular to the loading axis. The failure load is decreased as the hole diameter is increased. Compressive strength results for all specimen configurations tested statically at room temperature conditions are summarised in TABLE 3; the coefficient of variation is less than 3%. Strength results are also displayed in Figure 2, where the remote failure stress normalised by the unnotched strength of the laminate is plotted against the hole diameter (a) normalised by the width (W) of the specimen. Strength values are based on the gross-sectional area of the test piece. It is clear that the compressive strength is reduced by 30-70% by the presence of the hole. The measured notched strengths are bounded by simple criteria based on notch sensitivity [13]. If the material is ideally notch insensitive (ideally ductile), the failure stress is proportional to net sectional area, while if the material is ideally notch sensitive (brittle) then the plate fails when the local stress at the edge of the hole equals the failure stress of the material The notch sensitive curve in Figure 2 refers to the case of a quasi-isotropic finite plate; the stress concentration factor (SCF) for

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a finite plate with a hole can either be determined analytically or from a twodimensional finite element stress analysis. Material orthotropy alters the position of the curve. Since the data points for all specimen configurations tested lay above the notch sensitive curve, the T300/924C system is not ideally brittle and some stress redistribution occurs around the hole before failure.

Examination of the damaged laminates reveals that fibre microbuckling in the 0° plies, surrounded by delamination, occurs in the vicinity of the hole prior to catastrophic fracture. The microbuckled zone extends like a line-crack for 2-3 mm along the specimen width before final failure; its critical length depends on specimen geometry and decreases with increasing hole size, leading to a more brittle behaviour, see Figure 2. The stress concentration factor and local stresses near the hole edge increase with increasing hole size and therefore, it is less likely that local stress relief or stress redistribution can occur in such plates [14] resulting in lower strengths. During the compression test of specimens with a large hole failure occurred at about the same time as initial cracking sounds were heard. Figure 2 indicates that the notched compressive strength decreases as the specimen width is increased. The open hole compressive (OHC) strength for the specimen with a 50 mm x 50 mm gauge section is lower than the 30 mm x 30 mm specimen with the same a/W ratio. However, no significant width effect is observed for specimens with W>50 mm, TABLE 3.

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Strength Analysis of Open Hole Specimens

4.1 STRESS DISTRIBUTION NEAR AN OPEN HOLE IN A FINITE WIDTH PLATE In order to verify the hole size effect the stress distribution near the hole is required. In this study, it is obtained by performing a two-dimensional finite element analysis using the FE77 package [15] and also an approximate analytical solution by Savin [16]. Figure 3, illustrates the stress distributions along the transverse axis (y-axis) for a T300/924C laminate with a gauge section of 30mm x 30mm, containing a 3mm, 6mm, 9mm, 12mm and 15mm diameter hole, derived by the two methods. The agreement between the numerical and analytical solution is acceptable except for a/W >0.4 at y/W>0.7 due to finite width effects that are not accurately captured by Savin’s method. The results indicate that the stress concentration factor increases with increasing a/W and for bigger hole sizes the material near the hole experiences higher local stress that may cause failure at lower applied loads, see experimental observations above. The SCF values could be used with the maximum stress failure criterion to estimate damage initiation. The stress distributions could be employed with the point stress or average stress failure criteria to predict the failure load of the laminate. However, these criteria require the knowledge of a characteristic length, which is used as a free parameter to be fixed by best fitting the experimental data. In the following paragraph a cohesive zone model [17] is employed to estimate the notched strength using the unnotched strength and the laminate in-plane fracture toughness

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4.2 COHESIVE ZONE MODEL Soutis and coworkers [11, 17] have developed a crack bridging model for the initiation and growth of compressive damage from the edge of a hole. The microbuckled region (and associated deformation in the off-axis plies and delamination between plies) is treated as a compressive mode I crack with a cohesive zone at its tip. Damage is modelled by a linear softening spring law within the cohesive zone: the crack bridging normal traction is assumed to decrease linearly with increasing crack closing displacement (CCD) from a maximum value (equal to the unnotched compressive strength of the composite) to zero at a critical crack face overlap The critical CCD can be estimated from a fibre microbuckling analysis [18] or determined experimentally by independent fracture toughness tests [11, 17]. The approach has been applied to a wide range of specimen geometries and has been used to examine the effect of lay-up upon notched strength of carbon fibre-epoxy laminates. In the present study the measured unnotched strength and a fracture toughness were used to estimate the notched strength of the T300/924C laminate. The theoretical predictions (dotted lines) are in a good agreement with the experimental measurements, difference is less than 10%, see Figure 2.

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5. Concluding Remarks The question of the existence of a size effect in filamentary composite materials is of great importance. If there were a size effect, it would mean that use of standard test specimens to establish design allowables for large structures could be very nonconservative. Further, it would be necessary to analyse the strength of large composite structures and components using statistical methods, as is done for ceramic materials. In this study the effect of gauge section (length x width) was investigated on the static compressive strength of a currently used carbon fibreepoxy system; the lay-up was a quasi-isotropic and the specimen thickness was kept constant. From these limited data, it appears that the unnotched strength does not depend on specimen size. Failure is matrix dominated and occurs due to fibre microbuckling in the 0° plies. The initiation of this failure mode is greatly dependent on initial fibre waviness, which highlights the importance of manufacturing processes. Fibre waviness may get worsen with increasing thickness [19, 20] so further work is required to investigate this effect. The introduction of open holes causes severe reduction in the compressive strength of the T300/924C laminate. However, the composite system is not ideally brittle and some stress redistribution occurs around the hole before failure. X-ray radiography reveals that damage in the form of fibre microbuckling, delamination and matrix cracking, occurs in the vicinity of the hole prior to catastrophic fracture. This damage reduces the stress concentration factor at the edge of the hole and delays final failure to higher applied stresses. The results indicate that the notched compressive strength decreases with increasing hole diameter or specimen. However, no significant width effect is observed when W>50 mm. The effects of the hole size and specimen width upon the compressive strength are predicted with reasonable accuracy by the Soutis et al [11, 17] cohesive zone model. The model takes as its input the compressive unnotched strength and the fracture toughness of the laminate that can be measured or estimated from a fibre microbuckling model [18]. The local stress concentration arising from the open hole dominates the compressive behaviour of the laminate rather than the stresses in the bulk of the structure making the size issue and the application of statistical methods to predict notched strength less relevant. Further work is required to examine the effect of laminate thickness on the unnotched strength since this depends on initial fibre waviness, which may increase with thicker laminates.

6.

Acknowledgements

This work was carried out with the financial support of the Procurement Executive of the Ministry of Defence (Structural Materials Centre, DERA, Farnborough, UK). The authors are grateful for many useful discussions with Professor G. A.O. Davies of the Department of Aeronautics, Imperial College and Professor P.T. Curtis of the Defence & Evaluation Research Agency (DERA).

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7.

References

1. 2.

Zweben, C. “Is there a size effect in composites?”. Composites, 25(6), 1994,451-454. Zweben, C., Smith, W.S. and Wardle, M.W. “Test methods for fiber tensile strength, composite flexural modulus and properties of fabric-reinforced laminates”. Composite Materials: Testing & Design, Conf. ASTM STP674 (American Society for Testing and Materials, Philadelphia, 1979), 228-262. Herring, H.W. “Selected mechanical and physical properties of boron filaments”. NASA TN D-3202, 1966. Bullock, R.E. “Strength ratios of composite materials in flexure and in tension”. J. Composite Mater., 8, 1974, 200-206. Berg, K.R. and Ramsey, J. “Metal aircraft structural elements reinforced with graphite filamentary composites”. NASA CR-112, 62, August 1972. Wisnom, M.R. “The effect of specimen size on the bending strength of unidirectional carbon fibre-epoxy”. Composite Structures, 18, 1991, 47-63. Waddoups, M.E., Eisenmann, J.R. and Kaminski, B.E. “Macroscopic fracture mechanics of composite materials”. J. Composite Mater., 5, 1971, 446-454. Lavoie, J.A., Soutis, C. and Morton, J. “Apparent strength scaling in continuous fiber composite laminates”. Composite Science & Technology, 60, 2000, 283-299. Airbus Industrie Test Method, AITM-1.0008, Issue 2, June 1994 Haberle, J.G., “Strength and Failure Mechanics of Unidirectional Carbon fibre-Reinforced Plastics under Axial Compression”, PhD Thesis, University of London, December 1991. Soutis, C., “Compressive failure of notched carbon fibre-epoxy panels”, PhD Thesis, University of Cambridge, October 1989. Lagoudas, D.C and Saleh, A.M., “Geometry and Loading Effects on the compressive Strength of fibrous composites”, J. Reinforced Plastics & Composites, 12, 1993, 1016-1023 Mikulas, M.M.. “Failure prediction techniques for compression loaded composite laminate with holes”, NASA CP-2142, 1980. Rhodes, M.D., Mikulas, M.M, and McGowan, P.E. “Effects of orthotropy and width on the compression strength of graphite-epoxy panels with holes”, AIAA Journal, 22(9), 1984, 1283-1292. Hitchings, D. “FE77 Users Manual”, Imperial College, Department of Aeronautics, 1995 Savin, G.N., “Stress concentration around holes”, Pergamon Press, 1961. Soutis, C., Fleck N.A. and Smith, P.A. “Failure Prediction Technique for compression loaded in carbon fibre-epoxy laminate with open hole,”J. Composite Mater., 25, 1991, 1476-1498. Soutis, C. and Curtis, P.T. “A method for predicting the fracture toughness of CFRP laminates failing by fibre microbuckling”, Composites, Part A, 31, 2000, 733-740. Componechi, E.T., Gillespie, J.W., and Wilkins D.J., “Kink-band failure analysis of thick composites in compression”, J. Composite Mater., 27, 1993, 471-490 Daniel, I.M. and Hsiao, H.M. “Is there a thickness effect on compressive strength of unnotched composites”, Int. J. Fracture, 95, 1999, 143-158.

3. 4. 5.

6. 7.

8. 9. 10.

11. 12.

13. 14. 15. 16. 17.

18. 19. 20.

INTERFACIAL STRENGTH AND TOUGHNESS CHARACTERIZATION USING A NOVEL TEST SPECIMEN

G. P. TANDON, R. Y. KIM and V. T. BECHEL Structural Materials Branch Air Force Research Laboratory Wright-Patterson AFB, OH 45433

Abstract In this study, a cruciform-shaped test specimen has been utilized to characterize the fiber-matrix interface under transverse loading. Initiation and growth of interface debonds are detected optically by observation of variations in the intensity of light reflected from the surface of the fiber during loading. Using the measured values of applied stress at debond initiation, and debond length and shape as a function of the applied loading, the tensile strength and the critical value of energy release rate of the interface, are estimated. Finally, an off-axis cruciform geometry, in which the wings of the cruciform sample are inclined at an angle with respect to the loading direction, is introduced to characterize the fiber-matrix interface under combined transverse and shear loading. 1.

Introduction

A method that has often been used in the past to determine the interfacial normal strength of fiber-reinforced composites has been to load a straight-sided specimen, with fiber ends exposed to the free edges, in a direction transverse to the fiber axis. The basic problem with the straight-sided configuration is that an exposed fiber specimen contains a stress singularity [1-2] at the fiber-matrix interface location where the fiber intersects the free surface. Consequently, in straight-sided specimens, the free edge is a favored site for interface debonding because of stress intensification at this location. In order to remove the influence of the free edges from the test, a cruciform specimen geometry was introduced by Gundel et al. [3], as shown in Figure 1. In this configuration, the composite sample is in the shape of a cross with an extremely large width in the gage section. Under tensile loading perpendicular to the fiber axis, the central portion of the fiber-matrix interface is highly stressed, whereas the interface has negligible loading in the wing region. Consequently, the free-edge singularity is rendered ineffective for the cruciform geometry. In this work, single fiber cruciform specimens are incrementally loaded in tension to failure, and damage initiation and propagation are studied using the reflected light 163 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 163–174. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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method [4]. The cruciform specimens are analyzed using 3-D FEM to estimate the interfacial stresses in terms of constituent material properties and specimen geometry.

These analytical results are then used in conjunction with the experimental observations to extract important interfacial parameters, such as tensile strength, shear strength and the fracture toughness of the fiber-matrix interface. Two major advantages of the cruciform geometry are highlighted. Firstly, as mentioned before, interface debonding is shown to occur in the central region of the cross (region free of initial stress singularities), so that the estimated bond strength is free of edge effects. Secondly, the cruciform geometry is shown to produce very stable debond growth from which the entire debond profile could be obtained at several applied loads. Thus, all the geometric information necessary for calculating the potential energy release rate for the interface crack is made available. Thus, by also using the cruciform geometry for energy release rate measurement, it is now possible to characterize both initiation (interface strength) and propagation (energy release rate) from the same sample or set of samples. 2.

Specimen Design and Fabrication

For this work, the composite reinforcement consisted of an uncoated SiC fiber (SCS-0 from Textron) with a diameter of chosen primarily because of its large size. The matrix was an epoxy resin (Epon 828 from Shell Chemical Co.) cured with a polyetheramine (Jeffamine D-230 from Texaco, Inc.) for a minimum of 3 days at ambient temperature (most samples were cured 1-2 weeks prior to testing). The transparent matrix allows visual observation of the debonding process which occurs at the fiber-matrix interface.

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A silicon rubber mold with the desired cruciform shape was first fabricated from an aluminum pattern with a slot centered at the end of each wing for aligning the fiber. Model single-fiber composite specimens were cast in the cruciform-shaped silicone rubber mold with the following dimensions (see Figure 1 for notation): 2a=14.8 mm, 2h=3.44 mm, 2l=27.94 mm, t=0.9 mm, 2g=50.8 mm and R=6.35mm. The SCS-0 fiber was cleaned with acetone to remove impurities from handling. A single fiber was placed in the mold from the end of one wing to the end of the opposite wing, supported only at the ends of the wings, and epoxy resin was cast around them. The two flat surfaces of the cruciform specimen were progressively ground to the desired thickness and then polished using successively smaller diameter alumina polishing powder (final size was 0.3 micrometer) in order to enhance the microscopic image for interfacial debond detection. Fiberglass/epoxy end tabs were then adhesively bonded on the upright portion of the specimen, i.e., in the loading arm region, in order to prevent specimen failure in the grip region. 3.

The Reflected Light Method

The reflected light technique has been successfully employed to study the initiation and growth of interface debond in model single-fiber composites. The technique is described in detail in Bechel et al. [4] but is briefly reiterated here. An intense light source is used to illuminate the fiber surface to the verge of being shiny, as shown in Figure 2. With the fiber surface in this condition, a small increase in the ability of the interface to reflect light results in the appearance of a bright or white region. Initiation and growth of interface debonds are observed as variations in the intensity of the light reflected from the surface of the fiber during loading. A CCD camera and VCR are used to capture a video of the fiber surface at 30 frames/second. After completion of the test, a video capture card is used to select frames from the video tape record. From these images, it is possible to determine the load when debonding initiates, the angular swath of the debond wake, and the axial debond length to within ± 0.1 fiber diameters.

4.

Interfacial Strength Characterization

The single-fiber cruciform specimen was analyzed using the 3-D finite element method (FEM) employing the ABAQUS code [5]. The model was simplified by using

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symmetry planes so that only an eighth of the total specimen was required to be modeled. The reinforcement and the epoxy matrix were treated as three-dimensional eight-node brick elements with elastic properties listed in Table 1. Further, the fibermatrix interface was assumed to be perfectly bonded since the stress distribution in the specimen was examined prior to damage.

Figure 3 shows the variation of the normal stress at the fiber-matrix interface in the loading direction as a function of distance from the center of the cruciform specimen. The normal stress at the interface is normalized with respect to the far-field applied stress, and thus represents a stress concentration factor (SCF), while the distance is normalized with respect to the fiber length. The SCF remains reasonably constant (within 10 percent) over two-thirds of the loading region and approaches zero value in the arms before increasing marginally near the free edge. It is the large value of SCF in the specimen center which is responsible for debonding in that region.

Figure 4 is a photomicrograph of the cruciform specimen which had been subjected to transverse loading and is illuminated by a light source. What is interesting to note in this figure is the white line in the central region of the cruciform sample which is the light reflected by the debonded fiber-matrix interface. This micrograph is therefore a visual confirmation of fiber-matrix debonding which takes place in a cruciform

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specimen, and the length of the illuminated region is an approximate measurement of the extent of debonding.

Figure 5 is a photograph of the failed specimen and clearly shows fiber-matrix debonding in the central region. The debond is seen to propagate along the fiber in the loading arm region and then runs parallel to the interface in the matrix. Eventually, the crack front bows away from the interface near the fiber end. Note that even at failure the fiber ends remain in contact with the matrix. For this particular sample, first occurrence of acoustic emission activity takes place at a stress level of 29.8 MPa, which in conjunction with the calculated radial SCF of 1.387, results in a debond strength of 33.8 MPa (after accounting for 7.5 MPa shrinkage-induced stresses). This value is in reasonably good agreement with the average value of the predicted bond strength of 29.2 ± 5.2 MPa using an older geometry [6], considering possible batch-to-batch material nonuniformity.

5.

Interfacial Toughness Characterization

The reflected light technique can also be employed to measure both the location and length of the debond as a function of applied load [7,8]. Images showing debond initiation and debond growth from a representative test are shown in Figure 6. A

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transparent scale has been attached to the sample to aid in measuring debond length. Image (a) of Figure 6 shows 0.3 mm of a debond that is approximately 6.4 mm in length in an SCS-0/epoxy sample at initiation. The initial debond was too long to image entirely. The debond initiated and immediately extended across the full field of view. The camera was then quickly moved axially along the fiber until the crack tip was reached. After the crack tip was reached, further crack growth could be followed at a leisurely pace. An assumption of symmetric axial debond growth was made since all measurements could only be made on one side of the center of the gage length.

Figure 6 also contains images of subsequent debond growth. In each image the fiber is dark except for an area on the bottom right portion of the fiber that reflects light with a greater intensity. The load increases from 186.1 N in image (a) to 196.9 N in image (d), while the debond grows an additional 0.35 mm (2 fiber diameters). Surface defects captured in each image can be used as position markers to compare the changing debond length. At a higher magnification additional information can be obtained from images obtained during the SCS-0 cruciform tests. In Figure 7, the shape of the debond

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can be seen more clearly without a scale attached. The debond front is not straight and perpendicular to the fiber axis but has a more complicated curved shape that extends further in the axial direction on the bottom of the fiber. As illustrated in Figure 7, the lateral height of the debond wake can be measured and used to infer the angle, that the debond wraps around the fiber.

The measured debond angle, varies as a function of applied load, position along the fiber, and debond length. As can be seen in Figure 7, reaches a constant value within a distance of approximately 3 fiber diameters of the debond front. The full debond could not be imaged because it was longer than the field of view. Consequently, as a compromise, the debond angle was measured at a position four fiber diameters in the wake of the debond front for each debond length. Figure 8 is a plot of applied load and the measured debond angle as a function of half debond length for the test specimen. The measured debond angle ranged from 135.3° to 157.4° and was scattered about an average value of 143.4°.

Interfacial debonding in the cruciform specimen was also analyzed using 3-D finite element methods [5]. To simplify the model and to reduce the size of the problem, it was assumed that the debond was symmetrically located. Consequently, only an eighth

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of the full specimen was modeled. Further, even though the experimental observations indicate that the debond front had a curved shape, we assumed as a first approximation, that the debond front was straight and perpendicular to the fiber axis, and it propagated with a constant debond angle This approximation simplified the model considerably. The reinforcement and epoxy matrix were modeled with three-dimensional, eight-node, brick elements (C3D8), while the debond faces were modeled using interface contact elements (INTER4). Since the contact area changed as a function of applied load, an incremental process was followed to attain a solution. Friction was not modeled on the crack surface so the energy release rate (ERR) could be calculated by obtaining the J integral. The energy release rate is maximum for the crack tip in the loading direction (point in Figure 7) and decreases with angle (measured from the loading direction) along the fiber-matrix interface. It is the maximum value of the energy release rate which is plotted in Figure 9. This value represents the total critical energy release rate since the contributions from each mode were not separated. The ERR was calculated for debond angles of 135.3°, 143.4°, and 157.4° to determine whether the average debond angle value could be used effectively. Based on these results, the debond angle averaged from a measurement at a few debond lengths (<10 measurements should be sufficient) is considered adequate to obtain an accurate ERR. Also, from Figure 9, it is apparent that the ERR for a fixed debond angle is fairly uniform with respect to debond length. For example, for an angle of 143.4°, the scatter in the ERR during the first 1.2 mm of stable debond growth is approximately ±0.75 J/m2 (<3%) after which the ERR begins to drop significantly. The relatively low variation during initial growth is very encouraging and is an indication that the value of ERR derived from the initial debond growth will be useful for predicting debond growth under mode I loading.

6.

Failure Under Mixed Modeloading

In this section, the cruciform specimen design was modified [9] to characterize the fiber-matrix interface under a combined state of transverse and shear stresses. This was achieved by utilizing a cruciform specimen in which the wings of the cruciform

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specimen were not perpendicular to the loading direction. Instead, the ratio of the normal to shear loading was governed by the amount of off-axis angle the fiber (oriented along Z') made with the loading direction (X-axis), as seen in Figure 10.

Figure 11a is a photomicrograph of a cruciform specimen with the fiber oriented at 45 degrees with respect to the loading direction. Note the white line indicating a debond in the central region of the cruciform sample in the gage area. Figure 11b is a magnified view of the central region of the sample and clearly shows the illuminated white area corresponding to the debonded interface. The micrographs in Figure 11 are, therefore, a visual confirmation of fiber-matrix debonding which takes place in an off-axis cruciform specimen.

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Table 2 lists the average value (over all specimens) of externally applied stress at debond initiation measured for the specimens tested along with the maximum radial and shear stress concentration factors evaluated at the fiber-matrix interface as a function of off-axis angle. In absence of precise measurement of the failure initiation site, the maximum values of stress concentration factors will serve as useful approximations. The maximum stress criteria [10] can be employed to predict failure initiation in the offaxis cruciform specimens, since we have demonstrated that debonding initiates in the interior region of the specimen, which is free of initial stress singularities. The average applied stresses at debond initiation for different off-axis angles are multiplied with their corresponding stress concentration factors, and the results plotted in the normalshear stress space, as shown in Figure 12.

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A quadratic failure envelope given by

fits the experimental data rather reasonably well. The constants and are evaluated using the measured value of interfacial normal strength and the 45-degree off-axis cruciform data. The constant is estimated as 1.5, while the shear strength of the interface is extrapolated as 34.5 MPa. The empirical curve given by Equation 1 is found to be a good fit to the remainder of the off-axis data (i.e., 30- and 60-degree measurements). Note that for this part of the study we have neglected the cure shrinkage-induced stresses since no data was available for Epon 828 cured with polyetheramine. Current effort is underway to estimate these quantities, and when properly accounted for, it would result in a horizontal shifting of the failure envelope to the left. Nevertheless, the proposed off-axis specimen design promises to be an elegant method for extrapolating the shear strength of the fiber-matrix interface, although an independent assessment of the shear strength would be a valuable test of the proposed failure criteria. 7.

Summary

In this work, a novel test specimen in the shape of a cruciform has been utilized to characterize the fiber-matrix interface in single-fiber composites. The experimental part of this study clearly establishes that the cruciform geometry is successful in forcing debond initiation in the central region, which is free of initial stress singularities, and therefore provides valid interfacial tensile strength data. Such data are critical for establishing design methodologies based on micromechanical failure theories so that empirical failure diagrams obtained through extensive composite testing can be avoided. Using the reflected light technique, debond initiation, location, length, and shape are measured as a function of applied load; thus, all the required parameters for calculating the normal strength of the interface as well as the interfacial energy release rate are acquired. Finally, an off-axis cruciform geometry, in which the wings of the cruciform sample are inclined at an angle with respect to the loading direction, is introduced to characterize the fiber-matrix interface under combined transverse and shear loading. The proposed specimen design promises to be an elegant method for extrapolating the shear strength of the fiber-matrix interface. Moreover, the evaluated strength values are not influenced by the free-edge effects which seem to dominate some of the more common test methods, such as slice compression, micro-indentation, push-out and fragmentation, used to characterize the fiber-matrix interface. 8.

Acknowledgements

The authors would like to thank Erik Ripberger of Wright State University (SOCHE program) and Ron Trejo of the University of Dayton Research Institute for specimen preparation and conducting the necessary mechanical testing. This research was performed in part under U. S. Air Force Contract No. F33615-00-D-5006.

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References

Hu, S., Karpur, P., Matikas, T. E., Shaw, L. and Pagano, N. J., "Free Edge Effect on Residual Stresses and Debond of a Composite Fiber/Matrix Interface," Mechanics of Composite Materials and Structures, 1995, 2, 215-225. 2. Tandon, G. P., Kim, R. Y., Warrier, S. G. and Majumdar, B. S., "Influence of Free Edge and Corner Singularities on Interfacial Normal Strength: Application in Model Unidirectional Composites," Composites, Part B: Engineering, 1999, 30, 115-134. 3. Gundel, D. B., Majumdar, B. S. and Miracle, D. B., "Evaluation of the Transverse Response of Fiber-Reinforced Composites Using a Cross-Shaped Sample Geometry," Scripta Metall., 1995, 33, 2057-2065. 4. Bechel, V. T. and Tandon, G. P., "Characterization of Interfacial Failure Using a Reflected Light Technique," to appear in Experimental Mechanics, 2001. 5. ABAQUS/Standard User’s Manual, Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, RI, 1997. 6. Tandon, G. P., Kim, R. Y., and Bechel, V. T., "Evaluation of Interfacial Normal Strength in a SCS-0/Epoxy Composite Using Cruciform Specimens," Composites Science & Technology, 2000, 60, 2281-2295. 7. Bechel, V. T., Tandon, G. P., and Kim, R. Y., "Fiber/Matrix Interface Debond Length Measurements Under Transverse Loading in the Cruciform Test," Proceedings of the American Society of Composites, 14th Technical Conference, 1999, 367-376. 8. Tandon, G. P., and Bechel, V. T., "Estimation of Interface Toughness in a Cruciform Specimen Under Transverse Loading," Proceedings of the American Society for Composites, 15th Technical Conference, 2000, 1035-1042. 9. Tandon, G. P., Kim, R. Y., and Bechel, V. T., "Mixed-Mode Failure Criteria Using Cruciform Geometry," presented at 16th Annual Technical Conference of the American Society for Composites, Virginia Tech, Blacksburg, VA, Sept. 9-12, 2001. 10. Whitney, J. M. and Nuismer, R. J., "Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations," Journal of Composite Materials, 1974, 8, 253-265.

A MODEL FOR THE ACCURATE PREDICTION OF THE RESIDUAL STRENGTH AFTER DAMAGE DUE TO IMPACT AND EROSION OF FRPs G. C. PAPANICOLAOU, G. SAMOILIS, S. GIANNIS Composite Materials Group Department of Mechanical and Aeronautical Engineering University of Patras, Patras 265 00, Greece

N.-M. BARKOULA, J. KARGER-KOCSIS Institute for Composite Materials Ltd. University of Kaiserslautern POB 3049, D-67653 Kaiserslautern, Germany Abstract This paper discusses the prediction of the residual compressive strength after low velocity impact for different CFRP material systems. Several material systems were examined using each time different types of epoxy matrices and carbon fibers. Experimental results concerning residual compressive strength after impact for all the material systems are compared here to values predicted by the model developed by the CMG group. Predicted values agree very well with respective experimental results. In addition, in the present work, the influence of stacking sequence, existence and position of interleaves on the solid particle erosion in carbon fiber reinforced epoxy composites (CF/EP) are investigated. The erosive wear behavior was studied in a modified sandblasting apparatus at 90°-impact angle. The erosion behavior was considered as a repeated impact procedure (impact fatigue). Predictions made by the same model for the residual tensile strength after erosion give excellent results. 1.

INTRODUCTION

All structures and components made of composite parts are subjected to impacts during their service. Fiber reinforced polymer composites, especially carbon fiber reinforced plastics (CFRP), are very susceptible to accidental impact damage. Impact damage is a combination of delamination, matrix cracking and fiber breakage. The first two types of failure are sensitive to matrix properties, whereas fiber breakage is more sensitive to properties of the fibers. Improvements in the properties of fiber and matrix lead to improved impact behaviour of the laminate, as many studies have shown. Experimental results reported [1] indicate that matrix properties govern the damage threshold and determine the impact damage size. Fiber properties, on the other hand, control the penetration resistance. Extensive studies [2] show that an increase in the strain to failure of the matrix will result in improved residual strength of the composite after impact, due to a better resistance to delamination and to matrix cracking. However, the development of higher strain capability of the matrix is limited by the need to maintain satisfactory performance at high temperatures and under aggressive environmental conditions. 175 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 175–184. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Several authors [3-8] have discussed the introduction of a discrete layer of high strain resin for increased toughness, which creates what is called an interleaf system. In general, such a system delivers significantly higher compression strength after impact as a result of having less delamination, but it has the inconvenience of raising the resin content of the laminate which results in a lower modulus and more sensitivity to environmental conditions. Fiber strain energy has been pinpointed as one of the fiber parameters that would most significantly improve the properties of the composite. Higher fiber failure strains, with the same elastic modulus, will result in higher energy absorption, especially since the energy absorbed by the matrix represents a large portion of the total strain energy. For the same impact energy, higher capacity to adsorb energy results in less fiber breakage and a higher residual tensile strength. Secondary matrix damage, which occurs after initial fiber failure, will also be reduced so that residual compressive strength is also increased. Cantwell et al [9] conducted tests using high strain carbon fibers. Results were compared to similar tests on specimens with high strength carbon fibers. Both types of fibers had the same modulus, so that an increase in strain to failure corresponded to a similar increase in strength. Low velocity impact tests showed that with the same failure modes, less damage was induced in the laminates with high strain to failure fibers.

2. Theoretical Background 2.1 MODELING OF THE RESIDUAL TENSILE STRENGTH AFTER IMPACT Papanicolaou et al. [10-15] adopted an approach to describe the residual strength of impacted fiber reinforced laminates (FRP). However, a limitation of that model was its prerequisite of evaluating laminates including always ±45° oriented plies. In order to overcome this limitation, a new model was recently developed by Papanicolaou which takes into account the quasi-statically response of the impacted structures. The theoretical background of this model is analytically described elsewhere [16]. Visco-elastic behavior of fiber and matrix materials is not the only mechanism for the structural damping in composite materials but appears to be the dominant mechanism in undamaged polymer composites vibrating at small amplitudes. This is also the case in solid particle erosion. The degradation of the mechanical strength due to impact damage is assumed to follow an exponential decay law of the form:

where u is a function of the impact energy as well as of the energy absorption capacity of the material expressed through Thus, the strength degradation after low energy impact can be described by a differential equation of the type:

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where residual tensile strength at high impact energy / tensile strength before impact

where U = the impact energy = the impact energy threshold related to the onset of strength degradation. For impact energy values no interior damage is induced; the impact energy causes the laminate to deform elastically. Once the impactor ceases to exert load on the plate, the latter recovers its original shape and retains its nominal strength in compression/tension. Solving equation (2) we obtain:

According to relation (3) for the prediction of the residual strength after impact, one only needs to know and which can easily determined from simple experiments. In addition, as it is stated in ref. [16], starting from physical considerations, the value of the strength degradation impact energy threshold, can be calculated by:

where = is the effective longitudinal Young’ s modulus of the laminate V = the total volume of the specimen = loss factor at the of the non-impacted material, m = is the mismatching coefficient between adjacent layers due to the difference in their fiber orientation angle [10-14], defined as follows:

Here

is the mean value for the bending stiffness mismatching coefficient

of the the distance of the

is the x-direction stiffness matrix term of the

is

from the middle plane of the laminate and n is the total

number of plies in the laminate. The mean value of

is defined as follows:

178

where

G. C. PAPANICOLAOU, et al.

refers to

and

and

refer to the interfaces of

the adjacent layers and The above-mentioned m-parameter depends on the laminate material system elastic properties, lay-up, stacking sequence and individual lamina thickness. Relation (4) has been successfully applied to the experimental results presented in ref. [16].

3.

Model Verification

RESULTS AND DISCUSSION The model described in the previous section has been applied in order to predict the residual compressive strength after low energy impact of a series of CFRP's. Characteristic properties, dimensions, compression strengths of unimpacted specimens and respective experimental results [10] are given in Tables 1-10.

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Next, both predicted values for the residual compressive strength and respective experimental data were plotted against impact energy for each type of material for comparison (Figs. 1 - 9). From these figures it is clear that predicted values agree well with experimental findings.

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Next, the efficiency of the model has been verified in the case of a series of eroded UD CF/EP (AS4/3501-6; BASF) composites having different stacking sequences [16]. In this case, the residual tensile strength after erosion of these materilas has been predicted. Erosion tests were performed in a sandblasting chamber by sharp, angular corundum with a particle size between 60-120 at 90°-impact angle while their viscoelastic response of both eroded and virgin laminates was studied by DMTA. Figure 10 shows a characteristic example of this prediction. It is clear that predicted values are very close to respective experimental results. The same agreement was found to exist in all types of composites studied and for every stacking sequence [16].

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4. Conclusions In the present work a new model for the prediction of the residual strength after impact was used in order to study the compressive behavior of impacted CFRP laminates. The theoretical predictions were compared with experimental results. In all cases a good agreement was observed. In the case of eroded CFRP's the model predicts both the impact energy threshold and the residual strength after solid particle impact. The proposed model seem to be a promising design tool since it can be applied to any type of impacted material and for its application a minimum number of experimental input data is needed.

5. References 1. 2.

3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

14.

15.

16.

Elber W. Effect of matrix and fiber properties on impact resistance. Tough Composite Materials: Recent Developments, Noyes Publ., Park Ridge, NJ, pp. 89-110. Evans R E, Masters J E., (1987) A new generation of epoxy composites for primary structural applications: materials and mechanics, ASTM STP 937, pp. 413-436. Hirschbuehler K R., (1985), An improved performance interleaf system having extremely high impact resistance, SAMPE Quarterly, 17(1), pp. 46-49. Singh S, Partridge I K. (1995), Mixed mode fracture in an interleaved carbon fibre/epoxy composite, Comp Sci and Tech, 55(4), pp. 319-327. Yuan Q, Karger-Kocsis J. (1995), Effects of interleaving on the impact and impact fatigue responses of cross-ply carbon fibre/epoxy laminates, Polymers and Polymer Composites, 3(3), pp. 171-179. Yuan Q, Friedrich K, Karger-Kocsis J. (1995), Low energy Charpy impact of interleaved CF/EP laminates, Applied Composite Materials, 2, pp. 119-133. Karger-Kocsis J, Yuan Q. (1994), Damage growth in interleaved CF/EP composites from instrumented impact fatigue data, Advanced Composites Letters, 3(4), pp. 123-126. Ozdil F, Carlsson L A. (1992), Mode I interlaminar fracture of interleaved graphite/epoxy, J Comp Mat, 26(3), pp. 432-459. Cantwell W, Curtis P, Morton J. (1986), An assessment of impact performance of CFRP reinforced with high-strain carbon fibers, Comp Sci and Tech, 25(2), pp. 133-148. Papanicolaou G C, Stavropoulos C D. (1995), New Approach for Residual Compressive Strength Prediction of Impacted CFRP Laminates, Composites, 26(7), pp. 517-523. Stavropoulos C D, Papanicolaou G C. (1997), Effect of Thickness on the Compressive Performance of Balistically Impacted CFRP Laminates, Journal of Materials Science, 32, pp. 931-936. Papanicolaou G C, Stavropoulos C D, Mouzakis D E, Karger-Kocsis J. (1997), Residual tensile strength modelling of Polymer-Polymer Microlayer Composites after low energy impact, Plastics Rubber and Composites, Processing and Applications, 26(9), pp. 412-417. Papanicolaou G C, Blanas A M, Pournaras A V, Stavropoulos C D. (1998), Impact Damage and Residual Strength of FRP Composites, In Kim J K, Yu T X, editors, Key Engineering Materials, Impact Response and Dynamic Failure of Composites and Laminate Materials, Part 1: Impact Damage and Ballistic Impact, Trans Tech Publications, pp. 127-148. Stavropoulos C D, Papanicolaou G C, Karger-Kocsis J, Mouzakis D E. (1998), Effect of ± 45° Layers on the Residual Compressive Strength of Impacted Glass Fibre/Polyester Laminates, In Gibson A G, editor, “FRC” Seventh International Conference on Fibre Reinforced Composites Conference Proceedings (April 15-17, 1998, Newcastle upon Tyne), pp. 327-337. Papanicolaou G C, Pournaras A V, Mouzakis D E, Karger-Kocsis J, Bofilios D A.(1998), Probabilistic Approach for Residual Compressive Strength of CFRP Laminates After Low Velocity Impact, In Reifsnider K L, Dillard D A, Cardon A H, editors, Progress in Durability Analysis of Composite Systems, Balkema, Rotterdam, pp. 325-330 Barkoula N -M, Papanicolaou G C, Karger-Kocsis J., Prediction of the Residual Tensile Strength after Solid Particle Erosion of CF/EP Composite Laminates with and without Interleaves, Composites Science and Technology, In Press.

3. Fracture and Fatigue

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THE ORIGIN AND INCEPTION OF FATIGUE IN STEEL – A PROBABILISTIC MODEL SIDNEY A. GURALNICK and JAMSHID MOHAMMADI Department of Civil and Architectural Engineering Illinois Institute of Technology Chicago, Illinois 60616

Abstract

This paper presents a simple model to simulate the fatigue behavior of materials. The model consists of a system of parallel springs, each with a non-linear stressdeformation behavior. The applied stress is modeled as a random variable. Using a probabilistic analysis, the system is subject to a series of stress applications. Upon each stress application, a certain number of springs fail. When a sufficiently large number of springs fail, the entire system is considered to have failed. The number of stress cycles corresponding to this condition is considered to represent the fatigue life. This system’s behavior is similar to fatigue damage accumulation in materials and fatigue failure. The description of the model is presented in the paper along with a numerical illustration to demonstrate its applicability. The paper also discusses the areas where the model can be further developed to obtain a broader understanding of the fatigue behavior of materials. 1.

Introduction

Various attempts [1,2,3,4] have been made in the past to use mechanical models to explain the interlinked set of phenomena which comprise metal fatigue. Previous attempts have been unsatisfactory to a greater or lesser degree because the models used simulated only one or two of the gross macroscopic aspects of metal fatigue but could not be extended to cover the entire spectrum of observable phenomena. The current understanding of the evolution of the fatigue process is outlined in Fig. 1. That is, the evolution of fatigue is assumed to proceed through four stages prior to fracture or rupture. These four stages are: (1) the inception of isolated microscopic zones of plasticity, (2) the “organization” of the microscopic zones of plasticity into macroscopic plastic regions, (3) the initiation of cracks and (4) complete separation or rupture. Stage 1 probably occurs at very low stress levels and, if stresses remain low, fatigue failure in steels with a definite endurance limit does not occur. At higher stress levels than those needed to initiate microscopic yielding, stage 2 occurs and this is illustrated by the curve marked P in Fig. 2. If the stresses are raised to still higher levels, then stage 3, or crack initiation, occurs as illustrated by the curve marked F in Fig. 2. This curve is sometimes called, “French’s line of damage.” When the stresses are raised to even higher levels, the cracks which first appeared in stage 3 propagate and eventually cause rupture as illustrated by the uppermost curve in Fig. 2. This last curve is the conventional S-N (or Wöhler) curve. It is also clear from Fig. 2 that the entire fatigue process (stage 1 through stage 4) is quite different in the low-cycle region from that which occurs in the high-cycle region. 187 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 187–196. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Guralnick [5,6] has previously proposed a model for fatigue phenomena based on the incremental collapse of a simple portal frame composed of an elastic perfectlyplastic material. At best, however, this model can only simulate the first two stages of the progression leading to fatigue fracture shown in Fig. 1. The model proposed below, however, among its other virtues, overcomes this difficulty by providing a pathway through all four of the stages culminating in rupture or failure. This new model is an extension and modification of the model originally proposed by Jenkin [7] to simulate the elastic-plastic mechanical behavior of materials. Our extension of Jenkin’s model utilizes probabilistic arguments to simulate fatigue behavior which, of course, was not contemplated by Jenkin.

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The model proposed herein is based on the behavior of a system of parallel springs which undergo a random stress cycle. Failures among the springs occur at random and can be used as a means to simulate fatigue damage and fatigue behavior. In this paper, however, only a simple application is presented to demonstrate the method’s applicability. Further studies will be needed to fully develop the model for a broad application in the study of fatigue behavior in materials.

3. Model Description Fatigue behavior is modeled with m springs working in parallel to carry the stress F (see Fig. 3). Each spring is a non-linear element with the characteristics shown in Fig. 4. Although the springs are considered to be identical, their behavior will differ from one another if one defines one or more of the spring parameters as random variables. In the most general case, the stress F is assumed to be a random variable. Furthermore, all of the other parameters describing the spring behavior may also be treated as random variables.

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3. Basic Formulation

Considering m identical springs, at the start of the loading and immediately after the first stress cycle, the stress in any spring i is,

Failure in a spring will occur when the stress exceeds cycle, the probability of failure in any one spring is,

Let

After the first stress

then

From Fig. 4,

can be written as

and as such,

where

Parameters F, and may all be treated as random variables. For simplicity, we can assume that only F, and are random variables. It is further assumed that these are independent random variables, that follow the normal distribution. Then will also follow the normal distribution. Thus,

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in which and are the mean and standard deviation of respectively; and is the normal probability distribution function. If the mean and standard deviation of individual random variables in Eq. 5 are known then and can be computed. Upon the application of the first stress cycle, the estimated number of springs that survive the load is After the second cycle, the probability of failure of a given spring is where,

where,

and and are the mean and standard deviation of respectively. The number of springs that survive after the second stress cycle is In general, after the nth stress cycle, if is the failure probability of a given spring, then,

where,

and and product cycle. If F,

are the mean and standard deviation of respectively. Note that the is the number of springs surviving after the nth stress and are independent random variables, then it can be shown that,

and

in which and and

and are the mean values of the random variables F, are their respective standard deviations.

and

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Fatigue Failure – Fatigue failure occurs when a sufficiently large number of springs have failed. For example, one may assume that when at least 25% of the springs have failed, the system can no longer carry the load. This is tantamount to fatigue failure. The estimated fatigue life is the value of n at which at least 25% of the springs have failed. Solution Process – Equations 9 and 10 can be solved in a step-by-step manner starting with and continuing the computations until system failure occurs. The estimation of the mean and standard deviation of will be based on the first order approximation. In this process, will be assumed to be a random variable following a normal distribution. Furthermore, using the Taylor series, Eq. 10 may be expanded into a polynomial, of which only the linear terms are used (first order approximation). This approximation will provide for a straightforward and convenient way of computing the mean and standard deviation of when parameters F, and are all assumed to be random variables. It is further noted that the first order approximation provides satisfactory results when the variations in these parameters are small. In a special case where only F, and are treated as random variables, and they are considered to be independent of one another, then Eqs.l1 and 12 can be used to compute the mean and standard deviation of An alternative solution method for Eqs. 9 and 10 is by means of simulations using a random number generator. Determination of Spring Parameters -- Spring parameters depicted in Fig. 4 can be determined from simple tension tests of materials. This, however, requires that a value for m be decided. Generally, m will be proportional to the inverse of the probability values One can begin with a large m as a starting value and revise it later if such revision becomes necessary. The stress that causes yielding and subsequently the failure of the material in a simple test can be used to estimate the mean values for the spring constants. For example, if the yield strength of the material is then the constant in Fig. 2 can simply be estimated as and the uncertainty in can be estimated based on the coefficient of variation in 4. Special Cases The following two special cases represent conditions where no fatigue failure occurs or when a low-cycle fatigue failure develops. Special Case 1 -- The value of does not change. This case will occur when F is very small and is large. And as such, will be large and results in a nearly zero failure probability. If after a very large number of cycles, there is still no change in failure is unlikely. This case corresponds to a very low stress range smaller than the “endurance” limit. Special Case 2 --The value of changes dramatically and approaches a large value rapidly after only a few stress cycles. This case will occur when F is very large or

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is small. This condition corresponds to low-cycle fatigue. After only a few stress cycles, failure occurs. 5. Numerical Illustration

Suppose in a given material, m is assumed to be 10,000. The variabilities in force and spring parameters are assumed to be those in Table 1. After the first stress application,

At the start of the second cycle, springs survive. Table 2 summarizes values obtained for the failure probability of any given spring and the percentage of failed springs after each stress cycle. The system failure criterion is taken to be 25%; i.e. when at least 25% of springs fail, system failure occurs. As is evident from the results, this happens after 153 cycles of stress.

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To construct an S-N (or Wöhler) curve for this example problem, the value of applied stress is changed and the number of cycles to failure recorded. In each stress application, a standard deviation equal to 20% of the mean stress was used. The results are summarized in Table 3 and depicted graphically in Fig. 5.

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6. Discussion and Conclusions

The model presented in this paper can be used to study the entire fatigue process in materials without an excessive appeal to empiricism. The behavior of the model is demonstrated for a simple problem and it shows how the S-N relationship can be developed through this analytical approach. To apply the model to a broader fatigue study, several additional developments will be necessary. The model needs to be expanded to include stress reversal and mean stress effects. The number of random variables in the model can also be increased to better portray the scatter of the data inherent in observed S-N relationships. The following are the conclusions of this study: A model made of non-linear parallel springs can be used to simulate the fatigue behavior of materials. The parameters needed to define the model’s behavior are analogous to those determined by mechanical tests of the material. The model can easily be modified to simulate most of the observable fatigue phenomena in materials including hysteretic and Banschinger effects.

7. References 1. 2. 3.

4. 5. 6. 7.

Timoshenko, S.P.: Strength of Materials. Vol. II, Third Edition, D. Van Nostrand Company, Inc., Princeton, N.J., 1956, pp. 413 and 514. Jackel, H.R.: Simulation, Duplication and Synthesis of Fatigue Load Histories. Sound and Vibration, Vol. 4, March, 1970, pp. 18 to 29. Martin, J.F., Topper, T.H., and Sinclair, G.M.: Computer-Based Simulation of Cyclic StressStrain Behavior with applications to Fatigue. Materials Research and Standards, Journal ASTM, Vol. 11, No. 2, February 1971. Wetzel, R.M.: A Method of Fatigue Damage Analysis. Ford Motor Company Scientific Research Staff Technical Report, September, 1971, privately published. Guralnick, S.A.: Incremental Collapse Under Conditions of Partial Unloading. Publ. Intern. Assoc. of Bridge and Structural Engineers, Vol. 33, part II, Zurich, 1973. Guralnick, S.A.: An Incremental Collapse Model for Metal Fatigue. Publ. Intern. Assoc. of Bridge and Structural Engineers, Vol. 35, part II, Zurich, 1975. Jenkin, C.F., Engineering, Vol. 114, 1922, p. 603.

FATIGUE DAMAGE TOLERANT ANALYSIS USING THE FATIGUE DAMAGE MAP C. A. RODOPOULOS and J. R. YATES SIRIUS, Department of Mechanical Engineering University of Sheffield Mappin Street, S1 3JD, UK

Abstract Based on the principles of microstructural fracture mechanics the current work demonstrates the physical relationship between the size of crack plasticity and the five stages of crack growth from catastrophic crack initiation to failure. The relationship allows the continuous evaluation and distinction of the five stages of fatigue damage in the form of the Fatigue Damage Map (FDM). The FDM is used to determine the limits of damage tolerance by providing information regarding stress and the crack length for near-threshold and unstable propagation.

1.

Introduction

Damage tolerant design requires accurate knowledge of the size and the stress field characterising a fatigue crack from near-threshold to catastrophic failure conditions. According to Linear Elastic Fracture Mechanics the above limits are defined by the threshold stress intensity factor and toughness respectively. During the last two decades, extensive research in the fields of near-threshold crack propagation and short crack growth, concluded that the use of LEFM can dangerously underestimate the fatigue life of a component leading to dangerous designs [1,2]. In this work, the determination of the damage tolerant design limits is achieved by exploiting the relationship between the size of the crack and the size of the crack tip plasticity throughout the stages of fatigue damage.

2.

The Fatigue Damage Map

Once the fatigue crack is formed, the movement of dislocations (which defines the movement of crack tip plasticity) ahead of the crack tip increases. This increase will continue until the dislocations would encounter some constraining feature (e.g. grain boundary, twin boundaries, pearlite zones, fibres, etc) [3,4]. At this point, the dislocations start to pile-up against the constraining feature and their density increases with decreasing gliding activity and decreasing crack velocity. At a particular distance between the crack tip and the constraining feature, the dislocation density becomes critical (corresponding to the critical resolve shear stress / critical elastic strain energy) and the stress intensification, also known as constraint effect, due to pile-up is high enough to activate a new dislocation source in a grain that is: a) adjacent; b) large and c) preferably oriented [5]. At this point, 197 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 197–208. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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the constraint effect relaxes and the velocity of the crack increases. It should be noted that, once a new dislocation source is activated within a grain, then the whole grain undergoes slip. Such statement is sound considering that PSB formation takes place at low stress levels. This process will continue to recur until the crack is long enough to produce a near tip stress field which is high enough to partially dismiss the pile-up process. This point represents the end of the Stage I crack growth also known as short crack growth [3]. It is worth noting that during Stage I growth, even though the crack is subjected to Mode I far-field stress, the strain is mostly Mode II (shear). The observations that grain boundaries retard or even arrest the growth of short fatigue cracks [3], prompted Navarro and de los Rios [6] to develop a Microstructural Fracture Mechanics (MFM) model, known as the NR model, which describes microstructural sensitive crack propagation. According to the model, the crack tip plastic displacement and hence the crack velocity, changes in value in an oscillatory manner every time dislocations are emitted into a new grain. This oscillation represents the blocking and unblocking of the crack tip plasticity, modelled as a persistent slip band (PSB), by the grain boundary. The amplitude of the oscillation can be illustrated by plotting the crack velocity against the crack length. Briefly, the amplitude of the oscillation is reported to decrease with crack length as the propagation behaviour of the crack changes from Stage I (short crack) to Stage II (long crack). The above is illustrated in Figure 1.

According to the NR model, which is based on the continuous distribution of dislocations as proposed by Bilby et al [7], the fatigue damage under the operation of a far field stress can be represented by three zones as shown in Figure 2. The first zone represents the crack, the second the crack tip plasticity and the third a microstructural barrier (grain boundary). In terms of the materials resistance to crack propagation, the crack is considered as stress free unless some closure stress, is acting on the crack flanks, the stress at the plastic zone is equal to the resistance of the material to plastic deformations (cyclic yield stress), and at the

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grain boundary, of width the stress developed due to the constraint exerted on the plastic zone is This stress represents a measurement of constraint effect developed on the barrier due to the PSB blocking.

Based on the equilibrium of dislocations along the three zones, the constraint effect ahead of the slip band for an opening mode loading, is given by [5],

The parameters and represent in a dimensionless form the crack length and the fatigue damage size respectively. According to the model, grain boundaries are located at the crack tip at the end of the fatigue damage is at and the end of the three zones at It is worth noting, that the effect of the width of the boundary becomes less important for high values of i and therefore can be neglected However, such simplification should be avoided in the case of twin boundaries, pearlite zones or fibres where the width of the constraining feature could be substantial.

2.1 CRACK ARREST AND NEAR-THRESHOLD PROPAGATION Using the concept of equilibrium (Equation 1), the boundary conditions for negligible crack tip plasticity are [8],

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where is the value of the constraint effect that corresponds to the critical resolve shear stress. For the above boundary conditions, Eq.(1) yields,

where

is the crack arrest capacity or threshold stress of the material and is the fatigue limit of a smooth specimen at a given stress ratio. The parameter is the grain orientation factor and for aluminium alloys is given by [9],

It should be noted that increases monotonically with i from a value of 1 until reaches the saturated Taylor value of 3.07 (truly polycrystalline behaviour). The hypothesis is rationalised by the fact that, in many cases, crack nucleation takes place initially in grains that are most favourably oriented so that the resolved shear stress can easily reach maximal. According to the boundary conditions of Eq.(3), the threshold stress in terms of crack length can be expressed as,

The threshold stress of 2024-T3 with is shown in Figure 3.

(for stress ratio,

and

The threshold stress defined by Eq.(5), identifies two controlling parameters: a) the strength of the grain boundary which is part of the factor (fatigue limit) and b) the effect of the grain orientation, Both parameters reflect the influence of microstructure on crack arrest. The first, by relating the strength of the boundary to

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the threshold stress for crack propagation and the latter, by incorporating the effect of the increasing number of grains transversed to the crack front as the crack grows. The above reflects the increasing probability of a “hard” grain being included in the plastic zone [11].

2.2 THE LIMITS OF STAGE I CRACK PROPAGATION Immediately after the formation of the catastrophic crack and at stress levels described by Eq.(5), the size of the crack tip plasticity is about one grain in size. The above reveals the existence of two important cases. At threshold stress levels slightly above the fatigue limit and for crack lengths equal to the grain diameter, the crack and the crack tip plasticity are approximately equal in size (one grain). In this case, the similitude (LEFM) concept is seriously violated and hence significant oscillation of the crack velocity should be expected (microstructurally short crack propagation). At lower threshold stress levels, the catastrophic crack is characterised by longer lengths and therefore the one grain crack tip plasticity can be several orders smaller. Here, two different conditions can be identified: a) if the crack tip plastic zone is larger than 1/50 of the crack length [11], conditions of physically short crack propagation prevail [12] and b) if the size of the crack tip plasticity is smaller than 1/50 of the crack length the propagation is characterised as pseudo Stage I crack growth. The pseudo Stage I crack growth behaviour indicates the possible use of LEFM growth models immediately after the formation of the catastrophic crack. In [13] it was suggested that the Stage I crack growth terminates when the crack tip stress field is able to initiate persistent slip bands on two successive grain without further growth of the crack (two grain crack plasticity). This assumption is rationalised considering that when the crack is small a single family of slip planes can accommodate the crack tip plasticity. At longer crack lengths, crack tip plasticity is more intense and can only be accommodated by multiple slip. The above was validated by Yoder et al [14] who examined the transition from Stage I to Stage II for a number of materials. In [13] the two grain crack tip plasticity was mathematically expressed as,

For the case of negligible crack closure, Eq.(6) can be further simplified as,

To express the transition in terms of stress versus crack length, knowledge of the fatigue damage at any given stress level is required. An accurate prediction of can be attained by employing the following boundary conditions,

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In Figure 4, the results of an iterative solution between Eq.(7) and Eq.(8) are shown.

The limits of the Stage I growth can be determined by utilising both Eq.(5) and Eq.(7) as shown in Figure 5.

The transition from Stage I to Stage II crack growth depends on parameters such as: a) the stress ratio, plasticity induced crack closure, the mechanical properties of the material, the crack/loading geometry, stress concentrations, etc. More details can be found in [11 ].

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2.3 THE LIMITS OF STAGE II CRACK PROPAGATION Once the crack propagates within the area defined as Stage II crack growth, the Paris linearity can effectively predict the propagation rate of the crack. The accuracy of such application will be terminated when the crack propagation rate would start to deviate from its linear pattern (Stage III crack growth). During Stage III, it is believed that the stress field ahead of the crack tip is strong enough to break second phase particles or other microstructural defects. The above process, results into the development of gaps/holes, principally, inside the crack tip plastic zone. The consequence of such phenomenon is twofold: a) the flow resistance of the material decreases and b) the elastic strain energy increases. In both cases, crack plasticity would have to expand to a size larger than that defining by steady-state propagation. Under such conditions, the fracture surface of the fatigue damage would exhibit characteristics similar to those found in static fracture where cleavage, intergranular separation and fibrous failure are dominating. The degradation rate of is not instantaneous. The breaking of a single weak point will overstrain the area ahead of the crack tip and consequently will increase the probability of additional failures triggering a “chain reaction”. The onset and duration of the “chain reaction” will depend on mechanical, as well as, on material parameters. The first will control the spread of crack plasticity and the latter the presence and density of weak points.

2.4 THE LIMITS OF STAGE III CRACK PROPAGATION From the above, it is clear that definition of the onset of Stage III crack growth requires a complex interdisciplinary analysis and extensive statistics. Though, it is feasible to overcome such complexity by employing certain simplifications. Herein, we assume a) that there is an elastic relationship between the crack tip opening displacement, and the crack tip plasticity and b) that there is a resemblance between Stage III crack growth and the monotonic elongation to failure, of the material. Even though the first assumption might not be trustworthy especially for very short cracks, the assumption has been proven to provide good predictions as demonstrated by the crack path models [15]. The second assumption can be considered as trustworthy considering the resemblance between Stage III crack growth and static fracture. It is important however to make clear that represents the final stage of monotonic failure and thus it overestimates the onset of Stage III crack growth. According to the above Stage III crack growth takes place when,

where by,

is the crack tip plasticity. The crack tip opening displacement,

is given

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where G is the shear modulus. Considering that during Stage II crack growth, crack tip plasticity will continue to increase in multiples of D, the parameter can be written as,

where x is the number of half grains constituting crack tip plasticity. The boundary condition is used to differentiate from the transition from Stage I to Stage II crack growth. Using Eqs.(9-11) the onset of Stage III crack growth is given by,

It should be noted that the shear modulus in Eq.(10) has been replaced by the elastic modulus E to conform with the opening mode characterising Stage II crack growth. Solution of Eq.(12) using an iteration method is shown in Figure 6.

Crack propagation within the Stage III area will results into catastrophic failure. For conditions of cyclic loading catastrophic failure is defined either in terms of general yielding (also know as crack instability) or in terms of fracture toughness. Whether general yielding or toughness failure will prevail depends on several parameters, including the shape of the crack, the dimensions of the specimen/component, the stress level and state, the ductility of the material, the

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density of weak points, etc. From a theoretical point of view, we can suggest that that toughness failure will surpass general yielding when the conditions are such that they promote the coalesce of micro-cracks ahead of the crack tip with the main crack in a rate faster than that use to promote the expansion of crack tip plasticity. In [16] it was suggested that conditions of general yielding prevail when the crack tip plasticity has reach “infinite” size. In this case the contribution of the crack length in fatigue damage is minimum Substitution of the above boundary condition into Eq.(1) results to,

or,

for negligible crack closure. To express crack instability in terms of crack length, the “infinite” size of the crack tip plasticity has be related to some characteristic dimension of the component. In the case of a component of width W this can be expressed by the following equation system,

Solution of the above system shows that for components with tends to the cyclic yield stress. Deviations can be experienced in some unrealistic cases where is a first order fraction of W. From the above it is clear that toughness failure will surpass crack instability in almost every engineering component. Under plain strain conditions and considering the application of LEFM as credible, toughness failure can be expressed as,

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In Figure 7, crack initiation, the three stages of crack growth and the conditions of failure are combined to what is known as the Fatigue Damage Map (FDM).

3. Discussion - The FDM and Damage Tolerant analysis Damage tolerant design is divided into four areas: a) material selection; b) initial crack size; c) final crack size; and d) fatigue life.

3.1 MATERIAL SELECTION Material selection is an interdisciplinary process which involves the accuracy of the selected non destructive technique (NDT), weight limitations, design requirements, cost, etc. From the FDM the following information can be extracted regarding the material selection: a) Considering that more than 60% of the life of a component is consumed by the propagation of a Stage I crack, the transition line from Stage I to Stage II crack growth can provide information regarding the required accuracy of the NDT at any given operational stress level. For the case of the 2024-T3, at MPa the NDT accuracy has to be smaller than b) The transition from Stage I to Stage II crack growth can provide an insight over the susceptibility of the material to an overdesign (underestimation of fatigue life). In Figure 8 the transition from Stage I to Stage II for AISI 4340 is shown.

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By examining Figure 7 and Figure 8 it is clear that the susceptibility of AISI 4340 to Stage I crack growth is minimum and consequently the underestimation of the fatigue life.

3.2 INITIAL CRACK SIZE The determination of the initial crack size from a metallurgical point of view requires a complex statistical analysis over the size and distribution of inclusions. To overcome such difficulty, the engineer should choose materials with high grain boundary angle. This will increase the difficulty of achieving slip in a “hard” grain and thus the extension of the crack. Additionally, the material would be able to maintain its crack arrest capacity (non catastrophic crack) for longer crack lengths. For example, the 2024-T3 will loose 20% of its crack arrest capacity when the crack is In contrast, similar reduction in crack arrest, in the case of the AISI 4340, results into a crack length of Such information are valuable in the case of fail safe criteria.

3.3 FINAL CRACK SIZE To date, the determination of final crack size was determined through the value of fracture toughness. However, such approach can lead to dangerous results in the case of materials with low elongation to failure values. A typical example is the 18 Ni maraging steel with and [16]. According to the FDM, the limits of damage tolerant design should be set by the transition from Stage II to Stage III crack growth.

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3.4 FATIGUE LIFE The FDM can be used in collaboration to fatigue life models to provide information regarding the fatigue life of every stage of crack growth. Such information is valuable in the designing of non destructive inspection schedule and the determination of the operational life of the component.

4. References 1. 2.

3. 4. 5. 6. 7. 8.

9.

10.

11. 12. 13. 14.

15. 16.

Pearson S. (1975) Initiation of fatigue cracks in commercial aluminium alloys and the subsequent propagation of very short cracks, Engng. Fract. Mechan., 7, 235-247. Kitagawa H. and Takahashi S. (1976) Applicability of fracture mechanics to very small cracks or cracks in the early stage, 2nd Int. Conf. On Mechanical Behaviour of Materials, ICM2, ASM Metal Park, Ohio, 627-631. Miller K. J. (1993) Materials science perspective of metal fatigue resistance, Mater. Sci. Techn., 9, 453-462. Gdoutos E. E., Rodopoulos C. A. and Pilakoutas K. (2000) Failure Analysis of Industrial Composite Materials, McGraw-Hill, NY. Navarro A. and de los Rios E. R. (1992) Fatigue crack growth modelling by successive blocking of dislocations, Proc. Royal Soc. London, A437, 375-390 Navarro A. and de los Rios E. R. (1987) A model for short fatigue crack propagation with an interpretation of the short-long crack transition, Fatigue Engng. Mater. Struct., 10, 169-186 Bilby B. A., Cottrell A. H. and Swinden K. H. (1963) The spread of plastic yielding from a notch, Proc. R. Soc. London, A272, 304-314 de los Rios E. R. and Navarro A. (1997) Microstructural fracture mechanics in high-cycle fatigue, In: High Cycle Fatigue of Structural Materials, W. O. Soboyejo and T. S. Srivatsan eds, TMS, USA. Curtis S. A, Solis Romero J, de los Rios E. R, Rodopoulos C. A. and Levers A. (2001) Predicting the interfaces between fatigue crack growth regimes in 7150-T651 aluminium alloy using the fatigue damage map, Mater. Sci. Engng, to be published. Kitagawa H. and Takahashi S. (1976) Applicability of fracture mechanics to very small cracks or cracks in the early stage, 2nd Int. Conf. On Mechanical Behaviour of Materials, ICM2, ASM Metal Park, Ohio, 627-631. Rodopoulos C. A. (2001) A philosophical study on fatigue damage, Theor. Applied Fract. Mechan., submitted. Miller K. J. (1997) The three thresholds for fatigue crack propagation, STP 1296, ASTM, 267-286. Rodopoulos C. A., de los Rios E. R, Yates J. R. and Leevers A. (2001) A fatigue damage map for the 2024-T3 aluminium alloy, Fatigue Fract. Engng. Mater. Struct., in press. Yoder G. R., Cooley L. A. and Crooker T. W (1982) On microstructural control of nearthreshold fatigue crack growth in 7000-series aluminium alloys, Scripta Metall., 16, 10211025. Hahn G. T and Rosenfield A. R. (1975) Metallurgical factors affecting fracture toughness of A1 alloys, Metall. Trans., 6A, 653-670. Boyer H. E. and Gall T. L. (1985) Metals Handbook/Desk Edition, American Society for Metals, Metals Park, Ohio.

CRACK GROWTH BEHAVIOR AND SIF’S AS OBSERVED BY OPTICAL METHODS C. W. SMITH Department of Engineering Science and Mechanics Virginia Polytechnic Institute and State University Blacksburg, VA 24061 Abstract A laboratory based experimental method consisting of a marriage between frozen stress photoelasticity and the near tip equations of linear elastic fracture mechanics is applied to two three dimensional cracked body problems. Stress intensity factor distributions were determined for cracks in models of nuclear pressure vessels and rocket motor grain providing validation of numerical results.

1.

Introduction

In the late 1960’s, the writer and his graduate students began to develop a marriage between the near tip equations of linear elastic fracture mechanics and frozen stress photoelasticity. The original goal was to provide a cost effective laboratory based experimental method for validating numerical determinations of distributions of stress intensity factor (SIF) values along the border of cracks in three dimensional problems. It became apparent quite early in the study that, although no plastic zone existed at the crack tip, a small non-linear zone was present but the overriding feature in most of these problems was the complex geometry and loading orientations which meant that local data for evaluating the SIF values were concentrated very close to the crack tip. In fact, the concept of local Mode I data for a growing crack was clearly demonstrated by Cotterell in the chronologically first [1] of two papers [1], [2] which were published in the first two issues of the International Journal of Fracture in 1965 and 1966 for the two dimensional case. He found that, for growing cracks, only two terms in the field equations for a cracked body were required to analyze his near tip two dimensional photoelastic data. He also postulated that his theory for the paths and stability of cracks could be extended to the three dimensional case. Our goal was to develop a simple experimental based method for treating a wide variety of complex three dimensional problems. For simplicity, we focused on a two parameter algorithm. The analytical framework for our algorithm is described briefly in the Appendix. We verified the algorithm by applying it to planar part through cracks in plates where all crack growth was planar and pure Mode I [3] and also on such cracks loaded in Mixed Mode [4] (but not grown). During the 1970’s the concept of crack growth, where a shear mode existed initially was that the crack would kink into a new direction and then proceed under pure Mode I. Concepts were basically two dimensional. During this period we were performing some 209 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 209–216. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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experiments for the Heavy Section Steel Technology Program at Oak Ridge on cracked stress frozen models of reactor vessel nozzles under internal pressure. In this study we encountered cracks which grew along gently curved paths which led to previously unreported changes in the SIF distributions along the flaw border. Perhaps even more significant, from a practical point of view was that, when we tested scale models of test vessels containing nozzle cracks, we found that crack shapes deviated from those assumed in analysis and broke though to the outer surface at the same location as full scale tests on steel models under tension-tension fatigue tested at Oak Ridge. More recently, we have found other instances where cracks have kinked and grown along curved surfaces from rocket motor fin tips under internal pressure. Until very recently, however, we have seen no experimental evidence of a shear mode once the original crack has turned. In very recent tests, on a scale model of a solid rocket motor under internal pressure where we inserted a starter crack at the location on a fin tip where the large tip radius coalesces with the much smaller edge tip radius. we have encountered the presence of a shear mode which varies along the crack front shortly after turning begins. In the sequel, we shall describe results obtained from the two test geometries noted above.

2. Tests Results

2.1 MODELS OF REACTOR VESSELS WITH NOZZLE CRACKS [5] Fig. 1a shows a photoelastic model of the reactor vessel constructed in four parts; two end domes and two semi-cylindrical side walls, each containing two integrally cast nozzles diametrically opposite to one another. Plane, starter part through cracks were placed at one of the orientations in Fig. 1b in each nozzle and the vessels were stress frozen under internal pressure. Several tests were run for each orientation in order to monitor

the progress of each crack. Crack shapes and SIF distributions for the orientation (Fig. 1b) are shown in Fig. 2. In Fig. 2a, the flaw shapes, all of which remained

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planar, exhibited non-self similar flaw growth, deviating from the semi-elliptic shapes which were assumed by analysts and shown as dashed curves. As shown in Fig. 2b, the SIF distribution changed from dish shaped for shallow flaws to inverted dish shape for the deeper flaws, reflecting changing boundary influences. Only one slice value was available for

When the starter cracks were moved to (Fig 1b), several interesting results were observed. Figure 3a shows the shape taken by these cracks. The starter cracks were again planar with curved border nm. However, upon growth, the part of the cracks near the inner nozzle surface remained in their starter planes when growing (n to n’) but at point a kink occurred and the crack then proceeded along a curved path with a crease where the planar crack became a curved surface. Corresponding normalized SIF distributions for these cracks are given in Fig. 3b. These curves reveal an interesting three dimensional phenomenon. Clearly, the maximum value of the SIF along the crack front is a maximum near and increases with crack depth up to Then, between and 0.46, a radical redistribution of the SIF results moving the maximum SIF to near This redistribution was observed to be accompanied by an elimination of the crease or three dimensional kink in the crack surface. A notable feature of the above result is that the maximum normalized SIF after crease or kink removal is nearly double that before the kink removal.

2.2 MODELS OF SOLID ROCKET MOTOR GEOMETRY In the late 1980s, the writer and his associates began a study of cracked rocket motor grain geometries where the cracks emanated from the tips of fins and grew under internal pressure. Beginning with a generic model geometry [6] to validate the method for such a geometry, the work proceeded to include cracks initiated at an angle to the fin tip [7] and then to scale models of rocket motors [8]. For all of the starter cracks which were inclined to the fin axis, the starter cracks always kinked prior to growing but never indicated a shear mode once growth began.

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It was noted that the kinking or turning of the part through cracks began at the point of maximum depth and then spreading along the entire crack front. During this process, very high values of were measured in the unturned part of the cracks. Once completely turned, the cracks followed slightly curved paths towards the fin axis and the boundary always in pure Mode I. Very recently, we have encountered a very different behavior in scale models of a specific motor grain geometry (Fig. 4). Starter cracks were inserted either along a fin axis, or at the locus of the coalescence of a large fin tip radius with a much smaller radius at the end of a fin edge. The symmetric cracks on the fin axis grew readily under internal pressure but the off-axis cracks initiated at the coalescence of the two radii were very resistant to growth even though fringes around a fin tip with no crack suggested that the points of origin of the off-axis cracks were somewhat more highly stressed than on the fin axis. Three models have been tested to date, each containing cracks in two separate fins separated by uncracked fins. Table I presents the results to date from these tests. An end load was superimposed on the pressure to simulate field conditions. The off-axis crack in Model 2 grew sufficiently to established Pure Mode I around the crack front but the crack was not planar, and it grew on a slightly curved path. The shape of this crack initially and after growth is shown in Fig. 5. The left off-axis crack in Model 4 did not grow sufficiently to escape the shear mode measured during kinking on both cracks. This shear mode appears to be a three dimensional extension of the two dimensional prediction which was made analytically by Cotterell and Rice in 1979 [9]. Rubenstein [10] has suggested that sharp crack kinking occurs only in very brittle materials and that cracks in all other materials undergo a more gradual change in direction controlled by the shear mode and suggests a two dimensional rationale for this behavior. These ideas appear to be borne out especially in the off-axis motor grain cracks. As can be seen in Table I, the symmetric cracks grew readily and two of them penetrated the outer wall of the cylinder.

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When penetration occurred, the pressure was dropped to stress freezing pressure and maintained there throughout the tests.

3. Summary Two model configurations were used to illustrate the complexities involved when measuring SIF’s and crack paths in three dimensional problems. In the first case, all cracks were either planar or grew on slightly curved paths but indicated a radical change in SIF values after removal of the crease or kink. The second model revealed plane extensions of cracks which were located symmetrical both with respect to load and geometry and grew much more readily than off-axis

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cracks even though the latter emanated from a more highly stressed locus. Also, off-axis cracks were much more difficult to grow and were non-planar initially. However, their growth resistance appears related to a shear mode which persisted during the kinking process. Further studies are underway on this latter problem.

4. Acknowledgements The writer wishes to thank the Oak Ridge National Laboratory and Sparta, Inc. (Through the Air Force Research Laboratory) for the financial support for parts of this work. He also wishes to acknowledge the contributions of Dr. Dan M. Constantinescu who conducted the experiments on the motor grain models. Finally, the writer wishes to congratulate Professor Isaac M. Daniel, an esteemed colleague for over 40 years on an outstanding career in the field of Mechanics to whom this symposium is dedicated.

5. References 1. Cotterell, B., “On Brittle Fracture Paths,” International Journal of Fracture Mechanics, Vol. 1, pp. 96-103, (1965). 2. Cotterell, B. “Notes on Paths and Stability of Cracks,” International Journal of Fracture Mechanics, Vol. 2, pp. 526-533, (1966). 3. Schroedl, M. A., McGowan, J. J. and Smith, C. W., “Determination of Stress intensity Factors from Photoelastic Data with Applications to Surface Flaw Problems,” Journal of Experimental Mechanics, pp. 392-399, (1974).

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4. Smith, C. W., Peters, W. H. and Andonian, A. T., “Mixed Mode Stress Intensity Distributions for Part Circular Surface Flaws,” Journal of Engineering Fracture Mechanics, Vol. 13, pp. 615-629, (1979).

5. Smith, C. W., “Crack Tip Stress Fields Under Complex Loads: Application to Pressure Vessel Problems,” International Journal of Pres. Ves. & Piping, Vol. 12, pp. 43-60, (1983).

6. Smith, C. W., “Fracture Parameter Measurements for Cracks Emanating from Curved Surfaces by the Frozen Stress Method,” Proc. of 9th International Conference on Experimental Mechanics, Vol. 5, pp. 1776-1785, (1990). 7. Smith, C. W. and Wang, L., “Stress Intensity Factor Distribution and Crack Shapes in 3-D Problems Using Frozen Stress,” AMD-V37, Advances in Fracture/Damage Models for the Analysis of Engineering Problems, ASME, pp. 109-120, (1992). 8. Smith, C. W., Wang, A. L. and Liu, C. T., “Stress Intensity Distributions and Crack Growth Concepts for a Basic Motor Grain Shape with Application to a Scaled Model Geometry,” Recent Advances in Structural Mechanics: Fracture Mechanics, ASME-PVP-V295 NE-V26, pp 1-14, (1994). 9. Cotterell, B. and Rice, J. R., “Slightly Curved or Kinked Cracks,” International Journal of Fracture, Vol. 16, No. 2, pp. 155-169, (1980). 10. Rubenstein, A. A. “Mechanics of Crack Path Formation,” International Journal of Fracture, Vol. 47, pp. 291-305, (1991).

Appendix - Algorithms Mode I Beginning with the Griffith-Irwin Equations, we may write, for Mode I, for the homogeneous case,

where are stress components, is the SlF, are non-singular stress components. The first terms represent the singular stresses and the second terms represent a Taylor Series Expansion about the crack tip of non singular stresses. Notations refer to Fig. A1. By truncating the second terms to the leading term in the Taylor Series Expansion, evaluating the result along computing the maximum shearing stress in the plane, and normalizing with respect to the result is:

where the apparent SIF and is the maximum shear stress in the plane, and is a constant over the data range. Moreover is proportional to the stress fringe order. Since Eq. 2 predicts a linear variation of vs.

the normalized SIF can be estimated as shown in Fig.

A-2. Mixed Mode The mixed mode algorithm was developed (see Fig. A-3) by requiring that:

which leads to:

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By measuring which is approximately in the direction of the applied load. can be determined. Then writing the stress optic law as:

one may plot

vs.

obtain

may be written as:

Now

as before, locate a linear zone and extrapolate to

to

Knowing and can be determined from Eqs. 4 and 5. Details are found in Smith and Kobayashi (1993). When using the two parameter method for mixed mode analysis, test data are always taken from the forward leading fringe loop (Fig. A-3). In the event that the two parameter model must be expanded for reasonably accurate results.

A MODEL FOR FAILURE INITIATION IN DUCTILE MATERIALS JIANZHENG ZUO, MICHAEL A. SUTTON, XIAOMIN DENG Department of Mechanical Engineering University of South Carolina Columbia, SC 29208

Abstract Initial results from a set of three-dimensional unit cell simulations clearly demonstrate the nature of the relationship between local constraint and local failure initiation processes. The simulations show that the deformations are substantially different under high and low levels of stress constraint; the effective plastic strain at failure initiation is quite small under high triaxial stress constraint, increasing rapidly as constraint decreases. Through extensive parametric studies, the numerical results demonstrate that two-parameters—triaxial stress constraint and effective plastic strain—are sufficient for development of a three-dimensional ductile fracture criterion that is consistent with void growth concepts. Additional support for the proposed criterion is provided by experimental evidence that indicates the effective plastic strain at failure initiation is a monotonic function of stress constraint.

1.

Introduction

Ductile fracture of metals and alloys has been observed to be the result of nucleation, growth and coalescence of microscopic voids. The process of ductile fracture can be divided into three stages. The first stage consists of the nucleation of micro-voids that initiate at inclusions and second phase particles in matrix, when the deformation in the matrix has elevated the matrix/particle interface stress to the level where the particles either crack or de-bond from the matrix [1-4]. The second stage consists of the dilatational and extensional growth of the micro-voids. The process continues until the inter-void matrix reaches its maximum load state and plastic deformation is localized [5-7]. This stage depends upon the applied triaxial stress constraint [8-10]. The third stage consists of the failure of the inter-void matrix, across a sheet of micro-voids, which results from microscopic necking and localized plastic failure [11-13]. Experimental, analytical and numerical results [8, 14-16] show that the main macroscopic parameters influencing microscopic void nucleation, growth and coalescence, and hence ductile fracture, are triaxial stress constraint and effective plastic strain. Hancock and Mackenzie [14] experimentally studied the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress states and evaluated the effect of triaxial stress constraint factor on the average plastic strain to failure initiation by use of Bridgman’s equation. Subsequently, Walsh et al [16] investigated the ductile fracture of 2134 type Al alloys. Results indicated the effects of triaxial stress constraint on void nucleation and void growth are quite 217 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 217–224. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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different; the rupture ductility of material was observed to decrease with increasing triaxial stress constraint. In the study of three-dimensional ductile fracture of materials, there are two major approaches in the literatures. One approach utilizes macroscopic crack tip parameters (e.g., the J-Integral [17], the HRR singular fields [18, 19], mixed mode COD [20, 21]) to predict crack growth. Another approach is based on the constitutive model initially proposed by Gurson [22], improved by Tvergaard [12, 13], and extended by Needleman and Tvergaard [23-25] to account for the rapid loss of carrying capacity during void coalescence by incorporating the effects of void nucleation, growth and coalescence. Different kinds of void nucleation models have also been proposed to supplement the Gurson model [26, 27]. To include the effects of initial void shape, void shape evolution on void coalescence, the extended Gurson model has been proposed by employing numerical and micro-mechanical analyses under axi-symmetric conditions [28, 29]. Though this contribution to the Gurson model makes it more robust, it also makes the Gurson model more complicated making it difficult to determine the influence of various effects, such as (a) micro-void coalescence and (b) micro-void interaction and (c) micro-void size and shape under different levels of triaxial stress. Based on results obtained through a series of unit cell analyses investigating the effect of triaxial stress constraint on effective plastic strain to failure initiation, we present initial findings related to the development of a "unified" three-dimensional ductile fracture criterion. In Section 2, the basic framework is outlined, with special emphasis on the inter-relationship of macroscopic effective plastic strain and constraint. In Section 3, the numerical analysis of the unit cell model used in the paper is presented and results presented. In Section 4, a brief discussion of the results is provided, with emphasis on the significance of the findings.

2. A model for failure initiation As noted above, metallurgical research has clearly established that the link-up of micro-voids results in the local failure of a ductile material. The link-up of microvoids occurs when the inter-void matrix can no longer support the equilibrium loads that are being applied by the stress and plastic strain-rate fields. In order to determine the critical load at which the link-up of micro-voids occurs, the plastic limit-load model was proposed [6]. This model suggests that the plastic limit-load stress corresponds to the stress at which plastic slip-lines undergo incipient link-up between two adjacent voids, a concept that seems appropriate for a nonstrain hardening or perfectly plastic material. However, in general, the inter-void matrix may not break upon initiation of localization between adjacent voids, with the applied stress increasing due to strain hardening effects. Another model for the link-up process [30] assumes the existence of a critical plastic strain, within the remaining material so that the inter-void matrix fails when the magnitude of minimum plastic strain in the localized plastic band reaches Conceptually, the critical plastic strain should be dependent on material properties, stress state and void volume fraction. For voided materials, suppose is the initial void volume fraction in the matrix, the macroscopic (some refer to these as meso-scopic) strains and stresses in the study of the cell model (matrix +void) were defined based on previous work [31]. Previous numerical studies have shown that the relationship of macroscopic effective stress to macroscopic effective plastic strain will be a function of

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constraint [32]. Furthermore, as shown in Figure 1, experimental results [15, 16] for notched specimens for three low-alloy steels show that the effective plastic strain, at failure initiation is a monotonically decreasing function of the level of constraint, where the constraint parameter is in which is the hydrostatic mean stress.

Denoting

as the failure plastic strain for a constant stress constraint one has the following inequality based on results in Figure 1;

Thus, the functional relationship should have the form shown in Figure 2, where the dashed vertical line denotes the maximum value of The decrease in beyond its maximum in Figure 2 is generally associated with a significant increase in damage (e.g., void initiation and growth). Based on these results, the following local failure initiation criterion for ductile materials is proposed. Failure initiation occurs when the macroscopic effective stress, reaches its maximum in the loading history under a constant level of constraint defined by Using the hypothesis that there exists a one-to-one relationship between the critical plastic strain in the remaining material, and macroscopic constraint, recent work [30] has shown that there exists a unique functional relationship at failure initiation. Conversely, assuming that there exists a unique functional relationship at failure initiation, it can be shown [30] that this

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implies a unique functional relationship that is similar to the form shown in Figure 1. A schematic of the key result is shown in Figure 3.

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In the following section, unit-cell simulations for a model material are described. The results provide quantitative data for the functional form of while demonstrating that the form of the computed functional is consistent with existing data.

3. Numerical studies using unit-cell model The model of a porous ductile material to be studied here is defined by a threedimensional periodic array of spherical micro-voids in a strain-hardening elasticplastic matrix. Figure 4 shows the geometry of a model. The initial void radius and the void spacing are and respectively. Denoting the initial void volume fraction as then

In these studies, the model is subjected to a prescribed tensile or compressible loading in the direction(s) resulting in a deformed, three-dimensional array of voids that remains periodic.

Specifically, the conditions imposed on the 1/8 model are (a) zero traction rates in all three directions along the void boundary, (b) zero normal displacement rate and zero shear traction rates along the three planar surfaces through the origin and (c) specified normal displacement rates and zero shear traction rates along the three planar surfaces through the point The specified normal displacement rates are defined to simulate different levels of constraint. Since deformations at failure initiation are quite large, (a) the plastic strain components are much larger than the elastic strain components so that (b) the macroscopic strain rate in the direction is defined to be the ratio of the

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specified displacement rate to (c) the macroscopic applied traction in the direction on the three planar surfaces through is defined to be the integrated average over the surface. All analyses were performed using the general-purpose commercial finite element program ANSYS Version 5.7. Three-dimensional ten-node tetrahedral elements and finite deformations were used in all analyses. The uniaxial true stresslogarithmic strain curve for the matrix material is represented by the piecewise power law

where is the initial uniaxial yield stress, E is the Young’s modulus and is the strain hardening exponent. Properties for the matrix material used in this work are and In order to investigate the effect of initial void volume fraction on the failure initiation, the analyses were performed for several values of ranging from 0.005 to 0.08; in this work, results for are reported. To obtain the function form for a series of analyses were performed with a range of applied values for and Link-up of the periodic voids was assumed to occur when a plastic strain contour of magnitude extends from the void boundary to an outer boundary of the model in Figure 4. By symmetry, the contour will extend to a neighboring void also. This process gives initiation values for for each value of selected. By plotting for each critical strain, the outer envelope of the data can be obtained; the envelope represents the unique set of stresses for volume fraction, Figure 5 presents results from our numerical analyses for the functional form of at failure initiation for void volume fraction Similar results were obtained for and 0.005, with the magnitude of the mean stress at initiation increasing with decreasing In addition to determining the functional form for at failure initiation, the values for and at failure initiation were also determined. Figure 6 presents the computed critical plastic strain versus constraint at failure initiation. A direct comparison of Figures 1 and 6 indicates that they have the same trends; decreasing with increasing Furthermore, the trends are consistent with the conclusion derived from experimental results of notched specimens [15]. Though not shown, similar results were obtained for and 0.005, with increasing as decreases.

4. Conclusions Using previous experimental and analytical results that show the critical plastic strain at failure initiation in a ductile material is a function of constraint, a stressbased failure initiation criteria is proposed that has the potential for use in threedimensional ductile fracture studies. Based on the assumption that failure initiation

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occurs at a maximum effective stress under conditions of constant constraint, a failure initiation criterion of the form is demonstrated through numerical simulations employing a three-dimensional unit cell model under different levels of constraint. In addition, simulation studies confirm that the functional form of is consistent with experimental observations. Thus, the results from the study indicate that two parameters, and can also be used as the basis for developing improved fracture criteria, such as COD or CTOA, by including the effects of constraint on the critical value.

5. References 1. 2. 3. 4.

Goods, S.H. and Brown, L.M. (1979) The nucleation of cavities by plastic deformation, Acta Metall. 27, 1-15. Argon, A.S., Im, J. and Safoglu, R. (1975) Cavity formation from inclusions in ductile fracture, Metall. Trans. 6A, 825-837. Argon, A.S. and Im, J (1975) Separation of second phase particles in spheroidized 1045 steel, Cu-0.6%Cr alloy, and maraging steel in plastic straining, Metall. Trans. 6A, 839-851. Brown, L.M. and Embury, J.D. (1974) The initiation and growth of voids at second-phase particles, in Proc. 3rd Int. Conf. Strength of Metals and Alloys, London, pp. 164-169.

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5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31. 32.

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Thomason, P.F. (1981) Ductile fracture and the stability of incompressible plasticity in the presence of micro-voids, Acta Metall, 29, 763-777 Thomason, P.F. (1985) Three-dimensional models for the plastic limit-loads at incipient failure of the inter-void matrix in ductile porous solids, Acta Metall, 33, 1079-1085 Thomason, P.F (1990) Ductile Fracture of Metals, Pergamon Press, Oxford Rice, J R . and Tracey, D M . (1969) On the ductile enlargement of voids in triaxial stress fields, J. Mech. Phys. Solids, 17, 201-217. Fleck, N.A., Hutchinson, J.W. and Tvergaard, V. (1989) Softening by void nucleation and growth in tension and shear, J. Mech. Phys. Solids, 37, 515-540 Huang, Y. (1991) Accurate dilation rates for spherical voids in triaxial stress fields, J. Appl. Mech., 58, 1084-1086 Hayden, H.W. and Floreen, S. (1969) Observations of localized deformation during ductile fracture, Acta Metall., 17, 213-214. Tvergaard, V. (1981) Influence of voids on shear band instabilities under plane strain conditions, Int. J. Fracture, 17, 389-407 Tvergaard, V. (1982) On localization in ductile materials containing voids, Int. J. Fracture, 18, 237-251. Henry, B.S. and Luxmoore, A.R. (1997) The stress triaxiality constraint and the Q-value as a ductile fracture parameter, Eng. Frac. Mech., 57, 375-390. Hancock, J.W. and Mackenzie, A.C. (1976) On the mechanisms of ductile failure in highstrength steels subjected to multi-axial stress-states, J. Mech. Phys. Solids, 24, 147-169 Walsh, J.A., Jata, K.V. and Starke Jr., E.A. (1989) The influence of Mn dispersoid content and stress state on ductile fracture of 2134 type Al alloys, Acta. Metall., 37, 2861-2871. Rice, J.R. (1968) A path independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 35, 379-386. Hutchinson, J.W. (1968) Singular behavior at the end of a tensile crack in a hardening material, J. Mech. Phys. Solids, 16, 13-31. Rice, J.R. and Rosengren, G.F. (1968) Plane strain deformation near a crack tip in a power law hardening material, J. Mech. Phys. Solids, 16, 1-12. Ma, F., Deng, X., Sutton, M.A. and Newman, J.C.Jr (1999) A CTOD-based mixed-mode fracture criterion, ASTMSTP 1359 on mixed mode crack behavior, 1359, 86-110. Sutton, M.A., Deng, X., Ma, F., Newman, J.C. Jr. and James, M. (2000) Development and application of a COD-based mixed mode fracture criterion, Int. J. Fracture, 37, 3591-3618. Gurson, A.L. (1977) Continuum theory of ductile rupture by void nucleation and growth, Part I — Yield criteria and flow rules for porous ductile media, J. Eng. Mater. Tech., 99, 2-15. Needleman, A. and Tvergaard, V. (1984) An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids, 32, 461-490. Tvergaard, V. and Needleman, A. (1984) Analysis of the cup-cone fracture in a round tensile bar, Acta metal., 32, 157-169. Needleman, A. and Tvergaard, V. (1987) An analysis of ductile rupture modes at crack tip, J. Mech. Phys. Solids, 35, 151-183. Chu, C.C. and Needleman, A. (1980) Void nucleation effects in biaxially stretched sheets, J. Eng. Mater. Tech., 102, 249-256. Saje, M., Pan, J. and Needleman, A. (1982) Void nucleation effects on shear localization in porous plastic solids, Int. J. Fracture, 19, 163-182. Gologanu, M.., Leblond, J.B. and Devaux, J. (1993) Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities, J. Mech. Phys. Solids, 41, 1723-1754. Pardoen, T. and Hutchinson, J.W. (2000) An extended model for void growth and coalescence, J. Mech. Phys. Solids, 48, 2467-2512 Zuo, J., Sutton, M.A. and Deng, X. (in review) Basic studies of ductile failure processes and implications for fracture prediction. Bishop, J.F.W. and Hill, R. (1951) A theory of the plastic distortion of a polycrystalline aggregate under combined stresses, Phil. Mag., 42, 414-427. Kuna, M. and Sun, D.Z.. (1996) Three-dimensional cell mode) analyses of void growth in ductile materials, Int. J. Fracture, 81, 235-258.

CRACK PATHS IN ADHESIVE BONDS

LESLIE BANKS-SILLS AND JACOB SCHWARTZ The Dreszer Fracture Mechanics Laboratory Department of Solid Mechanics, Materials and Systems The Fleischman Faculty of Engineering Tel Aviv University 69978 Ramat Aviv, Israel

Abstract The reliability of a joint subjected to mechanical and thermal loads during processing and service constitutes a major technical problem. Joints contain flaws. The observed strength of a joint depends upon the location and size of the flaws, as well as the crack path through the joint. The aim of this investigation is to examine the path of a crack in an adhesive bond. Sandwich Brazilian disk specimens made of two aluminum adherends joined by a thin layer of epoxy are employed in the testing. This specimen allows for a wide range of mixed mode loading. A paraffin pre-crack is located within the adhesive layer. During testing, all cracks divert from within the layer and grow toward and into the interface. Comparison of crack path direction is made to two theories. 1. Introduction There have been many studies which have considered crack path direction in homogeneous materials. These investigations begin with works of Cotterell [1,2]. In the first study, Cotterell shows analytically that the sign of the T-stress determines crack path stability; for a positive T-stress he crack will deviate from its original path, whereas if it is negative, the crack will continue along a straight path. In the second study, Cotterell observes that this criterion does not correctly predict path stability for the compact tension specimen. For this specimen T > 0, yet experience shows the path to be stable. Cotterell and Rice [3] showed that a curved or kink crack will continue to diverge from the main crack direction when T > 0. The angle diverges further as increases. For T < 0, the kink tends toward the main crack direction. Other researchers considered the problem of crack instability. Leevers et al [4] carried out experiments with a biaxially loaded specimen. For negative T-stress values, cracks were stable. The positive T-stress values they were unstable. But the compact tension specimen, examined again (Leevers and Radon [5]; Sumi, et al. [6]) remains a problem for the T-stress criterion. Selvarathinam and Goree [7] extended the Cotterell and Rice [3] model and calculated the T-stress at a small branch of the main crack in various directions in an isotropic material. They define a value which is a material 225 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 225–234. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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parameer and which they obtain from the fracture tests. If the crack path is unstable. If it is greater than it is stable. Fleck et al. [8] developed a criterion for crack path direction of a crack in a joint. Accordingly, a stable path within an adhesive can exist only if a pure mode I crack path exists and if T < 0. The crack seeks the direction of Thus, it is possible to consider the sign of the derivative of with respect to its position within the layer to determine crack path. Investigations of the fracture toughness of bonded joints are numerous. Only those studies in which attention was paid to crack path are mentioned here. Cao and Evans [9] tested four specimen types to obtain a wide range of mixed mode deformation. Both glass and aluminum adherends were glued by means of a thermoplastic adhesive. They observed that crack extension within the adhesive occurred only when there was mode I deformation. For mixed modes, the crack always propagated along the interface. Brazilian disk specimens with aluminum, steel, brass and plexiglass adherends glued by epoxy were employed by Wang and Suo [10]. A starter notch was placed between one of the adherends and glue, creating an interfacial pre-crack between the two materials. In that investigation, asymptotic relations for a thin layer between two adherends were employed to determine the critical interface energy release rate and phase angle A wide range of mode mixities was attained. It was found for small values of the phase angle that the crack sometimes propagated within the adhesive and, for these specimens, values were higher than those obtained when the shear component of loading was somewhat increased and the crack propagated along the interface between the epoxy and metal. Three types of sandwich specimens were employed by Akisanya and Fleck [11] to obtain values over a wide range of phase angles and to investigate the dependence of crack path on mode mixity. The specimens they employed were the Brazilian disk specimen and symmetric and asymmetric double cantilever beam specimens. The adherends were aluminum joined by a layer of epoxy. A notch was placed between the epoxy and the upper adherends. Different crack paths were obtained by changing the amount of residual stress present in the epoxy layer. In the double cantilever beam specimens, four different crack paths were obtained. The most common was along the interface. For the Brazilian disk specimens, either the cracks ran along the interface, or jumped to an opposite interface and then propagated. Sandwich scarf joint specimens made of mild carbon steel adherends glued by means of epoxy were tested by Wang [12]. The layer was placed at six different angles relative to the loading direction. In each specimen, a notch was located in the middle of the epoxy layer. As opposed to other investigations, most cracks propagated within the layer. In this investigation, the crack path in an epoxy bond between two aluminum adherends is considered. The Brazilian disk specimen exhibited in Fig. 1 is employed in the testing. The pre-crack is placed nearly in the center of the bond and experiments are carried out for various mixed mode ratios. The specimens and tests are described in Section 2. Calculated stress intensity factors and the T-stress are presented in Section 3. In Section 4, the crack path direction for each specimen is presented, together with crack path predictions. These are based on two theories: (1) the T-stress for bonds

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(Fleck, et al. [8]) and (2) the maximum tangential stress criterion of Erdogan and Sih [13]. These are compared to test results.

2.

Specimens and Testing

The Brazilian disk specimen was chosen in this investigation because of its ability to produce a wide range of mode mixity with one geometry. Two aluminum disks (A17075-T7351) were bonded together with the epoxy EPON-815 (produced by Shell) containing 26% hardener HY-491. This is a brittle epoxy. Some material properties are given in Table 1.

A special mold and device to form the notch were employed; these will be described elsewhere. The epoxy layer thickness was between 0.45 mm and 0.90 mm. A paraffin strip 0.12 mm thick was employed to form the notch. The notch length was mm and was placed nearly in the center of the epoxy layer. The epoxy was polymerized for 72 hours at 25°C. The specimen radius and thickness were nominally 20 mm and 8 mm, respectively. An example of a specimen is exhibited in Fig. 2. Tests were carried out with an Instron (model no. 1341) uniaxial loading machine. The specimen was placed in a special loading device which enables accurate measurement of the loading angle in Fig. 1. A load/crack sliding displacement record

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was made and the load at fracture was measured for each specimen. Since our interest here is the crack path, these will be reported elsewhere.

Four possible crack paths exist for a sandwich specimen including propagation within one of the adherends, the layer, the interface or an alternating path. Since the epoxy layer is much weaker than the aluminum adherends, the crack will not propagate within the aluminum. Crack paths were examined for eight specimens. For six, the crack ran immediately within the interface; for two, it began to run within the glue for a short distance (tenths of a millimeter) and then went into the interface. These latter two were at loading angles of and For these angles, there is predominately crack opening deformation. A similar phenomenon was observed by Wang and Suo [10] in which some interface starter cracks ran into the adhesive. In this investigation, after fracture, all specimens appear as in Fig. 3.

Two crack path criteria are examined. The first was presented by Fleck et al. [8] for a sandwich structure and is based upon the T-stress. The second is the maximum tangential stress criterion of Erdogan and Sih [13] for mixed mode cracks in

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homogeneous materials. For the first criterion, the sign of the T-stress and the derivative are considered; the length c is illustrated in Fig. 4. Thus, both the T-stress and stress intensity factors are required.

3.

Stress Intensity Factor and T-stress Calculations

The stress intensity factors and T-stress were calculated by means of the finite element program ADINA (Bathe [14]) and conservative integrals. The M-integral (Yau et al. [15]) was employed to calculate and and the Eshelby method (Kfouri [16]; Chen et al. [17]) was employed to calculate the T-stress. It is clear that there are two components to the T-stressL one from the mechanical loading denoted as T and one from the residual stresses. The total T-stress is given

by

where is the residual stress in the x-direction referring to Fig. 4. Recall that polymerization of the specimens takes place at 25°C. The residual stress may be calculated by elementary strength of materials considerations or a finite element analysis. Results are presented in Table 2 for the eight specimens. The loading angle and the layer thickness h are presented. The temperature at which each test was carried out is denoted by For each specimen, the mechanical T-stress is calculated numerically for the load and crack length at failure. A finite element mesh containing 24,688 eight nodded isoparametric elements and 74,244 nodal points was employed to determine the displacement field for each specimen. There were ten elements through the layer thickness. Convergence was examined with a finer mesh. It may be observed that the total T-stress is negative for most specimens. Next, the stress intensity factors are determined as a function of crack location c within the layer. A typical layer thickness mm and normalized crack length a/R were chosen to study. The same finite element mesh described previously for loading angles of 0°, 5° and 10° is employed here. The crack position is varied such that Values of the normalized stress intensity factors and are illustrated in Figs. 5 and 6 respectively.

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The normalized stress intensity factors are given by

where a is half crack length,

P is applied load, R and t are specimen radius and thickness, respectively. The T-stress is normalized as

and, for completeness, is presented in Fig. 7.

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It may be observed in Figs. 5 and 6 that the stress intensity factors are not a function of normalized distance c/h from the lower adherend (see Fig. 4). For all loading angles, that is, is nearly constant regardless of the relative position of the crack within the epoxy layer. On the other hand, the T-stress is rather sensitive to crack position within the layer, especially for loading angles and 10°. In fact, for and c/h > 0.75, the sign of the T-stress changes from negative to positive.

4.

Crack Path Criteria

From the earlier work of Cotterell [1,2] and Cotterell and Rice [3], it was determined that a positive T-stress leads to unstable crack propagation, whereas a negative T-stress leads to stable crack propagation. It has been seen, however, that positive T-stress does not always imply an unstable crack path (see Leevers and Radon [5], for example). Thus, one may add a condition, that the crack searches for a position in which Thus, the sign of is also of interest.

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Before considering the behavior of specific specimens, a summary of the experimental results is presented in Table 3. For each specimen, the sign of is taken from Table 2. The sign of the derivative is taken from the analyses with layer thickness and normalized crack length For all specimens, recall that layer thickness 0.45 mm < h < 0.9 mm. The sign of for is taken from the calculations of and that of from Although from Fig. 6, it appears that is constant, one may see from the numerical values that there are slight changes in different ranges of c/h, so that may be positive, negative or zero. If for example one considers specimen No. 167 for which the loading angle with a layer thickness mm, and MPa > 0. This would imply from the criterion of Fleck et al. [8] that the crack will begin to propagate along the layer centerline; but the positive T-stress will cause it to be unstable. This is precisely what occurred. Next, consider specimens No. 170 and 165. Here but For both specimens, the crack is situated nearer to the lower adherend. Decreasing c implies decreasing which would appear to imply that the crack runs into the lower interface, although the value of never approaches zero. Since is nearly constant with respect to crack position within the layer, this criterion does not seem suitable for determining crack path for the majority of these specimens.

It may be noted further that if crack path stability is analyzed by means of the Tstress only as outlined earlier for homogeneous specimens, negative T implies a stable crack path. Analysis of the homogeneous Brazilian disk specimen for 0.35 reveals that the T-stress is negative for loading angles Only for 0° is crack propagation self-similar.

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Next, the maximum tangential stress criterion of Erdogan and Sih [13] is considered. This criterion was developed for homogeneous material. Since the crack is within the homogeneous layer, it will be applied here. It is assumed that the layer is sufficiently thick and brittle so that it predicts crack propagation direction. According to this criterion, the crack will propagate in a radial direction measured from the crack tip satisfying the relation

For each of the specimens, and at failure were determined and substituted into Eq. (5). Two values of are found. By obtaining the maximum value of one is eliminated. The values of for each specimen considered here are presented in Table 4. It may be observed that for all loading angles except the crack propagation angle This implies that the crack runs into the lower interface as shown in Fig. 4. Thus, the general crack path direction is predicted by this criterion. Unfortunately, crack direction angles were not measured. Also note that the absolute value of the crack propagation angle increases as the loading angle 6 increases.

5.

Discussion and Conclusions

The crack path direction was considered for bonded sandwich Brazilian disk specimens. A pre-crack was placed nearly in the middle of the bond. For most loading angles, the crack propagated into the lower interface. For two loading angles in which at failure, the crack began to propagate within the glue, but then ran into the lower interface. This short propagation did not appear to change the critical energy release rate at failure for these specimens. Tests by Wang and Suo [10] showed an increase in fracture toughness when the crack propagated within the adhesive.

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The T-stress criterion proposed by Fleck et al. [8] for cracks in bonded joints does not appear appropriate for the Brazilian disk specimen since is nearly constant with respect to position of the crack within the bond. The maximum tangential stress criterion of Erdogan and Sih [13] predicted well the general direction of crack propagation. Further experiments are required to measure the exact direction of crack propagation in order to compare to this theory. Tests with other specimens would be useful, as well. 6.

Acknowledgement

We would like to thank Mr. Rami Eliasi for his assistance with the finite element calculations and Dr. Victor Fourman for his assistance with specimen fabrication and testing. 7. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

References Cotterell, B., Notes on the paths and stability of cracks, International Journal of Fracture Mechanics, 2, (1966), 526-533. Cotterell, B., On fracture path stability in the compact tension test, International Journal of Fracture Mechanics, 6, (1970), 189-192. Cotterell, B. and Rice, J. R., Slightly curved or kinked cracks, International Journal of Fracture, 16, (1980), 155-169. Leevers, P. S., Randon, J. C. and Culver, L. E., Fracture trajectories in a biaxially stressed plate, Journal of the Mechanics and Physics of Solids, 24, (1976), 381-395. Leevers, P.S. and Radon, J. C., Inherent stress biaxiality in various fracture specimen geometries, International Journal of Fracture, (1982), 19, 311-325. Sumi, Y., Nemat-Nasser, S. and Keer, L. M., On crack path stability in a finite body, Engineering Fracture Mechanics, 22, (1985), 759-771. Selvarathinam, A. S. and Goree, J. G., T-stress based fracture model for cracks in isotropic materials, Engienering Fracture Mechanics, 60, (1998), 548-561. Fleck, N. A., Hutchinson, J. W. and Suo, Z., Crack path selection in a brittle adhesive layer, International Journal of Solids and Structures, 27, (1991), 1683-1703. Cao, H. C. and Evans, A. G., An experimental study of the fracture resistance of bimaterial interfaces, Mechanics of Materials, 7, (1989) 295-304. Wang, J. S. and Suo, Z., Experimental determination of interfacial toughness curves using Brazilnut-sandwiches, Acta Metallurgica, 38, (1990), 1279-1290. Akisanya, A. R. and Fleck, N.A., Brittle fracture of adhesive joints, International Journal of Fracture, 58, (1992), 93-114. Wang, C. H., Fracture of interfacial crack under combined loading, Engineering Fracture Mechanics, 56, (1997), 77-86. Erdogan, F. and Sih, G. C., On the crack extension in plates under plane loading and transverse shear, Journal of Basic Engineering, 85, (1963), 519-527. Bathe, K. J., ADINA – Automatic Dynamic Incremental Nonlinear Analysis System, Version 7.3, Adina Engineering, Inc., (1999), USA. Yau, J. F., Wang, S. S. and Corton, H. T., A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity, Journal of Applied Mechanics, 47, (1980), 335-341. Kfouri, A. P., Some evaluations of the elastic T-term using Eshelby's method, International Journal of Fracture, 30, (1986), 301-315. Chen, C.-S., Krause, R., Petit, R. G., Banks-Sills and Ingraffea, A. R., Numerical assessment of Tstress computation using a p-version finite element method, International Journal of Fracture, 107, (2001), 177-199.

EXPERIMENTAL DETERMINATION OF FRACTURE PARAMETERS FOR PREDICTING CRACK GROWTH IN VISCOELASTIC POLYMERS

D. H. ALLEN, J. J. WILLIAMS Texas A&M University Dept. of Aerospace Engineering College Station, TX 77843

Abstract

A micromechanical model for a viscoelastic cohesive zone has been previously formulated and based on a continuum mechanics model of the damage zone ahead of the crack tip in a polymer solid. The scale of the cohesive zone model is quite small, thus rendering it difficult to obtain the cohesive zone parameters experimentally. Presented herein is an experimental procedure for measuring the crack tip opening, damage zone profile ahead of the crack tip, crack tip opening rate, critical fibril breaking length and diameter. It is then shown that these parameters can be used to characterize the viscoelastic cohesive zone model. 1. Introduction

In the past several researchers have developed experimental procedures for determining physical parameters near crack tips. Kramer et al. [1,2,3,4] have worked with thin films of polymers bonded to annealed copper grids. These grids were used to support the polymers while the whole structure is stressed and plastically deformed. The stiffness of the copper grid holds the stressed polymers so that they can then be placed in a scanning electron microscope. While the grid does hold the deformations of the polymers in place so that the developed fibrils and other cohesive zone parameters are viewable this procedure does not allow for in-situ real-time examination of the development and evolution of the different crack tip parameters. Bradley et al. [5] developed a unique technique in which polymeric composites are marked by an electron beam to create a grid of trackable dots. The dot mapping technique is described as a grid of dots that are produced by burning a whole in the gold-palladium sputter coating with electron beam concentration. In essence the sputter coating leaves a layer of surface contamination that can be utilized without effecting the characteristics of the material. The size of the hole or dot can be controlled to work well with desired strain fields. This technique could prove useful for digital tracking of particular areas on a specimen during real-time experimentation. Pandya and Williams [6,7] have developed another experimental technique for studying medium and high-density pipe grade polyethylene stress crack properties. 235 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 235–244. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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These experiments use a notched sample with an initially square cross sectional area. The samples are turned on a lathe and a circular cross sectional area is notched out with a new area to original area ratio of 1:10. The samples are tested on an Instron MTS machine at varying rates, from rather rapid rates of 50 mm. per min to nearly quasistatic rates of 0.005 mm. per min. After the samples pass peak loads the samples are sliced open so that the cross sections could be examined by an electron microscope. A vice type clamp has been developed to reopen the samples during viewing. This reopens crack tips and extends fibrils. Once again this method is postmortem viewing, but has proven to be a good method for gaining quality images of cohesive zone parameters. Sue [8] developed a method for the study of fracture toughness measurements in polymeric alloys. This method consists of using a four-point double notched sample where forces are applied a distance of L apart on one side and L/3 apart on the other. On the side of the forces separated by a distance of L, the notches are milled and precracked at a distance of L/9 apart. Testing samples in this manner allows for pure mode I failure. Ultimately one crack will fail before the other. This leaves a very nicely developed cohesive zone in front of a crack tip. These samples can then be thinly sliced and examined both optically and in an electron microscope. They produce excellent views of the damage in front of the crack tip and details like crack tip opening and fracture toughness. Unfortunately, once again the details are all examined postmortem and details about the evolution of the crack tip and cohesive zone are not revealed. Allen and Searcy [9,10] have postulated a micromechanical model. This is based on a continuum mechanics model of the damaged region ahead of the crack tip with a zone of non-zero tractions ahead of the crack tip. This model is particularly beneficial due to its ability to incorporate history and rate dependence in the critical energy release rate in viscoelastic media. Because of the extremely small scale of many cohesive zones, it is very difficult to experimentally determine parameters to be used in the model. In the past the cohesive zone properties have been postulated in order to make some qualitative predictions. Although the resulting predictions have been somewhat successful, it is important to experimentally determine what the actual cohesive zone geometry is. 1.1 THE MICROMECHANICAL MODEL Allen and Searcy developed the micromechanical model described below in a traction, T(t), equation 1 [9,10]. This utilizes a nonlinear and history-dependent tractiondisplacement relation, where is the viscoelastic relaxation modulus of the undamaged material, are the opening displacements of the cohesive zone also in the local coordinates of the macroscale crack, and are the components of a material constant representing the fracture mode mixity.

In this equation is the Euclidean norm of the opening displacements. It is defined in equation 2, where and are measured directly from experimental data. These

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represent the crack opening displacements due to pure mode I, mode II, and mode III failure respectively. and are material-specific length scale parameters that are scaled according to the experimental results.

In this model is the area of free space between fibrils. Under opening displacements this area of free space will increase as the mean fibril diameter decreases. Fibril failure occurs when a fibril’s cross-sectional area falls below a critical value, The damage parameter, is related to the representative volume element (RVE). In these experiments A is equal to the planform area of the RVE and thus equal to the length of the cohesive zone multiplied by the sample thickness. is the planform area of the pth fibril, and P is the number of fibrils, see Figure 1. A, and P will all be measured directly from the experimental data.

2.

Fibrils

Many thermoplastic materials, especially those with higher molecular weights, will tend to undergo extensive drawing and plastic deformation. This drawing causes molecular orientation in the direction of or parallel to the drawing. The orienting of these long chain molecules results in the material being stronger in this direction. This is because the covalent bonds within the polymer chains are significantly stronger than that of the van der Waal’s forces between whole chains [11].

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Suresh explains that cyclic deformation or advancing fatigue failure is due to nucleation, growth and breakdown of crazes [13]. If we go to a smaller scale it can be seen that within the craze zone the material is being drawn out more extensively in one direction and it tends to fibrillate, see Fig. 4. It is important to note that the craze zone will be normal to the uniaxial tensile stress within the material and the fibrils will be parallel to it. As the fibrils reach their maximum length or critical breaking length, they each individually fail. This is the propagation of the crack tip. The voids and holes produced during fibrillation within the craze can reduce the material volume by as much as 50% or more from that of the parent matrix. [14]

It is important to also discuss the size and length of the fibrils. Fibrils will vary in length and size depending on the material. Certain thermoplastics, like polystyrene, produce very small fibrils, on the order of 200 to 400 Å. [15] Other materials may produce fibrils much larger in scale. For example, rubber cement tends to fibrillate on the order of millimeters. 3.

Experiments and Analytical Procedures

The following section is a combination of many of the procedures and research that has been performed in the past with some new ideas for producing not just images of cohesive zones and fibrils, but real time video of crack tip evolution. The following describes two ways of producing video images of cracks on two different length scales. The first is performed with optical microscopy. This is useful for larger crack tips and cohesive zones. The second is the same experiment, but performed inside a Environmental Scanning Electron Microscope (ESEM), and produces images of much smaller cohesive zones and fibrils.

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3.1 EXPERIMENTAL PROCEDURE In order to create a cohesive zone ahead of the crack tip a compact tension test is performed. The same type of material is tested twice in order to produce data on two different length scales. Compact tension tests are performed under a stereomicroscope and also in an environmental scanning electron microscope (ESEM). The optical experiments produce data larger than the wavelength of light and the ESEM tests will produce data that are smaller than the wavelength of light. The small scale of all the parameters that are measured in this experiment require the highest amount of precision possible. All measurements are on the micron and sub-micron level. Because of the small scale, the slightest movement in the crack causes the points of interest to move out of the field of view. Therefore, a precision tension rack must pull symmetrically from both sides. In both optical and ESEM experiments the samples are under tension and experience displacements equally on both sides of the sample. All the specimens are cut in accordance with the latest ASTM standard D5045. Specifically, the standards for compact tension samples are used. In the optical experiments the compact tension samples are displaced with the Melles Griot Nanomotion II actuators. These motors are capable of steps as small as ten nanometers. The Panasonic digital cameras receive the image and send it through a color sync and stereoscopic multiplexer. Only the video from one eyepiece is sent to a video recorder through S-video output. Crack tip opening, and damage zone profile is viewable through the stereomicroscope. All of the videos are recorded on videotape and then digitized through a Belkin USB Videobus II device. A similar experiment is performed in the Environmental Scanning Electron Microscope (ESEM). In these experiments a tension stage was custom built to perform compact tension experiments inside the ESEM. The ESEM has stepper motors that drive a shaft connected to gears that drive two counter threaded screws. This is the actuation that displaces two clamps that are connected to the samples. Ultimately, the same compact tension experiment that was performed optically previously is now being viewed on a much smaller length scale. Of course the polymers will limit magnification. Electron beam concentration has been known to melt or deform polymers at high magnification levels if the beam focus is left in one spot for too long of a period of time. This very property may be utilized to our advantage if the tracking of points on the video proves difficult. In the same manner in which Bradley demonstrated the grid tracking method on gold-palladium sputter coating we may create a grid with a fine beam concentration on the polymers. The electron beam could be concentrated to a very fine point and briefly turned on. This will develop a grid of points that may be more easily tracked in the video. The experiments are recorded on VHS tape and edited in the same software and hardware that is used in the optical experiments. The benefit of using the digital camera and computer to do this is that the computer can add up to another 50x or more of magnification to the images. Theoretically infinite magnification is possible, but in reality the number of pixels produced by the camera limits image magnification. A high-resolution video camera may allow for better digital magnification and image

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quality. From these images, measurements of the crack tip and cohesive zone are made. The video imaging is a key for the determining the time dependent measurements. 3.2 OPTICAL EXPERIMENTAL EQUIPMENT The optical experiments are less expensive due to the great cost of an Environmental SEM. It is important to note that the magnification of the stereomicroscope is not the actual final magnification. The camera’s images pass through the multiplexer and then are recorded on the video camera through its S-video input. This changes the final magnification of the image. Of course the initial magnification of the image is reliant on the power and quality of the microscope objective.

The strain stage was designed to be symmetric, thus allowing equal and opposite forces to be applied on each of the ends. This is necessary to keep the crack tip from moving out of the field of view of the cameras. The motors push a translation stage that slides the ends of the samples apart. The translation stages are built to a very high tolerance, and even though there is a small moment created by the separation of the axis of motor extension and the axis of testing, it is negligible. Drivers for the Nanomotion II motors are provided by Melles Griot, unfortunately they did not provide satisfactory control for the experiments. The generic drivers provided were used as a foundation for a custom program. Most importantly the ability to start and stop the motion of the motors simultaneously was added to the drivers. The load cell output was also added to the driver, along with the ability to run cycles of load for fatigue testing. The driver was written in LabVIEW 6i and programmed to output a text file of the load and time every second of the test. Of course, the acceleration and velocity of the motors is changeable within the program, as well as the frequency of data points for the output.

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3.3 ESEM EXPERIMENTAL EQUIPMENT The ESEM experiments required a special tension stage to be built in order to apply a load to a sample while under the electron beam, and can be seen in figures 5 and 6. This stage was built out of 316 steel. 316 steel has very low magnetic qualities, and thus will reduce the amount of interference the metal will have with the electron beam. The Electroscan ESEM has stepper motors with drive shafts built into the door of the microscope. These motors are normally used to rotate or tilt a sample pedestal. This drive shaft is connected to the tensile stage gearbox and drives the opening or closing of the grips. The amount of tensile or compressive force that can be applied to a sample is limited by the amount of torque the motors can produce. At this point the controlling of the motors is limited to an on/off fixed speed.

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3.4 CALIBRATION OF MOTORS Calibration of the stepper motor displacement is an important factor in the accuracy of the tests. To do this the microscope is focused on the tip of one of the stepper motors. In the same manner in which the crack tips are monitored, the motor is monitored as it is extended. The computer is used to tell the motor how much to displace, and then this is compared to the actual amount of displacement. Actual displacement is measured with a caliper micrometer. Once this is done for both motors, it will determine the accuracy and percent error of the experiment. The video taken of this will also allow for the determination of the pixel accuracy. For the ESEM tests, the calibration is performed with a caliper micrometer. Each rotation of the drive shaft of the tensile stage is equivalent to a certain amount of travel of the grips of the stage. This travel is measured with the micrometer. As the computer rotates the drive shaft of the grips, the rotations of the shaft are counted, thus producing strain data for the tests. In both test, the same load cell is used. The manufacturing company, Transducer Techniques, publishes the calibration and accuracy of the load cell. The load cell produces excellent results, and is highly accurate.

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3.5 DATA ACQUISITION AND ANALYSIS Data Acquisition for these experiments comes in two parts. The Load cell provides highly accurate data about the load being applied to the sample. Granted there is some translation or movement of the load cell as it is compressed, but the movement of the load cell is several orders of magnitude smaller than the translation of the stage, and therefore is considered to be negligible. The same load cell will be used for both optical and ESEM experiments. The strain of the samples is determined through the video imaging. The video of the samples under tension causes the deformation, and this is recorded on videotape. This tape is then digitized on the computer, and the net change in length of the sample fibrils, crack tip opening, etc. is determined through pixel counting from frame to frame of the video. The video image produced is 3.33 inches by 2.5 inches, or 320 by 240 pixels, as viewed on screen. This provides a resolution of one pixel, which is approximately 0.01 inches on screen. When magnification of the stereomicroscope is at a maximum, the field of view is approximately of an inch high (2.5 inches). That translates to one pixel (0.01 inches on screen) representing inches or a magnification of 150x. Unfortunately, the field of focus is very limited with the stereomicroscope at that magnification. When this same technique is used with the ESEM, the magnification is significantly higher, on the order of thousands of x. 3.6 CONCLUSION In conclusion, this experiment is proving to be a useful way to gather geometric measurements of specific cohesive zone parameters in viscoelastic materials. From the video collected the crack tip opening, damage zone profile ahead of the crack tip, crack tip opening rate, critical fibril breaking length and diameter can all be measured. This information is critical in characterizing and using the viscoelastic cohesive zone model. However, it is not valid for all materials. It has proven to be very difficult, even with this procedure, to produce quality video of fibrils and cohesive zones. As previously mentioned, fibrils in certain materials are extremely small, and may not be seen with this procedure. Highly ductile, viscoelastic materials tested at quasi-static velocities seem to produce the best results. The next step in this experiment is to develop an effective method for fatigue testing of materials. Fatigue testing of samples seems to develop more pronounced or definitive cohesive zones. The more pronounced the cohesive zone is the easier it will be to produce a video of cohesive zone ahead of a propagating crack tip. 4.

References

1. Wang, W. V., Kramer, E. I, 1982, “The micromechanics and microstructure of CO2 crazes in polystyrene,” Polymer, Vol. 23, pp. 1667-1674.

2. Wang, W. V., Kramer, E. J., 1982, “A distributed dislocation stress analysis for crazes and plastic zones at crack tips,” Journal of Material Science, Vol. 17, pp. 2013-2026.

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6. 7.

8.

9. 10. 11. 12.

13. 14. 15.

