Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3

Conference Proceedings of the Society for Experimental Mechanics Series

For other titles published in this series, go to www.springer.com/series/8922

Tom Proulx Editor

Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3 Proceedings of the 2011 Annual Conference on Experimental and Applied Mechanics

Editor Tom Proulx Society for Experimental Mechanics, Inc. 7 School Street Bethel, CT 06801-1405 USA [email protected]

ISSN 2191-5644 e-ISSN 2191-5652 ISBN 978-1-4614-0212-1 e-ISBN 978-1-4614-0213-8 DOI 10.1007/978-1-4614-0213-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929868 © The Society for Experimental Mechanics, Inc. 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials represents one of eight volumes of technical papers presented at the Society for Experimental Mechanics Annual Conference & Exposition on Experimental and Applied Mechanics, held at Uncasville, Connecticut, June 13-16, 2011. The full set of proceedings also includes volumes on Dynamic Behavior of Materials, Mechanics of Biological Systems and Materials; MEMS and Nanotechnology; Optical Measurements, Modeling and, Metrology; Experimental and Applied Mechanics, Thermomechanics and Infra-Red Imaging, and Engineering Applications of Residual Stress. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics. This collection, from the “Challenges in Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials,” track was organized by: Richard B. Hall, Air Force Research Laboratory; H. Jerry Qi, University of Colorado; Peter Ifju, University of Florida; Gyaneshar P. Tandon, University of Dayton Research Institute; Bonnie R. Antoun, Sandia National Laboratories; Hongbing Lu, University of Texas-Dallas; Y. Charles Lu, University of Kentucky The papers in this volume address constitutive, time (rate)-dependent constitutive and fracture/ failure behavior of a broad range of materials systems, including prominent researchers in both applied and experimental mechanics. Solicited papers involve non-negligible time-dependent mechanical response in cases incorporating non-mechanical fields. Papers in the following general technical research areas are included:  Effects of interfaces and interphases on the time-dependent behaviors of composite, hybrid and multifunctional materials  Effects of inhomogeneities on the time-dependent behaviors of metallic and polymeric materials

vi

    

Environmental and reactive property change effects on thermomechanical and multifunctional behaviors Challenges in time-dependent behavior modeling in metallic and polymeric materials at low, moderate and high strain rates Challenges in time-dependent behavior modeling in composite, hybrid and multifunctional materials - effects of strain rate and damage Modeling and characterization of fabrication processes of conventional and multifunctional materials Time dependent behaviors at the nanoscale

The track organizers thank the authors, presenters, organizers and session chairs for their participation and contribution to this track. We are grateful to the SEM TD chairs who cosponsored and organized sessions in this track. The opinions expressed herein are those of the individual authors and not necessarily those of the Society for Experimental Mechanics, Inc. Bethel, Connecticut

Dr. Thomas Proulx Society for Experimental Mechanics, Inc

Contents

1

Using Remendable Polymers for Aerospace Composite Structures T. Duenas, J. VanderVennet, A. Jha, K. Chai, NextGen Aeronautics Inc.; C. Nielsen, University of California, San Diego; A.J. Ayorinde, Sade Inc.; A. Mal, University of California, Los Angeles

1

2

Study of the Effect of Interface Enthalpy on Nanocomposite Viscoelasticity B. Natarajan, M.Krein, C.M. Breneman, Rensselaer Polytechnic Institute; L.C. Brinson, Northwestern University; L.S. Schadler, Rensselaer Polytechnic Institute

7

3

Sterilization Effect on Structure, Thermal and Time-dependent Properties of Polyamides G. Kubyshkina, Elektromaterial Lendava d.d.; B. Zupanči č, M. Štukelj, D. Groš elj, L. Marion, I. Emri, University of Ljubljana

11

4

Microstructural Evolution of Nafion During Uniaxial Deformation Monitored by X-ray Scattering M.N. Silberstein, Massachusetts Institute of Technology; J.D. Londono, Dupont Experimental Station; M.C. Boyce, Massachusetts Institute of Technology

21

5

Coupled Non-fickian Diffusion and Large Deformation of Hydrogels H. Lee, University of Illinois at Urbana-Champaign/Massachusetts Institute of Technology; J. Zhang, Arizona State University; J. Lu, J.G. Georgiadis, University of Illinois at Urbana-Champaign; H. Jiang, Arizona State University; N. Fang, Massachusetts Institute of Technology

25

6

Accelerated Testing Methods for Oxidative Aging of Polymeric Composites N. An, K.V Pochiraju, Stevens Institute of Technology; G.P. Tandon, U.S. Air Force Research Laboratory/University of Dayton Research Institute

29

7

Degradation of Shape Memory Polymer Due to Water and Diesel Fuels M.N.H. Nahid, M.A. Wahab, K. Lian, Louisiana State University

37

8

Structural Enhancement of Framing Members Using Polyurea D.J. Alldredge, J.A. Gilbert, H. Toutanji, University of Alabama in Huntsville; T. Lavin, Soems, Inc.; M. Balasubramanyam, University of Alabama in Huntsville

49

9

Experiments and Models for the Time Dependent Mechanics of Nanoscale Polymeric Structures and Nanocrystalline Metal Films I. Chasiotis, University of Illinois at Urbana-Champaign

65

viii 10

Study of Damage Evolution in High Strength Al Alloys Using X-ray Tomography H. Jin, W.-Y. Lu, A. Mota, J. Foulk, Sandia National Laboratories; G. Johnson, University of California; N. Yang, J. Korellis, Sandia National Laboratories

11

Corrosion Behavior of SS 304 with Ball Milling and Electrolytic Plasma Treatment in NaCl Solution J. Liang, M.A. Wahab, S.M Guo, Louisiana State University

12

Experimental Studies of Phase Transformation in Shape Memory Alloys K. Kim, S. Daly, University of Michigan

81

13

Measurement of Energy Loss in Thin Films Using Microbeam Deflection Method F.-C. Hsu, C.-J. Tong, M.-T. Lin, Y.-C. Cheng, National Chung Hsing University

89

14

Multiscale Characterization of Water-, Oil- and UV-conditioned Shape-memory Polymer Under Compression J.T. Fulcher, H.E. Karaca, University of Kentucky; G.P. Tandon, Air Force Research Laboratory/University of Dayton Research Institute; D.C. Foster, Air Force Research Laboratory; Y.C. Lu, University of Kentucky

15

Shape Memory Polymer Based Cellular Materials D. Restrepo, Purdue University; N.D. Mankame, General Motors Global Research and Development; P.D. Zavattieri, Purdue University

16

Influence of Mechanical Properties and Loading Conditions on the Recovery of Shape Memory Polymers R. Xiao, X. Chen, T.D. Nguyen, Johns Hopkins University

17

Fatigue Cycling of Shape Memory Polymer Resin A.J.W. McClung, G.P. Tandon, J.W. Baur, Air Force Research Laboratory

119

18

Higher Rate Testing of Long Fiber Filled Polypropylene S.I. Hill, P. Phillips, University of Dayton Research Institute

129

19

Characterization of Elastomeric Composite Materials for Blast Mitigation K. Schaaf, S. Nemat-Nasser, University of California, San Diego

137

20

Experimental Arrangement for Measuring the High-strain-rate Response of Polymers Under Pressures M. Alkhader, W.G. Knauss, G. Ravichandran, California Institute of Technology

21

Simulation of Impact Tests on Polycarbonate at Different Strain Rates and Temperatures J.L. Bouvard, C. Bouvard, B. Denton, M.A. Tschopp, M.F. Horstemeyer, Center for Advance Vehicular Systems

145

22

Experimental Investigation of Dynamic Mechanical Properties of Polyurea-fly Ash Composites A.V. Amirkhizi, J. Qiao, W. Nantasetphong, K. Schaaf, S. Nemat-Nasser, University of California, San Diego

149

23

Damage & Fracture of High-explosive Mock Subject to Cyclic Loading C. Liu, P.J. Rae, C.M. Cady, M.L. Lovato, Los Alamos National Laboratory

151

69

73

97

105

113

139

ix 24

An Evaluation of A Modified Iosipescu Specimen for Measurement Of Elastic-Plastic-Creep Properties of Solder Materials S. Mukherjee, A. Dasgupta, University of Maryland, College Park

25

Temperature Effect on Poisson’s Ratio of Woven Composites Y. Budhoo, Vaughn College of Aeronautics and Technology

163

26

Detection and Damage Monitoring in Composite Structures Using Piezoelectrics H.P. Konka, M.A. Wahab, K. Lian, Louisiana State University

171

27

Representative Volume Element Analysis for the Evaluation of Effective Material Properties of Fiber and Particle Loaded Composites with Different Shaped Inclusions V.K. Srivastava, Institute of Technology, Banaras Hindu University; U. Gabbert, H. Berger, Otto-von-Guericke University of Magdeburg

28

Time Dependent (Creep) Deformation of Thin Elastomers at Cold Temperature and Effective Strain Analysis of Their Laminates J.-M. Adkins, P. Majumdar, K. Reifsnider, University of South Carolina

193

29

Time and Temperature Response of Composite Overwrapped Cylinders J.T. Tzeng, U.S. Army Research Laboratory

207

30

High Temperature, Non-contact, Electro-magnetic Mechanical Apparatus for Creep Testing S. Gangireddy, J.W. Halloran, University of Michigan; Z.N. Wing, Advanced Ceramics Manufacturing

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31

Experiments and Predictions of Large Deformation and Failure in Thermomechanical Loading Environments B.R. Antoun, J.F. Dempsey, G.W. Wellman, Sandia National Laboratories

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32

Compliance Plot Analysis of Nonlinear Response of PMMA During Nanoindentation R.J. Arenz, S.J., Loyola Marymount University

225

33

An Incremental Formulation for the Linear Analysis of Viscoelastic Beams: Relaxation Differential Approach Using Generalized Variables C. Chazal, Limoges University; R. Moutou Pitti, A. Chateauneuf, Université Blaise Pascal

231

34

Modeling the Nonlinear Viscoelastic Behavior of Polyurea Using a Distortionmodified Free Volume Approach G. Chevellard, K. Ravi-Chandar, K.M. Liechti, The University of Texas at Austin

239

35

An Incremental Constitutive Law for Damaging Viscoelastic Materials O. Saifouni, R. Moutou Pitti, J.-F. Destrebecq, Université Blaise Pascal

241

36

Reliability Analysis of Mixed Mode Cracking with Viscoelastic Orthotropic Behaviour R. Moutou Pitti, A. Chateauneuf, Université Blaise Pascal; C. Chazal, Limoges University

249

37

Long-term Life Prediction of CFRP Structures Based on MMF/ATM Method Y. Miyano, M. Nakada, Kanazawa Institute of Technology; H. Cai, Xi'an Jiaotong University

257

38

Non-local Solutions to Direct and Inverse Problems in Mechanics: A Fractional Calculus Approach C.S. Drapaca, The Pennsylvania State University; S. Sivaloganathan, University of Waterloo

267

159

185

x 39

Study on Crystallinity Dependency of Creep Deformation on GFRTP of Polyoxythlene (POM) S. Somiya, K. Yamada, Keio University; T. Sakai, Metropolitan Tokyo University

40

Numerical Simulation of hot Imprint Process of Periodical Lamellar Microstructure into Polycarbonate R. Gaidys, B. Narijauskaite, A. Palevičius, G. Janušas, Kaunas University of Technology

41

Rapid Characterization of Visco-elastic Properties of Polymeric Materials Y. Kim, B. Han, University of Maryland

42

Estimation of Fatigue Life of Cortical Bone Considering Viscoelastic Properties and Damage Mechanics T. Sakai, K. Yasui, S. Wakayama, Tokyo Metropolitan University

43

Crack Initiation and Viscoplasticity in Polyethylene Joint Replacement Components J. Furmanski, Los Alamos National Laboratory; P.A. Sirimamilla, C.M. Rimnac, Case Western Reserve University

275

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289 295

Using Remendable Polymers for Aerospace Composite Structures Dr. Terrisa Duenas [email protected] NextGen Aeronautics, Inc. 2780 Skypark Drive Suite 400 Torrance, California 90505 Ms. Jennifer VanderVennet, Dr. Akhilesh Jha, and Ms. Karen Chai NextGen Aeronautics, Inc. Christian Nelsen University of California, San Diego Dr. A. John Ayorinde Sade Inc. Dr. Ajit Mal University of California, Los Angeles ABSTRACT The work described in this research focuses on the development of a single-component remendable polymer suitable for integration into aerospace composite structures. The structural polymer can be used as a replacement matrix material in fiberreinforced composites and allows for in-situ site-specific healing of delamination and matrix cracking when combined with a small volume fraction of heat-assisting materials such as magnetic particles. Whereas previous studies focused on the optimal volume fraction and composition of magnetic particles, and the healing of cracks in carbon-fiber composite coupons, current studies focus on the barriers to adopt this material in a aircraft manufacturing environment. These barriers include the (1) the extensive labor involved in producing a limited quantity of Mendomer (1-3 grams), (2) the evolution and entrapment of voids for all but exquisitely controlled environments, and (3) the low melting temperature (ca. 125°C) of the material when compared against high-temperature matrix systems. Proprietary steps have been formulated to increase raw material yield, reduce viscosity, and increase the glass transition temperature. Interests have also been motivated by reducing costs and adhering to conventional composite fabrication techniques. Research has also involved the investigation of automated damage detection to locate the site of damage for further healing, followed by automated healing since the ultimate goal of this research is to develop an autonomously healing composite system. However, a remendable composite is in itself valuable where its successful incorporation will reduce the need for part replacement and maintenance as well as increase the longevity and reliability of the structure. MOTIVATION Microcracking, followed by delamination caused by aging, accidental impact, and other damage events can compromise the integrity of a composite structure and lead to required maintenance or in some cases structural failure. The ongoing work presented addresses this issue by suggesting Mendomer as the matrix in fiber composites [Duenas 2006, Duenas 2009B]. Remendable materials re-heal during the application of heat and when used as a composite matrix, can heal cracks in the composite to prevent delamination and further failure. Site-specific heating can be accomplished by embedding heatassisting materials such as carbon-fiber for healing by resistive heating, or magnetic particles for healing by induction heating [Duenas 2010]. Embedded magnetic particles are suitable for non-conducting systems such as fiberglass composites. Maintenance cost savings are expected where rather than removing and replacing damaged components, remendable composites can be healed repeatedly in place. T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_1, © The Society for Experimental Mechanics, Inc. 2011

1

2 COMPARISON TO CONVENTONAL SELF-HEALING MATERIALS Passive self-healing material systems that use microcapsules are superior to remendable systems in their autonomous response to damage where in the case of remendable systems only active self-healing facilitates an autonomous response; active implies the presence of a system that is actively monitoring the composite for damage to be followed by remending. However, there are several advantages to remendable materials: (1) They have the capacity to heal repeatedly without significant change in material properties [Murphy 2008]. (2) Where as it was previously shown that remendable membranes can self-heal after bullet penetration [Duenas 2010], Mendomer may similarly self-heal during similar heat-generated damage events. (3) Mendomer also exhibits a shape memory effect (SME) where after deformation, followed by heating, the material returns to its original shape [Duenas 2006, Murphy 2008]. This observed SME may assist in self-healing where previously severed bonds return to their original positions during heating and facilitate re-linking of the polymer. The remendability and SME of Mendomer is attributed to its highly cross-linked properties where thermally reversible linkages are the result of multiple Diels-Alder (DA) cycloadditions. Mendomer is a Furan-Maleimide crosslinked solid that can be regarded as a thermoset that softens with temperature [Chen, Duenas 2006, Murphy 2008]. MENDOMER LIMITATIONS To be sure, there are significant limitations to Mendomer that limit its commercial use and range of applications. Since 2005, there has been an effort to scale up the synthesis process to larger quantities (i.e., produce 1 kg product instead of 1 gram product) by increasing yield [Murphy, Westscott-Baker]. Additional barriers exist for using this material in a fiber-composite matrix in aerospace applications including the frequent evolution of voids during processing and their subsequent entrapment, and the low glass transition temperature of Mendomer (150° C [Duenas 2006]) and associated healing temperature when compared with the operational temperature of conventional fiber-composite matrix systems. Voids appear in neat Mendomer as well as when a low-volume fraction of magnetic particles are added. The persistence of voids appears sensitive to material age, sample geometry, quantity, and processing parameters--such as temperature profile and severity of temperature excursions and whether vacuum is used during preparation. To address the non-autonomous limit of the material, detection methods [Duenas 2009C] and automated healing systems have been investigated for a commercially available remendable material (Surlyn 8940) to demonstrate the concept. VOID ENTRAPMENT AND REDUCTION Because of the repeated appearance and entrapment of voids when processing Mendomer at the aircraft manufacturing facility at NextGen with limited oven temperature control and the absence of vacuum and an inert gas supply, the team visited a university (University of California, Santa Barbara, UCSB) to process the materials there. The two samples in Figure 1 were heated for 16 hours at 140°C in Argon in a preheated oven.

5mm

MAY NOT BE TO SCALE

7mm

(a)

(b)

Figure 1. Samples fabricated by NextGen at UCSB fabricated. (a) 5mm-diameter tube leaned on its side, (b) 7-mm diameter upright tube.

3 Both were placed inside a beaker before placed into the oven. The sample in Figure 1a was prepared in a tube leaned on its side while the sample in Figure 1b was upright. Voids appeared in both samples, but fewer were visible in the 7-mm diameter sample. The samples were cured without the presence of a vacuum. Mendomer was also cured independently at another university (University of California, San Diego, UCSD) as shown in Figure 2. Approximately 200mg of Mendomer was polymerized in a glass vial. The vial was heated to 175 °C in a silicon oil bath over a period of approximately 20 minutes. High vacuum was applied to the vial while the monomer melted to remove trapped bubbles. The final sample is a transparent orange solid with no observable voids.

25 °C

120 °C

124 °C

130 °C

143 °C

172 °C

10 mm

Figure 2. Polymerization of Mendomer. In an effort to attribute void content to material composition and the presence of impurities when compared, Differential Scanning Calorimetry (DSC) tests were performed independently at different entities (UCLA, Sade, Inc., and UCSB) and compared to previously published results. A subset of these results is provided in Table 1. 7DEOH0HQGRPHUSURSHUWLHVDVJDWKHUHGE\VHYHUDO'6&SORWV '6&3ORW 7Pƒ&  (QWKDOS\-J  7Fƒ&  (QWKDOS\-J  >0XUSK\@ ƒ&  ƒ&                 The differences in the melting temperature, enthalpy and polymerization temperature differed from published results, but were credited to systematic error and not attributed to the purity of Mendomer. The differences in DSC peaks are attributed to the difference in heating rates of Mendomer during the independent DSC analyses. MENDOMER IMPROVEMENTS The current synthetic method for the Thiele’s acid and the Mendomer 400 monomers only result in a yield of 53 percent and 38 percent respectively. This translates to a 20 percent overall yield for Mendomer 400. Obviously this level of yield is low; but effort recently conducted by one of our team members suggests that another method of synthesis could provide an overall yield close to 90 percent. This method would ultimately replace the current process and make the commercial production of the Mendomer family of resin a realty. The basic technology behind the remending ability of the Mendomer is not in question. The ability to use the monomer in a manufacturing environment, however, must be addressed. Synthesis of a low viscosity version of the monomer is straight forward to facilitate in processes such as filament winding and resin transfer molding. A highly viscous version for hot melt use such as in facilitating carbon, glass and aramid fibers impregnation is also possible because the healing mechanism of the Mendomer is not based on its ability to melt such as the common thermoplastic material but the fundamental re-arrangement of the backbone by heat. The current low glass transition temperature of Mendomer is part due to low crosslink density. Increasing the service temperature is commonly achieved by replacing the aliphatic moiety with phenyl radicals. For instance the 1, 4-butanediol could be replaced with the likes of alpha,alpha'-Dichloro-p-xylene which should provide a higher glass transition temperature. There numerous other ways to increase the Tg of the Mendomer that are being considered at this time. PROGRESS TOWARDS ACTIVE SELF-HEALING

4 A feasibility study was conducted to investigate the potential of an automatically healable carbon-fiber composite structure. In this study, carbon-fiber composite coupons were fabricated with embedded disks made of a commercially available remendable ionomeric material due to the limited quantities of Mendomer available. The composite coupons were a sandwich structure consisting of 2x2 twill woven carbon fiber plies, a traditional 2-part resin epoxy (2/3 Unibond 1070 and 1/3 601 hardener), and Surlyn 8940 (ca. 0.33mm thick in a disk shape). This sandwich structure involved 3 iterations, totaling 4 carbon-fiber plies and three ionomeric remendable disks. The sandwich structure is shown below in Figure3a, and final composite panel in Figure 3b.

a)

b)

Figure 3. a) Sandwich layup for remendable carbon-fiber coupon and b)the final 4inchx4inch carbon fiber composite with embedded disk-shaped remendable material. Characterization of the composite was implemented by use of a non-destructive evaluation (NDE), ultrasonic guided wave method using PZT transducers. The testing process included 1) characterizing the Surlyn embedded coupons after fabrication, 2) damaging the coupons, 3) re-assessing the coupons, 4) healing the samples via heat or induction heat, and 5) re-examining the coupons to determine the amount of healing. It was seen that after applying a 5 ft-lb point load and subsequent healing cycle, the ionomer-embedded composite panel regained an average of 10% of the material’s dynamic wave propagating properties. For this study, healing was initiated using direct heat as applied using a powder coating oven. Figure 4 shows the different signals obtained from a pure Surlyn sample using the ultrasonic guided wave method before damage, after damage, and after healing. The source wave propagated (not shown) for the test performed in Figure 4 had frequency of 250 MHz and amplitude of 10V.

Figure 4. Results of the wave propagation tests performed on carbon-fiber panels embedded with pure Surlyn disks: before damage, after damage, and after healing to determine the amount of signal recovery obtained from the healing cycle.

5 The design metrics of this approach of nondestructively examining a carbon-fiber panel pre- and post-damage, along with those of the healing cycle are potentially automatable. To most conveniently accomplish this automation, the same system that identifies the presence of damage will also deliver healing to the location of damage. By converging the two functions of damage detection and panel healing in one package, locating and addressing damage are streamlined and maintenance turnover rate reduced. CONCLUSIONS As mentioned there are several advantages to Mendomer: (1) It has the capacity to heal repeatedly with minimal change to material properties, though the number of allowable re-healing events before material degradation must be further investigated and quantified. (2) Where as it was previously shown that remendable membranes can self-heal after bullet penetration, Mendomer may similarly self-heal during similar heat-generated damage events. (3) Mendomer exhibits shape memory properties where after deformation, followed by heating, the material returns to its original shape. The authors have observed a vice versa effect in shape-memory polymers (SMPs) where SMPs exhibit self-healing properties, but this must be further investigated. This observed SMP effect may assist in self-healing where previously severed bonds, return to their original positions during heating and facilitate re-linking of the polymer. Several disadvantages were articulated including the limited availability of the material, evolution and entrapment of voids, the low glass transition temperature of Mendomer, and the need for self-healing autonomy. While proof that void-free neat Mendomer coupons can be fabricated, the limited size and required control during processing remain outside the manufacturing requirements of aircraft composite skin fabrication. Preliminary, but proprietary steps have been formulated to increase raw material yield, reduce viscosity to minimize void evolution, and increase the glass transition temperature but were only briefly discussed. The identification of heat-generating damage events such as during certain phases of projectile flight have been investigated, but were also not been discussed. The ultimate vision of using this material is the harnessing of energy from damage events and the direction of this energy towards healing. This could be solved by first solving the problem on the macroscale using an active self-healing system as the one discussed and then ultimately scaling this active self-healing system to the material level. Multiscale modeling is expected to assist in this ultimate goal. However, while currently not an autonomous system, remendable composites provide a valuable alternative to part replacement where rather than replacing entire parts, damaged areas can be left in place and healed in-situ. REFERENCES Chen, X., et al., “New Thermally Remendable Highly Cross-Linked Polymeric Materials,” Macromolecules, Vol. 36, Pg. 1802, 2003. Duenas, T., et al., “Multifunctional Self-Healing and Morphing Composites,” Conference Proceedings for the 25th Army Science Conference, Paper No. GP-03, 2006. Duenas, T., et al., “Ballistic Missile Defense System (BMDS) Solutions Using Remendable Polymers”, SEM Annual Conference & Exposition on Experimental and Applied Mechanics, Indianapolis, Indiana, June 7 - 10, 2010, Paper No. 53. Duenas, T. et al, "Self-healing shape-memory carbon-fiber composites", 2nd Int'l Conference on Self-Healing Materials ICSHM2009. Duenas, T., et al., "Remendable Materials Using Structural Health Monitoring (SHM) to Solve Aerospace Problems, 7th International Workshop on Structural Health Monitoring, Sept. 9-11, Stanford, CA, Vol. 2, Pg. 2203, 2009. Duenas, T., et al., “Smart Self-Healing Materials Systems Using Inductive and Resistive Heating,” Smart Coatings III, American Chemical Society Symposium Series, Vol. 1050, Chapter 4, pp 45-60, Nov 10, 2010. Murphy, E.B., et al., “Synthesis and Characterization of a Single-Component Thermally Remendable Polymer Network: Staudinger and Stille Revisited," Macromolecules, Vol. 41, Pg. 5203, 2008. Murphy, E.B., "The Synthesis and Characterization of Single-Component Thermally Remendable Polymer Networks", PhD Thesis, University of California, Los Angeles, 2009. Westcott-Baker, L.A, "Synthesis and Characterization of Thermally Remendable Diels-Alder Polymers", PhD Thesis, University of California, Los Angeles, 2009.

Study of the Effect of Interface Enthalpy on Nanocomposite Viscoelasticity B. Natarajan, Rensselaer Polytechnic Institute, Troy, NY; H. Deng, D. Gai, Northwestern University, Evaston, IL; M. Krein, C. M. Breneman, Rensselaer Polytechnic Institute, Troy, NY; L.C. Brinson, Northwestern University, Evaston, IL ; L.S. Schadler, Rensselaer Polytechnic Institute, Troy, NY

ABSTRACT: Polymer Nanocomposites (PNC) are a fascinating category of material systems, in which nanoscopic fillers are found to induce remarkable improvements in the properties of the polymeric matrix.[1] This improvement is often determined by the interfacial energetic interactions which dictate the matrix chain mobility, filler dispersion, and distribution. The PNC community has widely studied the glass transition temperature (Tg) with emphasis laid primarily on the effects of confinement on the matrix polymer.[2, 3, 4] Dispersion/flocculation has also received attention in the elastomeric nanocomposite community in relation to the wetting behavior of fillers. [5] A consolidated view of both these phenomena is yet to be presented in terms of the interfacial energetics. In this study we investigate bare or short chain silane modified filler systems, in which enthalpic effects dominate. This is a first step towards understanding the effect of surface energetics on the thermomechanical properties of PNCs, such as the Tg and viscoelasticity. We subsequently seek to develop a predictive model for the same invoking an informatics based approach [6], which employs Materials Quantitative Structure-Property Relationships (MQSPR) coupled with a Finite Element Method, to link disparate length scales, reducing the need for detailed calculations. It was noted by Wang [7] that the thermodynamic driving force for flocculation is determined by the relative attraction of filler to filler ((Wc(FF))) and polymer to polymer (Wc(PP)) over the attraction of filler to polymer ((Wa(FP)) expressed as a nett work of adhesion 'Wa. The larger the value of 'Wa the worse the flocculation. This very same rationale may be used in explaining equilibrium morphologies in nanocomposites.

'Wa = Wc (FF) + Wc (PP) - 2Wa (FP)

(1)

On the other hand we hypothesize that the polymer mobility in the interphase, is dictated by the work of spreading (Ws) which is the difference between the work of adhesion ( Wa (FP) ) of the polymer to particle and the work of cohesion of the

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_2, © The Society for Experimental Mechanics, Inc. 2011

7

8 polymer to itself ( Wc (PP) ). It however does not depend on the attraction between particles. The more positive the value of Ws ( = Wa (FP) - Wc (PP) ) the larger the decrease in the mobility of the polymer and therefore more positive the Tg change. This suggests that for every polymeric matrix, there exists a threshold surface energy of the filler below which Ws is negative (Tg decreases) and above which Ws is positive (Tg increases).

b.

a.

c.

Fig.1a The schematic represents a wettability-adhesion map for Polystyrene. The X and Y axes represent the polar component and the total surface energy of the filler respectively. The curved lines represent iso-contact angle lines of the wetting species (filler) on the wetted polymer. The straight lines represent the iso-work of adhesion lines of polymer and particle 1b TEM Micrograph of 3% Loading of 3-Aminopropyldimethylethoxysilane modified particles in Monodisperse PS 190,000g/mol ('Wa=11.8 mN/m) 1c TEM Micrograph of 3% Loading of n-Butyldimethylmethoxysilane modified particles in Monodisperse PS 190,000g/mol ('Wa=8.64 mN/m). Scale bars represent 1Pm Note in Fig.1a,b,c that the larger the contact angle the poorer the wetting i.e. the poorer the dispersion. Also, the higher the Wa line, the larger the attraction of particle to polymer and therefore the more positive the change in Tg for the same dispersion state. This implies that beyond a threshold work of adhesion line the Tg always increases. Since the wetting behavior above this line may be good or bad it is possible to have poor dispersions with an increase in Tg. Likewise systems may display excellent dispersions with Tg drops, below the threshold line. The four colored regions in the graph represent all

9 possible scenarios. It must be noted that for the same Ws, the system with a better dispersion will show a larger Tg change as the surface area to volume ratio is larger. In order to confirm the aforementioned hypothesis, polymer nanocomposites with three different matrices having surface energies ranging from polar to non polar ( Poly(2-VinylPyridine), Polymethylmethacrylate and Polystyrene), filled with colloidal silica nanoparticles surface modified with four different monofunctional silanes of surface energies again varying from polar to nonpolar (amino, butyl, octadecyl, heptafluoro), are being studied. 14±4nm Colloidal Silica particles from Nissan Chemicals are functionalized by refluxing with the silanes in an anhydrous inert environment at 75ºC, overnight. The PNC samples are made by solution mixing of particles and polymer and casting the mixture. The surface energies are characterized quantitatively by static contact angle measurements on silane modified silica surfaces [8] and spin coated polymer thin films using the method outlined by van Oss et al [9]. The change in time dependent behavior is monitored by subjecting annealed samples to quantitative dynamic mechanical analysis and by measuring the glass transition temperature independently using modulated differential scanning calorimetry. The degree of nanoparticle clustering is quantified using TEM micrographs and correlated with the interface enthalpy as in Fig .1. We then invoke a finite element model of a 2D representative volume element of the PNC that includes the degree of clustering (related to the 'Wa) and a gradient of polymer mobility in the interaction zone (related to Ws). Through a comparison to experiment, the parameters used in the FEM model are correlated to the interface enthalpy. MQSPR is then employed to establish quantitative relationships between chemical properties and molecular structure represented by a set of descriptors, which capture the effects of filler functionality. This novel technique combines statistical models of the nanoscopic interactions between polymer molecules and particles, FEM and experimental data for the prediction of PNC thermomechanical properties to enable high throughput materials design. 5HIHUHQFHV

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Sterilization effect on structure, thermal and time-dependent properties of polyamides G. Kubyshkina1, B. Zupančič2, M. Štukelj3, D. Grošelj4, L. Marion4, I. Emri2 1

2

Elektromaterial Lendava d.d., Slovenia Center for Experimental Mechanics, Faculty of Mechanical Engineering, University of Ljubljana, Slovenia 3 Veterinary Faculty, University of Ljubljana, Slovenia 4 Faculty of Medicine, University of Ljubljana, Slovenia

ABSTRACT This article studies the effect of different sterilization techniques on structure, thermal and time-dependent mechanical properties of polyamide 6 materials. We used two different types of polyamide 6 material: conventional polyamide 6 with monomodal molecular mass distribution and modified with bimodal one. Samples were prepared and properly shaped in order to observe the mentioned features before and after sterilization with three different techniques, namely sterilization with autoclave, ethylene oxide, and hydrogen peroxide plasma. Optical microscopy, differential scanning calorimetry and torsional creep testing were used for the properties investigation. As a main criterion to evaluate the effect of sterilization durability of materials was used. There was observed significant difference between monomodal and bimodal PA6 materials regarding their morphology and consequently thermal and mechanical properties: bimodal material exhibits more complex structure than chemically identical monomodal one. The results of analysis did not show considerable influence on both materials structure and thermal properties. However, there was observed strong effect on creep behavior. The most significant effect is detected for monomodal material sterilized by ethylene oxide, while creep compliance of the bimodal material did not change much due to any of applied sterilization techniques. Thus, modification the complexity of the material inherent structure improves sensitivity to sterilization.

KEYWORDS: sterilization, ethylene oxide, hydrogen peroxide plasma, autoclave, polymer, polyamide, nylon, bimodal molecular mass distribution, time-dependent mechanical properties, creep compliance.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_3, © The Society for Experimental Mechanics, Inc. 2011

11

12 INTRODUCTION Nowadays the use of polymeric materials in medicine is growing steadily. Polymers occupy leading position in industry and commonly applied for medical equipment and devices, instruments, packaging, sutures etc. On top of it all, recent years’ research showed that polymers have a number of advantages for wide application in implantology branch. Compared to metals and ceramics polymeric materials provide better mechanical compatibility, allow bone growing into the implant and may be absorbed by body over time. For the most part, polymers are chemically inert, not toxic, exhibit good hydrophilic properties. Finally, they provide reasonable cost, safety, lighter weight and manufacturability. Thus, polymers are currently replacing traditional materials used for implants production. It was reported on possible use of different polymers for dental implants [1]. In this work we propose the use of polyamide 6 (PA6) as an option. This material is known to be biocompatible, exhibits good time- and temperature stability, has excellent mechanical properties and exceptional capability of moisture absorption. In addition, use of PA has already been introduced into dentistry practice in form of sutures material and dentures production. Another reason is that current progress in science allows production of so-called intelligent polyamides (I-PA) with modified initial kinetics. Such modification leads to formation of different structure and, consequently, to different macroscopic properties of final product. Thus, deliberate modification of material inherent structure allows tailoring unique properties which can prove better behavior under certain conditions than standard chemically identical materials [2]. Sterilization is essential for all implanted materials. However, currently used methods of sterilization alter properties of polymers and consequently have influence on their functionality and durability [3]. Furthermore, as sterilization affects surface properties it may change biocompatibility of the product [4] what is crucial for medical applications. Unfortunately, methods of sterilization are paid relatively little attention in respect to their effect on materials used, and existing sterilization techniques assumed to be appropriate for any product. It is, therefore, important to understand how the polymer will behave post-sterilization [5], with respect to the technique used at that. Currently the following techniques are mainly used in medical branch: γ-irradiation, dry heat, autoclave, ethylene oxide (EtO) and hydrogene peroxide (H2O2) plasma. In this paper the effect of three last mentioned methods on PA6 with monomodal and bimodal molecular mass distribution is studied. Characterization of materials was performed in respect to their structure, thermal and time-dependent mechanical properties. 1

MATERIALS

Materials tested on sterilization effect were commercially available PA6 BS400N and multimodal I-PA6, described by Emri et. al [2], both produced by BASF. Conventional PA6 has monomodal molecular mass distribution, whereas the new I-PA6 has bimodal one. The average molecular weight of both materials is roughly the same.

(a)

Fig. 1 Morphology of non-sterilized monomodal PA6 (a) and bimodal I-PA6 (b) Axioskop 2 MAT (Carl Zeiss), polarized transmitted light, 200x

(b)

13 Structure of both materials, observed by means of optical microscopy, is shown in the Fig. 1. By comparison one may infer significant difference in morphology of non-sterilized standard material and modified one, prepared by the same technology. In the case of monomodal PA6 obtained image indicates formation of coarse spherulites that can be observed as Maltese cross patterns (Fig. 1a), while bimodal I-PA6 forms homogeneous fine grain structure when exposed to the same processing conditions (Fig. 1b). From both types of polyamide materials samples were prepared to observe changes in the properties of interest after exposure to three different sterilization techniques: autoclave sterilization, EtO sterilization, and H 2O2 plasma sterilization. 2

SAMPLES PREPARATION

From both types of PA6 cylindrical samples were prepared using the equipment illustrated in Fig. 2a by means of compression molding procedure, which temperature profile is shown in Fig. 2c. PA granules were placed in a cylindrical glass tube with an inner diameter of 3mm for creep test and 6mm for structural and thermal analysis. Following this, the heater was placed on the bottom of the glass tube, and a pressure was applied from above to prevent material adhesion to the tube inner surface (Fig. 2b). The heater temperature was maintained at 270°C, and it was moved up at a relatively slow rate (2.25 mm/min), giving the pellets enough time to melt. As a consequence, the applied pressure pushed above pellets into the melt, removing any possible trapped air. After sliding over the whole tube, the heater was removed, and the specimen was left to cool down naturally to room temperature.

Fig. 2 Scheme of cylindrical rods preparation by compression molding procedure Finally, received cylindrical rods were extracted from the glass tube. For creep tests the cylinders were cut down to length 60mm in order to fix the sample for creep measurement. For optical microscopy and DSC analysis the 6mm in diameter rods were cut using the Rotary Microtome HM 355 S (MICROM International GmbH) into slices with 10 μm and 100 μm thickness, respectively. 3

METHODS OF STERILIZATION

The listed sterilization techniques were applied to obtained rod specimens and slices, which before the sterilization itself were put and sealed in sterilization bags. Steam sterilization was realized in autoclave EUROKLAV 23-VS (Melag, Germany) for 75 min at 121 °C and 0.1 MPa. After the procedure the specimens were allowed to dry for 2 hours. EtO sterilization was performed in DLOG chamber (De-Lama, Italy). The samples were sterilized for 1 hour, then they were aerated for 2 hours inside of the chamber and 10 days before further investigation.

14 H2O2 plasma sterilization was performed on STERRAD 100S system (Johnson & Johnson, USA). The operation temperature and time were 50°C and 55 min, respectively. 4

METHODS OF CHARACTERIZATION 4.1

OPTICAL MICROSCOPY

The morphology of PA samples was studied with an optical microscope Axioscop 2 MAT (Carl Zeiss, Germany), appointed with polarized light equipment. Prepared slices with thickness ~10μm were observed with polarized transmitted light under magnification of 200-x at room temperature. The analyzer was installed perpendicularly to polarizer. All the pictures were taken by AxioCam HRc digital camera with resolution 1300x1030 pixels. 4.2

DIFFERENTIAL SCANNING CALORIMETRY

Thermal properties of investigated samples were defined by DSC. All measurements were carried out on DSC7 instrument, produced by Perkin Elmer, USA, in nitrogen atmosphere at a flow rate 50ml/min. Mass of used specimens ranged from 5 to 7mg. Each test was performed at a nominal scanning rate of 20°/min in the following order: holding the sample at 0°C for 10min, heating from 0 to 280°C, holding at 280°C for 2min, cooling from 280 to 0°C, holding at 0°C for 10 min, heating from 0 to 280°C. For each specimen we performed 6 repetitions of measurement. All reported results are averaged over the number of samples examined. To determine temperature and enthalpy of crystallization the cooling curve of the first scan was used, whereas, to determine melting temperature and enthalpy of melting the second scan heating curve was used. Melting and crystallization temperatures were defined as the melting and crystallization peak temperatures, respectively. Enthalpies of transitions were obtained by integrating the area under the corresponding peaks. The materials crystallinity, c, was defined as a ratio between 0

experimental heat of fusion, ΔHm, and the theoretical value for 100% crystalline PA6 ΔH m , which is 230 J/g:

c=

4.3

ΔH m 0

ΔH m

⋅ 100, %

(Eq. 1)

TORSIONAL CREEP MEASUREMENTS

Since properties of polymers may considerably change with time, which means that functionality and stability of polymeric products may be affected over time of their exploitation, we investigated how the sterilization affects mono- and bimodal polyamide time-dependent (long term) mechanical properties [6, 7]. In order to characterize the effect of different sterilization techniques on materials’ long-term behavior, shear creep compliance measurements were performed according to the procedure described below. Shear creep measurements of investigated PA materials were performed with the Apparatus for Torsional Creep Measurements that has been designed and manufactured at the Centre for Experimental Mechanics at the University of Ljubljana, Slovenia [8 - 12]. Iitial part of the creep measuring procedure started with the annealing phase at higher temperature (110°C) that should erase mechanical stress-strain history of the material. After the annealing phase the torsional creep measurements were performed in segmental form at 8 different elevated temperatures, nominally at 30°C, 40°C, 50°C, 60°C, 70°C, 80°C, 90°C, and 100°C in the time interval of 3 hours, under loading in shear by a constant torque, Mt. The response of the material was measured in terms of rotational angle of the specimen, φ(t), as a function of time and then recalculated to the material creep compliance:

J (t ) =
(t ) r / l 0

(Eq. 2)

where r denotes radius of the circular cross section of the sample and l - length of the sample. τ0 is the magnitude of constant shear stress loading, calculated as:

15

0 = Mtr / I p

(Eq. 3)

with polar momentum Ip for circular cross section geometry of cylindrical samples. The magnitudes of applied shear stress loading were sufficiently low to ensure that the entire measurement procedure remained within the scope of the theory of linear viscoelasticity. Following the time-temperature superposition principle, measured segments were shifted along the time-scale in relation to the segment, measured at the nominal reference temperature, Tref = 50ºC. Obtained shift factors were then modelled by WLF equation in order to obtain material constants, C1 and C2. By knowing these two constants we calculated shift factors that corresponded to the shifting of master curves in order to represent the creep behaviour at the reference temperature of 37ºC. 5

RESULTS 5.1

STRERILIZATION EFFECT ON MATERIALS STRUCTURE

Regarding sterilization-related changes, no strong effect on both examined materials after applying different sterilization techniques was observed by means of optical microscopy. Sterilization processes with mentioned above conditions did not change the morphology neither for standard nor for modified PA6. However, discoloration effect took place. The color faded to a brownish hue from being exposed to each of three sterilization techniques. 5.2

STRERILIZATION EFFECT ON MATERIALS THERMAL PROPERTIES

The results of DSC measurements, i.e., melting temperature Tm, temperature of crystallization Tc, and enthalpies of corresponding transitions ΔHm and ΔHc, are presented in Tab. 1. Tab. 1 DSC measurements characteristic data for two types of PA6 material

Monomodal PA6

Bimodal PA6

Sterilization technique Non-Sterilized Autoclave EtO H2O2 plasma Non-Sterilized Autoclave EtO H2O2 plasma

TmL, °C

TmH, °C

ΔHm, J/g

Tc, °C

ΔHc, J/g

c, %

216.6 216.9 216.9 215.8

221.3 221.2 221.2 220.9 220.0 220.8 221.0 219.8

54.9 53.9 54.1 53.6 63.9 63.6 64.5 64.1

173.2 170.6 173.0 172.5 187.2 187.5 187.4 187.2

-64.7 -63.5 -63.6 -63.6 -73.8 -74.3 -75.7 -75.2

23.9 23.4 23.5 23.3 27.8 27.7 28.0 27.9

As before, first we dwell on non-sterilized monomodal and bimodal PA6 comparison first. It is evident that materials exhibit significantly different behavior during melting and crystallization. Fig. 3 shows non-isothermal melt-crystallization exotherms and subsequent melting endotherms for both materials. Clearly, a single crystallization peak was observed for each of them (Fig. 3a). However, monomodal PA6 exhibits gently sloping broad exothermal peak, while bimodal PA6 reveals sharp and lofty one. Such behavior corresponds to materials’ structure. Growth of big spherulites takes more time and heat is released bit by bit in this case, while small grains formation happens fast, and heat is released stepwise. Despite both peaks are situated in the same temperature range, crystallization temperatures, which are defined as peak temperatures, are different, being approximately 173.2°C for monomodal material and equal to 187.2°C for bimodal. The second heating scan was used for melting peaks investigation. Such ones of mono- and bimodal PA6 are presented in the Fig. 3b. One may note that bimodal material melting peak is seemingly composed of two overlapped peaks with different peak temperatures: lower TmL and higher TmH. Double melting behavior was observed on overwhelming majority of first scans as well, especially for samples with low mass. In contrast with it, standard material exhibits a single melting peak. Most probably, such double-melting behavior corresponds to two different lamellar thicknesses that appear as a consequence of bimodal molecular mass distribution. Thus, the heating curve reflects the material nonuniform structure. Also it should be also noted that bimodal material indicates approximately 4% higher crystallinity than monomodal one.

16

(a)

(b)

Fig. 3 Crystallization exotherms (a) and subsequent melting endotherms (b) for monomodal and bimodal PA6 materials To all appearance, the thermal properties of both materials were not significantly affected when samples were exposed to different sterilization methods. No additional transitions appearance due to sterilization was observed. Both materials exhibit the same crystallization and melting behaviour as in non-sterilized state. 5.3

NON-STERILIZED MONOMODAL AND BIMODAL MATERIALS COMPARISON

Performing creep tests according to the procedure described above we firstly investigated the difference in time -dependent mechanical behaviour of non-sterilized monomodal and bimodal PA6. In the concerned case it is appropriate to present the long-term creep behaviour at the reference temperature close to the temperature of a human body with respect to the proposed conditions of use. Fig. 4 shows comparison of time-dependent creep compliances for both non-sterilized PA6 materials for the reference temperature 37°C. 0,007 Tref = 37°C

mono-modal PA 6, Non-sterilized bi-modal PA 6, Non-sterilized 0,006

0,005

0,002

t ≈ 30 years

Δ J t monomodal

0,003

Δ J t bimodal

Jref (1 / MPa)

0,004

0,001

0 0

1

2

3

4

5

6 log t (s)

7

8

9

10

11

12

Fig. 4 Comparison of creep-compliances of non-sterilized monomodal PA6 and bimodal PA6 at reference temperature 37°C

17 Comparison of creep-compliances may be interpreted in terms of long-time stability for the products made of such materials. In the diagram it is clearly seen that creep compliance of both materials increase with time, but, nevertheless, creep compliance of bimodal material is lower than such of monomodal within all the time scale of experiment. From the point of view of desired life-span of dental implant it seems to be reasonable to analyse mechanical properties and their change in the time period of 30 years ≈ 109s. In considered time period it can be observed that creep compliance of bimodal material is approximately 25% lower relative to the creep compliance of monomodal PA6 (Fig. 4) which indicates better creep resistance (long-term stability) properties of bimodal PA6. As it was observed from the optical microscopy pictures above (Fig. 1), bimodal PA6 has finer structure in comparison to monomodal PA6, which can consequently lead to better creep resistance properties of bimodal material. It is evident that both materials become less stiff with time, but for monomodal material this process is much faster than for bimodal one. The absolute change in creep compliance, ΔJt, which represents the magnitude of total deformation when the implant is exposed to constant load over certain period of time, for both materials may be estimated from the diagram as well. For monomodal material this change is bigger than for bimodal one. Thus, one may summarize that bimodal PA6 creeps much less in the same time period in comparison with monomodal PA6 both in absolute and relative numbers. 5.4

STERILIZATION INFLUENCE ON MATERIALS LONG-TERM MECHANICAL PROPERTIES

After exposure of both materials to three mentioned sterilization techniques it was investigated how much the process of sterilization affected their mechanical properties. Fig. 5 presents the comparison of creep compliances of non-sterilized and sterilized monomodal PA6 materials at reference temperature 37°C, whereas Fig. 6 shows the same comparison for bimodal PA6. 0,007

mono-modal PA 6, Non-sterilized mono-modal PA 6, Ethylene Oxide (EtO) mono-modal PA 6, Steam (autoclave) mono-modal PA 6, H2O2 plasma

0,006

Tref = 37°C

Jref (1 / MPa)

0,005

0,004

0,003 non-sterilized material

0,002

t ≈ 30 years

0,001

0 0

1

2

3

4

5

6 log t (s)

7

8

9

10

11

12

Fig. 5 Creep behaviour of non-sterilized and sterilized monomodal PA6 at reference temperature 37°C From Fig. 5 one may observe all sterilization techniques have noticeable effect on long-term stability of monomodal material and all exhibit a tendency to deteriorate it at that. From the diagram it is clearly seen that EtO has the most severe effect on monomodal material time stability within whole time scale. The diagram in Fig. 6 shows that in case of bimodal PA6 sterilization effect is not significant: the creep compliance curves of sterilized material stand much less apart from the one of non-sterilized than in case of monomodal material. At the same time it is also observed that EtO has the strongest effect on both materials.

18 0,007

bi-modal PA 6, Non-sterilized bi-modal PA 6, Ethylene Oxide (EtO) bi-modal PA 6, Steam (autoclave) bi-modal PA 6, H2O2 plasma

0,006

Tref = 37°C

non-sterilized material

Jref (1 / MPa)

0,005

0,004

0,003

0,002

0,001

t ≈ 30 years

0 0

1

2

3

4

5

6 log t (s)

7

8

9

10

11

12

Fig. 6 Creep behaviour of non-sterilized and sterilized bimodal I-PA6 at reference temperature 37°C CONCLUSIONS From performed analysis of both monomodal and bimodal PA6 materials properties in non-sterilized state and after exposure to three different sterilization techniques, the following conclusions can be made: 1)

Non-sterilized bimodal material yields more complex structure (finer and more homogeneous) than non-sterilized standard PA6 material, when exposed to the same processing conditions during solidification (Fig. 1)

2)

No strong effect on the material structure was observed by means of optical microscopy after applying different sterilization methods to both materials. However, discoloration effect took place.

3)

Non-sterilized monomodal and bimodal materials exhibit significantly different crystallization and melting behaviour (Fig. 3). Both materials show single crystallization peak. Subsequent melting endotherms showed single melting peak for monomodal material and double melting peak for bimodal. Such double melting behavior of bimodal PA6 is most probably caused by presence of two populations of lamellar thickness that sequent from bimodal molecular mass distribution. Crystallinity of bimodal material is approximately 4% higher than such for monomodal material.

4)

Thermal properties of both materials were not affected when samples were exposed to different sterilization methods (Tab. 1). Both materials exhibit the same crystallization and melting behaviour as in non-sterilized state. No additional transitions were observed.

5)

Characterization of time-dependent behaviour in terms of the creep compliance has shown that in non-sterilized state bimodal PA6 has better time stability than monomodal, since it creeps less (Fig. 4).

6)

Applied sterilization techniques had more significant influence on the monomodal PA6 than on bimodal one (Fig. 5 and Fig. 6).

7)

Within the whole investigated time scale the most significant effect on creep behaviour of monomodal PA6 as well as bimodal PA6 was observed after EtO sterilization.

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Metlikovič P, Emri I. ’Naprava za merjenje lezenja torzijsko obremenjenih polimernih preizkušancev. Stroj vestn 1989; 35: 56- 58

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Metlikovič P, Emri I. ’Naprava za merjenje lezenja torzijsko obremenjenih polimernih preizkušancev’. Stroj vestn 1990; 36: 101- 104

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Coupled Non-Fickian Diffusion and Large Deformation of Hydrogels

Howon Lee1,2, Jiaping Zhang3, Jiaxi Lu1, John Georgiadis1, Hanqing Jiang3, and Nicholas Fang2* 1

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 2 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 3 Department of Mechanical Engineering, Arizona State University, Tempe, AZ 85287

ABSTRACT Solvent migration in swelling polymer shows complex behaviour, as the interface of wet (rubbery) region moves along with solvent diffusion into the dry (glassy) region, which is accompanied by local deformation. This extrinsic mechanism has led to novel three-dimensional (3D) polymer micro-actuators using direct solvent delivery via microfluidic channels. Here we present experimental techniques to quantify the non-Fickian diffusion in the swelling polymer in an attempt to predict the dynamics of local deformation in such solvent driven micro-actuators. We recorded the evolving diffusion front of solvent in poly(ethylene glycol) diacrylate (PEG-DA) hydrogel upon wetting. In order to measure diffusivity of solvent in the polymer, magnetic resonance imaging (MRI) was used. Simulation result from the theory shows good agreement with Case II non-Fickian swelling experiment. We expect that our experimental methods for such coupled diffusion and deformation will help better capture the underlying physics of hydrogel behaviour and provide fundamental basis in exploration of various hydrogel applications. INTRODUCTION Hydrogel is a network of polymer chains which allows the diffusion of solvent in the network. Upon solvent diffusion, hydrogel undergoes significant volumetric change. This unique process has become increasingly important in many applications ranging from micro-actuators [1] to tissue engineering [2] to drug delivery [3]. Recently, we reported 3D hydrogel micro-actuators driven by capillary network [4, 5]. We embedded in the device microfluidic channels through which solvent is directly supplied to specific locations for swelling (Fig. 1). Therefore, complex 3D motion of the micro device was enabled by designing capillary network, unlike other hydrogel actuators where actuation relies mostly on simple expansion and shrinkage of the whole device. Since the actuation of our device is triggered by direct supply of solvent to dry polymer, study for solvent migration into glassy polymer and associated swelling deformation is important to understand and exploit this new class of micro actuation. It is well accepted that for most glassy polymers, at temperatures far above Tg (glass transition temperature), the diffusion follows Fick’s law, where the flux increases linearly with the gradient of solvent concentration. However, near or below Tg, more complicated non-Fickian behavior is observed. One particular instance of non-Fickian diffusion is called Case II diffusion. Because of the complex physical situation including moving interface between dry (glassy) and wet (rubbery) regions and strong coupling between solvent migration and polymer deformation, the coupled Case II diffusion and large deformation has not been fully explained by a mathematical model [6]. Here we present experimental techniques to characterize physical properties of hydrogel and to quantify the non-Fickian diffusion in the swelling polymer for better prediction of the dynamics of local deformation in solvent driven micro-actuators.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_5, © The Society for Experimental Mechanics, Inc. 2011

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b

Fig. 1. Hydrogel actuator driven by embedded microfluidic channel for direct solvent delivery (a) Actuation of the device. Microfluidic channel is embedded off the center of the beam to the left side. When the tip of the channel touches solvent, solvent fills the channel via capillary force and subsequent local swelling around the channel makes the beam bend. (b) Schematic of polymer network in dry state and in swollen state

EXPERIMENTAL BACKGROUND In our model for Case II diffusion coupled with a large deformation of polymer, the polymer network is modelled as a viscoelastic material characterized by Maxwell model [7] in the large deformation theoretical framework. Swelling of polymer by solvent sorption is modelled using free energy function by Flory [8], where Flory’s interaction parameter Ȥ represents enthalpic contribution from mixing solvent and polymer. Once elastic modulus is known, Ȥ can be calculated using the experimentally measured equilibrium swelling ratio Ȝeq. The diffusion kinetics states that the flux depends on the gradient of solvent concentration and the viscoleasticity of the polymer. In our study, a phenomenological relation is used to link the diffusion coefficient and the solvent concentration by following the Thomas-Windle model [9]. Self-diffusion coefficient of solvent molecules in polymer network is strongly dependent on solvent concentration in polymer. To quantify concentration dependent diffusion coefficient of water in PEG-DA hydrogel, we used magnetic resonance imaging (MRI) [10]. Coupling between mass transport of solvent and large deformation of polymer is described by molecular incompressibility and momentum balance equation. More detailed theoretical development of the model will be described elsewhere [11]. CHARACTERIZATION OF PHYSICAL PROPERTIES OF HYDROGEL In this study, porous PEG-DA hydrogel was synthesized by mixing PEG-DA prepolymer with PEG in a weight ratio of 1:3 followed by addition of 2%wt. of photo-initiator for UV crosslinking. Not being polymerized, PEG contributes to reducing crosslinking density by occupying intermolecular space between PEG-DA during photo-polymerization, resulting in low modules and large swelling ratio. In addition, this polymer dramatically changes optical property from transparent to opaque as it swells. This facilitates visualization of the interface between dry and wet regions in experiment. Swelling ratio of the hydrogel was measured in equilibrium state. Hydrogel disks were prepared and the diameter in dry state and fully swollen state in water was measured to obtain swelling ratio Ȝeq in length. Ȝeq =1.7, which corresponds to 400% volumetric increase upon swelling. Viscoelastic parameters of the hydrogel were obtained from compression test. We applied strain at constant rate and measured stress change over time. Parameters were extracted by fitting the data to Maxwell model (Table 1). Flory’s interaction parameter obtained using viscoelastic parameters and Ȝeq is Ȥ = 0.45. Hydrogel samples with different water concentration were prepared and diffusion coefficient of water in each sample was measured by MRI. Concentration dependent diffusivity obtained by curve-fitting to Thomas-Windle model is D

D0 expad v 0.0019 u exp17.6 u v u10 9 m 2 /s

where v is water volume fraction.

27 Table 1. Viscoelastic parameters Er (MPa)

Em (MPa)

IJm (s)

Ș (×106 N·s/m2)

Ȟ

2.90

0.58

6.77

3.91

0.45

EXPERIMENT FOR COUPLED DIFFUSION AND SWELLING OF HYDROGEL Non-Fickian diffusion and associated swelling of hydrogel was quantified by directly measuring diffusion front propagation and swelling length. In Case II diffusion, there is always a sharp boundary between dry and wet regions because diffusion is limited by slow mechanical relaxation of polymer. Since our porous PEG-DA hydrogel changes optical properties upon swelling, this sharp diffusion front can be clearly visualized and easily tracked while propagation. For onedimensional (1D) diffusion experiment, a PEG-DA rod was prepared and made in contact with a water droplet on the tip (Fig. 2). Then, water starts to diffuse into the polymer network, creating a visible boundary indicating the location of diffusion front. This experiment was carried out in oil to prevent possible evaporation through the side wall of the rod, ensuring that solvent migration is primarily in the direction along the axis only. Diffusion front propagation was measured over time and compared to numerical simulation of the model using the parameters measured above. The result showed the presence of sharp diffusion front and its linear propagation trend, indicating Case II diffusion. In both diffusion front propagation and polymer deformation, simulation result agreed well with 1D swelling experiment of a PEG-DA rod. We expect that our theoretical model and experimental method for Case II diffusion will help better understand the underlying physics of hydrogel behaviour and provide fundamental basis in exploration of various hydrogel applications. a

b

c

Fig. 2 (a) 1D Case II diffusion experiment configuration (b) PEG-DA rod in initial state (left) and during experiment (right) Diffusion front propagation is measured by tracking the sharp boundary. (c) Swelling-induced deformation of hydrogel from experiment and simulation REFERENCES

 [1] Beebe, D. J. et al, Functional Hydrogel Structures for Autonomous Flow Control inside Microfluidic Channels, Nature, 404, 558, 2000 [2] Griffith, L. G. and Naughton, G., Tissue Engineering – Current Challenges and Expanding Opportunities, Science, 295, 1009, 2002 [3] Kost, J. and Langer, R., Responsive Polymeric Delivery Systems, Adv. Drug Delivery Rev., 46, 125, 2001 [4] Lee, H., Xia, C. and Fang, N. X., First Jump of Hydrogel; Actuation Speed Enhancement by Elastic Instability, Soft Matter, 6, 4342, 2010

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 [5] Xia, C., Lee, H. and Fang, N. X., Solvent-Driven Polymeric Micro Beam Device, J. Mircomech. Microeng., 20, 085030, 2010 [6] Kee, D. D., Liu, Q. and Hinestroza, J., Viscoelastic (Non-Fickian) Diffusion, Can. J. Chem. Eng., 83, 913, 2005 [7] Ward, I. M. and Sweeney, J., An Introduction to the Mechanical Properties of Solid Polymers, Wiley, 2004 [8] Flory, P. J. and Rehner, J., Statistical Mechanics of Cross-linked Polymer Networks II. Swelling, J. Chem. Phys., 11, 521, 1943 [9] Thomas, N. L. and Windle, A. H., A Theory of Case II Diffusion, Polymer, 23, 529, 1982 [10] Raguin, G. et al, Magnetic Resonance Imaging (MRI) of Water Diffusion in Hydroxyethyl Methacrylate (HEMA) Gels, Mater. Res. Soc. Symp. Proc., 930, 2006 [11] Manuscript is in preparation

Accelerated Testing Methods for Oxidative Aging of Polymeric Composites

Nan An1 and Kishore V. Pochiraju2 1

Graduate Research Assistant 2

Associate Professor Department of Mechanical Engineering Stevens Institute of Technology Hoboken, NJ 07030 USA Gyaneshwar P. Tandon3,4 3

US Air Force Research laboratory, Wright-Patterson Air Force Base, OH 45433, USA 4

University of Dayton Research Institute 300 College Park, Dayton, OH 45469-0168, USA

Abstract Controlling damage progression in oxidative environments is critical for enhancing the long-term durability of polymeric resins and composites. Traditional methods for characterization of these materials for their practical service life require several thousands of hours of isothermal aging. Therefore, there is a need for accelerated testing methods in order to reduce the prohibitive cost in this testing process. Both elevated temperatures and pressure environmental conditions can accelerate oxidative aging of the materials. This paper presents methods to characterize oxidation progress and damage growth in a polymeric matrix composite based on stress-assisted diffusion and sample miniaturization. In this approach, microscale specimens are fabricated using micro-fabrication techniques. The specimens are isothermally aged at controlled stress levels that accelerate both the oxidation and damage growth in the specimen. Coupling effects of temperature and stress on the oxidative aging are investigated based on the presented method. Due to the small scale of the specimen, the number of specimens that can be tested in parallel grows significantly. Micro-fabrication techniques also allow integration of instrumentation for measurement of the specimen response during aging, thereby reducing the additional effort, time and expense of acquiring and processing the data from the specimens during the traditional long-term aging tests.

1.

Introduction

Accelerated aging methods are needed to evaluate materials which are to be used under long-term exposure to elevated temperature in oxidative environments. The need for accelerated test methods for High Temperature Polymer Matrix Composites (HTPMC) stems from the requirements for characterization of these materials for their expected service life which can be several thousands of hours. The cost of aging these materials for this long period is often prohibitive. Thermooxidative stability of the HTPMC is typically determined in practice by weight loss behavior of the composites specimens. In contrast, Pochiraju, et al. [1, 2] consider the chemo-mechanics based mechanisms for modeling oxidation and damage growth in HTPMCs. In these efforts, oxygen diffusivity, rate of oxidation reaction and damage evolution kinetics of the materials were determined for high temperature resins and the oxidation behavior of the composite was simulated from the constituent behavior. Composite when subjected to oxidative environments absorbs the oxygen at the gas-solid interface and the dissolved oxygen diffuses deeper into the solid. The exposure and absorption at the surface is first of several interacting

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_6, © The Society for Experimental Mechanics, Inc. 2011

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30

mechanisms leading to thermo-oxidative degradation. The anisotropic diffusion and the reaction of the dissolved oxygen with the polymer substrate is the mechanism driving morphological changes. The conversion of the polymer into oxidation products is generally accompanied by oxidation-driven strain and damage evolution is the next phase of oxidative degradation. The discrete damage (crack faces) form new surfaces from which oxygen is further absorbed into the structure. Damage and oxidation layer growth are strongly coupled and lead to accelerated degradation of the material. In order to observe the long-term behavior in shorter time scales, the diffusion/reaction and damage growth behavior must be accelerated in a controlled and coupled manner. Elevated temperature and oxygen partial pressures are commonly used to accelerate the oxidation growth by speeding up diffusion and reaction behaviors. Use of stress assisted acceleration is typically complicated by the associate damage growth processes. In this paper, we examine the feasibility for miniaturization and parallelization of the oxidative aging testing, thus compressing the time required to carry out a comprehensive aging study. As the measurement of weight loss, oxidation growth and damage evolution using the methods used in previous studies [3, 4] require optical observation of morphological changes, features that assist in characterizing the oxidation and damage growth are built into the design of the specimens. The paper focuses on specimen design and fabrication techniques.

2.

Accelerating Aging and Oxidation Processes

A good accelerated test method neither introduces extraneous damage/degradation mechanisms nor omits any actual mechanisms at its use temperature. The three primary mechanisms traditionally used for accelerating aging mechanisms are elevated temperature, elevated pressure or increased partial pressure of the oxygen in the environment and stress assisted aging.

2

2.1 Elevated Temperature Aging

800 700 600 500 400 300 200 100 0

o

Air, 343 C

Weight loss/area, g/m

All thermally activated rate processes are accelerated by PMR-15 increasing the temperature. Acceleration by temperature occurs by reducing the activation energy of chemical bond rupture in the polymer macromolecule. Unfortunately, elevated temperature may o promote degradation processes that do not occur at application Argon, 343 C o temperatures. Temperature can also affect the rate of degradation by Air, 316 C Air, 288oC increasing the thermal stress in polymer composites caused by o Argon, 288 C differences in the thermal expansion coefficient of the constituents. Figure 1 shows that for PMR-15 specimens aged at 288ºC, which is 0 1000 2000 3000 4000 5000 near the use temperature, the majority of the weight loss is due to oxidation since the weight loss in an inert argon environment at this Aging Time, hrs temperature is minimal. However, for specimens aged at 343ºC, a Figure 1: Weight loss in inert and oxidative substantial weight loss percentage is attributed to the non-oxidizing environments at three temperatures thermal aging (aging in an inert argon environment). The aggressive 343ºC aging temperature is near the glass transition temperature of the material. Thermal aging in an inert environment is believed to involve chemical changes associated with chain scission reactions, additional cross-linking, or reduction of cross-link density, etc. that can result in changes in the molecular weight of the polymer, altering the physical and mechanical properties. Thus, there is likely a change in the thermal aging mechanism between the specimens aged at 288ºC and 343ºC. One therefore needs to be extremely careful in using elevated temperature for accelerated aging since the materials already operate near the Tg and one must be certain that either the aging mechanisms do not change at the elevated temperature or be able to account for changing mechanisms. 2.2 Elevated Pressure Aging The rate of oxidation is sensitive to the partial pressure of oxygen at the composite surface and acceleration can be achieved by increasing the partial pressure of oxygen within the aging chamber. Increasing the partial pressure of oxygen can be achieved in one or two ways. Firstly the partial pressure of oxygen can be increased by increasing the mole fraction of the gas in the aging chamber. For example pure oxygen ( O2 ) can be used instead of ambient air. However, the use of pure oxygen in a high temperature aging chamber may be a safety concern. Secondly the partial pressure of oxygen in air can be increased by increasing the total pressure since the partial pressure is directly proportional to the total pressure. By studying the dependence of the flexural strength of glass-reinforced epoxy resin on temperature and oxygen pressure, Ciutacu, et al. [5] demonstrated the importance of oxygen pressure as an accelerating factor in thermo-oxidative degradation. The results show that the same thermo-oxidative degradation mechanisms for glass-reinforced epoxy resin occur both in air and oxygen,

31 400 2

350

Weight loss/area, g/m

at the pressure and temperatures used. Subsequent studies conducted by Tsotsis, et al. [6, 7] demonstrate that higher pressures of air or oxygen tend to increase the rate of degradation of polymeric composites. Recent work by Tandon et al. [8] examines the use of elevated pressure in conjunction with a realistic use temperature to accelerate the rate of thermo-oxidative degradation in PMR-15 resin. The effect of aging in 0.414 MPa (60 psi) pressured air further results in nearly a two-fold increase in the rate of volume change and increases the weight loss rate of neat resin specimens by approximately a factor of two (see Fig. 2). Further, mechanical testing reveals that the specimens aged in the pressurized air environment have by far the lowest failure strain and that the strength reduction rate is large for short aging times. Since the oxidation process in PMR-15 resin is diffusion limited versus reaction rate limited, the oxidation process is accelerated leading to significant increases in mechanical property degradation.

o

PMR-15 @ 288 C

300 250

0.414 MPa, dry air

200

Ambient pressure, lab air

150 100 50 0

Ambient pressure, dry air 1000 2000 3000 4000 5000

0

Aging Time, hr Figure 2: Effect of elevated pressure on oxidation behavior in a high temperature resin

2.3 Stress-Assisted Aging

Oxidation Thickness (Pm)

The effect of mechanical stress on long-term thermal aging was investigated in NASA’s HSR program [9]. It was observed that the addition of mechanical stress has an accelerating effect on changes in the glass transition temperature in IM7/K3B composites. However, after 10,000 hours of aging at 177ºC under a constant axial load (loaded to 3000 Pin/in strain level), the axial stress applied during aging has little or no effect on aged unnotched tensile properties of quasi-isotropic lay-ups for both IM7/K3B and IM7/PETI-5 composites. To investigate the coupling effects of aging time, temperature, and stress, a lowcost pretensing fixture (inspired by the preload fixture design under the HSR program) was developed [10] that allows thermal aging of neat resin specimens under applied load at elevated temperatures. A photograph of the test fixture is shown in Fig. 3(a). The specimen is securely tightened in between the two cross-heads with the lower end kept fixed while the upper end is spring loaded to the desired stress level. The fixture assembly is then placed in the oven which is then brought up to the specified aging temperature, and the specimen is allowed to age for the specified time. Fig. 3(b) compares the total thickness of the oxidized region measured [10] in toughened polyimide neat resin at 177ºC for accelerated aging environments measured using optical methods, as discussed earlier. Similar to the case of PMR-15 resin, aging under pressure [0.414 MPa (60 psi)] results in far greater oxidation zone thicknesses than are achievable in ambient air pressure environments. Additionally, using the tension aging test fixture shown in Fig 3(b) [loaded to a stress level of 13.79 MPa (2 ksi)], results in a small increase in oxidation zone thickness at longer aging times compared to an unloaded specimen in an ambient lab air environment. The stress level of 13.79 MPa is chosen based on uniaxially tension-loaded specimens tested at (177ºC) to ensure that the mechanical response of the neat resin is limited to elastic behavior at the elevated temperature. Similar trends are observed for the mechanical behavior of the resin, with the tensile strength reducing considerably under accelerated pressure aging, while only minor deterioration is observed under stress-assisted aging when comparing the performance to ambient air pressure-aged specimens.

160 120 80

0

(a)

Ambient pressure, no load 0.414 MPa, no load Ambient pressure, 13.79 MPa load

40 0

200

400

600

800 1000 1200 1400

Aging Time, hr

(b) Figure 3: (a) The preloading fixture used to induce stress into the specimen during aging. (b) Effect of ambient pressure and load on oxidation growth for a polyimide.

32

3.

Miniaturization and Parallelization of Stress Assisted Aging Studies

As pressure and stress-assisted acceleration of oxidation and oxidation induced damage are more effective than elevated temperature acceleration, we designed a specimen and loading frame using MEMS fabrication technology. The design concept enables introducing multiple but controlled stress levels into miniaturized specimens. The design takes advantage of the coefficient of thermal expansion mismatches to pre-strain upon the polymer tensile sample. A typical sample is fabricated using photolithography and etching processes. A sample design is shown in Figure 4. A dog-bone shaped polymer specimen (shown in gray) is attached to an actuation layer (shown in blue) and the strain is induced due to the thermal expansion mismatch at the aging temperature and process induced residual strains. The magnitude of the strain is controlled by the geometry of the actuation beam and the process parameters. The test stage consists of a pair of symmetric silicon nitride actuation beams and a test specimen in the A A middle. Internal stress is generated during the Low Pressure Chemical Vapor Deposition (LPCVD) of silicon nitride on the substrate. The release of the actuation beam after attaching the test specimen induces a strain upon the test specimen. This strain can be controlled by using different l l 0 act aspect ratios of the actuation beam and sample beam. Detail A-A The pre-strain mechanism utilizes the internal stress induced from the difference in the thermal expansion coefficients of Silicon substrate and Silicon nitride during synthesis of Silicon Nitride film at high temperature (800°C) [11]. The internal stress generated is of the order of 700~800 MPa and the contraction rate of the silicon nitride actuation beam is characterized around 0.33% after release. Figure 4: Geometry of the specimen and actuation beams. The release step is carried out at room temperature and the high temperature polymer sample is not exposed to the LPCVD process temperatures. The corresponding tensile loading upon the sample is given by the following equation: ି୳ (1) ൌ ୟୡ୲ ୟୡ୲ ሺ୐బ െ ɂ୫୧ୱ ୟୡ୲ ሻ ౗ౙ౪

where ୟୡ୲ ǡ ଴ୟୡ୲ ǡ ɂ୫୧ୱ ୟୡ୲ and ୟୡ୲ are, the cross section, length, mismatch strain and Young’s modulus of the actuator beam, respectively. The displacement u after the release step can be measured using the displacement of cursors embedded in the actuator and reference fixed cursors (triangular features in silicon nitride) as shown in figure 4. The value of the estimated mismatch strain is around 0.0033 [11]. The strain of test sample ɂ can be written as: ୳ ɂ ൌ ୐బ െ ɂ୫୧ୱ (2) ୫୧ୱ with the mismatch strain for test sample of ɂ ൌ ሺȽୱ୳ୠ െ Ƚሻο. Since the interaction forces between actuator beam and sample are equal and opposite, we have: ି୳ ୳ ୫୧ୱ ୟୡ୲ ୟୡ୲ ቀ బ െ ɂ୫୧ୱ ሻ (3) ୟୡ୲ ቁ ൌ ሺ బ െ ɂ ୐౗ౙ౪

୐

where Ƚ and Ƚୱ୳ୠ are the thermal expansion coefficients of the test sample and substrate, respectively, and ο is the difference between the deposition and test temperatures. ଴ and ɂ୫୧ୱ are the initial length and mismatch strain(developed during the deposition process) of the sample, respectively, and S is the section of the test sample. From this equation, as the initial length and cross section of the actuator beam and test sample are known, then displacement u can be estimated. When the test sample is heated up to 177°C, due to the thermal expansion of both test sample and actuator beam, —୭୶ ൌ Ƚο ୭୶ ሺ଴ ൅ —ሻ ୭୶ ଴ —୭୶ (4) ୟୡ୲ ൌ  Ƚୟୡ୲ ο ሺୟୡ୲ െ —ሻ ୭୶ ୭୶ where — and —ୟୡ୲ are the displacement of the test sample and the actuator beam at 177°C, respectively, and ο ୭୶ is the temperature change from room temperature to 177°C. Then we can obtain the new length of the test sample and the actuator beam ୭୶ ᇱ ൌ  ଴ ൅ — െ ሺ—୭୶ (5) ୟୡ୲ െ — ሻ ଴ ୭୶ ᇱ ୟୡ୲ ൌ ୟୡ୲ െ — ൅ ሺ—ୟୡ୲ െ —୭୶ ሻ Also we can calculate the final strain on the test sample ୐ᇲ ି୐బ

ɂ୲୭୲ୟ୪ ൌ బ (6) ୐ By changing the length of the test sample, different strain levels can be achieved. The relative material properties for both Silicon and Silicon nitride are listed in Table 1. In the present study, three different samples are considered with details listed in Table 2.

33

Table 1 Material properties of the stress-assisted oxidation stage Si

SiNx

Polyimide

CTE (C-1)

2.49E-06

3.20E-06

44E-6

E (GPa)

112.4

260~320

4.6

T (C )

800

800

177

Table 2 Sample design for three pre-strains. All dimensions are in Pm. Test #

Actuation beam (l x w x t)

Sample ( l x w x t)

Strain

A

1000 x 200x1

200 x 50 x 100

2%

B

1000 x 200 x 1

400x 50x 100

1.3%

C

1000 x 200 x 1

600x 50x 100

0.8%

The miniaturized size of the specimens and the loading mechanism enables fabrication of multiple specimens on a single 100mm wafer. The aging process can be parallelized very economically as all the specimens are subjected to the same environmental conditions, (partial pressure and temperature). A typical 100 Pm thick specimen will oxidize in 200 hours under ambient pressure and operating temperatures. Lowering the oxygen partial pressure can reduce the oxidation rates so that the long-term damage growth can be examined in the oxidation layers. With comprehensive planning and assistance from oxidation models [1], experiments can be designed with numerous specimens to measure both the oxidation and damage growth. For each specimen with different pre-strain condition, the oxidation layer growth can be obtained by Scanning Electron Microscopy or nanoindentation techniques. Use of the pre-fabricated displacement cursors can allow real time image-based monitoring of the stiffness changes in the specimens. 3.1 Micro-Fabrication Processes The fabrication of the wafer with multiple testing specimens requires Figure 5: Typical wafer layout with a 3x3 several lithography, deposition and etching steps. The fabrication process array (3 samples x 3 Pre-strains) on a single 100 steps for the strain actuation stage are shown in Figure 6. A SOI (siliconmm wafer. on-insulator) wafer is used as the substrate. Then, symmetric thin silicon nitride actuation beams are deposited using LPCVD technique as shown in Figure 6 (b). High temperature (800 °C) is used to create the process-induced strain due to thermal expansion mismatch between the silicon nitride layer and the substrate. CHF3-based plasma is used with mask #1 to pattern the actuation beam as shown in Figure 6 (c). Then the test sample, BMI resin in the present study is machined and attached. Alternatively, the polymer can be masked, deposited and lifted-off as shown in Figure 6 (d) and (e). This method was demonstrated feasible by our experimental study. Finally, the test stage is released from the substrate by removing the sacrificial layer of silicon using hexafluoride-based RIE with mask 3 as shown in Figure 6(f). In order to demonstrate the feasibility of this process, we successfully patterned the PDMS resin with lift off process as shown in Figure 7. The PDMS pattern deposited had thickness around 40 Pm. The figure shows a 6x8 array of polymer test specimens on the wafer. By changing the mask design, the length of the specimen can be altered. The thickness of both the sample and the actuation beam can be changed by controlling the deposition process steps.

34

Figure 6: Schematic of process and corresponding masks for fabrication of the stress-assisted oxidation samples.

Figure 7: Images of patterned PDMS test specimens on a SOI wafer as realized from process shown in Fig. 6.

35

4.

Concluding Remarks

In this effort, we explore the feasibility of miniaturizing the aging test specimens using MEMS fabrication techniques and introducing controlled pre-strain into the specimens for stress assisted acceleration of the aging process. After ensuring the feasibility of the fabrication process and understanding the size and shape limitations, we considered the challenging task of attaching a HTPMC sample to the actuation beam. Work is currently in progress to realize high temperature polymers such as polyimides as the test specimens instead of the PDMS used for demonstration purposes. Machining the test samples and attaching the specimens to the substrate before the release step, and molding the test samples with the right geometry, are two options that may be feasible. While the advantages of parallelization are readily obvious, the advantages of miniaturization or length scaling need to be ascertained. Reducing the specimen thickness leads to complete oxidation of the specimen in a shorter time interval and lead to acceleration of the onset and growth of damage processes. The large scale specimens currently being used require more than 500 hours of aging before the damage related mechanisms are observed and oxidation-damage coupling becomes evident.

5.

References

[1] Pochiraju K.V., Tandon G.P. (2006) Modeling thermo-oxidation layer growth in high temperature resins. Journal of Engineering Materials and Technology 128:107–116 [2] Pochiraju, K.V., Tandon, G.P., Schoeppner, G.A. (2008) Evolution of stress and deformations in high-temperature polymer matrix composites during thermo-oxidative aging. Mech Time-Depend Mater 12: 45–68 [3] Pochiraju, K.V., Tandon, G.P. (2009) Interaction of oxidation and damage in high temperature polymeric matrix composites. Composites Part A: Applied Science and Manufacturing 40 (12), pp. 1931-1940 [4] Tandon, G.P., Ragland, W.R., Schoeppner, G.A. (2009) Using optical microscopy to monitor anisotropic oxidation growth in high-temperature polymer matrix composites. Journal of Composite Materials 43 (5), pp. 583-603 [5] Ciutacu S, Budrugeac P, Niculae I (1991) Accelerated thermal aging of glass-reinforced epoxy resin under oxygen pressure. Polymer Degradation and Stability 31:365–372 [6] Tsotsis TK, Keller S, Bardis J, Bish J (1999) Preliminary evaluation of the use of elevated pressure to accelerate thermo-oxidative aging in composites. Polymer Degradation and Stability 64:207–212 [7] Tsotsis TK, Keller S, Lee K, Bardis J, Bish J (2001) Aging of polymeric composite specimens for 5000 hours at elevated pressure and temperature. Composites Science and Technology 61:75–86 [8] Tandon GP, Ripberger ER, Schoeppner GA (2005) Accelerated aging of PMR-15 resin at elevated pressure and/or temperature. In: Proceedings of the SAMPE 2005 Symposium and Exhibition, Seattle, WA [9] Accelerated Test Methods for Durability of Composites, High Speed Research Materials Durability (Task 23), High Speed Research Program, March 1998 Treloar, L.R.G., The absorption of water by cellulose, and its dependence on applied stress ,Trans. Faraday Soc. 49 (1953), p. 816 [10] Schoeppner, G. A., Tandon, G. P., Pochiraju, K. V. (2008) Predicting Thermo-Oxidative Degradation and Performance of High Temperature Polymer Matrix Composites in Multiscale Modeling and Simulation of Composite Materials and Structures, Y. Kwon, D. Allen and R. Talreja, eds, ISBN:978-0-387-36318-9, Springer Verlag, pp 359-462. [11] Boe, A., Safi, A., Coulombier, M., Fabregue, D., Pardoen, T., Raskin, J. P. (2009) MEMS-based microstructures for nanomechanical characterization of thin films. Smart Mater. Struct. 18:115018

Degradation of Shape Memory Polymer Due to Water and Diesel Fuels

M.N.H. Nahid1, M. A. Wahab1*, Kun Lian2 1

Mechanical Engineering Dept., Louisiana State University (LSU), Baton Rouge, LA 70820, USA 2

Center for Advanced Microstructures & Devices, LSU, Baton Rouge, LA 70820, USA. *1 Corresponding Author’s E-Mail: [email protected], Tele: (1) 225-578-5823

ABSTRACT Shape-Memory-Polymers (SMP) is smart materials have the ability to memorize original shape; and can be used as sensors, transplants, and structural materials in engineering applications. Despite its practical importance, limited work is available on its degradation behavior. This study has been carried out to evaluate degradation behavior of Veriflex-SMP on exposures to water and diesel fuels separately. It is found that “glass transition temperature, Tg” decreases due to immersion in liquids; and immersion facilitates acceleration of breakage of the existing bonds and the formation of new bonds, thereby increasing the mobility of polymeric chains; and the immersion medium effectively plasticize SMPs by reducing storage modulus and decreases structural integrity of SMP. Upon exposure to diesel fuel for several weeks, “Tg” goes below room temperature; and SMP goes back to its original shape without the need of the application of external energy. Stress relaxation tests shows that stress decay is found to be much faster and to a much lower value in degraded samples and increases with increase of immersion time. From Fourier Transform Infrared tests formation of hydrogen bonding between SMP and the solvents in which it is immersed, is the main reason for SMP degradation; although hydrogen has a minor effect on SMP structure but has an obvious influence on glass transition temperature, Tg. Keywords: SMP (Shape-Memory Polymer), degradation behavior, glass transition temperature, bond formation, polymeric chains, stress relaxation, Fourier Transform Infrared test

1.0

INTRODUCTION

Shape memory polymer (SMP) is a type of shape memory material, which has been investigated in this research. The SMP is a smart material whose property can be altered in a controlled way and has the capability of recovering large deformation. Smart material has the ability of sensing, processing, actuating, self-diagnosing, and self-recovering. They can sense the change in environment and act accordingly to minimize the action of the changes in environment; similar to a living being that can sense, make decisions, and take actions. Their physical and chemical properties are sensitive to the change in environment such as pressure, temperature, electric field, pH level, magnetic field, etc. They utilize their own properties and functions to achieve smart action. This material has the ability to return from its temporary deformed shape to a permanent shape upon the application of external stimulus such as heat, i.e., it remembers its original shape. They are stimuli-responsive T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_7, © The Society for Experimental Mechanics, Inc. 2011

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materials; for example, a temperature responsive shape memory material is the one that undergoes a structural change upon reaching a certain temperature, called the “transition temperature”, where the polymer goes from a brittle or glassy state to a flexible or rubbery state. The main advantage of shape memory polymer is the ability of recovering a large amount of strain (usually >400%) in comparison to shape memory alloys (only up to 15%) and shape memory ceramics (2 - 3%) [Hu, 2007]. 1.1

Structure of Shape Memory Polymer

Generally, SMPs are phase-segregated linear block copolymers having a “hard segment” and a “soft segment” in terms of molecular structure (Fig.1). Hard segments (or crystalline segments) are responsible for maintaining the permanent or original shape through intra or inter polymeric chain attractions such as hydrogen bonding or dipole-dipole interaction together with the physical or chemical cross-linking. Whereas, the soft segments (or amorphous segments) are responsible to freely absorb external stress by unfolding and extending their molecular chains.

Hard

Fig.1: (see left): Network structure of SMP showing hard and soft segments

Soft

(i) Glass Transition Temperature, (Tg) is above the transition temperature of the soft or amorphous segment (Tgs) but below the transition temperature of the hard or crystalline segment (Tgh). At this transition temperature (Tg,,amorphous < Tg < T

g,,crystalline),

the crystalline segment will not be affected

by the temperature whereas the amorphous segment will be melted and will absorb the deformation. In the case of amorphous based shape memory polymer, when the temperature of the SMP will go below the glass transition temperature, the amorphous segments will begin to rearrange themselves but due to the presence of crystalline segment they will not be able to go back to the original shape and consequently, the deformation will be locked in. The shape memory effect not only depends on the specific material property of a single polymer but it results from the combination of the polymer structure and morphology together with the applied processing and programming technology [Hu, 2007]. It is known that the macroscopic properties of polymer can be controlled by specific variation of molecular structure. This can be used to tailor the specific properties of shape memory polymer by slight variation in molecular structure [Lendlien & Kelch, 2002]. (iii) Thermo-mechanical Programming: In order to use the shape memory functionality of SMP, we need thermomechanical programming (see Fig. 2). For programming, the shape memory polymer is heated above the transition temperature, which can be glass transition or melting transition. At this temperature the shape memory polymer is in rubbery state and it can easily be deformed i.e. it can accommodate deformation. Now at this stage, a force is applied on SMP, which induces strain in the polymer (process-1). After deforming, it is cooled below the transition temperature, at which the switching segment becomes inactive and the polymer is in frozen or glassy state (process-2). After that, the applied load has been removed. On the removal of load, the polymer recovers some of its deformation due to elasticity behavior; but most of the deformation has been locked- in and the polymer remains in the deformed state because of the decrease in temperature from above the transition to below the transition temperature (process-3). The resulting deformation creates a residual stress/strain which remains in the polymer until the recovery. Now at this stage if we heat the polymer above its glass transition temperature and it will go back to its original shape (process-4) [Lendlien & Behl, 2007]. Polymer Properties: (i) Visco-elastic Properties: Polymers are neither perfectly elastic nor perfectly viscous material, but they show deviations from idealistic behavior; such as: (i) The strain (in a solid) or the rate of strain (in a liquid) is not

39

directly proportional to stress. In fact, there exists no clear relationship; (ii) Stress may depend on both the strain and the rate of strain as well as higher time derivative of strain. Because of these deviations, polymer posses both solid as well as liquid like characteristics and are termed as visco-elastic material their relationship between stress and strain depends on time.

Original shape

Temporary/Deformed shape

Recovered/Original shape Fig. 2: Thermo-mechanical cycle of a thermally induced shape memory polymer [Lendlien & Behl, 2007] The most important visco-elastic properties of polymer are: (a) Storage Modulus (Ec), (b) Loss Modulus (Es), Tan delta (Tan G), (d) Complex Modulus (E*): and (e) Transition Temperature, Tg- As the temperature of polymer increases most of the polymer chains begin to relax and move. As a result, free volume of polymer increases significantly which is accompanied by drastic change in property. The glass transition temperature is the temperature at which the polymer undergoes a drastic change in the property such as mechanical, electrical, thermal, or other electrical property. The glass transition temperature is dependent on the degree of crystallinity of the polymer. In this study, the glass transition temperature has been determined by DMA (Dynamic Mechanical Analyzer). In order to characterize the shape memory properties of polymer a set of parameters such as (i) Strain Recovery Rate, (ii) Shape Fixity Rate, (iii) Recovery Rate, and (iv) Recovery Stress are determined. 1.2 Research Objectives: Shape memory polymers exhibit a unique combination of low density, higher recoverable strain, and shape memory behavior that gives them a wide range of potential applications. Earlier works have considered the effect of limited environmental factors on shape memory polymers, and focused on polyurethane based shape memory polymers. This study is to provide an additional insight of the effect of environmental factors on SMP such as the thermo-mechanical cycle, water absorption, moisture, heat treatment, and dynamic loading. This study examines the degradation behavior of Veriflex-SMP, (from Corner Stone Research Group) due to prolonged immersion in water and diesel fuel separately. 2.0

BACKGROUND RESEARCH

The shape memory effect was first reported by Chang and Read in 1932 [Chang & Read, 1951]. Thermally responsive shape memory materials are the most notable one, in which the recovery takes place upon heating it above the transition

40

temperature. According to Sakurai et al. [Sakurai & Takahashi, 1989] the first SMP, developed by the French CDF Chimie company and commercialized by Nippon Zeon Co. in Japan, was polynorborene based SMP with the glass transition temperature, Tg ranging from 35°C to 40°C, which was suitable for developing apparel for textiles. However, the molecular weight of this polymer is very high which makes it difficult for processing. Later Kurray company in Japan developed a transpolyisoprene based SMP with a Tg of 67qC in 1987. Asahi Company also introduced a styrene-butadiene based SMP with the Tg ranging from 60°C to 90°C [Shizuo & Nobuo, 1991]. The high transition temperature, Tg of this two types of SMP serves them as a good candidate for high temperature application but limit their usefulness in textile apparel. Finally, Mitsubishi Heavy Industry developed a polyurethane based thermoplastic polymers (SMPU) with transition temperature, Tg ranging from -30qC to +65qC. Apart from that, the processibility of SMPU is also easy [Hayashi & Shirai, 1988]. Since the discovery, SMP have drawn increasing attention because of its high recoverable ratio, low cost, low weight, and wide range of mechanical properties as compared to SMAs or ceramics [Liu et al., 2007]. To fully utilize the shape memory functionality of shape memory polymer and explore new area for application of shape memory polymer demands thorough characterization of SMP. A comprehensive description of the recent development of SMP has been reported by Nahid (2010). In order to explore possible new applications of SMP, it is necessary to determine the material resistance against various environmental factors. Huang et al., 2005 examined the effect of moisture on the Tg and shape memory property of SMP. Yang et al., 2006 found that the Tg of the polymer decreases and ultimately the SMP losses its shape fixing capability if it is left in air at room temperature for several days. 3.0

EXPERIMENTAL METHODS

A brief description of experimental methods is given in this section. Materials: In general, the SMP can be manufactured by mixing one reagent containing two active amino hydrogen or two active phenolic hydrogen with at least one multifunctional cross-linking reagent which contains at least three or more active amino or phenolic-hydrogen. The mixture is further mixed with at least one difunctional epoxide. In this case the SMP is the product of reaction between (a) styrene, (b) a vinyl compound other than styrene, (c) a cross linking agent, and (d) an initiator. By varying the composition of each component, the Tg of SMP can be tailored to the required application. 3.1 Properties of Material: A few important properties of styrene based Veriflex- SMP supplied by manufacturer are: Ultimate Tensile strength (23 MPa), Flexural strength (37.1 MPa), Compressive strength (32.4 MPa), Thermal conductivity (0.17 W/(m-°K), Tensile elongation to break (3.9%), Glass transition temperature (143°F ( 62°C)), Thermal conductivity (0.17 W/( m-qK), Material density (0.92 g / cm3). 3.2 Experimental Program: The general degradation process of polymer is of two types: (i) Physical, and (ii) Chemical. To investigate the effect of degradation we used only two different types of solvent: water and diesel fuel: (i) Water: In this research distilled water at Room temperature has been used. Moisture varies from 10% to 100% depending on the weather condition. According to the knowledge of the authors, no research has been carried out on Styrene based SMP. (ii) Diesel fuel: Automotive diesel fuel has been used as another solvent, which is a strong solvent. For designing new application of SMP the effect of diesel fuel on its property must be considered as exposures to diesel fuel is quite common. Four experiments were performed: Thermomechanical cycle, Dynamic mechanical thermal analysis, Stress relaxation test, Fourier transform infrared test.

41

Sample Preparation: Commercial styrene based SMP samples were received as thin sheet. Then the samples of desired dimensions were cut according to the test requirements. After the cut, the samples have been polished using 1200 grit sandpaper to remove surface roughness. Equipments: In order to measure the effect of degradation due to water and diesel fuel several tests such as: shape memory capability, recovery force, visco-elastic properties, and change in molecular vibration tests several equipments are used: Dynamic Mechanical Analyzer (DMA -2980) is an analytical instrument used to measure the physical properties, mainly visco-elastic properties of many different types of material. The tensile clamps are used which consists of two clamps: movable and fixed clamp. The movable upper clamp applies sinusoidal force on the sample with the help of a variable speed motor. Due to the application of applied force, the sample undergoes sinusoidal deformation, which is not in the phase of applied force because of visco-elasticity property. Rheometric Scientific Analyzer, RSA-III (Fig. 3 -shown left) with the tensile clamp shown in figure has been used in this study. It uses a servo linear actuator to impose an oscillatory deformation or strain, mechanically upon the material to measure the dynamic mechanical properties. Fourier Transform Infrared (FTIR) Spectroscopy: To determine the root cause of a degradation process in a polymer it is necessary to identify chemical properties of an unknown substance or contaminants. The identification becomes more challenging when the contaminant is organic in nature because of the presence of large number of organic compounds. The basic principle of FTIR lies on the absorption of infrared photons, which excite vibrations of molecular bonds. The amplitude of vibrations or a rotation of chemical bonds increases at specific frequencies corresponding to discrete energy levels due to the interaction of infrared light with the matter. This frequency is related to the shape of the molecular potential energy surfaces, the masses of the atoms, and the vibronic coupling. In this way, the presence of a functional group in a sample can be identified by the correlation of the bond wave number position with chemical structure. Dynamic Mechanical Thermal Analysis (DMTA) Tests: To investigate the change in dynamic mechanical properties of virgin SMP samples and the one degraded by water and another one by diesel fuel as a function of time, DMTA tests (Fig. 4) have been performed on RSA- III, in straincontrolled dynamic temperature ramp mode at a constant frequency of 1.0 Hz in the temperature range of 30qC to 100 qC at a heating rate of 3 qC/min. The tensile clamp has been used and the gaps between the clamps are 15 mm. (Fig. 4: Experimental Setup-shown left). Recovery Force Measurement: In order to measure the recovery force and time, a two- stage process was developed. In the first stage, the temperature of the sample has been raised to above the glass transition temperature to accommodate deformation. Then the sample has been deformed to a set strain (50% strain). Finally, the deformed sample has been cooled down below the room temperature while keeping the strain locked-in. This stage deforms the sample into a ‘reconfigured’ state. In the second stage the deformed sample is actuated while the machine is set to prevent strain recovery in the sample. On DMA, this test was performed in the iso-strain mode (the material is given a certain amount of constant strain and the force required to maintain that strain is measured as a function of temperature). During the strain recovery test, the temperature was ramped at 5qC/min between 30q C and 100qC.

42

Stress Relaxation Tests: The stress relaxation properties of SMP have been determined using RSA- III (Fig. 4). In this experiment, the force required to maintain a constant tensile strain of 0.4% was monitored as a function of time, at a constant temperature of 30 qC. The gap between the clamps was kept 15 mm, as before. Fourier Transform Infrared (FTIR) Spectra: Fourier transform infrared spectra were obtained using an Attenuated – Total- Reflectance (ATR) cell covering the 650 to 4000 cm-1 spectral range using Bruker-Tensor- 27 FTIR system. Each FTIR-ATR spectrum was the average of 64 scans at 4cm-1 of nominal spectra resolution, using air as reference. The FTIR data were analyzed using OPUS- 6.5 data Collection Program software. 4.0 RESULTS AND DISCUSSION This section focuses on the results obtained from various tests on the degradation of Veriflex- SMP due to water and diesel fuel. The result of the ambient scan of dry SMP is shown in Figures 5 to 7. “Dry SMP” is defined as the SMP that was

1.03u109Pa

1.20E+09 1.00E+09 8.00E+08 6.00E+08 4.00E+08 2.00E+08 0.00E+00 0

50

100

Temperature(qqC) Fig. 5: The storage modulus as a function of temperature for untreated SMP

LossModulus(Pa)

StorageModulus(Pa)

equilibrated at the room temperature conditions.

150

1.40E+08 1.20E+08 1.00E+08 8.00E+07 6.00E+07 4.00E+07 2.00E+07 0.00E+00 0

50

100

150

Temperature(qqC) Fig. 6: The loss modulus as a function of temperature for untreated SMP

From the behavior of storage modulus curve (Fig. 5), we see that at the initial stage, the storage modulus of the polymer is 1.03u109 Pa which represents its load bearing capacity at room temperature whereas at high temperature, the storage modulus drops almost to zero. Macroscopically, this means that the phase angle between stress and strain approaches 90q. That is, the energy stored by the sample per cycle of deformation becomes negligible in comparison to that energy dissipated as heat. In the transition region, the storage modulus decreases with the increase of temperature. As a result, it can withstand a large amount of load. With the increase of temperature, the hard segments of shape memory polymer do not undergo any kind of change. However, the mobility of soft segments increases due to the lifting of external constraint which results in the increase of heat dissipation. As a result, the storage modulus of shape memory polymer decreases with the increase of temperature. From Fig. 6, we can see that at the beginning with the increase of temperature, the loss modulus increases. This is because with the increase of temperature, the molecular mobility of soft segment increases which, in turn, increases the heat. After going through the maximum attainable temperature limit, the loss modulus begins to decrease and ultimately, at higher temperature it goes to zero. From the behavior of tan delta curve (Fig.7), we have seen that in the transition zone between glasslike and rubberlike consistency, the tangent delta value goes through a pronounced maximum. This is associated with configurational rearrangement of the strands of the network structure of the soft segment. The storage modulus, loss modulus,

43

and tangent delta values are the three different methods to determine the transition temperature, since each property is

tan G

dependent on the molecular movement of the polymer.

2 1.5 1 0.5 0

72.2 qC

0

50

100

150

Temperature (qqC) Fig. 7: The behavior of tanG as a function of

Fig. 8: Thermally driven shape memory cycle of SMP

temperature for untreated SMP From our results, we see that DMA spectrum of storage modulus curve of SMP shows a drop of storage modulus from 1.03u109 to 0.0 Pa in the temperature region between +30qC and +70q C. The loss modulus and Tan G shows peak at +50qC and +72.2qC, respectively. For DMA, the Tg is taken as the average of the Es and Tan G peak temperatures measured at 1.0 Hz. In our case, the glass transition temperature, Tg of our Veriflex-SMP, according to our test is: 61.382qC, which is close to the value 62qC supplied by the Veriflex’s manufacturer. This confirms that the correctness of our experimental procedure. 4.1 Shape Memory Property Testing: To determine the shape memory phenomena of the SMP, the thermo-mechanical programming cycle has been done on the specimen and the results are shown in Fig. 8. The important processes are: (i) The specimen is heated to 100 °C, which is above the glass transition temperature, Tg = 62°C.; (ii) At 100 °C the specimen has been given a 50% tensile deformation by applying a tensile force on it; (iii) After the deformation, the temperature of the sample has been cooled below the room temperature while maintaining the load on the specimen; (iv) At temperature below the glass transition temperature, the load is removed and the specimen is removed from the fixture. The resulting shape is called the deformed or temporary shape. In this condition, the length of the sample has been measured. (v) Then it is heated again to 100 °C, and no-constraint was imposed on the specimen to recover the original shape; (vi) Finally, it is cooled to room temperature and the length of the sample has been measured from which the shape memory property of the sample has been calculated. The following results are obtained from the thermo-mechanical cycle on the specimen: Shape fixity (98.07%), Shape recovery ratio (96.9%), and Recovery time, 10s. 4.2 Effect of Degradation due to Water and Diesel Fuel: In this study, the degradation of the SMP due to water and diesel fuel has been examined. At first, the intensity of degradation causes by water and diesel fuel has been compared. Then the progress of degradation with time due to water and diesel respectively, for the duration of one month has been investigated. Comparison of the Degradation due to Water and Diesel Fuel with Untreated Samples: As stated earlier, the effect of water and diesel fuel on styrene based Veriflex- SMP due to immersion for 1- week has been investigated and compared with respect to each other and with un-degraded samples.

44

Change in Storage Modulus: The behavior of storage modulus as a function of temperature of the untreated sample and the sample immersed in water and diesel fuel for 1 week has been shown in Fig. 9. 1.20E+09

2 Treated in Water for 1 week Untreated Untreated

Storagemodulus(Pa)

1.00E+09

1.5 Treated in Diesel Fuel for 1 week

Treated in

8.00E+08

1

tan G

6.00E+08

4.00E+08

0.5 0 25

Treated in diesel

2.00E+08

-1 20

65

85

105

-0.5

0.00E+00 0

45

40 60 80 Temperture(0C)

Fig. 9: Comparison of storage modulus between Untreated and treated Veriflex- SMP

100

120

Temperature (qqC)

Fig. 10: Comparison of tan delta curve between untreated and treated Veriflex-SMP

From the curves shown in the Fig.9 we see that the value of storage modulus, the load bearing capacity, of the material at room temperature decreases due to exposure to solvents. We also observe a greater decrease in the value of storage modulus for the case of SMP degraded by diesel fuel in comparison to water. We found that that due to immersion in diesel fuel the storage modulus decreases by 96.1% whereas, the decrease of storage modulus due to immersion in water is around 8.85% in comparison to untreated samples. Change in Glass Transition Temperature, Tg: The behavior of tangent delta as a function of temperature due to immersion in water and diesel fuel solvents for 1- week and that of the untreated one has been shown in Fig. 10. From these curves we see that the temperature at the peak of the tangent delta, i.e., the glass transition temperature of the material decreases due to exposure to water and diesel fuel. We also observe that a greater decrease in the value of glass transition for the case of SMP degraded by diesel fuel in comparison to that of water. It is an indication that the molecular structure of the polymer has been greatly affected by diesel fuel at room temperature. We found that due to immersion in diesel fuel the tg decreases by 52.2% whereas the decrease of tg in water is only 1.3%. Normalized Force Decay Curves: When a constant strain is applied to a polymer at a constant temperature, the force required to maintain the strain decreases with time. This behavior is called stress relaxation. The stress relaxation, which depends upon the time scale, is usually the result of both physical and chemical processes. For short time and/or at room temperature, usually physical processes are responsible for the cause of stress relaxation. But in degradative environments or at higher temperature and/or for long times, chemical processes are more significant than physical processes. Figure 11 shows the decay of normalized force as a function of time for the untreated samples and the treated one with water and diesel fuel for one week. At the initial stage, the force required to maintain the deformation has decreased significantly. After the

45

initial decrease of force, the force required to maintain the deformation remain almost constant with the passage of time and ultimately it reaches a plateau. The decay in force is the indicative of the onset of the physical and chemical relaxation 500 450 400 350

Strees

300

Untreated

250 200

Treated in water for 1 week

150 100 50

Treated in diesel fuel for 1 week

0 0

1000

2000

3000

4000

5000

6000

Time

 Fig. 11: Stress decay due to immersion in water & diesel Fig. 12: SMP Shape memory phenomena due to water processes within the SMP. It is also evident from Fig. 11 that the rate of decay in force is greater for the degraded ones. We can also see that due to immersion in diesel fuel the force required to maintain the deformation becomes almost zero. This is an indication of the change of state of the polymer from the glassy state to the rubbery state due to the immersion in diesel since; we know that in the rubbery state, the polymer is able to accommodate a large amount of deformation. Effect of Degradation due to Water: After comparison of the degradation due to water and diesel fuel with the un-degraded one, now we investigate the effect of water SMP alone with the increase of immersion time. Shape Memory Phenomena The thermo-mechanical cycle to investigate the shape memory phenomena of SMP due to immersion in water has been shown in Fig. 12. The processes of the thermo-mechanical cycle have been described below: (i) Initially, the SMP was in brick shape, which is its original shape. (ii) At this stage, the temperature of the SMP has been increased above the transition temperature and it has been deformed to a certain amount of tensile deformation (about 50%). After the application of deformation, the SMP has been cooled below the transition temperature while maintaining the constant load. At temperature below the transition temperature, the load has been removed. At the glassy state, the polymer is not able to reconfigure itself and the deformation has been locked- in. This shape is known as deformed or temporary shape. (iii) The deformed shape has been immersed in distilled water for up to 12 weeks to investigate the effect of water on the shape fixing capability of SMP. It has been shown that water does not have much effect on the shape fixing property of styrene based Veriflex- SMP. On the other hand, the color of SMP has been changed from transparent to opaque, ivory white (Fig. 12). Change in Glass Transition Temperature From Fig. 13 we see that water does not have significant change on the Tg of SMP. The percentage change in the transition temperature with respect to the untreated one after immersion for 4 weeks the change in transition temperature is only 4.66%. Change in Stress Relaxation The change in stress relaxation due to immersion in water has been shown in Fig. 15. It is seen that due to immersion in

46

2

500

untreated-1

1.8

450

untreated-2

1.6 1.4

400

treated in water for 1 week treated in water for 2 week treated in water for 3 week

1

0.8 0.6

Stress

Tan G

1.2

350

250 200 150

0.4

100

0.2

50

0

0

25

45

65 85 Temperature (qqC)

Untreated

300

Treated in water for 1 week

0

105

1000

Treated in water for 3 weeks

2000

3000

Treated in water for 2 weeks

4000

5000

6000

Time (sec)

Fig. 13: tan delta as a func. of temperature due to water

Fig. 14: Stress decay of Veriflex-SMP due to water

water, the relaxation of force occurs at a faster rate in comparison to the untreated one. It has also been found that there is no difference in stress relaxation between 1 week and 3 weeks duration of immersion time. After duration of immersion in water for 3 weeks the polymer does not lose its capacity to withstand the load. 2.5 2 Treated in diesel fuel for 1 week

1.5 1

tan G

0.5 0 20

30

40

50

60

70

80

90

100

110

-0.5 Untreated -1 -1.5

Treated in diesel fuel for 2 weeks

-2 Temperature (qqC)

Fig.15: Phenomena of SMP due to diesel

Fig. 16: Change in transition temp. due to immersion in diesel

Effect of Diesel Fuel: Shape Memory Phenomena Shape memory phenomena of SMP due to immersion in diesel fuel by going through the thermo-mechanical cycle which is shown in Fig. 15. The processes of the thermo-mechanical cycle have been described as: (i) Initially, the SMP was in brick shape, which is termed as its original shape; (ii) At this stage, the temperature of the SMP has been increased above the transition temperature and it has been deformed to a certain amount of deformation (about 50%). After the application of deformation, the SMP has been cooled below the transition temperature while maintaining the constant load. At temperature below the transition temperature, the load has been removed. At the glassy state, the polymer is not able to reconfigure itself and the deformation has been locked-in. This is known as deformed or temporary shape. The deformed shape has been immersed in diesel fuel to investigate the effect of diesel fuel on the shape fixing capability of SMP. It is observed that due to

47

immersion in diesel at room temperature, Veriflex- SMP has been recovered its original shape after two weeks. The shape recovery of the SMP is almost 99%. It indicates that diesel fuel has a great influence on the shape fixing property of styrene based Veriflex- SMP. On the other hand, the color of SMP has been changed from transparent to opaque, yellowish (Fig. 15). Change in Transition Temperature The effect of diesel fuel on the glass transition temperature, Tg SMP is shown in Fig. 16. It is seen that due to the immersion in diesel fuel for 1- week, the transition temperature of SMP has decreased to 34qC whereas due to immersion for 2 weeks, the transition temperature goes below the room temperature. The findings exactly verify the result obtained from thermomechanical cycle. It has also been observed that there is a considerable fluctuation in the behavior of tan G curve, which also indicates that the material goes into the rubbery state. As a result, the material is not able to withstand any amount of load and the fluctuations arrived from the frequency of the equipment. 500

Fig. 17: Stress decay of Veriflex- SMP due to diesel fuel

450

(shown left)

400

Change in Stress Relaxation

350 Stress

The effect of diesel fuel in stress relaxation of SMP is shown

Untreated

300

in Fig. 17. It is shown that due to the immersion in diesel fuel

250 200

even for 1- week the stress relaxes drastically and ultimately

150 100

it goes to zero. From the results, we can conclude that due to

Treated in diesel fuel for 1 week

immersion in diesel fuel the SMP has gone to rubbery state

50 0 0

1000

2000

3000 Time (sec)

4000

5000

6000

since its load retention capacity goes to zero. We can also conclude that besides physical process, chemical process also

occurs during the immersion period of SMP into diesel fuel, which breaks the crystalline portion of soft segment completely. Discussion: Shape memory polymer’s hard and soft phases are incompatible. These phases are randomly distributed along the polymer chain forming a structure of separated micro-phases. The plasticization effects of water and diesel fuel also have a significant influence on the deterioration of property of SMP. Plasticizers are low molecular weight compound due to which the rigid polymeric materials become soften. They penetrate into the polymeric matrix and interrupt the chain-to-chain secondary bonding. As a result, the glass transition temperature of the material decreases. Usually, the Tg of a polymer, in contact with a plasticizer, depends on the Tg of the plasticizer. In this study, the Tg of the solvents used that is, water and diesel fuel, is much below in comparison to that of Veriflex-SMP, 62qC, and acts as a good plasticizer. As a result, the mechanical property of shape memory polymer has been degraded because of the plasticization effect of both solvent. We also see that diesel fuel acts as a better plasticizer for shape memory polymer in comparison to water since the load bearing capacity of shape memory polymer goes to almost zero due to immersion in diesel fuel. 5.0 CONCLUSIONS The main aim of this research is to study the effects of degradation due to water and diesel fuel on the properties of styrene based Veriflex-SMP. The general key conclusions are summarized below: (i)

Veriflex- SMP immersed in water and diesel, showed the similar tendency in terms of degradation of property.

48

(ii)

Due to immersion in diesel fuel, the appearance of Veriflex- SMP changes from transparent to opaque, yellowish

one. On the other hand, due to immersion in water, the appearance of Veriflex- SMP changes to ivory white. (iii)

Due to degradation by diesel fuel significant reduction in glass transition temperature, Tg has been achieved;

whereas the change due to degradation by water is less significant. (iv)

It is found that the Veriflex- SMP is able to recover the original shape from the deformed shape upon immersion in

diesel fuel for 2- weeks; but the extent of recovery depends on the thickness of the sample. Thus, shape recovery can be triggered not only by applying energy but also without energy. (v)

Micro molecular solution absorbed in the Veriflex- SMP weakens the elasticity modulus, which causes a significant

decrease in the Tg. As a result, the SMP is able to recover the original shape. It has been found that hydrogen bonding is mainly responsible for the changes in properties of Veriflex- SMP. 6.0 ACKNOWLEDGMENTS This study is based upon work supported by the NASA/EPSCoR under grant number NASA/LEQSF (2007-10)-Phase3-01. 7.0 REFERENCES: 1.

Chang L.C., ReadT.A., (1951), “Plastic deformation and diffusionless phase changes in metals. The Gold-Cadmium Beta Phase”, Trans. AIME, 189, pp.47-52.

2.

Hu, J., (2007). Shape Memory Polymers and Textiles, Boca Raton, FL: CRC Press LLC

3.

Huang W. M., Yang B., An L., Li C., Chan Y.S. (2005), “Water driven programmable polyurethane shape memory polymer: Demonstration and mechanism”, Appl. Phys. lett., 86, 114105.

4.

Lendlien A., Kelch S. (2002), “Shape memory polymers”, Angew. Chem. Int. Ed., 41(12), pp.2034-2057.

5.

Lendlein A., Behl M. (2007), “Shape memory polymers”, Materials Today, 10(4), pp.20-28.

6.

Liu C., Qin H., Mather P.T., (2007), “Review of progress in SMP”, J. Mater. Chem., 17(16), pp.1543-1558.

7.

Nahid, M.N.H. (2010), Degradation Behavior of Shape Memory Polymer Due to Water and Diesel Fuel, Master of Engineering Thesis, Louisiana State University, Baton Rouge, LA 70803, U.S.A.

8.

Sakurai K., Takahashi T., (1989), “Strain-induced crystallization in polynorbornene”, J. Appl. Polym. Sci., 38(6), pp.1191-1194.

9.

Shizuo K., Nobuo, N., (1991), “Cross-linked polymer having shape memorizing property. Method of its use and molded article having shape memory”, US patent 5043396.

10.

Tobushi H., Matsui R., Hayashi S., Shimada D. (2004). “The influence of shape-holding conditions on shape recovery of polyurethane-shape memory polymer”, Smart Mater. Struct., 13(4), pp.881-887.

11.

Yang B., Huang W. M., Li C., Li L. (2006), “Effects of moisture on the thermo mechanical properties of a polyurethane shape memory polymer”, Polymer, 47(4), pp.1348-1356.

Structural Enhancement of Framing Members Using Polyurea

David J. Alldredge Department of Mechanical and Aerospace Engineering University of Alabama in Huntsville Huntsville, Alabama 35899 John A. Gilbert Professor of Mechanical Engineering Department of Mechanical and Aerospace Engineering University of Alabama in Huntsville Huntsville, AL 35899 Houssam Toutanji Professor of Civil Engineering Department of Civil and Environmental Engineering University of Alabama in Huntsville Huntsville, AL 35899 Thomas Lavin President Soems, Inc. Watchung, NJ 07069 Madhan Balasubramanyam Research Engineer Propulsion Research Center University of Alabama in Huntsville Huntsville, Alabama 35899 ABSTRACT This paper demonstrates the potential for using field applied structural coatings to reinforce traditional framing members and standard building ties, thereby providing an improved and continuous foundation to roof load pathway. Tension tests are performed on rafter top plate model joint connections, some of which were reinforced with a hurricane tie, to establish how much of a difference a polyurea coating made as the joints between the stud and top plate, and top plate and rafter, were loaded to failure. Polyurea provides universal strengthening compared to hurricane ties with the added advantage that members and joints can be protected from a multitude of threats including corrosion due to moisture, damage due to flood; and, with selfextinguishing properties, fire. Results show that the failure mode of the structures tested can be controlled by using different types of field or factory applied polymer coatings. The addition of the coating allowed both unreinforced and reinforced configurations to withstand higher loads (200-400% more). In general, the polyurea delayed the onset of failure and significantly strengthened every configuration by increasing T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_8, © The Society for Experimental Mechanics, Inc. 2011

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50 the amount of work/energy required to pull it apart; in some cases, by almost 800%. INTRODUCTION It is evident that significant problems exist in coastal buildings and there is a need for new water resistant building materials and techniques that can reinforce new and existing structures while providing safety for building occupants. FEMA technical notes and bulletins outline many of these problems including flood hazard information and recommendations for reconstruction practices [1], the structural needs for occupant safety and the criteria for designing hurricane and tornado safe rooms [2], material ratings for common building materials and evidence of the need for Class 5 water resistant materials [3], and recommendations for home construction in storm pathways [4]. In their Home Builder’s Guide to Costal Construction [4], FEMA also stresses the need for wall to sill plate reinforcement, foundation to shear wall reinforcement, and the development of continuous foundation to roof load pathways. They highlight problems with practices used to tie and strengthen framing members to the foundation or roof, framing member to framing member, etc. The goals of this research were to develop new composite building materials and construction techniques involving coatings containing polyurea or polyurethane or hybrids thereof; and, test the effectiveness of using field applied polymer coatings to solve issues of roof joist to wall framing attachment and foundation-rim joist wall framing to produce a continuous foundation to roof load pathway and to supplement standard hurricane and construction ties. POLYUREA Polyurea is a high strength polymer with scalable and predictable material characteristics that can be sprayed onto a substrate to make it waterproof. The polymer has a variable gel time; tensile strengths in commercially available materials vary from 13.8 to 34.5 MPa (2000 to 5000 psi) with inversely related elongation rates. The polymer can be field applied with a brush, a high temperature pump applicator, or a low temperature low pressure dispenser. Polyurea is currently used for truck bed liners and for explosive blast resistant walls. Thin interlayers of polyurea have been shown to increase blast resistance of carbon fiber foam composites [5]. Because polyurea strain hardens under load [6], the US Army uses polyurea to coat and harden field buildings against explosive blast; and, it is likely used to retrofit Government offices. Polyurea adheres well to concrete, metal and to wood. However, to our knowledge, prior to this investigation, polyurea had never been tested as a field applied structural adhesive for framing in combination with standard framing techniques. TEST PROGRAM We selected a framing connection for testing and had a licensed carpenter make several rafter top plate model joint constructions (subsequently referred to as the “configuration”) from Southern Pine. As illustrated in the photo shown to the left in Fig. 1, he nailed a vertical 2 by 4 stud to the bottom of a doubled top plate using two, 16d x 3.5” nails. Then, he toe nailed a 2 by 8 rafter having a bird’s mouth to the top plate using four, 16d x 3.5” long nails. In some cases, the configuration was reinforced by using a Simpson LTS12 hurricane tie fastened between the stud and the rafter with 14 (fourteen) 4d x 1.5” nails (see the photo shown to the right in Fig. 1). We used fourteen 4d nails, as opposed to twelve 10d nails as recommended, to anchor the tie in order to avoid splitting the stud. For reference, according to the manufacturer, when properly installed, the LTS12 is designed to withstand a maximum allowable lift load of 3.2 kN (720 lb). We applied loads in the vertical direction in an attempt to either pull the stud away from the top plate or fail the toe nail joint between the rafter and the top plate. During construction, no attempts were made to control the orientation and coarseness of the grain structure in the configurations and these varied widely for different structural members. In the rafter joint shown in the photo to the left in Fig. 1, for example, the gain runs in a direction perpendicular to the applied load, whereas the grain structure in the vertical stud runs parallel to it. In selecting our mode of loading, we assumed that the rafter lifts directly upward from the vertical stud and top plate, thereby placing both joints in tension. By positioning the hurricane strap where we did, its outer surface is subjected to an additional tensile load due to eccentric loading.

51 As described below, we tested standard nailed configurations with and without the metal hurricane tie and used the results to define our control standards. While testing the configurations, we neglected the reinforcement effects caused by plywood sheathing that may be nailed to the top plate and the stud.