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Yang, A. C.M., Kramer, E. J., Kuo, C. C, and Phoenix, S. L., 1986, “Craze Fibril Stability and Breakdown in Polystyrene,” Macromolecules, Vol. 19, pp. 2010-2019. Washiyama, J., Creton, C., and Kramer, E.J., 1992, “TEM Fracture Studies of Polymer Interfaces,” Macromolecules, Vol. 25, pp. 4751-4758. Corleto, C.R., Bradley, W.L., Brinson, H.F., 1996, “An experimental micromechanics measurement techniques for submicrometre domains,” Journal of Material Science, Vol. 31, pp. 1803-1808. Pandya, K.C., and Williams, J.G., 2000, “Measurement of Cohesive Zone Parameters in Tough Polyethylene,” Polymer Engineering and Science, Vol. 40, pp. 1765-1776. Pandya, K.C., and Williams, J.G., 2000, “Cohesive zone modelling of crack growth in polymers Part 2-Numerical simulation of crack growth,” Plastics Rubber and Composites, Vol. 29, pp. 447452. Sue, H.J., 1991, “Study of Robber-Modified Brittle Epoxy Systems. Part 1: Fracture Toughness Measurements Using the Double-Notch Four-Point-Bend Method,” Polymer Engineering and Science, Vol. 31, pp. 270-288. Allen, D.H. and Searcy, C.R., 2001a “A Micromechanically-Based Model for Predicting Dynamic Damage Evolution in Ductile Polymers,” accepted by Mechanics of Materials, 2001. Allen, D.H. and Searcy, C.R., 2001b “A micromechanical model for a viscoelastic cohesive zone,” International Journal of Fracture, Vol. 107, pp. 159-176. Hull, D., 1999 “Fractography observing, measuring and interpreting fracture surface topography,” Cambridge University Press, pp219-257. Hibbs, M.F. and Bradley, W.L., 1987 “Correlations Between Micromechanical Failure Processes an the Delamination Toughness of Graphite/Epoxy Systems,” Fractography of Modern Engineering Materials: Composites and Metals, ASTM STP 948, pp. 17, 68-97. Suresh, S, 1991 “Fatigue of Materials,” Cambridge University Press, pp. 456-498. Hertzberg, R.W., 1987 “Fracture Surface Micromorphology in Engineering Solids,” Fractography of Modern Engineering Materials: Composites and Metals, ASTM STP 948, pp. 5-36. Beahan, P., Beavis, M. and Hull, D. 1971 “The Morphology of Crazes in Polystyrene,” The Philosophical Matazine, Vol. 24, No. 192, pp. 1267-1279.

FAILURE OF SPOT WELD: A COMPETITION BETWEEN CRACK MECHANICS AND PLASTIC COLLAPSE

YUH J. CHAD Department of Mechanical Engineering University of South Carolina Columbia, SC 29208

Abstract

Spot weld made by resistance welding has been widely used in joining sheet metal of auto body since 1950’s. Every modern car has over 2000 spot welds. Failure of the spot weld is therefore an important concern to auto body durability and safety design. Spot weld can fail in two completely distinct modes, namely, nugget pullout failure and interfacial failure. In this paper, we show that the nugget pullout failure is caused by plastic collapse and the interfacial failure is governed by crack or fracture mechanics. These two failure mechanisms compete each other and failure of a spot weld occurs as one of the failure criterions is first satisfied. Test data from General Motor Corporation, Daimler-Chrysler, The Welding Institute and the University of South Carolina are used to validate the theoretical prediction. Recommendation is made for minimum weld nugget size for a given sheet metal thickness so that nugget pullout failure, the acceptable mode of failure in industry, is ensured. 1.

Introduction

Spot weld made by resistance welding has been widely used in joining sheet metal of auto body since 1950’s. A modern vehicle typically contains 2000 to 5000 spot welds. For a given sheet metal thickness, correct choice of weld size is critical. A sub-sized spot weld may present inadequate strength under overload or crash scenario and reduced fatigue life under normal operation of the vehicle. An over-sized nugget requires large scale welding machinery and thus higher cost to fabricate. Various industry standards recommend the optimum or minimum size of the spot weld for a given sheet metal thickness. For example, American Welding Society (AWS), Society of Automotive Engineering (SAE) and ANSI [1], together, recommend a weld nugget diameter for steels where d and t are the weld nugget diameter and thickness, respectively, in mm. A minimum nugget diameter and nominal nugget diameter where d and t are in inches, are often used by auto industry [2] as well. Note that all these recommended formulae in industry standards are empirical nature derived from extensive experimental testing. An interesting point for these empirical formulae is that they are independent of the varying properties of the materials to which they are applied, as long as the material of the sheet metal is steel. We will show later in the paper that this is indeed the case. 245 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 245–256. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Spot weld can fail in two markedly different modes. As shown in figure 1, the fracture by crack propagation through the weld nugget, often occurring in smaller weld for a given sheet thickness, is called the interfacial failure. In larger weld, nugget pullout failure may occur in which the weld nugget is pulled from one piece of the sheet metal leaving a circular hole on the other sheet. Interfacial failure, which has a less load carrying capacity, is considered unsatisfactory and industry standards [1,2] are often designed to avoid its occurrence.

Detailed stress analysis of spot weld was recently published by Radaj [3] and Zhang [4]. Failure mechanisms in spot weld under monotonic load were studied by Zuniga and Sheppard [5]. Mixed mode testing and analysis was performed by Lin, et al [6]. Smith [7] used a novel, yet simple, approach to investigate the competition between the two failure modes of spot weld. Test data from industry are presented by Kim, et al [2], Lee, et al [8] and VandenBossche [9].

In the strength or weldability test of spot welded joints, lap shear sample or cross tension sample, as shown in figure 2, are often used. The test on lap shear sample is to

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provide the shear strength and the test on cross tension sample the tensile strength of the joint. In the current paper, we analyze the failure of the cross tension sample. The approach by Smith [7] was followed and extended. The two failure modes, i.e. interfacial and nugget pullout, are first studied individually using fracture mechanics and plastic collapse, respectively. The competition of these two failure modes is then investigated. Based on this analysis, minimum weld nugget diameter for a given sheet metal thickness to ensure a nugget pullout failure, as contrast to interfacial failure, is obtained. Experimental data are provided to verify the analytical formula. It is further shown that the results can be applied to lap shear sample test data and a variety of steels as well. Finally, discussion relative to current industry standards for sizing the weld nugget is given. 2.

Governing Equations for Interfacial Failure

Interfacial failure of spot weld, as demonstrated in figure 1, is essentially crack propagation or fracture of weld nugget under mode I conditions using the terminology in fracture mechanics. A precise relation, which links the applied load to a stress

intensity factor essential to a fracture mechanics study, for such a geometry is not readily available. Smith [7] adopted the stress intensity factor relation from Tada, et al [10], as shown in figure 3, for the spot-weld geometry. In figure 3, t is used to represent the sheet thickness, d the weld nugget diameter and P the remotely applied tensile load to the weld. The functional relationship between the applied load P and the stress intensity factor is then

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Note that the function is highly non-linear with respect to t and d, which renders a simple relation of the nugget diameter to sheet thickness impossible. Smith [7], however, observed that the function when plotted against is nearly linear as shown in figure 3. As such, Smith then chose an approximate, linearized form which is also plotted in figure 3. It can be seen that this approximation is fairly accurate especially in the range to which most of the spot weld in auto body applies. Incorporating this approximation, equation (1) becomes

At failure, the stress intensity factor equals to the critical stress intensity factor or fracture toughness. And, equation (2) becomes

where is the fauilure load and material. 3.

is the fractrure toughness of the weld nugget

Governing Equations for Nugget Pull-Out Failure

In steel, a spot weld nugget typcailly has a hardness value two to three times of the base metal. As a consequence, mechanically, the nugget functions like a rigid button embedded in a ductile metal sheet. The pullout failure of spot weld in cross tension

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sample is thus predominantly plastic shear or plastic collapse around the circumference of the weld nugget. The stress distribution for such a geometry is very complex. A simplified engineering analysis is recently proposed by Chao [11] which assumes a rigid weld nugget and considers only the predominant (shear) stress around the nugget. As shown in figure 4, using the assumed stress distribution, the far field applied tensile load is related to the local stresses by

where P is the applied tensile load and is the maximum shear stress at the weld nugget. At the intitiation of fracture, equation (4) becomes

where 4.

is the failure load and

is the material’s fracture stress in shear.

Competition of the Two Failure Modes

Now we have two distinct failure modes, interfacial mode governed by equation (3) and nugget pullout mode governed by equation (5). For a given spot weld, the failure mode will depend on which condition is met first. As demonstrated in figure 5, for a given

sheet thickness the failure would be interfacial if the nugget diameter is small and pullout if it is large enough. The changeover takes place as equation (3) equals to equation (5) or at the intersection of the two curves shown in figure 5. Equating (3) to (5) and letting yields

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where is the critical weld nugget diameter for a given sheet thickness. The fracture criterion of the spot weld can therefore be stated as, for a given sheet thickness t, interfacial failure would occur if the diameter and nugget pullout would occur if Note equation (6) contains two material properties, fracturing stress under shear and fracture toughness By properly varying the welding schedule to achieve different weld diameter, running the tensile failure test, and recording the failure load, can be determined using equation (5) and test data from those spot welds fail by nugget pullout, which is generally true for large weld nugget. can be determined using equation (3) and test data from those welds failed by interfacial mode, which is generally true for small weld nugget. If the exact stress intensity factor equation (1) is used in conjunction with (5), the critical weld nugget diameter for a given sheet thickness is then obtained as

5.

Test Data

Rivett of The Welding Institute preformed a series of test to study the failure of spot weld [12] on a cold rolled mild steel, 1.18 mm thick, tensile yield stress and UTS (Ultimate Tensile Strength). The carbon equivalent content was 0.12. The test data (failure load and nugget diameter) containing interfacial and pullout as well as mixed failure are reproduced and plotted in figure 6.

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Using mm, KN from the data in figure 6 and equation (5), the fracturing stress in shear is determined as Using mm, KN from figure 6 and equation (3) the fracture toughness is calculated as The critical weld nugget diameter is therefore, using equation (6),

As mm, the critical nugget diameter is mm per (8) and accordingly the failure load is 4.36 KN using either equation (3) or (5). These critical nugget diameter and failure load are included in figure 6 which clearly distinguishes the two failure modes. It is noted that the fracture toughness value is obtained as if the exact equation (1) is used. The critical nugget diameter for this particular material using equation (7) is then

Both equation (8) and (9) are plotted in figure 7. Note that (9) is very complex in d since d is also embedded in the function in addition to the appearance at the left hand side of the equation. To plot (9) in figure 7, MathCAD was used. On the other hand, equation (8) is, much simpler than (9) and the difference between the two is negligible as shown in figure 7. For practical applications, the approximate formula (8) is more than adequate and therefore recommended. To further validate the model, test data from Ewing, et al [2] are reproduced and plotted in figure 8. Shown in figure 8 are from two materials; SAE-1006 (a mild steel, Galvanized, 1.3 mm thick, 252 MN/m2 tensile yield stress and UTS) and

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SAE-960X (a high strength steel, Galvanized, 1.4 mm thick, tensile yield stress and UTS). Test results from static and impact tests on three standard spot weld test sample geometries, i.e. lap-shear, cross-tension and coach-peel, are reported [2]. The data in figure 8 are from figure 9 of Ewing, et al [2] and are those obtained from cross tension sample, quasi-static and low rate impact testing. In designing the weld, Kim, et al [2] used a nominal welding schedule to produce the “nominal” weld, shown as schedule #2. The welding schedule was intentionally changed to a higher current to produce an over-sized weld nugget, reported as schedule #3, and a lower current to produce an under-sized weld nugget, as schedule #1. All welds from schedule #1 failed by interfacial mode, while welds from both schedules #2 and #3 fail by nugget pullout mode, under quasi-static as well as impact loading for all test sample geometries. Following the same procedure used for Rivett’s data, we obtain the fracturing stress in shearing as using KN from figure 8 and equation (5), and the fracture toughness as using 2.60 KN and equation (3). The critical weld nugget diameter using equation (6), is

therefore

Note that two materials are together here because the failure loads are nearly identical as shown in figure 8. Using mm, the critical nugget diameter is therefore 5.5 mm per (10) and accordingly the failure load is 6.16 KN using either equation (3) or (5). These critical nugget diameter and failure load are included in figure 8, which, again, clearly distinguishes the two failure modes.

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Discussion

6.1 FAILURE MODE TRANSITION IN LAP-SHEAR SAMPLE GEOMETRY The analysis for the failure mode transition from interfacial to nugget pullout discussed so far in this paper is for the cross tension sample geometry. In practice, spot welds are subjected to other types of loading, such as remote shear. An obvious question is therefore “Is equation (6), adequate in predicting failure under other types of loading ?” To answer this question, we compare our prediction with test data from lap-shear samples in this section.

Vanden Bossche [9] of Daimler-Chrysler performed a series of test to study the failure of spot weld using lap-shear test samples. Fifteen different steel sheet materials with yield strength ranging from 206 MPa (30 ksi) to 655 MPa (95 ksi), and thickness from 0.64 mm to 2.26 mm were tested. The critical nugget diameters in twenty test groups are obtained and plotted in figure 9 as well as the predictions based on equations (8) and (10). As shown in figure 9, the prediction is remarkably good. The comparison shown in figure 9 demonstrates that equations (8) and (10) can also be used in predicting the failure mode transition from nugget pullout to interfacial for both tensile and shear loads of spot weld. This conclusion is further corroborated by the test results from Ewing, et al [2], in which all welds fabricated using welding schedule #1 (#2 and #3) failed under interfacial (nugget pullout) mode in all lap-shear, cross tension and

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coach peel sample geometries. Furthermore, similar failure modes are observed under impact conditions for all these sample geometries [2]. 6.2 MATERIAL DEPENDENCE OF THE PREDICTION It is observed from figure 9 that predictions from equations (8) and (10) are very close to each other, despite that they are derived from three different steels. Furthermore, test data from VandenBossche [9] contain fifteen different steel sheet materials with yield strength ranging from low carbon mild steel, e.g. 206 MPa (30 ksi) to very high strength steel, e.g. 655 MPa (95 ksi). It may be concluded from this comparison that the prediction based on (6) is weakly dependent upon the grades of steels. As such, either (8) or (10) can be used for most of the steels. This conclusion could be due to that the effect of material property is eliminated through the term in equation (6). In fact, this conclusion coincides with the industry practice, i.e. the recommended empirical formulae in various industry standards are independent of the varying properties of the materials [1,2].

6.3 FURTHER COMPARISON WITH TEST DATA AND WITH INDUSTRY STANDARDS Shown in figure 10 are the predictions from the current work (equations (8) and (10)), the recommended spot weld size from American Welding Society/Society of Automotive Engineering/ANSI [1], minimum/nominal weld diameters from [2], and test data from Rivett [12], Ewing, et al [2] and the University of South Carolina. Solid

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symbols represent nugget pullout and hollow symbols interfacial failure, unless otherwise specified. In some cases, one data point in figure 10 represent more than ten test samples since an average nugget diameter is often used for a batch of test samples having the same welding schedule. Test data from USC are under either quasi-static or impact conditions and from either cross-tension or lap-shear sample geometry made of low carbon to high strength steels. Figure 10 shows that the prediction using the results from this work is excellent. When compared with industry standards, figure 10 indicates that while these industry standards are relatively conservative in predicting the failure mode for thin gage steel sheet, i.e. t < 1.2 mm, they are not sufficient to predict the interfacial failure for thick steel sheet, i.e. t > 1.2 mm. 7. Conclusion

An analysis combining the fracture mechanics and plastic collapse is presented to predict the failure mode of resistance spot weld. The analytical results are validated through comparison with available test data. Spot weld under either tensile (cross tension samples) or shear (lap-shear samples) load is considered. Excellent prediction is obtained. In addition, through comparison it is concluded that the prediction may be applied to many different grades of steels ranging from low carbon mild steel to high strength steel as well as under either quasi-static or impact loading conditions. When compared with industry standards for sizing weld nugget, the result obtained in this paper provides a simple, more accurate and better prediction for the transition of failure mode from pullout to interfacial. With further validation, e.g. more test data in thin ( t < 1.0 mm) gage steels, the results, e.g. equations (8) or (10), could be recommended for sizing the spot weld in steel sheet for auto industry. 8. Acknowledgement

The author acknowledges the partial support of this work by NSF through grant CMS0116238 and the encouragement from the Program Director, Dr. Kenneth P. Chong. Valuable discussion with Dr. P. C. Wang of General Motor Corporation is specially noted. 9. References 1.

2. 3. 4. 5.

6.

Section 5.7, Weld Button Criteria, Recommends Practices for Test Methods for Evaluating the Resistance Spot Welding Behavior of Automotive Sheet Steel Materials, ANSI/AWS/SAE/D8.997, An American National Standard (1997). Ewing, K. W., Cheresh, M., Thompson, R., and Kukuchek, P., “Static and Impact Strengths of Spot-Welded HSLA and Low Carbon Steel Joints,” SAE Paper 820281 (1982). Radaj, D., “Stress Singularity, Notch Stress and Structural Stress at Spot-Welded Joints,” Engineering Fracture Mechanics, 34(2), (1989), 495-506. Zhang, S., “Approximate Stress Intensity Factors and Notch Stresses for Common Spot-Welded Specimens,” Welding Journal, (1999), 173-179. Zuniga, S. and Sheppard, S.D., “Resistance Spot Weld Failure Loads and Modes in Overload Conditions,” Fatigue and Fracture Mechanics: Volume, ASTM STP 1296, R.S. Piascik, J.C. Newman, and N.E. Dowling, Eds., American Society for Testing and Materials, (1997) 469-489. Lin, S.H., Pan, J., Wu, S., Tyan, T., and Wung, P., “Spot Weld Failure Loads under Combined Mode Loading Conditions,” SAE 2001-01-0428 (2001).

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Smith, R.A., “Sizing of Spot Welds by Elastic/Plastic Analysis”, Fracture, Radon, J.C., London, (1980) 49-56. 8. Lee, Y., Wehner, T., Lu, M., Morrissett, T., Pakalnins, E., “Ultimate Strength of Resistance Spot Welds Subjected to Combined Tension and Shear,” Journal of Testing and Evaluation, 26(3), (1998), 213-219. 9. VandenBossche, D.J. “Ultimate Strength and Failure Mode of Spot Welds in High Strength Steels,” SAE 770214 (1977) 10. Tada, H., Paris, P. and Irwin, G. The Stress Analysis of Cracks Handbook, 25.5, Del Research Corp., St. Louis, Missouri. (1973) 11. Chao, Y.J., “Failure of Spot Weld under Tensile, Shear and Mixed Tensile and Shear Loads” manuscript in preparation. (2001) 12. Rivett, R.M. Welding Institute Research Bulletin, 20, (1979), 235-239.

INVESTIGATING THE EFFECTS OF SPECIMEN THICKNESS AND PRESSURE ON THE CRACK GROWTH BEHAVIOR OF A PARTICULATE COMPOSITE MATERIAL

C. T. LIU AFRL/PRSM 10 E. Saturn Blvd. Edwards AFB CA 93524-7680

Abstract

In this study, the effects of specimen thickness and confined pressure on the crack growth behavior in a particulate composite material, containing hard particles embedded in a rubber matrix, were investigated. The experimental data were analyzed and the results are discussed. 1.

Introduction

An important engineering problem in structural design is evaluating structural integrity and reliability. It is well known that structural strength may be degraded during its design life due to mechanical or chemical aging, or a combination of these two aging mechanisms. Depending on the structural design, material type, service loading, and environmental condition, the cause and degree of strength degradation due to the different aging mechanisms differs. One of the common causes of strength degradation is the result of crack development in the structure. When cracks occur, the effects of crack sizes and the rate of growth on the fracture resistance of the material need to be investigated. In recent years, a considerable amount of work has been done studying crack growth behavior in particulate composite materials under different loading conditions at ambient pressure [1-4]. This work was based on linear fracture mechanics. The principles of classical fracture mechanics are well established for single-phase materials. However, experimental evidence indicates that linear fracture mechanics theories have been applied to particulate composite materials with varying degrees of success. In this study, pre-cracked specimens were used to study crack growth behavior in a particulate composite material, containing hard particles embedded in a rubbery matrix, under a constant strain rate condition at ambient and 8697 kPa confined pressure. The effects of specimen thickness and pressure on crack growth behavior was investigated and the results are discussed. 257 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 257–266. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The Experiments

In this study, single-edge notched tensile specimens made from polybutadiene rubber embedded with hard particles were used in crack propagation tests. The specimens were 1.0 in. wide, 3.0 in. high and the thicknesses of the specimens were 0.2in., 0.5 in., 1.0 in., and 1.5 in. The geometry of the specimen is shown in Fig. 1.

Prior to testing, a 0.76 cm. crack was cut at the edge of the specimen with a razor blade. The pre-cracked specimens were tested at an 8 cm/cm/min strain rate under ambient and 8697 kPa confined pressures. During the tests, a video camera was used to monitor crack growth. The raw data obtained from the tests were the crack length a; the time, t; and the load, p, corresponding to the measured crack length. These raw data were used to calculate the crack growth rate, da/dt, and the Mode I stress intensity factor . The recorded experimental data, a, t, and p, were used to calculate the Mode I stress intensity factor, and the crack growth rate, da/dt. In calculating the stress intensity factor, for a given set of values of a and p, a nonlinear regression equation, which relates the normalized stress intensity factor, to the crack length, a, was used. The values of for different crack lengths were determined from the ABAQUS computer program. In calculating the crack growth rate, da/dt, the polynomial method was used. In the polynomial method, a high order polynomial function was selected to fit the crack length versus time data, and the crack growth rate was computed by taking the derivative of crack length with respect to time at a given time. To avoid the timeconsuming process of data reduction, a computer program was written to calculate and da/dt.

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Results and Discussion

It is well known that, on the microscopic scale, a highly filled polymeric material can be considered an inhomogeneous material. When these materials are stretched, the different sizes and distribution of filled particles, the different crosslink density of polymeric chains, and the variation in bond strength between the particles and the binder can produce highly nonhomogeneous local stress and strength fields. Depending on the magnitude of the local stress and the local strength, damage can be developed in the material, especially near the crack tip region. The damage developed in the material may be in the form of microvoids or microcracks in the binder or dewetting between the binder and the filler particles. Damage growth in the material may occur as material tearing or as successive nucleation and coalescence of the microcracks. These damage processes are time dependent and are the main factor responsible for the time sensitivity of strength degradation as well as the fracture behavior of the material. A typical photograph showing the crack surface during the earlier stage of crack growth under ambient pressure is shown in Fig. 2. Experimental results indicate that crack tip blunting takes place both before and after crack growth. The material at the tip of the crack suffers very large elongation and is nearly straight. The highly strained or damaged zones extend ahead of the crack tip, appearing as an equilateral triangle with the crack tip as its base. This damage zone is known as the failure process zone, which is a key parameter in viscoelastic fracture mechanics [5-6]. When the local strain reaches a critical value, small voids are generated in the failure process zone. Due to the random nature of the microstructure, the first void is not restricted to the surface where the maximum normal strain occurs. Since the tendency of the filler particle to separate from the binder under a triaxial loading condition is high, it is expected that voids, or a damage zone, will also be generated in the specimen’s interior. Consequently, there are a large number of strands, essentially made of binder material, which separate the voids that form inside the failure process zone.

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As the applied strain increases with time, material fracture occurs at the blunted end of the crack tip. This will always be the location of the maximum local strain. The failure of the material between the void and the crack tip causes the crack to grow into the failure process zone. This kind of crack growth mechanism continues until the main crack tip reaches the front of the failure process zone. When this occurs, the crack tip resharpens temporarily. The above paragraphs discusses the damage and crack growth mechanisms when the specimen is strained under ambient pressure. The formation of voids in the material, especially in the highly damaged zone near the crack tip, is a typical damage mechanism observed at ambient pressure. In order to obtain a fundamental understanding of the effect of confined pressure on the crack growth behavior, the effects of pressure on damage initiation and evolution processes need to be investigated. The results of a preliminary study are shown in the following paragraph. The effects of confined pressure and applied strain on volume dilatation, which is due to the formation of voids in the material, in smooth specimens, are shown in Fig. 3. It is seen that the volume dilatation decreases as the confined pressure is increased. At 8697 kPa confined pressure, the volume dilatation approaches to zero. This phenomenon is mainly due to the suppression of the formation of voids and increase the debond stress at the interface between the filler particles and the matrix. Under this condition, it is conjectured that, instead of the development of voids, micro cracks are developed in the matrix material. The number of micro cracks increases with increasing applied strain, and eventually, a macro crack is formed as a result of the coalescence of the micro cracks. The propagation of the macro crack leads to the fracture of the specimen. Experimental data revealed that for specimens without pre-crack, micro cracks started to develop approximately at 30% applied strain and the number of the micro cracks increased significantly at 40% applied strain (Fig.4) and specimens fractured at 50% applied strain as a result of the fast propagation of a long macro crack.

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The above paragraph describes the damage mechanisms in smooth specimen under confined pressure. In the following paragraphs, the crack growth behavior in pre-crack specimen under ambient and 8697 kPa confined pressure is discussed. Under ambient pressure, experimental data reveal that the crack start to propagate when the applied stress reaches a critical value, which is in the neighborhood of the maximum stress, as shown in Fig. 5. Once the crack starts to grow, the high crack growth rate results in a fast fracture of the specimen. In other words, unstable crack growth occurs as soon as the crack starts to grow. This type of crack growth behavior is similar to the brittle fracture observed in metals. Therefore, in this study, there is no crack growth analysis conducted under ambient pressure, and only the critical Mode I stress intensity factor, for the onset of crack growth is calculated. The results of analysis are shown in Fig. 6. From Fig. 6, it is seen that the variations of among different specimen thicknesses are within experimental scatter. Therefore, as a first approximation and for engineering applications, it can be assumed that for the onset of crack growth is independent of specimen thickness. A similar conclusion can be draw for under 8697 kPa confined pressure as shown in Fig. 7. It is known that one of the most important practical aspects of linear elastic fracture mechanics for single-phase materials is the concept of the plain strain value of the critical stress intensity factor, This is because this value is perceived to be the minimum critical value and that, for a given material and test environment, it exhibits a constant value to within a certain experimental scatter. This value is usually obtained from a tensile test where the specimen size is adjusted to produce sufficient thickness for the through-the-thickness cracked specimen to induce a brittle fracture. Under this condition, the crack front will bow in the direction of the crack growth, creating a thumbnail shape. This suggests that there is a plane strain constraint in the center portion of the specimen, which diminishes near the side boundary. However, this may not be true for highly filled polymeric materials. Experimental data obtained from crack

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propagation tests on highly filled polymeric material specimen revealed that the crack front exhibited no “thumbnailing” both before and after growth. In other words, the crack front was ideally straight, but with local irregularities. The straight crack front observed in solid propellants is believed to be due to the development of a highly damaged zone at the crack tip. The development of the highly damaged zone together with the straight crack front suggests that, within the highly damaged zone, the transverse constraint is very small and that the plane strain fracture toughness does not exist for these materials. The independent of to the variation of specimen's thickness indicates that the highly filled particulate composite material behaves differently from metals, and concepts that appeared clear in one instance are only found to contradict physical reality in another.

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In order to obtain a fundamental understanding of the effect of damage at a crack tip on the fracture toughness, and also to determine damage initiation and evolution processes near the crack tip, three-dimensional numerical modeling analyses were conducted, based on a micro-macromechanical approach [7]. A detailed description of the numerical model and micro-macromechanical analysis are shown in Reference 7. Therefore, only the relevant results are presented in the following paragraph. The initiation and evolution of damage at the crack tip near the center and near the surface of the specimen were determined and the results are shown in Fig. 8. It is noted that damage initiated earlier near the center than near the surface of the specimen. Since the damage state is closely related to the stress state, the difference in the damage initiation processes is due to the differences between the stress states near the center and near the surface of the specimens. It is known that near the center of the specimen, the stress state is close to the plane strain stress state as a result of relatively high constraint developed near the center of the specimen. However, near the surface of the specimen, the stress state is close to plane stress conditions. It is known that under a triaxial loading condition, it is relatively easy to develop microcracks in the binder and/or debonds at particle-binder interface. Therefore, it is expected that damage will initiate earlier near the center of the specimen. When the material is damaged, the stiffness, the magnitude of the stress, and the constraint in the damaged region will be reduced. Consequently, redistribution of the stresses occurs and the material adjacent to the damaged material will be subjected to a higher stress that, in turn, will induce damage to the material. These stress redistribution and damage evolution processes continue, and eventually all of the material in the immediate neighborhood of the crack is damaged, i.e., the thickness of the damaged material at the crack tip is equal to the thickness of the specimen. Under this condition, the constraints, both in-plane and out of plane, become insignificant. The uniform distribution of the damage along the crack front will result in a uniform distribution of stress. Since the crack growth behavior is controlled by the

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local stress at the crack tip, the uniform distribution of stress along the crack front will also result in a relatively straight crack front, as observed experimentally.

Referring back to Fig. 8, it is seen that in the early stage of damage evolution, the damage rate, expressed as the damage parameter per the applied strain, or the slope of the damage parameter versus the applied strain curve, is higher near the center than that near the surface of the specimen. However, the damage rate near the center of the specimen starts to decrease when the applied strain reaches 3% and the opposite is true near the surface of the specimen. Finally, damage saturation occurs near the center and the surface of the specimen at 6% applied strain level, resulting in a uniform distribution of damage at the crack tip and a straight crack front. In order to investigate the effect of damage at the crack tip on the distribution of along the crack front a three-dimensional linear elastic finite element analysis was conducted by Liu [8]. The results indicate that if there is no damage at the crack tip, the value of at the center of the specimen is about 12% higher than that near the surface of the specimen. However, when through-thickness damage occurs at the crack tip, the distribution of along the crack front is uniform. Since crack growth rate is control by it is expected that a constant along the crack front will lead to a straight crack front as mentioned before. A typical plot of the stress–strain curve of pre-crack specimen under 8697 kPa confined pressure is shown in Fig. 9. It is noted that, unlike ambient pressure test, the crack starts to grow in the linear region of the stress-strain curve and a considerable amount of stable crack growth occurs after the crack propagates. Under this condition, we calculated the crack growth rate, da/dt, and the Mode I stress intensity factor, and determined the relationship between da/dt and The results are discussed in the following paragraphs.

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The determination of the crack growth rate requires an analysis of discrete data relating the instantaneous time, t, to the corresponding crack length, a. Due to the nonhomogeneous nature of the particulate composite material, the measured data shows a considerable scatter. Therefore, it is anticipated that a smooth and steadily increasing relationship between the crack growth rate and time is difficult to obtain, and the different methods of da/dt calculation may result in different solutions [ 9]. Therefore, when selecting a method to calculate the crack growth rate from the raw experimental data, the accuracy and the scatter that introduced into the calculated crack growth rate by the selected data reduction method should be considered. As pointed by Liu (9), the secant method introduces a pronounced fluctuation of da/dt whereas the polynomial method results in a continuous smooth crack growth velocity curve. It is important and interesting to note that the fluctuation of da/dt, calculated by the secant method, is consistent with experimental observation. Based on experimental evidence, in general, the crack does not grow in a continuous and smooth manner. During the crack growth process, crack growth rate both accelerates and decelerates. Therefore, the secant method appears to provide the best estimate of both the actual crack growth process and the actual crack growth rate. However, it was found that the crack growth rate calculated by the secant method oscillates approximately around the smooth crack growth rate curve, obtained by the polynomial method. Therefore, if one is interested in knowing the average crack growth behavior, the fluctuation in crack growth rate seems unimportant and the polynomial method can be use to calculate the crack growth rate. It is based on this argument the polynomial method was used to calculate the crack growth rate in this study. A typical plot of crack growth rate da/dt versus the stress intensity factor for initial crack lengths of 0.1 in and 0.3 in. is shown in Fig. 10. From Fig. 10, a power law relationship exists between log (da/dt) and log which can be mathematically expressed as:

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Conclusions

In this study, the effects of specimen thickness and confined pressure on damage mechanisms and crack growth behavior in a particulate composite material were investigated. Experimental results indicate that, the critical Mode I stress intensity factor, for the onset of crack growth is insensitive to the specimen's thickness. Therefore, on the first approximation, it can be assumed that is independent of specimen’s thickness and plane strain fracture toughness does not exist for this material. Experimental results also indicate that brittle fracture occurs under ambient pressure, whereas a considerable amount of stable crack growth occurs under 8697 kPa confined pressure. Under the stable growth condition, a power law relationship exists between the crack growth rate and the Mode I stress intensity factor. 5.

References

1.

Beckwith, S.W. and Wang, D.T (1978)., Crack Propagation in Double-Base Propellants,” AIAA Paper 78-170 Liu, C.T. (1990) Crack Growth Behavior in a Composite Propellant with Strain Gradients - Part II, Journal of Spacecraft and Rockets, 27, pp. 647-659. Liu, C. T. (1990) Crack Propagation in a Composite Solid Propellant, Proceedings of the Society of Experimental Mechanics, Spring Conference, pp. 614-620. Liu, C.T. and Smith, C.W.,(1996) Temperature and Rate Effects on Stable Crack Growth in a Particulate Composite Material, Experimental Mechanics, 36 (3), pp. 290-295. Schapery, R.A. (1973) On a Theory of Crack Growth in Viscoelastic Media, Report MM 2765-731, Texas A&M University. Knauss, W.G. (1970) Delayed Failure – The Griffith Problem for Linearly Viscoelastic Materials, International Journal of Fracture Mechanics, 6, pp.7-20. Liu, C.T. and Kwon, Y.W. (1999) Numerical Modeling of Damage Initiation and Evolution Processes in a Particulate Composite Material, International Conf. on Computational Engineering and Science. Liu, C. T. (1990) Three-Dimensional Finite Element Analysis of a Damaged Fracture Specimen, Paper No. AIAA 90-2088, AIAA/SAE/ASME/ASEE Joint Propulsion Conf. Liu, C. T. (1990) Critical Analysis of Crack Growth Data, Journal of Propulsion and Power, 6 (5), pp. 519-524.

2. 3. 4. 5. 6. 7. 8. 9.

DYNAMIC FRACTURE EXPERIMENTS USING POINT IMPACT

D. RITTEL Faculty of Mechanical Engineering Technion, Israel Institute of Technology 32000 Haifa, Israel

Abstract This paper summarizes experimental results on dynamic crack initiation, using onepoint impact technique. Two specific issues will be addressed here. The first concerns the determination of the mode I dynamic fracture toughness of small cracked beam specimens. Next, some basic results related to the testing of standard Charpy specimens at high impact velocities will be presented. All these tests have in common the fact that the impact is applied in one point to an unsupported specimen, so that fracture is triggered by inertial effects only.

1.

Introduction

The concept of dynamic fracture of unsupported specimens is not a new concept. In fact it has been noted, e.g. by Kalthoff et al. (1977, 1983), that an impacted specimen which is originally supported (such as three-point bend) may fracture at times where the contact with the supports is lost. This observation has been applied to the development of impact tests of unsupported specimens. Similar observations have been made by Rittel and Maigre (1996), who studied the dynamic fracture of Compact Compression Specimens using a (split Hopkinson) bar (Kolsky, 1963). In these experiments, it was noted that fracture proceeded before the stress wave induced on the impacted incident side had exited the specimen on the transmitter side. Such an observation led to the conclusion that the transmitter bar was not needed in that case. In general, as long as fracture originates before the stress wave reaches the supports, one point impact experiments can be performed. This also implies that the investigated material is sufficiently “brittle” to fracture at the first passage of the stress wave. Such experiments have been also performed by Giovanola (1986), and a number of analytical solutions have been proposed for cracked beams, e.g. by Kishimoto et al. (1990), and by Rokach (1994). In the present paper, we will briefly describe the application of the one point impact technique to mode I and Charpy fracture studies. 267 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 267–274. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Experimental Results

2.1 DYNAMIC MODE I CRACK INITIATION IN SHORT BEAM SPECIMENS

Dynamic fracture toughness testing is not a standardized procedure, by contrast with static fracture toughness. Yet, the typical specimens are generally of a finite size, reaching sometimes several tens on centimeters. However, the final dimensions of manufactured products may be such that the specimens that can be machined are of limited size. Weisbrod and Rittel (2000) investigated the problem of testing short beam specimens machined in the radial direction of 25.4 mm diameter extruded bars. The central assumption was that linear elastic fracture mechanics concepts can be used to determine the dynamic stress intensity factor. It was therefore suggested that the linearity of the problem could be used advantageously, once the response of the cracked structure to a unit impulse is determined (dynamic weight function). The stress intensity factor K(t) can be calculated by a time convolution integral of the applied load with the dynamic weight function,

In the present case, the finite geometry of the specimen precludes the use of analytical solution, and the dynamic weight function was calculated using the finite element method. The dynamic fracture toughness is the value of the stress intensity factor at the onset of fracture, as indicated by a single wire fracture gage. A great advantage of the one point impact technique is that the boundary conditions are extremely simple and they reduce to the applied load. This is a hybrid experimental-numerical approach in which measured loads are blended with the calculated response to yield the stress intensity factor. Several experiments were performed using an instrumented bar as shown in Figure 1. The specimen was instrumented using a single wire fracture gage on one side and a small strain gage on the other side, both positioned in the immediate vicinity of the crack-tip. The validity of the approach was assessed by comparing the measured crack-tip strains with those obtained through the hybrid experimental-numerical procedure outlined in eq. (1), using the relationship between strain and stress intensity factor. Fracture was assessed in a straightforward manner from the single wire fracture gage on the one hand, and interpreted as causing a significant change of slope in the strain gage signal. An excellent correlation was observed between these two way of assessing the onset of crack-propagation. Figure 2 shows the dynamic stress intensity factors (SIF) as mentioned above. One can note the marked similarity between SIF’s, determined directly and

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indirectly. This observation validates the overall approach, including the assumptions related to the applicability of linear elastic fracture mechanics.

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Consequently, it can be concluded that the above-mentioned approach is suitable for the testing of any geometry of specimens, provided that linear elastic fracture mechanics (lefm) applies. This includes the present case of the small specimens presented here.

2.2 ONE-POINT IMPACT OF CHARPY SPECIMENS One-point impact can also be applied to standard or precracked Charpy specimens, provided the material is sufficiently “brittle” to fracture. In this paper, we will address an experimental issue related to the test itself, that is the kinematics of the specimen. Specifically, the problem dealt here is to determine how long does the specimen remain in contact with the incident bar. Moreover, a related issue is that of the correlation of the specimen kinematics with the strain gage readings of the incident bar.

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To address these issues, a series of experiments were carried out with Charpy steel specimens tested at room temperature. The selected steel (A508) was tough enough not to fracture at room temperature. In this case, The motion of the specimen was recorded using a high speed Imacon 790 camera. Special care was paid to the synchronization of the strain gage signals and the firing of the camera. Figure 3 shows a typical record of these experiments. The specimen is noted to remain in contact with the incident bar during the first 4 frames, and careful examination of the frame number 5 shows a very faint gap between the specimen and the bar that grows in the subsequent frames.

In parallel, the incident and reflected signals, recorded by the strain gage on the bars are shown in Figure 4. It can be noted that the two signals reach a similar level at the same time, marked as time 2. Keeping in mind that the applied force, F(t), is proportional to the sum of the incident and reflected signals, the equality of these just indicates that the applied load This is consistent with a free end

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condition for the bar, that corresponds to the specimen “take-off”. One can now correlate the load drop to zero to the instant at which the specimen separates from the incident bar. Several tests showed that the characteristic “take-off” time is of the order of which is much larger than the typical needed for a roundtrip of the stress wave, based on one dimensional wave propagation in a steel specimen. At this stage, low temperature fracture experiments can be carried out, during which the fracture time is recorded, using fracture gages. It is interesting to note whether fracture occurred while the specimen was still in contact with the bar or during its flight. It must be emphasize that the presence or lack of contact between the specimen and the bar at fracture does not affect the validity of the results whatsoever. However, if a measure of the fracture energy is to be assessed, the contact case is preferable since kinetic energy may be neglected at this stage (Rittel et al., 2002).

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Discussion

We have presented various aspects of one-point impact loading experiments. While such experiments have been carried out in the past, the present work emphasizes the advantages of the method which is relatively simple to operate and model numerically as it overcomes all the contact related issues of other methods, such as Charpy testing. Concerning fracture mechanics, a very simple way to determine the dynamic fracture toughness has been shown, that also applied to the case of relatively small specimens. In this method, one takes advantage of the linearity of the problem to determine the stress intensity factor as a function of time. Yet, issues such as loss of contact between the specimen and the bar and accuracy of fracture time determination are important factors in such experiments. Consequently, additional experiments have been carried out to assess the duration of the contact phase between the specimen and the bar. It was observed that for a standard Charpy specimen, this time is of the order of Of course, additional work is needed to obtain a more general relationship between the specimen dimensions and contact time. 4.

Conclusion

One-point impact testing is a convenient technique for dynamic fracture studies of relatively brittle materials. For mode I dynamic fracture toughness characterization, the technique provides valuable results on the basis of simple lefm concepts. The determination of the stress intensity factor of the selected geometry is based on the prior calculation of the structural response to unit impulse. The fracture toughness is determined as the value of the stress intensity factor at the onset of crack propagation. Next, one point impact can be applied to testing of standard notched Charpy specimens, by extending the range of impact velocities. While restricted to the lower shelf domain of steels, this technique can provide reliable values of the fracture energy of brittle materials, which is known to be a delicate issue. 5.

Acknowledgement

This research was supported by the fund for promotion of research at the Technion (030-094). Mr. G. Weisbrod and Prof. A. Pineau are gratefully acknowledged for their contribution on dynamic fracture and Charpy testing respectively.

6.

References

1.

Giovanola, J.H., (1986), “Investigation and application of the one-point bend impact test”, Fracture Mechanics: Seventeenth Volume, ASTM-STP 905, (Edited by Underwood, J.H., Chait, R. et al.), 307-328.

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7. 8. 9.

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Kalthoff, J.F , Winkler, S. and Beinert J, (1977), “ The influence of dynamic effects in impact testing, International Journal of Fracture, Vol. 13, 528-531. Kalthoff, J.F., Winkler, S., Böhme, W., Shockey, D.A., (1983), in Proceedings, International Conference on the Dynamical Properties and Fracture Dynamics of Engineering Materials, Brno, Czechoslovakia. 12. Kishimoto, K., Fujino, Y., Aoki, S. and Sakata, M. , (1990), ” A simple formula for the dynamic stress intensity factor of an impacted freely supported bend specimen ”, JSME Int. J., Series I, Vol. 33, No.l., 51-56. Kolsky, H., (1963), Stress Waves in Solids, Dover Publications Inc., New York, NY. Rittel, D. and Maigre, H., (1996), "An investigation of dynamic crack initiation in PMMA", Mechanics of Materials, Vol. 28, 229-239. D. Rittel, A. Pineau, J. Clisson and L. Rota, “On testing of Charpy specimens using the one point bend impact technique”, (2002), in press, Experimental Mechanics. Rokach, I..V., “Comparison of simplified methods of dynamic stress intensity factor evaluation", (1994), Mechanika Teoretyczna I Stosowana, Vol. 1, No. 32, 203-212. Weisbrod, G. and Rittel, D., (2000), “A method for dynamic fracture toughness determination using short beams”, Intl. Journal of Fracture, Vol. 104, No. 1, 89-103.

EXPERIMENTAL AND NUMERICAL INVESTIGATION OF SHEARDOMINATED INTERSONIC CRACK GROWTH AND FRICTION IN UNIDIRECTIONAL COMPOSITES

A. J. ROSAKIS, C. YU, M. ORTIZ Graduate Aeronautical Laboratories, California Institute of Technology Pasadena, CA 91125 D. COKER Division of Engineering, Brown University Providence, RI 02912 A. PANDOLFI Dipartimento di Ingegneria Strutturale, Politecnico di Milano 20133 Milano, Italy

Abstract

Dynamic crack growth in unidirectional graphite/epoxy composite materials subjected to in-plane impact loading is investigated experimentally and numerically. The experiments are conducted using CGS (Coherent Gradient Sensing) Interferometry in conjunction with high-speed photography to visualize the crack growth events. Cracks are found to propagate at subsonic speeds in the Mode-I case, whereas in both mixed mode and Mode-II the crack tip speed clearly exceeds the shear wave speed of the laminate. For these intersonically growing shear (Mode-II) cracks a shock wave emanating from the crack tip is observed. This provides direct evidence that the cracks propagate faster than the shear wave speed of the composite. The crack tip speed is initally observed to jump to a level close to the axial longitudinal wave speed along the fibers (7500 m/s) and then to stabilize to a lower level of approximately 6500 m/s. This speed corresponds to the speed at which the energy release rate required for shear crack growth is non-zero as determined from asymptotic analysis. The CGS interferograms also reveal the existence of large-scale frictional contact of the crack faces behind the moving shear cracks. In addition high speed thermographic measurements are conducted that show concentrated hot spots behind the crack tip indicating crack face frictional contact. These experiments are modeled by a detailed dynamic finite element calculation involving cohesive elements, adaptive remeshing using subdivision and edge collapse, composite elements, and penalty contact. The numerical calculations are calibrated on the basis of fundamental material properties measured in the laboratory. The computational results are found to be in excellent agreement with the optical 275 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 275–288. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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experimental measurements (crack speed record and near tip deformation field structure). For shear crack growth, the numerics also confirm the optical observation of large-scale crack face contact. 1.

Introduction

Dynamic crack growth along weak planes is a significant mode of failure in composites and other layered materials. In the past years bimaterial fracture specimens, fabricated by the bonding of a stiff material to a compliant material (featuring a mismatch in wave speeds), have been used to demonstrate the importance of highly transient and dynamic crack growth in heterogeneous solids. It was observed that interfacial crack tip speeds rapidly approached and exceeded the shear wave speed of the more compliant material [1-5]. Thus they reached intersonic speeds with respect to the more compliant material in which intersonic crack tip speed is defined as the speed in the open interval between the shear wave speed and the longitudinal wave speed. For crack tip speeds above the shear wave speed a ray emanating from the crack tip representing a line of strong discontinuity (shear shock wave) was observed experimentally and also predicted theoretically [2]. These high crack tip speeds were obtained under loading conditions that promoted locally shear dominated deformations at the crack tip which were further enhanced by the stress wave mismatch due to the bimaterial nature of the solid [6]. In homogeneous materials and in the absence of a weak prescribed crack growth path, Mode-II crack propagation is not possible because the crack naturally kinks and propagates in a direction that causes a local Mode-I stress state around the crack tip. This is not necessarily true in solids that may be homogeneous regarding their properties but may involve preferable crack paths in the form of weak planes of lower toughness. Unidirectional composites may indeed fit this description if viewed from the "macroscopic" point of view of a homogenized anisotropic theory. Although from the microscopic viewpoint, unidirectional composite materials are locally inhomogeneous and are related to bimaterials, in the sense that they also involve a preferred crack growth direction, and are also composed of a stiffer material (the fiber) bonded together with a more compliant material (the matrix), from a macroscopic point of view such solids can be viewed as homogeneous anisotropic materials as far as their elastic properties are concerned. However, they are still inhomogeneous regarding their fracture toughness properties. Indeed, from both the macroscopic or microscopic view points the common characteristic between unidirectional composites and bimaterials which also seem to be the most relevant is the existence of a weak straight line crack path which may accommodate growing cracks of both modes and not the existence of the two phases. Because of the above observations and because of our previous discovery of intersonic crack growth in bimaterials, unidirectional composites seem to be natural candidates for studying maximum allowable crack tip speeds of cracks of both modes in a system that is macroscopically at least homogeneous. While the theory excludes intersonic growth of mode-I cracks, it has not excluded the possibility of intersonic mode-II dominated crack growth in isotropic or anisotropic homogeneous elastic solids in the case of a self similar crack growing in the direction of the plane of the crack [7]. Several researchers [7-11] calculated the critical speed at which intersonic crack growth is stable in isotropic materials to be For stable intersonic

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crack growth this is the only speed at which the energy release rate is finite. Analysis of the intersonic crack growth in anisotropic materials was developed by Piva and Hasan [12], Huang et al. [8] and Broberg [13]. They predicted that, just as in isotropic materials, intersonic shear crack growth is possible at a specific critical speed, above the shear wave speed. However, in the case of anisotropic materials, this speed is the product of a complex function of the material properties and the shear wave speed. To our knowledge, very few experimental studies of dynamic crack propagation in fiberreinforced composites have been performed. Liu et al. [14], investigated quasi-static fracture of polymer composites using the optical technique of CGS. Khanna and Shukla [15] extended the theoretical results of Piva and Viola [16] to in-plane stress and used it to analyze the results of strain gages and to thus determine mode-I stress intensity factors for cracks propagating in unidirectional glass-epoxy composite laminates at constant speed. Lambros and Rosakis [17], using CGS shearing interferometry technique in conjunction with high speed photography, looked at the initiation and growth in thick unidirectional graphite-epoxy composite plates under symmetric 1-pt impact loading. They observed crack tip speeds for mode-I loading which approached 900 m/s or where is the speed of the Rayleigh waves travelling in the direction of the fibers. In another investigation, Lambros and Rosakis [18] also studied the dynamic delamination propagation in real time of a cross-plied laminate subjected to out-of-plane impact. In this paper we summarize some recently obtained experimental results of dynamic crack propagation in unidirectional graphite-epoxy composite plates under mode-I and mode-II loading. Many of the experimental details have already appeared in an extensive article by the first two authors (Coker and Rosakis [21]) and will not be discussed again here. The optical technique of CGS was used in conjunction with high-speed camera to obtain interferograms during crack growth. Crack tip speeds are then computed from the interferograms and the limiting crack tip speeds are determined for both the mode-I and the mode-II cases. These experiments are modeled by finite element analysis involving cohesive elements and penalty contact. In addition, high speed thermographic measurements are conducted which show regions of contact behind the crack tip. The numerics demonstrate good agreement with the experiments for both crack tip speeds and predicted contact behavior at the crack faces.

2.

Material Characterisitcs

A cross-section of the composite material is shown in Fig. 1. The unidirectional graphite/epoxy composite plates were manufactured from 48 layers of graphite fiber and epoxy matrix pre-pregs laid up in the thickness direction to form a 6.3-mm thick plate. The fiber volume fraction in the prepreg is 65% while the fiber diameter is 7.3 The surface on one side of the composite plate was made optically flat by adding a thin layer of epoxy on one surface and then curing the composite specimen upon an optically flat glass plate. This surface was then made specularly reflective by coating with a thin layer of aluminum of 1-2 thickness in a vacuum deposition chamber. The specimen geometry is shown in Fig. 2. The orientation of the axes with respect to the composite plate is shown in Fig. 1. The axis is defined to lie along the fibers, the axes is perpendicular to the plane of the composite surface while the axis is perpendicular to

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the plane. The orthotropic elastic constants, the compliance matrix and stiffness matrix values for graphite/epoxy unidirectional composite material are given in Table 1.

In Table 2 denotes the dilatational wave speed parallel to the fibers while denotes the dilatational wave speed perpendicular to the fibers. The density is 1478 The shear wave speed as well as the longitudinal wave speeds parallel and perpendicular to the fibers are shown in Table 2. The Rayleigh wave speed parallel to the fibers is 1548 m/s.

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Experimental Technique

Schematic of the experimental set-up and the optical technique of coherent gradient sensing, CGS, is shown in Fig. 3. The specimen was subjected to impact loading through a projectile fired from a gas gun. The dynamic stress field produced by the impact loading wave leads to a changing out-of-plane deformation on the surface of the composite plate. The optical technique of reflection CGS, in conjunction with highspeed photography, was used to record in real time the slopes of the out-of-plane deformation field. CGS is a full-field, lateral-shearing interferometer. The details of CGS can be found in several articles [19, 20] and its application to crack initiation in composite materials has been demonstrated before [14, 17]. CGS, when used in reflective mode, measures the in-plane gradients of out-of-plane displacement.

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Experimental Results

Symmetric Mode-I crack tip deformations were attained by impacting the specimen symmetrically along the notch line by a steel projectile as shown in Fig. 3 with projectile speeds varying from 10 m/s to 57 m/s. A typical sequence of experimental CGS interferograms for mode-I crack initiation and propagation are shown in Fig. 4 for the highest impact speed of 57 m/s. The first frame shows the CGS interference fringe pattern around the notch tip just before crack initiation. The fringe pattern around the crack tip during propagation at times 1.8 and 3.0 is shown in Figs. 4(b) and (c). Using the above sequence of pictures the crack tip history was recorded as a function of time for Mode-I symmetric loading. The crack tip speed history was then obtained by using 3-point fit to the crack tip history. The crack started growing subsonically at 1350 m/s and accelerated to 1550 m/s which is the Rayleigh wave speed of the composite in the direction of the fibers. The crack tip speeds never exceeded the Rayleigh wave speed. Asymmetric 1-pt bend impact experiments were conducted at different impact speeds by impacting the specimen below the notch. Projectile speed varied from 21 m/s to 30 m/s. For an impact speed of 21 m/s a sequence of three frames taken with the high speed camera at an interframe time of 1.39 is shown in Fig. 5. The notch tip as well as the stress concentration at the notch tip is clearly visible in the first frame. The impact wave has propagated from one end of the plate to the other, was reflected as a tensile wave below the notch and loaded the notch tip in a predominantly shear mode. Notch-tip stress concentration is seen to increase as the interferogram fringes increase in size and number. The nature of the fringe pattern shows that the loading is primarily mode-II [14] featuring one fringe lobe in the back and two fringe lobes in the front. These pictures show the fringe patterns around the notch tip just before and after crack

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initiation (last photograph). However, in the last frame, a significant change takes place in which the fringe loops are squeezed backwards indicating very high initial crack tip accelerations. Indeed, the average crack tip speed between these two frames is 2100 m/s, which is above the shear wave speed 1800 m/s for this material. This dramatic change in shape is not observed in the mode-I crack growth experiments. The notch tip is loaded by the arrival of the loading wave initially at 11 after impact and is fully loaded by the reflected wave at 15 Fig. 6 shows a sequence of three CGS interferograms featuring shear dominated dynamic crack growth. The rear loop shape changes from rounded to a triangular wedge bounded by a line of high intensity fringes emerging from the crack tip at an angle. This line is caused by a steep change in the stress gradients in a localized area, which later forms a discontinuity in the shear stresses, i. e. a shear shock wave. Although CGS is sensitive to the sum of the normal stresses in homogeneous, isotropic materials, the normal stresses and shear stresses are coupled in stresses in our experiments. This would not hold for isotropic solids. Finally, this line broadens into two parallel lines (a double shock wave) and intercepts the crack surface over a finite area between the crack tip and 4-5 mm behind it. We propose for the reason of the double shock wave structure as the existence of a contact region behind the crack tip or alternatively as the cohesive zone in front of the crack tip. The slope of the shear shock wave changes as you move away from the crack tip which may be due to the unsteady growth of the crack tip. The crack tip location and speed history is shown in Fig. 7(a) and 7(b). The crack-tip speed approaches 5200 m/s, a speed 3 times higher than the shear wave speed and is clearly intersonic. In this experiment the crack tip jumps immediately from rest to 2100 m/s becoming intersonic in the first frame after initiation, thus not passing through the subsonic regime. The crack tip acceleration was also very high and was of the order of From Fig. 7 it can be seen that the crack propagation had not reached steady state within our window of observation. A separate experiment was conducted under similar conditions with the field of view further downstream of the notch in order investigate whether the crack tip speed eventually attains steady state conditions following the initial acceleration stage. In this case the impact velocity was 28 m/s and the interframe time was 0.83 Fig. 8 shows a selected set of CGS fringe patterns. As already noted from the previous case, shear shock waves are also observed. The first recorded crack tip speed is 4000 m/s as the crack appears in our field of view. The crack tip speed then climbs up and oscillates around and an average speed of 7000 m/s. This is the highest crack tip speed ever observed in laboratory setting (for details see reference [21]).

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Discussion

A view of the general trends in crack tip position history characteristics of many different experiments corresponding to very similar impact conditions have been collected and displayed in Fig. 9(a). This figure shows the reproducibility of the observed experimental trends (within the same geometry) and allows us to obtain an average sense of the trends in the variation of the intersonic crack tip speed for both mode-I and mode-II cases. Crack tip speed as a function of crack extension is shown in Fig. 9(b). Mode-I cracks initiate around 1300 m/s and subsequently the crack speed increases to the neighborhood of the Rayleigh wave speed of 1548 m/s. However, mode-II cracks initiate around 2000 m/s (intersonic) and rapidly accelerate above 6000 m/s. These cracks asymptotically approach a steady state speed of 6500 m/s. In order to interpret the observed phenomena theoretical analysis was developed by Huang et al. [8]. They obtained the asymptotic stress and displacement fields near an intersonically propagating crack tip in an orthotropic material under steady-state conditions. In this analysis, the composite was modeled as an elastic, orthotropic, homogeneous solid. For both mode-I and mode-II cases a prescribed straight-line crack path was assumed. Mode-I elastic asymptotic field yielded a negative, unbounded crack tip energy release rate in the entire intersonic crack growth regime. This is physically impossible since a propagating crack-tip cannot radiate out energy. Thus, a mode-I crack tip cannot propagate intersonically. This conclusion is supported by our experiments in which the crack tip speed never exceeded the shear wave speed regardless of how large the impact velocity of the projectile was (see Fig. 9b). For mode-II, the energy release rate supplied by the elastic asymptotic field is finite and positive only at a distinct critical crack tip speed [8]. Since a positive and finite energy supply is required to break the material bonds in front of the crack tip this speed corresponds to the only possible steady state intersonic crack growth speed according to the above steady state theory. Using the material properties given in Table 1, this critical crack tip speed is 6600 m/s.

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In our experiments the mode-II crack tip seem to eventually reach this value sometimes from below and sometimes from above [21].

6.

Finite Element Simulations

The shear dominated crack growth experiments were modeled with dynamic finite element analysis using cohesive elements [22,23]. An automatic self-adaptive procedure was used to explicitly reproduce crack propagation by inserting a new cohesive surface along previously coherent inter-element surfaces when the local traction reaches a limit value. The analysis consisted of cohesive elements and contact elements along the line of the initial crack. The numerical calculations were calibrated on the basis of fundamental material properties measured in the laboratory. The loading history and the mesh geometry using a minimum mesh size of 1mm, 44593 nodes and 24685 elements is shown in Fig. 10. The composite is modeled as a homogeneous orthotropic material with the principle axis of symmetry along the original crack. Assuming a cohesive model for the description of the progressive decohesion between two fracture surfaces, the characteristic strength of the cohesive law, as well as the fracture energy, is locally varied with the angle with respect to the fiber orientation. A bilinear cohesive model (see Fig. 10) relating an effective cohesive traction to an effective displacement is chosen to avoid instabilities. Following Camacho and Ortiz [24] the effective opening displacement is defined as where and are the normal and sliding displacements, respectively. The effective traction is given by where defines the ratio between the shear and normal tractions. If we take the shear and opening critical displacements to be the same, is roughly the ratio

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of to of the material. Upon closure the cohesive surfaces are subjected to a unilateral contact constraint and Coulomb’s friction law. The parameters in the model are obtained from fracture and strength experiments (surface energy N/mm, maximum cohesive stress, Mpa, critical displacement, and the weighting coefficient, The contours of constant shear stress obtained from the numerical simulations are shown in Fig. 11. The shear shock wave on the upper part of the crack line can be clearly distinguished in the numerical simulations. The contours resulting from the reflected wave due to impact can be seen to be effecting the lower part. The feature of a double shock wave is captured very well by the numerical simulations. The crack tip speed history obtained from the numerical runs is shown in Fig. 12 for an impact speed of 30 m/s. The crack tip speed history is obtained by differentiation by using a sectional quadratic three-point fit to the crack length history. The plot shows the experimental points (symbols) and the numerical simulations (lines) for different mesh sizes. There is very good agreement between the experimental and numerical results showing the initiation at intersonic speeds and steady climb to crack tip speed between vc and the longitudinal wave speed. 7.

High Speed Infrared Thermal Measurements

Since the events we are trying to observe occur over a few microseconds, a thermal camera designed to capture images at a rate of with a response time of 500 ns is used [25]. The camera features a square array of 64 detectors, in an 8 x 8 format. Each detector is 100 x 100 in size and is separated by a distance of 30 from its neighbor. The detectors are operated at a temperature of 77 K to maximize the signal to noise ratio. The camera is focused on 1.1 mm by 1.1 mm area ahead of the notch tip. Figs. 13(a) show contours of constant temperature lines. In the first image, the crack tip can be seen to be approaching from the left hand side. In the second image the crack has already left our field of view at after impact. Behind the crack tip we can observe local hot spots forming behind the crack tip due to crack surface sliding. These local hot spots of high temperature frequently jump around. The temperature increase is due to frictional sliding of the crack surfaces and later becomes a band of high temperature region enveloping several layers of fibers. This band grows in width and finally saturates the infrared detectors. The area where frictional sliding is taking place is of the order of However, the temperatures measured are being averaged over a area for each detector so that the recorded temperatures are upper lower bounds for the actual temperatures due to sliding. A sequence of numerically calculated temperature fields are shown in Fig. 13(b) showing localized high temperature fields indicating the existence of frictional sliding behind the crack tip. There is qualitatively very good agreement between the experimental and numerically generated temperature fields.

8.

Acknowledgement

This investigation was supported by the Office of Naval Research (Dr. Y. D. S. Rajapakse, Scientific Officer) through grant #N00014-95-1-0453 to Caltech and is gratefully acknowledged.

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2.

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References Lambros, J. and A. J. Rosakis, “Shear dominated transonic interfacial crack-growth in a bimaterial: 1. Experimental observations,” Journal of the Mechanics and Physics of Solids, 43(2), 1995, 169-188. Liu, C., Y. Huang, Rosakis, A. J., “Shear dominated transonic interfacial crack growth in a bimaterial: 2. Asymptotic fields and favorable velocity regimes,” Journal of the Mechanics and Physics of Solids, 43(2), 1995, 189-206. Singh, R. P., J. Lambros, Shukla, A. and Rosakis, A. J., “Investigation of the mechanics of intersonic crack propagation along a bimaterial interface using coherent gradient sensing and photoelasticity,” Proceedings of the Royal Society of London Series A, 453(1967), 1997, 26492667.

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Singh, R. P. and Shukla, A., “Subsonic and intersonic crack growth along a bimaterial surface,” Journal of Applied Mechanics 63, 1996, 919-924. Rosakis, A. J., Samudrala, O., Singh, R. P. and Shukla, A., “Intersonic crack propagation in bimaterial systems.” Journal of the Mechanics and Physics of Solids, 46(10), 1998, 1789-1813. Lambros, J. and Rosakis, A. J., “Development of a dynamic decohesion criterion for subsonic fracture of the interface between two dissimilar materials,” Proceedings of the Royal Society of London Series A, 451(1943), 1995, 711-736. Broberg, K. B., “The Near-Tip Field at High Crack Velocities.” International Journal of Fracture, 39, 1989, 1-13. Huang, Y., Wang, W., Liu, C. and Rosakis, A. J., “Analysis of intersonic crack growth in unidirectional fiber-reinforced composites,” to be published in Journal of the Mechanics and Physics of Solids, 1999. Burridge, R., “Admissible speeds for plane-strain self-similar shear cracks with friction but lacking cohesion,” Geophysical Journal of the Royal Astronomical Society, 35, 1973, 439-455. Freund, L. B., “The Mechanics of Dynamic Shear Crack Propagation,” J. of Geophysical Research, 84(B5), 1979, 2199-2209. Georgiadis, H. G., “On the stress singularity in steady-state transonic shear crack propagation,” International Journal of Fracture, 30, 1986, 175-180. Piva, A. and Hasan, W., “Effect of orthotropy on the intersonic shear crack propagation,” Journal of Applied Mechanics, 63,1996, 933-938. Broberg, K. B., “Intersonic crack propagation in an orthotropic material,” International Journal of Fracture, to be published, 1999. Liu, C., Rosakis, A. J., Ellis, R. W. and Stout, M. G., “A study of the fracture behavior of unidirectional fiber-reinforced composites using coherent gradient sensing (CGS) interferometry,” International Journal of Fracture, 90, 1998, 355-382. Khanna, S. K. and Shukla, A., “Development of stress-field equations and determination of Stress intensity factor during dynamic fracture of orthotropic composite materials,” Engineering Fracture Mechanics 47(3), 1994, 345-359. Piva, A. and Viola, E., “Crack-propagation in an orthotropic medium,” Engineering Fracture Mechanics, 29(5), 1988, 535-548. Lambros, J. and Rosakis, A. J., “Dynamic crack initiation and growth in thick unidirectional graphite/epoxy plates,” Composites Science and Technology, 57(1), 1997, 55-65. Lambros, J. and Rosakis, A. J., “An experimental study of dynamic delamination of thick fiber reinforced polymeric matrix composites,” Experimental Mechanics, 37(3), 1997, 360-366. Tippur, H. V., Krishnaswamy, S. and Rosakis, A. J., “A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results,” International Journal of Fracture, 48, 1991, 193-204. Rosakis, A. J., “Two optical techniques sensitive to the gradients of optical path difference: The method of caustics and the coherent gradient sensor (CGS),” Experimental Techniques in Fracture, J. S. Epstein, Ed., New York, VCH, 1993, 327-425. Coker, D. and Rosakis, A. J., “Experimental Observations of Intersonic Crack Growth in Asymmetrically Loaded Unidirectional Composite Plates,” GALCIT SM Report No. 98-16, Pasadena, California Institute of Technology, 1998 and Philosophical Magazine A, 81, 571-595, 2001. Ortiz, M. and Pandolfi, A., “Finite deformation irreversible cohesive elements for threedimensional crack propagation analysis,” International Journal for Numerical Methods in Engineering, 44, 1267-1282, 1999. Yu, C., Pandolfi, A., Ortiz, M., Coker, D. and Rosakis, A. J., “Three-dimensional cohesive modeling of impact damage in composites,” submitted to International Journal of Solids and Structures, 2001. Camacho, G. T. and Ortiz, M., “Computational modeling of impact damage in brittle materials,” Int. J. Solids and Structures, 33 (20-22), 2899-2938, 1996. A. T. Zehnder, P. R. Guduru, A. J. Rosakis and G. Ravichandran, “Million Frames per Second Infrared Imaging System”, Review of Scientific Instruments, 71 (10), 3762-3768, 2000.

4. Optical Methods

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MOIRÉ INTERFEROMETRY—PAST, PRESENT AND FUTURE DANIEL POST Virginia Polytechnic Institute and State University Prof. Emeritus, Engineering Science and Mechanics Blacksburg, VA 24061

Abstract Moiré interferometry is introduced as an outgrowth of moiré fringe multiplication. Coherent laser light enabled the practical implementation of virtual reference gratings and high-frequency specimen gratings. Reliable analysis of shear depended upon the ideas in a little-known paper. These three facets are the essence of current techniques. Present-day applications are exemplified by work in electronic packaging and composite materials. Several significant advances are forecast to further broaden the usefulness and appeal of moiré interferometry.

1.

Introduction

This paper is dedicated to Professor Isaac M. Daniel, on the occasion of the Symposium presented in his honor at the 14th U.S. National Congress of Applied Mechanics, June 2002. I feel honored, too, to participate in this symposium together with renowned colleagues, gathered to express our admiration and offer tribute to Isaac. His extensive contributions and his gentle, helpful nature are broadly recognized and appreciated. We have had numerous exchanges through many years, but Isaac and I have only one mutual publication [1], co-authored also by our colleague and friend, Bob Rowlands. That work is 30 years old, but it is a fitting entrance to the current topic—the past, present and future of moiré interferometry. 2.

A Perspective of the Past

Moiré interferometry is a special case of moiré fringe multiplication—it is fringe multiplication by a factor of two. In the past, only low frequency gratings could be applied to specimens. However, fringe multiplication utilizing a higher frequency reference grating increased the sensitivity, since sensitivity corresponds to the frequency of the reference grating. In the aforementioned paper with Isaac, fringe multiplication by a factor of 10 was achieved. The glass/epoxy specimen, with a 500 lines/inch specimen grating, was loaded and the deformed specimen grating was photographed. The image on the photographic plate was a transparent replica of the deformed grating. The optical system for fringe multiplication was similar to that of Fig. 1, where the photographic plate was the specimen and the frequency of the reference grating was 10 291 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 291–302. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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times higher. Note that fringe multiplication by a factor of 6 is shown, whereas in the case of multiplication by 10 the beams of +5 and –5 diffraction orders were isolated. It was exhilarating to see the enhanced sensitivity, with 10 times as many moiré fringes as conventional moiré. Several innovations evolved to link the technique of the 1970s to the present. A significant factor was the practical availability of the laser, which provided very high monochromatic purity and allowed high contrast two-beam optical interference even when the beams traveled substantially unequal path lengths. The laser led to a practical technique to produce high-frequency specimen gratings. A two-step process was developed. First, a high-frequency grating was made by a socalled holographic process. A high resolution photographic plate was exposed to two coherent collimated beams, incident at angles and After processing and drying, the emulsion shrank in the exposed zones, while shrinkage was restrained in the unexposed (destructive interference) zones by the silver grains. Thus, the plate had an undulating surface of ridges and furrows that formed the high-frequency phase grating.[2] Actually, two exposures were made, with the plate rotated 90° between exposures to produce a cross-line grating with ridges in orthogonal x and y directions. In the second step, the photographic plate was used as a mold to cast, or replicate, the phase grating on the specimen surface. Silicone rubber was used as the replication material. With this development, it was no longer necessary to use high diffraction orders to achieve high sensitivity. Instead, fringe multiplication by a factor of two could be used, whereby the beams of +1 and –1 diffraction orders were combined to form the moiré pattern.

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In addition, it was realized that the high-frequency reference grating was not required at all. Instead, the two beams indicated in Fig. 1 by rays labeled 0 and 1 could be replaced by two coherent beams generated by any optical system.[2,3] The two beams create a virtual reference grating. One such system is illustrated in Fig. 2(a) for a transparent specimen, where the corresponding beams are labeled 0 and 1. The symbol for an eye represents a camera. A similar scheme was used in reflection for opaque specimens. Figure 2(b) illustrates the arrangement used to analyze off-axis composite specimens of graphite/polyimide.[3].

Another important advance relied on an early paper addressing geometrical moiré, the moiré grid-analyzer method.[4] If instrumentation could be arranged for viewing the U and V fringe patterns without moving the sample or the interferometer, it becomes possible to extract shear strains without ambiguity. This realization led to the 4-beam arrangement illustrated schematically in Fig. 3. Numerous implementations of Fig. 3 have been produced, and several are illustrated in Ref. [5]. Improved techniques for specimen gratings evolved. These include molds that utilize photoresist to optimize their diffraction efficiency, and molds with a transferable film of aluminum to maximize reflectivity. The evolution continued with the development of apparatus and techniques for microscopic moiré interferometry, including a robust algorithm called O/DFM, the optical/digital fringe multiplication method.[5] Displacement fields with contour intervals as small as 17 nm were demonstrated.

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The foregoing is a brief perspective on moiré interferometry in terms of my own experience and that of my students. We knew that moiré interferometry would evolve into a practical technique as early as 1982, when Weissman demonstrated superb fringe patterns at 97.6% of the theoretical upper limit of sensitivity, with a virtual reference grating frequency of 4000 lines/mm (101,600 lines/inch).[6] Closely related work progressed elsewhere in the U.S., Europe and Asia. In his paper “A Historical Review of Moiré Interferometry,” Walker reviewed the chronology of many of the contributions.[7] He recounted the pioneering developments in Japan, which predated accomplishments in the Western world by several years, but was not widely known in the West. He outlined work by Sciamarella and by Post and his students in the U.S., and by several investigators in Europe—especially the outstanding work at Strathclyde University by Walker, McKelvie and their colleagues. Moiré interferometry has matured through worldwide efforts. It has gained extensive application in the micromechanics industry for the analysis of thermal stresses in electronic packages, and it serves in numerous other technologies as an important tool of experimental solid mechanics. It will continue to evolve through worldwide activity.

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3. The Present The current status of moiré interferometry is revealed in the open literature, and also in Ref. 5, which is a comprehensive volume on the subject. It has been used effectively in a highly diverse array of disciplines, especially electronic packaging, but also composite materials, fracture mechanics, biomechanics, metallurgy, ceramics, polymers, concrete, mechanical joints, adhesive joints, and undoubtedly other disciplines too. The examples selected here emphasize accommodation for special problems. In the first and third examples, the frequency of the virtual reference grating was 2400 lines/mm, providing contour intervals of 0.417 µm/fringe order. In the second example, the contour interval was 35 nm/contour. 3.1 REAL-TIME THERMAL DEFORMATION

The design of electronic devices is critically dependent upon knowledge and management of thermal deformations, strains and stresses. Figure 4 illustrates the complexity of such analyses. The device is composed of diverse materials, each with a different coefficient of thermal expansion and difference creep/relaxation behavior. It must resist temperature extremes in operation and in shipment without mechanical or electrical failure. The fringe patterns are contour maps of the U and V displacement fields occurring at different temperatures in a thermal cycle. The fringes are remarkably clear at this magnification, except for those in the printed circuit board (PCB) at 125°C; the PCB is a heterogeneous material—a composite of woven glass fibers in an epoxy matrix—and because of the weave its displacement contours are very complicated. Visual inspection of the V field shows opposite directions of curvature in the chip region before and after the 125°C temperature; this behavior results from creep of the solder and molding compound. Whereas extensive computational analysis is undertaken for the mechanical design of electronic packages, the complexities of geometry and materials necessitate experimental guidance and verification. For these tests, the specimen is cycled in an environmental chamber that provides controlled heating and cryogenic cooling. The specimen is not attached physically to the chamber, but instead it is supported by arms extending from the moiré interferometer through the chamber walls, and thus it is isolated from vibrations of the chamber.[8] 3.2 BITHERMAL DEFORMATION In this application of microscopic moiré interferometry, the specimen was a unidirectional boron/aluminum composite subjected to thermal loading. Bithermal analysis was applied, whereby a cross-line grating was replicated on the specimen at 132°C. The specimen was cooled to room temperature and then observed in a microscopic moiré interferometer. The reference grating frequency was f = 4800 lines/mm (122,000 lines/inch); the fringe multiplication factor (by O/DFM) was the contour interval was 35 nm per contour. [5] Figure 5 shows the combined effects of load-induced displacements plus free thermal expansion When is subtracted off for each material, the substantial range of strains occurring in the aluminum matrix becomes evident.

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3.3. CURVED SURFACES

Moiré interferometry has been developed for routine analysis of flat surfaces. However, certain accommodations can be made for curved surfaces when the problem is important enough to invest extra effort. Such a case is the ply-by-ply deformation at a central hole in a multi-ply composite plate.[9] The specimens were thick laminated composite plates, each with a 25.4 mm diameter central hole. The grating was applied to the cylindrical surface of the hole and replicas of the deformed grating were made with the specimen at different tensile load levels. First, a cross-line grating was formed on the cylindrical surface of a disk and it was used as a mold to apply the grating to the specimen. Then, with the specimen under load, the deformed grating was replicated (or copied) on another disk. The replica was inserted in a moiré interferometer and deformation data were recorded by a moiré interferometer as a series of narrow strips, each approximating a flat surface. Figure 6 shows a mosaic of such strips for a 90° portion of the hole in a cross-ply laminate of IM7/5250-4 L(graphite fibers in a bismalimide-cyanate ester matrix). An enlarged view of the fringe pattern at the = 75° location is shown in Fig. 6(d), together with a graph of the ply-by-ply distribution of shear strains. The patterns reveal the data with excellent fidelity, enabling dependable determination of strain distributions. The interlaminar shear strains at the 0/90 interfaces are about five times greater than the tensile strain at A primary purpose of the work was to provide experimental data for the evaluation of computational techniques of analysis for composite structures. 4.

The Future

I see moiré interferometry as an extraordinarily useful tool—a tool that will continue to expand in scope and utility as technology and science cope with evermore complex materials and structures. It is ideal to meet the challenges of small size and high-level dependability, among others. We can expect advances in several categories, whereby the technique will further broaden in usefulness and appeal. 4.1. REPLICATION OF DEFORMED GRATINGS

The experiments conducted with Isaac in 1972 utilized photographic replicas of the specimen grating, taken with the specimen at different load levels. Subsequently, replication of deformed phase gratings in plastic or elastomer became practical. Replication was used to measure the deformation of an electronic package at a cyrogenic temperature.[5] In the work described above, replication was used to measure deformations along curved surfaces of composite laminates.[8] In a classical paper by McKelvie and Walker, replication was advocated for remote sites and harsh environments, followed by analysis of the replicas in the laboratory.[9] Now, replication is envisioned as a routine practice. In this practice, a grating is applied to the specimen or workpiece in the usual way. Then, the workpiece is subjected to its working loads, for example in a mechanical testing machine, which deforms the specimen and the specimen grating. Replicas of the deformed grating are made at

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desired intervals in the loading process, and subsequently the replicas are analyzed in a moiré interferometer.

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Replication provides numerous advantages for typical applications. No limits are applied to the size of the workpiece. Familiar equipment can be used for loading, including large or special purpose machines. Vibrations and air currents, which otherwise might have to be suppressed for observations with a moiré interferometer, are inconsequential. The technique can be applied conveniently in the field, far from the laboratory, and in difficult environments. In many applications, replicas can be made for cases where the loading is not mechanical, but stems from changes of temperature, humidity, chemical or radiation environments, etc. The replicas can be made on transparent substrates, enabling use of transmission systems of moiré interferometry instead of the current reflection systems. When transmission systems are used, the camera lens can be located very close to the replica and, therefore, very high magnifications become convenient. Additionally, the replicas become permanent records of the deformation, so there is easy recourse to checking and extending an analysis after an initial investigation. What developments are required? Although the replication method can be practiced with current technology, it would be advantageous to find materials and techniques for quick and routine replication of deformed gratings. I believe these will be optimized and broadly implemented.

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4.2. DATA REDUCTION FROM SINGLE IMAGES For most macromechanics analyses, moiré interferometry provides a great number of stress-induced fringes, sufficient for a detailed analysis. Yet, in current practice, there is a tendency to use phase-stepping schemes in order to reduce these data to graphs of displacement and strain distributions. In many cases, the measurements are conducted at abnormally low load levels in order to reduce the number of moiré fringes in the field, since the phase-stepping algorithms are more effective with sparse fringe patterns. This is artificial suppression of data, and the practice seems counterproductive. Automated analyses do not recognize extraneous inputs like those from scratches or other imperfections of the specimen grating. Automated analyses do not cope well with rapidly changing displacement fields. Special cautions are required if the region of interest extends across dissimilar materials. Thus, for the majority of applications—those that exhibit numerous fringes— another scheme of data reduction would be beneficial. New schemes of computer aided analyses will be developed that produce displacement and strain graphs from a single pair of moiré fringe patterns, i.e., single images of the U and V fields. They will require user input, firstly to choose regions of interest and to determine whether carrier fringes should be used in these regions; and secondly to deal with possible imperfections in the fringe patterns. The computer techniques will likely quantify the location of fringes and graph displacement fields from these; apparent displacement from carrier fringes would be subtracted off. Strains would be calculated from computer measurements of the x and y components of distance between fringes, either the distance between neighboring fringes or between more distant fringes; the operator would decide. We can foresee the development of a variety of algorithms that reduce tedious aspects of the analyses, but retain the logical step-by-step human input that is present in manual analyses. Fourier methods utilize single fringe patterns and they offer current alternatives to phase-stepping analyses. However, they do not offer the high fidelity and user discretion and control that may be desired. As the technology evolves, Fourier techniques may be employed for a global view of the strain field, followed by a direct computer aided analysis for local regions of special interest. The key will be human understanding and control, replacing relatively blind automation. 4.3. INSTRUMENTATION Less complicated designs for moiré and microscopic moiré interferometer systems are anticipated. Instruments designed exclusively for transparent replicas will be the least complex, but those offering the greatest versatility will be simplified as well. Emphasis will be placed on variable image magnification so that regions of high fringe density can be resolved by using higher magnifications, and global patterns can be recorded with lower magnifications. Strong and uniform light distribution will be emphasized to achieve high quality fringe patterns at all magnifications. 4.4. MICROMECHANICS We can anticipate extensive use of microscopic moiré interferometry for micromechanics studies. Special instrumentation will continue to be developed.

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Techniques to enhance the measurement sensitivity will be advanced for microscopic moiré interferometry. Since the basic sensitivity depends upon the frequency, f, of the virtual reference grating, further increases of f will be introduced by utilizing light of shorter wavelengths. Ultraviolet light or immersion interferometry, or both, will provide the shorter wavelengths. The enhancement of basic sensitivity will help, but in most cases the number of fringes in the field will be too small for an accurate analysis. Additional data will be needed and these data will be obtained by phase-stepping. The uncertainties introduced by random electronic noise in the CCD camera and associated hardware will be minimized by repeating the measurements a number of times and averaging the results. Alternatively, electronic components of highest performance will be selected. The additional effort and expense would be justified for analyses involving few fringes, in contrast to those that exhibit an abundance of fringes. Most problems at the finest level of micromechanics—with the smallest region of interest—will utilize replicas of the deformed grating. Then, transmission type moiré interferometers will allow the highest magnification and spatial resolution. This scheme also provides the greatest stability, and thus, the optimum conditions for phase stepping. The algorithms that transform the fringe data to displacement and strain fields will advance too. They will become more interactive to allow compensation for local defects or special conditions. The traditional quasi-heterodyne method will be optimized for microscopic moiré interferometry. The optical/digital fringe multiplication method will be extended to produce a larger number of fringe contours from a given number of phase steps. 4.5. PROPAGATION INTO COLLEGE/UNIVERSITY CURRICULA The theory and practice of moiré interferometry will be taught at many colleges in engineering, science, and technology curricula. Instrumentation will become available for these educational purposes, and it will be more basic and economical. As the technology propagates, it is likely that new innovative designs will emerge. The more advanced courses will surely encounter computer-aided analysis. These experiences will inevitably lead to experimentation with alternate schemes of analysis and superior programs will evolve. Moiré interferometry will become familiar in engineering and science. It will be known by many as a tool for measurement and exploration. 5.

Concluding Comments

Whereas moiré interferometry and microscopic moiré interferometry are currently used in many fields of engineering and science, we can anticipate that they will be used much more extensively in the same fields, and applications will extend into additional fields. The unique combination of capabilities—whole field measurements, very high sensitivity, very high spatial resolution, excellent signal-to-noise ratio, extensive range, etc.—will be put to use for measurements and explorations, as engineering tools and research tools, in widely diverse applications. I am sure Professor Isaac Daniel would be especially pleased to find that these expectations are realized.

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References

Daniel, I.M., Rowlands, R.E., and Post, D., “Strain Analysis of Composites by Moiré Methods, Experimental Mechanics, 13(6), pp. 246-252 (1973). 2. Post, D., “Optical Interference for Deformation Measurements—Classical, Holographic and Moiré Interferometry,” Mechanics of Nondestructive Testing, W.W. Stinchcomb, Editor, Plenum Press, NY, pp. 1-53 (1980). 3. Post, D., and Baracat, W.A., “High-Sensitivity Moiré Interferometry—A Simplified Approach,” Experimental Mechanics, 21(3), pp. 100-104 (1981). 4. Post, D., “The Moiré Grid-Analyzer Method for Strain Analysis," Experimental Mechanics, 5(11), pp. 368-377(1965). 5. Post, D., Han, B., and Ifju, P. G., High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials, Springer Verlag, New York, (1994). 6. Weissman, E.M., and Post, D., “Moiré Interferometry Near the Theoretical Limit," Applied Optics, 21(9), pp. 1621-1623 (1982). 7. Walker, C.A., “A Historical Review of Moiré Interferometry,” Experimental Mechanics, 34(4), pp. 281-299 (1994). 8. Cho, S.M., Cho, S.Y., and Han, B., “Real-time Observation of Thermal Deformations in Microelectronics Devices: A Practical Approach Using Moiré Interferometry,” Experimental Techniques, (submitted for publication September 2001). 9. Mollenhauer, D.H., and Reifsnider, K.L., “Interlaminar Deformation along the Cylindrical Surface of a Hole in Laminated Composites—Experimental Analysis by Moiré Interferometry.” J. Composites Technology and Research, 23(3), pp. 177-188 (2001). 10. McKelvie, J., and Walker, C.A., “A Practical Multiplied Moiré-fringe Technique,” Experimental Mechanics, 18(8), pp. 316-320 (1978).

OPTICAL FIBRE BRAGG GRATING SENSORS IN EXPERIMENTAL MECHANICS OF COMPOSITE LAMINATED PLATES J. BOTSIS, F. BOSIA, M. FACCHINI, Th. GMÜR Laboratory of Applied Mechanics and Reliability Analysis (LMAF) Department of Mechanical Engineering CH-1015 Lausanne - Switzerland

Abstract Optical fibre Bragg grating sensors have proved in the past years to be extremely useful tools for internal strain measurements in layered composite materials. Two applications of this type of technology are presented in this paper. Firstly, embedded Bragg grating sensors are used to study the through-the-thickness strain field of laminated plates in various quasi-static loading cases. It is shown how in three-point bending the onset of non-linearity in the strains can be detected for decreasing span/depth ratios in the specimens. The strain response of the embedded sensors is also investigated in the presence of a non-uniform strain field generated by a concentrated load. Finally, the use of fibre Bragg gratings is demonstrated in dynamic conditions, both in impulsive transient tests and using harmonic excitation. 1. Introduction

In the early sixties the scientific progress in the optics domain brought important advances in the conception of laser sources and low-loss fibre light guides. As a direct consequence, the first experiments using optical fibre sensors were carried out in the early seventies [1]. Optical fibres show certain properties that make them unique in applications as sensors for single and multiplexed measurements of temperature, pressure, strain, etc. Compared to other traditional sensors, they are compact, lightweight, and minimally invasive due to their small size is the standard diameter of the fibre), furthermore they are immune to electromagnetic interference, have greater resistance to corrosion, higher temperature capacity (~200 ºC) and longer lifetime [2-4]. Thus, fibre optic sensors in general and optical Fibre Bragg Grating (FBG) sensors in particular seem to be ideal for so-called “smart structures” [5, 6], where they are used for process monitoring, non-destructive analysis and structural inspection. While FBG sensors are used in several engineering applications, they are also an important tool in experimental mechanics. They can be used to perform experiments that are very difficult or impossible with other standard techniques. 303 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 303–314. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Characterization of the deformation, damage and fracture characteristics of composite materials is often not trivial. From the experimental point of view, surface displacement or strain distributions can be easily determined by various full-field noninvasive methods (visual inspection, interferometric techniques, etc.), but the internal strains cannot be measured with sufficient resolution using existing non-destructive techniques (X-rays, ultrasonic probing, acoustic emission, etc.). Fibre optic sensors, instead, can be embedded in multi-layered composite materials during the fabrication process, and can provide accurate non-invasive internal strain measurements at selected locations. Thus, key experiments can be designed, and data from such experiments can be used to develop or validate pertinent analytical and numerical models. For example, FBG sensors can be used for direct measurements of bridging tractions in model systems [7], help investigate fibre-matrix interface [7], and most importantly characterize the deformation characteristics of layered polymer composite plates [8]. This report is centered on a study of the through-the-thickness deformation characteristics of layered composite plates using FBG sensors. Measurements are carried out a) on simply supported plates loaded under quasi-static line or concentrated load, and b) on a clamped circular plate subjected to dynamic loads. The surface displacements are simultaneously obtained by electronic speckle pattern interferometry. An introductory part on the working principles of FBGs is presented in section 3; materials and methods used in this work are outlined in section 4, and the results of the quasi-static and dynamic experiments are given in section 5. Finally, a summary of results is presented.

2. Fibre Bragg grating sensors From the practical point of view, a Bragg grating can be described as a spatial modulation of the refractive index created along the core of an optical fibre. The Bragg grating generation is based on the photosensitivity property of silica fibres doped, for example, with germanium when illuminated with UV light, and is usually obtained by two different solutions based on the two-beam interference technique or the phase mask method [9]. The operating principle of an FBG sensor [1, 2] can be summarized in the property of the grating to reflect from a broadband source only the wavelength that matches the grating pitch (Fig. 1) through the equation:

where is the Bragg wavelength, is the grating pitch and n is the mean refractive index of the core. The response of the sensor depends on its physical elongation (and the corresponding change in the grating pitch), and on the change in the fibre index due to photoelastic effects. Both parameters are influenced by strain and temperature. With respect to other widely used fibre optic sensors, FBG show an intrinsic self-referencing capability: the signal of interest is the Bragg wavelength and this output does not depend directly on parameters as the total light intensity, the losses in the connecting fibres and couplers, or the source power. Sensor systems using FBGs usually work by injecting a broad band source into the fibre and detecting the back-reflected Bragg wavelength. Such systems can work in

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transmission or reflection configurations and use as source and detector respectively a broad band source (e.g. a superluminescent LED) combined with a spectrum analyzer, or a tunable laser combined with a photodetector. While the response of FGBs is well understood when the strain field is homogeneous, its response on a heterogeneous field is not completely resolved. In such cases, special iterative procedures should be used to determine the strain distribution along the fibre [10]. In this work, the assumption is made that the axial strain experienced by the fibre is equal to the longitudinal strain in the host (good adhesion between the fibre and matrix is essential to fulfil this condition), and the ambient temperature is maintained constant during experiments.

3. Materials and methods 3.1. SPECIMEN PREPARATION AND CHARACTERIZATION For quasi-static tests, the specimens are in-house fabricated glass/polypropylene (glass/PP) laminated plates whose planar dimensions are 290x250mm. Two symmetrical cross-ply lay-ups are considered, and corresponding to thicknesses of 5 and 10mm, respectively. Similarly, 400x400mm, 5-mm thick, specimens are used for dynamic tests. The prepregs, supplied by Vetrotex (France), are 0.625mm thick and based on a 4/1 fibre weave (four times as many fibres in the weft direction as in the warp direction), which means that the single plies effectively behave as orthotropic unidirectional layers when loaded in the two principal directions. The laminates are autoclave-produced at 10 bar and 190°C, and the FBG sensors are included between plies prior to consolidation. The embedded fibres are 125 µm diameter, polyimide-coated, standard monomode optical fibres. Bragg gratings are written into the fibres by means of a phase mask technique, using a pulsed excimer laser. They are 3 mm in length and the wavelength is centred at 1530 nm. The stripped

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length of the optical fibre at the location of the grating is subsequently treated with silane to ensure strong adhesion with the PP matrix of the composite. The sensors are placed centrally in the specimens, as shown in Fig. 2. To derive strains at different locations through the thickness, various identical specimens are fabricated, differing only in the depth location of the sensor. Thus, in one specimen the FBG sensor is embedded between the first and second plies, in another between the second and third plies, and so on.

Standard quality checks are carried out to verify that the level of porosity and the thickness uniformity are acceptable in order to ensure reproducibility in the mechanical properties of identical specimens. Single-layer material constants are also determined for the glass/PP laminae and for the considered laminates by means of standard tensile tests and modal identification tests [11] The single-layer values are subsequently used in finite element simulations. Further characterization of the fabricated specimens includes checking the proper alignment of the embedded fibres and estimating the uncertainty on their depth location. Having observed various sections of the laminate through an optical microscope, the latter is estimated to be half a fibre diameter, i.e. 62.5 [8]. Significant errors can be made in strain measurements when the sensor location differs from that expected, especially in strongly non-uniform strain fields, as in the region surrounding the application point of a concentrated load (see section 5.1.2). Therefore, the exact position of the grating along the fibre is also checked once the latter is embedded in the laminate, using the Optical Low-Coherence Reflectometry (OLCR) technique [12]. A commercially available HP 8504B precision reflectometer is used for this measurement.

3.2 EXPERIMENTAL TECHNIQUES Experimental inspection of the specimens is carried out by combining different techniques: embedded FBGs for internal local strain measurements, Electronic Speckle Pattern Interferometry (ESPI) for the determination of distributed surface strain, and a laser vibrometer to monitor local surface response in dynamic tests.

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The measurements with the embedded fibre Bragg gratings are carried out using a tunable laser diode source, and the back-reflected signal is directed via a 2x2 coupler to a photodetector. The acquisition and treatment of the signals are done via in-house programmed software. The axial strain in the fibre, which corresponds to the longitudinal strain of the host material in the fibre direction and at the location of the sensor, is related to the Bragg wavelength shift through the well-known relation

where is a sensitivity factor of the fibre [9]. All sensors used in this study are calibrated previous to embedding, and an average value of is found. The system, although quite slow, allows the determination of the peak wavelength shift with a precision of 10 pm, corresponding to a strain resolution of l0µm and under. Electronic Speckle Pattern Interferometry is a powerful, non-contact tool to measure the distribution of displacements occurring on the surface of a specimen with an accuracy of fractions of micrometers [13, 14]. Two standard set-ups exist: the out-ofplane and the in-plane configurations, sensitive to normal and tangent components to the observed surface, respectively. Both are employed in this study. A common 0.632mm wavelength He-Ne laser is used, together with a standard Sony 768x572 pixel CCD camera. Acquisition and processing of the speckle patterns are carried out with commercially available software, and phase maps are derived by means of a standard phase-shifting procedure. In section 5.1, an in-plane configuration is used to measure surface displacements on the deforming specimens, and therefore to derive surface strains at the location where internal strains are measured via the embedded FBG. Instead, dynamic out-of-plane ESPI measurements are described in section 5.2. In this case, the shape of stationary vibration modes can be retrieved by using a special approach called “time-average” [15, 16]. Finally, a commercially available Polytec laser vibrometer is used to measure the surface velocity of specimens subjected to dynamic loading in section 5.2.

3.3 NUMERICAL SIMULATIONS The experimental results of the quasi-static experiments are compared to numerical predictions, using finite-element-method (FEM) models of the laminates in the two loading configurations considered herein. Simulations are carried out using the I-DEAS and ABAQUS codes. The specimens are modelled using solid-element meshes, experimentally-derived single-layer material constants for each layer, as well as the appropriate ply orientation and thickness (0.625mm). A typical model consists of a 54×50×n-element mesh, where n is the number of plies in the laminate considered. The mesh is sufficiently refined to avoid using excessively thin elements. Additionally, in simulations involving concentrated loads, the mesh is more refined in the region surrounding the load-application point. The simulations are carried out using linear hexahedral I-DEAS elements and a selective integration scheme for the computation of the stiffness matrices [17]. Due to some variation in the specimen thickness and possibly due to warping effects, contact between loading/supporting members and the plates does not run uniformly along the whole plate width. Therefore, these contact

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lengths are measured experimentally by using pressure-sensitive film and boundary conditions are chosen accordingly. These amount to enforcing zero z-displacements along the measured line of contact between the plate and the supports, and, in the case of three-point bending, distributing the total applied load uniformly along the measured line of contact between the plate and the loading cylinder. Having included these data in the FEM models, and thus employed the appropriate boundary conditions, the corresponding strain fields are then calculated for the entire structure. Thus, the strain distribution is evaluated at the centre of the plate, in the xy position corresponding to the location of Bragg grating sensors.

4. Experimental measurements 4.1. QUASI-STATIC BENDING TESTS An issue that is of great interest in composites is the study of the type of stress and strain distributions that occur through the thickness of a laminate subjected to flexural deformation. It is well known that because shear moduli are generally smaller than in isotropic materials, shear effects are in most cases non-negligible, and account for nonlinear distributions [18]. It is also known that this effect is all the more pronounced as the thickness of the laminate increases, though in the literature this has often remained a rather qualitative statement. The combined use of embedded FBG sensors and ESPI can therefore be used to experimentally highlight the type of nonlinearity present, and check the range of pertinent parameters in which this occurs.

4.1.1. Quasi-static three-point bending test The first loading configuration considered is a three-point bending set-up, as shown in Fig. 2a Measurements are carried out in a specifically designed loading frame that allows ESPI measurements on the specimen surface. Loading is applied by means of a displacement-controlled motor, and the force is measured by a load cell. The support span is variable, in order to provide the possibility to perform measurements at various span/thickness, or span/depth (s/d) ratios, a parameter used to classify plates as “thin” or “thick”. In this study, s/d varies between 50 and 10. Due to the difference in behaviour of the laminate in tension and compression, experiments are carried out with the FBG sensors situated in the laminate half subjected to tension only. For this reason, results are presented relative to only half of the laminate. All measurements are carried out in the linear elastic range, over two or three loading/unloading cycles. Typically, FBG and ESPI readings are taken at loading steps of about 100N, and a maximum total load of 1500N is reached. Due to the viscoelastic properties of the PP matrix, readings are taken once the load value has stabilized, after a time lapse at least equal to the relaxation time of the material. For each specimen and support span, a strain-load curve is constructed. The data show good linearity and are approximated by a linear fit. Thus, the slope of the fit, corresponding to the strain per unit load at a specific depth location, is derived. Figures 3a and 3b summarize results for s/d=50 and s/d=10, the two extreme cases considered. The experimental strain/load values as a function of normalized laminate thickness are shown, together with the corresponding FEM-calculated distributions. Vertical error bars on the FBG sensor data are due to the uncertainty on the depth location, while the horizontal ones are due to the uncertainty in the determination of the wavelength of the

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Bragg-peak shift. This uncertainty on the strain is in some cases greater than that due to the resolution of the tunable laser. This is because in woven composites, as in this case, the reinforcing fibres can induce micro-bending in the embedded optical fibre and a relatively large loss of signal, and as a consequence give rise to noisier reflected peaks. As for the error bars on ESPI values, these are mainly due to the uncertainty on the determination of the impinging angles of the illuminating beams. Good agreement between experimental and numerical data can be noted in Fig. 3, as well as a clearly nonlinear behaviour in the thicker plate. The experimental data in the latter case would seem to deviate slightly from the numerical curve and display a more marked nonlinearity. This could be due to the afore-mentioned difficulties in accurately modelling the boundary conditions experienced by the plates, especially in the thicker specimens.

4.1.2 Quasi-static concentrated loads The second experimental configuration considered is shown in Fig. 2b. The specimens are simply supported on two sides and are subjected to an out-of-plane concentrated load applied in the centre of the plate. From the experimental point of view, this loading configuration presents some new characteristics with respect to the previously considered one: firstly, the strain field in the vicinity of the loading point displays an abrupt variation, so that the type of FBG response must be monitored for signal modification due to the non-uniform field; secondly, the application of a point load, rather than a line load, eliminates the non-uniform contact problem encountered previously in the load application. The first remark to be made in these experiments is that the detected Bragg peaks do not undergo any broadening once loads are applied, nor is there any secondary peak formation. This is an indication that the strain field experienced by the sensor is

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sufficiently uniform, or in other terms, that the sensor length (3mm) is sufficiently small for the measurement to be considered point-wise. If this were not the case, a more elaborate analysis of the reflected peaks would be called for [10]. Results are shown in Fig. 4 for s/d ratios of 25 and 10, and once more experimental data are compared to FEM-calculated values. There is some discrepancy between ESPI-measured points and numerical predictions. This is probably because these measurements are affected by a systematic error which is hard to estimate, but which in all cases leads to an overestimate, due to the presence of a large out-of-plane displacement component. The latter influences the measurement because due to the curvature experienced by the plates in bending, the reference axes of the interferometric set-up no longer coincide with the directions which are normal and tangent to the observed surface. This effect is more pronounced in the case of concentrated loads, as compared to three-point bending with a line load. It is apparent how the nonlinearity of the through-the-thickness strain distributions increases with respect to three-point-bending. Furthermore, it can be noticed that even though the laminate is symmetrical, the neutral plane no longer coincides with the midplane, rather it is shifted towards the load-application point. These two facts can be explained by observing that the strain field results from two contributions, one deriving from the overall bending, the other from the local effect of the concentrated load [19]. However, it is hard to separate these two effects.

For this reason, it is of interest to monitor the through-the-thickness strain behaviour in other sections of the laminate, at a distance from the load-application point, where local effects are negligible. This can be done by shifting the specimens from their central position in the loading frame by fixed amounts in the x-direction. The resulting loading configuration remains identical, but for the location of the embedded FBG sensor.

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Thus, the x-dependency of the strain distribution can be determined for each layer and once again compared to numerical predictions. An example of such a derived strain distribution is given in Fig. 5. The measurements are relative to the FBGs embedded between the second and third plies in the laminate (1.25mm from the free surface), for a support span of 100mm (s/d=10). The location x=0 corresponds to the sensor placed centrally, and the specimen is shifted by 2-3 mm between successive measurements. There is good agreement between simulations and experimental values, except for the points in the vicinity of the supports This could be again due to nonuniform contact between the supports and the plates, an effect that is hard to model in FEM simulations. To extract the strain distribution through the thickness at various sections of the laminate, several curves such as that shown in Fig. 5 should be constructed for the sensors embedded at different depth locations. Such results will be reported elsewhere.

4.2. DYNAMIC TESTS Although quasi-static conditions can be observed in some cases in real applications, existing structures are most likely submitted to dynamic solicitations. Because health monitoring and reliability evaluation are some of the most important objectives for the application of sensors in composite materials, the potential of FBG sensors to carry out dynamic measurements is fundamental, and their applicability to this domain needs to be verified. Therefore, emphasis in research activities involving FBG sensors at LMAF, EPFL, is also on the study of the mechanical response of composite plates in dynamic conditions. Thus, a series of preliminary tests has been carried out to examine the reliability, advantages and performance of FBG sensors in dynamic tests. The use of FBG sensors for dynamic phenomena is documented in the literature and systems able to detect strain changes in the kHz range already exist. Most of them (MicronOptics and BlueRoad Technologies among others) are based on the ability to track the wavelength of the Bragg peak. The solution adopted in this study, instead, is

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simply based on the intensity change of a selected back-reflected wavelength. The working principle is shown in Fig. 6: once the response curve of the grating in quasistatic conditions has been obtained, the wavelength corresponding to the maximum slope is determined and the injected light is fixed at this value. This working point guarantees the highest sensitivity and the maximum dynamic range. From the plot it is clear that a small change in the Bragg wavelength corresponds to a shift of the complete curve, which in turn results in a change of the voltage read-out of the photodetector. The latter signal is then acquired with a digital signal analyser, and information on frequency and amplitude can be extracted.

As mentioned previously, the considered specimen is an eight-ply square plate of 400-mm side and 5-mm thickness. The plate is clamped between two steel blocks with a circular 300-mm diameter opening, so that the tested assembly can be assimilated to a 300-mm diameter clamped circular plate. The specimen contains a centrally embedded FBG sensor (Fig. 2c), whose grating wavelength is 1529.65 nm and maximum-slope wavelength is 1529.52 nm. The specimen surface is inspected at the same time by an ESPI system and by an interferometric Polytec vibrometer whose beam impinges on the centre of the plate, and the results are compared to the FBG sensor read-out. Two different solicitations are applied: harmonic excitation and impact loads. Firstly, the rear side of the specimen is exposed to spherical acoustic waves generated by a loudspeaker driven by a sinusoidal signal produced by an electronic function generator. The frequency and the amplitude of the signal are tuned and the signals obtained by the three different techniques are acquired and compared. The ESPI technique allows the acquisition of fringe patterns only in the proximity of stationary vibrations: resulting interferograms corresponding to the first four modes are presented in Fig. 7. The vibrometer is sensitive to the velocity of the targeted central point and the FBGs is sensitive to the longitudinal strain component in the composite at the location of the sensor (see Fig. 2c). The signals can be presented in the time domain (Fig. 8a) and,

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as expected, they are in phase opposition. Frequencies can be easily determined if the signal is represented in the Fourier domain (Fig. 8b). In the proximity of the stationary frequencies, detected via ESPI, the response amplitude of the FBG sensor and of the vibrometer increase, as expected. In a second experiment, the specimen is excited by impact loads at different locations. The signals “read” by the FBG system and by the vibrometer are acquired and then transformed in the frequency domain to obtain the transfer functions. A comparison between the FBG and vibrometer signals demonstrates that peak amplitudes occur at approximately the same frequencies, which also correspond to the stationary modes detected with the ESPI approach.

The adopted read-out method can probably give better resolution than those based on Bragg wavelength tracking, and the frequency range can be considerably higher (limited in fact only by the speed of the signal acquisition system). However, some problems related to amplitude fluctuations, probably due to interferometric reasons, are still under investigation.

5. Summary The use of FBG sensors for deformation measurements and inspection of layered composite materials in quasi-static and dynamic loading conditions has been presented. The through-the-thickness strain distribution of laminated plates subjected to bending

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loads has been derived and the feasibility of similar measurements in examples of dynamic solicitation has been demonstrated. While no attempt is made to describe all the multiple applications of FBG sensors, this article intends to show how the use of this type of sensor could be very important in experimental mechanics. Key experiments can be designed to investigate the mechanical behaviour of materials and structures and verify pertinent models.

6. Acknowledgements The authors wish to thank the Institute of Applied Optics, EPFL, for supplying the Bragg sensors used in this study and for support on related matters. The Laboratory of Composite and Polymer Technology, EPFL, is also gratefully acknowledged for advice and collaboration in the preparation of layered specimens. This work is supported by the Swiss National Science Foundation, Grant no. 21-52486.97.

7. References 1. 2. 3. 4. 5. 6. 7. 8.

9. 10.

11. 12. 13. 14. 15. 16. 17. 18. 19.

Culshaw, B. and Dakin, J. (1988-1997) Optical Fiber Sensors, Vol. 1-4, Artech House, Boston Grattan, K.T.V. and Sun, T. (2000) Fiber Optic Sensor Technology: an overview, Sensors and Actuators A, 82, 40-61. Collected papers of the international conferences on Optical Fiber Sensors (OFS), 1983-1997, CDROM, SPIE/IEEE, Washington Kersey, A.D. (1996) A review of recent developments in fiber optic sensor technology, Optical Fiber Technology 2 (3), 291-317. Udd, E. (1995), Fibre-optic smart structures, Wiley, New York. Measures, R. M. (1992) Smart composite structures with embedded sensors. Composites Engineering 2, 597-618. Studer, M. (2001) Étude des forces pontantes dans un matériau composite à 1’aide de réseaux de Bragg dans des fibres optiques. PhD Thesis n. 2451, EPFL. Bosia, F., Botsis, J., Facchini, M., and Giaccari, P. (2001) Deformation characteristics of composite laminates, part I: speckle interferometry and embedded Bragg grating sensor measurements. To be published in Composites Science and Technology. Hill, K.O. and Meltz, G. (1997) Fiber Bragg grating technology fundamentals and overview Journal of Lightwave Technology 15 (8), 1263-1276. Peters K, Studer M, Botsis J, Iocco A, Limberger H, Salathé R. (2000) Embedded optical fiber Bragg grating sensor in a nonuniform strain field: measurements and simulations, Experimental Mechanics 41 (1), 19-28. Cugnoni, J. (2000) Identification des propriétés constitutives des stratifiés par des méthodes mixtes numériques-expérimentales. Internal Report, LMAF, EPFL. Sorin, W. V. (1996) Fiber optic sensing using low-coherence interferometry, Proc. SPIE 2872, 4047. Jones, R and Wykes, C. (1983) Holographic and speckle interferometry, Cambridge University Press. Wykes, C. (1982) Use of Electronic Speckle Pattern Interferometry (ESPI) in the study of static and dynamic displacements), Optical Engineering 21(3), 400-406. Lu, B. et al. (1989) Time-Average Subtraction Method In Electronic Speckle Pattern Interferometry, Optics Communications, 69 (3), 214-218. Lokberg, O.J. and Hogmoen, K. (1976) Vibration Phase Mapping Using Electronic Speckle Pattern Interferometry, Applied Optics, 15, 2701-2704 Hughes, T. (1987) The Finite element method. Prentice-Hall Int Reddy, J.N. (1997) Mechanics of Laminated Composite Plates: Theory and analysis. CRC Press. Timoshenko, S.P. and Goodier, J.N. (1970) Theory of elasticity, 3rd Ed. McGraw-Hill Int.

OPTOELECTRONIC DISPLACEMENT MEASUREMENT METHOD FOR ROTATING DISKS

C.E. BAKIS, B.J. HALDEMAN, R.P. EMERSON Composites Manufacturing Technology Center The Pennsylvania State University University Park, PA 16802

Abstract

Radial displacements on an axial surface of a rotating disk can be measured with optoelectronic sensing devices consisting of infrared emitters and phototransistors. The aim of this investigation is to improve the stability and robustness of a previously described optoelectronic measurement system. Stability is improved by a computerized algorithm for the control of the emitter intensity based on feedback from a compensation portion of an optical pattern placed on the disk. Robustness is improved by using a laser emitter that allows for a larger standoff distance in comparison with the previously used light emitting diode. Radial displacements obtained with automatic intensity control of the laser emitter agreed with theoretical values in a glass/urethane composite disk to within ±1 µm at rotational speeds up to 3300 rpm. With further development, the proposed displacement method has application in the noncontact measurement of radial and hoop strains through the radial thickness of high-speed axisymmetric rotating machinery such as energy storage flywheels. 1.

Introduction

High-speed flywheels are currently attracting interest in energy storage applications where high rates of charge and discharge, high cycle life, and operation in a wide temperature range are required. Ongoing flywheel research activities are focusing on the improvement of specific energy and power density, the reduction of cost, the evaluation of performance, and the prediction of service life. A technical focus area associated with the last two activities mentioned is the measurement of deformation in flywheel rotors. It is beneficial to measure full-field deformations with high fidelity over extended periods of time so that time-dependent structural analyses can be validated. Various experimental methods have been used to measure radial and circumferential strains in high-speed flywheels. Examples of some of the better-known methods include x-ray diffraction, speckle interferometry, and electrical resistance strain gages. A specialized method of interest, attributed to Simpson and Welch [1], is known as the optoelectronic method since it uses simple electro-optical devices and electronic circuits to measure the real-time radial displacements of an optical pattern 315 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 315–324. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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applied on the axial end of a rotating object. Bakis and Emerson [2] subsequently developed a related optoelectronic method integrated with digital data acquisition and processing and demonstrated several key improvements: (i) an order of magnitude improvement in displacement sensitivity at rotor speeds twice as great as had been previously reported; (ii) the ability to separate rigid body vibration from axisymmetric radial displacement; (iii) a unique optical pattern that enabled compensation for changes of intensity and distance between the sensor and rotor during a test. The optoelectronic displacement method relies on the measurement of changes in angular width (duty cycle) of a reflective optical pattern applied to the axial surface of a rotor. An example of a multi-region reflective pattern is shown in Fig. 1, where the shaded triangular areas are considered reflective. In this illustration, four “lobes” of the pattern are evenly spaced at 90-deg. azimuthal intervals on the rotor. Each lobe contains two annular regions where displacements can be measured. For example, as the rotor spins at a known speed, a stationary sensor records angular duty cycles based on the time intervals spent “on” each individual reflective triangular patch encountered along the dashed path shown within one annulus. Since, in practice, the duty cycle versus radius relationship deviates from ideal, a unique relationship for each displacement patch needs to be determined by a calibration procedure with the rotor spinning at a very low speed – the “undeformed” reference condition. This relationship is stored in a look-up table. The duty cycle at each original radius on the rotor can be shown to remain constant for any rigid-body or flexible-body displacement of the patch. Thus, the radial displacement of the rotor at each nominal patch position is found based on the respective change in measured duty cycle. By measuring such total displacements, at 90-deg. (in this example) azimuthal intervals and fitting a sinusoidal function to the measurements as a function of azimuthal angle rigid body radial vibration, can be separated from axisymmetric radial displacement due to strain, u:

where is the angular phase of the vibration. Measurements can be made with sensors at a number of radii in order to calculate radial and hoop strains at several locations through the radial thickness of the rotor:

The main optical requirement of the pattern is that it has appropriately placed regions of contrasting reflectivity. The best radial displacement sensitivity is obtained when the duty cycle of the pattern changes most rapidly per unit change in radial position on the rotor. Additionally, the pattern could incorporate a means of correcting measurement errors that can arise if the intensity of the emitter changes or if the standoff distance between the sensor and rotor surface changes due to vibration of the rotor or thermal expansion of the test apparatus. Such corrections can be easily done if a compensation patch of fixed angular width as a function of radius is placed nearby each displacement patch. One lobe of such a pattern used in this investigation is shown

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in Fig. 2. A method for algebraically correcting a duty cycle based on the deviation of a compensation patch measurement from a presumed fixed value was described in Ref. [2].

An optoelectronic sensor aimed at the reflective pattern consists of an emitter projecting a small spot on the surface of the rotor and a phototransistor (detector) that provides an “on” or “off” logic signal according to the measured intensity of the reflected spot. An illustration of the sensor used in Ref. [2] is shown in Fig. 3. A 20 MHz digital counter triggered by the logic signal records the elapsed time that the sensor is on a reflective region as the target surface rotates. Using a tachometer, the elapsed times are converted to duty cycles. The collection of data and calculation of displacements is done with a personal computer equipped with a data acquisition card and commercially available software for data acquisition and process control. The objective of the present investigation is to explore improvements in the optoelectronic displacement measurement method reported previously [2] by (i) replacing the previously-used infrared light emitting diode (IR LED) emitter with an IR laser emitter and (ii) developing an algorithm for controlling emitter intensity with a computer. The laser emitter offers the potential advantages of higher overall intensity and less divergence, which lead to certain advantages in implementation and accuracy of the optoelectronic displacement measurement system such as larger standoff distance and potentially less sensitivity to out-of-plane vibration. Computer control of the intensity of the emitter simplifies operation of the displacement measurement system by automatically correcting for duty cycle fluctuations induced by external sources such as ambient temperature fluctuations during a test.

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Experimental Setup

Displacements were measured on a glass-fiber-reinforced polyurethane composite disk (3.8-cm ID, 32-cm OD, 25-mm thickness) attached via a ring-shaped aluminum hub (2.4-cm ID and 3.8-cm OD) to a 6-mm dia. steel shaft. The rotor was spun directly by a DC electric motor mounted via elastomeric vibration isolators to a massive steel base. The spin tests were done in open air at speeds as high as 3.5 krpm. The glass/urethane disk was fabricated by a filament winding process in which the fibers are oriented only circumferentially. Because the polyurethane matrix is very compliant, the hoop-direction modulus of elasticity is much higher than the radialdirection modulus. The material properties of the glass/urethane material are as follows: density, hoop modulus, MPa; radial modulus, kPa; major Poisson’s ratio, 0.36 [3]. For computing theoretical displacements in the rotor, a plane stress, anisotropic, axisymmetric, elasticity model was used [4]. The aluminum hub and glass/urethane disk comprised the rotor model. The reflective pattern was applied to the axial surface of the rotor using a standard black and white photographic process. First, the rotor was painted with sequential layers of reflective silver paint, polyurethane, and then (in a darkroom) a light-sensitive emulsion. The pattern, consisting of four lobes such as that shown in Fig. 2, was transferred to the emulsion using a contact negative that had been created with a photoquality ink-jet printer. Since the largest size negative that could be printed at once was smaller than the rotor, the four lobes were printed separately and then carefully aligned relative to each other and attached to a clear acrylic plate. This process admittedly allows the opportunity for errors in aligning the lobes on the rotor, even though much care was taken to minimize such errors. A final layer of polyurethane was applied to the developed rotor to help protect the pattern from scratches. The new emitter chosen for evaluation in this investigation is a 5-mW infrared (IR) laser micro-module with a wavelength specification of 904±10 nm. The IR light emitting diode (LED) used in previous research [2] was also used extensively in the experiments. The IR LED has power and wavelength specifications of 24 mW and 880±30 nm, respectively. The IR phototransistor used with either emitter has a 70 ns rise and fall time. The setup of the laser sensor, shown in Fig. 4, was similar to that shown for the LED in Fig. 3, except that the standoff between sensor and rotor surface was increased to 76 mm versus 4.5 mm in the case of the LED. The greater intensity of the laser enabled the larger standoff. One of the possible benefits of such an increase in standoff is the reduction in sensitivity of the measurements to minute variations in standoff caused by rotor vibrations or differential axial thermal expansions in the test apparatus, although this benefit was not explicitly investigated in the work reported here. 3.

Intensity Control

Given that inevitable changes in intensity of the IR radiation reaching the detector will cause changes in the measured duty cycles on a given optical patch even though there is no real change in displacement on the rotor, it is important to either account for such changes with additional measurements and data processing as was done in Ref. [2] or

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minimize the magnitude of change to an inconsequential amount, as was done in this investigation. The new intensity compensation method adopted in this investigation automatically adjusts the intensity of the emitter to ensure that a fixed angle is always measured on the compensation patch adjacent to the subject displacement patch. In this manner, intensity fluctuations caused by changes in ambient temperature or out of plane rotor displacement can be eliminated. Calibration of the displacement patch using intensity control proceeds as described earlier. A beneficial side effect of this approach for improving the long-term stability of the measurements is that small defects in the angular width of any compensation patch with respect to radius are corrected as well. In an experiment where the rotor undergoes real changes in radial displacement, the intensity control algorithm ensures that the duty cycle of the compensation patch remains constant, which ensures that the radial displacement versus duty cycle information stored in the lookup table is always correct. A flow chart of the intensity control algorithm for the laser sensor is shown in Fig. 5. The algorithm for the LED sensor is similar except for certain details such as the emitter voltage.

5. Results

Figure 6 shows the results of a low-speed calibration experiment relating relative radial position to duty cycle of the displacement patch while the intensity control algorithm enforces a 4.0000±.0001-deg. angle on the adjacent compensation patch using

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the IR LED. Ideally, the duty cycle versus radius relationship for a displacement patch is a smooth, slightly concave-up curve. The undulations in the calibration curve in Fig. 6 are attributed to pattern misalignment and edge roughness either in the displacement patch or the compensation patch that was used to control the intensity of the LED. As evidenced in Fig. 7, the emitter voltage needed to be changed substantially to maintain a constant compensation patch angle during this experiment. No voltage changes should be needed in the ideal case since the compensation patch supposedly has a constant duty cycle with respect to radius. It is demonstrated by the uniformity of the angular width of the compensation patch in Fig. 6 that the intensity control algorithm can effectively perform its intended function.

Spin tests carried out at a constant rotor speed (500 rpm) have been performed using the IR LED emitter to verify that the correction algorithm operates correctly for changes in standoff and in the ambient temperature. Figure 8 shows that over a very large (±200 µm) change in standoff, the apparent radial position of the IR LED sensor changed by only +1/-2 µm. Such a small apparent change could well be an actual change due to the difficulty in ascertaining with comparable or better accuracy the alignment of the instrumented stage used to move the sensor. Changes in standoff beyond ±200 µm cause much larger deviations in apparent radial position, possibly due

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to the illuminated spot leaving the field of view of the detector. Such a limitation can be envisioned by consideration of the triangulation involved in aiming the detector on the spot illuminated by the emitter (Figs. 3 and 4). Significant changes in standoff move the spot outside the field of view of the detector, although the problem is arguably less severe with the smaller included angle associated with larger standoff distances.

Figure 9 shows the effects of a sudden drastic heating of the IR LED sensor with a heat gun while using the intensity control algorithm to maintain a fixed compensation patch angle of 4.0000±0.0005 deg.. Within about 1 minute, the controlled voltage supplied to the LED returned the compensation patch angle to the specified value. Drastic temperature spikes such as this (estimated to be at least 5°C) are not typical in such short time intervals in flywheel testing. Smaller amplitude, longer-duration temperature changes encountered in this investigation were corrected without any outof-specification deviations in compensation patch angle.

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Calibration data for all four displacement patches in the 130-140-mm annulus on the glass/urethane rotor, recorded with the IR LED, are shown in Fig. 10 along with the theoretical relationship between angle and position found according to the algebraic equations given in Ref. [2]. The intensity of the LED for reading displacements in all four lobes was controlled in this experiment based on feedback from the compensation patch of only Lobe 1 rather than each respective lobe. Additional discrepancies among the curves can be caused by pattern roughness and misalignment of the individual lobes. These discrepancies are not of major significance assuming that the calibration curves are not speed dependent. Additional research is needed to determine if this assumption is in fact true. Regardless, since the look-up table created using data from Fig. 10 is used to interpolate radial locations based on measured duty cycles from a deformed rotor, it should be kept in mind that large discontinuities on a given calibration curve within the radii of interest might lead to a less reliable calculation of radial position. Based on the calibration data of Fig. 10, axisymmetric radial displacements along with rigid body vibrations were computed for various rotor speeds in two separate experiments (Fig. 11). Vibration was separated from axisymmetric displacement using eq. (1). Theoretical axisymmetric displacements were found using the plane stress elasticity equations for a dual material (aluminum and glass/urethane) polar orthotropic disk [4] and the material properties given earlier. There is some discrepancy between the theory and experiments, particularly above 2 krpm where the vibration, indicated by the vertical bars in the graph, increases dramatically as a resonant speed of the rotor was approached at 3500 rpm. Additional balancing of the rotor near the resonance could ameliorate this problem.