Fig. 1 A basic framing connection was constructed (left) and, in some case, reinforced with a hurricane tie (right) Then, as illustrated in the photos shown in Fig. 2, we sprayed polyurea on similar configurations continuously around the stud and up and around the top plate and rafter. For most retrofitting applications, the top plate to stud will not be available in the soffit. In this case, only the rafter to top plate can be easily coated. Two different types of polyurea were employed: 1) a black version having a relatively large elongation and low elastic modulus and 2) a white version having a relatively low elongation and high elastic modulus. The tensile strengths of these products were about the same. The thickness of the polyurea coatings varied throughout a configuration, and from configuration to configuration. An average value of the coating thickness was obtained for all of the coated configurations by making measurements: 1) across the thickness of the rafter at a point located midway between the centerline of the upper hole and the upper surface of the top plate, and 2) across the smaller dimension of the lower 2 x 4 at a point located midway between the centerline of the lower hole and the lower surface of the top plate. The average thicknesses for the black and white coatings were 2.0 mm (.078 in.) and 2.4 mm (.095 in.), respectively.

Fig. 2 Some configurations were coated with polyurea (black at left, white at right) prior to subjecting them to uniaxial tension (right) As illustrated in the photo shown to the right in Fig. 2, the configurations were initially placed in uniaxial tension by passing Kevlar straps through two 2.34 cm (15/16 in.) diameter holes drilled in the 2 by 4 and rafter. Each hole was reinforced using a 3.8 cm (1.5 in.) long section of aluminum pipe. The 98 kN (22 kip) capacity MTS testing machine used to conduct the tests

52 was equipped with a load cell; and, comparisons and observations were made between uncoated and coated specimens that were pulled in deflection controlled tests at a rate of 1.27 cm/min (0.5 in/min). A net deflection for each configuration was calculated by subtracting the deflection in the pull straps from that measured for the crossheads. The deflection in the pull straps was computed based on Fig. 3 which shows a load deflection plot obtained by placing a 50 cm (19.5 in.) long segment of one of the pull straps in tension.

(3.3.2)

Fig. 3 Load versus deflection plot for a 50 cm (19.5 in.) long Kevlar pull strap A calibration factor, C, of 6 με/N (26.8 με/lb) was obtained for the case when a 2.54 cm (1 in.) long segment of the strap was subjected to a 4.45 N (1 lb) load by fitting a linear curve through the data in Fig. 3 over the load range observed during our test program [0 to 9790 N (0 to 2200 lb)]. Referring to the configuration shown in the photo to the right in Fig. 2, each side of the upper and lower straps carries one half of the total load. Hence, the total deflection in the pull straps is: =

(1) 4

where P is the load and L is the total length of all straps used to pull on the configuration [approximately 55.9 cm (22 in.)]. The strain energy, U, is equal in magnitude to the area under the load/deflection plot. For the configuration that we used: =

8

(2)

Figure 4 shows plots of load versus total deflection (of the configuration and the pull straps) corresponding to three uncoated specimens. Two of the configurations were unreinforced (w/o tie) while one was reinforced with a hurricane tie. Note that the scale on the vertical axis was kept constant in all of the plots presented in this section so that direct visual comparisons could be made.

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Fig. 4 Load versus deflection plots for uncoated specimens See legend and Figs. 5 and 7 for details One of the uncoated configurations without the tie failed gradually as the nails in the top plate pulled out of the stud (see the photo shown to the left in Fig. 5). This configuration held a maximum load equal to 1850 N (416 lb); and, we considered this our “standard” for 2 by 4 end nail failure; i.e., pull out between the stud and top plate in an unreinforced configuration. In contrast, the other uncoated configuration without the tie represented our “best case” scenario. This configuration held a higher load [2562 N ((576 lb)] and failed relatively quickly as the toe nailed joint between the rafter and top plate fractured and gave way (see the photo shown to the right in Fig. 5). As explained later, we considered this as the “standard” for toenail rafter failure; i.e., failure of the toe nail joint between the top plate and rafter in an uncoated configuration. But, since nail pull out occurred in every other initial pull test except for this one; a forensic analysis was conducted to understand why. Inspection of the lower joints in each of the specimens shown in Fig. 5 revealed that all nails were driven vertically through the top plate into the stud. But Fig. 6 shows photographs taken of the grain structures in the studs for the configurations that experienced end nail failure (left) and toe-nail rafter failure (right). The ring patterns indicate that both studs were centrally cut; however, the one shown to the right had an unusually fine grain structure that resulted in toe-nail rafter failure as opposed to the end nail failure associated with the stud on the left. As described later, we subsequently tested the joint in the configuration shown to the right by placing pull straps over the top plate and through the hole in the stud. The partial configuration held 2998 N (674 lb), substantially higher than the load applied to fail the toe nail joint. Referring again to Fig. 4, the third plot is typical of an uncoated configuration that was reinforced with a Simpson LTS12 hurricane tie. As illustrated in the photos shown in Fig. 7, failure occurred when the tie deformed as the nail farthest away from it pulled out of the stud. The specimen held a maximum load equal to 2277 N (512 lb) which we considered “standard” for pull out failure of an uncoated, reinforced configuration.

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Fig. 5 The unreinforced configuration failed gradually as a result of 2 by 4 end nail failure (left) or, in one case, abruptly as a result of toe-nail rafter failure (right)

Fig. 6 When the grain structure in the stud was relatively fine (right), the joint was much stronger

Fig. 7 Failure of a reinforced configuration occurred when the nail farthest from the tie pulled out of the stud

55 Figure 8 shows the load/deflection plots obtained for the unreinforced standard along with those for typical configurations coated with black and white polyurea. The configuration that was coated with black polyurea held a maximum load of 3621 N (814 lb) and as seen from the plot, failed relatively slowly. As seen in the photo shown to the left in Fig. 9, wood fibers fractured in the top plate and the nails in the stud loosened and the coating stretched. In contrast, the configuration coated with white polyurea held a maximum load of 7624 N (1714 lb) and as seen in the photo shown to the right in Fig. 9, abruptly failed when the wood fibers fractured in the top plate. Although the white polyurea coating was on average slightly thicker than the black, the tensile strength of the black polyurea is slightly higher than that of the white. So, we attributed the superior performance of the white coating to the higher stiffness ratio between the polyurea and the wood and a better bond between the two. The higher stiffness ratio associated with the white polyurea forces more stress into the coating, thereby stiffening the overall configuration relative to that coated with the black version. This is evident in Fig. 8, where the slope associated with the white version is higher than the slope associated with the black version. The superior bonding associated with the white polyurea was attributed to greater penetration into the wood substrate. A comparison of the photos shown in Fig. 9 reveals that the fracture plane associated with the white polyurea is deep within the wood grain as opposed to the fracture plane associated with the black version which is very close to the surface.

Fig. 8 Load versus deflection plots for 2 by 4 end nail failure in the uncoated unreinforced standard and coated unreinforced configurations

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Fig. 9 The coated unreinforced configurations experienced 2 by 4 end nail failure relatively slowly (left) and quickly (right) when wood fibers fractured in the top plate An interesting observation was made during a forensic analysis of the coated unreinforced configurations. Figure 10, for example, shows the grain structure in the top plates of two different unreinforced configurations that were both coated with white polyurea. These configurations had very similar load/deflection plots but the configuration to the left held 7019 N (1578 lb) while the one to the right (described above and shown to the right in Fig. 9) held 7624 N (1714 lb). The difference is attributed to the orientation of the grain structure in the lower 2 by 4 that comprises the top plate. When the grain is oriented vertically (left) and placed in shear, the joint is relatively weak as compared to when the grain is oriented horizontally (right) and placed in tension.

Fig. 10 When the orientation of the grain in the lower member of the top plate was horizontal (right), the joint was stronger Figure 11 shows the load/deflection plot obtained for the reinforced standard along with those for typical configurations coated with black and white polyurea. The configuration that was coated with black polyurea held a maximum load of 5738 N (1290 lb). As illustrated in the photo shown to the left in Fig. 12, the wood fibers, diametrically opposed to the hurricane tie and along the sides of the stud, fractured in the lower member of the top plate. Then while the coating stretched, the nails in the stud loosened. In contrast, the sample coated with white polyurea held a maximum load of 9906 N (2226 lb). As illustrated in the photo shown to the right in Fig. 12, the configuration abruptly failed when the wood fibers, diametrically opposed to the hurricane tie and along the sides of the stud, fractured in the lower member of the top plate.

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Fig. 11 Load versus deflection plots for 2 by 4 end nail failure in the uncoated reinforced standard and coated reinforced configurations \

Fig. 12 The coated reinforced configurations failed relatively slowly (left) or quickly (right) when wood fibers fractured in the top plate Referring again to Fig. 11, it is clear from the slopes of the plots that that coatings stiffen the configuration, the white much more so that the black. Although the white coated configuration holds more load, the black coated configuration is able to sustain more deformation.

58 The secondary peak observed at a deflection of about 2.54 cm (1 in.) in the curve associated with the black coated configuration was seen in every test conducted on configurations of this type. The peak may correspond to the fracture that takes place in the wood and, if so, could provide some insight into the relative bond strength associated with the two polyurea coatings. Assuming that the wood fibers fracture in the white coated configuration at failure, the bond strength associated with the white coating is more than twice that associated with the black. During the initial pull test program we had two coated configurations that held substantially lower loads than those reported above: 1) an unreinforced configuration coated with black polyurea and 2) a reinforced configuration coated with white polyurea. The load/deflection curves obtained for these cases are plotted in Fig. 13 along with those corresponding to their stronger counterparts.

Fig. 13 Load versus deflection plots for 2 by 4 end nail failure in similar specimens that withstood very different peak loads During forensic analysis of the weaker specimens, we discovered that the polyurea felt sticky and failure occurred in the polyurea itself as opposed to within the wood fibers. We attributed this problem to incomplete mixing which most likely occurred when one of the canisters in the spray gun ran dry before the other. The interesting thing is that although the peak load is very different for the weak and strong specimens, for a given polyurea, the shapes of the curves are similar. Figure 14 includes load/deflection plots corresponding to pull out for all six cases tested: 1) an unreinforced configuration, 2) a configuration reinforced with a hurricane tie, 3) an unreinforced configuration coated with black polyurea, 4) an unreinforced configuration coated with white polyurea, 5) a reinforced configuration coated with black polyurea, and 6) a reinforced configuration coated with white polyurea.

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Fig. 14 Load versus deflection plots for 2 by 4 end nail failure in six different configurations Table 1 lists the maximum loads that these six different configurations took along with the value of total deflection at which they occurred and the total strain energy required to achieve the maximum load condition. The net deflection and strain energy for each configuration was obtained by subtracting the values associated with the pull straps (see Equations 1 and 2). Table 1 Tabulated results for 2 by 4 end nail failure in six different configurations Total Total Strain Configuration Maximum Load Deflection @ Strain Energy to Deflection @ Energy to Status (Pmax) (N) Pmax (mm) Pmax (N mm) Pmax (mm) Pmax (N mm) uncoated standard w/o 9.7 8699 1850 8.1 7231 tie uncoated standard 29.7 35814 2277 27.7 33668 w/tie coated w/black 19.8 28810 3621 16.8 23274 w/o tie coated w/white 26.4 81571 7624 20.1 57167 w/o tie coated w/black 41.4 113544 5738 36.6 99648 & tie coated w/white 31.8 121340 9901 23.4 80102 & tie Table 2 lists the multiplication factors associated with coated specimens computed by taking the value found for a given quantity and dividing it by the value determined for its standard counterpart. The net strain energy is equal to the work done as the load is slowly applied and represents the amount of energy required to bring each configuration to the maximum load condition.

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Table 2 Multiplication factors; 2 by 4 end nail failure in coated configurations Total Total Strain Deflection @ Configuration Load (Pmax) Deflection @ Energy to Pmax wrt to Status wrt Standard Pmax wrt Pmax wrt Standard Standard Standard coated w/black 2.05 3.31 1.96 2.07 w/o tie coated w/white 2.74 9.38 4.12 2.47 w/o tie coated w/black 1.39 3.17 2.52 1.32 & tie coated w/white 1.07 3.39 4.35 0.84 & tie

Strain Energy to Pmax wrt Standard 3.21 7.87 2.96 2.38

As mentioned previously, all specimens except for one failed as the nails in the top plate pulled out of the stud. As illustrated in the photo in Fig. 15, we subsequently tested the remaining joint on this configuration by placing pull straps over the top plate and through the hole in the stud. The partial configuration held 2998 N (674 lb) when it was reloaded.

Fig. 15 A partial configuration is retested for 2 by 4 end nail failure As illustrated in the photo shown to the left in Fig. 16, we tested the other partial configurations that did not include a hurricane tie by placing pull straps over the top plate and through the hole in the rafter. The photo shown to the right in the figure shows how the toe nail joint failed in the coated specimens as a crack developed and propagated along the rafter. This was notably different from the failure that occurred in the standard (described previously and shown in Fig 4) where the wood surrounding the nails fractured. Figure 17 shows plots of load versus total deflection (of the configuration and the pull straps) corresponding to toe nail fracture for an uncoated standard (described previously and shown in Fig. 4) and configurations coated with black and white polyurea.

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Fig. 16 When partial coated configurations were retested to evaluate toe-nail rafter failure (left) a crack developed in the rafter (right)

Fig. 17 Load versus deflection plots for toe-nail rafter failure of the uncoated unreinforced standard and unreinforced configurations coated with black and white polyurea

62 Table 3 lists the maximum loads that these three different configurations took along with the value of total deflection at which they occurred and the total strain energy required to achieve the maximum load condition. The net deflection and strain energy for each configuration was obtained by subtracting the values associated with the pull straps (see Equations 1 and 2). Table 3 Tabulated results for toe-nail rafter failure in three different unreinforced configurations Total Total Strain Configuration Maximum Load Deflection @ Strain Energy to Deflection @ Energy to Status (Pmax) (N) Pmax (mm) Pmax (N mm) Pmax (mm) Pmax (N mm) uncoated standard 18.3 20788 2562 16.3 18077 w/o tie coated w/black 55.1 155798 7949 48.5 129248 w/o tie coated w/white 36.3 119306 10164 27.7 75922 w/o tie Table 4 lists the multiplication factors associated with coated specimens computed by taking the value found for a given quantity and dividing it by the value determined for its standard counterpart. Table 4 Multiplication factors; toe-nail rafter failure in unreinforced coated configurations Total Total Strain Strain Deflection @ Configuration Load (Pmax) Deflection @ Energy to Energy to Pmax wrt to Status wrt Standard Pmax wrt Pmax wrt Pmax wrt Standard Standard Standard Standard coated w/black 3.01 7.49 3.10 3.00 7.17 w/o tie coated w/white 1.99 5.74 3.97 1.72 4.21 w/o tie DISCUSSION Referring to Tables 1 – 4, there are many factors that could have contributed to deviations and potential errors in the values determined for deflection and strain energy such as variations in wood grain and structure, kinks in the straps, elongation of the holes at the attachment points, variations in strap length, and differences in construction and the manner in which failure took place. But the results presented for peak load are indisputable. It is clear from the tabulated data that the addition of a polyurea coating allowed both the unreinforced and reinforced configurations to withstand a greater load. Results for nail pull out indicate that when compared to their uncoated counterparts, the black coated configurations were about twice as strong whereas the white coated configurations were four times as strong. The follow up tests conducted to evaluate toe nail fracture show that when compared to their uncoated counterpart, the black coated configurations were three times as strong whereas the white coated configurations were four times as strong. A review of the load/deflection plots revealed that the peaks corresponding to the maximum load in the unreinforced coated configurations always occurred at a greater deflection than those corresponding to their uncoated counterparts. Thus, the addition of a polyurea coating delayed the onset of failure allowing the unreinforced configurations to sustain more deflection before they reached their peak loads. Results for nail pull out and toe nail fracture indicate that when compared to their uncoated counterpart, the coated configurations can sustain anywhere from two to three times as much deflection. The fact that it can take almost eight times more energy to bring a coated unreinforced configuration to the peak load as compared to that required to bring an uncoated unreinforced configuration to the same condition is staggering. Even when the configuration is reinforced, the strain energy associated with the coated configurations is more than double that associated with their uncoated counterparts. Significantly, the polyurea significantly strengthened both the unreinforced and reinforced configurations by increasing the amount of work/energy required to pull them apart.

63 Since nearly all hurricane straps are made of galvanized steel, there is limited design flexibility from a materials standpoint. This makes it difficult to design a configuration that can readily adapt in real time to withstand sustained winds or wind gusts. Referring again to the plots, the differences in the peaks and shapes associated with the coated configurations show that the failure mode of the unreinforced and reinforced configurations can be controlled by using different types of polyurea. The configuration can be designed to fail at a higher peak load in a relatively brittle fashion or at a lower load in a relatively ductile manner. If this knowledge can be harnessed and used to advantage, polyurea coatings may revolutionize how construction in hurricane prone regions is done. The hurricane tie, as installed, only increased the peak load of the unreinforced configuration by about 25% and allowed the configuration to deflect three times as much before the maximum load was achieved. Significantly, we have shown that, when used in combination with a hurricane tie, a polyurea coating substantially enhanced the structural performance even when the strap alone does little to strengthen the uncoated joint. In contrast to existing metal hurricane ties that are designed to withstand a specific loading, a polyurea coating provides more universal strengthening with the added advantage that members and joints can be protected from a multitude of threats such as corrosion due to moisture and damage due to flood and, with self extinguishing properties, fire. CONCLUSIONS In conclusion, we had a licensed carpenter construct several rafter top plate model joint connections. We performed tension tests on these configurations, some of which were reinforced with a hurricane tie, to establish how much of a difference a polyurea coating made as the joints between the stud and top plate, and top plate and rafter, were loaded to failure. The addition of the coating allowed both unreinforced and reinforced configurations to withstand higher loads (200-400% more). In general, the polyurea delayed the onset of failure and significantly strengthened every configuration by increasing the amount of work/energy required to pull it apart; in some cases, by almost 800%. We showed that the failure mode can be controlled by using different types of polyurea and proved that the commercial applications of using polyurea to strengthen structures in hurricane prone areas are enormous. Polyurea provides universal strengthening compared to hurricane ties with the added advantage that members and joints can be protected from a multitude of threats including corrosion due to moisture, damage due to flood; and, with self extinguishing properties, fire. ACKNOWLEDGMENTS The authors would like to thank the U.S. Department of Commerce for supporting this research under NOAA SBIR Phase I and Phase II contract no. WC133R-09-CN-0108. They would also like to thank Mr. Rajesh Vuddandam for his help with running the pull tests and acquiring data and John Becker for helping to select the polyureas used in this study. Any opinions, findings, conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the Department of Commerce. REFERENCES [1] “Reconstruction Guidance Using Hurricane Katrina Surge Inundation and ABFE Maps,” See: http://www.fema.gov/pdf/hazard/flood/recoverydata/katrina/katrina_reconstruction.pdf [2] “FEMA 361, Guidance for Community Safe Rooms,” See: http://www.fema.gov/plan/prevent/saferoom/fema361.shtm [3] “Flood Damage-Resistant Materials Requirements for Buildings Located in Special Flood Hazard Areas in Accordance with the National Flood Insurance Program,” See: http://www.fema.gov/library/file?type=publishedFile&file=fema_tb_2.pdf&fileid=a4cd38c0-84db-11dd-8541-001185636a87 [4] “FEMA 499, Home Builder’s Guide to Costal Construction,” See: http://dnr.louisiana.gov/sec/execdiv/techasmt/programs/residential/coastal_construction/fema499.htm [5] Bahei-El-Din, Y.A., Dvorak, G.J., “Behavior of Sandwich Plates Reinforced with Polyurethane/Polyurea Interlayers under Blast Loads,” Journal of Sandwich Structures and Materials, Vol. 9, pp. 261-281, 2007. [6] Roland, C.M., Twigg, J.N., Vu, Y. Mott, P.H., “High strain rate mechanical behavior of polyurea,” Polymer, 48, pp. 574578, 2007.

Experiments and Models for the Time Dependent Mechanics of Nanoscale Polymeric Structures and Nanocrystalline Metal Films Ioannis Chasiotis Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

To date, micro and nanoscale experiments have been mostly focused on the length scale dependent mechanical behavior of nanostructures and nanostructured thin films but have not been able to address their time and rate dependence. This inefficiency stems from the use of high resolution electron microscopes which are slow imaging tools, and quite often are of detrimental effect to the integrity of polymeric materials. Optical methods have been revisited in the recent years and were modified to accommodate micro and nanoscale specimens in order to obtain high resolution deformations and their time evolution at time scales varying from microseconds to days [1-4]. This research, conducted at the University of Illinois, has developed new approaches to investigate the time-dependent mechanical behavior of metallic thin films for MEMS and polymeric nanostructures in an effort to understand the important deformation processes at small scales. The extended (internal) surfaces in nanocrystalline metal films and the large surface-to-volume ratios in polymeric nanostructures favor material transport mechanisms that are not important in bulk or large grain materials because they do not result in appreciable strains. On the contrary, in nanoscale polymeric fibers for instance such processes result in large material deformations and sustained ductilities in a large range of loading rates. This presentation will summarize experimental work conducted with polymeric nanofibers and nanocrystalline metals and some early modeling efforts to rationalize the measured time- and ratedependent mechanical behavior. 1.

Time-dependent Mechanics of Polymeric Nanofibers

Polymeric nanofibers have been of interest in the last two decades due to scalable manufacturing processes based on electrospinning. Although they are materials with clear time dependence, only recently their temporal mechanical behavior has been investigated in a relatively broad range of strain rates, varying from 10-4 to 200 s-1 [5,6] as the appropriate experimental tools have become available [1]. Using these experimental methods, the first creep experiments with nanoscale fibers were also conducted to determine simple viscoelastic laws which, when calibrated, could provide predictions for the rate-dependent response. At relatively low stresses (<40 MPa) compared to the yield strength (>100 MPa) of the nanofibers it has been shown that linear viscoelasticity may be applied and predictions of the strain evolution at slow strain rates could be made if the Figure 1. Engineering stress-strain curves of PAN viscoelastic time constants were calculated as a function of nanofibers at three strain rates. The curly brackets group the the nanofiber diameter. This size effect was explained by plots according to the engineering stress in the fiber during extensive experiments with amorphous polymeric drawing [5]. nanofibers, which have shown a strong dependence of the initial elastic stiffness and the yield and ultimate strengths on the nanofiber diameter. This behavior has been attributed to

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_9, © The Society for Experimental Mechanics, Inc. 2011

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molecular orientation and density variations across the nanofiber thickness [6]. Additionally, it was found that the nanoscale size of the polymer fibers gives rise to enhanced ductility and strength: Contrary to macroscale trends, the reduction in the nanofiber ductility with the applied strain rate is rather small while the nanofiber strength increases significantly with the loading rate [6,7]. The reduced molecular confinement in nanofibrous structures can account for these effects. Finally, an unusual and interesting mechanical behavior of polymeric nanomaterials has been shown to take place in polymeric nanofibers when structural heterogeneity is present: strain rate experiments have shown a reversal in the rate dependency of the flow stress at slow (10-4 s-1) as opposed to 10-2 s-1, or higher, strain rates. At slow rates the stress riser effect of structural heterogeneity was mediated by stress relaxation contrary to faster strain rates where structural heterogeneity resulted in deformation localization and periodicity.

2.

Strain Rate Behavior of Nanocrystalline Metallic Films

Equally interesting is the strain rate and creep response of nanocrystalline metallic films with grain sizes of 50 nm or smaller. Such films have received major attention in the recent years because of their large yield strength. Their small grain size promotes strengthening, which, in turn, results in high tensile strengths at the expense of reduced ductility and potentially toughness. An overlooked aspect of the mechanics of nanocrystalline metals is the increased strain rate sensitivity at reduced grain sizes and increased temperatures. The extended grain boundary network favors increased diffusion rates of defects and dislocations resulting in a significant and prolonged primary creep regime lasting for hours at pronounced primary creep rates [8]. This mechanical behavior has been the root cause for the long term performance degradation of metallic MEMS devices. Even when the latter are operated at otherwise elastic stresses, the appreciable primary creep rates result in considerable cumulative creep strain. The evolution of creep strain in nanocrystalline metals has been modeled via linear viscoelasticity in the past yielding satisfactory predictions [ 9 ]. More recently, such modeling based on creep experiments has provided predictions about the onset of plastic deformation at different strain rates [10]. Furthermore, relatively small increases in temperature have an equally significant, yet with opposite trends effects on the yield strength. At room temperature the effect of primary creep is evident in the mechanical response of films loaded at rates as high as 10-4 s-1. A small increase in temperature to 110°C further propels the effect of grain boundary induced creep affecting the mechanical response at rates as high as 10-1 s-1, which is faster that the rates encountered in most thin film applications. Finally, void nucleation, growth and microcracking have been shown to be rate dependent too demanding a three-dimensional description of damage nucleation and evolution. Experiments have shown that this evolution is dictated by the competition between creep/stress relaxation and the loading rate, which leads to a variety of failure modes due to distributed damage at slow strain rates, versus large void growth and localized damage at fast loading rates.

References [1] [2] [3] [4] [5] [6] [7]

M. Naraghi, I. Chasiotis, Y. Dzenis, Y. Wen, H. Kahn, “Novel Method for Mechanical Characterization of Polymeric Nanofibers,” Review of Scientific Instruments 78, pp. 0851081-6, (2007). K. Jonnalagadda, I. Chasiotis, J. Lambros, R. Polcawich, J. Pulskamp, M. Dubey, “Experimental Investigation of Strain Rate Dependence in Nanocrystalline Pt Films,” Experimental Mechanics 50 (1), pp. 25-35, (2010). M. Naraghi, I. Chasiotis, “Optimization of Comb-driven Devices for Mechanical Testing of Polymeric Nanofibers Subjected to Large Deformations,” Journal of Microelectromechanical Systems 18(5), pp. 1032-1046, (2009). M. Naraghi, T. Ozkan, I. Chasiotis, S.S. Hazra, M.P. de Boer “MEMS platform for on-chip nanomechanical experiments with strong and highly ductile nanofibers,” Journal of Micromechanics and Microengineering 20, pp. 125022-1-9, (2010). M. Naraghi, I. Chasiotis, Y. Dzenis, Y. Wen, H. Kahn, “Mechanical deformation and failure of electrospun polyacrylonitrile nanofibers as a function of strain rate,” Applied Physics Letters 91, pp. 151901-3, (2007). M. Naraghi, S. Arshad, I. Chasiotis, “Molecular Orientation and Mechanical Property Size Effects in Electrospun Polyacrylonitrile Nanofibers,” Polymer, (2011). M. Naraghi, I. Chasiotis, “Mechanics of PAN Nanofibers” in Major accomplishments in composite materials and sandwich structures. Editors: E.E. Gdoutos and I.M. Daniel, Springer, pp. 757-778, (2009).

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[8]

K. Jonnalagadda, N. Karanjgaokar, I. Chasiotis, J. Chee, D. Peroulis, “Strain Rate Sensitivity of Nanocrystalline Au Films at Room Temperature,” Acta Materialia 58, pp. 4674-4684, (2010). [9] S Hyun, T.K. Hooghan, W.L. Brown, R.P. Vinci, "Linear viscoelasticity in aluminum thin films," Applied Physics Letters 87, 061902 (2005). [10] F.V. Stump, N. Karanjgaokar, P.H. Geubelle, I. Chasiotis, “A multiscale model of rate dependence of nanocrystalline thin films,” in press in International Journal for Multiscale Computational Engineering (2011).

Study of Damage Evolution in High Strength Al Alloy using X-Ray Tomography Helena Jin1, Wei-Yang Lu1, Alejandro Mota1, James W Foulk III1, George Johnson2, Nancy Yang3, John Korellis1 1

Mechanics of Materials Energy Nanomaterials Sciences Sandia National Laboratories, Livermore, California 3

2

University of California, Berkeley, California

Many micromechanics-based damage models were developed to mimic the macroscopic response of materials through matching measures such as toughness and failure strain [1]. However, there is a lack of microstructural experimental data to identify the roles of the initiation, growth and coalescence of voids to damage and failure. This paper is aimed to experimentally investigate the microstructure of the material and understand the damage processes leading to failure. Experiments using X-Ray Computed Tomography (XRCT) 3D imaging technique with in-situ loading were conducted [2 - 5]. The material of interest is a rolled aluminum alloy, 7075-T7351, which shows the behavior of anisotropic ductile failure. To study the microstructure and damage evolution behind such phenomenon, smooth tensile specimens were extracted from three material primary directions, i.e. rolling (R), transverse (T), and short transverse (S). Figure 1 shows the aluminum plate and the original locations of the specimens. Figure 2(a) shows stress versus strain curves of these specimens. This alloy shows much more ductility in the rolling direction than in the transverse and short transverse direction.

Fig1: Specimen extrusion from different orientations;

F/A0(MPa)

400 300

UT_Roll UT_Trans UT_Short

200 100 0 0.00

0.04

0.08

0.12

ΔL/L0

Fig 2: (a) Stress versus strain curves; (b) A typical loading curve with marked scanned steps. X-Ray tomography scans were performed in-situ during the incremental steps of loading. In order to interrupt the test at the desired deformation states for scanning, two critical modifications were done to the existing beam-line loading frame. (1) A new laser confocal displacement sensor with submicron

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_10, © The Society for Experimental Mechanics, Inc. 2011

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resolution was installed to measure the crosshead displacement. (2) Tensile fixtures with ball joints were added for precise alignment. With these two improvements, we were able to take tomograms at a loading state very close to failure. Due to a large amount of time was needed per scan, only six tomograms were taken for each specimen. Typically, they were: the unloaded state, the yield point, one during hardening, the maximum load, one during necking, and after failure. The scanned states of the specimen loaded in the transverse direction are marked on the stress-strain curve shown in Fig 2(b). The tomograms were reconstructed using Octopus software. Figure 3 shows the void growth and coalescence from the horizontal slices for the transverse specimen, where the scan number corresponds to the marker in Fig 2(b). The voids growth and coalescence which leads to material failure can be viewed at steps of 5 and 6. Small sliced vertical sections were also selected to demonstrate various voids growth and coalescence mechanisms from specimens which are loaded in different directions, shown in Fig. 4. The voids initiation and growth appear to be closely associated with stringers. Scan 1: Original

Scan 2: Yield point

Scan 4: Maximum load

Scan 5: Necking

Scan 6: Failure

Fig 3: Void growth and coalescence from the horizontal view for the specimen loaded in rolling direction

 Rolling:

 













Short Transverse

 

Transvers









(a) (b) (c) Fig 4: Void growth and coalescence in the sliced vertical section for the specimens loaded in different directions; (a) selected small vertical section, (b) map of constituent particle, (c) map of voids

In summary, in-situ x-ray tomography tensile tests of uniform smooth specimens were performed to study the damage evolution of Al 7075-T7351. The data have provided insights to the voids growth

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and coalescence of the material. Comprehensive sets of tests will be performed to investigate the effects of stress triaxiality and loading orientation other than the material primary axes.

ACKNOWLEDGEMENTS Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000. The experiments were performed at Advanced Light Source, Lawrence-Berkeley National Laboratory.

REFERENCES 1. Lu, W.Y; Horstemeyer, M.F.; Korellis, J.S.; Grishabar, R.B.; Mosher, D., "High temperature sensitivity of notched AISI 304L stainless steel tests”, Theoretical and Applied Fracture Mechanics, v 30, p 139-152, 1998. 2. Mota, A., Jin, H., Lu, W. Y., Foulk, J. W., Johnson, G. C., “Quantifying the debonding of inclusions through tomography and computational homology”, SAND report, SAND2010-000UR. 3. Jin, H., Lu, W.Y., Mota, A., Foulk, J. W., Johnson, G. C., Yang, N., Korellis, J., “Investigation of the microscale damage evolution in high strength aluminum”, ASME International Conference of Mechanical Engineering and Exposition 2010, Vancouver, Canada. 4. Jin, H., Lu, W.Y., Mota, A., Foulk, J. W., Johnson, G. C., Yang, N., Korellis, J., “Experimental Study of Failure Mechanism in high strength aluminum”, SEM annual conference and exposition on experimental and applied mechanics, Indianapolis, Indiana. 5. Jin, H., Lu, W.Y., Mota, A., Foulk, J. W., Johnson, G. C., Yang, N., “Voids growth and coalescence study of Al 7075-T7351 using X-Ray tomography”, International symposium on Plasticity 2011, Puerto Vallarta, Mexico.

Corrosion Behavior of SS 304 with Ball Milling and Electrolytic Plasma Treatment in NaCl Solution

Mr. J. Liang, Dr. M. A. Wahab*, Dr. S.M. Guo Department of Mechanical Engineering, Louisiana State University, Baton Rouge, 70803 Email: [email protected] Abstract Electrolytic plasma process (EPP) was used as a fast annealing treatment on ball milled surfaces. With localized melting of EPP, the surface morphology and the chemical composition can be drastically changed. Annealed, ball milled and EPP treated stainless steel 304 samples were examined and EPP treatment has been shown to improve the overall corrosion and localized corrosion. Materials were characterized by X-Ray Diffraction, SEM. The 24 hours of exposure in NaCl solution, the sample open-circuit potential, potentiodynamic polarization, and electrochemical impedance spectroscopy were measured to estimate the anti corrosion properties. 1.

Introduction High strength and high corrosion resistance are the desired properties for modern engineered surfaces. Stainless steels are widely applied in industry owning to its good balance of strength and corrosion resistance. Recent studies indicate that the surface strength of stainless steel 304 could be significantly increased by heavy mechanical deformation [1-3]. The surface hardness increases with reduction of grain size, according to Hall-Petch effect. Stainless steel 304 surfaces also experience phase transformation from Austenite (Ȗ) to Martensite (Į’) during heavy surface deformation. These phases are metastable and transformable in SS304, based on the external conditions. The martensitie with b.c.c. structure acquires a much higher hardness than that of f.c.c. austenite and hence it is an important mechanism for surface hardening on SS304. Lee’s report [4] suggested that the heavy mechanical deformation on SS304 increased the surface roughness and the open-circuit potential drops. In the previous works, the annealing process was generally adopted to obtain nano-sized austenite for better corrosion resistance and reduce residual stresses from mechanical treatments. On the other hand, the annealing process affects the bulk material and consequently leads to the overall growth of grain and possible unwanted precipitation, which degrades the strength of the bulk material. Electrolytic plasma process (EPP) was developed from conventional contact glow discharge [5]. It’s fast heating and quenching mechanism was utilized to perform surface annealing treatment [6]. Comparing to other plasma related surface technologies, atmospheric working environment makes EPP an affordable surface cleaning and modification technique for large scale applications. In EPP treatment, a crucial step is to generate a gas layer on the metal surfaces by overcharging the target metal in an electrolyte. With large joule heating and chemical yield on metal, a continuous gas layer forms on the metal surfaces. Further increase the applied voltage, the electrons in the gas layer would be ionized and form low temperature plasma (> 2000K). Discharge occurs at a certain voltage threshold accompanied with a large current flow and heat dissipation. The surface heat rate was estimated to be around 300oC/sec during the plasma onset and discharge, but the period of plasma formation and discharge is about a few milliseconds [7]. The plasma sets in with radii in the order of microns [8]. Thus, the thermal effect on substrate material is much less, comparing to other annealing treatments. The open-circuit potential was improved by EPP upon the ball milled SS304 [6]. In current study, corrosion properties of ball milled and EPP treated SS304 surfaces were studied by long term open circuit potential scan, potentiodynamic polarization, and electrochemical impedance spectroscopy. Corrosion resistance analyses were conducted based on the experimental results from electrochemical tests and surface material characterization. 2.

Experimental The materials used in this study were sectioned from 3mm thick stainless steel 304 plate (wt%: 0.064C, 18.20Cr, 8.64Ni, 1.45Mn, <0.03S and Fe balance). The samples were machined into 1.5” diameter disks and initially polished by 400 grade emery paper to remove initial oxides and manufacturing marks. The polished samples were annealed at 1050oC for 30



T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_11, © The Society for Experimental Mechanics, Inc. 2011

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minutes and quenched in distilled water. By such treatment, austenite was retrieved and carbide precipitation was minimized. The surfaces of the samples were lightly polished by 600/1200 grade emery paper to remove the external oxides due to quenching. Sample surfaces were subjected to ball milling treatment in a SPEX 8000 mill for 40 min. The samples were then positioned at one end of a cylindrical cup (of same material) with 1.5” diameter and 2.5” height. 40 1/4” balls of 302 stainless steel type were used inside the cylinder. Electrolytic plasma processes were performed on ball milled samples. All EPP samples were cathodically connected with a DC power source (Magna PQA 6.6kW) in NaHCO3 electrolyte (7.5wt%, equivalent electrical conductivity of 43.1±0.5 Sƒcm-1), the anode was a much larger stainless steel plate. By using a large anode plate, the plasma is confined to the vicinity of the cathode [9]. The effective contact area between the cathode and the electrolyte is 1±5% inch2; the contact area varied during EPP due to the dynamic instability near liquid-air boundary. The applied voltage was set at 130V; the increase rate was default set at about 10V/sec. Current rises initially and then converges to a stable value (2±0.5A) after the plasma discharge is stabilized on the cathode surface. The EPP process time was 30 minutes. All solution temperature started from room temperature. It took approximately one minute to reach the stable plasma discharge. Afterwards, all samples were cleaned in ethanol alcohol with ultrasonic bath for 5 minutes. Material characterization and electrochemical based corrosion tests were performed on three groups of samples, including annealed, ball milled (40minutes) and ball milled (40 minutes) followed by 30 minutes EPP. Annealed, BM and BM+EPP are respectively referred for these groups in the following discussion. Surface microstructures were characterized by XRD (Rigaku, MiniFlex, Cu Kɲ, ʄ=0.154nm) and analyzed with JADE 5. The surface morphologies were studies by field emission scanning electron microscope (FE-SEM) and surface roughnesses were measured by TENCOR P11 surface profiler. The electrochemical studies were conducted with a CHInstruments electrochemical workstation 600C. Open circuit potential was measured over 24 hours for different groups. The potentiodynamic polarization were performed after 1-hour exposure of corrosion medium (3.5% NaCl) and electrochemical impedance spectroscopy (EIS) measurements were conducted at 0.5 hour, 1 hour, 2 hour, 5hour and 10 hour of immersion in order to examine the corrosion property evolution over time. The 3.5% NaCl solution for all the tests were deaerated by bubbling N2 gas for 1 hour. The testing procedures were also guided by ASTM G5-94 and G3-89. 3. Results and analysis 3.1 Material characterization (XRD) The annealed, ball milled and EPP treated samples were inspected with XRD and the diffraction patterns are shown in Fig 1. Assuming austenite (Ȗ) and martensite (Į’) are the only two phases in sample, the volumetric fraction of martensite could estimated by a direct comparison method [10]. The calculation for the volume fraction of martensite (fɲ’) is conducted through the following equation. σ ூംሺ೓ೖ೗ሻ σ ஼ ᇲሺ೓ೖ೗ሻ ഀ ሿିଵ ூ ᇲሺ೓ೖ೗ሻ σ ஼ംሺ೓ೖ೗ሻ

݂ఈᇱ ൌ ሾͳ ൅ σ

(1)

ഀ

Where Ij is the relative intensity jth phase and could be measured from the integrated area of the relevant peak. Cj is the factor related with Bragg angle of diffraction. hkl is the miller indices for the plane direction vector. In this calculation, only the following peaks were used (Ȗ (200), Ȗ (220), ɲ’ (200), ɲ’ (211)) in order to avoid the overlapping of peaks. Annealing treatment provided austenitic (Ȗ) initial state (fɲ’=0.15%). Ball milling effectively converted surface microstructure into large amount of martensite (fɲ’=40.2%). Also, very much widened peaks (Fig 1(b), Ȗ (200)) suggest ultrafine grain size. With further 30 minutes EPP treatment on ball milled samples, the martensite volume fraction reduced to 6.01%.

Fig 1 X-ray diffraction patterns: (a) annealed; (b) ball milled 40 minutes; (c) ball mill 40min + Epp 30min

 

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3.2 Open Circuit potential over time All the samples were exposed in 3.5wt% NaCl solution for 24 hours and the open-circuit-potentials (OCP) were recorded (Fig 2). All groups had 2 characteristic passivation cycle, indicated by arrows in Fig 2. The annealed sample had very quick passivation after immersion and OCP reached -0.15V at about 2 hours, then its OCP dropped at a rate of 0.0125 V/hr for 8 hours. The reduction of OCP on annealed sample indicated a depassivation process and such processes were attributed to the penetration of Cl- through initial chromium oxide layer. Following the slow depassivation, an accelerated breakdown of passivation film indicated by a steeper drop (0.04V/hr) in OCP proceeded until it reached a minimum level, -0.58V. The repassivation produced a second plateau (-0.52V). The second passivation period took 4 hours to decay. The BM sample had a similar pattern of evolution in OCP as the annealed one, but its first passivation kept at a much lower potential level (0.23V) and sustained for less time (3 hours). It’s believed that the surface roughness played an important role. It was shown in table 1 that the surface roughness of BM sample was five times larger than the annealed ones. The second passivation on BM sample also took place and kept at roughly the same level as the annealed one. The OCP for BM+EPP sample initially dropped at a very high rate. But, after surfing through a small repassivation-depassivation cycle (1 hour), it restored quickly after 2 hours. Although EPP generated the highest surface roughness level, BM+EPP samples manifested a sustainable passivation from 6th hour and high OCP was kept till the end of the test. The integrity and good sealing of protective layer on EPP treated sample are related to its special composition resulted from localized high temperature treatment.

Fig 2 Open-circuit-potential over 24 hours of immersion in 3.5% NaCl solution for anneal, ball milled and EPP treated samples Table 1 Surface roughness for different treatment groups. Scan speed: 400ȝm/sec, sampling rate: 100Hz Unit (ȝm) Ra Rq Rt

Annealed 0.385 0.472 2.565

BM 1.669 2.085 13.071

BM+EPP 1.888 2.345 13.272

3.3 Potentiodynamic polarization test The potentiodynamic polarization was utilized to examine the corrosion potential and protection potential (Ep). Tests were conducted after one hour of insertion into the solution. Figure 3 is a selected plot of current density vs. EOCP. The averaged results of potentiodynamic polarization tests are listed in Figure 4. The EPP treated sample exhibited highest averaged EOCP and Ep values, although its initial roughness is the highest. Such performance could be related to the high temperature treatment from EPP. On the other hand, BM samples yield the lowest EOCP and Ep values due to the larger surface roughness and possible galvanic effect between austenite and martensite phases [11]. But BM samples have the lowest corrosion current on average. The annealed sample did not have significant passivation but the protective layer could sustain till higher potential (Fig 3).

 

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Fig 3 Potentiodynamic polarization. Scanning range: -0.6 to 1.6 Volts; scan rate: 1mV/sec Since the protection potential could be used as a measurement of pitting corrosion, the initial and corroded sample surfaces from potentiodynamic polarization tests were examined by SEM. Figure 5(a), (c), (e) shows the initial surface morphology of annealed, BM and BM+EPP samples. Annealed samples remain clear polishing marks and BM sample surfaces have cracks due to heavy impingement of steel balls. EPP samples contain typical spheroidal extrusion and craters from melting and quenching process. Figure 5(b), (d), (f) are corroded surfaces. Low magnifications were used to view the whole picture of corroded morphologies. The annealed samples have most severe pitting after the polarization cycle. Exfoliation of the surface crust was clear on Fig 5(b). The amount of pits on BM sample is less than the annealed ones, the ball milling process effectively refined the surface grains and the increase of grain boundary density would promote the diffusion of Cr [12] and possibly insist the continuous formation of Cr oxide protective layer. Nevertherless, cracks were also observed on BM samples. On EPP samples, the pits were smaller but more in amount. It was reported that the light annealing or heat treatment on sandblasted steel could lead to increase of OCP [2], which was proved by this study (Fig 4). The EPP induces much higher temperature than conventional annealing temperature on the steel surfaces. Such high temperature leads to the precipitation and depletion of Cr during melting of surface materials. Tables 2 are the statistical atomic percentage distribution of surface chemical element for different groups. EDAX was used for this measurement. The BM sample has very similar element distribution as the annealed one, except chromium and oxygen percentage is slightly higher and matches the reported values [2]. Three selected measurements from different spots BM+EPP samples were listed. The bulk result of EPP sample shows there is a 50% increase in carbon and 18% decrease in chromium, compare to the annealed sample. On the spheroids, which are the solidified molten material, the depletion of chromium is very clear (-61%). In the valleys (Fig 5 (e)), the atomic percentage of chromium recovered to the similar level as annealed sample’s. From this observation, the pits on EPP treated samples could be considered to initiate from those chromium depleted molten spots.

Fig 4 Averaged corrosion potentials and current densities from Potentiodynamic polarization tests

 

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Fig 5 SEM for initial and corroded surface: (a-b) annealed; (c-d) BM; (e-f) BM +EPP

 

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Element C O Cr Mn Fe Ni

Table 2 EDAX results for element distribution on initial surfaces Annealed BM BM+EPP BM+EPP (bulk) (bulk) (bulk) (spheroid) 07.98 06.63 10.14 22.29 01.25 02.66 01.05 00.59 16.47 16.64 13.48 06.38 01.33 01.68 01.28 00.52 65.36 63.85 65.28 57.95 07.60 08.54 08.77 12.28

BM+EPP (valley) 05.85 01.04 16.34 01.40 66.10 09.28

3.4 Electrochemical Impedance Spectroscope (EIS) The electrochemical impedance test was performed at OCP ±0.002V for each indicated time point. Nyquist plots (Fig 6) are used to illustrate the evolution of corrosion properties. At initial exposure (0.5h) stage, annealed sample has a large capacitive loop (radius of 3.5 x103 Ÿ/cm2), which indicates a thicker protective layer. The loop radius reduced to half of its initial level after 2 hours. The reduction of corrosion resistance was less significant after 5 hour immersion. BM sample exhibited larger corrosion resistance (radius of 2.5x104 Ÿ/cm2), one order of magnitude large than the annealed sample. This behavior agrees with the suggestion of faster chromium diffusion towards surface after heavy mechanical impingement [12]. A small inductive loop was observed at 1h for BM sample. This indicates a fast breakdown of passive film. It could be explained that the refined surface provide larger amount of corrosion path than annealed case, once the initial chromium oxide film was penetrated, the consequent deterioration is very fast. It could also be seen from Fig 2, where the OCP decays much faster at the initial few hours of exposure. The corrosion resistance reaches minimum after 5 hour of exposure and the corrosion resistance restored at 10 h, which could also be confirmed from Fig 2. A second passivation starts about 10 hour. EPP treated samples had a very different behavior from the previous two. The initial corrosion resistance (0.5h) was the lowest (radius of 1.3 x103 Ÿ/cm2). It is because of the larger surface roughness and the chromium depletion at some spots like spheroids from melting. The corrosion resistance experienced a rise after 1 hour, and then the loop had a slight shrinkage after 2 hour, which corresponds to the first small peak of EPP curve on Fig 2. Significant rise of corrosion resistance after 5 hour could be seen from Nyquist plot, which could also be checked in Fig 2. The ball milled surface experience fast melting and quenching through EPP. It could be observed that the surface crust of EPP treated surface did not contain large cracks under anodic loading (Fig 5(f)). This implies good structural integrity of the top layer and secondary layer could be assumed to acquire a similar structure as BM sample. The good structural integrity of the top layer effectively sealed the chromium oxides underneath it. And hence, provides a sustainable protection against corrosion.

 

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Fig 6 Nyquist plots for annealed, BM and BM+EPP samples 4.

Conclusion In this study, the electrolytic plasma process on SS304 was proved to have longer sustaining corrosion resistance in 3.5% NaCl solution. EPP annealing could effectively convert surface martensite back to austenite. Although the EPP treatment yields higher surface roughness, its sealing mechanism on top surface effectively retain the substrate protective layer and hence provides a good corrosion resistance. EPP treatment also shows better pitting corrosion resistance than the annealed and BM cases because no significant cracking and exfoliation are found. An explanation for this improvement of corrosion resistance in NaCl solution was proposed. The special structure from ball milling and consequent EPP treatment provides a sustainable corrosion barrier on 304 stainless steel surface. The promising corrosion application of this surface modification treatment is worth for further study. Acknowledgements This project is funded by NASA EPSCoR NNX09AP72A and the first author is supported by LSU-EDA fellowship. Reference [1] Ni Z, Wang X, Wang J, Wu E, 2003, Characterization of the phase transformation in a nanostructured surface layer of 304 stainless steel induced by high-energy shot peenin, Phys B, pp. 221-228 [2] Zhang H, Hei Z, Liu G, Lu J, Lu K., Formation of nanostructured surface layer on AISI 304 stainless steel by means of surface mechanical attrition treatment, Acta Mater, Vol. 51, pp. 1871-1881, 2003. [3] Wang X, Li D., Mechanical and electrochemical behavior of nanocrystalline surfaceof 304 stainless steel. Electrochimca Acta , Vol. 47, pp. 3939-3947, 2002. [4] Lee H, Kim D, Jung J, Pyoun Y, Shin K., Influence of peening on the corrosion properties of AISI 304 stainless steel. Corrosion Science, Vol. 51, pp. 2826–2830, 2009. [5] Yerokhin X Nie, Leyland A, Matthews A, Dowey S., Plasma electrolysis for surface engineering, Surf Coat Technol, vol,122 pp.73̢93, 1999. [6] Liang J, Wang KY,Guo SM, Wahab MA., Influence of electrolytic plasma process on corrosion property of peened 304 austenitic stainless steel, Materials Letters, Vol. 65, pp. 510-513, 2010. [7] Gupta P, Tenhundfeld G, Daigle E, Ryabkov D. Electrolytic plasma technology: science and engineering ̢ an overview. Surf Coat Technol, vol.201, pp.8746̢60, 2007 [8] Dunleavy I Golosnoy, Curran J, Clyne T, Characterization of discharge events during plasma electrolytic oxidation, surfCoat Techol, vol. 203, pp.3410-3419, 2009. [9] Sengupta, S.K., Singh, O.P., Contact glow discharge electrolysis: a study of its onset and location, Journal of Electroanalytical Chemistry, vol. 301, no. 1-2, pp 189-197, 1991, [10] Cullity B, Stock S. Elements of X-Ray Diffraction, Prentice-Hall, 2001. [11] Mordyuk BN, Prokopenko GI, Vasylyev MA, Iefimov MO, Effect of structrure evolution induced by ultrasonic peening on the corrosion behavior of AISI-321 stainless steel, Materials Science & Engineering A, vol. 458. pp.253-261, 2007.

 

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[12] Wang ZB, Tao NR, Tong WP, Lu J, Lu K, Diffusion of chromium in nanocrystalline iron produced by means of surface mechanical attrition treatment, Acta Materialia, vol. 51, pp. 4319-4329, 2003.

 

Experimental Studies of Phase Transformation in Shape Memory Alloys

Kyubum Kim, Samantha Daly Department of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109

ABSTRACT This paper presents experimental studies to examine stress-induced martensitic phase transformation during the superelastic deformation of the shape memory alloy Nickel-Titanium. The phase transformation, which is solid-to-solid and diffusionless, occurs between austenite, a B2 cubic crystal structure, and martensite, a monoclinic crystal structure during loading and unloading at constant ambient temperature. To examine the complex local thermo-mechanical interactions that affect transformation behavior, we utilize a temporally- and spatially- simultaneous combination of strain and thermal mapping using three-dimensional digital image correlation and infrared thermography, respectively. This combined experimental approach enables full-field, quantitative maps of strain and temperature fields over the specimen surface, allowing the investigation of factors including cycling, strain rate, texture, and local temperature variations. The effects of these factors on fundamental transformation properties, such as the stresses required for phase nucleation and propagation, accumulated residual plastic strain, total strain accommodated by phase transformation, the evolution of martensitic volume fraction, and the amount of hysteresis, are discussed. INTRODUCTION Shape memory alloys (SMAs) have been widely studied because of their high work density. NiTi, CuZnAl, CuAlNi, AuCd and other alloys are classified as SMAs. Among them, NiTi (Nitinol) is under rigorous investigation, since it is suitable for many applications, notably in the biomedical, aerospace, and civil engineering fields. SMAs have two unique properties, known as the shape memory effect and superelasticity (pseudoelasticity). The shape memory effect refers to the reversion of the material to its original shape after deforming, releasing and heating it above a critical temperature, known as the austenite finish temperature (Af). Superelasticity is the ability of the material to recover large amounts of strain in an elastic manner. Both the shape memory effect and superelasticity are caused by a solid-to-solid and diffusionless phase transformation between a cubic austenite phase and a monoclinic martensite phase, and are the result of different transformation paths between the austenite and martensite phases. In spite of intensive research for more than 50 years, there is still much unknown about the complex local-thermo mechanical interactions that underlie phase transformation in this alloy. The ambient and local temperature, applied strain rate, and cycling show complex interactions with each other that are reflected in the macroscopic behavior. Many of the notable experimental results detailing various aspects of the behavior of this alloy are qualitative in nature and do not show the fullfield strains. Thus, quantifying important parameters such as velocity of the phase front and the martensitic volume fraction has been difficult. These results are meant to fill this gap through the combination of simultaneous full-field strain mapping by three-dimensional digital image correlation (DIC) and full-field thermal mapping by infrared thermography. DIC is an insitu optical technique used to quantify Lagrangian surface strains by tracking surface markers. The full-field strain map obtained by DIC facilitates quantitative characterization of the nucleation and propagation of the stress-induced martensitic phase transformation. In addition, the interaction between latent heat and the extent of transformation is observed by overlaying infrared thermography with local strain values. These results assist in understanding the phase transformation

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_12, © The Society for Experimental Mechanics, Inc. 2011

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82 mechanism, including the relation between stress, strain, and temperature during phase transformation, and how important factors, such as the strain rate and latent heat, affect transformation. LITERATURE BACKGROUND Previous research (see for example [1-3] and the references therein) shows that martensitic phase transformation in Nitinol is inherently sensitive to both local and global temperature in the material. At temperatures T < Af, the initial phase is martensite and the superelastic effect is not observed. At temperatures T > Af, the initial phase is austenite, and phase transformation from austenite to martensite can be stress-induced at a constant ambient temperature. As the temperature increases, the stress necessary to nucleate phase transformation increases. The latent heat that is released (austenite to martensite) and absorbed by (martensite to austenite) by the stress-induced martensitic phase transformation also affects transformation characteristics due to local self-heating. Tests on the effects of mechanical cycling have determined that hard cycling causes a decrease in the applied stress needed to nucleate the martensite phase (σNM) [4-7]. It is hypothesized that this decrease in σNM is caused by a change in dislocation structure and the retention of martensitic nuclei that helps form the stress-induced martensite in the subsequent cycles [8]. Thus, as cycling progresses, the stresses for the nucleation and propagation of the martensitic bands during loading decreases significantly, but the stresses for austenitic nucleation and propagation during unloading remain nominally the same. Another noticeable effect of cycling is the stabilization of the stress-strain curve [4]. One postulate for stabilization is that dislocations accumulate around defects as cycling increases. Accumulated dislocations also contribute to the accumulations of the residual strain and increase internal stresses during cycling [2]. Because the higher internal stress state helps transform the stressinduced martensite, the stress for the nucleation of martensite decreases as cycling increases. The degree of stability can be characterized by factors such as the amount of accumulated residual deformation, the critical stress for nucleation of the martensitic bands, and the amount of hysteresis. Those factors stabilize as cycling increases, and the macroscopic curve can be considered as nominally constant after a certain number of cycles. Though the amount of residual deformation increases as the number of cycles increases, and the amount of hysteresis decreases, the incremental rates of these factors decrease as cycling progresses. The applied strain rate is an important parameter in determining the quality of the phase transformation, and is closely related 5 1 to the local self-heating and heat transfer. At slow strain rates (e.g. H 10 05 s ), the stress plateau that the macroscopic stressstrain curve exhibits during phase transformation is relatively flat. Typically only one martensitic band nucleates and propagates at slow strain rates, because the temperature increase is small and the strain rate is sufficiently low to allow heat to 2 1 escape [1]. At a faster applied strain rate (e.g. H 10 02 s ), the stress plateau during martensitic transformation becomes inclined, and more than one martensitic band can nucleate and propagate due to a high local temperature increase and a buildup of thermal energy. Thus, the martensitic phase transformation is more homogeneous at faster strain rates. Previous research shows that, at very fast strain rates, the phase transformation changes from a solid-to-solid, diffusionless and shearlike mechanism to a slip mechanism that is based on dislocation slip. It has been shown that when Nitinol is subjected to an applied compressive strain rate of 4200 s-1, the constant stress plateau disappears and the sample behaves in a purely austenitic manner [4]. The stress required for martensite nucleation and the amount of dissipated energy increase, but the stress for reverse phase transformation and the strain energy decreases as the strain rate increases ( H t1.667 t1.667u103s1) [1, 9]. 4 1 The strain rate effect was not shown at H d3.333 d3.333u10 s [9]. EXPERIMENTAL METHODOLOGY A cold-rolled flat polycrystalline Nickel-Titanium thin sheet, which has dimensions of 63.5 mm wide x 3048 mm length x 0.254 mm thickness, was received by Nitinol Devices & Components. The composition of the sample was 55 wt% Nickel and 45 wt% Titanium. The sheet was manufactured to have an austenite finish temperature Af = 2°C to ensure that the material was austenite at the ambient testing temperature. Differential scanning calorimetry (model TA-Q200) was used to measure the Af independently, and resulted in a value of 1.76 ± 0.45 °C. The dominant plane of the rolled sheet is NiTi (110) and the grain size of the plane is 45.8 nm, which was measured by X-ray diffraction using a Rigaku Rotating Anode X-Ray Diffractometer and Jade software. Dog-bone shape specimens following standard ASTM E8 were cut from the sheet along the rolling direction by electric discharge machining (shown in Figure 1), which prevented any property change of the material due to residual stress and temperature.

83 Full-field and quantitative maps of local strain and local temperature were obtained by digital image correlation (DIC) and infrared thermography, respectively. DIC is an in-situ optical method used to calculate surface displacements, and thus Lagrangian strains, by tracking a highly random and contrasting surface pattern. In this experiment, air brushing (model Iwata Custom Micron B) produced a fine, random and isotropic pattern at the appropriate length scale. A thin white titanium oxide layer was first applied and then followed by a random speckling with carbon black in order to produce a high contrast pattern. Two five megapixel gray scale CCDs (model Point Grey GRAS-50S5C) were used, each with a 2048x2448 pixel field of view and a spatial resolution of 9 Pm/pixel at the chosen test field of view. An infrared camera, (model Inframetrics ThermaCam SC 1000) was placed between the two CCDs to get simultaneous and corresponding two-dimensional temperature map of the surface of the sample. Hard cycling tests were performed under a ramp profile using a 45-kip Fig. 1 Tensile specimen geometry Instron uniaxial load frame at three applied strain rates, H 10 ,10 and 10 s . The specimen was cycled 50 times and the 1st, 2nd, 5th, 10th, 25th and 50th cycles were recorded. The load values were obtained from the Instron load cell (model Instron 2525-805, maximum 1000 lb) through a custom LabVIEW data acquisition system collecting load, displacement, and the start time of CCD and IR images. The applied load, the acquisition time of the CCD images and the starting time of IR camera were recorded by the LabVIEW data acquisition system, with a trigger on the time domain. When CCD images were taken, the electric signal output dropped from 5 V to 0 V. When the IR camera began to capture thermal images, the trigger (connected between IR camera and LabVIEW) sent a signal to the LabVIEW data acquisition system to increase the signal output from 0 V to 5 V. The full-field strain was calculated by commercial VIC-3D software from Correlated Solutions, Inc. Through the use of this experimental setup, the full-field and quantitative maps of surface strain and temperature were obtained simultaneously and the properties of the stress-induced martensitic phase transformation were characterized. 2 2

3

4

1

RESULTS AND DISCUSSION Three macroscopic stress-strain curves of Nitinol during cycle 1, with simultaneous images of the strain and temperature fields at the gage section at H 10 ,10 and 10 s , are shown in Fig. 2. The strain values in the macroscopic stress-strain curve were calculated by averaging the values of approximately 600,000 pixels in the gage section. Fig. 2 shows the maps of local strain and temperature at eight points on the stress strain curve, during the first loading cycle, for three applied strain 2 2

3

4

1

rates ( H 10 ,10 and 10 s ). In Fig.2 (a), the initial phase of the specimen is austenite. When the applied load reaches the critical stress for phase transformation, a large localized martensitic band of strain nucleates at point 2. Once phase transformation begins, the martensitic band propagates throughout the entire specimen until point 3, during which time the applied stress stays constant. After point 4, the martensitic band has completely propagated through the specimen and we observe elastic behavior between points 3 and 4. As the applied load is decreased, the stress-induced martensite reverts back to austenite because of the instability of martensite at this temperature. Small peaks in the stress, which are shown in Fig. 2(a) during loading (point 2) and unloading (point 6), indicate the critical stresses for the nucleation of the martensite or austenite phases. During loading, the transformation of the austenite phase to the martensite phase is affected by intragranular constraints. When the stress becomes large enough to overcome these constraints, localized martensitic bands of strain can nucleate. The existence of small stress peaks during unloading is due to a similar mechanism. Note that stress concentrations at the grips can facilitate phase transformation at the grip, which can effectively hide the nucleation peak. For example, the macroscopic stress-strain curve of a straight Nitinol wire does not have the peak [10]. The stress peak is also affected by strain rate and cycling. As the strain rate increases, the stress peak disappears because of internal friction resistance that interrupts the propagation of martensitic band. In addition, the peak diminishes as cycling increases [9]. Because the phase transformation is 1st order, the martensitic band incurs latent heat, which is shown in the temperature map in Fig. 2. 2 2

3

4

1

4 1 4 At an applied strain rate of H 10 s , only one primary martensitic band nucleates and propagates and the slope of the transformation plateau is approximately 0 degrees as seen in Fig. 2(a). As the strain rate increases, more martensitic bands nucleate and the transformation plateau becomes increasingly inclined, as shown in Figs. 2(b) and (c). In Fig. 2(b), two martensitic bands nucleate and propagate during phase transformation in the strain field images. The temperature peaks are

84 observed at the end of martensitic bands in the thermal images and the peak temperature is higher than in Fig. 2(a). The phase transformation plateau in Figure 2(b) is inclined with the slope of approximately 1094 MPa. At an applied strain rate of 2 1 H 10 2 s , approximately five martensitic bands nucleate, the temperature peak is more than 50 °C and the phase transformation plateau is the most inclined with the slope of approximately 2939 MPa. Observation of the strain maps at each strain rate shows increasing homogenization, and temperature maps show higher local and global temperature as the strain rate increases.

Fig. 2 Macroscopic stress-strain curves with full-field maps of strain and temperature (a) (c)

H

H

4 1 10 4 s , (b) H

3 1 10 3 s and

2 1 10 2 s

The reason that different numbers of martensitic bands nucleate is closely related to the latent heat [1]. Latent heat has a major role in phase transformation, as it can increase or decrease the stress required for transformation. In Fig. 2, the local temperature during loading increases as the applied strain rate increases, due to the latent heat that is released during the martensitic phase transformation. In order to see the effect of latent heat, the average temperature on the surface of the specimen was calculated and plotted on the macroscopic stress-strain curve in Fig. 3. In Fig. 3, the red line with round dot is the average temperature of the gage section and the blue line with the triangle dot is the peak temperature, taken as the maximum temperature during loading and the minimum temperature during unloading. When the martensitic band nucleates, the released latent heat increases the temperature of whole specimen. However, at a slow strain rate (Fig. 3(a)), there is a relatively small amount of latent heat which escapes easily, and the average temperature drops even though phase transformation is proceeding. As the strain rate increases (Fig. 3(b) and (c)), more latent heat occurs and is not able to escape to the environment, thus the average temperature increases. The increment of temperature makes the critical stress for phase transformation high, as seen through the Clausius-Clapeyron relation. Thus, the critical stress increases during transformation and the slope of plateau becomes stiffer than at low strain rates. The phenomena of the strain rate effect on the critical stress and temperature is shown clearly in Fig. 3. From the result, the ratio of the critical stress to critical temperature is obtained as:

dV * dT *

12.79 MPa / qC at H

103s 1 (1)

dV * dT *

6.36 MPa / qC at H

102s 1 (2)

2 1 s in Fig. 3(c) should be ignored, Note that the small region marked by an arrow on the average temperature plot of H 10 2 as the highest temperature at this region is more than 50°C which cannot be measured because of equipment limitations.

85

Fig. 3 Macroscopic stress-strain curve with corresponding average temperature and peak temperature of the specimen at applied strain rates (a)

H

4 1 10 4 s , (b) H

3 1 10 3 s and (c) H

2 1 10 2 s

Six hard cycles (1st, 2nd, 5th, 10th, 25th, and 50th cycles) were recorded by CCDs and IR imaging for three globally applied strain rates, H 102,103 and104s1. Macroscopic stress-strain plots of these cycles at each strain rate are shown in Fig. 4. Note that the phase transformation becomes increasingly homogenized as cycling increases. In a recent paper by the authors [11], the effect of cycling is discussed in detail, particularly evidence of a microscale pattern memory in the manner that the martensite is accommodated, both within a loading cycle and from cycle to cycle. Comparison of Fig. 4(a), (b) and (c) shows the effect of strain rate, and each plot individually shows the effect of cycling at a chosen strain rate. Both faster strain rates and cycling result in increased homogenization of the phase transformation. In Fig. 4, the macroscopic stress-strain curve of 2 1 s has the smoothest shape, smallest stress required to nucleate phase transformation, the largest 50th cycle at H 10 2 amount of residual strain, and the largest amount of hysteresis. This is due to increased internal stress, accumulated dislocations around defects, and the highest local and global temperature at this strain rate because of the accumulation of released latent heat.

Fig. 4 Macroscopic stress-strain curves of all cycles at applied strain rates (a)

H

H

4 1 10 4 s , (b) H

3 1 10 3 s and (c)

2 1 10 2 s

One of the advantages of this experimental approach is that the nucleation stress of the martensitic band can be accurately obtained through full-field strain mapping. However, the exact nucleation stress cannot be determined at select points, due to the inherently fuzzy and criss-cross character of the martensitic band under certain loading conditions (for example, at a th 2 1 s and the 50 cycle, where transformation is highly homogenized). Instead of comparing the critical strain rate of H 10 2 stress, we therefore compare the amount of dissipated energy in a cycle, in Fig. 5. When the specimen is under applied load, energy is dissipated, and then absorbed in unloading as evidenced by the release/absorption of latent heat. The amount of dissipated (loading) and absorbed (unloading) energy is shown in Fig. 5(a) and (b). The hysteresis is obtained as the difference of the dissipated and absorbed energies, and is shown in Fig. 5(c). In the first cycle, the largest dissipated energy is 2 1 s and quickly drops as cycling increases. A significant amount of energy is dissipated by the latent heat observed at H 10 2 at this strain rate in the first cycle, but a large decrease in the stress for phase transformation dominantly affects the dissipated energy, lowering it in the fiftieth cycle due to accumulated martensite retention. Thus, in Fig. 5(a), temperature dominates the

86 stress required for transformation in cycle 1 at different strain rates, but the dominant reason for the decrease in nucleation stress within a strain rate can be attributed to the amount of accumulated martensite retention. These different dominant 3 1 3 2 1  4 1 s crossing the H 10 s and H 10 4 s lines. However, the dissipated energy at reasons result in the line of H 10 2  4 4  1 3 1 3 strain rates H 10 s and H 10 s has nearly same character. The absorbed energy at the three different strain rates 3 1 s is shifted during unloading shows a similar pattern, but note that the absorbed energy at an applied strain rate of H 10 3 vertically to a lower value. This corresponds with previous research, which has found that reverse phase transformation is 3 1 s is lower than other strain rates is that the maximum minimally affected by cycling. The reason that line of H 10 3 3 1 s should be vertically elongation is shorter. Because of this, the amount of both dissipated and absorbed energy of H 10 3 shifted upwards. However, because the strain level at which the specimen gage section can be considered as fully martensite depends on the applied strain rate, it is difficult experimentally to set the same maximum elongation. In Fig. 5(c), the amount 3 1 s can be compensated because the hysteresis is calculated by sum of them. of dissipated and absorbed energy at H 10 3 Thus, the overall tendency of the hysteresis is similar to the dissipated energy because of the minimal effect in the reverse phase transformation (unloading).

Fig. 5 Strain rate effect on hysteresis (a) energy dissipation during loading, (b) energy absorption during unloading, (c) hysteresis during one cycle

As seen in Fig. 4, the residual strain (ε P) accumulates as cycling increases. As cycling increases, dislocation hardening results in a rise of the stress required for slip (σS), but the transformation stress (σNM) decreases for the reasons previously discussed. Thus, the transformation stress becomes dominant as cycling increases. The increment of residual strain decreases as cycling increases, though the total amount of the residual strain increases. Secondly, the amount of incurred residual strain (εP) increases as the applied strain rate increases. Fast strain rates cause both an overall and local temperature increase of the specimen. By the Clausius-Claypeyron relation, the temperature increase causes the transformation stress to increase. However, the resolved shear stress for slip decreases as temperature increases because of thermal activation of dislocation motion. As a result, the relative difference between σS and σNM becomes lower as the strain rate increases [5, 10]. Consequently, cycling decreases the amount of residual strain, but increasing the applied strain rate acts to Fig. 6 Residual strain at applied strain rates 4 1 3 1 increase the residual strain. The cycling and strain rate effects on the (a) H 10 4 s , (b) H 10 3 s and (c) residual strain are shown in Fig. 6. In this figure, the increment of the 2 1 2 residual strain at each strain rate decreases as cycling increases, and a H 10 s larger amount of residual strain is evident as the strain rate is increased. CONCLUSIONS In this paper, the effects of cycling and strain rate in stress-induced martensitic phase transformation during superelasticity in shape memory alloy, Nitinol, are examined through full-field maps of Lagrangian strain and temperature fields by simultaneous digital image correlation and IR cameras. Different amounts of latent heat at various strain rates incur a global

87 and local temperature increase of the specimen and affect the stress for phase transformation, which leads to interesting results in the number of the martensitic bands, the slope of the plateau during phase transformation, the energy dissipation and the residual strain. The stress for phase transformation and the average and local temperature of the specimen increase as the strain rate increases, and the relation between the stress and temperature is coincident with Clausius-Clapeyron relation. 2 1 s , To examine both effects of strain rate and cycling, the amount of hysteresis and residual strain are calculated. At H 10 2 the largest hysteresis occurs and decreases rapidly as cycling increases. The residual strain increases as the strain rate increases, but decreases as the cycling increases due to changes in the stresses required for phase transformation and slip.

REFERENCE [1] Shaw, J.A., Kyriakides, S.: Thermomechanical aspects of NiTi. Journal of the Mechanics and Physics of Solids. 43, 8, 1243-1281 (1995). [2] McCormick, P., Liu, Y.: Thermodynamic analysis of the martensitic transformation in NiTi--II. Effect of transformation cycling. Acta Metallurgica et Materialia. 42, 7, 2407-2413 (1994). [3] Shaw, J.A.: Simulations of localized thermo-mechanical behavior in a NiTi shape memory alloy. International Journal of Plasticity. 16, 5, 541-562 (2000). [4] Nemat-Nasser, S., Guo, W.: Superelastic and cyclic response of NiTi SMA at various strain rates and temperatures. Mechanics of Materials. 38, 5-6, 463-474. (2006) [5] Strnadel, B. et al.: Cyclic stress-strain characteristics of Ti-Ni and Ti-Ni-Cu shape memory alloys. Materials Science and Engineering A. 202, 1-2, 148-156 (1995). [6] Strnadel, B. et al.: Effect of mechanical cycling on the pseudoelasticity characteristics of Ti---Ni and Ti---Ni---Cu alloys. Materials Science and Engineering A. 203, 1-2, 187-196 (1995). [7] Sehitoglu, H. et al.: Cyclic deformation behavior of single crystal NiTi. Materials Science and Engineering A. 314, 1-2, 67-74 (2001). [8] Brinson, L.C. et al.: Stress-induced transformation behavior of a polycrystalline NiTi shape memory alloy: micro and macromechanical investigations via in situ optical microscopy. Journal of the Mechanics and Physics of Solids. 52, 7, 15491571 (2004). [9] Tobushi, H. et al.: Influence of strain rate on superelastic properties of TiNi shape memory alloy. Mechanics of Materials. 30, 2, 141-150 (1998). [10] Shaw, J.A., Kyriakides, S.: On the nucleation and propagation of phase transformation fronts in a NiTi alloy. Acta Materialia. 45, 2, 683-700 (1997). [11] Kim, K., Daly, S.: Martensite Strain Memory in the Shape Memory Alloy Nickel-Titanium Under Mechanical Cycling. Exp Mech. (2010).

Measurement of Energy Loss in Thin Films Using Microbeam Deflection Method F.-C. Hsu, C.-J. Tong, M.-T. Lin*, Y.-C. Cheng Graduate Institute of Precision Engineering, National Chung Hsing University, Taichung 402, Taiwan, R.O.C. *E-mail: [email protected]

ABSTRACT A technique developed for studying the energy loss behavior of submicron to nanometer scale thin metal films on substrate is presented. The test microstructure was designed the triangular cantilever beam and fabricated by the standard CMOS processes, which can improve stress distribution non-uniform problem and the thickness regime of deposited metal thin film on its surface could reduce to several nanometers. In order to reduce the measure error and calculation complex due to the contact force, the driving system was used electrostatic force to making the paddle cantilever beam bend and the deflection of paddle cantilever beam due to the electrostatic force was measured by a capacitance change. The deflection of the paddle beam can be measured from the capacitance value. A force equilibrium calculate method (include sample compliance force, force due to the film, force due to the gravity and electrostatic force) could determine the stress and strain of the deposited films easily. The anelastic behavior and internal friction of 200~500 nm Al thin film were studied using the dynamic frequency response of the paddle structure generated by electrostatic force under vacuum pressure. The result show the measurement system used here can accurately measures the loss mechanism of thin film using dynamic response which give potential to study the grain boundary motion and dislocation motion in the nano-scale thin films. I.

INTRODUCTION

With the development of the micro-electro-mechanical systems (MEMS) technology, the system and device design required further miniaturization in order to increase the performance need and cost efficiency. As a result, the mechanical properties of sub micron and nano scale thin films have become one of the most important issues. In MEMS applications, the static mechanical properties such as residual stress, modulus and fractural toughness are keys for the design protocol. Moreover, dynamic properties of metal thin films as a function of the vacuum pressure can be pivotal. However, due to the difficulties on measurement techniques, a simple and accurate measurement arrangement cannot be fulfill [1]. Many methods used to measure the mechanical properties of the thin film have been proposed in previous studies. The results obtained from different measurement techniques were vary widely for nominally identical samples due to the difficulty with the techniques [2, 3]. Example of the traditional micro-beam bending test used nano-indentation measure the relation between the applied force and the deflection, but the indenter tip touching directly the sample surface may break the thin film. Each method has the difficult techniques on itself need to overcome. In the literature reviews on dynamic damping responds of materials, many anelastic mechanisms had investigated in bulk material [3], but rarely in thin films. In order to understanding the accurate static and dynamic response of the thin film materials, the energy dissipation study through simply damping response of thin film on substrate was performed. Here we developed a method that can be used to measure the energy loss of thin metal films with thickness less than 100 nm. The test specimen was designed to deposit on a novel triangle shape “paddle” beam in order to provide uniform plane strain distribution. When the sample reached the desired thickness, the tested thin film on the top surface can then be tested for measuring its static and dynamic mechanical properties. II.

EXPERIMENTAL

Previously, a paddle cantilever beam was proposed [4, 5]. This paddle like beam is different from the traditional parallel-sided cantilever beam; it was designed to have triangular side shape to provide a

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_13, © The Society for Experimental Mechanics, Inc. 2011

89

90 uniform stress distribution. The connected square plate is used as an electrode to induce the electrostatic force to make the deflection during the tests. The schematic and dimensions of paddle sample is shown in figure 1.

Figure 1- Schematic and dimensions of paddle sample The sample fabrication procedure was using standard clean room processing and the sample frame dimensions are 20mm square, the length of triangular beam from the fixed end to the free end connected to the paddle plate is 3mm. The area of the paddle plate is 25mm2. The thickness of constant stress beam is ȝPDIWHUFRPSOHWH IDEULFDWHSURFHVVLQJ(DFKRIWKHSDGGOHVDPSOHV ZHUH IDEULFDWHGWKURXJK VWDQGDUG semiconductor fabricate processing. A four inches double sides polished silicon wafer using the RCA clean process for removing particles and organisms due to environment then grown silicon nitride about 200nm on both wafer surfaces Using low pressure chemical vapor deposition (LPCVD). Photoresist layer on both sides of wafer were patterned using two aligned mask. Anisotropic etching (ICP-RIE) was used to etching silicon nitride and to remove the photoresist. The un-protect regime on the silicon wafer was etched in 30%wt. KOH solution at 85ɗ XQWLO WKH EHDP WKLFNQHVV UHDFKLQJ ȝP )LQDOO\ UHPRYH WKH etching barrier layer. The metal film, serve as conductive layer, was then deposit on the bottom surface of the silicon. The metal film that interested to be measured was then deposit on the top surface of the sample. The complete fabrication process flow is shown in figure 2.

Figure 2- The fabrication flow of the paddle sample

91 The experiment is to deflect “paddle” cantilever beam sample using the electrostatic force at bottom and then using capacitance measurement on the other side of the deflected plate to measure its deflection with respect to the force and so on. This technique has allowed testing of thin films at the desired length scales with the thickness as few hundred nanometers to less than 10 nanometers and also maintained consistent preparation and experimental procedures. The electrical driving circuits system was based on the charging parallel-plate principle when the two parallel planes shift produces the electrical field. Here, the two charging plates in the system were both the bottom surface of the sample and the electrode underneath the paddle plate. It is defined the electrostatic force 1 V2 (1) Fe H0 A 2 d2 where, Fe is the electrostatic IRUFHİ0 is the dielectric constant in the vacuum, V is the applied voltage, d is the distance between the two parallel plates and A is the effective area of the parallel plates. When the paddle beam is bending, the distance between the two plates is not the constant but is a function of x. Therefore, equation (1) is rewritten as equation (2).

Fe

H 0l p l 2

³

0

p

V 2 dx (d e  yb  slope ˜ x) 2

(2)

where Fe LV WKH HOHFWURVWDWLF IRUFH İ0 is the dielectric constant in the vacuum, lb is the length of the paddle cantilever beam, lp is the length of the paddle plate, V is the applied voltage, de is the distance between the under surface of the sample and the electrode, and yb is the position of the paddle cantilever end. The symbol in the cantilever beam and the test sample structure are shown in the figure 3.

Figure 3- The symbol definition of optical measure system. Integrating equation (2) and rewriting it as equation (3), the electrostatic force is defined as

Fe V2

§ ¨ H 0l p lb ¨ 1 1  ¨ 2y l 4 yb d e  yb d e  yb  b p ¨ lb ©

· ¸ ¸ ¸ ¸ ¹

(3)

During the experiment, we apply an electrostatic force to the specimen and measure the capacitance change. The deflection of the paddle beam can be measured from the capacitance value. The sample chip is mounted together with a guard-ringed capacitor electrode as shown in Figure 4. A spacing of 25 to 125um to the window frame chip surface around the paddle structure is defined by a metallic spacer.

92

Figure 4- The electrostatic deflection “paddle” A second electrode is mounted below the paddle plate. This electrode is used for electrostatic deflection of paddle. The whole chip is at DC ground but driven at 100 kHz with amplitude of a few volts. That provides a displacement current to central electrode of the capacitor plate which is proportional to the capacitance, and hence inversely proportional to the gap. Depending on the spacing selected, the capacitance is between 2 and 4 pF. The measurement of the capacitance can be made to a precision of approximately 0.1 fF so that paddle spacing changes of 50 nm are readily determined. The paddle can be pulled up with a DC voltage on the guard-ringed electrode or pulled down with a DC voltage on the lower electrode. The capacitance measurement can be made with a time resolution of r 10 msec. For the electronic setup of the capacity measurement, a sine-wave generator at 100 kHz is applied to the film simultaneously measuring capacity of the paddle capacitor while a second generator drives at the same frequency for a test capacitor which has a known capacity. The two units are coupled (one master, one slave) and have a phase shift of 1800. Figure 5 show the circuits. The 1800 out of phase currents from the two generator-capacity pairs are summed at input of change sensitive preamplifier. The amplified sum is measured with lock-in amplifier with the reference signal from one of the frequency generators.

Figure 5- Electronic setup for the capacity measurement The measurement is controlled by PC through National Instrument LabVIEW program. The control electronics include a controller, amplifier and waveform generator. Monitored signals are conditioned and then fed into an A/D board which is located in a PC. Data acquisition is performed with LabVIEW software. During sample testing, it is placed inside the vacuum chamber. After the sample is being locked inside the system, then wait until the system is reaching the thermal equilibrium and the capacitor read out is clear, the test can be perform. III.

RESULTS AND DISCUSSION

The internal friction of film energy loss can be measured from the dynamic response of the paddle structure provided while testing it in high vacuum (10^-6 torr). In the absence of the air damping effects, measuring the resonant response from forced vibrations can thus reveal the dynamic inelastic behavior of materials. During the experiment, a step voltage is applied to the bottom deflection electrode, and the samples elastic decay behavior of the materials is observed. We verified that the paddle returns to its initial position by observing free damping when the electrostatic force was abruptly removed, and the amplitude of the damping response decays with time. It was using an applied step voltage on the

93 deflection electrode and then the elastic aftereffect behavior was observed. We observed the paddle sample returns to its initial position with free damping when the electrostatic force was suddenly removed from the paddle sample and the amplitude of the damping response was decay with time. Figure 6 shows the results of the damping behavior for the paddle sample tested in the vacuum (1.6E-6 torr). Internal friction, į, is defined as the ratio of the energy dissipated per cycle and the maximum elastic energy stored in one cycle. The dissipation of elastic energy in a vibrating sample caused by stress induced defect motion is rate-limited by a kinetic process. Usually, internal friction of vibrating samples can be determined from the rate of decay of the amplitude. From the frequency and free decay data, we can calculate the internal friction from a simply equation:

y t

y0 e G ˜ fot

y0 e  n

(4)

where y(t) is the vibration amplitude, yo is the initial amplitude, n is the decrement of damping, įLV the internal friction parameter, fo is the damping frequency.

Figure 6 -the damping behavior for the paddle sample tested in the vacuum (1.6E-6 torr). Moreover, the resonance-frequency excitation test can be used to find the decay rate of the sample as the consequence of the internal friction. We excite the paddle at its resonance frequency, and abruptly change the paddle excitation, from its stable state and observe its response until the paddle again achieves an equilibrium. The decay rate could be found and the loss angle could be calculated using Equation derived from [2, 3]

G ln  A A n

n 1

I G S

(5)

Here, An represents the amplitude of the n cycles after removal of the excitation and An+1 is just n+1 cycles. į is the natural logarithm of amplitude in two successive vibration cycles. Equation above describes to the internal friction with the loss angle, Ȝ LQ WHUPV RI į DQG ʌ In addition, a sweep frequency excitation from the bottom deflection electrode can also show the response due to sweep frequency excitation versus time. Using Fast Fourier Transform (FFT) conversion of the time domain into the frequency domain produced the FFT power spectrum gives us several peaks representing the paddles resonant frequency occurred. We can thus evaluate the energy loss contribution to the resonant frequency by repeating the sweep frequency experiment at different nanocrystalline film structures and conditions. Following equation shows that the loss angle Ȝ, which is obtainable directly from the width of the resonant peak at half-maximum height in the FFT power spectrum:

94

I Z  Z Z 2

1

r

(6)

The (Ȧ2-Ȧ1) term is the full width at half-maximum height, and Ȧr is the resonant frequency. Using this measurement, the point defect relaxation, dislocation relaxation, grain boundary interface relaxation, thermoelastic relaxation, phase transformation, dislocation motion of the film, if any, can be observed from the resonant frequency and drive amplitude (loss mechanism) in each tested cycle. Thus allow us to measure mechanical behavior of the sample with respected to load, time, temperature, resonant frequency, drive amplitude and microstructure. Figure 7 show the results of the damping behavior for the paddle sample. This resonance experiment was use the sine wave on the deflection electrode and the frequency was sweep from 50Hz to 350Hz, sweep rate was 16 stepping per second and increasing 0.005Hz per step, the integration time of lockin amplifier is 300us. In figure 7, we can clearly obtain the there are two larger response at 1222.638 seconds (max) and 3072.67 seconds (2nd) respectively. The second response we have never seen in former measure results that using the low conductance paddle sample with a thin metal layer because of narrower frequency range (50-150Hz). We calculated the frequency of excitation the maximum response to be 147.81Hz, and the second response is 296.05Hz. Figs 8 shows the FFT spectrum tested in Figure 7. In frequency spectrum also show two larger peaks and the maximum one is stand on 216.97Hz, second peak is on 295.27Hz.

Figure 7- the results of the resonance damping behavior for the paddle sample.

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Figure 8 - the FFT spectrum for the tested in figure 7 Figure 9 shows the frequency scan of the tested paddle sample with 500nm Al film deposited on top. This result shows the shift comparison of resonance frequency triggered by the electrostatic force. Fig. 10 shows the frequency scan of the tested paddle sample with four different thickness of Al film deposited on top.

Figure 9.

The frequency scan for the tested paddle sample with 500nm Al film deposited on top.

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Figure 10.

The resonant frequencies scan using sweep frequency for four different samples.

These results show the measurement system used here can accurately measures the loss mechanism of thin film using dynamic response which give potential to study the grain boundary motion and dislocation motion in the film. IV. CONCLUSION A technique developed for studying the energy loss behavior of submicron to nanometer scale thin metal films on substrate is presented. The anelastic behavior and internal friction of 200~500 nm Al thin film were studied using the dynamic frequency response of the paddle structure generated by electrostatic force under vacuum pressure. The result show the measurement system used here can accurately measures the loss mechanism of thin film using dynamic response which give potential to study the grain boundary motion and dislocation motion in the nano-scale thin films.

REFERENCES [1] C. M. Zener, Elasticity and Anelasticity of Metals. Chicago, IL: Univ.of Chicago Press, 1960. [2] B. S. Berry, “Anelastic relaxation and diffusion in thin layer materials,in Diffusion Phenomena in Thin Films and Microelectronic Materials D. Gupta and P. S. Ho, Eds. Park Ridge, NJ: Noyes, 1988, vol. 73, pp.73–145. [3] A. S. Nowick and B. S. Berry, Anelastic Relaxation in Crystalline Solids. New York: Academic, 1972. [4] C-J Tong, M-T Lin “Design and development of a novel paddle test structure for the mechanical behavior measurement of thin films application for MEMS” Microsystem technologies, Vol. 15, Issue 8, 1207-1216, 2009. [5] C-J Tong, Y-C Cheng, M-T Lin, K-J Chung, J-S Hsu, C-L Wu, “Optical Micro-Paddle Beam Deflection Measurement for Electrostatic Mechanical Testing of Nano-Scale Thin Film Application to MEMS” microsystem technologies ,DOI: 10.1007/s00542-009-0999-7 [6] H. Huang, F. Spaepen, “Tensile testing of free-standing Cu, Ag and Al thin films and Ag/Cu multilayers,” Acta mater., Vol. 48 (2000) pp. 3261-3269. [7] K. Kusaka, T. Hanabusa, M. Nishida and F. Inoko, “Residual stress and in-situ thermal stress measurement of aluminum film deposited on silicon wafer”, Thin solid films, 290-291(1996), pp. 248-253.

Multiscale Characterization of Water-, Oil- and UV-Conditioned Shape-Memory Polymer under Compression J. T. Fulcher1, H. E. Karaca1, G. P. Tandon2, 3, D.C. Foster2, Y.C. Lu1 1

University of Kentucky, Lexington, KY 40506 Air Force Research Laboratory, Materials and Manufacturing Directorate, AFRL/RXBC, Wright-Patterson AFB, OH 45433 3 University of Dayton Research Ins., 300 College Park, Dayton, OH 45469-0060 2

ABSTRACT Shape memory polymers (SMPs) are an emerging class of active polymers that may be used for reconfigurable structures on air vehicles. In this study, epoxy-based SMPs were conditioned separately in simulated service environments relevant to Air Force missions, namely, (1) exposure to UV radiation, (2) immersion in jet-oil, and (3) immersion in water. The mechanical properties and shape recovery abilities of the unconditioned and conditioned SMPs were examined using large-scale compressive tests and small-scale nano-indentation tests. Results show that all the conditioned SMPs exhibit a decrease in Tg as compared to the unconditioned one. Under compressive loading, the SMPs undergo significant plastic deformation prior to fracture, except the UV conditioned sample. Environmentally conditionings generally result in higher moduli and yield strength of the SMPs. Environmental conditionings, in particular the UV exposure and water immersion, also affect the shape recovery abilities of the SMPs if the recovery temperatures are set low. 1. INTRODUCTION Shape memory polymers (SMPs) are novel class of active polymers that have been considered for the development of reconfigurable air vehicles. SMPs have the ability to change shape in a predefined way from a temporary shape to a permanent shape when activated at the so-called the activation temperature (Td), usually above their glass transition temperatures (Tg). The temporary shape is obtained by mechanical deformation and subsequent fixation of that deformation. Upon application of an external stimulus such as heat, the polymer starts to recovers. The temperature at which the material is made to recover will be called as the recovery temperature (Tr). The typical thermomechanical cycles used to quantify the shape-memory behavior of the SMPs have been described by Lendlein and Kelch [1], Beloshenko et al. [2], Liu, et al. [3-4], Behl and Lendlein [5], Ratna and Karger-Kocsis [6], Wei et al. [7], Schmidt et al. [8], Gall et al. [9], Tobushi, et al. [10-11], and Atli, et al. [12]. So far, little work has been carried out to investigate the durability of the SMPs in anticipated service environment. Like conventional polymers, the shape memory polymers will undergo physical and chemical aging when exposed to simulated service environments. Failure of the morphing materials in aggressive environments will have a direct impact on reconfigurable ability of the aircrafts and fleet readiness of aerospace missions. It is thus critical to examine the shape recovery ability and mechanical properties of SMPs conditioned at relevant service environments so that the true reconfigurable ability of the SMPs can be predicted. This paper presents multiscale mechanical characterizations of the SMPs conditioned at water, jet oil, and UV environments. 2. EXPERIMENTATION 2.1 Materials The SMP chosen for this study was an epoxy-based thermosetting resin (Veriflex-E) developed by Cornerstone Research Group, Inc. (CRG). Two types of samples were fabricated: thin square plaques measuring 300 mm by 300 mm with an T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_14, © The Society for Experimental Mechanics, Inc. 2011

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average thickness of 3.2 mm and cylindrical buttons with nominal diameter of 25.4 mm and nominal thickness of 12.7 mm. The plaque samples were further cut into smaller specimens (22 mm length x 10 mm width x 3.2 mm thickness) with waterjet machining. The SMP specimens were environmentally conditioned as follows. Conditioned in water In order to evaluate the hot-wet performance of SMPs, specimens were immersed in water at 49qC for four days (following the water immersion procedure described in MIL-PRF-23377H paragraph 4.6.6.). Care was taken to ensure that the water used for the immersion met ASTM D 1193 Type IV water with a minimum of 0.2 MOhm electrical resistivity. Conditioned in lube oil SMP specimens were immersed in MIL-L-23699 lubricating oil at room temperature for 24 hours to isolate the influence of lubricating oil alone. Conditioned by UV radiation The accelerated weathering conditioning was performed in a Xenon-Arc exposure chamber. The button samples were placed in a specialized basket to allow the majority of the surface to be exposed to the UV radiation. The standard exposure cycle is 102 minutes of light only followed by 18 minutes of light and water spray. High purity deionized (DI) water with a minimum of 16.5 MegOhms-cm resistivity was used. During the light only, the black panel temperature in the exposure chamber was controlled to 63qC r 2.5qC, and the intensity of the spectral irradiance was controlled to 0.3 to 0.4 watts/meter2 at a wavelength of 340 nm. Humidity in the chamber was not controlled. Exposure length was limited to 125 cycles. 2.2 Transition temperature The glass transition temperatures of the SMPs were characterized with a dynamic mechanical analyzer (DMA), Model Q800 from TA Instrument. Conditioned and unconditioned SMPS cut from the thin plaques were tested in torsion mode. Tests were scanned from 25oC to 130oC with a heating rate of 2oC/min. The applied strain was 0.1% and the oscillating frequency was 1Hz. 2.3 Mechanical Characterization Large-Scale Compression Tests The mechanical properties of the SMPs were first examined through large-scale compression tests at ambient temperature (25°C). A MTS LandMark with FlexTest 40 machine was used to perform all of the compression tests for the current study. All test were done utilizing displacement control. A capacitive displacement sensor was used to measure the displacements for all ambient compression tests. Each specimen was loaded to a level of strain of 10% of the sample initial undeformed height. A loading time of 5 minutes was used for each test, which resulted in a strain rate of 0.00034/s. The specimen was initially loaded with a 1?? MPa stress to ensure the sample-plate contact. Once loaded, the initial capactive displacement was recorded. The specimen was then loaded to the prescribe level of strain and then unloaded. Nanoindentation Tests The mechanical properties of the SMPs were further evaluated through nanoindentation tests. The Nano Indenter XP (MTS NanoInstruments, Oak Ridge, TN) equipped with a Berkovich diamond indenter was used in the experiments. The indenter has a nominal tip radius <20 nm and an inclined angle of 65.3 (degree). For nanoindentation studies, the surface of SMP specimen was prepared by successive polishing, the final polishing being an alumina compound with an average particle size of 0.3 μm. The “held-at-the-peak” method was used, which is a special technique that was proposed for testing polymer and soft materials [13]. The nanoindentation tests were conducted at a loading rate of 5 mN/s. The indenter holding time at the maximum load was 5 s. A total of 10 measurements were conducted on each specimen. 2.4 Shape Recovery Ability Large-Scale Compression Tests The shape memory behaviors of the SMPs were first examined on cylindrical button specimens. For these experiments, the oven and liquid nitrogen cooling system were incorporated into the same MTS Landmark machine. To help ensure even loading and proper deformation two metal plates were used to augment the contact area between the mandrels and specimen. The specimen was inserted into the machine and loaded initially with a 0.02 MPa stress to help ensure contact before being heated. A 30 minute ramp time was used to heat the specimen to 130°C and then a 30 minute dwell time was used to ensure thermal equilibrium. Once the specimen was heated the specimen was deformed to

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a level of 30% strain from its original undeformed height. The 30% strain level with a 5 minute loading resulted in a strain rate of 0.001/s. Once the apporiate strain level was reached the displacment was held constant and the specimen was cooled at a ramp time of 30 minutes. Each specimen was measured and weighted before beginning the recovery process. For recovery measurements, a MTI KSL 1100X oven was used to heat each specimen to the prescribed recovery temperatures (60°C, 93°C, 98°C, 125°C, 135°C, 150°C). A dwell time of 8 minutes was used for each stage of recovery and at each stage the samples were measured and photographed to help quantify and qualifiy the shape recovery ability of the SMP. Nanoindentation Tests The shape memory behavior of the SMPs was further examined at small scales by using the high temperature nanoindentation apparatus. A spherical indenter of 300 ȝm tip diameter was used. First, the specimen was heated to a deformation temperature 130°C. The heating rate was approximately 2oC/min. The actual temperature in the specimen was monitored through thermocouples placed near the specimen. The specimen was kept at this temperature for at least 1-hr to allow the temperature in the specimen reach equilibrium. The specimen was then deformed with the indenter with a maximum load of 50 mN at a loading rate of 5 mN/s. While the indenter was maintained at that load, the temperature in the sample was brought to room temperature and then the indenter was withdrawn from the specimen. It took approximately two hours for the sample to reach the room temperature after the heater was turned off. The surfaces of the specimens were then examined with a Wyko Optical Surface Profiler (Veeco Instruments, Inc.) for the initial indentation profile and indentation depth measurements. Subsequently, the specimen was placed back on the indenter microheater stage for recovery experiments. The recovery experiments were conducted at various temperatures (Tr): 60oC, 75oC, 93oC, 98oC, 125oC. A 2 minutes dwell time was used for all experiments. The indentation profile at each recovery temperature was next obtained by scanning the surface of the recovered specimen with the Wyko Optical Surface Profiler.

3. RESULTS AND DISCUSSION 3.1 Glass transition temperatures Effect of environmental conditioning on the physical properties of the SMPs was first examined. Figure 1 depicts the Tan(į) curves of the unconditioned and conditioned SMP specimens measured using a DMA (Dynamic Mechanical Analyzer). From the curves, the glass transition temperature (Tg) of each SMP was determined (the peak of the Tan(į) curve). It is seen that the transition temperature of the unconditioned SMP occurs at 105oC while the onset of glass transition is at approximately 90oC (from the temperature dependence of storage modulus not shown here). After conditioning, all SMPs are seen to exhibit a decrease in Tg in comparison to the unconditioned one. Among them, the UV exposed SMP has the largest decrease in Tg (101oC). 3.2 Mechanical properties The mechanical behaviors of the SMPs as results of environmental conditioning were examined using large scale compression tests and small scale indentation tests. Figure 2 shows the stress-strain responses of the unconditioned and conditioned SMPs from the compressive tests at ambient temperature. Under compressive loading, the SMP resins undergo significant plastic deformation following yielding. These observations are in contrast with the tensile testing results reported earlier, where the SMPs exhibit primarily liner elastic deformation [14]. Environmental conditionings have affected the deformation of the SMPs. The UV exposed SMP is seen to have limited ductility; the specimen became brittle and broke into pieces before the prescribed strain of 10% was reached. Environmental conditionings have also increased the modulus and yield strength of the SMPs. The UV exposed sample was seen to have the greatest increase in yield strength (121 MPa), in comparison with the unconditioned one (110 MPa). Figure 3 shows the elastic moduli of the unconditioned and conditioned SMPs obtained from large scale compression tests and small scale indentation tests. The compressive moduli were determined by the slopes of the initial linear portion of the stress-strain curves of the SMPs (Figure 2). The indentation moduli were determined from the parameters obtained on the load-depth curves, as documented in reference [13]. Nanoindentation test was conducted since it had been proven to be a useful technique for examining the local deformation of polymers as a result of environmental aging [15, 16]. Like conventional polymers, the shape memory polymers will undergo physical and chemical aging when exposed to simulated service environments. Detailed surface examinations have shown that the UV exposure SMP specimens have developed yellow tints throughout the entire specimens and exhibited noticeable surface roughness compared to an unconditioned specimen. All conditioned specimens are seen to exhibit higher modulus than the unconditioned specimen, indicating the onset of brittleness when exposed to radiation and fluids.

100

(103oC)

o

(101 C)

(102oC)

(105oC)

Figure 1. Tan(G) curves of the unconditioned and conditioned SMPs measured using the DMA in torsion mode. Numbers shown in parentheses are the glass transition temperatures.

140 fracture

120

Stress(MPa)

100 80 60 Unconditioned ExposedatUV WaterImmersed OilImmersed

40 20 0 0

0.02

0.04

0.06

0.08

0.1

Strain Figure 2. Stress-strain curves of the unconditioned and conditioned SMPs from large scale compression tests.

101 5

Compression 4.4

nanoindentation 4.3

ElasticModulus (GPa)

4.1 4 3.49 3.02

4.2 3.61

3.06

3

2

1 Unconditioned

UVExposed

WaterImmersed

OilImmersed

Figure 3. Elastic modulus of the unconditioned and conditioned SMPs obtained from large scale compression tests and nanoindentation tests.

3.3 Shape recovery ability Effect of environmental conditionings on the shape recovery ability of the SMPs was also examined at various scales. The shape recovery experiments were conducted according to the standard shape memory cycle. First, specimens were heated to a deformation temperature Td=130oC (>Tg) and then deformed (large compression or small indentation). When deforming, the temperatures in the sample were brought to room temperature and then the constraints were removed from the specimens. Subsequently, the specimens were placed back on the heating stage for recovery measurements. For large scale shape recovery experiments, the large button specimens were used (25.4 mm diameter x 12.8 mm thickness). The prescribed compressive strain at fixing was ~40%. At each recovery temperature (25oC, 60°C, 98°C, 125°C, 135°C, 150°C), the specimen was photographed and height measured to quantify and qualify the shape recovery process of the SMP. For each of the recovery temperatures a dwell time of 8 minutes was used. Figure 4 shows the typical photographs representing the shape recovery profiles of the unconditioned SMP. The shape recovery of the SMP was seen to be very limited at temperatures below the Tg. Once the temperature exceeded the transition temperature, recovery was seen to increase. The UV exposed and water immersed samples had the lowest level of recovery when the unconditioned specimen achieve almost full recovery. However, all specimens were within 5% of achieving full recovery. For small scale shape recovery experiments, the newly modified high temperature nanoindentation was used. The high temperature nanoindenter consisted of the Nano Indenter XP equipped with a microheater assembly. Smaller specimen (10 mm length x 5 mm width x 3.2 mm thickness) was used. During activation, a spherical indenter of 150 Pm radius was used to indent the specimens, which resulted in the prescribed strain of approximately 6%. Recovery experiments were performed at several different temperatures (25oC, 60oC, 98oC, and 125°C). A 2-minute dwell time was used for all experiments. As each SMP specimen was re-heated, a microscope incorporated with the nanoindenter equipment was used to closely monitor the recovery process. The SMP surface was then examined using the Wyko Optical Surface Profiler to obtain the corresponding indentation profile and dimensions. Results show that the indent recoveries are negligible when the material is in the glassy state (25oC, 60oC). As the recovery temperature is increased above the onset of the glass transition (98oC), the indents begin to show significant recoveries. However, at this temperature, the UV exposed and water immersed SMPs have smaller recovery ratios in comparison to the unconditioned resin. Finally, as the SMP resin is re-heated above its Tg to 125oC, the indents are nearly 100% recovered for all specimens. Overall, the material’s ability to regain shape remains relatively unchanged with conditioning, as long as the recovery temperature is sufficient high. Larger specimens require longer hold times at recovery temperatures.

102

Figure 4. Large scale shape recovery experiments for the unconditioned SMP sample.

25°C

60°C

20X

20X

(a)

(b)

98°C

125°C

20X (c)

20X (d)

Figure 5. Small scale shape recovery experiments for the unconditioned SMP sample.

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4. CONCLUSIONS Multiscale experiments were conducted to characterize an epoxy-based SMP separately exposed to moisture, lubricating oil and UV radiation. Results show that environmental conditioning affects the glass transition temperature, the mechanical properties, and the shape recovery abilities of the SMPs. It appears that all the conditioned SMPs exhibit a decrease in Tg as compared to the unconditioned one. The environmental conditioning also affects the modulus and yield strength of the SMPs. Most conditioned SMPs exhibit higher yield strength than the unconditioned SMP. In particular, the UV conditioned specimen is seen to have the highest increase in yield strength indicating onset of brittleness with exposure to radiation. The shape recovery ability of the SMP was assessed through both large and small recovery experiments according to the standard shape memory cycle. Both large-scale and small-scale testing show that the recovery is limited below the glass transition temperature (Tg), but the amount of recovery increases as the recovery temperatures are increased above Tg. Overall, the UV exposed and water-immersed SMPs exhibit lower shape recovery ratios as compared with the unconditioned one.

REFERENCES 1. A. Lendlein and S. Kelch, Shape-Memory Polymers, Angew Chem. Int. Ed., 41 (2002) 2034-2057. 2. V. A. Beloshenko, V. N. Varyukhin and Y. V. Voznyak, The shape memory effect in polymers, Russian Chemical Review, 74(3) (2005) 265-283. 3. C. Liu, H. Qin and P. T. Mather, Review of progress in shape-memory polymers, Journal of Materials Chemistry 17 (16) (2007) 1543-1558. 4. Y. Liu, K. Gall, M. L. Dunn, A. R. Greenberg and J. Diani, Thermomechanics of Shape Memory Polymers: Uniaxial Experiments and Constitutive Modeling, International Journal of Plasticity, 22 (2006) 279-313. 5. M. Behl and A. Lendlein, Shape-Memory Polymers, Materials Today, 10(4) (2007) 20-28. 6. D. Ratna, D. and J. Karger-Kocsis, Recent Advances in Shape Memory Polymers and Composites: A Review, Journal of Materials Science, 43 (2008) 254-269. 7. Z. G. Wei, R. Sandstrom and S. Miyazaki, Review Shape-Memory Materials and Hybrid Composites for Smart Systems, Part I Shape-Memory Materials, Journal of Materials Science, 33 (1998) 3743-3762. 8. C. Schmidt, K. Neuking and G. Eggeler, Functional Fatigue of Shape Memory Polymers, Advanced Engineering Materials, 10(10) (2008) 922-927. 9. K. Gall, P. Kreiner, D. Turner and M. Hilse, Shape-Memory Polymers for Microelectromechanical Systems, Journal of Microelectromechanical Systems, 13(3) (2004) 472-483. 10. H. Tobushi, H. Hara, E. Yamada and S. Hayashi, Thermomechanical Properties in a Thin Film of Shape Memory Polymer of Polyurethane Series, Smart Materials and Structures, 5 (1996) 483-491. 11. H. Tobushi, T. Hashimoto, N. Ito, S. Hayashi and E. Yamada, Shape Fixity and Shape Recovery in a Film of Shape Memory Polymer of Polyurethane Series, Journal of Intelligent Material Systems and Structures, 9 (1998) 127-136. 12. B. Atli, F. Gandhi and G. Karst, Thermomechanical Characterization of Shape Memory Polymers, Journal of Intelligent Material Systems and Structures, 20(1) (2009) 87-95. 13. J. T. Fulcher, Y. C. Lu, G. P. Tandon, D. C. Foster, Thermomechanical characterization of shape memory polymers using high temperature nanoindentation, Polymer Testing, 29 (2010) 544–552. 14. Y. C. Lu, G. P. Tandon, D.C., Jones, G. A. Schoeppner, Elastic and viscoelastic characterization of thermally-oxidized polymer resin using nanoindentation, Journal of Mechanics of Time-Dependent Materials, 13: 245–260 (2009). 15. Y.C. Lu, G. P. Tandon, S, Putthanarat, G. A. Schoeppner, Nanoindentation strain rate sensitivity of PMR-15 Polyimide, Journal of Materials Science, 44:2119–2127 (2009). 16. G.P. Tandon, K. Goecke, K. Cable, J. Baur, Durability assessment of styrene- and epoxy-based shape-memory polymer resins. J. Intell. Mater. Sys. Struct., 20 (2009) 2127-2143.

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T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_15, © The Society for Experimental Mechanics, Inc. 2011

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Influence of Mechanical Properties and Loading Conditions on the Recovery of Shape Memory Polymers

Rui Xiao, Xiang Chen, Thao. D. Nguyen Department of Mechanical Engineering Johns Hopkins University, Baltimore, MD,21218

Abstract: This work presented a parameter study to investigate the influence of material properties and loading conditions on the recovery performance of amorphous shape memory polymers using a recently developed thermoviscoelastic model. The model incorporated the time-dependent effects of both structural relaxation—using a nonlinear Adam-Gibbs model—and viscoelasticity. The model can predict well the unconstrained strain recovery response and stress evolution during constrained recovery process. The materials properties and the loading parameters, including the cooling rate, the annealing time, and the heating rate, were varied one by one to compare the effects on the start and end temperatures and recovery time of the unconstrained recovery response and on the stress hysteresis of the constrained recovery response. The results confirmed experimental observations that unconstrained strain recovery response was mostly influenced by viscoelasticity, while the constrained recovery response resulted from the interaction of many different mechanisms, including structural and stress relaxation, thermal expansion, the modulus of rubbery and glass state. The results also showed that the cooling and heating rates had the largest influence on both recovery responses.

1.

Introduction

Shape memory polymer (SMP) amorphous networks offer a broad range of mechanical, chemical, and biological tailor-ability.

These features make SMP materials attractive for biomedical applications, such as implants and devices for

minimally invasive surgery [1,2].

An example of thermally active, amorphous SMP materials is a family of random

copolymers synthesized from tert-butyl acrylate monomers and poly (ethyleneglycol) demethacrylate cross-linkers. The thermomechanical properties of the tBA-co-PEDGMA networks can be tailored by varying the weight fraction and the molecular weight of the cross-linkers [3]. Nguyen et al. [4] has developed a finite deformation thermoviscoelastic model for amorphous SMP networks that incorporates the mechanisms of structural and stress relaxation to model the dependence of the recovery response on the loading rate and cooling rate of the shape memory programming cycle and the heating rate during deployment. The model can reproduce the temperature dependence of the unconstrained recovery response and the hysteresis in the stress-temperature curve of the constrained recovery response. Based on this model, a parameter study was performed to characterize the effect of mechanical properties of SMPs and loading conditions on the unconstrained and constrained strain recovery performance. 2.

Model Formulation and simulation process

A detailed description of the thermoviscoelastic model was presented in Nguyen et al.[4]. Here we present only a brief

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_16, © The Society for Experimental Mechanics, Inc. 2011

113

114

Fig 1 Rheological representation of the constitutive model description of the model. Fig 1 shows a linear rheological model analogy of the generalized finite deformation thermoviscoelastic model. The total deformation is decomposed into an isotropic thermal part and a mechanical part, F=

Fm, where the mechanical deformation gradient is further split into elastic and viscous decompositions Fm=Fe Fv.

The

constitutive relations of the model for the thermal dilatation
T, the internal deformation components, and stress response are summarized in Table 1. The important innovations of the model include the incorporation of the nonlinear Adam-Gibbs [5,6,7] model for structural relaxation and the glass transition in a finite-deformation thermoviscoelastic framework and the inclusion of both viscoelastic and structural relaxation processes to model the relaxation exhibited by the materials.

The parameters in the model listed in Table 2 were determined by a series of thermomechanical experiments with detailed information in Nguyen et al[4]. Here the parameters were varied one to one from the baseline values as shown in the paper. The effect of the loading conditions on the recovery behavior was also investigated by varying the loading parameters in Table 3.

115

3. Results and Discussions: Fig 2 shows the results of the unconstrained and constrained response calculated for the baseline parameters. The recovery rate was defined by the maximum slope of the unconstrained recovery curve, kmax ,. The peak stress overshoot of the constrained recovery response was defined as the difference between the maximum stress of the heating and loading stages. Fig 3 shows the influence of thermoviscoelastic properties and loading conditions on the behavior of unconstrained

116

Fig 2 (a) The strain-temperature curve of the unconstrained recovery response and (b) the stress-temperature of the constrained recovery response calculated for the baseline material properties and loading conditions in Table 2 and 3. recovery. The rubbery and glassy coefficients of thermal expansion r and g had a negligible influence on the unconstrained recovery response. A larger structural relaxation time produced a faster recovery rate. A larger structural relaxation time caused the stress relaxation time to respond more slowly to a temperature change thus shifting the onset of strain recovery to a higher temperature. A smaller stress relaxation time provided a narrower glass transition temperature range, which also resulted in a faster recovery rate. Both the rubbery and glassy modulus had a noticeable influence on the slope of the unconstrained recovery. A higher rubbery modulus and smaller glassy modulus resulted in a higher recovery rate. The WLF constants, C1 and C2, directly influenced the glass transition temperature range and had the most significance effect on the slope value. The recovery rate was influenced most by the cooling, heating rate and annealing time, which indicated that structural relaxation was important to the unconstrained recovery behavior. A slower cooling rate and a longer annealing time gave the material more time to evolve to an equilibrium configuration during cooling. response during reheating and led to a

This results in a more initially sluggish

higher onset temperature and a faster recovery rate. In contrast, a slower heating rate

allowed strain recovery to start at a lower temperature and led to a slower recovery rate.

117

Fig 3 Effects of the (a) thermoviscoelastic material properties and (b) loading conditions on the recovery rate of the unconstrained recovery response. The recovery rate was influenced most by the cooling, heating rate and annealing time, which indicated that structural relaxation was important to the unconstrained recovery behavior. A slower cooling rate and a longer annealing time gave the material more time to evolve to an equilibrium configuration during cooling. response during reheating and led to a

This results in a more initially sluggish

higher onset temperature and a faster recovery rate. In contrast, a slower heating rate

allowed strain recovery to start at a lower temperature and led to a slower recovery rate. Fig 4 shows the influence of the thermoviscoelastic properties and loading conditions on the stress overshoot of the constrained recovery response. The peak stress overshoot is greatly influenced by glassy thermal coefficient g and non-equilibrium modulus μneq, which indicated that the stress overshoot was caused by the constrained thermal expansion in the heating process. A higher g provided a larger thermal strain, while a higher μneq produced a stiffer material. Both led to a larger stress overshoot. The WLF constants and relaxation time influenced the stress overshoot by influencing the glass transition . The annealing time, cooling rate, and heating rate had the most significance effect on the stress overshoot. A longer annealing time, smaller cooling rate, and faster heating rate all produced a stiffer material during heating process which will produce a higher overshoot.

118

Fig 4 Effects of the (a) thermoviscoelastic material properties and (b) loading conditions on the peak stress overshoot of the constrained recovery response.

4. Conclusion: The simulation results revealed the unconstrained strain recovery response was influenced mainly by viscoelastic and structural relaxation. In contrast, the constrained recovery response resulted from the interaction of many different mechanisms, including structural and stress relaxation and thermal expansion. The result also revealed that the cooling and heating rate had the largest influence on constrained recovery responses. Currently we are working on incorporating multiple viscoelastic and structural relaxation processes to model the broad relaxation spectrum exhibited by the materials. This will improve the prediction of the unconstrained recovery rate. Reference: [1] Yakacki, C. M., Shandas, R., Lanning, C., Rech, B., Eckstein, A., Gall, K., Biomaterials, Unconstrained recovery characterization of shape-memory polymer networks for cardiovascular applications.28, 2255. 2007 [2]Behl M., Razzaq, M. Y., Lendlein, A. , Advanced Materials, Multifunctional Shape-Memory Polymers 22, 3388. 2010 [3]Yakacki, C. M., Shandas, R., Safranski, D., Ortega, A. M., Sassaman, K., Gall, K.,. Advanced Functional Materials. Strong, tailored, biocompatible shape-memory polymer networks. 18,2428.2008 [4] Nguyen, T., Qi, H. J., Castro, F., Long, K. N., Journal of the Mechanics and Physics of Solids. A thermoviscoelastic model for amorphous shape memory polymers: Incorporating structural and stress relaxation. 56, 2792. 2008 [5] Adam, G., Gibbs, J. H., Journal of Chemical Physics. On the temperature dependence of cooperative relaxation properties in the glass-forming liquids. 43, 139. 1965 [6] Scherer, G. W., Journal of American Ceramic Society. Use of the adam-gibbs equation in the analysis of structural relaxation. 67, 504. 1984 [7] Hodge, I., Macromolecules. Effects of annealing and prior history on enthalpy relaxation in glassy polymers: Adam-gibbs formulation of nonlinearity. 20, 2897–2908. 1987




Fatigue Cycling of Shape Memory Polymer Resin A.J.W. McClung Air Force Research Laboratory Materials and Manufacturing Directorate Wright-Patterson AFB, OH 45433, USA National Research Council, USA

G.P. Tandon Air Force Research Laboratory Materials and Manufacturing Directorate Wright-Patterson AFB, OH 45433, USA University of Dayton Research Ins. 300 College Park, Dayton, OH 45469, USA

J.W. Baur Air Force Research Laboratory Materials and Manufacturing Directorate Wright-Patterson AFB, OH 45433, USA

Abstract Shape memory polymers have attracted great interest in recent years for application in reconfigurable structures (for instance morphing aircraft, micro air vehicles, and deployable space structures). However, before such applications can be attempted, the mechanical behavior of the shape memory polymers must be thoroughly understood. The present study represents an assessment of viscoelastic and viscoplastic effects during multiple shape memory cycles of Veriflex-E, an epoxy-based, thermally-triggered shape memory polymer resin. The experimental program is designed to explore the influence of multiple thermomechanical cycles on the shape memory performance of Veriflex-E. The effects of the deformation rate and hold time at elevated temperature on the shape memory behavior are also investigated. Introduction Shape memory polymers (SMPs) can change their shape in a predefined way from a locked-in (deformed) shape to their original shape when exposed to an appropriate stimulus, as illustrated with heat as the stimulus in Figure 1. The material begins at state A1 with a relatively high “glassy” modulus. Heat is applied to the sample which causes the modulus to drop by several orders of magnitude to its “rubbery” modulus. While in this high temperature state (B in the figure), the sample is deformed into its new shape (in the present study it is deformed in axial tension as signified in C). The deformed shape is held in place while the sample is cooled back to its “glassy” modulus. Once it is cooled, the sample is in the locked-in state D. When heat is reapplied to the locked-in sample, the material reaches its “rubbery” modulus again and the unconstrained sample returns to its original “memorized” shape at state E. Finally, the sample is cooled to state A2 which is close to state A1. The better the “memory” properties of the SMP, the closer state A2 is to state A1. In the current work, the pathdependent behavior of Veriflex-E is evaluated by means of a shape memory cycle similar to the schematic in Figure 1. For an ideal shape memory material, the original state is exactly achieved. However, in reality, shape memory materials achieve a state close to their original shape. The shape recovery parameter is a measure of how closely the material returns to that original shape in the shape memory cycle (A2 comparison to A1 in Figure 1). Various researchers have established performance parameters to compare the response of SMP materials. Tobushi et al. [1] defined a set of performance parameters termed strain fixity and strain recovery. The strain recovery term was defined specifically to evaluate material performance throughout multiple shape memory cycles by comparing the material strain following recovery (state A2) in the current cycle to the strain following recovery of the previous cycle. Many researchers have employed Tobushi’s definition. The drawback of this definition is that it does not give a full picture of the material capability. For example, in [1] the permanent strain following the shape memory cycle increases from cycle 1 to 10 (see Fig 5b in [1]), Tobushi’s shape recovery parameter indicates a recovery performance which begins low and increases toward 100% as the cycling is continued. This increase leads the reader to believe that the material recovery is improving while in fact it is simply stabilizing to a constant state of performance which is worse than that in the first shape memory cycle. T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_17, © The Society for Experimental Mechanics, Inc. 2011

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120

Fig. 1 Schematic of shape memory cycle with free recovery for heat activated SMPs

Schmidt et al. [2] examined functional fatigue of Veriflex (a styrene based thermally triggered SMP). They used their own definition of shape recovery which compares the recovered strain to the maximum programmed strain (and therefore the programmed length in addition to the sample original length) and found that the styrene based resin exhibited recovery values between 65 and 85% with a large decay from the first cycle to the 18th. The main experimental setup difference between the current work and Schmidt et al. is that they had no way to directly measure the strain in the sample. So they consider the cross-head displacement to be equivalent to the displacement in the gage section of the sample. While the cross-head displacement can give an approximation of the sample gage displacement, they are not equivalent as variations in the sample shape near the clamped section are evident in the photograph that Schmidt et al. present of a deformed sample (Fig 2b “after programming”[2]). They do not adjust for the grip thermal expansion and contraction while the specimen is cycled through the heating and cooling stages of the shape memory cycle (the grips are heated/cooled along with the specimen). In addition, they did not study the influence of the deformation rate on the shape memory response of the SMP. Liu et al.[3] also defined a set of performance parameters including a shape fixing parameter and a shape recovery parameter from the displacement at various points in the shape memory cycle. Ratna and Karger-Docsis [4] defined two similar parameters, also measured from similar shape memory cycles, which measure the performance of the material when subjected to multiple cycles. These various pre-defined parameters are useful in comparing the performance of candidate SMP materials. 7KHSUHVHQWVWXG\IRFXVHVRQWKHGXUDELOLW\SHUIRUPDQFHXWLOL]LQJWKHVKDSHIL[LW\DQGOLQHDUVKDSHUHFRYHU\SDUDPHWHUV GHILQHGE\7DQGRQHWDO>@7KHVKDSHIL[LW\5ILVGHILQHGDV  HX 5I u  

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Fig. 2 Baseline shape memory cycle with free recovery. The circled numbers correspond to the second column in Table 1.