Although the results shown above demonstrate the robust characteristics of the IR LED sensor with intensity control, similar robustness was realized with the laser in similar types of tests, with the main exception being the amount of electrical noise in the laser measurements. This noise was easy to eliminate using data sampling and averaging, however.

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Figure 12 shows radial displacement versus speed for the same 130-140-mm annulus of the glass/urethane rotor as in Fig. 11, with the laser replacing the LED in this case. As in the LED experiment, the intensity for measurement of all four lobes was controlled based on the compensation patch of only Lobe 1. The measurements are within ±1 µm of the theoretical displacements for all rotor speeds evaluated. A direct comparison of the two types of emitters on a smaller displacement scale in Fig. 13 reveals that the theoretical curve is closer to the laser results than the LED results.

5.

Conclusions

An algorithm has been developed that controls emitter supply voltage using feedback from a compensation patch. In controlling the measured angular width of the compensation patch to remain constant during a test, consistent displacement measurements can be made on the nearby displacement patch regardless of external effects such as changes in ambient temperature and sensor standoff distance that would otherwise adversely influence the measured displacements. Preliminary results show that an IR laser emitter can provide slightly more accurate results in comparison to the previously used IR LED emitter. When using the laser, measured axisymmetric displacements agree to within ±1 µm of theoretical values in a glass/urethane composite rotor spun as fast as 3300 rpm. Implementation advantages of the laser over the LED accrue mainly to its greater intensity and collimation, which allow a larger standoff between the rotor and sensor. Greater standoff distances should reduce the sensitivity of the sensor to out-of-plane displacements that can occur during a spin test due to temperature changes in the support apparatus or out-of-plane rotor vibration. The use of a laser in addition to an intensity control algorithm shows strong potential in the creation of a more robust and accurate optical strain measurement technique for high-speed flywheels. Further investigation is recommended to verify the

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full potential of this improved optoelectronic strain measurement system. In particular, the intensity control feedback signal should be taken from the compensation patch located closest to each individual displacement patch. Doing this could further reduce the potential for intensity- or speed-dependent changes in the calibration curves of the displacement patches. 6.

Acknowledgements

The authors acknowledge the support of the US Army Research Laboratory, Aberdeen, MD, for portions of this research. Dr. Jerome Tzeng was the project monitor. 7.

References

1.

Simpson, M .L., and Welch, D.E.: “Optoelectronic strain measurement system for rotating disks,” Experimental Mechanics, 27 (1987), 37-43. Bakis, C.E., and Emerson, R.P.: “Optoelectronic radial displacement measurement on rotors,” Proc. Annual Conference on Experimental and Applied Mechanics, Soc. Experimental Mechanics, Bethel, CT, (2001), 411-414. Gabrys, C.W., and Bakis, C.E.: “Design and testing of composite flywheel rotors,” Composite Materials: Testing and Design, 13-th Vol., STP 1242, S. J. Hooper, Ed., American Society for Testing and Materials, Conshohocken, PA (1997), 3-22. Genta, G.: Kinetic Energy Storage: Theory and Practice of Advanced Flywheel Systems, Butterworth & Co. Ltd., London, 1985.

2.

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DEFORMATION MEASUREMENT OF SHEET METAL FORMING USING PHOTOGRAMMETRY K. ANDRESEN Dr.-Ing., Acad. Director Center of Mechanics, Technical University D-38102 Braunschweig, Germany

Abstract A measuring machine based on a stereo device is developed for analyzing large sheet metal parts deformed by deep drawing. A cross grating on the surface is subdivided into meshes which are uniquely encoded by circular marks. This provides absolute indices for each grating point when taking segments of the surface by the stereo device. Hence overlapping areas can be used to fit together the segments with its spatial coordinates. By means of a theory of large deformation the principal strain and the principal directions are calculated. 1. Introduction

In industry sheet metal forming by deep drawing is simulated with commercial software packages [l]. Since material parameters, friction coefficients and boundary conditions are to be assumed, often the results are not satisfying. To improve this issue, experimental investigations are necessary parallel to the numerical analysis. Therefore the geometry and the flow of material must be measured on the surface of the object. The resulting surface coordinates then can be compared with numerical results and hence the before mentioned parameters may be adapted. Usually a regular grating (circles or crossing lines) is fixed on the undeformed plane surface and its coordinates are measured in the deformed state mainly in critical areas by optical methods. In this paper the method is extended to measuring the whole surface of the object by fitting small local segments into a global coordinate system. This is performed by a measuring machine which scans the surface in overlapping areas with a stereo device, consisting of a stiff framework with 3 CCD-videocameras. In each segment local grating coordinates are determined [2]. which are related to the position of the device. If the position and its orientation is known exactly, the coordinates of each segment can be transformed into a global 325 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 325–334. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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coordinate system. Two different procedures were applied for this function. If the device is moved only in direction of the 3 spatial axes, the shifting coordinates are given very accurate by calibration. Then they can directly be added to the local coordinates to derive global coordinates. If the device is rotated, no adequate calibration is performed. Then overlapping coordinates are used to calculate a transformation matrix fitting neighboring segments successively into the global system. 2. Fitting together of segments

The surface of the object carrying a fixed grating is scanned by the stereo-cameras in segments of about which provides the required accuracy of about 1/10 mm for the spatial coordinates. The device is positioned nearly perpendicular to each segment by a 5 axes measuring machine in a distance suited to the depth of focus of the cameras. Within the images the coordinates of grating points are calculated by picture processing with an accuracy of about 1/10 pixel. From adjoined points in the images the spatial coordinates are determined by ray intersection with the above mentioned accuracy. The camera device is shifted by stepping motors in the 3 spatial directions (X. Y, Z) and rotated around the Y, Z axes by angles and resp. According to the quality of the stepping motors and due to a calibration process, the shifting position is measured with high accuracy (1/100 mm). Therefore its coordinates can be added directly to the local point coordinates in the segment yielding global coordinates. When the stereo device is rotated its related position and orientation are known only roughly. Therefore overlapping points in neighboring segments are taken to calculate a transformation matrix. which transforms the local coordinates into the global system. The search for overlapping points is supported by a global index for every grating point. This is provided by subdividing the whole surface into meshes of grating points. Each mesh carries a unique pattern of circular points. The L–like patterns in Fig. 1 define local grating directions The top of an L represents the origin and the smaller arm of the L identifies the x direction. An index pair (k, l) of each mesh is encoded binary by the points parallel to the x, y axis. eg. in the upper left mesh. Thus local indices (i, j) of grating points will be transformed to global ones by

These global indices are used in two ways. First adjoined points within the stereo images of one device position are supplied directly. They are used to calculate the 3D coordinates by ray intersection. Furthermore the global indices are needed to search for overlapping points in neighboring segments. If at least 3 points not on a straight line are overlapping in segment 0 and 1 resp., there exist a translation vector and a rotation matrix depending on 3 rotation angles (around X–axis). (around Y–axis), (around Z axis)

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which transform the coordinates segment 0.

of segment 1 into

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of the neighboring

The two upper left indices, eg 10, mean a transformation from segment 1 to 0. In Eq. (3) are measured values, while are unknown transformation parameters, which explicitly depend on and the rotation angles Putting these values into a parameter vector the nonlinear Eq. (3) can be written symbolically as

This system will be solved iteratively starting with a suitable initial vector

yielding residuals

not equal to zero.

will be improved by linearization

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which evolves to

or

This describes an overdetermined system of linear equations for the increments written as

Application of a least squares method yields

which is a linear equation for the increments

The improved solution then is

This procedure will be repeated, now being the initial vector, until the increments are smaller than a given limit 3. Global transformation The transformation between neighboring segments with overlapping points can be applied successively to all overlapping segments. Only one reference segment must be fixed. because it defines the global coordinate system. The disadvantage of this technique becomes obvious, when looking at an actual segment far from the reference segment. Then the transformation into the reference system consists of a combination of all transformation parameters of the segments lying between the actual and the reference segment. This gives rise to error propagation from segment to segment with decreasing accuracy to the outer segments. Therefore a more global approach was chosen, which directly determines the transformation parameters from each segment i to reference segment 0. This means for overlapping areas in segment i and j:

where are the unknown global coordinates of These equations are setup simultaneously for all overlapping segments considering that for a reference segment the parameters must be given. Then each segment is transformed with the same accuracy, and a higher global accuracy is reached.

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When using the foregoing linearization and the least squares method, it has to be considered, that the overlapping coordinates now are unknown too and hence are included in the parameter vector p. If the values are put into the first part of p, called and the transformation parameters in the second part and the related right hand sides into then one gets the following structure of the normal equations:

Fortunately the very large matrix A, related to the coordinates form and hence easy to invert. Thus the system is split into

is of diagonal

From the first equation one gets

inserted into the second equation yields a linear equation system

Its order is only six times the number of segments, while the order of the original system in Eq.(14) mainly depends on the number of overlapping points, which generally is much larger than the number of segments. This techniques saves up to 90% of computing time. 4. Measuring machine

For scanning curved surfaces of sheet metal parts the stereo device has to be moved along 3 orthogonal axes (X, Y, Z) and it must be rotated around 2 axes, e.g. Y, Z. For these functions a measuring machine was built with standard commercial components as shown in Fig. 2. The components in the upper part of the image realize the linear X, Y–moving and the rotation around the Y–axis, also using a linear device acting on a lever. The lower part shows two linear devices for the Z axis and a table rotating around the Z axis. The linear devices are controlled by stepping motors providing an accuracy of about 1/100 mm. But by assembling the components the orthogonality of the 3 axes does not reach a comparable accuracy. Therefore the deviation from the orthogonal directions will be determined in a calibration process. For this purpose a reference grating on a high quality transparent film (about accuracy 1/100 mm) was fixed on a plane glass plate. The grating is encoded by unique mesh indices as described in section 2. In a first step the grating is used to calibrate the 3 cameras in the stereo device [3]. This defines a projection center and a rotation matrix R for each camera with reference to the global coordinate system, X, Y in the plane of the film and Z perpendicular to it. This system also may be assumed to be fixed to

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the framework of the stereo device. If the device then is moved to a measuring position, the coordinate system is moved likewise, supplying now local stereo coordinates for each segment on the surface. After calibration of the stereo device, the axes of the measuring machine are calibrated. The device is moved successively from the origin into the X, Y and Z direction. Each time pictures are taken from the reference grating and its 3D coordinates are calculated. Then from different positions of the device the difference between two points on the grating is determined. The same difference is known by the grating pitch and the related indices. Usually there is a small

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deviation between these two difference vectors, which are used to calculate two correcting angles for each axis. In practice a large number of points and their differences are evaluated to reduce possible errors. Statistical investigations for the actual setup show the following results for the X-axis.

The angles describe the deviation of the actual X-axis from the global X-axis by a rotation around the Y,Z-axes. is the actual shift in X-direction. Measuring the distance of a grating point from the origin on the reference grating yields the following results:

E is the unit matrix and is the rotation matrix calculated from the rotation of the X-axis by the angles These results show that especially the components in Y, Z direction are significantly improved. Similar calibration in Y-Z-directions show comparable improvements.

4.1. DISPLACEMENT AND STRAIN For the calculation of large strain a thin surface with no extension perpendicular to it’s local plane is assumed. Hence only strain within this plane is available from the grating. The strain perpendicular to the surface, describing the reduction of thickness, is determined afterwards from the condition of volume constancy, which is valid for plastic deformation of metals. The translation of any point x from the plane undeformed surface into its deformed state is described by 3 displacement functions in directions (x, y, z) resp., written in vector form by

An incremental vector into

in the undeformed plane state is deformed by

where the transformation matrix F is known as deformation gradient

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Since F describes a deformation U as well as a rotation

one has to eliminate the rotation

From the Cauchy-Green tensor G

it can be seen, that the square root of G yields the deformation tensor U. According to Ref. [5] one has

with

and det G the related determinant. The elements of the deformation tensor U

include the large strain components in the tangential plane. The displacement functions will be developed similar to Ref. [4]. The four neighboring meshes around a grating point form a basic element, which will be measured in the undeformed state and the deformed state, resp. Both elements will be moved into a virtual origin, where the axes of the deformed state are rotated as to coincide approximately with the undeformed ones. Then for each point except for the origin a displacement vector

is given. Its three components are used to calculate approximating displacement polynomials by a least squares method, e.g. in x–direction

Similar equations with parameters tions the gradients in the origin

Then the deformation gradient is

hold for are

From these approxima-

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According to Eq.(27) two-dimensional strain tensor

is calculated. The principal strain and the related principal direction are derived from the eigenvalues and the eigenvectors of S. The volume constancy then requires

which is used to calculate 5.

the reduction in thickness.

Results

By use of the measuring machine the surfaces of different, parts of sheet metal were measured mainly for comparison with numerical simulations. A project is going on to analyze the spring back of parts when leaving the mould. As an example in Fig. 3 a connecting piece of a tank filling device is shown after the first state of deep drawing (pitch: 2.5 mm, line width: 0.4 mm, sheet metal thickness: 1 mm). The 3D-coordinates of the object are shown in Fig. 4 together with the global reference marks derived from 30 overlapping segments. The grey values (alternative in colors) represent the strain which decribes the reduction in thickness of the sheet metal. In this example also the limits of the method become obvious. Grating lines in regions of large curvature or in areas of deep grooves cannot be determined, because the distance between the lines in the images becomes too small. This results in low contrast, where the picture processing algorithm fail.

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Acknowledgements The research project was partly granted by EFB/AIF 12275 BG. Thanks to the following coworkers. Mr. Hübner constructed and built the mechanical part of the measuring machine. Mr. Algermissen programmed the calibration of the measuring framework and Mrs. Hentrich developed software packages for 3Dmeasuring. Etching of the objects was performed by Mr. Schatz, Techn. University Dresden.

References 1. 2.

”Pam Stamp” from Esi Group France: http://www.csi.fr/products/ Lci,Z., Andreson, K.: Subpixel grid coordinates using line following filtering. Optik Vol. 100 (1995), 125-128 3. Andresen,K., Hentrich K., Hübner. B.: Camera Orientation and 3D-deformation Measurement by Use of Cross Gratings. Optics and Lasers in Engineering Vol 22 (1995), 215-226 4. Andresen, K., Hentrich, K.: 3D Strain measurement from gratings on both surfaces of sheet metal. Arch.Appl. Mech. Vol. 70 (2000) 443-452 5. Stickforth, .J.: The square root of a three-dimensional positive tensor. Acta Mech. Vol. 67 (1987) 233-235

FRACTURE PROCESSES OF QUASI-BRITTLE MATERIALS STUDIED WITH DIGITAL IMAGE CORRELATION J. S. LAWLER AND S.P. SHAH Center for Advanced Cement Based Materials Northwestern University 2145 N. Sheridan Rd. Evanston, IL 60208

Abstract

An understanding of the fracture process is necessary for design and utilization of quasibrittle materials since cracking characterizes their failure. Subregion Scanning Computer Vision (SSCV) is an effective tool for observing these processes in an experimental setting. SSCV, based on Digital Image Correlation (DIC), is a full-field technique for quantitatively examining the development of cracks, with 0.5 micron resolution. DIC uses correlation matching to measure sub-pixel displacement in a sequence of digital images captured of a specimen undergoing failure. Using this technique, the complete behavior of specimens with multiple cracks at incremented levels of fracture can be investigated. SSCV improves on single-image DIC by dividing the specimen into 56 smaller subregions, each of which is imaged separately and so much improves measurement resolution. In this paper, examples of the results obtained in studies of crack growth in a model concrete under compressive loading and in a fiberreinforced mortar under tensile loading are presented. 1.

Introduction

Failure of quasi-brittle materials is governed by the development of cracks. In the case of concrete, cracking follows a tortuous path dictated by material features such as aggregates and pores. While crack development is a three-dimensional phenomena, much can be learned about the fracture process by focusing on cracking visible on the concrete surface. This surface information can be used to form hypotheses about the behavior of the entire concrete specimen. Direct observation of the fracture process is difficult because of the small scale at which microstructural features interact with the failure process. When cracks first initiate, their openings may be less than a micron. In addition, the development of microcracks occurs in a widely distributed manner so a large field of view must be examined. This requires the use of techniques with a much greater resolution than possible with the human eye. Moiré Interferometry [1-2], various forms of Electronic Speckle Pattern Interferometry [3-4] and Laser Holographic Interferometry [5-7] have 335 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 335–344. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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been used to measure deformation with extremely high (sub-micron) sensitivity using interference patterns produced with laser light. These methods, while sensitive, can be difficult to apply and require a vibration free-environment, which is often not easy to achieve around mechanical testing machines. A more robust method for studying fracture processes is Digital Image Correlation (DIC), which has been successfully applied to detect cracks in concrete [8] and other materials [9]. With DIC, full-field surface displacements can be measured with high accuracy for specimens with multiple cracks at incremented levels of fracture. This technique can be used to monitor the testing of a range of specimen sizes and conditions. Subregion Scanning Computer Vision (SSCV) improves on single-image DIC by computing displacement in a mosaic of images of the field of interest. Since the physical features in these images are captured with higher pixel resolution, the measurement resolution achieved using the SSCV technique is improved significantly. This paper describes in detail the SSCV method and its application. In addition, to demonstrate the type of information that may be obtained, results from two separate investigations of the fracture process of concrete materials using SSCV are presented. In both studies, this technique provided insight into the mechanism by which inclusions affect the development of cracks. Such insight is necessary for the efficient and rational design of these materials. 2.

Subregion-scanning Computer Vision (SSCV)

Digital Image Correlation (DIC) is a Computer Vision technique that relies on the optimization of the cross-correlation coefficient to measure the displacement of a portion of an image with respect to the frame of reference for that image. This technique is flexible in its application and lends itself to testing a range of specimen sizes and conditions, requiring only a digital camera capable of imaging the surface of the specimen undergoing deformation [10-11]. Since the analysis is performed by comparing subsequent images of the deformed specimen to an initial image, crack growth can be monitored at numerous selected stages in the loading history without interfering with the fracture process itself. The process of measuring deformations is based on tracking a portion of an image, known as a subimage, as the pattern of pixels it describes moves in a sequence of images of a concrete surface (Figure 1). The displacement field is calculated for a regularly spaced grid of nodes arbitrarily positioned on the reference image. For each node location, the surrounding subimage, typically 48 pixels on a side, is sampled. This subimage is then compared to various subimages from the deformed image in the neighborhood of the original node by calculation of the correlation coefficient. The correlation coefficient is a mathematically defined measure of similarity between two variables, which in this case are the pixel intensity patterns of a sampled subimage from the reference image and from the deformed image at the location being evaluated. The location of highest correlation in the deformed image is determined to be the new location of the subimage and the deformation that occurred is simply the difference between the original and new locations. To account for variations in overall image brightness, the correlation coefficient is normalized and is calculated by equation (1),

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where N is the number of pixels in the subimage, M is the model, undeformed subimage, I is the subimage taken from the deformed image being searched and the summation is over each member pixel of the image. While the correlation coefficient can only be calculated at discrete pixel coordinates, displacements are measured with sub-pixel accuracy by locating the peak on a surface fit to the correlation scores in the neighborhood that likely contains the best match. An example of a displacement field measured with DIC is shown in Figure 2a in vector form, where the circular dots represent the nodes in the arbitrary grid and the lines depict the direction and magnitude of the displacement occurring at the nodes. Figure 2b shows contours of constant displacement that correspond to the vectors displayed in Figure 2a. Discontinuities in the displacement field, represented by tightly packed contours, reveal the location, shape and size of developing cracks.

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The accuracy for DIC is given in terms of pixel size so the resolution of measurement of specimen deformation depends on the amount of physical space imaged by each pixel in the digital images. This is determined by the resolution of the chargecoupled device (CCD) camera and the size of the field of view being imaged. The resolution of this technique is improved significantly in the work presented here through the use of Subregion Scanning Computer Vision (SSCV) [12] and recently updated search algorithms. In SSCV, the digital camera is mounted on a precisely controlled two-dimensional motion stage permitting a set of 56 images, called subregions, to be captured, which together represent the full specimen surface (Figure 3). Since each image describes a smaller area, increased resolution of measurement is achieved. The deformation is measured at each image location separately and these displacement fields are combined in a mosaic fashion to represent the full specimen area.

For the work presented here, an 8-bit camera with 1035 x 1317 pixel resolution was used in combination with a 105 mm macro lens. The correlation search was performed by an algorithm from the Matrox Imaging Library v6.1, which provides a measurement accuracy of approximately 1/20th of a pixel. This combination enables displacements to be measured to with 0.5 µm accuracy for a 75 x 75 mm specimen. Before testing, a fine speckle-pattern is applied to the surface of the specimen using black spray paint, which produces a many-featured and unique pattern, ensuring that a high correlation is found only at the correct location. Note that during testing, the test must be paused to allow the camera to traverse the full specimen area, a process requiring approximately 2 minutes. Since displacement is measured at sub-pixel resolution and this resolution is better than the repeatability of the stage movement, reassembly of the subregion set is a nontrivial problem. To minimize the amount of rigid body motion in the displacement field, a new method has been developed that measures directly the displacement between adjacent images using overlapped areas approximately 100 pixels wide. This relative motion is then accounted for in the displacement calculations. Because this procedure is additive, error accumulates and becomes visible in the form of vertical

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and/or horizontal concentrations in the contour maps further away from the central subregion that is used as the anchor for the assembly process. This full procedure is detailed elsewhere [13]. Through the use of the updated Subregion Scanning system, a resolution was achieved that was approximately 20 times higher than previously possible with single-image DIC. This technique may be easily applied to simultaneously study crack evolution of a specimen during mechanical testing, provided the area examined is a flat surface. Figure 4 shows the arrangement of the testing system as applied to visualize the fracture process in concrete under compression. The CCD camera on the transitional stage was fixed to table approximately 30 cm in front of the specimen. Two white light sources were placed on the left and right of the specimen to produce sufficient lighting of the specimen surface. A automated calibration operation was performed to determine the resolution of the images and to ensure that the movement of the stage was in a plane parallel to the specimen surface [13].

3.

Experimental Results

To illustrate the usefulness of the SSCV technique, experimental results obtained from two previous studies are summarized, along with some conclusions that were made about the relationship between the observed fracture development and mechanical response. The first study examined the effect of aggregate inclusions in twodimensional concrete during compressive failure. The second investigated the role of discontinuous, reinforcing fibers in a mortar matrix under tensile loading. 3.1

MODEL CONCRETE IN COMPRESSION

In the investigation pictured in Figure 4 above, the effects of aggregate spacing and properties on the fracture process in a two-dimensional model concrete were examined. Regularly spaced cylindrical aggregates of either granite or limestone, were imbedded in mortar specimens, 25 x 75 x 75mm, in a predefined arrangement. The aggregate

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cylinders, which had a diameter of 12.7 mm, numbered 13, five and one in the configurations tested (Figure 5). These specimens were then loaded in compression, using a closed-loop control signal based on a combination of both axial and lateral deformation [12].

In Figure 6, contour maps, constructed from displacements obtained using SSCV, that describe the evolution of cracking of a specimen containing five granite aggregates is pictured. Cracking begins around the aggregates as a result of tensile and shear stresses generated by differences in the geometric and material properties of the two phases. These interfacial cracks then link from aggregate to aggregate as the load increases, eventually forming continuous cracks that traverse the entire specimen at peak load.

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The crack patterns for all three aggregate configurations at the peak load are pictured in Figure 7 for specimens containing both granite and limestone aggregate. It is clear that the aggregate material had a strong influence on the way cracks developed. The crack path splits the weaker limestone while the stronger granite forced the cracks to travel around the aggregate particles. The stronger bond formed between the porous limestone and the mortar matrix also reduced the likelihood of this type of interfacial cracking. Note also that for both aggregate types, in the specimens containing one and 13 aggregates, one main crack developed, which determined the strength of that specimen. However, for the specimens containing five aggregates, distributed cracking was visible at this point. This is significant because the development of more distributed cracking consumes a greater amount of energy, since more cracks must be formed. This means that a material in which distributed cracking occurs is likely to exhibit a greater amount of ductility than a similar material in which only one crack develops.

3.2

HYBRID FIBER-REINFORCED MORTAR IN TENSION

In a separate investigation, the role of reinforcing fibers in a mortar matrix during failure was examined with the SSCV technique [13]. In this study, hybrid blends of different size fibers were tested since fiber size was believed to be the main factor in

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determining how the fibers interacted with the fracture process. To investigate this hypothesis, mortar specimens containing various dosages and types of fibers were loaded in uniaxial tension (Figure 8). Specifically, PVA microfibers (22 µm in diameter and 12 mm long) were blended with steel macrofibers (500 µm diameter, and 30 mm long). This combination of reinforcement resulted in greater strength and toughness than was achieved with either fiber type individually.

The development of cracking can be traced in Figure 9, which shows contours of vertical displacement obtained with SSCV for a mortar specimen without any fibers. A plot of stress versus the nominal strain is also included in the figure. While two cracks initially open in this specimen, it is clear that only one develops into the post-peak region. The peak load in the specimen coincides with a crack crossing the full specimen width. Since this specimen does not contain fibers, once this crack occurs there is little left to hold the specimen together and the stress in the specimen drops accordingly. Note that while stress relaxation occurred as the axial displacement was held constant during the image capture process, the measured specimen response quickly returned to the envelope of the stress-strain curve once loading was resumed.

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In Figure 10, the crack pattern at the peak load of mortar specimens containing various fiber types are visible. While one single crack dominates the failure of the unreinforced specimens and those with only macrofibers, those with hybrid fiberreinforcement exhibit multiple wide cracks, also known as macrocracks. This increases the deformation capacity of the material and delays the development of the critical through-specimen crack that determines the strength of the specimen.

In the course of this second study, a means for estimating the crack width based on the measured displacements was devised to provide additional insight into the process of crack formation. Since the tests performed were conducted under uniaxial tension and it was observed that cracks formed largely perpendicular to the loading direction, it was assumed their width could be determined based on the measured vertical displacements. The relative displacement of vertically adjacent nodes was calculated

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and when this displacement was higher than a user-defined threshold, this displacement was defined as a crack width. From these results, color coded images of the crack pattern were produced, in which color represents the crack widths at that location. While these figures can not be reproduced here, the crack width information obtained in this manner was used to successfully estimate the water permeability of the mortar based on the assumption of laminar flow occurring through the cracks [13]. These estimates closely matched direct measurements performed on similar specimens. In this way, data obtained using the SSCV technique was used to quantitatively describe cracking and predict material behavior beyond mechanical response. 4.

Conclusions

Failure of brittle materials is governed by the development of cracking. In most cases, this process occurs at a scale below what is visible to the human eye. SSCV is an effective full-field method for detecting crack development because of its robust application and the high resolution with which it measures surface displacement (0.5 µm). SSCV utilizes a DIC search algorithm for determining displacement but improves on single-image DIC by using a mosaic of 56 images to describe the specimen surface. To show how SSCV may be applied in an experimental setting, studies where this technique was used to examine the fracture process in concrete and mortar during loading were presented. In these studies, the topology of cracking was strongly connected to the specimen behavior, particularly the mechanical performance. 5. 1. 2.

References

Dally, J.W., and Riley, W.F.: Experimental Stress Analysis, McGraw-Hill, Inc., New York, 1991. Shao, Y., Li. Z., and Shah, S.P.: Matrix Cracking and Interface debonding in Fiber-reinforced Cement Matrix Composites, Advanced Cement Based Materials 1 (2) (1993) 55-66. 3. Jia, Z., and Shah, S.P.: Two-dimensional Electronic-speckle-pattern Interferometry and Concretefracture Processes, Experimental Mechanics 34 (3) (1994) 262-270. 4. Creath, K.: Phase Measurement Interferometry Technique, in E. Wolf (ed.), Progress in Optics XXVI, Elsevier Science, New York, 1988, 349-393. 5. Miller, R.A., Shah, S.P., and Bjelkhagen, H.I.: Crack Profiles in Mortar Measured by Holographic Interferometry, Experimental Mechanics 28 (4) (1988) 388-94. 6. Mobasher, B., Castro-Montero, A. and Shah, S.P.: A Study of Fiber-reinforced Cement-based Composites Using Laser Holographic Interferometry, Experimental Mechanics 30 (3) (1990) 28694. 7. Castro-Montero, A., Jia, Z., and Shah, S.P.: Evaluation of Damage in Brazilian Test Using Holographic Interferometry, ACI Materials Journal 92 (3) (1995) 268-75. 8. Choi, S., and Shah, S.P.: Measurement of Deformations on Concrete Subjected to Compression Using Image Correlation, Experimental Mechanics 37 (3) (1997) 307-13. 9. Han, G., Sutton, M.A., and Choa, Y.J.: A Study of Stationary Crack-tip Deformation Fields in Thin Sheets by Computer Vision, Experimental Mechanics 34 (2) (1994) 125-40. 10. Chu, T.C. Ranson, W.F., Sutton, M.A. and Peters, W.H.: Applications of Digital Image Correlation Techniques to Experimental Mechanics, Experimental Mechanics 25 (3) (1985) 23244. 11. Pratt, W.K.: Digital Image Processing, 2nd edition, John Wiley & Sons, Inc., New York, 1991. 12. Choi, S., and Shah, S.P. Propagation of Microcracks in Concrete Studied with Subregion Scanning Computer Vision (SSCV), ACI Materials Journal 96 (2) (1999) 255-60. 13. Lawler, J.S.: Hybrid Fiber-reinforcement in mortar and concrete, Ph.D. Thesis, Northwestern University, Evanston, IL, 2001.

ON THE USE OF DIFFERENT WAVELENGTHS TO DIGITALLY DETERMINE THE ISOCHROMATIC FRINGE ORDER

T. Y. CHEN, Y. C. CHOU, H. L. LEE, and S. H. TSAO Department of Mechanical Engineering National Cheng Kung University Tainan, Taiwan 701, R.O. China

Abstract Four approaches, a novel, two modified and a previous developed approaches, are compared for the determination of the total fringe orders using different wavelengths. The results show that an error of 0.05 fringes can be achieved by all methods. The MIN-MAX method gives the best result for fringe orders greater than 0.6, a better result can be given by the rest methods for fringe orders less than 0.5. The three-wavelength criterion could yield better result than that of the two-wavelength for determination of the total fringe order. Owing to various noises, both criteria cannot assure to obtain correct fringe order. Therefore two-wavelength method would be sufficient to determine the total fringe orders with a correctly determined total fringe order.

1.

Introduction

As an experimental technique for stress and strain analysis, photoelasticity is very useful for problems in which stress or strain information is required for extended regions of the structure or member. The birefringence, or fringe order, in a stressed photoelastic model is controlled by the state of stress at each point in the model. Conventionally the measurement of the fringe order is done manually, which can be time-consuming, and require a well-trained operator [1]. By using the computers and digital image processing techniques, the developments in digital photoelasticity have made photoelastic analysis faster and more reliable for solving engineering problems [2]. Chen reviewed the approaches for digital determination of photoelastic fringe order [3]. Earlier developments in digital analysis of photoelastic fringe orders mainly involve the determination of the location of fringe points and interactively assigning the fringe orders from one image only [4, 5]. Recent studies propose to process intensity data of the images recorded to obtain the whole-field total fringe orders automatically with a PC-based image processing system [6-10]. Recent developments using two-load [11], three-load [12], and the combination of load-stepping to phase-stepping methods [13] for more easily or accurately determination of the total fringe order are also reported. Basically one fringe pattern does not provide sufficient information to determine the total fringe orders. Thus most methods mentioned above acquire more information by using either different wavelengths or different loads to determine the total fringe orders. 345 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 345–352. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The use of different wavelengths of light to determine the isochromatic fringe order has been reported. Umezaki [14] and Plouzennec [15] identified the zero order fringes through an image processing process in which two patterns of different wavelengths were superimposed on each other. The computer was then able to identify the other fringes from the located zero order fringes. However, this method fails when no zero order fringes are present. Chen [7] proposed a scheme that uses two light-field isochromatic fringe patterns with different wavelengths to determine the fringe order automatically. The fringe order is determined regardless of whether zero-order fringes are present or not. Kihara [16] and Buckberry [17] used the image data from three different wavelengths to determine the total fringe orders. Use of two-wavelength or three-wavelength method generally involves two steps. The first step is to determine the fractional fringe orders from two or three sets of images obtained from different wavelengths. Then the second step is to search for the total fringe orders by setting a minimum criterion. Since the accuracy of the intensity data might be affected by various factors, such as, non-uniform distribution of light, material inhomogeneities, spatial quantization, gray-level discretization, and variation of light due to rotation of the optical element, the effectiveness of various approaches for determination of the fringe order are studied and compared with an approach developed previously. The principles of the methods are described. Test results from the experiments are given and discussed.

2. Theoretical Considerations In photoelasticity, with a monochromatic light of wavelength the stress-optic law can be written for a two-dimensional plane-stressed photoelastic body as where and are the principal stresses at point; is the total fringe order, is the relative retardation, and is the relative phase shift; is the material fringe value, and C is the relative stress-optic coefficient, which is wavelength dependent. Using the circular polariscope arrangement as shown in Fig. 1, the intensity of light emerging from the analyzer is given by

where is the maximum dark-field intensity of light emerging from the analyzer. Rotating the analyzer 90°, the intensity of light emerging from the analyzer becomes where is the maximum light-field intensity of light emerging from the analyzer. As the total fringe order of the isochromatic fringe pattern is determined, the principal stress difference, can be calculated from eq (1). Since the light intensity varies according to the or function, the intensity of one wavelength can only provide the fractional fringe order of the fringes. Use of different wavelengths to determine the total fringe orders generally involves two steps. The first step is to determine the fractional fringe orders from a set of images obtained from each wavelength used. Then the second step is to search for the total fringe orders from the determined fractional fringe orders having different wavelengths by setting a minimum criterion. Using a circular polariscope

USE OF WAVELENGTHS TO DETERMINE ISOCHROMATIC ORDER 347 setup, the principles of the approaches used for the two steps are described in the following sections.

2.1 DETERMINATION OF FRACTIONAL FRINGE ORDERS FROM ONE WAVELENGTH Four approaches are considered here to determine the fractional fringe orders. Basically the methods proposed or described below allow a normalized isochromatic fringe pattern to be obtained: (1)Use of a light-field unloaded image and a light-field loaded isochromatic fringe pattern (LF-ADJ). This method, proposed by Chen, generates a normalized light-field intensity using an unloaded and a loaded light-field isochromatic fringe pattern of a photoelastic model. After linearly adjusting the normalized intensity to get rid of the background/stray light effect, the fractional fringe order, n, can be calculated from:

where and are the light-field intensities of the isochromatic fringe patterns for the unloaded and loaded model, respectively. (2) Use of a dark-field unloaded image, a light-field unloaded image, and a light-field loaded isochromatic fringe pattern (LF-DF). This approach resembles to the phase shifting method. In this method, the intensities of light emerging from the analyzer of the dark-field unloaded, light-field unloaded, and light-field loaded photoelastic model can be represented by

where a accounts for the amplitude of light and b denotes the background/stray light intensity. From eqs (5)-(7), the fractional fringe order can be calculated from the following equation: (3) Use of the scanned minimum and maximum light-field intensities of the model under various loads and an light-field isochromatic fringe patterns of the model under a specific load (MIN-MAX). This novel approach is similar to method (2). Instead of using the unloaded dark- and light-field image intensities, they are replaced by the minimum and maximum intensities scanned from

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the loaded light-field fringe patterns under various loads. In this case, the variation of light due to rotation of polarizer is avoided. The fractional fringe order thus can be calculated from the following equation: (4) Use of the dark-field and light-field intensities of the polariscope without the model, and a light-field isochromatic fringe patterns of the loaded model (LF-DF-NM). This method may be considered as an alternative of method (2) or method (1), and may be used for analyzing three-dimensional photoelastic slice, in which the unloaded model image is not obtainable. If the maximum fringe order in the field of interest does not exceed 1/2, the fringe order can be determined without ambiguity using the above methods. If the maximum fringe order in an isochromatic fringe pattern greater than 1/2, the total fringe order, N, should be one in a set of multiple-valued fringe orders given by:

or 2.2

DETERMINATION OF TOTAL FRINGE ORDERS USING DIFFERENT WAVELENGTHS

For two different wavelengths and orders are obtained from eq (10) as

used, two sets of multiple-valued fringe

or where the subscript i (= 1, 2) corresponds to the wavelength Since the state of stress at the same point on a model is the same for different wavelengths, the following relationship can be obtained from the stress-optic law as where and are the material fringe values with respect to wavelengths and Therefore, the total fringe order of and at any point must satisfy eq (12), and can be determined by finding a pair that minimizes the error function, D, as follows: Similarly, if three wavelengths are used, the total fringe order of a point can be determined by using the error function, E, defined as:

2.3

and

RANGE OF MEASUREMENT

Since the light intensity function has a period of the difference of the two fringe orders between the two images is limited to 1/2. Hence the range of measurement for using two wavelengths can be derived as

For three wavelengths used, the range of the measurement becomes the lowest common multiple of which is much larger than that for using two wavelengths.

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3. System Configurations and Experiments The system used, as shown schematically in Fig. 1, consists of a diffused-light circular polariscope for producing isochromatic fringe patterns using a white light, three narrow-band filters with a band width of 10 nm for obtaining monochromatic fringe patterns, a Pulnix TM745 CCD camera, a Matrox frame grabber with four frame buffers, a PentiumIII PC, and other peripheral devices. Each frame buffer has 256K bytes of memory that is organized as 512x512 pixels with 256 gray levels. The specimen used is a circular disk (made of photoelastic material, PSM-1), 50.8 mm in diameter and 5 mm thick. The wavelength of the three narrow-band filters used are and As in any photoelastic analysis, the specimen was calibrated and the material fringe values are determined for the three wavelengths. Figures 2(a)-2(c) show the fringe patterns of the disk under a diametral compression of 500N observed through the three filters, respectively. Using the four approaches, the fractional fringe orders were determined for each wavelength. A gray-level representation (0-255 correspond to the orders 0-0.5) of the determined fractional fringe orders of the disk using different methods is shown in Figs. 2 (d)-2(g) for A comparison of the gray-level represented fractional orders along a horizontal line across the center of the disk for the four methods is shown in Fig. 3. Then the total fringe orders were determined using the criteria for two-wavelength and three-wavelength, respectively. A gray-level representation (0-255 correspond to the orders 0-5) of the determined total fringe orders of the disk using two-wavelength method is shown in Fig. 4 for A comparison of the total fractional orders determined by the four methods to the theoretical result along a horizontal line across the center of the disk is shown in Fig. 5.

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4. Discussions Comparing the fringe patterns in Fig. 2, it can be observed that the fractional fringe patterns agree well to the original ones. Further examining the plots in Fig.3, it can be seen that the gray-level values do no reach 0 or 255, which correspond to the integral and the half fringe order for all methods. The maximum gray-level values

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reach almost the same level for all methods, but the minimum gray-level values show a little different for different approaches. The MIN-MAX method reaches the lowest value and the error is less than 6 gray levels, and the errors for the rest methods are almost the same, which are between 0-20 gray levels. The effect of these errors may also he observed in the total fringe orders plot shown in Fig. 5.

We observe in Fig. 4 that the gray-level (fringe order) varies in most parts of the image except for the regions close to the load point where an error is evident. The error is caused mainly by the resolution of the image processing system. Enlarging those regions would solve this problem. There is not much difference among the four methods. From Fig. 5, it can be seen that the determined fringe orders agree well with the results determined from the theory for all methods. The difference among them is less than 0.05 fringes except for some points at or near the integral or half-fringe order locations. Use of a higher resolution camera or digitally filtering the results may alleviate the errors. Averaging speaking, MIN-MAX method gives the best result for fringe orders higher than 0.6, a better result can be given by the rest methods for fringe order less than 0.5. Owing to the various noises contained in the digital values, it is noted that the total fringe orders may not be those with a minimum difference. Present studies show that the correctness of the determined total fringe orders is about 55% by using the two-wavelength criterion, and that is 70% for using three-wavelength criterion. To insure the determined total fringe orders is correct, a two-wavelength digital procedure [7] that uses 5x5 cross-shaped points can be used no matter the zero-order fringes are present or not in the fringe pattern. If zero-order fringes exist

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in the fringe pattern, they can be identified by superimposing two fringe patterns having different wavelengths. Thereby two-wavelength method is sufficient to determine the total fringe orders.

5.

Conclusions

Use of different wavelengths to determine the isochromatic fringe orders has been presented. Four approaches, a novel, two modified, and a previous developed approaches, are compared to determine the total fringe orders. The results show that an error of less than 0.05 fringes can be achieved by all methods. The MIN-MAX method gives the best result for fringe orders greater than 0.6, a better result can be obtained by the rest methods for fringe orders less than 0.5. Although the three-wavelength criterion yields better result than the two-wavelength criterion for determination of the total fringe order, both criteria cannot assure their correctness with existing noises. Therefore two-wavelength method would be sufficient to determine the total fringe orders if a total fringe order can be correctly determined.

6. Acknowledgment The author is thankful to the National Science Council of the Republic of China for supporting this research under grant NSC89-2212-E-006-116.

7. References 1. 2. 3 4. 5. 6.

7. 8. 9.

10. 11. 12. 13. 14.

15. 16. 17.

Dally, J.W. and Riley, W.F. (1978) Experimental Stress Analysis, McGraw-Hill Co.. Chen, T.Y. (1999) Digital photoelasticity, in P.K. Rastogi (ed.), Photomechanics, Springer-Verlag, pp. 197-232. Chen, T.Y. (2000) Recent advances in digital photoelasticity, in P. Rastogi and D. Inaudi (eds.), Trends in Optical Nondestructive Testing and Inspection, Elsevier Science, pp. 533-544. Chen T.Y., Taylor C.E. (1989) Computerized fringe analysis in photomechanics, Exp. Mech. 29, 323-329. Ramesh K., Ganesan V.R., and Mullick S.K.. (1991) Digital image processing of photoelastic fringes - a new approach, Exp. Tech. 15,41-46. Voloshin, A.S. and Burger, C. P. (1983) Half-fringe photoelasticity: a new approach to the whole field stress analysis, Exp. Mech. 23, 304-313. Redner, A.S. (1985) Photoelastic measurements by means of computer aided spectral-contents analysis, Exp. Mech. 25, 148-153. Patterson, E.A. and Wang, Z.F. (1991) Toward full-field automated photoelastic analysis of complex components, Strain 27, 49-56. Ajovalasit A., Barone S., and Petrucci G (1995) Toward RGB photoelasticity; full-field automated photoelasticity in white light, Exp. Mech. 35, 193-200. Chen, T.Y. (1997) Digital determination of photoelastic birefringence using two wavelengths, Exp. Mech. 37,232-236. Chen T.Y. (2000) A simple method for the digital determination of the photoelastic fringe order, Exp.Mech. 40, 256-260. Asundi, A. Liu, T. and Chai, G.B. (2000) Determination of isoclinic and isochromatic parameters using three-load method, Meas. Sci. Technol. 11, 532-537. Ramesh, K and Tamrakar (2000) Improved determination of retardation in digital photoelasticity by load stepping, Optics and Lasers in Eng. 33, 387-400. Umezaki, B., Tamaki T. and Takahashi, S. (1984) Automatic stress analysis from photoelastic fringes due to image processing using a personal computer, Proc. Soc. Photo. 504, 127-134. Plouzennec, N. and Lagarde, A. (1999) Two-wavelength method for full-field automated photoelasticity, Exp. Mech. 39, 274-277. Kihara T. (1994) Automatic whole-field measurement of principal stress directions using three wavelengths, in S.Gomes (ed.) Recent Advances in Experimental Mechanics, Balkema, pp.95-99 Buckberry, C. and Towers, D (1996) New approach to the full-field analysis of photoelastic stress patterns, Optics and Lasers in Eng. 24, 415-428.

THE APPLICATION OF SPECKLE METROLOGY TO HEART MECHANICS

G. R. GAUDETTE, E. U. AZELOGLU, J. TODARO, L. KEENE, I. B. KRUKENKAMP, F. P. CHIANG State University of New York at Stony Brook Division of Cardiothoracic Surgery Department of Mechanical Engineering Stony Brook, NY 11794-2300

Abstract Commonly used techniques for analysis of regional heart function include sonomicrometry, implanted markers, and surface markers. However, these techniques cannot offer the high spatial resolution needed to define regional abnormalities in the heart. We have recently applied a computer aided speckle interferometry technique (CASI) with high spatial resolution to measuring the deformation of heart muscle. We applied silicon carbide particles to the surface of the isolated heart and loaded the heart by increasing the intracavitary pressure. The movement of the speckle pattern was tracked with a CCD camera. This technique produces equivalent results to that of the gold standard in heart mechanics, sonomicrometry, but with three orders of mapitude higher spatial resolution. We have used this technique to determine changes in myocardial deformation of perfused, ischemic and reperfused rabbit hearts. 1. Introduction The heart tissue, myocardium, is anisotropic, inhomogeneous, and has a non-linear material properties [1-3]. The heart can experience a reduction in blood flow due to partial or total occlusion of the vessels that supply blood to the region. In general, this reduction in blood flow (ischemia) is a regional phenomenon resulting in changes in the material properties of the heart. These changes can occur over a 10-20 millimeters length during regional ischemia in hearts of dogs or pigs[4-6]. Other clinically relevant scenarios can occur which also result in abrupt changes in regional myocardial function, such as pacing from various locations [7] and tissue engineering procedures [8]. It is therefore necessary to measure the mechanical response of the heart over a large region with high spatial resolution. Currently employed techniques used in determining regional function of the heart include sonomicrometry, surface attached or implanted markers, echocardiography, and magnetic resonance imaging (MRI). Sonomicrometry is the most commonly used technique for measuring regional function in the laboratory setting. This technique involves the implantation of two (or more) transducers in the heart wall (myocardium). One transducer send an ultrasonic signal while the second transducer receives the 353 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 353–364. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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signal. This technique can only provide a one-dimensional change in length over a relatively large gauge length (generally > 10 mm). To determine 2-D deformation, investigators have placed paper markers [5, 9] or titanium dioxide particles [10] on the surface of the heart and tracked their movement with a video camera. By using 40-60 paper markers some investigators have been able to obtain a spatial resolution of approximately 5 mm[5]. Although this technique offers higher spatial resolution compared to sonomicrometery, important deformation information may be lost in the borderzone between normally perfused tissue and ischemic tissue. In addition, this resolution may not be sufficient in smaller hearts, such as that of rabbits and rats. Clinically, echocardiography is generally used to determine regional function [11, 12]. This technique can determine the velocity of sound inside the tissue at various locations and determine the approximate displacement from the sequential acquisition of the velocity. However, orientation of the echocardiography probe may generate errors and some areas of the heart are difficult to image [13]. In addition, the spatial resolution of this technique is lacking. MRIs can offer higher spatial resolution and determine 3-dimensional deformation [14, 15]. However, this technique can not acquire a full heartbeat in one data acquisition set and must rely on steady state conditions. Furthermore, it is a very expensive technique and requires extensive computational time. Computer aided speckle interferometry (CASI) is a technique which was recently developed in our laboratory [16-20]. This technique evolves from the earlier works on laser and white light speckle methods [21, 22]. It offers high spatial resolution, is nondestructive, relatively inexpensive, and is a whole field technique. In this paper we have applied this technique to determining myocardial function in hearts exposed to ischemia, which results in a decrease in heart function.

2. Materials & Methods 2.1. ANIMAL CARE All animals received humane care in compliance with the "Principles of Laboratory Animal Care" formulated by the National Society for Medical Research and the "Guide for the Care and Use of Laboratory Animals" prepared by the National Academy of Sciences and published by the National Institutes of Health (NIH Publication No. 85-23, revised 1985). Furthermore, the Institutional Animal Care and Use Committee at Stony Brook reviewed and approved the protocol followed in this study (IACUC # 0834). In this study rabbits were chosen because their hearts could easily be isolated and they are commonly used in a Langendorff apparatus, which is a well-designed model for myocardial function. For all studies involving rabbits, New Zealand rabbits, weighing 2.5-3.5 kg, were anesthetized with sodium pentobarbital (30 mg/kg) and anticoagulated (1,000 units sodium heparin) by an ear vein. The chest was opened via bilateral thoracotomy, the heart rapidly excised and placed in iced Krebs-Henseleit solution.

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2.2. CASI ANALYSIS One of the major advances of experimental strain measurement techniques is the advent of speckle interferometry (also referred to as speckle photography) [18, 19]. Heretofore, displacement/strain measurements were done by reading the change of distances between well-defined geometric markers. The speckle technique on the other hand employs a pattern of random markers called speckles as displacement transducers. Instead of following the movement of each and every speckle, the movement of a cluster of speckles is monitored through a correlation calculation before and after the movement. In the earlier stages of its development coherent laser speckles (which are the natural result of multiple interference of wavelets reflected from an optically rough, laser illuminated, surface) were exclusively used [21]. Later incoherent white light speckles were employed [22]. With the rapid advance of CCD (charge coupled device) cameras and computer technology, a digital version of the method, i.e. CASI, was developed [20]. In the following section the essence of CASI, as evolved in references [17, 20, 23, 24], is briefly described. First a random speckle pattern is created on the surface of a specimen, typically using a combination of white and dark particles to obtain a high contrast light intensity distribution. This speckle pattern is recorded by a CCD camera and is digitized. Upon deformation of the specimen, the speckle pattern is digitally recorded again. The two speckle patterns are subdivided into corresponding subimages, each consisting of 32 x 32 pixels (other choices such as 64 x 64 or 16 x 16 pixels can also be used depending on the situation). Let (x,y) and (x,y) denote the intensity distributions of speckle patterns before and after deformation, respectively, and:

in which u and v are the average displacement components along the x and y directions, respectively, as represented by the cluster of speckles within the subimage. By maximizing the cross correlation coefficient of the two speckle patterns,

the displacement components, u and v, can be obtained. In the earlier days, correlation was obtained optically by either pointwise or full field spatial filtering techniques [21]. Direct numerical calculation of the cross correlation coefficient [25, 26], however, is very time consuming. Instead a Fourier Transform is applied to the speckle functions and

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In the spectrum domain an interference function is introduced:

where and are phase angles of and * denotes the complex conjugate. It can be shown [20] that:

respectively, and

By performing another Fourier Transform, we obtain the so-called halo function, G:

which is an expanded impulse function located at (u,v). By detecting the crest of this function, the displacement vector between the two speckle patterns is uniquely determined. Strains can be calculated by taking derivatives of the displacement components (u and v) and using an appropriate strain-displacement relationship (be it linear or non-linear). The resolution of the speckle technique largely depends on the size of speckle employed, the resolution of the camera, and the magnification factor. In this study we typically used 40 µm particles with a 1,024 x 1,024 pixel resolution, and 20X magnification. 3. Validation of CASI with Sonomicrometry 3.1. INRODUCTION In order to establish CASI as a useful technique to measure myocardial deformation, the results obtained with CASI must be compared to those obtained with an accepted technique, such as sonomicrometry. The use of sonomicrometry in myocardial mechanics involves the implantation of two ultrasonic transducers. One transducer sends an ultrasonic signal and the second transducer receives the signal. The time it takes for the signal to travel between the two transducers is proportional to the distance between them. This technique is essentially the “gold standard” in myocardial deformation measurements. Urheim and coworkers have used sonomicrometry to validate determination of regional myocardial strain using echocardiography[27], similarly, Omens and colleagues have used sonomicrometry to validate their use of epicardial markers to determine regional myocardial deformation[28]. Herein, sonomicrometry will be used to validate the CASI technique.

3.2. EXPERIMENTAL PROTOCOL In this study CASI was used to measure deformation in the epicardium of rabbit hearts (n=3). After the heart was isolated, a latex balloon was placed in the left ventricle of the heart, and the pressure in the ventricle was adjusted by changing the pressure head of a static column of water attached to the intracavitary balloon. Small silicon carbide

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particles (approximately 40 microns in diameter) were dispersed in an area on the anterior surface of the heart to serve as the speckle pattern. A schematic diagram of the test arrangement is shown in Figure 1. Two ultrasonic transducers (Iowa Doppler Products, Iowa City, IA) were implanted into the epicardium at the boundary of the speckled region. Three hearts were subjected to various protocols to determine the relationship between results obtained with sonomicrometry and CASI. All hearts were conditioned three times prior to protocols by inflating the intracavitary balloon to approximately 25 mmHg, and then deflating. Subsequently the heart was subjected to incremental intracavitary pressure increases of 10 mmHg, from 0 to 30 mmHg. A second heart was allowed to passively contract, with data acquired at 0, 2, 4, and 6 minutes after conditioning. In the final heart, the strain determined by sonomicrometry was incremented from 0 to 1% and 2% strain (based on the original length between the sonomicrometry transducers at intracavitary pressure being equal to zero). Both sonomicrometry data and the speckle image were acquired at each load and digitally stored for analysis.

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The use of silicon carbide particles provided for a good speckle image. These particles adhered well to the myocardium. Immediately after application, any loose particles were removed by gently bathing the heart in Krebs-Henseleit solution or by applying gentle air currents over the surface of the heart. The speckle patterns before and after the heart deformation were recorded using a high resolution charge coupled device (CCD) camera (Kodak Megaplus camera Model 4.2) with a 2029 x 2048 pixel array and 8 bit gray level resolution. The images were stored on a workstation for subsequent analysis using CASI. 3.3 STRAIN DETERMINATION For the purpose of comparison, normal strains were determined from displacement data obtained by both sonomicrometry and CASI using the following equation:

In the case of sonomicrometry, and are the initial and final distances between the sonomicrometry transducers. For corresponding strains determined by CASI, displacements from the two subimages closest to each sonomicrometry transducer were used. Because CASI gives both the u and v displacements (i.e. displacement components in the x and y axes, respectively), the magnitudes of these displacement components resolved along the axis of the sonomicrometry transducers were used to calculate the CASI strain data The algebraic sum of the resolved displacement components is simply and is the distance between the centroids of the two subimages most adjacent to the transducers.

3.4. RESULTS For the three hearts, the region of interest between the transducers was covered by 8001200 pixels in the x direction, and 200-400 pixels in the y direction. With a 32 x 32 subimage size and a shift of 16 pixels between subimages, matrices between 50 x 12 “points” and 75 x 24 “points” representing 600-1800 displacement vectors were generated. This is compared to only one value (the displacement component along the direction of the transducers) determined by sonomicrometry. In addition, with displacements vectors two-dimensional strain tensor distribution can be calculated using appropriate strain-displacement relationships. Figure 2 shows approximately 25% (for reasons of visual clarity) of the displacement vectors obtained from heart #1 using CASI in the region between the two transducers. The strains obtained from CASI and sonomicrometry from all three hearts are shown in Figure 3. A solid line with slope of one is also shown. These data clearly show that the two methods produce equivalent results. However, the spatial resolution of CASI is some three orders of magnitude higher than sonomicrometry.

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Deformation in the Ischemic Heart

4.1. INTRODUCTION Although CASI has been established as a technique that can be used to measure 2-D strain distribution in the myocardium, its ability to measure deformation in clinically relevant scenarios must be established. One scenario that occurs during cardiac surgery is global ischemia in the arrested (non-beating) heart. This results in no blood flow being supplied to the entire heart. We therefore used CASI to determine regional differences in myocardial deformation that results from global ischemia. 4.2. MATERIALS & METHODS In addition to the procedures previously described in the rabbit model, a cannula was inserted into the aorta to perfuse the myocardium with Kreb’s Heinselett solution via the coronary arteries. The coronary arteries were perfused at a constant perfusion pressure head equal to 75 mmHg. An intracavitary balloon was inserted into the left ventricle. The balloon was filled with water and attached to a vertically adjustable reservoir used to change the intracavitary pressure (see Figure 1). Small silicon carbide particles were dispersed randomly on the left ventricle to create a speckle pattern. After a 15-minute equilibrium period, the hearts were then infused with Krebs-Henseleit solution plus high potassium (approximately 20 mM) to stop the heart from beating. The intracavitary load was adjusted from 0 to 20 mmHg, and data acquired at 0, 10, and 20 mmHg. The flow was then discontinued to the hearts (global ischemia). After 5, 10, 15 and 30 minutes of global ischemia, at which the speckle data were recorded, the left ventricle was again incrementally loaded by increasing the intracavitary pressure. The hearts were then reperfused for 15 minutes and incrementally loaded a final time. 4.2.1.

CASI ANALYSIS

The images of the region of the myocardium that is covered with speckles are recorded at different pressures using a 1,024 x 1,024 pixel CCD camera. The subimage size for data analysis in these hearts was 64 x 64 pixels, and the shift size was 4 pixels. The following equations were used to determine strain:

A linear fit to several data points is determined from the data obtained with CASI. As the approximate resolution of the system in this protocol was 20 µm/pixel, and the shift size was 4 pixels between subimages, a linear fit was employed over 13 subimages (slightly larger than 1 mm). Therefore, to approximate the partial differentials, a least squares linear regression was fit to 13 points in their respective directions. The principle strains and were determined and the first invariant of the 2D strain tensor was also computed using the following equation:

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These variables provided valuable information to determine the perfusion status of the heart.

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RESULTS

Figure 4 shows the u and v displacement contour plots of a typical rabbit heart under the status of perfusion, global ischemia and reperfusion. It is seen that more deformation is experienced by the ischemic myocardium when the heart is subjected to the same intracavitary pressure. In the small region that we investigated the strain is essentially uniform. Thus we calculated the average strain to show the difference of myocardial response between perfused and ischemic states of the heart. In Figure 5 the average maximum principle strain and the first strain invariant of the epicardium are plotted as a function of time as the heart goes through the different stages of perfusion, ischemia, and reperfusion. It is seen that when the hearts were made globally ischemic, both the maximum principle strain and the first invariant significantly increased. Upon reperfusion (restoration of flow), the maximum principle strain and the first invariant both decreased compared to the ischemic case. Indeed it is even slightly lower than the original perfused state. Clearly the deformation is significantly higher (about seven times) in the ischemic condition than in the perfused or reperfused conditions. This indicates that the

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tissue is significantly weakened by the lack of blood flow. While both and indicators of epicardium deformation, seems to be slightly more sensitive.

are good

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Conclusion & Discussion

We have demonstrated that CASI, a full field strain measurement technique that heretofore has been used in measuring strain in engineering materials, can also be used to measure deformation of myocardium. It has a spatial resolution at least three orders of magnitudes higher than that of sonomicrometry, the “gold standard” in heart mechanics. We have shown that the effect of ischemia significantly weakens the heart muscle. However, after reperfusion, the strength of the muscle is restored. In work to appear elsewhere we have also demonstrated that CASI can be applied to in-vivo studies as well. 6. Acknowledgements

The financial support of this work by the National Science Foundation through Grant number BES – 9903516 is gratefully acknowledged. 7. References 1. 2. 3.

4.

5.

Fung YC. Biomechanics, Mechanical Properties of Living Tissues. 1981, New York: SpringerVerlag. Spotnitz HM. Macro design, structure, and mechanics of the left ventricle. Journal of Thoracic & Cardiovascular Surgery, 2000.119(5): p. 1053-77. Guccione JM, O'Dell WG, McCulloch AD, Hunter WC. Anterior and posterior left ventricular sarcomere lengths behave similarly during ejection. American Journal of Physiology, 1997. 272(1 Pt 2): p. H469-77. van Leuven SL, Waldman LK, McCulloch AD, Covell JW. Gradients of epicardial strain across the perfusion boundry during acute myocardial ischemia. American Journal of Physiology, 1994. 267: p. H2348-2362. Prinzen FW, Arts T, Hoeks APG, Reneman RS. Discrepancies between myocardial blood flow and fiber shortening in the ischemic border zone as assessed with video mapping of epicardial deformation. Phlugers. Arc (Europ. J. Physiol.), 1989. 415: p. 220-29.

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Gallagher KP, Gerren RA, Stirling MC, Choy M, Dysko RC, McManimon SP, Dunham WR. The distribution of functional impariment across the lateral border of acutely ischemic myocardium. Circulation Research, 1986. 58: p. 570-583. Prinzen FW, Hunter WC, Wyman BT, McVeigh ER. Mapping of regional myocardial strain and work during ventricular pacing: Experimental study using magnetic resonance imaging tagging. Journal of the American College of Cardiology, 1999. 33: p. 1735-1742. Chatterjee S, Stewart AS, Bish LT, Sweeney HL, Garder TJ. Gene transfer of HGF is superior to VEGF in inducing coronary angiogenesis and preservation of myocardial contractility. Surgical Forum, 2001.52: p. 67-69. Augustijn CH AT, Prinzen FW, Reneman S. Mapping the sequence of contraction of the canine left ventricle. Pflugers Arch, 1991. 419: p. 529-533. Karlon WJ, McCulloch AD, Covell JW, Hunter JJ, Omens JH. Regional dysfunction correlates with myofiber disarray in transgenic mice with ventricular expression of ras. American Journal of Physiology - Heart & Circulatory Physiology, 2000. 278(3): p. H898-906. Derumeaux G, Ovize M, Loufoua J, Andre-Fouet X, Minaire Y, Cribier A, Letac B. Doppler tissue imaging quantitates regional wall motion during myocardial ischemia and reperfusion. Circulation, 1998. 97(19): p. 1970-7. Heimdal A, Stoylen A, Torp H, Skjaerpe T. Real-time strain rate imaging of the left ventricle by ultrasound. Journal of the American Society of Echocardiography, 1998. 11(11): p. 1013-9. Castro PL, Greenberg NL, Drinko J, Garcia MJ, Thomas JD. Potential pitfalls of strain rate imaging: angle dependency. Biomedical Sciences Instrumentation, 2000. 36: p. 197-202. Moulton MJ, Creswell LL, Downing SW, Actis RL, Szabo BA, Vannier MW, Pasque MK. Spline surface interpolation for calculating 3-D ventricular strains from MRI tissue tagging. American Journal of Physiology, 1996. 270: p. H281-H297. Scott CH, Sutton MS, Gusani N, et al. Effect ofdobutamine on regional left ventricular function measured by lagged magnetic resonance imaging in normal subjects. American Journal of Cardiology, 1999. 83(3): p. 412-7. Chen DJ, Chiang FP. Optimal sampling resolution and range of measurement in digital speckle correlation 2: white speckle method. Proc SEM Annual Spring Conf on Exp Mech, 1989. Chiang FP, Wang Q, Lehman F. New development's in full field strain measurements using speckles., in Nontraditional Methods of Sensing Stress, Strains, and Damage in Materials and Structures., C.F. Lucas and D.A. Stubbs, Editors. 1997, ASTM. p. 156-169. Erf RK. Speckle Metrology. 1978: Academic Press. Chiang FP. Speckle Metrology, in Metals Handbook. 1989, SSM-International. p. 432-437. Chen DJ, Chiang FP, Tan YS, Don HS. Digital speckle-displacement measurements using a complex spectrum method. Applied Optics, 1993. 32(11): p. 1839-49. Chiang FP. A family of 2D and 3D experimental stress analysis technique using laser speckles. Solid Mechanics Archives, 1978. 30: p. 1-32. Asundi A, Chiang FP. Theory and Application of White Light Speckle Methods. Optical Engineering, 1982. 24(4): p. 570-580. Chen DJ, Chiang FP. Computer aided speckle interferometry using spectral amplitude fringes. Applied Optics, 1993.32: p. 225-36. Chen DJ, Chiang FP. Range of measurement of computer aided speckle interferometery (CASI). Proc 2nd Int Conf on Photomechanics and Speckle Metrology, 1991.1554A. Peters WH, Ranson WF. Digital imaging techniques in experimental stress analysis. Optical Engineering, 1982. 21: p. 427-31. Lu H, Vendroux G, Knauss WG. Surface deformation measurements of a cylindrical specimen by digital image correlation. Experimental Mechanics, 1997. 37: p. 433-39. Urheim S, Edvardsen T, Torp H, Angelsen B, Smiseth OA. Myocardial strain by Doppler echocardiography. Validation of a new method to quantify regional myocardial function. Circulation, 2000. 102(10): p. 1158-64. Omens JH, Farr DD, McCulloch AD, Waldman LK. Comparison of two techniques for measuring two-dimensional strain in rat left ventricles. Am. J. Physiol, 1996. 271: p. H1256-61.

5. Non-Destructive Evaluation

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LINE FOCUS ACOUSTIC MICROSCOPY FOR THIN-FILM MEASUREMENTS ZHIQI GUO Sonoscan Inc. 2149 E. Pratt Blvd. Elk Grove Village, IL 60007 USA JAN D. ACHENBACH Center for Quality Engineering and Failure Prevention Northwestern University Evanston, IL 60208 USA

Abstract A thin-film coating generally is a single or multiple layered thin film deposited on a component to extend its life and performance. As discussed in this paper the V(z) effect in line-focus acoustic microscopy is a very suitable technique for the quantitative nondestructive determination of thin-film elastic constants and the bond quality at interfaces. After a brief introduction to acoustic microscopy and a discussion of calibration for frequency dependence, the modeling of multilayered configurations is summarized. The modeling results are used in a measurement model for the V(z) effect which can be used to select a frequency range suitable for a particular configuration. In combination with measurements the model is subsequently employed to determine elastic constants and interface stiffnesses. Experimental results are presented for single, multilayered, isotropic and anisotropic films. The effect of imperfect interfaces on strength considierations is also discussed. 1.

Introduction

Thin film coatings can significantly extend the life or enhance the performance of components of turbine engines and machine tools. In fact, approximately 75% of all the components in an aircraft engine are now coated with a different material [1]. For example, transition metal carbides and nitrides of the IV to VI group of the periodic table have extremely high melting points and extreme hardness, excellent hightemperature strength and good corrosion resistance. They are widely used for thin-film coatings to produce wear resistant surfaces, to provide corrosion protection against harsh environments, and for other applications in surface engineering [2]. Other coatings such as multilayered ceramic and metallic coatings can offer unique physical and mechanical properties due to the refined microstructure. Such coatings have a wide range of uses, including for optical devices, for corrosion protection, as thermal barriers, for wear resistance, and for tribological applications. Examples are the layered coatings for cutting tools [3], and the layered Diamond-Like-Carbon (DLC)/Si coatings for biomedical and aerospace applications [4]. 367 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 367–380. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Basically, a coated component, here referred to as a film/substrate system, is a layered structure, with a single or multiple layered thin film deposited on a substrate. Film/substrate systems are usually prepared by deposition techniques such as physical vapor deposition (PVD) or chemical vapor deposition (CVD). The mechanical properties of the thin film and the adhesion quality between coating layers and between the thin film and the substrate determine its functional characteristics. These properties are very sensitive to a number of factors, which are determined by the deposition technique and by processing parameters. For essentially the same material, the thin-film properties may be quite different from bulk properties. The mechanical properties of thin films need to be measured and analyzed after deposition for quality assurance of the coated components and for the optimization of coating design and development. Currently, only limited destructive methods, such as the scratch test and the nanoindentation test [5], are used to measure the mechanical properties of thin films and to tune the deposition processing parameters. A suitable nondestructive technique that is capable of providing reliable and reproducible quantitative information for thin films will have obvious advantages. Acoustic Microscopy (AM) is a very suitable technique for the quantitative nondestructive determination of material properties such as elastic constants and the bond quality of thin films because it can be utilized to generate surface acoustic waves (SAW) at very high frequencies to obtain quantitative information. The generated high frequency SAWs that propagate in a thin surface layer make them ideally suitable for thin film characterization. Acoustic Microscopy has been used in various fields including biology, material science, and quantitative NDE. It can be broadly divided into two classes, the scanning mode (SAM) for imaging [6] and the quantitative mode (QAM) for material property measurements. This paper is concerned with QAM, which has the capability to measure acoustic properties with high accuracy, especially with line-focus acoustic microscopy (LFAM). This technique, which has been discussed in great detail in Ref. [7], provides for directional generation of SAWs as well as more accurate analysis of the acoustic properties, as compared with point-focus acoustic microscopy. In earlier work, LFAM has been used to determine properties, such as the thickness, the elastic constants and sometimes-even the densities of bulk materials and thin films. Generally, only one thin film layer on a substrate has been considered and it has been assumed that the interface between the thin film and the substrate is perfect. In certain frequency ranges it has been difficult to obtain a useful material acoustic signature for thick or very stiff thin films. Very little work has been done for the case of multilayered thin films. Quantitative acoustic microscopy for the evaluation of thin film interface conditions has also been relatively unexplored. This paper is concerned with these topics. 2.

Line-Focus Acoustic Microscopy

An acoustic microscope consists of four main components: the acoustic probe, the pulse-mode measurement system for transmitting and receiving signals, the mechanical system for alignment and movement of the specimen and a computer for controlling the system and processing the recorded wave forms.

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The water-coupled line-focus acoustic lens allows the measurement of the SAW velocity in specified directions Hence line-focus acoustic microscopy (LFAM) can be used to determine the elastic constants of anisotropic materials. The usual technique measures the so-called V(z) curve, which is a recording of the voltage output of the transducer when the distance between the lens and the specimen is decreased. Figure 1 shows the configuration. For the focal line is on the upper surface of the specimen. As z is decreased, oscillations appear in the V(z) curve due to interference between the rays specularly reflected by the specimen and the rays associated with propagation of a leaky SAW on the specimen. The phase velocity of a leaky surface wave, can be determined from the period, of the periodic oscillations, using a well-known relationship based on ray considerations:

where

is the wave velocity in the coupling fluid, and

Equation (1) implies that the measured

is the operating frequency.

must satisfy

The period can be accurately extracted from the measured V(z) curve by a spectrum analysis, as discussed in detail in Ref. [8]. Therefore, by measuring V(z) curves for a specimen at different frequencies or at the same frequency but different

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wave propagation directions, the dispersion curves or the directional variations of the leaky surface wave velocity can be measured. By numerically simulating the V(z) measurement procedure, a V(z) measurement model can be developed to calculate theoretical V(z) curves for a layered specimen of arbitrary configuration for use with a line-focus acoustic microscope. In this paper, we use the approach proposed by Achenbach et. al. for synthesizing V(z) curves [7]. In that approach, the output of the line-focus acoustic probe is expressed as a Fourier integral over the product of the characteristic functions of the acoustic lens, and and the reflectance function of the fluid-loaded specimen

where

defines the wave vector, and

As shown by Eq. (3), the theoretical measurement model for the V(z) curve contains the reflectance function of the film/substrate system as a principal component. Wellknown theoretical methods are available to determine reflectance functions. See, for example, Ref. [9]. By using the theoretical reflectance function model for parametrical studies, the phase velocities of the possible SAW modes and their corresponding mode reflection coefficients can be predicted for an arbitrary film/substrate system. Therefore the reflectance function can be used as a predictive tool to estimate in which frequency range a wave mode may be picked up by acoustic microscopy. By applying the V(z) measurement model, the phase velocities can be calculated for a film/substrate system at different frequencies. For trial values of the unknown elastic constants of the thin film, these theoretically calculated phase velocities have been used in an iterative manner together with phase velocities obtained from V(z) measurements to inversely determine the elastic constants of thin films by minimization of the deviation between theoretical and experimental results at N points. The deviation is defined as

where 3.

and

are the measured and calculated values of the leaky SAW velocities.

Calibration for Frequency Dependence

The lens of a line-focus acoustic microscope optimally operates at a fixed center frequency. The lens used in this work is designed to be used at 225 MHz. For some applications, for example for isotropic materials, measurements at a number of

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frequencies are required. The measurements at frequencies other than 225 MHz have to be corrected by the use of a calibration factor. Thus, it is necessary to calibrate the LFAM system at each used operating frequency to eliminate system errors. Using a standard specimen for calibration provides a reasonable approach. In principle, any material could be used as the standard specimen if all the elastic constants necessary for the theoretical calculation of the leaky SAW characteristics can be determined with high accuracy. However, a specimen with the following properties is preferred: (1) non-piezoelectric, homogeneous, with a small number of independent elastic constants (isotropic or cubic); (2) small attenuation at higher frequencies, such as 150~300 MHz, and (3) having a suitable leaky SAW velocity quite close to those of the materials to be investigated. A silicon single crystal wafer was taken as the standard specimen, because its properties are well known, and its direction-dependent leaky SAW velocities cover a suitable velocity range. For a silicon single crystal wafer, the directional variations of the leaky SAW velocities, displayed as curves, were measured at different frequencies in the range from 135 MHz to 255 MHz using the nominal 225 MHz LFAM system. In Fig. 2 the measured results are plotted in dotted lines with symbols and the theoretical ones in solid lines. The theoretical curves are calculated by using the known material properties of silicon. The difference between the measured and calculated curves at each frequency indicates the measurement accuracy at that frequency. The results show that the measured curve at each frequency generally shifts up or down a fixed value relative to the theoretical curve, this shift, here defined as a correction factor can be used for calibration at this frequency. By performing

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the same procedure for all the frequencies to be used, a set of correction factors corresponding to a set of operating frequencies can be obtained. For later measurements on other specimens, these correction factors are used to compensate the measured velocities to obtain more accurate velocities namely, For example, the measured curve at 225 MHz is very close to the theoretical one, thus there is no need to compensate at this frequency, The measured curve at 210 MHz shifts down 11(m / s) on average, relative to the theoretical one, thus Next a uniform aluminum specimen with a polished surface was used to check the obtained correction factors. The leaky SAW velocities at different operating frequencies for the aluminum body, which should be constant, were measured and calibrated using the obtained correction factors. After calibration, the error of the velocity measurement by LFAM is estimated to be less than ±0.5%.

4.

Modeling

Figure 3 shows the configuration of a layered medium with N layers on a substrate. A time-harmonic plane wave is incident from a coupling fluid, at arbitrary incident The response of the layered medium to the incident time-harmonic plane wave can be expressed in terms of the reflectance function. This reflectance function carries information on the layered medium, including its mechanical properties, the thickness of each layer, and the interface conditions. A general theoretical model has been developed which quantitatively relates the reflectance function to these characteristic properties of the layered specimen. See Ref. [9]. It has been shown that the reflectance function is of the general form

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where

Here and are the density and the velocity of the coupling fluid, and and the frequency and the angular frequency of the incident wave with incident angle and is the tangential component of the wave vector, where

are

and are the elements of the global transfer matrix [T]. For the numerical calculations, the delta-matrix reformation is introduced to avoid the “precision problem,” see Ref. [10]. The reflectance function given by Eq. (4) can be used for wave mode identification. The vanishing of its denominator, i.e.,

is exactly the characteristic equation of the layered medium. Solutions for the velocity and the amplitude of the possible wave modes can be obtained by numerically solving this equation. Setting takes out the effects of the fluid loading, and thus the equation can be used to identify the free wave modes. For a given frequency, the reflectance function is a function of the incident angle, The reflectance function shows distinct behavior when the incident angle equals the critical angle associated with a possible wave mode. By virtue of the relations between the incident angle and the phase velocity c shown in Eq. (8), the reflectance function can be used to obtain the phase velocities of the wave modes for the layered system at a fixed frequency. The reflectance function undergoes a phase shift and its real part undergoes a jump decreasing its magnitude as the incident angle passes a critical angle which corresponds to a wave mode. Using this criterion to determine the critical angles, the phase velocity of a wave mode can be calculated as

As discussed by Guo et al. [11], the magnitude of the reflectance function at the critical angle indicates how strongly the wave mode will be reflected. For convenience that magnitude will be referred to as the mode reflection coefficient. A strongly reflected SAW mode with a large mode reflection coefficient can be picked up easily by acoustic microscopy. The results show that the theoretical mode reflection coefficient of a true surface wave mode is unity and that of a pseudo-SAW mode that radiates energy into the substrate is somewhere between 0 and 1.

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Selection of Frequency Range

Theoretical calculations and experimental observations show that the LFAM can not pick up any SAW mode with a mode reflection coefficient smaller than around 0.15. A strongly reflected SAW mode can be easily picked up by the LFAM for generation of a V(z) curve, and its velocity can be extracted with high accuracy. The V(z) measurement will be less precise for a weakly reflected surface wave mode. There is a significant decrease in the number and magnitude of the oscillations of the V(z) curve for a weakly reflected SAW mode. Comparisons of theoretical results for the phase velocity for a slow-on-fast system calculated both directly from the reflectance function and from the V(z) curve via Eq. (1), presented by Guo et al. [11], illustrate the significance of an appropriate selection of the frequency range for modes with a sufficiently large value of the reflectance function.

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For a fast-on-slow system of an isotropic (TiN) film on an isotropic (M2-HSS) substrate the material constants have been determined from the experimental V(z) results by the iterative optimization method mentioned in Section 2. After determination of the material constants the corresponding phase velocities were recalculated to verify the procedure. Results are shown in Fig. 4. The best results are obtained in the frequency range defined by i.e. before the cut-off of the true SAW mode. In that range the magnitude of the reflectance function is unity. Based on the excellent agreement displayed in Fig. 4 it may be expected that the material constants have been obtained with good accuracy. A detailed discussion of the accuracy can be found in the paper by Guo et. al. [11]. 6.

Two-Layer Thin Film

A two-layer thin film, was deposited on an M2-HSS substrate. The earlier discussed procedure for measurement and data inversion was applied to determine the material constants, namely, Young’s moduli and Poisson’s ratios of the Titanium and Aluminum coatings layers. Known material properties of the substrate and known values of the densities, and the thicknesses, d, of the two layers were used. The determined elastic constants are listed in Table 1.

After the material constants had been determined, the corresponding leaky SAW phase velocities were recalculated to verify the procedure. The recalculated phase velocities are shown in Fig. 5 as a solid line, together with the measured values in empty circles for various frequencies.

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In order to investigate the uniqueness and accuracy of the data inversion method to determine the four unknowns, a convergence and sensitivity study was performed. The contour plots in Fig. 6(a) and Fig. 6(b) show the deviations due to variations of and respectively. The deviation is defined in terms of N-point measured and calculated phase velocities as in Eq. (5), here N=11. The elastic constants and were varied ±20% around the determined values. The individual cross-sectional variations of the deviation with respect to and shown in Fig. 6(c) and Fig. 6(d), illustrate the sensitivity of the deviation to variations of those constants. The more sensitive the deviation, the more accurate the constants can be determined. The results show that the expected accuracy of is better than that of, and the expected accuracy of is better than that of. This is so because the Aluminum layer has greater thickness and is more effective than the Titanium layer. The expected accuracy of Young’s modulus is better than that of Poisson’s ratio.

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Anisotropic Thin Film/Substrate Systems

For an anisotropic film/substrate system composed of anisotropic thin-film layers on an anisotropic substrate, the determination of material constants can be based on the directional variations of the leaky SAW velocities. For each specimen, the directional variation of the leaky SAW velocities were measured by operating the LFAM system at 225 MHz and measuring V(z) curves for different wave propagation directions on the specimen. The phase velocities were extracted by spectral analysis of the corresponding V(z) curves to determine and by subsequent use of Eq. (1). The measured phase velocities were then used together with theoretical predictions to inversely determine the elastic constants of the thin film, based on Eq. (5). Four specimens with different cubic-crystalline thin films deposited on the (100) plane of an MgO substrate were tested. These specimens include two Tungsten (W) and Molybdenum (Mo) single-layer thin films and two double-layer thin films with a Niobium Nitride (NbN) layer on a Tungsten layer or a Molybdenum layer. The material properties of the MgO substrate were determined separately on a polished MgO specimen. Known values of the densities and the thickness of the thin films were used. The values of the densities on MgO: 3,598; Tungsten (W): 18,700; Molybdenum (Mo): 10,200; and Niobium Nitride (NbN): 8,430, all in

The determined elastic constants and are listed in Table 2 together with the global errors for the specimens. The global error is the value of the deviation function when a best fit has been found for each specimen. The comparisons between the determined values and literature values (from Ref. [12]) are listed in parentheses. After the material constants had been determined, the corresponding directional variation of the leaky SAW velocity was recalculated for each specimen to verify the

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data inversion procedure. For the different specimens the recalculated phase velocities versus the wave propagation direction are shown in Figures 7(a)-(f) in solid circles with the measured values in empty circles. As shown in Fig. 7, the presence of a single-layer or a multilayered thin film generally causes the variation with direction of the leaky SAW velocity to shift and to change the shape of the curve. The determined elastic constants are consistent for different specimens and they agree with values in the literature within less than ±5%. Different deposition conditions are thought to lead to different values of the elastic constants of the coatings layers.

4.

Effect of Imperfect Interfaces

The paper is completed with some comments on the effects of imperfect interfaces. In the theoretical approach an imperfect interface is replaced by a layer of extensional and shear springs. Such a layer of springs can easily be included in the method of calculation of the reflectance function that was earlier presented in Section 4. In Ref. [13] it has been attempted to correlate calculated values of the spring constants with

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results obtained by a destructive scratch test for strength determination. Specimens with different and interface conditions were prepared by using different lengths of time for etching of the substrate surface. For film deposition by the reactive sputtering technique, etching of the substrate for different time periods has an observable effect on the stiffness constants and of the interface, as shown in Fig. 8. The determined constants and show that for 10 minutes etching (specimen 1) they are about 2.7 times the values for the case of 20 minutes etching (specimen 2). The scratch test results indicate that the adhesion of specimen 1 is much better than that for specimen 2. These limited experimental results show better adhesion for a higher stiffness interface.

However, it is important to note that there is no theory which shows that nondestructive techniques can directly measure interface strength. At the present time, only destructive techniques, such as the scratch test and the pullout test, can provide us with a measure of the interface strength. In Ref. [13] the LFAM at high frequency was used to measure the interface stiffness. For most of the film/substrate systems studied in Ref. [13] the interface stiffness parameters seem to have a strong correlation with the destructively measured interface strength, such as the critical normal force of a scratch test. Therefore, measured interface stiffness parameters may provide an indication of the bond quality between a film and a substrate.

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Acknowledgement

The results reported here were obtained in the course of research sponsored by the Office of Naval Research under Grant N00014-89-J1362. References 1.

2. 3. 4. 5. 6.

7. 8. 9.

10. 11. 12.

13.

Wolfe, D. E., Movchan, M B. and Singh, J.: Architecture of functional graded ceramic/metallic coatings by electron beam-physical vapor deposion, in Advances in Coatings Technologies for Surface Engineering, C. R. Clayton, J. K. Hirvonen and A. R. Srivatsa, Eds., TMS, New York (1997), 93-128. Toth, L. E.: Transition Metal Carbides and Nitrides, 7, Academic Press, New York and London, 1971. Prengel, H. G.: Coating carbide cutting tools, Manufacturing Engineering, 117 (1996), 82. Ianno, N. J., Dillon, R. O., Ali, A. and Ahmad, A.: Deposition of diamond-like carbon on a titanium biomedical alloy, (in press). Valli, J.: A review of adhesion test methods for thin hard coatings, J. Vac. Sci. Technlogy., A4(6) (1986), 3007-3014. Briggs, A.: An Introduction to Scanning Acoustic Microscopy, Oxford U. Press, New York, 1985. Achenbach, J. D., Kim, J. O. and Lee, Y.-C: Measuring thin-film elastic constants by line-focus acoustic microscopy, in Advances in Acoustic Microscopy I, A. Briggs, Ed., Plenum Press, New York (1995), 153-208. Kushibiki, J. and Chubachi, N.: Material characterization by line-focus beam acoustic microscopy, IEEE Trans. Sonics Ultrason, SU-32, (1985) 189-212. Nayfeh, A. H. and Chimenti, D. E.: Reflection of finite acoustic beams from loaded and stiffened half-spaces, J. Acoust. Soc. Am., 75, (1984) 1360-1368. Pestel, E. G. and Leckie, F.A.: Matrix methods in elasto-mechanics, McGraw-Hill Book Company, New York, 1963. Guo, Z., Achenbach, J. D., Madan, A, Martin, K. and Graham, M. E.: Integration of modeling and acoustic microscopy measurements for thin films, J. Acoust. Soc. Am., 107(5), (2000) 24622471. Anderson, O. L.: Determination and some uses of isotropic elastic constants of polycrystalline aggregates using single-crystal data, in, Physical Acoustics, W. P. Mason and R. N. Thurston, Eds., Academic Press, New York and London (1965), 43-95. Guo, Z., Achenbach, J. D., Madan, A., Martin, K. and Graham, M.E.: Modeling and acoustic micrscopy measurements for evaluation of the adhesion between a film and substrate, Thin Solid Films (2001), in press.

RECENT ADVANCES IN ACOUSTOGRAPHY-BASED NDE

J. S. SANDHU and H. WANG Santec Systems, Inc. 716 South Milwaukee Avenue Wheeling, IL 60090 Abstract Acoustography is the ultrasonic analog of radiography and photography in that a 2D area detector is used to form full-field ultrasonic images. This unique detector, called the acousto-optic (AO) sensor, is capable of directly converting ultrasound into a visual image; much like a fluorescent screen is able to convert x-rays into visual image. It offers an exceptionally high pixel resolution since ultrasound field is detected by a continuous layer of liquid crystal material with molecules that are only 20 Angstroms in size. The AO sensor can be fabricated to have a large sensing area that allows image formation either through simple shadow casting (analogous to x-ray image formation) or with acoustic lenses (analogous to photographic or video camera). In this paper, we report on recent progress that is making acoustography a viable ultrasonic testing method both for through-transmission and reflection modes. 1.

Introduction

Acoustography [1] is a full-field ultrasonic imaging process where an acousto-optic area detector (AO Sensor) is employed to convert the ultrasound into a visual image in near real time. The physical principle by which the AO sensor converts ultrasound into visual images is based upon the birefringent properties of a proprietary liquid crystal layer contained in the AO sensor [1]. The layer exhibits no birefringence in the absence of ultrasound and exhibits a uniform dark field under cross-polarizer viewing [1]. Upon exposure to ultrasound, the layer becomes birefringent, showing a brightness change (i.e. optical density change) that is related to the ultrasonic intensity by the AO sensor's acousto-optic transfer curve [1]. To date, acoustography has been used analogously to x-ray imaging where images were formed in through-transmission via simple shadow casting [2-5]. However, recently progress has been made toward developing photography analogy where images are formed using acoustic lenses [6]. Regardless of how the images are formed, geometrical and lateral resolution in acoustography, like other ultrasound techniques, is diffraction-limited and is governed by the ultrasound wavelength in the test specimen [1]. The pixel resolution, on the other hand, is not usually an issue because of the high pixel density of the AO detector. For example, the ultrasound image is converted into a visual image by countless number of minute (~ 2 nanometer) liquid crystal molecules, and correlation between molecules persists for only a few micrometers [1]. Therefore, 381 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 381–388. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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the pixel size in the AO detector can be expected to be on the order of a few micrometers. The contrast resolution in acoustography depends on two factors: 1) acoustic contrast between anomaly and the normal component structure; 2) acoustooptic transfer curve of the AO detector which expresses the relationship between ultrasonic intensity (exposure level) and the corresponding optical density (brightness level). In this paper, we will report on recent advances in acoustography including improvement in AO sensor acoustic detection sensitivity, correlation between C-scan and acoustography ultrasonic data, and development of reflection-mode acoustography. 2.

Acoustography-based Ultrasonic Testing

Acoustography has been used for ultrasonic testing of materials mostly in throughtransmission mode, where ultrasound is passed through the test component. As it propagates, the ultrasound wave is absorbed, reflected, refracted and scattered by material structure and any anomalies therein. Upon exiting the test specimen, the ultrasound wave creates a projection image of the material structure and anomalies, which is received by a wide area AO sensor that converts the ultrasound image into a corresponding visual image instantly (Figure 1). Once the image appears on the AO sensor, a video camera can be used in combination with a frame grabber to acquire and digitize the image for computer storage and processing.

3.

Recent Advances

3.1 ACOUSTIC DETECTION SENSITIVITY

Although acoustography was introduced more than a decade ago [2,3], its applicability for nondestructive testing has remained limited due to numerous technical drawbacks. One of the major drawbacks has been the low acoustic detection sensitivity (i.e. high

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threshold) of the AO sensor. This shortcoming has recently been overcome through improved understanding of the acoustic coupling mechanism in LC materials, as discussed below. A theoretical model has been developed [6] to show that the torque (Figure 2), responsible for the reorientation of LC molecules in a LC layer exposed to ultrasound is given by:

where

d=LC layer thickness =Acoustic attenuation anisotropy c= Sound velocity I=Acoustic Intensity =Incident angle of the ultrasound beam

The molecular reorientation, in the LC layer (Figure 2) due to the acoustic torque, produces an optical output (acoustic-to-optic conversion) that can be described for normal-incident viewing by [7]:

where is the angle between the polarization axis of the incident light beam and projection of the optic axis of the liquid crystal and known as the retardation, is the phase difference between the ordinary ray and the extraordinary ray and is given by:

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where and are refractive indices of the extraordinary and ordinary rays respectively, is the light wavelength, and is the LC molecule tilt angle. Using the above model as a guide, new LC materials with physical properties that enhance the acoustic detection sensitivity have been developed. Figure 3 shows the performance of new materials compared to the prior LC materials. These proprietary LC materials have improved acoustic detection sensitivity of AO sensors from prior

3.2 CORRELATION BETWEEN ACOUSTOGRAPHY AND C-SCAN To test validity of the acoustography approach, a side-by-side acoustography and Cscan study was conducted of a graphite/epoxy composite panel standard. The panel was 10" x 12" in size with a thickness of approximately 1/8 in. It contained intentionally introduced defects of various sizes, ranging from 3/8 x 3/8 sq. in to 1/8 × 1/8 sq. in. The map of the general arrangement of defects is shown in Figure 4.

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Figure 5 shows the acoustographic image and ultrasonic C-scan of the graphite/epoxy composite panel standard. Only four rows of the intentionally placed defects are shown because of the limited tank size of the acoustography system (see Figure 4). As seen in Figure 5, the ability of acoustography to detect all defect sizes is comparable to the conventional C-scan method. However, definition of the defects seemed better in the acoustography method. The fibrous nature of the composite was also more evident in the acoustography method. The superior definition of defects and the fibrous nature of the specimen may be attributed to the superior pixel resolution of the AO detector used in acoustography. We should note that the C-scan was generated with a 3.5 MHz focused transducer, where the 0.02” (0.5mm) indexing defined the lateral resolution. In comparison, lateral resolution in acoustography depends on the pixel molecules in the AO detector are on the order of 20 angstroms and the correlation between molecules persists only for a few micrometers. Therefore, the pixel size on the AO detector is on the order of a few micrometers. However, the pixel resolution of the video camera and graphic monitor is much lower compared to that offered by the AO detector. Accounting for the lower resolution of the video camera and monitor and the magnification factor, we determined the actual resolution of the acoustographic image to be around 0.23mm. We should note that the field of view of the acoustography system used was 5cm x 5cm, therefore, the part was moved to image 5cm x 5cm contiguous areas. The collaging is apparent due to the inaccuracy of the manual part manipulation. This can be overcome by employing a mechanical manipulator that allows repeatable position of the test areas. Moreover, the use of large area AO detectors (e.g. 6”× 6” or larger), currently under development, will reduce the number of images required to inspect the entire part as well as produce dramatic improvements in inspection speed (e.g. 6”× 6” area in only seconds).

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3.3 REFLECTION MODE ACOUSTOGRAPHY A major limitation of acoustography has been its through-transmission mode of operation, which does not permit the method to be applied for NDT in the field. However, this limitation has recently been overcome. Several reflection mode concepts have now been explored. The first concept to be explored is shown in Figure 6.

The sound source and the AO sensor are placed on the same side of the test part. A water-filled prism constructed from acoustically transparent materials was used to interface the sound and the AO sensor assembly to the test part. Sound was received by the test part at an incident angle, where the front face (water/part interface) reflection was much lower than that from the back face (part/air interface) reflection. Since the AO sensor is an intensity detector, and ultrasound speed is much greater than the AO sensor response time, a summation image of all the transmitted, reflected and scattered rays is displayed on the AO sensor. The transmission, reflection and scattering of ultrasound waves is affected by the presence of flaws in the test component. Therefore, an image of the defect can be made. th Figure 7 shows the plastic sheet specimen that was inspected using reflection-mode acoustography. The specimen was a l/8th in. thick plastic sheet. Three plastic pieces of different thickness (1/16th in., 1/8 in. and ¼” ) were adhesively bonded on the back surface of the plastic sheet specimen. In addition, a blob of silicone adhesive was placed on the back surface approximately in the middle of the three plastic pieces. The bonded pieces on the back surface and silicone blob are evident in the acoustographic

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image. However, the image quality is not as high as the through-transmission acoustography-based UT because of three reasons. First, the image geometrically distorted since the AO sensor is at an angle with respect to the test specimen. Second, the standoff distance between the AO sensor and the specimen is large compared with the through-transmission case. Third, the sound source was monochromatic (single frequency) causing interference effects.

To improve the image quality of reflection-mode acoustography, a new approach was explored [8]. In this approach, the AO sensor is placed directly over the test component with an acoustic coupling layer (Figure 8). Ultrasound passes through the AO sensor and on into the test component. Regions containing defects absorb, reflect and scatter ultrasound differently than the normal material, resulting in a differential attenuation that creates the defect image on the AO sensor. The acoustographic image of the plastic sheet specimen produced using this reflection mode approach is shown in Figure 9. The improvement in image quality was found to be significant. Note the sharpness with which the bonded plastic pieces and silicone blob are imaged compared to the first reflection mode approach.

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4. Conclusions The recent advancements in acoustic detection sensitivity of the AO sensor used in acoustography are reported. The correlation of data between acoustography and conventional C-scan is excellent, lending support to the efficacy of acoustography as an ultrasonic NDE method. Finally, the feasibility of acoustography to be implemented in reflection mode has been demonstrated, which could lead to significant applicability of the method for NDE in the field where access is limited only to one side. 5.

Acknowledgements

This material is based upon work supported in part by the National Science Foundation (Grant # DMI-9407745) and Department of Defense (Contract # DAAD17-00-C-0014). The authors would also like to acknowledge Ms. Noreen Cmar-Mascis, U.S. Army Research Laboratory Vehicle Structures Directorate, and Dr. Pat Johnston, NASA Langley Research Center for their assistance in performing ultrasonic C-scans of the materials tested for this effort. 6. 1.

2. 3. 4. 5. 6. 7. 8.

References J. S. Sandhu, "Acoustography," Special nondestructive testing methods, Nondestructive Testing Handbook, 2nd. Edition, Editors: P.O Moore and P. Mclntire, vol.9, pp. 278-284, American Society for Nondestructive testing, Columbus OH, 1995. J. S. Sandhu, "Acoustography: A new imaging technique and its applications to nondestructive testing," Materials Evaluation, vol. 46, No. 5, pp. 608-613, 1988. I. M. Daniel, S.C. Wooh, J.S. Sandhu and W.A. Hamidzada, “Acoustographic Nondestructive Evaluation of Composites,” Review of Progress in QNDE, Eds. D.O. Thompson and D.E. Chimenti, Plenum Press, Volume 7A, pp. 325-332, 1988. J. S. Sandhu, H. Wang and W.J. Popek, "Acoustography for rapid ultrasonic inspection of composites," Proceedings SPIE conference on NDE, vol. 2944, pp. 117-124, AZ, Dec. 1996. J. S. Sandhu, H. Wang and W.J. Popek, "Ultrasonic Inspection of tight-radii in composites using acoustography," Proceedings SPIE conference on NDE, vol. 3396, pp. 169-179, San Antonio, TX, Mar. 1998. J. S. Sandhu, H. Wang and W.J. Popek, "Liquid crystal based acoustic imaging,” in Liquid Crystal Materials, Devices, and Flat Panel Displays, Ranganathan Shashidhar, Bruce Gnade, Editors Proceedings of SPIE Vol. 3955, pp. 94-108, 2000. M. Born and E. Wolf, “Principles of Optics,” Pergamon Press, Chapter XIV pp665, 1987. J.S. Sandhu and H. Wang, “Reflection Mode Device” worldwide patents filed and in process.

EXPERIMENTAL LIMITATIONS TO GUIDED WAVE GENERATION IN ELASTIC MATERIALS D. A. SOTIROPOULOS and E. BABATSOULI College of Science, Math & Technology The University of Texas at Brownsville & TSC Brownsville, Texas 78520, USA

Abstract

In generating small amplitude guided waves in elastic materials for diverse experimental investigations, one is faced with several limitations. This paper examines the limitations arising from the inherent nature of materials not to withstand, in some cases, the existence of guided waves either in certain finite frequency ranges or in the whole frequency spectrum. The material constants as well as the stress conditions in the materials play a key role in allowing the experimental generation of guided waves. The model examined in this paper incorporates a small layer either imbedded in a host material or a small surface layer overlying a host material, both materials in general being pre-stressed. The analysis defines limitation regimes where experimental generation of guided waves would not be possible. These regimes are given explicitly in terms of material constants and stress conditions. 1. Introduction

The generation of guided elastic waves in layered structures is of practical significance both for experimental investigations and for in-situ diagnostics. The areas of applications include the ultrasonic non-destructive evaluation and characterization of materials, their interfaces and stress conditions, the design of vibration isolators, and the processing of electronic signals. Of interest, here, is the generation of guided waves both in anisotropic layered materials and in stressed non-linear elastic layered materials. Of particular interest is the existence of frequency regions in which guided elastic waves cannot be generated as an inherent characteristic of the structure and its elastic properties. Previous studies, see for example [1-3], have demonstrated this. However, from the practical point of view a simple, if possible, rule is needed as an experimental guide to wave generation limitations. A simple rule that defines the structural elastic properties which permit guided wave generation in layered structures. In the present paper, simple formulae are presented which define structural properties regions in which guided waves can be generated. This is done for structures that consist of a small layer either surrounded by a host material or as a surface layer under-laid by a host material. Several cases are considered depending on whether the materials are compressible or incompressible, isotropic or anisotropic, and stressed or unstressed before the waves are generated. The limitations of guided wave generation are obtained by considering the dispersive wave characteristics. The simplicity of the 389 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 389–396. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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limitations, as far as their dependence on the structural properties is concerned, depends on the state of the underlying stress as well as on the direction of propagation of the generated waves with respect to material axes of symmetry and geometry of the structure. In the present paper the underlying stress is homogeneous in both materials with common principal strain axes, one axis being normal to the planar interfaces and guided wave propagation along a principal axis. 2. Dispersive Characteristics

The structure considered in the present paper consists of a small layer either imbedded in or overlying a host material. Guided waves are considered propagating along the planar surfaces separating the layer from the host material. Therefore, the host material can be considered infinite without loss of generality. Since the layer considered is small, it suffices to take kh small, where k is the wave number and h is the layer thickness. The generic structure consists of a pre-stressed layer of non-linear elastic material imbedded in an infinite pre-stressed different non-linear elastic material. The underlying stress is homogeneous in the two materials with the finite strains having common principal axes, one axis being normal to the planar interfaces. For the case that both materials are compressible, the dispersive characteristics are described by the dispersion equation which can be derived (the details of the derivation are outside the scope of the present paper) as

where

in which the elastic constants are given by

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are the principal Cauchy stresses and are given as

with being the strain energy density function after deformation. The elastic constants satisfy the conditions:

In equation (2), the star refers to the layer properties which are defined analogously to the above. The bars are defined as

Also, in equations (1, 2) the following definitions are pertinent:

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3. Limiting Conditions Most cases of practical interest (in this is also included the case of no pre-stressing) fall under the category and Then, for small kh, can be written as where is obtained from equation (1) as

By definition, must be positive in order for the guided wave to decay away from the interface. Therefore, guided waves cannot be generated (or exist) when

or, equivalently, when

Necessary conditions to satisfy (16) or (17) are

The above simple material and pre-stressed parameter conditions define the limitations to experimental investigations as far as the generation of guided waves is concerned. For equibiaxial in-plane (along and perpendicular to the layering) deformations as well as for materials that are not pre-stressed, so that R > 1 or r > 2 guarantee the feasibility of generating guided waves. When there is no pre-stressing the material constants become

and J = 1. Then, we conclude that materials with being the shear moduli of the host material and layer respectively), or allow the propagation of guided waves. The limiting condition (16) for the non-existence of guided elastic waves is shown graphically in Figure 1 that follows. In the figure, for simplicity, is represented by whose value affects the existence of condition (16) for small r depending on whether it is smaller, equal or larger than one.

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Naturally, the value of (in the figure) depends on the type of material and its prestress condition. For example, for Varga or Blatz-Ko materials where and are the principal stretches of the interlayer along and perpendicular to the interface respectively. 4.

Incompressible Materials

The limiting condition in the case of incompressible materials for the non-existence of guided waves can be obtained from (16) by setting and The resulting condition is:

where and The constants are related to the strain energy density function of the material, W, and the principal stretches by

The limiting condition (20) for the non-existence of guided elastic waves is shown graphically in figure 2 that follows. In the figure, for simplicity, is represented by and by The value of (in the figure) affects the existence of condition (20) for small r depending on whether it is smaller, equal or larger than one. This value (of is given by equations (21) as independent of the type of incompressible material. 5.

Incompressible Layer in a Compressible Material

For the case that the layer is incompressible and the host material is compressible, the limiting condition for the non-existence of guided elastic waves becomes

where From the generic cases described in sections 3, 4, and 5 follow results for a specifically stressed surface layer overlying the host material as well as for orthotropic materials. These will be discussed following Figure 2.

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A Thin Surface Layer Over a Host Material

The results obtained above as far as material properties not withstanding guided wave generation are applicable also to the case that a surface layer is stressed perpendicular to the interface by a Cauchy principal stress equal to / J. In other words, effectively, this stress replaces the host material above the layer in the cases considered in sections 3, 4 and 5. The limiting materials region is, then, given by (16), (20) or (23) depending on whether the surface layer and the host material are compressible or incompressible. 7.

Anisotropic Layer and Host Material

When both the layer and the host material are not stressed, the results obtained above can be used to obtain limiting conditions for orthotropic materials. In other words, the stressed isotropic materials possess a stress-induced anisotropy, consequently, including results for stress-free orthotropic materials. The limiting condition for orthotropic materials with the direction of guided wave propagation along a material axis of symmetry is obtained from (16) as

where The constants are the shear moduli for the host material and layer respectively in the plane of guided wave propagation, i.e. along the interface and perpendicular to it. 8.

Composite Layer and Host Materials

The layer, considered in this paper, is small as compared to the guided wave length. For this reason, the experimental limitations derived above for generating guided elastic waves are applicable to the layer and host material being composite materials as well. In this case, the composite materials can be effectively represented by homogeneous materials. For more details on this, the reader is referred to [4]. 9.

References

1.

Ogden, R. W. and Sotiropoulos, D. A.(1995) On interfacial waves in pre-stressed layered incompressible elastic solids, Proceedings of the Royal Society of London A, Vol. 450, pp. 319341 Sotiropoulos, D. A. and Sifniotopoulos, C. G. (1995) Interfacial waves in pre-stressed incompressible elastic interlayers, J. Mech. Phys. Solids 43, 365-387 Sotiropoulos, D. A. and Sifniotopoulos, C. G. (2001) The effect of stress on interfacial waves in elastic compressible interlayers, in Solid Mechanics and Its Applications Vol. 92: Mechanical Waves for Composite Structure Characterization, D. A. Sotiropoulos (ed.) Kluwer Academic Publishers, 169-185 Daniel, I. M. and Ishai, O. (1994) Engineering Mechanics of Composite Materials, Oxford University Press

2. 3.

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NONDESTRUCTIVE TESTING USING SHEAROGRAPHY MICHAEL Y. Y. HUNG Department of Building and Construction City University of Hong Kong Tat Chee Avenue, Kowloon, Hong Kong The author is presently on leave from Oakland University Rochester, Ml 48309. Abstract This article reviews shearography and its applications in nondestructive testing. Shearography is an interferometric technique for full-field and non-contacting measurement of surface deformation (displacement or displacement derivatives). It was invented to overcome some limitations of holographic interferometry by eliminating the reference beam, resulting in having much higher tolerance to environmental disturbances. Consequently, shearography can be practiced in a typical industrial setting, and it has already received considerable industrial acceptance, in particular, for nondestructive testing. One major difference of shearography from other NDT techniques is the mechanics of revealing flaws. Shearography reveals defects in an object by identifying defect-induced deformation anomalies which are more relevant to structural weakness. Other applications of shearography include strain measurement, material characterization, residual stress evaluation, leak detection, vibration studies and 3-D shape measurement.

1. Introduction Shearography is an optical method for measuring surface deformation. Unlike traditional measurement techniques, shearography does not require the laborious task of mounting a large number of strain gages or transducers. It is a noncontacting method that yields full-field information about surface displacement or displacement derivatives. Shearography was developed to address several limitations of holography. Its significant advantages include (1) not requiring a reference light-beam, thus leading to simple optical setups, reduced coherence length requirement of the laser, and lax vibration isolation; and (2) direct measurement of surface strains (first-order derivatives of surface displacements). These distinct advantages have rendered shearography as a practical measurement tool and it has already gained wide industrial acceptance in nondestructive testing. For instance, the rubber industry routinely uses shearography for evaluating tires, and the aerospace industry has adopted it for nondestructive testing of aircraft structures, in particular, composite structures. Other applications of shearography include: measurement of strains, material properties, residual stresses, 3-D shapes, vibrations, as well as leakage detection. 397 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 397–408. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Three versions of shearography are in existence which are based on different recording media : photographic[1]; thermoplastic[2]; and digital[3]. In this paper, only the digital version is presented.

2. Principles of Digital Shearography

Description of shearography.

A schematic diagram of digital shearography is shown in Fig. 1. The object to be evaluated is illuminated with laser light radiating from a point-source, and it is imaged by an image-shearing camera connected to a microcomputer for recording and processing. The camera comprises a CCD image sensor, a lens and an image shearing device. The image shearing device consists of a double-refractive prism and a polarizer. The function of the image shearing device is to produce a pair of laterally displaced (sheared) images, and hence the technique is named as shearography.

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The principle of image-shearing device using a doubly-refractive prism is illustrated in Figure 2. A light beam passing through a doubly refractive prism such as a Wollaston prism is split into two angularly separated beams. Conversely, as shown in the figure, two non-parallel beams of light scattered from two different object points are combined by the prism and become nearly collinear. Since the spatial frequency of the interference fringe pattern formed by two beams is proportional to the sine of the half angle between the interfering beams, two nearly collinear beams produce a very low frequency interference pattern that is comfortably resolved by a video image sensor such as CCD. The two sheared wavefronts transmitted by the two axes of the image-shearing device, however, are orthogonally polarized, hence they will not interfere with each other. To enable interference, a polarizer with its polarization axis oriented at an azimuth of 45° is required. As an object surface is generally optically rough, interference of the two sheared wavefronts will result in a speckle pattern embedded in the shearographic image. The speckle pattern is slightly altered when the object is deformed. Two speckle patterns of the test object, one before and another after it is slightly deformed, are digitized into a microcomputer via a frame grabber. As will be shown below, the difference of the two speckle patterns will enable reconstruction of a visible fringe pattern that depicts displacement-derivatives with respect to the direction of image-shearing. Fig. 3 shows a fringe pattern depicting the deflection derivatives of a rectangular plate clamped along its boundaries and loaded by uniform pressure.

Principles of Fringe Formation A shearographic image contains a speckle pattern that may be mathematically represented as follows.

where I is the intensity distribution of the speckle image received at the image plane of the camera; is the intensity of the laterally-sheared images (which may be perceived as the d.c. term); is the amplitude of modulation of the speckle pattern; and is a random phase angle that accounts for the random interference

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pattern. After the object is deformed, the intensity distribution of the speckle pattern is changed slightly to I’, which is described by the following equation.

where denotes phase-change due to surface deformation. Numerically, the difference of intensities of the two speckle patterns (Eqs. (1) and (2)) yields the following equation.

where

manifests as a fringe pattern in which the dark fringes correspond to with n = 0,1,2,3,... being the fringe orders, and the bright fringes correspond to half fringe orders. The fringe pattern of Fig. 3(left) is for a fully clamped rectangular plate under uniform lateral load when image-shearing is along the x-direction, and Fig. 3(right) shows the corresponding fringe pattern when image-shearing is along the y-direction. Note that the absolute value of is used in the display of the fringe patterns, as an image cannot have negative value.

Fringe Interpretation The phase-change in Eq. (2) is induced by the change in the relative optical path length of light scattered from two neighboring object points, P(x,y,z) and considering that the direction of image-shearing is parallel with the x-axis and the amount of shearing is is related to the relative displacement of the two neighboring points as follows.

where (u,v,w) and are, respectively, the displacement vectors of P(x, y, z) and is the wavelength of the laser, and A, B, C are sensitivity factors that are related to the positions of the illumination point and the camera in the following manner.

with It is noted that z = z(x,y) describes the object surface and is not an independent variable when surface points are considered. Equation (4) may be rewritten in the following form.

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If the amount of shearing, is small, the relative-displacement terms in Eq. (6) may subsequently be treated as the partial derivatives of u, v and w with respect to the direction of image-shearing x. Thus, Eq. (6) becomes

Rotating the image-shearing device (Figure 1) about its optical axis will alter the direction of image-shearing, resulting in fringe patterns depicting displacement derivatives with respect to the current direction of image-shearing. Hence, if imageshearing is along the y-axis by an amount Eq. (7) is modified to the following.

The fringe patterns in Figures 2(a) and 2(b) for a laterally deflected plate therefore represent, respectively, displacement derivatives, namely,

and

Equation (7) contains three and

so that three measurements

with different sensitivity factors A, B, C, are required to separate these displacement derivatives. In NOT applications to be presented below, only the derivative of out-of-plane displacement is needed. In this case, the setup parameters A and B are selected to be zero.

Automated Fringe Phase Determination Traditional methods of shearographic analysis rely on the reconstruction of visible fringes and the correct identification of fringe orders (Eq. (3)). Therefore, only at locations where the dark and bright fringes appear can the fringe orders be correctly determined; elsewhere the fringe orders are generally estimated using linear interpolation. Because of the page limit, readers are referred to Ref (3) for an automated process in which the phase-change is directly determined at every digitized points without having to rely on fringe reconstruction and fringe order identification. Fig 4 shows a 3-D plot depicting the deformation of Fig 3 obtained by the automated fringe phase determination technique

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3. Applications Shearography is a full-field optical sensor based on the principles of optical interference. It measures the relative phase change between interfering wavefronts. The phase change can be induced by surface displacements, strains and refractive index change. Applications of shearography that have been explored so far include measurement of strains, material properties, residual stresses, 3-D shapes, vibrations,leakage detection, and nondestructive evaluation. This paper reviews its application for nondestructive evaluation, in particular for composites structures.

How does shearography detect flaws? When an object containing a flaw is loaded, strain concentration at the vicinity of the defect is induced. If the flaw is not too remote from the object surface, the induced strain concentration would cause anomalies in the surface strain distribution. Subsequently, these anomalies are translated into fringe-anomalies if two speckle patterns, taken one before and another after loading, of the object are compared. Thus, shearography reveals flaws, both surface and internal, through identification of anomalies in the fringe pattern. Shearographic nondestructive inspection is full-field and non-contacting.

Shearography vs Ultrasound Shearography has found wide applications in nondestructive testing and it has already received industrial acceptance as a useful NDT tool, particularly for composite structures such as tires and honeycomb structures. It is particularly effective in revealing delaminations in laminated composites. Fig 5 shows a comparison of test result obtained using digital shearography and C-scan ultrasonic testing on a composite sample. Both techniques readily reveal an edge pullout and a delamination. With digital shearography, the flaws are revealed in less than one second, whilst the ultrasonic technique requires point-by-point scanning along the test surface and at the same time, proper fluid coupling between the transducer and the test surface is to be ensured. A limitation of shearography is the need to apply suitable stress-increments to the test object during inspection, since the underlying principle of this technique is based upon the response of defects to stresses. However, the mechanics of revealing flaw in shearography yields information more relevant to the structural weakness.

Methods of stressing The development of shearographic NDT procedures has therefore essentially become the development of a practical means of stressing the object that would readily reveal flaws. Ideally, the stress-increment should be similar to the service stresses so that flaws that are critical and detrimental to the service life of the object would be revealed, and cosmetic flaws that do not undermine the structural integrity of the test object can be ignored. This would minimize unnecessary rejects during inspection. In this regard, shearography has an advantage over ultrasonic testing, as the latter detects flaws by identifying inhomogeneities in the object and does not provide direct information on the criticality of the flaws. Exact duplication of the actual stress-increment for shearographic testing, however, is generally difficult. Therefore, various practical means of stressing the object must be developed. In

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developing these methods, an important precaution to be taken is restricting rigidbody motion of the object during stressing, as excessive rigid-body motion would cause speckles de-correlation, resulting in degradation of fringe quality. The methods that currently used that do not cause intolerable rigid-body motion of the test object include the use of pressure, vacuum, thermal, and acoustical and mechanical excitations. The use of microwave that excites water molecules is particularly effective for detecting moisture in plastics and non-metallic composites, but safety precautions must be taken seriously. The following figures illustrate some examples of the shearographic NDT application.

Samples of NDT applications Some examples of nondestructive testing using shearography are illustrated in Fig 5 through Fig 13.

4. Conclusion A review of shearography and its applications in nondestructive testing is given in this paper. Unlike holography, shearography does not require a reference beam and hence it is more tolerable to environmental disturbances. Indeed, shearography can be employed in field/factory settings. This technique is still relatively young and its full capability awaits further exploration.

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5. Acknowledgement The work presented in this paper was supported by a National Science Foundation grant (9601778).

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6. References 1. 2.

3.

Hung, Y.Y. “Shearography: A New Optical Method for Strain Measurement and Nondestructive Testing”, Optical Engineering, pp.391–395, May/June, 1982. Hung, Y.Y. and J.D. Hovanesian, “Fast Detection of Residual Stresses in an Industrial Environment by Thermoplastic-based Shearography”, Proceedings of the 1990 SEM Spring Conference on Experimental Mechanics, pp769-775, Albuquerque, New Mexico, June4-6, 1990. Hung, Y.Y. “Digital Shearography and applications” Trends in Optical Non-destructive Testing and Inspection, pp287-308, Elsevier, 2000 (Editors: Pramond K. Rastogi and Daniele Inaudi).

THEORETICAL AND EXPERIMENTAL STUDY OF LASER-ULTRASONIC SIGNAL CHARACTERISTICS ENHANCED BY WETTING THE SURFACE SHI-CHANG WOOH Department of Civil and Environmental Engineering Massachusetts Institute of Technology Cambridge, Massachusetts

Abstract This paper deals with a theoretical and experimental study to predict the characteristics of ultrasonic waves irradiated from a wet surface illuminated by a laser beam. The theoretical model is developed by assuming normal surface tractions distributed elliptically over the line-focused illumination source. The solutions for both longitudinal and shear waves are obtained using Fourier analysis, in which the transform integral is evaluated asymptotically. The theory is evidenced by the experimental study which allows for quantitative comparisons between the theoretical and experimental directivity patterns. The effects of beam width and frequency of the propagating wave, which are the most influential parameters on the characteristics of the generated wave, are investigated. The experimental results are in strong agreement with the theoretical predictions.

1 Introduction The work described in this article is inspired by the need for developing a testing method enabling non-contact detection of rail flaws, particularly those which are not easily detectable by other methods. In order to avoid catastrophic failure due to the growth of defects, the running rails should be inspected periodically using one or more nondestructive evaluation (NDE) techniques. Among the many types of flaws introduced to rail tracks from a variety of sources, the most common and pernicious kinds are transverse defects (TD) which may cause derailment if they go undetected [ 1 ]. These flaws are mostly detectable using high-angle ultrasonic wheel transducers. However, they are sometimes masked by other defects, such as longitudinal shells. Traditional ultrasonic techniques (e.g., those that impinge ultrasound from the top surface of the head) are intrinsically blind in dealing with such situations because the propagating beam is blocked off by the shell. The increased number of undetected defects reduces the overall reliability of detection. In order to resolve this problem, it may be necessary to access the rail specimen on its sidewall. In this way, it is possible to detect the hidden TD because the area 409 E.E. Gdoutos (ed), Recent Advances in Experimental Mechanics, 409–420. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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underneath the shell is directly insonified. However, it is not practical to use traditional wheel transducers because the accessible space on the side is t y p i c a l l y less than 5 cm. The use of laser generation and detection of ultrasound allows for easy access to the sidewall and enables totally non-contact operations. This paper deals with the theoretical and experimental study which may be used to find the optimum design of laser ultrasonic rail defect detection. Figure 1 shows a laser-ultrasonic system scanning a rail to detect internal defects. In principle, acoustic wave energy is generated by illuminating a point on the surface of the railhead using a laser beam, and the traveling wave arriving at another location on the head is recorded. The use of optical laser beams for both the generation and detection of ultrasound could completely eliminate the need for contact. If there exists a discontinuity that blocks the acoustic beam pathway, it may change the characteristics of the propagating wave and the signals may be interpreted to identify and characterize the discontinuity or rail defect. Figure 2 shows the laser-ultrasonic rail flaw detection scheme, in which an excitation laser (Q-switched Nd:YAG laser) shoots a pulsed laser beam at a point (Point A) on the front wall of the specimen. At the same time, a detection laser beam (Photo-EMF detector with a continuous probe laser) is aimed at another point (Point ) on the same surface, separated by a fixed distance Both lasers are operating in synchronization to generate an ultrasonic pulse and measure the surface displacements at the corresponding points. The waves irradiated from the source point A travels forward into the rail covering a wide range of angles and they are reflected off after hitting the back wall. As shown in Fig. 2(a), one or more rays of the reflected waves may continue traveling all the way to the detection point if no flaw exists along its pathway. By contrast, if there is a discontinuity (Fig.2(b)). the rays cannot reach the detection point , since they are blocked by the

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flaw. This results in the formation of a shadow zone behind the flaw created by a bundle of such rays. The shadow pattern may be used not only to detect the defect but also to measure its size and location. While maintaining the distance between the generation and detection lasers at the system moves together along the specimen (from Point A to D) to scan the area of interest. Figure 3(a) shows a typical waveform obtained from an unflawed zone. As noted in the figure, the laser source beam produces both body waves (L and T-waves) and strong surface waves (R-waves). In Fig. 3(b), a noticeable reduction of shear wave amplitude is observed at a shadowed region. These observations give us a clue that the most efficacious wave mode is the shear wave propagating strongly at high angles, which promotes good interaction between the wave and the flaw. The characteristics of lasergenerated ultrasound are dependent on various factors such as the size and shape of the illumination spot, frequency, elastic and thermal properties of the target materials, and the surface condition. It is known that a surface modified by grease, water or other liquid can significantly boost the signal amplitude [2,3]. Since the body waves must travel relatively long distances in the specimen, the signal amplitudes produced by the laser irradiation on a dry

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surface may not be high enough to be detected. Thus, it is preferred in our case to wet the surface before activating the generation laser, which can be achieved in-situ using water sprays. It is the objective of this paper to study the behavior or ultrasound radiated from a wet surface.

2 Theory The basis of our theory and experimental work taking into account the wet surface condition was reported in our previous articles [4, 5], in which a laser beam of circular cross section was modeled. In this paper, we extend our study by considering a line-focused laser source. The fundamental difference between the circular and line sources is that the former produces axisymmetric conical waves, whereas the latter produces cylindrical waves. Most of the derivations are similar to those for circular sources except for the choice of coordinate systems and the use of the double Fourier transform (FT) instead of the Fourier-Bessel transform in order to take into account the rectangular source geometry.