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Fig. 3 Baseline shape memory cycle in temperature-stress-strain space. The numbers correspond to the second column in Table 1.

Fig. 4 Shape fixity, shape recovery, and maximum stress for ten shape memory cycles (deformation rate 50 mm/min followed by 5 min hold time)

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Shape memory response— influence of deformation rate A previous study [7] revealed that Veriflex-E exhibited strain rate sensitivity in axial tension to failure at both room and elevated temperatures. To explore the strain rate sensitivity of the shape memory response of the Veriflex-E, specimens were subjected to shape memory cycles with deformation rates of 0.5, 5, and 50 mm/min from state 1 to 2 in Table 1. The stressstrain curves generated as the sample is strained to 60% at the three deformation rates followed by stress relaxation at a constant strain of 60% (state 2 to 3) were evaluated. The stress-strain results (given in Figure 5a) exhibit rate dependence of the initial linear elastic modulus during loading consistent with the previous study (the modulus increases with an increase in deformation rate). Strain hardening was observed in the previous study beginning near 60% strain, this was a major factor in the selection of 60% as the maximum strain in the current study. The plots of stress drop during relaxation versus relaxation time in Figure 5b show that the change in stress during relaxation is strongly influenced by the prior deformation rate. A larger magnitude drop in stress is observed in relaxation following loading at the deformation rate of 50 mm/min than following loading at 5 mm/min. This larger drop is consistent with results found for other polymers [8, 9] (not exhibiting shape memory capabilities). In contrast, no stress drop is observed during relaxation following loading at 0.5 mm/min.

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Fig. 5 Tension to 60% strain at deformation rates of 0.5, 5, and 50 mm/min followed by 5 min stress relaxation periods at constant 60% strain a) stress-strain curve and b) change in stress versus relaxation time during relaxation period

The effects of the prior deformation rate on the shape fixity, shape recovery, and maximum stress were also evaluated. The results shown in Figure 6 represent a single test for the two slower rates and the average of three tests at the 50 mm/min rate. At all three rates, the material exhibits excellent shape fixity (near 100%). There is a slight increase in shape recovery and slight decrease in the maximum stress as the deformation rate is increased. The less time the material is given for configurational changes to take place at elevated temperature (a faster rate deformation takes less time), the less resistance the material produces against locking in the strain and the better the subsequent shape recovery. More repeated testing at the two slower repeats are planned to verify the patterns in this data. Shape memory response— influence of hold time after deformation The influence of the hold time following deformation to 60% strain (duration of step 2 to 3 in Table 1) on the shape memory performance was also investigated. The stress relaxation following loading at 50 mm/min is shown in Figure 7 for hold times of 5, 30, and 60 min. The rate of stress drop has already begun to slow by the end of 5 min relaxation time and by 10 min relaxation time a steady stress magnitude has been reached. The effect of this hold time on the shape fixity, shape recovery, and maximum stress were evaluated and shown in Figure 8. There is a slight decrease in shape recovery and slight increase in the maximum stress as the hold time is increased. Once again, the less time the material is given for configurational changes to take place at elevated temperature, the less resistance the material produces against locking in the strain and the better the subsequent shape recovery. More repeated testing at the two slower repeats are planned to verify the patterns in this data.

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Fig. 6 Shape fixity, shape recovery, and maximum stress for deformation rates of 0.5mm/min, 5 mm/min, and 50 mm/min (5 min hold time)

Fig. 7 Stress decrease versus relaxation time during relaxation periods of 5, 30, and 60 min (50 mm/min deformation rate)

Fig. 8 Shape fixity, shape recovery, and maximum stress for hold times of 5 min, 30 min, and 60 min (50 mm/min deformation rate)

The shape recovery and maximum stress results from Figures 4, 6, and 8 give evidence that the longer amount of time (slower deformation rate and longer hold time) that the Veriflex-E is subjected to deformation at high temperature, the lower its shape recovery capability and the higher the stresses that are generated during locking in the intended deformation. Concluding remarks The shape memory recovery of Veriflex-E degrades with repeated cycles (the residual strain after the cycle increases) and the maximum stress built up during the cycle increases with repeated shape memory cycling (at 50 mm/min deformation rate and 5 min hold time). These changes during repeated cycling would be amplified at slower deformation rates and longer hold times. The shape recovery values are observed to decrease as the deformation rate is decreased and as the hold times are increased. The recovery of the material is faster and more complete for the SMP subjected to the fastest deformation rate up to 60% strain and the shortest hold time following deformation. In addition the maximum stress levels slightly increase as the deformation rate is decreased and as the hold times are increased. Moreover, the elastic modulus increases with an increase in deformation rate during loading to 60% strain, and the stress drop during relaxation is higher for a higher prior deformation rate. The material demonstrates excellent shape fixity regardless of number of cycles, deformation rates, or hold times at high temperature.

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Higher Rate Testing of Long Fiber Filled Polypropylene Susan I. Hill Senior Research Engineer University of Dayton Research Institute Peter Phillips Associate Research Engineer University of Dayton Research Institute 300 College Park Dayton, OH 45469 ABSTRACT: The properties of long fiber-reinforced thermoplastics at high rates are needed to model the impact behavior of materials and structures. Testing at high rates introduces several issues, namely, bulk material property measurement, equipment capacity due to the higher failure loads, proper load introduction into the material before failure, achieving dynamic equilibrium within the timeframe of the test event, and natural oscillatory vibrations within the specimen. This paper describes a program to develop a specimen suitable for higher rate testing of long fiber reinforced thermoplastics. Two general configurations, with variations, were analyzed using finite-element analysis methods: (1) a radial shoulder-loaded configuration and (2) a mechanical wedge grip-loaded configuration. The selected configuration minimized stress concentrations and had a high probability of failure in or near the gage section. Long glass fiber filled polypropylene was tested at rates from 0.4 to 400/s (0.004 to 6.4 m/s) using variations of the selected specimen configuration. The gage widths were 5, 10, and 15mm. The tensile strength increased with increasing strain rate for all three sizes. The strengths at a given rate were similar for all three sizes. The failure strain of the 15mm wide specimen was lower than the 5mm and 10mm at 0.4/s and 40/s. The stiffness increased with specimen width and rate. The maximum practical strain rate decreased with increasing specimen width. Quality data were obtained from 5mm wide specimens at rates up through 400/s. However, the stiffness and failure strain varied from those of the wider specimens. The 15mm specimen configuration produced data at rates above 100/s, although some issues still remain regarding data interpretation. 1.0 INTRODUCTION Stronger and lighter-weight automotive components are being used to increase fuel efficiency by minimizing vehicle weight. Some of the materials currently under investigation are long fiber-filled polymers and composites. It is critical to understand the change in the material response and energy absorption of these materials under impact conditions if they are to be considered in the design of automotive components and structures. Small specimens are needed to generate useable data at impact conditions. The small size minimizes inertial and natural oscillatory system vibrations while still achieving rates above 200/s. The specimen configuration is usually in direct conflict with the size needed to represent bulk material properties, especially for filled polymers. The gage length and cross-sectional area of current high rate specimens can be of the same scale as the fiber length (~10 mm or greater). In addition, the fiber distribution and orientation within the test section can greatly affect the measured strength and failure strain. This paper describes a program funded by the American Chemistry Council (ACC) to develop a specimen suitable for higher rate testing of long fiber reinforced thermoplastics (LFRT) and long glass fiber-filled polypropylene (LGFPP), in particular. The program included a literature survey of current dynamic specimen configurations, LRFT and LGFPP mechanical properties, finite element analysis of various specimen configurations, and higher rate experimental tests using three different specimen sizes. 2.0 LITERATURE SURVEY Literature, technical standards, and current practices were reviewed to identify potential specimen configurations and material property data. The reported mechanical properties for LGFPP varied based on fiber length distribution and orientation (FLD and FOD) [1], loading level [2-5], and processing [6], amongst other parameters. T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_18, © The Society for Experimental Mechanics, Inc. 2011

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Injection-molded (IM) properties of LGFPP are highly dependent on both the material loading and processing parameters because of the fiber orientation and the high probability of fiber damage. IM LGFPP tends to form a skin: core: skin structure with varying fiber orientations and mechanical properties within each layer. The fibers in the skin tend to align with the flow direction and the fibers in the core tend to be random or transverse to the flow direction. Very little published data are available for the dynamic tensile response of glass fiber-reinforced thermoplastic (GFRT) within the rates of interest (0.1/s to 1000/s). Most of the data are at quasi-static rates (below 0.001/s) or at rates above 1000/s using the Split Hopkinson Bar (SHB). In addition, the identified dynamic data are for short fiber-reinforced polymers or directional composites. The initial search for dynamic data for LGFPP resulted in only two related articles [7, 8]. These articles summarized the results from a SAE round-robin performed in support of the J2749 standard on high strain rate testing of polymers. The literature search was expanded to focus on identifying the current specimen designs used for highly filled materials at dynamic rates. Of particular interest were data generated using a specimen which had at least one of the following attributes: a maximum straight gage length of 30 mm or a minimum width of 3 mm. The 30mm gage length limit was selected to minimize issues related to wave propagation and the natural oscillatory ringing of the test system (resonant ringing) at the higher rates. A width of at least 3 mm was needed to ensure the integrity of the fiber length in various orientations since the fiber length distribution was expected to contain fibers greater than 1mm after injection molding. Results from the literature survey were used to down-select specimen shapes for further detailed analysis. 3.0 FINITE ELEMENT ANALYSIS (FEA) OF SPECIMEN CONFIGURATIONS 3.1 FEA Methodology - Results from the literature survey were used to down-select specimen shapes on the basis of gage width and overall length. Information regarding the differences in the tensile strength, elastic modulus, and failure strain gathered from the literature were used as a guideline for the specimen modeling and optimization. FEA was performed for two different tensile test configurations: 1) a radial shoulder-loaded (RSL) configuration, where loading was introduced through contact of the shoulder grips on an edge of the specimen, and 2) a mechanical wedge griploaded configuration. The tensile specimen designs were evaluated on property prediction capability, failure location, size, and weight. The FEA work was done using the commercial software package, ABAQUS Version 6.8-2. Generation of the models was done within the ABAQUS/CAE platform and analysis was performed using both ABAQUS/Standard and ABAQUS/Explicit. Attempts were made to use the most simplified models that could still accurately depict the key responses of the specimens being evaluated. Due to contact conditions present in all models, nonlinear analyses were used. Twodimensional (2D) plane stress models were utilized for the majority of the shoulder-loaded configurations; three-dimensional (3D) models were analyzed using ABAQUS/Explicit for those models incorporating damage initiation and evolution behavior. The goal was to recommend a specimen design for a general high strain rate specimen for LFRTs. Since the actual material properties were not specifically known, the material models were based on estimates of the properties for fiber-filled thermoplastics [5, 9]. Initial studies were done using isotropic material properties to develop an understanding of the stress concentrations and general specimen behavior. Further analyses were conducted using an orthotropic material model allowing for orientation dependent Young’s moduli, Poisson’s ratios, and shear moduli. Because LFRT can be highly orthotropic due to forming procedures, the orthotropic model was used to represent a general worst-case scenario in which the strong and weak axis moduli values differed by a factor of two. For all material definitions, plasticity was included to account for the ductility of the thermoplastics. Damage initiation and damage evolution criteria were combined with the orthotropic material model during explicit analyses to determine the location of initial failure and how propagation of failure spread throughout the specimen. Each specimen was evaluated using two sets of defined criteria. The first evaluated the specimen’s ability to accurately predict material properties. By employing standard property measurement techniques on the specimen, comparisons were made between the predicted specimen properties and the FEA input properties. That is, the FEA model was treated as if it were a test specimen, and properties (Young’s modulus and Poisson ratio) were computed based on displacements and loads in similar fashion to experimental methods. The ideal specimen would exactly predict the same set of properties used as input to the FEA model. Several width variations were used to identify what differences in the measured properties, if any, were noted due to the specimen shape.

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The second analysis criterion deals with the location of failure and the response of the gage section during failure. Optimum specimens would have failure occurring in the gage section. However, for some specimens, this is not feasible due to the geometry, loading conditions, and stress concentrations. For these cases, the equivalent plastic strain within the gage section was analyzed to determine the effect of the premature failure on the specimen’s predicted strains at failure. Attempts were made to not only meet the evaluation criteria but also to minimize the overall size and weight of the specimens. The University of Dayton Research Institute’s (UDRI) servo-hydraulic test setup was also modeled. Various actuator velocities were used in explicit analyses to determine the stress and strain response of the specimen. The stress and strain responses that would be computed if load and displacement data were reduced using the methodology typically used in testing were compared to axial stress and strain for elements in the gage section. The velocities at which the two sets of data showed large differences was indicative of a maximum useable test speed for the given specimen size. Results from this program were compared to the FEA predictions regarding the maximum machine rate for valid data generation. 3.2 FEA Results and Final Specimen Configuration – With the specification of proper manufacturing tolerances, two RSL designs met the design criteria for a tensile specimen to be used with highly orthotropic material that exhibits moderate ductility during high strain rate testing. Using Johnson-Cook failure modeling, equivalent plastic strains (PEEQ) in the gage section of an RSL at failure were predicted to be 70 – 80% of the material’s maximum PEEQ strain. This prediction was acceptable, given that material imperfections were not considered. A straight tensile specimen with bonded tabs significantly under-predicted the failure strains of an orthotropic material with moderate ductility. This specimen was not recommended for high strain rate testing of orthotropic material. However, analytical stress concentrations were present in the model; thus, the model represented a worst-case scenario for the straight tensile specimen. The recommended RSL specimens were capable of accurately predicting material behavior for orthotropic materials in which the strong and weak axis moduli varied by less than a factor of two. The selected specimen was a radial shoulder loaded specimen, shown in Figure 1. The test program included three variations on the gage width (W = 5 mm, 10 mm, and 15 mm), as summarized in Table 1. A slight taper was added to the straight section (L) to encourage failure in the center. The 5 mm wide specimen (5W) was chosen because its size was similar to other specimens used at rates above 200/s. The disadvantage was the possibility of not including the longer-length fibers in the tested section and perhaps not accurately representing the bulk material property. The 15 mm wide specimen (15W) was chosen in order to maximize the likelihood of retaining the long length of the fibers. The disadvantage was its length and the maximum speed one could achieve and still generate useable data. 4.0 TEST PROGRAM 4.1 Material – Injection molded LGFPP panels were provided by SABIC Germany. The panels were 2mm thick, and contained 30wt% fiber. They were molded with a slow fill speed and low back pressure in order to minimize fiber breakage. The original fiber length was 12.5 mm. The final fiber distribution had a median length of less than 3.5mm with 90% of the fiber below 10mm [10]. Each panel was X-rayed to identify anomalies and fiber clusters. The panels were screened to removed any panel which had fiber clusters or anomalies in the region used for specimen extraction. Only one specimen was taken from each panel. The specimen was located perpendicular to the radial flow and at a fixed distance from the injection gate end. The specimens had relatively thick core sections. Cutting the specimens perpendicular to the flow provided the maximum number of fibers in the core layer oriented parallel to the test direction.

Fig. 1 Radial Shoulder-Loaded Tensile Specimen Showing Geometric Parameter Labels [Figure 34 Reference 11]

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Table 1. Specimen Geometric Parameters Parameter (mm) G L LO R RO T W WO TL

Specimen Width 5 mm 10 mm 15 mm (5W) (10W) (15W) 0.85L 0.85L 0.85L 10 20 30 42 83 124 10 20 30 2 2 2 2 2 2 5 10 15 15 30 45 7 14 21

4.2 Test Matrix and Procedures – Tests were performed at room temperature ambient conditions on a MTS servo-hydraulic station equipped with a 14.7 KN (3,300 lbf) actuator. The three specimens sizes, described in Table 1, were tested at nominal strain rates of 0.4/s, 4/s and 125-400/s. The nominal rate is defined as: x

Hnom G where

x

G

(1)

l

s

is the actuator displacement rate and ls is the straight gage length (10, 20, or 30 mm). The equation assumes that all

of the actuator displacement is translated into a corresponding displacement of the material. This is a valid assumption once the material has plastically deformed. At this point the measured strain rate will be similar to the nominal rate. However, the measured strain rate for materials with limited deformation before failure will be lower than the nominal rate. The measured strain rates of the LFGPP in the program were lower than the nominal rate by a factor of 3 to 4. A close-up of the test setup is shown in Figure 2. The specimens were loaded on the shoulders, or blend radii, using an UDRI designed lightweight grip. The same gripping method was used at all rates. A slack adapter allowed the actuator to attain test speed before applying load to the specimen. Load was measured with a load washer dynamically calibrated at 5 Hz to 444, 890, and 2224 N full scale. Actuator displacement was measured using a linear variable differential transformer. A computer equipped with a National Instruments PCI 6110E high speed data acquisition card was used to capture the test data. 4.2.1 Strain Gages - Back-to-back strain gages were used to capture the response on a select number of specimens. Each gage was centered on the straight section and wired as a quarter-bridge. Data were collected separately for each gage and averaged to compensate for potential bending. The strain gages were Vishay MicroMeasurements EP 08-031-DE120 (5W specimens) or EP-08-125AC-350 (10W and 15W) bonded with Vishay MicroMeasurements M-Bond AE-10. The EP gages were annealed constantan foil with a high-elongation polyimide backing and the specimen surface was lightly sanded to improve the adhesive adherence. The full scale setting was 5% strain. 4.2.2 Digital Image Correlation System (DIC) with ISTRA Software - Full-field 3D deformation was measured with two Phantom V710 high speed cameras and Dantec Dynamic ISTRA digital image correlation (DIC) software. The ISTRA software tracked the motion of a random pattern on the specimen through to failure (Figure 3). Three-dimensional analysis of the pattern movement was used to calculate the net displacements and strains of the features of the pattern. Several sources are available for additional information regarding DIC measurements [12, 13].

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Fig. 3 DIC pattern

Fig. 4. Analyzed DIC Image

Fig. 2 Specimen and Fixturing The DIC image size varied with the filming rate and the specimen size. The typical framing rates for this program were 2000 frames per second (fps) at a test rate of 0.00254 mm/s and 200K fps at 6 m/s. The corresponding region of interest was approximately 400x200 pixels down to 200x96 pixels, respectively. The actual number of pixels across the specimen was less. A central larger polygon was used for all specimens to track a global strain throughout the test (Figure 4). The approximate size of the polygons varied with each size specimen, but was approximately 30 to 40% of the straight gage section. Figure 3 shows a typical DIC image with the random pattern and the measured region. 5.0 RESULTS 5.1 Mechanical Properties – An example of the material behavior across the tested rates is shown in Figure 5. The tensile strength increased about 10% with each decade increase in nominal strain rate, as seen in Figure 6. The measured strength was similar for all three sizes 1 at a given rate. The variability increased with rate and was of similar magnitude for all three sizes. The breaking strain does not appear to change with rate (Figure 7) for a given size. The 15mm wide specimens had a lower breaking strain than the 10mm or 5mm specimens at 0.4/s and 40/s. In contrast, the measured stiffness was different for a given size and with increasing rate. There was a similar positive trend in the stiffness with increasing rate for all three sizes. The measured stiffness also increased with specimen width at a given rate. The increased stiffness with specimen size may be due to a higher number of long length fibers in the test section. This would be consistent with results from other studies [5, 14]. 5.2 Maximum Practical Test Speeds –The natural resonant frequency of the test system, approximately 7.5 to 9 kHz, was dependent on the fixture weight and load train length. Increasing the specimen size increased both the fixture weight and load train length and decreased the natural resonant frequency. . As the test speed increased, the test duration approached the inverse of the natural frequency and increased the likelihood of a resonant response. Figure 9 shows two 10W specimens tested at 6.4 m/s; one shows small amplitude stress waves corresponding to the natural frequency and one does not. Increasing the specimen size, lengthening the load train, and increasing the grip weight at 6.4 m/s, i. e, testing the 15W resulted in higher amplitude stress waves. Modulus determination for the 15W was difficult and data filtering was needed to determine the failure stress and strain. The maximum rate predicted by FE analyses of the test system was between 3 and 5 m/s [11] using the 10mm wide specimen. Figure 10 compares the predicted stress reduced from the EA data (Stress-Test) to that being experienced by an element in the gage section (Stress-Element). The high amplitude waves of the Stress-Test are from resonant ringing of the load washer, grip, and rod assembly. Actual tests showed that the maximum test speed of 6.4 m/s still provided clean, useable data, as shown in Figure 9. Simplifications of the test setup in the FEA model and lack of material damping are two major contributors to the difference in experimental and analytical test speed limits. 1

Statistical significance was determined from the two-sided Student’s t-distribution using an alpha of 0.05 and assuming unequal variances.

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Fig. 5 Stress-strain Response of 5mm Wide Specimen Across Tested Rates

Fig. 6 Strength of 2mm Thick, 30 wt% LGFPP Across Tested Rates

Fig. 7 Breaking Strain of 2mm Thick, 30 wt% LGFPP Across Tested Rates

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Fig. 8 Measured Stiffness of 2mm Thick, 30 wt% LGFPP Across Tested Rates

Fig. 9 Comparison of Response of Two 10mm Wide Specimens Tested at 6.4 m/s 140

0.035

Stress(MPa)

120

Velocity=5.0m/s 0.03

100

0.025

80

0.02

60

0.015

40

Strain(mm/mm)

StressͲTest StressͲElement StrainͲTest StrainͲElement

0.01 Approximate StrainRate:77

20

0.005

0

0 0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Time(10Ͳ3 sec)

Fig.10 Stress in Specimen (Stress Element) versus Measured Stress (Stress Test) at a Velocity of 5.0m/s as Predicted with FE Analyses

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6.0 ACKNOWLEDGEMENTS AND DISCLAIMER The authors would like to express their appreciation to Mr. James Kolb and the Plastics Division of the American Chemistry Council (ACC) for their support and funding for this program. They are also grateful to the helpful guidance of Dr. Michael Wyzgoski (ACC technical point of contact) and Vlastimil Kunc (Oak Ridge National Labs) and to Dr. Geoffrey Frank, who aided in the finite element analyses. 7.0 REFERENCES 1. Nguyen, B.N., S.K. Bapanapalli, J.D. Holbery, et al., “Fiber Length and Orientation in Long-Fiber Injection-Molded Thermoplastics — Part I: Modeling of Microstructure and Elastic Properties ", J Composite Mater, 42, 2008, 1003-1029. 2. Thomason, J.L., “The Influence of Fibre Length and Concentration on the Properties of Glass Fibre Reinforced Polypropylene: 5. Injection Moulded Long and Short Fibre PP, Composites Part A, 33, 2002, 1641-1652. 3. Thomason, J.L., “The Influence of Fibre Length and Concentration on the Properties of Glass Fibre Reinforced Polypropylene. 6. The Properties of Injection Moulded Long Fibre PP at High Fibre Content, Composites Part A, 36, 2005, 995-1003. 4. Thomason, J L. and M.A. Vlug, “Influence of Fibre Length and Concentration on the Properties of Glass Fibre-Reinforced Polypropylene: 1. Tensile and Flexural Modulus”, Composites Part A, 27, 1996, 477-484. 5. Thomason, J.L., M.A. Vlug, G. Schipper, and H.G.L.T. Krikor, “Influence of Fibre Length and Concentration on the Properties of Glass Fibre-Reinforced Polypropylene: Part 3. Strength and Strain at Failure”, Composites Part A, 27, 1996, 1075-1084. 6. Short, W.T. and E.J. Wenzel, “Effects of Molding Process on Residual Fiber Length of Long Fiber Polypropylene Composites", Paper 0082, Society of Plastics Engineers - 65th Annual Technical Conference of the Society of Plastics Engineers, Plastics Encounter at ANTEC 2007, Mar 6-11, 2007,Cincinnati, OH. 7. Pinnell, M., Hill, S. and Minch, A.,"Special Concerns in High Strain Rate Tensile Testing of Polymers", SAE 2006 Transactions Journal of Materials and Manufacturing V115 5, April 3-6, 2006, Detroit, Michigan. 8. Hill, S., Pinnell, M., Minch, A., “Standardization of High Strain Rate Tensile Testing of Polymers”, ANTEC 2005, Boston, MA, May 3, 2005, Paper No. 101529, Transactions pp. 2729-2733. 9. “Predictive Engineering and Mechanical Performance of Injection-Molded, Short-Fiber-Filled Thermoplastic Parts,” American Plastics Council Auto Learning Center, Dec. 2006. 10. Warren, C., Schutte, C., “Engineering Property Prediction Tools for Tailored Polymer Composite Structures”, Lightweigth Materials FY 2010 Annual Report, Contract No. DE-AC05-00OR22725 and DE-AC06-76RLO1820 11. Hill, S. Phillips, P., “Optimization of Specimens Used for High-Rate Testing of Long Fiber-Filled Polymers,” UDR-TR-2009-161, American Chemistry Council, 2009. 12. Wang, Y. H., Jiang, J. H., Wanintrudal, C., et. al., “Whole Field Sheet-metal Tensile Test Using Digital Image Correlation”, Experimental Techniques, March/April 2000, pp 54-62 13. Sutton, M., Orteu, J-J., Schreirr, H., Image Correlation for Shape, Motion and Deformation Measurements, Springer Publishing, 2009 14. Kumar, K.S., Bhatnagar, N. and Ghosh, A.K., “Development of Long Glass Fiber Reinforced Polypropylene Composites: Mechanical and Morphological Characteristics”, J Reinf Plast Compos, 26, 2007, 239-249

Characterization of Elastomeric Composite Materials for Blast Mitigation

K. Schaaf and S. Nemat-Nasser Department of Mechanical and Aerospace Engineering, Center of Excellence for Advanced Materials, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0416, USA ABSTRACT In this work, we seek to develop elastomeric composite materials capable of shock mitigation through material design by small-scale heterogeneity. The host elastomeric material is a polyurea system that is a lightly cross-linked two-phase polymer, which consists of the diamine component Versalink P-1000 as the soft segment and the diisocyanate component Isonate 143L as the hard segment. This study evaluates the impact of additives in the form of untreated and surface treated milled glass fibers. The properties of the resultant elastomeric composite materials are mechanically and thermally characterized using durometer testing, dynamic mechanical analysis (DMA) testing, and differential scanning calorimetry (DSC) testing in order to determine the hardness, storage and loss moduli, and glass transition temperature of the composites, respectively. Preliminary results indicate that the dynamic mechanical properties of the material can be significantly altered through such modifications. The work described here is part of an ongoing effort to understand the impact of additives on the ultimate properties and performance of the host elastomeric material. Keywords: Elastomeric composite, shock mitigation, polyurea, storage and loss moduli, glass transition temperature 1. INTRODUCTION Block copolymers are a unique subset of copolymers that have the ability to microphase separate. Phase separation occurs because of the chemical dissimilarity of the soft and hard phases resulting in a hard phase that is dispersed in a soft matrix. Additionally, the hard phase has the ability to reinforce the soft matrix and to create a cross-linked backbone, thus enhancing the mechanical properties of the polymer. Polyurea is created by reacting a difunctional amine, Versalink P-1000 [1], with Isonate 143L [2], which is a mixture of difunctional and trifunctional isocyanates. The reaction occurs at room temperature forming a lightly cross-linked and twophase polymer.

Fig. 1 Reaction of Isonate 143L and Versalink P-1000 to form polyurea [3]

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_19, © The Society for Experimental Mechanics, Inc. 2011

137

138 2. POLYUREA COMPOSITES The elastomeric material system can also be modified through the integration of additives, both those that are covalently bonded and those that are simply mechanically bound in an effort to tailor the ultimate properties and performance. Silica, in the form of micro-scale milled glass fibers is an attractive additive. The milled glass fibers are functionalized with surface treatments (hydrophobic, hydrophilic, covalent bonding promoters, etc.) to tailor the interfacial properties. Carbon nanotubes (CNTs) have a higher modulus than milled glass fibers and serve as further reinforcements. Milled glass fibers are functionalized using silanes with aromatic rings, resulting in a surface that will naturally attract the CNTs. In this scenario, the integration of carbon nanotubes serves as an adhesion modifier between the polymer matrix and glass fibers.

Fig. 2 (left) The length scales of the silane surface treatments, surfactants, polyurea elastomeric matrix, carbon nanotubes, and milled glass fibers are identified (right) Schematic of functionalized milled glass fiber/CNTs/polyurea matrix. 3. RESULTS AND DISCUSSION Durometer testing indicated that the integration of milled glass fibers had a negligible effect on the hardness of the host elastomeric matrix. DSC testing demonstrated that the integration of the milled glass fiber additives, regardless of the surface treatment, did not significantly affect the glass transition temperature of the polyurea. DMA testing concluded that the integration of milled glass additives with silane and amino surface treatments are capable of increasing both the loss and storage moduli by more than 100% Work is currently in progress to assess the impact that CNTs serving as further adhesion modifiers in the polyurea matrix can have on the ultimate properties and performance of the elastomeric material. ACKNOWLEDGEMENTS This research has been conducted at the Center of Excellence for Advanced Materials (CEAM) at the University of California, San Diego. This work has been supported by the Office of Naval Research (ONR) grant N00014-09-1-1126 to the University of California, San Diego, under Dr. Roshdy Barsoum’s Explosion Resistant Coating Joint Enhanced Explosion Resistant Coatings Exploitation (JEERCE) Advanced Concept Technology Demonstration (ACTD) research program. REFERENCES [1] Air Products Chemicals, Inc., Polyurethance Specialty Prodcuts (Air Products and Chemicals, Allentown, PA, 2003). [2] The Dow Chemical Company, Isonate 143L, Modified MDI (Dow Chemical, Midland, MI, 2001). [3] Fragiadakis, D., Gamache, R., Bogoslovov, R.B., Roland, C.M. Segmental dynamics of polyurea: Effect of stoichiometry, Polymer, 51, 178-184, 2010.

Experimental arrangement for measuring the high-strain-rate response of polymers under pressures

Maen Alkhader, Wolfgang Knauss and Guruswami Ravichandran Graduate Aerospace Laboratories, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA e-mail: [email protected]

ABSTRACT This study aims to investigate the high-strain-rate shear response of viscoelastic elastomeric coatings at large strains and under elevated levels of hydrostatic pressure. Results of this study shed light on the combined effects of deformation rate and pressure which might promote a transition from viscoelastic to glassy behavior. This work utilizes a Split Hopkinson Pressure Bar (SHPB) apparatus in conjunction with a customized version of the recently proposed Shear Compression Specimen (SCS) which consists of a polymer gage section with two metal ends that remain essentially rigid during deformation. Detailed finite element simulations were used to customize the adopted specimen, to determine its proper dimensions and promote its functionality. The customized specimen permits subjecting the tested specimen to a state of uniform pressure and shear stress, while allowing for measuring pressure, shear stress and shear strain directly. Results obtained using the customized specimen, which are included in this paper, illustrate its usefulness in measuring the effect of high-strain-rate, large strain and hydrostatic pressure on the shear stress-strain response of viscoelastic elastomers. INTRODUCTION Viscoelastic elastomers have been surprisingly effective in enhancing the resistance to penetration and fragmentation of armor when applied as armor coatings. During such processes they experience strains, strain-rates and complex states of stress under which their response might not be fully described by the commonly used constitutive models based on linear viscoelasticity or quasi-linear viscoelasticity, which were utilized in describing the constitutive behavior of polyurea in [1] and [2], respectively. The latter assumes the response to be a superposition between linear viscoelasticy and hyperelasticty. In this superposition, large strains are described by a hyperelastic potential while time-dependence follows linear viscoelastic principles. The deviation from linearviscoelasticy is anticipated because one observed that under high-strain-rates and when deformations achieve large metrics, elastomeric coatings provide excellent stiffness, highlighting the possibility of a deformation induced transition to the glassy state [3-5]. Theoretically and as presented in [6,7] such transition is generated by impeding relaxation mechanisms which can be achieved by negative dilatation derived from low temperature, high pressures or simply by deforming the material at a time scale shorter than the relaxation times of the material [5]. Although a detailed understanding of the mechanisms behind the effectiveness of elastomeric coatings is imperative to the development of such coatings, for design purposes and in order to employ coatings in engineering structures reliably, it might be sufficient to establish empirical constitutive models capable of predicting the response under any complex loading. Accordingly, the constitutive response should rely on experimental observations that cover the full range of the anticipated behavior and should represent all the principal parameters and their interactions. Hence, experimental data covering the full range of the strain, strain rates and pressures existing during impact or blast loadings are required for both developing constitutive models and understanding the different mechanisms. So far experimental methodologies have been used effectively to provide significant amount of data. Split Hopkinson Pressure Bar (SHPB), which is a standard technique for extracting the high-strain-rate response of materials, was used to obtain the high-strain-rate response of polyurea and polyurethane at strain rates [8]. However, since viscoelastic elastomers are pressure dependent, with SHPB, the effect of strain-rate and mean stress are not easily separable and pressure-induced effects might be mistaken for strain-rate-sensitivity. Alternatively, one can directly measure the shear response independently of pressure by using a torsional Kolski apparatus or by using the specialized shear specimen proposed in [9], in which the T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_20, © The Society for Experimental Mechanics, Inc. 2011

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specimen is placed at an angle of 45º with respect to the load. This specimen was used successfully in conjunction with SHPB to measure the response of polyurea at moderate to high strain rates. On the other hand, the effect of pressure at low strain rates was measured for polyurea by superposing a torsional strain field on confined specimens preloaded with an equitriaxial state of stress [10]. At the extreme end, the combined effect of pressure and high-strain-rates was measured for polyurea using the plate-impact experiment [11]. Although, the aforementioned experiments provided valuable data, they covered important but limited segments of the space defined by the potentially important strain-rates and pressures. Accordingly, an experimental arrangement is needed to measure the response of elastomeric coatings at moderate to high strain-rates and pressures. The present study aims to address this gap by merging two of the aforementioned techniques, namely the shear specimen [9] and plate impact experiment [11]. The remainder of this paper is organized as follows: We first present the experimental arrangement, then, a finite element verification is presented, followed by a presentation of the results and their discussion, with global conclusions presented last. EXPERIMENTAL ARRANGEMENT The shear-compression specimen which was originally proposed for polymers in [9] consists of two steel or aluminum blocks with the polymeric specimen (2mm by 2mm cross-section and 15.6 mm in length) placed between the blocks at an angle of 45 degrees as seen in Fig. 1. The polymeric specimen is glued to the aluminum blocks to apply the intended shear loading. The blocks, which remain essentially rigid during the specimen deformation, are then subjected to a compressive force, P(t) , which can be decomposed into compressive and shear stresses using

V xy =

Pt cosT .sin T w'

V yy =

(1)

Pt cos 2 T w'

(2)

were w and ¨ are known geometric parameters and ș in [9] is set to 45º. In this work, ș is varied so as to subject the sample to various levels of compressive stress. Metal blocks

Specimen w

ο 2mm X 2mm cross-section

45 deg

ș

Fig. 1 Shear compression specimen [9], used to measure the quasi-static and high strain rate shear response of polyurea. During deformation, the polymeric specimen is subjected to a strain field that can be described analytically as [9]

H xy =

1 2w

u

2



§ §u ··  v 2 cosT ¨¨ arctan¨  T ¸ ¸¸ ©v ¹¹ ©

(3)

such that u and v are the relative horizontal and vertical displacements of the blocks, respectively. Although in [9] the plane stress assumption was utilized since the out of plane thickness was small, which was also justified by way of finite element analysis, equations 1-3 are derived from force equilibrium and basic kinematics of deformation in the deformation (x-y) plane independently from plane stress assumption.

141

Although this shear-compression specimen was utilized successfully in [9], it inherently has two issues that conflict with the objectives of the current study. First, at small strains the polymer specimen is subjected to a plane state of stress but at lager strain a triaxial state of stress exists, however, only the normal stress component can be measured directly. More importantly, computing shear strains using Eqn. 3 requires knowing both components of displacements. But while the vertical component is easily measured, in [9] it was suggested to use optical techniques or a vibrometer to measure the horizontal component of relative displacement. In this work, we modified the geometry of the specimen by increasing its area normal to the applied load and decreasing its thickness, to generate a nearly equitriaxial state of stress in the polymeric specimen; hence the measured normal force component is enough to describe the state of stress in the specimen. The modified specimen, see Fig. 2, mimics specimens used in plate-impact experiments, except that loading is applied using SHPB for high strain rates and universal testing machines for quasi-static loading rates. 20

10 Steel Specimen thickness ranges between 0 .2

h

Specim

Fig. 2 Modified shear-compression specimen. The incompressible nature of the tested specimens provides an easier alternative for determining the horizontal relative displacements of the blocks. Due to incompressibility and the geometrical properties of the modified specimen, the thickness (h) of the tested polymeric specimen can be assumed to remain unchanged during deformation. Accordingly, the relative deformation of the blocks has to satisfy

tanT =

v u

(4)

this equation is used in this work to compute the horizontal displacement instead of measuring it. Accordingly, by measuring the normal force and displacement either while using universal testing machines or SHPB, one can fully describe using Eqns. 1-4: the shear stress, pressure and shear strain in the tested polymeric specimen. Therefore, the new modified arrangement can be effective in quantifying the combined effect of pressure and strain rate on the shear stress of viscoelastic elastomers. FINITE ELEMENT VERIFICATION To ensure the accuracy of the measured response, finite element simulations were used to verify the assumptions behind this new configuration and establish the range of applicability of Eqns. 1-4. finite element verification was performed using two inclination angles 9 and 18, such that the smaller angle results in larger pressures. The metal blocks were made of steel and had a cross section of 20*10 mm2 as described in Fig. 2. The specimen thickness was 0.2 mm. Multiple constitutive models were employed in the finite element-based verification to ensure that the new specimen and Eqns. 1-4 were independent of the tested material. However, these constitutive models were incompressible, pressure independent and have a stress-strain behavior that lies within the bounds of the expected response of the tested materials. Although the different constitute models confirmed the validity of the assumptions, here we present the results obtained using a bilinear elastic-plastic constitutive model. This model had a stiffness of 10GPa, a yield stress of 2 MPa and a hardening modulus that allows the flow stress to attain the value of 30 MPa at a strain value of 1.0. Poisson ratio during the elastic phase was set to 0.33, while during the plastic phase it took the value of 0.5, following plasticity principles. Load was applied by means of displacement controlled boundary condition and friction was neglected at the top and bottom faces of the metallic blocks; since lubricant is present at these interfaces during experiments. The commercial FE software ABAQUS was used to perform all simulations. For both angles, finite element results confirmed that the new modified geometry is effective in subjecting the specimen to a state of uniform hydrostatic pressure in conjunction with a uniform shear stress as can be seen in Figs. 3 and 4. Figure 3 illustrates the evolution of the normal components of stress with the applied prescribed displacement, while Figure 4 compares between the shear stress-strain relation reported by the finite element software and the values predicted by Eqns 14.

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stress xx stress yy -120 stress zz

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0

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-60 -80

stress xx -100 stress yy -120 stress zz -140 -160

-160

vertical displacement (mm)

Vertical displacement (mm)

Fig. 3: Normal stress generated in the specimen. a) The inclination angle is 9º, while for b) angle is 18 º. As expected in both cases, the normal stresses are equal and are smaller for the larger angle.

0 -0.4

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Fig. 4: Shear stress-strain response. Showing a comparison between the stress-strain response reported by the finite element software and that computed by Eqns. 1-4. For a) the inclination angle is 9º, while for b) that angle angle is 18 º.

Figure 4 illustrates the ability of Eqns. 1-4 to determine the shear stress-strain relation accurately up to strains of at least 50%. Accordingly, in the following experimental work, the modified specimen is used to measure the shear stress-strain response of polyurea for strains that do not exceed 0.5. EXPERIMENTS MEASUREMENTS Steel blocks identical to those used in the simulations were machined. The face of each block at which specimens were to be glued was polished with 400 grit paper to increase the strength of the bond to the polyurea specimen. The process of creating the polyurea specimens commences by casting polyurea between two Teflon blocks to form 0.5 to 1.0 mm thick sheets. Specimens are then cut from the sheets and glued with superglue between the steel blocks. A 24 hours curing period was maintained consistently before testing to permit the glue at the interface to achieve its maximum strength. Specimens were then cleaned from any excess glue. Special attention was given to faces at which load was to be applied. These faces were cleaned thoroughly and lubricated with grease to minimize friction. Figure 5 presents one of the specimens.

Fig. 5 A specimen with a 9 inclination angle

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RESULTS AND DISCUSSION Quasi-static tests were performed using a 14000N capacity MTS system. Specimens with 9 º inclination angles were tested at shear strain rates of 5 x 10-3 s-1 and 0.5 s-1, while specimens with 18º inclination angles were tested at shear strain rates of 0.03 s-1 and 0.3 s-1. Although these shear strain rates are small, their influence -when accompanied by large hydrostatic pressure- on the shear response of viscoelastic elastomers is of interest. Experimental results illustrating the measured shear stress-strain with respect to the accompanying pressure are presented in figure 6.

a

5x10-3 s-1 4x10-1 s-1 5x10-1 s-1

b

5x10-3 s-1 4x10-1 s-1 5x10-1 s-1

Fig. 6 Quasi-static response, showing: a) shear stress-strain response and b) the accompanying pressure. Figure 6 shows that even at very slow rates (10e-3) the shear stress of polyurea monotonically increased and does not achieve a plateau flow stress at 1 to 2 MPa as was found in [12]. This suggests that the accumulated pressure during deformation promotes stiffening of the shear response of polyurea; this is in agreement with the observations presented in [10]. This figure also illustrates that the two orders of magnitude difference in strain rate has negligible effect within the range of quasi-static strain rates tested. One can argue that polyurea at room temperature behaves as an elastomer, the relaxation times of which are smaller than the time scale characterizing the tested quasi-static strain rates.

Fig. 7 Dynamic response: showing shear stress-strain response and the accompanying pressure for two dynamic tests performed at a shear strain rate of approximately 5000s-1. Two specimens (9º and 18º) were used the sample with smaller angle produced the larger pressure Figure 7shows the dynamic response of polyurea at a shear strain rate of (5000s-1). During both tests, the glued interface was compromised at strains of about 0.15. Postmortem analysis of the specimens revealed that the specimens sustained damage during the test. This damage was repeatedly localized at the center of tested specimens. Figure 7 not only illustrates the strain-rate sensitivity of polyurea but also highlights the important role of the accompanying pressure; higher pressures stiffen the response, which agrees with the quasi-static tests.

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CONCLUSIONS In this work the shear compression specimen proposed in [9] was modified to investigate the shear response of polymers under a wide range of strain-rates and pressures. Principles behind the proposed modifications were outlined and the effectiveness and accuracy of the modified specimen was verified using FE simulations. Experiments were performed on polyurea under a wide range of strain rates (5x10-3 to 5x103 s-1) and pressures (reaching 90 MPa). Results illustrated the strain-rate sensitivity of polyurea, but more importantly it highlights the effect of pressure on the shear stress-strain response.

REFERENCES [1] Amirkhize, A., Isaacs, Mcgee, J., Nemat-Nasser, S., An experimentally-based viscoelastic constitutive model for Polyurea, including pressure and temperature effects. Philosophical magazine, 86, 36, 5847-5866, (2006). [2] Li, C., Lua, J., A Hyper-viscoelastic constitutive model for polyurea, Materials Letters, 63, 877-880, (2009). [3] Bogoslovov, R.B., Roland, C.M., Gmache, R., Impact-induced glass transition in elastomeric coatings, Applied physics letters, 90,22, 221910-221913, (2007). [4] Lee, g., Mock, W., Feddrly, J., Drotar, J., Balizer, E., Conner, M., The effect of mechanical deformation on the glass transition temperature of polyurea. Schok Compression of Condensed Matter, CP955, (2007). [5] Fragiadakis, D., Gamache, R., Bogoslovov, R.,B., Roland, C.,M., Segmental dynamics of polyurea: Effect of stoichiometry, Polymer, 51, 178-184, (2010). [6] Ferry, Viscoealstic Properties of Polymer’s, 3rd ed. Wiley and Sons, New York, 1980 [7] Tschoegel, N., Knauss, W., Emri, I., The effect of temperature and pressure on the mechanical properties of themoand/or Piezorheologically simple polymeric materials in thermodynamic equilibrium- A critical review. Mechanics of TimeDependent Materials, 6, 53-99, (2002). [8] Yi, J., Boyce, M.C., Lee, G.F., Balizer, E., Large deformation rate-dependent stress-strain behavior of polyurea and polyurethans, Polymer, 47, 319-329, (2006). [9] Zhao, J., Knauss, W.G., Ravichandran, G., A new Shear-Compression-Specimen for Determining quasistatic and dynamic polymer properties. Experimental Mechanics, 9, 49,427-436, (2009) [10] Alkhader, M., Knauss, W.G., Ravichandran, G., The influence of pressure on the large deformation shear response of a Polyurea. Proceedings of SEM conference, Indianapolis, (2010). [11] Jiao, T., Clifton, R., Grunschel, S., Pressure-sensitivity and tensile strength of an elastomer at high strain rates. Shock compression of condensed matter. CP955, (2007). [12] Chakkarapani, V., Ravi-Chandar, K., Liechti, K., Characterization of multiaxial constitutive properties of rubbery polymers. Journal of engineering materials and technology, 128, 489-494, (2006).

Simulation of impact tests on polycarbonate at different strain rates and temperatures J.L. Bouvard, C. Bouvard, B. Denton, M.A. Tschopp, M.F. Horstemeyer Center for Advance Vehicular Systems 200 Research Boulevard, Starkville, MS 39759 ABSTRACT The use of lighter and impact resistant materials, such as polymers, in vehicular systems is an important motivation for the automotive industry as these materials would make vehicles more fuel-efficient without compromising safety standards. In general, polymers exhibit a rich variety of material behavior originating from their particular microstructural (long molecular chains) behavior that is strongly temperature, pressure, and time dependent. To capture such intricate behavior, a number of polymer constitutive models have been proposed and implemented into finite element codes in an effort to solve complex engineering problems (see [1] for a review of these models). However, developing improved constitutive models for polymers that are physically-based is always a challenging area that has important implications for the design of polymeric structural components. The work describes the extension and the application to temperature dependence of a time dependent material model developed for amorphous polymers in [2]. The modeling approach follows current methodologies used for metals [3] based on a thermodynamic approach with internal state variables. Thus, the material departs from spring-dashpot based models generally used to predict the mechanical behavior of polymers. To select the internal state variables, we used a hierarchical multiscale approach for bridging mechanisms from the molecular scale to the continuum scale. First, molecular dynamics (MD) simulations were used to study deformation mechanisms during uniaxial tensile deformation of an amorphous polyethylene polymer [4]. The examination of the MD simulations stress-strain curves showed specific features characterized by elastic, yield, strain softening and strain hardening regions that were qualitatively in agreement with experimental results on amorphous polymers. Factors such as the energy contributions from the united atom potential, change of free volume and chain entanglements were also calculated as a function of strain to help elucidate the inherent deformation mechanisms within the elastic, yield, and strain hardening regions. Information obtained from the molecular dynamic scale was next used to build a continuum level constitutive model [2]. The continuum constitutive model applied a formalism using a threedimensional large deformation kinematics and thermodynamics framework. The model predictions are then compared to compression, tensile and impact test data for a thermoplastic polycarbonate deformed at different strain rates and temperatures. Fig. 1 presents the comparison between the model predictions and the experimental data for uniaxial compression at different strain rates and temperatures. All the curves show the expected features of the mechanical response for PC at temperatures below the glass transition: an initial linear elastic response followed by a non-linear transition curve to global yield, then strain softening and subsequent strain hardening. The model predicts well the mechanical behavior for both loading and unloading.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_21, © The Society for Experimental Mechanics, Inc. 2011

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Fig. 1 Comparison between model predictions and experimental data of PC in uniaxial compression for different temperatures at: (a) 0.0005/s, and (b) 0.007/s. Fig. 2 displays the model predictions on tensile tests at 0.001/s for three different temperatures (-20, 25 and 100˚C). We can notice that the numerical simulations predict qualitatively the test results for the different regimes.

Fig. 2 Comparison between model predictions and experimental data of PC in uniaxial tension at 0.001/s for different temperatures. Fig. 3 displays a comparison of the force-displacement curve between the model predictions and the impact test for a velocity of 3 mm/s and 300 mm/s at different temperatures. As depicted in the figure, the numerical simulations results are in good agreement with the test data.

(a)

(b)

Fig. 3 Comparison between model predictions and impact data for different temperatures at ; (a) 3 mm/s , and (b) 300 mm/s Acknowledgements

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The authors would also like to thank the Center for Advanced Vehicular Systems (CAVS) at Mississippi State University and American Chemistry Council for their support. References [1] Bouvard, J.L., Ward, D.K., Hossain, D., Nouranian, S., Marin, E.B., and Horstemeyer, M.F., Review of hierarchical multiscale modeling to describe the mechanical behavior of amorphous polymers, JEMT, DOI: 10.1115/1.3183779, 2009. [2] Bouvard, J.L., Ward, D.K., Hossain, D., Marin, E.B., Bammann, D.J. and Horstemeyer, M.F., “A General Inelastic Internal State Variables Model for amorphous glassy polymers”, 2010, Acta Mechanica, 213(1), 71-96., 2010. [3] Bamman, D.J., Chiesa, M.L., and Johnson, G.C., Modeling Large Deformation and Failure in Manufacturing Processes, In: Tatsumi, T., Wanatabe, E., and Kambe, T. (Eds), Theoretical and Applied Mechanics, Elsevier Science, 359-376, 1996. [4] Hossain, D., Tschopp, M.A., Ward, D.K., Bouvard, J.L., Wang, P., Horstemeyer, M.F., “Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene”, Polymer, 51(25), Pages 6071-6083, 2010.

Experimental Investigation of Dynamic Mechanical Properties of Polyurea-Fly Ash Composites

Alireza V. Amirkhizi*, Jing Qiao, Wiroj Nantasetphong, Kristin Schaaf, and Sia Nemat-Nasser Center of Excellence for Advanced Materials, Department of Mechanical and Aerospace Engineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0416, USA * [email protected]

ABSTRACT Polyurea has been the material of choice in many applications due to its thermomechanical properties. These applications span a wide spectrum from abrasion-resistant coating to reinforcement against blast damage in structures, ships, and vehicles. The improved observed performance is linked to its microstructure, a lightly crosslinked (elastomeric) block copolymer. The constitutive modeling of such materials is generally based on a wide variety of experimental measurements, including dynamic mechanical analysis. Furthermore, the response under stress-wave propagation may be measured through ultrasonic tests. In this work we present our research towards improvement and modification of the dynamic mechanical properties of polyurea through inclusion of various size and distributions of fly ash hollow spherical particles. The extensive experimental results are reported and a micromechanical homogenization model is presented. The extent of application of such model will be established and alternative modeling techniques which include the possible inertia effects at high rates of deformation will be surveyed.

Introduction The constitutive modeling of polyurea, and elastomeric polymers in general, is an inherently difficult problem. The significant difference between the shear and volumetric response, both in magnitude and mechanism, is a main reason for such difficulty. The shear response is generally very compliant and lossy, hence a viscoelastic model based on tests performed in larger deformation levels are required. On the other hand, the volumetric behavior is comparatively very stiff, nonlinear and generally not very lossy. A complete model created for polyurea can be found in literature [1]. Here we will present our latest experimental results regarding polyurea-based composites, especially those involving fly-ash hollow particles [2]. The dynamic and quasi-static response of the composite under a variety of pressure and temperature conditions will be presented. At the end we will discuss a micromechanical model for the overall properties of hollow-sphere reinforced composites and compare the predictions of the model with experimental data.

Experimental Setup The experimental setup is shown in Figure 1. Longitudinal and shear ultrasonic waves are sent through the sample and a confining chamber and measured on the opposite side. The sample may be loaded axially to very high

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_22, © The Society for Experimental Mechanics, Inc. 2011

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pressures and also is held in a temperature-controlled chamber. By measuring the time of travel through the setup and comparing it with expected time of travel in a setup with known properties, the wave speed for shear and longitudinal waves are measured as functions of temperature and pressure.

Fig. 1 The ultrasonic setup for high-pressure experiment. The transducers and securing chambers are shown on the left while the pressure cell, the sample and the loading pistons are shown in detail on the right.

Micromechanical Modeling Two micromechanical models have been used to estimate and predict the overall properties of composites. In the first one a dilute and random distribution of fly ash particles is analyzed and the elastic problem is solved. Based on the solution of the elastic problem of concentric spherical shells, the overall properties are bounded [3]. In the second model, suitable for higher concentrations, the overall properties are estimated by solving a 3D infinite periodic elastic system [4].

ACKNOWLEDGEMENT This work has been conducted at the Center of Excellence for Advanced Materials (CEAM) at the University of California, San Diego. This work has been supported in part through ONR under Grant No. N00014-09-1-1126 to the University of California, San Diego.

REFERENCES [1] Amirkhizi A. V., Isaacs J., McGee J., Nemat-Nasser S., An experimentally-based viscoelastic constitutive model for polyurea, including pressure and temperature effects, Philosophical Magazine, 86, 5847-5866, 2006. [2] Qiao, J., Amirkhizi, A. V., Schaaf, K., Nemat-Nasser, S., Dynamic mechanical analysis of fly ash filled polyurea elastomer, Journal of Engineering Materials and Technology, 133, 011016 (7 pages), 2011. [3] Lee, K. J., Westmann, R. A., Elastic properties of hollow-sphere-reinforced composites, Journal of Composites Materials, 4, 242-252, 1970. [4] Nemat-Nasser, S., Hori, M., Micromechanics: Overall Properties of Heterogeneous Materials, 2nd Edition, Elsevier, 1998.

Damage & Fracture of High-Explosive Mock Subject to Cyclic Loading C. L IU † , P.J. R AE‡ , C.M. C ADY† , AND M.L. L OVATO† † Materials

Science & Technology Division and ‡ Weapons Experiments Division Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

Abstract We use four-point bend specimen with a single shallow edge notch to study the fracture process in Mock 900-21, a PBX 9501 high explosive simulant mock. Subject to monotonic loading we determine quantitatively the threshold load for macroscopic crack initiation from the notch tip. The four-point bend specimen is then subject to cyclic loading in such a way that during the Àrst cycle, the applied force approaches but does not exceed the threshold load determined from the monotonic loading test and in the subsequent cycles, the overall maximum deformation is maintained to be equal to that of the Àrst cycle. It is expected and is also conÀrmed that no macroscopic damage and cracking occur during the Àrst cycle. However, we observe that sizable macroscopic crack is generated and enlarged during the subsequent cycles, even though the applied force never exceeds the threshold load. Details of the process of damage formation, accumulation, and crack extension are presented and the mechanical mechanism responsible for such failure process is postulated and discussed.

1 Introduction Heterogeneity is one of the dominating microstructural characteristics of the high explosives and their mock simulants. Such heterogeneity also plays a paramount role in the process of mechanical damage, damage accumulation, crack formation, and crack extension in these materials. In this investigation, we study the effect of cyclic loading on the formation of macroscopic cracks and their subsequent extension. We use the four-point bend specimen with a single shallow edge notch. First we subject the sample to monotonic, displacement-controlled loading till Ànal failure. We determine the threshold loading, at which macroscopic cracking initiates. Then, we subject the specimen to cyclic loading in such a way that during the Àrst loading cycle the maximum applied force is close but does not exceed the threshold load determined from the monotonic loading test, and for each of the subsequent cycle, the maximum value of overall deformation approximately equals to that of the Àrst cycle. We found that the maximum applied force for each of the subsequent cycle never reaches and exceeds that of the Àrst cycle. In fact, this maximum applied force monotonically decreases remarkably as a function of number of cycles. Using the technique that we developed for quantitatively identifying the location and extent of macroscopic cracks in heterogeneous high explosive and mock material [1], we conÀrm that during the Àrst cycle, there is no macroscopic crack formation and extension from the notch tip. However, during the subsequent loading cycles, we observe that macroscopic crack extends from the notch tip and propagates into the specimen. This is due to the accumulation of damage, which in turn, is the result of heterogeneity of the material and the non-uniform stress state near the notch tip. In this paper, we Àrst brieÁy describe the testing material Mock 900-21, the sample geometry, and the loading conÀguration of our experiments. Then we present experimental results of both monotonic loading and the cyclic loading, and the macroscopic crack formation and extension associated with these two loading conÀgurations. Finally, we discuss the mechanical mechanism that might be responsible for the failure process under cyclic loading and the signiÀcance of the Àndings to the safety of weapons and munitions systems.

2 Description of experiments The heterogeneous material studied in this investigation is Mock 900-21, which is formulated as a density mock of the PBX 9501 high explosive. Its mechanical responses, under various loading conditions, also emulate that of the PBX 9501. The composition of the Mock 900-21 is listed in Table 1. Barium nitrate [Ba(NO3 )2 ] has the cubic crystalline structure with the lattice parameter a = 8.11 Å and density W = 3.24 g/cm3 . Pentek or pentaerythritol [C5 H12 O4 ] has the ditetragonal crystalline structure with density W = 1.399 g/cm3 . The polymeric binder of the heterogeneous composite is an equal-weight mixture of a polyurethane (Estane 5703) and a nitroplasticizer (BDNPA-F), with density W = 1.283 g/cm3 . Estane is copolymer made

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_23, © The Society for Experimental Mechanics, Inc. 2011

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152 from a combination of soft segments and hard segments bonded by ester and urethane linkages. The nitroplasticizer (BDNPAF) is a mixture of bis(2,2-dinitropropyl)acetal and bis(2,2-dinitropropyl) formal. The addition of NP would lower the glass transition temperature (Tg ) of the polymer binder to about −50 ◦ C. Table 1: Composition of Mock 900-21 Constituent wt.%

Ba(NO3 )2 44.65%

pentek 49.35%

estane 3.0%

BDNPA-F 3.0%

To make the Mock 900-21, the binder materials (Estane and nitroplasticizer) were Àrst dissolved in methyl ethyl ketone (MEK) to make a lacquer. The solid ingredients (barium nitrate, pentek, and red dye, for distinguishing the mock from real energetics) were mixed together. The lacquer was then added to the solids and the batch (about 500 pounds) was blended until the temperature reaches 65 ◦ C. During this blending, the solvent was pulled off and recovered. The blended part was pressed at 20,000 psi (or 137.9 MPa) and at the temperature of 90 ◦ C for 5 minutes. The average Ànal density of the batch is W = 1.837 g/cm3 . Note that the Ànal density of the heterogeneous material is lower than the theoretical density (1.861 g/cm3 ). One may calculate that the volume fractions of each constituent are: 25.3% of barium nitrate, 64.8% of pentek, 8.6% of the estane/BDNPA-F binder, and Ànally, 1.3% of voids within the material. The sample geometry was the four-point bend specimen with a single shallow edge notch at the tensile side of the specimen. The four-point bend specimen and associated loading are presented in Fig. 1. The nominal dimensions shown in the Àgure are: L = 25.4 mm, H = 25.4 mm, a = 3.0 mm, the sample thickness W = 12.7 mm. The notch angle is about 40 ◦ . P/2

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Figure 1: Four-point bend specimen with a single shallow edge notch. We used a conventional Instron 1125 screw-driven loading frame to load the specimen at a constant crosshead speed. We chose a crosshead velocity to be 0.25 mm/minute, and all tests were conducted in ambient temperature (∼ 22◦ C). We monitored and recorded the applied load and the crosshead displacement during the tests. We also used the digital image correlation (DIC) technique to obtain the full-Àeld displacement and strain on the surface of the sample. For applying the DIC technique, a random speckle pattern was made on the sample surface by Àrst painting a thin layer of white background and then by spraying a black paint onto the surface. A CCD camera, with resolution of 1628 × 1236 pixels, was setup in front of the specimen. A series of images was captured during the test. The random speckle image has a spatial resolution of about 16 R m/pixel.

3 Monotonic loading We Àrst subject the four-point bend specimen to a displacement-controlled monotonic loading, by that we meant the crosshead of the load frame continuously moving down at a constant speed until the specimen completely failed. The variation of the applied force as a function of time is shown in Fig. 2. The applied force Àrst monotonically increases as the crosshead displacement proceeds, it then reaches the maximum value and starts to decrease and Ànally the applied force drops signiÀcantly as the sample breaks into two pieces along the segment ahead of the edge notch. The peak load is believed to be associated with the occurrence of localized deformation and cracking within the specimen. However, from the load versus time curve shown in Fig. 2, one cannot tell at what exact moment that the macroscopic crack starts to initiate from the notch and propagate into the sample. Also careful inspection of the random speckle images acquired during the test reveals very little regarding the initiation moment and the proÀle of the macroscopic crack. This has been a major obstacle in studying and quantifying the fracture process in high explosive materials and their mocks.

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Figure 2: Experimental results of four-point bend specimen subject to monotonic loading. We have developed a technique recently for quantitatively identifying the location and extent of macroscopic cracks in heterogeneous high explosive and mock material [1, 2]. This technique is based on local measurement of the deformation Àeld using digital image correlation (DIC). To determine the motion and distortion of a small region, DIC uses the correlation coefÀcient, designated as c, as a discriminating parameter. The correlation coefÀcient is a function of the two random speckle images before and after deformation, and it also depends on the changes of the small region. When the movement and distortion of the small region is such that the features of the two random speckle images match each other, the correlation coefÀcient reaches a minimum. As a result, the displacement of the small region is determined. The process is repeated for all the data points within the deformation region and the displacement Àeld of the sample surface is thus obtained. When damage or cracks develop in the small region during the deformation, the fundamental assumptions of DIC that allow for the analysis of deformation are no longer strictly true, since the damage or cracks will alter the local characteristics of the speckle image. However, the numerical scheme in DIC calculation is tolerant and the correlation coefÀcient may still be able to attain a minimum. The absolute value of this minimized correlation coefÀcient becomes much larger, however, than the one calculated for regions where no damage or cracks are present. Also, as the cracking in a small region becomes more severe, the increase in magnitude of the correlation coefÀcient becomes more signiÀcant. Therefore, one may use the distribution of the minimized correlation coefÀcient at each location on the sample surface to reveal and quantify the location and extent of cracks in a test specimen. The extent of cracked region, indicated by the dashed lines, in the four-point bend specimen at two selected moments of time is also shown in Fig. 2, the Àrst moment is when the load reaches the maximum and the other is when the applied load is on the decreasing stage. The colored plots in Fig. 2 represent the contours of the normalized correlation coefÀcient c Àeld within the specimen. The dashed line is just one contour with the critical parameter ccrit . During the deformation, there is a critical moment at which the maximum normalized correlation coefÀcient starts to exceed the critical parameter ccrit . The macroscopic crack starts to extend from the notch tip at that critical moment. We take the farthest point on the contour ccrit in the vertical direction as the location of the macroscopic crack tip, then, the history of the crack growth )a, as function of time, can be determined and this is shown in Fig. 2 as open circles. The growth of the macroscopic crack can be divided into three stages. In the Àrst stage, the growth is quite slow, after which the crack growth become steady. In the last stage, the crack growth accelerates. The transition from steady crack growth to accelerating propagation occurs when the macroscopic crack tip approaches and then crosses the neutral axis of the bend beam, as if the bending beam did not have the shallow edge notch. By combining the loading history and the crack growing history, we can identify the threshold load, under which macroscopic crack will initiate from the notch tip, and this threshold load for crack initiation is indicated in Fig. 2 to be approximately 380 N.

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We then subject the four-point bend specimen through a series of cyclic loadings. The loading proÀle is designed as the following. The loading is still displacement-controlled through the crosshead motion at constant velocity. As shown in Fig. 3, during the Àrst cycle, when the applied force approaches but does not exceed the threshold loading for crack initiation in monotonic loading, which is approximately 380 N, the crosshead motion was reversed and the sample was unloaded until the applied force was reduced to zero. The specimen was then reloaded. However, for the subsequent loading cycles, we controlled the crosshead motion in such a way that the maximum movement of the crosshead was approximately equal to that of the Àrst cycle, as shown in Fig. 3(A), where the four-point bend specimen was loaded through ten cycles. 0.25

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Time, t (second)

Figure 3: Experimental measurements of four-point bend specimen with shallow edge notch subject to cyclic loadings: (A) Time history of crosshead motion; (B) Time history of applied force; (C) Time history of the parameter cmax , characterizing evolution of damage. Since the crosshead moves at constant speed during loading and unloading in each cycle, the motion history of the crosshead is the composition of piece-wise linear segments, as shown in Fig. 3(A). However, one can see that the crosshead motion never returns to its original location before the deformation starts, i.e., zero displacement, which means that after each cycle, there is permanent overall deformation when the applied force is completely removed. Figure 3(B) presents the corresponding applied force of the deformation process. According to the loading proÀle, the maximum load during the Àrst cycle approaches but does not exceed the threshold loading for crack initiation in monotonic loading. The dash-dotted lines in Fig. 3 indicate the moment of transition from loading to unloading. Note that the crosshead displacement is symmetric about these dash-dotted lines due to the constant crosshead moving velocity, while the applied force is not symmetric about these lines. The applied force drops much faster after the transition point than that when it approaches the transitions. This corresponds to the hysteresis phenomenon, which attributes to the viscous properties of the material. Because the applied force never exceeds the threshold loading for crack initiation in monotonic loading, we would expect that no macroscopic crack should be generated during the cyclic loadings. But one observes that the applied force at the transition from loading to unloading decreases drastically after each loading cycle and during the tenth cycle this force is only

155 a small fraction of the Àrst one. So what is behind this radical reduction in load carrying capacity of the four-point bending specimen? In Fig. 3, we also present the contour plots of the normalized correlation coefÀcient Àeld within the gage section of the specimen at the transition points of the fourth and the eighth cycles. Surprisingly, we see that a sizable macroscopic crack has formed and propagated into the specimen and the presence of this macroscopic crack reduces the load-carrying capacity of the four-point bend sample, especially, at the moment corresponding to the fourth cycle (moment d), very little change in the optical image can be observed by the human eye. To further examine the formation of the macroscopic crack, we turn again to the correlation coefÀcient. Note that the correlation coefÀcient c is a scalar Àeld and will evolve as a function of time. As a simple measure, at any given moment of time, we select the maximum c in the entire Àeld and designate it as cmax (t) and we monitor the variation of cmax (t) as deformation proceeds. We also normalize cmax (t) such that during the entire deformation process, the normalized correlation coefÀcient varies between 0 and 1. In Fig. 3(C), the variation of the normalized parameter cmax as function of time is shown. Like in the monotonic loading case, initially, the parameter cmax remains a positive but small constant. This indicates that no degradation of correlation was detected, and thus no damage or cracking was occurring in the sample. As the cyclic deformation continues, the parameter cmax starts oscillating but the overall trend is increasing when the specimen experiences more loading cycles. At the tenth cycle, the parameter cmax has approached the maximum value of 1, indicating severe degradation of correlation that associated with damage and cracking. In Fig. 3(C), two particular regions, I and II, are identiÀed. The variations of the normalized parameter cmax in those regions are magniÀed and shown in Fig. 4. In region I, we see a sharp transition of the parameter cmax from a small positive constant to monotonically increasing showing that degradation of image correlation has been detected, such a degradation is the result of damage developing within the sample that alters the characteristics of the speckle image, probably in a very subtle way since human vision fails to identify the changes. From this sharp transition, we can identify two pieces of information, one is the moment, tcrit , at which such transition occurs and the other is the critical level of the normalized correlation coefÀcient, ccrit , which identiÀes quantitatively the threshold above which correlation degradation, thus the damage or cracking, does occur. It is interesting to note that from Fig. 4, we have tcrit = 58.0 second, while during the Àrst cycle, unloading occurs at 55.2 second. This observation conÀrms two facts, one is that during the loading stage of the Àrst cycle, no damage or cracking has happened in the sample, which is expected since the applied load never crosses the threshold loading for macroscopic crack initiation in monotonic test, the second is that the degradation of correlation or emergence of damage happens during the unloading stage of the Àrst cycle. The plot shown in Fig. 4 for region II also indicates that after reaches a maximum, the parameter cmax turns around and decreases, and at some point its value is below the critical value, ccrit , suggesting that any damage incurred has healed itself near the end of the Àrst loading cycle. Region I

Region II

0.08

Cycle No.1

0.07

Parameter, c−max

Parameter, c−max

0.20

c−crit

0.06 0.05 0.04

0.15 Region I 0.10

−

ccrit

0.05

tcrit

Loading

0.

0.03 45.

50.

55.

60.

Time, t (second)

65.

Cycle No.2

0.

Unloading 50.

Loading

100.

Unloading

150.

200.

Time, t (second)

Figure 4: Evolution of parameter cmax in region I and region II indicated in Fig. 3(C). As the applied force starts to increase again in the second cycle, the parameter cmax still remains below the critical value, ccrit , and only after some period of time (approximately 15 second), it starts to increase and crosses the critical value, ccrit , and for all the subsequent cycles and deformations, parameter cmax never falls below the critical value, ccrit , again. Indeed, as shown in Fig. 3(C), near the end of the second cycle, parameter cmax becomes quite small but still it is slightly above the critical value, ccrit . Therefore, only after the second cycle, part of the damage becomes permanent. Of course, as the loading cycles increase, such accumulated damage becomes so severe that the contour plots in Fig. 3 show the formation of a sizable macroscopic crack in the gage section of the sample.

156

5 Discussions and concluding remarks The cyclic loading experiment designed and conducted on the four-point bend specimen made of the heterogeneous high explosive simulant, Mock 900-21, exhibits some very interesting phenomena. Although the contour plot of the so-called normalized correlation coefÀcient and the variation of the parameter cmax provide quantitative description of the process of damage and cracking in the specimen, the mechanism behind such a process remains elusive. Here we only speculate the processes occurring at lower length scale that might be responsible to the phenomena we saw in the cyclic loading experiment on the four-point bend specimen. We claim that there is no macroscopic damage and cracking occurring during the loading stage of the Àrst cycle and this is conÀrmed by the facts that applied force never exceeds the threshold loading for macroscopic crack initiation determined in monotonic loading test and that the parameter cmax remains below the critical value, ccrit , which is the threshold value indicating correlation degradation, or damage and cracking. Only during the unloading stage of the Àrst cycle, parameter cmax exceeds the critical value, ccrit , however, near the end of the Àrst cycle, parameter cmax falls below the critical value. This process may suggest the following scenario. Mock 900-21 is a heterogeneous composite made of hard particles, over 90% in volume fraction, and a soft polymeric binder. As the four-point bend specimen was loaded, the material in the region near the shallow edge notch deforms the most due to stress concentration. At the local level, the deformation within the binder and the crystal is extremely non-uniform because of the high contrast of the mechanical properties of the binder and the crystal. The polymeric binder is stretched and as a result, some of the crystals rotate and deviate from their original conÀgurations. Nevertheless, at the macroscopic level, all these are part of the continuous deformation so that the parameter cmax , according to the DIC calculation, detects nothing that the speckle image is altered. Now the applied force starts to decrease. As the stress level near the notch tip is lower, the material in that region will try to recover to it original conÀguration. However, due to the visco-elastic nature of the polymeric binder, the crystals that have rotated further away from their initial position react slower than the change of stress imposed to that region. So some of them were locked temporarily into an “awkward” position, such that the changing of the speckle image cannot be treated as a continuous deformation. This may explain why the parameter cmax exceeds the critical value, ccrit , after the applied force starts decreasing. But given enough time, or because that the rotation of some of the crystals is still not large enough, eventually, those crystals can return to either their original conÀguration or positions very close to the original conÀguration, so that the changing speckle pattern is no longer regarded as being degraded and the parameter cmax falls below the critical value. During the second cycle, the same process will repeat but additional mechanisms may get involved since the parameter cmax becomes larger than the critical value when the applied force is still increasing. The additional mechanisms may include interfacial debonding between the crystal and the binder [3, 4], or the opening of pre-existing micro-cracks. These additional damage mechanisms must be aided by that we have seen during the Àrst cycle. Otherwise, the second cycle would just repeat exactly what is happening during the Àrst cycle. As the applied force starts to decrease, the parameter cmax still increases before it reaches maximum and decreases, further illustrating the temporary – eventually this might become permanent – locking mechanism we just mentioned above. The disturbance and recovery, completely or partially, of the local microstructural conÀguration seem to be a persistent process during the cyclic loadings, for one can see from Fig. 3(C) that across each transition moment, the applied force decreases while the parameter cmax continues to increase before it also decreases, so that the parameter cmax lags behind both the crosshead motion and the applied force. Note that parameter cmax only represents the state of the material element that suffers the worst damage, or the upper bound of the current state of damage of the entire gage section, at any given moment. A more appropriate measure is the evolution of the area, where the normalized correlation coefÀcient c has exceeded the critical value, ccrit . We are continuing our analysis along this line of thinking. The observations presented in this investigation can lead to some signiÀcant consequences in weapons or munitions systems. Suppose that we can determine the threshold of loading at which the energetic material will develop macroscopic cracks, and accordingly, we design the system so that the loading on the energetics will never exceed that threshold loading. However, according to our Àndings in this study, because of the heterogeneous microstructure of the energetic material and with the help of non-uniform stress state, if the applied load cycles enough times, macroscopic cracks can still form and propagate within the energetic material. Such a scenario may affect the safety and performance of the weapons or munitions system and deserve further investigation.

Acknowledgments Los Alamos National Laboratory is operated by the Los Alamos National Security (LANS), LLC for the National Nuclear Security Administration (NNSA) of the U.S. Department of Energy (DOE) under contract DE-AC52-06NA25396. This study

157 was supported by the Joint DoD/DOE Munitions Program (JMP) and the High Explosive Science and Engineering program.

References [1] C. Liu, C.M. Cady, M.L. Lovato, and P.J. Rae, “A technique for revealing and quantifying cracks in heterogeneous solids,” Los Alamos National Laboratory Report LA-UR-09-02906, 2009. [2] C. Liu, C.M. Cady, P.J. Rae, and M.L. Lovato, “On the quantitative measurement of fracture toughness in high explosive and mock materials,” Proceedings of the 14th International Detonation Symposium, Coeur d’Alene, Idaho, USA, April 11–16, 2010. [3] C. Liu, “Fracture of the PBX 9501 high explosive,” In Shock Compression of Condensed Matter – 2003, edited by M. D. Furnish, Y. M. Gupta, and J. W. Forbes, American Institute of Physics, pp.786–791, 2004. [4] H. Tan, C. Liu, Y. Huang, and P.H. Geubelle, “The cohesive law for the particle/matrix interfaces in high explosives,” Journal of the Mechanics and Physics of Solids, 53, pp.1892–1917, 2005.

An Evaluation Of A Modified Iosipescu Specimen For Measurement Of Elastic-PlasticCreep Properties Of Solder Materials Subhasis Mukherjee, Abhijit Dasgupta CALCE Electronic Products and Systems Center Mechanical Engineering Department University of Maryland, College Park, MD 20742 USA Phone: 301- 405-5231 Email: [email protected]

ABSTRACT There are various specimen configurations available in the literature for characterizing the mechanical behavior of solder interconnect materials. An ideal test specimen should use a simple geometry which produces a reasonably uniform material response throughout the gage zone and minimizes the complexity of data reduction to extract material model constants. In the thermomechanical micro scale (TMM) test used in this study, we use a simple, notched shear specimen, based on a concept originally proposed by Iosipescu [1967] [1], which produces a reasonably uniform shear stress field throughout the solder joint [Reinikainen et al.,1998] [2]. Our modified Iosipescu specimen comprises of two oxygen free, high conductivity (OFHC) copper platens soldered together and loaded in simple shear, as in a lap-shear specimen. The solder joint in this specimen is only 180 microns wide to capture the length scale effects of functional solder interconnects. This study examines the effects of dimensional variabilities of this modified Iosipescu test setup on the shear stress and strain distributions in the solder specimen. Variabilities encountered in these specimens include: (i) fillets at the V-notches, caused by excess solder; (ii) offset between the two copper platens along the loading direction; (iii) taper of the solder joint due to lack of parallelism of the edges of the copper platens; and (iv) misalignment between the specimen centerline and loading axis of the TMM test frame due to mounting variability. Detailed parametric studies of these four dimensional variations in the TMM specimen are conducted using a simple two-dimensional finite element model for measurement of elastic-plastic-creep properties of solder materials.

were conducted on monolithic Iosipescu specimens, and only a few [2] have addressed multi-component structures such as the solder specimen we have used in our study.

misalignment Test frame centerline

Specimen centerline

Fig 1: Misalignment between loading axis and specimen centerline

Fillet

Fig 2: Fillet due to excessive solder volume

Taper

INTRODUCTION & MOTIVATION Typical field conditions seen in electronic packages include thermal cycling, mechanical cycling, vibration loading and drop, to name a few. Solder joint integrity is critical to functionality since they not only form the electrical interconnection but are also the load bearing mechanical interconnection. Hence accurate experimental methods to characterize the material response of the solder are very important. In this study we examine the accuracy of a selected mechanical test method, by examining the sensitivity of the extracted elastic-plastic stress-strain curves and constitutive creep properties to the dimensional, processing and loading variabilities of the test setup. Past studies confirm that geometric variabilities in the test specimen such as notch angle, root depth, misalignment between the specimen centerline and the loading axis of the test setup can significantly alter the constitutive and durability properties of the solder interconnects. Most of the studies reported in the literature

Fig 3: Tapered solder joint due to lack of parallelism between edges of mating copper platens

Offset

Fig 4: Offset between horizontal centerlines of mating copper platens

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_24, © The Society for Experimental Mechanics, Inc. 2011

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160 Our solder test specimen consists of two different copper platens soldered together, which creates opportunities for additional dimensional, processing and loading variabilities, that are examined in this paper: (i) misalignment between the specimen centerline and loading axis of the TMM test frame due to mounting variability; (ii) fillets at the V-notches, caused by excess solder; (iii) taper of the solder joint due to lack of parallelism of the edges of the copper platens; and (iv) offset between the two copper platens along the loading direction. These variabilities are schematically illustrated in Figures (1-4). Fabricated specimens considered in the present study have a solder joint thickness of 180 Pm, which is of the same length scale seen in functional solder interconnects. This is important to replicate microstructural length-scale effects that are known to exist in functional solder joints, especially for Pb-free, Sn-Ag-Cu (SAC) solders. The specimens are deformed in mechanical shear. A schematic of the test specimen considered in the 2D finite element elastic-plastic analysis is shown in Fig 5. The TMM specimen considered here consists of a thin layer of solder connecting two copper platens. The solder joint is approximately 180 microns wide, 3.0 mm long and approximately 1 mm thick. The specimen is originally 1.5 mm thick and reduced to approximately 1mm thickness using standard grinding and polishing processes. Due to this planestress configuration, 3D and 2D models are found to provide similar parametric sensitivities. To minimize computing time in nonlinear parametric sensitivity studies, the model considered for study here is a plane stress 2D model.

90º

Cu 5mm

3mm

solder Cu

(12 + joint thickness) mm Fig 5: Schematic of TMM specimen The main objective of this paper is to study the effects of selected dimensional variabilities of this modified Iosipescu specimen on the shear stress and strain distributions and constitutive creep properties in the solder joint, using elastic-plastic finite element analysis (FEA) and comparing them with the distributions in a nominal specimen with no geometric imperfections in it. The range of different variabilities considered for the modeling has been decided on the basis of the measured variations in different batches of our fabricated test specimens.

APPROACH The 2-D FEA [3] model considered here consists of 8noded 183, quadrilateral, plane-stress elements in a mapped mesh configuration. The elastic-plastic behavior is modeled with a Ramberg-Osgood model, as presented in Equation (1) ε = σ/E + K(σ/E)ⁿ

(1)

where, ε is von-Mises strain, σ is von-Mises stress, E is elastic Young’s modulus, K & n are the elastic-plastic Ramberg-Osgood constants for the solder considered. An image of the finite element model showing mesh, boundary conditions & loads is shown in Fig 6. The copper platen in the left is constrained in both x and y directions but the platen in the right is constrained only in x-direction and given a displacement of 90 microns in y-direction. The mesh sensitivity of the model accuracy was studied by gradually increasing the density of mesh from the center of the specimen towards the root of the notch where stress concentrations are expected. The current mesh was found to be acceptable from the

perspective of reducing the effect of the mesh, element distortion and element aspect ratio, on the accuracy of the output.

Fig 6: Meshed elastic plastic creep model Parametric FEA studies are conducted by systematically varying the geometric parameters listed in Section 1 and shown in Figures 1-4. The effect of these variations on the accuracy of the TMM test is examined by comparing the elastic-plastic material properties (E, K, n) and secondary creep strain rate (dγ/dt) extracted from these errorseeded FEA models, to those from a reference model which has the nominal geometry and loading. Secondary creep strain rate is calculated using sine-hyperbolic Garofalo secondary strain rate equation as presented in Equation (2). Rate dependent plasticity study was done for a single stress level (10 MPa) for two different temperatures 298 & 398 K. The extracted material constants are based on averages of the deformation and stress fields over the entire solder volume, since typical experimental measurement capabilities are based on similar averages. Localized averages of these constants are not investigated since they cannot be experimentally measured and are of limited value due to local microstructural inhomogeneties. dε/dt = A[sinh(ασ)]n* exp[-Q/RT]

(2)

where, dε/dt = von Mises strain rate, σ = von Mises stress, α = Stress constant, n = stress exponent, Q = Activation energy, R=Universal gas constant, T = Temperature in Kelvin The range of variabilities considered here for the relevant geometric parameters in this study is decided based on experience gained during fabrication & testing of TMM specimens. Misalignment between the centerlines of the specimen & test frame (shown in Figure 1) is varied up to 560 microns. Offset between the horizontal centerlines of the copper platens (shown in Figure 2) is varied up to 125 microns, based on the asymmetry check done on the copper platens. The asymmetry is inspected using an optical microscope and ESEM (Environmental Scanning Electron Microscope) imaging techniques. The taper of the joints (shown in Figure 3) is varied upto 50 microns over the length of the joint. Taper variation is inspected in the ESEM after the solder specimen is aged, ground and polished. Fillets produced at the notch of the solder joint due to overflow of excess solder (shown in Figure 4) are varied by 240 microns. These fillets can create localized stress inhomogeneties in the solder joint during the test, and cause anomalous recrystallization. .

RESULTS SUMMARY & DISCUSSIONS The variation of the extracted values for elastic shear modulus (G), with respect to previously mentioned variabilities, is evaluated. Corresponding variations in the extracted values for plastic Ramberg-Osgood model constants are also evaluated as a function of the mentioned variabilities. Results show that the parameter that has the strongest effect on the accuracy of the extracted shear stiffness is the lateral misalignment between the specimen and test frame centerlines. The next most influential parameter is solder joint taper. The top two parameters (in descending order of severity) to affect the accuracy of the plastic constants (hardening exponent n, and strength coefficient K) are solder joint taper and fillet size at the notch root. The accuracy of the extracted plastic Ramberg-Osgood constants is

161 therefore found to be more sensitive to the solder geometry variations than the geometric misalignment generated due to faulty positioning of the specimen with respect to the test frame. Rate dependent von Mises strain rate is found to be sensitive to the fillet of the solder joint In general, the elastic shear stiffness (G) is found to be more sensitive than the plastic Ramberg-Osgood constants (K, n), to the geometric error seeding study here. In rate dependent model study, von Mises strain rate is found to be more sensitive than elastic & rate independent plastic parameters. This is due to the fact that plastic deformations reduce the local stress gradients caused by the various geometric errors. The results obtained from this study can be used to specify tolerance limits for the geometric attributes studied here. Finally, the results of this study can be used to calculate correction factors to improve the accuracy of the extracted material constants, based on the extent of the observed geometric variability in each specimen. These corrections will reduce the scatter usually observed in the measured results.

ACKNOWLEDGMENTS This work is sponsored by the members of the CALCE Electronic Products and Systems Consortium at the University of Maryland, College Park.

REFERENCES [1] Iosipescu, N., September 1967, “New Accurate Procedure for Single Shear Testing of Metals,” J. Materials, Vol. 2, No. 3, pp. 537566. [2] Reinikainen, T., Poech M., Krumm M., Kivilahti J., March 1998, “A finite-element and experimental analysis of stress distribution of in various shear tests of solder joints,” J. Electron Packaging, Vol. 120, Issue 1, 106 (8 pages). [3] Ansys” Release 11.0 UP20070125 ;ANSYS Academic Research, ANSYS Inc., a UL registered ISO 9001:2000 Company. [4] Zhang, Q., Dasgupta, A., and Haswell, P., 2004, “Partitioned Viscoplastic Constitutive Properties of Pb-Free Sn3.9Ag0.6Cu Solder,” Journal of Electronic Materials, Vol.33, No.11, pp 13381349 [5] Haswell, P., 2001,“Durability Assessment and Microstructural Observations of Selected Solder Alloys”, Ph.D. Dissertation, University of Maryland, College Park, MD, USA., Chapter 3., Experimental and analytical approaches., page 2-11. [6] Zhang Q., 2003, “Viscoplastic constitutive properties and reliability of lead-free Sn3.9Ag0.6Cu solder”, ASME proceedings., IMECE2003-41840, page 4.

SEM 2011 Annual Conference & Exposition on Experimental and Applied Mechanics, June 13– 16 2011, Uncasville, Connecticut, U.S.A.

Temperature Effect on Poisson’s Ratio of Woven Composites Yougashwar Budhoo* Department of Engineering and Technology, Vaughn College of Aeronautics and Technology, 86-01, 23 Ave, East Elmhurst, Queens, NY 11369, USA ABSTRACT Monotonic tensile tests were conducted following ASTM Standards D3039 (Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials) and D3518 (Standard Test Method for In-Plane Shear Response of Polymer Matrix Composite Materials by Tensile Test of a r45° Laminate), on non-hybrid plain weave composite materials. Strips (6.35mmu25mmu250mm) of non-hybrid IM-7 Graphite/SC-79 epoxy called GR for short, non-hybrid S-2 Glass/SC-79 epoxy called GL for short specimens were tensile tested. The tests were conducted at 60qC, 20qC, 75qC and 125qC. The Poisson’s ratios were measured using strain gages. It was observed that temperature had a small effect on the Poisson’s ratio.

INTRODUCTION Due to the increasing use of polymer composites, a greater understanding these materials are necessary. Numerous researches have been conducted on composite in general to study their material properties [1-4]. Studies have also been done to study their responses under various types and rate of loading [5-8] and environmental conditions [9-10]. In this research woven composited will be studied. Experiments will be carried to study the effect of temperature on the Poisson’s ratio of these composites. Woven composites are known for their excellent dimensional stability and impact properties.

EXPERIMENTAL PROCEDURE Materials: The individual constituent materials combined to form the composite material used in this research are, IM-7 graphite (IM7GP 6000) and S2-glass (S2-4533 6000) woven fabrics placed in SC-79 toughened epoxy resin matrix. The IM-7 graphite woven fabric and SC-79 epoxy matrix form the composite called GR. The S2-Glass woven fabric and SC-79 epoxy matrix form the laminate called GL. S2-glass fabrics and IM7-graphite fabrics were supplied by the Hexcel Corporation. The matrix, SC-79 toughened epoxy resin, which has Part A (Batch number: SC79A012307) and Part B (Batch number: SC79B012507), was supplied by Applied Poleramic Inc. The manufacturing of the composite was done by EDO Fiber Innovations. The vacuum assisted resin transfer molding (VARTM) technique was used to stack the plain woven fabrics together. The specimens were cured at 1770 C. Fiber volume fraction for all types were 55%.

Experimental setup: Strips of GL and GR specimens, both of dimensions 6.35mmu25.4mmu254mm were tested under uniaxial tension at 125 °C, 75 °C, 20 °C and 60 °C following ASTM Standards D3039 (Standard Test Method for Tensile Properties of Polymer Matrix Composite Materials). The GL laminate consisted of 39 laminas stacked together, while those of GR consisted of 28. During the experiments, the desired temperatures were achieved using an environmental chamber, where each specimen was allowed to soak at the required temperature for thirty minutes before testing. *Corresponding author. Tel.: +1 646 496 6102; Fax: +1 212 650 8013.

E-mail address: [email protected] T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_25, © The Society for Experimental Mechanics, Inc. 2011

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The chamber utilizes a heating coil for achieving high temperatures and liquid nitrogen for low temperatures. During this soaking period, one end of the specimen was clamped while the other end was free; this was done in order to allow the specimen to elongate freely. The tests were conducted using a universal testing machine also known as MTS while specimens were still in the environmental chamber. Figure 1 shows a schematic of the tensile specimen.

Figure 1 Tensile test specimen dimensions In figure 2, the tensile testing machine, environmental chamber and liquid nitrogen used to achieve low temperatures are shown. All tests were displacement controlled. Specimens were clamped two inches each, thus the resulting gage length was six inches.

Figure 2 MTS machine with environmental chamber and liquid nitrogen tank In order to obtain the Poisson’s ratio of the composite, the strain along the longitudinal, transverse and thickness direction was needed. These strains were obtained by the use of strain gages (CEA-13-062UW-350). The output data obtained from the stain gages were then used to calculate the Poisson’s ration of each specimen.

EXPERIMENTAL RESULTS Before plotting the data obtained from the experiment and finding the Poisson’s ratio, it was necessary to correct the lateral and thickness direction strain for transverse sensitivity. During tensile testing, the longitudinal strain in the specimen is several times larger than the lateral and thickness direction strains. The strain gages placed in the thickness and transverse directions will therefore be strained in a direction which is not their primary sensing direction. As a result of the finite width of the grid lines in the gage, the presence of end loops connecting the grid lines, strain gages are generally sensitive not only to strain parallel to the grid direction, but also to strain perpendicular to the grid direction. This property of strain gages is referred to as “transverse sensitivity”. Transverse and thickness direction strains were corrected for transverse sensitivity following guidelines from the manufacturer. Figure 3 below shows the placement of strain gages on the specimen.

165

Graphite (b)

Composite Specimens

Glass (a)

Figure 3 Location of strain gages to measure the Poisson’s ratio of woven glass and woven graphite toughened epoxy composites

H 1 vs.H 2 , H 1 vs.H 3 , H X vs.H Y and H X vs.H Z , respectively, for GL specimen at various temperatures from whichQ 12 ,Q 13 Q 23 , Q XY and Q XZ Q YX of GL are determined. Figures 8 to 11 shows the H 1 vs.H 2 , H 1 vs.H 3 , H X vs.H Y and H X vs.H Z , curves respectively, for GR specimens at various temperatures, from whichQ 12 ,Q 13 Q 23 , Q XY and Q XZ Q YX of GR are determined. Figures 4 to 7 show the

Figure 4 Poisson ratio Q 12 of woven GL

Figure 5 Poisson ratio Q 13 of woven GL

166

On each of the curves, an arrow bar is places to show the variation of the Poisson’s ratio from specimen to specimen. For each case, a minimum of three tests were done. The subscripts 1, 2, and 3 represent the longitudinal, transverse and thickness directions of the strips respectively. In this case, the fibers in the composite strips are parallel and perpendicular to the direction of loading. The subscripts x, y, and z represent the longitudinal, transverse and thickness direction of the strips respectively, however, in this case the fibers are at r45° angle with respect to the direction of loading. In figure 4 it can be seen that Q 12 increase with a decrease in test temperature whereQ 12



H2 . On the other hand, Q 13 decrease with a decrease H1

in test temperature.

Figure 6 Poisson ratio Q XY of woven GL

Figure 7 Poisson ratio Q XZ of woven GL

167

Figure 8 Poisson ratio Q 12 of woven GR

Figure 9 Poisson ratio Q 13 of woven GR

Figure 10 Poisson ratio Q XY of woven GR

168

Figure 11 Poisson ratio Q XZ of woven GR Table 1 below shows the Poisson’s ratios obtained from the above graphs for the GL and GR specimens respectively. The Poisson’s ratio for room temperature was taken from [] Table 1 Poisson’s ratio of woven glass and graphite fibers-reinforced toughened epoxy

Composite

Test Temperature

Non- Hybrid Glass

60 °C (76 °F) 20 °C (4 °F) R T (R T) 75 °C (167 °F) 125 °C (257 °F)

Non- Hybrid Graphite

60 °C (76 °F) 20 °C (4 °F) R T (R T) 75 °C (167 °F) 125 °C (257 °F)

Q12

Q13=Q23

Qxy

Qxz=Qyz

0.152 0.145 0.13 0.128 0.121

0.370 0.361 0.41 0.436 0.453

0.487 0.515 0.578 0.656 0.702

0.246 0.214 0.184 0.153 0.147

0.140 0.129 0.121 0.116 0.099

0.382 0.462 0.427 0.627 0.649

0.726 0.754 0.794 0.846 0.921

0.170 0.162 0.102 0.085 0.081

As seen in table 4.5, the Poisson’s ratio appears to be a function of temperature. As the temperature increases the Poisson’s ratios Q12 and Qxz= Qyz decrease for both GR and GL specimens while the Poisson’s ratios Q13=Q23 andQxz = Qyz increase. The Poisson’s ratioQ13 tends to increase with an increase in temperature.

CONCLUSION

a) As the test temperature increases, the Poisson’s ratio Q12, decreases where as Q13=Q23 increases. b) As the test temperature increases, the Poisson’s ratio Qxy, increases where as Q13=Q23 decreases.

169

REFERENCES [1] Okoli, O., Smith, G.F., 2000, “Effect of strain rate and fiber content on the Poisson’s ratio of glass/epoxy composites”, Journal of Composite Structures, Vol. 48, pp. 157-161 [2] Sevkat, E., Liaw, B., Delale, F., 2010 “tensile behavior of woven composites”, Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition IMECE2010 [3]Scida D., Aboura Z., Benzeggagh M. L. and Bocherens E., 2003,“Prediction of the elastic behavior of hybrid and nonhybrid woven composites”, Composites Science and Technology, Volume 57, Issue 12, 16 January 1998, Pages 1727-1740 [4] Huang Z. M., “The mechanical properties of composites reinforced with woven and braided fabrics”, Composites Science and Technology, Volume 60, Issue 4, 1 March 2000, Pages 479-498. [5] Budhoo, Yougashwar, Delale, Feridun and Liaw, Benjamin. “Effect of temperature on tensile testing of woven composites”. To be published (SEM 2011 Annual Conference & Exposition on Experimental and Applied Mechanics) [6] Dlouhy I., Chlup Z., Boccaccini D. N., Atiq S. and Boccaccini A. R., 2003, “Fracture behavior of hybrid glass matrix composites: thermal ageing effects”, Composites Part A: Applied Science and Manufacturing, Vol 34, Issue 12, pp 11771185. [7] Sevkat E., Liaw B., Delale F., Raju B.B., 2009, “Drop-weight impact of plain-woven hybrid glass–graphite/toughened epoxy composites”, Composites Part A: Applied Science and Manufacturing, Vol. 40, Issue 8, pp. 1090-1110 [8] Naik N. K., Sekher C.Y. and Meduri S., 2000, “Damage in woven-fabric composites subjected to low-velocity impact”, Composites Science and Technology, Vol 60, Issue 5, pp 731-744 [9] Budhoo, Y., Liaw, B., Delale, F., 2010 “Effect of temperature on the impact damage of composite materials”, Proceedings of the ASME 2010 International Mechanical Engineering Congress & Exposition IMECE2010 [10] Im K.H, Cha C.S, Kim K.S, Yang I.Y., 2001, “Effect of temperature on impacts damages in CFRP composite laminates”. Composites Part B, Vol. 32, pp. 669–82.

Detection and Damage Monitoring in Composite Structures Using Piezoelectrics

H. P. Konka1, M. A. Wahab1*, and K. Lian2 1

Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803, USA.

2

Center for Advanced Microstructures & Devices, 6980 Jefferson Hway, Baton Rouge, LA 70806, USA.

*1 Corresponding Author’s E-Mail: [email protected], Telephone: (1) 225-578-5823.

ABSTRACT Piezoelectric-fiber-composite Transducers (PFCT) are an ideal choice for composite structures application, as they are highly flexible, easily embeddable, and have high compatibility with composite structures; and they provide manufacturing flexibility. The major objective of this research is to use PFCTs as an embedded sensor within the composite structures. As embedded sensors PFCTs are able to perform important functions, such as: (i) monitor the stress/strain levels inside the structures continuously and (ii) detect the damages inside the composite structures. Initially, tensile tests are carried out to investigate the effect of the embedded PFCT sensor on the tensile strength of an impregnated glass fiber-epoxy prepreg composite laminate. It is found that by embedding a PFCT sensor inside the composite structure reduces the ultimate strength and modulus of the overall structure about 2.5% and 7.5% respectively. Then tests are performed to investigate the ability of the embedded PFCT sensor to detect the changes in the applied stress/strain. It was found that these sensors are effectively able to detect the changes in the applied stress/strain. A linear relationship has been observed between the applied input mechanical stress and the sensor generated voltage output. Finally, experiments are performed using the embedded PFCT sensors to detect the damages inside the composite structure using the modal analysis and impact methods. From the results of these experiments, it is concluded that embedded PFCT sensors are able to detect the damages using the above methods effectively. Keywords: Embedded sensor, transducers, composites, piezoelectric fiber, glass fiber-epoxy prepreg, modal analysis, impact, voltage, damage, and delamination.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_26, © The Society for Experimental Mechanics, Inc. 2011

171

172 1.0 INTRODUCTION Composite structures have been developed and used for modern aviation, military, and civil applications for over 50 years because of their inherent advantages over metals, such as: (i) higher stiffness to weight or strength-to-weight ratio, (ii) higher resistance to fatigue damage and resistance to harsh environments, (iii) repairable, (iv) provides design flexibility, and (v) lighter in weights. Typical aircraft composites are not ductile; they are brittle, which means they undergo relatively minor permanent deformation prior to final failure. Composite structures are also subjected frequently to external excitations over a variety of vibration frequency ranges. Such dynamic interference may cause the structures to suffer from fatigue damage and/or catastrophic failures if the excitation frequency approaches to the natural frequency of the structures, causing resonances in the structure [1-2]. Hence composite structures require a Health Monitoring System, which, not only monitor the pre-existing defects but also predict the occurrence of damages inside the structure. Nondestructive evaluation (NDE) techniques have been developed to detect the internal or invisible damages inside the structure. Traditional NDE techniques are ultrasonic scan, an eddy current method, X-ray radiography, an acoustic emission method, and passive thermography. These NDE techniques are effective in detecting damages in materials and structures, but it is difficult to use them in operation due to the size and weight of the devices. Therefore, there is strong interest in development of smart composite structures with integrated sensors which would allow in-situ monitoring of both the manufacturing process and service life. Compared to traditional NDE techniques embedded sensors offer unique capabilities: monitoring the manufacturing process of composite parts, performing nondestructive testing once fabrication is complete, and enabling real time health monitoring and structural control [2-3]. A multi-functional sensor that can measure multiple parameters would offer significant economic advantages and end-user appeal. Furthermore, the ability to monitor multiple parameters simultaneously would be of significant benefit to material and structural engineers. Sensors embedded within the structural materials add intelligence to structures and enable real-time monitoring at some critical non-accessible locations. These sensors can be used to gain data for validating or improving designs during the prototype stage or to obtain information on the performance and structural integrity of functional components in service. The capability to obtain such information is important to many industries. Examples include the aerospace industry (components of jet engines), the power industry (vessels and pipes), the automotive industry (components of motors), and the construction industry (structural components in buildings), the oil industry (drilling equipment), manufacturing industry (molds, dies, drilling bits, etc.). Optical fibers, strain gauges, piezoelectric (PZT) sensors, thermocouples and thin films are some of the common sensors that were used as embedded sensors for the composite structures applications [4].

2.0 PIEZOELECTRIC COMPOSITE TRANSDUCERS There are numerous types of sensors that can be used for structural health monitoring applications. The three most common smart sensors are Fiber-Optic Interferometers, Piezoelectric Ceramics, and Strain Gauges. Piezoelectric materials are solids, which generate a charge in response to a mechanical deformation or excitation. In addition, when an electric field is subjected to the material, it mechanically deforms. This allows the piezoelectric materials to be employed in smart structures as either actuators or sensor. Resistive strain gauges operate on the principle that there is a change to the electric resistance of the gauge when subjected to mechanical deformation. A strain gauge is made by bending a conduction wire back and forth over a very small surface that is then bonded to the structure being measured. It exhibits a change in electrical resistance when subjected to strain and this change in resistance is typically measured using a Wheatstone bridge arrangement. They are small in size and low in mass. The fiber optic interferometer involves embedding fiber optics into a structure. A device is used to split a coherent beam of light and transmit the light through the fibers, which are recoupled at the other end. If there is deformation of the structure, the fibers are deflected, which causes the length of the optical path to change and produces an interference pattern at the recoupler. This interference pattern provides a measure of the structure’s mechanical strain. Strain gauges are conventional strain/force measuring devices and are mostly used to measure the static-forces. They come under the category of active sensors, which requires an external excitation source. The disadvantages of the strain gauges include the cumbersome and lengthy installation process, low sensitivity at low strain domains, and require signal conditioning/amplification and noise cancellation circuitry. Piezoelectric sensors are passive (or self-generating) sensors and are able to generate their own electrical output signals without requiring external excitation source [4-5]. Piezoelectric sensors can be used to measure the dynamic-forces (such as oscillation, impact, or high speed compression or tension) acting on a structure. Some of the advantages of piezoelectric sensors are their superior signal to noise ratio, compactness, they require no signal conditioning circuitry and have very high sensitivity even at low strain domains, selfexcitation (no cumbersome electrical excitation devices are required), low acoustic impedance, have a broad dynamic

173 response. But the major drawbacks of these ceramics are their high brittleness, low flexibility, and low tensile strength of piezoelectric ceramics have blocked their extensive application in engineering field. Due to their high brittleness, the piezoelectric ceramics cannot withstand bending and also exhibit poor conformability to curved surfaces [4-5]. For the last 20 years, piezo-composite materials have been developed to overcome this problem by combining piezo-ceramics with passive non-piezoelectric materials. A typical piezo-composite transducers is made of an active layer (PZT) sandwiched between two soft thin encapsulating composite layers. A typical piezo-composite transducers is made of an active layer (PZT) laminated between the two soft thin encapsulating composite layers (sheets of polymer printed circuitry). Superior properties have been achieved by these composites by taking advantage of the most profitable properties of each of constituents and great varieties of structures have been made. This provides the much robustness, reliability and ease of use. Piezo-composite transducers are highly flexible and can be easily used as an embedded sensor [5-8]. The piezoelectric fiber composite transducer (PFCT) comes under the category of piezo-composite transducers, which are manufactured by embedding piezoelectric fibers into a composite matrix along one specific direction only. However, the preparation process for such a structure is delicate and time - consuming because of the necessity of handling large numbers of fragile ceramic fibers. The PFCTs will be an ideal choice for many of composite structures applications, as they are highly flexible, easily embeddable; provide manufacturing flexibility, and more importantly, it is expected that they will produce less interfacial stresses when embedded inside the composite structures. The following Figure -1 shows the schematic details of the two PFCT sensors: (a) MFC (from Smart Materials Corp.) and (b) PFC (from Advance Cerametrics Inc.) used in this research. PFC (Figure 1(a)) contains the circular cross-sectional PZT fibers, whereas MFC (Figure 1(b)) contains the square cross-sectional PZT fibers. (a)

(b)

Polyimide film with interdigitated electrodes

Polyimide film with interdigitated

Epoxy Matrix PZT fibers

PZT fibers

Epoxy Matrix

Figure 1: Schematics of piezoelectric fiber composite transducer (PFCT) - (a) MFC and (b) PFC. The major objective of this research is to investigate the applicability of PFCTs as an embedded sensor inside the composite structures. As embedded sensors PFCTs should be able to perform important functions, such as: (i) monitor the stress/strain levels inside the structures constantly and (ii) detect the damages inside the composite structures. Hence a set of carefully planned experiments were carried out to test the ability of embedded PFCTs sensors to perform the above mentioned functions. 3.0 EXPERIMENTS Initially, tensile tests are carried out to investigate the effect of the embedded PFCT sensor on the tensile strength of an impregnated glass fiber-epoxy prepreg composite laminate. Then tests were performed to investigate the ability of the embedded PFCT sensor to detect the changes in the applied input mechanical stress/strain. Finally, experiments were performed using the embedded PFCT sensors to detect the damages inside the composite structure through the modal analysis and impact methods. The details of experiments and their results are illustrated briefly in the next few sections.

174 3.1 Effect of Embedded Piezoelectric-Fiber-Composite Transducers (PFCT) Sensor on the Tensile Strength of the Composite In this experiment the effect of the embedded PFCT sensor on the tensile strength of an impregnated glass fiber-epoxy prepreg composite laminate is investigated. The DA409U/S2-unidirectional glass fiber-epoxy prepreg sheets (from www.prepregs.com) were used for the composite laminate sample preparation. The DA409U/S2-glass is a tough, versatile, modified epoxy resin prepreg that cures at 250ºF. The processed prepreg sheet was cut into strips of dimensions (260 mm × 27 mm × 0.5 mm). Three such strips were glued together to form one sample. The sensor was embedded inside the center strip as shown in the following Figure 2. A groove is made on the center strip using the filer manually; the dimensions of grove are the same as the dimension of the sensor. The sensor is embedded inside the groove. Tabs were also attached in order to prevent the gripping damage while performing the test. Three types of samples were made with different types of embedded sensors: (i) with no-embedded sensor (PURE), (ii) with MFC as an embedded sensor (MFC), and (iii) with PFC as an embedded sensor (PFC). All the embedded sensors were of the same dimensions (30.1 × 9.7 mm × 0.07mm). A total of 12 specimens were made; Table 1 shows the details. All the specimens were tested in an MTS servo-hydraulic testing machine. In this test the in-plane tensile properties of the specimens were determined. The ASTM standard D 3039/ D 3039 M was used for fabricating the specimen and to perform tensile tests. The specimen is mounted in the grips of a mechanical testing machine and monotonically loaded in tension while recording load and strain data (see Figure 3). The ultimate strength of the material can be determined from the maximum load carried before failure. From this test the stress-strain response of the material, the ultimate tensile strain, tensile modulus of elasticity, Poisson’s ratio, and transition strain can be derived. The specimens were monotonically loaded in tension until failure. The average values of the ultimate strength and modulus of the specimens with and without PFCT sensors were within 4% of each other in both the cases. These observed variations are comparable to the previous studies [910]. A comparison of the stress-strain curves for the specimens with and without the embedded sensors is presented in Figure 4. The stress-strain curves for the specimens with PFC and MFC sensors mimic each other closely, and bending results almost along the same path. The most noticeable difference between the specimens with embedded sensors and specimen without the sensor is the ultimate strength values. The PURE sample (with no embedded sensor) has higher ultimate strength and modulus values when compared with specimens (MFC and PFC) with embedded sensors. The ultimate strength value for PURE sample (no embedded sensor) is about 321.7 MPa, whereas for the specimens MFC and PFC with embedded sensors are found to be about 314.75 and 313.3 MPa respectively. Overall a reduction of about 2.16 and 2.61 % in ultimate strength value is observed in the samples with embedded MFC and PFC sensors respectively. A reduction of about 3.4 and 5.57 % in modulus values is observed for the samples with embedded MFC and PFC sensors. The bar graph plots in the figure 5 compares the ultimate strength and modulus values of the specimen with (MFC and PFC) and without (PURE) embedded sensors. From this figure we can conclude that embedding a sensor inside the composite structure reduces the ultimate strength and modulus of the structure. Table 1: Test Specimen Details. No

Embedded Sensor Type

Specimen ID Number

1

No Embedded Sensor (PURE)

P1

P2

P3

P4

4

2

PFCT Sensor (MFC)

M1

M2

M3

M4

4

3

PFCT Sensor (PFC)

PF1

PF2

PF3

PF4

4

Total Specimens

Total Number of Specimen

12

175

Figure 2: Details of the test specimen.

MTS Crosshead

Sample

MTS Crosshead

Figure 3: Sample mounted on MTS machine.

176 350

PURE (MPa)

300

MFC (MPa)

Stress (MPa)

250

PFC (MPa)

200 150 100 50 0 0

0.5

1

1.5

2

2.5

3

3.5

Strain (%)

Ultimate Strength (GPa)

350 300 250 200 150

321.7

314.75

313.3

100 50

Young's Modulus (GPa)

Figure 4: Stress vs. Strain response for the specimens. 40 30 20

32.3

31.2

30.5

MFC

PFC

10 0

0 No sensor

MFC

PFC

Specimens

No sensor

Specimen

Figure 5: Ultimate tensile strength and Young’s modulus of the composite specimen with and without embedded sensors.

3.2 Monitoring Stress Changes using PFCT sensors In this experiment the ability of the embedded PFCT sensors (MFC and PFC (Dimensions: 30.1 × 9.7 mm × 0.07mm)) to monitor the changes in the applied input stress/strain on the composite structure is investigated. The ASTM standard: D 3039/ D 3039 M was used for fabricating the specimen and performing relevant tests (similar to the specimen in the previous section). The specimens were tested in a servo-hydraulic MTS universal testing machine. The specimen is mounted in the machine and output voltage was recorded from the sensor under the tension-tension loading condition at a frequency of 10 Hz. The capacitance of the sensor was also periodically checked. Generally, if the PFCT sensor is damaged or depoled, then the capacitance should drop. The capacitance of the PFCT sensor is about 100 nF. The specimen was cycled briefly at 20 Hz in tension-tension, while maintaining σmin = 1.0 MPa (constant) and varying σ max (10 MPa to 250 MPa) and simultaneously recorded the sensor voltage output and capacitance. Figure 6 shows the experimental setup. Voltage output response from embedded PFCT sensor (MFC), when the specimen is loaded at different

177 stress ratio is presented in Figure 7. It is observed that with the increase in the loading amplitude (σamp) there is an increase in the amplitude of the voltage output response.

Composite specimen

Figure 6: Experimental setup used for monitoring stress changes in composite specimen using PFCT sensors.

Figure 7: Voltage output response from embedded PFCT sensor (MFC), when the specimen is loaded at different stress ratio. Figure 8 shows the relationship between the amplitude of the output voltage response of the PFCTs sensors to the applied input stress amplitude. The sensitivities of the MFC and PFC sensors as obtained from experiment are 0.0417 and 0.0431 Volts/ MPa respectively. A linear relationship has been observed (Figure 8) between the changes in the output voltage response of the sensor and changes in the input stress amplitude. This means that by continuously monitoring the output response of the embedded PFCT sensors, one could effectively monitor the magnitude of stress/strain acting on the structure. Hence these sensors could be embedded in the critical locations/components of airplanes, spacecraft, buildings, and bridges to monitor the stress/strain levels continuously. Understanding the stress/strain levels in the critical locations/components of the structure will help the design engineers to work on a better and durable design of structure components.

178 12

σmin =10 MPa (constant) and varying σmax MFC y = 0.0417x - 0.6599

10

PFC y = 0.0431x - 0.5184

Volts (V)

8 6 4 2 0 0

25

50

75 100 125 150 175 Maximum Stress (σ max) MPa

200

225

250

Figure 8: Voltage output response from embedded PFCT sensors (MFC and PFC), when the specimen is loaded at different σmax and maintaining σmin = 10 MPa (constant). 3.2 Damage Detection using Embedded PFCT Sensors This section presents the experiments performed to explore the ability of the embedded PFCT sensors (MFC and PFC) to detect the damages in the structures using modal analysis and impact methods. 3.2.1 Damage Detection using Modal Analysis 3.2.1.1 Concept All structures have the dynamic parameters (i.e., natural frequency ( Y n ), Damping ratio (ζ)) that relates the structure’s stiffness (k) and mass (m)). The natural frequency is expressed as:

Yn

k/m

(1) When a composite structure is damaged due to the crack formation, delamination, fracture or loose bolts changes could occur to the structure’s mass and stiffness. Changes in the stiffness and mass of the structure, changes the dynamic characteristics of the structure. Some of the specific changes in the observed dynamic characteristics include changes in natural frequencies, changes in mode shapes, and changes in the damping ratios. These changes can be used to identify the existence, location and magnitude of damage before they can grow to their critical sizes. The natural frequencies of a structure can be measured using modal analysis techniques and can be compared with analytical predictions using finite element analysis. Through comparison of the actual and predicted values of the structure’s natural frequencies, the extent of damage can be determined and decisions made on the integrity of the structure. The three modal analysis techniques to be used in the investigations are random, known, and impact vibration tests. The random vibration test, theoretically, excites all frequencies in a structure, including its natural frequencies. The known vibration test is performed by exciting a structure with a known frequency and determining the magnitude of the structure’s frequency response. When the magnitude reaches a peak at a certain frequency, then this is determined as one of the structure’s natural frequencies. The impact vibration test applies an impulse load to the structure, which excites a range of frequencies in a structure, including the natural frequencies [11-17]. 3.2.1.2 Experiment Experiments were performed to investigate the ability of the PFCT sensors to detect the changes in the dynamic parameters of the structure due to the damages of various levels when introduced in the structure. Figure 9 shows the experimental setup. The PFCT sensors are glued on to a cantilever beam made of glass fiber composite laminate (l=30cm, w=3.2 cm, t=2.1cm) and their output response is monitored and recorded using Hewlett Packard 54603B oscilloscope. Figure 10 shows the typical output response of the PFCT sensor (MFC), when the tip of the cantilever beam is displaced for about 0.5 inches

179 and subjected to free vibration. On performing the Fast Fourier Transformation (FFT) to the output signal of the PFCT sensor, the dynamic parameters of the cantilever beam are evaluated. Damages (such as, de-lamination and cracks) of various levels were introduced on the cantilever composite beam and the natural frequencies of beam were analyzed from the FFT of the output signal of PFCT sensors. Figure 11 and 12 shows the changes in the first two natural frequencies of the beam when de-lamination and cracks of various levels were introduced. Lead wires from sensor Oscilloscope

Composite beam

Embedded PFCT sensor (red)

Figure 9: Composite cantilever beam with embedded PFCT sensor.

Figure 10: The typical output response of the PFCT sensor (MFC) captured using oscilloscope when the tip of the cantilever is displaced to about 0.5 inches. 200 Mode-2 natural frequency (Hz)

Mode-1 natural frequency (Hz)

30

25

20

15

10 0

10

20

30

40

Delamination (%)

50

60

180 160 140 120 100 80 0

10

20

30

40

50

60

Delamination (%)

Figure 11: Changes in the Mode-1 and Mode-2 natural frequencies of composite beam with the various levels of delamination assessed by the PFCT sensor output response.

180

190

30

Mode-2 natural frequency (Hz)

Mode-1 natural frequency (Hz)

35

25 20 15 10 5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

185 180 175 170 165 160 0

Normalized crack length (lc/l)

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Normalized crack length (lc/l)

Figure 12: Changes in the Mode-1 and Mode-2 natural frequencies of composite beam with the cracks of various lengths assessed by the PFCT sensor output response. From these results it was found that with the increase in delamination (%) there is a decrease in the natural frequencies of the composite beam. It was also found that with the increase in the crack length (lc), there is a decrease in natural frequency as well. No significant changes in the natural frequencies have been observed for the crack of length up to 2.0 cm (normalized length =0.05). It is observed that mode 2 natural frequencies are more sensitive to the damages like delamination and cracks, when compared to mode 1 natural frequency. From the results of this experiment, it is evident that damages in the structures can cause the changes in the dynamic parameters of the structure. The PFCT sensors were effectively able to detect the changes in these parameters. 3.2.2 Damage Detection using Tapping Method 3.2.2.1 Concept The method presented in this section is similar to the coin- tap method (which involves tapping on structure and listening for a hollow sound indicating the presence of the damage in the structure). In this method the PFCT sensors are embedded inside the various layers of the glass fiber-epoxy prepreg composite laminate sheets as shown in the Figure 13. This method is based on the energy transfer between the layers of the composite laminate. By monitoring the embedded PFCT sensors output electrical signal the mechanical energy transfer between the composite laminate layers can be calculated. In the Figure 13, the block diagram of the experimental setup and detection technique concept is presented. The test specimen contains 4 layers of unidirectional glass fiber-epoxy prepreg sheets, glued together using the epoxy adhesive. The PFCT sensors were embedded in layers 1 and 4 and their output response is recorded using the data acquisition box. Small Impact Energy

E Top sensor output

V

T

Layer 1 Layer 2

E1

PFCT Sensor

Glass fiber-epoxy prepreg composite laminate

Layer 3 Layer 4

VB E2

Bottom sensor output

PFCT Sensor

Figure 13: Block diagram of the experimental setup and concept.