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Excitation Source As shown in Fig. 4(a), we consider an infinitely long line-focused source of width loaded by a normal stress traction. It is reasonable to assume that the rapid evaporation of fluid will cause high reaction pressure acting in the direction normal to the surface. This assumption is supported by [3], in which it is shown that the experimental directivity patterns for a wet surface are close to those of the ablation source. Studies involving the range of laser intensity below the evaporation level lie beyond the scope of this paper. The problem is treated using Cartesian coordinates where the is normal to the surface and the is parallel to the lateral direction of the line strip. Due to the cylindrical symmetry of the propagating wave, the problem is simplified as a two-

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dimensional elastic half-space excited by the surface traction, as illustrated in Fig. 4(b). It is also assumed that the vertical displacement component and its derivatives vanish due to the uniform loading along the

Boundary Conditions Fora line source of width . the boundary condition can be written by prescribing the time harmonic surface tractions distributed arbitrarily within the source width as:

where is the position of a point on the surface measured from the source origin is the excitation frequency, and is the unit imaginary number. It is assumed that the shear traction is zero everywhere on the surface because we only consider the reaction forces acting normal to the surface. We are interested in obtaining the field equations and studying the directivity patterns to study the angular distribution of acoustic wave energy propagating in the medium. Since the directivity can be directly obtained from the steady-state solution, it is sufficient to consider only the amplitude in the analysis. We consider two ideal tractions representing dry and wet surfaces (Case I and II) shown in Figs. 5(a) and 5(b), respectively. For dry surfaces, the traction is idealized as a vertical load of magnitude distributed uniformly over the illuminated area. On the other hand, the wet surface is modeled by assuming elliptical distribution of traction with an aspect ratio of The tractions can thus be written in the following forms:

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The elliptical distribution model is based on the same physical insights that we explained in our previous articles [4, 5]. If a wet surface is struck by a pulsed laser beam, it is likely that the reaction forces at the center, resulting from the evaporation of water particles, are significantly greater than those near the boundary of the illuminated area, which is surrounded by colder neighboring molecules. In order to represent this physical phenomenon, an elliptical function is chosen primarily for mathematical convenience.

Field Equations The following equation of motion for elastic isotropic solids governs the behavior:

where u is the displacement vector, is the mass density, and are the Lame constants of the medium, and is the gradient operator. Note that this equation is written in terms of displacements and should be solved to satisfy the stress boundary conditions given earlier. By defining the scalar and vector potentials and the wave equations, Eq. (3), can be written as

so that

where

and are the time-harmonic displacements, W is the component of W in the and and are the wavenumbers of the longitudinal and shear waves, which are respectively given as

Using Hooke’s law, the stress boundary conditions can be written in terms of potentials as:

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within the region where The Fourier transformation can be used to eliminate from these equations. For example, the FT of the elliptical loading given in Eq. (2) can be simply written as

After carrying out the transforms of Eqs. (4), (5), (6), (7), (9) and (10), the FT of the potentials are obtained. Then, the displacements and are obtained by solving for and and asymptotically evaluating their inverse FT [4, 6]. The longitudinal and transverse displacement components can be derived by decomposing the displacement vector using the relationship

The subscript is used to distinguish the line source from the circular source. The radial and tangential displacement amplitudes for a line source of width can thus be written for the two difference loading cases as follows:

Case I. Dry surface (Uniformly distributed load) The solutions for the corresponding displacement amplitudes for this case are already derived by Miller and Pursey [6] as

where

The functions and denote the longitudinal and transverse displacement amplitudes in the far-field for a hairline source, in which the width approaches zero. They are respectively given as

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where

Note that the displacement amplitudes hairline sources.

and

are the same for both point and

Case II. Wet Surface (Elliptically distributed load) The solutions for wet surfaces are obtained as follows, using the aforementioned elliptical loading model:

where and are the far-field displacement amplitudes of the compressional and shear waves radiated from a hairline source given by Eqs. (17) and (18).

3 Experimental Results and Discussion

Experimental Setup As we have done in our previous work [5] for convenience, the out-of-plane displacements are measured in this study to validate and compare the models, instead of measuring the directivities directly. Figure 6 illustrates the schematic of the experimental setup used to measure the displacements and their angular dependence. First, a laser beam is fired to illuminate a source point on the aluminum test block of thickness Then, an ultrasonic ray irradiated at an angle from the source point reaches the point on the opposite surface of the specimen. The displacement in the direction normal to the surface is then detected by either a laser interferometer or a polyvinylidene fluoride (PVDF) piezo-polymer transducer. By scanning the surface with the excitation laser beam, it is possible to obtain a displacement profile along the or as a function of corresponding angle for the known thickness The detailed experimental

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test procedure can be found in our earlier article [5]. The basic difference between the two studies is the laser illumination shape. While using the same laser source of circular beam cross-section, the beam shape was transformed into a line segment using an assembly of spherical and cylindrical lenses. With this setup, the rectangular illumination shape of the line segment is provided by the cylindrical lens while its illumination width was controlled by the optical characteristics of both lenses.

Results Figure 7 shows the displacements of transverse waves plotted as a function of propagating angle The circular symbols represent the normal (out-of-plane) displacements measured experimentally, while the solid lines represent the corresponding theoretical displacements for the same experimental conditions. The curves displayed on the left-hand side of the figures are the displacements for dry surface (Case I); the ones on the right-hand side are those for wet surface (Case II). The signals for wet surface were obtained by wetting the illumination surface using a water spray gun for each measurement. The tests were carried out for four different conditions: and 3.5 mm and and 5 MHz. The line width was varied by changing the distance between the lens assembly and the specimen surface so that the beam is defocused to the desired degrees. For this, a calibration paper was used to measure the dimensions. On the other hand, the different frequencies were obtained by using different detection techniques: A Photo-EMF laser detection system was used to attain signals of 1 MHz

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and a PVDF film transducer was used for 5 MHz. The validity of using these techniques are again discussed in our previous article [5] where the frequency spectra of the signals are shown.

4 Conclusions The models for both dry and wet surfaces are considered for various conditions. From a theoretical point of view, the two solutions are similar, except that the equations are modified by different terms. In describing the directivities for a line source loading, the corresponding modification terms for wet and dry surfaces are and respectively. The experimental results are surprisingly consistent and predictable by the theory, even with the assumption that the source is infinitely long in the From the results, we can observe that the propagation angles where the amplitude reaches its maximum are almost the same for both wet and dry surfaces (approximately at 35°), which can be used for optimum operation. For the lower frequency of 1 MHz, the change of beam diameter does not alter the characteristics of transverse waves much at low angles (below 30°). But the curves at high angles are noticeably squeezed to the left. On the other hand, the changes due to the beam size for the higher frequency (5 MHz) are dramatic in both high and low angles. It is interesting to note that wetting the surface introduces stronger shear waves propagating at low angles. For the wet condition for a large beam at high frequency, the amplitude shows more uniform and periodic patterns.

5 Acknowledgment This work was supported by the National Science Foundation under the Contract Number CMS-9733157. We are grateful for their support and encouragement.

6 References 1. Sperry Rail Service (1989) Rail Defect Manual. 2. Fox, J. A. (1974) Effect of water and paint coatings on laser-irradiated targets, Applied Physics Letters, 24(10), pp. 461–464. 3. Hutchins. D. A.. Dewhurst, R. J., and Palmer, S. B. ( 1 9 8 1 ) Laser generated ultrasound at modified metal surfaces, Ultrasonics, 19(3), pp. 103-108. 4. Wooh, S. C. Wooh and Zhou, Q. (2001) Behavior of laser-induced ultrasonic waves radiated from a wet surface. Part I: Theory. Journal of Applied Physics, 89(6). pp. 3469–3477. 5. Wooh, S. C. Wooh and Zhou, Q. (2001) Behavior of laser-induced ultrasonic waves radiated from a wet surface, Part II: Experimental Work, Journal of Applied Physics, 89(6), pp. 3478–3485. 6. Miller, G. F and Pursey, H. (1954) The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid, Proc. Royal Society (London), A223, pp. 521–541.

DEFECT DETECTION BY THE SCATTERING ANALYSIS OF FLEXURAL WAVES PAUL FROMME and MAHIR B. SAYIR Institute of Mechanical Systems ETH Zürich - Swiss Federal Institute of Technolog, CH-8092 Zürich, Switzerland

Abstract The propagation and scattering of flexural waves by obstacles in plates is studied experimentally and theoretically. When the guided wave hits a discontinuity like a hole, a typical scattered displacement field is obtained. Defects like a notch or a fatigue crack at the hole boundary change the scattered field significantly. The first antisymmetric Lamb wave mode is excited selectively by means of a piezoelectric transducer. The scattered field around undamaged and damaged holes is measured on a grid around the hole using a heterodyne laser interferometer. The out-of-plane displacement is measured with good spatial resolution, accuracy, and repeatability. Applying fast Fourier transform, the amplitude and phase values of the scattered field are extracted, overcoming the typical problems associated with the measurement of dispersive waves. The experimental results are compared with theoretical calculations. The wave propagation is studied using classical plate theory and Mindlin’s theory of plates. The scattered field around the circular hole is calculated analytically and good agreement is found for the range of validity of the used theories. The scattering at the hole with a defect is calculated numerically, using a finite difference scheme. The method can be applied for the detection of cracks at rivet holes in aircraft fuselage, an important nondestructive testing application. Guided waves allow a fast inspection of large areas, reducing the need for time-consuming scanning. The minimum detectable crack length and the sensitivity of the method are discussed.

1. Introduction Technical machinery, systems, and components, e.g., airplanes, automobiles, and pipes, are subject to varying or cyclic service loads and environmental influences. Such operation conditions can lead to wear, corrosion, and damaging of the components. The problem is relevant in aircraft industry, where a common maintenance problem is the development of fatigue cracks during the service life span of the aircraft. The fuselage and wings of planes consist of large plate-like parts, connected with lines of rivets and fasteners. Due to the stress concentration at these connections, fatigue cracks start to grow from the fastener holes, and must be detected before reaching a critical length (Fig. 1). 421 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 421–432. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Mechanical waves, as used in ultrasonic testing (UT) have a well-established performance for the detection of such defects. However, in classical UT the used wavelength of the mechanical waves is in the order of the crack length one aims to detect and thus small compared to the thickness of the structure. Waves at such rather high frequencies dampen out quickly and good signal strength of either transmitted or reflected wave can usually only be achieved working through the thickness of the structure. Therefore, classical UT involves manual scanning around the area of suspected defects, which is time-consuming and therefore cost-intensive, as it increases the downtime of the plane. An alternate, more elegant and promising approach is the use of guided waves, resulting in a propagation direction along the structure and performing such checks automatically over large parts of the structure. Placing a transducer on the specimen, a guided wave can be excited, that travels along the plate and interrogates a whole line of rivets. From the measurement of the guided wave at a few points on the surface of the structure, it is possible to detect defects in a large area with a fast and cost-effective method [1]. Rose and Soley [2] showed the practical applicability of guided waves for a variety of problems in aircraft components, e.g., crack detection in helicopter blades. In complicated systems, as in an aircraft, there are many difficult to access places like the interior of the fuel tanks in the wings. Automated built-in measurement systems will reduce the laborious and hazardous checking by personnel. Using low cost piezoelectric transducers for the excitation of the guided waves, they can be integrated into the structure. This way a smart structure is fabricated, where the damage checks can be performed automatically while being in service. The problem arising from the use of guided waves is the fact that the wavelength is usually of the order of magnitude or larger than the thickness of the structure and hence large compared to the typical flaw size one aims to detect. The large ratio between wavelength and defect length reduces the sensitivity and makes an accurate study of the propagation and scattering characteristics necessary, to determine experimental constraints and gain a well-founded theoretical understanding of the interaction between wave and flaw.

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423

Background

The propagation of guided waves in isotropic, homogeneous plates is described by the theory of Lamb waves [3]. The first antisymmetric mode is a flexural wave, studied in the work reported here. In contrast to the bulk waves used in UT, the propagation of the guided wave modes is dispersive, i.e., wavelength and propagation velocity depend on the frequency. This distortion of the signal poses some experimental challenges, but can be overcome by using either narrowband excitation or advanced signal processing [4]. Due to the two-dimensional propagation of waves in plates, the amplitude of the wave decreases with distance from the source. This decrease in amplitude can lead to problems with the signal to noise ratio, as the wave travels from the excitation transducer to the defect and then further on to the measurement spot. Therefore, measurement methods that are sensitive to small excitations like laser interferometry are advantageous. Flexural waves were successfully employed for the measurement of the material properties of composite plates [5]. The displacement of the flexural wave is primarily a bending of the plate. For low frequencies, i.e., when the wavelength is large compared to the plate thickness, the propagation can be described approximately using classical plate theory (CPT), taking only bending stiffness and inertia into account (see e.g. [6]). For higher frequencies, shear and rotatory inertia have to be considered according to the theory of Mindlin [7]. The model system investigated here is a circular through hole in an aluminum alloy plate with a notch at an arbitrary angle. This interesting theoretical problem has only partially been studied in literature. Analytical solutions can be found for the geometrically simpler problem of the scattering of flexural waves at a circular cavity [8]. Only few, mostly numerical, studies exist on the scattering characteristics of flexural waves at a notch or a crack in a plate [9, 10].

3.

Experimental setup

Flexural waves can be excited rather easily by means of a piezoelectric transducer and measured using a heterodyne interferometer, but are highly dispersive in the frequency range of interest here, between 20 kHz and 200 kHz. Applying Fourier transform for the data evaluation and studying amplitude and phase variations instead of time of flight measurements, this problem can be overcome. The dispersive nature of the excited pulse can be further used to measure material properties. Applying the known dispersion properties, a desired signal shape in the measurement area can be achieved, e.g., a contraction of the signal in the time domain. The geometry of the setup (Fig. 2) used in this investigation and further described in Ref. [11] is not a conventional C-scan, as used in UT. The scattered field of the flexural is measured on a grid around the hole, while the excitation transducer is fixed in one position. Therefore, as assumed in the analytical calculation, the incident wave for each measuring point is the same and the scattered field can directly be compared with the calculations. The first antisymmetric Lamb wave mode is excited selectively, working in a frequency range below the cutoff frequencies of the higher Lamb modes. The velocity of the out-of-plane displacement is measured point-wise on a grid around the hole using a heterodyne laser interferometer. At each measuring point, a time signal of the velocity of the plate surface is stored. Amplitude and phase of the scattered wave are extracted by fast Fourier transform (FFT) and analyzed.

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The specimen is a thin aluminum alloy plate (1 mm thick), having a size of 1000 mm by 1000 mm. The plate material is a commercially available aluminum alloy with a Young’s modulus E of 6.9 and a density of 2700 Poisson’s ratio v is assumed as 0.31. A circular hole with a radius of 10 mm is drilled through the plate. Static bending of the plate is avoided by suspending it vertically. A piezoelectric disc, polarized for thickness extension mode, of 10 mm diameter and 1 mm thickness is glued to the plate 300 mm away from the hole using a two-component epoxy glue. For the simulation of a defect at the hole boundary, a through thickness notch of about 0.2 mm width, 2 mm length, and a blunt tip is cut into the hole boundary at an angle to the vertical (see Fig. 2), using a fine saw blade. The excitation signal is chosen as a sinusoid multiplied by a Hanning window to achieve a short time, narrowband signal with the energy concentrated around the center frequency The short time duration of the excitation pulse allows a time gating between the measurement signal (the interference of incident wave and wave scattered at the hole and the notch) and the reflection from the plate boundaries, arriving some time later at the measurement spot. The arrival times of the different pulses are calculated from the theoretical group velocities. The narrow bandwidth avoids extensive signal distortion due to the dispersive character of the flexural wave. A typical measured time signal with a center frequency of 50 kHz and 20 cycles is shown in Fig. 3. The excitation signal is generated in a programmable function generator and amplified to 200 V peak to peak. Applying the voltage to the piezoelectric transducer, the disc contracts and expands. This way a vertical force to the plate surface is generated and primarily the first anti-symmetric Lamb wave mode is excited. For the frequencies range of this study, the energy transferred to the longitudinal mode and the shearhorizontal mode SH is negligible. Working below the cutoff frequencies for the

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higher wave modes, only the desired mode is excited, as no selection between different modes is necessary. The used frequencies of up to 200 kHz are well below the eigenfrequencies of the piezoelectric disc (ca. 1 MHz), thus working in the frequency range of the transducer with a linear transfer curve. However, the amplitude of the excited wave is rather small, below 1

The resulting flexural wave propagates radially outwards from the piezoelectric disc. The distance between the transducer and the hole is selected as large compared to the hole radius. Therefore, the wave front can be assumed to be planar when it reaches the hole, simplifying the theoretical simulation. The incident wave is scattered at the stress-free boundaries of the hole, and a scattered wave is generated. The scattered wave consists of three parts: a flexural wave propagating radially outwards from the hole, a flexural boundary layer close to the hole, and a shear boundary layer. Near the hole, incident and scattered wave overlap in time. Therefore, only a single pulse is visible in Fig. 3. The specimen size is selected large enough that the reflections from the plate boundaries arrive at the measurement area some time later, allowing a time gating of the measurement pulse. The scattered field on a measurement grid around the hole is recorded using a commercially available heterodyne laser interferometer. The demodulator output is a voltage signal proportional to the velocity of the out-of-plane component of the displacement of the plate surface. The measurement spot is less than 0.1 mm in

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diameter, defined by the laser beam. This achieves a point-wise measurement of variations in the scattered field, as no implicit average over a rather large surface of the measuring transducer is made. The laser interferometer is moved parallel to the plate using a positioning system. A radial measurement grid is used, moving the laser beam spot on concentric circles around the hole and recording a time signal with 10’000 values at every measurement point. The measurements are very repeatable, with the largest cause of variation due to an inaccurate positioning of the laser beam relative to the hole center, which could only be achieved with an accuracy of about 0.1 mm. The voltage signal is band pass filtered around the center frequency and averaged over 25 measurements in a digital storage oscilloscope. The function generator triggers the oscilloscope, so that excitation and measurement start at the same time. The measured time series are evaluated on the computer. Fast Fourier transform (FFT) is applied and the amplitude and phase values at the center frequency are calculated. These values are the equivalent of the theoretical results, derived in section 4, where an infinite sinusoid is assumed for the incident wave.

4. Theoretical calculations The propagation and scattering of flexural waves in isotropic, homogeneous plates can be described by various approximate solutions. To achieve a fast computation, different degrees of simplification have been used. The simplest approach is using classical plate theory (CPT), taking only inertia and bending stiffness into account according to

in which is the specific mass, h the plate thickness, and w the out-of plane displacement of the plate. The flexural rigidity D is given by

where G is the shear modulus and v Poisson’s ratio. This approach is valid only at low frequencies, when the wavelength is large compared to the plate thickness. This was shown by Niordson [12], using an asymptotic expansion. For higher frequencies, corresponding to shorter wavelengths, shear and rotatory inertia have to be considered. Therefore, the theory of Mindlin [7] can be used

The term denotes a dimensionless correction factor. One can choose the value of this factor so that waves at very high frequencies propagate with the Rayleigh velocity of surface waves. According to this condition, the value of the correction factor is obtained as a function of v, given by Mindlin [7]. With the theories described above, the scattering of a plane incident flexural wave by a circular cavity with radius is studied [11]. The incident wave is taken as an infinite sinusoid, planar wave, propagating in the y direction, and is given by

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is the wavenumber of a propagating flexural wave. The origin of the coordinate system is chosen at the center of the hole, cylindrical coordinates are introduced. To satisfy the stress-free boundary conditions at the hole

a scattered wave must occur. This problem was studied by Norris and Vemula [13], and Staudenmann [14] using CPT. Following their approach, the scattered wave is assumed in the form of

The first part describes a wave propagating outwards (Hankel function of the second kind), and the second part a boundary layer around the hole (Hankel function of the first kind). The angular dependence is given by the sum over the cosine functions, is the wavenumber of the nonpropagating flexural mode. With this approach, the boundary conditions can only be fulfilled in the Kirchhoff approximation

satisfying a combination of vertical force and the derivative of the twisting moment. Using Mindlin’s theory and following the work of Pao and Chao [8], the boundary conditions can be fulfilled as an average over the plate thickness with

The scattered wave consists of the same flexural wave as in Eq. (6) and an additional shear boundary layer with the two components

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Evaluating the boundary conditions in polar coordinates, the coefficients of the scattered waves are calculated. is the wave number of the shear wave, and give the relation between the wave numbers, as defined in Ref. [8]. Finite difference methods (FDM) are used to calculate the propagation and scattering characteristics of flexural waves in a plate with a more complicated geometry, i.e., the scattering at a hole with a notch. Mindlin’s equations of motion are discretized on a Cartesian, staggered grid, as described in Ref. [15]. To implement the scattering at a circular hole in the plate, the hole is approximated by a right-angled contour and the stress-free boundary conditions are implemented. A crack or notch at the hole boundary, through the thickness of the plate, is implemented in a similar way. The boundary conditions are applied to two parallel lines and one point at the tip of the notch.

5. Measurement of the scattered field A typical measured scattered field around a hole in an aluminum alloy plate for an excitation with a center frequency of 50 kHz can be seen in Fig. 4a. The measurement is made on a circular grid around the hole, with a signal recorded every 5 degrees on radii 0.5 mm apart. The incident wave with a nearly straight wave front propagates from the direction of the positive y-axis and is scattered at the hole boundary. Behind the hole, an area with low amplitude can be seen, the socalled shadow area. Directly at the hole, a high amplitude results from the scattering at the free surface. Further outwards a characteristic hill and valley pattern develops due to the constructive and destructive interference of incident and scattered waves. In Fig. 4b the measured and calculated amplitude of the scattered field on a circle around the hole is shown. Measurement and analytical description using Mindlin's theory show good quantitative agreement. The calculation using CPT, neglecting the effects of shear and rotatory inertia, has a slight deviation around 180°, but the main features of the scattered field are described accurately enough. After the first measurement (Fig. 4a), a notch of 2 mm length is cut at an angle of 90° on the hole boundary using a fine saw blade. The scattered field is measured again and the difference in amplitude due to the notch is shown in Fig. 4c. At the free surfaces of the notch an additional scattered wave is generated, which changes the scattered field significantly up to about 30% in amplitude and allows the detection of such a notch from an amplitude measurement. The calculation of this difference in amplitude with FDM (Fig. 4d) shows good agreement with the measurement, and it can be concluded that we can accurately model the scattered field around a hole with a notch. More complicated geometries, like a line of holes in a plate, have also been studied with this method [16].

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6. Application for nondestructive testing The growth of fatigue cracks in tensile specimen was monitored as a nondestructive testing application in cooperation with the fatigue engineering laboratory of RUAG Aerospace, Emmen, Switzerland. Tensile specimen 3.17 mm thick, 40 mm wide, and 500 mm long, made of Al-7075 PL-T3 were used. A fatigue grown crack at the hole boundary was generated by cyclic tensile loading in a servohydraulic testing machine (Fig. 5). A small starter notch was cut to prescribe the position of the crack. The setup described in section 3 was slightly modified to automatically monitor the motion of a single spot close to the crack during the cyclic tensile loading. The laser interferometer was affixed to the testing machine and the measurement was triggered on the maximum tensile force. This assures a measurement at always the same spot and when the crack faces are open. An excitation frequency of 40 kHz was used, resulting in a wavelength of the flexural wave of 26 mm. The amplitude of the measured movement of the specimen surface in the vicinity of the crack was extracted from the time series by FFT and plotted. Additionally the crack length was measured optically using a microscope.

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After the crack initiation of several thousand cycles, the crack started to grow as a quarter-elliptical crack before reaching through the thickness and increasing further in length. Monitoring measurements during crack growth were made at several specimens. Typical measured signals can be seen in Fig. 6, where the crack started to grow around 100’000 cycles and went through the thickness of the specimen at 124’000 cycles. The amplitude measured at an excitation frequency of 40 kHz (Fig. 6a) starts to rise around 122'000 cycles, or a crack length of ca. 2 mm, and increases smoothly with the crack growth. A significant increase in amplitude could be seen in all measurements for a crack length of ca. 2.5 mm, and therefore a crack of this length can be securely detected. The variation among the measured amplitude curves is rather small and the curves agree well with the FDM calculation, allowing a sizing of the crack length.

7. Conclusions Flexural waves in isotropic, homogeneous plates are generated using piezoelectric transducers. Driven below the cutoff frequencies of the higher wave modes, the first antisymmetric Lamb wave mode is excited selectively. The scattered field on a measurement grid around a hole is measured pointwise using a heterodyne laser interferometer. Amplitude and phase of the scattered field are extracted from the measured time series using FFT. Good agreement between the measurements and analytical calculations of the scattered field at a circular cavity is found. Mindlin’s theory of plates describes the propagation and scattering very accurately, while a slight divergence can be seen for CPT, neglecting the influence of shear and rotatory inertia. The measured scattered field around the hole changes significantly with the presence of a notch or a crack, much smaller than the wavelength. The change in the measured amplitude is modeled using finite difference methods and good agreement is found. The significant change in amplitude due to a defect at the hole allows the use of the proposed method for the nondestructive testing of large structures. A flexural wave traveling along the structure can be excited, and its scattering measured. From changes in the measured amplitudes, the development of defects, like fatigue cracks in aircraft structures, can be detected. As a first application, the monitoring of fatigue cracks in tensile specimen is shown. Significant changes in the measured amplitude of the flexural wave can be seen for crack lengths smaller than the specimen thickness.

8. Acknowledgements We would like to thank Daniel Gsell and Tobias Leutenegger for their help on the FDM coding, and Bernard Masserey for his help on the analytical calculations. Our thanks go to the fatigue engineering center of RUAG Aerospace, Emmen, Switzerland for the possibility of conducting the cyclic tensile loading in their laboratory. References 1. 2.

E.V. Malyarenko, M.K. Hinders, Ultrasonic Lamb wave diffraction tomography, Ultrasonics 39, 269-281 (2001) J.L. Rose, L.E. Soley, Ultrasonic guided waves for anomaly detection in aircraft components, Mater. Eval. 58(9), 1080-1086 (2000)

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3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16.

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H. Lamb, On waves in an elastic plate, Proc. R. Soc. London, Ser. A 93, 114-128 (1917) D. Gsell, D. Profunser, J. Dual, Measurement of the dispersion relation of guided nonaxisymmetric waves in filament-wound cylindrical structures, Ultrasonics 38, 517-521 (2000) M. Veidt, M.B. Sayir, Experimental evaluation of global composite laminate stiffnesses by structural wave propagation, J. Composite Mater. 24, 688-706 (1990) K. F. Graff, Wave motion in elastic solids, Dover Publications, New York (1991) R.D. Mindlin, Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech. 18, 31-38 (1951) Y.-H. Pao, C.C. Chao, Diffractions of flexural waves by a cavity in an elastic plate, AIAA J. 2(11), 2004-2010 (1964) R. Paskaramoorthy, A.H. Shah, S.K. Datta, Scattering of flexural waves by a crack in a plate Eng. Fract. Mech. 33(4), 589-598 (1989) G.C. Sih, J.F. Loeber, Flexural waves scattering at a through crack in an elastic plate, Eng. Fract Mech. 1, 369-378 (1968) P. Fromme, M.B. Sayir, Measurement of the scattering of a Lamb wave by a through hole in plate, J. Acoust. Soc. Am. 111(3), (2002) F.I. Niordson, An asymptotic theory for vibrating plates, Int. J. Solids Struct. 15(2), 167-1 (1979) A.N. Norris, C. Vemula, Scattering of flexural waves on thin plates, J. Sound Vib. 181(1), 1: 125 (1995) M. Staudenmann, Structural waves in nondestructive testing, PhD. Thesis, Diss. ETH No. 113 (1995) P. Fromme, M.B. Sayir, Monitoring of fatigue crack growth at fastener holes using guided Lamp waves, to appear in Review of Progress in Quantitative Nondestructive Evaluation 21, ed. by D. Thompson and D.E. Chimenti P. Fromme, M.B. Sayir, Experimental detection of cracks at rivets using structural waves propagation, in Review of Progress in Quantitative Nondestructive Evaluation 20 B, ed. by D. Thompson and D.E. Chimenti, AIP Conference Proceedings 557, New York, 1626-1633 (2001)

EVALUATION OF FIBER WAVINESS IN THICK COMPOSITES BY ULTRASONIC TEST

H.-J. CHUN School of Electrical and Mechanical Engineering Yonsei University 134, Shinchon-dong Seodaemun-gu, Seoul, 120-749, Korea

Abstract An analytical model, based on the ray and plane wave theories, was proposed to understand the ultrasonic wave propagation in thick composites with uniform fiber waviness. In the analysis, the composites were assumed to have continuous fibers with sinusoidal fiber waviness in a matrix and were modeled as stacks of infinitesimally short off-axis subelements with varying fiber orientation along the lengthwise direction. From the analysis, it was found that the converted wave of the same mode as that of incident wave carried most energy and this tendency was sustained throughout the thick composites. The path and traveling time of ultrasonic wave in the subelement were obtained by computing the velocity and direction of group wave. The predicted results showed that the ray paths of quasi-longitudinal and quasi-transverse waves were highly affected by the degree of fiber waviness and had tendencies to trace toward the fiber direction and to converge to the adjacent peak of fiber waviness. Some parameters that showed strong effects on the wave propagation in the composites were identified for both insonified longitudinal and transverse waves. The experiments were also conducted on the specially fabricated thick composite specimens with various degrees of uniform fiber waviness using the conventional through-transmission method to verify the predicted results. Then, the analytically determined values were compared with the actual measurements obtained from the test specimens. Good agreements were observed among them. 1.

Introduction

Fiber waviness is one of the manufacturing defects frequently encountered in thick composite structures. It results from local buckling of prepreg or wet hoop-wound filament strands under pressure exerted by the overwrapped layers during the filament winding process or from lamination residual stress built up during curing. Its characteristic can be represented by the through thickness undulation of fibers within thick composite laminates. According to a number of studies on the behavior of thick composites with fiber waviness, fiber waviness causes degradation of strength and stiffness significantly[12]. Therefore, nondestructive evaluation technique that can detect fiber waviness of thick composite is very important for the integrity of structures. For this purpose, there have been investigations on the thick composites with fiber waviness by means 433 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 433–442. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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of ultrasonic waves. Wooh and Daniel[3] attempted to explain how ultrasonic wave propagates in thick composites with fiber waviness by using discrete ray tracing method. Mclntyre et al. [4] studied amplitude of ultrasonic wave using finite difference method. Kim et al. [5] investigated ray paths and traveling time using geometrical acoustics. In this paper, both theoretical and experimental investigations were conducted to evaluate uniform fiber waviness in thick composites nondestructively. Efforts were made to fully understand ultrasonic wave propagation in thick composites with fiber waviness by considering various parameters that represent degree of fiber waviness. The ray path, time of flight and refracted waves at imaginary interface between subelements in the composites with fiber waviness were calculated. The experiments were also conducted to prove predicted traveling time and wave paths. 2.

Analysis

Figure 1 shows the geometry of a representative volume of uniform fiber waviness. It is assumed that all the fibers are parallel to each other and have sinusoidal curvature along one spatial coordinate direction (x axis).

The angle between the tangent to fiber and the x axis is given by,

where a and are the amplitude and wavelength of fiber waviness, respectively. is the maximum amplitude of fiber waviness. is defined as the angle between the tangent to fiber and the x axis. Off-axis stiffness referred to the xyz coordinates is obtained from the transformation relation with on-axis stiffness and fiber

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orientation If ultrasonic wave propagates in the x-y plane, Christoffel's equation is express as follows

and

where is wave normal vector. For a given wave normal direction, phase velocities v and its polarization vector are obtained as below from Eq 2.

Due to shear coupling in anisotropic media, the wave polarized in the x-y plane has pure-mode only if wave normal is parallel to the principal axes of materials. Hence, in the case of wave normal lie in the x-y plane, the longitudinal and transverse waves polarized in the x-y plane have quasi-mode coupled each other. But, the transverse wave vertically polarized to the x-y plane has pure-mode. Waves that have quasi-mode actually propagate along the direction of group wave with group velocity. The group velocities of quasi-longitudinal and quasi-transverse waves can be determined by known phase velocity and polarization vector [7].

where is slowness vector. Because the elastic properties change continuously as wave propagates, an analytical model is proposed to consider the refraction of wave as below.

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For incident wave with the elastic properties of the ray point, two refracted waves and two reflected waves at the ray point are considered to calculate the propagation direction and energy of wave. Based on Snell's law, the wave normal of refracted or reflected wave is obtained from Christoffel's equation in terms of slowness[8]. The group waves of all waves are also obtained using Eqn 3 at the ray point with each phase velocity and its polarization vector. Finally, the ray path is given by

where superscript i is the step number and is a discrete ray increment. angle of group velocity determined by fiber orientation angle. 3.

is the

Experimental Procedures

The materials investigated in this study were DMS 2224 graphite/epoxy composites (Hexcel, Inc.). Ultrasonic inspections were conducted on specially fabricated thick composite specimens with uniform fiber waviness. The fiber waviness ratios were 0.011, 0.034 and 0.059. The standard composite specimens without fiber waviness were fabricated for comparing experimental results with those with fiber waviness and also used for the characterization by conventional wave velocity measurement method. The elastic constants of standard specimen listed in Table 1 were used for input data for the numerical ray tracing model as on-axis stiffnesses.

Pulser/reciver (5072PR, Panametrics Inc.) and longitudinal transducers (10MHz central frequency, Panametrics Inc.) were operated in through-transmission mode as shown in Figure 3. The captured ultrasonic signals in oscilloscope (ScopeStation 140, LeCroy Inc.) were stored into the personal computer via GPIB bus.

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For a given position of transmitting transducer, waveforms were recorded as receiving transducer moved along x axis at 1mm interval. The energy P of received wave was obtained by integrating the magnitude of signal s(t) as below.

For a particular position of transmitting transducer, received energy of wave at various positions were represented as relative values to the maximum value of P.

4.

Results and Discussion

The numerical simulations were conducted for the waviness model with various fiber waviness ratios. Incident wave was normal to the surface of model and wave source was considered as a point-like source. The ray paths were traced for each wave mode. Numerical results are shown in Figures 4 and 5. As the results of simulation, insonified waves at different locations propagate toward adjacent concave region of fiber waviness.

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Waves insonified at various bottom positions were detected at the opposite side of the specimen. It was observed that the energy of wave converged to the adjacent peak of fiber waviness. For the standard specimen without fiber waviness, waves with the maximum energy were detected at the same location in the x-axis as that of transmitting transducer, and energy distribution of received energy was symmetric to it. The location where maximum energy of wave was detected moved toward the concave region of fiber waviness as the degree of fiber waviness increased. It is expected from the predicted results. This tendency is obviously shown in the contour plots of energy distribution of received wave. The regions checked by ellipse in Figure 6 are the locations where maximum energy of wave is detected for various positions of transmitting transducer.

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The predicted traveling time and measured traveling time of wave are shown in Figure 7. The traveling time of wave was measured at the location where the maximum energy was detected. As the degree of fiber waviness increased, variation of traveling time was also increased. Some discrepancies were found between the experiments and prediction, but tendency of traveling time obtained by the experiments was similar to that of predicted results. These discrepancies might be caused by assumption that wave-source is point-like source.

5.

Conclusions

Wave propagation in thick composites with fiber waviness was studied by considering wave path and traveling time. From the numerical simulations, it was predicted that insonified waves at different locations propagated toward the adjacent concave region of fiber waviness. This tendency was confirmed by ultrasonic tests in through-transmission operation. If the maximum energy of received waves is detected at the same locations in the x-axis for both transmitting and receiving transducers, it indicates convex or concave peak of fiber waviness. The wavelength of fiber waviness is determined quantitatively by the relative distance between the peaks. By

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considering the energy distribution of received waves for a position of transmitting transducers, concave or convex region of fiber waviness is distinguished. The predicted traveling time also showed good agreement with the experiments and can be correlated with the degree of fiber waviness. This study of wave propagation in thick composites with fiber waviness can suggest possibilities to detect the presence and degree of fiber waviness in composites nondestructively. 6.

References

1.

H. -J. Chun, J. -Y. Shin and I. M. Daniel : Nonlinear Behaviors of Thick Composites with Uniform Fiber Waviness, AIAA Journal, Vol. 38, No. 10 (2000), 1949-1955. Heoung-Jae Chun : Flexural behavior of thick composites with fiber waviness, proc. of first Asian-Australasian Conference on Composite Materials(ACCM-1) (1998), 421-424. Shi-Chang Wooh, Isaac M. Daniel : Wave propagation in composite materials with fibre waviness, Ultrasonics, Vol. 33, No. 1 (1995), 3-10. Joseph S. McIntyre, Charles W. Bert, Ronald A. Kline : Wave propagation in a composite with reinforcing fibers, Review of Progress in Quantitative Nondestructive Evaluation, Vol. 14 (1995), 1311-1318. Kwang Yul Kim, Wei Zou, Wolfgang Sachse : Wave propagation in a wavy flber-epoxy composite material : Theory and experiment, J. Acoust. Soc. Am., Vol. 103, No. 5 (1998), 22962301. Isaac M. Daniel, Ori Ishai: Engineering mechanics of composites materials, Oxford University Press, 1994. Fedor I. Fedorov : Theory of elastic wave in crystals, Plenum Press, New York, 1968. Edmund G. Henneke II : Reflection-refraction of a stress wave at a plane boundary between anisotropic media, J. Acoust. Soc. Am., Vol. 51 (1972), 210-217. S. I. Rokhlin, T. K. Bolland, Laszlo Adler : Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media, J. Acoust. Soc. Am., Vol. 79, No. 4 (1986), 906-918.

2. 3. 4.

5. 6. 7. 8.

9.

DEVELOPMENT OF A DRY-CONTACT ULTRASONIC TECHNIQUE AND ITS APPLICATION TO NDE OF IC PACKAGES H. TOHMYOH and M. SAKA Department of Mechanical Engineering Tohoku University 01, Aoba, Aramaki, Aoba-ku, Sendai 980-8579 Japan

Abstract

A dry-contact technique for transmitting higher frequency components of ultrasonic wave is proposed to accomplish an inspection which requires higher resolution without wetting the tested parts in water. In this technique, a plastic film is inserted between the water and the tested parts, and the interfacial continuity is improved by decreasing the pressure between the film and the tested parts. From both the experiment and the calculation used acrylic resin, we verified that the dry-contact technique achieves higher resolution similarly to the conventional immersion technique. The dry-contact technique was applied to the inspection of a delamination in IC package, and the delamination was imaged clearly without wetting the package. 1. Introduction

Sensitive nondestructive testing methods that can achieve higher resolution are strongly demanded in the electronic industry with miniaturization of the integrated circuit (IC) packages [1]–[3]. In point of higher resolution, ultrasonic method is superior to other nondestructive testing methods. However, conventional ultrasonic testing is performed by immersion of tested parts in water, using various liquid and semiliquid couplants [4]. The use of such couplants is not allowed for IC packages that can absorb moisture and be contaminated. By this reason, X-ray method is widely used to the nondestructive testing of the 1C packages [5]–[8]. Although the X-ray method enables us to test with non-contact, it is difficult to detect a plane defect, such as crack, delamination and so on. On the other hand, laser ultrasonic method [9]–[11] and microwave method [12] also enable us to test with non-contact. But the laser ablation with the generation of the bulk acoustic waves is destructive in the laser ultrasonic method, and the microwave method is not available for the test object through the metallic conductor. Recently, air-contact ultrasonic technique that transmits an ultrasound by the medium of air has been studied, but higher frequency components of ultrasound are not available by the technique under the existing conditions [13]. 443 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 443–454. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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In this study, a new dry-contact technique for transmitting higher frequency components of ultrasound is proposed. In this technique, by inserting various thin plastic films, such as polyvinylidene chloride, polyvinyl chloride and polyethylene, between the water as the medium and the tested parts, ultrasonic testing is performed without wetting the tested parts in water. In this technique, it is a key to remove an air gap on the contact area between the plastic film and the tested parts to achieve higher resolution. The air gap is formed by the surface roughness of the tested parts and the elastic deformation of the film and the tested parts due to ultrasonic transmission. The air gap was removed by decreasing the pressure between the film and the tested parts with a vacuum pump. At last, we succeeded in transmitting higher frequency components of ultrasound into acrylic resin by the new dry-contact ultrasonic technique, similarly to the conventional immersion technique. Furthermore, the lateral resolution on the back-wall of the acrylic resin was estimated by calculating the beam intensity formed at the focal plane by the dry-contact technique. The calculation results and the experimental ones showed that the drycontact technique has high lateral resolution as the conventional immersion technique. The dry-contact ultrasonic technique was applied to nondestructive evaluation (NDE) of the IC package. The delamination in the IC package was detected clearly without wetting the IC package. 2. Proposal of a Dry-Contact Ultrasonic Technique

To carry out ultrasonic testing without wetting the tested parts, some attempts have been made [14]. In these studies, by inserting various elastic membranes, such as rubber sheet, between the water as the medium and the tested parts, ultrasonic testing was performed [15]–[17]. However, signal intensity was not sufficient, and the testing which requires high resolution is not yet accomplished. The main reason is considered to be due to the ultrasonic attenuation in the elastic membrane and disagreement of the acoustic impedance between the water and the membrane. In addition to these two factors, we took note of that the interfacial continuity between the membrane and the tested parts must be kept during ultrasonic transmission to achieve higher resolution. Based upon the above-mentioned knowledge, a new dry-contact ultrasonic technique (DCUT) is proposed. Figure 1 illustrates the principle of DCUT. In this technique, by inserting various thin plastic films, such as polyvinylidene chloride (PVDC), polyvinyl chloride (PVC) and polyethylene (PE), between the water as the medium and the tested parts, ultrasonic testing is performed without wetting the tested parts in water. Furthermore, to improve the interfacial continuity between the film and the tested parts, the pressure of the interface between the film and the tested parts was decreased by a vacuum pump. In the vacuumizing process, it is important on the uniform interface formation to control the exhaust path of the air. The exhaust path was controlled with a path control layer between the film and the tested parts. Details of the layer are shown in Figure 2. The path control layer has eight radial holes on the circumference and eight perpendicular holes, and the exhaust path was controlled by the radial holes. The tested parts were kept under the path control layer by the perpendicular holes.

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3. Selection of Plastic Film

3.1. CHARACTERISTICS OF PLASTIC FILMS

To achieve higher resolution by DCUT, selection of plastic film inserted between the water and the tested parts is important. Optimum plastic film was considered from the viewpoint of resolution. A PVDC, a PVC and two kinds of PE films were used. One PE film was not added additive, and another was added additive which promoted cohesion between the film and the tested parts. In the following, the former is denoted by PE1, and the latter is denoted by PE2.

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On the selection of plastic film, two experiments were carried out. Firstly, a reflective plate of stainless steel was placed in a water bath, and the frequency spectrum of the surface echo from the reflective plate, was recorded in the case of inserting a plastic film between the transducer and the reflective plate. Measurement setup used in this experiment is shown in Figure 3. Broadband ultrasonic transducer V390-SU/RM (Panametrics, Inc.) was used, where it had the nominal center frequency 50.0MHz, the diameter of the piezoelectric element and the focal length The frequency spectra of the surface echo with inserting PVDC, PVC, PE1 and PE2 films, and the frequency spectrum without inserting plastic film are shown in Figure 4. In addition, the characteristics of the plastic films are summarized in Table 1. Figure 4 shows that the decreasing in spectrum intensity caused by inserting PVDC film is remarkable in comparison with the cases of inserting the other plastic films. Here, signal loss caused by inserting plastic film, is calculated as a function of the frequency, by the following equation:

The quantity represents the sum of the signal loss caused by the ultrasonic attenuation in plastic film and the signal loss caused by the difference of the acoustic impedance between the film and the water. Figure 5 shows for various plastic films. The absolute value of of two PE films and PVC film are smaller than that of PVDC film in available frequency range, and that of PE1 film is the smallest in the four films.

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Next, back-wall echo of the acrylic resin with 2.0mm in thickness was recorded by DCUT. Back-wall echoes of the acrylic resin received by DCUT using various plastic films, and its frequency spectra, are shown in Figures 6 and 7, respectively. Figure 6 shows that the signal with sufficient intensity, which is similar to the conventional immersion technique, is received by DCUT using various plastic films. However, fairly large discrepancy is notable among the frequency spectra for every plastic film. Here, actual signal loss at the back-wall of acrylic resin, [dB], is calculated by the following equation:

The quantity represents the actual signal loss in case of DCUT. Figure 8 shows for various plastic films. Especially, the decrease of higher frequency components in PE1 film whose absolute value of was the smallest in the four films is remarkable. The fact indicates that the plastic film which is superior on acoustical characteristics, for example, ultrasonic attenuation and acoustic impedance, cannot necessarily be superior on transmitting higher frequency components of ultrasonic wave to the tested parts in DCUT. It is suggested that the condition of the interface between the plastic film and the tested parts affects significantly the transmission of higher

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frequency components. The decrease of higher frequency components may be called frequency filter effect. Figure 9 illustrates the frequency filter effect. The ultrasonic waves can well pass through the continuous interface between the plastic film and the tested parts [Figure 9 (a)], but the waves hardly pass through the interface with air gaps. The air gaps are formed by the surface roughness of the tested parts [Figure 9 (b)] and the elastic deformation of the film due to ultrasonic transmission [Figure 9 (c)] and so on. The frequency filter effect between the film and the tested parts is considered due to a disorder of the interfacial continuity. In fact, Figure 8 shows that the absolute value of of PE2 which is added additive for promoting cohesion is much smaller in higher frequency range than that of PE1 without additive, and it is clear that the disorder of the interfacial continuity between the plastic film and the tested parts is improved by the additive.

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3.2. LATERAL RESOLUTION OF DRY-CONTACT ULTRASONIC TECHNIQUE

By using the experimentally recorded frequency spectra, let us now consider on the lateral resolution on the back-wall of the acrylic resin theoretically, and select a plastic film to achieve higher resolution. The lateral resolution is estimated by calculating the beam intensity formed at the focal plane. In case of pulse echo mode, the normalized beam intensity formed at the plane focused by a spherical lens to a spot, is given by [18]

where is Bessel function of the first kind and first order, the wavenumber, C the ultrasonic velocity ( in water), the radial distance from the center axis of the lens and calculated from Equation (3) is valid when In case of 50MHz transducer, at a plane focused in water is shown in Figure 10. On the other hand, by using lateral resolution, is defined as the separation distance between the two sound sources which can be distinguished clearly, and is given by the following equation empirically:

where is the wavelength, and In case of using a broadband transducer, the transformation of the frequency components in ultrasonic waves at the focal plane must be considered. This was done by multiplying by the frequency spectrum at the focal plane [19], The improved beam intensity at the focal plane, is given by the following equation:

where is the lower frequency limit and the upper frequency limit. As an example, in case of DCUT using PVC film is shown in Figure 11. On the other hand, according to the definition of must satisfy the following equation:

The response of ultrasound to two sound sources in case of DCUT using PVC film is shown in Figure 12. When the separating distance between two sources equals to each sources is barely resolved [Figure 12 (b)], while the separation distance less than makes it impossible to resolve each sources [Figure 12 (a)]. If the separating distance is greater than each sources is clearly resolved [Figure 12 (c)].

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The lateral resolution on the back-wall of the acrylic resin in case of using the broadband transducer was estimated by Equations (5) and (6). In addition, by inserting the peak frequency of the frequency spectrum at the focal plane into Equation (4), empirical resolution for reference was calculated, too. Both values of calculated lateral resolution are shown in Table 2 respectively. The lateral resolution achieved by DCUT using PVDC and PVC films are 68 and 66µm, respectively. Each lateral resolution is near to that of the conventional immersion technique (53µm), and enough high for applying to NDE of the IC package.

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3.3. CONFIRMATION OF LATERAL RESOLUTION

To confirm the lateral resolution calculated from Equations (5) and (6), the groove pattern introduced by a slicing machine to the back-wall of the acrylic resin with 2.0mm in thickness was imaged by DCUT. The dimensions of the groove pattern at the backwall of the acrylic resin and the optical view are shown in Figures 13 (a) and (b) respectively. C-scan mode image of the groove pattern obtained by the conventional immersion technique is shown in Figure 14 (a), and C-scan mode images of the groove pattern by DCUT using PVDC and PVC films are shown in Figures 14 (b) and (c), respectively. The scan pitch was 0.01mm.

Figure 14 (a) shows that the narrowest projection that is clearly resolved by the conventional immersion technique is of 60µm in width. Figures 14 (b) and (c) show that the same projection as the above is clearly resolved by DCUT using PVDC and PVC films. From the experimental facts, it was confirmed that DCUT has high lateral resolution as the conventional immersion technique. In addition, the value of 60µm in width resolved by DCUT and the conventional immersion technique agrees with the lateral resolution estimated in the previous section, and the validity on the estimation of lateral resolution using Equations (5) and (6) is confirmed. Based on the discussion made in the previous and this sections, PVDC and PVC films were selected for the use in DCUT to NDE of IC package. These films are two which can achieve higher resolution in four films examined.

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4. Application to NDE of IC Package DCUT was applied to the inspection of a delamination in a quad flat package (QFP). The examined QFP contained a delamination between the silicon chip and the metallic lead frame in the plastic encapsulant. The delamination was inspected from the side of silicon chip in C-scan mode imaging. C-scan mode image of the delamination in the QFP obtained by the conventional immersion technique is shown in Figure 15 (a), and C-scan mode images of the delamination by DCUT using PVDC and PVC films are shown in Figures 15 (b) and (c), respectively. The scan pitch was 0.20mm. Figure 15 (b) and (c) show that the delamination part is clearly discriminated from the part without delamination. Especially, in the C-scan mode image by DCUT using PVC film shown in Figure 15 (c), the lead frame pattern is also imaged clearly, and the image closely resembles the one by the conventional immersion technique shown in Figure 15 (a). At last, by DCUT achieved higher resolution, we succeeded in testing an IC package without wetting the package in water. On the other hand, while DCUT using PVDC film discriminates the delamination part from the part without delamination, the lead frame pattern is not so clear. The cause is considered due to a disorder of the interfacial continuity between the PVDC film and the package.

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5. Conclusions A dry-contact technique for transmitting higher frequency components of ultrasonic wave was proposed. In this technique, a plastic film was inserted between the water and the tested parts, and the interfacial continuity was improved by decreasing the pressure between the film and the tested parts. From both the experiment and the calculation used acrylic resin, it was confirmed that the dry-contact technique achieves higher resolution similarly to the conventional immersion technique. Finally, the dry-contact technique was applied to the inspection of a delamination in IC package, and the delamination was imaged clearly without wetting the package in water. 6.

Acknowledgements

The authors wish to acknowledge Mr. M. Mikami of Tohoku University for his help in the experiment. This work was partly supported by Japan Society for the Promotion of Science under Grant-in-Aid for Scientific Research (B)(2) 12450041. 7.

References

1. Tummala, R.R. (2001) Fundamentals of Microsystems Packaging, McGraw-Hill, New York, 44–79. 2. Baba, S., Tomita, Y., Matsuo, M., Matsushima, H., Ueda, N. and Nakagawa, O. (1998) Molded chip scale package for high pin count, IEEE Trans. Comp., Packag., Manufact. Technol.–Part B, 21, 28–34. 3. Amagai, M. (1999) Characterization of chip scale packaging materials, Microelectronics Reliability, 39, 1365–1377. 4. Gilmore, R.S. (1996) Industrial ultrasonic imaging and microscopy, J. Phys. D: Appl. Phys., 29, 1389–1417. 5. Ikeda, Y., Mizuta, Y. and Hirashima, R. (1999) Radiography testing for micro defects with a psudo-3D X-ray TV system, Proceedings of the Fifth Far-East Conference on Nondestructive Testing, 361–368. 6. Rooks, S.M., Benhabib, B. and Smith, K.C. (1995) Development of an inspection process for ball-gridarray technology using scanned-beam X-ray laminography, IEEE Trans. Comp., Packag., Manufact. Technol.–Part A, 18, 851–861.

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7. Neubauer, C. (1997) Intelligent X-ray inspection for quality control of solder joints, IEEE Trans. Comp., Packag., Manufact. Technol.–Part C, 20, 111–120. 8. Sankaran, V., Kalukin, A.R. and Kraft, R.P. (1998) Improvements to X-ray laminography for automated inspection of solder joints, IEEE Trans. Comp., Packag., Manufact. Technol.–Part C, 21, 148–154. 9. Yamanaka, K. (1997) Precise measurement in laser ultrasonics by phase velocity scanning of interference fringes, Jpn. J. Appl. Phys., 36, 2939–2945. 10. Nishino, H. and Tsukahara, Y. (1993) Excitation of high frequency surface acoustic waves by phase velocity scanning of a laser interference fringe, Appl. Phys. Lett., 62, 2036–2038. 11. Lafond, E., Coulette, R., Grand, C., Nadal, M.-H., Dupont, B., Lepoutre, F., Balageas, D. and Petillon, O. (1998) Application of a two-layer semi-analytical model for the improvement of laser-ultrasonic generation, NDT&E International, 31, 85–92. 12. Ju, Y., Saka, M. and Abé, H. (2001)NDI of delamination in IC packages using millimeter-waves, IEEE Trans. on Instrum. Meas., 50, in press. 13. Manthey, W., Kroemer, N. and Magori, V. (1992) Ultrasonic transducers and transducer arrays for applications in air, Meas. Sci. Technol., 3, 249–261. 14. Rogovsky, A.J. (1991) Development and application of ultrasonic dry-contact and air-contact C-scan systems for nondestructive evaluation of aerospace composites, Materials Evaluation, 1491–1497. 15. Liaw, P.K.., Hsu, D.K.., Yu, N., Miriyala, N., Saini, V. and Jeong, H. (1996) Investigation of metal and ceramic-matrix composites moduli: experiment and theory, Acta mater., 44, 2102–2113. 16. Im, K.-H., Hsu, D.K. and Jeong, H. (2000) Material property variations and defects of carbon/carbon brake disks monitored by ultrasonic methods, Composites: Part B, 31, 707–713. 17. Roth, D.J., Riser, J.D., Swickard, S.M., Szatmary, S.A. and Kerwin, D.P. (1995) Quantitative mapping of pore fraction variations in silicon nitride using an ultrasonic contact scan technique, Res. Nondestr. Eval., 6,125–168. 18. Canumalla, S. (1999) Resolution of broadband transducers in acoustic microscopy of encapsulated ICs: transducer selection, IEEE Trans. Comp. Packag. Technol., 22, 582–592. 19. Bechou, L., Ousten, Y., Tregon, B., Marc, F., Danto, Y., Even, R. and Kertesz, P. (1997) Ultrasonic images interpretation improvement for microassembling technologies characterisation, Microelectron. Reliab., 37, 1787–1790.

6. Neutron Diffraction and Synchrotron Radiation Methods

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HIGH-RESOLUTION NEUTRON DIFFRACTION TECHNIQUES FOR STRAIN SCANNING P. MIKULA, M. VRANA, P. LUKAS Nuclear Physics Institute 250 68 Rez, Czech Republic V. WAGNER Physikalisch-Technische Bundesanstalt Bundesallee 100, 38116 Braunschweig, Germany Abstract Several high-resolution neutron diffraction techniques using Bragg diffraction optics based on cylindrically bent perfect crystals which have been tested with the aim of their efficient exploitation for strain/stress are summarized. Presented experimental results demostrate their experimental abilities for nondestructive testing and evaluation of a phase and deformation response of polycrystalline materials on residual stress or deformation loading. Due to the unique resolution and good luminosity achieved by focusing in real and momentum space, besides the macrostrain scanning, the presented techniques are also qualified for microstrain/stress studies by an analysis of the diffraction profiles. At the medium flux reactor LVR-15 in NPI there are operating two focusing double axis strain/stress diffractometers. Good luminosity of the diffractometers and a sufficiently high-resolution (FWHM of the instrumental - profile can be less than at nm) permit investigations of both the macro- and microstrains in the sample gauge volumes of several cubic millimetres with a sensitivity in the relative peak shift of several rad.

1. Introduction Residual stresses or their development under applied external force can have a strong influence on their basic mechanical properties. They displace atoms from their original positions in a crystalline material and neutron diffraction along with X-ray diffraction can nondestructively reflect the resulting change of lattice constants with a sufficiently high precision by using the Bragg Diffraction Angle Analysis (BDAA) and/or by Energy-Dispersive Neutron-Transmission Diffraction (EDNTD). Conventional neutron strain/stress scanners using the BDAA method (based on Bragg condition lattice distance, angle, wavelength) are in fact powder diffractometers equipped with a mosaic monochromator and a system of Soller collimators and for high-resolution measurements they usually suffer from required luminosity. However, the dedicated diffractometers using Bragg diffraction optics provide luminosity by one order of 457 E. E. Gdoutos (ed. ), Recent Advances in Experimental Mechanics, 457–466. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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magnitude higher with a substantially better resolution (FWHM of the instrumental can be of the order of [1-7]. The EDNTD method is based on the measurement of a decrease of a beam intensity transmitted through a sample in the dependence on the wavelength in a -range in the vicinity of the Bragg cut-off. The edge in this case can be observed when passing through the limit ( is the lattice spacing), below which particular reflection planes (hkl) begin to scatter neutrons. It means that no angular dependence of scattering is measured and only integrated intensity of this particular reflection can be determined for the transmitted beam for each value of Mathematically, edge profile can be described by a Gaussian cumulative function (FWHM of the instrumental can be of the order of As will be presented, using the Bragg diffraction optics the EDNTD measurements, usually carried out only at the pulsed neutron beams [8], can be effectively carried out even at a medium power reactors [3, 5, 9-12]. The excellent properties of focusing strain scanners permits one investigations of both the macrostrains with the sensitivity to changes down to as well as microstrains by a peak profile or Bragg edge analysis in the gauge volumes of several mm 3 . In principle, the substructural changes of e. g. dislocation density and average grain size in engineering components after use can be investigated in situ, under loading by an external force. The stresses are not measured directly by diffraction techniques, but one measures strains defined as corresponds to the unstrained material), which are then converted to stresses using appropriate moduli. Then, the shift in the Bragg angle or of the position of the Bragg edge with respect to the strain-free material permits one to determine the average macrostrain over the irradiated gauge volume. Information on the microstrain (in the form of as well as dislocation density and grain size present in the irradiated gauge volume can be determined from a change of the width and the form of the diffraction peak or edge profile [10, 13-17]. During the last years we tested several modifications of the high resolution BDAA and EDNTD performances based on Bragg diffraction optics, which were developed within the collaboration between NPI Rez and PTB Braunschweig. In what follows, their advantages, drawbacks and properties are introduced.

2. Two axis performance, BDAA method The most spread high-resolution neutron diffraction two-axis performance (see Figure 1) consists basically of the following steps and properties [7, 18]: a. Monochromatic neutrons selected by a bent monochromator from the white spectrum are focused on a sample (real space focusing) making the luminosity of the device high. b. The monochromatic beam is diffracted by a chosen volume element (determined by a pair of input and output slits) into the angle c. There is a strong correlation between the divergences and as

which all can be easily manipulated by changing the monochromator radius is the total change of the angle of incidence (exit) over illuminated crystal

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length and value of the radius

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is the dispersion parameter). By setting a proper

and a quasiparallel and highly luminous detector signal can by adjusted for a chosen scattering angle Contrary to the conventional powder diffractometers, the minimum resolution e. g. at can be achieved even for monochromator take off angles far below 90°. d. The quasiparallel diffracted beam is directly analyzed by using a 1D-PSD. There are small resolution uncertainties and influencing the instrumental resolution which come from a nonnegligible thickness of the monochromator and from the finite width w of the irradiated volume of the sample determined by the input and output slits. Thus, for a point like sample, the monochromator of the thickness and the width of the sample w,

Both of them bring about the diffracted beam slightly divergent (quasiparallel) and directly determine the instrumental resolution. It is clear that it can be easily estimated and adjusted according to the experimental requirements [5]. e. No Seller collimators are required.

Of course, the total resolution of the instrument is strongly dependent on the spatial resolution of the 1D-PSD. The employment of the Bragg diffraction optics resulted in an improvement of the luminosity and/or resolution of our dedicated strain scanners several times in comparison with the conventional counterparts. Moreover, recently, one of our two axis strain scanners has been equipped with a two-crystal-slab sandwich monochromators (combinations of Si(111)+Si(220) or Si(111)+Ge(311) reflections) which permit us to work simultaneously with two neutron wavelengths of 0. 27 nm + 0. 165 tun or 0. 27 nm + 0. 143 nm, respectively [19]. This fact enables us to investigate more sample-reflection profiles under the same experimental conditions within a range of covered by one angular setting of the PSD. Thanks to a remote control of the bending device and a remote

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manipulation of the crystal curvature, the optimum focusing conditions can be achieved practically for any scattering angle As a result of a compromise between the resolution and the luminosity, the two axis performances used in our laboratory work with the resolution derived from the FWHM of the diffraction profiles taken with a well annealed standard samples.

3. Three axis performance, BDAA method The three axis performance is along with the cylindrically bent perfect monochromator equipped also with a cylindrically bent analyzer set either in symmetric/slightly asymmetric diffraction geometry (see Figure 2) [1-3, 5, 6] or in the fully asymmetric diffraction geometry in combination with 1D-PSD (see Figure 3) [2, 9, 20]. The analysis of the beam diffracted by the irradiated sample volume is in these cases performed in momentum space.

In the performance sketched in Figure 2 a high-resolution can be achieved when the analyzer is optimally curved, thus, all correlated rays coming within from the sample fulfil the Bragg condition on the analyzer simultaneously (focusing in momentum space). Then, the dispersion over the whole diffraction device is minimized. In case of the slit-like sample it can be simply expressed in the simple form [6]

where is the monochromator-sample distance and is the sample-analyzer distance. If the three-axis setup is optimized for the slit-like sample a non-negligible width of the sample Y introduces a resolution uncertainty

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In comparison with the two axis performance, the resolution of the detector has no influence on the resolution device and a high resolution can be achieved with the analyzer situated relatively close to the sample (about 30-40 cm). Thanks to a high peak reflectivity of the bent perfect crystals luminosity of the performance is still high. The drawback of the performance consists in a necessity of the step-by-step scanning (by rocking the analyzer) of the diffraction profiles. The performance sketched in Figure 3 overcomes the problem of the step-bystep scanning. A slightly divergent output beam from the sample is analyzed also by the optimally chosen bent analyzer [9]. However, thanks to the special asymmetric diffraction geometry of the crystal, the angular scale is transformed on the linear scale of the 1D- PSD. Such analysis in the momentum space makes possible to place the analyzer close to the sample and to cover relative large area of the Debye Scherrer cone without affecting the angular resolution which results in a favourable detector signal. Contrary to the two axis counterpart, in the case of three axis performances, there are much smaller blurring effects influencing the instrumental resolution. It has been demonstrated that the three axis performances can work with the resolution even better than For optimization of this three axis performance a more detailed treatment of individual rays coming from the sample with respect to the two dimensional lattice deformation in the cylindrically bent crystal is required. A detailed procedure can be found in refs. 2, 9 and 21.

4. One axis performance, EDNTD method The performance sketched in Figure 4 is the simplest one and by adjusting the monochromator take-off angle it can be used in a large range of the neutron wavelengths. A strong angular-wavelength correlation in the beam passing through the slit can be expressed in the form as [12, 18]

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where

is an angular deviation of a neutron trajectory from the central beam and is the value of the edge position (on the related to the lattice planes (hkl). The resolution uncertainties and which come from the effective mosaicity of the bent perfect crystal and the width of the slit w, respectively, can be derived in the form [12]

They bring about the correlation imperfect and thus have an influence on the total smearing of the Bragg edge as seen by PSD. The parameter was derived for the case of the monochromator crystal thickness For an effective slit width should be introduced. The parameter was derived for the case of an unlimited collimation of the incident polychromatic beam. However, in practice its impact can be considerably reduced for well collimated beam in a guide tube having a divergence For we arrive at the estimation Using a well annealed Fe(321) standard sample, a 4 mm thick monochromator and 2 mm wide slit, the instrumental resolution (represented by the width of the edge profile) was about of

5. Two axis performance, EDNTD method Similarly to the case of BDAA method sketched in Figure 5 the analyzer in fully asymmetric diffraction geometry can be also effectively used for EDNTD scanning [5, 11, 15]. Its advantage consists in a possibility to place the analyzer close to the sample and also in minimizing the influence of the slit width w and spatial resolution of the PSD on the FWHM of the edge profile. The drawback of this performance is in the necessity of using a special crystal cut for each mean wavelength For obtaining maximum amplitude we investigated the Bragg edge corresponding to reflection with a high multiplicity. To achieve a sufficiently high resolution with a reasonable counting rate the reflection combination Si(220)/polycrystalline working in the vicinity of nm was chosen. Using the standard sample and the 3 mm slit

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width, for crystals of the thickness of 5 mm and bending radii of m and m the instrumental resolution was of about Recently performed Monte Carlo simulations indicate that using properly chosen bent crystals, the -resolution of can be easily reached. High resolution and good luminosity of this performance permits us macrostrain scanning derived from the of the Bragg edge with the sensitivity of 10-5 [11].

The principal drawback of the previous performance can be overcome by using the dispersive double-bent-crystal setting as schematically sketched in Figure 6 [3, 22]. It can be adjusted practically for any mean and the Bragg edge scanning is performed either by rocking the analyzer scan) or by simultaneous rocking of both crystals with respect to the incident polychromatic beam [3, 22]. However, step-by-step scanning makes the measurement rather long.

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In all cases of the EDNTD method the following characteristics have been observed: a. The amplitude of the Bragg diffraction edge depends on the thickness of the sample. b. The steepness and the shape of the Bragg diffraction edge is determined dominantly by the distribution. c. Steepness of the edge can change with the thickness of the sample due to the multiple reflection effect. d. The resolution of the instrument permits to study macrostrain effects even below the value of in as well as some of the microstrain characteristics from the edge profile analysis. e. The EDNTD techniques are limited to measurements of only one strain component parallel to the axis of the incident monochromatized beam.

6.

Discussion

This paper briefly reports about achievements of neutron diffraction groups of NPI and PTB Braunschweig in the field of residual stress/strain instrumentation developments. All presented performances are based on Bragg diffraction optics with cylindrically bent perfect single crystals. In comparison with the conventional strain scanner equipped with mosaic monochromator and a system of Soller collimators its employment enabled us to improve the instrumental resolution and increase the detector signal, simultaneously. Thus, at present, the strain/stress measurements can be efficiently carried out even at the medium power reactors. The only parameters which influence the instrumental resolution are the effective mosaicity (linearly dependent on the thickness) of the bent perfect monochromator, widths of the slits and the spatial resolution of the position sensitive detector. Therefore, the resolution measured by FWHM of the diffraction profile or Bragg edge is a matter of a choice of these parameters with respect to the flux of the neutron source and the resulting detector signal. For further increase of the detector signal, in some cases a doubly-bent monochromator using also vertical focusing (which has no influence on the resolution) can be used [23]. As can be seen from presented sketches the BDAA as well as EDNTD performances can be easily adapted to any conventional neutron diffractometer providing the appropriate wavelength. Monte Carlo simulations of resolution function and a detector signal can provide a great help in the course of the optimization of the experimental performance [24]. Common feature of all performances is a sufficiently high resolution permitting evaluation of microstrains in plastically deformed polycrystalline samples. Of course, information about the strain state of art averaged within the irradiated sample volume is provided. Depending on the reactor power and the sample thickness, the area of the sample slits can be reduced to less than 10 (eventually to less than 1 for high flux neutron sources) still keeping reasonable counting times. Bragg diffraction optics investigations and instrumentation developments are supported by the Grant Agency of the Czech Rep. No. 202/97/K038, Grant Agency of CAS No. A1048003 and the TMR-Network ERB FMR XCT 96-0057.

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7. References 1. 2.

3.

4. 5. 6.

7.

8. 9. 10. 11.

12. 13. 14.

15. 16.

17. 18. 19.

Kulda, J., Mikula, P., Lukas, P., and Kocsis, M.: Utilisation of Bent Si Crystals for Elastic Strain Measurements, Physica B 180&181(1992) 1041-1043. Mikula, P., Lukas, P., Vrana, M., Klimanek, P., Kschidock, T., Macek, K., Janovec, J., Osborn, J. C., and Swallowe, G. M.: Advanced Neutron Diffraction Techniques for Strain Measurements in Polycrystalline Materials, Journal de Physique IV, Collogue C7, 3 (1993) 2183-2188. Vrána, M.., Mikula, P., Lukas, P., Saroun, J., and Strunz, P.: High-Resolution Diffraction Techniques for Strain/Stress Measurements at a Steady State Reactor, Acta Physica Hungarica, 75 (1994) 305-310. Mikula, P., Vrana, M., Lukas, P., Saroun, J., and Wagner, V.: High-Resolution Neutron Powder Diffractometry on Samples of Small Dimensions, Mat. Science Forum 228-231 (1996)269-274. Vrana, M., Klimanek, P., Lukas, P., Mikula, P., Saroun, J., and Wagner, V.: Neutron Optics for Micro- and Macrostrains Investigations, Proc. of the 4th European Conf. on Advance Materials and Processes, EUROMAT 95, 25. -28. September 1995, Padua/Venice, p. 35-40. Vrana, M., Lukas, P., Mikula, P., and Kulda, J.: Bragg Diffraction Optics in High Resolution Strain Measurements, Nucl. Instrum. Methods A 338 (1994) 125-131. Mikula, P., Vrana, M., Lukas, P., Saroun, J., Strunz, P., Ullrich, H. J., and Wagner, V.: Neutron Difftactometer Exploiting Bragg Diffraction Optics - A High Resolution Strain Scanner, Proc. of the 5th Int. Conf. on Residual Stresses ICRS-5, June 16-18, 1997, Linköping, eds. T. Ericsson, M. Oden and A. Andersson, Vol. 2, p. 721-725. Priesmeyer, H. G.: "Transmission Bragg-Edge Measurements", Measurement of Residual and Applied Stress Using Neutron Diffraction, Proc. NATO ARW Oxford, eds. Hutchings, M.T., and Krawitz, A. D., Kluwer, Dordrecht, 1992, p. 389-394. Lukas, P., Kouril, Z., Mikula, P., Saroun, J., Strunz, P., Vrana, M., and Wagner, V.: Bent Crystal Analyzer in Fully Asymmetric Diffraction Geometry for Neutron Scattering Instrumentation, J. Phys. Soc. Japan 65 (1996)241-244. Lukas, P., Kouril, Z., Strunz, P., Mikula, P., Vrana, M., and Wagner, V.: Microstrain Characterization of Metals Using High-Resolution Neutron Diffraction, Physica B 234-236 (1997) 956-958. Wagner, V., Kouril, K., Lukas, P., Mikula, P., and Vrana, M.: Residual Strain/Stress Analysis by Means of Energy Dispersive Neutron Transmission Diffraction, Proc. of 5th Int. Conf. on Applications of Nuclear Techniques "Neutrons in Research and Industry", 9. 6. -15. 6. 1996, Crete, SPIE 2867 (1997) 168-171. Mikula, P., Wagner, V., and Vrana, M., Bragg Diffraction Optics for Energy-Dispersive Neutron-Transmission Diffraction, Physica B, 283 (2000) 403-405. Klimanek, P., Kschidock, T., Mikula, P., and Vrana, M.: Substructure Analysis in Polycrystalline Materials by Means of Neutron Diffraction, Physica B 234-236 (1997), 965966. Lukas, P., Tomota, Y., Harjo, S., Ono, M., Vrana, M., Strunz, P., Kouril, Z., and Mikula, P.: Neutron Diffraction Investigation of Residual Strains Fe-Cr-Ni Alloys, Proc. of the 5th Int. Conf. on Residual Stresses Stresses ICRS-5, June 16-18, 1997, Linköping, eds. Ericsson, T., Odén, M., and Andersson, A.: Vol. 2, p, 880-885. Strunz, P., Lukas, P., Mikula, P., Wagner, V., Kouril, Z., and Vrana, M.: Data Evaluation Procedure for Energy-Dispersive Neutron-Transmission-Diffraction Geometry, Ibidum p. 688-693. Lukas, P., Zrnik, J., Ceretti, M., Vrana, M., Mikula, P., Strunz, P., Neov, D., and Keuerleber, J.: Neutron Diffraction as a Tool for Microstrain Characterization of Polycrystalline Ni-Base Superaloys after Thermal Fatigue, Mat. Science Forum 321-324 (2000) 1048-1050. Neov, D., Sittner, P., Lukas, P., Vrana, M., and Mikula, P.: In Situ High Resolution Neutron Diffraction Study of Stress Induced Martensitic Transformation in CuAlZnMn Shape Memory Alloy, Mat. Science Forum, 347-349 (2000), 334-339. Mikula, P., Kulda, J., Lukas, P., Ono, M., Saroun, J., Vrana, M., and Wagner, V.: Bragg Diffraction Optics in Neutron Diffractometry, Physica B 283 (2000) 289-294. Vrana, M., Mikula, P., Lukas, P., and Wagner, V.: Two Wavelength Sandwich Monochromator for Materials Research Experiments, Physica B 241-243 (1998) 231-233.

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20. Vrana, M., Mikula, P., Lukas, P., Dubsky, J., and Wagner, V.: New Developments in 21. 22. 23.

24.

Instrumentation for Strain Scanning in NPI Rez, Mat. Science Forum 321-324 (2000) 338344. Saroun, J., Lukas, P., Mikula, P., and Alefeld, B.: Optimization of a Double Bent Crystal Diffractometer for Neutron Small-Angle Neutron Scattering Experiments, J. Appl. Cryst. 27 (1994) 80-88. Mikula, P., Vrana, M., Lukas, P., Saroun, J., Strunz, P., Wagner, V., and Alefeld, B.: Bragg Optics for Strain/Stress Measurement Techniques, Physica B 213-214 (1995) 845-847. Wagner, V., Mikula, P., and Lukas, P.: A Doubly Bent Si-Monochromator, Nucl. Instrum. Methods in Phys. Research A 338 (1994) 53-59. Saroun, J., and Kulda, J.: RESTRAX - A Program for TAS Resolution Calculation and Scan Profile Simulation, Physica B 234-236 (1997) 1102-1104, and http: //omega. ujf. cas. cz.

DRAFT STANDARD FOR THE MEASUREMENT OF RESIDUAL STRESSES BY NEUTRON DIFFRACTION G. A. WEBSTER Mechanical Engineering Department, Imperial College Exhibition Road, London, SW7 2BX, UK A. G. YOUTSOS Joint Research Centre, Institute for Energy, High Flux Reactor Unit PO Box 2, 1755 ZG Petten, The Netherlands C. OHMS Joint Research Centre, Institute for Energy, High Flux Reactor Unit PO Box 2, 1755 ZG Petten, The Netherlands R. C. WIMPORY Mechanical Engineering Department, Imperial College Exhibition Road, London, SW7 2BX, UK Abstract Residual stresses can have important consequences for the load carrying capacity and safety of engineering components. Neutron diffraction is a non-destructive method for determining residual stresses in crystalline materials. It is a relatively new technique and no standard is currently available for making these measurements. This paper gives the background to research that has been carried out to develop a standard. It outlines the main findings and indicates the precautions that are required to achieve accurate positioning and alignment of specimens (and components) in a neutron beam and the analysis required to obtain reliable results. It also shows that special attention is needed in dealing with near-surface measurements because of surface aberration. It is demonstrated that, provided the recommended procedures are followed, a positional tolerance of ± 0.1mm can be achieved with an accuracy in strain of to give a resolution in residual stress of ± 7 to 20 MPa in most materials of practical interest.

1. Introduction Residual stresses can be introduced into engineering components during manufacture as a result of, for example, forging, bending and welding processes. They can also be caused by the forces and thermal gradients imposed during operation. These stresses can affect the load carrying capacity and resistance to fracture of components. In order to quantify their effect it is necessary to know their magnitude and distribution [1, 2]. Several destructive and non-destructive techniques are available for determining residual stresses. They include X-ray diffraction [3], neutron diffraction [4, 5], hole drilling [6], slicing [7] and boring [8] methods. In all cases, strains are measured and stresses calculated. However, only the neutron diffraction method is capable of obtaining these stresses non467 E. E. Gdoutos (ed. ), Recent Advances in Experimental Mechanics, 467–476. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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destructively within the interior of components. It is similar to the X-ray technique for surface determinations, but because neutrons are not charged, neutron diffraction can be used to obtain residual stresses non-destructively to a depth of several centimetres in most materials of practical relevance. There is wide spread interest throughout the world in exploiting the technique to examine relatively large components where the consequences of failure could be catastrophic. Neutron diffraction is a relatively new technique and no standard or code of practice is available for making stress measurements by this method. As a consequence two international projects were initiated to carry out the under-pinning research necessary to develop a standard. The first, under the auspices of VAMAS (Versailles Agreement on Advanced Materials and Standards), Technical Working Area 20 (TWA 20) was initiated in January 1996. The second study was supported by a European (EU) project RESTAND (Residual Stress Standard using Neutron Diffraction) which was started in December 1997. The objectives of the two projects were to: -establish accurate and reliable procedures for making non-destructive residual stress measurements by neutron diffraction, -examine a selection of samples in which residual stresses had been introduced by different techniques, -conduct inter-laboratory comparisons to establish reproducibility, -assemble the necessary information for preparing a draft standard for making the measurements. The VAMAS TWA 20 activity involved most of the neutron sources worldwide, which are capable of making the measurements. A series of ‘roundrobin’ specimens including a shrink-fit aluminium alloy ring and plug assembly, a ceramic matrix composite, a nickel alloy shot-peened plate and a ferritic steel weldment were examined. An additional objective of RESTAND was to demonstrate the usefulness of the technique to a range of practical applications and to develop confidence in the method for industry. This paper presents the findings of these studies.

2. Principles of the Technique Neutron diffraction can be used to measure components of strain from changes in lattice spacings in crystalline materials. When illuminated by radiation of wavelength similar to interplanar spacings, crystalline materials diffract this radiation as distinctive Bragg peaks. The angle at which any given peak occurs can be calculated using Bragg’s law of diffraction,

where is the wavelength of the radiation, d is the lattice plane spacing responsible for the Bragg peak and is the Bragg angle. The peak will be observed at an angle of from the incident beam, as shown in Figure 1. Neutrons can be generated in a reactor by fission or at a spallation source. Normally at reactor sources, a continuous monochromatic beam of neutrons is used to determine lattice spacings whereas at spallation sources usually a pulsed polychromatic beam is employed.

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When a monochromatic beam of neutrons of constant wavelength is used, the lattice strain in the direction of the scattering vector Q (Figure 1) from Bragg’s law is given by,

where is the change in lattice spacing from its strain free value and and is the change in Bragg angle from its strain free value When a pulsed polychromatic beam of neutrons is used, measurements are made at a constant angle (usually 90°) and the time of flight of the neutrons is recorded. From de Broglie’s relation,

where h is Planck’s constant, m is the mass of a neutron and L is the path length of the neutrons. Therefore substitution of this equation into eq. (1) and differentiation at constant gives lattice strain as,

where is the change in time-of-flight from its strain free value Consequently either eqs. (1) or (4) can be employed for determining lattice strains.

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As neutron diffraction only measures elastic strains, stresses can be obtained from the elastic properties of materials. Since both stress and strain are tensors, in general measurements in six directions at a point are required to completely define the stress state there. However, when the principal stress directions are known 3 orientations will suffice. In this case, when the principal directions coincide with the coordinate measurement directions x, y and z the principal stresses and in terms of the coordinate strains become,

where E is the elastic modulus and v Poisson’s ratio. In practice, it is often possible to infer the principal directions from a knowledge of the fabrication procedures or loading conditions imposed on a component.

4. Round Robin Measurements The round robin samples were chosen to represent a range of situations of practical importance. Ring and plug sample: The ring and plug sample (Figure 2) was chosen as the first to be measured because of its axial symmetry and the fact that residual stresses could be produced elastically by the adoption of an interference fit. It also had the attraction of introducing a relatively simple residual stress pattern that has an analytical solution. Two nominally identical samples were prepared, one for distribution predominantly through Europe #1 and the other #2 elsewhere. Measurements were made at both reactor and pulsed neutron sources to investigate, respectively, the benefits of using a monochromatic or polychromatic beam of neutrons. In all cases an experimental protocol was specified for obtaining the strains and eqs. (5) used for calculating stresses from the strain measurements. The findings are contained in VAMAS report no. 38 [9] and are summarised here. The studies on the ring and plug assembly established the basic procedures that should be followed. Measurements were made across the diameter of each sample as shown in Figure 2. It was found that it is essential to ensure accurate positioning and alignment of a specimen in the neutron beam for reliable results to be obtained. A suitable shape and size of ‘gauge volume’ over which individual measurements should be made to achieve adequate resolution in regions of strain gradients has been identified. It is recommended that a minimum of is adopted to encompass sufficient grains and to give neutron count times that are not excessive. For the measurements in the axial orientation, it has been found that a cube shaped gauge volume is required whereas a ‘match-stick’ shape can be employed for the hoop and radial orientations (as indicated in Figure 2) because of the lack of a strain gradient in the axial direction. A Bayesian analysis has been used to interpret the results [9]. From this analysis, it has been established that equally reliable data are obtained from a continuous monochromatic neutron beam source or from a pulsed polychromatic

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source. The best estimates of the three principal strains determined from the Bayesian analysis are shown in Figure 3. The corresponding residual stress distributions calculated from eqs. (5) are presented in Figure 4. It is evident from these figures that excellent agreement is attained with theory when allowance is made for averaging of the stress over the whole of the volume of material sampled for an individual measurement. The results were obtained for sampling volume cross-sections which ranged from From this study it has been found that, provided the experimental procedure specified is followed, strains can be determined to an accuracy of which corresponds to a resolution in stress of between ± 7 to 20 MPa in most engineering materials (depending on the elastic modulus for the material being measured).

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Ceramic Matrix Composite: In this investigation three separate specimens were examined. Each specimen was in the form of a disc, 30mm in diameter and nominally 3mm thick. Two of these specimens were provided to enable the strain free lattice parameter values for the base materials to be obtained. One of the specimens consisted of sintered alumina only and the other of silicon carbide powder only. The third specimen was the ceramic matrix composite (CMC) which contained a mixture of 25% silicon carbide powder in a matrix of alumina. The aim of the neutron diffraction measurements was to establish whether the strains in each component of the CMC could be determined reliably using the individual base material specimens as references. Each participant was allowed to choose their own (hkl) crystallographic planes on which to make measurements. They were also able to select their own gauge volume sizes and shapes. This study high-lighted a number of potential problems. In some instances, positioning uncertainties resulted in the gauge volume not being totally immersed in a specimen, which could lead to undesirable surface aberration complications. Also any divergence of the neutron beam could exaggerate this effect. It was found that the main problem was experienced with overlapping diffraction peaks as illustrated in Figure 5. In this case the background and peak positions become more difficult to estimate once there is more than one peak to fit. This affects the value of uncertainty estimations produced by peak fitting routines. A criterion has been developed for dealing with this situation. It provides guidance on how close interfering peaks can be for reliable results to be obtained. Shot-Peened Plate: Shot-peening introduces steep stress gradients close to surfaces. In order to determine these gradients, it is necessary to traverse the gauge volume through a surface as indicated in Figure 6. This results in a change in shape and size of volume of material being irradiated for each measurement. To deal with

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this situation, it has been found necessary to define two gauge volumes. One termed the instrumental gauge volume (IGV) is the volume in space which is illuminated by the neutron beam (taking into account any beam divergence). In Figure 6 this corresponds with the square cross section shape. The other volume is called the sampled gauge volume (SGV) which is defined as the volume in a material over which a strain measurement is being made and is averaged. For partial immersion the SGV will always be less than the IGV. For complete immersion the two gauge volumes are identical. In all cases it is recommended that the position at which the average strain is recorded corresponds to the centroid of the neutron distribution in the SGV. This will be nearer to the surface than the centroid of the cross-section for highly absorbing materials.

Results that have been obtained through the thickness of a shot-peened plate of a nickel base alloy that had been subjected to different peening intensities on opposite faces are shown in Figure 7. A gauge volume with a ‘match-stick’ shape was used. It has been found that when the above precautions are taken into account residual stress gradients as steep as 2000MPa/mm can be determined reliably.

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Ferritic Steel Weldment: The ferritic steel weldment examined is shown as an insert in Figure 8. It was produced by a manual metal arc (MMA) process. Twelve passes were used which introduced a 7° angular distortion. The weldment consisted of parent material, heat affected zone (HAZ), and weld metal. This example was chosen to investigate the feasibility of making neutron diffraction residual stress measurements through regions of variable microstructure and chemical composition. The main concern was to establish the appropriate strain free crystallographic lattice spacing reference for use in making the residual stress calculations. To aid in this process a thin slice was removed from the end of the weldment into which cuts were made to produce 'stress-free' prongs. These were situated in the parent material, HAZ and weld metal regions to enable the values of to be obtained for each zone. It has been shown elsewhere [11] that by ignoring the spatial and directional variation of which is exhibited throughout the weld pool and the heat affected zones, erroneous strain data can be derived leading to non self-equilibrating internal stress estimates. This is particularly true for austenitic and austenitic/ferritic welds. An illustration of the longitudinal, transverse and normal residual stresses measured through the weld centre line using the relevant values is shown in Figure 8. These results indicate approximately zero normal residual stresses as is to be expected, and transverse residual stresses which satisfy force equilibrium. Taken in combination, these observations demonstrate that reliable estimates of had been obtained. Also presented for comparison in Figure 8 are the residual stresses that were measured in a thin slice (without the cut prongs). These show no change from those in the plate in the transverse and normal stresses but relaxation of the longitudinal stresses to zero. This observation indicates that slices can be removed from samples without the in-plane residual stresses being affected.

4. Discussion The results of an investigation that was carried out to develop a draft standard under the auspices of VAMAS TWA 20 and RESTAND have been presented. In the RESTAND study measurements have been made additionally on felt and fibre

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reinforced composites for heat insulation and thermal shock resistance, on deep rolled crankshafts to represent complex shapes, a quenched component and through a variety of welds for power generation and aerospace applications. It has been demonstrated with complex shapes that care is needed to avoid orientations, which involve long neutron path lengths to minimise attenuation. Similarly it has been found that curved surfaces can exaggerate surface aberrations. Nevertheless, it has been shown that neutron diffraction can be used to measure residual stresses reliably in components for a wide range of industrial applications.

Based on the outcome of VAMAS TWA 20 and of RESTAND, an International Standard Organisation Technology Trends Assessment document (ISO/TTA) [12] has been issued which provides guidance for making reliable measurements of residual stresses by neutron diffraction. When these procedures are followed it is anticipated that positional tolerances to ± 0. lmm can be achieved and that strains can be measured to an accuracy of giving a resolution in residual stress of ± 7 to 20 MPa in most materials of practical interest. This document is now being used as a draft for the preparation of an international “technical specification” within a joint CEN/TC 138- ISO/TC 135 group of experts (AHG 7). Finally, ASTM E28. 13 is concerned with the preparation of this standard.

5. Acknowledgements The authors would like to acknowledge the contributions of their colleagues from the following institutions and neutron facilities around the world: Chalk River (Canada); ISIS (UK); ILL, LLB (France); JRC (Netherlands); HMI, GKSS (Germany); NPI (Czech Republic); RISOE (Denmark); Studsvik (Sweden); KUR, JAERI (Japan); KAERI (Korea); AEC (South Africa); LANSCE, ORNL, NIST and MURR (USA); Salford and Manchester University (UK). They would also like to thank the European Commission for the financial support provided for RESTAND.

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6. References 1.

Webster, G. A.: Role of residual stress in engineering applications (Eds Bottger, A. J., Delhez, R. and Mittermeijer, E. J. ) Matls. Sci. Forum, trans tech publications. Switzerland, 347-349 (2000), 1-9. 2. Youtsos, A. G. and Ohms. C.: Towards an evaluation of residual stress analysis based on neutron diffraction, Proceedings of the 3rd Size-Strain Conference “Analysis of microstructure and residual stress diffraction methods” (SS-III), Trento, Italy, December 2001. 3. Hughes, H..: X-ray techniques for residual stress measurements, Strain 3 (1967) 26-31. 4. Allen, A., Andreani, C., Hutchings, M. T., and Windsor, C. G.: Measurements of internal stress within bulk materials using neutron diffraction, NDT International, 14 (1981), 249254. 5. Stacey, A., Macgillivray, H. J., Webster, G. A., Webster, P. J. & Ziebeck, K. R. A.: Measurement of residual stresses by neutron diffraction, J. of Strain Analysis 20 (1985), 93100. 6. Beaney, E. M. and Proctor, E.: A critical evaluation of the centre hole technique for the measurement of residual stresses, Strain 10 (1974), 7-14. 7. Prime, M. B.: The contour method: Simple 2-D Mapping of Residual Stresses, Proceedings of ICRS-6, Oxford, UK, July 2000. 8. Sachs, G.: Der nachweis immerer spannungen in stangen und rohren, Zeits. Metall 19 (1927), 352-357. 9. Webster, G. A. (Ed)., Neutron Diffraction Measurements of Residual Stress in a Shrink-fit Ring and Plug, VAMAS Report No. 38 ISSN 1016-2186, January 2000. 10. Withers, P. J and Pang, J. W. L (private communication). 11. Ohms, C. Youtsos, A. G., Idsert, P. v. d., and Timke, Th., Residual Stress Measurements in Thick Structural Weldments by Means of Neutron Diffraction (Eds Bottger, A. J., Delhez, R. and Mittermeijer, E J. ) Matls. Sci. Forum, trans tech publications, Switzerland , 347-349 (2000), 658-663. 12. Polycrystalline materials - Determination of residual stresses by neutron diffraction ISO/TTA 3, ISO, Geneva, Switzerland September 2001.

MICROSTRESSES DETERMINED BY NEUTRON DIFFRACTION AND SELFCONSISTENT MODEL

A. BACZMANSKI WFTJ, Akademia Górniczo-Hutnicza, al. Mickiewicza 30, 30-059 Kraków, Poland C. BRAHAM LMMM, URA-CNRS 1219, Ecole Nationale Supérieure d’Arts et Metiers, 151, Bd de l’Hopital, 75013 Paris, France R. LEVY-TUBIANA LLB, CEA-CNRS, CEA Saclay, 91191 Gif-sur-Yvette, France A. LODINI LACM Université de Reims Champagne Ardenne UFR Sciences 51100 Reims, France LIB, CEA-CNRS, CEA Saclay, 91191 Gif-sur-Yvette, France K. WIERZBANOWSKI WFTJ, Akademia Górniczo-Hutnicza, al. Mickiewicza 30, 30-059 Kraków, Poland

Abstract Neutron diffraction was successfully applied to measure the lattice strain distribution in polycrystalline materials. The experimental data were studied with help of selfconsistent model. Using a non-standard method of analysis the values of macro and microstresses in textured ferritic steel were estimated. The validity of self-consistent model for prediction of mismatch stresses in two phase materials was examined. The theoretical results were successfully compared with the diffraction data for "in situ" tensile test. The parameters characterising elastoplastic deformation of polycrystalline grain were determined.

1. Introduction Diffraction technique is commonly used for the determination of the lattice elastic deformation and distortion i. e., macro and microstrains from the displacement and broadening of diffraction peaks [1, 2]. This method allows the measurement of stresses and elastic properties of polycrystalline materials. The stress state can be examined for near-surface volume by X-ray radiation [1] and inside the sample up to several 477 E. E. Gdoutos (ed. ), Recent Advances in Experimental Mechanics, 477–486. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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centimetres in depth, by synchrotron beam [3] or neutron radiation [4]. The diffraction methods have several advantages such as non-destructive character, the possibility of macro and microstress analysis for composites and anisotropic materials. Different methods can be used for the interpretation of diffraction measurements. The standard method of stress determining is based on measurement of interplanar spacing for various directions of the scattering vector which orientation is characterised by the and angles [1]. Using the simple linear regression procedure the macrostress field can be studied. Additionally, the analysis of diffraction peak broadening allows the observation of the microstrains evolution caused due to dislocations inside polycrystalline grains [2]. In the present work the non-conventional methods for interpretation of the experimental data are described. The methods, based on the Eshelby type model [5], are used for the study of mismatch stresses created due to incompatibilities of grains in their boundary regions. The model calculations are based on the prediction of interaction of ellipsoidal inclusion with a homogeneous matrix. The behaviour of different groups of grains (inclusions) can be directly compared with the diffraction data. The neutron diffraction and self-consistent model [6-8] of elastoplastic deformation were applied for analysis of the anisotropic mismatch stresses in steel sample subjected to plastic deformation. As the result both the macro and the mismatch stresses were found. Another analysis was performed for two phase materials for which the lattice strains were measured independently for each component. Thus the evolution of mismatch stresses (between different) phases were directly observed. Neutron diffraction experiments were performed on the G5. 2 spectrometer at the LLB, Saclay, France, while the X-ray measurement was done at the LMMM, Ecole Nationale Supérieure d’Arts et Metiers, France.

2. Self-consistent model The elastoplastic model [6-8] was used for interpretation of stress measurements obtained using diffraction technique. This model treats a given number of polycrystalline grains having different lattice orientations. The calculations are based on the modelisation of the processes occurring inside and between grains. As the result, the internal stresses for different grain orientation and for the whole sample are predicted. The deformation models, used in this work, are based on the prediction of the elastoplastic behaviour of a crystal grain inside the polycrystalline material under applied external stress If the local stress (at the particular grain) is large enough the plastic deformation occurs due to slip phenomenon. According to the Shmid’s law, the slip can be activated only on this slip system [uvw] (hkl) (the slip direction and plane are specified) for which the resolved shear stress exceeds some critical value

During plastic deformation, the multiplication of dislocations and evolution of their spatial distribution inside a grain leads to the hardening of slip systems ( increases with deformation). If we are interested only in the kinetic description of the active slip systems behaviour, their latent hardening can be described with some approximation by

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a matrix reflecting the interaction between the slip systems (the work hardening matrix). Consequently, the rate of the critical shear stress on the g-th system is equal to [6-8]:

where: is the rate of the critical stress in the g-th system, is the rate of the plastic glide in the h-th system and dot denotes the time derivative. The interaction between two active slip systems depends on their relative geometrical orientation. The weak hardening relations are represented by terms in the work hardening matrix, while the strong hardening relations are given by terms. Hence, the hardening matrix can be constructed using two independent parameters, i. e., To predict the plastic deformation, all the mentioned physical phenomena and quantities should be considered at the grain-scale. Moreover, the influence of the macroscopic quantities ( and rates characterising the sample) on the behaviour of the grain ( and rates) must be established. According to the self-consistent approach [6-8], the polycrystalline material is approximated by a macro homogeneous medium at the macroscopic scale and micro heterogeneous medium at microscopic scale. The relation between the stress rate and strain can be written for both scales:

where: is the local elastoplastic tangent modulus for the grain and is the macroscopic tangent modulus for a fictionally assumed average homogeneous medium. In general the properties of the grain and assumed matrix are different and the local stress rates can be obtained from the relations:

where while T is the interaction tensor calculated assuming the polycrystalline grain as an ellipsoidal inclusion embedded into the homogeneous matrix represented by L [6-8]. However, the A tensor can be calculated only if the L tensor is known, and inversely. To find both tensors the self-consistent procedure must be used [6-8].

3.

Anisotropy of plastic mismatch stresses

The average elastic lattice strain (noted by direction of scattering vector can be defined as:

measured by diffraction in

where the measured interplanar spacing are averaged only for the reflecting grains which possess the scattering vector normal to the crystallographic

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planes is measured and is the interplanar spacing for {hkl} planes in the stress-free material. On the other hand, the strain can be expressed through residual macrostresses and mismatch microstrains, i. e.:

where: are the macrostresses, are the diffraction elastic constants and is the elastic strain caused plastic incompatibilities of grains, defined for the reflection hkl and characterises orientation of the scattering vector. The aim of this part of work is to present a method for experimental determination of the macrostresses and the mismatch plastic incompatibility microstresses for textured polycrystalline sample. Using the self-consistent model the theoretical values of stresses can be predicted. Moreover, knowing the elastic constants for single grain, the average strain can be calculated. Since now, the model-predicted (theoretical) quantities will be denoted by bar. Let us assume that the second term in the Eq. 6 can be approximated by:

where: is calculated from the model and q - is a constant scaling factor. The factor q has been introduced to find the real magnitude of "plastic incompatibility" strains, assuming that their variation with the and angles is correctly described by models. Hence, for the hkl reflection, Eq. 6 takes the following form (see also Eq. 5):

For known diffraction elastic constants theoretically predicted strains and measured spacings the other quantities from Eq. 8 can be determined using a fitting procedure [9, 10]. The unknowns are q and the macrostresses Knowing the value of the q parameter from Eq. 8 (by applying a fitting procedure) the plastic incompatibility stresses can be determined for all grain orientations g:

where are the model predicted values. Thus, the macrostresses the mismatch microstresses can be determined simultaneously. The neutron diffraction experiment has been performed for two cold rolled samples i. e.: ULC90 (90% of reduction) and LC (70 % of reduction) [10]. The spacings have been measured inside the samples. The theoretical values of the strains were predicted by the self-consistent model [9, 10], Applying Eq. 7 and fitting the results obtained from the models to the experimental data, the

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macrostresses the interplanar spacing for a stress-free material and the q-factor have been found. In Figs. 1 and 2 the results of fitting are compared to the results of measurement. The theoretical curves approaches experimental points when the influence of plastic incompatibility strains (stresses) is taken into account in calculations (i. e., assumption in Figs. 1 and 2).

To show the level of the second order stresses, the average effective residual stress was calculated:

where the effective stress calculated according to the von Mises formula is integrated over the whole orientation space E.

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The results are given Tables 1. For neutron diffraction in the interior, the three dimensional stress state must be considered. In fitting procedure the and and components are not independent. Consequently, the absolute values of the principal stresses can not be determined if the is not known precisely. Hence, at this stage of study we show only the and differences instead of and absolute values (see Table 1).

4.

Mismatch stress in two phase materials

4.1. DUPLEX STEEL Duplex stainless steel with equal fractions (50%) of austenite and ferrite (50%) was studied [11]. The X-ray diffraction method was used to determine the internal stresses for the “in situ” deformed samples (tensile test). The interplanar spacing were measured separately for each phase, i. e.: using Cr radiation for phase, using Mn radiation for phase. From the experimental strains the surface stresses were calculated using the standard analysis [1]. The measured initial stresses of the both phases were an input data for the self-consistent model [6-8]. In calculations, the optimal model parameters of elastoplastic deformation were chosen in order to fit to the experimentally determined stresses for the and phases respectively (Fig. 3a) as well to the macrostress obtained from mechanical test (Fig. 3b). The mechanical tensile curve was obtained as the function of surface macrostress vs. macrostrain measured by the electrical gauge. It should be stated that the surface macrostress is taken as the measured stress applied to the sample shifted down by the value of initial residual stress (average for both phases) obtained from X-ray diffraction. In Fig. 3, the predicted stresses are compared with the experimental ones. The parameters of elastoplastic deformation are presented in Table 2. The results presented in Fig. 3a show that the model calculations agree very well with the experimental points for the deformation lower than 3%. For larger deformation the values obtained from X-ray diffraction are over the predicted curves because of relaxation of the surface compressive macrostress due to plastic deformation of the sample. This effect is not seen in the Fig. 3b where the experimental macrostress

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corresponds to the value applied to whole cross section of the sample. The initial parts of the curves obtained from X-ray diffraction are sufficient for determination of critical resolved shear stresses and amplitude of hardening matrixes for the both phases. The self-consistent calculations performed for these parameters agree very well with the result of mechanical test in which the relaxation of the surface stress does not occur.

4.2. METAL MATRIX COMPOSITE (Al/SiCp) The material was a 2124 Al alloy reinforced with 17% vol. SiC particles, produced by Aerospace Metal Composites, by a powder metallurgy route. A plate of composite was solution heat treated at 505°C for 2 hours and next quenched in cold water [12].

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The neutron diffraction technique was applied for lattice strains measurement. The sampling volume of mm was placed 1mm below the upper surface of the "in situ" bent bar. The tensile strains, induced by bending deformation, were measured along direction parallel and perpendicular to the applied stress As the reference value the measured interplanar spacing of initial (not deformed) sample was used for calculations, i. e.:

where: is the interplanar space measured for the initial sample, i. e. quenched but not deformed, is measured for the “in situ” bent bar and ph indicates the phase for which the interplanar spaces were determined (Al or SiC) Four points bending applied different deformations, while the surface strain was controlled using the stain gauge connected to the top of the bar. The total strain of the material in the region of neutron measurement (1 mm below the surface) was calculated from the surface strain assuming symmetrical bending. Simple tensile test was simulated and the lattice elastic strains and see Eq. 10) were calculated independently for the Al and SiC phases. The average strain values corresponding to those measured by diffraction were determined from elastoplastic model:

where the average (or ) lattice strain is calculated for the phcrystallites having the scattering vector parallel to the [111] crystallographic direction and being rotated by the angle about it. The theoretical results were compared with the phase strains measured by diffraction for the "in situ" bent samples (Fig. 4). In modelling the single crystal elastic constants were used for the both phases of the composite (Table 3). Simultaneously, the calculated function of total stress vs. total strain was compared with the results of mechanical tensile test (Fig. 5). In this case the macroscopic quantities were calculated as the average over volume of all composite grains. Purely elastic properties were assumed for the SiC component, while the plastic properties of the Al matrix were changed in order to find the best agreement of the measured and theoretical strains (see Figs. 4 and 5). The optimal model parameters of plastic deformation for Al (i. e.: critical shear stress, H - rate of work hardening and A- hardening anisotropy [6-8]) are given in Table 3. It should be stated that the theoretical values of and H could be effected by the residual stresses created due to quenching. In this case, both parameters should be treated as the effective ones defined for equivalent stress-free material representing Al matrix, which can not be directly compared to the thermally treated Al 2124 alloy. An excellent agreement between experimental data and model results was obtained for the neutron diffraction as well as for simple tensile test (Figs. 4 and 5). The agreement was obtained simultaneously for three different curves when plastic

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parameters of only one component (Al) were modified. It proves that the elastoplastic self-consistent model predicts very well the relation between macro stresses and the elastic strains (and stresses) measured for two phases in the elastic and elastoplastic ranges of deformation. Moreover, in the elastic range the single crystal elastic constants (Table 3) were used as the input data of model and any free parameters were optimised.

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5. Conclusions The diffraction methods and self-consistent model are particularly convenient tools for determining of the incompatibility stresses in polycrystalline materials. Using both methods the stress state for different groups of grains can be analysed. The comparison of theoretical prediction with experimental data allows us to understand the phenomena occurring during elastoplastic deformation. The advantage of the methods presented in this work is that the qualitative as well as quantitative analysis of stress evolution can be performed. The first presented method enables to evaluate the magnitude of the macrostress tensor and the residual plastic incompatibility stresses using some additional information from theoretical models. The variation of the microstress in function of the crystal orientation is supposed to be known from a theoretical model; its magnitude, however, is fitted from experimental results. Finally, the level of mismatch stresses for the plastically deformed sample can be estimated. The study of microstresses created in the two phase polycrystalline material can be easily done using diffraction methods. The stresses are measured independently for each phase and compared with self-consistent prediction. As the result the model parameters describing behaviour of crystallites can be estimated. For the optimal parameters the model stresses fit very well to the experimental data obtained from the mechanical test as well as measured by diffraction.

6. Acknowledgements The authors thank to the Komitet Bada Naukowych (Polish Committee of Scientific Research ) for financial support and the LLB, CEA Saclay (France) for enabling them to use neutron diffraction. 7. References 1. Noyan, I. C., Cohen, J. B. (1987) Residual Stresses - Measured by Diffraction and Interpretation, Springer-Verlag, Berlin, 1987.

2. Warren, B. E. (1969) X-ray diffraction, London: Addison-Wesley. 3. Daymond, M. R. & Withers, P. J. (1996) Scripta Mater. 35, 1229-1234. 4. Allen, A. J., Hutchings, M. T., Windsor, C. G & Andreani,C. (1985) Adv. Phys., 34, 445-473.

5. Eshelby, J. D. (1957) Proc. Roy. Soc. A241, 376-396. 6. 7. 8. 9.

Lipinski, P. and Berveiller, M. (1989) Int. Journ. of Plasticity, 5, 149-172. Lipinski P., Berveiller M., Reubrez E. and Morreale J. (1995) Arch of Appl. Mech. 65, 291-311. Zattarin, P., ., and Wierzbanowski K. (2000) Arch. Metall. 45, 163-184. Baczmanski, A., Wierzbanowski, K., Lipinski, P., Helmholdt, R. B., Ekambaranathan, G., & Pathiraj, B. (1994) Phil. Mag., A 69, 437-449. 10. Baczmanski, A., Wierzbanowski, K., Tarasiuk, J., Ceretti, M. and Lodini, A. (1997) Rev. de Metall., 94, 1467-1474. 11. Baczmanski, A., Wierzbanowski, K.., Braham Ch. and Lodini, A. (1999) Arch. Metall., 44, 39-50. 12. R. Levy-Tubiana, A. Baczmanski, M. Ceretti, M. Fitzpatrick, A. Lodini and K. Wierzbanowski (2000) Mat. Sci. Forum, 347-349, 510-515.

RESIDUAL STRESS MEASUREMENTS AT THE METAL/CERAMIC INTERFACE USING MODELLING OF NEUTRON DIFFRACTION SPECTROMETER ADELE CARRADO L. L. B. - Laboratoire Léon Brillouin, CEA-Saclay 91191 Gif sur Yvette, France L.A.C.M. - Université Reims Champagne Ardenne France

JEAN-MICHEL SPRAUEL LM3 UPRESA-CNRS 8006, IUT, 2 av. Gaston Berger F-13625 Aix en Provence LAURANT BARRALLIER E.N.S.A.M. - Laboratoire MécaSurf, 2, Cours des Arts et Métiers 13617 Aix en Provence, France ALAIN LODINI L.L.B. - Laboratoire Léon Brillouin, CEA-Saclay, 91191 Gif sur Yvette, France L.A.C.M. - Université Reims Champagne Ardenne, France Abstract The aim of this work is to improve some experimental techniques dedicated to the evaluation of residual stress (RS). These methods have been applied to a sample consisting of a glassy ceramic coating moulded on a metallic substrate (palladium-silver alloy) used in dental applications. Knowing the residual stress distributions is very important to determine the lifetime of the sample. We will describe a new approach to evaluate the RS at interfaces and in the bulk of materials, using neutron diffraction techniques. The RS in a glassy ceramic coated on metallic substrate are generated by the manufacturing process. They are present in the two materials. The RS depend principally on the thermal treatments imposed to the sample and they could have a very strong influence to the lifetime of the sample and in particular for the existing bounding between the metal and the ceramic. Moreover, the mechanical proprieties of glassy ceramic are known to be affected by proprieties of particle dispersed throughout the glassy matrix. The difference in the coefficients of thermal expansion (CTE) that exists between a ceramic and a metallic material could also be at the origin of a stress field. The first part of this paper concerns a development of a numerical simulation [1], [2] of whole two-axis neutron spectrometers. This programme allows correcting systematic errors due to the great parasitic peak shifts which appear when the measurements are carried out at the interface between two different materials [3]. 487 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 487–494. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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The second part of this paper leads on the experimental determination of RS by neutron diffraction in the ceramic and metallic materials. These measurements were performed using different European facilities: D1A spectrometer at ILL (Grenoble, F) and E3 at BENSC-HMI (Berlin, D) for the analysis of the palladium substrate, and G5. 2 spectrometer at LLB (CEA- Saclay, F) for the characterisation of the glassy ceramic coating. Key Words: Neutron Diffraction, Residual Stress, mechanical behaviour, and numerical simulation.

1. Introduction The control of residual stress obtained during the manufacturing of a ceramic coating applied to a metallic substrate is very delicate. The PFM (Porcelain- fused- to- metal) technology of dentistry industry is well adapted for this aim [4]. For a successful operation, the coupled materials are selected in relation to their physical propriety and their biocompatibility. Alloys intended for use as base for porcelain have special requirements, because of the need to develop and maintain strength at the temperature involved in porcelain applications and to provide a firm bond to the applied porcelain. The PFM is the manufacturing process most widely used in commercial laboratories. Rational selection of a casting alloy for porcelain- fused- to- metal restorative should be based on the following: 1. Physical proprieties 2. Chemical propriety 3. Biocompatibility 4. Laboratory workability 5. Porcelain compatibility [5]. Thermal expansion and bond strength are also important proprieties to consider when choosing among PFM restoration alloys. These characteristics determine the porcelain / metal compatibility. For our ceramic, it was chosen Leucite which is a reaction product of potassium feldspar and glass. It is a particularly important component in dental porcelain because it affects the optical properties, thermal expansion, strength and hardness of the porcelain [5]. Due to its relatively inert behaviour in aggressive environments, it has a high hardness and a good wear resistance. The ability of ceramic materials, to resist higher temperatures than metals, offers the potential for major improvements in component design for a wide range of applications. However, ceramics typically exhibit statistically variable brittle fracture, environmentally enhanced sub critical crack growth, sensitivity to machining damage, and creepdeformation behaviour at elevated temperatures. The leucite crystals serve to increase the thermal expansion of the porcelain to bring it closer to that of the metal substrate. The leucite prevents stresses occurring, due to a thermal mismatch, which could lower the strength. An opaque ceramic such as a titanium oxide glass frit has to be applied as the first layer of veneer, to mask the metallic substrate in the PFM systems. Although the PFM systems have high strength, the opacity of the metal substructure has encouraged the

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development of all-ceramic core materials containing crystalline components which are stronger than the traditional (predominantly glassy amorphous) feldspathic porcelain. This type of core material can then be veneered with a more translucent ceramic material. The CTE of the porcelain must be suitably matched with that of the alloy and the melting range of the alloy must be raised sufficiently above the fusion temperature of the porcelain for a successful enamelling operation. Dental alloys must be inert, rigid and strong. They must be rigid in thin section to resist bending and they assist in the formation of a chemical bond with the porcelain. They have a similar CTE to the porcelain, so that when the metal/porcelain combination cools high interfacial stresses (which could dislodge the porcelain) are not created.

2. Material and Methods Sample A porcelain/palladium alloy was prepared following standard routines practiced by dental technicians in construction of PFM crowns. This particular ceramic/metal combination has low thermal mismatch, avoiding the generation of significant RS in the coating. Palladium alloy casting ingot was casted at 1450°C into a substrate block of 20 x 60 x1.8 mm (TAB LE 1).

The block was then ground flat on opposite faces, and sandblasted with alumina to to provide good bounding for the ensuing coating and finally oxidised 10 minutes at 980°C at 101325 Pa. A first coating was prepared by opaque powder, which was mixed in slurry and applied to the whole surface of the substrate with a paintbrush. It gives a regular layer of opaque (approximately 100 after heating). The layer was sintered according to the following firing cycle: heat to 980°C in vacuum and hold for 1 minute, it produces the real first layer of opaque that will hide the alloy. The top surface of the porcelain coatings was then sandblasted (alumina, A second opaque layer is normally optional. It has been used for this sample and its role is to complete to hide the metallic effect of the substrate, as the size of the samples was particularly large. The second opaque layer was heated to 980°C for 1 minute under vacuum. Concerning the dentin layer, this layer was just heated to 950°C for 1 minute. First Part: The Model The simulation program accounts for all the major components of the neutron spectrometer: the characteristics of the neutron guide or tube, a double curved mosaic monochromator, up to hundred primary and secondary slits, a classical or position sensitive detector. An integrated feature oriented CAD drawing module defines the

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sample. This allows the description of very sophisticated specimens (Figure 1). The software accounts for the horizontal and vertical divergence of the incident and diffracted beams, for the local conditions of diffraction in the monochromator and the sample, and for the absorption by the monochromator and the sample.

The simulation program first computes the distribution of intensity and wavelength across the incident beam. The precise shape and size of the probe volume is then calculated. We have also defined the centre of diffracting volume, which has to account for the absorption in the material To define precisely the position of the neutron probe close to the surface we have performed a scanning along the direction normal to the surfaces. So we can plot the evolution of diffracted intensity versus the position Z of the geometric centre of the neutron probe (Figure 2).

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Classically, it is considered that at half of maximum of this curve, exactly half of the neutron probe is immersed in the material. In our case, this assumption is not valid, because of the strong absorption of palladium. A non-negligible shift exists for the determination of the true analysed depth. In addition, the program allows also computing the intersection between the neutron probe and the sample, thus defining the diffracting volume. A theoretical diffraction peak is finally calculated through a Monte Carlo simulation method. The whole simulation program allows thus to optimise the experimental conditions and to predict all parasitic shifts of the diffraction peak.

Second Part: Experimental Methods and Measurements Stress evaluation Method The diffraction method for the determination of residual stress is based on the measurement of the interplanar spacing in various directions (Figure 3).

In a diffraction experiment, the mean interplanar spacing be measured (Figure 4). The mean value of the strain scatting vector Q is equal to:

of the reflecting grains can in the direction of the

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where is the lattice spacing corresponding to the {hkl} planes of a stress free specimen, is the local strain in the direction of the scattering vector. V is the whole volume of the reflecting grains and the brackets denotes, the average over the diffraction grains of the hkl reflection [7]. Due to the big difference between the lattice parameters of the two materials, it was not possible to do all the measurements on one single instrument (TABLE 2). For that reason, neutron diffraction experiments were performed at LLB (CEA-Saclay, F), on G52 two-axis diffractometer using cold neutrons to evaluate the RS in the glassy ceramic coating. For the palladium, the measurements were realised on D1A spectrometer at ILL (Grenoble, F) and on E3 equipment at HMI-BENSC (Berlin, D). The experimental conditions of these neutron diffraction measurements are reported in TABLE 3 [8]. In figure 4, the sample geometry and the co-ordinate systems are shown.

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Results

The RS profiles are represented as a function of the distance to the metal- ceramic interface. In figure 5, for both the materials, the residual stress distributions are presented. - The residual stresses observed in the palladium alloy substrate are tensile but their magnitude is very low. - In the ceramic coating, the stresses are mainly compressive. Due to the low diffracted intensity, linked to a high amount of amorphous phase, only the component was determined. The stress profile in the ceramic shows low compressive values, but significant stress may exist near the interface.

4. Discussion and Conclusions In this paper, we have presented residual stress profiles at the interfaces of a glassy ceramic coated onto a palladium alloy substrate. Measurements by neutron diffraction techniques have been realised in different zones of the sample. In particularly, we have employed the neutron diffraction to analyse the surface and the bulk of glassy ceramic and the core of the metallic substrate. This technique, largely employed, leads to very

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good results. It has permitted us to obtain a precise and non-destructive evaluation of the residual stress in the core of the materials. We have also improved the experimental techniques dedicated to the determination of residual stress. This has required a precise and refined data processing, which accounts for different physical phenomena and for geometrical aberrations, which appear in the measurements. New data treatment procedures, based on a numerical simulation of the neutron instruments, have also been developed. These methods allow accurate and non-destructive evaluations of the in-depth residual stress profiles of surface or interface layers. In particularly, we have employed the neutron diffraction to analyse the surface and the depth of the glassy ceramic, and the bulk of the metal. It would however be very interesting to evaluate the stress profile starting from the metal/ ceramic interface and going inside the first of the ceramic (opaque interface ceramic).

5.

Acknowledgements

The experiments at BENSC in Berlin were supported by the European Commission through the TMR/LSF Access Programme (Contract: ERBFMGE CT950060). Sincere thanks are due to the staff of E3 at HMI Berlin, of D1A at ILL (Grenoble, F) and of G52 at LLB (CEA- Saclay, F) without whose help the measurements would not have been possible.

6. References 1. 2.

3. 4. 5.

6. 7. 8.

Pluyette E., Sprauel J. M., Lodini A., Perrin M., Todeschini P.: Residual stresses evaluation near interfaces by means of neutron diffraction: modelling a spectrometer, ECRS4, Cluny: pp. 153-163, 1996. Sprauel J. M.: Stress evaluation by neutron diffraction: Modelling of a two axis spectrometer, Journal of Neutron research, in press, 2001. Webster P.J., Mills G., Wang W. P., Holden T. M.: Impediments to efficient through surface scanning, Journal of neutron research, 3, p. 223-240, 1996. McLaren EA.: All-Ceramic Alternatives to Conventional Metal/ceramic Restorations. Compendium 307-325, March 1998. Van Vlack L. H.: Materials for Engineering, Addison - Welsley AB 92 - chapter 11, 12, 13 and 15, 1982. Doremus RH, Review bioceramics, Journal of Materials Science Vol. 27, 285-297, 1992. Noyan I. C., Cohen J. B.: Residual Stress Measurement by Diffraction and Interpretation (Materials Research and Engineering), 1987. Carradó A., Sprauel J. M, Barrallier L., Lodini A., Neutron and Synchrotron evaluation of residual stresses in coatings, Journal of Neutron research, in press, 2001.

ELASTOPLASTIC DEFORMATION OF TWO PHASE STEELS STUDIED BY NEUTRON DIFFRACTION AND SELF-CONSISTENT MODELLING M. R. DAYMOND ISIS Facility, Rutherford Appleton Laboratory Chilton, Didcot, Oxon. OX11 0QX, UK H. G. PRIESMEYER Institut für Experimentelle und Angewandte Physik Christian-Albrechts-Universität, Kiel, Germany A. M. KORSUNSKY Department of Engineering Science University of Oxford, Parks Road Oxford, OX1 3PJ, UK

Abstract In situ neutron diffraction experiments allow the measurement of phase specific strain, and thus stress, in a multi-phase system. Further, monitoring of multiple hkl lattice planes provides information as to the response of differently oriented grains within the polycrystal. These experimental results can be compared directly with predictions from a self-consistent Hill-Hutchinson model modified to take into account the presence of the second phase. The techniques and ideas are demonstrated with recent tests on two practical engineering steels (1) ferritic steel, with a small volume fraction of carbon in the form of cementite and (2) duplex steel, consisting of approximately equal amounts of austenitic and ferritic phases. Good qualitative agreement was obtained between model and experimental data in each case. Issues affecting the use of such a model for two phase systems are discussed. The interpretation of residual stress measurements, for both single peak and Rietveld multi-peak measurements is considered.

1.

Introduction

It has been recognised for some time that internal and residual stresses in materials can have a considerable effect on material properties, including fatigue resistance, fracture toughness and strength. Such stresses can vary greatly as a function of position within engineering components, due to manufacturing processes during the production route. Their measurement and interpretation is thus of considerable interest to the engineer. Monitoring of phase specific internal strains can also provide a direct insight into the mechanical behaviour of multi-phase materials. The strength of a multi-phase material 495 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 495–506. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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depends on the level of in-situ matrix strengthening (e. g. due to increased dislocation density from the presence of a secondary phase), but also on the efficiency of load transfer between the two phases. The distribution of load between the phases is dependent on relative volume fraction, shape and orientation and on the relative elastoplastic properties of the phases [1]. The partitioning ratio of applied load between the two phases will remain constant with increasing applied load provided that both phases remain elastic. However, once the stress levels in a multi-phase material are high enough for relaxation or inelastic deformation processes to occur in one or more of the phases, the load partitioning ratio changes. The mechanics in the case of the classic “composite”, where a small volume fraction of hard reinforcement is dispersed uniformly in a more compliant matrix, has seen considerable discussion [1]. The distinction between reinforcing phase and matrix is more complicated in the case of a two phase system with approximately equal volume fractions of the two phases. In this case an interpenetrating microstructure may well exist. This paper addresses a number of issues relating to mechanisms of materials deformation, and to the practical measurement of stress in components by diffraction from two common two phase steels; the system, and an austenitic-ferritic duplex stainless steel. Ferritic steels are extremely common, having good mechanical and forming characteristics as well as being relatively cheap. Small volume fractions of dispersed secondary phases such as cementite are found in many practical ferritic steels, commonly found in the case of the pearlitic microstructure. Duplex stainless steels, consisting of approximately equal amounts of austenite and ferrite, often combine the best features of these constituent phases. They generally have good mechanical properties, including high strength and ductility, and are increasingly used as an alternative to conventional austenitic grades. Neutron diffraction is an important experimental technique, ideal both for profiling macro-strain variations as a function of position, and for measuring phase specific load transfer. It relies on the same physics as the analogous measurement of stress using xray diffraction [2], however with the principal benefit for the engineer or materials scientist of increased penetration depth compared to traditional x-rays. Since neutrons interact primarily with the nucleus, rather than the electron cloud as x-rays do, the penetration depth of neutrons is very large compared to x-rays. For example, the penetration depth (1/e) for steel is ~1cm for thermal neutrons, but less than for xrays of comparable wavelength [3]. This results in neutrons being a probe suitable for bulk average measurements of material properties. For many practical applications, it is bulk averaged properties that the engineer or material scientist is primarily interested in. Diffraction measurement of strain involves the monitoring of changes in separation of one or more suitably orientated crystallographic lattice planes. It is thus a direct measure of the elastic strain in the material. In a polycrystal a diffraction peak represents the average lattice separation over all the grains in the irradiated volume which are suitably oriented to diffract. Thus each diffraction peak is produced by a different family of grains within the polycrystal, giving a direct insight into the grain orientation dependence of elasto-plastic deformation. For instance in the presence of an applied stress during elastic loading, the strains observed for measurements on individual diffraction peaks will typically differ in magnitude with respect to each other, and to the continuum macrostrain (e. g. [4]), a phenomena commonly dealt with during

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engineering stress measurements through the use of a ‘diffraction elastic constant’. The importance of understanding the mechanisms of interaction between differently orientated crystallites is highlighted when one considers that not only do individual grains exhibit elastic stiffness anisotropy, but also that plastic relaxation occurs preferentially on certain slip systems (plastic anisotropy). Despite increasingly sophisticated models (e. g. [5]) the current state of the art is far from quantitatively predicting the evolution of hkl dependent strains or the implications of their spread on failure. The use of self-consistent models does however give a direct insight into plasticity in polycrystals; the onset of deformation in differently oriented crystallites and the importance of hardening mechanisms. Systems where an elastic phase interacts with a ductile phase have been treated extensively using continuum mechanics arguments [1], but are relatively unexplored using crystal plasticity arguments. This lack of data is even more clear for the more complicated case where both phases are ductile. This paper applies a self consistent model to the deformation of two phase steels and compares the results with lattice reflection data from neutron diffraction experiments.

2. Uniaxial tensile tests on steel A series of uniaxial tensile loads were applied sequentially to specimen in situ on the ENGIN instrument of the PEARL beamline at the ISIS pulsed neutron facility, Rutherford Appleton Laboratory. The load frame is purpose built for use within the neutron beam. The loading axis is horizontal and typically at 45° to the incident beam, allowing simultaneous measurement of lattice plane spacings both parallel and perpendicular to the loading direction in the opposing 90° detector banks (Fig. 1). Universal joints were used for these tensile tests in order to maintain uni-axiality of load during the test. The material used was a 38MnS6 steel consisting mainly of a pearlitic microstructure, with larger scale ferrite veins and cementite particles, indicating a

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hypereutectoid structure, The material was nominally untextured, with a volume fraction of cementite ~5%. A circular sample of 8mm diameter cross section was tested, with a nominal gauge volume (neutron sampling volume) 8mm high, 5mm wide defined by incident slits, and 1. 5mm wide outgoing defined by radial collimators. The duplex steel was cold rolled SAF2304, consisting of approximately equal volumes of austenite and ferrite phases. The duplex steel was received as 1. 5mm thick plate, with a heavily banded microstructure of elongated austenite islands in a ferrite matrix [6]. The material was strongly textured, with texture varying as a function of depth; from ~2.5x random at the surface of the plate to ~5x random at the centre of the plate for the austenite, and correspondingly from ~5x to ~25x random for the ferrite [7]. The sample was cut with tensile axis parallel to the transverse plate direction, and measured with the rolling direction vertical, thus strains were determined in the transverse and normal plate directions. A nominal gauge volume 3 x 15 x 1. 5mm was defined for this test. The austenite grains were approximately equiaxed with dimensions the ferrite grains around in the rolling and transverse directions, in the normal direction [6].

3.

Self-consistent modelling

The elastic-plastic properties of a polycrystalline aggregate have been described using the Hill self-consistent approach [8], which was first implemented by Hutchinson [9]. A population of grains is chosen with a distribution of orientations and volume fractions that match the measured texture. Each grain in the model is treated as an ellipsoidal inclusion, though spherical grains were used for all the results presented in this paper. Each grain is attributed anisotropic elastic constants and slip mechanisms characteristic of a single crystal of the material under study. Interactions between individual grains and the surrounding medium, (which has properties of the average of all the grains) are performed using an elasto-plastic Eshelby type self-consistent formulation. Since the properties of the medium derive from the average response of all the grains, it is initially undetermined and must be solved by iteration. Small total deformations are assumed (typically less than 4%), and no lattice rotation or texture development is incorporated in the model. The model used is described in more detail in [10]. The single crystal elastic constants used in the model are shown in Table 1. Plasticity in the austenite was assumed to take place on <110> (111) systems. Since there is no close packed plane in bcc materials, slip is generally considered to take place on many slip systems, with the <111> slip direction in common. In order to mimic this simply in the model, three slip systems were included, all with the same critical resolved shear stress and hardening behaviour, namely the {110}<111>, {112}<111>, and {123}<111> [11]. The critical resolved shear stress and exponential hardening coefficients in each case were fitted to give optimum agreement with the macroscopic stress-strain curves, and are listed in Table 2. The hardening function used for each system is described by Equation 1,

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where is the accumulated shear strain in the grain. The threshold stress in Equation 1, describes (in an average way) the resistance to activation that the deformation modes experience; it usually increases with deformation due to strain-hardening. Equation 1 represents an extended Voce law [12] which, instead of stress saturation, exhibits an asymptotic hardening rate For the strains used in this work it is an adjustable hardening parameter. In addition, we allow for ‘self’ and ‘latent’ hardening by defining coupling coefficients which account for the obstacles that dislocations in system represent to the propagation of s dislocations. The increase in the threshold stress of a system due to shear activity in the grain systems is calculated as:

When 'self' and 'latent' hardening are indistinguishable, Equations 1 and 2 permit a description of the high hardening rate observed at the onset of plasticity, and its decrease towards saturation at large strains. Linear hardening is a limiting case of this law, and takes place when In this paper an isotropic hardening model was used with latent hardening equal to self hardening.

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In order to compare the model with the experiment a subset of the total population of grains used in the model is identified for each diffracting family defined by the condition of having an hkl plane lying within ±5 degrees of the loading axis, or perpendicular to it. These are the grains that would contribute to a diffraction measurement of strain in that direction, and an average over each subset of the strains across the diffracting plane is compared with the neutron results below. Each diffraction peak is produced by a different population of grains. Although the self-consistent model has been used extensively on single phase materials, some simple modifications are required for multiple phases. We here simply create a population of grains, with the appropriate volume fraction with properties of the individual phases; the model has already been applied to both fcc and bcc materials [5, 13]. For the ferritic steel / cementite system we take advantage of the fact that cemenite has approximately the same stiffness as ferrite [14], as has also been observed by diffraction measurements [15]. Therefore without introducing a new crystal structure into the model, we treat the cementite as having the elastic properties of iron crystallites but constraining these crystallites not to slip, and providing the appropriate volume fraction. Since the self-consistent model treats the homogenous matrix as having the average elastic properties of all the grains, provided we a) do not have texture in the cemenite, and b) require only an average phase strain (rather than lattice reflection strains) for the cementite, this is a good approximation within the self-consistent model. It is a valid approach in considering the reinforcing effect of the cementite particles on the iron lattice reflection strains in the plastic regime. It should be noted of course that the model therefore does not take into account the complexities of microstructure actually observed. The model was run with 1000 randomly oriented grains representing the iron with weight 0. 95, and 1000 randomly oriented grains representing the cementite with weight 0. 05. Both phases were therefore represented with sufficient grain populations for random textures, and with appropriate volume fractions. For the duplex steel, the measured texture was converted to a grain population (1152 grains per phase), with equal weighting, and again the appropriate elasto-plastic properties applied to each phase. The incident spectra at time-of-flight neutron sources are polychromatic, thus a wide range of possible lattice planes are recorded in each measurement. The scattering vectors for all reflections recorded in one detector lie in the same direction, and thus indicate the strain in that direction. Each reflection is produced from a different family of grains that are oriented such that a specific hkl plane diffracts to the detector. We have therefore analysed the experimental data in two methods, both carrying out fits to individual peaks within this spectrum, and fitting the whole spectrum simultaneously using a Rietveld refinement [16]. The use of this technique is discussed in detail in [17], but in brief has been shown to be a good representation of the mean phase elastic strain. Thus we can compare both the hkl specific and average phase strains obtained from the model directly with strains obtained from analysis of the diffraction data.

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4. Results The agreement that can be achieved between the model and experimental macroscopic curves is reasonable (Fig. 2); it should be remembered that this agreement has been achieved by choice of the parameters in Table 2. However important discrepancies exist, which further modifications to the hardening parameters do not remove. For instance, in the (Fig. 2a) a reverse yield can be seen at the base of the experimental unload, which is not captured in the model. Secondly, the measured

macroscopic response immediately after yield is somewhat softer than shown in the model; the plastic hardening curve is initially shallower in the experimental data. It is not possible to fully capture this in the model, even if a very low hardening rate is used. This fact can be partially understood from the mechanisms of the model, in which each grain is modelled as being a single, independent ellipsoid within a medium which has properties of the average of all the grains. When the first optimally oriented grain in the modelled population reaches yield and plasticity occurs, the relaxation caused has only a small effect on the stiffness of the medium due to the small volume fraction represented by each grain. This will initially have a relatively small effect in bringing other grains to their yield surfaces. This constrained transfer of relaxation effects between differently oriented grains via the interaction of the medium limits how steep the initial part of the macro plastic curve can be, given that the value of the asymptotic slip hardening parameter (Eqn. 1) is constrained by the requirement to match the observed macrostrain gradient at large plastic strains. It should be noted however that if a prior deformation or thermal treatment had created residual stresses a more rapid switch from elastic to long-term plastic behaviour is indeed possible. This is because the prior deformation biases the state of individual grains such that more reach their yield surfaces at a given macroscopic stress, most easily imagined in the case of an interrupted tensile test, though the effect is also possible in for example rolling followed by tension [18]. Correspondingly, in the duplex data (Fig. 2b), some slight discrepancies also exist. Initial departure from the elastic limit is seen experimentally perhaps at an applied stress of as low as 320MPa, but more significant yielding occurs

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at 450MPa; this two stage yielding behaviour is not fully captured. Further, the high plastic strain gradient is slightly too low in the model. The strains obtained from a Rietveld refinement on the phases, indicative of the ‘bulk’ or average strain in each phase are shown in Figure 3. These are compared with the mean phase elastic strain obtained from the model. Considering first the ferrite /

cementite system, the experimental cementite strains (Fig. 3a) show a large scatter due to the small volume fraction and broad peaks, but within experimental error there is good agreement. In both axial and transverse directions the cementite elastic response is the same as the iron, with larger strains generated in the cemenite once plasticity occurs and load transfer initiates from iron to cementite. The iron strains, with uncertainties of the same size as the symbols, are in excellent agreement with the model which captures the small but measurable deviation from linearity in both axial and transverse directions. This is a reassuring observation for stress measurement in bcc materials, since in common with fee steels it suggests that the Rietveld strain is indeed a good representation of the bulk macro strain, even in the plastic regime. However it does highlight the importance of considering even relatively minor volume fraction phases when making stress measurements. For the duplex system, again the initial elastic response in both axial and transverse directions are very similar for the two phases, indicating the similar ‘bulk’ moduli of the phases; as would be hoped this is correctly captured by the model. Experimentally there is then a gradual load transfer from the austenite phase to the ferrite, as the austenite initially yields (starting just above 300MPa). At an applied stress of around 500MPa the strain increment in the austenite with increasing applied stress changes again, presumably due to yield in the ferrite phase. This effect is seen in both the axial and transverse directions in the austenite, though the effect on the measured ferrite strains is small. The model qualitatively captures the load redistribution between the phases but does not correctly describe the magnitude of the effects. In particular the model has the austenite phase with nearly zero strain increment for increasing applied loads between

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300 and 450MPa, whereas experimentally this is still significant. Further the reverse transfer of load from the ferrite phase as it yields is much larger in the model than seen experimentally. The anisotropic response of individual lattice reflections during in situ loading continues to be a subject of considerable experimental interest, since it provides an excellent test of crystal plasticity models. Figure 4a shows the response of four (first

order) lattice reflections plotted against the macroscopic applied stress, parallel to the loading direction for the ferrite in the ferrite / cementite system. These strains represent the values determined by single peak fits. It should be remembered that each line thus represents the response of a different family of grains, oriented such that the given hkl lattice direction is parallel to the loading direction, respectively. Elastic moduli show generally good agreement between model and experimental results, though the differences are somewhat large to be due to experimental scatter alone. Deviations from an elastic linear response occur in the single peak strains close to the onset of macroscopic plasticity, but at different macro stresses for different grain populations. This is because once plastic deformation initiates, the yield of preferentially oriented grains relative to their neighbours causes strain redistribution, and a divergence from the hitherto linear response in other peaks. As would be expected the 110 and 211 peaks have the same elastic modulus [e. g. 17], but here can be seen to have different plastic anisotropy strains. The qualitative behaviour of the peaks in the plastic regime is well captured. There are some quantitative differences, notably the exact macro stress at which yield occurs in the different grain populations, though this is not perhaps overly surprising given the differences seen in the macroscopic response. Secondly, it appears that the initial yield is sharper in the experimental data than in the model; the departure from elastic slope to plasticity is experimentally nearly vertical in the axial direction, while the model shows a more gradual yielding. This suggests a very rapid initial transfer of load from the iron to the cementite. This observation is based on a small number of data points, but the

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effect seems consistent in the different individual peak strains, as well as the mean phase strains. It also ties in with the evidence from the macroscopic curve that we are not fully capturing the speed of onset of plasticity. It is interesting that the model 002 and 310 peak strains show a deviation first one way then the other from the elastic gradient, becoming softer than elastic, then harder. Correspondingly the 110 and 211 do not show this behaviour, deviating immediately to stiffer behaviour. This also appears to be the case in the experimental data, for instance in the 002, though these changes are at the level of experimental uncertainty. This is demonstrating that the 110 and 211 directions are preferentially oriented for slip (i. e. higher Schmid factor), and hence yield first whilst the 002 and 310 grain directions are poorly oriented for slip and hence some small degree of load redistribution is occurring onto these planes as slip initiates from the former grain sets to the latter. As larger macroscopic plastic strains are generated, the 002 and 310 oriented grains also start to slip, and the load transfer to the cementite dominates. The axial strains observed in the duplex austenite phase are shown in Figure 4b, and illustrate a number of points. The initial yield behaviour, shown as a reduction in the gradient of the hkl strains is captured. As with the mean phase strains (Fig. 3b), the model predicts a smaller strain increment with applied stress than is experimentally observed for the 311 and 200 peaks. Interestingly, the model appears to under predict the softening in the 111 peak; the experimental strain appears to be nearly vertical between 350 and 500MPa. While the effect of the initiation of ferritic yielding at ~500MPa on individual grain family strains can be seen in both model and experimental data, again the magnitudes are not correctly captured. The ferrite single peak strain data has not been shown due to lack of space, however we note that (a) the 200 peak is extremely weak in this direction due to the strong texture, and therefore useful strains could not be obtained, (b) the 211 and 110 peaks provide indistinguishable strain data in both the elastic and plastic regime (c. f. Fig. 4a), (c) as in Fig 3b, the qualitative behaviour of ferrite single peaks below 450MPa is well captured by the model, with strain magnitudes again also well captured. However there is no experimental evidence in the experimental single peak strains of the reverse load transfer which is seen in the modelled ferrite response as a reduction in gradient (in both phase and single peak strains)

5. Discussion From the results detailed above it is clear that, while the model of the duplex steel has captured some key features of the phase specific and grain orientation dependent elasto-plastic deformation, the model of the ferrite/cementite system has been considerably more successful, achieving excellent quantitative agreement between model and experiment. It is important to address the reasons for this discrepancy beyond the obvious issue of either one or both phases yielding, as this highlights the successes of this modelling approach, as well as present problems. Firstly, when modelling a two phase system where only one phase is elastic, as described above, the plasticity hardening parameters of the plastic phase are chosen to provide ‘best agreement’ with the macroscopic response. Comparisons are then made with the internal phase strains. The choice of ‘best agreement’ has some subjectivity,

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which could be removed for instance by incorporating the process in a least squares’ approach. However, the effect on the internal hkl strains of small changes in the hardening parameters to ‘tune’ agreement with the macroscopic response is small. However, when both phases can yield there is clearly opportunity for more flexibility in choice of yield stress and hardening parameters to obtain the macroscopic response. In this case, since it was known that the austenite would yield before the ferrite phase [6], and further this yield was identified with the slight initial change in macroscopic slope (at ~300MPa), sufficient extra constraints are introduced into the model to limit the range of feasible hardening parameters. While an appropriate through-thickness average texture was incorporated in the model, a number of important microstructural and processing issues were not included in the model. For example, the texture varied as a function of depth through the plate. This will result in a variation of mechanical properties with depth, and thus some level of load transfer between material at different depths, although the neutrons do produce a result averaged over the entire thickness of the plate. Further, the ferrite grains were treated as spheres, while they in fact have a distinct aspect ratio – this will affect the Eshelby stiffness of the grain in the model. While the austenite grains are indeed equiaxed, making a sphere approximation appropriate, they are found in elongated islands within a ferrite matrix. The self-consistent model treats each grain in a medium consisting of the average properties of the entire grain population, and therefore does not take into account the environment of grains. While this ‘dilute’ approximation has been successful with the ferrite/cementite system, it is possible that failure to capture these geometric effects is worsening agreement in the duplex system. Finally, and perhaps most importantly, some residual strains from the processing of the duplex steel will almost certainly exist. These strains will effect the yielding behaviour of the phases, as well as for given grain orientations. The generation of these strains is extremely complicated to model, since the process includes texture development as well as recrystallisation. Attempts have been made to model the development of internal strains due to processing in a single phase steel prior to such a uniaxial loading test [18] and this avenue needs to be explored with the duplex steel results described above.

6.

Conclusions

Good qualitative agreement has been obtained between experimental and model data for uniaxial loading of the duplex steel, where both phases were modelled with elastoplastic behaviour. Discrepancies between model and experiment were attributed to microstructural characteristics and processing induced residual strains. Excellent agreement has been achieved between experimental and model data for uniaxial loading of a simple two phase iron-iron carbide system, in which the elastic properties of the two phases are the same, while only one phase has been modelled as exhibiting plastic slip. The strains obtained from a Rietveld refinement have been shown to provide a good description of the mean behaviour of the iron. The importance of considering the influence of even a small volume fraction of secondary phase in the plastic regime has been demonstrated.

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7. Acknowledgements Diffraction experiments were carried out at the ISIS pulsed neutron facility at the Rutherford Appleton Laboratory, Oxon., U. K., under support from the Engineering and Physical Sciences Research Council. The authors would like to acknowledge DaimlerChrysler and Magnus Odén, Linköping University for supplying the and duplex samples respectively.

8. References 1. 2.

3. 4. 5.

6. 7. 8. 9. 10. 11. 12. 13.

14. 15. 16. 17. 18.

Clyne, T. W. and P. J. Withers, An Introduction to Metal Matrix Composites. Cambridge Solid State Science Series, ed. E. A. Davis and I. M. Ward. 1993, Cambridge: Cambridge University Press. 509. Noyan, I. C. and J. B. Cohen, Residual Stress - Measurement by Diffraction and Interpretation. Materials Research and Engineering, ed. B. Ilschner and N. J. Grant. 1987, New York: SpringerVerlag. 272. Hutchings, M. T. and C. G. Windsor, Industrial Applications of Neutron Scattering, in Neutron Scattering (Ch. 25), K. Skold and D. L. Price, Editors. 1986, Academic Press: Orlando, Fla. p. 405482. MacEwen, S. R., J. Faber, and A. P. L. Turner, The Use of Time-of-Flight Neutron Diffraction to Study Grain Interaction Stresses. Acta metall., 1983. 31(5): p. 657-676. Clausen, B., T. Lorentzen, M. A. M. Bourke, and M. R. Daymond, Lattice Strain Evolution during Uniaxial Tensile Loading of Stainless Steel. Mat. Sci. Eng., 1999. 259(1). p 17-24. Johansson, J., M. Oden, and X. H. Zeng, Evolution of the Residual Stress State in a Duplex Stainless Steel during Loading. Acta Mater, 1999. 47(9): p. 2669-2684. Moverare, J. J. and M. Oden, Deformation Behaviour of a Prestrained Duplex Strainless Steel. Acta Mater., 2001 (submitted). Hill, R., Continuum Micro-Mechanics of Elastoplastic Polycrystals. J. Mech. Phys. Sol., 1965. 13: p. 89-101. Hutchinson, J. W., Elastic-Plastic Behaviour of Polycrystalline Metals and Composites. Proc. Roy. Soc. Lond. A, 1970. 319: p. 247-272. Turner, P. A., N. Christodoulou, and C. N. Tome, Modelling of the Mechanical Response of Rolled Zircaloy-2. Int. J. Plasticity, 1995. 11(3): p. 251-265. Chin, G. Y. and W. L. Mammel, Computer Solutions of the Taylor Analysis for Axisymmetric Flow. Trans. TMS-AIME, 1967. 239: p. 1400-1405. Kocks, U. F., C N. Tomé, and H. -R. Wenk, Texture and Anisotropy. 1998, Cambridge: CUP. Pang, J. W. L., T. M. Holden, and T. E. Mason, The Development of Intergranular Strains in a HighStrength Steel. J. Strain Analysis, 1998. 33(5): p. 373-383. Kagawa, A., T. Okamoto, and H. Matsumoto, Young's modulus and thermal expansion of pure iron-cementite alloy castings. Acta mater., 1987. 35(4). p. 797-803. Acker, K. V. and P. V. Houtte. Macro- and Microstresses in the Cementite and Ferrite Phases of Cold Drawn Steel Wires Measured with XRD and Neutron Diffraction, in Proc. of 4th Eurpoean Conf. on Residual Stresses. 1996. Cluny en Bourgogne, France. Rietveld, H. M., A Profile Refinement Method for Nuclear and Magnetic Structures. J. Appl. Cryst., 1969. 2: p. 65-71. Daymond, M. R., M. A. M. Bourke, R. B. Von Dreele, B. Clausen, and T. Lorentzen, Use of Rietveld Refinement for Residual Stress Measurements and the Evaluation of Macroscale Plastic Strain from Diffraction Spectra. J. Appl. Phys., 1997. 82(4): p. 1554-1562. Daymond, M. R., C. N. Tomé, and M. A. M. Bourke, Measured and Predicted Intergranular Strains in Textured Austenitic Steel. Acta Mat., 2000. 48: p. 553-564.

RESIDUAL STRESSES AND ELASTIC CONSTANTS IN THERMAL DEPOSITS

THOMAS GNÄUPEL-HEROLD National Institute of Standards and Technology Center for Neutron Research, Gaithersburg, MD 20899, U. S. A. University of Maryland, College Park, MD 20742-2115 HENRY J. PRASK National Institute of Standards and Technology Center for Neutron Research, Gaithersburg, MD 20899, U. S. A. FRANK S. BIANCANIELLO National Institute of Standards and Technology, Metallurgy Division MD 20899, U. S. A.,

Abstract

In this study we investigate experimentally and theoretically the influence of various aspects of the pore structure on residual stress, coating adhesion to the substrate and elastic properties of the coating. For that purpose, feedstock powders of Inconel 625 were prepared with four different particle size distributions. The coatings were prepared by air-plasma spraying under nearly identical spray conditions on grit blasted steel substrates. Neutron diffraction was used to determine the average in-plane residual stresses in the coatings. The in-plane Young’s modulus was determined using a fourpoint bending apparatus. The porosity and the pore distribution were characterized using small-angle neutron scattering as well as precision density measurements. It was found that both the residual stresses and the elastic modulus of the coatings sprayed with coarse powder were considerably lower than the stresses in the coatings sprayed with the smaller particles. As the particle size decreases, a rising oxide content in the coating as well as a change in the pore distribution elevate the elastic modulus and, as a consequence of that, the residual stresses. The most pronounced effect on the pore distribution is a lower fraction of connected porosity which effectively decreases the pore aspect ratio. With no ductility left due to the embrittlement effect of the oxide particles, these coatings exhibit a low strain tolerance and the residual stresses are close to their maximum level. 507 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 507–514. © 2002 All Rights Reserved. Printed in the Netherlands.

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1. Introduction Metallic coatings are used in a variety of applications ranging from protecting surfaces from corrosion, providing wear resistance, bond-coats for thermal barrier coatings or restorative layers for machine parts. Unlike with ceramic coatings, for metallic coatings there are comparatively little data available regarding the relationship between microstructure, residual stress and mechanical properties. Differences with respect to these properties between metallic and ceramic coatings can be expected due to the ductility of metals. For example, there is evidence that the elastic anisotropy between Young’s modulus in the plane of the surface, and normal to the surface, is different for metallic coatings where is found. For ceramic coatings it is usually found that The elastic anisotropy is rooted in the preferred orientation of elongated voids among which two main populations can be distinguished: horizontal (x, y), pennyshaped voids and vertically (z) elongated cracks. Depending on their concentration and their aspect ratios, pores can substantially reduce the elastic moduli and they can create large elastic anisotropy. It is easy to understand that, for example, if more flat voids are oriented horizontally (i. e. the two long axes are horizontal) than there are vertically oriented cracks then because the vertical cross sectional area is smaller. Spherical pores are also present but they simply decrease the effective elastic modulus without adding to anisotropy. The measurement of the elastic moduli is commonly done by four-point bending, indentation or sometimes by ultrasonic resonance. More recently, diffraction has also been used. Each of these methods comes with its own restrictions but all of them have in common that they provide results only for Young’s modulus. So far, there are very few data available about Poisson’s ratios or the shear modulus. With respect to the elastic modulus two different values have to distinguished: the “mechanical” value that relates macroscopic strain to macroscopic stress, and the diffraction modulus which relates lattice strain to macroscopic stress. The latter can be quite different from the first because in a diffraction measurement only lattice strain is recorded while a macroscopic load is applied. However, the straining of the lattice of the grains in a coating is still anisotropic and it depends on the reflection (hkl) used in that measurement. There are two main conditions that determine the level of residual stress in a coating: the coating elastic constants and thermal strains that originate from temperature differences and different coefficients of thermal expansion (CTE). Variations in certain spray parameters like particle velocity in the plasma jet can cause critical changes in the bonding of particles and in the pore structure of the coating. The coating elastic constants are very sensitive to the microstructure and can be up to a factor ten smaller than bulk (zero porosity material) values. Thermal strain develops first due to the rapid quenching from the splat solidification temperature to the substrate temperature. The second thermal strain contribution develops when the substrate-coating sandwich cools down to ambient temperature with different contraction strains due to different CTEs of substrate and coating. The most commonly used methods for stress determination in coatings are diffraction, deflection methods and material removal. For several reasons we choose neutron diffraction for the strain measurements. First, we deemed X-ray diffraction

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unsuitable because of the large surface roughness of the coatings and because of the possibility of stress gradients that would lead to misinterpretation by a surface technique like X-rays. Also, with neutrons it is easy to measure at different spots on the coating in order to check the uniformity of the bonding to the substrate. Second, neutron diffraction is especially suitable for strain measurements under applied load. In particular, the high penetration of neutrons becomes very useful for measuring the inplane strain in a four-point bending apparatus. For this application it is a valuable feature of diffraction methods that only elastic strain can be measured. 2.

Experimental Procedure

The coatings were prepared by atmospheric plasma spraying at the thermal spray facility of the NIST Metallurgy Division. The substrates were low carbon steel slabs of dimensions 2. 9(t) × 29. 5(w) × 120(l) mm with corund grit blasted surface. The particle size distributions of the Inconel 625 feedstock powders were set by sieving. The details are listed in Table 1.

The composition if the powder is listed in Table 2.

The crystal structure of the material is face centered cubic with a lattice parameter of approximately 3. 6 Å. The neutron diffraction measurements were done using the residual stress diffractometer (BT8) at the NIST Center for Neutron Research, Gaithersburg, MD, U. S. A. The technique is explained in detail by Allen et al and for coatings by Kesler et al so that we will restrict ourselves to the details of a stress measurement on medium thickness coatings. The coating thickness was found to be somewhat non-uniform because of the large surface roughness of the outer surface. The thickness measurements were done using a caliper on several points on the coating. The average values were from 0.4 mm up to 0.7 mm with an uncertainty of 0.1 mm whereas neutron beams usually have sizes of 0.5 mm or larger. Fig.1 shows how the neutron measurement is done.

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Neutron diffraction requires an unstressed d-spacing as a reference value to calculate strain and stress. An equally viable alternative is the use of mechanical boundary conditions which is a method of choice for coatings. Here, the condition is fulfilled through the thickness of coating and the unstressed d-spacing can be eliminated from the equations which is basically equivalent to the technique in Xray diffraction stress analysis (Hauk (1997)). The wavelength was chosen such that the (311) reflection could be measured at Once the strains are obtained, the specific elastic constants for each coating are needed to calculate the stress. These measurements were done both with freestanding coatings and with substrate-coating composites using a four-point bending apparatus. We also conducted density measurements by immersing small pieces of coating into ethanol under vacuum to achieve the best possible penetration. Subsequently the weight of the immersed and the dry coating was measured. The density measured that way includes the enclosed (not connected to the surface) pores.

3. Results and Discussion The effect of the feedstock powder particle size on the elastic moduli, residual stresses and porosity were investigated and the results are presented in the following. The results in Tab. 3 show that the particle size has a distinct effect on the density of the coatings. Also, preliminary results from small angle neutron scattering show that the surface area ratio of surface connected porosity to enclosed porosity is about four times bigger for the coarse coating than for the medium coating (Barker (2001), making this coarse coating more like a network than a porous solid. However, X-ray diffraction measurements also indicate that the lower densities of the fine, medium and mixed coating are mostly due to a high content of nickel oxide and chromium oxide that are

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formed in-flight especially on small particles during spraying. Clearly, formation is promoted by the higher surface to volume ratio of the smaller particles.

Four-point bending measurements on free standing coatings were performed with the outer surface in compression and in tension. The results in Fig. 2 indicate that the elastic modulus is consistently higher if the outer surface is in a tensile mode. This indicates that the pore structure through the thickness of the coatings is inhomogeneous. It is known that coatings are stiffer in compression than in tension because a compressive stress closes voids with high aspect ratios while tensile stresses open even more pores. It is likely that the inner layers of the coatings have an increased porosity compared to the outer layers, thus making it easier to strain the inner layers in compression with the outer layers in tension.

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The diffraction moduli are up to a factor of two bigger than the mechanical values. The reason for this behavior is the same for all coatings: the lattice of the grains is strained only to a fraction of the strain that deforms the coating as a whole. The “rest” of the strain goes into the deformation of the pores and cracks. That means that only some microscopic regions in the bulk of the coatings deform purely in tension or compression. Other region will be subjected to bending or twisting, both of which produce less strain than pure compression or tension. The combined effect of both reduces the average lattice strain as measured in the diffraction experiment so that approximately only 50 % of an applied strain are measured as lattice strain. Using these measured diffraction moduli assures that we obtain the correct stresses from the lattice strain measurements. The very low modulus of the coarse coating can be explained as follows. Preliminary results from small angle neutron scattering experiments show that the surface area of the medium coating is slightly higher than that of the coarse coating. The connected porosity, on the other hand, is approximately five times higher for the coarse coating. A connection between elongated voids of the same orientation effectively increases the aspect ratios of the average void which has a significant impact on the elastic constants as shown in Figure 3. The self-consistent model used to calculate the values in Figure 3 approximates the coating as a two-phase composite consisting of a crystalline phase and ellipsoidal voids with elastic constants close to zero (see Willis (1981), (Matejicek et al (2001)).

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The most significant impact of the pore aspect ratio on the elastic moduli is caused if the aspect ratios are in the range from 0.1 to 0.01 for horizontal voids and from 10 to 100 for the vertical cracks, respectively. The 50% drop in elastic constants can be well explained by assuming that the average pore aspect ratio for the coarse coating is five times larger than that of the medium coating due to increased connection between pores. The residual stresses in the coatings are presented in Fig. 4. These values were obtained from the measured strains using the diffraction modulus shown in Figure 2. Poisson’s ration was measured on the coarse coating to be 0.3. This value was also used for the other coatings.

There are only small differences among samples that were sprayed with the same particle size. The small variations in coatings thickness do not cause large changes in stress for the same particle size. However, the results for different particle size distributions are a clear indication that the actual stresses are mostly controlled by the elastic modulus because both the stress and the moduli follow the same behavior. The only source of the tensile coating strains are quenching strains and thermal strains which are approximately the same for all coatings. Other processes as sliding between splats due to imperfect bonding or plastic deformation can only decrease these strains. The contribution from plastic deformation is very small because especially the medium and the fine coating contain a substantial fraction of oxides. From X-ray measurements we estimate the oxide content to approximately 20 % for the medium and the fine coating. Such a high fraction of non-shearable particles effectively suppresses any ductility. The consequence is that plastic deformation can no longer contribute to stress relaxation. As a result, both stresses and elastic moduli are the highest for these coatings. Both effects can be detrimental for the adherence of the coating because there is very little strain tolerance left.

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4. Acknowledgement

The authors would like to thank Dr Jiri Dubsky from the Institute for Plasma Physics in Prague for allowing us to use his four-point bending apparatus, and Drs Steven Mates, Steven Ridder and Rodney Jiggets from the NIST Metallurgy Division for their help in spraying the coatings. 5. References 1. 2.

3.

4.

5. 6. 7.

Barker, J. (2001), NIST Center for Neutron Research, unpublished results Allen, A. J., Mulchings, M. T., Windsor, C. G. and C. Andreani (1985), Advances in Physics, 34 (4), 445 Gnäupel-Herold, T, Matejicek, J, Prask, H J (2000), Mechanical Properties of Plasma Sprayed Coatings - Measured by Diffraction, Proc. of the 9th International Metallurgical Conference Metal 2000, Ostrava, Czech Republic, paper no. 508 Hauk, V. (1997), Structural and Residual Stress Analysis by Nondestructive Methods, Elsevier Sciece Pub., Amsterdam Kesler, O., Matejicek, J., Sampath, S., Suresh, S., Gnaeupel-Herold, T., Brand, P. C., Prask, H. J. (1998): Measurement of Residual Stress in Plasma-Sprayed Metallic, Ceramic and Composite Coatings; Mater. Sci. Eng. A Vol. 257, No. 2, p. 215-224 Matejicek, J, J, Gnäupel-Herold, T (2001) Neutron scattering in studies of complex anisotropic microstructures, Int. Conf. on Materials Structure & Micromechanics of Fracture, MSMF-3, Brno, Czech Republic, in press Willis, J. R. (1981), Advances in Applied Mechanics 21, 1-78

NEUTRON DIFFRACTION ASSISTED RESIDUAL STRESS ANALYSIS IN WELDED STRUCTURES C. OHMS and A. G. YOUTSOS Joint Research Centre of the European Commission Westerduinweg 3 1755 LE Petten, NL Abstract Welding residual stresses in structural components can significantly compromise their performance and lifetime. Prediction of welding stresses based on numerical modeling has not yet proven to be reliable, while measurement of such stresses based on NDT remains a challenging task. It is shown in this paper that neutron diffraction is a reliable non-destructive method for residual stress analysis in structural weldments. The Large Component Neutron Diffraction Facility (LCNDF) at the High Flux Reactor (HFR), Petten has facilitated residual stress measurements in various weldments, including large steel piping welds. A key issue in applying neutron diffraction to welds is the reliable estimation of the stress-free lattice distance in the heat affected zone and weld pool and in all directions of interest. Results of numerous investigations at HFR show that this is achievable by testing small coupons, cut from a companion weld specimen, which are nearly free of macro-stresses and consequently it is reasonable to be used as reference specimens. In fact, the feasibility of this approach has been demonstrated in monolithic and bimetallic welds. In this paper residual strain/stress data in five welded specimens, based on neutron diffraction, are presented. The results presented in this paper consistently show that, by ignoring the spatial and directional variation of the reference lattice distance, which is exhibited throughout the weld pool and the heat affected zones, erroneous strain data can be derived leading to non self-equilibrating internal stress estimates. Keywords: Residual stress, welding, neutron diffraction, large components, coupons

1. Introduction Neutron diffraction is a powerful technique for non-destructive analysis of residual stresses deep within crystalline materials at reasonable spatial resolution. The HFR provides two facilities dedicated to this test method. One of these is the Large Component Neutron Diffraction Facility (LCNDF) capable of handling specimen weights up to 1000 kg. In testing thick structural weldments and their respective heat affected zones (HAZs) by neutron diffraction, one has to take into account variations of the stress free reference parameter throughout the weld and the HAZ. These are due to plastic 515 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 515–526. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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anisotropy, changes in chemical composition and texture formation caused by the welding process. In neutron diffraction this problem can be handled by using coupons carefully cut to the appropriate size as free of macroscopic stress representatives of the measurement locations that are going to be investigated. For the economy of the paper, due to its nature and also for the sake of the reader, we present together in Section 2 the experimental campaigns based on neutron diffraction and the respective results. Two brief Sections follow this on discussion of the results and conclusions respectively. In this paper residual strain/stress data, based on neutron diffraction, in five welded specimens are presented. The first specimen is a section cut from an austenitic steel piping butt weld of 68-mm thickness. A large campaign of reference measurements was conducted, which was based on a large number of coupons cut from a thin companion specimen. This has shown the extent of potential errors, which are introduced in the strain analysis, if reference variations are neglected. The second specimen is a 25-mm thick post-weld heat-treated austenitic/ferritic steel fusion welded piping section. In this case "comb" type reference specimens were used, and the derived stress data were found to be in good agreement with computational results based on numerical modeling techniques. The third sample is a 65-mm thick section girth weld joining two ex-service AISI type 316H austenitic stainless steel forgings. Due to the high wall thickness, residual strains were measured adequately only in the radial direction in this case. This is confirmed by comparison of measured strains with computational data based on FEM techniques. Measurements were made before and after PWHT in order to evaluate the efficiency of the applied stress relief process. The fourth specimen is a ferritic steel alloy fusion welded plate of 12. 5-mm thickness. Residual stresses in this specimen have been investigated at various neutron diffraction facilities worldwide to establish feasibility of neutron diffraction. For the HFR investigation reported in this paper, reference measurements based on “comb” type reference specimens, which have revealed negligible reference variations. Finally, the fifth sample is an aluminum alloy friction welded plate of 6-mm thickness, representative of an aircraft wing welded section.

2. Experimental Campaigns and Results Based on Neutron Diffraction 2. 1. AUSTENITIC STEEL BUTT WELD SECTION Measurements have been performed on a thick nuclear piping austenitic steel butt weld. Fig. 1 shows the location and size of the coupons, which were cut from a companion specimen for the investigation of the macro-stress free lattice parameters [1]. All coupons are 5-mm cubes with their centres at the strain measurement locations. Measurements were made at all locations of the weld transverse direction - line G, which coincides with the piping axial direction, and all locations of the weld axial direction - column 6, which coincides with the piping radial direction. For each of the investigated coupons the lattice parameter was measured using the 113- and 002-refleciton planes in the normal, axial and weld directions. The sampling volume chosen for all tests was 3x3x3 and at a fixed wavelength of 1. 528 Å using the Ge-113 neutrons of the monochromator at the E3 neutron beam at former CRNL, CAN. This wavelength gives

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rise to a value near for the 113-reflection, which renders a cubical sampling volume, but only for the 002-reflection.

Figs. 2 – 3 and 4-5 show the measured diffraction angles in all the coupons examined for the weld axial and normal directions respectively. It can be readily observed that in all cases reported in Figs. 2-5 the values within the weld are much higher than in the parent material. In fact for the 113-reflections varies by as much as and for 002 by about resulting into compressive pseudo strain levels in the range of 2200 and 2800 respectively. Strain measurements were made in the weld specimen and in its normal and axial directions using both 002 and 113.

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Fig. 6 shows the resulting residual strains in the axial direction along the transverse axis of the weld based on the 002-reflection and taking correctly into account the

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macro-stress free lattice parameter variation as shown in Fig. 3. Fig. 6 also shows the pseudo strain data, which one would obtain if this variation were neglected. The plotted results in this Figure suggest that the parent material is undergoing large compressive strains while the strains in the weld are tensile and relatively low. Fig. 7 shows strains measured in the weld axial direction and along its axis of symmetry. These results show, as those plotted in Fig. 6, that the weld material is exhibiting low tensile strains along the weld axis. Position M6 appears as an exception due to its position with respect to the geometry of the weld (see Fig. 1).

2.2. BIMETALLIC WELD SPECIMEN The specimen investigated is a tube of 393 mm length, 168 mm outer diameter and 25 mm wall thickness. The weld has been applied circumferentially at mid-length. On the outer and inner surfaces the transition from the austenitic to the ferritic material is clearly visible. Measurements were performed at various locations within the weld, the buttering layer and the HAZs in three ortho-normal directions, i. e., tube hoop, axial and radial directions. The measurement locations are shown in Fig. 8. In order to facilitate

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these measurements windows had to be cut into the component for providing access for the neutron guides. From the thus removed material a slice was then cut in order to provide for the reference coupons; these were wire eroded in form of 6 x 6 columns. Thus two reference specimens were made, one representing the locations within the buttering layer and the ferritic steel HAZ and the other representing the locations within the weld and the austenitic steel HAZ [2]. Based on the assumption of macro-stress relief in the columns through cutting, these were used for reference measurements at each location in each direction. The most appropriate test procedure was found to be operating at a fixed scattering angle 29, while varying the neutron wavelength, using an adjustable double-monochromator, for the various material phases.

First tests revealed a strong (200)-texture within the weld and buttering layer, while in the austenitic and ferritic HAZs the (111)- and (110)-reflections could be used, respectively. In order to illustrate the necessity of measuring location dependent references, Fig. 9 shows apparent strains derived based on a common reference

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value for each of the four measurement regions (ferritic HAZ, buttering, weld and austenitic HAZ). It can be seen that apparent strains of more than 1000 can be introduced by ignoring the reference parameter variation. This effect was found to be most significant within the weld. Nevertheless it is also present in the other measurement regions. The measurements were performed using sampling volume of 4x4x5 to 4x4x10 scattering angle of 76.150° and the following crystallographic reflections: ferrite (110), austenite (200) in weld and buttering, and austenite (111) in HAZ austenite. From the thus derived strains, residual stresses were calculated. The elastic constants related to the various material reflections were taken from the literature [3]. Figures 10-12 show the residual stresses that were found in the hoop, axial and radial directions, respectively. These show compression in the ferritic steel, and tension in the austenitic phase in the hoop direction, while in the axial direction high stresses are present closer to the outer and inner specimen surfaces and low stresses near the middle of the piping wall. Radial stresses remain within ±100 MPa.

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2. 3. AUSTENITIC STEEL GIRTH WELD The spatial distribution of residual stress in a thick section girth weld (432 mm outside diameter, 65 mm thick) joining two ex-service AISI type 316H austenitic stainless steel forgings was characterized by both neutron diffraction testing and numerical modeling [4]. A neutron wavelength of 2. 8 Å was employed rendering a scattering angle of about 84. 5° with the austenitic (111) reflection plane. A large sampling volume was selected for measurements in the specimen radial, longitudinal and hoops directions. Seven locations through the piping wall in the HAZ were measured (nearly 20 mm from the weld centerline). Further measurements were made 6 and 20 mm below the outer specimen surface at various distances from the weld centerline. All hoop, longitudinal and radial direction strain measurements at mid-thickness of the specimen wall necessitated a total neutron path length within the steel of about 90 - 95 mm. This proved to be beyond the capabilities of the diffractometer because of the resulting severe neutron beam intensity attenuation. The above experimental campaigns were repeated following completion of the PWHT of 2. 5 hours at 750°C. The strains presented in Fig. 13 are based on a unique reference value derived from measurements in the parent material far from the weld. This Figure shows the apparent radial strains before and after PWHT measured through thickness in the HAZ 20 mm from the weld centre line. The strain relief is positive for the test locations closer to the outer surface whereas is becomes negative at the inner surface. The difference in measured radial strains, that is the strain relief, is in good agreement with equivalent results based on FE simulation performed by British Energy, as Fig. 14 clearly shows. Additional tests indicate that measurements, based on the selected sampling volume and wavelength, would have been feasible for up to 25-28 mm below the specimen surface locations, while beyond this depth the deteriorating quality and reliability of data seems to be prohibitive for neutron diffraction stress testing. This means that, and in agreement with previous experience, reliable neutron diffraction data could be obtained for a piping wall thickness of up to 50 - 55 mm.

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2. 4. FERRITIC STEEL WELD Residual strains and stresses were measured at mid-length of a ferritic steel plate of 200x150x12.5 mm. A 12-pass TIG-weld was applied to the plate at mid-width. Various neutron diffraction laboratories worldwide tested the specimen and a comprehensive report on the obtained results is due to appear shortly. At JRC tests were performed using a gauge cross section and a neutron wavelength of 2. 57 Å. Reference coupons in this case did not reveal significant variations in the stress free lattice parameter thus an average was used as reference value for all tests. Figs. 15-16 show the strains and stresses measured at the HFR-Petten at the plate mid-thickness in the weld longitudinal (X), weld transverse (Y) and plate normal (Z) directions.

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2. 5. ALUMINUM ALLOY FRICTION STIR WELD

Two Al-7010 plates of 6 mm thickness and 120 mm width were joined through friction stir welding. ILL, Grenoble, and HFR-Petten performed strain measurements based on neutron diffraction. The aluminium (111) reflection was chosen and an incident beam wavelength of about 2. 99 and 2. 57 Å, respectively, giving rise to Bragg angles of about 79. 4° and 67°. Both facilities employed gauge volumes of cross section. An eventual variation in the reference parameter was not taken into account in this case. Whereas the agreement between ILL and JRC results in the weld transverse direction is remarkably good (see Fig. 17), in the plate normal direction only a reasonable qualitative agreement could be established (Fig. 18) due to the unfavourable texture of the material. The alignment discrepancy between ILL and JRC results is 0. 5 to 0. 75 mm. Fig. 19 shows the weld transverse strains obtained by JRC at the midlength of the specimen compared to test results from other measurement locations. The

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good agreement in the data suggests uniformity of the stress distribution over a large range of the specimen.

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3. Discussion of Results The austenitic steel butt weld section analysis reveals significant spatial and directional variations in the reference lattice spacing. The bimetallic weld exhibits, as expected, an asymmetric stress pattern in the hoop direction, whereby the ferritic part is predominantly under compression, while the austenitic HAZ is mainly under tension. The transition from the compression-dominated to the tension-dominated zone takes place at the ferritic steel buttering interface, which exhibits a very significant stress gradient. In radial direction only minor stresses were found varying between +100 and -100 MPa. Axial stresses are tensile at the outer specimen surface and compressive at the inside. In both cases the stresses tend to be higher in the ferritic than in the austenitic HAZ. Slightly lower stresses were found in the weld itself. It seems that the stresses found in the axial direction do not balance everywhere. Both the bimetallic weld stress tensor data and the austenitic steel girth weld radial strain relief data are in good agreement with similar results based on numerical modelling. In the ferritic weld case the test data, for strain and stress alike, clearly show symmetry where expected. Finally, strain/stress data for the ferritic steel weld and strain data for the aluminium alloy weld are in very good agreement with similar results based on neutron diffraction testing carried out at other than the HFR facilities.

4. Conclusions Neutron diffraction proves to be a very suitable tool for non-destructive determination of residual stress in structural welds. It was shown that the HFR LCNDF could accommodate efficient strain/stress testing of steel welded structures with up to 50-mm wall thickness. Measurement of the reference scattering angle at each location and in all measurement directions proved to be necessary in austenitic and austenitic/ferritic welds for accurate strain estimates. The statistical error of uncertainty associated with all neutron strain/stress analyses is, generally, quite low, i. e., 75-150 and 5-20 MPa.

5. Acknowledgements This research was carried out within the European Commissions’ Research and Development Programme. The authors wish to thank Messrs. Th. Timke and P. van den Idsert for their valuable contributions to the above work. The contributions of Messrs. Th. Holden (ex-CRNL) and Th. Pirling (ILL) are greatly acknowledged.

6.

References

1. 2. 3. 4.

C. Ohms and A. G. Youtsos, J. of Textures and Microstructures, 33, (1999) 243-262. C. Ohms, A. G. Youtsos, Materials Science Forum, 347-349 (2000) 658-663. B. Eigenmann, E. Macherauch, Mat. -wiss. U. Werkstofftech., 27 (1996) 426-437. P. J. Bouchard, S. K. Bate, D. George, R. H. Leggatt and A. G. Youtsos, Proceedings of the Conference on residual stress, Vol. 2, IOM Communications Ltd., London, 2000, 972-979.

Int.

SYNCHROTRON RADIATION IN-SITU ANALYSES OF AA 6061 + DURING TENSILE DEFORMATION AT AMBIENT AND ELEVATED TEMPERATURE A. PYZALLA TU Berlin, Institute for Materials Science and Technology Sekr. BH 18, Ernst-Reuter-Platz 1, D-10587 Berlin, Germany B. REETZ TU Berlin, Institute for Materials Science and Technology Sekr. BH 18, Ernst-Reuter-Platz 1, D-10587 Berlin, Germany A. JACQUES L.P.M. (UMR CNRS 7556), Ecole des Mines de Nancy Parc de Saurupt, F-54042, Nancy, France J.-P, FEIEREISEN L.P.M. (UMR CNRS 7556), Ecole des Mines de Nancy Pare de Saurupt, F-54042, Nancy, France O. FERRY L.P.M. (UMR CNRS 7556), Ecole des Mines de Nancy Parc de Saurupt, F-54042, Nancy, France T. BUSLAPS European Synchrotron Radiation Facility (ESRF) BP 220, F-36043 Grenoble, France Abstract High energy synchrotron radiation enables the in-situ determination of the elastic strain developing during deformation of metals and especially of metal matrix composites. By using a white beam the strain response can be characterised simultaneous with texture development. The high photon flux of the synchrotron beam allows short data acquisition times and thus an in-situ combined stress and texture determination is possible at high temperature. Here results of investigations of the tensile load stress – elastic strain response of aluminium metal matrix composites are presented. Keywords: synchrotron radiation, high energy, metal matrix composite, aluminium oxide, tensile deformation, elevated temperature.

1. Introduction Metal matrix composites (MMCs) combine the beneficial properties of their constituents. The embedding of suitable ceramic hard phases in an aluminium base alloy produces light weight wear resistant materials. These materials are used in 527 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 527–534. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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aerospace and automotive applications, e. g. in automobile motors. In the automobile motors the aluminium metal matrix composites are subjected to elevated temperatures. Thus, the combination of the aluminium alloy matrix with alumina particles is beneficial with respect to the weight as well as with respect to the overall thermal expansion of the component. But, the differences in the physical properties and the mechanical behaviour of the constituents result in a non-uniform load uptake of the thermal and the mechanical loading of the component and also in the formation of residual stresses after cooling. In order to determine the elastic strain and the load sharing between the aluminium matrix and the particles at elevated temperature, in-situ tensile tests of AA6061 + - samples in a furnace with a loading device were performed using white synchrotron radiation [1, 2]. At ambient temperature in-situ stress analysis in composites can well be done using neutron diffraction [e. g. 3, 4]. Typical data acquisition times are in the range of several minutes. At elevated temperature during this time interval severe stress relaxation or creep effects may occur. Thus, at high temperatures the use of synchrotron radiation with its high flux and, thus, a short acquisition time is preferable. The high photon flux at the sample position enables the recording of a spectrum with sufficient counting statistics within a few ten to a few hundred seconds. By using an energy dispersive set-up simultaneous to the information about the elastic strain in both phases of the composites also information about the texture development in the composites is available.

2. Experimental Details

Material The AA 6061 + – composites were manufactured by hot extrusion. From the extrusion products samples with a diameter of 4, 4 mm and an overall length of 45 mm were manufactured. The active length amounts to 25 mm. The longitudinal axis of the samples corresponds to the longitudinal direction of the extrusion profiles. The microstructure of AA6161 + in the as - manufactured condition is characterised by a homogeneous distribution of the alumina particles in the aluminium matrix. (Fig. 1).

The ceramic – particles have a rectangular shape and a size of The grain size of the AA6061 matrix is 40 At the interface between the

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AA6061 – matrix and the particles a fringe containing an aluminiummagnesium - spinell - phase is present (Fig. 2).

Experiment Set-Up The experiments were performed at the High Energy Beamline ID 15A of the European Synchrotron radiation facility (ESRF) in Grenoble, France. For the experiments an energy dispersive arrangement using the white beam was chosen. This set-up was described in detail in [1, 5]. For the experiments a diffraction angle of was chosen as a compromise between maximising the resolution respectively the photon flux available. In order to define the gauge volume a slit with variable horizontal and vertical gap as well as 2 slits with 60 horizontal gap and 10 mm vertical gap were inserted in front of the sample. In the diffracted beam two further slits with gaps of the same size 60 x 10 mm were placed. The first of these slits was directly attached to the surface of the furnace. The second slit was mounted closed to the detector. The size and the shape of the gauge volume was determined experimentally using a thin aluminium foil. It was found to be 70 wide and 1200 long, approximately. The gauge volume was centred in the sample using maximum integrated intensity values of translation scans. In order to increase the number of scattering grains the sample was continuously oscillated by ± 1° in during the measurements. An energy– dispersive Ge-detector was calibrated in an energy range of 25 keV to 250 keV using the characteristic lines of a source.

Furnace with Integrated Tensile Test Device The furnace allowing in-situ tensile tests (Fig.. 3) with a load up to 5 KN at elevated temperatures up to 1200° was developed at L. P. M. (Ecole des Mines Nancy). The sample is heated-up by two pairs of graphite resistors. The combined tensile test – heating device was set-up on the sample stage. The sample was installed horizontally, so that the longitudinal axis of the sample pointed in the direction of the scattering vector, that is nearly perpendicular to the incoming beam. The samples were heated-up in vacuum. Their temperature during the tensile deformation was measured using three thermocouples. Using the local temperature recorded at the different sample positions by the thermocouples the temperature of

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the samples could be controlled. For experiments scheduled at a temperature of 200°C the temperature of the sample could be kept constant within a temperature range of approximately ± 20K

Data Evaluation Diffraction spectra were taken from the unloaded samples and after defined loading steps up to a maximum load stress of 150 MPa. For all spectra an acquisition time of 300s was used. From the energy dispersive spectra the line position, the integrated intensity and the full width at half maximum of different reflections were obtained by fitting the reflection profiles using a Gaussian function. The energy value representing the line position corresponds to the lattice spacing including the lattice strain. Thus can be calculated according to Bragg’s law. Then the stresses are obtained by Hooke’s law taking account of the phase and reflection specific XECs. For the determination of the strain-free lattice distance of the particles, powder was chemically extracted from the samples. A spectrum of the powder was recorded at ambient and at elevated temperature.

3. Results A typical high energy synchrotron radiation spectrum of AA6061 + 15vol. -% is shown in Fig. 4.

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Obviously in the energy range between 50 keV and 150 keV a number of reflections of the aluminium matrix is accessible. Due to the low – content of the metal matrix composite and due to the hexagonal structure of the there is a multitude of reflections with significantly lower intensity. The ratio between the intensity of the reflections and the background of the spectrum obtained justifies the short acquisition time of 300s chosen for both the aluminium matrix and the particles.

Tensile deformation at ambient temperature The elastic strain – load stress - curve obtained for the particles (Fig. 5) shows initially compressive residual strains which result from the cooling of the AA6061 – alloy after the extrusion processes and the heat treatment due to the differences in the thermal expansion of the particles and the aluminium matrix.

During the tensile loading the elastic strain in the particles changes into tensile strain values which increases steadily with increasing tensile external load. This indicates that part of the load stress during the deformation is taken up by the particles. The spectra reveal further, that the escalation of the external load does not only result in increasing strain of the particles but also in changes of the reflection intensities. As an example the development of the intensity of the 0 2 10 reflection of the – particles due to increasing loading is shown in Fig. 6. This particle behaviour is quite different from our earlier observations on an Al-MMC with globular silicon particles [6]. The values presented are peak maxima normalised by the total photon count rate and based on the peak intensity obtained in the initial state, before loading. The change of the reflection intensity monitored reveals that the particle orientation changes slowly at external stress below 150 MPa approximately and increase strongly if the sample is subjected to higher external tensile stresses. The change of

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the particle orientation is accompanied by an increase in the full width at half maximum (FWHM) of the reflections of the – particles (Fig. 7).

This indicates that strong micro strains are present in the particles due to inhomogeneities in their shape and orientation as well as their individual strength. Microscopical investigations of the composites even reveal that some of the particles show damage effects (Fig. 8).

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The aluminium matrix contains tensile residual strains before loading, which balance the compressive residual strains in the particles. With increasing tensile external load also the tensile strains in the aluminium matrix increase. This increase is linear up to an external strain of MPa which corresponds to the start of the formation of strong micro stresses in the – particles. The matrix reflections also show that the aluminium matrix tends to enhance its <100> /<111> double fibre texture, especially the <111> - fibre component. This texture formation during the in-situ tensile deformation, however, is weak, which is due to the <100>/<111> - fibre texture being present even before loading due to the manufacturing of the profiles by hot extrusion. Tensile deformation at 200°C The lattice distances of powder and the particles in the composite reveal that at 200°C, before loading, due to stress relaxation effects during the heating of the sample the particles and thus also the aluminium matrix are strain-free within the limits of experimental accuracy. The load stress – elastic strain curve of the – particles obtained during tensile loading at 200°C is shown in Fig. 9.

A comparison of this curve with the curve obtained at room temperature (fig. 5) reveals that at 200°C the strains of the – particles are higher at equivalent load stresses than at room temperature. Further on the elastic strain in the aluminium metal matrix in case of the deformation at 200°C is higher than it is at equivalent load stresses in case of deformation at room temperature. This is due to the decrease of the modulus of elasticity at increasing temperature. As was observed during room temperature deformation also during deformation at 200°C the <111> - fibre component of the aluminium metal matrix increases in strength.

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4. Conclusions and Outlook Tensile deformation of AA6061 + 15 vol. -% was carried out at ambient and at elevated temperature using white high energy synchrotron radiation. The results of the experiments show the development of the lattice strain in the aluminium matrix and the particles with increasing tensile load stress. At high external stresses the – particles tend to assume a preferred orientation and this orientation change is accompanied with the formation of micro stresses in the particles. Further work will concentrate on the investigation on the load transfer between the phases and its relation to particle damage when the sample is subjected to high external strains and stresses.

5. Acknowledgements The authors gratefully acknowledge the allocation of beamtime and the use of the beamline ID 15A at the ESRF Grenoble, France. A. Pyzalla and B. Reetz would like to thank the Deutsche Forschungsgemeinschaft for the grant Az. Py 9/1-1.

6. References 1. 2. 3. 4.

5. 6.

Reimers, W., Pyzalla, A., Broda, M., Brusch, G., Dantz, D., Liss, K.. -D., Schmackers, T., Tschentscher, T.: J. Mat. Sci. Letters 19 (1999) 581 – 583. Pyzalla, A., Reimers, W.: Mat. Sci. Forum Vols. 347 – 349 (2000) 34 – 39. Seol, K., Krawitz, A. D.. Mat. Sci. Eng. A127 (1990), 1-5. Korsunsky, A., Daymond, M. R., Wells, K. E.: Mat. Sci. Forum Vols. 347– 349 (2000) 492-497 Pyzalla, A . : J. Nondestructive Evaluation 19 (2000) 21-31. Pyzalla, A.; Jacques, A.; Feiereisen, J. P.; Buslaps, T.; D’Almeida, T.; Liss, K. In - situ Analyses of the Microstrains during Tensile Deformation of an AlSi – MMC at Room Temperature and Elevated Temperature, J. Neutron Diffraction, in print.

7. Hybrid Methods

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REFLECTIONS ON THE IMPORTANCE OF EXPERIMENTAL RESULTS TO ALL MECHANICISTS, ESPECIALLY THEORETICIANS ROBERT M. JONES Emeritus Professor of Engineering Science and Mechanics Virginia Tech Blacksburg, Virginia 24061-0219 Abstract All too often, mechanicists divide themselves into two camps, theoreticians and experimentalists. This unfortunate state of affairs denies us the essential two-sided, realistic view of important physical problems. We must realize that theory cannot exist without suitable experiments, and that experiments cannot be properly designed or used in engineering practice without some form of theory. Thoughtful, penetrating interaction between theoretician and experimentalist is crucial. The truly effective experimentalist must have a strong theoretical background. The truly effective theoretician must have at least a rudimentary, if not a strong, experimental background. Of course, the ideal mechanicist would be neither a theoretician nor an experimentalist, but both simultaneously! The important factor is cross-talk between the two points of view with the objective of achieving a unified view of a phenomenon and its model that can be used to treat behavior beyond that which was observed in experiments. This paper is about the elements that go into making up the unified viewpoint. Those elements include appreciation of the actual characteristics of the phenomenon studied as determined by experimental techniques and how those charcteristics help develop and refine every theory. The point is that experiment is the driving force behind theory.

1. Introduction We will consider examples of agreement between theoretical predictions and experimental measurements of behavior of two complicated mechanical systems. They are the illustrations of the usual comparison between theory and experiment. But, what more do we want? Can we perceive a stronger useful purpose for experimental measurements than verification of theory? Of course we can! To perform any theoretical prediction well, we must have a lot of information about the behavior of the widget we are investigating. Without that information, we are totally in the dark about how to model the widget. Or else we’re kidding ourselves that we can construct a useful model without extensive fundamental (experimental) information. However, some people attempt to do just that! They seem to drastically oversimplify the problem so they can use a convenient overly simplistic mathematical model. This paper is addressed to those of us who are more rational and realize, if we only thought about it, that theoreticians need experimentalists much more that we might admit. This paper has four major parts: (1) examples of good agreement and interaction between theory and experiment, (2) how experiments are used more than ‘just’ to verify theory, (3) how experiments are the very basis of the presumptions we make in every theory to effectively capture the essence of the observed behavior, and (4) why the term ‘presumption’ should be used in place of assumption because of a clear relation to experimental evidence. 537 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 537–550. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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2. Examples of Good Agreement between Theory and Experiment Two examples that are representative of the complexities involved in contemporary behavioral modeling will be discussed. One is the interlaminar stress phenomenon found in laminated fiber-reinforced composite materials, and the other is the high-temperature behavior of particulate composite materials under extreme thermal gradients. The fundamental modeling of complex material properties is described along with how the final behavior is predicted and measured.

2.1 INTERLAMINAR STRESSES IN FIBER-REINFORCED COMPOSITE LAMINATES In classical lamination theory for fiber-reinforced composite laminates, no account is taken of stresses such as and in the coordinates of the laminate in Figure 1. Those stresses are called interlaminar stresses, and they exist on surfaces between adjacent layers, although they also exist within the layers but are usually largest at the layer interfaces. Thus, only the stresses in the plane of the laminate, and are considered; that is, a plane-stress state is presumed to exist. Unfortunately, classical lamination theory often implies values of and where they cannot possibly exist, namely on the unloaded edge or free edge of a laminate. Accordingly, classical lamination theory includes some stresses that cannot exist and does not include some of the stresses that do exist and actually cause failure of a composite laminate. High interlaminar stresses are the basis for one of the failure mechanisms uniquely characteristic of composite laminates, namely, free-edge delamination and subsequent delamination growth as shown in Figure 1. There, the laminae could come apart in the as shown, or they could also merely experience a crack between them and then slide along the crack in either the or

The contribution of Pipes and Daniel [1] was to firmly establish that the theoretical stress distribution predicted by Pipes and Pagano [2] was indeed correct. Their approach was the Moiré fringe technique to examine the surface displacements of a symmetric angle-ply laminate under axial extension. At various load levels on long, flat graphiteepoxy specimens, Moiré fringes were photographed on the top surface of the upper angleply lamina. That lamina is one lamina thickness away from the interlaminar plane where both laminae must be a rectangle at their interface. The farther away from that interlaminar plane, the more the top lamina tends to deform into a parallelogram. The S-shaped Moiré fringes associated with the axial displacements are shown along with the elasticity

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solution of Pipes and Pagano in Figure 2. If an orthotropic lamina is loaded off-axis with a tensile stress, then shear-extension coupling exists, leading to an originally rectangular shape both elongating and shearing into a parallelogram. That shape is the natural shape toward which even a lamina in a laminate is tending as we observe the behavior as we go away from the interface between the two top layers. Thus, a line drawn across the specimen perpendicular to the axial load before loading tends to deform into a diagonal line indicating shear deformation. However, the influence in the top layer of the shear stress which is high at the free edge and quickly decreases in the direction away from the edge and in the direction toward the top surface, is to deform the diagonal line more at the free edge than as the middle of the laminate is approached. Thus, the predicted surface deformation is the somewhat S-shaped divergence from a straight line. That predicted divergence is plotted with the measured deformation in Figure 2 where we see excellent agreement. Thus, the physical existence of interlaminar stresses has been clearly demonstrated. One observation is that the ’weird’ shape of the theoretical displacement results was confirmed by the identically ’weird’ shape of the measured displacements. Thus, both theory and experiment must be correct!

2.2 THERMALLY STRESSED ATJ-S GRAPHITE DISKS An annular disk, similar in shape to a lifesaver as seen in Figure 3, is exposed to very high transient radially inward induction heating on its outer perimeter as a test of how well some materials behave in thermal shock conditions. In this Southern Research Institute thermal stress disk test, a high outside-to-inside temperature gradient results. The outside portion of the disk tends to expand more than the inside, so very severe thermal stresses are generated. The stresses in the circumferential direction are compressive near the outer diameter and tensile near the inner diameter. The slanted wedge disk has an inclined inner

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diameter surface to provide a well-defined target for a laser diameter-measuring device. All the many references are contained in the summary paper for this problem by Jones and Starrett [3].

2.2.1 ATJ-S MATERIAL CHARACTERISTICS ATJ-S graphite is a transversely isotropic granular or particulate composite material made in cylindrical billet form, as shown in Figure 4. The flake-like graphite particles are aligned in planes of isotropy because of billet compaction in the The resulting material stress-strain behavior is highly nonlinear, bimodular (quite different in tension than in compression), and strongly temperature-dependent, as displayed for the plane in Figures 5 and 6. There, the boxes are actual experimental data reported by Starrett and Pears, and the curves are the Jones-Nelson nonlinear material model fits to the data. In Figure 5, the tension behavior becomes stiffer as the temperature approaches 2000°F (1100°C), and even stiffer at 3000°F (1650°C). However, the stress-strain curve at 3000°F (1650°C) is slightly lower for high strains than that at 2000°F (1100°C). The compression behavior monotonically becomes more flexible as the temperature increases from 3000°F (1650°C) to 5000°F (2800°C) in Figure 6. Moreover, the stress-strain behavior is different under tension loading than under compression loading at every temperature, although the two behaviors are shown here only for3000°F (1650°C).

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2.2.2 INNER DIAMETER CHANGE PREDICTIONS Times late in the test run are selected for correlation of predicted and measured diameter change results. By then, the disk should be deforming nonlinearly, i.e., the stresses should be inelastic. The measured inner diameter changes are shown in Figure 7 for two orthogonal directions as a function of time. The change of diameter in the two directions is measured with two electronically equivalent circuits. The measurement in the y-direction is much less noisy than in the However, both measurements are sufficiently accurate for the present correlation effort without calibration. The difference in measured deformations in the and also can be attributed to the fact that the disk hole

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does not remain perfectly circular. Of course, the hole should remain nominally circular because the material is nominally isotropic in the plane of the hole, and the temperature distribution is nominally axisymmetric about the perpendicular to the plane of the disk. However, the material does not have perfect transverse isotropy, nor is the temperature distribution perfectly axisymmetric. The disk inner diameter change is predicted with the nonlinear material model embedded in a version of the SAAS III finite element computer program. The predicted diameter change is remarkably close to the measured diameter change (within 3%) in Figure 7. The nonlinear predictions are consistently from 2.2 to 3.3% below the average measured values. However, the elastic predictions are 5% lower than the nonlinear predictions at sec, 3.6% lower at sec, but 3.4% higher at Thus, the nonlinear predictions for this body with a strong nonhomogeneous character (because of the temperature gradient and temperature-dependent mechanical properties) are more consistent (qualitative) and more accurate (quantitative) relative to the measured deformations than are the elastic predictions. However, the percentage differences between the various approaches are not high enough to warrant strong claims of accuracy for the present approach. The reported agreement between predictions and measurements is a necessary but not sufficient condition for testing the accuracy of the nonlinear model.

2.2.3 STRESS, STRAIN, AND TEMPERATURE PREDICTIONS The predicted stresses and are shown, along with the corresponding temperature distribution at in Figure 8. Stresses of course cannot be measured; however, their predicted values are examined to determine the degree of stress-strain curve nonlinearity at various times in the thermal stress disk test. Although we expect to predominate, substantial values of tensile radial stress, do exist in Figure 8. Despite the fact that the surfaces have zero radial stress, can have substantial values elsewhere because of the small disk inner diameter and the high circumferential stresses to which is inversely proportional and proportional, respectively. Thus, the disk stress state consists of areas of biaxial tension near the inner diameter. Accordingly, the bimodulus capability of the Jones-Nelson model is essential to accurate stress analysis of the disk.

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2.2.4 SUMMARY The Jones-Nelson nonlinear material model is verified for rapid thermal loading problems involving irregularly shaped nonhomogeneous bodies. The nonhomogeneity results from a temperature gradient over a body with strongly temperature-dependent mechanical properties. Moreover, the model is shown to be valid for materials with highly nonlinear stress-strain behavior that is different under tension loading than under compression loading. The vehicle for the verification of the model is the Southern Research Institute thermal stress disk test. The inner diameter changes of this annular wedge-shaped disk made of ATJ-S graphite are predicted with the model to within about 3% over a relatively wide range of temperatures and hence material behavior. This close agreement is a vivid demonstration of the good performance of the nonlinear material model in an environment with temperatures that change rapidly in space and time, a complex geometry, and a material with properties that are a strong function of temperature level, stress level, and stress sign. The irregularly shaped body is treated with a finite element model in which the material model is embedded. Because of the agreement between nonlinear predictions and measurements (a necessary but not sufficient condition for validity of the model), the present results are an important step in the qualification of the model for general use in nonlinear material deformation problems. How were we able to develop such an accurate model of material and structural behavior? Answer: by integrating our knowledge of many behavioral characteristics obtained solely by experimental observation! The credit goes to the superb high-temperature and high-temperature-gradient property measuring skills, equipment, and understanding of the people at Southern Research Institute.

2.3 EFFECT OF GOOD AGREEMENTS BETWEEN THEORY AND EXPERIMENT When we get good agreement between theory and experiment, we tend to feel we understand the phenomena involved, and we congratulate each other. However, getting to the point where we agree on how to model a phenomenon as well as how to measure that phenomenon might take years of effort by both experimentalists and theoreticians.

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3. Experimental Results are Used More than ”Just” to Verify Theory There is an absolute need for experimental results to go beyond mere verification of theory and for theory to extend the domain of usefulness of experimental results that obviously cannot be obtained for every conceivable circumstance. We considered two examples of good correlation between theory and experiment in the previous section. The point was that both theory and experiment had become refined enough that clearly superior results were obtained, and they were in agreement. Here, we address further uses of experiment in the all-important theory-experiment interaction that must exist for mechanicists to make progress. Josef Singer describes eight motives for experiments in the area of shell buckling, and their interpretation in light of more general mechanics topics should be fairly obvious:

1. “Better understanding of buckling and postbuckling behavior and the primary factors affecting it. 2. To find new phenomena. 3. To obtain better inputs for computations. 4. To obtain correlation factors between analysis and test and for materials effects. 5. To build confidence in multipurpose computer programs. 6. To test novel ideas of construction or very complicated elements of a structure.

7. For buckling under dynamic loading and in fluid-structure interaction problems. 8. For certification tests of full-scale structures.”

Singer originated those points in 1982, and he and his coauthors, Arbocz and Weller, go on to explain each of them in their remarkable book Buckling Experiments: Experimental Methods in Buckling of Thin-Walled Structures [4]. Drucker hit the alarm bell in 1962 when he lamented that too many students were choosing to study theoretical approaches to mechanics problems rather than taking up the experimental reins. In fact, he was so upset that he said “Unless appreciable numbers of the most qualified students aim at combined experimental and theoretical research, the storehouse of physical information will be depleted by the tremendous emphasis on analysis and theory, and the theorist will be reduced to playing useless games. Experiment is essential, it is vital, and it is creative. Over the years, experiment provides the basis for the refinement of existing theory and the development of new theory” [5]. That these two luminaries of mechanics put so much emphasis on the importance of experimental work ‘speaks volumes’. Dan Drucker spoke those words in 1962, and his laments and cautions are probably more applicable today than when he first spoke them! Joe Singer spoke twenty years later, and addressed the still-needed work in experimental research, especially in shell buckling. The combination of those two points of view is a powerful commentary on the status of mechanics today. Our focus on theory at the expense of experiment has ‘led us down a garden path’ to overreliance on the computer to solve our problems to the extent that we hardly appreciate experimental work, and that state of affairs is a real professional shame.

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4. Experimental Results are the Very Basis of all Theoretical Models That experimental results are the fundamental basis of all theoretical models is a truism known to all (“Whatever you do, try to have a reason to do it”, said Forrest Gump [6]). Experimental results should be the driving force for theory! Experiments should be revealing new behavioral characteristics and motivating ever-advanced theory until the phenomenon is well understood. In the theoreticians’ world of curly overbars, underbars, various squiggles, bold-faced symbols, and other manifestations of what we often (questionably) regard as elegant, theoreticians must recognize that experimental work provides the soul behind all theoretical work. ‘Theory is important, yet theory without practice is worthless’. Sounds correct, but you wouldn’t ordinarily guess who said it! Nikita Khrushchev was actually discussing Marxism-Leninism at the time [7], but his remark holds true for us, too! Experimental investigation is necessary to provide the observations of specific behavioral features that must be modeled with a theory. Without the observations, there is absolutely no basis for a model! That is, there must be a firm foundation of multiple observations of how a material or structure behaves under various loading conditions to serve as the evidence to be modeled. Sometimes, it is very difficult to perform or conduct a proper experiment because of inadequate or severely challenged instrumentation, difficult to simulate events, or even impossibility or improbability of causing certain events to happen. Witness the severe trials of the astrophysicist who must rely on bits and pieces of information to construct or verify a theory. Similarly, a structural mechanicist involved in analyzing and designing structures to resist earthquakes has ground-motion spectra from past earthquakes to guide the loading definition but little except a few earthquake simulator “tables” to determine or confirm what happens to actual structures. Moreover, heating and support of plates for thermal buckling experiments are two challenging topics in structural mechanics. Fortunately, many structural mechanics simulations are performable. The reality of “how things work” is quite hard to determine in some cases. For example, shell buckling under axial compression loading occurs far too rapidly for the human eye to perceive all that is happening. It took the amazing experiments and high-speed motion pictures of Almroth, Holmes, and Brush [8] to bring those facts to our attention and to help us understand how shells really buckle. Note that Almroth and Brush were primarily theorists, although Almroth would have considered himself a practical engineer. An engineer should seek “clues” in the form of revealing information from critical experiments to develop an appreciation for the behavior of a particular type of mechanical system. With that appreciation, an engineer is prepared to model the behavior with the appropriate physical laws and principles. If the model is good and the analysis correct, then the behavior might be predicted well. If the model is lacking in essential features, then the behavior cannot be adequately predicted. The point is that the actual behavior provides modeling clues that must be addressed. Thus, we must observe the behavior before we can model it! I am personally disappointed when I see some try to develop a model with an obviously poor understanding of the phenomenon or phenomena addressed. In some cases, the purported elegance of the mathematical approach is used to “cover up” the fundamental fact that the theory has little basis in fact, namely the fact of the actual behavior. Some (fortunately few) theoreticians quite apparently choose to totally ignore all experimental

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data related to the phenomenon addressed with their theories! Seemingly, they do not use experimental results to justify their assumptions (presumptions, as we will see in the next section) or even to use as a comparison for their theoretical results! How do they ever know if what they did has any meaning? Or, was the work done to satisfy the insatiable proclivities of a dean or department head with minimal ability to think about the significance of scholarly work, but seemingly maximal ability to count papers? That is the min-max problem of academia! That unfortunate problem is enormously compounded when a book is regarded by those who have lost their sense of scholarship as ‘worth’ only a paper or two. Recall Drucker’s prediction that ‘the theorist will be reduced to playing useless games’ [5].

5. Clarification of the Terms Assumption, Presumption, and Restriction and Their Relation to Experiments When theoreticians ‘assume’, ‘presume’, or ‘restrict’ at various stages in their analyses, two areas of concern are apparent. Just what do they mean by, one, presumptions as opposed to assumptions, and, two, restrictions as opposed to presumptions. The distinctions are related to the reasons why we perform certain acts in analysis. Again, “Whatever you do, try to have a reason to do it”, said Forrest Gump [6].

5.1 PRESUMPTION VERSUS ASSUMPTION Too often, a theoretician ’assumes’ that a certain class of behavior exists and proceeds to develop a model and theory based on that assumption. However, the fundamental concept in analyses of mechanical behavior of materials and structures is not as simple as merely making an assumption because to assume is ‘to take for granted; suppose (something) to be a fact’ [9]. Thus, an assumption is arbitrary, not fact! Accordingly, the word assumption leaves out entirely whether the (something) has any reason whatsoever to be considered. We in mechanics and more generally in engineering cannot be that illogical! We are not interested in arbitrarily taking something for granted! Our concern is with finding why we should consider some line of reasoning or approach. The far more appropriate word for the mechanics analysis is presume because it ‘implies taking something for granted or accepting it as true, usually on the basis of probable evidence in its favor and the absence of proof to the contrary:ehp 1.’ [9]. Thus, presume (presumption) is a far stronger and, indeed, far more appropriate word for use in analysis than assume (assumption). For example, we could assume something that violates equilibrium, but certainly we cannot presume something that violates equilibrium! Assuming is poor practice, and violating equilibrium is unacceptable. An engineer should never make an arbitrary decision, i.e., one without a reason based in fact. Instead, an engineer makes ‘presumptions’. Clearly, the physical reason that is an essential part of every presumption must come from personal observation of the real behavior of the system addressed. In fact, some engineers use the term assumption in place of the far more appropriate term presumption, i.e., they say assumption, but really mean presumption as described in the foregoing discussion. A crucial part of almost any mechanics analysis is the identification of the probable physical evidence that leads us to the correct presumption. That is, the steps often called assumptions in a mechanics analysis are actually presumptions because they are not arbitrary assumptions (a redundant term), but are presumptions based on evidence that we

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can examine, and we often critically evaluate them, although sometimes not until after the analysis is completed. Our concern in mechanics analyses is with the probable evidence of material and structural behavior, so we deal with presumptions, not assumptions, and we must identify the probable behavior. That is, we must have a reason to justify our presumption, i.e., to make it plausible enough to consider. Thus, every presumption has two elements: (1) what? and (2) why? Or, (1) what is the presumption? and (2) why is the presumption believed to be true? What is the evidence to support the presumption? Without some form of evidence, we have no justification to make the presumption! The obvious basis for every presumption is experimental fact! The classical Euler-Bernoulli beam-bending hypothesis in elementary mechanics of materials is a good illustration of the two elements of a presumption. In accordance with that theory, we presume that (1) planes originally perpendicular to the neutral axis1 of an undeformed beam remain perpendicular to the deformed neutral axis after bending because (2) we can see from the deformation of a rubber beam (a material that is easily and visibly deformable) that vertical lines drawn on the side of the beam of rectangular cross section in Figure 9a remain essentially straight during pure bending as in Figure 9b. Those vertical lines are traces in the plane of Figure 9 of planes that are perpendicular to the beam neutral axis. However, those originally vertical lines do rotate to a position such that, if extended, they all pass through a common point, namely the center of curvature. Thus, plane sections remain plane after bending. During bending, the horizontal lines in Figure 9b below the neutral axis stretch, and the horizontal lines above the neutral axis compress, describing arcs of circles with centers at the center of curvature through which the originally vertical lines pass after bending. Note that we are relying on the resolution capability of the naked eye in our observations. That is, at the level of our vision, we see that plane sections remain plane. However, if we were to bend a much deeper, shorter beam, we would observe that plane sections do not remain quite plane because shear deformation of the original planes is evidenced by some curvature of the deformed ‘planes’. Alternatively, if we used a refined measurement technique such as

1 The neutral axis of a beam is the originally straight horizontal line that does not change length (strain) when pure bending moments are applied at each end of the beam. The neutral axis is at the mid-height of a rectangular cross-section beam.

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Moiré interferometry, we would detect non-planeness of the deformed ‘planes’. Most engineers do not distinguish between assumptions and presumptions. They typically use the word assume (or assumption), but generally mean presume (or presumption) because good engineers always have a reason for doing anything (just like Forrest Gump). In contrast, a presumption (assumption) might be quite different from an approximation. An approximation is often more related to the mathematics of treating the model of a physical situation than to the actual physical situation itself. In mathematics, we might consider only the (apparently) dominant terms in a series as an approximation (and we ‘assume’ that the rest of the terms are small enough to ignore, although often we are not certain). This approach is a definitive and well-demonstrated mathematical approach to analysis. A presumption (or assumption) is often more related to the physical side of a problem than to the mathematics. We attempt to simplify the process of solving a physical problem, or the model thereof, by making certain simplifying presumptions about the modeling of the physical behavior based upon physical observations. Those simplifying presumptions might involve the perceived nature of the physical response that occurs, e.g., a deformation.

5.2 PRESUMPTION VERSUS RESTRICTION The seemingly dual terminology of restrictions and presumptions is used because the terms have fundamentally different meanings that unfortunately are not always acknowledged. Restrictions are limitations on the use of the theory that are obviously either satisfied or they are not. Thus, restrictions are concerned with the known. For example, a theory for square plates does not apply to round plates. Presumptions are limitations on the theory that have a nature of uncertainty to them. That is, presumptions are concerned with the unknown. For example, stresses perpendicular to the surface of a plate, but within the plate, are commonly presumed to be small enough to be regarded as zero, or assumed to be zero; however, we do not know for certain just how small the stresses are unless we appeal to a more accurate theory. Also, displacements might be presumed to be small to enable certain approximations. However, whether the displacements actually are small can be determined only when the final results are known. In summary, the difference between restrictions and presumptions is that restrictions involve the known and presumptions involve the unknown (about which we wish to speculate). The following restrictions and presumptions provide further opportunity to clarify the difference between the two classifications for the common example of a plate as shown in Figure 10. Recall that presumptions in engineering must be justified, i.e., we must know why we believe the presumption to be true from a physical standpoint.

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Restrictions The plate is isotropic, linear elastic, and of constant thickness. The plate thickness is very small compared to its length and width (such a configuration is commonly called a thin plate, although the name plate itself implies such a geometry). No body forces exist.

Presumptions Stresses acting in the plane (the plane of the plate) dominate the plate behavior. Then, and are presumed to be zero such that an approximate state of plane stress is said to exist (wherein only and are considered). The Kirchhoff hypothesis of negligible transverse shear strains, and and negligible transverse normal strain, constitutes a statement of nondeformable normals to the middle surface although there is an inherent, but commonly ignored, conflict with the presumption of zero transverse normal stress, Displacements and are small compared to the plate thickness (generally, although not necessarily, indicative of small-deflection theory). Strains,

and

are small compared to unity (small-strain theory).

Rotatory inertia terms are negligible. Some engineers assume something that can be clearly determined as fact or fallacy if they would just look at the phenomenon. For example, some engineers ‘assume a rectangular plate’ when all you have to do is look at the plate to determine whether it is rectangular or in fact some other shape. Those engineers really mean to say they restrict their attention to a rectangular plate. That is, they confuse restriction with assumption. Clearly, there is no need or place for such confusion. Engineers are responsible for communicating their opinions, results, designs, etc. in an absolutely unambiguous manner. Otherwise, the possibility certainly exists that serious misinterpretation will occur. As an example of misinterpretation, in the 1920s, a building was designed to be built just off Red Square in Moscow. The architect expressed the possibility of two different (alternative) exterior designs in his plans by drawing a single end view of the building with one design on the left half of the building and the other design on the right half of the building, all on one sheet. No note was placed on the drawing to explain that there was a choice of exterior designs to be made, so the constructor built the building just as he saw it on the plans. After all those were Stalinist times, so asking too many questions could get a person into serious trouble. That unusually nonsymmetric building stands to this day! Obviously, engineers cannot afford to be ambiguous when we present our results!

6. Concluding Remarks We have seen examples of good agreement between theoretical predictions and experimental measurements of behavior for two complicated mechanical systems, and other

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examples abound. But, what do they mean? They mean that we have been able to successfully perform two challenging tasks: (1) develop a material model and a structural model that are capable of high-fidelity prediction of behavior and (2) measure behavior in a complex environment in which the many involved behavioral aspects of the widget are excited. But is our sole interest obtaining verification of theoretical predictions by performing useful measurements? No! Absolutely not! We must have experiments to help us understand the behavior of our widget so that we can determine the very basis for any model that we might develop. To properly do so, we must first establish the behavior and its characteristics, and then we’ll see if we can develop useful approximations to capture the essence of the behavior of the widget. Those approximations must be both realistic and useful. The approximations are decidedly not ‘assumptions’ because they are not arbitrary as is inherent in the definition of the word assumption. The approximations must have a solid, reasonable basis in fact, so they are ‘presumptions’. The distinction between assumptions and presumptions (and restrictions) is extremely important, yet tends to be glossed over by all too many. The most important message to theoreticians is: Seek and rely on experimental evidence to motivate, develop, verify, and extend your theories. Work closely with an experimentalist to maximize your opportunity to capture and to understand the phenonmenon. Mere information is not knowledge. It takes much thoughtful work to mold information into a body of knowledge – it’s called research! For some complex phenonomena, there can’t be enough information. “Too much ain’t enough” said Willie Nelson.

7. References 1. R. Byron Pipes and I. M. Daniel, Moire Analysis of the Interlaminar Shear Edge Effect in Laminated Composites, Journal of Composite Materials, Volume 5, April 1971, pp. 255–259.

2. R. Byron Pipes and N.J. Pagano, Interlaminar Stresses in Composite Laminates Under Uniform Axial Extension, Journal of Composite Materials, Volume 4, October 1970, pp. 538–548.

3. Robert M. Jones and H. Stuart Starrett, Nonlinear Deformation of a Thermally Stressed Graphite Annular Disk, AIAA Journal, Volume 15, Number 8, August 1977, pp. 1116–1122.

4. J. Singer, J. Arbocz, and T. Weller, Buckling Experiments: Experimental Methods in Buckling of Thin5.

6. 7. 8. 9.

Walled Structures, Volume 1, Basic Concepts, Columns, Beams and Plates, 1998, Wiley, Chichester, United Kingdom. D. C. Drucker, On the Role of Experiment in the Development of Theory, Proceedings of the 4th U. S. National Congress of Applied Mechanics, 1962, pp. 15–33. Forest Gump ref. personal communication daily. Nikita Khrushchev, Khrushchev Remembers – The Glasnost Tapes, Translated and edited by Jerrold L. Schecter with Vyacheslav V. Luchkov, Little, Brown and Company, Boston, 1990, p. 3. B. O. Almroth, A. M. C. Holmes, and D. O. Brush, An Experimental Study of the Buckling of Cylinders Under Axial Compression, Experimental Mechanics, Volume 4, Number 9, September 1964, pp. 263– 270. Webster’s New World Dictionary, Second College Edition, David B. Guralnik, Editor in Chief, Prentice Hall Press, 1984, pp. 84 and 1126.

MIXED NUMERICAL-EXPERIMENTAL TECHNIQUES : PAST, PRESENT AND FUTURE A.H. CARDON, H. SOL, W.P. DE WILDE, J. DE VISSCHER, K. HOES, D. DINESCU Dept. Mechanics of Materials and Constructions (MEMC) Faculty of Applied Sciences and Engineering (FTW) Free University Brussels (V.U.B.) Pleinlaan 2, B1050 Brussels, Belgium, EU Abstract Direct methods in continuum mechanics start from the thermomechanical characteristics and by solving the equilibrium or motion equations, satisfying boundary and critical conditions, result in stress, strain and/or displacement fields. Inverse methods are starting from the obtained solutions in order to compute a priori unknown characteristics or parameters. Direct experimental measurements of some characteristics or parameters are sometimes very difficult, if not impossible, to perform. Special applications of inverse methods combining some experimental global results with computed results are known as mixed numerical-experimental techniques or hybrid methods. Those MNET can be applied in order to obtain thermomechanical characteristics of material systems with elastic, viscoelastic, elastoplastic or viscoplastic behaviour. They can also be applied for damage, interphase, damping and processing characterisation. The MNET-approach can also be used in many other fields out of mechanics if a good theoretical model exist and a sensitive “experimental” method is available.

1. Introduction Classical definitions of stress and strain fields in continuum mechanics are based on some homogenisation procedure over a representative volume element (RVE). That RVE has to be small enough to avoid smoothing of high gradients but large enough to represent an average of the microbehaviour, J.Lemaitre, [1], Direct experimental measurements of stresses are only possible in some very special situations due to the definition of the stress vector. For the direct experimental measure of strains, strong “uniformity” conditions are necessary. For classical materials those conditions can be satisfied but with composite or multimaterial systems such direct experimental measurements of stress and strain fields are often difficult, if not impossible. In many other domains also the theoretical basic elements are not accessible for a direct “experimental” measure. In such a case it is possible, after definition of a set of control parameters, to start with estimated values of those control parameters, to compute some general effect obtained by application of the 551 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 551–560. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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theoretical model and compare that computed result (CR) with a direct experimental measure (EM) of the same global effect. If the “distance” between CR and EM is small enough, norm to be defined, the estimated set can be considered as acceptable, even if it cannot be proven that this set is unique. If the “distance” between CR and EM is too large an iteration procedure has to be developed till an acceptable set of control parameters is obtained. Perturbations of the so obtained acceptable control parameters have to be performed in order to check if the set can be considered as unique and to validate the results. This is the general basis of mixed numerical-experimental techniques (MNET) closely related to the inverse methods in model theory. In solid mechanics those methods were developed for the characterisation of material systems, for the characterisation of the interaction regions between basic materials in a multimaterial system, for the “measure” of some damage parameters and for the “measure” of residual stresses. In many other domains such as processing, general production and manufacturing methods and fluid-solid interactions those MNET can be applied. Those methods are not limited to mechanics and biomechanics, but can be used in any domain where some models describe a global evolution or behaviour starting from some locally defined parameters. For the application of MNET it is necessary to have a good model in order to obtain an accurate computed value of some global result and a sensitive experimental technique for the direct measure of that same global result.

2. Short history of the basic aspects and developments of MNET In the years following world war II, with the development of photoelasticity, it was necessary to arrive at the separation of the principal stresses. A first generation of hybrid experimental-numerical stress analysis methods were proposed such as the shear difference method and the Filon’s method, as described by A.J. Durelli and W.F. Riley, [2]. This approach was later developed and formalised by K.H. Laermann, [3], and G.V. Rao, [4], An overview of this first period, 1950-1985, is given by A.S. Kobayashi, [5]. In fact inverse methods are applied since a long time in order to obtain the value of some material characteristics directly on construction components under loading instead of using separate test coupons. If we consider the 4-point bending of an elastic isotropic beam we have as approximation for small displacements w such that equation :

With the boundary conditions w = 0 for

we obtain

the equilibrium

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and in the centre of the beam, x = 0, As consequence we obtain an estimation of the elastic modulus of the material of the beam by measuring w; M, and I given, as :

In this example we use for the computation an “analytical” solution and for the experimental measure an explicit measure of the displacement w for different values of the bending moment M. The relation between CR and EM is linear and a direct computation is possible. In a more general case the relation between CR and EM is nonlinear and an iteration cycle has to be applied. The number of analytical available solutions and approximations is limited, especially for composite materials and non-elastic behaviour, but we can use also the results of asymptotic solutions obtained for thin walled elastic and viscoelastic bodies as shown by A.H. Cardon et al., [6], It is not absolutely necessary to start from the estimation of the complete set of control parameters of a given phenomenon. It is perfectly possible to use a direct experimental measurement of the values of a subset of control parameters, the MNET being only applied in order to obtain the other control parameters. If we consider the characterisation of long fiber reinforced polymer matrix lamina assumed as transversely isotropic we are able to measure directly 4 of the material characteristics and we only need to apply a MNET for the fifth caracteristic as shown by A.H. Cardon et al., [11]. In the seventies with the development of modern computers and computational facilities based on finite and/or boundary element methods, see e.g. K.T. Kavanagh, [7], we saw a new start of MNET-applications as developed by W.P. De Wilde et al., [8], L.R. Deobald et al.,[9] and H. Sol, [10].

3. Characterisation of material systems In the period 1982-1990 a central problem was the measure of the material constants of fibre reinforced composites and especially long fibre reinforced lamina. Those lamina are strongly anistropic and, function of the relation between the fibre diameter and the thickness of the laminated plate, we can assume the material system as being orthotropic or transversely isotropic. Considering the complete set of material constants we have in general the possibility to obtain some of those constants by a direct experimental measure as e.g. the longitudinal modulus, and the transverse modulus In an elastic situation those two moduli are real. In case of a viscoelastic situation we have complex moduli and for a fibre reinforced polymer matrix will be real and complex. We start from the direct measured values of and we introduce estimations for

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Based on this completed set a global characteristic is computed by an appropriate numerical method, or from some analytical or asymptotic solution, if available, CR. The same global characteristic is measured directly, EM. Finite element or boundary element methods are very convenient for the computation and different programs are available especially for thin and moderate thick plates. The global characteristic can be a static one, but in that case iterations are not easy and the experimental technique is not very sensitive. Following different results as obtained by H. Sol, [10], a dynamic analysis based on resonant frequencies or a more general modal analysis is much more sensitive and efficient. After definition of a norm for the measure of the distance between CR and EM, we consider If this distance is smaller than the defined norm, we have an acceptable set of characteristics If for a first computation the distance is too large we have to change the values of the estimated subset and compute CR(1). The general MNET-scheme can be presented as follows:

to

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Different research groups developed this method in the period 1985-1990 and presented their results at the International Conference of Experimental Mechanics in August 1990. The group of the TUDenmark by P. Pedersen et al., [12]; the group of TUEindhoven by M.A.N. Hendriks et al., [13], and the group of Brussels by H.Sol, [14]. At the same conference some other presentations related to MNET were presented, e.g. by K.H. Laerman, [15], B.L. Agee, [16] and J.L. Jensen et al., [17]. The same year of the International Conference of Experimental Mechanics a European Mechanics colloquium (Euromech 269) was organised in St. Etienne by A. Vautrin and H. Sol on “Mechanical Identification of Composites”, [18]. Besides a number of papers on specific experimental methods, impact behaviour, damage characterisation and interfaces a large number of papers are applications and developments of MNET. On the basis of dynamic characteristics Hua Hongxing, [19], showed the possibilities of this scheme. The composite properties to be measured are assumed to be macroscopically homogeneous, globally averaged and elastic. The assumption of an elastic behaviour is convenient but seems not to be a limitation for the application of MNET. The plates used as test specimen are thin in order to be able to use the Love-Kirchhoff assumptions and from the stiffness coefficients the material engineering constants can be obtained. The dynamic behaviour, characterised by resonance frequencies and/or mode shapes, if the damping is very small, is sensitive to the plate rigidities. The measurement and the calculation of those dynamic characteristics are easy to be obtained by one test on the same specimen. A detailed discussion of the finite element method, the correlation criteria between numerical and experimental models and the updating methods are given in [19]. Applied to a bi-directional glass fibre reinforced 10-layers plate following MNET, results were obtained for the engineering constants :

Those results have to be compared with the direct measured values by static experiments on beams in tension, bending and torsion loadings :

A satisfactory agreement appears clearly, but a complete error analysis is necessary from the numerical modelling errors, over the classical measurement errors to the external induced errors and those coming from the numerical computations.

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4. More recent developments The material identification does not have to be limited to an elastic behaviour and it became possible to obtain the entire complex stiffness matrix of orthotropic materials by some generalised mixed numerical experimental technique, as shown by J. De Visscher, [20]. As expected an important number of papers presented at the ICEM, Lisbon, 1994, [21], were related to one of the aspects of the application of the mixed numerical-experimental methodology. The MNET was applied not only to problems of characterisation of material systems, but progressively to general nondestructive techniques, damage analysis, residual stresses, processing methods, interphase characterisation and was also applied to non-mechanical engineering problems and even to more general non- engineering fields. A large number of examples were presented during the European Mechanics Colloquium (Euromech 357), organised by H. Sol and C.W.J. Oomens in April 1997, [22], We also have to menition some papers presented at the International Conference on Experimental Mechanics, Oxford, 1998, [23], such as the contribution e.g. by K.H. Laermann, [24], that by T. Comlekci et al., [25] and by Inoue et al., [26]. As illustrated by those two last papers, inverse methods, or mixed numericalexperimental techniques, are the declared or hidden basis of many non-destructive testing methods. Any signal coming out of the volume of a construction component through the free surface contains some integrated information on the internal state. The reconstruction of the acceptable interior elements from the measured signal in combination with a model of the material system behaviour may be considered as a possible definition of an inverse problem and an application of some MNET. This was also illustrated recently by J.D. Achenbach, [27], as part of the general report of the US National Committee on Theoretical and Applied Mechanics “Research Trends in Solids Mechanics”, [28]. Many examples of the MNET applications in non-destructive testing can be found in the proceedings of the First Joint Belgian-Hellenic Conference on Nondestructive Testing, Patras, [29] and later on a lot of papers presented at NDT-NDE conferences are showing, explicitely or not, the application of the MNE concept. An immediate extension of the method out of the strict elastic behaviour is the measure of damping as shown by J. Vantomme, [30], and J. De Visscher et al., [31]. The MNET is a global method. The results are based on the comparison of integrated or averaged quantities. This is an advantage in some situations but if some localised information is needed a large number of specific iterations are necessary.

5. General applications of MNET 5.1. INTERPHASE In a multimaterial system such as a composite or an adhesively bonded joint one of the essential elements of the mechanical performance is the interaction level between the different basic materials. This is known as the problem of interfaces or interphases.

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This interaction region is very small and it is impossible to obtain any direct experimental measure of the characteristics of this interphase. The only possibility is to include this region in a multilayered plate or shell system and to apply a mixed numerical experimental technique. In order to work out a MNET, we have to model the interphase. This can be done by the introduction of imperfect interface conditions as proposed by J.D. Achenbach et al., [32], or by a continuum model and a plate type approximation as proposed by A.H. Cardon, [33]. The application of MNET for the characterisation of the interphase was discussed e.g. by A.H. Cardon et al., [34].

5.2. RESIDUAL STRESSES If

is the Cauchy stress tensor, we have for an elastic equilibrium problem : in the volume V, along S around V,

and Hooke’s law For residual stresses

we have :

in V, along S. In order to obtain information on the stresses we relax them by some destructive technique (e.g. drilling hole method or cutting techniques), measuring the resulting strain field and computing the related stress state. Instead of relaxing the existing residual stress state it is also possible to introduce a supplementary loading and applying a numerical-experimental scheme starting from some estimated values of the residual stress field and using some global characteristics, displacements, resonant frequencies or mode shapes, we can obtain acceptable values of the residual stresses.

5.3. DAMAGE Damage can be defined as the irreversible rupture of internal bonds in a continuum from micro- to macroscopic level. The characterisation of damage in a direct way is not easy and it is in general the influence of damage on the thermomechanical characteristics of the continuum that gives us some measure of the damage level. MNET can be applied for damage characterisation based on an estimation of the damage parameters in the stiffness characteristics and the comparison of computed results with experimentally obtained values.

5.4. PRODUCTION PROCESSES Many production processes of polymer matrix composites are the result of some flow over, or through, a reinforcement element. A good example is the resin

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transfer moulding (RTM) process. In all those flow related production processes the efficiency of the process is related to the velocity of the flow propagation around the reinforcement. That velocity can be calculated from some diffusion or permeability coefficients. Those coefficients are difficult to be measured experimentally or to be computed on the basis of non Newtonian fluid mechanics in interaction with some solid reinforcements and different surface conditions. It is possible to apply a MNET in order to obtain the permeability coefficients necessary to optimise the resin transfer moulding process as shown by the research group of H. Sol, [35],[36], in collaboration with the group of I. Verpoest, (University Leuven, KULeuven).

5.5. BIOLOGICAL MATERIALS The knowledge of the thermomechanical characteristics of living tissues is an essential element not only for healing procedures but also for any type of artificial elements to be introduced in the human body. For such prosthesis biological compatibility is essential but thermomechanical compatibility is also necessary in order to avoid any type of stress concentrations leading to singular material productions in reactive or living materials. In the characterisation of living tissues the most difficult problem is the fact that we are not able to obtain normalised test specimen and to use normalised test methods. We have to take the living material in the geometrical form and shape as it appears in nature. As consequence a direct experimental measure of the thermomechanical characteristics of living tissues, such as. e.g. tendons, muscles, bones, arterial wall and skin is very difficult if not impossible. Only the application of MNET can give us results obtained in vivo or on material elements with their initial shape. The research group of Eindhoven initiated such applications e.g. M. Van Ratingen, [37], and W.K.L. Van der Voorden et al., [38]. We have also to mention for bones the work of B. Van Rietbergen et al., [39] and many examples are discussed in the IUTAM Symposium on Biosolid Mechanics, [40].

6. Conclusions The mixed numerical-experimental techniques based on the adjusted comparison of computed results with direct measured experimental values of some global effect or behaviour have proven their efficiency in many applications where no direct experimental measure of the details of the basic elements are possible. MNET is a global method where integrated and averaged quantities are used. The “localisation” of some sources can be obtained by adjustments of the basic set of parameters. MNET need a good model to obtain the computational results and a sensitive experimental technique for the experimental results. Taking into account the important development of our computational facilities and the new possibilities of our experimental methods MNET is probably the most promising method for the future of experimental mechanics with applications in all domains where models are used in order to describe evolutions of phenomena and systems. MNET is not a miracle method and a carefully analysis is necessary on computational and experimental level, but the potentials are high.

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7. Acknowledgments Large parts of this research were made possible by financial support from the Research Fund of the Flemish Region of Belgium (FWO-Vlaanderen), from the Research Council of the Free University Brussels (OZR-VUB), from the Belgian Office of the Prime Minister for Scientific, Technical and Cultural Affairs (UIAPproject), and from the International Association for Promotion of the Cooperation with Scientists from the New Independent States of the former Soviet Union. (Intas-contract 96-2113).

8. References 1. 2.

Lemaitre J. (1966): A course on damage mechanics Edition), Springer. Durelli A.J. and Riley W.F. (1965): Introduction to Photomechanics, Prentice Hall, pp. 185186. 3. Laermann K.H. (1981): Recent developments and farther aspects of experimental stress analysis in the Federal Republic of Germany and Western Europe”, Exp. Mech., 21, pp. 4957. 4. Rao G.V. (1982): Experimental-numerical hybrid techniques for body-force and thermal stress problems – Applications in power industry, SESA Conference on Experimental Mechanics, pp. 398-404. 5. Kobayashi A.S. (1987): Hybrid experimental-numerical stress analysis, Handbook on Experimental Mechanics (Ed. A. Kobayashi), Prentice Hall, chapter 17, pp. 739-767. 6. Cardon A.H., Kossovich L., Kaplunov J., Emri I. and Gallegos C. (1999): Interactive theoretical-numerical-experimental methods for the characterisation of composite systems, Proc. SEM Annual conference 1999, SEM, pp. 825-827. 7. Kavanagh K.T. (1971): Finite element analysis in the characterisation of elastic solids, Int. Journal of Solids and Structures, vol. 7, pp. 11-23. 8. Hermans Ph., De Wilde W.P. and Hiel C.C. (1982): Boundary integral equations applied in the characterisation of elastic materials, Computational Methods and Experimental Measurements (Eds. Karamidas G.A. – Brebbia C.A.), Springer pp. 189-199. 9. Deobald L.R. and Gibson R.F. (1986): Determination of elastic constants of orthotropic plates by a modal analysis/Raleigh Ritz technique, Proc. Int. Modal Analysis Conference, pp. 682-690, Union College, Shenectady, NY, U.S.A. 10. Sol H. (1986): Identification of anisotropic plate rigidities using free vibration data”, PhDdissertation, VUB, Brussels. 11. Cardon A., De Muynck E., Hiel C.C. and Boulpaep F. (1979): Utilisation d’une méthode mixte, numérique et expérimentale pour la détermination des caractéristiques mécaniques de matériaux composites”, Proc. of the Canadian Congress of Applied Mechanics, pp. 105106. 12. Pedersen P., Frederiksen P.S. (1990): Sensitivity analysis for identification of material parameters, Proc. ICEM, Copenhagen, pp. 545-551. 13. Hendriks M.A.N., Oomens C.W.J., Jans H.W.J., Janssen J.D. and Kok J.J. (1990): A numerical-experimental approach for the mechanical characterisation of composites”, Proc. 9th ICEM, Copenhagen, pp. 552-561. 14. Sol H.: Identification of the complex moduli of composite materials by a mixed numericalexperimental method, Proc. ICEM, Copenhagen, pp. 562-571. 15. Laerman K.H. (1990): On a computer oriented numerical-experimental method in photoviscoelasticity, Proc. ICEM, Copenhagen, pp. 2059-2068. 16. Agee B.L. and Mitchell L.D. (1990): Frequency-dependent viscoelastic property measurement via modal analysis techniques, Proc. ICEM, Copenhagen, pp. 1978-1988. 17. Jensen J.L., Brincker R. and Rytter A. (1990): Uncertainty of modal parameters estimated by ARMA-models, Proc. ICEM, Copenhagen, pp. 2095-2104. 18. Vautrin A., Sol H. (Eds), (1991): Mechanical identification of composites", Elsevier Appl. Science.

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19. Hongxing Hua (1993): Identification of plate rigidities of anisotropic rectangular plates, sandwich panels and circular orthotropic disks using vibration data, PhD-thesis, VUB, Brussels. 20. De Visscher J. (1995): Identification of the complex stiffness matrix of orthotropic materials by a mixed numerical experimental method, PhD-thesis, VUB, Brussels. 21. Silva Gomes J.F. et al (Eds), (1994): Recent advances in experimental mechanics, A.A. Balkema Publishers. 22. Sol H. and Oomens C.W.J. (Eds), (1997): Material identification using mixed numerical experimental methods, Kluwer Academic Publishers. 23. Allison I.M. (Ed), (1998): Experimental mechanics – Advances in design, testing and analysis, A.A. Balkema Publishers. 24. Laermann K.H. (1998): Contribution to solving inverse problems in NDT of structures, Proc. ICEM 11 (Ed. I. Allison), A. A. Balkema Publishers, pp. 755-760. 25. Comlekci T. and Boyle J.T. (1998): A general hybrid numerical-experimental technique for thermoelastic stress separation”, Proc. ICEM 11 (Ed. I Allison), A.A. Balkema Publishers, pp. 463-468. 26. Inoue K., Kishimoto K., Hayakusa K. and Shibuya T. (1998): Stress separation in thermoelastic by means of boundary element inverse analysis, Proc ICEM 11 (Ed. I. Allison), A.A. Balkema Publishers. 27. Achenbach J.D. (2000): Quantitative non-destructive evaluation, Int. Journal Solids and Structures 37, pp. 13-27. 28. Dvorak G.J. (Ed), (2000): Research trends in solid mechanics, Pergamon Press. 29. Van Hemelrijck D., Anastassopoulos A. (Eds) (1996): Non-destructive testing, A.A. Balkema Publishers. 30. Vantomme J. (1992): A parametric study of material damping in glass fibre reinforced thermosetting plastics, PhD-thesis, VUB, Brussels. 31. De Visscher J., Sol H., De Wilde W.P., Vantomme J. (1997): Identification of the damping properties of orthotropic composite materials using a mixed numerical experimental method, Applied Composite Materials, vol. 4, n°1, pp. 13-33. 32. Achenbach J., Zhu H (1989): Effect of interfacial zone on mechanical behaviour and fracture of fibre reinforced composites, J. Mech. & Phys. Of Solids, 37, pp. 381-393. 33. Cardon A.H. (1993): From micro- to macroproperties of polymer based composite systems by integration of the characteristics of the interphase regions, Composite Structures, 24, pp. 213-217. 34. Cardon A.H., De Wilde W.P., Van Hemelrijck D., Van Vinckenroy G., Sol H., De Visscher J. (1994): Experimental characterisation of the interface-interphase properties in composite systems by mixed numerical-experimental techniques, Recent Advances in Experimental Mechanics, J.F. Silva Gomes (Ed), Proc. ICEM, A.A. Balkema Publishers, pp. 35-38. 35. Dinescu D., Sol H., Hoes K. (2001): A fast mathematical model for permeability identification in resin transfer moulding using a mixed numerical-experimental method, Proc. Of the 3rd Canadian Composite Conference – Cancom 2001, Montreal, pp. 109-117. 36. Hoes K., Dinescu D., Sol H., Luo Y., Verpoest I., Parnas R.S. (2001):New sensor based setup for permeability identification, Proc. Cancom 2001, Montreal, pp. 101-108. 37. Van Ratingen M. (1994): Mechanical identification of inhomogeneous solids – A mixed numerical-experimental approach, PhD-thesis, TUEindhoven. 38. Van der Voorden W.K.L., Douvers L.F.A. (1997): Characterisation of the in-plane mechanical behaviour of the human skin in vivo, Proc. Euromech 357, Material Identification using Mixed Numerical Experimental Methods (H. Sol, C.W.J. Oomens, Eds), Kluwer, pp. 173-182. 39. Van Rietbergen B., Kabel J., Odgaard A., Huiskes R. (1997): Determination of trabecular bone tissue elastic properties by comparison of experimental and finite element results, Proc. Euromech 357, Material Identification using Mixed Numerical Experimental Methods (H. Sol, C.W.J. Oomens, Eds), Kluwer Academic Publishers, pp. 183-192. 40. Pedersen P., Bendsoe M.P. (Eds), (1999): Iutam Symposium on Synthesis in Bio Solid Mechanics, Kluwer Academic Publishers.

A MOIRÉ-FE METHOD FOR INTERNAL CTOA DETERMINATION

J. H. JACKSON Idaho National Engineering and Environmental Laboratories P.O. Box 1625, Idaho Falls, ID 83415-2218 A. S. KOBAYASHI University of Washington Box 352600, Seattle, WA 98195-2600

Abstract A hybrid moiré-finite element (FE) analysis was used to determine the crack tip opening angle (CTOA) along a tunneling crack front in a three-point bend (SENB) specimen. The specimen was machined from a ductile 2024-T351 aluminum plate of 8.1 mm thickness and was pre-fatigued to an initial crack length to width ratio of The specimen was then subjected to stable crack growth of varying after which the specimen was post-fatigued to mark the final crack front and loaded to failure. A quarter segment of the SENB specimen was modeled with a truncated 3-D elastic-plastic FE model. Measured surface displacements, which were obtained by moiré analysis, and stable crack growth with increasing load were prescribed on the FE model. The CTOA was obtained from the computed crack opening displacement, approximately 1 mm behind, and normal to the crack front. The good agreement between the measured and computed surface CTOA are indicative of the accuracy of the procedure. 1. Introduction A ductile fracture criterion, which relates to the plastic strain field in the crack tip plastic zone, is the crack tip bluntness which was quantified by the crack opening displacement (COD) criterion advanced by Wells in 1963 [1]. This COD criterion, which was initially related to Irwin’s plastic zone estimate [2], and later to the Dugdale plastic zone [3], was promoted mainly for fracture assessment of thick-walled pressure vessels by The Welding Institute in the 60’s and 70’s [4]. Since the COD along a surface crack front in a thick plate cannot be readily measured and must be computed, its acceptance was hindered by the lack of easily accessible threedimensional elastic-plastic computer codes with the sensitivity to accurately map the crack tip bluntness. The CTOA, which uniquely characterizes the crack tip strain field [5], is another local field parameter which is measured as close to the crack tip as is feasible within the limits of experimental accuracy on the specimen surface. Shih et al [5] as well as Kaninnen et al [6] in the 70’s concluded that the CTOA was a computationally attractive operational parameter and an alternative to the J-integral 561 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 561–570. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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criterion. Unlike the J-integral criterion, which subsequently dominated the ductile fracture research of the 80’s and 90’s, the CTOD/CTOA criterion is also valid in the presence of unloading and crack extension. The CTOA criterion was used in the 80’s to analyze axial crack propagation in a subscale gas transmission line [7]. The rupturing profile of a propagating crack in a pressurized 2-in-diameter, schedule 10, carbon steel pipe was recorded with a highspeed framing camera and was processed through a special software [8] which yielded the CTOA and the flap motion. The CTOA criterion was also used in a ring model of the rupturing pipe and a thermal hydraulic depressurization code to simulate the largescale burst tests of 48-inch diameter x 0.720-inch thick, X70 line pipes [9]. The excellent agreement between the measured and the computed CTOA versus crack velocity relations demonstrated that the CTOA was an effective static and dynamic fracture parameter for pipe rupture [10]. A renewed research thrust in the CTOA criterion was undertaken in the early 90’s in response to the NASA Aircraft Integrity Program. The thin aluminum specimens considered for aircraft structure were ideally suited for the CTOA criterion. Newman et. al. showed that the CTOA criterion will predict the residual strength in laboratory specimens [11, 12] as well as in the FAA/NASA wide panels with multiple site damages [13]. Dawicke et. al. [14] and Gullerud et. al. [15] modeled the tunneling behavior in 2024-T3 specimens through a series of elastic-plastic FE analysis and found that during the initial, transient stage of crack propagation, the CTOA is on the order of a few degrees smaller than its steady state value in the center of the specimen and is a few degrees higher on the surface. When steady state (stable tearing) crack growth is reached, the two values become nearly equal and coincide with the typical steady state value. This study used a unique finite element model that incorporated crack tunneling during the initial stable tearing phase which was measured experimentally by fatigue marking the extent of tunneling after subjecting the specimen to various load levels and crack extensions. These crack front shapes were digitized from photographic images and fit with polynomial curves to describe the crack front shape as a function of through-thickness position. The elastic-plastic portion of the FE analysis relied on the incremental theory of plasticity, and was composed of several layers of elements to model the curved crack front. Gullerud et. al. [15] used steady state CTOA as a crack tip node release parameter for 3-D FE analysis. That is, nodes are released to allow crack extension when all values of CTOA are equivalent across the crack front. The FEA and experimental load versus displacement trends of 2024-T3 aluminum CT and MT specimens compared well. In the following, the authors’ finding on the utility of CTOA in a SENB aluminum specimen is presented. 2.

Method of Approach

A 3-D FE model of the tunneling crack was driven in the generation mode by the experimentally determined crack profile and applied load on 2024-T351 aluminum SENB specimen. The CTOA computed from the surface displacements determined by moiré interferometry was compared with FE computed CTOA for partial validation of the analysis.

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2.1 EXPERIMENTAL PROCEDURE Six 2024-T351 SENB specimens, shown in Figure 1, were monotonically loaded under displacement control, at a rate of 0.254 mm/min until stable crack growth initiated. Here the tests were stopped and the first displacement fields were captured via a CCD camera, connected to a frame grabber in a Windows NT computer. The load rate was then reduced to 0.127 mm/min to maintain stable crack propagation and the ensuing monotonic loading was stopped at crack extension intervals of 0.25 mm to record the and fields. Each test was stopped after a unique level of crack extension had been reached. Since the 3-D flaw analysis will rely on the tunneling present in the interior of the specimen, the experimental approach involved marking the crack front at various crack extensions by post-fatigue marking. Each of the specimens tested in this experimental work were loaded to a different crack extension length to provide a view of the tunneling behavior as the crack extends. Post-fatigue cycling was then used to mark the new crack front in a similar manner compared with pre-fatigue cycling. However, since the extent of tunneling was unknown prior to fatigue marking, an estimated average crack length had to be used for calculation of the values to grow the post-fatigue crack at an appropriate rate. This tunneling analysis is similar to the approach used by Dawicke [1], and Gullerud et. al. [15] in their 3-D CTOA analyses. The post-fatigue crack front profiles were digitized into a computer from photographic images and curve fit to allow it to be mapped in the FE model.

The moiré interferometery technique used for capturing surface displacements was developed by Wang et. al. [16] for measuring relatively large deformations. This method utilizes a low spatial frequency, steep grating of about 40 lines/mm on a highly polished surface to achieve very high contrast fringes. It combines the advantages of the geometric and traditional moiré interferometry methods to allow measurement of relatively large deformations. Here, two collimated beams are directed onto the surface of the specimen at a very shallow angle of approximately 6 degrees. A moiré pattern is visible to the naked eye and can be photographed without the need to capture the 1st order diffraction, which is the typical way to extract moiré data.

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2.2 NUMERICAL MODELING Only one-quarter of the SENB specimen was modeled, due to symmetry considerations. In addition, this model was further truncated at a specific distance from the crack plane for computational efficiency since a full 3-D elastic-plastic model can be very computationally expensive. The truncation distance was determined by an elasticplastic FE analysis of a full quarter model of relatively coarse mesh, i.e. 0.5 mm elements in the vicinity of the crack tip and transitioning to 1.0, and 2.0 mm elements further away. The coarse model, which contained only four elements through the half thickness of the specimen, was loaded to 3.6 KN at which crack extension was expected. The truncation distance for prescribing experimentally obtained surface displacements through the thickness of the FE model was set as the distance from the crack face where the von Mises stress was approximately constant through the thickness, implying a 2-D approximation. For this analysis, the truncation distance was set to y = 15 mm. The application of boundary conditions at a truncation point in the case of bending is a complex issue. The v-displacement (normal to the plane of the crack) varies linearly from top to bottom to ensure a bending load. In contrast, the u-displacement (parallel to the plane of crack) boundary condition is applied at only one point on the 15 mm truncation boundary as it is nearly constant in the vertical direction. To ensure that the prescribed u-displacement was prescribed to reflect reality, several different scenarios were explored in a simple 2-D model of one quarter of the specimen with actual bending employed. Numerical experiment snowed that u-displacement prescribed at the mid height of specimen offered the best comparison with the true shear stress distribution near the truncation boundary in the full model. The final model used in this analysis is shown in Figure 2.

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The extent of tunneling seen in the experimental analysis required a sufficient resolution of element layers through the specimen thickness. To accommodate the extreme slope of the tunneled crack front, thin (0.25 mm) layers were utilized in the areas of the steep crack front gradient with a transition to thicker, 0.5 mm element layers near the center of the specimen where the crack front gradient is not as large. The final model (Figures 2 and 3) consisted of ten, 0.25 mm thick layers near the outer surface followed by three, 0.5 mm thick layers to total the specimen half-thickness of 4.0 mm for a total of 13 element layers. This arrangement of layering also helps to accommodate the natural surface singularity that is expected near the free surface. Element sizes were 0.25 mm in the x and y directions near the crack tip area and transitioning to 0.5 mm and 1.0 mm further away. Figure 2 also shows the model ydirection truncation distance of 15 mm, with experimentally obtained displacement boundary conditions prescribed. The measured tunneling profiles were prescribed as boundary conditions defining the crack face in the final model as seen in Figure 3.