R= V / V T

B

Voltage output ratio

181 Consider a case when impact energy (E) is applied on the structure and let us assume that the impact mechanical energy transferred to the layer 1 and layer 4 are E1 and E2 respectively. Let us consider the sensor generated output signal to be VT1 and VB1 for the energies E1 and E2 respectively. Now consider there is a delamination between layers 2 and 3, and same amount of impact energy (E) is applied on the structure. The energy transferred to the layer 1 will be the same as the previous case (E1), but energy transferred to the layer 4 will be not same as E 2. Due to delamination between layers 2 and 3, more energy will be observed at delamination region. Hence the energy transfer to the layer 4 will be less than E 2 and the voltage generated by the sensor will also be less than V B1. This means that by continuously monitoring the energy levels in the various layers of composite layers, the presence of damage in the structure can be detected. 3.2.2.2 Experiment Figure 14 shows the experimental setup. Impact energy is imposed by dropping the small steel ball on to the composite laminate through PVC pipe. The PVC pipe guides the ball to fall on a certain desired area of the composite laminate. The output of the PFCT sensors is recorded through data acquisition interface. The impact energy is varied by varying the height of mass being dropped. Composite Laminate Top sensor electrodes

Bottom sensor electrodes

PVC pipe Sensor Electrodes connected to alligator clips

Figure 14: Experimental setup Figure 15 shows the typical voltage ouput response of the embedded PFCT sensor, when an impact energy of 0.25 Joules is applied on the composite laminate. A sharp spike of amplitude about 0.735 volts is observed at the time of impact. When an external load is applied on the composite structure the PFCT sensors in the top (layer- 1) and bottom (layer- 4) layers of composite laminate structure generate voltage output. When a delamination is produced in between the composite layers 2 and 3, most of the externally applied mechanical impact energy is absorbed by the delamination areas. Hence less energy is transformed to the bottom sensor layer, when compared to the case of no damage present. This will result the changes in the voltage output ratio (R). If this ratio (R) is monitored continuously the intensity and presence of damage could be estimated.

0.8

Voltage peak generated due to impact

Voltage ouput (volts)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5 6 7 Time (seconds)

8

9

10

11

Figure 15: Typical voltage ouput response from PFCT sensor.

12

182

Figure 16 shows the voltage ratio (voltage output of top to bottom layer sensor) vs. potential energy when there is no damage within the composite laminate structure. The voltage output ratio is almost constant (1.85 for PFC sensor and 1.77 for MFC sensor) for the various external impact load of various levels. De-lamination is created between the layer- 2 and layer- 3 of the structure artificially. The external impact load is applied on the structure and the voltage ratio is monitored continuously. Figure 17 shows the changes in voltage output ratio with the increase in the de-lamination area when the potential energy of about 0.07 Joules is applied on the composite laminate structure. It was observed that with the increase in the de-lamination area there is an increase in the voltage output ratio. PFC sensors were more sensitive to the delamination damage, with voltage ratio sensitivity of about 1.4862 per 1cm increase of delamination length. MFC sensors have the voltage ratio sensitivity of about 1.0285 per 1cm increase of delamination length. Hence by continuously monitoring the voltage ratio parameter, the presence of the damages in the composite structure could be identified. 1.86 1.85

Ratio (Vt/Vb)

1.84

y = 0.0109x + 1.8485

1.83

MFC

1.82

PFC

1.81

Linear (MFC)

1.8

Linear (PFC)

1.79

y = 0.0011x + 1.7661

1.78 1.77 1.76 0.02

0.045

0.07

0.095 0.12 0.145 Potential Energy (Joules)

0.17

0.195

Figure 16: Voltage output ratio of the top and bottom layers sensor vs. Potential energy when there is no damage in the structure. 10

MFC

9

PFC

8 Ratio (Vt/Vb)

y = 1.4862x + 1.923

Linear (MFC)

7

Linear (PFC)

6 5

y = 1.0285x + 1.5355

4 3 2 1 0 0

0.5

1

1.5 2 2.5 3 Delamination Length (cm)

3.5

4

4.5

Figure 17: Changes in the voltage o/p ratio with the increase in the de-lamination area.

183 4.0 CONCLUSIONS The feasibility of using piezoelectric fiber composite transducers as an embedded sensor inside the composite structure is studied. From the experiments it was found that PFCT sensors can monitor effectively the changes in the input dynamic loads. The sensitivities of the MFC and PFC sensors as obtained from experiment are 0.0417 and 0.0431 Volts/ MPa respectively. A linear relationship has been observed between the changes in the output voltage response of the sensor and changes in the input stress amplitude. Hence these sensors could be embedded in the critical locations/components of airplanes, spacecraft, buildings, and bridges to monitor the stress/strain levels continuously. Understanding the stress/strain levels in the critical locations/components of the structure will help the design engineers to work on a better and durable design of structure components. These sensors were effective in detecting the damages in the composite structure using the modal analysis and impact method. From the results of modal analysis experiment, it is evident that damages in the structures can cause the changes in the dynamic parameters of the structure. It is observed that mode 2 natural frequencies are more sensitive to the damages like delamination and cracks, when compared to mode 1 natural frequency. By constantly monitoring PFCT sensors output response, the changes in the natural frequencies of the structure could be effectively calculated. From the impact method experiment, it was observed that with the increase in the de-lamination area there is an increase in the voltage output ratio. PFC sensors were more sensitive to the delamination damage, with voltage ratio sensitivity of about 1.4862 per 1cm increase of delamination length. MFC sensors have the voltage ratio sensitivity of about 1.0285 per 1cm increase of delamination length. Hence by continuously monitoring the voltage ratio parameter, the presence of the damages in the composite structure could be identified. ACKNOWLEDGMENTS This study is based upon work supported by the NASA/EPSCoR under grant number NASA/LEQSF (2007-10)-Phase3-01.

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Representative Volume Element Analysis for the Evaluation of Effective Material Properties of Fiber and Particle Loaded Composites with Different Shaped Inclusions V.K. Srivastava1, U. Gabbert2, H. Berger2, Institute of Technology, Banaras Hindu University, Varanasi - 22105, India 2 Institute of Mechanics, Otto-von-Guericke University of Magdeburg, Universitaetsplatz 2, D-39106 Magdeburg, Germany 1

ABSTRACT In this paper effective material properties are predicted for composites with different shape and size of inclusions such as cylindrical fibers, spherical and elliptical particles and cylindrical fibers with hemispherical ends. The analysis is based on a numerical homogenization technique using finite element method in connection with three-dimensional representative volume element models. Investigations are carried out to study the influence of various parameters like volume fraction, aspect ratio and particle distribution. Results are discussed and compared. Keywords: Composites, Homogenization, Effective mechanical properties, Finite element analysis 1 Introduction The development and characterization of multifunctional composite materials have been the focus of the research community in recent years with emphasis on potential applications and benefits in various industries. Composites constitute an important class of materials with a wide variety of applications ranging from aerospace structures and electronic printed circuit boards to recreational and commercial equipment. Some of the most important and useful properties of these composites are lightweight, high strength and stiffness, excellent frictional properties, good resistance to fatigue and retention of these properties at high temperatures. Many authors have developed techniques to study the elastic behavior of fibrous composites. Several homogenization techniques have been developed to obtain a suitable constitutive model to be inserted at the macroscopic level [3,7,8,10]. Homogenization is a mechanics based modeling scheme that transforms a body of a heterogeneous material into a constitutively equivalent body of a homogeneous material, where the total energy stored in both systems is approximately same. The macroscopic properties are determined by a homogenization process, which yields the effective stresses and strains acting on the effective, homogenized sample of material. The sample of material is often called as statistically representative volume element (RVE) [5,10]. Analytical and empirical models provide an effective way for predicting the overall properties of composites from the known properties of its constituents, especially for simple configurations of the phases. Even if analytical and semi analytical models have been developed to homogenize fiber composites, they are often reduced to specific cases. Finite element analysis was attempted to describe the behavior of these materials, because there is no restriction on the geometry, size, material properties and number of phases in the composite [4,5]. Kari et al. [6] have studied the influence of randomly distributed spherical and cylindrical particle reinforced composites using numerical homogenization techniques with periodic boundary conditions. In order to determine the best predictions of the elastic properties of an aligned unidirectional fiber reinforced composite, spatial distribution of the reinforcing fibers should be considered. The random nature of the transverse packing of the composite has a significant influence on the transverse elastic constants; and to predict them accurately, this microstructure aspect have to be considered. Several authors have reported the behavior of the composite materials with multi-scale approach. The overall properties of the composites that are obtained from the experimental results are well fitted with the investigations made either through the micro or macro mechanical analysis. Due to lack of structural simplicity of continuous fiber composites or laminates, the finite element method is often more suitable than analytical modeling for multiphase materials. Therefore, finite element method is very suitable to determine the effective properties of multi-shaped fiber composites. In this paper, the effective material properties of randomly distributed particle loaded composites are calculated by considering a representative volume element approach. The main goal of this paper is comparison of the behavior of composites with different types of inclusion. For these investigations inclusions of type sphere, ellipsoid, cylinder and cylinder with hemispherical ends are considered (see Fig. 1). Based on appropriate periodic boundary conditions and load

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_27, © The Society for Experimental Mechanics, Inc. 2011

185

186 cases the finite element method can be used to calculate the full set of effective coefficients of the composite. A general numerical homogenization concept was used, which can be extended for composites with arbitrary inclusions. 2 Basic equations The generalized Hooke’s law can be formulated to correlate the effective stiffness matrix Cij , average stresses Tij and strains S ij , for homogenized composites as

ªT11 º « » «T22 » «T » « 33 » «T23 » « » «T31 » «T » ¬ 12 ¼



ªC11 C12 C13 C14 « C22 C23 C24 « « C33 C34 « « C44 « symm. « « ¬

C15 C16 º » C25 C26 » C35 C36 »» C45 C46 » » C55 C56 » » C66 ¼

ªS11 º « » «S22 » «S » « 33 » . «S23 » « » «S31 » «S » ¬ 12 ¼

(1)

The above stiffness matrix contains 21 independent constants with no symmetry. But majority of the engineering materials exhibits one or more symmetry. The considered composites in this paper have aligned inclusions in direction 3 (see Fig. 1). Therefore the effective coefficients can be characterized by a nearly transversely isotropic behavior. For transversely isotropic material the number of independent constants reduces to five. With coordinate plane 1-2 as transverse plane the following five independent constants remain

C11

C 22 , C12 , C13

C 23 , C33 , C 44

C55 and C66

1 (C11  C12 ) . 2

(2)

With the following relations these coefficients can be expressed by engineering constants Young’s moduli E, shear moduli G Poison’s ratios v and bulk modulus K

E33

C33 

2(C 13 ) 2 ; E11 C11  C12

C11 

C13 (C 23 ) 2  C33C12 ; v12 C11  C13 (C 23 ) 2  C33C11 1 1 (C11  C12 ) ; K 33 (C11  C12 ) C44 ; G12 2 2 v13

G23

(C 23 ) 2 (C11  2C12 )  C33 (C12 ) 2 (C 23 ) 2  C33C11 (3)

For presenting and comparison of results these constants are used. 3 RVE model 3.1 Periodic boundary conditions and load cases Three-dimensional composite materials can be represented as a periodic array of the RVEs. To ensure this periodicity also after deformation appropriate periodic boundary conditions must be applied to the RVE model. The main idea of homogenization procedure consists in the assumption that the average mechanical properties of the RVE must be equal to the average properties of the particular composite. By constructing six special load cases – three pure tension cases and three pure shear cases – the full set of effective coefficients can be calculated [1,2,5]. Every load case leads to an average strain vector where only one strain component is non-zero and all others are zero. Then by using this non-zero strain all coefficients in the corresponding column of stiffness matrix can be calculated in a simple manner. E.g., for the calculation of C11 only

187

S11 may be non-zero. Then C11 can be calculated from first row of constitutive equations by the ratio of T 11 / S 11 and C21 from the second row by the ratio of T 22 / S 11 and analogous C31 etc. The average strains and stresses are calculated from averaged finite element values by

S ij where Vk is the finite element volume and

1 V

¦S V ij

k

, T

ij

k

1 V

¦T V ij

(4)

k

k

V is total volume of the RVE.

The periodic boundary conditions are applied as constraint equations between corresponding nodes on opposite surfaces of the RVE. The six load cases are realized by appropriate displacement differences between opposite RVE surfaces. To guarantee identical mesh configurations on opposite surfaces special mesh copy procedures were used. Furthermore algorithms for automatic selection of corresponding nodal pairs and applying the constraint equations were developed. 3.2 Geometrical modeling The simplest RVE is a cube with one inclusion. There is no special effort necessary to create such models. But for models with more and also random distributed inclusions special generation algorithms are required. In our modeling procedure a modified random sequential adsorption algorithm (RSA) is used [5,9]. Certain parameters like desired volume fraction of inclusion, starting size of inclusion, minimal distance of inclusions, minimal distance to cube surfaces etc. can be given as input to control the generation process. Minimum distances are necessary to ensure an adequate finite element mesh [5]. The particles are placed at a random location. If a new generated particle intersects with another or with surfaces it will be deleted and a new location is tried to find. If no more inclusions with the previous size can be placed the size is step wise reduced. This process will be repeated until the desired volume fraction or the jamming limit is reached. In our models the inclusions are placed fully inside. The algorithm can be extended to allow also intersection with RVE surfaces also under ensuring periodicity [5]. The RSA algorithm with the combination of the above conditions is used to generate the inclusion volumes up to a desired volume fraction. But according to the type of inclusion different limits for maximum volume faction exist. The volume fraction was varied from 10% to 50% for cylindrical fibers, 10% to 50% for spherical inclusions 5% to 25% for elliptical fibers and 1% to 5% for cylindrical fibers with hemispherical ends. For the combination of different inclusion types the volume fraction was varied from 3% to 15%. In all cases the RVE size is kept constant (a unit length of 1 is used for the cube edges) and the inclusions are varied to achieve the different volume fractions. All finite element calculations were performed with the commercial FE package ANSYS. The matrix and the fibers were meshed with 10 node tetrahedron elements with full integration [5]. The ANSYS parametric design language (APDL) is used to calculate the average strains, stresses and effective material properties. The developed APDL scripts in combination with the ANSYS batch processing, provide a powerful tool for the fast calculation of homogenized material properties of composites. 4 Results and discussion Three-dimensional RVE models of fiber inclusions were created containing different shape, size and distribution of inclusion. Fig. 1 shows RVE examples for the investigated shapes of inclusions: sphere, ellipsoid, cylinder , cylinder with hemispherical ends. With the RSA algorithm different volume fractions with certain steps were generated. For better visibility the figures show examples with low volume fraction. Furthermore various combinations of inclusion types were investigated. Fig. 2 shows the used combinations: cylinders and spheres, cylinders and ellipsoids, spheres and ellipsoids. In all models perfect contact between matrix and inclusion is assumed. For all calculations a composite consisting of aluminum alloy with embedded silicon carbide (SiC) particles was used. The constituents have isotropic material behavior. The used values are listed in Table 1. Table 1: Material properties for the constituents of the composite Material Fiber / Particle (SiC) Matrix (aluminum alloy)

Young’s modulus (GPa) 450 70

Poisson’s ratio 0.17 0.3

188

a)

b)

c)

2

1 3

d)

e)

Figure 1: RVE models for different types of inclusions: a) sphere, b) ellipsoid), c) cylinder, d) cylinder with hemispherical ends, e) meshed example

a)

b)

c)

Figure 2: RVE models for different combinations of inclusions: a) cylinder and sphere, b) cylinder and ellipsoid c) sphere and ellipsoid First the results for uniform inclusions are discussed and then the behavior for merged inclusions is considered. Because of the random distribution of the inclusions but aligned in direction 3 a nearly transversely isotropic overall behavior was

189 observed. That's why for discussion and comparisons of effective material behavior the engineering constants for transversely isotropic material according to Eq. (3) were used.

200

350 300

160

E11 (GPa)

E33 (GPa)

180

400

Spheres Cylinders Ellipsoids

140 120

G23 (GPa)

50

0.1

0.2 0.3 0.4 Volume fraction

0.5

50 0.0

0.6

140

Spheres Cylinders Ellipsoids

120

40 30 20 0.0 0.5 0.4

200

100

G12 (GPa)

60

250

150

100 80 0.0

Spheres Cylinders Ellipsoids

0.1

0.2 0.3 0.4 Volume fraction

0.5

0.6

0.5

0.6

0.5

0.6

Spheres Cylinders Ellipsoids

100 80 60 40

0.1

0.2 0.3 0.4 Volume fraction

0.5

20 0.0

0.6

0.6

Spheres Cylinders Ellipsoids

0.5

0.1

0.2 0.3 0.4 Volume fraction

Spheres Cylinders Ellipsoids

0.4

Q 13

Q 12

0.3 0.2

0.3

0.1

0.2

0.0 0.0

0.1

0.2 0.3 0.4 Volume fraction

0.5

0.6

0.1 0.0

0.1

0.2 0.3 0.4 Volume fraction

Figure 3: Variation of selected effective engineering constants over volume fraction for three different types of inclusions 4.1 Effective material constants for uniform inclusions The volume fraction of inclusions was varied and the size of the RVE was kept constant. Two achieve the desired volume fraction the size, position and number of inclusions were varied using the RSA algorithm.

190 Fig. 3 shows the variation of effective material properties over volume fraction. For spheres and cylinders the volume fraction ranges from 0.1 to 0.5 plotted in steps of 0.1. For ellipsoidal inclusions only a volume fraction up to 0.25 could be reached. Therefore the range is from 0.05 to 0.25 plotted in steps of 0.05. For every volume fraction step several calculations with randomly distribution and size of inclusions were carried out. The plotted points show the averaged values from every sample series. 95

150

Cylinders with hemispherical ends

140 130

85

E11 (GPa)

E33 (GPa)

90

80

34

0.01

0.02 0.03 0.04 Volume fraction

0.05

60

Cylinders with hemispherical ends

G12 (GPa)

G23 (GPa)

0.02 0.03 0.04 Volume fraction

0.05

0.06

Cylinders with hemispherical ends

40 30

0.01

0.02 0.03 0.04 Volume fraction

0.05

20 0.00

0.06

110

Cylinders with hemispherical ends

0.45

0.01

0.02 0.03 0.04 Volume fraction

0.05

0.06

Cylinders with hemispherical ends

100 K33 (GPa)

0.40 Q 12

0.01

50

31

0.35 0.30

90 80 70

0.25 0.20 0.00

100

70 0.00

0.06

32

0.50

110

80

33

30 0.00

120

90

75 70 0.00

Cylinders with hemispherical ends

0.01

0.02 0.03 0.04 Volume fraction

0.05

0.06

60 0.00

0.01

0.02 0.03 0.04 Volume fraction

0.05

0.06

Figure 4: Variation of selected effective engineering constants over volume fraction for inclusion type cylinder with hemispherical ends Like expected the Young's moduli, shear moduli and consequently bulk modulus increase with increase of volume fraction. It is observed that cylindrical fiber inclusions give higher values. The results for the spherical and elliptical inclusions are

191 nearly identical in the overlapping range of volume fraction. The reason lies in the comparable geometry. The values for the Poisson’s ratios decrease with the increasing of volume fraction of inclusion. Interesting is the fact that for the considered composites the Young's modulus in transverse plane (E11) is higher than in longitudinal direction (E33) which is more obvious for higher volume fraction. This comes from short length of the inclusions and the high density in packing. Similar is the behavior for the in-plane shear modulus (G12) in comparison to the out-ofplane shear modulus (G23) In Fig. 4 the behavior of composite with cylindrical inclusions and hemispherical ends is shown. Here only very low volume fractions could be generated, range 0.01 to 0.05 in steps of 0.01. Therefore the results are plotted in separate graphs. With this type of inclusion singularities at the ends of simple cylinders can be removed. Typically real fibers have not exact hemispherical caps but it can be considered as a minor change in shape compared to plane cylindrical approach. Here also Young’s modulus, shear modulus and bulk modulus increase with increase of volume fraction. Like in the previous consideration constants for transverse direction are higher than for longitudinal direction. 4.2 Effective material constants for combination of inclusions Finally, three different pair inclusions are considered: cylinders and spheres, cylinders and ellipsoids, spheres and ellipsoids like shown in Fig. 2. The volume fraction ranges from 0.03 to 0.13 and 0.03 to 0.15 only for the combination spheres and ellipsoids. Different models with random merge, size and distribution of inclusions were calculated for every volume fraction. The averaged results are plotted in Fig. 5. The highest overall stiffness can be reached with the combination cylinders and spheres. But in comparison to uniform inclusions the stiffness is lower in the comparable range of volume fraction. Furthermore the maximum volume fraction due to the jamming limit is also lower. Because of this limitation to low volume fraction the stiffness behavior in transverse and longitudinal direction is only slightly different.

80

75

70

0.03

34

0.06 0.09 0.12 Volume fraction

75

0.03

76

K33 (GPa)

28

0.06 0.09 0.12 Volume fraction

0.15

Cylinders & Spheres Cylinders & Ellipsoids Spheres & Ellipsoids

74

30

26

Cylinders & Spheres Cylinders & Ellipsoids Spheres & Ellipsoids

80

70

0.15

Cylinders & Spheres Cylinders & Ellipsoids Spheres & Ellipsoids

32 G12 (GPa)

85

Cylinders & Spheres Cylinders & Ellipsoids Spheres & Ellipsoids E11 (GPa)

E33 (GPa)

85

72 70 68

0.03

0.06 0.09 0.12 Volume fraction

0.15

66

0.03

0.06 0.09 0.12 Volume fraction

0.15

192 0.32 0.31 0.30

0.30

0.29

0.29

0.28

0.28

0.27

0.27

0.26

0.03

0.06 0.09 0.12 Volume fraction

0.15

Cylinders & Spheres Cylinders & Ellipsoids Spheres & Ellipsoids

0.31

Q 12

Q 13

0.32

Cylinders & Spheres Cylinders & Ellipsoids Spheres & Ellipsoids

0.26

0.03

0.06 0.09 0.12 Volume fraction

0.15

Figure 5: Variation of selected effective engineering constants over volume fraction for different combinations of inclusions 5 Conclusions Numerical homogenization tools have been developed for the evaluation of the effective material properties of the short fiber composites. The results showed that the effective material properties depend mainly on the volume fraction. Also, Young’s modulus, shear modulus and bulk modulus properties can be improved with the increase of fiber volume fraction. Volume fraction for cylindrical and spherical inclusions cannot be increased more than 50%, where as in the combined cases, it should be only up to 15% to avoid the jamming effect. The results show that it is possible to improve the overall behavior of composites with desired properties by changing shape type, size and distribution of inclusions. In this paper the effect by various variations can be seen in the plotted graphs. With the presented general numerical homogenization procedure for calculation of effective material constants a powerful tool is given to the engineer for design of such optimal composites. References [1] Berger H, Kari S, Gabbert U, Ramos-Rodriguez R, Guinovart-Diaz R, Otero J A, Bravo-Castillero, J, An analytical and numerical approach for calculating effective material coefficients of piezoelectric fiber composites, Int. J. Solids Struct., Vol. 21–22, pp 5692–5714, 2005. [2] Berger H, Kari S, Gabbert U, Ramos-Rodriguez R, Guinovart-Diaz R, Otero J A, Bravo-Castillero J, Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties, Smart Mater. Struct., Vol. 15, pp 451–458, 2006. [3] Böhm H J, Eckschlager A, Han W, Multi-inclusion unit cell models for metal matrix composites with randomly oriented discontinuous reinforcements, Comput. Mater. Sci., Vol. 25, pp 42-53, 2002. [4] Kari S, Berger H, Rodriguez-Ramos R, Gabbert U, Computational evaluation of effective material properties of composites reinforced by randomly distributed spherical particles, Composite Structures, Vol. 77, pp 223–231, 2007. [5] Kari S, Micro-Mechanical Modelling and Numerical Homogenization of Fiber and Particle Reinforced Composites, Reihe 18, Mechanik, Nr 309, VDI Verlag GmbH, Germany, Ph. D. Thesis, 2006. [6] Kari S, Berger H, Gabbert U, Numerical evaluation of effective material properties of randomly distributed short cylindrical fiber composites, Computational Materials Science, Vol. 39, pp 198–204, 2007. [7] Ramos-Rodriguez R, Sabina F J, Guinovart-Diaz R, Bravo-Castillero J, Closed- Form Expressions for the Effective Coefficients of Fiber-Reinforced Composite with Transversely Isotropic Constituents. I: Elastic and Square Symmetry, Mech. Mater., Vol. 33, pp 223-235, 2001. [8] Torquato S, Random heterogeneous media: Microstructure and improved bounds on the effective properties, Appl. Mech. Review, Vol. 44, pp 37-76, 1991. [9] Wang, J S, Random sequential adsorption, series expansion and Monte Carlo simulation, Physica A; Vol. 254, pp 179184, 1998. [10] Zihui X, Yunfa Z, Fernand A, Unified periodic boundary conditions for representative volume elements of composites and applications, Int. J. Solids. Struct., Vol. 40, 1907-1921, 2003.

Time Dependent (Creep) Deformation of Thin Elastomers at Cold Temperature and Effective Strain Analysis of Their Laminates

Jon-Michael Adkins, Dr. Prasun Majumdar & Prof. Kenneth Reifsnider Department of Mechanical Engineering University of South Carolina 300 Main St. Columbia, SC 29208

ABSTRACT Rolling resistance from tires accounts for 20% of the fuel used in the average car. In applications like automobile tires, heterogeneous “inner liners” are being used as barriers to retain air and to reduce rolling resistance. However, there is great uncertainty in estimating the life of those components under extreme, synergistic environments (e.g., rate and time dependent, cyclic, thermal, and mechanical loading). In the present work, the behavior of such thin films under extreme cold temperature (-40C) has been investigated experimentally. The expected life was estimated both experimentally and by using a parametric creep model. Failure mechanisms were studied extensively and interesting phenomena (such as time-delayed longitudinal fracture) are documented. In order to develop a better understanding of the mechanisms driving crack formation in field tested tires of inner liners, actual tire specimens were also tested. An experimental Arcan fixture was used in conjunction with digital image correlation (DIC) to determine the full-field strain state of the inner liner material under simulated loading conditions. The normalized component sums of the effective strains were computed, and it was found that these strain resultant vectors correspond to directions of observed cracks in field testing.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_28, © The Society for Experimental Mechanics, Inc. 2011

193

194

Background The main goal of this study is to determine the 3D strain state of the inner most lamina of a tire side wall, an elastomeric laminate, under combined loading. This is important because the innermost lamina has been known to develop surface ruptures under conditions of high applied shear making traditional strain gages unsuitable for the study. Also, the fact that the material is elastomeric means that strain gages could be stiffer than the surrounding material; another reason they could not be used for the study. Third, it is highly anisotropic, and one lamina contains reinforcement (Illustrated in Figure 1) so that even if a gage could be mounted effectively it would provide limited information about the laminate behavior.

Figure 1-Tire Sidewall Section (Approximate Geometry). Fourth, the lamina itself is a heterogeneous composite containing two materials in proportions of approximately 50% volume fraction each. Lastly, it is a nonlinear material with unusual geometry as seen in Figure 1. We will demonstrate how this material behaves and deforms accordingly later. Given the preceding information, the need for a method of measurement that can work with all of these material characteristics is clear. Digital image correlation is a speckle method of experimental strain measurement. When applied in a stereo or two camera configuration the movements and deformations of speckle groups can be tracked in three dimensions in a way that is analogous to depth perception being achieved from two eyes being focused on the same object. This technique has been used before to verify FEA model results for the global deformations of similar laminates used in the same applications, but not to examine the local strain of a particular lamina as in this study [1]. A big advantage is that no contact is required; providing a capability to take measurements that cannot be recovered with surface mounted gages clear need of a method of measurement

195

that can work with all of these material characteristics. Second, this method can track very large strains (well over 1000%) that are also not possible with traditional gages. Lastly, depending on the setup, image analysis software can output deformed and undeformed coordinates of speckle groups in three dimensions; allowing calculation of shear and normal strain components over a large area of the sample unlike gages. It is also important to note that given this material’s characteristics, any 3D strains calculated are only close approximations to conditions at the surface, and would not be representative of the whole sample. Fortunately, the lamina of interest is thin (0.1mm) compared to the material thickness; making the surface strain approximations reasonable estimates throughout the lamina’s cross section. Possible disadvantages include the need for a speckled pattern to be fixed to the surface to be imaged. Depending on a sample’s traits, it may be difficult to apply this pattern; known issues include sample reflectivity or color (due to light reflecting into cameras) as well as surface chemistry preventing pattern adhesion. Fortunately, the surface of the present material is dark enough to avoid contrast and reflectivity issues, and can be easily cleaned with acetone or alcohol to remove surface contaminants that affect speckle adhesion [2]. It can be shown that simple extensions of this material lead to significant out-of-plane bending as well as in plane shear responses (shown later). This means that if data sets from a simple extension test are to be examined along with combined loading, there is need of a fixture that can support both. The Arcan fixture has been used before to determine the behavior of fiber reinforced composites under combined loading at various orientations. The main characteristic of the fixture that makes this work is a series of user-selectable gripping patterns that allow different loading directions depending on how a particular Arcan configuration was made. This fixture seems to yield itself well to different configurations depending on the type of test desired; like the common “butterfly” shaped specimen shown to maintain good uniform stress in isotropic specimens via photoelastic methods. However, a rectangular sample was used for this study to optimize gripping. Any gripping effects were compensated for by performing analysis on areas of test images in the far-field away from the gripping areas [3].

Fixture Development The original Arcan fixture produced for this study was designed to test only the lamina of interest, or more specifically, a thicker version of it. The disadvantage of that approach was that any test done on the lamina alone would not be a true reflection of in service conditions in the tire section. At the same time, the greatest advantage is the ability to apply a

196

combined normal and shear loading that is similar to those in which the highest strains are believed to occur in the problem of interest. When testing of the entire laminate was attempted, two issues became clear. The first was that the gripping altered specimen curvature thus detracting from the realism of the test. The second observed effect was due to the softness of the material. During testing, a large 3D Poisson effect caused an unacceptable amount of slipping due to a reduction in sample thickness. Both of these problems were solved by encasing the ends of the specimens in epoxy. Next, it was found that this made the specimen ends too large to fit into the original fixture. There was need of a revised design to avoid a compromise on specimen gage size (38mm sq gage). The benefits of redesign were two-fold; a refined fixture would fit the larger epoxy coated ends and provide a chance to implement further proposed changes. One such change, an improved knurl pattern for specimen gripping is shown in Figure 2. It should also be noted that the new fixture has more/larger clamping fasteners spaced evenly on either side of the knurl to provide a more uniform and firm grip when equivalent torque is applied to all fasteners. Figure 3 shows the new Arcan with a loaded sample after a test.

Figure 2-Original gripping pattern (left) compared to the refined one (right).

197

Figure 3-New Arcan loaded with coated sample (pattern damaged during test). Finally, the following procedure was isolated for producing quality samples. The first step was to coat the ends in epoxy for gripping as follows: 1. Cut a section of material 38 x 135mm 2. Trim both ends such that surface ridges are aligned in the desired direction and ends are parallel 3. Place ends in mold one at a time, and fill with fast setting general purpose epoxy The laminate had irregular geometry, including two axes of curvature and ridges along its surface. This, combined with the laminate’s overall softness suggested the use of a less traditional method of strain measurement; digital image correlation in the present case. Before such a correlation could be performed, it was important to find a repeatable procedure for applying a speckled pattern to the samples. The procedure was as follows: 1. Set sample in fume hood running on “high” with imaging side up and longest side parallel with direction of painting 2. Place a 13mm tall barrier on an epoxy coated end between the area to be imaged and the nozzle on a white spray paint can. 3. Spray over the barrier parallel to specimen length until top end of imaging area is speckled, then quickly rotate and repeat to speckle the bottom half of the imaging area. This process allows paints droplets of optimal size to be applied to the surface by causing droplets that are too large and have higher aerodynamic drag to collide with the barrier while undersize droplets go past the sample leaving ideal droplets on target.

198

Results The images were analyzed using VIC 3D software developed by researchers at the University of South Carolina. This program outputs discrete displacements in three dimensions for points within a user-defined area of interest and can calculate all in plane strains, principal strains and local rotations in a user or automatically defined coordinate system by tracking the movement and deformation of speckle groups. Additional information about this testing system can be found at a company called Correlated Solutions in West Columbia, SC, or by visiting http://www.correlatedsolutions.com/. The goal of this study was to use these displacements to learn the 3D strain state along the top and bottom of surface ridges to learn what the inner-most laminate of this material may be experiencing during in service conditions [4]. During analysis, deformed and undeformed coordinates from the software were used to calculate local displacements in the x, y and z directions (u, v and w respectively) at each point. These values were then fit to the model in Equation 1 with a least squares method for the purpose of finding continuous displacement gradients. Note that since u, v and w are each fit as functions of undeformed coordinates X, Y and Z the resulting models are all four dimensional. The model in EQN (1) is only second order since higher order models quickly become cumbersome in four dimensions; meaning due to the size of surface features, two small areas were analyzed instead of one large one to ensure a quality curved fit. The first section examined is the area between the surface ridges; the second is the area at the top of a ridge. Both are shown in Figure 4. These areas were chosen to be in the far-field to avoid any gripping effects and in locations where the image analysis yielded low errors to ensure high quality data; the idea being that a known strain state in both of these areas yields a good idea as to that of the rest of the lamina. This concept is illustrated in Figure 5. Note that when the two areas of interest are combined, strain data are obtained for an entire surface ridge as well as the area between.

EQN(1)

199

Figure 4-Area between ridges (left) and area across top of ridges (right).

Figure 5-Cross-section of laminate illustrating area of interest coverage (not to scale). The presence of quality data was further confirmed by generating surface plots with the deformed and undeformed coordinates to make sure we get a result that looks like what you may expect from those surfaces. Figure 6 shows areas of interest at each load (all units in mm). Even with good data, some rough places are still visible around the edges. These rough places in the raw data are believed to be discontinuities from the numerical analysis that generates the displacements. It is clear that for the area on top of the ridges at 53.5 N correlation was unsuccessful. The problem is attributed to the way light reflected off of that area at that particular loading. This figure also verifies flattening and out of plane bending of the material under a load; particularly for the area between ridges. Finally, note that all plots for each area of interest have the same axis orientation to help the reader visualize the process.

200

Figure 6-Between ridges for unloaded, 53.5, 98 and 133 N cases.

These plots can also confirm the presence of an in-plane shear response when examined in another orientation (Figure 7). Note that the y-axis has the same orientation in all of these plots, but each image had to be rotated along the y-axis to show the full magnitude of the shear response. Additionally, the need for this adjustment serves as evidence of out-of-plane rotation under applied load.

201

Figure 7-Illustration of shear response between ridges for unloaded, 53.5, 98 and 133 N cases (Arrows indicate loading direction).

Displacements calculated from the data can now be used to find supporting evidence for the previously mentioned non-linear response. So, far everything presented has come from 45° combined loading tests, but the following figures will also look at the 0° configuration; meaning the sample is mounted straight up and down in the Arcan fixture as if you were performing a standard tensile test. This will provide data for the case of a simple extension of the material. Figures 8 and 8 show the presence of non-linear response (particularly along the vertical axis) in combined loading and simple extension tests. Note that only the area between ridges is shown due to the failed correlation in one of the cases during the 45° test.

202

Figure 8-Load vs. average displacement between ridges for 45° test.

Figure 9-Load vs. average displacement between ridges for 0° test.

Next, the calculated displacements were curve fit using the model in Equation 1. According to the fit statistics (r squared values) in Table 1, the model was found to be a good means of curve fitting displacements to find continuous displacement gradients for all loadings in both regions except where correlation was unsuccessful. In all other cases the model explains no less than 84% of the data’s variability.

203

Table 1-Displacement curve fit statistics. Quality of Fit (r-squared Between Ridges Top of ridges -

Load(N) 53.5 98 133 53.5 98

u 0.998690 0.999028 0.999205 0.233338 0.999437

v 0.999574 0.999567 0.999554 0.747354 0.999639

w 0.973673 0.994938 0.996452 0.066588 0.885872

133

0.999240

0.999628

0.841439

These continuous displacement gradients were evaluated at every point and plugged into the Green-Lagrangian strain tensor; where u1, u2, u3, X1, X2 and X3 are displacements and undeformed coordinates respectively [5]. After looking at all of the strain magnitudes, it is clear that the greatest strains occur in the low places between ridges. This is shown in Table 2, where mean and median effective strains are related to fixture displacement as well as applied global normal and shear strains in the fixture for the 45° configuration. Table 2-Relationship between global strain and fixture displacement in 45° configuration.

Displacement(mm)

Total Load(N)

6.63

53.5

-

53.5

13.26

98

-

98

19.89

133

-

133

Region Between ridge Top of ridge Between ridge Top of ridge Between ridge Top of ridge

Applied nor. strain (global)

Applied shear strain (global)

Avg. effective strain

Median eff. strain

12.30%

10.90%

47.27%

47.70%

12.30%

10.90%

11.04%

10.94%

24.61%

19.50%

108.19%

100.59%

24.61%

19.50%

83.11%

82.57%

36.91%

26.33%

125.71%

122.66%

36.91%

26.33%

85.58%

83.34%

With effective strain magnitudes known, it was decided that their directions should also be computed to look for anything that may yield further insight into material behavior. This was done by summing the respective components of each of the principal directions, normalizing the result and generating a 3D vector plot with the resultant vectors to visualize the effective strain field in each region. Figure 10 is one particular result of great interest to us because it shows effective directions

204

parallel to surface cracks observed in the product during field tests. This occurs between the ridges at the highest loading of 133 N in the 45° combined loading test. Figure 10 is a 3D vector plot of the effective strain directions at each point on the deformed surface of the material. The points at the vector bases represent the deformed surface, the double-ended arrow is the direction of observed failures from field tests, the green line running through the plot is the lowest point across the area of interest and the arrows along the corners represent the direction of applied load. The three vectors that do not conform to the others are believed to be due to numerical issues described earlier.

Figure 10-Vector plot of effective direction in area between ridges (133 N load 45° test). Conclusions A modified Arcan fixture and digital image correlation were used to measure the strain state of the inner-most lamina of a material in conditions similar to those seen in service. It was found that the surface geometry played a significant role in the local strain state, e.g., the general surface topology was found to introduce significant non-uniformity and nonlinearity into the point-wise measured strains in the presence of a uniform global shear deformation. This is most evident in Table 2, where it was found that the overall state of strain is greater between the surface ridges, and the tops of the ridges are generally less sensitive to load, e.g., the average effective strain on the top of ridges changes very little from 98 to 133 N (2.47%) when compared to the area between them (17.52%).

205

The Arcan fixture modifications discussed proved to provide reliable gripping for DIC strain measurement in this combined loading. It was also shown that full field 3D strain measurements can be done successfully with the outputs from VIC 3D software to get a better understanding of what is happening in a loaded anisotropic material with irregular geometry. Finally, the effective direction plot in Figure 16 yields insight into observed failures by showing a correlation between the effective strain directions and crack formation observed during field testing.

206

References 1.

Dally, James W. Experimental Stress Analysis. 4th ed. Knoxville, Tennessee: College House Enterprises, LLC, 2005.

2.

Moser, R., and J. G. Lightner III. "Techniques." USING THREE-DIMENSIONAL DIGITAL IMAGING CORRELATION TECHNIQUES TO VALIDATE TIRE FINITE-ELEMENT MODEL (2007).

3.

El-Hajjar, Rani, and Rami Haj-Ali. "In-plane Shear Testing of Thick-section Pultruded FRP Composites Using a Modified Arcan Fixture." Composites: Part B 35 (2004): 421-28. Science Direct. Web.

4.

Correlated Solutions. Web. http://www.correlatedsolutions.com/.

5.

Reddy, J. N. An Introduction to Continuum Mechanics. New York, NY: Cambridge UP, 2008.

6.

Chapra, Steven C., and Raymond P. Canale. Numerical Methods for Engineers. 5th ed. New York, NY: McGraw Hill, 2006.

Time and Temperature Response of Composite Overwrap Cylinders Jerome T. Tzeng US Army Research Laboratory Weapons and Materials Research Directorate Attn: AMSRD-ARL-WM-MB Aberdeen Proving Ground, MD 21005-5066

ABSTRACT Stress relaxation and creep of composite cylinders are investigated based on anisotropic viscoelasticity. The analysis accounts for ply-by-ply variation of material properties, ply orientations, and temperature gradients through the thickness of cylinders subjected to mechanical and thermal loads. Fiber reinforced composite materials generally illustrate extreme anisotropy in viscoelastic behavior. Accordingly, viscoelastic characteristics of composite cylinders are quite different from those of isotropic cylinders. Viscoelastic effects of the composite can result in a drastic change of stress and strain profiles in the cylinders over a period of time, which is critical for structural durability of composite cylinders. The developed analysis can be applied to composite pressure vessels and flywheels. 1 INTRODUCTION Composite materials are currently used for lightweight pressure vessels and highly efficient rotors for energy storage. For both applications, pre-stresses are built in during fabrication of the cylinders through a "press-fit" procedure to enhance the mechanical performance. Composite overwrap pressure vessels are prestressed; therefore, the liner is in a state of hoop compression and the composite overwrap is in tension. For the other application, the rotors are subjected to a radial compression prior to operation. Accordingly, centrifugal force resulting from the rotation of the rotors generates tensile stresses in the radial and circumferential directions. Since the composite rotors are mainly reinforced in circumference (filament wound cylinders), the radial tensile stress is critical to the ultimate performance of the rotors. Accordingly, it is essential to design and build the rotors with radial pre-compression. However, polymer matrix composites generally creep over a long period of time, especially at an elevated temperature [1]. The associated stress relaxation in the composite will result in the loss of the pre-stresses and lead to a potential failure. The objective of this investigation is to develop an analytical method to study the viscoelastic behavior of thick-walled composite cylinders. The analysis can be applied to the design of flywheel machinery and composite pressure vessels.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_29, © The Society for Experimental Mechanics, Inc. 2011

207

208

Figure 1: Cylindrical coordinate system in a laminated composite cylinder To date, activities in the research of viscoelasticity have mainly concerned isotropic materials including the studies by Muki and Sternberg [2]; Schapery [3]; Williams [4]; and Christensen [5]. These basic theories of viscoelasticity were then extended to the area of heterogeneous and anisotropic materials for a variety of applications. Hashin [6] used the effective relaxation moduli and creep compliances to define the macroscopic viscoelastic behavior of linear viscoelastic heterogeneous media and its implementation in viscoelastic modeling. The general formulation of linear viscoelastic boundary value problems of composite materials, including the thermal viscoelastic problems for thermorheologically simple materials, and the applications of the correspondence principle were examined by Schapery [7]. Rogers and Lee [8] investigated the viscoelastic behavior of an isotropic cylinder. In addition, finite element packages such as ABAQUS, ANSYS, DYNA3D are not suitable for the viscoelastic analysis of composite cylinders because of the lack of anisotropic viscoelastic elements.

2 VISCOELASTIC FORMULATION In the following research, the linear quasi-static viscoelastic behavior of a thick laminated composite cylinder with an elevated temperature change is studied. The analysis accounts for ply-by-ply variation of properties, temperature changes, and fiber orientations. The thick cylinder is assumed to be in the absence of thermomechanical coupling and to be in the state of generalized plane strain such that all the stress and strain components are independent of the axial coordinate (Tzeng and Chien [9,10]). Moreover, due to the nature of axisymmetry, all the stress and strain components are also independent of the circumferential coordinate. The mechanical responses of this thick composite cylinder will, therefore, only have to satisfy the governing equation in the radial direction. Invoking the Boltzmann superposition integral for the complete spectrum of increments of anisotropic material constants with respect to time, the thermoviscoelastic constitutive relations of the anisotropic composite cylinder can be derived in integral forms. Since the thick composite cylinder is subjected to a

209

constant elevated temperature, and boundary conditions are all independent of time, formulations of the linear thermal viscoelastic problem can have forms identical to those of the corresponding linear thermoelastic problem by taking advantage of the elastic-viscoelastic correspondence principle. In other words, all of these integral constitutive equations reduce to the algebraic relations, which are very similar to those developed for thermoelastic media when they are Laplace-transformed by means of the rule for convolution integrals. The thermoelastic analysis can thus be used to derive the transformed thermal viscoelastic solutions in the frequency domain. The Boltzmann superposition integral of stress Vij (i, j = 1, 2, 3) and strain Hij (i, j = 1, 2, 3) relation for an isothermal viscoelastic problem with a constant temperature increase, T and the thermal expansion coefficient Dkl(T) is t

V ij (t) = ³ 0 Cijkl (T,t - W )

w H kl ( W ) dW - E ij (T,t)'T , wW

(1)

kl ij

where C (T,t) is the relaxation modulus dependent on temperature T and time t,

Cijkl (T(t),t) = Cijkl ( T 0 , O (t)) t O (t) = , aT (T(t)) Here, T0 is the base temperature, and aT is the temperature shift factor. Eij(T,t) is given by Bij(T,t) = Cijkl(T,t)xDkl(T). It is often desirable to use the inverse form of the constitutive relation (1), w V kl ( W ) t dW + \ ij (T,t)'T , H ij (t) = ³ 0 Aijkl (T,t - W ) wW (2) where Rij(T,t) is the tensor product of the creep compliance Aijkl (T,t) and the thermal creep coefficient I kl . Since the elevated temperature change, 'T, is constant above some reference value in time, the relaxation moduli and creep compliance are evaluated at that reference temperature regardless of whether or not the material is thermorheologically simple by employing the temperature shift-factor. The Laplace transform of a function f(t) is defined as f

f = f (s) = ³ 0 e-st f(t)dt ,

(3)

where s is the Laplace transform variable. Applying (3) with the convolution rule to (1) and (2) reduces the integral constitutive equations to the following algebraic relations: ~ kl ~ 'T (4) , V ij = C ij H~ kl - E ij s and the inverse form 'T ~ kl (5) , H~ ij = A ij V ij + \~ ij s

210

~ ~ ~ ~ ~ respectively, where C ijkl = s C ijkl , A ijkl = s Aijkl , D~ ij = s D ij ,I ij = s I ij , E kl = C ~ kl ~ ~ kl ~ kl -1 and \~ kl = A ij x I ij . Furthermore, it can be shown that A ij = C ij .

kl ij

x D~

ij

,

> @ > @

Consider a filament-wound axisymmetric thick composite cylinder consisting of N layers with the axial coordinate z, the radial coordinate r, and the circumferential coordinate ĭġŢŴġ shown in Figure 1. The composite cylinder has the inner radius a, the outer radius b, and the length L. Accordingly, there is a corresponding thermoelastic problem with the transformed displacement components u, v, and w in the axial direction, the circumferential direction, and the radial direction, respectively, in each layer. The axisymmetric character of the thick composite cylinder along with the assumption of the state of generalized plane strain leads to a simplified displacement field, which reflects the circumferential independence and only radial dependence of w ,

u(r,T , z) =u(r, z), v(r,T , z) =v(r, z), and w(r,T , z) =w(r). (6) Since each layer of the thick laminated cylinder is cylindrically monoclinic in reference to the global coordinates, there is no coupling between transverse shears and other deformations. Accordingly, the vanishing shear traction boundary conditions and interface continuity conditions generate zero out-of-plane shear traction and shear strains for each layer. Moreover, owing to the absence of torsional deformation, the transformed displacement components u and v become 0

u = H z and v = 0, (7) 0 where the constant quantity H has the physical interpretation of transformed axial strain of a 0 layer. In fact, H , according to the present formulation, also represents the transformed axial 0 strain of the entire composite cylinder. The calculation of H requires the knowledge of end boundary conditions and will be given later. Likewise, solving for w(r) requires the information of transformed strain components, the constitutive equations, as well as the equilibrium equations. The previously transformed displacement field gives the transformed strain components in cylindrical coordinates

H rr =

du w(r) d w(r) 0 = H , and , H TT = , H zz = dz r dr H Tr = H zr = H z = 0 .

The unabridged form of the constitutive equation (4) for each layer in cylindrical coordinates with the radial coordinate r normal to the plane of symmetry is expressed as

(8)

211

~ ­ V zz ½ ª C 11 ° ° «~ «C 12 ° V TT ° ~ «C °° V rr °° 13 « = ¾ ® « 0 ° V Tr ° « 0 ° V zr ° «~ ° ° °¯ V zT °¿ «¬C 16

~ C 12 ~ C 22 ~ C 23 0 0 ~ C 26

~ C 13 0 ~ C 23 0 ~ C 33 0 ~ 0 C 44 ~ 0 C 45 ~ C 36 0

0 0 0 ~ C 45 ~ C 55 0

~ C 16 º ~ » C 26 » ~ C 36 » » 0 » 0 » ~ » C 66 »¼

­ ° ° °° ® ° ° ° °¯

H zz H TT H rr H Tr H zr H zT

~ ½ ­ %zz ½ ° °~ ° ° ° %TT ° ~ °° '7 °° %rr °° ¾. ¾ ® s 0 ° ° ° ° 0 ° ° °~ ° ° °¯ %zT °¿ °¿

(9)

Furthermore, from the previous discussions, it can be shown that two of the three equilibrium equations are satisfied automatically. The only nontrivial equilibrium equation is the one in the radial direction: w V rr V - V TT = 0. + rr r wr (10) and V TT are Substituting (6), (7), and (8) into (9), the transformed stress components V rr obtained in terms of the transformed radial displacement w . Incorporating the resulting V rr and V TT functions with (10) gives a non-homogeneous Euler differential equation of w for a layer

r ª 'T ~ dw d 2w 2 ~ ) - ( ~ - ~ ) oº , ( %rr - % - O w = ~ r C 13 C 12 H » TT 2 + r « dr dr C 33 ¬ s ¼ 2

where

O Solving (11) for w yields

2

~ C = ~22 . C 33

~p , w = A1r O _ + A2 r - O _ + w

where

(11)

(12) (13)

~ o ~p = ~ w f 1H r + f 3r ~ ~ ~ C 12 - C 13 f1 = ~ ~ C 33 - C 22 ~ (14) S ~ f3 = ~ ~ C 33 - C 22 'T ~ ~ ~ S = > %rr - %TT @ s and A1 and A2 are coefficients to be determined from boundary and continuity conditions. Finally, it is understood that the initial condition of the original thermoviscoelastic problem is displacement-free state of rest. The boundary condition is of free traction and, hence, of free transformed traction on both inner and outer circular surfaces:

212

V rr = V Tr = V zr = 0 at r = a, b .

(15)

On both end surfaces, stress resultants are zero: ro N (16) ¦ ³ V zz rdr = V zr = V zT = 0 at z = 0, L , k 1 ri The ri and ro are inner and outer radii, respectively, of the kth layer. The continuity conditions at each interface between two adjacent layers require continuous radial traction and continuous radial displacement at any instant as shown in Figure 2. Thus, when written in the transformed form, they become (k) (k+1) V rr,o - V rr,i = 0

(17)

and

w,(k) = w,(k+1) , (18) o i where k = 1, , N - 1; and subscripts i and o denote inner and outer surfaces, respectively.

Figure 2: Radial displacement and stress continuity in the laminated cylinders Accordingly, the formulation accounts for ply-by-ply variations of material properties and temperature change. The matrix form numerical solution procedure with parallel computing techniques resolved the complexity and time-consuming calculation procedures in Laplace transform of a multi-layered composite cylinder [10]. 3 RELAXATION OF THERMAL STRESSES The time-dependent thermal viscoelastic behavior of a 100-layer AS-4/3502 (graphite/ epoxy) composite cylinder subjected to a temperature increase 'T = 150q C is examined. Accordingly, initial residual stress built up in the cylinder due to the 'T. The composite cylinder has an inner radius a = 3.5 in, an outer radius b = 4.1 in, and a thickness of each layer h = 6.0 u 10-3 in. Stacking sequence is given as [0/30/60/90]25 from inside out with

213

the 0q direction coinciding with the axis of the cylinder. The creep property of an AS-4/ 3502 graphite/ epoxy composite with a fiber volume fraction of 0.67 was measured at different temperatures by Kim and Hartness[11]. The study shows increase of compliance with time due to creep behaviors of material was found at elevated temperatures. A least-squares curve fitting was used to express the transverse and shear creep from the original AS-4/3502 data in power law forms as follows:

>

@

(19)

@

(20)

0.1954 + 1 S 022 S 22 (t) = 1.7051  t

and

>

0.2771 + 1 S 066 , S 66 (t) = 11.3076  t

where S 022 = 7.5328 u 10-7 / psi S 066 = 1.3834 u 10-6 / psi .

The compliance in the fiber direction, S 11 = 5.9 u 10 -8 / psi , and Poisson's ratios, Q 12 = Q 13 = 0.3,Q 23 = 0.36 , are assumed to be time-independent. Thermal expansion coefficients of the composite in three principal directions are D 11 = - 0.5 u 10-6 / qC and D 22 = D 33 = 40.0 u 10-6 / qC , where the negative value indicates shrinkage with temperature increase. Figures 3 and 4 show radial displacement and radial stress profiles across the thickness of the cylinder at three instants, instantaneous (initial stress), two years, and infinite time. The radial traction and displacement satisfied the continuity conditions at every interface of layers at all instants. The radial displacement, w(t) , will reach a steady state over a long period time (infinite time) because of the creep behavior of composites. In fact, the radial displacement of most layers approaches to a constant value, except at the innermost and outermost portions of the cylinder. The free traction boundary at the surface of cylinders causes the gradients in the radial displacements. A similar phenomenon is also observed in the radial stress profile, which approaches to a constant over a long period of time. This long-term creep which approaches to a constant over a long period of time. This long-term creep characteristic reflects the power law form, equation (19) and (20), of the creep compliance. The "saw" shaped radial stress distribution is the result of variation of fiber orientations through the thickness of cylinder. The radial stress is continuous but the stress gradient is not. Accordingly, the stress profile illustrates the “saw” shape.

214

Radial Displacement (inch)

5.0E-03

4.0E-03

3.0E-03

2.0E-03

Initial

1.0E-03

2 Infinite 0.0E+00 3.5

3.6

3.7

3.8

3.9

4

4.1

Radius (inch)

Figure 3: Radial displacement profiles in a cylinder subjected to thermal loads.

0.0E+00

Radial Stress (psi)

-5.0E+01

-1.0E+02

-1.5E+02

-2.0E+02 Initial

-2.5E+02

2 Infinite

-3.0E+02 3.5

3.6

3.7

3.8

3.9

4

4.1

Radius (inch)

Figure 4: Radial stress profiles in a cylinder subjected to thermal loads. The hoop stress, V TT (t) through the thickness of the cylinder is illustrated at three instants in Figures 5. There exist two distinct values (discontinuity) of V TT (t) across each interface of two adjacent layers due to the various fiber orientations through the thickness. The hoop stress profile also shows a trend of relaxation over a period of time. The hoop stresses in 60q and 90q layers show a fairly steep gradient across the cylinder thickness initially. However, the gradient gradually disappears as time approaches infinity.

215 3.0E+04 Initial

2

Infinite

Hoop Stress (psi)

2.0E+04

1.0E+04

0.0E+00

-1.0E+04

-2.0E+04 3.5

3.6

3.7

3.8

3.9

4

4.1

Radius (inch)

Figure 5: Hoop stress profiles in a cylinder subjected to thermal loads. 4 RELAXATION OF MECHANICAL STRESSES In the following study, creep and stress relaxation of a composite cylinder subjected to internal pressure is investigated. The calculation is performed using a 0.6 inch thick composite cylinder with a lay-up construction of [0/30/60/90]25. Calculations have been performed using similar basis graphite/epoxy composite materials. The material properties are same as described in the previous section. A pressure load of 1,000 psi was applied at the inner radius of the cylinder. Some selected displacement and stress profiles from the model prediction are illustrated and discussed in the following sections. Figure 6 shows the radial displacement through the thickness of cylinder at three instants, instantaneous (initial stress), two years, and infinite time. The radial displacement profile clearly illustrates the creep behavior of cylinder. The inner radius of the cylinder increases over a period of time because of the application of inner pressure. The outer radius actually shrinks down because of creep characteristics. The radial strain, which is equal to the gradient of radial displacement increases over a period of time.

216

Radial Displacement (inch)

8.0E-03 Initial 2

6.0E-03

Infinite

4.0E-03

2.0E-03

0.0E+00 3.5

3.6

3.7

3.8

3.9

4

4.1

Radius (inch)

Figure 6: Radial displacement profiles in a cylinder subjected to internal pressure.

Figure 7 illustrate the relaxation of radial stresses through the thickness at three instants. The radial stress at the inner radius is 1000 psi, equivalent to the pressure applied. The radial stress is zero at the outer surface of the cylinder since it is traction free. The curve is constructed by connecting the stress value at all the interfaces of layers. The “saw shape” of curve is due to the variation of fiber orientations through the thickness. Significant relaxation occurs over a period of time as shown in the radial stress profiles. 0

Radial Stress (psi)

-200

-400

-600 Initial 2

-800

Infinite -1000 3.5

3.6

3.7

3.8

3.9

4

4.1

Radius (inch)

Figure 7: Radial stress profiles in a cylinder subjected to internal pressure. The hoop stress profiles at three instants are illustrated in Figure 8. The stress is not continuous from layer to layer due to the change of fiber orientation. The gradient of the stress profile is mainly due to curvature of cylinder. The 90q layers have higher stresses because they are stiffer in the circumference direction. Accordingly, they carry more loads. The viscoelastic effect is quite interesting as observed in the hoop stress profiles changed over a period of time. The hoop stress at the inner radius increases while it decreases at the outer radius. The integration of hoop stress through the thickness should be balance with the

217

inner pressure applied if a free body is taken from the cylinder. Since the stress gradient increases over a period of time, the hoop stress will also increase at the inner radius of cylinder. 5.0E+04 Initial 2 Yrs

Hoop Stress (psi)

4.0E+04

Infinite 3.0E+04

2.0E+04

1.0E+04

0.0E+00 3.5

3.6

3.7

3.8

3.9

4

4.1

Radius (inch)

Figure 8: Hoop stress profiles in a cylinder subjected to internal pressure. 5 CONCLUSIONS An analysis has been developed for viscoelastic behavior of laminated composite cylinders with ply-by-ply variation of anisotropic viscoelastic properties, which cannot be studied using an isotropic model. Stress relaxation and creep are properly determined in a cylinder subjected to a thermal and mechanical loads. The anisotropic viscoelastic behavior of the composite causes interesting characteristics in cylinders, which are critical for the durability of the structure. Creep and stress relaxation could exist in the fiber direction even though the fiber dominant properties are elastic. This is mainly due to the contribution of the Poisson's effects of transverse and shear properties. Viscoelastic characteristics are critical to the service life cycle of applications such as pressure vessels and composite rotors designed with built-in pre-stress to achieve desired mechanical performance. REFERENCES 1. Ferry, J. D., Viscoelastic Properties of Polymers, John Wiley & Sons, Inc., New York, NY, 1961. 2. Muki, R., and E. Sternberg. "On Transient Thermal Stresses in Viscoelastic Materials With Temperature Dependent Properties." J. of Applied Mechanics, pp. 193-207, 1961. 3. Schapery, R. A. "Application of Thermodynamics to Thermomechanical, Fracture, and Birefringent Phenomena in Viscoelastic Media." J. of Applied Physics, vol. 35, no. 5, pp. 1451-1465, 1964. 4. Williams, M. L. "Structural Analysis of Viscoelastic Materials." AIAA Journal, vol. 2,

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no. 5, pp. 785-808, 1964. 5. Christensen, R. M. Theory of Viscoelasticity, An Introduction, 2nd edition, New York: Academic Press, Inc., 1982. 6. Hashin, Z. "Viscoelastic Behavior of Heterogeneous Media." J. of Applied Mechanics, vol. 32, pp. 630-636, 1965. 7. Schapery, R. A. "Stress Analysis of Viscoelastic Composite Materials." J. of Composite Materials, vol. 1, pp. 228-267, 1967. 8. Rogers, T. G., and E. H. Lee. "The Cylinder Problem in Viscoelastic Stress Analysis." Quarterly of Applied Mathematics, vol.22, pp. 117-131, 1964. 9. Tzeng, J. T., and L. S. Chien. "A Thermal/Mechanical Model of Axially Loaded ThickWalled Composite Cylinders." J. of Composites Engineering, vol. 4, no. 2, pp. 219-232, 1994. 10.Chien, L. S., and J. T. Tzeng. "A Thermal Viscoelastic Analysis for Thick-walled Composite Cylinders." J. of Composite Materials, vol. 29, no. 4, pp. 525-548, 1995. 11.Kim, R. Y., and J. T. Hartness. "Time-dependent Response of AS-4/PEEK Composite." Proc. of the 19th Inter. SAMPE Conference, pp. 468-475, October 1987.