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Results

3.1 TUNNELING CRACK PROFILE Since the crack extension behavior is somewhat difficult to control, only four of the six tests yielded usable results. Images of these four tunneling profiles are shown in Figure 4. Each of these images was calibrated to known dimensions, and the pre-crack and tunneling crack front profiles were digitized using Sigma Scan software. The digitized crack fronts were then fit to polynomials of varying orders from 4th order to 6th order depending on the level of complexity. The polynomial curve fit served to fill in sparse data at regular (0.25 mm) increments through the thickness (z) direction and to provide a means for calculating local crack front tangents via differentiation for use with the numerical analysis. Since the amount of stable crack propagation is difficult to control in an experimental environment, tunneling profiles at very sparse points were obtained. To compensate for the sparseness of data without performing what could amount hundreds of different tests, a linear interpolation scheme was written using Matlab to obtain the intermediate tunneling profiles and increase the data resolution by two times. Figure 5 shows the interpolated crack fronts after two interpolations, and Figure 6 is a comparison of the raw, digitized data to the polynomial fit data.

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Although a fairly insignificant specimen shear lip was expected, and observed in this thick specimen configuration, it is worth noting its value. For this specimen configuration, a shear lip of approximately 0.5 mm per side was measured. This amounts to a total shear lip of 1.0 mm, or 13% of the specimen thickness. After approximately 2.0 mm of surface crack extension, the crack is observed to propagate in a relatively self-similar manner. After several levels of crack extension however, the near mid-plane crack front flattens as it approaches the externally applied mid-span load. This will affect the tunneling as well as the numerical analysis of these crack fronts.

3.2 MOIRÉ FRINGE PATTERN Figure 7 shows typical moiré fringe patterns corresponding to the and v-displacement fields observed in the SENB tests. A program was written in Matlab to compute the and v-displacements from digitized moiré fringe patterns. This allowed easy extraction of relevant displacement quantities at chosen nodal points along the truncated boundary of the FE model as well as along crack faces for CTOA measurement. 3.3 EXPERIMENTAL CTOA The CTOA was calculated on the surface of three specimens labeled T3-3PB, T4-3PB, and T5-3PB. The CTOA calculation consisted of extracting the v-displacement at approximately 1.0 mm behind the current crack tip, obtaining the inverse tangent, and

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multiplying by 2.0 (symmetry). Figure 8 is a plot of the resulting CTOA for these three experiments.

3.4 CALCULATED CTOA Figure 9 shows the CTOA calculated at three locations through the thickness at approximately 1.0 mm behind and normal to the crack tip for the surface, the quarter point (midpoint of numerical model), and the specimen mid-point. The surface CTOA trend shows the typical sharp increase at the beginning of crack growth, followed by a decline to a fairly steady state value of approximately 7-8 degrees. The quarter point and mid-point trends exhibit an interesting slow rise as the crack extends. This behavior is due to the rapid crack propagation in the center (tunneling) near the beginning of the test, which will produce a small amount of crack tip blunting and hence low CTOA. As

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the tunneling slows, the CTOA on the inner layers should increase as seen here due to increasing amounts of plastic deformation and crack tip blunting. It is observed that the CTOA roughly follows the same trend as the near-field calculated on an mm contour. Both the CTOA and the near-field show a decreasing trend followed by a fairly sharp increase at local crack extensions longer than approximately 6.0 mm.

4. Conclusions

A hybrid experimental-numerical procedure was used to determine the CTOA of a tunneling crack in a plane strain, SENB specimen. Moiré interferometry was used to obtain prescribed displacement boundary conditions for application in the numerical model as well as crack opening displacement values for obtaining the surface CTOA. The interior CTOA was lower than the surface CTOA at the initial phase of crack extension due to the difference in plane stress state on the specimen surface and plane strain state in the interior. As the crack extension approaches the specimen thickness, the interior CTOA approaches the surface CTOA thus indicating that a pseudo plane stress state is achieved.

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Acknowledgement

This work was sponsored by U.S. Department of Energy Grant #03-97ER14770. 6. References 1. 2. 3. 4.

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6.

7.

8.

9.

10. 11.

12.

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14. 15.

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Wells, A.A. (1963) Application of Fracture Mechanics at and Beyond General Yielding, British Welding Journal, pp. 563-570. Irwin, G.R., Kies, J.A. and Smith, H.L. (1958) Fracture Strengths relative to Onset and Arrest of Crack Propagation, Proc. of ASTM, pp. 640-657. Goodier, J.N. and Field, F.A. (1963) Plastic Energy Dissipation in Crack Propagation, Fracture of Solids, eds. D.C. Drucker and J.J. Gilman. Interscience, pp. 103-118. Harrison, J.D., Dawes, M.G., Archer, G.L., Kamath, MS. (1979) The COD Approach and Its Application to Welded Structures, Elastic-Plastic Fracture, eds. J.D. Landes, J.A. Begley and G.A. Clarke. ASTM STP 668, pp. 606-631. Shih, C.F., deLorenzi, H.G. and Andrews, W.R. (1979) Studies on Crack Initiation and Stable Crack Growth, Elastic-Plastic Fracture, eds. J.D. Landes, J.A. Begley and G.A. Clarke. ASTM STP 668, pp. 65-120. Kanninen, M.F., Rybicki, E.F., Stonesifer, R.B., Broek, D., Rosenfield, A.R., Marshall, W.C. and Hahn, G.T. (1979) Elastic-Plastic Fracture Mechanics for Two-Dimensional Stable Crack Growth and Instability Problems, Elastic-Plastic Fracture, eds. J.D. Landes, J.A. Begley and G.A. Clarke. ASTM STP 668, pp. 121-150. Kobayashi, A.S., Emery, A.F., Love, W.J. and Chao, Y.H. (1988) Subsize Experiments and Numerical Modeling of Axial Rupture of Gas Transmission Lines, ASME Journal of Pressure Vessel Technology, 110, pp. 155-160. Emery, A.F., Kobayashi, A.S., Love, W.J., Place, B.W., Lee, C.H. and Chao, Y.H. (1986) An Experimental and Analytical Investigation of Axial Crack Propagation in Long Pipes, Engineering Fracture Mechanics, 23, pp. 215-226. Sugie, E., Matsuoka, M., Akiyama, T., Tanaka, K. and Kawaguchi, Y. (1985) Notch Ductility Requirement of Line Pipes for Arresting Propagating Shear Fracture, Proceedings of the 1985 Pressure Vessels and Piping Conference, ASME PVP, 98(8), pp. 53-61. Emery, A.F., Chao, Y.H., Kobayashi, A.S. and Love, W.J. (1992) Numerical Modeling of FullScale Pipe Rupture Tests, ASME Journal of Pressure Vessel Technology, 114, pp. 265-270. Newman, J.C., Dawicke, D.S. and Bigelow, C.A. (1992) Finite-Element Analyses and Fracture Simulation in Thin-Sheet Aluminum Alloy, Durability of Metal Aircraft Strucutres, eds. S.N. Atluri, C.E. Harris, A. Hoggard N. Miller and S.G. Sampath, Atlanta Technology Publication, pp. 167-186. Dawicke, D.S., Plascik, R.S. and Newman, J.C. (1997) Prediction of Stable Tearing and Fracture of a 2000-Series Aluminum Alloy Plate Using a CTOA Criterion, Fatigue and Fracture Mechanics: 27th Volume, eds. R.S. Plascik, J.C. Newman and N.E. Dowling. ASTM STP 1296, pp. 90-104. Newman, J.C. (2000) Advances in Fatigue and Fracture Mechanics Analysis for Aircraft Structures, Structural Integrity for the Next Millennium, eds. J.L. Rudd and R.M. Bader, 1, pp. 342. Dawicke, D.S., Newman, J.C. Jr., and Bigelow, C.A. (1995) Three Dimensional CTOA and Constraint Effects During Stable Tearing In A Thin-Sheet Material, NASA TM-109183. Gullerud, A. S., Dodds, R. H. Jr., Hampton, R. W., and Dawicke, D. S. (1998) 3-D Finite Element Modeling of Ductile Crack Growth in Thin Aluminum Materials, Fatigue and Fracture Mechanics: 30th Volume, Jerina, K. L., and Paris, P. C., Eds., ASTM. Wang, F.X., May, G.B. and Kobayashi, A.S. (1994) Low-spatial Frequency Steep Geometric Grating for Use in Moiré Interferometry, Optical Engineering, 33 (4), pp. 1125-1131.

PATTERNS OF MODERN EXPERIMENTAL MECHANICS Principles of Modeling of Actual Responses of Materials. Machines and Structures

JERZY T. PINDERA Department of Civil Engineering University of Waterloo. Waterloo. Ontario. Canada. N2L 3G1 Abstract The paper presents – in a very condensed form – origin, consequences, and a suggested resolution of the existing dichotomy in applied mechanics, which eventually causes unnecessary failures of machines and structures. The major contributing factor is the fuzzy distinction between the rigorously defined properties of hypothetical abstract mathematical constructs used in applied mechanics, and experimentally determined responses of simplified, approximate mathematical models of real engineering objects. Four figures illustrate this dichotomy. Only selected references are given. A reader is referred to the original publications. 1. Introduction - Frame of Reference 1.1. GENERAL FRAME OF REFERENCE Experimental mechanics is a component of applied mechanics, within the general framework of engineering mechanics. Engineering mechanics is understood here as synergy of applied mechanics and applied mathematics, numerical mechanics and mathematics, experimental mechanics and physics, materials science with physics and chemistry, and computational materials science. All activities occur within a framework provided by society. It is impossible to discuss in a short paper all major aspects of modern experimental mechanics. Thus, out of necessity, the theme of the paper is limited to the issue that – in the author’s judgment – is of a paramount theoretical and practical importance: How to assess reliability and applicability of particular engineering theories, particular design methodologies, and evaluated results of engineering experiments, which could be safely used in real design, including the requirement of a controlled failure. References contain data on major original contributions. In accordance with the author’s academic and professional background in aeronautical engineering, the following criteria are accepted as axioms: Theory and practice may be compatible. Theory must be tested experimentally. No engineering theory must violate any accepted scientific theory. Safety first, the stress is put on rational strength optimization and on controlled failure (in engineering parlance “an airplane is build around a pilot”). 571 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 571–584. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Dichotomy in Engineering Applied Mechanics

2.1. BRIEF CHARACTERIZATION

For the purpose of this paper engineering applied mechanics (EAM) is understood in the narrow sense, as a discipline encompassing analytical mechanics (AM) intertwined with experimental mechanics (EM) and materials science (MS). EAM is based on mathematics and utilizes natural sciences to deal with responses of hypothetical bodies that are assumed to reliably represent real physical bodies. However, the methodology of approach accepted in contemporary EAM often deviates from that accepted in science and in mathematics. This results in dichotomy between very technological methodology that does not require any comprehensive theoretical background, and methodology based on the present state of knowledge in natural sciences that does not accept convenient mathematical simplifications. In science a theory is accepted only when it is tested by experiment or observation - no discrepancy between a theory and an experiment is accepted. This rule is accepted, as an axiom, in aeronautical-aerospace engineering, but is not accepted in general engineering mechanics. Thus, in natural sciences a series of experiments are needed to confirm or to refute the assertions. The purpose of scientific experiments is to determine various responses of a tested body to various energy inputs, in accordance with the obvious truism that no energy, no information. To describe correctly an information-producing energy flow, it is necessary to introduce the concept of a dynamic system response that can degenerate to a static system response. A testing system consists of the tested object, of a loading and measurement subsystem, of a theory of experiment, and an experimenter. The role of experimenter is essential. The notion of system response encompasses mathematical tools, such as transfer functions, and physical parameters, such as impedances and parameters of the flow of various forms of energy through the system [4, 12]. In mathematics only arguments and rigorous computation are needed. In mathematics theories pertain to properties of mathematical objects that exist in a hypothetical, abstract space, and are unequivocally defined by a set of axioms. Mathematics deals with abstract concepts and their interrelations [1]. Computations in mathematics are rigorous – no convenient simplifications are permitted. No energy flow is needed to investigate properties of mathematical constructs, and no analysis of system response is needed. The only unresolved intellectual problem is the surprising effectiveness of mathematics when applied to describe the real, perceived world [35]. Engineering applied mechanics uses the concepts of abstract mathematical objects, or constructs, which can be rigorously presented analytically, and uses the concepts of physical bodies, which are described by physical responses to energy flow. (Unlike natural sciences and mathematics, EAM often uses physical terms to denote abstract bodies, usually without satisfactory experimental evidence that an assumed abstract mathematical object really represents a real physical body, and describes responses of abstract bodies and systems to various energy inputs by relations that contain arbitrary mathematical simplifications, without any proof that the influence of the eliminated mathematical terms is negligible). It is commonly disregarded that mathematics is only a language of a book. Such a situation leads to a dichotomy in EAM – dichotomy between a simple phenomenological approach to the real engineering tasks that accept

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unproven assertions, and a physical approach to the real engineering tasks that complies with scientific requirements [16, 17, 20, 25, 26]. This dichotomy exists at both the analytical and the experimental levels, both in teaching and in engineering practice. Quite often analytical relations for stress/strain/temperature/time states are used that may be considered rigorous in the abstract mathematical space, but are crude when used to represent real physical processes. Some examples are given below. The problem is rooted in curricula. The engineering curricula are usually constructed in such a manner that the students believe the abstract mathematical constructs denoted as physical objects – beam, plate, or column – correctly and completely represent the behavior of real beams, plates or columns. As a result, a number of engineering students request Do not teach me formulas based on assumptions – give me the correct formula, or state As an engineer I do not care where my formulas came from. All I am interested in is how to use them. Such requests are inconceivable in sciences and in aeronautical-aerospace engineering. The notion of idealization, a common term in EAM, does not comply with the Platonic notion of idea – it usually represents a crude simplification that is often not justified [8, 15, 18]. At the experimental level it is often ignored that the reliability of experimental results depend on whether or not the experimenter is satisfactorily conversant with all involved physical theories related to the experiment. There was a time when measurement problems limited reliability of engineering measurements. This problem does not exist any longer in theories and techniques of engineering measurements – (the sky is the limit). However, this incredible progress exposed major weaknesses in applied mechanics: the theoretical framework of experiments, which is commonly and uncritically based on relations of abstract analytical mechanics, and the very simplified relation of physics, including materials science that are used as components of the theoretical foundation , are often unnecessarily deficient as illustrated below. Consequences of this dichotomy are not only intellectual or esthetical but also most practical –many costly failures of engineering structures were caused not by fabrication errors, but by theoretical mistakes made by the designers or by the experimental consultants. Only a few classical examples on unnecessary failures are mentioned here: the collapse of the Tacoma Narrows bridge caused by the well known but neglected aeroelastic interaction (flutter); the failures of the first passenger jet Comet caused by the known but neglected fatigue mode loading; the damages to the John Hancock Tower in Boston that was caused by well known but neglected torsional aeroelastic interaction; or the massive collapse of the power network in Quebec that was caused by neglecting the well known principle of a controlled failure, which was accepted in the aeronautical engineering almost a century ago. Interestingly, the extensive experimental model studies of the Tacoma Narrows Bridge were performed before construction started, but the investigator was not aware of existence of the wellknown flutter, designed a wrong theory of experiments, and could not predict the failure caused by flutter. Also, it is common to neglect the influence of thermoelastic coupling at static and dynamic loadings, which results is misinterpretation of test results: a small increase of the value of apparent elasticity modulus, but a significant increase of value of the dynamic strain wave released by a fracture front [35]. The neglected heat production at plastic deformations in steel often results in apparently mysterious fatigue tensile fractures in compressively loaded machine components.

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This discussed dichotomy is intellectually unpleasant and leads to unnecessary engineering failures, which are often costly. It is possible to eliminate this dichotomy by stressing the importance of scientific methodology in engineering theories, that is, by accepting the same patterns in the development of a theory as accepted in sciences and mathematics. This would necessitate changes in engineering curricula. 2.2. A FEW EXAMPLES OF DICHOTOMY IN EAM 2.2.1. PROBLEMS IN ENGINEERING APPLICATIONS OF SOLUTIONS OF THE THEORY OF ELASTICITY The singular solutions in elasticity theory are very popular, but they must be applied very cautiously as mentioned in [31] and illustrated by Figure 1, [20]. In a large region of a circular disk loaded diametrically the pertinent elegant singular solution yields wrong results, both regarding values and signs of stresses. An analogous situation exists in the isothermal mathematical theory of plasticity [36] Practically all solutions of the mathematical theory of elasticity, which are accepted in engineering design, pertain to a plane stress state, including the notions and analytical relations of fracture mechanics. However, it has been well known for more that seventy years that the stress states in the region of notches in plates are noticeably three-

dimensional, [33]. Extensive quantitative data on the actual three-dimensional stresses (3D) in notches and cracks are given in [15-17, 19-26]. Compatible experimental

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evidence shows that the actual maximal stress in notches and cracks are up to 30% higher that those evaluated analytically or by means of averaging photoelasticity. Local loads also produce 3D stresses in prismatic beams, as shown in [18]. The problem is not only limited do the stress magnitude, but directly pertains to our definition of the stress concentration factor and of the stress intensity factor. Local effects are more significant than it is commonly believed [19]. As a result, two incompatible models of t stress state in a plate with symmetric notches, concurrently used in engineering mechanics and in design procedures: a plane model and a 3D model. Both models lead to different notions of the stress concentration factors and stress intensity factors and yield different vales of maximal boundary stresses [20]. The 3D stress distribution along the crack tip in a beam is quite interesting. The maximal stress occurs at the beam middle plane, and the maximal thickness stress is not negligible, see Figure 2, [20].

The actual distribution of the stress intensity, presented as a function of the distance r from the crack tip, shows that this function is not monotonic and it does significantly depend on the thickness ordinate, Figure 3, [26]. 2.2.2. DICHOTOMY IN MATERIALS CHARACTERIZATION AND IN SYSTEM RESPONSE ANALYSIS Engineering materials can be roughly divided into two classes: time-independent and time-dependent materials. The time-independent materials, such as steel, usually are

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tested at arbitrary strain rates, with disregard of thermoelastic coupling that influences indicated strain values and slightly alters the evaluated modulus of elasticity [9, 14]. It is common to neglect the existence of the huge thermal strain that is produced by heat

generated during plastic deformation of metallic and plastic material, and its influence on the actual stress state in machines and structures. Figure 4, [26], presents the essential difference between two interpretations of the same tensile test for steel: a common phenomenological interpretation, and an advanced engineering physical interpretation. Obviously, the test diagram does not represent material response to a tensile load; it represents response of the system consisting of tested specimen-testing machine, as it is indicated in the testing standards [2] and in bibliography [30]. The evaluation depends on the theoretical level of the accepted models of material and system responses. The typical evaluation of the test results in introduction of artifacts that do not exist, such as notions of the upper and lower yield stresses, or the notion of isothermal plastic deformation processes. Time-dependent materials are of major engineering interest [6, 29, 32, 34]. A very interesting situation exists in the testing of mechanical and optical responses of high polymers. Time-dependent responses to loads of plastic are commonly described by linear ordinary differential equations leading to hereditary integrals, what often is quite satisfactory. The coupling between the mechanical and optical responses is commonly assumed to be proportional, what is often wrong, and the optical responses at different spectral frequencies are assumed to be proportional to ratios of corresponding wavelengths, what violates basic theory and results in large errors [13, 20, 32]. The accepted mathematical models of viscoelastic responses require that the test results be presented as isochronal diagrams. However, it is a common practice to present a three-

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parameter stress-strain-time relations by strain-stress relation at a constant strain rate, and on that basis to decide whether or not the tested material is linearly or nonlinearly viscoelastic. Figure 5, [26], illustrates the issue, and shows that only an isochronal presentation allows for determination of the range of linear response and its change with time [13, 20, 26]. That figure presents graphs of the typical isochronous relations in isothermal conditions at uniaxial loads, which characterize mechanical and birefringence responses of various polymeric materials. Birefringence response is spectral frequency dependent. The isochronous relations show the existence of two linear response ranges, limited by the linear limit stress.

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A very confusing situation exists in applications of photonic radiant energy to the determination of mechanical quantities such as stress, strain or displacement. It is common to apply the most elementary relations of optics and of photonics, disregarding classical textbooks [3, 28]. This eliminates possibility of optimally utilizing actual patterns of interaction between radiation and the deformed body for determination of mechanical quantities of interest. Figure 6 presents, briefly, the existing dichotomy between the traditional engineering understanding of optics and the needed understanding [20]. It should be noted that the theoretically correct understanding of the patterns of propagation of photons inside stressed body resulted in the development of theory and techniques of the strain-gradient stress analysis [7, 26], and of the theory and techniques of the three-dimensional, tomographic isodyne stress analysis – the only nondestructive 3D stress analysis. 3. Basic Patterns of Modern Experimental Mechanics 3.1. VARIOUS NOTION OF A THEORY It is recognized that in engineering mechanics a balance should be maintained between reality and complexity, but this requirement does not justify unnecessary and destructive simplifications and unnecessary vagueness. In engineering mechanics the term theory quite often denotes intellectual structures formulated in language of mathematics, which are supposed to represent quite accurately responses to real loads of real materials and real bodies of various shapes.

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Very often the term rigorous solution is used, which implies that behavior of a the body in questions is presented completely and accurately. Usually no information is given whether the system in question, consisting of a body and of acting loads, actually exists in the physical space, or whether it only exists as a hypothetical system in an abstract space. It is uncommon to consider the influence of the thermo-elastic and thermo-plastic coupling what results in a drastic violation of the conservation of energy principle [9, 34]. This approach contrasts with approaches accepted in natural sciences and in mathematics. In natural sciences it is common to distinguish between a physical theory and a phenomenological theory [10, 15, 27]. A physical theory gives a deep insight into the mechanism of a phenomenon. A phenomenological theory in sciences in based on a set of assumed relations that must be compatible with all known pertinent facts. When a theory is phenomenological, the author states this fact. A basic theory of stress-induced birefringence presented in [28] is a scientific phenomenological theory. Patterns of development and testing of scientific theories are elegantly and succinctly presented by Feynman: guess – compute consequences – compare with experiment. If it disagrees with experiment, it is wrong [5]. Interestingly, theorists in theoretical physics accept this approach, but theorists in applied mechanics, both analytical and experimental, mostly reject it. Evidently, mathematics is a tool, the language of the book. Mathematics deals with mathematical constructs, or object, which exist in a hypothetical, abstract space – elliptic curves, matrices, or the mathematical theory of elasticity [1]. One can denote parameters of those mathematical constructs using terms taken from a real world, but this changes nothing – a beam defined as a mathematical construct still only exists in an abstract space and it must be analytically proven that such a beam partially simulates some patters of behavior of a real beam. Computations in mathematics are rigorous, without exceptions. Thus, mathematics is about mathematical objects, and mathematical theories are satisfied in mathematical models as it is said. There is no uniform approach to engineering mechanics in engineering sciences [11]. Different approaches in various branches of engineering are highly visible. This results in drastically different levels of engineering mechanics, so the levels of corresponding theories in EAM are different. For example, some publications discuss experimental impact problems as obviously isothermal solid body responses to an applied delta-function force, when some other researchers present analogous problems within a framework of a comprehensive system response, involving processes in solid and liquid bodies, patterns of energy transfer and flow, heat generation resulting in major temperature increase, and characterized by varying velocities of elastic waves. The common publications in EAM are usually written within the framework of a particular kind of phenomenological theory, which often violates one, or more, basic physical principles. It is common to use terms that are only commensurate with abstract mathematical constructs, but are misleading when understood as responses of real bodies. The comment presented in [29] about the philosophical flavor of the term “property” is instructive. The influence of the prevailing paradigm [10] in common EAM is strong, despite the incredible progress in technology and sciences.

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3.2. MODELING OF REALITY IN ADVANCED EXPERIMENTAL MECHANICS Reality is understood here as a perceived reality that is a reflection of a real, objective world [27]. It is realized that this is a strong philosophical statement, but it is unavoidable. Reality is perceived by means of energy flow. No energy flow, no perception. Thus we deal with a system, with its transfer functions and impedances of various kinds, and with all related constrains. The levels of perception of an observer and of his knowledge are major components of the perception system. The perceived reality must be simplified to basic models that can be presented in an analytical form, preferably as systems of linear differential equations. Presently, there are two approaches in modeling reality: a simple phenomenological approach based on utilization of the concept of “black boxes”, and a physical approach based on a deeper insight into the mechanism of modeled body responses, or processes. Figure 7 presents a simplified diagram of both modeling approaches. Both approaches result in development of basic physical models and of representative mathematical models, but the scientific levels of both kinds of models are drastically different [4, 8, 15,]. Advanced experimental mechanics is based on the physical methodology that excludes internal and external incompatibilities. Practical experience shows that the physical approach in development of physical and mathematical models is superior, and opens new research horizons [15, 17, 20, 25, 26]. Perhaps it should be noted that the intellectual level and elegance of analytical solutions in applied mechanics that deal with abstract mathematical constructs, and the physical reliability of the developed relations, are not causally connected. 3.3. ADVANCED EXPERIMENTAL MECHANICS: DEVELOPMENT PATTERNS Particular measurement procedures and their application to determine responses of materials, bodies, and systems no longer represent the main tasks of experimental mechanics. New measurement theories and techniques allow performing any measurement with accuracy surpassing engineering requirements. Experience shows that physicists are better equipped than engineers to develop advanced measurement systems and procedures. Theories and techniques of measurement have already developed as separate disciplines. Consequently, the major problem of applied mechanics became the reliability of the analytical and experimental procedures. One major task is to produce physical data on the actual behavior of objects, materials, and systems to be used as the basis for constructing more advanced and more reliable pertinent analytical relations, in particular for constructing reliable constitutive relations. The basic underlying issue is a rational designing of pertinent physical models starting from first principles, such as mechanical, thermodynamic, and photonic interactions at the atomic, molecular, and crystalline levels during the processes of deformation and fracture. Models must be physically admissible, thus energy and power must be finite. The other major task is to produce reliable experimental data. Thus, experiments in AEM must be based on correctly developed physical models that are used as a basis for a rational theory of experiments.

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These features allow characterizing the developing AEM by two principles: No theory – no experiment, No experiment – no theory. The major features of Advanced Experimental Mechanics (AEM) are the theoretical compatibility of applied theories and techniques, and the compatibility of theories, observation and experiments. These features allow for eliminating the unnecessary and destructive dichotomy in contemporary applied mechanics. 4. Concluding Remarks.

The major problems in contemporary experimental mechanics are of a theoretical nature. Disregarding approaches accepted in scientific theories and in mathematical theories mostly causes them. It would be useful if authors of papers presenting results of research in applied mechanics – including analytical and experimental mechanics, together with computational fracture mechanics and computational materials science would clearly state when they deal with the hypothetical mathematical constructs that may, or may not, bear some resemblance with responses of real physical objects, and when they attempt to assure a reliable correspondence between the developed system of analytical expressions and the selected responses of real bodies, supported by a discussion of experimental evidence. In other words, it would be useful to have a clear distinction between the responses of constructs existing within an applied mathematics realm, and the responses of models of bodies, existing within a physical realm. Evidently, the meaning of the same differential equations and their parameters depend on the general frame of reference. Modern engineering is an interdisciplinary field of intellectual and technological activities, involving physics, chemistry, biology, mathematic, economic, societal issues, political issues, and ethical issues. Thus, to satisfy societal expectations, education patterns of leading engineers should be commensurate with the notion of a modern Renaissance man. 5. 1.

References

Aleksandrov, A. D., Kolgomorov, A. N., Lavrent’ev (eds.): Mathematics. Its concepts, Mthods, and Meaning, The M.I.T. Press, Cambridge, Massachusetts, 1963. 2. ASTM,: 1969 Book of ASTM Standards, The American Society for Testing and Materials, Philadelphia, Pennsylvania, 1969. 3. Born, M. and Wolf, E.: Pricciples of Optics, Pergamon Press, Oxford – New York, 1975 4. Doeblin, E. O.: Measurement Systems: Application and Design, McGrow-Hill Book Co., New York, 1983. 5. Feynman, R.: The Character of Physical Law, MIT Press, Cambridge, Massachusetts,, 1993. 6. Haddad, Y. M.: Viscoelasticity of Engineering Materials, Chapman& Hall, Lomdon- New York, 1995. 7. Hecker, W. F. and Pindera: General relations of strain-gradient stress analysis, Theoretical and Applied Fracture Mechanics 25 (1996), 233-261 8. Kac, M.:Some Mathematical Models in Science, Science 166 (1969), 695-699. 9. Kestin, J.: A Course in Thermodynamics, McGraw-Hill Book Company, New York, 1979. 10. Kuhn, T. S.: The Structure of Scientific Revolution, University of Chicago Press, Chicago, 1970. 11. Leipholz, H. H. E.: On the Role of Analysis in Mechanics, Transactions of the CSME 7 (1983), 3-7. 12. Natke, H. G.: About the Role of Mathematics in Engineering – Thoughts of a Scientist between both Branches, GAMM Mitteilungen 1966, Akademie Verlag, Berlin 19 (1996), 121-131.

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13. Pindera, J. T. and Straka, P.:On physical measures of rheological responses of some engineering materials in wide ranges of temperature and spectral frequencies, Rheological Acta 13 (1974), 338-351. 14. Pindera, J, T., Straka, P. and Tschinke, M. F.: Actual Thermoelastic Responses of some Engineering Materials and its Applicability in Investigations of Dynamic Responses of Structures, VDI-Bechichte 313 1976), 579-584 15. Pindera, J. T.: Foundations of Experimental Mechanics: Principles of Modelling, Observation and Experimentation, in J. T. Pindera (ed.), New Physical Trends in Experimental Mechanics (CISM Courses and Lectures No. 64), Springer-Verlag, New York (1981) 199-327. 16. Pindera, J. T.: Advanced Experimental Mechanics in Modern Engineering Science and Technology, Transactions of the CSME 11 (1987) 125-138. 17. Pindera, J. T.: Advanced Experimental Mechanics and its Components: Theoretical, Physical, Analytical and Societal Aspects, in A. P. S. Selvadurai (ed.), Development in Engineering Mechanics, Elsevier, Amsterdam – New York, (1987) 347-414. 18. Pindera, M.-J., Pindera, J. T. and Ji, X.: Three-dimensional Effects in Beams – Isodyne Assessment of a Plane Solution, Experimental Mechanics 29 (1989), 23-31. 19. Pindera, J. T.: Local Effects and Defect Criticality in Homogenous and Laminated Structures, Trans. ASME, J. Pressure Vessel Technology 111 (1989), 136-150. 20. Pindera J. T. and Pindera M.-J.: Isodyne Stress Analysis, Kluwer Academic Publishers, Dordrecht, 1989. 21. Pindera, J. T., Hecker, F. W. and Wen, B.: Testing theoretical bases of caustic method in fracture mechanics and stress analysis, Theoretical and Applied Fracture Mechanics 15 (1991), 11-33. 22. Pindera, J. T. and Liu, X.; On the Actual Three-dimensional Stresses in Notches and Cracks, Composites Engineering 1 (1992), 281-301. 23. Pindera, J. T. and Wang, G.: Isodyne Stress Analysis of Adhesively Bonded Symmetric Joints, Experimental Mechanics 32 (1992), 348-356. 24. Pindera, J. T., Josepson, J. and D. B.: Electronioc Techniques in Isodyne Stress Analysis. Part 1: Basis Relations. Part 2: Illustrating Studies and Discussion, Experimental Mechanics 37 (1097), 33-38, 110-114. 25. Pindera, J. T.: Principles and Approaches of Advanced Experimental Mechanics in Service of Modern Technology, in Y. M. Haddad (ed.), Advanced Multilayered and Fibre Reinforced Composites, Kluwer Academic Publishers, Dordrecht, 1998. 26. Pindera, J. T.: Techniques of Tomographic Isodyne Stress Analysis, Kluwer Academic Publishers, Dordrecht, 2000. 27. Popper, K. R.: The Logic of Scientific Discovery, Harper and Row, London, 1977. 28. Ramachanidran, G. N. and Ramaseshan, S.: Crystal Optics, in S. Flügge (ed.), Encyclopedia of Physics, volume XXV/1 Crystal Optics. Diffraction, Springer-Verlag, Berlin, (1961) 1-592. 29. Reiner M.: Rheology, in S. Flügge (ed.), Encyclopedia of Physics VI, Springer-Verlag, Berlin, (1958), 434-520. 30. Siebel, E. and Ludwig, N.: Handbuch der Werkstoffprufung (Material Testing Manual), SpringerVerlag, Berlin, 1958. 31. Sokolnikoff, I. S.: Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956. 32. Stuart, H. A. (ed.): Die Physik der Hochpolymeren, Dritter Band (Physics of Highpolymers, Third Volume), Springer-Verlag, Berlin, 1955, 33. Thum, A. et all.: Verformung, Spannung und Kerbwirkung (Deformation, Stress and Notch Action), VDI-Verlag, Düsseldorf, 1960. 34 Valanis, K. C. (ed.):Constitutive Equations in Viscoplasticity: Phenomenological and Physical Aspects, The American Society of Mechanical Engineers, United Engineering Center, New York, N. Y., 1976 35. Wigner, E. P.: The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Communications on PURE AND APPLIED MATHEMATICS XIII (1960) 1-14. 35. Zehnder, A. T. On the temperature distribution at the vicinity of dynamically propagating crack in 4340 Steel, J. Mech. Phys. Solids 39 (1991), 385-415. 36. M.: Combined Loadings in the Theory of Plasticity, PWN, Warszawa, 1981.

INVERSE METHODS IN EXPERIMENTAL MECHANICS J. F. DOYLE School of Aeronautics and Astronautics Purdue University West Lafayette, IN 47907 Abstract This paper provides a framework for the active use of experimental mechanics in solving partially specified (or inverse) problems. It is based on a wavelet (or unit load) representation of the unknown applied loading that allows the FEM analyses to be performed as an external process and thus utilize the power and versatility of commercial codes. As a demonstration of the method and its attributes, different structures are studied experimentally using strain gage and whole-field data. 1.

Introduction

Experimental analyses are regularly combined with finite element analyses to give a deeper understanding of a problem. More often than not, however, these analyses are used passively in the sense that the separately generated results are simply compared on some common basis; a simple example is comparing measured strain gage results with those predicted using FEM. What this paper addresses is the active use of experimental mechanics as an integral part of an FEM analysis and to explore some of the new analysis opportunities made available by this merger. A fully specified problem in FEM analyses is one where all the material properties, geometries, boundary conditions, loads, and so on are known and the displacements, stresses and so on are required. A partially specified problem is one which deals with an existing structure or prototype where some information is missing. An example is modeling a structure with a loose bolt or one with wind loading. A common practice is to try to make all problems fully specified by invoking reasonable modeling assumptions. For example, it would be reasonable to assume an elastic joint for the bolt and report the results for joint stiffness ranging from very stiff to somewhat flexible. Because assumptions are used, the results must be reported over the full possible latitude of the assumptions. This adds considerably to the total cost of the analysis. Additionally, and ultimately more important, the use of assumptions make the results uncertain and even unreliable. On the face of it, the most direct way of narrowing the uncertainty in the results is to simply measure the unknown. For a dimension this is probably straightforward but what about the force due to the impact of hale on an aircraft wing? This is not something that can be measured directly. The primary concern of research reported in this paper is the establishment of formal methods for including measurements as part of the analysis of partially specified problems. 585 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 585–594. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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One key to our approach is to take the applied loading as the fundamental unknown even if we are not specifically interested in it. To see this, consider a linear complex structure subjected to an excitation, the response is governed by the equations

Within a structural context, therefore, system changes such as fracture, erosion, bolt loosening, and so on, can only appear as changes of stiffness and/or mass (damping is usually an unknown anyway). The changed condition is represented as a perturbation of the stiffness and mass matrices from the original state as and the applied loading as equations can be rearranged as

the governing

That is, the changed structure can be thought of as the original structure (and hence assumed completely known) plus a set of additional loads. The vector clearly has information about the changed (unknown) state of the structure; therefore, if somehow, we can determine then it is a direct (but sometimes subtle) process to extract the unknown structural and/or loading information. Thus, the essential problem is to determine an applied load from some appropriate measurements. This is a problem in force identification and References [1-8] can be used as an entry to some of this literature. The example problems discussed here are chosen to illustrate just some of the possibilities of new analyses and questions that can be explored although they are all restricted to being static and linear. 2.

Computer Implementation for Static Problems

The formulation of the problem is detailed in Reference [9] and will not be repeated here; instead we just summarize the approach as implemented in the computer code Wavelet.

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The governing equations for a static linear system can be discretized using the finite element method as

In this, is the vector of all the free degrees of freedom and is of size [K] is the banded symmetric stiffness matrix, is the vector of all applied loads (some of which are zero), is the vector subset of that contains the non-zero applied load scales, and is the matrix that associates these load scales with the degrees of freedom. In the inverse problem of interest here, both the applied loads and displacements are unknown but we know some information about the responses. In particular, assume we have a vector of measurements of size which are related to the structural DoF according to where the logic matrix [Q] is of size We want to find the forces that make the system best match the measurements. The unknown forces are represented as the collection

where is a distribution of forces and the corresponding scale. There are unit distributions. The key to understanding the computer implementation is this series of load vectors in – each load case can be performed externally by a commercial FEM code. The basic relationship is shown in Figure 1. One of the roles of the inverse program is to prepare the scripts to run the external programs. Each causes a response such that the total response (because of the linearity of the system) is

Because the responses are obtained by the FEM, they will have imbedded in them the effects of the complexity of the structure. Furthermore, it is important to note that these responses are not necessarily just displacement, they could be strain or principal stress difference as required by the experimental method, but in each case it would be precisely the same force scales Consequently, the inverse program does not need to perform any operation such as differentiation of displacements in order to get strains. In other words, anything particular to the modeling or mechanics of the structure is handled by the FEM program. The measured data at a collection of locations are related to the computed responses as where [Q] only symbolically plays the role of selecting the subset of participating in forming the data. That is, the arrays are established directly from the participating components of and The minimizing principle is

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where [W] is a diagonal data weighting array, and [H] is one of the regularization forms discussed in Reference [9, 8]. A readable discussion of regularization is given in Reference [10] and a more extensive mathematical treatment can be found in Reference [11]. This latter reference has an extensive bibliography. The least squares minimization leads to

If supplementary information is available, it is incorporated simply by appending to the matrix [A] before the least squares procedure is used. This system of equations is symmetric of size and can be solved by the standard techniques such as LU decomposition [10]. The purpose of and the regularization term is to overcome the ill-conditioning usually associated with least squares solutions. A reasonable beginning value of is to try where Tr is the trace of the matrix computed as the sum of diagonal components. Three cases are implemented in Wavelet: zero, first, and second order. 3.

Experiment I: Displacement Data

Figure 2 shows a double exposure holographic fringe pattern for the out-of-plane displacement of a pressure loaded plate. The displacements are related to the fringe order by

where N are the black fringes, is the wavelength of light, and illumination and viewing, respectively.

and

are the angle of

The fringe pattern is manually digitized along the center of some of the black fringes and the data stored as pixel location and fringe order. This is then scaled to the physical coordinates of the mesh. Figure 3 shows the limited number of discretized data points taken from the image. The empty circles shown were used to scale the data from pixel coordinates to physical coordinates. Figure 3 also shows the FEM mesh used.

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The data were relocated onto the nearest nodes of this mesh by the scheme discussed in Reference [9].

In formulating the inverse solution for this problem, there is only one unknown and that is the pressure. That is, is comprised of a single vector representing a uniform pressure of unit magnitude. (Because of the different element sizes, does not necessarily contain constant values.) This is a case where the script is for a uniform pressure and the specific contents of is determined by the Mesh/FEM program. The nominal measured pressure is 7.63 kPa and computed pressure is 7.77 kPa. The comparison is very good. Figure 2 shows a comparison of the reconstructed fringe pattern with that measured. Since the pressure comparison is good, it is not surprising that this comparison is also good. A theme running through the inverse solutions of this paper is that once the unknown loads have been determined, then the apparatus is already available for the complete stress analysis. Figure 4 shows, for example, the and stress contours. What is significant about these reconstructions is that (in comparison to traditional ways of manipulating such holographic data) the process of directly differentiating experimental data is avoided. Note that for plate problems this would require double differentiation. Furthermore, it is because of this attribute, that only a limited number of data points needed to be recorded.

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Experiment II: Strain Gage Data

Figure 5 shows a photograph of a 3-D thin-walled reinforced box-beam structure. The vertical loading is applied through a turn-buckle which has a force transducer. The dial gauges can be used to locate the shear center (the load position which causes no rotation). Figure 6 shows the geometry and the mesh used in the modeling. The reinforcing steel bars are attached to a thick end-plate; the thin aluminum skin is riveted to these bars at every 25 mm (1 in.). The box-beam structure is modeled as a 3-D shell.

Figure 6 shows the locations of three delta rosettes placed mid-way along the beam; the shear strain is easily obtained from a delta rosette as [12]

where the gage is oriented along the x-axis. Figure 7 shows the recorded shear strains for six load increments when the nominal load position is at x/W = 0.71 (this location is actually very close to the shear center); the data trends for the other load locations are similar. These data were fitted with straight lines and then the values adjusted to the origin. The maximum shear strains at the nominal maximum load of P = 60 were used in the inverse solutions.

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In the inverse problem, both the load and its position are assumed unknown. Rather than treat the unknown position as a variable (which would require an iterative solution) the problem instead is converted to a load problem by considering two unknown loads at the lower corners. These two forces are equipollent to the single force at position x and by taking moments about the load point, gives or Figure 7 shows a plot of the computed load position against its nominal value. The comparison is quite good. This experiment shows that while an instrumented structure can be its own load transducer, it is also capable of determining where the load is applied. As a consequence, this provides more flexibility in designing the load scheme.

5.

Experiment III: Photoelastic Data

The previous sections dealt with experimental methods where there is a linear relation between the data and the mechanical quantity. Photoelasticity is different because it has a nonlinear relationship between the observed fringes and applied loads. While it is true that a doubling of the load causes a doubling of the fringe orders, this is not true for non-proportional loading cases. That is, the fringe pattern caused by two loads acting simultaneously is not equal to the sum of the fringe patterns of the separate loads. Since our inverse method is based on superposition, we must modify it to account for this nonlinear relation. The measured fringe data are related to the stresses by [12]

where N are the black fringes, is the material fringe constant, and h is the model thickness. As is usual in nonlinear problems, we begin with a linearization about a known state. That is, suppose we have a good estimate of the stresses as then the true stresses are only a small increment away. Do a Taylor series expansion about this known state to get

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Let the stress increments be computed from the scaled unit loads so that for a particular component of force it is given by

This must be computed for each component of force. In anticipation of using regularization, it is best to deal directly with the force distributions. Noting that we write the approximate relation as

with and

The minimum principle can now be established leading to

This is solved repeatedly using the newly computed to replace In terms of the computer implementation of Figure 1, the unit solution is determined only once (for each load), but the updating of the stress field requires an FEM call on each iteration. Figure 8 shows the loading arrangement for a photoelastic model with a hole. The specimen is essentially under three-point loading. The inset shows the light field fringe pattern. The fringes were recorded using a diffuse light polariscope with white light and the grey scale image shown is that recorded directly by the camera. The material fringe value was calibrated with this arrangement. Although there is some wash-out of the fringes near the concentration points (load and support point plus edge of hole), the fringe patterns are deemed adequate because we will mostly use data away from these locations.

In this experiment, the contact load was distributed by placing a soft layer between the model and the applied load; consequently, there was no possibility of knowing a priori the distribution. It is not the intention here to determine the precise distribution of this load – to do that would require collecting data close to the distribution. Rather, we are concentrating on doing a stress analysis in the region close to the hole. The fringe data of Figure 8 were discretized in a manner similar to that of the holographic fringes. That is, the center of the dark fringe was manually digitized at a number of points and this data relocated to the nodes in its vicinity. Only fringes that are clearly visible were discretized.

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In all iterative schemes, it is important to have a method of obtaining good initial guesses. Figure 9 shows the results of finding force pairs at neighboring nodes at many node locations. No regularization was used and convergence occurred in about five iterations for each of these two-force problems. Shown is the sum and difference of the forces. The sum gives an idea of the resultant force of the actual distribution while the force difference gives an idea of the location of the resultant. The separation between the force pairs depends on the ability of the data to resolve the forces without regularization. In the present test, the separation used was four node spacings. This information can now be used to make a reasonable initial guess.

The initial guess for the iterative scheme was taken as the resultant load with all other fourteen being zero. Figure 9 shows a typical reconstructed distribution, where about ten iterations was required for convergence. The precise distribution depends on the amount of regularization used. A key point, however, is that the solution in the vicinity of the hole does not depend sensitively on the precise nature of the distribution of applied stress. That is, all reconstructed distributions gave nearly the same solutions in the vicinity of the hole. Figure 8 shows a comparison of the measured and the reconstructed fringe patterns. The patterns close to the hole are almost identical. The patterns near the distribution show a difference which might be due to the presence of a shearing traction. Figure 10 shows the reconstructions for the and stress contours.

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6. Discussion This paper provides a framework for the active use of experimental mechanics in solving partially specified problems. That the FEM analyses can be arranged as an external process is a significant attribute because it means the power and versatility of commercial codes can be leveraged and this greatly extends the range and type of problems that can be solved. A point worth reiterating is the significant side benefits of using an FEM-based modeling. First, there is a consistent underlying framework for handling the experimental problem. That is, the same mesh is used for the manipulation of the experimental data (e.g., smoothing, relocating, cropping). Second, the postprocessing apparatus is already in place for obtaining information such as stress, as well as simply presenting the results. The example problems discussed were for linear, static problems – the emphasis was on structural complexity as represented by the box-beam structure. The underlying formulation, however, is also applicable to dynamic problems as illustrated in References [13,8]. It is also applicable to nonlinear problems. 7. Acknowledgements This work was supported in part by a U.S. Army Multi-Disciplinary University Research Initiative (United States Army Grant No. DAAH04-96-10331) awarded to Purdue University. 8. References 1. 2. 3. 4.

5. 6.

7. 8. 9. 10. 11. 12. 13.

K. K. Stevens, “Force Identification Problems: An Overview,” Proceedings of SEM Spring Meeting, Houston, 1987, pp. 838-844. M. T. Martin and J. F. Doyle, “Impact Force Identification from Wave Propagation Responses,” International Journal of Impact Engineering, 18, 1996, pp. 65-77. A. S. Kobayashi, “Hybrid Experimental-Numerical Stress Analysis,” Experimental Mechanics, 23, 1983, pp. 338-347. C.-H. Haung and W.-Y. Shih, “An Inverse Problem in Estimating Interfacial Cracks in Bimaterials by Boundary Element Technique,” International Journal for Numerical Methods in Engineering, 45, 1999, pp. 1547-1567. T. H. Baek and R. E. Rowlands, "Hybrid Stress Analysis of Perforated Composites using Strain Gages,” Experimental Mechanics, 41, 2001, pp. 195-203. T. Nishioka, “An Intelligent Hybrid Method to Automatically Detect and Eliminate Experimental Measurement Errors for Linear Elastic Deformation Fields,” Experimental Mechanics, 40(2), 2000, pp. 170-179. S.-M. Cho, A Sub-Domain Inverse Method for Dynamic Crack Propagation Problems, M.S. Thesis, Purdue University, 2000. J. F. Doyle, “Reconstructing Dynamic Events from Time-limited Spatially Distributed Data,” International Journal for Numerical Methods in Engineering, to appear, 2001. U. Kang and J. F. Doyle, “An Inverse Method for Static Problems using Whole-field Data,” Experimental Mechanics, submitted, 2001. W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling, Numerical Recipes, ed., Cambridge University Press, Cambridge, 1992. A. Neumaier, “Solving Ill-conditioned and Singular Linear Systems: A Tutorial on Regularization,” Society for Industrial and Applied Mathematics, 40(3) 1998, pp. 636-666. J. W. Dally and W. F. Riley, Experimental Stress Analysis, Ed., McGraw-Hill, New York, 1991. J. F. Doyle, “A Wavelet Deconsolution Method for Impact Force Identification,” Experimental Mechanics, 37, 1997, pp. 404-408.

COMPLEX STIFFNESS IDENTIFICATION BY INVERSE METHODS H. SOL and W.P. DE WILDE Vrije Universiteit Brussel (V.U.B.) Department Mechanics of Materials and Constructions Pleinlaan, 2 1050 Brussels, Belgium Abstract The basic principle of an inverse method for material stiffness identification is to compare measured values of a test specimen with computed values from a numerical model of the test specimen. The parameters in the numerical model are the unknown material stiffness properties and they are iteratively updated starting from an initial set of parameters until the computed output matches the measured values. This paper first outlines the general procedure used for material stiffness identification with inverse methods. The procedure is next illustrated with an example on the identification of complex engineering constants of composite materials. In this example the vibration behaviour of a freely suspended test plate is used as the information source. The measured quantities are the resonance frequencies of the test plates and the damping ratios of their associated mode shapes. The resonance frequencies are used to tune the real part of the complex moduli, while the damping ratios are used to identify the imaginary part.

1.

Introduction

Making models is very common in engineering science. With a model one tries to simulate the behaviour of a physical reality, but it is clear that even the best model will always have its limitations [13], [10]. The appearance of computers has increased the possibility of making accurate numerical models to simulate the static and dynamic behaviour of engineering constructions. The most powerful and popular numerical method for simulation of the behaviour of constructions is the finite element method. Since its appearance in 1960 [6], many good textbooks have been written about finite element modelling e.g. [19]. A finite element model of a construction contains geometrical and constitutive parameters. For an assumed known geometry and constitutive parameter values, the construction model allows the computation of the response ('Output') for given excitation forces ('input'). It must be however clear that one can only expect a physically reliable output if the model is sufficiently accurate and if the inputted parameter values are sufficiently correct (if not: garbage in, garbage out!). The values of most constitutive or material parameters involved in a model can be measured experimentally by appropriate standard testing. The ASTM society [2] has developed state of the art procedure for the identification of many material properties. Unfortunately, in some cases it is not possible to apply standard testing procedures. Sometimes, the material shows such 595 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 595–608. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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complex constitutive behaviour that standard testing can not give satisfying solutions. In such cases it can be considered to use an inverse method. In other cases, inverse methods sometimes just offer an advantageous alternative for standard testing. The principle of an Inverse Method for Material Identification (IMMI) is to compare a measured output from an experiment on a test specimen with the computed output of a numerical model of the same specimen loaded with the same input values (Figure 1). The difference between the measured and computed output is called "the residual". The residual can be incorporated in a user defined cost function that describes the goodness of fit of the numerical model with the test specimen. A simple example of a cost function is the summation of the squared differences between the corresponding output components.

The - a priori unknown - material properties in a IMMI are the parameters in the numerical model. These material properties are (usually iteratively) tuned in such a way that the computed output matches the measured output. The parameter modifications are computed by minimisation of the cost function. The final result of the output matching procedure is an estimate of the parameters values and eventually an estimate of their error bounds.

2.

Working principles of material identification by inverse methods

All a priori unknown material parameters of the numerical model can be stored in a parameter column {p}. The computed output column {y} contains the corresponding components of the measured output column {m}. The relation between the output column {y} and the parameter column {p} can be linear or non linear. The two cases will be considered in the next two paragraphs.

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2.1. LINEAR RELATION BETWEEN THE OUTPUT COLUMN AND THE PARAMETER COLUMN The relation (2.1) gives a general expression for a cost function with the parameter column {p} as an independent variable (a repeated index indicates a summation) and thus also as a function of the residual {m} - {y}:

In (2.1) is a Cost function yielding a scalar value, (or alternatively written as or ) is a column containing the NP unknown material parameters, contains initial user estimates for the material parameters, {m} contains the (NM x 1) measured output values, {y} is a (NM x 1) column containing the NM computed output values using {p} in the numerical model, (or or is a (NM x NM) weighting matrix applied on the difference between the measured column and the output column, and is a (NP x NP) weighting matrix for the difference between the initial parameter column and the parameter column {p} . Because the output column must contain enough information to evaluate properly the parameter column, NM > NP in a IMMI. The Cost function C(p) has a minimal value for the optimal parameter values column . It can be seen easily that (2.1) reduces to a weighted least squares cost function of the residual if is set to zero. Optimisation algorithms for minimising the cost function can be categorised in two classes: (i) algorithms that use function evaluations only (direct search algorithms) and (ii) algorithms that use output gradient information. An overview of algorithms from both classes can be found in [1], [14], [12]. The algorithms of the second class are more efficient for both IMMI and will be illustrated in the following text. Let be an initial estimate of the parameter values. The value of the Cost function in an arbitrary position {p} close to the position can be found with a Taylor expansion:

or where

is the (symmetrical) Hessian matrix evaluated at

The partial derivative of (2.3) with respect to the parameter component

is:

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For

the above expression becomes zero, hence:

thus: The partial derivative in expression (2.6) can be evaluated by taking directly the partial derivative of (2.1) with respect to component and evaluating the result at

with: For

the (NM x NP) sensitivity Matrix. (2.7) becomes:

The sensitivity matrix [S] is constant because of the assumed linear relation between {y} and {p}. The Hessian matrix can be obtained by taking the partial derivative of (2.7) with respect to component and evaluating the result at

Thus with (2.9) and (2.8) in (2.6):

The formula (2.10) is called the estimator'. In the linear case, (2.10) gives directly a value for starting from an arbitrary initial value (2.10) contains an expression that must be inverted. can be used to obtain the necessary numerical stability in case the sensitivity of the output {y} for parameter variations is poor.

2.2.

NON-LINEAR RELATION BETWEEN THE OUTPUT COLUMN AND THE PARAMETER COLUMN

Unfortunately, the difficulty with most IMMI is that the relation between the output vector {y} and the parameter vector {p} is nearly always non-linear. This means that updating the parameter values {p} from an initial value to a final optimal value has to be done in an iterative way. Expressions (2.2) - (2.9) hence need to be evaluated (linearised) in intermediate positions between and

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(2.6) becomes:

with:

and: or, if the non-linear part of the sensitivity matrix is neglected:

(2.12) and (2.14) in (2.11) yields:

The estimator (2.15) is now a recurrent formula that updates {p} from an intermediate value towards a better value with j = 0 , 1, 2, ....till convergence is reached at the value The linearised value of the sensitivity matrix must be re-computed in every iteration step. The consecutive values of {p} iterate towards if convergence occurs and if the convergence is not blocked at a local minimum of the Cost function (instead of in the desired global minimum). In (2.15) the weighting matrices and and the column with the initial parameter estimates remain constant during the consecutive iterations. The 1 second term in the general cost function (2.1) hence tends to act as a kind of 'spring' that prevents the parameter column of evolving too far from the initial estimate The 'stiffness' of this 'spring' is dependent on the relative values of the weighting matrices. It is also possible to re-evaluate the cost function (2.1) in each iteration by replacing the initial estimate by

In that case, the last term in (2.15) vanishes and the estimator becomes:

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Illustration: the resonalyser procedure

3.1. GOAL OF THE RESONALYSER PROCEDURE The resonalyser procedure is a IMMI that aims the identification of complex orthotropic engineering constants by the measurement of the vibration behaviour of rectangular test plates. The procedure is based on the above described inverse method and results in averaged material stiffness values over the whole plate area. The obtained material stifnesses are therefore ideally suited as input values for finite element models. The visco-elastic behaviour of orthotropic materials in a plane state of stress can be characterised by 4 independent engineering constants. The constants appear as complex values in the compliance matrix that describes the relation between the strain column and stress column:

In (3.1) the used symbols are: : normal strain component in the i-direction : shear strain component in the 12-plane : normal stress component in the i-direction : shear stress component in the 12-plane : Main Poisson's ratio in the 12-plane : Young's modulus in the i-direction : Shear modulus in the 12-plane The complex engineering constants can be written as:

The real part of the engineering constants describes the pure elastic behaviour of the material while the imaginary part describes the dissipative or damping behaviour. The dissipative part is often called 'the tangents delta'. The characterisation of orthotropic material hence requires the evaluation of 8 independent constants: 4 real values and 4 'tangents delta' values.

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The resonalyser procedure proceeds in two parts: the first part estimates the real part of the engineering constants using measured resonance frequencies and the second part estimates the imaginary part by measured modal damping ratios. The second part is optional and will be only executed by people who are also interested in the damping behaviour of the examined material.

3.2. ESTIMATION OF THE REAL PART OF THE ENGINEERING CONSTANTS The principle for the determination of the real part is to compare measured resonance frequencies of a rectangular test plate with frequencies computed with a numerical model of the test plate (Figure 2).

A thin plate model, based on the Love Kirchhoff theory, is used for the numerical model of the test plate [15], [16]. Extensions of the resonalyser procedure use thick plate models with linear and higher order shear corrections [11]. The resonance frequencies and mode shapes of the numerical model can be found by the solution of an eigenvalue problem. The parameter column contains the 4 real parts of the orthotropic engineering constants (hence NP = 4):

The 5 first resonance frequencies of the freely suspended test plate are stored in the output column {y} and the measurement column {m} (hence NM = 5). The relation between {p} and the computed resonance frequencies in the numerical

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model is non-linear. The estimation of {p} is thus obtained using the iterative estimator (2.17). The formula yields good results if the parameters can be observed through the measured resonance frequencies. This means that at least one of the five measured resonance frequencies must sufficiently change for variations of each of the 4 material parameters. It can be shown [16] that this demand requires a length/width ratio a/b of the test plate close to the value given by:

The preparation of such a test plate however requires the knowledge of the ratio which is basically unknown. The value of this ratio can be found by cutting two test beams in perpendicular directions from the test plate (Figure 3).

The measurement of the fundamental (first) resonance frequencies and of this two beams allows the computation of fair values for and with (3.5) and hence of the ratio a/b using (3.4):

(EI is the bending stiffness, the weight per unit length, L the beam length). The obtained values for and provide at the same time excellent starting values for the inverse identification procedure. A freely suspended test plate according to (3.4) has a torsional mode shape associated to the first resonance an anticlastic bending mode shape associated to the second resonance and a synclastic bending mode shape associated to the third resonance The stress pattern of the torsion mode shape is dominated by shear stresses and hence the resonance frequency is very sensitive for variations of The difference between the resonance frequencies and is mainly due to the value of Poisson's

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ratio. (for the imaginary case of zero Poisson's value, both mode shapes will have the same resonance frequency). These observations provide an easy way of finding good starting values for and

in which: The first eigenvalue of the test plate The second eigenvalue of the test plate The third eigenvalue of the test plate The sensitivity of the eigenvalue associated with the torsional mode shape for

variations

The sensitivity of the difference of the synclastic and anticlastic eigenvalues for variations of Poisson's ratio The empirical evaluation of

and

is explained in [16].

The identification of the elastic parts of the engineering constants is performed using the 5 plate frequencies:

The components of the (5 x 4) sensitivity matrix [S] are recomputed numerically in every iteration step (see e.g. [16], [9], [11]):

in which is the i-th computed mode shape and is the stiffness matrix of the numerical model of the test plate (both computed with the value of the parameters at a given j-th iteration step). The parameter value in the final iteration is the result of the first stage of the resonalyser procedure.

3.3. ESTIMATION OF THE COMPLEX PART OF THE ENGINEERING CONSTANTS The complete identification of the complex moduli requires also the knowledge of the imaginary (or dissipative) part of the engineering constants. In the orthotropic material case this includes the determination of 4 tangents delta. These tangents values can be identified by the measurement of the modal damping ratios associated with the 5 plate resonance frequencies. The (4x1)

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parameter column {p} and the (5x1) measurement column {m} for the second stage hence are:

The modal damping ratios can be evaluated by curve fitting the decaying sinusoidal signal in the time domain after an excitation of the test specimen with the envisaged resonance frequencies (see Figure 4). This signal can be curve fitted with the formula (4.11):

in which

is the modal damping ratio and is the circular resonance frequency The procedure must be repeated for each of the five resonance

frequencies.

The curve fitting in the time domain allows to obtain accurate results even for lightly damped test specimen. The main problems for the measurement of the modal damping ratios are situated on the experimental level [9]. Small variations in the boundary conditions (suspension threats, accelero meter wiring,..) induce external damping that can not be distinguished from the internal material damping. Therefore it is strongly advised to generate and measure the signals without physical contact. The excitation can be performed acoustically with a loudspeaker and the measurement of the decaying signal can be done with a laser vibrometer. The influence of the boundary conditions can be minimised by the suspension of the test specimen in the nodal points of the mode shapes. A simplifying element in the second stage is that the relation between the parameters and the measured modal damping ratios is linear:

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The coefficients in (3.12) can be computed with the numerical plate model from the modal strain distributions and modal damping energy expressions [9]. The sensitivity matrix [S] can be derived easily from (3.12):

The linear character of the equation (4.12) allows to start from arbitrary initial values for the parameters. They can thus be taken all as zero. The least squares estimator (3.10) can be used with and and will yield immediately the optimal parameter values.

3.4. PRACTICAL PROCEDURE AND AN EXAMPLE The above described resonalyser procedure is automated using a MS-windows PC software programme. The PC is equipped with a data acquisition card for the vibration measurements. The practical measurement procedure proceeds as follows: Machine two test beams from the original test plate as indicated in figure 3 Measure the first resonance frequency of both test beams and compute initial values for and Determine the dimensions of the test plate according to (3.4) and machine the test plate Measure the first 5 frequencies of the test plate Start the first stage of the resonalyser procedure for the identification of the real parts using (2.17) Measure the modal damping ratios associated with the measured frequencies of the test beams and plate Start the second stage of the resonalyser procedure for the identification of the imaginary parts using (2.10) At the end of the procedure the complete set of orthotropic complex engineering constants are identified. Table 1 gives an example of the results of the resonalyser procedure on a Carbon/Epoxy [0°, 90°]s test plate with dimensions (0.29 m, 0.26 m, 0.0018 m). Other examples from the resonalyser and validation of

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these results with results from traditional testing (ASTM tensile and DMA tests) is given in [9], [15], [16].

3.5. ERROR DISCUSSION Error discussion of the final results of material identification using inverse methods is a difficult matter because the physical definition of the desired material properties is given outside the framework of the identification procedure (e.g. a modulus is defined as a ratio of strain and stress while in the numerical model it acts as a model parameter). Inside the IMMI framework and with the assumption that the numerical model reflects perfectly the behaviour of the test specimens, the standard variation on the final results can be computed [5], [7] based on the standard variation of the measurements. In the resonalyser procedure the estimation of the real part is based on the measurement of resonance frequencies. The experimental error on measured frequencies is small, even if non-sophisticated measurement equipment is used. The results of the IMMI with high quality test plates (homogeneous elastic properties and constant thickness and thus an accurate numerical model) are accurate. A detailed error discussion on the real part errors can be found in [16]. The estimation of the complex part is based on the measurement of the modal damping ratios. This part is more sensitive to measurement noise and non-linear behaviour of the damping as a function of the excitation amplitude. The results are hence less accurate. A detailed discussion on the complex part errors can be found in [9]. More examples of IMMI procedures on different physical problems can be found in [17], [18].

5. Discussion: Resonalyser procedure versus standard testing It is useful to compare the resonalyser procedure with standard procedures to measure material stiffnesses like tensile and shear testing as described in ASTM procedures [3], [4]. A basic difference between standard testing and the resonalyser procedure is that standard testing aims to create a uniform field (uni-axial stress fields), while the resonalyser procedure analyses the complex multi-axial stress fields occuring in mode shapes. Indeed, standard testing is based on analytical processing of the measured results and thus requires data that can be described with a 'simple' formula. Uniform response fields are usually very difficult to obtain correctly from an experimental point of view and require often expensive equipment. IMMI in general can deal with complex heterogeneous fields containing

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a lot of information. Heterogeneous fields occur usually in a natural way and are thus easier to obtain experimentally than uniform fields. An advantage of standard testing is that the error can be observed straightforwardly thanks to the direct character of the procedure, while IMMI is indirect and thus makes the error more difficult to estimate. A important advantage of IMMI is that it is possible to identify many material properties simultaneously from one test specimen, while standard testing is limited to the determination of one material property from one test specimen. This can lead to considerable time savings with IMMI. Probably the biggest advantage of IMMI is that it can be tailored to any test specimen geometry and boundary conditions. Only the imagination of the experimentalist is the limit for the conception of a good experiment with IMMI.

6.

Conclusion

IMMI is an identification technique that offers a uniform approach for different physical systems. The most important advantages of the method are the freedom in experiment conception and the possibility of simultaneous identification of set of parameters from the same test-system A disadvantage is the indirect character of the method and thus the increased sensitivity to errors. IMMI also requires a sound understanding of the relation between the desired parameter values and the measured response and a good insight in the nature of experimental errors and the numerical model limitations. It can be foreseen that in the future the sources of errors of IMMI will become smaller and smaller with the rapid evolution of computer power (more accurate numerical models) and the technical evolution in the measurement techniques of field information (data acquisition and scanning techniques). This evolution, in combination with a better understanding of the principles of mixed methods, will without any doubt lead to improved results and new applications of IMMI in the future.

7. References 1. 2. 3.

4. 5. 6. 7. 8. 9.

Atrek, A. 1984. New directions in optimum structural design, John Wiley and Sons, ISBN 0471-90291-8. ASTM, American Society for testing Materials, http://www.astm.org. ASTM, Designation D 3039, Standard Test Method for Tensile properties of Fiber resin composites, revised 2000. ASTM, Designation E 756-83, Standard Method for Measuring Vibration Damping properties of Materials, 1983. Catin, D.E. 1989. Estimation Control and Discrete Kalman Filter, Springerverlag. Clough, R.W. 1960, The finite element in plane stress analysis, Proc. 2nd.ASCE conf. on electronic computation. Collins, J.D. & Hart, G.C. & Hasselman, T.K. & Kennedy B. 1974. Statistical Identification of Structures, AIAA Journal, 12(2), pp.185-190. Cottin, N & Natke, H.G. 1986. On the Parameter Identification of Elasto-mechanical Systems using Weighted Input and Model Residuals, Ingenier. Archiv.56, pp.106-113. De Visscher, J., 1995, Identification of the complex stiffness matrix of orthotropic materials by a mixed numerical experimental method, Ph.D. Thesis presented at the Vrije Universiteit Brussel, Belgium.

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10. De Wilde, W.P. 1992, Numerical Modelling of linear elastic and visco-elastic response of 11. 12.

13. 14. 15. 16. 17. 18. 19.

composite structures, Computational Methods in Applied science, Elsevier Science publishers, pp. 233-245. Hua, H. 1993. Identification of Plate Rigidities of Anisotropic Rectangular Plates, Sandwich Panels and Orthotropic Circular Disks using Vibration Data. Ph.D. Thesis presented at the Vrije Universiteit Brussel, Belgium. Mottershead, J.E., Friswell, M.I. 1993, Model updating in structural dynamics: a survey, Journal of sound and vibration 167 (2), pp 347-375. Natke, H.G. 1995, What is a True mathematical model?, Inverse problems i n engineering, Vol.1, pp.26 7-272. Reklaitis, G.V., Ravindran, A., Ragsdell, K.M. 1983, Engineering Optimisation, John Wiley and Sons ISBN 0-471-055794. Sol, H. et all., 1996, La procedure resonalyser, La revue des laboratoires d'Essais, n° 46, pp. 10-12. Sol,H., 1986, Identification of anisotropic plate rigidities using free vibration data, Ph.D.Thesis presented at the vrije Universiteit Brussel, Belgium. Sol, H. & Oomens, C. Editors 1997. Material Identification Using Mixed Numerical Experimental Methods, Kluwer Academic Publishers. Vautrin, A. & Sol, H Editors 1991 Mechanical Identification of Composites., Proc of Euromech 269, St.Etienne, France. Zienkiewicz, O.C. 1977, The finite element method, McGraw-Hill, ISBN 0-07-084072-5.

CONSIDERATIONS OF A FLUTTER PREDICTION METHODOLOGY USING A COMBINED ANALYTICAL-EXPERIMENTAL PROCEDURE

PIERGIOVANNI MARZOCCA Virginia Polytechnic Institute and State University Blacksburg, VA 24061 LIVIU LIBRESCU Virginia Polytechnic Institute and State University Blacksburg, VA 24061 WALTER A. SILVA NASA Langley Research Center 100 NASA Road Hampton, VA 23681-2199

Abstract A non-destructive procedure enabling one to predict the flutter instability boundary from the data acquired in the subcritical flight speed regime is presented. The proposed technique combines an analytical approach with the experimental tests carried out in flight or in a wind tunnel. The expected outcomes of this study are: a) to reduce the risks of flying in the proximity of the flutter critical boundary, and b) to reduce significantly the amount of flight tests required in any flight clearance test program, that are both time consuming and costly. 1. Flutter Prediction Using Combined Analytical-Experimental Procedure

1.1 INTRODUCTIVE CONSIDERATIONS One of the most important phases involving the aeroelastic design is the evaluation of the flutter instability boundary by means of flight and tunnel tests at subcritical speed [1,2]. The objective of the flutter analysis is to quantify the critical boundaries of flutter [3-5]. Even at the present time, when analytical predictive methodologies have become increasingly useful toward this goal, flight and wind tunnel testing in conjunction with data analysis are always required for the verification of the theoretical predictions. Attempts to formulate adequate methods that permit a reliable prediction of the flutter boundary at speeds well below the flutter speed were made in the past by several investigators [3,6-8] and recently by Lind [5]. As it was pointed out, in order to reduce the number of flight or wind tunnel tests, a reliable technique for real-time monitoring of the aeroelastic stability of the aircraft is required [9]. A systematic approach should 609 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 609–618. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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be adopted toward the accurate estimation of aeroelastic structural parameters (mass, damping, stiffness, etc.) and to use these findings to effectively predict flutter boundary. In [1] an analytical approach was developed to address this problem. At the same time, the possibility of obtaining the flutter boundary from the aeroelastic response was indicated in [10] for 2-D lifting surfaces, and in [11,12] for aircraft swept wings in a compressible flow. One of the key elements of the methodology presented in this paper consists of the use, in combination with a real-time parametric identification procedure, of the coefficients of the aeroelastic governing equation. These parameters will be used for an exact evaluation of the aeroelastic response from the combined analytical procedure with the real time experimental results. Only when the certitude of the correctness of the coefficient of the equation of motion is obtained, we can extrapolate the effective value of the flutter speed. In addition, the analytical procedure of determining the subcritical aeroelastic response can be applied at any flight speed regime, i.e. subsonic, transonic and supersonic, via the inclusion in the model of the pertinent aerodynamics based on indicial functions [11-13]. 1.2 FLIGHT AND WIND TUNNEL TESTS A particularly dangerous aeroelastic instability that one must avoid is the flutter, which results from the interaction of structural, inertial and aerodynamic forces. Flutter is a special case of self-excited oscillation in which the wing exposed to the airflow, absorbs energy from the airstream in such a way that the added energy overpowers the natural damping of the structure, and causes the amplitude of the oscillations to increase exponentially at time unfolds. Flight tests are performed in order to evaluate the characteristics of an aircraft and determine a range of flight conditions in which an aircraft can operate without the occurrence of any aeroelastic instability. The totality of flight conditions fulfilling this requirement constitute the flutter flight envelope, see Fig. 1.

Within the traditional methods of flight testing, the flight envelope is expanded successively via a series of test points, using measurements generated in response to some excitation. This procedure involving flight flutter testing is based on the

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experimental determination of the modal damping coefficients and on their variation with the flight speed [14]. Such dynamical properties as damping are estimated from the measurement data, and trends are used to determine whether the envelope may be further expanded. As is well known [3], the procedures based on the determination of the damping are efficient in the case of mild or moderate flutter. On the other hand, the flutter phenomena can be violent, i.e. explosive, and therefore it may exhibit, with the increase of the flight speed, a rapid deterioration of the modal damping. In addition, in order to be able to use the flight test data, one must ensure that the data points are closely spaced, although the trends do not guarantee what increases in flight conditions may be safely considered without the sudden occurrence of the flutter instability. In this sense, flight flutter testing is potentially dangerous because an unexpected flutter instability, namely one not predicted from the trends can occur during a transition to a new test point. On this basis, a prediction of the flutter speed can be hazardous with the risk of a loss of the vehicle and even of human lives. It clearly appears that these methods are not reliable enough. Several approaches have been applied to improve the deficiencies in the flutter testing. Roy and Walker [9], have formulated an improved method for the evaluation of damping, Zimmerman and Weissenburger [3], and Houbolt [7], have tried, with certain success, of predicting the flutter speed at a speed well below the flutter dynamic pressure. However these methods are working well for system up to two degrees of freedom (2-DOF). Nissim and Gilyard [15], have developed an identification procedure as to identify the coefficients of the equations of motion of M-DOF dynamic systems. Such a procedure can be used in conjunction with the present one as to arrive to a better prediction of the flutter dynamic pressure. The procedure presented here provides the possibility to establish the stability boundary in a direct way from both the flight tests and the theoretical prediction of the flutter speed, see Fig. 2. In addition, having in view the fact that the proposed procedure is based on data that are elaborated in real-time, an expanded flutter flight envelope, is likely to be obtained. The prediction of the flutter boundary is accomplished via its identification from the subcritical aeroelastic response of the wing structure to pulse loads applied in a fixed position, such as the wing tip, and for selected flight speeds. On the other hand, transfer functions between accelerometers and the excitation can be determined and theoretical models predicting the flutter speed can be improved based on the deviations between the aircraft vibration test and model transfer function results [16]. Flutter predictions via subcritical responses and eigenvalues analyses can be performed with the updated and adjusted aeroelastic model parameters. Wind tunnel and flight tests are efficient ways enabling one to quantify in a rather exact way the aeroelastic instability parameters of lifting surfaces [17,18]. In the general case, using the wind tunnel tests it is possible to arrive close to the critical value of the flutter speed without the loss of the model [19]. The advantage of this new proposed method and its potential application carried out in flight or in a wind tunnel toward exactly determining the flutter boundary is presented next.

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1.3. COMBINED ANALYTICAL –EXPERIMENTAL PROCEDURE 1.3.1. PRELIMINARIES

In this section, the problem of the determination and study of the aeroelastic response behavior and the use of these predictions towards determination in flight of the flutter critical speed is addressed. The procedure presented in this section is applicable to the full aircraft, in general, and to advanced aircraft lifting surfaces and empennage incorporating composite materials [20] toward determination of the flutter dynamic pressure, in particular. In the most general form, the aeroelasticity of dynamic systems can be written symbolically as [21]:

Herein S , A and I denote the structural, aerodynamical and inertial operator, respectively, whereas Q denotes the external disturbance forces, assumed to be known explicitly, while t is the time variable. In our procedure it is represented by a pulse load. Its expression can be found in [10]. Equation (1) represent the functional relation between the generalized displacement u, at a specified point and in a specified direction, and the external disturbance generalized forces.

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SUGGESTED PROCEDURE FOR THE DETERMINATION OF THE FLUTTER SPEED

The proposed testing procedure avoids the risk of flying in the vicinity of the flutter speed and assists the tests toward determination of the flutter speed. A chart depicting the procedure enabling one to predict the flutter speed from this combined analytical – experimental procedure is included (Fig. 3). The theoretical analysis of the aeroelastic response to a delta pulse acting on the wing will provide the possibility to determine the critical flutter speed. At the same time, from the analysis it is possible, for selected values of the flight speed, to obtain information on the response behavior of the aeroelastic system. In order to supply the necessary elements enabling one to substantiate the method that will be described in the sequel, a simple application of the flutter instability and aeroelastic response of a swept wing operating in an incompressible flow and exposed to blast pulses is presented in Fig. 4. To do this, in the linear formulation, the aeroelastic governing equation (1) specialized for a swept aircraft wing [11,12], can be converted into the Laplace transformed space and solved for the unknown generalized displacements in bending and twist. Taking the inverse Laplace transform of these quantities, one obtain the bending and twist time-histories due to the external pressure pulse applied at the wing tip. Notice that, well established procedures for the identification of aerodynamic indicial Functions using flight test data [13] and of aerodynamic impulse response using digital filter technique [22], can be adopted for the evaluation of the aerodynamic load acting on the aircraft or on the lifting surface. This procedure can provide a model superior to the aerodynamic derivative model [11,12]. An accurate approach enabling one to evaluate the flutter speed is based on the recursive model parameter evaluation that can be extracted directly from system identification or modal test data [16,17,23], in conjunction with the analytical model. All the information extracted from the subcritical response can be used in the process depicted in Fig. 4 that highlights the aeroelastic response of the swept wing (the sweep angle being ), when the flight speed increases, in the subcritical speed range. The phase portraits, at the top of the graph and in the chart of Fig. 4, show how the plunging response time-history to a unitary pulse (at the zero time and located at the typical section of the wing) evolves with the increase of the flight speed V . Corresponding to the flutter speed, that coincides with that obtained from the eigenvalue analysis, the trajectory of motion describes an orbit with constant amplitude, the so called center. For as time unfolds, a decay of the amplitude is experienced, which reflects the fact that in this case a subcritical response is involved (stable focal point), while for the response becomes unbounded, implying that the occurrence of the flutter instability is impending (unstable focal point).

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Figures 5a,b and c illustrate the effect that the airflow might have on the vibrational behavior of the wing structure. Figure 5a shows how, when the aircraft is flying at a subcritical speed, the vibrations usually damp out. But, when the flow speed is increasing, the vibration amplitudes can remain constant, Fig. 5b, or increase without limit, Fig. 5c. This phenomenon is extremely dangerous and can result in the catastrophic failure of the structure.

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The critical value of the flutter speed in Fig. 4 is obtained here from both the aeroelastic response and the eigenvalue analysis of the homogeneous system of equations. As a preliminary requirement, it is necessary to choose Mach number that is kept constant throughout the procedure. During the flight tests, exciting the wing by a unitary pulse at selected values of the flight altitude [24], keeping the Mach number constant, is possible to measure in each specific condition of flight, the maximum acceleration or deflection at the wing tip. This procedure can be implemented via the use of the structural acceleration measures that can easily be evaluated with real time measurements via accelerometers placed on the aircraft wing. The data, corresponding to the acceleration/deflection at the wing tip, obtained from the test can be included in the analytical loop, Fig. 4, where the theoretical maximum acceleration ratio (normalized by the acceleration of gravity g) or deflection at the wing tip for selected sweep angle are indicated. When the values of the deflection provided via flight tests coincide with those in the plot, Fig. 4, it becomes clear that the real flutter speed is that obtained from the analytical prediction. Supposing that the predictions via experimental procedures provide different values of the maximum acceleration/amplitude for selected values of the flight speed, the input data have to be updated as to yield, for two flight speeds and the same pressure signature, correct values of the aeroelastic response. In this way, we can avoid the problem of the uncertainties that are involved in any aeroelastic analysis, theoretical or experimental. In such conditions, using the methodology presented in this section, a new line connecting the maximum amplitudes of the aeroelastic response, that is determined uniquely by two values of the flight speed can be traced. This line is labeled as TL (Theoretical Line). It is clearly seen that corresponding to EL1 (Experimental Line) and EL2, the wing will feature lower and larger stiffness characteristics, respectively. At the same time, the change in the slope of EL1 or EL2 as compared to that of TL, is due to a different damping or mass parameter as compared to that of the theoretical model. We assume that the coefficient of equations of motion can be reasonably identified, using for example the Nissim’s method [15], so that these parameters should be updated as to arrive to the convergence of the theoretical prediction with the experimental ones. The slope of the experimental line, the amplitude envelope of the response that bounds the oscillation in time, and a parametric study of the response in terms of the structural parameter will provide the possibility to update these coefficients. With an automatic iterative procedure, updating the parameter at each loop, the new theoretical line of the maximum amplitudes that coincide with the experimental ones can be determined. Consequently, the exact value of the flutter speed of the aeroelastic structure will be supplied from extrapolation and intersection with the updated maximum acceleration or deflection line. As clearly appears from this paper, the issue related with the test technique and the excitation system as to obtain a pulse was not addressed. This excitation can be produced via a control surface pulse, that can approximate a delta function at high frequency content and due to its short test duration can be repeated many times at each test point, or by thrusters, ballistic exciters or impulse generator, simple devices that can produce short transient response [24].

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2. Concluding Remarks

This paper suggests an alternative approach for the prediction of the flutter speed via a combined experimental-theoretical analysis. The approach presumes that modal parameters, such as frequencies, damping ratios, and aeroelastic responses in terms of deflections or accelerations (at sensor locations), can be identified during the subcritical flight speed range. The proposed approach is based on the possibility of updating a theoretical model in terms of its mass, damping, and stiffness, that enables an extrapolation of the flutter speed. Among the advantages of the proposed procedure there are: i) possibility of a systematic and accurate flutter prediction on the basis of the on-line real modal parameters; ii) can contribute to the expansion of the flight envelope without weight penalties; and, iii) possibility of reducing significantly the amount of flights required in the flight test program. In addition, this methodology may significantly improve the procedures for the flight flutter testing and reduce its cost. 3. Acknowledgment

The partial support of this research by the NASA Langley Research Center through Grant NAG-1-01007 is acknowledged. 4. 1. 2.

References

Houbolt, J.C. (1974) Subcritical flutter testing and system identification, NASA-CR-132480. Rosenbaum, R. (1974) Survey of aircraft subcritical flight testing methods, NASA-CR-132479, ARAP-218. 3. Zimmerman, N.H. and Weissenburger, J.T. (1964) Prediction of flutter onset speed based on flight testing at subcritical speeds, J. Aircraft 1 (1), 190-202. 4. Lind, R. and Brenner, M. (1999) Robust aeroservoelastic stability analysis, Springer-Verlag, London. Lind, R. (2001) Flight testing with the flutterometer, presented as AIAA Paper 2001-1654 at the 5. 42nd AIAA/ASME/ASCE/ASC Structures, Structural Dynamics, and Materials Conference, Seattle, WA, 16-19 April. 6. Richardson, J.R. (1965) A more realistic method for routine flutter calculations, AIAA Symposium on Structural Dynamics and Aeroelasticity, Boston, Massachusetts, August 30 - September 1, 1017. 7. Houbolt, J.C. (1975) On identification frequencies and damping in subcritical flutter testing, NASA SP-415, Flutter testing techniques, A conference held at Dryden Flight Research Center, Edwards, CA, 9-10 October. 8. Bucharles, A., Cassan, H., and Roubertier, J. (1990) Advanced parameter identification techniques for near real time flight flutter test analysis, AIAA/SFTE/DGLR/SETP 5th Biannual Flight Test Conference, Ontario, CA, May 22-24. 9. Roy, R., and Walker, R. (1985) Real-time flutter identification, NASA-CR-3933. 10. Marzocca, P., Librescu, L., and Silva, W.A. (2000) Aerodynamic indicial functions and their use in aeroelastic formulation of lifting surfaces, presented as AIAA paper 2000-WIP at the 41st AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Atlanta, GA, 3-6 April.

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11. Marzocca, P., Librescu, L., and Silva, W.A. (2001) Aeroelastic response of swept aircraft wings in a compressible flow field, presented as AIAA Paper 2001-0714 at the 39th AIA A Aerospace Sciences Meeting, Reno, NV, 8-11 January. 12. Marzocca, P., Librescu, L., and Silva, W.A., (2002) Unified approach of aeroelastic response and flutter of swept aircraft wings in an incompressible flow, AIAA Journal, in press. 13. Gupta, N.K. and Iliff, K.W. (1982) Identification of aerodynamic indicial function using flight data, presented as AIAA-82-1375 at the 9th AIAA Atmospheric Flight Mechanics Conference, San Diego, California, 9-11 August. 14. Russo, M.L., Richardson, P.T., and Perangelo, H.J. (1983) Identification of linear flutter model, presented as AIAA-83-2696 at the AIAA/AHS/IES/SETP/SFTE/DGLR 2nd Flight Testing Conference, Las Vegas, Nevada, 16-18 November. 15. Nissim, E. and Gilyard, G.B. (1989) Method for experimental determination of flutter speed by parameter identification, NASA-TP-2923. 16. Lacabanne, M. and Esquerre, J.P. (1995) Correlation between theoretical flutter models and tests for civil aircraft, Aerospatiale – Aircraft Division TR. 17. Dat, R., Dunoyer P. (1981) Identification des modes propres d’une structure a partir des reponses a une excitation non appropriee, ONERA TP-1981-25, 52ème réunion de la Comission Structures et Matériaux de l'AGARD, Cesme-Izmir, 6-11 April, in French. 18. Smith, M.S. (1999) Analysis of wind tunnel oscillatory data of the X-31A aircraft, NASA-CR1999-208725. 19. Klein, V. and Noderer, K.D. (1994) Modeling of aircraft unsteady aerodynamic characteristics, Part 1 - Postulated models, NASA-TM-109120. 20. Karpouzian, G. and Librescu, L. (1994) Comprehensive model of anisotropic composite aircraft wings suitable for aeroelastic analyses,” J. of Aircraft, 31 (3), 703–712. 21. Bisplinghoff, R.L. and Ashley, H. (1996) Principles of Aeroelasticity, Dover Publications, Inc. 22. Silva, W.A. (1997) Identification of linear and nonlinear aerodynamic impulse response using digital filter techniques, NASA-TM-112872, presented as AIAA Paper 97-3712 at the AIAA Atmospheric Flight Mechanics Conference, New Orleans, LA, 11-13 August. 23. Gaukroger, D.R., Skingle, C.W., and Heron, K.H. (1980) An application of system identification to flutter testing, J. Sound and Vibration 72 (2), 141-150. 24. Kehoe, M.W. (1995) A historical overview of flight flutter testing, NASA-TM-4720.

DISPLACEMENT-BASED SMOOTHING HYBRID FINITE-ELEMENT REPRESENTATION FOR STRESS ANALYZING PERFORATED COMPOSITES

K. Y. HE and R. E. ROWLANDS Department of Mechanical Engineering University of Wisconsin Madison, WI 53706

Abstract A reliable and effective displacement-based smoothing finite-element representation is demonstrated for determining stresses on, and near, the edge of a geometric discontinuity in finite, orthotropic composites. Relatively little measured displacement data are necessary and they originate away from the cutout. Data acquisition positions can be selected at the user’s discretion and scatter in the measured input information is automatically filtered.

1. Introduction Obtaining accurate stresses at holes or notches in finite orthotropic materials by purely theoretical or numerical techniques can be difficult and/or undependable, particularly when far-field loading and geometry are complicated or unknown. Although numerous experimental techniques are available for obtaining stress (strain, displacement) information, such measurements are often unreliable on the edge of a discontinuity. One superior approach is to evaluate stresses on the edge of a hole or notch from measured information away from the cutout. Rhee [1, 2] determined the stresses associated with cutouts in composites utilizing concepts from Gerhardt’s hybrid element [3]. However, Rhee’s method necessitates interpolating displacements at the external nodes of the hybrid element from adjacent moiré fringes. One has only finite flexibility as to where to locate these nodes and reliability of that approach is further eroded if fringe quality or density near a node is inferior. Such shortcomings are overcome here by actually embedding, numerically and experimentally, the external portion of the hybrid element amongst the measured displacements. Data acquisition positions now become quite arbitrary and can be selected at the user’s discretion (e.g., use input data having high resolution and reliability). The present technique enjoys the additional advantage of automatically smoothing experimental scatter from the measured input data. 619 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 619–628. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Analytical Background

2.1 GENERAL COMMENTS

Consider a loaded finite composite component containing an arbitrarily-shaped hole, Fig. 1. The edge of the hole is denoted by Moreover, (line DEF) is the external boundary of a hybrid element (region S) which surrounds the cutout, and the area between contours ABC and GHI forms a data acquisition region, R, throughout which one collects measured displacements. Stresses are be evaluated here throughout S, including along the edge of the hole, from the measured displacements in region R.

2.2 HYBRID ELEMENT

The stresses and displacements in a loaded homogeneous orthotropic material under plane stress are given by [4]

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Re denotes the real part of a complex number, complex material properties are the roots of the characteristic equation associated with the compatibility relationship, and and are

Satisfying the boundary conditions on the edge of the cutout of Fig. 1 is aided by mapping its edge, in the physical plane into the unit circle, in the where (j = 1 and 2) maps the unit circle and its exterior in the into the edge of geometric discontinuity and its exterior in the physical plane, and and are related to material properties and location within S [1-3]. Traction-free conditions on the edge, of the cutout can be implemented using the principle of analytic continuation, such that [3]

where and are real numbers, n and m are positive integers, and B and C are material properties [3]. Combining Eqs. 1 through 5 gives the stresses and displacements within the hybrid element, i. e.,

where is the coefficient vector of Eqs. 4 and 5, and [U] and [S] depend on the cutout shape, material properties and k [5]. Terms with k = 0 contribute to rigid body motion and can consequently be omitted. As occurs here, for geometrical, mechanical and material symmetry about the x- and y-axes, Eqs. 4 and 5 involve only real and odd coefficients Taking m = n such that results in a total of (m+1) real coefficients. The elastic strain energy of a region, S, (hybrid element) surrounding a traction-free boundary, is related to {c} and the nodal displacement vector, on the external boundary, of the element [3,5]:

with

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and ds is an arc length of integral path along (DEF), Fig. 1. Matrix [L] of Eqs. 8 and 9 relates displacements, on the external boundary of the hybrid element to the nodal displacements, on and and are the direction cosines of the normal to A quadratic interpolation is utilized here for [L] [5]. Minimizing the potential energy in the hybrid element of Eq. 11 gives the following relationship between and

Matrices [H] and [G] depend on material and geometry of a given hybrid element. Combining Eqs. 6 through 10 gives

Quantity of Eq. 11 can be treated as a shape function similar to those in structural FEA. Once is known, Eq. 10 provides and the stress functions, stresses and displacements (hence strains) are available throughout S by Eqs. 1 through 11, including on Determining smoothed experimentally-based data, will now be discussed. 2.3 SMOOTHING HYBRID FINITE-ELEMENT REPRESENTATION

References 1 and 2 used the foregoing concept to determine stresses associated with cutouts from nodal values of on by interpolating from adjacent moiré fringes. That concept is enhanced here by embedding, numerically and physically, the outer boundary of S, within the data-acquisition region, R, of Fig. 1. The smoothing hybrid finite-element representation (SHFER) is well suited for smoothing measured displacements and providing accurate at the nodes along [5]. In addition to filtering noisy experimental input, SHFER enables the user to employ measured data from arbitrary locations throughout R and to accommodate a region S of Fig. 1 of widely-varying shape and/or size. The relative impact of individual measured input values can also be controlled numerically through a weighting factor. Denote the in-plane displacement vector of the two components u(x, y) and v(x, y) in region R of Fig. 1 as Measured displacements at locations in R are used to quantify the displacement representations throughout R and particularly to evaluate reliable on of region S of Fig. 1. It is advantageous to employ a predicting representation in R that does not necessarily agree exactly with the measured displacements at each input location but smoothes the measured information. Such a smoothing function can be obtained by minimizing a functional. Using concepts from Refs. 6 and 7, and discretizing region R, one obtains [5]

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where are the local coordinates of the locations of measured input displacements in the surrounding isoparametric elements (between and ABC of Fig. 1), are the global coordinates of the input locations in that portion, of region S between curves GHI and of Fig. 1, J is the number of input locations in a surrounding element and H is the number of input locations in Vectors and represent measured input displacements in the surrounding elements and in respectively. Integer M is the total number of surrounding elements, and and are the nodal displacements of a surrounding element and on the external boundary of the hybrid element, respectively. Shape functions and (the latter given by Eq. 11) are evaluated at input locations in the surrounding elements and in respectively. Equation 12 can be rewritten as

or

where is the assembly of and and is the assembly of and in region R. Equation 14 resembles that for structural FEA, but the individual terms are physically different. Pseudo stiffness of Eq. 14 involves geometry, material properties, number of terms and coordinates of measured input values in region R. In addition to material properties, pseudo load vector, involves the locations and magnitudes of the input displacements in R. Solving Eq. 14 for gives the nodal displacements throughout R, including those on the external boundary of S, The total amount of measured input data utilized throughout region R should be no less than the total d.o.f. of the elements which make up region R.

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4. Experimental Results

The method was used to determine the tangential stress on the edge of the central circular hole in the graphite/epoxy composite tensile member of Fig. 2. Elastic properties are and The strong, stiff orientation of the orthotropic composite of Fig. 2 is parallel to the applied load, P. The hole has diameter plate width W = 38.1 mm (1.5 in.), length of the plate between loading grips is 38.1 cm (15 in.) and composite thickness t = 2.2 mm (0.085 in.).

Four-beam interferometric moiré was used to record the u- and v-displacements. The specimen contained a 1200 lines/mm (30,480 lines/inch) crossed diffraction grating and the virtual analyzer had a frequency of 2400 lines/mm (60,960 lines/inch), for a displacement resolution of (16.4 per fringe. The moiré system, composite plate and loading frame were all supported on an isolated table. Moiré fringes associated with each of the x- and y-displacements were recorded throughout the shaded area of Fig. 2, and those for the v-field are shown in Fig. 3 at P = 1334.4 N (300 Ib) Figure 4 shows the configuration of the hybrid element (region S) and the dataacquisition region R utilized, with c = 2r = 11.25 mm. (0.443 in). Geometry and material symmetry enable one to work with only one quarter of the composite member. Both u- and v-displacements were employed at 69 locations in region R (of Fig. 4) of a quarter-member in Fig. 2 to provide 138 input values with which to evaluate the 76 unknown nodal displacements at the relevant 38 nodes of region R. However, measured displacements u and v were recorded at all 276 associated locations within the four quadrants around the hole and their values at corresponding locations were averaged to eliminate any non-symmetry. Fringe orders were obtained at the respective positions

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and at several load levels. Symmetry was imposed and 10 real and odd coefficients were used, i.e., m = n = 9. The 69 input locations were somewhat uniformly distributed throughout region R associated with Figs. 1 through 4. If uniformly distributed, and based on a total area of region R of Fig. 4 of this would represent an average input spacing of 1.4 mm (0.05 in.), corresponding to ~7 input readings per cm on the actual composite.

The 38 nodes of region R consist of 13 nodes (N1 through N13) on the outer boundary, of the hybrid element, S, (region denoted by DEFIQPGD of Fig 4) plus 25 nodes associated with the seven surrounding isoparametric elements contained within lines DEF and ABC of Fig. 4.

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Table 1 contains the stress concentration factors in the graphite/epoxy as obtained from the measured displacements and the smoothing hybrid finite-element representation, SHFER. These results are based on and various values of The locations of measured input displacements used closest to the hole are approximately 0.5r = 2.8 mm (0.11 in.) from its boundary, whereas the most distant measured input displacements originate at approximately 2.5r = 1.4 cm (0.55 in.) from the edge of the hole. Based on the geometry of Fig. 2 (r = 0.563cm), the external horizontal boundary, FE, of region S (the hybrid element) of Fig. 4 coincides approximately with the top edge of the fringe pattern of Fig. 3, whereas its vertical boundary, DE, is somewhat inside the right edge of this fringe photograph.

Current results are compared in Table 1 with those by Tan [8] and ANSYS. One quarter of the composite plate of Fig. 2 was modeled for the latter. The normalized tangential stresses, are plotted around the boundary of the hole in Fig. 5.

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Complete details, including those for the area integrals of the pseudo stiffness of that portion of S which helps form region R, are contained in Reference 5. One quarter of the composite plate of Fig. 2 was modeled (ANSYS) using from 3766 eight-node isoparametric elements (90 evenly spaced elements around the quarter-hole boundary and a total of 11,583 nodes) to 7374 elements (180 evenly spaced elements around the quarter-hole and a total of 22,567 nodes). These FEA models give a tensile stress concentration factor of to 7.741, respectively. 5.

Summary, Discussion and Conclusions

A simple, reliable and effective displacement-based technique is demonstrated for determining stresses on the boundary, and in the neighborhood, of geometric discontinuities in orthotropic composites from distant measured displacements. The method synergizes analytical, numerical and experimental techniques, and incorporates some previous concepts by Rhee et al. [1,2] and Feng [7]. Relatively few measured input data are needed and they originate away from the cutout. Whereas Rhee’s technique involved interpolating displacements at the nodes on the external boundary of a hybrid element from neighboring fringes, this shortcoming is overcome here by embedding the external boundary of hybrid element amongst the measured displacements. Moreover, the present technique filters experimental scatter and data acquisition positions can be selected largely at the user’s discretion. The latter permits one to omit questionable or unreliable data. Although Baek and Rowlands [9] used technology associated with Eqs. 1 through 5 to evaluate edge stresses at a hole from distance strain-gage readings, the present approach does a better job of filtering the input scatter. Unlike Refs 1, 2 and 9, the relative contribution of individual input values can also be controlled here through a weighting factor. These features enhance reliability. Only the tangential stress on the hole boundary is shown here. Once one knows the displacements of the external nodes of the hybrid element, individual components of stress, strain and displacement are available throughout the hybrid element. Input displacements range here between 3mm to 14 mm from the hole boundary. Although not assessed quantitatively, one could probably omit some of the current input values, particularly those closest to the hole, without degrading the results. In addition to employing fewer input values (locations) per node, one might consider reducing the number (increasing individual size) of the surrounding elements, thereby decreasing the number of nodes on of Fig. 1 and 4. Present analyses were conducted for various values of smoothing parameter, but no attempt was made here to evaluate an optimum value of As with most experimental methods, one neither needs to know the far-field physical boundary conditions nor analyze the entire structure. Moiré is used here, but displacements could also be recorded using methods such as speckle, holography or image correlation. The approach is applicable to multiple cutouts and/or other geometries.

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Acknowledgement

Ms. Lynda Litzkow proficiently prepared this manuscript. 7. References 1. 2.

3. 4. 5. 6. 7. 8. 9.

Rhee, J., (1995), Geometric Discontinuities in Orthotropic Composites, Ph.D. Thesis, University of Wisconsin, Madison. Rhee, J., He, S. and Rowlands, R. E.: Hybrid moiré-numerical stress analysis around cutouts in loaded composites, Experimental Mechanics 36(4) (1996), 379-387. Gerhardt, T. D.: A hybrid finite element approach for stress analysis of notched anisotropic materials, Jour, of Applied Mechanics 51 (1984), 804-810. Lekhnitskii, S. G., (1968), Anisotropic Plates, Gordon and Breach New York. He, K. Y., (2000), Hybrid Stress and Fracture Analysis of Orthotropic Media, Ph.D. Thesis, University of Wisconsin, Madison. Segalman, D. J., Woyak, D. B. and Rowlands, R .E.: Smooth spline-like finite element differentiation of experimental vector data over arbitrary geometry, Experimental Mechanics 19 (1979), 429-439. Feng, Z. and Rowlands, R. E.: Continuous full-field representation and differentiation of threedimensional vector data, Computers and Structures 26 (1987), 979-990. Tan, S. C., (1994), Stress Concentrations in Laminated Composites, Technomic Publishing, Co, Inc., Lancaster, PA. Baek, T. H. and Rowlands, R. E.: Hybrid stress analysis of perforated composites using strain gages, Experimental Mechanics 41(2) (2001), 195-203.

8. Composite Structures

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FUTURE EXPERIMENTAL METHODS NEEDED TO VERIFY COMPOSITE LIFE-CYCLE SIMULATIONS

CHRISTOS C. CHAMIS NASA Glenn Research Center Cleveland, OH 44135, USA LEVON MINNETYAN Clarkson University Potsdam, NY 13699, USA

Abstract

The future experimental methods needed for composite life-cycle are identified by computationally simulating the fracture of an integrally stiffened composite structure. The simulation describes events occurring during the fracture progression at all composite structure scales, the fracture modes that contribute to those events and the respective local failure mechanisms. The fracture modes in their respective scales provide opportunities to suggest future testing techniques to measure them. For example, energies emitted can be calibrated to identify non-destructive techniques to measure corresponding energies such as acoustic, thermal and even optical. Successful testing methods can then be used to implement monitoring systems for in-service structural life-cycles. 1.

Introduction

The life-cycle of composite structures can be determined by evaluating its damage tolerance. One way to a-priori computationally simulate damage tolerance is by progressive structural fracture, which includes all failure mechanisms that contribute to progressive structural fracture. Since composite structures have a multitude of failure mechanisms, the simulation must combine composite structural analysis, composite laminate theory, composite macromechanics and composite micromechanics. This combination will allow the synthesis of local behavior and respective fabrication process through the various scales up to structural behavior (scale telescoping) and the decomposition of structural response through the various scales down to individual constituents (scale tunneling through point and deplying through layers). The quantification of composite scale telescoping and composite scale tunneling provides description of the various events that occur at each scale during the progressive fracture of the composite structure. Since each scale represents a respective physical entity, it is rather reasonable to design experiments to measure the events that occur at that scale 631 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 631–644. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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during progressive structural fracture. The objective of this paper is to computationally simulate the events that occur in each scale during the evaluation of the life-cycle of a select composite structural component. Other objectives are to identify the fracture modes that contribute to those events and to suggest possible non-destructive experiments to measure them. 2.

Select Composite Structural Component

Laminated composite structures are used in many aerospace applications such as rotorcraft components, advanced aircraft fuselage, rocket motor cases, pressure vessels, containment structures, and other components with various shapes and sizes. In these applications composite structures are required to withstand significant in-plane loads. Additionally, composite structures are required to possess sufficient bending stiffness to resist buckling. The component selected for this evaluation is an integrally stiffened composite structures subjected to in-plane loads and internal/external pressures. Damage initiation, growth, accumulation, and propagation to fracture are simulated for composite panels and cylindrical shells with and without integrated stiffening layups. The influence of integrated stiffeners is examined with regard to out of plane stiffness contribution as well as damage progression and structural durability assessment under applied loading. Changes in the damage initiation load and the structural fracture load are quantified due to the presence of integrated stiffeners in order to identify viable nondestructive test methods. Recent developments in computational simulation technology have made it possible to evaluate the details of progressive damage and fracture in composite structures. Computational simulation enables assessment of the damage initiation and propagation loads. A damage energy release rate is evaluated globally during simulation by computing the work done per unit damage created. The damage energy release rate is used to quantify the structural damage tolerance at different stages of degradation. The influence of local defects or flaws and effects of the fabrication process in terms of residual stresses are taken into account. An important feature of computational simulation is the assessment of damage stability or damage tolerance of a structure under loading. At any stage of damage progression, if there is a high level of structural resistance to damage progression under the service loading, the structure is stable with regard to fracture. The corresponding state of structural damage is referred to as stable damage. On the other hand, if damage progression does not encounter significant structural resistance, it corresponds to an unstable damage state. Unstable damage progression is characterized by very large increases in the amount of damage due to small increases in loading, enabling measuring of the events that cause those large damage volumes. Whereas, during stable damage progression, the amount of increase in damage is consistent with the increase in loading, providing challenging opportunities for measurement. 3.

Methodology

Computational simulation is implemented via the integration of three modules: (1) composite mechanics, (2) finite element analysis, and (3) damage progression tracking.

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The overall evaluation of composite structural durability is carried out in the damage progression module (Refs. 1 and 2) that keeps track of composite degradation for the entire structure. The damage progression module relies on the composite mechanics code (Ref. 3) for composite micromechanics, macromechanics and laminate analysis, and calls a finite element analysis module that uses anisotropic thick shell elements to model laminated composites (Ref. 4). CODSTRAN Computational Cycle is shown in Figure 1.

Details of the methodology are described in the references cited. Herein its application to special types of composite structures and relevance to nature experimental methods needed are described. 4.

Integrally Stiffened Panel

A graphite/epoxy laminated composite plate with integrated ±45° intermittent lattice stiffeners was investigated with damage and fracture propagation due to tension, compression, and in-place shear loads. The response of the integrally stiffened composite panel was compared with that of an unstiffened skin plate. The unstiffened plate was given additional skin thickness such that the material volume was the same as the material volume of the integrally stiffened plate. The additional plies of the unstiffened plate were given fiber orientations of ±45°, same as the fiber orientations of the integrated stiffeners. Both the unstiffened and integrally stiffened panels were made of AS-4 graphite fibers in a high modulus high strength epoxy matrix (AS-4HMHS). Ply layup of the unstiffened panel was The stiffened panel had a skin layup of The stiffener plies were added to the top of the skin as or as Ply thickness was 0.127 mm (0.005 in.). The fiber volume ratio was and the void volume ratio was The cure temperature was (350° F).

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The fiber and matrix properties were obtained from a databank of composite constituent material properties resident in the composite mechanics module (Ref. 3). The fiber and matrix properties corresponding to this case are given in Tables I and II, respectively.

The HMHS matrix properties are representative of the 3501-6 resin. These in-situ properties are similar to those identified by (Ref. 5) with experimental correlation. The panels had identical planar geometry with a width of 305 mm (12.0 in.) and length of 457 mm (18 in.). Each finite element model contained 117 nodes and 96 uniformly sized square elements. Figure 2 shows the finite element model with diagonal lines along the ±45° stiffener bands.

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Numbers at nodes indicate the laminate type number at that node for the integrally stiffened panel. Laminate type 1 had a layup of representing the skin. Laminate type 2 had the skin layup plus representing the +45 stiffened nodes. Laminate type 3 had the skin layup plus representing the –45 stiffened nodes. Laminate type 4 had the skin layup plus representing intersection nodes for +45 and –45 stiffeners. Panels were assumed to be simply supported for structural response; i.e. the left end nodes of the panel were restrained against all displacement components and the right end nodes were restrained against displacement normal to the plane of the panel via duplicate node specifications.

Structural response characteristics of the integrally stiffened panel were evaluated in terms of the buckling load and the bending stiffness. Analysis of the integrally stiffened panel under uniaxial compression indicated a buckling load of 1,114 N (250 lbs). The buckling load of the unstiffened panel was only 213 N (48 lbs). Therefore the buckling resistance of the integrally stiffened panel was 5.2 times that of the unstiffened panel with the same amount of composite material. Similarly, the bending rigidity was significantly improved due to integrated stiffeners. Figure 3 shows a comparison of the midspan deflections of unstiffened and integrally stiffened panels due to uniform bending moment applied to the simply supported right end of the panel. Figure 3 shows that the integrally stiffened panel was 7.2 times stiffer than the unstiffened panel in

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bending. The future experiments needed are those that can measure incipient buckling and the respective resistance of stiffeners and skin at that instant.

Next, the in-plane progressive damage and fracture responses were compared for the unstiffened and stiffened panels. Figure 4 shows damage progressions of unstiffened and integrally stiffened panels subjected to tension.

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Damage initiation for the integrally stiffened panel occurred in the 90° skin plies due to transverse tensile fractures. Damage initiation for the unstiffened panels was also in the 90° plies due to transverse tensile fractures. In both cases the combined stress failure criterion was activated. However, the damage initiation load for the integrally stiffened panel was approximately one third of the damage initiation load for the unstiffened panel with the same material volume. The transverse tensile fracture in the 90° plies is mainly controlled by the matrix and the interface. Microstresses are all predicted from ply stresses as well as energies emitted during fracture. Local experimental methods are needed to identify which failure mechanism was activated first. Figure 5 shows the displacements in tension.

The uniaxial stiffnesses of the unstiffened and integrally stiffened panels are approximately the same prior to damage initiation. However, stiffness of the integrally stiffened panel degrades quickly due to damage initiation and progression. The respective stresses at damage initiation are predicted and the corresponding fracture modes as well as energies emitted and temperature rises. Unique experiments are needed to measure them directly. Figure 6 shows damage progressions of the unstiffened and integrally stiffened panels under uniaxial compression. Damage initiation for the integrally stiffened panel occurred at the edge of the panel at midspan where +45 and -45 stiffeners converged (laminate type 4). The damage initiation modes included the transverse tensile and longitudinal compressive failures of the 90° skin plies, in-plane shear failures of the ±45° skin plies, as well as the in-plane shear failures of the +45 stiffener plies near the skin. Damage initiation for the unstiffened panels under compression was in the 0° plies due to longitudinal compressive fractures. The damage propagation load of the unstiffened panel was more than five times that of the integrally stiffened panel. The compression fracture modes predicted provide ample opportunities for innovative non-destructive experimental

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techniques to measure their magnitudes in terms of energies emitted in the form of temperature changes, or acoustic signatures or even optical changes.

The initial stiffnesses of the unstiffened and integrally stiffened panels in compression were the same as observable from Figure 7. However, the stiffness of the integrally stiffened panel degraded at a much lower loading compared to degradation load of the unstiffened panel. Figure 8 shows the damage progressions of unstiffened and integrally stiffened panels under in-plane simple shear loading.

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Damage initiation for the integrally stiffened panel was due to longitudinal compressive failure of a +45 skin ply at a type 3 stiffened node. On the other hand, damage initiation for the unstiffened panel under shear was due to transverse tensile failures of the 90° plies. As it was in tension and compression, also in shear the integrally stiffened panel degraded at a lower load compared to the unstiffened panel. Figure 9 shows the load-displacement relationships in shear. Computational simulation results depicted in Figure 9 indicate that in shear, the initial stiffness of the unstiffened panel was slightly higher than that of the integrally stiffened panel.

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Shear loadings offer challenging opportunities to measure fracture modes and respective failure mechanisms non-destructively. The predicted magnitudes and locations definitely provide guidance. 6. Integrally Stiffened Composite Shell A composite cylindrical shell with and without integrated ±45° intermittent lattice stiffeners was investigated with damage and fracture propagation due to axial tension, as well as internal and external pressure loads. The response of the integrally stiffened shell was compared with that of an unstiffened shell. The unstiffened shell was given additional skin thickness such that the material volume was the same as the material volume of the integrally stiffened shell. The additional plies of the unstiffened shell were given fiber orientations of ±45°, same as the fiber orientations of the integrated stiffeners. Both the unstiffened and integrally stiffened shells were made of AS4/HMHS composite. Ply layup of the unstiffened shell was The stiffened shell had a skin layup of The stiffener plies were added to the top of the skin as or as Ply thickness was 0.127 mm (0.005 in). The fiber volume ratio was and the void volume ratio was The cure temperature was (350° F). Their respective properties are listed in Tables I and II, as mentioned previously. Each cylindrical shell finite element model contained 400 nodes and 384 uniformly sized square elements. Figure 10 shows the finite element model for a cylindrical shell.

The shell was simulated subject to increasing levels of internal and external pressurizations as well as axial loading. To represent the axial stresses produced in the closed end pressure vessel, boundary nodes at one end of the cylinder were subjected to

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force resultants in the axial direction, whereas the boundary FEM nodes were restrained in the axial direction at the opposite end. The uniformly distributed axial tension was such that the generalized axial stresses in the shell wall were half those developed in the hoop direction. Figure 11 shows a comparison of damage progressions for the integrally stiffened and unstiffened composite shells under axial tension. Figure 11 indicates that ultimate strength of the integrally stiffened cylindrical shell is approximately 60 percent of the ultimate strength of shell without integral stiffeners.

Additionally, damage initiation for the integrally stiffened shell begins at only 10 percent of the ultimate load. On the other hand, damage initiation for the unstiffened shell corresponds to a loading level that is approximately 97 percent of its ultimate load. The integrally stiffened shell provides ample opportunity for innovative non-destructive methods that are energy related. While the unstiffened panel provides a very challenging task to devise a test that will capture the insipient fracture which is that close to structural fracture. Figure 12 shows the exhausted cumulative damage energy based on depleted energies of the local failure mechanisms. The exhausted damage energy represents a similar pattern of damage progression as the percent damage volume depicted in Figure 12. Figure 13 shows the damage energy release rates (DERR) based on the incremental work done by applied loading during damage progression of the integrally stiffened cylindrical shell.

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Peak values in the DERR levels indicate significant damage events. The first peak in DERR corresponding to damage initiation for the integrally stiffened shell occurred due to transverse tensile fractures in the 90° skin plies. The steep increase in DERR at approximately 11 kN (2.5 k) axial tension corresponds to a partial separation between skin and stiffener plies. The results in Figures 12 and 13 show the amount of detailed information that is obtained from the computational simulation. This type of

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information is valuable to guide and even identify non-destructive test methods to measure it. Figure 14 shows comparison of damage progressions for integrally stiffened and unstiffened cylindrical shells subjected to external pressures.

The ultimate pressure for the integrally stiffened shell is approximately 75 percent of the ultimate pressure for the unstiffened shell. Damage initiation for the integrally stiffened shell occurs at 23 percent of its ultimate pressure. On the other hand damage initiation for the unstiffened shell occurs at approximately 94 percent of its ultimate pressure. Also, the volume of damage in the integrally stiffened shell at ultimate pressure is much greater than the volume of damage in the unstiffened shell at ultimate pressure. The damaged volume difference provides opportunities for non-destructive test methods to track damage tolerance in the presence of considerable damage and very little if at all.

6. Conclusions On the basis of the results obtained from the investigated flat composite plate, integrally stiffened panel, integrally stiffened and unstiffened composite cylindrical shell structure examples, experimental methods, and from the general perspective of the available computational simulation method and suggested future test method needs, the following conclusions are drawn:

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1.

Computational simulation can be used to track the details of damage initiation, growth, and subsequent propagation to fracture for unstiffened and integrally stiffened composite structures, as well as provide quantifiable information for implementing non-destructive test methods to measure it.

2.

For the considered integral stiffening structures, out-of-plane structural response characteristics such as the buckling load and the bending stiffness are significantly improved. Respective test methods needed to evaluate the load sharing between panel and stiffeners.

3.

In-plane load carrying capability of a composite panel is reduced due to damage initiation and progression processes caused by the presence of integrated stiffeners. Unique non-destructive test methods are needed to isolate the fracture modes that are active during the fracture progression.

4.

Computational simulation, with the use of established composite mechanics and finite element modules, can be used to predict the influence of composite geometry as well as loading and material properties on the durability of composite structures. This type of simulation also provides energy emitted during the fracture progression which can be converted to acoustic, thermal or optical in order to measure it by using non-destructive testing techniques.

5.

The demonstrated procedure is flexible and applicable to all types of constituent materials, structural geometry, and loading. Hybrid composites and homogeneous materials, as well as laminated, stitched, woven, and braided composites can be simulated. Appropriate innovative and inventive nondestructive test methods need to be developed to measure the damage accumulated for life-cycle verification of predictions.

6.

Computational simulation by CODSTRAN represents a new global approach to progressive damage and fracture assessment for any structure. Complementing this with suitable non-destructive testing will complete quantifiable in-service monitoring systems for structural life-cycles.

7. 1. 2. 3. 4. 5.

References L. Minnetyan, P. L. N. Murthy and C. C. Chamis, "Progression of Damage and Fracture in Composites under Dynamic Loading," NASA TM–103118, April 1990, 16pp. L. Minnetyan, P. L. N. Murthy and C. C. Chamis, "Composite Structure Global Fracture Toughness via Computational Simulation," Computers & Structures, Vol. 37, No. 2, pp. 175180, 1990. P. L. N. Murthy and C. C. Chamis, Integrated Composite Analyzer (ICAN): Users and Programmers Manual, NASA Technical Paper 2515, March 1986. S. Nakazawa, J. B. Dias, and M. S. Spiegel, MHOST Users' Manual, prepared for NASA Glenn Research Center by MARC Analysis Research Corp., April 1987. L. Sobel, C. Buttitta, and J. Suarez, "Probabilistic & Structural Reliability Analysis of Laminated Composite Structures Based on IPACS Code," Proceedings of the 34th SDM Conference, LaJolla, California, April 19-22, 1993, Vol. 2, pp. 1207-1217.

EXPERIMENTAL OBSERVATIONS ON THE DELAMINATION BEHAVIOR IN COMPOSITE STRUCTURES

G. A. KARDOMATEAS, V. LA SAPONARA* and G. J. SIMITSES School of Aerospace Engineering Georgia Institute of Technology Atlanta, GA 30332-0150, USA *Currently at the School of Civil and Environmental Engineering Georgia Institute of Technology Atlanta GA 30332-0355

Abstract

The objective of the current paper is to present experimental data of interface cracks (delaminations) in layered Glass/Epoxy and Graphite/Epoxy composites under cyclic compressive (fatigue) and static (Mixed Mode Bending and Double Cantilever Beam) loads. The growth behavior ranges from self-similar growth along the interface in the unidirectional configuration to branching out of the interface in 90 deg. and cross-ply configurations. In the case of cyclic compressive loading, the specimens undergo repeated buckling/unloading of the delaminated layer with a resulting reduction of the interlayer resistance. Also, for this case, equations describing the growth of the delaminations under cyclic loads are obtained on the basis of a combined delamination buckling/post-buckling and fracture mechanics model. The latter is based on a modedependent critical fracture energy concept and is expressed in terms of the spread in the energy release rate in the pre- and post- buckling state. The growth laws developed in this manner are integrated numerically, in order to produce the delamination growth vs. number of cycles curves. An important characteristic in all the configurations tested is that the state of stress near the delamination tip is of mixed mode, I and II. In the cases where branching (intra-layer cracking) occurs, there is some self-similar crack growth from the initial delamination (simulated as a Teflon sheet inserted in the plate) followed by branching of the crack through the thickness of the plate. Observed behavior ranges from sudden and catastrophic failure induced by the intra-layer crack in the 90 deg. specimen to the alternate successive initiation of intra-layer cracks and secondary interlayer cracks (delaminations) in the cross-ply specimens. Experiments and statistics show that there is a critical branching angle above which crack growth is greatly accelerated. The particular details of the experimental study, including the difficulties and challenges, are discussed. 645 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 645–660. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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1. Introduction One of the most serious problems associated with laminated composites is the formation of an internal delaminated zone, i.e. two adjacent layers are partially debonded at their interface. This occurs most commonly as a consequence of low-velocity impact, but it may also be due to pre-existing manufacturing imperfections. There may be other reasons that cause delaminations, such as vibrations excited by the propulsion systems. In particular, under compression loading, the delaminated layer may buckle. This local instability does not necessarily lower the ultimate load, and usually the laminate is capable of carrying on in the post-buckling phase under higher loading. However, the buckling-induced stresses at the tip of the delamination may cause growth of the interlayer crack and eventual complete separation of the layer. Besides strength and stiffness, delaminations can influence other performance characteristics, such as the energy absorption capacity of composite beam systems [1]. The prospect of local delamination buckling, and especially the possibility of growth, limits the high potential of composites and needs to be well-understood so that composites can be safely used in compressive load bearing applications. This is one of the composite failure modes under compression loading, other frequently observed modes include micro-cracking in the matrix and kinking zones. Self-similar growth was observed in the compression fatigue unidirectional experiments, [2] and [3]. Another type of growth behavior observed was crack branching in static compressive, static Double Cantilever Beam (DCB) or compressive fatigue tests of cross-ply configurations, [4]-[7]. A crack growing along an interface is said to branch when it deviates from the interface into the contiguous ply. Typically it returns to grow along the original interface. In the case of compression loading and the ensuing buckling, a geometrically nonlinear model should be adopted to determine the post-buckling deflections of a thin debonded layer when a plate element is subjected to cyclic compression. The explicit presence of additional parameters with the dimension of the length (e.g. the thickness of the separated layer) makes this a nontraditional problem of fracture mechanics. Additional complications are introduced by the circumstance that the process of separated-layer growth may be accompanied by the phenomenon of elastic instability of the entire structural element. The initial post-buckling behavior of general delaminated composites (i.e. with no restrictive assumptions on the delamination dimensions) was studied in [8], by using a perturbation procedure based on an asymptotic expansion of the load and deformation quantities in terms of the distortion parameter of the delaminated layer, the latter being considered a compressive elastica. The analysis leads to closed form solutions for the load versus applied compressive displacement and the near tip resultant moments and forces. In these studies, the bimaterial interface crack solutions for the mode mixity and the energy release rate in terms of the resultant moments and forces, as derived in [9], were subsequently employed. The growth law proposed is a function of the spread in the energy release rate in the pre- and postbuckled states and the maximum value of the energy release rate in the cycle, both normalized with the mode-dependent interface fracture toughness. The exponent and the constant of the growth law are also taken to be mode-dependent. The delamination

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growth vs. number of cycles curves are produced by integrating numerically the growth laws developed in this manner. Regarding crack branching, data for cross-ply specimens of Glass/Epoxy and Graphite/Epoxy under static and fatigue loads will be presented in this paper. Cracks branched with angles that could be as big as 90 deg., and they crossed a distance that was of the order of 1-2 ply thickness, [6] and [7]. The mechanism of this failure is far from being well understood and predicted, but these experiments provide much needed information on the crack branching behavior. Statistical analysis based on the experiments will also be discussed. 2.

Self-Similar Growth

The experimental study was conducted first on unidirectional graphite/epoxy, procured from Amoco in the form of prepreg tape (commercial specification T50 6K ERL 19393) with about 33 per cent resin content and ply thickness of about 0.10 mm. Test panels were laid up by hand and cured in the autoclave according to the manufacturer's recommended cure cycle, i.e. at a temperature of 177C (350F) and a pressure of 414 KPa (60 psi). The material was put in a vacuum bag prior to the curing cycle. The delaminations were introduced by placing at the desired location, through the width, a Dupont Teflon film of 0.025 mm (0.001 in) thickness. Since the curing process affects the final dimensions of the specimens due to resin bleeding, the thicknesses of the specimens were measured after curing with a micrometer. The thickness measurements were taken at different points through the width and length to insure overall uniformity of the thickness and they were found to be satisfactory. The cured panels were carefully inspected for possible abnormalities, and were subsequently cut to specimen size using a silicon carbide blade. The equipment used consists of an Instron 8501 dynamic testing machine and a Questar remote video measurement system. The system allows tri-axial motion of the microscope and a simultaneous digital measurement of distances. The cyclic compression tests were conducted on these unidirectional graph ite/epoxy specimens at a frequency of 1 Hz, and the delamination growth was monitored using the Questar. The experiments were at a constant maximum strain and the characteristic properties of the specimens were found to be as follows: longitudinal modulus of elasticity GPa; critical energy release rates, exponent ratio (see growth law below) constant ratio, 10.01. These constants, which are typical of graphite/epoxy, were found from independent Mode I and Mode II tests. The specimens had a width w = 12.7 mm and a half-length between the grips of L = 50.8 mm. A mode-dependent cyclic growth law is suggested in [2] as follows:

This growth law is expressed in terms of the spread release rate g in the pre- and post-buckling state:

of the normalized energy

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is the mode-dependent fracture toughness. Now the

exponent and the constant are also mode-dependent. The exponent's dependence is according to:

and a similar formula can be written for the constant Five delamination configurations were tested. In the following list, h/T is the ratio between h, the number of plies in the delaminated plate, and the total number of plies T. a) 30 plies, specimen thickness T = 2.87 mm, delamination of half-length 21.25 mm, between fourth and fifth ply, hence h/T = 4/30, and at a maximum compressive strain This specimen configuration is denoted as 4/30(A). b) 15 plies, specimen thickness T = 1.55 mm, delamination of half-length 25.86 mm, between fourth and fifth ply, hence h/T = 4/15, and at a maximum compressive strain (denoted as 4/15). 30 plies, specimen thickness T = 2.75 mm,delamination of half-length c) 26.15 mm, between sixth and seventh ply, hence h/T = 6/30 and at a maximum compressive strain (denoted as 6/30). 30 plies, specimen thickness T = 2.87 mm, delamination of half-length d) 18.75 mm, between fourth and fifth ply, hence h/T = 4/30, and at a maximum compressive strain This specimen configuration is denoted as 4/30(B). e) 36 plies, specimen thickness T = 3.00 mm, delamination ofhalf-length 33.02 mm, between tenth and eleventh ply, hence h/T = 10/36 and at a maximum compressive strain (denoted as 10/36). These five test configurations exhibited different mode mixities and energy release rate spreads; the 10/36 and 6/30 specimens exhibited the highest Mode I components. All specimens were subjected to less than 20 per cent of the critical energy release rate, with the 10/36 and 6/30 specimens under a larger spread Two data points, (l; N), from the 4/30(A) specimen were used to obtain the constants mI and CI. These data points are: (30.48 mm; 231,354 cycles) and (30.734 mm; 253,155 cycles). The values obtained are: and Based on these values, Fig. 1 shows the actual experimental data and the predicted cycles for all five specimens configurations on a semi-logarithmic plot. In all the experiments the delamination grew straight along the interface and the delamination length was measured with the help of the Questar. It should be first emphasized that the same exponent and constant in the growth law are used for all delaminations, although each delamination configuration is characterized by a different location through the thickness, different initial length, different applied peak strain and different mode mixity. Since measurements were performed at discrete points, the measured data are given by discrete data points, whereas the predictions are represented by the lines. It is

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seen that the data form five distinct groups, each for each specimen type. First, an immediate observation can be made that the closer the delamination to the surface (i.e. the lower h/T), the slower the growth. Second, the experimental data seem to correlate adequately with the predicted values. The proposed fatigue growth law is shown to correlate well with the experiments, in spite of the different geometry (location of the delamination through the thickness, initial delamination length, plate thickness) and applied loading in each of the five delamination configurations for each material tested (Fig.1).

Of the five specimen configurations, the mode mixity is fairly similar between the 4/30(A) and 4/15 specimens, but distinctly different between the 4/30(A) and the 6/30 or 10/36 or 4/30(B) specimens. In this context, in order to examine the importance of assuming a mode-dependent growth law, the predicted number of cycles for the corresponding mode-independent version of the growth law was calculated and compared to the experimental data. The following conclusions were made: if modeindependence was to be assumed in the growth law, which means that and then fitting the two constants, m and C, from the 4/30(A) data would fail to adequately predict the 6/30 data. Specifically, this would predict a number of cycles less than half those predicted by the present mode-dependent approach and the ones obtained from the experiments. Subsequently, the same study was conducted on unidirectional Glass/Epoxy specimens. The Glass/Epoxy used was the S2/SP250 made by 3M Co., and was supplied in the form of prepreg tape by NASA Langley. The average ply thickness of the S2/SP250 was 0.2413 mm and the mechanical properties of this material are as

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follows: moduli (in GPa) in-plane Poisson’s ratio resin content by weight 33±3 percent. The specimens were made by hand lay-up and curing in the autoclave according to the cure cycle provided by the manufacturer of the prepreg tape: a cure temperature of 120°C (250°F) and a pressure of 345 KPa (50 psi). The cured laminates which were 304.8 by 50.8 mm (12 by 2 inches), were carefully inspected for possible abnormalities and were subsequently cut into 152.4 by 12.7 mm (6 by 0.5 inch) pieces using a tungsten-carbide tipped tool. The delaminations were introduced by placing at the desired location, through the width, a Dupont Teflon film of 0.0254 mm (0.001 in) thickness. Since the curing process affects the final dimensions of the specimens due to resin bleeding, the thicknesses of the specimens were measured after curing with a micrometer. At this point it should be mentioned that the preparation of the Glass/Epoxy specimens required more effort and care than the Graphite/Epoxy ones because of the larger ply thickness and the different resin flow properties. The testing of the specimens was in compression at moderate load and displacement levels. Compressive testing is often done by using tabs or an interleaf between the specimen and the jaws of the testing machine in order to better transmit the applied load to the specimen through friction. However, neither tabs nor interleafs were found to be needed in these experiments. The tab-less specimens used in this experimental study proved to be very effective, besides being less expensive and requiring less time. Testing in compression fatigue was carried out in the Instron 8501 mentioned above. The experiments were done in displacement control. A sine wave of 5 Hz was applied. Regarding the toughness properties and fatigue growth parameters, these were as follows: critical energy release rates, exponent ratio, constant ratio, The specimens had a width w = 12.7 mm and a slightly varying half-length between the grips, L, between 50 and 60 mm, as reported below. The specimens consisted of 24 plies with the delamination implanted either between the fourth and fifth ply (4/24) or between the fifth and sixth ply (5/24). Different applied maximum compressive strains and different initial delamination lengths were used, which resulted in the following test configurations: a) 5/24 A: 24 plies, specimen thickness T = 4.98 mm, specimen half-length between grips L = 57.6 mm, initial delamination of half-length mm, between fifth and sixth ply, hence h/T = 5/24, and at a maximum compressive strain The delamination extended in fatigue up to half-length of l = 37.173 mm requiring 281,371 cycles. b) 5/24 B: 24 plies, specimen thickness T = 4.98 mm, specimen half-length between grips L = 54.6 mm, initial delamination of half-length mm, between fifth and sixth ply (h/T = 5/24), and at a maximum compressive strain Delamination extension up to l = 37.871 mm in 54,675 cycles. c) 5/24 C: 24 plies, specimen thickness T = 4.98 mm, specimen half-length between grips L = 58.4 mm, initial delamination of half-length mm, between fifth and sixth ply, hence h/T = 5/24, and at a maximum compressive strain Delamination extension up to l = 31.255 mm in 71,504 cycles.

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d) 4/24 A: 24 plies, specimen thickness T = 4.98 mm, specimen half-length

between grips L = 55.6 mm, initial delamination of half-length mm, between fourth and fifth sixth ply (h/T = 4/24), and at a maximum

compressive strain

In this configuration the delamination

extended in fatigue up to l = 27.775 mm in 252,260 cycles. 4/24 B: 24 plies, specimen thickness T = 5.10 mm, specimen half-length e) between grips L = 56.0 mm, initial delamination of half-length mm, between fourth and fifth sixth ply (h/T = 4/24), and at a maximum compressive strain In this configuration the delamination extended in fatigue up to l = 30.099 mm in 406,724 cycles. Two data points (l ; N) from the 5/24 A glass/epoxy specimen were used to obtain the constants mI and These data points are: (29.604 mm; 51,601 cycles) and (37.173 mm; 281,371 cycles). The values obtained are: and m/cycle. Results are shown in the following Fig. 2.

3. Non Self-Similar Growth Patterns When the composite construction is other than unidirectional, non self-similar growth patterns involving crack branching are observed. Specifically, experiments with growth of delaminations under fatigue or monotonic loading in cross-ply composite specimens revealed that intra-layer angle cracks often emanate from the delamination tip and propagate into the layer until they meet the next interface and then they may turn again

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into delaminations. The development and growth of the delamination/intra-layer crack system was examined under (i) axial compressive cyclic (fatigue) loading, (ii) off-axis, i.e. out-of-plane or transverse, monotonic, tensile loading of Glass/Epoxy and Graphite/Epoxy cross-ply laminates. Fig. 3 shows a sketch with the nomenclature that will be used to describe crack branching. Intra-layer and branching crack are two equivalent terms; kink angle is equivalent to branching angle.

In [5], symmetry was observed in crack branching in Glass/Epoxy specimens subject to fatigue loading, as shown in Fig. 4-5. This symmetry was not observed in [6] and [7], in Glass/Epoxy and Graphite/Epoxy specimens, subject to both static and fatigue loading. It should be pointed out that it is very difficult to calculate fracture toughness from the experiments, due to the presence of non self-similar crack growth. As it will be seen later on, several branchings occur, typically as soon as the crack grows from the delamination starter. For this reason, it is not possible, at this state, to present a mode mixity analysis nor to give an estimate of fracture toughness. However, experimental data and statistical analysis based on them will be presented and discussed. The tests performed in [6] and [7] were of different types: a) static Mixed Mode Bending (MMB) on: unidirectional 90 deg. S2/ SP250 Glass/Epoxy (24 plies, h/T=12/24); cross-ply S2/ SP250 Glass/Epoxy (lay-up h/T=15/30; 0], h/T=15/29). The average dimensions were 25.5 x 254x4.67 mm, with ply thickness = 0.205 mm. The average delamination starter length was mm. b) compressive fatigue tests on T7G145/F1914 Graphite/Epoxy specimens. The material was donated by Hexcel. The dimensions were 12.6 x 152.4x1.93 mm, the specimens had lay-up and h/T=4/15. The average delamination starter length, ao, was 50 mm and was located in the middle of the specimen. Crack growth could occur on either side of ao. The average ply thickness was 0.129 mm.

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static DCB test on cross-ply TOHO UT500 Graphite/Epoxy specimens. The specimens had dimensions 25.4 x 137x2.55 mm and the delamination starter lengths were ao= 46 mm and 90 mm, and were located either in the mid-plane (h/T=10/20) or two plies away from it (h/T=12/20). The material has properties

The tests were performed using the Instron 8501 machine and the Questar microscope. Moreover, the Mixed Mode Bending (MMB) and the fatigue tests were video-recorded, and the images had been analyzed on a Silicon Graphics O2 computer. All the specimens were coated with white primer for visualization of the crack. This did not allow the identification of the plies during the test. The DCB specimens were polished after the tests and observed with an Olympus BH2-UMA microscope connected with a CCD camera, a monitor and a Polaroid-type printer. A fixture has been used for the MMB tests, as described in [10] and [11]. In the static DCB tests and in the fatigue tests, the piano hinges (for the DCB tests) and the specimens (for the fatigue tests) were directly clamped in the grips of the testing machine. The delamination starter consisted of a Teflon film inserted during the hand lay-up preparation of the specimens, as in the previous cases. In the MMB tests, a rate of 0.00254 mm/s (0.0001 in./s) was used. The Mode I and Mode II components of the applied load P, called and depend on two specific lengths in the MMB fixture. An expression for the ratio of the strain energies of Mode I and Mode II, i.e. under the simplification of negligible weight of the fixture, is also given in the previously mentioned references. By changing the fixture’s lengths, it is possible to have predominance of Mode I or Mode II. The work described in [10] and [11] is applied to unidirectional specimens as self-similarity of the crack growth is sought. An estimate of the mode mixity for multi-directional specimens requires a numerical solution and is not available at this point. In the MMB tests, crack branching was observed in all the specimens. In the unidirectional 90 deg. specimens, the crack branched away from the interface between two contiguous 90 deg. plies, and broke the specimen (Fig. 6). The branching angles, measured with respect to the original delamination direction, varied between 70 deg. and 110 deg. In the cross-ply specimens, the crack branched away from the interface between a 0 and 90 deg. ply, went into the 90 deg. and then turned parallel to the original delamination direction. The phenomenon was confined into an area of the order of 1-2 ply thickness, as the crack never crossed a 0 deg. ply. Several branching were observed, and the angles varied between 29 and 55 deg. (Fig. 7). The fatigue tests were done in displacement control, with a 0.0381 mm (0.0015 in.) sine wave and a frequency of 5 Hz. The twelve specimens tested were gripped and brought to buckling in static condition. Then, cycling was started. The branching angles varied between 12 deg. and 90 deg. (Fig. 8).

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Statistical analysis was needed to understand the effect of branching on life and crack propagation, as the scatter did not allow to draw conclusions. The statistical technique used belongs to the field of Exploratory Data Analysis, and consists of the socalled smoothing by running median of 3 repeated, introduced in [12], and applied in [6]. Medians were chosen for two reasons: they are more robust than mere averages, and the Exploratory Data Analysis technique is based on medians. Each specimen was associated to: a) a median branching angle (median of the branching angles observed in that specimen); b) a median crack growth, measured as projection of the crack along the direction of the starter. As mentioned earlier, the delamination starter was located in the middle of the specimen, hence crack growth could occur on either side of the starter, on both sides, or on neither. Consequently, a median of these growths was calculated. Zero crack growths and zero branching angles were considered in the calculations when present. The crack growth data at 100,000 cycles (or the available closest number from the experiments) were chosen. The number of cycles for the twelve specimens ranged from 25,267 (corresponding to catastrophic crack growth) to 250,000. 100,000 cycles represented a reasonable number to compare different specimens, as branching occurred before 100,000 cycles, and there was enough crack growth data for the statistical analysis. Table 1 reports the original fatigue data for the twelve specimens. “Total cycles” is the total number of cycles for the specimen, “Kink angle (side 1)” and “Kink angle (side 2)” are the branching (or kink) angles observed on either side of the delamination. “Cycles kink” is the number of cycles at which a specific kink occurred. Table 2 shows the data used for the statistical analysis, ordered in ascending crack growths.

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The results of the smoothing by running median of 3 repeated are the following: there is a critical branching angle above which crack growth is greatly accelerated [6]. If the median of all the branching angles in a specimen is above such critical branching angle, then branching will considerably affect the life of the specimen and accelerate its

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failure. If the median of the branching angles in a specimen is below such value, branching is not an important failure mode. In the twelve specimens analyzed, this critical angle is 33 deg. This behavior may be explained with the presence of contact inside the branching crack: it is hypothesized that for angles below the critical branching angle, contact in the branching crack slows down crack propagation; for angles above the critical angle, the branching crack opens and therefore crack propagation is accelerated. Verification of this hypothesis on the amount of contact zone would require detailed investigation of the branching crack, which was not possible with the current capabilities and resolutions. Moreover, a similar study was performed at the onset of crack branching, in the same set of twelve specimens. Crack growth was analyzed around 50,000 cycles, half of the life considered earlier. It is found that the critical branching angle is the same, 33 deg. Hence, this phenomenon seems not to be affected by the specimen’s life [13]. Finally, eight DCB static specimens were tested in displacement control with a rate of mm/s (0.02 in./min). Fig. 9 shows one of the specimens, polished and observed with an Olympus BH2-UMA microscope after the test. Several branchings occurred per specimen, with a considerable scatter of data: from 11 deg. to 78 deg. The DCB specimens were designed according to a 2-level 2-parameters, factorial design of experiments based on a 8x8 Hadamard matrix. The method is described in [14], and was utilized in [7]. It was chosen in order to minimize the number of specimens to prepare and test, while obtaining statistically valid results. The effects of two delamination lengths (46 and 90 mm) and two positions of delamination in the specimen (h/T=10/20 and h/T= 12/20) were studied, and crack growth and branching angles were monitored in the experiments.

The results were obtained with 90% confidence and an estimate of the variance with four degrees of freedom. They can be applied to a population (theoretically, an infinite number) of specimens with the same characteristics. Moreover, the same smoothing technique utilized for the fatigue tests was applied to the DCB data: the goal was to identify the trend of median crack speeds (total crack growth divided by total

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time of the experiment in a specimen, calculated at the rate of

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and

median branching angle. The outcome of both design of experiments analysis and smoothing is the following: there is a critical branching angle that affects the behavior of the population of DCB specimens. Such angle is between 39 and 40 deg. In addition, when the median branching angle (median of all the branching angles in a specimen) is above such critical angle, crack growth is greatly accelerated. This behavior is similar to what was found in the analysis of the fatigue data, with the following differences: in the T7G145/F1914 Graphite/Epoxy specimens subject to compressive fatigue tests, the critical angle was 33 deg., while in the TOHO UT500 Graphite/Epoxy subject to static DCB tests, the critical angle was 39-40 deg. Geometry and loading conditions seems not to significantly affect the critical branching angle, if one considers the amount of scatter present in such experiments. A future goal is to establish whether this behavior could be repeated in other specimens’ configurations and other materials. Scatter of data seems to show that the crack growth is greatly influenced by the local conditions around the crack tip and heterogeneities like fiber clusters, inclusions and so on. Also, a study is required to observe the amount of roughness-induced closure in the branching crack, and correlate it to the branching angle. This can be done numerically with an accurate micro-mechanical model and a statistical model to describe the heterogeneities in the composite. 4.

Conclusions

This paper presented a review of experimental data and modeling in delaminated unidirectional and cross-ply composites. In unidirectional Glass/Epoxy and Graphite/Epoxy composites, a growth law was proposed, which is based on the spread in the energy release rate in the pre- and post-buckled states and the maximum value of the energy release rate in the cycle. The exponent and the constant of the growth law are mode-dependent. The crack growth vs. number of cycles curves given in the paper show a good correlation between analytical prediction and actual crack growth. Delaminated cross-ply specimens exhibited non self-similar crack growth, which did not allow a measurement of fracture toughness and mode mixity. However, experimental data were presented in the paper for Glass/Epoxy and Graphite/Epoxy specimens under static and fatigue loading. Scatter greatly affected the results. Two statistical techniques were utilized: an Exploratory Data Analysis technique called smoothing by running median of 3 repeated; a 2-level 2-parameters, factorial design of experiment based on a 8x8 Hadamard matrix. A critical branching angle is obtained as a result of the analyses, above which crack growth is greatly accelerated. Hence, branching could considerably affect the life of a multi-directional composite structure.

5.

Acknowledgements

The financial support of the Office of Naval Research, Ship Structures S&T Division, Grants N00014-90-J-1995 and N00014-00-10323, and the interest and encouragement of the Grant Monitor, Dr. Y.D.S. Rajapakse, is gratefully acknowledged.

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7. 8.

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References Kardomateas, G.A. and Schmueser, D.W. (1988) Buckling and Post-buckling of Delaminated Composites Under Compressive Loads Including Transverse Shear Effects, AIAA Journal 26, 337-343. Kardomateas G.A., Pelegri A.A. and Malik B. (1995) Growth of Internal Delaminations Under Cyclic Compression in Composite Plates, Journal of the Mechanics and Physics of Solids 43, 847-868. Kardomateas G.A. and B. Malik B. (1997) Fatigue Delamination Growth Under Cyclic Compression in Glass/Epoxy Composite Beam/Plates, Polymer Composites 18, 169-178. Chai H. (1984) The characterization of Mode I delamination failure in non-woven, multidirectional composites, Composites 15, 277-290. Pelegri A.A., Kardomateas G.A. and Malik B.U. (1997) The Fatigue Growth of Internal Delaminations Under Compression in Cross Ply Composite Plates, Composite Materials: Fatigue and Fracture (Sixth Volume), ASTM STP 1285, E.A. Armanios Ed., American Society for Testing and Materials, 143-161. La Saponara, V. and Kardomateas, G. A. (2000) Statistical considerations in the analysis of data from fatigue tests on delaminated cross-ply graphite/epoxy composites, ASME Journal of Engineering Materials and Technology, 122, 409-414. La Saponara, V. and Kardomateas, G.A. (2001) Crack Branching in Layered Composites: An Experimental Study, Composite Structures 53, 333-344. Kardomateas G.A. (1993) The Initial Post-buckling and Growth Behavior of Internal

Delaminations in Composite Plates, Journal of Applied Mechanics (ASME) 60, 903-910. 9. 10. 11. 12. 13.

14.

Suo, Z. and Hutchinson, J.W. (1990) Interface crack between two elastic layers, International Journal of Fracture 43, 1-18. Reeder, J. R. and Crews Jr., J. H. (1990) Mixed Mode Bending Method for Delamination Testing, AIAA Journal 28, 1270-1276. Reeder, J. R. (1992) An evaluation of mixed-mode delamination failure criteria, NASA Technical Memorandum 104210. Tukey, J. W. (1977) Exploratory Data Analysis, Addison-Wesley. La Saponara, V. (2001) Crack Branching in Cross-Ply Composites, Ph.D. Thesis in Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0150. Diamond, W. J. (1989) Practical Experiment Designs for Engineers and Scientists, John Wiley & Sons.

DIRECT IDENTIFICATION OF ELASTIC PROPERTIES OF COMPOSITE STRUCTURES – A WAVE-CONTROLLED IMPACT APPROACH

SERGE ABRATE Department of Technology Southern Illinois University Carbondale, IL62901-6603

Abstract

The determination of the four elastic constants for an orthotropic layer of composite materials usually requires that specimens be prepared and several tests be performed. This can be costly and time consuming and a number of investigators have sought to infer those properties from the free vibration characteristics of a single specimen. While this approach is successful tests are still performed on a specimen that must be assumed to be representative of the properties of the actual structure. In this paper, we investigate the possibility of generating impact that produce very localized bending deformations in the actual structure and monitoring the sound generated during the contact phase of the event. This approach could be used for in-situ determination of material properties, which is important with composite materials since the properties of a specimen can differ from those of an actual structure. 1. Introduction

Fiber reinforced composite materials are characterized by four independent elastic constants. These constants can be determined from static tests performed on specially prepared specimens. This approach has two drawbacks: (1) is requires that several tests be performed; (2) the specimens are usually made separately and have different layups than the actual structure. There is interest in developing a procedure to determine the material properties from tests on a single specimen that has the same layup as the actual structure, or, if possible, directly on the actual structure. In recent years several investigators used experimentally determined natural frequencies of composite plates to estimate the elastic properties of the material. In many cases the specimen is excited through an impact generated by an instrumented hammer or by a spherical projectile. The response of the plate can be sensed by a surface mounted accelerometer or by non-contact means: fiber optic sensor, holographic interferometry, optical interferometry [1], microphone [2]. Impacts can be classified into several categories [3]. With high-velocity impacts only a very small region near the projectile is under stress while the projectile is in contact with the target and bending motion of the target has not been established. With wave-controlled impacts, bending deformation takes place in a zone surrounding the 661 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 661–670. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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point of impact but does not reach the boundaries during the contact phase. For boundary-controlled impacts, waves have reached the boundary and have been reflected back and forth several times so that the entire plate participates in the response. Both wave-controlled and boundary-controlled impacts are called low velocity impacts. The response of a structure to an impact by a projectile generally consists of two phases: (1) the contact phase during which the projectile is applying a load on the structure; and (2) the subsequent free vibration phase. Previous investigators measure the response of a specimen during the free vibration phase in order to determine its natural frequencies and mode shapes. In this paper we investigate the possibility of using information from the contact phase of the impact in order to determine the elastic properties of the composite. Tests are designed to induce only local deformations to that they can be performed on actual structures regardless of the complexity of their geometry or their support conditions. 2.

Impact Dynamics

Several approaches are available for studying the dynamics of foreign object impacts with varying degrees of approximation to the dynamics of the target [3,4]. Three such approaches will be considered here. In the first one the deformation of the target is neglected, the second assumes the plate to be of infinite extend, and the third model accounts for the full dynamics of finite-sized plates.

2.1. IMPACT ON HALF-SPACE If the overall deformations of the plate are negligible, the impact causes only local deformations in the contact zone. The relationship between the contact force F and the indentation is governed by Hertz's contact law [3, 4]

where the contact stiffness k is given by

where R is the radius of the impactor, E is the modulus of elasticity and is Poisson's ratio. The subscripts p and s refer to the plate and the spherical indentor respectively. The equation of motion and the initial condition of the projectile are

The maximum contact force

and the contact duration

are given by

Eqs. (5,6) provide a quick means for estimating the magnitude of the maximum contact force and the contact duration.

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2.2. IMPACT ON INFINITE PLATE

Zener [5] showed that the dynamics for foreign object impacts on infinite isotropic plates is governed by the single differential equation

which is written in terms of the non-dimensional indentation time that are defined as

and the non-dimensional

with the time constant

Eq. (7) depends on the single non-dimensional parameter

where D is the bending rigidity and m is the mass per unit area of the plate. The initial conditions are

Note that Eq. (3) is a special case of Eq. (9) for Solving Eqs. (7, 12) for several values of shows the effect of the different factors on the qualitative response of the plate. As shown in Fig. 1, when the maximum non-dimensional force is the largest and the loading and unloading phases are symmetrical (Fig. 1). When the maximum contact force decreases, the contact duration becomes longer, and the unloading phase is longer than the loading phase. Olsson [6] extended this model to orthotropic plates and showed that in that case the equivalent rigidity of the plate should be

where with isotropic or quasi-isotropic laminates and can vary between 0 and 3 for general layups [7]. The contact force and the deflection of the structure is obtained from the solution of Eq. (7) using

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During impact, the wave front is assumed to be elliptical and the analysis is based on expanding the transverse displacements in terms eigenfunctions of a rectangular simply supported plate extending from –a/2 to a/2 and –b/2 to b/2 impacted in the center

Both i and j must be odd numbers and i=j. The axes of the ellipse are given by

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More recently, Olsson [8] proposed a mass criterion to discriminate between wavecontrolled and boundary-controlled impacts. Elliptical bending waves propagate away from the impact point and eventually reach the boundary of the plate as illustrated in Fig.2. The shaded area in that figure represents the portion of the plate in which bending waves can propagate freely response before reaching the boundary. If is the mass of that portion of the plate, wave-controlled impacts occur when <0.29.

2.3. IMPACT ON FINITE PLATES According to the classical plate theory, the equation of motion for an orthotropic plate subjected to a force F(t) applied at is

For a rectangular plate with simple support along the edges, the displacements can be expanded as

This leads to a system of p.q uncoupled equations of the form where

The motion of the projectile is governed by

Eqns. (22, 25) represent a total of equations differential equations that must be solved to determine the contact force history and the motion of the plate and that of the projectile. This model accounts for the full dynamics of a simply supported rectangular plate.

2.4. CALCULATION OF ACOUSTIC PRESSURE The acoustic pressure at a point M(x,y) is calculated using Rayleigh’s formula

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where is the density of air, m/s is the speed of sound in air at 20°C, and d is the distance from M to an arbitrary point on the plate. The term d/c is the time it takes for sound waves to travel a distance d in air. Therefore, in Eq. (26), d/c accounts for the time delay between the sound pressure at the microphone location and the motion of the plate that created it. The d term in the denominator indicate that elements of the plate located farther away from the microphone have diminishing influence on the sound pressure level. 3.

Impact of Steel Ball on Aluminum Plate

Troccaz et al. [9] studied the impact of a 7.9 mm diameter steel sphere on a 405 x 2225 x 3 mm aluminum plate. The initial velocity of the plate is 1.4 m/s, the material properties of the projectile are GPa, and those of the plate are GPa, The impact on a half space model (Eqs. 5,6) predicts a maximum force of 302.58 N and a contact duration of The maximum indentation can be calculated from Eq. (1). From Eq. (16), the deflection of the plate at that time is given by

In this case, the deflection of the plate is m when the indentation reaches its maximum of 1.583 x m. Therefore, the deflection of the plate is not negligible as assumed in Section 3.1 and therefore the results predicted using the impact on a half space model are not expected to be accurate. Using the impact on infinite plate model, it is found that for this example which confirms that plate deflections have a significant effect on the impact dynamics. To apply the Olsson’s mass criterion we compare the mass of the projectile (1.955 g) to that of an aluminum disc 325 mm in diameter and 3 mm thick (672 g). Accordingly, the present impact is expected to be wave controlled since 1.955/672=.00291<0.29. Using the impact on infinite plate model, we find that N and The values reported by Troccaz et al. [9]: 191N and Since the plate is isotropic, the wave front is circular and, according to Eq. (15), its diameter at the end of the impact will be mm. It is interesting to note that in this case, according to the mass criterion, the diameter of the wave-affected zone is 57.62 mm which is very close to the actual size. The complete model described in section 3.3 gives a maximum contact force of 202.6 N and an impact duration of 37.5 This is a harder impact than predicted by the impact on finite plate model. Eq. (16) implies that, once the contact ends, the displacement of the plate at the point of impact should remain constant until a wave is reflected from the boundary and travels back to that point. Eq. (18) predicts that mode 11 will not reach the boundary until The acceleration at the impact location is expected to be zero during that period. Fig. 3 shows that instead the acceleration starts to increase again after about When a 1000 x 1000 mm aluminum plate with the same thickness is subjected to the same impact, the displacement and the

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acceleration at the impact location remain constant (Fig. 4). This suggests that for the 325 x 405 mm plate, higher order modes traveling faster that predicted with in Eq. (18) reached the boundary and traveled back during the first 500

Fig. 5 shows a strong reflection from the boundary when and that right after the impact ended the deformed region is larger than that As expected, the deformation of the larger plate has not reached the boundary during the same time period (Fig. 6). Figure 7 shows that the acceleration is more severely affected by reflections of wave from the boundary.

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Troccaz et al [9] measured the acoustic pressure generated at various points as a result of the impact. The location of the microphone (point M) is defined by the radius r and the angles and (Fig. 8). is the angle between the radius r and the vertical. is the angle between the projection of the radius r onto the xy plane and the x axis. For the first measurement the microphone is located along the normal to the plate at the impact point The microphone should receive its first

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signal after 291 as the wave induced by the motion at the point of impact has traveled 10 cm in air. During the contact phase, the deformed region of the plate grows and at the end of the contact phase, the diameter of the contact zone estimated by in Eq. (18) is 98.2 mm. The effect of the motion of the edge of the deformed zone reaches point M when Therefore a pressure pulse is expected between 291 to 363 after the impact begins. This is in good agreement with the results of Troccaz et al. [9] in terms of the arrival time and the duration of the pulse. For later times, the noise due to bending waves reflected from the boundaries cause additional sound pressure variations as expected from Fig. 7.

When the microphone is located at cm, a different scenario occurs. Travel time from the impact point O to the microphone location M (Fig. 9) is 308.4 Experimental results show the arrival of a pulse at that time but other signals reach the microphone earlier. For mode 11 (as defined in section 2.3), the radius of the deformed zone, OE= 28.3 mm after 39 . The time needed to travel from E to M is 230.8 Therefore, the contribution from this mode reaches M after 269.8 Similarly, the contribution of modes 55 and 77 reaches M after 216 and 182 respectively. The dispersive nature of the wave propagation in the plate and the fact that the disturbance travels faster in the plate than in air results in a more complex signal in which a single pulse cannot be identified. However, the sound pressure is sensitive to the motion of points located closer to the measurement point. It should be possible to use the sound pressure measured near the surface of the plate, away from the line of impact, to monitor the progression of the bending deformation. 4.

In-situ identification of elastic properties

To use low velocity impacts for in-situ identification of the elastic properties of plates three issues have to be addressed: how to generate wave-controlled impacts, how to measure the propagation of bending waves, and how to extract the desired information from measurement data. For the dynamic response of the plate to be unaffected by the presence of boundaries, stiffeners, and other discontinuities, the mass of the sphere should be kept small. It can be shown that if is the mass of the wave-affected area and is

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the largest plate mass that can remain unaffected by boundaries, the mass criterion can be written as

where in Eq. (17). When the right hand side is 0.29 as in Olsson (2000). This extended mass criterion can be used to select the size of the sphere. Monitoring the progression of bending waves during the contact phase of the impact can be used to calculate two of the four bending rigidities of the plate and but the other two parameters and cannot be determined individually by this approach. However the ratio A in Eq. (14) could be determined. 5. Conclusion This paper presents a study of the sound radiated by a plate during impact by a foreign object. The analysis of the impact is performed using a model in which the plate is assumed to be of infinite extent. This analysis clarifies the noise generation mechanism and a new mass criterion for classifying impacts has been extended. The possibility of using sound pressure signals measured during wave-controlled impacts to identify the rigidities of the plates has been examined. 6.

Acknowledgements

This material is based upon work supported in part by NSF grant No. 9850351. The paper was completed while the author was on sabbatical leave from Southern Illinois University. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. 7. 1.

2. 3. 4.

5. 6. 7. 8. 9.

References Huang C.H., Ma C.C. (2001) Experimental Measurement of Mode Shapes and Frequencies for Vibration of Plates by Optical Interferometry Method. Journal Of Vibration And Acoustics. 276, 276. Araujo A.L., Mota Soares C.M., Moreira de Freitas M.J., Pedersen P., Herskovits J. (2000) Combined numerical-experimental model for the identification of mechanical properties of laminated structures. Composite Structures 50, 363-372. Abrate S. (1998) Impact on Composite Structures, Cambridge University Press. Abrate S. (2001) Modeling of Impacts on Composite Structures. Composite Structures 51 129138. Zener C. (1941) The intrinsic Inelasticity of large plates. Physical Review 59, 669-673. Olsson R. (1992) Impact response of orthotropic composite plates predicted from a oneparameter differential equation. AIAA J. 30, 1587-1596. Abrate S. (1993) On the Use of Levy's Method for Symmetrically Laminated Composite Plates. Composites 24, 659–661. Olsson R. (2000) Mass criterion for wave controlled impact response of composite plates. Composites Part A 31A, 879-888. Troccaz P., Woodcock R., and Laville F. (2000) Acoustic radiation due to the inelastic impact of a sphere on a rectangular plate. J. Acoust. Soc. Am. 108, Pt. 1, 2197-2202.

STATIC BEHAVIOUR OF PRE–STRESSED POLYMER COMPOSITE SANDWICH BEAMS R. A. W. MINES, Q. M. LI, R. S. BIRCH Impact Research Centre Department of Engineering (Mechanical) University of Liverpool Brownlow Street, Liverpool, L69 6GH, UK R. RIGBY, M. AL-KHALIL, A. TANNER Airbus U.K.,NTC Module D1 Dept B45 Filton,Bristol, BS99 7AR,UK

Abstract The paper describes an experimental and numerical study of the effect of pre-stress on the progressive collapse of polymer composite sandwich panels subjected to a central line load. The beams were made from carbon pre-preg laid up in a quasi-isotropic form for the skins and Rohacell 51WF foam for the core. Prestress loads were 0kN, 5kN, 15kN – the latter load being 15% of the in-plane failure load. Experimental results included force-deflection data and photos of the progression of skin and core damage. Experimental tests were simulated by the finite element code LS-DYNA. The focus for material modelling was the crush and failure behaviour of the polymeric foam core.

1. Introduction The use of polymer composite sandwich construction for primary structures in airframes is increasing. For example, sandwich construction has been used in small business jet fuselage structures [1,2], where primary considerations are damage tolerance and crashworthiness. There is general interest in sandwich structures by the Federal Aviation Authority in USA, and a specific area of consideration is damage tolerance [3]. The situation under consideration here is a structure that is under service loading and which is then subject to localised impact. This is a complex problem given the multiple material and structural failure modes, and so the simple case of a pre-stressed panel subject to a central line load is considered. This case has been considered for laminated panels [4], and it has been shown that the failure mode of the panel changes with inplane load. The numerical simulation of progressive damage in polymer composite sandwich construction is complex. Material models include progressive damage in the skin and crush and failure behaviour in the core. Not only does initiation behaviour need to be 671 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 671–682. © 2002 Airbus UK Ltd 2001 -reproduced with permission. Printed in the Netherlands.

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characterised, but also the progression of damage through the structure needs to be modelled. There is a need to balance the complexity of the models between the simulation of real structures [1,2] and ensuring rigorous and fundamental material models.

2.

Experiments

Figure 1 shows the test rig for the sandwich panel. The sandwich panel skins were made from a uni-directional carbon fibre pre-preg based on Fibredux 924 (924CT300(6k)-5-34%), which is used for primary aerospace structures subject to elevated temperatures [5]. The properties of the UD material are GPa and The UD material was laid up with eight plies in a quasi-isotropic form, i.e. [+45,-45,0,90]s. The foam core used was Rohacell 51 WF [6], which is a closed cell polymethacrylimide foam, and this was supplied in 10mm thick sheet. The 300mm by 300mm panels were made in a purpose built pressclave, and details of the cure cycle are given in [4]. An adhesive film was used between the skin and the core, and the core was pricked every 2.5mm prior to the application of the adhesive film in order to ensure the best possible bonding. There was a problem with the cure pressure, as this caused crush of the core during cure, and so spacers had to be used to prevent this from happening. The skin thickness was 1mm (each ply being 0.125mm thick) and the core thickness was 9mm. Thus a pre-crush in the core of 1mm occurred during manufacture.

The cured panel was cut into two 250mm by 100mm panels using a diamond saw. Each panel was then bonded to aluminium tabs using an epoxy adhesive, and the bond surfaces were reinforced with three large countersunk screws for each end. The specimen was then placed into clamps, which allowed the panel to be loaded in-plane but which constrained the ends to remain horizontal. The local geometry at the clamps was carefully designed so as to avoid the clamp edges cutting into the panel at large panel lateral deflections. The in-plane load was achieved using a hydraulic jack and a piezoelectric load cell was used to measure the in-plane load. Three in-plane loads were

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considered, namely 0, 5 and 15kN. The lateral load was achieved using a 15.9mm diameter cylinder attached to a servo-hydraulic machine, which ran at a displacement rate of 0.0167mm per second. As far as in-plane displacement was concerned, one end of the specimen was clamped and the other end was adjusted so that the in-plane load was constant at a pre-set value as the panel was loaded laterally. This means that the central load moves slightly from the central span during a test. The in-plane displacement was measured using a linear displacement transducer. The numerical simulation of panel behaviour requires material properties as input. Given the complexity of panel response (see below), it was decided that the progressive damage in the skin would not be modelled and that efforts would be focussed on the crush and failure of the core. Also, for simplicity it was decided to use a piecewise linear plasticity model (DYNA Material 24) with parameters based on measured tensile data for a complete skin lay-up. In other words, compression properties were assumed to be the same as tensile properties and properties in the 8 ply quasi-isotropic laminate were assumed to be ‘smeared’. As far as the mechanical properties of the foam core are concerned, the measurements for Rohacell 51WF have been discussed extensively in [7]. DYNA uses the Fu-Chang foam material model (Material 83) with tensile properties, and it was decided to fully validate this foam model using a combined tension and shear test [7]. In this, four blocks of foam are subject to a shear load and then the blocks are crushed [7]. The validation and calibration of the foam model is discussed in the simulation section. Figures 2 and 3 give typical traces for the 0, 5 and 15kN in-plane load, respectively. Three repeat tests were conducted for each pre-stress condition, and results were shown to be reproducible. Three parameters are shown in each figure, namely lateral load, inplane load and in-plane displacement. As far as the 0kN in-plane load is concerned, three panel failure modes occur in the order of core shear at a deflection of 3mm, upper skin failure at a deflection of 33mm and lower skin failure at a deflection of 44mm. Figure 4 shows a photo for the OkN in-plane load at a deflection of 34mm, i.e. including core shear and upper skin failure. As far as the 5kN in-plane load is concerned, Figure 3 shows failure modes of core shear at a deflection of 4mm and core skin debond at a deflection of 25mm. Figure 4 gives a photo of the 5kN in-plane load case at a deflection of 38mm, i.e. including core shear and core-skin debond. As far as the 15kN in-plane load is concerned, Figure 3 shows failure modes of core shear at a deflection of 3mm and upper skin failure at a deflection of 24mm. Figure 4 gives a photo of the 15kN in-plane load case at a deflection of 16mm,i.e.with core shear failure only. From Figure 4, it can be seen that panel deformation adjacent to the clamps is large. Hence the boundary conditions may influence panel behaviour. However, the aim of this work is to validate computer simulations. Figure 4 also shows a vertical line on the foam core, which denotes the central span. The displacement of this line with respect to the loading cylinder during a test can be seen. From this data, it can be seen that the mode of beam failure is dependent on in-plane stress. However, core shear occurs at similar deflections for all three cases. One issue is the energy absorbed for a given lateral beam deflection. The energy absorbed by the beam is given by the work done by the lateral load minus the work done by the in-plane load [4]. It can be shown that the energy absorbed by the panels to a given lateral deflection of 20mm are for the 0kN case – 10 J , for the 5kN case – 23 J and for the

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15kN case – 35 J, i.e. as the in-plane load increases so the ability of the panel to absorb energy increases. Figure 4 shows that this is achieved by increasing core crush with inplane load. From Figure 3 , it can be concluded that the core-skin debond does not greatly change the rate of beam energy absorption with increasing lateral load.

3. Numerical Simulation The numerical simulation of panel behaviour was achieved using LS-DYNA Version 950 [8]. It should be noted that this is an explicit finite element code suited to transient impact problems. The code is not suited to static work. However, the long term aim of the research was to simulate the drop weight impact behaviour of pre-stressed sandwich panels, and so it was decided that, for compatibility, the static experimental results should be simulated with DYNA. This means that, in order to avoid excessive computer run times, a beam of only 20mm width (not 100mm) was analysed. Also, the beam was loaded at a displacement rate of 40mm per second as compared with the experimental value of 0.0167mm per second. In order to reduce inertia effects, the density of the beam was increased by a factor of 16. As stated previously, a large amount of effort was expended to properly model the multi-axial crush and failure of the foam. The material model selected was Material 83, which is based on theory developed by Fu-Chang et. al. [9]. Data input required is mass density, Young's Modulus, viscous coefficient, and pressure versus volumetric strain curve (see [8]). The basic model was then enhanced by including the tension part

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of the nominal stress-strain curve, by including tensile failure of the foam as a function of maximum principal strain and by including an eroding single surface for the central portion of the foam and for top and bottom skins, which means that if an element deletes as a result of failure then the gap so created can close with contact between the faces. The resultant foam model was investigated and calibrated using the combined shear and compression test mentioned in the experimental section [7]. Figure 5 shows the finite element model of the multi-axial specimen, and shows experimental and numerical results compared for a ratio of shear stress to shear failure stress of 0.52, i.e. a transverse stress of 0.40 MPa. In the finite element model, elements 2,5,10 and 13 were the foam specimens and all other elements were steel. Elements 4 and 12 were pulled outwards, and the complete assembly was compressed [7]. It can be seen that the actual trace departs from the finite element model without foam failure, showing that the real material fails at this combined level of compression and shear stress. It was found that a maximum tensile principal strain value of 0.0168 allowed compression of the foam to proceed, without foam failure, for transverse stresses of less than 0.32 MPa. Above this value, the foam has deemed to have failed and the element is deleted. The core was modelled using solid elements [8].

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The skin was modelled as a piecewise linear plasticity curve (DYNA Material 24) as previously mentioned. The skin elements were Hughes Liu shell elements [8], and two integration points occur through the element thickness. Initial stiffness was 38GPa up to a strain of 0.91%, and the stiffness was then 32 GPa up to a failure strain of 1.22%. When an integration point exceeds the critical strain value, the stiffness of the element is degraded. When all the integration points exceed the critical strain then the element is deleted. Failure was specified for a plastic strain greater than 0.31%, i.e. for a total strain of 1.22%. The loading cylinder was modelled as a geometrical cylindrical stone wall, which allows forces exerted on the panel to be easily computed. Contact surfaces were placed between the loading cylinder and the top skin, and also between the end supports and the bottom skin. There was also an initial 0.3mm gap between the loading cylinder and the top skin. Figure 6 compares force-deflections for experiment and simulation for in-plane loads 0kN, 5kN and 15kN. For the 0kN case, the simulation terminated at a lateral deflection of 40mm. For the 5kN case, this value was 26mm and for the 15kN case the value was 14.5mm. Figure 7 shows that only half the beam was modelled and that the skin mesh was refined near the indenter and that the core mesh was also refined in this area. Figure 7 also shows that the skin mesh was refined at the support, as was the clamp mesh, for the 15kN in-plane load case.

4. Comparison between simulations and experiments Figure 4 and Figure 7 compare failure modes for the 0kN in-plane load case. In the simulation, foam element deletion has occurred for the shear crack near the indenter and at the lower core-skin interface. This has reproduced the actual failure shown in Figure 4. From Figure 6, it can be seen that the progression in damage in the simulation is similar to the experimental case up to upper skin failure (II). The simulation under predicts upper skin beam failure load and deflection. This is probably due to the simplified material model used here – in reality, a complex stress distribution exists across the skin and damage progresses through the thickness of the skin [10]. Also, in the simulation at II, two of the four skin elements under the indenter have failed. This means that subsequent behaviour assumes two elements in the upper skin are still effective. It should also be noted that more foam failure and element deletion occurs in the simulation as compared with experiment, showing that material behaviour in the simulation is more prone to damage. The simulation also shows that some foam elements adjacent to the top skin have markedly compressed. This is a result of strain softening and localisation in the Rohacell foam [11], in which the volume of foam, in which crush is initiated, continues to crush to lock up before initiating crush in an adjacent volume. Figures 4 and 7 compares failure modes for the 5kN in-plane load case. The simulation duplicates the shear crack near the indenter. However, the simulation does not model the core-skin debond. This is because it was felt that the debond may be due to a manufacturing problem and hence was not a valid mode of failure. Also, modelling the core-skin debond proved to be difficult. In the simulation, foam elements have been

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deleted near the upper skin and core elements have been crushed locally. The progression of damage shown in Figure 6 agrees well with experimental results. The numerical simulation terminates at deflection of 26mm, as a result of the skin shell elements nearest the indenter failing. In the experiment, the upper skin did not fail at this point. Finally, Figures 4 and 7 compares results for the 15kN in-plane load case. Foam failure and element deletion has occurred at three places along the beam in the core whereas in Figure 4, shear cracks have occurred in the foam core at one third span to the left of the indenter and at one quarter span to the right of the indenter. Hence, the simulation is more sensitive to foam failure. The progression of damage shown in Figure 6 is similar to the experimental results. The simulation terminates at a transverse displacement of 14.5mm,again due to top skin failure.

5. General Discussion There are two main points of interest in this paper. The first is a series of experimental results showing the effect of in-plane pre-stress on the progressive collapse of a sandwich beam. The results show that for in-plane loads up to 15kN, the main failure mode is core shear. This mode of failure initiates at similar deflections for the three inplane load cases and the mode markedly changes the force-deflection characteristic of the structure. Subsequent core damage does not markedly change the beam forcedeflection characteristics. Upper skin failure occurs in the experiment for the 0 kN and 15kN in-plane load case. However, upper skin failure did not occur for the 5 kN in-plane load case. Thus , there is a complex interaction between local indentation and global loads, which is dependent on the relative stiffnesses of the skin and the core. Core-skin debonding occurred for the 5kN in-plane load case. This may be due to a problem in manufacturing and needs to be investigated further. It is proposed that this mode of failure should be suppressed by improving the core-skin bond strength. It should be noted that the above comments are for a specific experimental set-up , e.g. beam lay-up, beam span, beam boundary conditions etc, and results may changed for other configurations. However the main aim of this work was to develop and validate a computer simulation. This brings us to the second main point of interest in the paper. A failure model for the foam which takes into account not only failure due to compression, but also failure as a result of tensile stresses has been developed. This means that failure in the foam core resulting from shear (tensile principal) stresses can be modelled and that the progression of damage through the foam core can be simulated. Results have shown good agreement between experiment and simulation. There is a tendency for the simulations to show more damage than the experiment. Hysteresis should be included in the Fu-Chang material model – problems were sometimes encountered when some foam bricks re-expanded after being compressed. Also, some foam brick elements compressed more than others, for similar stresses, giving rise to large distortions and negative volumes. The reason for this, as stated before, could be due to softening and localisation effects in the foam [11].

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The failure models for the skins need to be improved, to include compressive effects and ply by ply progressive damage, e.g. see [10]. Also, once the upper skin fails, the model terminates here whereas the panel continues to be loaded in the experiment. Hence the numerical model needs to be improved at this point.

6. Conclusions Experimental results have shown that beam response is similar for in-plane loads in the range of 5 to 15 kN. The main mode of beam failure is core shear, and after this occurs the force – deflection characteristics of the beams are similar. Also, for a given deflection, a beam with a larger in-plane load has absorbed a larger amount of energy due to the crush of the core. A foam failure model, based on maximum principal tensile strain, has been developed and implemented into beam simulations. The failure model has allowed foam failure to be monitored during the progressive collapse of the beams. Good agreement has been shown between experiment and simulation. Progressive damage in the skin was not modelled, but this was perceived to be a secondary effect. There is now a need to extend the modelling to more complex three dimensional cases.

7.

Acknowledgements

© Airbus UK Ltd 2001 –reproduced with permission. Thanks are due to Dr Thomas Muenz of CADFEM, Germany, for help in the development of the LS-DYNA model. Experimental work was conducted in the Impact Research Centre, The University of Liverpool. The research was supported by ‘CRASURV – Design for Crash Survivability of Commercial Aircraft’ which was a RTD project partially fiinded by the European Union under the Aeronautics Area of the programme on Industrial and Material Technology (BRITE/EURAM)

8. References 1. Fasanella E.L., Jackson K.E.,Analytical and experimental evaluation of scaled composite energy

2. 3. 4.

5. 6. 7. 8. 9.

10. 11

absorbing subfloor concepts, AHS National Technical Specialists Meeting on Rotorcraft Crashworthiness, Phoenix, AZ., September 14-16,1998 Jackson K.E., Fasanella E.L., Impact testing and simulation of a crashworthy composite fuselage structure, AHS 56th Annual Forum, Virginia Beach, Virginia, May 2-4,2000 Tomblin J., Lacy T., Smith B. Hooper S., Vizzini A.,Lee S., Review of Damage Tolerance for Composite Sandwich Airframe Structures,FAA Report No. DOT/FAA/AR-99/49, August 1999 Mines R.A.W., Li Q.M., Birch R.S., Static behaviour of transversely loaded CFRP laminates subject to in-plane tension, STRAIN, Vol.36 No.2,(2000), 21-30 Hexcel Composites, Fibredux 924 Data Sheets, Hexcel Composites, Duxford, Cambridge,CB2 4QD,UK,1997 Roehm , Rohacell 51WF, Roehm Ltd, Bradbourne Drive, Tilbrook, Milton Keynes, Bucks., MK7 8AU Li Q.M., Mines R.A.W., Birch R.S., The crush behaviour of Rohacell 51WF structural foam, International Journal of Solids and Structures, Vol. 37, (2000),6321 -6341 LS-DYNA, Users manual Version 950, 2000 Fu Chang S., Song Y., Lu D.X., DeSilva C.N., Unified constitutive equations of foam material, Journal of Engineering Materials and Technology (ASME), Vol. 120 (1998), 212-217 Mines R.A.W., Alias A., Numerical simulation of the progressive collapse of polymer composite sandwich beams under static loading. To be published Composites Part A Li Q.M., Mines R.A.W., Strain localisation in rigid crushable foam during uniaxial compression, University Of Liverpool Impact Research Centre Report No. IRC/183/99 (1999)

ON DEBOND FAILURE OF FOAM CORE SANDWICH

L. A. CARLSSON Department of Mechanical Engineering Florida Atlantic University Boca Raton, FL33431

Abstract Face/core debond fracture toughness has sandwich beams with a foam core made from cross-linked PVC has been examined experimentally and analytically using a recently proposed test method called Tilted Sandwich Debond (TSD) test. In this paper we will present a brief summary of research accomplishments in this area. 1. Introduction A sandwich structure provides very high stiffness per unit weight due to the combination of stiff face separated by a low-density core. If a debond is introduced in such structure, its structural stiffness may be substantially reduced because of the loss of shear and tension transfer between face and core [1,2]. The face sheet over a locally debonded region under compression may undergo buckling which may lead to further propagation of the debond [3]. If the sandwich is loaded in shear, a core crack may initiate at the tip of the bond and propagate through the entire core between the face sheet [3]. Consequently, face/core debonds may pose a substantial threat to the integrity of a sandwich structure. In this paper we will present a summary of research performed to characterize face/core debond failure of foam core sandwich structure since the late 1990's. The research revolves around a debond fracture test specimens, called Tilted Sandwich Debond (TSD) specimen. Test procedures and analysis methods will be described, and fracture toughness data will be highlighted. 2. The Tilted Sandwich Debond (TSD) Specimen Figure 1 shows the principle of the TSD specimen. This specimen was proposed by Grenestedt [4] who, however, did not publish any report on the method. The specimen has an initial crack introduced at the interface between the top face and core. A vertical load, P, is applied at the end of the debonded top face until the crack propagates. The specimen is tilted with respect to the horizontal by an angle, i.e. the "tilt angle". The purpose with the tilting was to promote continued growth at the face/core interface of the artificially introduced debond, see Fig. 1. As discussed by Prasad and Carlsson [5,6], a face/core debond commonly propagates into the core after "klinking", i.e. deflection of the crack plane as discussed by He and Hutchinson [7]. By 683 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 683–688. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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tilting the TSD specimen, a tensile load component acts on the debond face sheet which would promote a state of stress at the crack tip less favorably for kinking [8]. A detailed analysis of the stress intensity factors at the crack tip, however, reveals that the stress intensity factors are not sensitive to the tilt angle, and that other factors govern the details of the crack propagation path [9]. This analysis will be described later in this paper.

2.1 ANALYSIS OF TSD SPECIMEN The TSD specimen was first analyzed using an elastic foundation model [10]. In this analysis the TSD specimen, Fig. 1, is considered as consisting of a end-loaded cantilever beam supported by the core represented by an elastic foundation. The analysis is based on a superposition of the solutions for a end-force-loaded cantilever beam and a force and moment loaded beam on an elastic foundation. For TSD specimens with a reasonably long crack length, a, it is possible to express the load-point compliance, C, and strain energy release rate, G, as,

where

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P is the load applied to the TSD specimen, Fig. 1. K is the foundation modulus, is the core thickness, is Young's modulus of the core, I is the moment of inertia of the face sheet, and is the flexural modulus of the face sheet. Later, more detailed numerical finite element analysis was performed to determine compliance, energy release rate, and stress intensity factors for the TSD specimen [9]. Figure 2 shows compliance of a TSD specimen plotted versus crack length as determined experimentally, from elastic foundation model (EFM); eq. (1), and finite elements (FE). The sandwich consisted of 3.6 mm thick glass/vinylester face sheets over a 50 mm thick H200 PVC foam core. Mechanical properties of the face sheets and core are provided in Ref. [9].

Figure 2 shows that both the elastic foundation model and finite element predictions are in good agreement with experimental data. Stress intensity factors characterizing the opening and shearing modes of fracture, and were determined from the finite element model [9], The results show that the presence of a stiff face sheet over a compliant core influences and significantly as compared to a homogeneous specimen. Also, the exact crack position influences and It was found that is positive when the crack is at the face/core interface. This indicates tendency for klinking of the interface crack, which is indeed observed in many TSD specimens [8]. For a crack parallel to the interface, located in the core, it was found that crack tip loading is essentially mode I. This is consistent with the experimental observation of face/core cracks growing in the core parallel to the interface [8]. Parametric finite element analyses further revealed that kinking is less likely for sandwich beams with high modulus cores.

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2.2 EXPERIMENTAL STUDIES OF FACE/CORE DEBOND FRACTURE Careful inspection of the crack propagation in foam core sandwich specimens with adequate face/core bond reveals that an artificially introduced face/core crack tends to deflect into the core, normally to a quite small depth, and then grow in the core parallel to the interface. The depth, i.e. distance from the actual face/core interface, is about 1 2 mm [8, 11], although commonly there is some meandering of the crack. Only in sandwich specimens with high-density tough cores it has been possible to grow the crack at or very near the actual face/core, or core/resin layer interface [11]. Figure 3 shows a scanning electron micrograph of the edge of a fractured TSD specimen with R75 core [11]. An interphase region consisting of resin-filled surface cells can be seen in the lower region of the micrograph. The debond fracture obviously occurs in a plane outside this region. The fracture plane involves fracture of cell walls of the foams only, which would be less energy consuming than fracturing the resin and cells in the proximity of the actual face/core interface.

Based on the observation that face/core debond fracture actually involves fracture of the foam core, it would seem reasonable to expect that the inherent toughness of the core would govern the measured debond toughness, expressed as the critical energy release rate, For a given polymer, the fracture toughness of the foamed polymer is increasing with the effective density, of the foam [12], where and are the densities of the foam and dense polymer. Figure 4 shows face/core debond toughness for sandwich specimens with PVC foam cores over a range of foam densities [12]). The data approximate a linear relationship except for the H200 foam where the toughness falls above the line.

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The straight line in Fig. 4 may be expressed as,

This equation provides an interpolation equation for face/core debond toughness of well-bonded sandwich specimens with cross-linked PVC foam cores in the density range from 36 to [11]. 3.

Summary

The TDS specimen has been examined herein. In particular, we have examined its ability to achieve face/core debonding fracture in foam core specimens without the undesired kinking mode often observed. Analysis of compliance, strain energy release rate, and fracture modes has been reviewed, and fracture toughness data has been presented. Overall, the experience with the TSD specimen shows that this test is a very promising candidate for determining the face/core debond toughness of sandwich specimens with cross-linked PVC foam cores under mode I dominated conditions. Its applicability to other core materials remains to be examined. 4.

Acknowledgements

This research was sponsored by the Office of Naval Research (Grant #N00014-90J1995) with Dr. Yapa D. S. Rajapakse as the program manager through a sub-contract from Tuskegee University. We are grateful for the support by Prof. Mahfuz and his

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students at Tuskegee for the manufacture of sandwich panels, and to Chris Kilbourn of DIAB, Texas for supplying of foam materials. The drawings were prepared by Ms. Shawn Pennel. Typing by Mrs. Sara Martinez is gratefully acknowledged. 5. 1.

References

D. Zenkert, "Damage Tolerance of Foam Core Sandwich Constructions," Ph.D thesis, The Royal Institute of Technology, Stockholm, Sweden, 1987 2. L. A. Carlsson, L. S. Sendlein and S. L. Merry, "Characterization of Face Sheet/Core Shear Fracture of Composite Sandwich Beams," J. Compos. Mater., Vol. 25, 1991, pp. 101-116. 3. A. Sipsha, "Failure of Sandwich Structures with Sub-Interface Damage," Doctoral Thesis Report 2001-13, Department of Aeronautics, Royal Institute of Technology, Stockholm, May 2001. 4. J. L. Grenestedt, Department of Mechanical Engineering, Lehigh University, personal communication, 1996. 5. S. Prasad and L. A. Carlsson, "Debonding and Crack Kinking in Foam Core Sandwich Beam-I: Analysis of Fracture Specimens, " Eng. Fracture Mech, Vol. 47, 1994, pp.. 813-824. 6. S. Prasad and L. A. Carlsson, "Debonding and Crack Kinking in Foam Sandwich Beam-II: Experimental Investigation, " Eng. Fracture Mech, Vol. 47, 1994, pp. 825-841. 7. M. He and J. W. Hutchinson, "Kinking of a Crack Out of an Interface," J. Appl. Mech.,Vol. 56, 1989, pp. 270-278. 8. X. Li and L. A. Carlsson, "The Titled Sandwich Debond (TSD) Specimen for Face/Core Interface Fracture Characterization," J. Sandwich Struct. Mater., Vol. 1, 1999, pp. 60-75. 9. X. Li and L. A. Carlsson, "Fracture Mechyanics Analysis of Titled Sandwich Debond (TSD) Specimen," to appear in J. Compos. Mater., 2002. 10. X. Li and L. A. Carlsson, "Elastic Foundation Analysisi of Titled Sandwich Debond (TSD) Specimen," J. Sandwich Struct. Mater., Vol. 2, 2000, pp. 3-32. 11. G. M. Viana and L. A. Carlsson, "Core Density Influence on Debond Toughness of Sandwich Specimens with PVC Foam Core," to be published 12. L. J. Gibson and M. F. Ashby, "Cellular Solids - Structure and Properties," 2nd ed., Cambridge University Press, Cambridge, 1997.

CORE CRUSH MECHANISMS AND SOLUTIONS IN THE MANUFACTURING OF SANDWICH STRUCTURES

H. M. HSIAO, S. M. LEE AND R. A. BUYNY Hexcel Corporation Research and Technology Dublin, CA 94568

Abstract Core crush is a manufacturing defect occurred during the autoclave curing process of composite honeycomb sandwich structures. It usually leads to costly part rejections since the defect is non-repairable. In addition, this problem has posted constraints on aircraft engineers by limiting the ranges of core density and core thickness that could be used when designing these types of structures. In commercial production, several techniques (e.g., ply tie-down, pre-cured adhesive over the core) have been applied to restrain the core from collapsing inward; however, these approaches require additional labor, time and material cost and thus may not be the best solutions. This paper discusses the recent understandings in core crush mechanisms and the subsequent developments in core crush resistant prepreg based on that foundation. It shows that the prepreg frictional resistance is the key factor in controlling core crush. While past research in the scientific community has mainly focused on resin effects in core crush, studies conducted in Hexcel show that the core crush can also be significantly reduced by controlling construction of the fiber tow shape and fabric architecture. Rounder fiber tow or more open fabric produces rougher prepreg surface and thus yields higher prepreg frictional resistance to reduce core crush. Experimental results show that, without changing the resin, the developed core crush resistant prepreg increases the prepreg frictional resistance and effectively reduces the core crush.

1. Introduction Composite honeycomb sandwich structures are widely used in the aerospace industry as panel parts in various aerospace structural applications such as ribs, flaps, spars, and rudders. They are formed from a lay-up of prepreg skin plies encompassing a honeycomb core, the latter typically having beveled edges that taper from full thickness to none with a constant chamfer angle. Despite of long production experience in the aerospace industry worldwide, manufacture of these structures is still plagued by significant reject ratio, generating substantial quantities of unusable scrap and impacting negatively on production cost. One particular reason for rejection of cured parts is known in the industry as "core crush", the collapse of the honeycomb core in its weak lateral directions during autoclave curing especially when the core density is low 689 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 689–700. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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(Figure 1). Honeycomb core crush is among the costliest manufacturing problems in composite fabrication: the crush of the panel is generally so extensive that it is beyond repair. Over the years several techniques have been developed to restrain the core from collapsing inward. For example, Corbett and Smith [1] used tie-down plies in contact with the core to prevent core crush in sandwich structures. These tie-down plies were extended outwardly from the part beyond the trim line of the finished product and secured to the layup mandrel with tape. Hopkins and Hartz [2] discovered that, by using Corbett's tie-down method, core crush still occasionally occurred since these tiedown plies could pull away from the tape. They later developed an improved tie-down method. It should be noted that these approaches usually require additional labor, time, and material cost and thus may not be the best solutions.

Core crush occurs during the autoclave curing when the honeycomb sandwich structure is subjected to pressure and heat. The pressure difference between the autoclave pressure and vacuum in core provides the mechanical driving force for core crush; while the resisting forces developed to prevent the core from collapse are internal core pressure, the lateral compressive stiffness of the skin/core combination, and the prepreg frictional resistance [3-4]. When core crush occurs, the prepreg plies are observed to move inward with the core and slippage is initiated either between the prepreg plies or between the prepreg/tool interface. The elevated temperature during curing facilitates the slippage of prepreg plies by lowering the resin viscosity and providing the lubricating effect [5]. In the next section the core crush mechanisms and the entire crush sequence will be explained in much greater details from both the viewpoint of the structure as a whole (as typically described in most references) and the viewpoint of each individual ply on the ply level. Previous research studies in the scientific community have mainly concentrated on the resin effects in core crush [6-7]. The leading conclusion from these works indicated that resin is the predominant factor in causing core crush: at similar levels of impregnation, a highly rubberized, low-flow, elastomer modified epoxy prepreg showed consistently greater core crush than a standard high-flow epoxy prepreg. On the other

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hand, this and previous studies [8-9] conducted in Hexcel has demonstrated the core crush can also be reduced by controlling construction of the fiber tow shape and fabric architecture. It should be emphasized, however, that core crush is a complicated phenomenon and every component and step from raw material (fiber, resin) to fabric/prepreg processing (tow forming, weaving, impregnation, post-impregnation) and part manufacturing (curing cycle, vacuum level, layup, bag/tool surface) plays certain role in it. 2. Core Crush Mechanisms

Core crush occurs during the autoclave curing when the honeycomb sandwich structure is subjected to pressure and heat. As the autoclave is pressurized, the bag applies a compacting pressure P that is perpendicular to the structure's surface. A horizontal resultant force created by the difference between the horizontal component of this compacting pressure and the vacuum in core provides the mechanical driving force for core crush. From the point of view of the structure as a whole (shown schematically in Figure 2), this horizontal force is opposed by three independent forces. These three resisting forces are the frictional force developed by the prepreg ply of top skin against the bagging material, the frictional force developed by the prepreg ply of bottom skin against the tool plate, and the combined stiffness of the core and the uncured prepreg plies.

It should be noted this is a problem dealing with the equilibrium of a structure made of a honeycomb core and several connected prepreg plies. Since the frictional resistances between various interfaces of these connected plies are different (e.g., prepreg-to-bag, prepreg-to-prepreg, prepreg-to-tool), this problem calls for the determination not only of the external forces acting on the structure but also the internal forces which hold together the various plies of the structure. In this case the internal forces, such as the frictional force developed by the prepreg plies against each other, did not appear in the above equilibrium diagram of the structure. However, if the structure is dismembered and a free-body diagram is drawn for each of its component plies, the forces holding these plies together become external forces from the point of view of each component ply. Figure 3 shows the free-body diagram for each component ply of the structure in Figure 2 after it is dismembered. To simplify the diagram, only six prepreg plies are included in the figure to represent the real case. Using the notation in Figure 3, the core crush mechanisms are described as follows:

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1. When the autoclave pressure is applied, the honeycomb core tries to crush and drag its two adjacent prepreg plies (ply 3 and 4) with it. Here ply 3 and 4 are treated as one free body. 2. This movement is resisted by the frictional forces developed between ply 2 and 3 and between ply 4 and 5 3. At the interface between ply 4 and 5, in order to reach the force equilibrium, a reaction force equal and opposite to the friction force within the same interface develops according to the Newton's third law. This reaction force tries to drag the next neighboring ply, ply 5, inward by shear and again a frictional force develops between ply 5 and 6 in response to that. Similar scenario occurs on the prepreg plies of top skin with the friction/reaction forces of instead. 4. The crushing force applied to the honeycomb core therefore reaches the outermost plies (ply 1 and 6) through a series of internal force interaction and transfer across the prepreg skin plies. It should be noted that, if viewed from the point of the structure as a whole, these internal forces developed inside skins cancel out each other and Figure 3 becomes identical to Figure 2. 5. Assuming the stiffness of the uncured prepreg plies is comparatively weak, once the internal driving force and exceeds the frictional resistance of any interface at the top and bottom skin, respectively, the honeycomb core is basically ready to crush. However, there are three types of interface between these prepreg plies (prepreg-to-bag, prepreg-to-prepreg, prepreg-to-tool) and their frictional resistances are quite different. Therefore, it becomes very critical in determining the order of and so the slip planes at both top and bottom skins during core crush can be identified. Friction tests were conducted using the test method of Martin et al. [4] and the order of and was found to be Therefore, when the honeycomb core crushes, slip occurs between the prepreg and prepreg interface at the top skin and between the prepreg and tool plate interface at the bottom skin. In fact, this is exactly what has been observed in numerous core crush test panels through visual inspection and microscopy. One important implication here is that, unlike what was plotted in the free body diagram of Figure 2, the critical frictional resistance for the top skin is actually instead of during core crush.

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6. Based on the above information, core crush can be prevented when the combined resisting forces exceeds the horizontal component of the net crushing force

7.

3.

It should be noted that at the very beginning stage of the curing cycle, this relationship holds true and the left-hand side of the equation is much greater in magnitude than the right-hand side. However, all these forces continue to change throughout the curing cycle. As the resin softens due to heating and starts to act like lubricant, both frictional resistances and ) and the stiffness of the skin/core combination decrease significantly. At certain temperature before the resin viscosity bottoms out, these combined resisting forces drop to a magnitude so low that the crushing force could no longer be held. As discussed previously, the crushing force applied to the honeycomb core is transmitted to the outermost plies through shear forces. During this process, those two prepreg plies next to core are the most key element since they are the ones that react to the core movement and trigger the subsequent mechanisms. If these two specific plies were "tied-down" and not allowed to move, the core crush sequence described in 1-6 will be disrupted and core crush will unlikely to occur. Conceptual Approaches for Solutions

Since prepreg frictional resistance is the key factor in controlling honeycomb core crush, development of core crush resistant prepreg becomes possible if its frictional resistance can be increased. According to the principles of friction, higher frictional resistance can be achieved by increasing the undulation or irregularity degree of the surface topography in contact, reducing the amount of lubricant on the dry surface, or selecting a different lubricant system. When translated into prepreg characteristics, these parameters correspond to rougher dry fabric surface, less amount of resin on prepreg surface, and a resin system with different rheology and advancement, respectively. Based on these understandings, it is reasonable to deduce that rounder fiber tow as shown in Figure 4 produces rougher dry fabric surface (higher undulation degree) and thus yields higher prepreg frictional resistance to reduce core crush. In addition, rounder fiber tow creates more open fabric that helps promote mechanical interlocking or nesting between prepreg plies and further increases prepreg frictional resistance, as will be shown in the following section. Another surprising benefit of the rounder tow is that it allows better impregnation during prepregging and thus leaves less resin on prepreg surface. This leads to greater contact between fabrics and also increases the frictional resistance. Various methods can be used to manipulate or modify the fiber tow shape, and therefore the fabric architecture of a prepreg to the desired levels. Examples of such methods include, but are not limited to, sizing the tows prior to weaving, tow twisting, tow twisting and untwisting, varying the cross sectional shapes of the individual filaments in the tows, or employing various other modifications of the tow forming, weaving, prepregging, or post-impregnation process.

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One method for modifying the construction of fiber tow shape is by mechanically twisting or twisting untwisting the tow to a desired extent. A "twisted tow" means a tow that is subjected to a twisting process during the tow forming. Typically, twisted tows have a rounder shape fiber tow following weaving. The degree of twisting, i.e., the number of turns per unit length, can vary with the tow shape desired. An "untwisted" tow can be formed by first twisting a filament bundle to a desired degree and then untwisting, i.e., winding the twisted fiber filament bundle or tow in the opposite direction, to a desired degree. Twisting or untwisting can be done either before or after the carbonization of the precursor filaments. For example, precursor filaments can be carbonized after twisting, and then untwisting thereafter. Alternatively, a precursor filament bundle can be carbonized in the never twisted condition. Thereafter, twisting can be optimally performed on the carbonized tow, as desired. Another method of arriving at a desired cross sectional shape of a fiber tow strand is by chemically sizing the tow. Sizing refers to a process which includes coating or impregnating a tow with an epoxy-based or epoxy-compatible sizing agent, and drying the sizing agent, thus fixing or substantially fixing the shape of the cross sectional shape of the tow. Various drying methods can be used after the sizing agent is applied, including, e.g., drum drying, air drying, and air blowing. It is noted that drying methods may also affect the cross sectional shape of a fiber tow. Typically, sizing is done after the carbonization process. In addition, a predetermined, substantially stable cross sectional shape can be imparted to the tow by passing the tow during the tow preparation process, through a specially designed shaping die. The shape of the die can be designed to be the shape of the desired tow cross section. Prior to contacting the tow with the die, the tow can be subjected to a sizing operation so that the shaping operation will substantially fix the cross sectional shape of the tow. It has been found that irregularities in cross sectional shapes of the individual fibers or filaments in the tow can increase the entanglement of the filaments and change the

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tow shape. In particular, filaments having specific cross sectional shapes can be selected to vary the tow shape. For example, it has been found that individual filaments having a kidney or pea shape in cross section will generally form a less-compacted and bulkier fiber tow following weaving and impregnation as compared to otherwise identical tow in which each filament has a round cross section. 4.

Experimental

4.1. MATERIALS

Various techniques such as twisting, twisting untwisting, and sizing were used to manipulate or modify the fiber tow shape to the desired levels during this study. After tow forming, different drying methods such as drum drying, air drying, and air blowing were used for evaluation. Plain weave fabrics made from these 3K carbon fiber tows were impregnated by the solution method. 3K means each fiber tow strand has a total filament count of 3,000. Fiber areal weight of all fabrics studied was from about 180 to about 205 The resin system used in this study was a rubber-modified epoxy with relatively low flow. 4.2 CHARACTERIZATION OF FIBER TOW SHAPE

As a result of rounder fiber tow, prepreg generally becomes opener, thicker, and less tacky. Therefore, three parameters were found to be good indicators for core crush prediction and can be used as a set of core crush requirements: average fiber tow aspect ratio in prepreg, prepreg openness, and prepreg thickness. It is believed that these three parameters well represent the fiber tow shape and thus have strong correlation with the prepreg frictional resistance and core crush. When they are maintained within certain ranges, sufficient frictional resistance can be generated between prepreg plies such that the innermost prepreg plies, adjacent the honeycomb core, are restrained from slipping during the curing process to thereby eliminate or minimize core crush. These three parameters were evaluated using the test methods developed in [8-9] by which prepregs can be screened to determine their tendency to cause core crush during manufacture. Several pieces of prepreg were randomly chosen from different locations of a prepreg roll and several measurements were taken for each piece. Final number was based on the average of all measurements. 4.3 CORE CRUSH DISCRIMINATOR PANELS

In order to evaluate the core crush degree, test panels as shown in Figure 5 were produced. A core crush discriminator panel consists of 711 mm x 610 mm (28 in x 24 in) composite skins and a 610 mm x 508 mm (24 in x 20 in) 48 (3 pcf) Nomex core (Hexcel HRH-10 or equivalent) with a 20° chamfer angle. The types, directions, and dimensions of prepreg plies, as well as those of the honeycomb core are specified in [10]. To cure the panel, the panel was first placed in a vacuum bag. The vacuum bag and the panel were then placed in an autoclave. The bag was evacuated and cured under pressure at an elevated temperature. The detailed curing cycle can be found in [8-9]. The dimensions of the cured core crush panel were measured as shown in Figure 5. X is the displacement of the center of the core side from its original position. L

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represents the original length of core side. The crushed area A was calculated according to the formula:

The degree of core crush in percentage was determined by the following formula:

5.

Results and Discussion

5.1 FIBER TOW SHAPE IN PREPREG VS. PREPREG OPENNESS Figures 6a and 6b show the photomicrographs of a "rounder" tow versus a flat tow and their respective openness degree. The rounder tow is thicker near the tow centerline and narrower in the tow width when compared to the flat tow, resulting in rougher surface topography and thus higher frictional resistance. Rounder fiber tow also creates opener fabric that promotes mechanical interlocking or nesting between prepreg plies and further increases prepreg frictional resistance. Such phenomenon can be observed in Figure 7 where the openness areas of a specific ply, as indicated by arrows, lock and trap the peak/valley points of the weave architecture (at the fiber tow cross) of its two neighboring plies. However, it should be noted this correlation between tow thickness and width is not necessarily true in every case. Through tow shape manipulation by some special techniques, it is possible to create the tow shape that is sufficiently wide enough in shape and yet sufficiently thick enough near the tow centerline to achieve high friction while reducing openness.

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5.2 FIBER TOW SHAPE IN PREPREG VS. PREPREG SURFACE RESIN Prepreg surface resin was observed under SEM and the resulting photomicrographs are shown in Figure 6c. It suggests that the resin in the "rounder" tow case is forced into the openness areas as well as the fiber bundles, leaving the prepreg surface drier and allowing greater fiber contact between adjacent prepreg plies to increase friction. In contrast, flat tow results in reduced degree of openness that in turn creates a slip plane on the prepreg surface due to excess amount of surface resin. Because of the different degrees in surface resin, prepregs made from flat tows are tackier than those made from rounder tows.

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FIBER TOW SHAPE IN PREPREG VS. CORE CRUSH

Figures 8-9 shows the correlation between fiber tow shape in prepreg (average tow aspect ratio, prepreg thickness) and core crush. As illustrated in these figures, both average tow aspect ratio and prepreg thickness correlate well with the degree of core crush, which means, the rounder the fiber tow, the less the core crush. For various prepregs studied here with the same resin and fiber type, the degree of core crush decreases almost linearly with the decrease in tow aspect ratio (Figure 8) or with the increase in prepreg thickness (Figure 9). It should be noted that various approaches (not identified for proprietary reasons) were used to manipulate the tow shape. These figures indeed demonstrate the core crush has a strong connection with the fiber tow shape regardless which tow manipulation techniques are applied. The tow and fabric architectures are physically related to the prepreg surface roughness controlling prepreg frictional resistance and thus core crush. As long as the final fiber tow shape is maintained at certain levels, the possibility of core crush can be significantly reduced. It should be noted that, although the prepreg thickness may vary as shown in Figure 9, the difference in total thickness of final cured laminates (thus the deduced per ply thickness) is not significant. This is likely due to the meshing or nesting effects of weave peak/valley between adjacent prepreg plies. The dependence of core crush on tow aspect ratio also has important implication on fiber tows larger than 3K. Such tows (e.g., 12K, 48K) are expected to give substantially higher core crush than the 3K fiber if the resin were kept the same. Limited tests performed in this study show that 12K fibers (aspect ratio of 60) with the same resin produced high core crush of 45%. This proves again that the tow and fabric architectures are truly important factors in controlling core crush.

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Conclusions

This paper discusses the recent understandings in core crush mechanisms and the subsequent developments in core crush resistant prepreg based on that foundation. It shows the prepreg frictional resistance is the key factor in controlling core crush. The conceptual approaches for solutions are to make rounder fiber tow or more open fabric to produce rougher prepreg surface and thus yields higher prepreg frictional resistance to reduce core crush. Results show the developed core crush resistant prepreg increases the prepreg frictional resistance and effectively reduces core crush (Figure 10).

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References

Corbett, D.H. and Smith, S.A.: Tie-down Ply for Reducing Core Crush in Composite Honeycomb Sandwich Structure, U.S. Patent 5,895,699, U.S. Patent and Trademark Office, Washington, D.C., 1999. 2. Hopkins, W.B. and Hartz, D.E.: Adhering Tie-down Plies in Composite Construction, U.S. Patent 5,685,940, U.S. Patent and Trademark Office, Washington, D.C., 1997. 3. Brayden, T.H. and Darrow, D.C.: Effect of cure cycle parameters on 350°F co-cured epoxy honeycomb core panels, Proc. 34lh Int. SAMPE Symposiumn and Exhibition, Anaheim, CA, May 1989, 861-874. 4. Martin, C.J., Seferis, J.C., and Wilhelm, M.A.: Frictional resistance of thermoset prepregs and its influences on honeycomb composite processing, Composites Part A, 27A (1996), 943-951. 5. Buehler, F.U., Seferis, J.C., and Zeng, S.: Consistency evaluation of a qualified glass fiber prepreg system, J. Advanced Materials, 33 (2001), 41-50. 6. Martin, C.J., Putnam, J.W., Hayes, B.S., Seferis, J.C., Turner, M.J., and Green, G.E.: Effect of impregnation conditions on prepreg properties and honeycomb core crush, Polymer Composites, 18 (1997), 90-99. 7. Martin, C.J. and Seferis, J.C.: Effect of prepreg resin composition on honeycomb core crush, Proc. 43rd Int. SAMPE Symposium and Exhibition, Anaheim, CA, May 1998, 366-375. 8. Hsiao, H.M., Lee, S.M., Buyny, R.A. and Martin, C.J.: Core-crush Resistant Fabric and Prepreg for Fiber Reinforced Composite Sandwich Structures, U.S. Patent 6,261,675, U.S. Patent and Trademark Office, Washington, D.C., 2001. 9. Hsiao, H.M., Lee, S.M., Buyny, R.A. and Martin, C.J.: Development of Core Crush Resistant Prepreg for Composite Sandwich Structures, to be published in the Proc. Int. SAMPE Technical Conference, Seattle, WA, November 2001. 10. Boeing Company, Boeing Material Specification: Epoxy Preimpregnated Carbon Fiber Tapes and Woven Fabrics - 350°F Cure, BMS 8-256 (1999).

DISPLACEMENT FIELDS AROUND A CIRCULAR HOLE IN COMPOSITE LAMINATES S. M. CHERN Department of Civil Engineering Chung-Cheng Institute of Technology Taoyuan, Taiwan, 3305 ROC M. E. TUTTLE Department of Mechanical Engineering M/S 352600 University of Washington Seattle, WA 98195-2600

Abstract Moiré patterns representing in-plane displacement fields induced near a circular hole in graphite/epoxy composite laminates subjected to uniaxial tension are presented. The measured moiré patterns were obtained using an eight-mirror moiré interferometer. The moiré patterns are compared with predictions obtained using a combination of lamination theory and a reformulated version of the Savin elasticity solution. Predicted and measured displacement fields are in excellent agreement, although the magnitude of measured in-plane displacements transverse to the loading was slightly higher than predicted. It is speculated that this difference may occur because of the very different Poisson’s ratio exhibited in tension and compression by polymeric composites. 1.

Introduction

Advanced continuous-fiber composites are widely used in weight sensitive structures, such as aircraft or spacecraft, mainly due to their high stiffness-to-weight and strengthto-weight ratios. Also, composites can be tailored to match the stiffness or strength needed in a particular application. However, due to their anisotropic nature composite materials exhibit a more complicated response to loading than more conventional isotropic materials. The theoretical study of stress concentrations near holes began in about 1898.[1] Afterward, using the concept of complex stress variables, Savin [2,3] and Lekhnitiskii [4,5] developed a systematic formulation and solved several important problems for infinite anisotropic laminates containing elliptical, circular, square, triangular, or rectangular holes. Bonora et al. [6] proposed a closed form solution for the stress field around holes in orthotropic composite plates which is very similar to Savin’s solution. The Savin solution presented in [2,3] may be used to predict the stress and strains fields near a notch quite well, but in-plane displacement fields are improperly predicted except in a few special cases. 701 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 701–712. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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Chern and Turtle [7] recently reformulated Savin's solution such that the displacement, strain, and stress fields around elliptical holes in orthotropic laminates under in-plane loading are all properly predicted. The stress and strain fields surrounding a circular hole in composite plates have also been measured experimentally by Daniel and his colleagues.[815] A variety of experimental techniques have been used during these studies, including strain gages, birefringent coating, and moiré methods. In this work, moiré interferometry was used to measure fringe patterns around a circular hole in symmetric composite laminates subjected to uniaxial tensile loading. Laminates with two different stacking sequences were studied. Moiré fringe patterns measured experimentally are compared with predicted patterns based on the reformulated Savin solution developed by the authors [7]. 2. Summary of the Savin Solution A complete description of the reformulated Savin solution is lengthy and will not be presented in detail here. Briefly, Hooke's law for a two-dimensional orthotropic laminate can be written (neglecting body forces):

where are elements of the reduced compliance matrix. Stresses are assumed to be related to an Airy stress function, U(x,y) as follows:

Enforcing the compatibility condition results in the biharmonic equation:

The general solution of Eq. (3) depends on the roots of the characteristic equation. Assuming the roots have the general form F(x + sy), the characteristic equation becomes The characteristic equation can alternatively be written as:

where Through a consideration of potential energy, the roots of the characteristic equation are found to be always complex and are of the form:

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where and are real constants and The Airy stress function may therefore be expressed using two distinct roots, and has the general form:

where Re[ ] denotes the real part of the quantity in brackets. For further simplification, let:

where and and are complex functions. Thus, the stress and displacement components may be expressed as:

where: are complex constants, defined as follows

Assuming the laminate is specially orthotropic (i.e., then the form of the roots of the characteristic equation may be grouped into two possible cases: Case I: In this case, complex and of the form:

Case II: In this case imaginary and of the form:

and the roots are

and the roots are purely

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The solution for a thin orthotropic plate subjected to in-plane loading is completed by identifying complex functions and which satisfy the biharmonic equation, Eq.(3), and the prevailing boundary conditions. Once these functions are found the problem is solved. Savin identified functions that lead to the solution for an orthotropic plate with elliptical hole.[2,3] However, during application of these functions in Savin did not account for rigid body motions. Hence, the solutions presented in [2,3] for the stress fields surrounding an elliptical opening are correct, but in-plane displacement fields inferred from these solutions contain an undefined level of rigid body rotations and are therefore not directly comparable to displacement fields measured experimentally. The reformulated solution presented in [7] does account for rigid body rotations, and will be used to predict displacement fields in the present study. The expressions involved are lengthy and will not be presented here; the interested reader is referred to [7] for details. It is mentioned in passing that either Case I or Case II may be encountered during the study of composites. Which case is applicable to a given laminate depends on both the material system used as well as on stacking sequence. Daniel and his colleagues have studied the displacement fields surrounding a hole in laminates fabricated using two different material systems but with identical stacking sequences: in [9] they considered a boron/epoxy laminate, whereas in [14] they considered a glass/epoxy laminate. Although these laminates possess identical stacking sequences, the material properties involved are such that the boron/epoxy laminate corresponds to Case II, whereas the glass/epoxy laminate corresponds to Case I. In the present study laminates prepared using a single graphite/epoxy material system but with two differing stacking sequences were studied. As will be seen, one stacking sequence results in a laminate of Case I, whereas the second stacking sequences results in a laminate of Case II. 3.

Test Specimen Preparation

Specimens were made using 305 mm (12-in) wide Toray P725AW-15 (T700S) unidirectional prepreg tape. A series of uniaxial tension tests of eight-ply 0-, 90- and 45degree coupons were performed to determine the elastic ply properties of this material. Properties measured are summarized in Table 1. Symmetric laminates with a central 12.7 mm dia (0.50 in) hole and stacking sequences of and were tested. Laminates were prepared as follows. First, a 305 mm x 305 mm (12 in x 12 in) stack of pre-preg was prepared using hand lay-up according to the desired stacking sequence. Porous release plies and bleeder cloths were then positioned adjacent to the upper and lower surfaces of the laminate, and the entire assembly was vacuum bagged (Figure 1). A vacuum was drawn and the laminate cured using a hot-press according to manufacturer’s suggested curing cycle. Following cure the laminate was trimmed to nominal in-plane dimensions of 254 mm x 254 mm (10 in x 10 in) using a diamond-coated abrasive disk. Finally, the laminate was clamped between two sheets of aluminum and the central 12.7 mm diameter hole was machined through the thickness of the laminate using a vertical end mill. Clamping the laminate between two sheets of aluminum was required to avoid delamination around the edge of the hole.

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A crossed diffraction grating was then bonded to the specimen surface to accommodate measurement of in-plane displacements using moiré interferometry. The steps followed to create and bond the diffraction gratings have become more-or-less standard in recent years and are described in detail in several sources (see, for example, Chapter 7 of Reference [16]). The crossed gratings used in this study had a frequency of 600 l/mm.

4. Experiment Arrangement The loading frame shown in Figure 2 was used. This frame was built in-house during an earlier study, and can be used to apply biaxial loading to a flat plate using two pairs of hydraulic cylinders. Since only a uniaxial tensile load was applied in this study, only the vertical pair of hydraulic cylinders was used. A strain gage-based load cell (not shown in Figure 2) was mounted between the hydraulic cylinder and specimen grips to monitor applied loading throughout each test. The composite laminate was loaded via the notched aluminum grips shown in Figure 3. The grips were notched to avoid constraining the laminate against Poisson contractions during uniaxial tensile testing. The moiré interferometer used was the eight-mirror system shown in Figure 4. The laser beam emerging from a 35 mW He-Ne laser was passed through a spatial filter and then expanded to an eleven-inch-diameter collimated beam by means of a large collimating lens.

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The eight-mirror system caused four collimated beams to be incident upon the crossed diffraction grating bonded to each laminate. The whole system was mounted on an optical table with pneumatic legs, to isolate the system from vibrations from the surrounding environment. The moiré fringes patterns for u- and displacement fields were taken in turn by blocking out the appropriate set of mirrors. The displacements were defined to be in the direction of loading, whereas the u-field was transverse to the load.

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Results

Laminates with two different symmetric stacking sequences were tested: and . Elastic properties for the two stacking sequences are summarized in Table 2. Note that these stiffnesses were not measured directly. Rather, they were predicted using standard lamination theory [17] and the measured ply properties listed in Table 1. Also included in Table 2 are the roots of the characteristic equation corresponding to each laminate, calculated in accordance with Eq 4(a). As indicated, the roots for the laminate are purely imaginary, and hence this laminate belongs to Case II. In contrast, the roots for the laminate are complex, and hence this laminate belongs to Case I. The measured and predicted and displacement fields induced by a remote tensile stress of 34.5 MPa (5000 psi) in the laminate are shown in Figures 5 and 6, respectively. The agreement between measurement and prediction is considered to be excellent, especially since the prediction was based on laminate stiffnesses which were themselves predicted using lamination theory. The measured v-displacement moiré pattern is essentially identical to the predicted field. The measured (u-displacement pattern is similar to the predicted pattern, although the measured field contains a greater number of fringes than predicted. A similar comparison between measured and predicted and displacement fields induced by a remote tensile stress of 34.5 MPa applied to the laminate are shown in Figures 7 and 8, respectively. Once again, excellent agreement was achieved. As before, the measured v-displacement moire pattern is essentially identical to the predicted pattern. The measured u-field is similar to the predicted pattern, except the measured u-field contains a greater number of fringes than predicted.

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Summary and Concluding Remarks

A comparison between measured and predicted in-plane displacement fields induced near a central circular hole in symmetric composite laminates subjected uniaxial tension has been presented. Experimental measurements were obtained using moiré interferometry, while predictions were obtained using a combination of lamination theory and a reformulated version of the closed-from solution developed by Savin.

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Overall, excellent agreement between measurements and prediction were obtained. It is interesting to note that the numbers of v-field fringes (representing displacement in the direction of loading) were well predicted, but the numbers of measured u-field fringes (representing displacement transverse to the load) were modestly greater than predicted. It is speculated that this discrepancy may be due to differences in Poisson's ratio in tension and compression. The laminate Poisson's ratios (listed in Table 2) were calculated using lamination theory, and were fundamentally based on the ply Poisson ratio measured in tension (listed in Table 1). It is well known that for many composite materials Poisson's ratio differs substantially in tension and compression. This is especially true for graphite/epoxy (see, for example, Appendix C in Reference [17]). Since the stress induced transverse to the direction of loading near the hole is largely compressive, this may account for the modest discrepancies between measured and predicted transverse displacement fields. 7.

References

1. 2.

Savin, G. N., Stress concentration Around Holes, Pergamon Press, New York (1961). Savin, G. N., Kosmodamianskii, A. S. and Guz, A. N., “Stress Concentrations near Holes,” Soviet Applied Mechanics, 3(10), 23-37 (1967). Savin G. N., Stress Distribution around Holes, NASA Technical Translation F-607 (1970). Lekhnitskii, S. G., Theory of Elasticity of an Anisotropic Body (in Russian), Moscow: Gostekhizdat (1950). Lekhnitskii, S. G., “Some cases of the elastic equilibrium of a homogeneous cylinder with arbitrary anisortopy.” Applied Mathematics and Mechanics (in Russian), 2, 345-367 (1938). Bonora, N., Costanzi, M., and Marchetti, M., “On Closed Form Solution for the Elastic Stress field around Holes in Orthotropic composite Plates under In-plane Stress Conditions,” Composite Structures, 25, 139-156 (1993). Chern, S. M. and Turtle, M. E., “On Displacement Fields in Orthotropic Laminates Containing an Elliptical Hole”, J. of Applied Mechanics, Vol. 67, No. 3, pp 527-539 (2000). Daniel, I. M. and Rowlands, R. E., (1971). “Determination of Strain Concentration in Composites by Moire Techniques,” J. Composite Materials, 5(4):251:254. Daniel, I. M., Rowlands, R. E. and Whiteside, J. B., (1973). “Deformation and Failure of BoronEpoxy Plate with Circular Hole,” Analysis of the Test Methods for High Modulus Fibers and Composites, ASTM STP 521, American Society for Testing and Materials, 143-164. Daniel, I. M., (1978). “Strain and Failure Analysis of Graphite/Epoxy Plates with Cracks,” Experimental Mechanics, 18(7):246-252. Daniel, I. M., (1980). “Behavior of Graphite/Epoxy Plates with Holes under Biaxial Loading," Experimental Mechanics, 2 0( 1): 1 -8. Daniel, I. M., (1981). “Biaxial Testing of Graphite/Epoxy Laminates with Cracks,” Test Methods and Design Allowables for Fibrous Composites, ASTM STP 734, C. C. Chamis, Ed., American Society for Testing and Materials, 109-128. Daniel, I. M., (1982). “Biaxial Testing of Graphite/Epoxy Plates with Holes,” Experimental Mechanics, 20(5): 188-196. Rowlands, R.E., Daniel, I.M and Whiteside, J B., (1973). “Stress and Failure Analysis of a GlassEpoxy Composite Plate with a Circular Hole,” Experimental Mechanics, 13( 1 ):31-3 7. Rowlands, R. E., Daniel, I. M. and Whiteside, J. B., (1974). “Geometric and Loading Effects on strength of Composite Plates with Cutouts,” Composite Materials: Testing and Design (Third Conference), ASTM STP 546, American society for Testing and Materials, 361-375. Handbook on Experimental Mechanics, edited by Albert S Kobayashi, 2nd Edition, Society for Experimental Mechanics, Bethel, CT (1993). Jones, R. M., Mechanics of Composite Materials, Scripta Book Co., Washington D.C. (1975).

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9. Structural Testing and Analysis

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RECENT ADVANCES IN LONG-TERM MONITORING OF BRIDGES JOAN R. CASAS School of Civil Engineering, Technical University of Catalonia Jordi Girona 1-3, 08034 Barcelona (Spain) Abstract The paper deals with the importance of long-term bridge maintenance to keep bridges safe along their service-life at a moderate price. The basis for a correct management policy is to know (via updating) the actual bridge resistance and applied loads. This is possible by regular inspections, but the final and best solution is to integrate permanent sensors to continuously monitor the structure. The final step in the process is to have a smart bridge. A new series of advanced sensors and techniques for long-term monitoring of bridges is becoming available thanks to the development of these techniques in other fields of engineering. The most suitable techniques are based on the use of optical fiber sensors and more specifically the use of Bragg grating sensors. The paper shows different experiences to show how this new technique can be successfully applied to the accurate long-term monitoring of bridges.

1. Introduction Structural aging and resistance degradation of highway bridges under environmental stressors as sulphate or clorhide attack, carbonation, corrosion of reinforcing and prestressing steels and others are becoming more and more maintenance problems. As an example, the 2001 American Society of Civil Engineers Report Card for America's Infrastructure [1] indicates that USA needs to spend $1.3 trillion over the next five years to overcome the current infrastructure deficiencies. In bridges alone, 29 % have been classified as functionally deficient and it will require $10.6 billion per year over the next 10 years to remedy the situation. In Western Europe, the annual civil infrastructure repair/maintenance costs were estimated in 1995 to be around the 35 % of the total investment in building and construction. As infrastructure deteriorates over time, there is an increased risk to society and a limited amount of money to solve the problem. The risk can be reduced but eventually a point of diminished marginal returns is reached where minor reductions in risk require unjustified costs. Ideally, engineers want to spend the money most efficiently on the projects that pose the greatest degree of risk. Because acceptance of risk requires that uncertainty be quantified, reliability methods are useful and appropriate. Reliability methods can be used to optimize the management of critical structures. In fact, in the last years, the reliability management of deteriorating concrete bridges has shown the benefits of preventive maintenance based 715 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 715–726. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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on system reliability in highway bridges [2-5]. These methods require the use of deterioration models for evalution of strength and durability along time and evolution of loads. Therefore, long-term monitoring of structures, and bridges in particular, can help to up-date the strength and load models and hence the corresponding safety level along the structure's life. For this reason, the development of sensors and techniques for longterm monitoring is of great interest. There are two motivatings taks for developing monitoring and reliability management of deteriorating highway bridges: (a) to maintain the overall safety of these structures above a minimum acceptable target level and (b) to optimize the allocation of limited financial resources. Long-term monitoring of bridges will help in both aspects.

2. Advanced techniques for long-term monitoring of bridges The appearance of widespread failures in bridges which are 20 years of age or older has highlighted the importance of effective monitoring systems which are able to identify structural problems at an early stage, guaranteeing in this way the public safety. Apart from the safety concern, the aging of the structures is also generating a financial problem. The potential for monitoring systems to reduce operational maintenance costs by identifying problems at an early stage and by verifying the effectiveness of repair procedures, is clearly significant. Since now, this effective monitoring system was not available due to the deficiencies and long-term behavior of existing sensing technologies. But nowadays, new emerging technologies developed in other fields is giving the possibility to develop new sensors for civil engineering structures. Among them, for instance, the emerging fiber optic sensing technology overcomes some deficiencies of the standard sensors normally used in bridge monitoring: (a) it is free from corrosion, (b) having long-term stability and allowing continuous monitoring, (c) it is free from electromagnetic interference, avoiding undesirable noise, (d) it has a very low signal transmission loss, allowing a remote monitoring and (e) the cabling and sensors are very small and light, making it possible to incorporate then into the structures permanently. For these reasons the fiber optic sensors are considered one of the best candidates to monitor the state of a structure throughout its working life. However, there are other sensors recently developed and tested that are being also incorporated to monitoring systems. Recently, the European Commission has funded a research project, denominated Smart Structures [6], to develop an integrated monitoring system to measure, monitor and assess the conditions of bridges. The project has identified the parameters required for monitoring the most common deterioration mechanisms (freeze/thaw, carbonation, chloride attack, alkali-silica reaction, corrosion) and the developed sensors cover the monitoring of these parameters (moisture, temperature, chloride content, pH, corrosion risk, crack widths, deflections and vibrations). The project comprises laboratory tests and the long-term monitoring of a real bridge in Denmark as well. The sensors for the long-term monitoring of bridges can be divided in two main categories: portable and embedded sensors. In the first group we can place those sensors than do not remain permanently in the structure and are transported there just when a new measurement is needed (Galvanostatic Pulse Method for measuring half-cell potential and corrosion rates in just 10 seconds, measurement of polarization resistance

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to obtain corrosion intensity). The second group comprises the systems which sensors are permanently placed in the structure and are the most suitable for long-term monitoring because they provide a continuous monitoring. In the European Research Project already mentioned several embeddable sensors have been developed: corrosion risk sensors, chloride content sensors, humidity sensors and pH sensors [6]. However, as mentioned, the most promising embedded type sensors are those based on the fiber optic technology. Since now, the field where the optic sensors were more developed was Aeronautics. In this field, extremely small sensors with a high precision are needed. A complete state of the art concerning the different fiber optic sensor types may be found in [7]. In the civil engineering field, the technology is not so developed, but many sensor configurations have appeared in the last 5 years. However, most of them are not adequate, because they are excessively precise and thus they are also too expensive. Also, a disadvantage of fiber optic sensors claimed by some researchers is their high fragility what makes difficult to embed them into the concrete without breaking the sensor. This fact plays a negative role in the possibility of using such sensors in real concrete structures, where the manipulation and operations encountered in the normal casting process can invalidate a large number of sensors. Other arguments against use in concrete structures concerns their long-term behavior and reliability when affected by the different environmental attacks that concrete suffers during its service life. To answer to these questions, different experimental works have been outlined which are explained in the following section. Different optical fiber sensors are available to monitor civil engineering structures. The most simple fiber optic sensor is the one measuring the intensity of light transmitted through a fiber. For instance, integrating optical fibers in a concrete structure and determining when light is not transmitted anymore, one can infer the existence of cracks in the structure. With the optical fiber technique it is possible to create pressure, water and cracking detection sensors, corrosion detection and also acceleration transducers. However, the most widely use of optical fiber sensors in the civil engineering field is to measure strain and displacement. Different techniques based on the use of optical fibers are used to measure strains [7]. But, from all available fiber optic sensors, one of the most robust, accurate and cheap to be used in civil engineering structures is the fiber Bragg grating (FBG). For this reason, this is the type of sensor used in the tests shown in the next section. A FBG is a periodic perturbation of the refractive index along the fiber length, which is formed by exposure of the core to an intense optical interference pattern. The phase mask technique has the advantage of greatly simplifying the manufacturing process for Bragg gratings, yet yielding gratings with a high performance. Moreover, offers the possibility of manufacturing several gratings at once in a single exposure by irradiating parallel fibers through the phase mask. This technique makes feasible to manufacture high-performance gratings at a low cost per unit. This is critical for the economic viability of using gratings in civil engineering structures and bridges. The most successful technique for interrogating FBG sensors is based on the use of a tunable narrow-band filter for tracking the FBG signal. The most commonly used filter is a Fabry-Pérot (FP) filter [7]. This sensor is very attractive for the long-term monitoring of structures because of the following advantages: (a) it can be multiplexed, (b) it allows absolute measurements without taking reference measurements, (c) it allows dynamic monitoring, (d) it has excellent long-term stability

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due to its spectrally encoded output and (e) it does not present any electromagnetic interference.

3.

Experimental research

Besides the experimental work on Smart Bridges [6], where fiber optic sensors are developed and tested among other type of sensors, other experiences have been carried out to show the robustness, accuracy and efficiency of the FBG sensors. For instance in Ghent University a group of researchers have performed comparative measurements between Bragg grating sensors and standard strain sensors (normally electrical strain gauges) placed either in a reinforcing rebar or a reinforced concrete specimen subjected to flexural loading [8]. They show how the average difference between the two sets of readings is less than 3 The optical fiber sensor showed higher stability when several cycles of load-unloding were applied. The experience shown in [9,10] demonstrates how the FBG can be applied not only to monitor new structures, but also to monitor repaired old structures. In fact, the low size of FBG makes possible embedding them in laminates of fiber reinforced plastics (FRP) without impairing their mechanical properties. The use of Bragg gratings to measure the strain field in a FRP repair in a concrete structure can be used, on the short term, to check the integrity of the repair, the load transfer from the concrete to the reinforcing patch, and the validity of the models used in the design of the strengthening. On the long term, it can be used to permanently monitor the behavior of the repair, with its degradation producing a warning when a very low level of safety is reached. A reinforced concrete continuous beam with two spans 7.20 m each, representative of current in-service bridge structures, was loaded up to failure and repaired. In figure 1, the elevation and cross-section dimensions of the tested beam are shown. The cross- section is a box-girder with constant depth of 0.6 m.

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The dimensions of the specimen makes it one of the largest in-laboratory tests using advanced fiber repairs. The objective of this set of tests was the comparison of the ultimate load bending of the original and repaired beam in order to derive the efficiency of the repair technique used. The tests were aimed to check the feasibility of CRFP repairs and their continuous long term monitoring with fiber optic sensors. The original beam was fully damaged in the sections at mid-span and over intermediate supports by application of external load as shown in figure 1. The damaged beam was repaired, substituting the crushed concrete by new microconcrete, sealing and filling the reamining cracks and placing Carbon Fiber Reinforced Plastic (CFRP) in the parts where normal reinforcing steel was more severe damaged (figure 2). Fiber Bragg gratings as strain sensors were placed with the fabrics, both in the internal and external surfaces. The repairing patches and Bragg gratings locations (A,B,C, D) are shown in Fig. 2.

After placement of the first layer of epoxy resin the sensor A was placed just before placing the CFRP sheet over the resin layer and application of a second layer of resin over the sheets. In the second layer, the sensors B,C, and D were placed. In this way, sensor A is measuring the strain at interface between concrete and carbon fiber. Sensors B,C, and D were aimed to measure the strain in the carbon layer itself. Additional and redundant strain gauges were located in the external surfaces of the CFRP sheets close to the optical fiber sensors B,C and D in order to compare and check the experimental data provided by the two measurement techniques. Load vs. Strain is plotted in Figure 3, obtained with strain gauges and FBG. Coincidence is very satisfactory, even at the non-linear region of the concrete beam. Visible non-linearity starts near 80 kN, still with considerable residual strength. Small differences between strain gauges and Bragg gratings are due to creep of concrete. In fact, measurements in the gauges and FOS were made at the same load level but at different times. Therefore, because of concrete creep small differences in the strain are feasible. Results for Bragg grating sensors A and B are shown in Figure 4. They were at the same position, at one side and the other of the composite patch. A shear lag due to the finite thickness of the adhesive layer can be seen. The strain measured by sensor A, located in the interface, is higher than that measured by sensor B, at the same location but on top of the composite patch. This behaviour should be expected, as an adhesive joint, but has never been experimentally demonstrated previously and should be considered when designing these repairs.

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These experiences show how FBG have been successfully applied to measure strains when structures are subjected to external loads, even in the monitoring of structures strenghtened with carbon fiber composites. However, most of the criticism regarding the use of such sensors is based on the fact that they could hardly survive to the pouring operations when embedded in a concrete structure and also that their longterm performance in presence of agressive environments (chlorides, sulfates, acids,...) which normally affect concrete structures, could be critical. A research work to see if this is the case has recently started at the Structures Laboratory of the School of Civil Engineering of Technical University of Catalonia (UPC). The objective of the experimental work is to show the feasibility of embedding fiber optic sensors in concrete structures at the time of their construction and to investigate the reliability and behavior along time of such sensors when placed in aggressive environments such those produced by chloride or sulfate attack [2]. To this end, a set of reinforced concrete beams with a total length of 1.75 m and rectangular cross-section shape with dimensions 0.15m (width) and 0.25 m (depth) have been manufactured. Before casting of the beam, different sensors based on FBG technology were placed in the beams. In figure 5, the reinforcing (longitudinal and transversal) bar arrangement and the location of some sensors are shown.

In order to see the performance of the sensors when used in aggressive environments, the instrumented beams will be subject to accelerated tests. Some beams will be placed in an atmosphere with high concentration of for 3 different times (6 months, one year and two years). Other beams will be submerged in a tank with water with high concentration of chlorides (5 % in weight of ion chloride) and sulfates for the same periods of time. The objective is to see the evolution of corrosion or sulfate attack with time, to check if the optic fiber sensors survive to the attack and if they are able to continuously monitor the attack. Because corrosion can not advance without oxygen, the attack is performed by cycles. The beams remain submerged for one week and after

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that they are removed and exposed to the normal atmosphere. In order to accelerate the attack, the beams are pre-cracked by an external load before to be submitted to the attack. The objective of the sensors is to monitor the corrosion of the reinforcing steel. To this end, the FBG are used to measure the strain produced by the corrosion products. In fact, the corrosion causes an increment in volume of the reinforcing bar and the corresponding confinement stresses. The increase in volume produces a strain field around the bar. To measure this increment of strain with the FBG, different possibilities have been tested, deriving in different FBG sensors. Sensor type 1 consists on a FBG disposed around a n on-corrugated reinforcing bar (figure 6, right). The sensor should be placed close to the reinforcing bar where the measurement of corrosion effects is foreseen (figures 6). With this disposition, the sensor measures the angular strain produced around the bar. The sensor type 2 consists on a FBG disposed in contact with the rebar (figure 6, center). In this case, the sensor measures the strain in the concrete around the bar in the radial direction due to the expansion of the bar diameter caused by corrosion. Figure 6 (left) shows another type of sensor developed in the work. The purpose of sensor type 3 is to measure in a way similar to sensor 1. In this case, the fiber optic is mounted directly around the bar where corrosion has to be measured (figure 6). The inconvenience of this system is that the bar must be smoothened in a part to fix the sensor and also, if the diameter of the bar is small, it is extremely difficult to place the fiber around the bar without breaking the grating. For this reason, the alternative sensor type 1 was developed.

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Another option to detect the corrosion process is trying to measure directly the stress increment around the bar due to corrosion products instead of the strain. In this case, the sensors used are pressure gages based on the piezoelectric activity. The idea is to place the sensor directly attached to the reinforcing bar, which should be previously smoothed (figure 7). In figure 7, one of these sensors is placed close to a sensor type 2 in order to compare their accuracy and long-term performance in detecting corrosion effects.

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As mentioned, the beams are exposed to different environmental attacks. The beams are disposed in containers, submerged in special tanks with water and the appropriate concentration of chlorides and sulfates to accelerate the corrosion (Figure 8). Apart from the experimental research carried out in the laboratory, different “in situ” application of Bragg grating sensors to real bridges are being developed. Within the objectives of the research work foreseen in [6] it is the application of the developed sensors in the continuous long-term monitoring of a Danish Bridge. In [8], a new bridge over the ring canal around the city of Ghent was instrumented with 18 Bragg grating sensors, placed in 6 points of the three most critical cross-sections. The aim is to measure the strain during prestressing of the tendons and incremental construction of the bridge. The measurements taken and compared with standard techniques have shown how the sensors have survived succesfully to the pouring operations and will allow future monitoring of the bridge.

4. Cost of long-term monitoring of a bridge One of the criticism against the implementation of a long-term monitoring system in a bridge based on fiber optic sensors is the high cost of instrumentation. This is not the case of very large bridges (cable-stayed, arch or cable supported) where the cost of the monitoring is marginal if compared with the importance of the bridge, the high investment in the structure and the maintenance costs during service-life. The discussion could emerge from the normal bridges, where the cost of the monitoring could be claimed to be extremely high compared to the value of the bridge itself. In [7], a complete monitoring system for a standard highway bridge as presented in figure 9 has been studied, resulting in the monitoring of 7 different sections by 114 fiber optic sensors of 4 different types (see figures 9 and 10): 28 strain sensors (FBG), 28 temperature sensors (FBG), 30 cracking sensors (intensity modulation technology) and 28 corrosion sensors (FBG). The cracking sensors are totally independent from the rest because they use a different technology and thus cannot be placed in the same fibre leads. Moreover, these sensors must be read by a different optic system (OTDR). As demonstrated in [7]. the cost of the instrumentation (sensors and fiber) is in the range of 13.391 Euro, and the cost of the demodulation system about 64.909 Euro, being the cost of the complete system 78.300 Euro.

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The cost of the instrumentation must be compared with the cost of the structure to decide if it is affordable. The instrumented bridge would have a construction cost of 600,000 Euro. Therefore, the designed instrumentation costs 2.2 % of the bridge cost. The cost of the complete equipment (sensors + demodulator systems) is 13 % of the bridge cost. It is clear that 13 % of the bridge cost to be invested in long-term monitoring is too much because it exceeds the normal cost of maintenance. However, the value of 2.2 % is a reasonable amount compared to the amount of money to be invested in the bridge maintenance. The conclusion is that, taking into account that the designed monitoring system would reduce significantly this maintenance cost, it is affordable to instrument the bridge, but the demodulator systems must be shared with other bridges, as the cost is too high. This is not a problem for an Administration managing an important number of bridges. Nowadays, only the singular bridges (cablestayed, cable- supported,...) and the singular structures (dams, large roofs,...) could

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afford this permanent monitoring. But the price of the demodulation systems has decreased up to 50 % in the last 3 years. Thus, in a few years, the remote permanent sensing system will be affordable even for any medium-size bridge.

5. Conclusions The paper shows that the long-term monitoring of bridges is affordable both from the technical and the economic points of view. In fact, many new advanced sensors, mainly based on the use of fiber optic technology, are able to measure the parameters of interest for bridge maintenance and they seem to be robust and accurate even in the presence of agressive environments. On the other hand, the cost of a complete long-term monitoring system (instrumentation plus demodulation systems) for a medium size bridge ( more than 50 meters of total-length) seems to be not yet affordable by Administration. However, the cost of the instrumentation alone, sharing the demodulation systems with other bridges in the same network, looks low enough to be implemented in the near future.

6. Acknowledgements The partial financial support of the Commission for Cultural, Educational and Scientific Exchange between the United States of America and Spain and of the Spanish Ministry of Education (Research Project TAP99-1079-C03-01) is gratefully acknowledged.

7. References 1. American Society of Civil Engineers (2001), 2001 Report Card for America's Infrastructure, Washington D.C.

2. Casas, J.R. and Frangopol, D. (2001) Monitoring and reliability management of deteriorating 3.

4.

5. 6. 7.

8.

9.

10.

concrete bridges, in Proceedings of the 2"d International Workshop on Life-cycle Cost Analysis and Design of Civil Infrastructure Systems,. UBe (Japan). In press. Estes, A.C., and Frangopol, D.M. (2001) Minimum expected cost-oriented optimal maintenance planning for deteriorating structures: Application to concrete bridge decks, Reliability Engineering & System Safety, 73(3), 281-291. Frangopol, D.M., Kong, J.S., and Gharaibeh, E.S. (2000) Bridge management based on lifetime reliability and whole life costing: The next generation, in , M.J. Ryall, G.A.R. Parke and J.E. Harding (eds.), Bridge Management 4., Thomas Telford, London, pp. 392-399. Enright, M.P., and Frangopol, D.M. (1999) Maintenance planning for deteriorating concrete bridges, Journal of Structural Engineering, ASCE, 125(12), 1407-1414. BRITE/EURAM: Smart Stuctures. Research Project N. BRPR-CT98-0751. In progress. Perez, A. (2001), Fibre optic sensors: application to the long-term monitoring of civil engineering structures. Master Thesis. School of Civil Engineering. Technical University of Catalunya. Barcelona. Moerman, W., Taerwe, L.,De Waele, W.,Degrieck, J., Baets, R. (2001) Application of optical fibre sensors for monitoring civil engineering structures, Structural Concrete, 2 (2), 63-71. Casas, J.R. (1999) Experimental study on the repair and strengthening of existing concrete bridges by CFRP sheets and external prestressing, in P.C. Das,D.M. Frangopol and A.S. Nowak (eds.), Current and Future Trends in Bridge Design Construction and Maintenance, Thomas Telford, London, pp. 368-379. Casas, J.R., Ramos, G., Diaz, S., Guemes, J.A. (2002) Intelligent repair of existing concrete structures, Computer-Aided Civil and Infrastructure Engineering, 17 (1), pp. 41-50.

AN EXPERIMENTAL MECHANICS APPROACH TO STRUCTURAL HEALTH MONITORING FOR CIVIL AIRCRAFT Nano Measurements on Biologically Inspired Structures E. W. O'BRIEN Airbus UK Experimental Mechanics. Bristol BS9 3TR UK. Abstract The design philosophy for modern Civil Aircraft demands a structure that is free from propagating fatigue damage for at least the first half of its design fatigue life. Subsequently, well behaved growing fatigue damage is permitted providing that it can be detected by NDT techniques and repaired within certain well defined inspection periods. This approach allows an economic airframe structure to be designed and maintained such that it 'grows old gracefully - but safely ' and is not over designed to remain 'perfect' on its last day of operation. To achieve this design philosophy very strict NDT procedures are instigated on the aircraft structure at well defined check periods. This is time consuming - particularly in the case of an inspection where no detectable damage is present. The maintenance industry is expressing interest in the concept of ‘condition monitoring’ for cost reduction reasons - which is marvellous providing a way of determining true 'structural condition' could be found. Any such method must give higher confidence and be more economic than current NDT. This paper describes the development of a detection system that enables remote Structural Health Monitoring which has been developed from an Experimental Mechanics approach. By applying sensors to a component a ‘biologically inspired’ structure is produced effectively giving it a 'nervous system' that can detect the ‘pain’ of crack initiation and growth. Very precise laser measurements show that surface displacements of less than a 'nanometer' accompany the stress wave detected by piezo-electric sensors. This combination is used to produce what is in effect a remote ‘Crack Growth Monitor’ that benefits from an Experimental Mechanics approach. The development has produced a very advanced remote crack growth monitoring system for ground fatigue tests and opens up the possibility for application in flight. The issues that have to be addressed for flight application are discussed.

1.

Introduction

Many attempts have been made to remotely identify the onset of fatigue damage that is normally found by NDT by aircraft inspections. The motivation for this is to reduce the cost of maintenance by moving from 'scheduled maintenance' to 'condition maintenance' with no effect on the existing high safety levels achieved. Since safety is in no way negotiable for civil aircraft operations a monitoring 727 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 727–736. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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system must be based on principles that yield confidence and reliability. The economic benefits are considered to be real and in demand by a number of airline operators and the maintenance industry in order to reduce operating costs. The perceived requirement has sponsored research into achieving a system that could be offered for application to future aircraft.

2. Background Attempts to gain more precise data that would enable assessment of aircraft structure to be based on its actual exposure to fatigue loading have been problematic for Civil Aircraft although they are used successfully in Military applications. The actual loading experienced is unique to each aircraft and is dependent on the routes flown and precise flying styles etc. This 'operational loading' (OLM) is almost certainly a little different to the book values used for design which must cover a range of possibilities applied to the whole fleet. Early OLM methods were based on recording either strain or acceleration measurements at critical points on an aircraft structure to determine the historical exposure of the aircraft to service loading. The results were then related back to fatigue tests or numerical analyses in an attempt to estimate the actual fatigue exposure as opposed to the tested or calculated performance. The weakness of these methods was that they were only as good as the fatigue test or the numerical calculation and the ability to correlate data along with an allowance made for material and test scatter factors. There was also a possibility of missing vital data if the power to the electronics was lost when critical loading was experienced by the structure. Some attempts have also been made to detect minute changes in stiffness and strain distribution due to the presence of detectable fatigue damage. These methods are problematic and not very practical in service due to the very low signal levels requiring exotic procedures such as wavelet analysis. Thus a direct and powerful method that is insensitive to loss of power is required in order to satisfy the maintenance customer and the controlling Airworthiness Authorities. The economies result from addressing specific experience of individual aircraft rather than theoretical requirements that cover the whole fleet.

3.

Development

In the drive to produce a successful flying AHUMS system there are several progressive stages that must be pursued including a clear identification of the issues involved. The first major step in producing a useful fatigue damage detection system is to develop a system for ground based use in such areas as laboratory fatigue tests. A further stage is the development to include the requirements for in-flight operation that address integration into aircraft systems and analysis of data.

4.

Focus

An approach to the subject from the Experimental Mechanics point of view introduces some very valuable focus on the key issues that are necessary for designing a successful remote fatigue damage detection and AHUMS system. Some of the fundamental issues are identified and discussed as follows :Detectable Fatigue Damage. Fatigue damage as detected by NDT means 'cracks' for metallic structures.

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Recognition of this simple fact means that the system should focus on detecting 'cracks' rather than strains or accelerations. Detection of 'cracks' is also the most 'direct' method of monitoring. The Experimental Mechanics approach would generally encourage the engineer to acquire desired measurements in the most direct manner possible to maximise efficiency and reduce errors. Detection of crack growth in the most direct possible way demands an assessment of the mechanism of fracture under fatigue loading to guide an appropriate means of monitoring:Growing cracks are a feature of cyclic fatigue loading. The natural duty cycle of service aircraft causes cyclic loading, i.e. take off, landing, gusts, taxi etc. Growing cracks propagate by small steps at each striation. The physics of the advance of each striation is such that a plastic zone builds up ahead of the crack tip. Further cyclic loading causes the tip to advance through the plastic zone in an almost explosive manner. The sudden change of strain energy into fracture and distortion is accompanied by a strain wave emanating from the from the crack tip as a result of the energy release. This strain wave travels into the structure and is dissipated as heat. The strain wave is known as 'acoustic emission' despite the fact that the vibration is above 20 kHz which is the normal upper limit of hearing. By use of the acoustic emission phenomenon a growing crack effectively 'speaks to you' with the structure itself being the conductor of the message to the piezo-electric sensors.

The acoustic emission from crack advance is a direct evidence of crack growth which can be pin-pointed by triangulation if three or more sensors are used to detect the strain wave.

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During fatigue cycles many small acoustic emissions are excited at a crack site but the explosive advance through the plastic zone ahead of the crack tip causes massive acoustic emissions or bursts - it is these bursts that are the most useful in a crack detection system. Many sources of vibration or noise are present in operational aircraft structures such as rivets, material features, pumps, air flow, controls etc. Crack-like signals must be filtered from all the other sources to make accurate crack detection possible. Most structures particularly of metallic or composite construction are very good conductors of vibrations such as acoustic emissions. Thus the structure itself is part of the transducer system for detecting cracks and as such the structure can speak to us regarding its own health in a very direct and efficient manner.

5. Structural Health Monitoring Having explored the various parameters associated with growing cracks a system for Structural Health Monitoring can be defined focusing on the following main criteria :Only growing fatigue cracks will be monitored.

Growing fatigue cracks generate many low level acoustic emissions during the damage build up and propagation, however, massive bursts of acoustic emission accompany the progress of the crack tip through the plastic zone at the crack tip. These bursts are the identifiers of crack growth that form the basis of the monitoring system. A micro-graph of a typical crack is shown in Figure 2. Using the structure as a conductor of acoustic emission signals means that the site of the crack does not have to be known in advance of it occurring. Location of the crack tip can be achieved by analysing the time of flight of the stress waves arriving at a minimum of three sensors using a triangulation procedure.

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Knowing the advancing position of the crack tip enables an assessment of crack growth to be made. A knowledge of crack growth in each individual structure due to actual loading that it has experienced provides a data base that could enable the management of a ‘condition monitoring’ regime or assist in the reduction in cost of a traditional ‘scheduled maintenance’ regime by enabling directed maintenance to identified damage zones. The surface displacement due to a stress wave from a crack growth measured 30 mm from the crack tip has been measured by laser metrology to be approximately 0.01 of a nano-meter as shown in Figure 3.

6. Aircraft Health and Usage Monitoring (AHUMS) Certain important considerations to ensure the integrity of the data must be made when developing general Structural Health Monitoring into flying AHUMS which are reviewed below :-

Sensitivity to power loss. The major reason that previous strain gauge and accelerometer methods of Operational Load Measurement did not succeed for Civil Aircraft use was the fact that loss of vital data could occur if the recording devices became inoperative due to power loss etc. Using the modified acoustic emission technique means that all the fatigue damage data resides within the crack emanating as energy release from the crack tip. Subsequent to restoration of any loss of power or recording capability the crack tip acoustic emissions announce their new position which results from

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what ever duty cycle the aircraft has experienced. Providing the down time of the system is consistent with normal airline operational maintenance the crack growth will remain well within the critical length criterion. Sensitivity to loss of a sensor. One of the most perverse system failures in AHUMS would be the loss of a sensor. In this situation a crack could occur and the system still indicate a healthy structure. In a negative approach adjacent sensors would still pick up the stress waves due to any propagating crack. In a positive approach a method of self diagnostics has been developed whereby each sensor is programmed to emit a vibration signal in turn such that each of the surrounding sensors can verify its operational condition. Sensitivity to broad band noise/vibration. In an aircraft structure many sources of acoustic emission such as rivet slip, bearings, hydraulics, air flow which can contain frequencies over a range zero to over a MHz. To extract crack related signals from such broad band data would be difficult in the extreme, although, some systems attempt to do it! A simple review of the frequency content of typical crack growth vibrations reveals characteristic content above 200 kHz whilst much unwanted acoustic emission occurs below this level. This leads to the choice of a primary filter by the use of tuned sensors that respond to narrow band excitation above 200 kHz - the value of 300 kHz is particularly applicable to steel and aluminium materials. Sensitivity to non-crack acoustic emissions. Having selected a narrow band tuned sensor system that captures crack related acoustic emissions there is an issue with the non crack related acoustic emission that is detected by the tuned sensors. The Experimental Mechanics approach to the resolution of the difficult problem of background noise simply asks the question what is the feature we wish to find and then what is the most direct method of measurement. The answer must be - to detect and record acoustic emissions that result from crack growth despite a cacophony of emissions even in the narrow band selected. The most direct measurement technique comes from an appreciation of what happens in material that contains a growing crack which has led to what might be called a ‘phenomenological filter‘ described in more detail below.

7. Phenomenological Filter Despite sensing with narrow band tuned piezo-electric sensors there are a vast number of acoustic emission signals occurring at 300 kHz that come from structural fasteners, material effects, systems and air flow etc. These emissions also contain emissions from any crack growth that may be in the region. A filtering technique has been developed that is able to discriminate the wanted emissions crack sources and the remaining unwanted emissions based on fracture and experimental mechanics using the following logic sequence. A simple analysis concludes that the stress waves emanating from the crack tip only occur when a crack is growing and that apart from the minimal shift due to the crack growth the signal occurs at a regular interval within an aircraft duty cycle and from the same position each time. Emissions from other sources due to the nature of structures come from sources that do not repeat in the same manner as cracks. Effectively non- crack sources will arise in a non repeating manner which leads to the

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conclusion that there are certain phenomena associated with the regular performance of civil aircraft that may be used to extract data that is solely associated with crack growth. Further comparisons of wave forms etc. can be used to resolve any residual ambiguity. To elaborate the principle of the phenomenological filter assuming that a monitored structure contains a growing fatigue crack. On the first fatigue cycle acoustic emissions will be recorded from a number of fasteners, interstitial movements, mechanical systems and air flow as well as from the crack tip. On the second cycle acoustic emissions will come from a different set of fasteners, material features, etc will be recorded including emissions from the crack tip which retains its original location. On the third and subsequent cycle a similar process takes place again recording the crack tip information from its unchanged location. Of course the crack may not grow on every cycle but for a structure undergoing an organised loading plan such as a service aircraft any crack will emit signals on a regular basis from the crack tip. By a process of ranking signals that repeat in terms of location are retained and signals that do not repeat are discarded. Within a relatively short number of duty cycles, say 50 -100, a strong pattern emerges from the data indicating the presence and position of a growing crack. The use of the phenomenon of crack tip acoustic emissions emanating in strong bursts from the same geometric position and non-crack acoustic emissions presenting themselves in a manner that does not repeat on a regular basis forms an effective filter. The sequence of the filter is shown in the Figure 4 sequence. This filter is one of an array of methods used to overcome the issue of ‘background’ noise which has been problematic in acoustic emission for many years.

In addition to the phenomenological filter other attributes of the stress wave emitted from a growing crack are reviewed in the data such as the frequency content, rise and decay profile and amplitudes. Some of these features are material

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dependent and must be established as part of the design set up for an effective AHUMS system. In the case of composite materials that do not exhibit cracking in the same way as metals the system has proved effective although the higher attenuation in the composite means that sensors have a shorter range and the effects of directionality in the material must be considered. The effect of fatigue in composite materials is often to propagate physical damage that has come from an impact or a manufacturing weakness. The original structural damage often occurs instantaneously and is ignored by the AHUMS system, however, the subsequent propagation usually contains a significant amount of fretting which is detected by the AHUMS system.

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8. Development The primary objective in the development collaboration in the UK was to provide a sophisticated crack growth monitoring system for structural fatigue tests related to Civil Aircraft as well as remote NDT crack detection for marine structures such as ships and off shore structures. The development is complete to the stage of providing an intrinsically safe monitor that can avoid the need of saturation diving to conduct NDT on oil installations on the sea bed . It is also able to supplement remote NDT for crack growth with surveillance being conducted on Airbus fatigue tests over distances of well over 1000 km. The potential has been demonstrated for application to Civil Aircraft structures as an Aircraft Health and Usage Monitoring System (AHUMS) should this be

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required in the future to aid reduction in maintenance costs by a form of condition monitoring. The system has been successfully demonstrated in the laboratory for a number of years and also on a large jet aircraft during ground testing with full engine and systems being operated. All simulated crack ‘noises’ were detected during the test as well as all the background noise being eliminated.

9. Seismic Scale of Acoustic Emissio n Magnitude of acoustic emission

where

On this scale a 1 fracture is designated and a fracture event on the acoustic emission scale. The corresponding Richter scale for seismic events is more energetic. A fracture on the Richter scale and a fracture on the Richter scale.

10. Application to a Typical Wing Skin. See Figure5

11. Conclusion A remote crack growth monitoring system has been developed and tested for applications to laboratory fatigue testing. Many of the additional aspects for a flying AHUMS system have been explored in anticipation of the requirement being specified. The objective being reduction of maintenance costs whilst retaining the current high safety levels.

12. Acknowledgements The views expressed in this paper are those of the author and not necessarily his employer. Airbus UK are gratefully acknowledged for making the research subject possible.

SMART STRUCTURES APPLICATION TO AIRWORTHINESS AND REPAIR R. JONES DSTO Centre of Expertise for Structural Mechanics Department of Mechanical Engineering, PO Box 31 Monash University, Vic. 3800, Australia I. H. McKENZIE DSTO Centre of Expertise for Structural Mechanics Department of Mechanical Engineering, PO Box 31 Monash University, Vic. 3800, Australia S. GALEA

Air frames and Engines Division DSTO, PO Box 4331, Melbourne, Vic. 3001, Australia S. PITT A irframes and Engines Division DSTO, PO Box 4331, Melbourne, Vic. 3001, Australia DSTO Centre of Expertise for Structural Mechanics Department of Mechanical Engineering, PO Box 31 Monash University, Vic. 3800, Australia Abstract The present paper summarizes recent work, undertaken as part of a collaborative research program between DSTO and the DSTO Centre of Expertise in Structural Mechanics (CoE-SM), for development and assessment of new techniques for the insitu health monitoring of structures and any associated repairs.

1.

Introduction

The Aloha incident focused public and government attention on the problem of aging aircraft and highlighted the problems of both multi-site damage (MSD) and wide spread fatigue damage WSFD [1]. It also underlined the need for a strong science base to support the management of aging structures. One objective, central to this goal, is the development of techniques that both locate and quantify damage states. The requirement for continued airworthiness of aging aircraft has also fueled the development of repair technology. The application of bonded composite patches to repair and reinforce damaged metallic structures is one such development that is widely becoming recognised as a versatile and effective repair procedure [2]. It has been applied to the primary structure of the F-111C, to repair cracking in the lower wing skin, and to damage in civil transport aircraft [2,3]. The Aloha accident also 737 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 737–748. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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revealed that multiple mechanical repairs, in close proximity, can compromise structural integrity [1]. The challenge is to develop a methodology whereby the damage tolerance of a repaired structure can be assessed (in-situ). The present paper summarises recent work, undertaken as part of a collaborative research program between DSTO and the DSTO Centre of Expertise in Structural Mechanics (CoE-SM) at Monash University, for development and assessment of new techniques to meet these challenges, i.e. to enable the in-situ health monitoring of structures and any associated repairs. To ensure continued airworthiness it is not only important to find a flaw, the severity of the flaw also needs to be assessed. To this end the present paper first evaluates the ability of compliance measurements, via either piezoelectric or fibre optic sensors, for assessing both isolated cracks and repairs. It is shown that, in both cases, this requires a knowledge of the critical locations within a structure together with an estimate of the optimal sensor lengths. Attention is then focused on composite repairs to cracked metallic structural components [2]. In this work attention is focused on a fiber optic in-situ monitoring technique to detect and monitor crack growth and debonding under bonded repairs.

2. Compliance Measurements For Assessing Multiple Interacting Cracks Although the phenomena of multi-site damage was first seen in civilian aircraft [1] recent full scale testing in Australia and the US in support of the F/A-18 revealed that the fatigue critical location contained several hundreds of small cracks, [1]. Localised strain sensing is one method that could be used to sense MSD. However, as shown in [5] this procedure may not be the optimum. To explain this, and to understand this phenomenon and how sensor readings can be used to assess structural integrity, reference [5] considered a simple test problem. This problem involved an array of sixteen identical cracks placed in a symmetrical fashion around the origin of the coordinate system, see Figure 1. In this investigation the length of each crack was defined as 2a, the horizontal distance between the crack centres was defined as 2b and the vertical spacing between the rows was defined as c, see Figure 1. For cases where the a/b ratio was less than 0.75 [5] found that the value of where is the value of the stress intensity factor for a single crack, lay within the range 0.8 to 1.2. It thus appeared that, as a first approximation, for these crack configurations the maximum stress intensity factor behaves as though it is being produced by a single (i.e. "just one") crack. It was only when the cracks became very closely spaced that the local strain field could sense the presence of other flaws. As a result of this analysis it was thought that, monitoring of the near tip stresses and strains may not be particularly sensitive. At first glance it may appear that monitoring of the near tip strain field, and hence the stress intensity factor, should be sufficient. The problem in this approach is that the presence of MSD cannot be seen until immediately prior to link up. Furthermore, once link up has occurred the life to failure is generally quite small. The F/A-18 aircraft in service with the RAAF is a good example of this phenomenon. Here the requirement for maintaining continued airworthiness mandates the development of both adequate assessment tools, for assessing flaw criticality, and non-destructive inspection

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methodologies, This means that, prior to "link up", it is desirable ( indeed essential) to detect/sense these very small flaws (cracks). The challenge is therefore to develop fracture mechanics based tools, which will allow the accurate and rapid assessment of structural integrity. In [5] attention was focused on the use of compliance measurements, via fibre optic or mechanical sensors, with the hope that they may provide a useful tool for assessing structural integrity. In service aircraft this could be achieved using either extensometers, fibre optics or piezoelectric sensors. With this in mind the test case used in reference [5] simulated a typical fibre optic sensor output signals. Here the difference in displacements at the end points, divided by the length (L) of the sensor (or extensometer), was computed for a range of different (sensor) lengths.

A graph of the resulting sensor readings, the displacement per unit length versus the "measurement" length (L) over which the displacement was computed, is shown in Figure 2. In all cases the vertical spacing between the rows of cracks was kept constant. Each curve represents a different horizontal spacing between the cracks. For the cases where the cracks were closely spaced it was found that there was an optimal measurement length, of approximately 12mm, after which the signal became relatively constant. (This length was thought to be a function of the particular problem.) From this study it was concluded in [5] that the use of fibre optical sensors, or other methods for compliance measurement, may have potential for monitoring the cracking process. Furthermore, if as in the 488 aft bulk head, a prior knowledge of the size and location of the critical region exists then there may exist an optimal sensor size. Whilst [5] dealt with through the thickness cracks we also need to assess surface flaws. Thus the challenge is to extend the investigation presented in this earlier work to more representative flaw configurations. In this context Pitt and Jones [6] investigated the optimal sensor length for detecting small surface flaws. In this study it was found that as the displacement field decays rapidly to zero outside the extent of the cracked region it would appear, see Figures 3, that it is only viable to place a sensor directly over the

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cracked region itself. This study also pointed to an optimum length sensor. However, in this instance the length of the sensor is directly related to the size of the flaw that is needed to be detected. Thus if we are to find and assess small flaws we will need small sensor. Indeed, given the likely level of uncertainty in the precise location of any (small) flaw/crack this will necessitate an array of small sensors. One potential solution to this requirement is to use a number of optical fibres each with a number of Fibre Brag Gratings (FBG’s). This approach also has the potential to determine the power/work flowing through a closed contour surrounding the damage. The change in the power flow with crack length can then be related to the severity of the flaw.

3.

Optical Fibre Techniques

Having examined the ability to sense cracking in structural components let us now turn out attention to the ability of optical fibre techniques to assess composite repairs to cracked metallic structures. To this end an initial experimental program was conducted on cracked aluminium skins 3.14 mm thick. Each plate had an edge crack 10mm long, which was repaired with a semicircular unidirectional boron/epoxy patch. The aluminium skins were tested back to back separated by a honeycomb sandwich core, see Figure 4. The initial crack was then grown under constant amplitude fatigue loading to a length of 40mm.

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In this test program an array of surface mounted optical fibers, OF1 – OF4, was used, see Figure 4. Each fibre contained a number of Fibre Bragg Gatings (FBG’s), see Figure 4. To give a far field reading fibre OF1, containing a single sensor S1, was bonded to the aluminium skin using a cyno-acrylate adhesive. The remaining three optical fibers were bonded to the top surface of the boron repair also using a cynoacrylate adhesive. These sensors were then completely buried beneath a thick layer of room temperature curing adhesive so as to protect the optical fibers and also provide a smooth surface for making the eddy current measurements of crack length. These gratings had high reflectivities (80-90%), FWHM of approximately 0.5nm, and a gauge length of around 5mm. The precise measurement of wavelength is therefore the key to strain measurement using FBGs. For this work a tuneable Fabry-Perot (FP) filter was used to track the Bragg wavelength. It was selected as it gives reasonable resolution +/and a large dynamic range. In this test program the crack was grown under a FALSTAFF fatigue load spectrum that attempts to simulate flight loads experienced by a military aircraft. As the crack progressed a series of static strain readings were periodically undertaken at load levels of 0 kN and 70 kN, see Figures 5 - 8. These results clearly illustrate the ability of an optical fiber sensor array to monitor crack growth under a bonded repair. This can be seen in the results from OF2 in Figure 6, located 25mm from the panel edge. The strain can be seen to initially increase as the crack approaches the sensors S2 to S5. It continues to increase as the crack passes these sensors, with the exception of S4 which experiences a decrease as the crack passes directly under this sensor. This is to be expected, as a valley of low strain exists directly over the crack with ridges of high strain on either side. Such a shear lag effect is caused by the load flowing up into the patch causing local bending.

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Sensors S6 – S9, located 30mm from the edge of the panel, exhibited a similar trend as can be seen in Figure 7. Here sensor S7 was located over the crack and experienced low strain readings as the crack passes underneath it. Sensors S6, S8 and S9 continued to increase as expected. OF4 also exhibited a similar trend as the crack approaches the optical fiber sensors located 40mm from the edge of the patch the strain increases. The initial decrease in S12 was due to debonding of the sensor. However, as the crack passes beneath S10 the sensor reading begins to decrease.

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3.1 MONITORING VIA CHANGES IN THE RESIDUAL STRAIN FIELD Residual strains within the patch system can also be used as an in-situ structural health monitoring technique. This passive technique relies on the fact that when a composite repair is applied to a metallic substrate the difference in the coefficient of thermal expansion (CTE) between the composite and the metal results in significant residual thermal stresses in the patch system at room temperature. When disbonding or damage in the patching system occurs then redistribution in the residual strains occurs. Therefore monitoring the residual strains in the component gives an indication of the presence of damage. The magnitude of the change in residual strains depends on the type, severity and location of damage, the curing temperature, and the relative stiffness between the patch and the substrate. In this study it was found that as the crack grew those sensors over the crack had a negative residual strains while those to the sides of the crack had a positive residual strain, see Figure 8 which shows the change in the residual strain as the crack grows under the sensors.

3.2 MONITORING VIA ENERGY FLOW By using an array of optical fibres we also have the ability to monitor the flow of energy into the patch. Let us therefore assess the ability of power flow, and energy, techniques to assess the integrity of composite repairs. We know from [2] that for this problem the stress intensity factor K rapidly becomes a constant, and is independent of the crack length. We thus expect that any energy related measurement, which reflects the change in the energy in the structure, should be linearly related to the crack length. From conservation of energy considerations we know that the energy flowing across

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any contour surrounding the crack tip can be evaluated by considering the strain energy in the region enclosed by the contour. To evaluate this concept a 3D finite element analysis of the repaired structure was performed and crack lengths of 2.5, 5, 10, 15, 20, 25, and 30 mm were evaluated. In the present analysis four contours were considered, see Figure 9. The contours lay on the upper surface of the patch with path 4 which completely enclosing the repair, and path 1 surrounding the elements near the crack. The results of this analysis can be seen in Figure 10, where we see that, as anticipated, as the crack length increased the rate of change of the total strain energy, which directly relates to the power flow across the various contours/paths, with crack length rapidly became constant. Indeed, once the crack had reached approximately 10 mm, the slopes of the energy versus crack length curves were essentially constant, see Figure 10. Since the slope of the curves and the stress intensity factor are both constant the slope can be directly related to the stress intensity factor.

It thus appears that energy measurements, such as power flow, directly reflect the energy state at the crack tip, i.e. the stress intensity factor. As such it appears that by using an array of optical fibres we have the potential to not only monitor crack growth beneath a repair but to also assess the structural integrity of the cracked component. In this case the strains could either be measured using conventional strain gauges or by fibre optic techniques.

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4. Structural Health Monitoring Of Joints Using Residual Stress Following this work an experimental study was undertaken into the use of residual strain to monitor the structural-health of bonded composite repairs/joints. The specimens consisted of a 6.35mm thick aluminium inner adherend with two 140 mm long 11-ply uni-directional boron/epoxy doublers, see Figure 11. The doubler taper was 1 step (of 2 plies) every 4 mm, with the first step consisting of a single ply. The uni-axial loading was applied at a frequency of Hz. The doublers were bonded using FM73 at a cure temperature of 80°C. More details on these specimens can be found in [8]. Two strain gauges, with a gauge length of 2mm, were bonded onto the 2nd step on either sides of the boron epoxy laminate, so as to monitor the residual strain. In Figure 11 the location of these strain gauges is labeled as P3. The experimental program consisted of static loading the specimen to undertake a strain survey, in order to compare strain gauge and optical sensor measurements with finite element predicted strains, and then a limited fatigue study under a FALSTAFF load spectra to evaluate the performance of these sensors in monitoring damage growth. Residual strain measurements were undertaken during the fatigue test and the change in the residual strain with increasing number of loading cycles is shown in Figure 12. From this work we observed that, due to the disbonding process, the residual strain approached an asymptotic of ~300-400 Higher residual strains, and hence increased sensitivity, to disbonds are expected for regions on the outer, thinner, portion of the joint. These results, when taken together with the work reported in Section 3, have illustrated the potential for residual strain measurements to be used as an alternative method for monitoring the structural health of adhesively bonded joints and composite

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repairs. The absolute magnitude of the change in the residual strains will be dependent on: the nature and location of the damage, the cure process and cycle, and the relative stiffnesses of the adherends.

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Conclusion

This study has demonstrated the ability of an optical fiber sensor array to monitor crack growth and delamination under a bonded repair. We have also shown that residual strain measurements can be used to assess degradation in adhesively bonded joints and repairs. Whilst strain gauges can be used to measure residual strain optical fibres containing FBG’s have a improved sensitivity. This paper has focused on the use of FBG’s. However, other research activities under way as part of this collaborative research program involves the use of piezoelectric sensor technology.

6.

Acknowledgements

This work was performed as part of a collaborative project between DSTO and the CoE-SM. The authors would particularly like to acknowledge the input of Dr.’s A. A. Baker and S. D. Moss at DSTO. Assistance in specimen manufacture from Mr. Ivan Stiyanovski and John van den Berg, and in specimen testing from Mr. Rowan Geddes, Carlos Rey and Brian Jones is gratefully acknowledged. The authors would also like to acknowledge Drs Steve Collins and Peter Farrell for allowing them to use the Bragg grating writing facility at Optical Technology Research Laboratory, Victorian University.

7.

References

1.

R. Jones, L. Molent and S. Pitt, “Studies in Multi-site damage of fuselage lap joints”, Journal Theoretical and Applied Fracture Mechanics, 32, 18-100, 1999. A. A. Baker and R. Jones, Bonded Repair of Aircraft Structure, Martinus Nijhoff Publishers, The Hague, 1988. R. Jones, R. Bartholomeusz, R. Kaye and J. Roberts, Bonded-Composite Repair of Representative Multi-site Damage In A Full-Scale Fatigue-Test Article, Journal Theoretical and Applied Fracture Mechanics, 21,41-49, 1994. L. Molent, R. J. Callinan and R. Jones, “Structural aspects of the design of an all boron epoxy reinforcement for the F l 1 1 C Wing Pivot Fitting: Final Report”, ARL, Aircraft Structures Report 436, September 1992. Jones R, Hammond (Pitt) S and Williams J. F., “A numerical study of interacting cracks in aluminium alloys”, Computers and Structures, Vol. 55, pp. 178-183, 1995. S. Pitt and R. Jones, “Compliance measurements for assessing structural integrity”, Engng Failure. Analysis, 8,4,371-388, 2001. Galea S. C,. “Monitoring damage in bonded composite repairs of cracked metallic components using surface strain measurements”, Proceedings ICCM-11, 14th-18th July 1997, Edited by M. L. Scott, Publushed by Australian Composite Society, Melbourne, ISBN 185573, pp 707-722, 1997. S. Whitehead, S.C. Galea and I. McKenzie, 2000 “In-Situ Health Monitoring of Bonded Composite Repairs Using Embedded Optical Fibre Sensors” Proceedings of the International Conference on Computational Engineering & Sciences 2000 (ICES'2K).

2. 3. 4.

5. 6. 7.

8.

EXPERIMENTAL INVESTIGATION OF SHRINKAGE STRAINS IN MULTILAYERED STEREOLITHOGRAPHY PARTS

D. E. KARALEKAS University of Piraeus Karaoli & Dimitriou 80 Str. GR-185 34 Piraeus, Greece

Abstract The shrinkage characteristics of stereolithography built square laminate plates using an acrylic based photopolymer were studied after they have been post-cured under ultraviolet and thermal exposure. The specimen plates consisted of a resin plate laser cured on an identical plate of the same material that already had been cured and postcured. The assembled laminate was then cured, and the resulting out-of-plane displacement (warpage) due to shrinkage was recorded by means of the shadow moiré method. The exhibited warpage of the plates was related to the polymerization or crosslinking shrinkage strains through the elastic lamination theory, which was implemented to calculate the magnitude of the resulted shrinkage strains. 1. Introduction Stereolithography (SL) is one of the most popular rapid prototyping processes used to cure successive thin layers of liquid photopolymer resin to build each on top of the previous layer until a 3D part is fabricated. It uses an ultraviolet (UV) laser to cure the associated photopolymer [1]. Most parts fabricated by the SL process are not fully cured by the irradiation of UV light. They need to undergo through a post-curing process to fully cure the photopolymer, and, hence, increase their mechanical properties. Experimental results have shown that the degree of curing of photopolymer is proportional to the mechanical properties of the final part [2]. During the last few years, significant research work has been performed to widen the applications of SL built prototypes. Impact testing of SL models has been examined as a tool for the prediction of natural frequencies of prototype aluminum parts [3]. Such predictions may be used directly in the design process, or to validate and refine finiteelement models. Other recent work in rapid prototyping has been focusing on the fabrication of functional or semi-functional parts. However, the poor strength and dimensional accuracy can reduce the applicability of the final prototype part, hence hindering the concept of functional or semi-functional rapid prototypes [4,5]. Work has been carried out to improve the strength of stereolithography parts by embedding continuous fiber tows into the fabricated part. The effects of the addition of short glass fibers on the mechanical properties and dimensional accuracy of prototypes built by the 749 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 749–756. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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stereolithography method have also been investigated [6]. Tensile tests performed at the University of Piraeus on SL built specimens reinforced with various types of nonwoven fiber mats have shown improvements in stiffness and strength over the pure polymer specimens (Figure 1). It is also seen that there is a decrease in the elongation to fracture encountered by the fiber-reinforced specimen, implying that the specimens have become more brittle with the addition of fibers.

Dimensional accuracy and strength of SL created prototypes have always been a major problem, especially in the fabrication of thin-walled hollow structures. The weak mechanical properties and high volume of polymerization shrinkage associated with the photopolymer cause the prototype part to warp and distort during laser solidification process. These inaccuracies are further magnified during the post-processing stage resulting to higher internal stresses and strains. Previous investigations have shown that shrinkage in parts on a layer-by-layer basis depends on many processing factors (laser scan orientation, density and speed, layer thickness, hatch spacing etc.) that must be carefully balanced to produce a strong and accurate prototype [7]. Various studies have been undertaken to investigate the curing characteristics of photopolymers used in the SL process and the mechanisms of the resulting shrinkage. Fuh et al. [8-9] investigated the curability and mechanical properties of an acrylic-based photopolymer. They have found that the uncured and partially cured resins trapped within the photopolymer resulted in shrinkage and distortion at the post-curing stage. Also they have reported that, the higher the laser density and shorter the layer thickness, the higher the curing percentage of the resin and the better the mechanical strength of the fabricated part. Many researchers investigated or modeled the effect of SL processing parameters on the fabricated part [10-12]. Konig et al. [13] concluded that when layers are scanned in only one direction, shrinkage forces occur mainly in the scanning direction which results in one-sided curling of the parts. Karalekas et al. [14] carried out experiments to study the induced residual stresses resulted during the laser solidification of an epoxy-based photopolymer as a function of hatching space and curing depth.

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In this paper, an experimental investigation to determine the magnitude of the polymerization shrinkage strains resulted during the laser solidification of an acrylic based photopolymer is presented. In previous investigations raman spectroscopy, differential scanning calorimetry (DSC) and differential scanning photo-calorimetry (DSP) techniques were employed to investigate the extent of curing and shrinkage in acrylic based photo-sensitive resin systems. A more direct method was followed in this paper for the determination of polymerization shrinkage. Some preliminary results of the discussed methodology have been presented at [15]. The method is based on measuring the warpage of SL cured and post-treated, under ultraviolet (UV) and thermal exposure, resin plates and then using elastic lamination theory to calculate the resulted shrinkage strains. The method was first developed by Daniel [16] to obtain the chemical cure shrinkage in epoxy matrix laminates. 2.

Analysis of SL Resin Laminate Shrinkage

The inhomogeneous character of the curing process and of the process of building the models, layer upon layer, has as a result the introduction of material anisotropies in the final parts. The fabricated laminate plates used in the shadow moiré tests were analyzed as laminates consisting of two parallel orthotropic layers, of equal thickness and thermal and mechanical properties under conditions of plane stress. The first layer-plate is considered fully cured and flat when polymerization shrinkage starts taking place in the second half. During curing of the second layer-plate, initially in liquid form, both halves undergo the same thermal deformations, but only the new half undergoes the additional polymerization deformation. Thus, any warpage observed at the end of the solidification process is due to polymerization shrinkage. After subjecting the laminate plate to the second curing and post-curing process the polymerization shrinkage strain in layer 1 is assumed to be zero:

The unrestrained polymerization shrinkage strain in layer 2 would be

assuming that x- and y- axes coincide with the principal material axes, obtained along and perpendicular to the scan direction. The shrinkage strains in the x- and y- axes of the model laminate are related to the laminate curvatures as follows [16]:

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Integration of the curvature displacements relations

and ignoring rigid body displacements, we obtain for the out-of-plane deflection w at any point on a square laminate of side dimension a:

where, and are the mid-plane curvatures of the laminate plate. Thus, contours of equal deflection are ellipses as shown by the moiré fringe pattern. Each elliptical fringe corresponds to a deflection For an elliptical fringe with semi-axes a and b along the x- and y- axes and corresponding to a deflection w, it is obtained:

Solving equation (4) for

and

we obtain:

Thus, the components of polymerization shrinkage determined by the method followed here are independent of the material properties of the layers, since they are the same for both layers. 3. Experimental Procedure

The resin material investigated was an acrylic based photopolymer (Allied Signal Exactomer 2202 SF). An EOS Stereos Desktop Stereolithography system using an UV laser beam was employed for the fabrication of the test specimens. Liquid thin resin layers of 0.156 mm thickness were laser cured (all layers were scanned in the same direction) during the SL layer-by-layer building process to form square laminate plates (10 x 10 x 3.3 cm). Then, the plates were fully polymerized (post-cured) by placing half of them in an UV chamber for 8 hours and the remaining ones in an oven at 80° C for the same length of time. One surface of the post treated plates was sanded lightly and coated with a thin layer of primer to help the adhesion of the second resin layer to it. A second layer of the same material, thickness and laser scan orientation was stacked on top of the initially fully cured plate. The whole assembly was then post treated in an UV or thermal chamber. Assuming that both cured layers exhibit the same degree of resin polymerization at the end of the curing/post-curing process and undergo the same thermal expansions, any warpage observed after the selected post treatment process of the two layer laminate is due to the additional polymerization shrinkage that only the new half undergoes.

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The out-of-plane deflection (warpage) of the stereolithography formed resin plates was measured by means of projection or shadow moiré [16]. This method is ideal for obtaining full-field contour maps of large surfaces. A master grating is placed in front of the resin specimen and illuminated by a collimated light beam at an angle to the grating. The master grating and its shadow on the surface of the specimen then produce a fringe pattern viewed normally to the master grating. The out-of-plane displacement is given by

where p is the grating pitch, n is the fringe order and « is the angle of incidence of the oblique light beam. In the present work discussed here a grating of 0.56 pitch (18 lines/cm) was obtained in the form of transfer sheets. Transparent replicas of the grating were made and taped flat to a glass sheet. The sheet with the grating was held flat in front of the laminate specimen at a short distance from its surface. A 35 mm slide projector was used as a light source, illuminating the specimen at 45 degrees to the normal of the grating. Fringe patterns were viewed normally to the grating and recorded with a digital camera. The experimental setup is shown in Figure 2 while typical moiré fringe patterns in Figures 4-5. The fringes obtained represented loci of points of constant out-of-plane deflection such that all points on a given fringe corresponded to a deflection of one grating pitch relative to the points of the neighboring fringe. The obtained fringe patterns were analyzed according to the theory described next. For a fringe pattern containing several elliptical contour fringes, the determination above can be made for various fringe orders (ellipses) to increase the accuracy.

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4. Results and Discussion The presented fringe patterns were analyzed by measuring the major and minor axes of the concentric elliptical fringes corresponding to integral fringe order (dark fringes) and to integral plus one-half fringe orders (light fringes). These values (a and b) and the corresponding fringe order, were used in equations 7 and 8 to obtain several sets of values for the polymerization shrinkage strains, and for the grating (ruling) used. Laser curing of the second resin layer on top of the first layer resulted to significant outof-plane deflection of the laminate plate as seen in Figure 3. Projection moiré of the curved laminates gave fringe patterns consisting of concentric elliptical black and white lines (Figures 4-5). Distinct differences were noted for the viewed fringe patterns before and after post treatment. During post-curing of the two layer laminate the second layer shrinks additionally, as it is further polymerized, leading to an increase of the exhibited out of plane deflection. Post-curing of the examined laminate plates under UV exposure resulted to the formation of circular fringes, indicating that the laminate plate shrinks proportionally along its plain directions. When post-curing took place in the thermal chamber the viewed patterns were of elliptical shape, exhibiting higher shrinkage in the y- axis of the plate.

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It must be stressed that the analysis followed in the present work, based on the elastic lamination theory, could only be implemented for the determination of chemical shrinkage strains resulted after post-curing of the resin laminates, since only then both layers would exhibit the same mechanical properties. Typical results of measured shrinkage strains for the additionally polymerized test specimens are tabulated in Table 1.

5. Conclusions

An experimental investigation combined with a theoretical analysis based on elastic lamination theory was undertaken to determine the magnitude of the resulted shrinkage strains in a laser solidified acrylic based photopolymer. Standard two layer square samples were fabricated and subsequently subjected to UV and thermal post treatment. The resulting warpage, consisting of elliptical equal deflection contours, was recorded experimentally using the shadow (or projection) moiré technique and correlated to the chemical shrinkage by means of lamination theory. The test specimens were theoretically analyzed as a laminate plate consisting of a liquid resin layer applied and cured on an identical layer of the same resin material, which has already been cured and post-cured. It was found that test specimens post-cured under intense UV light exhibited a uniform shrinkage resulting in concentric circular moiré patterns. However, both post treatment processes lead to the generation of polymerization shrinkage strains of considerable magnitude. The followed experimental methodology and the characteristics of the moiré fringe patterns allow one to investigate in a simple and

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direct way the shrinkage characteristics of the photopolymer resin systems used in stereolithography. As a result, it can be implemented for comparison reasons between pure photopolymer specimens and their fiber-reinforced counterparts for the determination of the shrinkage encountered by both types of specimen during the postcuring process. 6.

Acknowledgements

The contribution of Mr. A. Aggelopoulos in the presented experimental work is greatly appreciated. 7. 1.

References

Jacobs, P.F.: Rapid Prototyping & Manufacturing: Fundamentals of Stereolithography, Society of Manufacturing Engineers, Dearbon, 1992. Cheah, C.M., Fuh, J.Y.H., Nee, A.Y.C., Lu, L., Choo, Y.S., Miyazawa, T.: Characteristics of 2. photopolymeric material used in rapid prototypes – Part II. Mechanical properties at post-cured state, J. of Materials Processing Technology 67 (1997), 46-49. Mahn, J.P., and Bayly P.V.: Impact testing of stereolithographic models to predict natural 3. frequencies, J. of Sound and Vibration 224(3) (1999), 411-430. 4. Jacobs, P.F.: Stereolithography and Other RP&M Technologies, Society of Manufacturing Engineers, Dearbon, 1996. Onuh, S.O., and Hon, K.K.: Improving stereolithography part accuracy for industrial 5. applications, Int. J. of Advanced Manufacturing Technology 17 (2001), 61-68. 6. Cheah, C.M., Fuh, J.Y.H., Nee, A.Y.C., and Lu, L.: Mechanical characteristics of fiber-filled photo-polymer used in stereolithography, Rapid Prototyping Journal 5(3) (1999), 112-119. 7. Wiedemann, B., Dusel, K.H., and Eschl, J.: Investigation into the influence of material and process on part distortion, Rapid Prototyping Journal 1(3) (1995), 17-22. 8. Fuh J.Y.H., Lu, L., Tan, C.C., Shen, Z.X., and Chew S.: Curing characteristics of acrylic photopolymer used in stereolithography process, Rapid Prototyping Journal 5(1) (1999), 27-34. 9. Fuh J.Y.H. et al.: Processing and characterizing photo-sensitive polymer in the rapid prototyping process, J. of Materials Processing Technology, 89-90 (1999), 211-217. 10. Chen, C.C., and Sullivan, P.A.: Predicting total build-time and resulted cure depth of the 3D stereolithography process, Rapid Prototyping Journal 2(4) (1996), 27-41. 11. Li, J.H.: Improving stereolithography parts quality – practical solutions, Fourth International Conference on Rapid Prototyping, (1993), 171-178. 12. Bugeda, G., Cervera, M., Lombera, G., and Onate, E.: Numerical analysis of stereolithography processes using the finite element method, Rapid Prototyping Journal 1(2) (1995), 13-23. 13. Konig, W., Celi, L., and Nokan S.T.: Stereolithography process technology, Proceedings 3nd European Conference on Rapid Prototyping and Manufacturing, (1994), 191-208. 14. Karalekas, D., Rapti, D., Kontomitros, C., Zacharopoulos, D., and Gdoutos, E.E.: Investigation of residual stresses in SL Built models as a function of process parameters, Proceedings 32nd International SAMPE Technical Conference, (2000), 68-75. 15. Aggelopoulos, A., and Karalekas, D.: Determination of cure shrinkage in SL layer build plates using lamination theory, Advanced Composites Letters 10(1) (2001), 7-12. 16. Daniel, I.M., Wang, T.M., Karalekas, D., and Gotro, J.T.: Determination of chemical cure shrinkage in composite laminates, J. of Composites Technology & Research, 12(3) (1990), 172176.

SUPPRESSION OF DIMPLING IN SHEET METAL PARTS FORMED ON DISCRETE TOOLING

ROBERT C. SCHWARZ, JERRELL NARDIELLO, and JOHN M. PAPAZIAN Northrop Grumman, Technology Development Bethpage, New York, 11714-3581

Abstract Discrete-element reconfigurable tools are attractive for some low-volume metal forming processes. However, the use of these “pin” dies, where the ends of the pins are hemispherically shaped, can cause unacceptable dimpling of the finished parts. Thus, polymeric interpolators are used between the die and the part to suppress dimpling. This paper presents the results of experiments performed to characterize the dimpling process and to establish the requirements for useful interpolator materials. 1. Introduction The use of reconfigurable tooling for the production of-sheet metal parts is an attractive concept in low production rate environments. Reconfigurable tooling can obviate the need to fabricate a forming, tool for each new shape to be produced. The cost of designing and manufacturing these tools is considerable, however, a suitable reconfigurable tool that could eliminate the need to manufacture and store traditional tools, would, over a reasonable period of time, reduce the cost of manufacturing parts. The most common of these low-rate environments is the repair of out-of-production aircraft still in service. In repair facilities, the original tooling is often not available, making reconfigurable tooling even more attractive. The fabrication of new aircraft is also attractive, cost-wise, because the total number of aircraft of a given type produced rarely exceeds 100 per year. Typical automotive production runs of greater than 10,000 per year do not offer the same economic advantages for reconfigurable tooling, although limited volume or prototype models may be suitable. The use of reconfigurable tooling for sheet metal forming has been the subject of prior research [1-8], and several patents have been issued, e.g. [9-12]. The concept derives from the use of “discrete-dies” composed of a dense array of square pins with hemispherical tips. The ends of the pins are hemispherical in order to enable production of parts of many shapes. However, forming directly on the pins would result in severe dimpling of the sheet metal, as shown in Figure 1. In order to preclude dimpling, a smoothing layer of an elastomeric material is required between the pins and the sheet metal. The properties of the elastomeric interpolating layer are critical to the suppression of dimple formation and to the success of the overall forming process. Thus, the objective of this work is to describe the experiments used to characterize the response of a sheet of metal to the discrete tooling and its “interpolator” during forming. 757 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 757–768. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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2. Experimental Results 2.1. TEST SET-UP The first experiments were conducted with a 152-mm by 152-mm tool containing thirtysix 25.4 mm by 25.4 mm pins, as seen in Figure 2. The pins were positioned such that the hemispherical tips were tangent to a 152-mm radius cylinder. The sheet metal chosen was 2024-O aluminum. This alloy was selected because it is commonly formed into aircraft parts. The reconfigurable tool was installed on a swing—arm press as shown in Figure 3. The two arms of the swing arm press are swung forward until, when the aluminum material is placed in the grips, the sheet is straight and tangent to the surface of the pin die with the interpolator material in place. A small “snug” force is applied to the sheet metal by the tension cylinders on the arms. A snug load that produced a tensile stress in the sheet metal of 9 MPa was typical. The sheet metal is now loaded past its yield point into a state of plastic strain. For the 2024-O aluminum material, this typically required a stress of 73.6 MPa. With the sheet well into the regime of plastic deformation, the arms of the press are directed to rotate backward, wrapping the sheet metal over the die. The “wrap” phase is accomplished at constant force. Upon completing the wrap, the sheet metal part may or may not be subjected to an additional strain to cause it to more completely retain the shape of the die. Following the verification of the swing arm press’ suitability as a forming machine, the press was modified to include the installation of precision 22,730-kgf load cells on each tension cylinder. The jaw assemblies shown in Figure 3 were replaced with 406-mm wide sheet metal grips. The modification also included the installation of high precision displacement transducers on the swing arms, to improve the repeatability and quality of the forming experiments.

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Initially, two methods were used to measure the average dimple depth in parts formed using the procedure described above. The first was an optical method, shadow moiré, Figure 4. The second was a Browne & Sharpe Coordinate Measuring Machine (CMM). In the shadow moiré method, a 102 mm by 102 mm shadow moire grille with a 7.87 lpmm ruling was set at an angle of 40 degrees, and the system was calibrated by recording the fringe pattern produced on a truncated cone with a height change of 2.46mm, Figure 5. The average fringe count on four orthogonal radial lines was found to be

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15.88 fringes. The resulting sensitivity was calculated to be 0.147-mm per fringe. The CMM is a known, reliable, precise method of measurement, but took much longer than the optical moiré method.

2.2 COMPARISON OF SHADOW MOIRE AND CMM MEASUREMENTS

To confirm the results obtained from the shadow moiré method, a Brown & Sharpe CMM was used to measure the profile of several parts produced. The CMM was directed to scan along the line of maximum dimple amplitude. The results of the shadow moiré measurements and the CMM measurements are presented in Table 1.

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It can be seen that the average difference between the techniques was 0.018 mm. These four tests were also examined visually for the presence of dimpling. As was seen earlier in Figure 1, when no interpolating layer was used the dimples were very visible. When a 6.35-mm interpolating layer was used, the dimples were only slightly visible, and when 25.4 mm or more was used, the dimples were not visible, Figure 6. We concluded from these experiments that the shadow moiré process is accurate, and will make the presence of even very shallow dimples (less than 0.069-mm) known.

2.3 CHARACTERIZATION TESTS 2.3.1. EFFECT OF INTERPOLATOR THICKNESS ON DIMPLE FORMATION A series of tests was conducted to understand the influence of interpolator thickness on dimple suppression. A series of parts was formed on the pin die in which the interpolator thickness was increased from zero (no interpolator) to as much as 30 mm. The average dimple depths were determined along the line of maximum dimple depression. This typically occurred over the centerline of a column of pins. For a representative interpolator, the influence of interpolator thickness on dimple formation

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is shown in Figures 7 and 8. The number of fringes between dimples is proportional to their depth.

2.3.2.

EFFECT OF INTERPOLATOR HARDNESS ON DIMPLE FORMATION

The characterization process continued with a series of tests to evaluate the effect of interpolator hardness on the formation of dimples on the formed part. The earliest dimpling tests had been performed with an interpolator composed of a thermoplastic copolymer of ethylene and vinyl acetate. The major drawback of this material was its sensitivity to temperature change. As the ambient temperature varied, so did its mechanical properties. Its ability to suppress dimpling was not a constant. Since this project was to lead to the manufacture of a full-size, computer-driven, reconfigurable

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forming tool that would be used on a factory floor, the variability of this copolymer’s mechanical properties with temperature was not acceptable. A search was made to find other polymers with sufficient toughness to be suitable for use on a pin die, and which were thermally stable through the range of ambient temperatures common in manufacturing plants. Each of these materials was characterized by determining their ultimate tensile strength, their compressive and tensile modulus of elasticity, and their Shore hardness. The forming tests were conducted for each new material with the interpolator thickness varied as before. Each test piece was examined in the shadow moiré set-up, and the average dimple depth determined as a function of interpolator thickness. It soon became apparent that the thickness of an interpolating material needed to suppress dimpling was a function of its hardness. The shadow moiré images of the dimples formed by a soft, an intermediate, and hard, 12.7-mm thick interpolators are presented as Figure 9. It can be seen that the harder an interpolator is, the thicker the interpolator must be to suppress dimpling. These data are presented in Figure 10.

2.3.3.

EFFECT OF SHEET METAL THICKNESS ON THE FORMATION OF DIMPLES

In a similar fashion, the effect of the sheet metal thickness was examined. For these tests 2024-O aluminum alloy was once again used as the test material. It was found that the thinner the sheet metal, the greater the thickness of interpolator material needed to suppress dimpling. Figure 11 presents the moiré images for three thickness of 2024-O aluminum on a 6.35-mm thick, intermediate stiffness interpolator. Counting fringes between dimples shows that the dimple depth increases as the sheet metal thickness decreases. These data are summarized in Figure 12.

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THE EFFECT OF SHEET METAL YIELD STRENGTH ON DIMPLE FORMATION

The last investigation performed with the 152 mm by 152 mm pin die was to study the effect of sheet metal yield strength on dimple formation. The test samples were cut from 2024-O and 2024-T3 aluminum alloy sheet. The results followed a trend similar to the earlier tests. In Figure 13, we see that the formation of dimples in the higher

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yield strength, 2024-T3 material, is not as influenced by the hardness of the interpolator as the lower yield strength, 2024-O alloy.

2.4. THE FULL-SIZE TOOL

These and many other sub-scale tests were intended to demonstrate the feasibility of scaling up to a full-size reconfigurable sheet metal forming tool. The full-size tool was to measure 1.22 m by 1.83 m, and was to demonstrate the ability of a densely packed reconfigurable pin die to produce aircraft components. The characterization of the

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interpolating material and the response of both soft and tempered aluminum to the interpolator/pin tips led to the selection of a single, soft, 25.4-mm thick interpolator material for the forming of parts on the full-size tool. In Figure 14, we see the full-size reconfigurable tool configured to manufacture an aircraft fuselage skin panel. The interpolating material is supported above the tool during the pin setting process. In Figure 15 we see the interpolator on the tool, and the sheet metal installed in the press during the forming process. In Figure 16 we show several different parts formed with different aluminum thickness. In all cases the parts were dimple free.

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3. Summary Experiments on the forming of parts using a reconfigurable "discrete-element" die have shown that dimpling can be completely suppressed by judicious selection of interpolator materials. The primary characteristics of the interpolator are thickness and hardness. It was found that harder interpolators require greater thickness before dimpling is absent, and conversely, thinner layers of softer interpolators are adequate. The thickness of the sheet metal being formed also plays a role, with thicker materials being more resistant to dimpling. Limited testing of the effect of the yield strength of the material being formed on dimpling was not conclusive. 4. Acknowledgements This work was partially supported by the DARPA Flexible Fabrication Program through ONR Agreement N00014-95-2-0003: DARPA Program Manager – Dr. W. Coblenz; ONR Program Manager – Dr. George Yoder. Additional support was provided by Northrop Grumman, MIT, and the Cyril Bath Corporation.

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5. References 1.

D.E. Hardt and D.C. Grossard, “A Variable Geometry Die for Sheet Metal Forming,” Proceedings of Joint Automatic Control Conference, San Francisco, CA, 1980. 2. D.E. Hardt, B.A. Olsen, B.T. Allison, and K. Pasch, “Sheet Metal Forming with Discrete Die Surfaces,” Proceedings of Ninth North American Manufacturing Research Conference, 1981. 3. D.E. Hardt, M.C. Boyce, K.B. Ousterhout, A. Karafillis, and G. Eigen, “A CAD Driven Flexible Forming System for Three-Dimensional Sheet Metal Parts,” Proceedings of SAE Sheet Metal Forming Symposium, Detroit, MI, 1993. 4. E.V. Finckenstein and M. Kleiner, in Annals of the CIRP (1990), pp. 311-314. 5. L.M. Kutt, P.L. Ogilvie, A.B. Pifko and J.M. Papazian, “Nonlinear Structural Analysis of Sheet Metal Forming with ABAQUS/Explicit”, ABAQUS Users’ Conference, May 27-29, 1998, Newport, RI. 6. L.M. Kutt, J.A. Nardiello, P.L. Ogilvie, A.B. Pifko and J.M. Papazian, “Finite Element Analysis of Sheet Metal Forming with Reconfigurable Tooling” Proceedings of the Sixth International Conference on Numerical Methods in Industrial Forming Processes-Numiform ’98, Enschede, Netherlands, June 22-25, 1998. 7. J. M. Papazian, E.L. Anagnostou, R.J. Christ, Jr., D. Hoitsma, J. Melnichuk, J.A. Nardiello, P.L. Ogilvie, A.B. Pifko and R.C. Schwarz, “Innovative Tooling for Sheet Metal Forming” Innovations in Processing and Manufacturing of Sheet Metals, Edited by M.Y. Demeri, The Minerals, Metals & Materials Society, 2001. 8. J. M. Papazian, D. Hoitsma, L.M. Kutt, J. Melnichuk, J.A. Nardiello, A.B. Pifko and R.C. Schwarz, “Reconfigurable Tooling for Sheet Metal Forming”, Sheet Metal Forming Technology, Edited by M.Y. Demeri, The Minerals, Metals & Materials Society, 1999. 9. G.T. Pinson, “Apparatus for Forming Sheet Metal,” United States Patent 4,212,188, July 15, 1980 10. E. Haas and M. Kesselman, “Adjustable Form Die,” United States Patent 5,546,784, August 20, 1996. 11. E.V. Sullivan, E.G. Haas, R.C. Schwarz, M. Kesselman, A.N. Peck and J.M. Papazian, “Individual Motor Pin Module”, United States Patent 6,012,314, January 11, 2000. 12. E.G. Haas, R.C. Schwarz, and J.M. Papazian, “Modularized Reconfigurable Heated Forming Tool”, United States Patent 6,089,061, July 18, 2000.

EFFECT OF LOADING RATE AND GEOMETRY VARIATION ON THE DYNAMIC SHEAR STRENGTH OF ADHESIVE LAP JOINTS

V. SRIVASTAVA, V. PARAMESWARAN, A. SHUKLA and D. MORGAN Dynamic Photomechanics Laboratory Department of Mechanical Engineering and Applied Mechanics University of Rhode Island Kingston, RI02881

Abstract Dynamic and quasi-static experiments were performed using a novel lap joint specimen to evaluate the shear strength of adhesive bonded lap joints at different loading rates, length to width ratios and lap areas. Dynamic shear strength was determined by subjecting the lap joints to stress wave loading in a Split Hopkinson Pressure Bar (SHPB) apparatus. All joints were bonded by a general-purpose epoxy adhesive (Armstrong A-12®). The shear strength of the joint was determined using maximum transmitted load through the joint, assuming that the load was predominantly transferred through shear. A series of tensile and compressive experiments were performed to determine the shear strength of a lap joint, for loading rates varying from quasi-static to 2500 The results indicated that as the loading rates are increased to 1000 the shear strength of the particular adhesive bonded lap joint increases to three times its static value, after which it stabilizes. The effect of bonded length to width ratio on the shear strength of similar lap joints was also experimentally investigated for both quasi-static and dynamic loading conditions. Experimental results showed that maximum dynamic shear strength is achieved for a length to width ratio of 0.8. Experiments based on a 3x3 factorial design indicated a statistically significant effect of bonded length to width ratio on the dynamic shear strength of lap joints. A disordinal interaction was observed for bonded length to width ratio and bonded area, implying that the main effects of length to width ratio and bonded area on the dynamic shear strength of joints are not separable. 1.

Introduction

Use of adhesive joints in civil, structural, aerospace and automotive applications is becoming a future trend because of their performance, manufacturing flexibility, and low cost. Adhesives, being polymeric materials, exhibit rate sensitive properties and require an assessment of their response under impact loading. This is a major consideration for designing new structural components involving adhesive joints economically and without risking human safety. As lap joints used in practical 769 E.E. Gdoutos (ed.), Recent Advances in Experimental Mechanics, 769–780. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

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applications are of various geometries, it is important to predict static as well as dynamic strength of these joints from the geometrical properties of bonded area. ASTM has defined procedures to determine the shear strength of lap joints subjected to quasi-static compressive (ASTM D905) [1] and tensile loading (ASTM D1002)[2]. While experimental methods for quasi-static testing of adhesive joints are well defined, little work has been done to study the dynamic strength of such joints. ASTM has defined an experimental method to determine the impact strength of adhesive bonded lap joints in terms of energy absorbed per unit bonded area, using a pendulum type impact tester (ASTM D950) [3]. But this method does not give any quantitative data suitable for design purposes. Prior studies have used instrumented Charpy impact tester to evaluate dynamic strength of lap joints. Kinloch and Kodokian [4] determined fracture properties and impact resistance of epoxy adhesives using an instrumented impact tester. The strength and energy absorption of bonded lap joints were measured using an instrumented impact tester at low impact velocity (1.34 m/s) by Harris [5]. Chao et al [6]&[7] investigated the strength of spot welded, adhesive bonded, and self piercing rivet lap joints using a modified impact tester. Although joint performance was assessed in terms of bond strength and energy absorption, only low to moderate loading rates were investigated and stress wave propagation was not considered. Recently, Yokoyama [8] evaluated tensile strength and energy absorption of adhesive butt joints under impact loading using a tensile Split Hopkinson Pressure Bar (SHPB). Srivastava et al [9] designed a novel specimen geometry, which can be used with the Split Hopkinson Pressure Bar in compression for evaluating the dynamic shear strength of adhesive lap joints. They established equilibrium conditions of the new specimen geometry during the dynamic event and determined the shear strength using the peak value of the transmitted load. For static loading several theoretical analyses have been performed to predict the strength and stress distribution of adhesive bonded joints. DeVries et al [10] studied adhesive fracture of lap joints and evaluated adhesive surface energy and maximum tearing stress for varying adhesive lengths. Chen and Cheng [11] presented a method for determination of stresses in adhesive-bonded single lap joints using two-dimensional elasticity theory in conjunction with variational principal of complimentary energy. Adams and Harris [12] used non-linear finite element technique to predict strength of bonded single lap joints. All these studies considered the effect of joint geometry on the strength of adhesive lap joints for the static loading of the joints and for the most part dynamic strength of joints has received only limited attention. This paper presents the effect of loading rate, bonded length to width ratio and bonded area variation on the shear strength of adhesive lap joints. The novel lap joint geometry proposed by Srivastava et al [9] was used in this study. Both quasi-static and dynamic experiments were performed on lap joints bonded with an epoxy adhesive. Quasi-static experiments were performed using an Instron Machine whereas the dynamic experiments were performed using SHPB apparatus. The effect of loading rate on the strength of a lap joint was determined by loading the specimen with a variety of compressive and tensile stress pulses. Similar specimen geometry was used in two compressive SHPB setups to investigate the effect of bonded length to width ratio (Ls/W) variation on the shear strength of lap joints. Since bonded area varied with Ls/W ratio variation, an experimental study based on a 3x3 factorial

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design of experiments was performed. Statistical significance of main, interaction as well as simple effects of Ls/W ratio and bonded area was obtained through 3x3 factorial analysis of variance and omnibus F tests [13]. 2.

Experimental Procedure

A special lap joint geometry, shown in Figure 1 was used to evaluate the shear strength of adhesive lap joints under dynamic loading. The specimen consisted of two cylinders each having an axial cut along a diametrical plane for part of its length, which formed the bonding surfaces. The specimens were fabricated from alloy steel, which has a tensile strength of 1770 MPa and yield strength of 1640 MPa. The two halves of the specimen were bonded with Armstrong A-12® general-purpose epoxy adhesive. It is a two-part epoxy adhesive which can bond almost all rigid to semi-flexible materials including ceramics, metals, woods and plastics. The mix ratio for all the experiments was 1:1 by weight. The bonding metal surfaces were degreased with methanol, mechanically abraded by using a 320-J grit paper and then re-cleaned again with methanol prior to bonding with the adhesive. A thin layer of premixed adhesive was smeared over the bonding surfaces. The two halves of the specimen were then assembled and secured in place. A small gap was provided on either side of the joint to ensure that the load transfer occurs only through the adhesive. After bonding, the specimens were cured for 1 hr at 200 °F. This procedure resulted in joints having an adhesive thickness of 0.1 mm.

Both quasi-static and dynamic compressive experiments were conducted on same type of joints. Quasi-static experiments were conducted using an Instron Machine (Model 1125) at a crosshead speed of 1.27 mm/min. Dynamic experiments were performed using the SHPB apparatus. A schematic of a typical compressive SHPB setup is shown in Figure 2. The SHPB set up consists of an incident bar and a transmitter bar, both instrumented with strain gages, with the specimen sandwiched

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between the two bars. A projectile fired from a gas gun impacts the incident bar, generating a stress pulse of finite length in the incident bar. Upon reaching the specimen, part of the stress pulse gets transmitted through the specimen into the transmitter bar while the remaining pulse gets reflected back into the incident bar due to impedance mismatch. The time resolved strain histories in the bars were measured by the strain gages. The dynamic stress and the load history at each end of the specimen were obtained from the strain histories using the theory of one-dimensional wave propagation [14] as explained in section 3.

2.1. LOADING RATE VARIATION EXPERIMENTS First, the effect of loading rate for a lap joint which had a width (W) of 12.7 mm and a bonded length to width ratio (Ls/W) of 1.20 was investigated. Dynamic experiments were conducted on a 12.7 mm bar diameter, compressive SHPB setup using various projectile lengths (10.0, 20.0 and 35 cm) and a pulse shaper made of a soft material (rubber) to obtain four different loading rates of 250, 600, 1100 and A series of three experiments was conducted for each loading rate. For these experiments, the ratio of the input pulse length projectile length) to the bonded length of the specimen (Ls) was in the range The same lap joint was investigated under tensile-shear to determine if a comparison could be made with the existing results of compressive-shear experiments at different loading rates. 2.2. Ls/W VARIATION EXPERIMENTS

In this series of experiments, Ls/W ratio was varied by changing the bonded length while keeping a constant width (W) of the specimens. Two different specimen widths of 12.7 mm and 50.8 mm were used in these experiments. Corresponding ratios

and bonded areas are given in Table 1.

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A set of two experiments for each Ls/W ratio was conducted to determine the effect of Ls/W on both dynamic and static shear strength of adhesive lap joints using two SHPB setups, which had bar diameters of 12.7 mm and 50.8 mm. For all dynamic experiments a long projectile of 35 cm length was used to obtain better equilibrium of the lap joint. For these experiments, the ratio of the input pulse length to the effective length of the specimen was 2.3. 3x3 FACTORIAL DESIGN EXPERIMENTS For the experiments in the previous section, changes in (Ls/W) resulted in a change in the bonded area also. In order to identify the independent effects of each of these factors, (Ls/W) and area, a factorial design of experiments was performed, in which the length and width (W) of adhesive lap joint specimens were varied through a 3x3 factorial design as shown in Table 2.

Minimum, maximum and an intermediate value of Ls/W ratio and their corresponding bonded areas from the previous set of experiments were selected. Two experiments were performed for each of the nine geometries of the lap joint specimens. For the specimens with diameter 12.7 mm, SHPB set up which had bar diameter of 12.7 mm was used and for the specimens which had 12.7 mm < diameter 50.8 mm, the 50.8 mm diameter SHPB setup was used.

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3.

Theory and Assumptions

For quasi-static experiments, the shear strength

where

is the maximum load and

of the joint was determined as

is the bonded area of the lap joint.

For dynamic experiments, the dynamic shear strength was calculated from the strain histories assuming that:

The wave propagation is one dimensional without any dispersion. Inertial effects in the specimen are negligible. The bars remain elastic. The joint transmits the load under pure shear and failure of the joint occurs at the maximum transmitted load. The joint remains in equilibrium until failure.

The gap shown in figure 1 has no effect on the shear strength of adhesive lap joints. Using the theory of one-dimensional wave propagation and the above assumptions, the dynamic loads at each face of the specimen can be obtained as [14]

where are the incident strain, reflected strain, transmitted strain, cross-sectional area of the bars, Young's modulus of the bar material (200 GPa), load at the incident bar end of the specimen and load at the transmitter bar end of the specimen respectively. The incident load was used to obtain the loading rates for the experiments. The dynamic shear strength of the adhesive joint is given by

Dynamic shear strength of joint,

where, is the maximum transmitted load through the lap joint and bonded shear area of the joint.

is the

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Experimental Results and Discussion

4.1. EFFECT OF LOADING RATE

Cohesive failure in the adhesive, as shown in Figure 3 was observed for all the dynamic experiments. The experimental results for the joints subjected to compressive loading indicated that the adhesive lap joint can transmit significantly higher loads when subjected to impact loading. Experimental results presented in Figure 4 show that the dynamic shear strength of the adhesive lap joint increased up to three times its static shear strength value for loading rates around 1000 after which it became constant. The normalized strength (ratio of dynamic strength to static strength) shown in Figure 5 presents a comparison of the experimental results for the tensile-shear experiments and the compressive-shear experiments. Both tensile-shear and compressive shear experiments showed same trend and a threefold increase in shear strength at

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EFFECT OF Ls/W RATIO

Figure 6 presents the variation of shear strength with bonded length to width ratio (Ls/W) for the dynamic and quasi-static experiments. The results indicated that for the dynamic loading of lap joints, shear strength of the lap joint increased with Ls/W ratio up to a ratio value of 0.8 and then it decreased. For quasi-static experiments, results indicated that the shear strength increased slowly with Ls/W ratio and it stabilized after Ls/W ratio of 1.0 for the range of experimental data. The results presented in Figure 7 show that, as the Ls/W ratio increased, the dynamic to static shear strength ratio increased from a value of 1.4 to a value over 3.0 at Ls/W = 0.8 and then it decreased. These results suggest that an Ls/W ratio of 0.8 is the optimum for achieving the maximum dynamic shear strength of an adhesive lap joint. It was also observed that the effect of Ls/W ratio variation was more prominent in the case of dynamic loading compared to quasi-static loading. The magnitude as well as the distribution of stresses varies inside the adhesive layer with varying bonded length (Ls) and width (W) of the lap joint, resulting in varying shear strength of lap joints. Shear strength may also vary with the change in bonded area as the probability of having flaws in the adhesive bonded joint increases with increasing bonded area.

4.3.

RESULTS OF FACTORIAL DESIGN EXPERIMENTS

In the previous set of experiments, as the bonded area varied with Ls/W ratio, a 3x3 factorial experimental design was used to study the effect of both Ls/W ratio and bonded area on the dynamic shear strength of adhesive lap joints. A 3x3 factorial analysis of variance was performed on the experimental data to test the hypothesis that the variation of Ls/W and bonded area will affect the dynamic shear strength of adhesive lap joints. Three levels of both Ls/W ratio and bonded area were selected for this study. Table 3 shows dynamic shear strength values for nine combinations of Ls/W and bonded area of lap joints. In this table, the experimental results are divided into

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different groups where each group is a particular geometry of the lap joint. For all statistical tables SS, MS and df denotes sum of squares of deviation, mean of sum of squares of deviation {= SS/df} and degree of freedom respectively. In general, within a group, mean square variability represents the experimental error. F is a ratio {= (factor variation effect + error)/(factor variation effect)} which when compared to a critical value obtained from a standard statistical table for a chosen significance level, provides statistical significance of the selected factor. If F value obtained for a factor, from the experimental data is higher than the critical value of F, we can interpret that there is a significant effect of that factor on the dependent variable. A significance level of 0.05 was used in the analysis, which implies that the probability of variation of the dependent variable (dynamic shear strength) due to chance alone will be less than 5%. Results of the omnibus F test (Table 4) indicated a statistically significant overall effect of bonded Ls/W ratio variation, F(2,9) = 6.15, p<0.05; bonded area variation, F(2,9) = 14.14, p<0.05; and an interaction between Ls/W and bonded area, F(4,9) = 4.02, p<0.05. A disordinal interaction for both the factors implied that the main effects of Ls/W and bonded area couldn't be independently interpreted. Statistically determined simple effects of Ls/W and area are shown in Tables 5 and 6 respectively. Significant simple effect of Ls/W was obtained at the areas of 130 F(2,9) = 13.83, p<0.05 and 440 F(2,9) = 7.59, p<0.05. A significant simple effect of bonded area was obtained at Ls/W ratio of 1.4, F(2,9) = 11.16, p<0.05.

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In general, for a given length (Ls), if the width (W) of a lap joint decreases, the area subject to high stress concentration at the edges decreases, resulting in higher shear strengths for narrow bonded lap joints. For a given width, a larger bonded length will cause the high stress concentration areas at the bonded edges to separate and will again result in a higher shear strength. After the length is increased more than the width the effect of overall bonded area becomes significant and as the increase in area results in more number of flaws, the shear strength decreases. It can be concluded that that in general the effect of bonded area on the dynamic shear strength of adhesive lap joints was not very significant compared to the effect of Ls/W. The significant simple effect of area, indicated by the statistical analysis, was primarily due to the high dynamic shear strength recorded for a single case in which the joint had a narrow width and Ls/W of 1.4. Figure 8 presents the dynamic shear strength of adhesive lap joints with varying

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Ls/W for all specimen geometries. Results indicate that in general the dynamic shear strength of bonded lap joints is higher for an Ls/W ratio of 0.8.

5.

Conclusions

The effect of loading rate, bonded length to width ratio and bonded area on the dynamic shear strength of adhesive lap joints was experimentally evaluated. Following is a summary of observations from this study: 1. The adhesive lap joints transmit significantly higher loads when subjected to impact loading. Dynamic shear strength of adhesive lap joints increased up to three times that of its static shear strength value for loading rates around l000N/µs and then remained constant. 2. With increasing bonded length to width ratio, the dynamic shear strength increased till a length to width ratio of 0.8 and then it decreased. Static shear strength slowly increased with increasing length to width ratio and for the values of this ratio higher than 1.0, static shear strength became constant for the range of experimental data. 3. With increasing bonded length to width ratio, the dynamic shear strength to static shear strength ratio increased from a value of 1.4 to a maximum value of over 3.0 at a length to width ratio of 0.8, after which the strength ratio decreased. The effect of lap joint geometry on the shear strength was observed to be more significant for dynamic loading compared to quasi-static loading. 4. A statistically significant main effect of bonded length to width ratio and an interaction between length to width ratio and bonded area on the dynamic shear strength of joints was observed indicating that, the effects of these two factors remains coupled and cannot be separated.

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6.

References

1.

"Standard test method for strength properties of adhesive bonds in shear by compression loading." Designation D 905-94, Annual Book of ASTM Standards. "Standard test method for apparent shear strength of single-lap-joint adhesively bonded metal specimens by tension loading." Designation D 1002-94, Annual Book of ASTM Standards. "Standard test method for impact strength of adhesive bonds". Designation D 950-94, Annual Book of ASTM Standards. Kinloch, A. J. and Kodokian, G. A., 1987, "The impact resistance of structural adhesive joints." Journal of Adhesion, Vol. 24, pp. 109-126. Harris, J. A., 1985, "An assessment of the impact performance of bonded joints for use in high energy absorbing structures." Proc. Inst Mech. Engrs, Vol. 199, No. C2. Chao, J. Yuh, Miller, K. W. and Wang, P. C., 13-16 Oct., 1998, "Impact strength of resistance spot welded joints." Sheet Metal Welding Conference, Detroit. Miller, K. W. and Chao, Y .J., 1-5 June 1998, "Performance comparison of spot welded, adhesive bonded, and self piercing rivet aluminum joints." 5th International Conference on Trends in Welding Research. Yokoyama, T., 21-24 July 1999, "Measurement of impact tensile properties of adhesive joints using the split Hopkinson bar." International Conference on Advanced Technology in Experimental Mechanics, Vol. 2, pp. 366-371. Srivastava, V., Shukla, A. and Parameswaran V., November 2000, “Experimental Evaluation of the Dynamic Shear Strength of Adhesive-Bonded Lap Joints.” Journal of Testing and Evaluation, Vol. 28, No. 6, pp. 438-442. DeVries, K. L., Williams, M. L. and Chang, M. D., March 1974, “Adhesive fracture of a lap shear joint,” Experimental Mechanics, pp. 89-97. Chen, D. and Cheng S., March 1983, "An analysis of adhesive-bonded single-lap joints." Journal of Applied Mechanics, Vol. 50, pp. 109-115. Harris, J. A. and Adams, R. D., April 1984, "Strength prediction of bonded single lap joints by non-linear finite element methods." International Journal of Adhesion and Adhesives, Vol. 4, No. 2, pp. 65-78. Keppel, G., 1991, Design and Analysis. 3rd edition, Prentice Hall, New Jersey. Kolsky, H., 1963, Stress Waves in Solids. New York: Dover Publications, Inc.

2.

3. 4. 5. 6.

7. 8. 9. 10. 11. 12.

13. 14.

Author Index Abdel-Tawah, K., 88 Abé, H., 443 Abot, J. L., 55, 56 Abrate, S., 661, 662, 663 Abu-Alrub, R. K., 109 Achenbach, J. D., 367, 368, 370, 373, 374, 375, 378, 379, 556, 557 Acker, K. V., 500 Actis, R. L., 354 Adams, R. D., 770 Agee, B. L., 555 Aggelopoulos, A., 751 Ahmad, A., 367 Ajovalasit, A., 345 Akisanya, A. R., 226 Akiyama, T., 562 Alefeld, B., 458, 463 Aleksandrov, A. D., 572, 580 Ali, A., 367 Alias, A., 682

Al-Khalil, M., 671 Allen, A. J., 467, 478 Allen, D. H., 235, 236 Allison, B. T., 757 Allison, I. M., 556 Allix, O., 109

Almroth, B. O., 545 Amagai, M., 443 Anagnostou, E. L., 757

Anastassopoulos, A., 556 Anderson, O. L., 377 Andonian, A. T., 209 Andreani, C., 478

Andre-Fouet, X., 354 Andreni, C., 467 Andresen, K., 325, 329, 332

Andrews, W. R., 561 Angelsen, B., 356 Aoki, S., 267 Araujo, A. L., 661 Arbocz, J., 544 Archer, G. L., 561 Arepalli, S., 72 Argon, A. S., 217 Armstrong, R. W., 99 Arts, T., 353 Ashby, M. F., 32, 33, 44, 55, 56, 57, 59, 61, 62, 686 Ashley, H., 612 Asundi, A., 345, 354, 355 Atrek, A., 597 Augustijn, C. H., 354 Ausman, K. D., 68, 71 Azeloglu, E. U., 353 Baba, S., 443 Babatsouli, E., 389 Baczmanski, A., 477, 479, 480, 482, 483 Baek, T. H., 586, 627 Baer, E., 44 Baets, R., 718 Bair, S., 19 Baker, A. A., 737, 738, 744 Bakis, C. E., 315, 316, 317, 318, 322 Balageas, D., 443 Banks-Sills, L., 225, 229 Barker, J., 510 Barkoula, N.-M., 175, 178, 183 Barone, S., 345 Barralier, L., 487, 492

782

AUTHOR INDEX

Barthelat, F., 75 Bartholomeusz, R., 737 Bate, S. K., 522 Bathe, K. J., 229 Bayly, P. V., 749 Bazant, Z. P., 13 Beahan, P., 238 Beaney, E. M., 467 Beavis, M., 238 Bechel, V.T., 163, 164, 165, 167, 170 Bechou, L., 449 Beckwith, S. W., 257 Beinert, J., 267 Bencher, C. D., 125 Bendsoe, M. P., 558 Benhabib, B., 443 Berg, K. R., 154 Bert, C. W., 434 Berveiller, M., 478, 479, 482 Bethune, D. S., 65 Beyers, R., 65 Bianas, A. M., 176 Biancaniello, F. S., 507 Bigelow, C. A., 562, 563 Bilby, B. A., 198 Bilkhu, S. S., 44 Binning, G., 67 Birch, R. S., 671, 672, 673, 677 Bish, L. T., 353 Bishop, J. F. W., 218 Bisplinghoff, R. L., 612 Bjelkhagen, H. I., 335 Bless, S. J., 4 Bofilios, D. A., 176 Böhme, W., 267 Boitnott, G., 3 Bonora, N., 701 Boracat, W. A., 293 Born, M., 383, 578 Bosia, F., 303 Botsis, J., 303, 305, 310 Bouchard, P. J., 522 Boulpaep, F., 553 Bourke, M. A. M., 497, 500, 501, 503, 505

Boyce, M. C., 757 Boyer, H. E., 205 Boyle, J. T., 556 Bradley, W. L., 235 Braham, C., 477, 482 Brayden, T. H., 690 Brenner, M., 609 Breuing, T. M., 109 Bridgman, P. W., 3 Briggs, A., 368 Brincker, R. 555 Brinson, H. F., 112, 235 Broberg, K. B., 276, 277 Broda, M., 528, 529 Broek, D., 561 Brown, L. M., 217 Bruck, H. A., 77 Brusch, G., 528, 529 Brush, D. O., 545 Bucharles, A., 609 Buckberry, C., 346 Buehler, F. U., 690 Bugeda, G., 750 Bullock, R. E., 154 Burger, C. P., 345 Burridge, R., 276 Buslaps, T., 527, 531 Buttitta, C., 634 Buyny, R. A., 689, 691, 695 Calver, L. E., 225 Camacho, G. T., 284 Campbell, J. D., 4 Camping, J. D., 4, 43, 46 Cantwell, W., 176 Canumalla, S., 449 Cao, H. C., 226 Cardon, A. H., 551,553, 557 Carlsson, L. A., 112, 176, 683, 684, 685, 686, 687 Carrado, A., 487, 492 Carruthers, J., 39 Carter, C. B., 66 Casas. J. R., 715, 716, 718 Case, S. W., 93, 94 Cassan, H., 609

AUTHOR INDEX

Castro, P. L., 354 Castro-Montero, A., 335 Catin, D. E., 606 Celi, L., 750 Ceretti, M., 458, 480, 483 Cervera, M., 750 Chaboche, J.-L., 111, 118 Chai, G. B., 345 Chai, H., 646 Chamis, C. C., 631, 633, 634 Chang, M. D., 770 Chao, J. Yuh, 770 Chao, P. P., 423, 428 Chao, Y. H., 562 Chao, Y. J., 245, 249, 250 Chatterjee, S., 353 Cheah, C. M., 749, 750 Chen, C.-S., 229 Chen, D. J., 354, 355 Chen, D., 770 Chen, E.-P., 13 Chen, T. Y., 345, 346 Chen, W., 34 Cheng, S., 770 Cheresh, M., 245, 251, 252, 253, 254 Chern, S. M., 701, 702, 704 Chew, S., 750 Chiang, C.-H., 147 Chiang, F. P., 353, 354, 355 Chimenti, D. E., 370, 372 Chin, G. Y., 498 Cho, S. M., 295, 296, 586 Cho, S. Y., 295, 296 Choa, Y. J., 336 Choi, S., 336, 338, 340 Choo, Y. S., 749 Chou, Y. C., 345 Choy, M., 353 Christ, R. J., Jr., 757 Christensen, R. M., 44 Christodoulou, N., 498 Chu,C. C., 218 Chu, T. C., 77, 33 Chubachi, N., 369 Chun, H.-J., 433

783

Chung, J., 56 Clausen, B., 497, 500, 503 Clough, R. W., 606 Clyne, T. W., 496, 497 Cohen, J. B., 477, 478, 492, 496 Coker, D., 275, 281, 284 Colaço, R., 56 Coleman, B. D., 88 Collins, J. D., 606 Comlekci, T., 556 Componechi, E. T., 161 Conrad, H., 75 Cook, W. H., 99 Cooley, L. A., 201 Corbett, D. H., 690 Cordes, R. D., 100 Corleto, C. R., 235 Corton, H. T., 229 Costanzi, M., 701 Cotterell, B., 209, 212, 225, 231 Cottrell, A. H., 198 Coulette, R., 443 Cousins, R. R., 55 Covell, J. W., 353, 354 Creath, K., 335 Creswell, L. L., 354 Creton, C., 235 Crews, Jr., J. H., 654 Cribier, A., 354 Crooker, T. W., 201 Cugnoni, J., 306 Culshaw, B., 303, 304 Curtis, P. T., 160, 161, 176 Curtis, S. A., 200 D’Almeida, 531 Dai, H., 69 Daigle, D. L., 33 Dakin, J., 303, 304 Dally, J. W., 335, 345, 590, 591 Daniel, I. M., 56, 60, 100, 133, 136, 138, 141, 147, 152, 161, 291, 381, 382, 396, 433, 434, 435, 538, 702, 704, 751, 753

784

AUTHOR INDEX

Danto, Y., 449 Dantz, D., 528, 529 Darrow, D. C., 690 Dat, R., 611, 613 Datta, S. K., 423 Dauskard, R. H., 125 Davies, R. M., 4 Dawes, M. G., 561 Dawicke, D. S., 562, 563 Daymond, M. R., 478, 495, 497, 500, 501, 503, 505, 528 de los Rios, E. R., 197, 198, 199, 200, 201, 202 De Muynck, E., 553 De Souza Neto, 121 De Visscher, J., 551, 556, 557, 603, 604, 605, 606 de Vries, M. S., 65 De Waele, W., 718 De Wilde, W. P., 551, 553, 556, 557, 595 Dee, A. T., 102, 103, 105, 106 Degrieck, J., 718 Dekker, C., 68 Deliktas, B., 110, 115, 117, 118, 119 deLorenzi, H. G., 561 Demosthenus, G. A., 39 Deng, H., 3, 4 Deng, X., 217, 218 Deobald, L. R., 553 Derumeaux, G., 354 DeSilva, C. N., 674 Devaux, J., 218 DeVries, K. L., 770 Dewhurst, R. J., 411, 413 Diamond, W. J., 658 Diaz, S., 718 Dillon, R. O., 367 Dinescu, D, 551, 558 Dinwoodie, J. M., 33 Dodds, R. H., Jr., 563 Doeblin, E. O., 572, 582 Don, H. S., 354, 355 Douvers, L. F. A., 558 Downing, S. W., 354

Doyle, J. F., 585, 586. 588, 589, 594 Dresselhaus, G., 66 Dresselhaus, M. S., 66 Drinko, J., 354 Drucker, D. C., 544, 546 Du, Y., 145, 146 Dual, J., 423 Dubsky, J., 458, 461, 463 Dufailly, J., 110 Dunham, W. R., 353 Dunoyer, P., 611,613 Dupont, B., 443 Durelli, A. J., 552 Dusel, K. H., 750 Dvorak, G. J., 556 Dyer, M. J.,68,71 Dysko, R. C., 353 Easterling, K. E., 32, 33 Edvardsen, T., 356 Ehr, J. R., 68, 71 Eigen, G., 757 Eigenmann, B., 521 Eisenmann, J. R., 154 Ekambaranathan, G., 480 Elber, W., 175 Ellis, J. R., 109 Ellis, R. W., 277, 279, 280 Embury, J. D., 217 Emerson, R. P., 315, 316, 317, 318, 322 Emery, A. F., 562 Emri, L, 19, 553 Enright, M. P., 716 Erdogan, F., 227, 228, 233, 234 Erf, R. K., 354, 355 Eschl, J., 750 Eshelby, J. D., 478 Espinosa, H., 75 Esquerre, J. P., 611, 613 Estes, A. C., 716 Evans, A. G., 226 Evans, R. E., 175 Even, R., 449

AUTHOR INDEX

Ewing, K. W., 245, 251, 252, 253, 254 Faber, J., 496 Facchini, M., 303 Farr, D. D., 356 Fasanella, E. L., 671, 672 Fátima, Vaz, 56 Fedorov, F. I., 435 Feiereisen, J.-P., 527, 531 Feng, Z., 622 Ferry, O., 527 Feyman, R., 580 Field, F. A., 561 Files, B. S., 72 Fillers, R. W., 19 Finckenstein, E. V., 757 Finnie, I., 55 Fischer, T. E., 75 Fitzpatrick, M., 483 Flannery, B. P., 588 Fleck, N. A., 17, 159, 160, 161, 217, 226, 228, 234 Floreen, S., 217 Follansbee, P. S., 4 Fortes, M. A., 56 Founas, M., 44 Fox, J. A., 411 Frangopol, D., 716 Frederiksen, P. S., 555 Freund, L. B., 276 Friedrich, K., 176 Friswell, M. I., 597 Fromme, P., 421, 423, 426, 428 Fu Chang, S., 674 Fuh, J. Y. H., 749, 750 Fujino, Y., 267 Fung, Y. C., 353 Gabrys, C. W., 318 Galea, S. C., 737, 746, 747 Gall, T. L., 205 Gallagher, K. P., 353 Gallegos, C., 553 Gallois, B., 75 Ganesan, V. R., 345

785

Gao, H., 13, 15, 17 Gao, Z., 96 Garcia, M. J., 354 Gaudette, G. R., 353 Gazit, S., 133, 152 Gdoutos, E. E., 55, 56, 60, 197, 750 Genta, G., 318, 322 George, D., 522 Georgiadis, H. G., 276 Gerhardt,T. D., 619, 621, 622 Gerren, R. A., 353 Gharaibeh, E. S., 716 Giaccari, P., 304 Giannis, S., 175 Gibbons, P., 71 Gibson, L. J., 32, 33, 44, 55, 56, 57, 686 Gibson, R. F., 553 Gillespie, J. W., 161 Gilletta, D., 109 Gillmore, R. S., 443 Gilyard, G. B., 611, 616 Giovanola, J. H., 267 Gmür, Th., 303 Gnäupel-Herold, T., 507, 512 Colbert, D. T., 69 Goldberg, H. A., 66 Gologanu, M., 21 Goodier, J. N., 310, 561 Goods, S. H., 217 Goree, J. G., 225 Gorman, G., 65 Gotro, J.T., 751, 753 Graff, K. F., 423 Graham, M. E., 373, 374, 375, 378, 379 Grand, C., 443 Grattan, K. T. V., 303, 304 Green, G. E., 690 Greenberg, N. L., 354 Greenfield, R., 106 Grenestedt, J. L., 683 Grossard, D. C., 757 Gsell, D., 423 Guccione, J. M., 353

786

Guduru, P. R., 285 Guemes, J. A., 718 Gullerud, A. S., 563 Gump, F., 545, 546 Gundel, D. B., 163 Guo, Z., 367, 373, 374, 375, 378, 379 Guralnick, S. A., 187, 188 Gurson, A. I., 218 Gurtin, M., 88 Gusani, N., 354 Guvenilir, A., 109 Guz, A.M., 701,704 Haas, E., 757 Haberle, J. D., 155 Haddad, Y. M., 576 Hafner, J. H., 69 Hahn, G.T., 203, 561 Hahn, H.T., 133, 152 Haldeman, B. J., 315 Hall, I W., 102, 105 Hamidzada, W. A., 381, 382 Hammond, S., 738, 739, 740 Hampton, R. H., 563 Han, B., 293, 295, 296, 297 Han, G., 336 Hancock, J. W., 217, 222 Harding, J., 4 Hardt, D. E., 757 Harjo, S., 458 Harris, J. A., 770 Harrison, J. D., 561 Harrysson, R., 32, 33 Hart, G. C., 606 Hartman, K. H., 4 Hartz, D. E., 690 Hasan, W., 277 Hasselman, T. K., 606 Hauk, V., 510 Haung, C.-H., 586 Hayakusa, K., 556 Hayden, H. W., 217 Hayes, B. S., 690 He, J., 75 He, K. Y., 619, 621, 622

AUTHOR INDEX

He, M., 683 Hecker, W. F., 574, 578 Heimdal, A., 354 Helmholdt, R. B., 480 Hendriks, M. A. N., 555 Henneke, E. G. II, 436 Henry, B. S., 217 Hentrich, K., 329, 332 Hermans, Ph., 553 Herring, H. W., 154 Herskovits, J., 661 Hertzberg, R. W., 238 Hiel, C. C., 553 Hill, K. O., 304 Hill, R., 218, 49 Hiltner, A., 44 Hinders, M. K., 422 Hirashima, R., 443 Hirschbuehler, K. R., 176 Hitchings, D., 159 Hoeks, A. P. G., 353 Hoes, K., 551, 558 Hogmoen, K., 307 Hoitsma, D., 757 Holden, T. M., 487, 500 Holmes, A. M. C., 545 Hon, K. K., 749 Hongxing, Hua, 555 Hooper, S., 671 Hopkins, W. B., 690 Hopkinson, B., 4 Hopkinson, J., 4 Horii, H., 3, 4 Hou, C.-K., 145, 146, 147, 150 Houbolt, J. C., 609, 610, 611 Houtte, P. V., 500 Hovanesian, J. D., 398 Hsiao, H. M., 100, 161, 689, 691, 695 Hsu, D. K., 44 Hu, S., 163 Hua, H., 601, 603 Huang, C. H., 661 Huang, H., 71 Huang, J. L., 68

AUTHOR INDEX

Huang, Y., 13, 15, 17, 217, 276, 283 Hubbard, R. F., 102 Huber, L., 44 Hübner, B., 329, 332 Hughes, H., 467 Hughes, T., 307 Huiskes, R., 558 Hull, D., 237, 238 Hung, M. Y. Y., 397, 398 Hunter, J. J., 354 Hunter, W. C., 353 Hutchings, M. T., 467, 478, 496 Hutchins, D. A., 411, 413 Hutchinson, J. W., 13, 15, 17, 49, 217, 218, 226, 228, 234, 646, 683 Ianno, N. J., 367 Ice, M., 75 Ichihashi, T., 65 Idsert, P.v.d., 474 Ifju, P. G., 293, 295, 296 Iijima, S., 65 Ikeda, Y., 443 Ilavsky.J., 512 Iliff, K. W., 610, 613 Im, J., 217 Im, K.-H., 444 Ingraffea, A. R., 229 Inoue, K., 556 Ioannidis, M. B., 39 Iocco, A., 305, 310 Irwin, G. R., 247, 561 Isaacs, J. B., 3, 4 Ishai, O., 133, 136, 138, 141, 152, 396, 435 Jackel, H. R., 187 Jackson, J.H., 561 Jackson, K. E., 671, 672 Jacobs, P. F., 749 Jacques, A., 527, 531 Janovec, J., 458, 460, 461, 463 Jans, H. W. J., 555 Janssen, J. D., 555

787

Jata, K. V., 217 Jenkin, C. F., 188 Jensen, J. L., 555 Jeong, H., 444 Ji, X., 573, 575 Jia, Z., 335 Jin, K., 75 Johansson, J., 498 Johnson, G. R., 99 Johnson, K. L., 55, 59, 61, 62 Jones, R., 307, 737, 738, 739, 740 Jones, R. M., 537, 540, 707, 712 Josepson, J., 574 D. B., 574 Ju, Y., 443 Kabel, J., 558 Kac, M., 573, 582 Kachanov, L. M., 91 Kagawa, A. T., 500 Kalthoff, J. F., 267 Kalukin, A. R., 443 Kamath, M. S., 561 Kaminski, B. E., 154 Kang, U., 588, 589 Kanninen, M. F., 561 Kaplunov, J., 553 Karafillis, A., 757 Karalekas, D., 749, 750, 751, 753 Kardomateas, G. A., 645, 646, 647, 652, 653, 656, 657, 658 Karger-Kocsis, J., 175,178, 183 Karlon, W. J., 354 Karpouzian, G., 612 Karpur, P., 163 Kattan, P. I., 110, 115, 117, 119 Kau, C., 44 Kavanagh, K. T., 553 Kawaguchi, Y., 562 Kaye, R., 737 Kear, B. H., 75 Kearns, K. M., 52, 53 Keene, L., 353 Keer, L. M., 225 Kehoe, M. W., 616

788

AUTHOR INDEX

Kelly, T. F.,71 Kennedy, B., 606 Keppel, G., 771 Kersey, A. D., 303 Kertesz, P., 449 Kerwin, D. P., 444 Kesselman, M., 757 Kestin, J., 576, 580 Kfouri, A. P., 229 Khanna, S. K., 277 Khrushchev, N., 545 Kiang, C. H., 65 Kies, J. A., 561 Kihara, T., 346 Kim, J. O., 368, 370, 373 Kim, K. Y., 434 Kim, P., 68 Kim, R. Y., 133, 152, 163, 167, 170 Kinloch, A. J., 770 Kinney, J. H., 109 Kiser, J. D., 444 Kishimoto, K., 267, 556 Kitagawa, H., 197, 200 Klein, V., 611 Kleiner, M., 757 Klimanek, P., 458, 460, 461, 462, 463 Kline, R. A., 434 Klintworth, J. W., 56 Knauss, W. G., 19, 259, 335 Knoell, A. C., 32 Knott, J. F., 44 Kobayashi, A. S., 552, 561, 562, 563, 586, 705 Kocks, U. F., 499 Kocsis, M., 458, 460 Kodokian, G. A., 770 Kok, J. J., 555 Kolgomorov, A. N., 572, 580 Kolsky, H., 4, 34, 267, 772, 774 Kong, J. S., 716 Konig, W., 750 Kontomitros, C., 750 Korsunsky, A. M., 495, 528 Kosmodamianskii, A.S., 701, 704

Kossovich, L., 553 Kouril, Z., 458, 461, 462, 463 Kraft, R. P., 443 Kralj, A., 19 Kramer, E. J., 235 Krawitz, A. D., 528 Kraynik, A. M., 44 Krishnaswamy, S., 277, 279 Kroemer, N., 443 Krukenkamp, I. B., 353 Kschidock, T., 458, 460, 461, 463 Kugler, H. P., 123, 127 Kuhn, T. S., 580 Kukuchek, P., 245, 251, 252, 253, 254 Kulda, J., 458 460, 464 Kunze, H. D., 4 Kuo, C. C., 235 Kushibiki, J., 369 Kutt, J., 757 Kutt, L. M., 757 Kwon, Y. M., 263 Kyriakides, S., 56 La Saponara, V., 645, 646, 647, 652, 653, 656, 657, 658 Lacabanne, M., 611, 613 Lacy, T., 671 Ladeveze, P., 109 Laerman, K. H., 552, 555, 556 Lafond, E., 443 Lagarde, A., 121, 346 Lagoudas, D. C., 157 Lamb, H., 423 Lambros, J., 276, 279 Lavernia, E. J., 75 Laville, F., 666, 668, 669 Lavoie, J. A., 154 Lavrent’ev, 572, 580 Lawler, J. S., 335, 338, 339, 341, 344 Leblond, J. B., 218 Leckie, F. A., 373 Lee, C. H., 562 Lee, H. L., 345

AUTHOR INDEX

Lee, J., 153 Lee, S. M., 689, 691, 695 Lee, S., 671 Lee, Y., 246 Lee, Y.-C., 368, 370, 373 Leevers, P. S., 225, 231 Leggatt, R. H., 522 Lehman, F., 354, 355 Lei, Z., 325 Leipholz, H. H. E., 580 Lekhnitskii, S. G., 620, 701 Lemaitre, J., 110, 111, 118, 551 Leon, G., 103 Lepoutre, F., 443 Letac, B., 354 Levers, A., 200, 201, 202 Levy-Tubiana, R., 477, 483 Li, J. H., 750 Li, Q. M., 671, 672, 673, 677, 681 Li, X., 684, 685, 686 Li, Z., 335 Liaw, P. K., 444 Librescu, L., 609, 610, 612, 613 Lieber, C. M., 68 Limberger, H., 305, 310 Lin, S. H., 246 Lind, R., 609 Lindholm, U. S., 4 Lipinski, P., 478, 479, 480, 482 Liss, K.-D., 528, 529, 531 Liu, C. T., 211, 257, 263, 264, 265 Liu, C., 276, 279, 280, 283 Liu, T., 345 Liu, X., 574 Lodini, A. 477, 480, 482, 483, 487, 492 Loeber, J. F., 423 Lokberg, O. J., 307 Lombera, G., 750 Lonborg, J. O., 33 Lorents, D. C., 66 Lorentzen, T., 497, 500, 503 Loufoua, J., 354 Lourie, O., 71

789

Love, W. J., 562 Lu, B., 307 Lu, D. X., 674 Lu, F., 34 Lu, H., 355 Lu, J. D., 66 Lu, L., 749 Lu, M., 246 Lu, X. K.,68, 71 Lukas, P., 457, 458, 459, 460, 461, 462, 463, 464 Luo, Y., 558 Luxmoore, A. R., 217 Ma, C. C., 661 Ma, F., 218 Macek, K., 458, 460, 461, 463 MacEwen, S. R., 496 Macgillivray, H. J., 467 Macherauch, E., 521 Mackenzie, A. C., 217, 222 Madan, A., 373, 374, 375, 378, 379 Magori, V., 443 Mahieux, C. A., 93, 94 Mahn, J. P., 749 Maigre, H., 267 Majumdar, B. S., 109, 163 Malik, B., 646, 647, 652, 653 Malukhin, K., 75 Malyarenko, E. V., 422 Mamalis, A. G., 39 Mammel, W. L., 498 Manolakos, D. E., 39 Manthey, W., 443 Marc, F., 449 Marchetti, M., 701 Marshall, W. C., 561 Martin, C. J., 690, 691, 695 Martin, J. E., 187 Martin, K., 373, 374, 375, 378, 379 Martin, M. T., 586 Marzocca, P., 609, 610, 612, 613 Mason, T. E., 500 Masters, J. E., 175 Matejicek, T., 512 Matikas, T. E., 163

790

AUTHOR INDEX

Matsumoto, H., 500 Matsuo, M., 443 Matsuoka, M., 562 Matsushima, H., 443 May, G. B., 563 Mayo, W. E., 75 McCulloch, A. D., 353 McGowan, J. J., 209 McGowan, P. E., 158 McIntyre, J. S., 434 McKelvie, J., 296 McKenzie, I., 737, 746, 747 McLaren, E. A., 488 McLeish, R. D., 32 McManimon, S. P., 353 McNeill, S. R., 77 McVeigh, E. R., 353 Measures, R. M., 303 Melnichuk, J., 757 Meltz, G., 304 Mendelson, A., 9 Merry, S.L., 683 Meyer, L. W., 4 Mikula, P., 457, 458, 459, 460, 461, 462, 463, 464 Mikulas, M. M., 157, 158 Miller, G. F., 416 Miller, K. J., 197, 198, 201 Miller, K. W., 770 Miller, R. A., 335 Mills, G., 487 Mills, N. J., 44 Minaire, Y., 354 Mindlin, R. D., 423, 426 Mines, R. A. W., 671, 672, 673, 677, 681, 682 Minnetyan, L., 633 Miracle, D. B., 163 Miriyala, N., 444 Mitchell, L. D., 555 Miyazawa, T., 749 Mizuta, Y., 443 Mobasher, B., 335 Moerman, W., 718 Mohammadi, J., 187 Molent, L., 737, 738

Mollenhauer, D. H., 296, 299 Moloni, K., 71 Moonan, W. K., 19, 28, 29 Moreira de Freitas, M. J., 661 Morgan, D., 769 Morreale, J., 478, 479 Morrissett, T., 246 Morton, J., 154, 176 Mota Soares, C. M., 661 Mottershead, J. E., 597 Moulton, M. J., 354 Mouzakis, D. E., 176 Movchan, M. B., 367 Moverare, J. J., 498 Mullick, S. K., 345 Murthy, P. L. N., 633, 634 Nadal, M.-H., 443 Nakagawa, O., 443 Narayan, J., 75 Nardiello, P. L., 757 Nardone, V., 105 Natke, H. G., 572, 595 Navarro, A., 197, 198, 199 Nayfeh, A. H., 370, 372 Nee, A. Y. C., 749, 750 Needleman, A., 218 Nemat-Nasser, S., 3, 4, 225 Neov, D., 458 Neubauer, C., 443 Neumaier, A., 588 Newaz, G. M., 109 Newman, J. C., 562, 563 Newman, J. C., Jr., 218 Nichols, M. C., 109 Nickolas, T., 4 Niordson, F. I., 426 Nishino, H., 443 Nishioka, T., 586 Nissim, E., 611, 616 Nix,W. D., 13, 15, 17 Noderer, K. D., 611 Nokan, S. T., 750 Norris, A. N., 427 Novak, D., 13 Noyan, I. C., 477, 478, 492, 496

AUTHOR INDEX

Nuismer, R. J., 172 Nusholtz, G. S., 44 O’Dell, W. G., 353 Oden, M., 498 Odgaard, A., 558 Odom, T. W., 68 Ogden, R. W., 3 Ogilvie, P. L., 757 Ohayon, R., 109 Ohms, C., 467, 474, 515, 516, 520, 522 Okace, B. W., 562 Okamoto, T., 500 Olsen, B. A., 757 Olssen, R., 663, 665 Omens, J. H., 354, 356 Onate, E., 750 Ono, M., 458 Onuh, S. O., 749 Oomens, C. W. J., 555, 556, 606 Ortiz, M., 275, 284 Osborn, J. C., 458, 460, 461, 463 Ousten, Y., 449 Ousterhout, K. B., 757 Overney, G., 66 Ovize, M., 354 Owen, Dr. J., 121 Ozdil, F., 176 Pagano, N. J., 163, 538 Pakalnius, E., 246 Palmer, S. B., 411, 413 Pan, J., 218, 246 Pandolfi, A., 275, 284 Pandya, K. C., 235 Pang, J. W. L., 473, 500 Pao, Y.-H., 423, 428 Papanicolaou, G. C., 175, 176, 178, 183 Papazian, J. M., 757 Papka, S. D., 56 Parameswaran, V., 769, 770 Pardoen, T., 218

791

Paris, P., 247 Park, T., 119 Parnas, R. S., 558 Partridge, I. K., 176 Pasch, K., 757 Paskaramoorthy, R., 423 Pasque, M. K., 354 Pastor, M., 87, 91 Patel, M. R., 55, Pathiraj, B., 480 Patterson, E. A., 345 Pearson, S., 197 Peck, A. N., 757 Pedersen, P., 555, 558, 661 Peimanidis, G., 133, 152 Pelegri, A. A., 646, 647, 652, 653 Perangelo, H. J., 611 Perez, A., 717, 724 Peric, D., 121 Perrin, M., 487 Pestel, E. G., 373 Peters, K., 305, 310 Peters, W. H., 77, 209, 355 Petillon, O., 443 Petit, R. G., 229 Petrucci, G., 345 Petters, W. H., 336 Phoenix, S. L., 235 Pifko, A. B., 757 Pilakoutas, K., 197 Pindera, J. T., 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 582 Pindera, M. J., 573, 574, 575, 576, 577, 578, 579 Piner, R., 71 Pinson, G. T., 757 Pipes, B. R., 112, 133, 538 Pitt, S., 737, 738, 739 Piva, A., 277 Place, B. W., 562 Planas, J., 13 Plascik, R. S., 562 Plouzennec, N., 346 Pluyette, E., 487 Popek, W.J., 381, 383, 384

792

AUTHOR INDEX

Popper, K. R., 580, 582 Post, D., 291, 292, 293, 294, 295, 296 Pournaras, A. V., 176 Powers, B. M, 102, 103, 104, 105 Prasad, S., 683 Prask, H. J., 507 Pratt, W. K., 336 Preissner, R. C., 105 Prengel, H. G., 367 Press, W. H., 588 Priesmeyer, H. G., 458, 495 Prime, M. B., 467 Prinzen, F. W., 353, 354 Proctor, E., 467 Prodan, T., 19 Profunser, D., 423 Prorok, B., 75 Pullman, D., 52, 53 Pursey, H., 416 Putnam, J. W, 690 Pyzalla, A., 527, 528, 529, 531 Quate, C. F., 67 Radaj, D., 246 Ragsdell, K. M., 597 Ramachanidran, G. N., 578, 580 Ramaseshan, S., 580 Ramesh, K., 345 Ramos, G., 718 Ramsey, J. 154 Randon, J. C., 225, 231 Ranson, W. F., 77, 336, 355 Rao, G. V., 552 Rapti, D., 750 Ravichandran, G., 31, 32, 33, 34, 40, 285 Ravindran, A., 597 Reddy, J. N., 308 Reeder, J. R., 654 Reetz, B., 527 Reifsnider, K., 87, 89, 91, 93, 94, 96, 296 Reimers, W., 528, 529 Reiner, M., 576, 580 Reklaitis, G. V., 597

Reneman, R. S., 353, 354 Reubrez, E., 478, 479 Rhee, J., 619, 621, 622 Rhodes, M. D., 158 Rice, J. R., 212, 217, 218, 225, 231 Richardson, J. R., 609 Richardson, P. T., 611 Richie, R. O., 125 Rietveld, H. M., 500 Rigby, R.,671 Riley, W. F., 335, 345, 552, 590, 591 Rinzier, A. G., 68, 69 Rittel, D., 267, 268 Rivett, R. M., 250, 254 Roberts, J., 737 Robinson, M., 39 Rodner, A. S., 345, 351 Rodopoulos, C. A., 197, 200, 201, 202 Roehm, R., 672 Rogovsky, A. J., 444 Rohrs, H., 65, 71 Rokach, I. V., 267 Rome, J., 3,4 Rooks, S. M., 443 Rosakis, A. J., 275, 276, 279, 280, 281, 283, 284, 285 Rose, J. L., 422 Rosenbaum, R., 609 Rosenfield, A. R., 203, 561 Rosengren, G. F., 218 Roth, D. J., 444 Roubertier, J., 609 Rowlands, R. E., 291, 586, 619, 621, 622, 627, 702, 704 Roy, A. K., 43, 46, 52, 53 Roy, R., 609, 611 Rubenstein, A. A., 212 Ruoff, R. S., 65, 68, 71, 72 Russo, M. I., 611 Rybicki, E. F., 561 Rytter, A., 555 Sachs, G., 467 Sachse, W., 434 Safoglu, R., 217

AUTHOR INDEX

Saini, V., 444 Saje, M., 218 Saka, M., 443 Sakata, M., 267 Salathe, R., 305, 310 Saleh, A. M., 157 Samoilis, G., 175 Samudrala, O., 276 Sandhu, J. S., 381, 382, 383, 384, 387 Sankar, B.V., 105, 106 Sankaran, V., 443 Saroun, J., 458, 460, 461, 462, 463, 464 Sata, T., 55, 57 Savin, G. N., 159, 701, 704 Savoy, R., 65 Sayir, M. B., 421, 423, 426, 428 Schapery, R. A., 121, 125, 259 Schmackers, T., 528, 529 Schmueser, D. W., 646 Scholtz, C. H., 3 Schroedl, M. A., 209 Schwartz, J., 225 Schwarz, R. C., 757 Sciammarella, C. A., 121, 127 Sciammarella, F. M., 121, 127 Scott, B., 103, 104 Scott, C. H., 354 Searcy, C. R., 236 Seferis, J. C., 690 Segalman, D. J., 622 Selvarathinam, A. S., 225 Sendlein, L. S., 683 Seol, K., 528 Shah, A. H., 423 Shah, S. P., 335, 336, 338, 340 Shao, Y., 335 Shaw, L., 163 Shaw, M. C., 55, 57 Shen, C. W., 133, 152 Shen, Z. X., 750 Sheppard, S. D., 246 Shibuya, T., 556 Shih, C. F., 561 Shih, W.-Y., 586

793

Shin, J.-Y., 433 Shockey, D. A., 267 Shukla, A., 276, 769, 770 Sifniotopoulos, C. G., 389 Sih, G. C., 227, 228, 233, 234, 423 Silva, Gomes J. F., 556 Silva, W. A., 609, 610, 612, 613 Simitses, G. J. 645 Simpson, M. L., 315 Sinclair, G. M., 187 Singer, J., 544 Singh, J., 367 Singh, R. P., 276 Singh, S., 176 Sipsha, A., 683 Sitner, P., 458 Skidmore, G. D., 68, 71 Skjaerpe, T., 354 Smalley, R. E., 68 Smiseth, O. A., 356 Smith B., 671 Smith, C. W., 209, 210, 211, 257 Smith, H. L., 561 Smith, K. C., 443 Smith, M.S., 611 Smith, P. A., 159, 160, 161, 246, 247, 248 Smith, S. A., 690 Smith, T. L., 127 Smith, W. S. 154 Smolley, R. E., 69 Sobel, L., 634 Soden, P. D., 32 Sokolnikoff, I. S., 574 Sol, H., 551, 553, 554, 555, 556, 557, 558, 595, 601, 602, 603, 606 Soley, L. E., 422 Solis Romero, J., 200 Song, Y., 674 Sorin, W. V., 306 Sotiropoulos, D. A., 389 Soutis, C., 153, 154, 155, 159, 160, 161 Spain, I. L., 66 Spinger, G. S., 133

794

AUTHOR INDEX

Spotnitz, H. M., 353 Sprauel, J. M., 487, 492 Springer, G. S., 152 Srivastava, V., 769, 770 Stacer, R. J., 123, 127 Stacey, A., 467 Starke, E. A., Jr., 217 Starrett, H. S., 540 Starrett, J. E., 4 Staudenmann, M., 427 Stavropoulos, C. D., 176, 178 Steimle, C., 123, 127 Stevens, K. K., 586

Stewart, A. S., 353 Stickforth, J., 332 Stinchcomb, W. W., 89, 96 Stirling, M. C., 353 Stock, R. S., 109 Stonesifer, R. B., 561 Stout, M. G., 277, 279, 280 Stoylen, A., 354 Straka, P., 576, 577 Stronge, W. J., 56 Strunz, P., 458, 461, 462, 463 Stuart, H. A., 576 Studer, M., 304, 305, 310 Suarez, J., 634 Subbash, G., 34 Subramanian, S., 96 Sue, H. J., 236 Sugie, E., 562

Sugihara, K., 66 Sullivan, E. V., 757 Sullivan, P. A., 750 Sumi, Y., 225 Sun, C. T., 99, 100 Sun, T., 303, 304 Suo, Z., 226, 228, 233, 234, 646 Suresh, S., 238 Sutton, M. A., 77, 217, 218, 336 Sutton, M. S., 354 Swallowe, G. M., 458, 460, 461, 463 Sweeney, H. L., 353 Swickard, S. M., 444 Swinden, K. H., 198

Szabo, B. A., 354 Szatmary, S. A., 444 Tada, H., 247 Taerwe, L., 718 Takahashi, S., 197, 200, 346 Tamaki, T., 346 Tamrakar,, 345 Tan, C. C., 750 Tan, S. C., 626 Tan, Y. S., 354, 355 Tanaka, K., 562 Tandon, G.P., 163, 164, 165, 167, 170 Tanner, A., 671 Tarasiuk, J., 480 Taylor, C. E., 345 Telegadas, H., 106 Teukolsky, S. A., 588 Thomas, J. D., 354 Thomason, P. F., 217, 218 Thompson, R., 245, 251, 252, 253, 254 Thorukpukuzki, S. V., 99 Thum, A., 574 Timke, Th., 474 Timoshenko, S. P., 187, 310 Tippur, H. V., 277, 279 Todaro, J., 353 Todeschini, P., 487 Tohmyoh, H., 443 Tomanek, D., 66 Tomblin, J., 671 Tomé, C. N., 498, 499, 501, 505 Tomita, Y., 443 Tomota, Y., 458 Topper, T. H., 187 Torp, H., 354, 356 Toth, L. E., 367 Towers, D., 346 Tracey, D. M., 217 Tregon, B., 449 Troccaz, P., 666, 668, 669 Tsai, C.-L., 133, 145, 146, 147, 152 Tsai, J., 100 Tsai, Y.-S., 145, 146,147,150

AUTHOR INDEX

Tsao, S. H., 345 Tschentscher, T., 528, 529 Tschinke, M. F., 576 Tschoegl, N. W., 19, 28, 29 Tsukahara, Y., 443 Tukey, J. W., 656 Tummala, R. R., 443 Turner, A. P. L., 496 Turner, M. J., 690 Turner, P. A., 498 Tuttle, M. E., 112, 701, 702, 704 Tvergaard, V., 217, 218 Tyan, T., 246

Udd, E., 303 Ueda, N., 443 Ullrich, H. J., 458 Umezaki, B., 346 Urheim, S., 356 Valanis, K. C., 576, 580 Valli, J., 368 Van der Voorden, W. K. L., 558 Van Hemelrijck, D., 556, 557 van Leuven, S. L., 353 Van Ratingen M., 558 Van Rietbergen B., 558 Van Vinckenroy, G., 557 Van Vlack, L. H., 488 VandenBossche, D. J., 246, 254 Vannier, M. W., 354 Vantomme, J., 556 Vautrin, A., 555, 606 Vazquez, J., 65 Veidt, M., 423 Vemula, C., 427 Vendroux, G., 355 Venema, L. C., 68 Venson., A.R., 109, 110, 115, 116, 117 Verpoest, I., 558 Vetterling, W. T., 588 Viana, G. M., 686, 687 Vinson, J. R., 99, 100, 102, 103, 104, 105, 106 Viola, E., 277

795

Vizzini, A., 671 Voloshin, A. S., 345 Von Dreele, R. B., 500, 503 Voyiadjis, G. Z., 109, 110, 115, 116, 117, 118, 119 Vrana, M., 457, 458, 459, 460, 461, 462, 463 Vural, M., 31, 32, 33, 40 Waas, A. M., 56 Waddoups, M. E., 154 Wagner, V., 457, 458, 459, 460, 461, 463, 464 Waldman, L. K., 353 Walker, C. A., 294, 296 Walker, R., 609, 611 Walsh, J. A., 217 Wang, A. L., 211 Wang, C. H., 226 Wang, D. T., 257 Wang, F. X., 563 Wang, G., 574 Wang, H., 381, 383, 384, 387 Wang, J. S., 226, 228, 233 Wang, K.-A., 56, 60 Wang, P. C., 770 Wang, Q., 354, 355 Wang, S. S., 229 Wang, T. M., 751, 753 Wang, W. P., 487 Wang, W. V., 235 Wang, W., 276, 283 Wang, Z. F., 345 Wardle, M.W., 154 Wardle, M., 103, 104 Warren, B. E., 477, 478 Warren, W. E., 44 Warrier, S. G., 163 Washiyama, J., 235 Webster, G. A., 467, 470, 487 Weeks, C. R., 99 Wehner, T., 246 Weisbrod, G., 268 Weissenburger, J. T., 609, 611 Weissman, E. M., 294 Weitsman, Y. J., 88

796

AUTHOR INDEX

Welch, D.E., 315 Weller, T., 544 Wells, K.E., 528, 561 Wen, B., 574 Wenk, H.-R., 499 Werkstofftech, M. U., 521 Wetzel, R.M., 187 Whitehead, S., 746, 747 Whiteside, J. B., 702, 704 Whitney, J. M., 133, 172 Widsor, C. G., 478 Wiedemann, B., 750 Wierzbanowski, K., 477, 478, 479, 480, 482 Wiesendanger, R., 67, 68 Wildoer, J. W. G., 68 Wilhelm, M. A., 690 Wilkins, D. J., 161 Williams, D. B., 66 Williams, J. F., 738, 739, 740 Williams, J. G., 235 Williams, J. J., 235 Williams, M. L., 770 Willis, J. R., 512 Wilsea, M., 55, 59, 61, 62 Wimpory, R. C., 467 Windsor, C. G., 467, 496 Winkler, S., 267 Wisnom, M. R., 154 Withers, P. J., 473, 478, 496, 497 Woldensenbet, E., 100, 105 Wolf, E., 383, 578 Wolfe, D. E., 367 Wood, E. O., 4 Woodcock, R., 666, 668, 669 Wooh, S. C., 133, 152,381, 382, 409, 412, 415, 416, 417, 418, 420, 434 Woyak, D. B., 622 Wu, S., 246 Wu, Z., 75 Wung, P., 246 Wykes, C., 307 Wyman, B. T., 353

Xiao, S., 99, 100 Xu, Y. L., 96 Yakobson, B. I., 66, 72 Yamanaka, K., 443 Yang, A. C. M., 235 Yaniv, G., 133, 152 Yates, J. R., 197, 202 Yau, J. F., 229 Yeakley, L. M., 4 Yoder, G. R., 201 Yokoyama, T., 770 Youtsos, A. G., 467, 474, 515, 516, 520 Yu, C., 275, 284 Yu, M.-F., 65, 68, 71, 72 Yu, N., 444 Yuan, Q., 176 Zacharopoulos, D., 750 Zattarin, P., 478, 479 Zehnder, A. T., 285, 572, 573 Zener, C., 663 Zeng, S., 690 Zeng, X. H., 498 Zenkert, D., 683 Zerilli, F. J., 99 Zhang, S., 246 Zhong, W., 66 Zhou, B., 34 Zhou, Q., 412, 415, 416, 417, 418, 420 Zhu, H. X., 44 Zhu, H., 557 Ziebeck, K. R. A., 467 Zienkiewicz, O. C., 595 Zimmerman, N. H., 609, 611 Zou, W., 434 Zrnik, J., 458 Zuniga, S., 246 Zuo, J., 217, 218 Zweben, C., 154 Zyczkowski, M., 574

Subject Index Acoustic microscopy, 367-380 Acoustography, 381-388

Crack problems, 209-216, 22-266, 275-288

Adhesive bonds, 225-234

Crack tip opening angle (CTOA), 561

Aircrafts, 727-748

Crush mechanisms, 689-700

Bragg gratings, 303-314

C-scan, 384

Brazilian specimen, 227

Damage tolerance analysis, 206

Bridges, 715-726

Delamination, 645-660

Bulk compliance, 25

Digital image correlation, 335-344

Bulk modulus, 25

Dilatometer, 21

Carbon nanotubes, 65-74

Dimpling, 757-768

Coatings, 75-84

Dynamic fracture, 267-274

Composite materials, 87-184

Dynamic testing, 3-12, 31-42, 75-84, 99-108

com pressive strength, 153-162 crack growth, 275-288 damage, 109-120 erosion, 175-184 high strain rates, 99-108

Erosion, 175-184 Failure initiation, 217-224 Fatigue damage map, 197-208 Fatigue, 187-208

hygric characterization, 133-144

Fiber waviness, 433-442

impact, 175-184

Finite element method, 561-570, 619-628

interfacial strength, 16-174 metal matrix, 109-120 particulate, 121-132,257-266 pneumatic behavior, 146-152 state variables, 87-98

Flutter, 609-618 Foams, 31-64, 686 Fracture, 209-288 Gradient theory, 13-18

thick, 433-442

Heart problems, 353-364

toughness, 163-174

Hopkinson bar, 3-12,34

Composite structures, 631-712

Hybrid methods, 537-628

Compression test, 57, 155

Impact, 175-184, 267-274, 661-670

798

SUBJECT INDEX

Indentation, 55-64

Porosity, 123

Intersonic crack growth, 275-288

Repair, 737-748

Inverse methods, 585-608

Residual stresses, 467-476, 487-494, 507-526

Isochromatic fringes, 345-352 Kolsky bar, 4, 34, 76 Laminates, 135-144, 153-162, 303-314, 701-712

Rocket motor, 210 Rotating disks, 315-324 Sandwich structures, 671-700

Lap joints, 769-780

Scaling, 13-18

Laser ultrasonic, 409-420

Scattering analysis, 421-432

Mechanical properties, 3-84

Shear relaxation modulus, 23

dynamic, 3-12, 31-42,75-84

Shear test, 43-54

foams, 43-54

Shearography, 397-408

nanomaterials, 75-84

Shrinkage strains, 749-756

static, 75-84

Shrinkage, 751

Microscale plasticity, 13-18

Smart Structures, 737-748

Microstresses, 478-486

Speckle metrology, 75-84, 353-364

Moiré interferometry, 291-302

Spot welding, 245-256

Moiré, 561-570, 753

Stereolithography, 749-756

Nanomaterials, 75-84 Nanotubes, 75-84

Stress intensity factors, 209-216, 230, 231, 263

Neutron diffraction, 457-526

Structural analysis, 715-780

Nondestructive evaluation, 367-454

Structural testing, 715-780

Nozzle cracks, 210

Synchrotron radiation, 527-534

Optical fiber sensors, 303-314

Tensile test, 112

Optical methods, 291-364

Thermal expansion coefficients, 27

Optoelectronic methods, 315-324

Thin films, 367-380

Perforated composites, 619-628, 701-712

Time-dependent materials, 19-30

Photoelasticity, 209-216

Triaxial compression, 3-12

Photogrammetry, 325-334

Ultrasonics, 382, 433-454

Plastic collapse, 24-256

Waves, 389, 421-432, 661-670

Plastic films, 445

Welded structures, 515-526

Time-dependent materials, 19-30