High Temperature, Non-contact, Electro-magnetic Mechanical Apparatus for Creep Testing

Sindhura Gangireddy, Doctoral Student, University of Michigan, Ann Arbor, MI John W. Halloran, Professor, University of Michigan, Ann Arbor, MI Zachary N. Wing, Director of Science & Technology, Advanced Ceramics Manufacturing, Tucson, AZ

Abstract We describe the design and implementation of a second generation Electromagnetic Mechanical Apparatus (EMMA-2) capable of Ultra High Temperature (UHT) creep testing. EMMA-2 uses a variable magnetic field to apply stress to ribbon specimens that are self-heated with DC current. EMMA-2 is capable of continuous non-contact creep testing and operates in a controlled atmosphere. Theoretical models are presented for mechanical stress and temperature, as related to specimen geometry, electrical current, magnetic flux density, and combined with creep models behavior to predict the accessible range of temperature and stress. This unique test apparatus allows for characterization of Ultra High Temperature Ceramics (UHTCs) without physically contacting test samples. Initial testing has been performed on a set of ZrB2-SiC (30 Vol.%) ceramics at temperatures ranging from 1700-2100°C. Introduction Ultra High Temperature Ceramics (UHTC’s) such as zirconium and hafnium diboride based composites are of particular interest for hypersonic leading edges due to their extreme refractoriness. Creep characterization is necessary for developing design and predictive lifetime estimates for hypersonic vehicles and rocket nozzles. Characterization is most commonly performed by axially loading samples in tension, compression, or bending and recording the evolution of strain as a function of time, temperature, and stress. The application of stress and the measurement of strain necessitates physical contact with the specimen. Non-contact optical methods may be used to measure strain, however, they require the use of “flags” attached to the sample. Secondary materials that contact the specimen must be at least as refractory as the test material, unreactive with the test material, and stable under test conditions. Therefore, the requirement of contact creates temperature limitations for materials tested at temperatures above ~1500ºC. For UHTCs, temperature regimes of interest may exceed 2000ºC. Within the last few years, the University of Michigan developed a rapid, table top apparatus for characterizing the oxidation behavior of UHTC’s [1]. This system exploited the electrical conductivity of ZrB2 based ceramics and small gauge test sections to resistively heat and oxidize specimens at ultra high temperature. Samples could be heated to temperature quickly (~seconds) while consuming ~100W of power. The oxidation system has been adapted to provide table top characterization of creep.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_30, © The Society for Experimental Mechanics, Inc. 2011

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Discussion Based on the oxidation ribbon apparatus and the need to conduct creep testing beyond 1500°C, an Electro-Magnetic Mechanical Apparatus and baseline theory were conceived. A prototype test system was developed (EMMA-1) to provide experimental verification. The theoretical basis, design, and test results were presented in detail in our prior work [2]. Briefly, the EMMA based testing approach exploits the resistively heated sample in the oxidation apparatus and adds a magnetic field. The high current flow heating the sample interacts with an applied magnetic flux. This interaction generates a non-contact Lorentz force which can be modeled as a uniformly loaded, rectangular beam. The 1st generation system (EMMA-1) exploited permanent magnets, tested samples in open air, and sample deflection was measured post test. Recently, our research group has refined the 1st generation apparatus and produced a more robust test system (EMMA-2) as shown in Fig. a. EMMA-2 incorporates an electro-magnet (which can readily tailor the magnetic flux), an environmental chamber, and a laser deflectometer to record beam strain. SideView Pyrometer Laser Deflectometer (detector)

Laser Deflectometer (source)

MagneticFlux Assembly

CurrentLeads



Fig. a (Left) EMMA-2 Schematic (Right) EMMA-2 Prototype.

With the revised EMMA-2 now online, serial creep characterization is being performed on a commercially produced ZrB2-SiC (30 Vol.% SiC) ceramic composite. Tests have been performed in air and in a nitrogen rich - low oxygen atmosphere (<0.25% O2), stress levels of 20-50 MPa, and temperatures up to 2100ºC. Initial analysis showed the creep activation energy for the UHTC ceramic to be 138.8 kJ/mol when tested in air and 148.3 kJ/mol when tested in the oxygen starved atmosphere. These results are comparable to values reported in the literature of 511 kJ/mol for ZrB2-SiC (50 Vol.%) [3].

References 1. Sigrun N. Karlsdottir and John W. Halloran, Rapid Characterization of Ultra-High Temperature Ceramics, J. Am. Cer. Soc. 90 [10] 323-3238 (2007). 2. S. Gangireddy, J.W. Halloran, Z.N. Wing, Non-contact mechanical property measurements at ultrahigh temperatures, J. Eur. Cer. Soc. 30 (2010) 2183–2189. 3. I. G. Talmy, J. A. Zaykoski, C.A. Martin, Flexural Creep Deformation of ZrB2/SiC Ceramics in Oxidizing Atmosphere, J. Am. Ceram. Soc.,Volume 30, Issue 4, March 2010, Pages 1021-1026.

Experiments and Predictions of Large Deformation and Failure in Thermomechanical Loading Environments

Bonnie R. Antoun1, J. Franklin Dempsey2, Gerald W. Wellman2 1

Sandia National Laboratories, Livermore, CA 94551-0969, USA 2 Sandia National Laboratories, Albuquerque, NM 87185, USA

The response of 304L stainless steel to combined mechanical and thermal loadings is studied to enable the development of validated computational simulation methods for predicting deformation and failure in coupled thermomechanical environments. Experimental coupling was accomplished on axisymmetric tubular specimens that were mechanically loaded by internal pressurization and thermally loaded asymmetrically by side radiant heating. Mechanical characterization experiments of the 304L stainless steel tube material was completed for development of a thermal elastic-plastic material constitutive model used in the finite element simulations of the validation experiments. The design and implementation of the experimental methodology and results of preliminary experiments were presented at 2010 SEM Annual Conference [1, 2]. The experiments were designed to have well-defined, controlled thermal and mechanical boundary conditions that could be accurately represented in the simulations. Experimental parameters studied include geometrical features, applied temperature rates, pressurization rates, maximum temperature and pressure, time at temperature and phasing of the thermal and mechanical loading. Specimens were made from 89 mm (3.5 in.) diameter 304L stainless steel tube with a wall thickness of 6.35 mm (0.25 in.), an overall length of 355 mm (14 in.) and a reduced wall thickness in sections of variable length. Specimens were instrumented with twenty-three Type K intrinsic thermocouples at locations around the circumference, mainly concentrated in the high heat region directly in front of the heat source, as shown in Figure 1. Full-field thermal measurements, applied internal pressure and specimen mechanical response were continuously monitored during the experiments and were used as direct input into the finite element simulations of each experiment. Specimen deformation and dimensional changes were measured in-situ with optical photography methods. A photograph of an instrumented specimen during an experiment is shown in Figure 2. The heat shroud is shown on the left surrounded by an insulating ceramic board, the shroud was heated by a double bank of quartz lamps. The specimen was internally pressurized using nitrogen gas. Both temperature and pressure were computer controlled to any ramp profile desired. Overall axial force and displacement Fig. 1. Thermocouple placement on 304L were controlled using MTS Flextest controller of the 220 stainless steel tube specimens. Kip test frame. The results from specimen #10 are shown in Figures 3 and 4. The center temperature of the specimen was ramped at 30 degrees/minute from room temperature to 650C. Once this temperature was reached it was held and the specimen was pressurized at a rate of 72 psi/minute until failure occurred at 645 psi. At failure, the specimen split vertically first, followed by horizontal splitting. Results from specimen #8 are shown in Figures 5 and 6. The temperature and pressure were ramped simultaneously, at 30 degrees/minute and 36 psi/minute, until failure occurred at 650C and 640 psi. In a similar experiment, specimen #9 was ramped at the same temperature rate, but at twice the pressure ramp rate. In that case, specimen failure occurred at 325C and 730 psi. Several

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_31, © The Society for Experimental Mechanics, Inc. 2011

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222 variations of temperature and pressure ramp rate were imposed, as well as heating to various hold temperatures prior to pressurizing. A photograph of several of the failed specimens is shown in Figure 7. In each case, the pressure, displacement, axial load and full-field temperature data was used as direct input into the finite element simulation of the experiment. Figure 8 shows an example of the simulation results for specimen #2 which had similar applied temperature and pressure profiles to specimen #10. Using the full-field thermomechanical loading data as input along with a tearing parameter based determination of failure [3], prediction of failure times have matched the experimental results very closely.

Fig. 2. Thermomechanical test specimen during experiment

Fig. 3. Temperature measurements from specimen #10

Fig. 4. Applied pressure and axial load response of specimen #10

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Fig. 5. Temperature measurements from specimen #8

Fig. 6. Applied pressure and axial load response of specimen #8

Fig. 7. Specimens failed under various thermomechanical loading conditions

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Fig. 8. Thermal-mechanical simulation using measured thermal field mapped onto finite element model of specimen

ACKNOWLEDGEMENTS Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

REFERENCES 1. Antoun, B. R., Dempsey, J. F., Wellman, G. W., Scherzinger, W. M., and Connelly, K., “Design and Implementation of Coupled Thermomechanical Failure Experiments,” Proceedings of the 2010 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Indianapolis, IN, June 7-10, 2010. 2. Antoun, B. R., Dempsey, J. F., Wellman, G. W., Scherzinger, W. M., Connelly, K., and Romero, V. J., “Coupled Thermal-Mechanical Experiments for Validation of Pressurized, High Temperature Systems,” Proceedings of the 2010 SEM Annual Conference and Exposition on Experimental and Applied Mechanics, Indianapolis, IN, June 7-10, 2010. 3. Wellman, G.W. and Salzbrenner, R., “Quasistatic Modeling and Testing of Exclusion Region Barrier Mock-Ups,” SAND92-0024, Sandia National Laboratories, Albuquerque, NM and Livermore, CA, March 1992.

Compliance Plot Analysis of Nonlinear Response of PMMA During Nanoindentation

R. J. Arenz, S.J., Jesuit Community, Loyola Marymount University, Los Angeles, CA 90045 ABSTRACT Application of nanoindentation testing to polymeric materials has produced linear viscoelastic properties with the hope of extending results to the nonlinear behavior range. Analysis has indicated that levels of stress well into the nonlinear range are present directly under the indenter. This paper examines how this high stress behavior can be characterized for PMMA material recently tested by nanoindentation. Although stress levels are low compared to many structural materials, a characterization by a compliance plot is made by analogy with the treatment used for metals and other elastic-plastic materials. It illustrates the nature of the stress build-up process during nanonidentation of this polymeric material.

Introduction. Nanoindentation testing has proven useful to determine material properties on the nanometer scale and has recently been applied to polymeric materials which exhibit time dependent response. Classical elasticity solutions of indentation have provided values of linear mechanical properties for elastic materials, but the viscoelastic nature of polymer response imposes added complexity to the analysis. Nanoindentation tests [1] of two polymers have utilized a Prony series to represent viscoelastic relaxation and provided evaluations of bulk and shear relaxation moduli in the relatively short time regions out to about 100 seconds. Recent analysis [2] has been made to consider the application to one of these materials (poly(vinyl acetate), PVAc) of the method of compliance plots sometimes utilized to characterize the indentation results for metals and other elastic-plastic materials. In this paper we consider such a compliance plot of “indentation stress” vs. “indentation strain” to the test results given by Huang and Lu [1] for the second polymer material, poly(methyl methacrylate), PMMA. Indentation Data. The results of tests using a spherical indenter by Huang and Lu [1] have been used to analyze the behavior of PMMA material. (These authors also employed a Berkovich three-sided indenter to obtain the needed independent data to evaluate the two separate bulk and shear relaxation moduli; however, the Berkovich indenter data are not suitable to calculate compliance plot data and so will not be used here.) The test temperature was 23qC, far below the glass transition temperature of approximately 105qC for PMMA. The indenter radius was R = 10 Pm and the force P was loaded at a constant (ramp) rate of 33 PN/s. The displacement h of the penetrator reached a maximum penetration value of 600 nm but the portion analyzed by Huang and Lu for modulus determination was 300 nm achieved in about 75 seconds. Thus, the test time was relatively short compared to the total relaxation period for the material to reach equilibrium or rubbery behavior at the test temperature. We present here only the experimental data graph of P vs. h in Fig. 1 (as presented in Fig. 8(b) of Ref. 1) since it was the curve approximated by a mathematical fit used in deriving relaxation modulus values for the time period of the test. Their results for bulk relaxation modulus K and shear relaxation modulus G are shown in Fig. 2 where the initial value shown corresponds roughly to a time of 10 seconds. Thus, the curves do not give the zero-time values of modulus as the initial process of indentation involves some friction that can distort the measurement, but the values deduced may be close to glassy behavior. (The difficulty of precisely determining the early-time indentation force and displacement poses problems for the

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_32, © The Society for Experimental Mechanics, Inc. 2011

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226 analysis in the next section of stress and strain near the start of the load-up process.) The experimental values are in approximate agreement with other data obtained by conventional relaxation testing. The results during indentation correspond to the time domain that is in the E-transition range for both bulk and shear response of PMMA at this temperature. (The characteristic of multiple transition processes for PMMA has caused the analysis of its relaxation to be more complicated than for many other polymers [3].) Accordingly, the test material has not yet entered the D-transition period of large decreases in modulus which for shear usually constitute about three logarithmic decades of decrease as it approaches rubbery behavior. As indicated in Fig. 2, the decrease over 100 seconds is 6% for both K and G. As such, the viscoelastic effect is quite small during this period, which Huang and Lu accommodated by a minimal three-term Prony series of small-coefficient exponentials in time. It is realized that the ramp force input means levels of the loading will occur at successive times and so the viscoelastic effect during the test will be an integrated response over time. But this condition of slowly-varying mechanical properties makes it reasonable to use an approximate approach for our further analysis of the indentation results by treating the time-wise changes as a succession of elastic states of varying moduli. Having the bulk and shear moduli obtained from the nanoindentation tests gives the possibility of examining the process further by calculating the indentation stress and indentation strain to form a compliance plot. Compliance Analysis. The nonlinear stress states undoubtedly present directly under a spherical indenter has led to analysis techniques to study this behavior in metals and other elastic materials. In parallel with this approach, we examine the use of the Hertz elastic solution for the load-deformation relations. Huang and Lu utilized the following Hertzian relation to relate force and displacement for a spherical indenter: P = (8/3)G(1-Ȟ -1(R)1/2h3/2

(1)

where P is the applied load, R is the radius of the indenter (10Pm), G is the shear relaxation modulus, Q is the Poisson ratio, and h is the displacement of the leading point of the indenter measured from the original surface of the test specimen. Coupled with a Prony series representation of viscoelastic relaxation giving the time-dependent behavior, they produced the calibration curves shown in Fig. 2. Since the rate of relaxation is slow, it is possible to use elastic material parameter relations to approximate the Poisson ratio as Q = 0.4 during times appropriate to the indenter test period. (Even if it is only of the order of 0.33 as indicated by other published data [3] when shifted to this time region, the compliance results we present would not be changed by an appreciable amount.) A corresponding Young’s modulus for early in the indentation period is E = 2.97 GPa; a gradual reduction in its value occurs (as with the decrease of K and G to approximately 2.88 GPa at 75 seconds. (These values correspond roughly to the calibration of PMMA obtained in other (conventional) tests [3]). Relation of penetration to the size of contact area for the spherical indenter as given by Fischer-Cripps [4] is uz = (3/2)(pm)(1 – Ȟ2) (S/4Ea) (2a2 – r2)

(r ”D 

(2)

where uz is the displacement downward of the surface of the specimen within the contact circle, r is the radial distance from the center of the indenter, a is the radius of the circle of contact between the sphere and the material, and pm = P/Sa2, the mean pressure applied by the indenter. At r = 0, uz corresponds to the penetration h. With the data obtained as a result of the Huang-Lu method for Fig. 2, it is possible to obtain plots of the time dependence of the significant factors P, h, a/R, and pm to use in the compliance analysis. These are shown in Fig. 3. Noticeable in these graphs is the tendency of the size of contact a/R to level off with increasing force application. However, the mean pressure pm continues to increase throughout the test period. This constitutes a notable difference between PMMA and metal products such as mild steel [4], where growth of the yield region to full plasticity within the specimen allows the mean pressure to approach a constant value; that is interpreted as the hardness of the material.

227 Figure 4 show pm vs. a/R, a graph called the compliance plot of “indentation stress” vs. “indentation strain” in usual indentation testing. The initial slope is interpreted [5] as representing elastic behavior; thus, for a rigid indenter the Hertz relation pm = (4/3S)[E/(1 -Q2)](a/R) provides an estimate of the Young’s modulus E of the material. In the case of PMMA the Poisson ratio Q = 0.4 is taken as noted above from the results of Huang and Lu [1]. For a brief time in the load-up of PMMA, the slope of Fig. 4 corresponds to the nearglassy Young’s modulus; the approximate value here is 3.2 GPa, which is close to the value the independent analysis method of Huang and Lu referred to above would provide for E at the start up of the indentation process. (The correspondence may be a bit fortuitous because of the difficulty, noted above due to friction effects, of the test method to produce precise values of force vs. displacement data in the first few seconds of the load-up process.) Beyond a test time of 10 seconds the value of pm continues to increase and seemingly at an increasing rate for a/R ! 0.12. This behavior of PMMA appears quite different from the usual compliance curves of metals and other “hard” materials. These materials display marked plastic behavior under indentation; Fischer-Cripps [4,5] has investigated two of these, a mild steel and a glass-ceramic. The experimental indentation stress-strain curves for these are shown in Fig. 5. They show that for these materials the process provides not only mechanical properties but illustrates the differences in behavior of the materials. Mild steel displays an approach to constant mean pressure with increasing load, indicating a fully developed plastic zone. The glass-ceramic shows a more continuing increase of mean pressure with load, which is also displayed in Fig. 4. For PMMA, however, no eventual downturn in slope is evident, suggesting an ongoing growth throughout the specimen of a nonlinear (yield-like) behavior (rather than definite plasticity) for PMMA during the indentation process. Indeed, Lu and coworkers [1,6] state that they limit the size of load on their specimens so that no permanent deformation remains after removal of the load and suggest that this makes it reasonable to use Hertz analysis as the basis for determining relaxation properties. In another test situation, it has been found [2] that some polymers do not exhibit specific plastic deformation during rapid load-up, but rather nonlinear response both of an elastic and then viscoelastic character. This may be the case for PMMA at 23qC responding in the E-transition region in the nanoindentation tests (which occur at a time prior to D-transition response). There may be greater nonlinearity rather than plastic deformation occurring which would result in higher indentation stress even as the area of contact increased; correspondingly, no change of slope appears in the early portion of the PMMA compliance plot to indicate a possible yield point as was shown for steel (Fig. 5). The possibility of obtaining information on yield stress for PMMA by study of the compliance plot seems to be obscured in the very early response period where accuracy of determining force and deformation values is limited. Indeed, care is needed in establishing these quantities to prevent wild scatter in the earlytime compliance plot. This appears to be an effect noticed in another viscoelastic property determination by other methods, such as converting uniaxial modulus and Poisson ratio relaxation data into bulk modulus values [3], where accuracy requirements limited the applicability of the process. Conclusion. Some comparisons have been shown between nanoindentation of PMMA and indentation testing of other engineering materials. PMMA reacts less in accordance with steel but has some similarity to a glass-ceramic substance (even though it is a fundamentally different material from a polymer). The standard indentation stress vs. indentation strain curve can provide an estimate of the initial modulus response of the material, although it is realized that early test data can suffer from an initial friction effect making accuracy requirements difficult to achieve. The seemingly unending rise in indentation stress as load is increased (Fig. 4) suggests a preponderance of nonlinear viscoelastic behavior in the body of the test specimen rather than concentrated plastic response directly under the indenter. There may be analysis techniques possible that will make nanoindentation testing helpful in determining nonlinear behavior of polymers, although the highly nonuniform stress state under the indenter greatly complicates the necessary analysis.

228 References [1] Huang, G. and Lu, H., “Measurements of Two Independent Viscoelastic Functions by Nanoindentation,” Experimental Mechanics, 47, 87-98, 2007. [2] Arenz, R.J., “Stress states and nonlinear mechanical behavior associated with nanoindentation testing of polymers,” Mechanics of Time-Dependent Materials, 13, 317-332, 2009. [3] Lu, H., Zhang, X. and Knauss, W.G., “Uniaxial, Shear, and Poisson Relaxation and Their Conversion to Bulk Relaxation: Studies on Poly(Methyl Methacrylate),” Polymer Engineering and Science, 37(6), 1053-1064, 1997. [4] Fischer-Cripps, A.C., Introduction to Contact Mechanics (Second Edition), Springer Science+Business Media, New York, NY, 2007. [5] Fischer-Cripps, A.C., “Elastic-plastic behavior in materials loaded with a spherical indenter,” Journal of Materials Science, 32, 727-736, 1997. [6] Lu, H., Wang, B., Ma, J., Huang, G. and Viswanathan, H., “Measurement of Creep Compliance of Solid Polymers by Nanoindentation,” Mechanics of Time-Dependent Materials, 7, 189-207, 2003.

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Fig 1 Nanoindentation load-displacement curve for PMMA at 23qC. (Obtained from Huang and Lu [1])

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Spherical Indenter P = load h = depth pm = mean pressure a = contact radius R = indenter radius

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Fig 3 Time dependence of load P, depth h, ratio a/R, and mean pressure pm during nanoindentation test at 23qC of PMMA by spherical indenter (R = 10 Pm). (Derived from data of Huang and Lu [1])

230 0.25 PPMA Compliance Plot pm vs. a/R

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Fig 4 Compliance plot of indentation stress (pm) vs. indentation strain (a/R) for PMMA under nanoindentation test at 23qC (R = 10 Pm)

2.5 Compliance Curves pm vs. a/R

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a = contact radius R = 3.18 mm

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Fig 5 Compliance plot of indentation stress (pm) vs. indentation strain (a/R) for tests under a spherical indenter (R =3.18 mm) of mild steel and glass ceramic material. (Derived from data of Fischer-Cripps [5])

An Incremental formulation for the linear analysis of viscoelastic beams: Relaxation differential approach using generalized variables Claude CHAZAL*, Rostand Mouto PITTI** and Alaa CHATEAUNEUF** * Heterogeneous Material Research Group, Civil Engineering and Durability Team, Limoges University, 19300 Egletons, FRANCE [email protected] ** Clermont Université, Université Blaise Pascal, Laboratoire de Mécanique et Ingénieries, EA 3867, BP 206, 63000 Clermont Ferrand, FRANCE

abstract This paper is concerned with the development of a new incremental formulation in the time domain for linear, non-aging viscoelastic materials undergoing mechanical deformation. The formulation is derived from linear differential equations based on a discrete spectrum representation for the relaxation function. The incremental constitutive equations are then obtained by finite difference integration. Thus the difficulty of retaining the stress history in computer solutions is avoided. The influence of the whole past history on the behaviour at any time is given by a pseudo second order tensor. A complete general formulation of linear viscoelastic stress analysis is developed in terms of increments of midsurface strains and curvatures and corresponding stress resultants. The generality allowed by this approach has been established by finding incremental formulation through simple choices of the tensor relaxation components. This approach appears to open a wide horizon (to explore) of new incremental formulations according to particular relaxation components. Remarkable incremental constitutive laws, for which the above technique is applied, are given. This formulation is introduced in a finite element discretization in order to resolve complex boundary viscoelastic problems. Keywords: Incremental formulation; Viscoelasticity; Discrete relaxation function; Generalized variables.

1 Introduction Viscoelastic materials are characterized by possessing infinite memory. Their actual mechanical response is a function of the whole past history of stress and strain. In most cases, the behaviour of any linear viscoelastic material may be represented by a hereditary approach based on the superposition principle of Boltzmann. This implies that stress and strain analysis of viscoelastic phenomena which can be observed in the behaviour of many real materials, presents many difficulties for real problems of complex geometry. The important use of viscoelastic materials in civil engineering structures requires understanding the behaviour of time dependent mechanics fields which can be lead to collapse of such structures. The main problem in computation mechanics is to know the response of viscoelastic materials taking into account its complete past history of stress and strain. An intensive research work is

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_33, © The Society for Experimental Mechanics, Inc. 2011

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investigated by a number of authors. Bozza and Gentili [1] use the theory of linear viscoelasticity to establish constitutive equations using relaxation functions. They seek solutions to the inversion problem of the constitutive equations. Aleksey Drozdov and AlDorfmann [2] derive constitutive equations for the nonlinear viscoelastic behaviour after performing tensile relaxation tests. Most of analytical solutions proposed in the literature can not deal with real and complex problems because these methods require the retaining of the complete past history of stress and strain in the memory of a digital computer. In this context, a number of theories have been proposed in the past in order to formulate incremental constitutive equations for linear viscoelastic behaviour. Among them, Kim and Lee [3] and Theocaris [4] have proposed incremental formulation and constitutive equations in the finite element context (see also Chazal and Pitti [5]). In fracture viscoelastic mechanics, Moutou Pitti et al. [6-7] have applied the incremental formulation in order to evaluate creep crack growth process in viscoelastic media. Krempl [8] and Kujawski et al. [9] have performed experimental study of creep and relaxation in steel at room temperature. Also, Godunov et al. [10] and Duffrène et al. [11] have proposed relaxation viscoelastic models in order to traduce the behaviour of the material. However, the formulation used is based on the spectral decomposition using a generalized Maxwell model. To avoid the use of generalized Maxwell model, we will develop in this paper an alternative incremental formulation. This new formulation is based on discrete relaxation spectrum and the finite difference method using generalized differential equations in the time domain. The incremental stress and strain constitutive equations are not restricted to isotropic materials and can be used to resolve complex boundary viscoelastic problems without retaining the past history of the material in the memory of a digital computer. First, we recall the discrete relaxation spectrum representation and its use in Boltzmann’s superposition principle [12]. The one dimensional linear viscoelastic behaviour is used to account for three dimensional responses. After that, we present the development of the generalized differential equations in terms of one dimensional stress and strain components. Finally, the incremental viscoelastic constitutive equations of the model are established.

2 Relaxation spectrum representation In this work, we will consider only small strains. According to the results obtained by Christensen [13], Mandel [14], Salençon [15] and Chazal and Dubois [16], the components of the relaxation tensor G(t ) can be represented in terms of exponential series: R ­ ρ G (t ) = ®G ∞ + ¦ G ρ e − tμ ρ =1 ¯

½ ¾ H (t ) ¿

(1)

where η ρ , ρ = 1,..., R , are strictly positive scalars and repeated indices do not imply summation convention. G ∞ and G ρ represent the equilibrium and the differed part of the relaxation function respectively and H (t ) is the Heaviside unit step function.

According to Boltzmann’s principle superposition in linear non-ageing viscoelasticity [12], the constitutive equations between the component σ (t ) of the stress and the component of the strain e(t )

233

for non-ageing linear viscoelastic materials can be expressed in the time domain by the hereditary Volterra’s integral equation: t

σ (t ) = ³ G (t − τ ) −∞

∂eγδ (τ ) ∂τ

dτ

(2)

We introduce stresses and strains in generalized variables according to Love’s first-order shell theory. The strain at any point of the beam may be given as

e(t ) = ε (t ) + ζ χ (t ) (3) where ε (t ) and χ (t ) are the middle surface extensional strain and curvature, respectively. If we consider a plane stress state, the non-vanishing resultant of stresses is then defined by

N (t ) =

+h / 2

+h / 2

−h / 2

−h / 2

³ σ (ζ , t ) dζ , M (t ) =

³ ζ σ (ζ , t ) dζ

(4)

N (t ) and M (t ) are the generalized stresses and h is the thickness of the beam assumed to be constant. In order to determine the constitutive equation in terms of generalized stresses and strains, we introduce generalized strains, given by equation (3), into equation (2). One find t ∂ σ N (t ) + σ M (t ) = ³ G (t − τ ) [ε (τ ) + ζ χ (τ )] dτ (5) ∂τ −∞

Note that the total stress σ (t ) is separated into two parts: normal stress σ N (t ) due to extensional strain and bending stress σ M (t ) due to curvature. The constitutive equations in generalized variables can now be obtained from behaviour equation (5). Using equation (4) and integrating equation (5) over the thickness, we find +h / 2 t ∂ N (t ) = ³ σ N (ζ , t ) dζ = h ³ G (t − τ ) ε (τ ) dτ ∂τ −h / 2 −∞ +h / 2

M (t ) =

M ³ ζσ (ζ , t ) dζ =

−h / 2

h3 ∂ G (t − τ ) χ (τ ) dτ ³ 12 − ∞ ∂τ

(6)

t

(7)

3 Formulation of differential equations When we apply the mechanical strain defined by the strain history {e(τ ), τ ∈ ℜ} , the response of the material is then given by the history of stresses {N (t ), M(t ), t ∈ ℜ} defined by the behaviour equations (6,7) in which the relaxation function is given by equation (1).

If the generalized strain {ε (t ), χ (t )} is applied to the material at time t, then the response in stresses can be obtained using the finite relaxation spectrum representation given by equation (1). This leads to ­ N (t ) ½ t ª ∞ R ρ − μ ρ ( t −τ ) º ∂ ® ¾ = ³ «G + ¦ G e » ρ =1 ¯M (t )¿ − ∞¬ ¼ ∂τ

­ h ε (τ ) ½ ° 3 ° ®h ¾ χ ( τ ) °¿ °¯12

(8)

234

Thus the stresses given by the last equation, and written as a function of equilibrium and differed part of the relaxation spectrum, can be rewritten in the following form

R ­ ∞ ½ N ( t ) + N ρ (t ) ° ¦ ° ­ N (t ) ½ ° ° ρ =1 ® ¾=® ¾ R ¯M (t )¿ °M ∞ (t ) + M ρ (t )° ¦ °¯ °¿ ρ =1

(9)

with t

N ∞ (t ) =

³hG

∞

−∞

N ρ (t ) =

t

³hG

−∞

ρ

∂ε (τ ) dτ = h G ∞ ε (t ) ∂τ e −μ

ρ

( t −τ )

∂ε (τ ) dτ ∂τ

(10a)

(10b)

M (t ) =

h 3 ∞ ∂χ (τ ) h3 ∞ G d τ = G χ (t ) ³ 12 ∂τ 12 −∞

(10c)

M ρ (t ) =

h 3 ρ − μ ρ ( t −τ ) ∂χ (τ ) dτ ³ 12 G e ∂τ −∞

(10d)

t

∞

t

It should be noted that N ∞ (t ) and M ∞ (t ) represent the equilibrium part of the mono-dimensional stress of the material while N ρ (t ) and M ρ (t ) represent the differed part of the same mono-dimensional stress. As we mentioned in the above section, a differential approach is used in order to establish the differential equations of the mechanical model. Thus, we need to express the viscoelastic response of the material as a function of stress and strain derivatives. For this reason, let us use equation (9), the rate of the total stress is determined by R ­ ∞ ½ N ( t ) + N ρ (t ) ° ¦ ° ∂ ­ N (t ) ½ ∂ ° ° ρ =1 ® ¾= ® ¾ R ∂t ¯M (t )¿ ∂t °M ∞ (t ) + M ρ (t )° ¦ °¯ °¿ ρ =1

(11)

The rate of the equilibrium part of the one-dimensional stress N ∞ (t ) and M ∞ (t ) is easy to be evaluated. According to equations (10a) and (10c), and after applying a time derivative operator, one find

∂N ∞ (t ) ∂ε (t ) = h G∞ ∂t ∂t

(12a)

∂M ∞ (t ) h 3 ∞ ∂χ (t ) = G ∂t 12 ∂t

(12b)

235

In other words, the equilibrium part of the one-dimensional stress is directly proportional to the total strain at time t . It is completely defined by the history of the applied strain. However, the rate of the differed part of the one-dimensional stresses N ρ (t ) and M ρ (t ) is more complicated to be determined. Using equations (10b) and (10d), and applying a time derivative operator, we can write ρ ρ 1 ∂ ρ ∂ ∂ N (t ) = G ρ e − μ (t −t ) ε (t ) − ³ G ρ μ ρ e − μ (t −τ ) ε (τ ) h ∂t ∂t ∂τ −∞

t

(13a)

ρ ρ 12 ∂ ρ ∂ ∂ M (t ) = G ρ e − μ (t −t ) χ (t ) − ³ G ρ μ ρ e − μ (t −τ ) χ (τ ) 3 ∂t ∂τ h ∂t −∞

t

(13b)

These integral equations give the total rate of the differed part of the mono-dimensional stresses. The main purpose of our development in this section is to establish differential equations between the total rate of the mono-dimensional stresses and the total strain. For this reason, we will transform last equations in differential type. Let us introduce the behaviour equations (10b) and (10d) in the integral equations (13). This leads to linear differential equations with constant coefficients and can be integrated analytically

∂ ρ ∂ N (t ) + μ ρ N ρ (t ) = h G ρ ε (t ) ∂t ∂t

(14a)

∂ ρ h3 ρ ∂ M (t ) + μ ρ M ρ (t ) = G χ (t ) ∂t 12 ∂t

(14b)

The solution of these linear differential equations gives the rate of the one-dimensional stresses N (t ) and M ρ (t ) . ρ

It should be mentioned that the non-ageing linear viscoelastic behaviour is completely defined by the differential equations (12) and (14). We note that this formulation, written in terms of generalized stresses and strains rates, is easily adapted to temporal discretization methods such as finite difference method.

4 Incrementalization of the differential equations Here we will describe the solution process of a step-by-step nature in which loads are applied stepwise at various time intervals. Let us consider the time step Δt n = t n +1 − t n . The subscript n and n + 1 refer to the values at the beginning and end of time step, respectively. This technique is successfully used by Chazal and Dubois in [16] in the case of viscoelastic structures. We assume that the time derivative during each time increment is. A linear approximation is used for strains and is expressed by ­ ε (t ) − ε (t n ) ½ ­ε (τ ) ½ ­ε (t n ) ½ τ − t n H (τ − t n )® n +1 ¾=® ¾ ® ¾+ ¯χ (τ )¿ ¯χ (t n )¿ Δt n ¯ χ (t n +1 ) − χ (t n )¿

(15)

236

This linear approximation leads to very accuracy results in finite element discretization as it is shown by Chazal and Dubois [16]. Thus we don’t need higher approximations for the strain during a finite increment of time load. This leads to a constant rate during each time increment:

∂ ­ ε (t n ) ½ 1 ­ ε (t n +1 ) − ε (t n ) ½ 1 ­ Δε (t n ) ½ ® ¾= ® ¾= ® ¾ ∂t ¯ χ (t n )¿ Δt n ¯ χ (t n +1 ) − χ (t n )¿ Δt n ¯Δχ (t n )¿

(16)

By integrating equation (11) between tn and t n +1 , it can be written in the following form ­ h Δε (t n ) ½ ­ ΔN (t n ) ½ R ­ ΔN ρ (t n ) ½ ° ∞° 3 ¾ ® ¾ = ¦® ¾ + G ®h ρ ¯ΔM (t n )¿ ρ =1 ¯ΔM (t n )¿ °¯12 Δχ (t n )°¿

(17)

In order to determine the generalized stress increments from this equation, we have to determine the generalized pseudo stress increments ΔN ρ (t n ) and ΔM ρ (t n ) during the time step Δt n . First, let us consider the differential equation (14). The analytical solution of this differential equation can be expressed as

[

N ρ (t n +1 ) − N ρ (t n ) = e − μ

[

ρ

M ρ (t n+1 ) − M ρ (t n ) = e − μ

Δt n

ρ

Δt n

]

− 1 N ρ (t n ) + h G ρ

]

− 1 M ρ (t n ) +

[

ρ Δε (t n ) 1 − e − μ Δt n ρ μ Δt n

[

]

ρ h 3 ρ Δχ (t n ) G 1 − e − μ Δt n ρ 12 μ Δt n

(18a)

]

(18b)

The incremental constitutive equations can now be obtained from the constitutive equation (17). Substituting equations (18a) and (18b) into (17), we find ~ 0 º ­ Δε (t n ) ½ ­ N (t n ) ½ ­ ΔN (t n ) ½ ªΠ (t n ) = − ® ¾ « ® ¾ ®~ ¾ Ξ(t n )»¼ ¯Δχ (t n )¿ ¯M (t n )¿ ¯ΔM (t n )¿ ¬ 0

(19)

where Π (t n ) and Ξ (t n ) are scalar tensors which can be interpreted as rigidity tensors in extensional and bending state respectively, they are given by ρ G ρ Δε (t n ) Π (t n ) = G + h¦ 1 - e − μ Δt n ρ ρ =1 μ Δt n

∞

Ξ (t n ) = G ∞ +

R

[

]

h 3 R G ρ Δχ (t n ) − μ ρ Δt n 1 e ¦ 12 ρ =1 μ ρ Δt n

[

(20a)

]

(20b)

~ ~ N (t n ) and M (t n ) are pseudo generalized stresses which represent the influence of the complete past history of extensional and bending generalized stresses. They are given by

237

~ ­ N (t n ) ½ R − μ ρ Δt n ®~ ¾ = ¦ 1− e ¯M (t n )¿ ρ =1

[

]

­ N ρ (t n ) ½ ® ρ ¾ ¯M (t n )¿

(21)

The incremental constitutive law represented by equation (19) can be introduced in a finite element discretization in order to obtain solutions to complex viscoelastic problems. Finally, in order to use the incremental viscoelastic formulation presented in this paper, we need to identify the relaxation components of the relaxation tensor. The experimental identification of viscoelastic properties is treated in details by Andreas and Lackner [17] and Herbert and Andreas [18]. The viscoelastic solution is obtained by the application of the method of functional equations to the elastic solution of the indentation problem and by means of torsional rheometry.

5 Conclusions The transformation in differential terms of the integral formulation of the viscoelastic continuum problem has been successfully achieved through the introduction of a discrete spectrum representation of the relaxation function. This leads to a new linear incremental formulation in the time domain for nonageing viscoelastic materials undergoing mechanical deformation. The formulation is based on a differential approach using discrete spectrum representation for the relaxation components. The governing equations are then obtained using a discretized form of the Boltzmann’s principle [12]. The analytical solution of the differential equations is obtained using finite difference discretization in the time domain. In this way, the incremental constitutive equations of linear viscoelastic material use a pseudo rigidity scalar tensor and the influence of the whole past history on the behaviour of the material at time t is given by a pseudo first order tensor. The generality allowed by this approach has been established by finding incremental formulation through simple choices of the function relaxation components. This approach appears to open a wide horizon (to explore) of new incremental formulations according to particular relaxation components. Remarkable incremental constitutive laws, for which the above technique is applied, are given. Among the numerous applications of the incremental formulations presented in this paper, is the numerical implementation in finite element software, thus the behaviour of complex boundary viscoelastic problems can be obtained. References 1- Bozza A. and Gentili G. (1995) Inversion and Quasi-Static Problems in Linear Viscoelasticity. Meccanica 30:321-335 2- Aleksey D. Drozdov and AlDorfmann (2004) A Constitutive Model in Finite Viscoelasticity of Particle-reinforced Rubbers. Meccanica 39:245-270 3. Kim K.S, Sung LEE H (2007) An incremental formulation for the prediction of two-dimensional fatigue crack growth with curved paths. Int J Num Methods Eng 72:697-721 4. Theocaris P.S (1964) Creep and relaxation contraction ratio of linear viscoelastic materials. J Mech Physics Solids 12:125-138

238

5. Chazal C, Moutou Pitti R (2009) An incremental constitutive law for ageing viscoelastic materials: a three dimensional approach. C. R. Mecanique 337:30–33 6. Mouto Pitti R, Chazal C, Labesse F, Lapusta Y (2011) A generalization of Mv integral to axisymmetric problems for viscoelastic materials, accepted for publication in Acta Mechanica, '2, V

7. Moutou Pitti R, Chateauneuf A, Chazal C (2010) Fiabilité des structures en béton précontraint avec prise en compte du comportement viscoélastique, Fiabilité des Matériaux et des Structures, 6èmes Journées Nationales de Fiabilité Toulouse, France 8. Krempl E, (1979) An experimental study of uniaxial room-temperature rate-sensitivity, creep and relaxation of AISI type 304 stainless steel. J Mech Physics Solids 27:363-375. 9. Kujawski D, Kallianpur V, Krempl E (1980) An experimental study of uniaxial creep, cyclic Creep and relaxation of AISI type 304 stainless steel at room temperature. J Mech Physics Solids 28:129-148. 10. Godunov S.K, Denisenko V.V, Kozin N.S, Kuzmina N.K (1975) Use of relaxation viscoelastic model in calculating uniaxial homogeneous strains and refining the interpolation equations for maxwellian viscosity. J Appl Mech Tech Physics 16:811-814. 11. Duffrène L, Gy R, Burlet H, Piques R, Faivre A, Sekkat A, Perez J (1997) Generalized Maxwell model for the viscoelastic behavior of a soda-lime-silica glass under low frequency shear loading . Rheologica Acta, 36:173-186. 12. Boltzmann L (1878) Zur Theorie der elastischen Nachwirkung Sitzungsber, Mat Naturwiss. Kl. Kaiser. Akad. Wiss 70, 275. 13. Christensen R.M (1971) Theory of Viscoelasticity: an Introduction. Academic Press, New York. ISBN 0-12-174250-4 14. Mandel J (1978) Dissipativité normale et variables caches. Mech Res Commun 5:225-229. 15. Salençon J (1983) Viscoélasticité. Presse de l’école nationale des ponts et chaussées, Paris. 16. Chazal C, Dubois F (2001) A new incremental formulation in the time domain of crack initiation in an orthotropic linearly viscoelastic solid. Mech Time-Depend Mater 5:229-253. 17. Andreas Jäger, Roman Lackner et al. (2007) Identification of viscoelastic properties by means of nanoindentation taking the real tip geometry into account. Meccanica 42:293-306 18- Herbert W. Müllner, Andreas Jäger et al. (2008) Experimental identification of viscoelastic properties of rubber compounds by means of torsional rhemetry. Meccanica 43:327-337

MODELING THE NONLINEAR VISCOELASTIC BEHAVIOR OF POLYUREA USING A DISTORTION MODIFIED FREE VOLUME APPROACH

G. Chevellard, K. Ravi-Chandar and K.M. Liechti1

Center for Mechanics of Solids, Structures, and Materials Aerospace Engineering and Engineering Mechanics The University of Texas at Austin 210 East 24th Street 1 University Station Austin, TX 78712-0235

Abstract The pressure-dependent behavior of polyurea was examined under monotonic loading in the confined compression configuration. Additional data from Arcan shear and uniaxial compression was used to respectively complete parameter selection for the linear and nonlinear behavior and then validate it. The distortion-modified free volume approach was examined as a model of nonlinear viscoelasticity for the polyurea. The bulk and shear relaxation behavior were both pressure dependent. Under ramp loadings, the shear and tensile responses were quite nonlinearly viscoelastic but the bulk response was almost linearly elastic.

1

Author for correspondence: [email protected]

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_34, © The Society for Experimental Mechanics, Inc. 2011

239

AN INCREMENTAL CONSTITUVE LAW FOR DAMAGING VISCOELASTIC MATERIALS Omar SAIFOUNI, Rostand MOUTOU PITTI, Jean-Francois DESTREBECQ Clermont Université, Université Blaise Pascal, Laboratoire de Mécanique et Ingénieries, EA 3867, BP 10448, F-63000 Clermont-Ferrand, France [email protected]

ABSTRACT. An incremental formulation suitable for modelling of materials with damaging viscoelastic behaviours is proposed in this work. A constitutive law based on linear viscoelasticity coupled with strain dependent damage is developed. The viscoelastic model is represented by a generalized Maxwell’s chain. It is governed by a set of internal stress variables attached to the branches of the Maxwell’s chain. The damage evolution is governed by values gained by a pseudo strain. The coupled law is turned into an incremental form suitable for the numerical analysis of damaging time dependent structures. Taking advantage of the incremental form, the coupled damaging viscoelastic law is implemented as a step-by-step procedure. The calculation procedure consists of a damaging elastic step followed by a number of damaging viscoelastic steps. The damage variable is adjusted at each step, according to the value gained by the pseudo strain. Exemplary calculations are worked out for two cases of uniaxial and biaxial variable or cyclic loadings. The results show the efficiency of the incremental model. It is worth noticing that the time increment used for the calculations is not necessarily small. As a consequence, precise analysis of damaging time dependent structures can be performed for low calculation cost. 1. Introduction A number of materials widely used in engineering applications exhibit viscoelastic behaviours, such as concrete, polymers, composites, wood, etc. [1]. From a general point of view, the viscoelastic behaviour can take different forms, including creep under constant load, stress relaxation under constant deformation, time-dependent recovery of deformation following load removal, time-dependent creep rupture [2]. Under complex loading and unfavourable physical environment, the mechanical properties of a viscoelastic material can gradually degrade in the course of time. Due to increasing deformations in the material, micro-defects existing originally or produced subsequently eventually grow, coalesce, and finally cause the fracture of the material. This phenomenon will affect the macroscopic behaviour and the durability of structures. It is therefore necessary to precisely account for the effect of damage on the time dependent behaviour of viscoelastic materials. A review of literature shows that different approaches have been proposed to address this problem. An analytical model was initially proposed by Simo [3] for damaging viscoelastic materials. Viscoelastic continuum damage models were used in order to solve linear and nonlinear viscoelastic problem [4-5]. Spectrum models for linear viscoelastic behaviours were used to describe the rate-dependent damage growth in a time dependent material under cyclic loading [6], environmental or corrosion effects [7], and with isotropic or anisotropic behaviour [8]. Also, finite element creep damage analysis have been used for the study of pressurised pipe bends [9], laminated composite bolted joints [10], or structures under multiaxial stress states [11]. A coupled damage viscoelastic model in the form of an incremental formulation is proposed in this work. The formulation is based on a description of the viscoelastic behaviour by means of a Maxwell’s chain, coupled with the damage model proposed by Simo [3]. The incremental formulation is based on an approach developed for viscoelastic composite structures [12]. One main advantage of this incremental formulation is that the finite time increment doesn’t need be necessarily small. This method was successfully applied to the analysis of time dependent effects in huge structures such as prestressed concrete bridges [13]. The main features of the damage law and its coupling with the viscoelastic law are described in the first part of the paper. Then, the coupled law is turned into an incremental form, according to the approach described in [12]. Taking advantage of the incremental formulation, the model is implemented as a step-by-step procedure. It is finally applied to the analysis of a damaging viscoelastic material subjected to a uniaxial cyclic stress or to a biaxial variable strain.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_35, © The Society for Experimental Mechanics, Inc. 2011

241

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Figure 1 – Flow chart of the step-by-step procedure for the incremental analysis of a damaging viscoelastic structure

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RELIABILITY ANALYSIS OF MIXED MODE CRACKING WITH VISCOELASTIC ORTHOTROPIC BEHAVIOUR

Rostand MOUTOU PITTIa, Alaa CHATEAUNEUFa, Claude CHAZALb. a

Clermont Université, Université Blaise Pascal, Laboratoire de Mécanique et Ingénieries EA 3867, BP 206, 63000 Clermont Ferrand, France. b GEMH Laboratory, Limoges University, Centre Universitaire de Génie Civil, 19300 Egletons, France. [email protected]

ABSTRACT. The reliability analysis applied to viscoelastic and orthotropic materials, in the case of mixed mode configuration, is studied in this work. The M integral, separating mixed mode during creep crack initiation in viscoelastic field, is used in the analytical approach. The main development, based on conservative law, and a combination of real and virtual displacement fields, is proposed. In order to provide mixed mode configuration, a Compact Tension Shear (CTS) specimen is used in the numerical process. Simultaneously the fracture and the viscoelastic procedures are coupled with reliability analysis in order to take account for model and parameter uncertainties. In this case, the random parameters related to model factors, elastic constants are defined in the reliability analysis of time dependent fracture materials subjected to complex loading. As results, the reliability levels are computed and discussed according to various mixed-mode loading scenarios 1. Introduction The mixed mode conditions often result from bending loads that are imposed on the structural component, and they are generally aggravated by the heterogeneous and orthotropic character, and the viscoelastic behavior of the material [1]. In the several cases, the negligence of mixed mode interaction in the design of composite structures may lead to significant errors in strength predictions, and mixed mode fracture criteria are thus of great importance for predicting failure of notched wood components. However, the mixed modes cracks, combining with the time depend behavior, are phenomena affected by high uncertainties, where deterministic methods fail to predict accurately the structural life. The objective of the current investigation is to apply the reliability model to the behavior of viscoelastic orthotropic material in order to estimate the uncertainties of the used fracture parameters. In viscoelastic approach, path independent integrals have been used in order to study the impact of the mixed mode ratio in crack initiation [2] and crack growth process in wooden material [3]. In the literature, several authors have applied the reliability theory to fracture mechanics problems [4]. In the past, the reliability approach has been coupled with the boundary element model for probabilistic fatigue life assessment in crack propagation mode mixty [5] and recently, random fatigue crack growth in mixed mode has been studied by stochastic collocation method [6]. However, these works don’t take into account the viscoelastic effects. In the first part of the paper, the reliability approach is recalled. In this case, the failure probability function and the First Order Reliability Method (FORM) are defined. In the second part, the conservative laws [7], combining the real and virtual mechanical fields [8] and the non-dependant integral parameter MT in crack initiation process, is recalled. The time dependent effects are introduced by the generalized Kelvin Voigt model resolving by an incremental viscoelastic formulation [2]. After, the CTS (Compact Tension Shear) specimen [9, 10] allows the mixed mode configuration is described. The subroutine of the crack initiation and reliability process is based on an energetic criterion. The random parameters used in the numerical model are fixed according to the critical values of energy release rate in opening and shear mode. Finally, the reliability analysis provides us to obtained the failure probability and the sensitivity of the fracture parameters regarding these complex solicitation are posted. T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_36, © The Society for Experimental Mechanics, Inc. 2011

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Long-term Life Prediction of CFRP Structures Based on MMF/ATM Method Yasushi Miyano* and Masayuki Nakada Materials System Research Laboratory, Kanazawa Institute of Technology 3-1 Yatsukaho Hakusan Ishikawa 924-0838, Japan * [email protected] Hongneng Cai School of Materials Science and Engineering, Xi'an Jiaotong University 28 Xianningxi Road Xi’ an Shaanxi 710049, China ABSTRACT The accelerated testing methodology (ATM) for the fatigue life prediction of CFRP laminates developed and verified theoretically and experimentally in the previous studies is expanded to the fatigue life prediction of the structures made of CFRP laminates in this paper. The procedure of MMF/ATM method combined with our developed ATM and the micromechanics of failure (MMF) by Professor Sung-Kyu Ha and others is proposed for the fatigue life prediction of the structures made of CFRP laminates. The time and temperature dependent MMF/ATM critical parameters of CFRP are determined by measuring the static and fatigue strengths at elevated temperatures in the longitudinal and transverse, tension and compression directions of unidirectional CFRP. The fatigue strengths of quasi-isotropic CFRP laminates with a central hole under compression load as an example of CFRP structures are measured at elevated temperatures, and these experimental data are compared with the predicted results by using the MMF/ATM critical parameters of CFRP. As results, it was cleared that MMF/ATM method has the possibility to be the strong tool to the fatigue life prediction of the structures made of CFRP laminates. Key words: CFRP, Structure, Life prediction, Viscoelasticity 1. Introduction The accelerated testing methodology (ATM) [1] was proposed for the prediction of long-term fatigue strength of CFRP laminates based on the time-temperature superposition principle (TTSP). Based on ATM, the long-term fatigue strength for CFRP laminates and structures can be predicted by measuring the short -term fatigue strengths at elevated temperatures. The applicability of ATM was confirmed for CFRP laminates and structures combined with PAN based carbon fibers and thermosetting resins [2-4]. Furthermore, the advanced accelerated testing methodology (ATM-2) was proposed in which the formulation for the master curves of time -temperature dependent fatigue strength was performed based on Christensen’s theory [5] which describes statistically the crack kinetics in viscoelastic body. The failure criteria of separated fiber and matrix in polymer composites have been developed and the failure of composite structures has been predicted based on the analyses on micromechanics, laminates and structure levels. Recently, the stress-based micromechanics of failure (MMF) have been proposed by Sung-Kyu Ha and others [6] for polymer composite with viscoelastic matrix. In this paper, the procedure of MMF/ATM method combined with our advanced ATM and MMF is proposed for the fatigue life prediction of the structures made of CFRP laminates. The validity of MMF/ATM metho d is confirmed through the following two steps. As the first step, the master curves of MMF/ATM critical parameters of CFRP are determined by measuring the static and fatigue strengths at elevated temperatures in the longitudinal and transverse, tension and compression directions of unidirectional CFRP. As the second step, the fatigue strengths of quasi-isotropic CFRP laminates with a central hole under compression load as an example of CFRP structures are measured at elevated temperatures, and these experimental data are compared with the predicted results by using the master curves of MMF/ATM critical parameters of CFRP based on MMF/ATM method.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_37, © The Society for Experimental Mechanics, Inc. 2011

257

258 2. Procedure of MMF/ATM method The procedure of proposed MMF/ATM method is shown schematically in Figures 1 and 2. Figure 1 shows the first step for the prediction procedure by MMF/ATM method that is the process of determination of MMF/ATM critical parameters. First, the viscoelastic modulus in the transverse direction of unidirectional CFRP is measured at various temperatures. The master curve and the time -temperature shift factor are determined by using these test data based on the TTSP. Second, the static and fatigue strengths in the typical four directions of unidirectional CFRP are measured at various temperatures at a single loading rate and single loading frequency, respectively. The strengths in four directions are longitudinal tension X, longitudinal compression X’, transverse tension Y and transverse compression Y’, respectively. Third, the master curves of these strengths are determined by using the measured data and the time -temperature shift factor for viscoelastic modulus. Fourth, the master curves of four MMF/ATM critical parameters , fiber tensile strength Tf, fiber compressive strength C f, matrix tensile strength Tm, and matrix compressive strength C m are determined through the method described in [7].

Figure 1 First step: Determination of MMF/ATM parameters Figure 2 shows the second step for the prediction procedure by MMF/ATM method that is the life determination of CFRP structures. With the master curves of the MMF/ATM critical parameters, long-term strength prediction of CFRP becomes possible. Three-step stress analyses are necessary to process the test result, including stress analysis for “homogenous” CFRP structures and CFRP laminates in macro level and stress analysis for the constituents in micro level by stress amplification. From the master curves of MMF/ATM critical parameters and failure criteria for fiber and matrix, the strength of CFRP structure under arbitrary time to failure and temperature can be determined.

259

Figure 2 Second step: Life determination of CFRP structures 3. Experimental procedure The test specimens were fabricated from unidirectional CFRP and quasi-isotropic CFRP laminate (QIL) [45/0/-45/90] 2s of MR60H/1053 which consists MR60H carbon fiber and epoxy resin 1053. The unidirectional CFRP were used to back-calculate the constituent properties. The QIL was used for strength prediction verification. All the laminates were made by the autoclave technique. The curing procedure includes 180o C for 2 hours and then postcured at 160 o C for 70 hours. The volume fraction of fiber is approximately 0.55. The laminates were cut by diamond-grit wheel to the specific size for the tests. The dynamic viscoelastic tests were performed for various frequencies and temperatures for the transverse direction of unidirectional CFRP. The shift factors for constructing master curve hold for strength master curve of CFRP and constituent critical parameters’ master curves. The static and fatigue tests for four directions of unidirectional CFRP were carried out to extract constituent critical parameters’ master curves by micromechanical amplification. Longitudinal tension tests under static and fatigue loadings were carried out at various temperatures according with ISO 527 to get the longitudinal tensile static and fatigue strengths. Longitudinal bending tests under static and fatigue loadings were carried out at various temperatures according with ISO 14125 to get the longitudinal compressive static and fatigue strengths. Transverse bending tests under static and fatigue loadings were carried out at various temperatures according with ISO 14125 to get the transverse tensile static and fatigue strengths. 20o off-axis tension tests under static and fatigue loadings were carried out at various temperatures to get the transverse compressive static and fatigue strengths. The compression tests for QIL under static and fatigue loadings were carried out at various temperatures using the open hole compression test specimens as shown in Figure 3. The e xperimental results have been already reported on [8-10].

Figure 3 Open hole compression (OHC) tests for QIL

260 4. Determination of MMF/ATM critical parameters 4.1 Creep compliance of matrix resin The left side of Figure 4 shows the master curve of the storage modulus E’ for the transverse direction of unidirection CFRP versus time t, where time t is the inverse of frequency. The right side shows the master curve of E’ which is constructed by shifting E’ at various constant temperatures along the logarithmic scale of t and logarithmic scale of E’ until they overlapped each other, for the reduced time t' at the reference temperature T0 =25o C. Since E’ at various constant temperatures can be superimposed so that a smooth curve is constructed, the TTSP is applicable for the storage modulus for the transverse direction of unidirectional CFRP. The time -temperature shift factor aTo (T) which is the horizontal shift amount is shown by rectangular symbols in Figure 5, and the temperature shift factor, b To(T), which is the amount of vertical shift, is shown by circular symbols in this figure. The creep compliance Dc of matrix resin is back-calculated from the storage modulus E’ for the transverse direction of unidirectional CFRP, and the master curve of Dc of matrix resin is shown Figure 6.

Figure 4 Master curve of storage modulus for transverse direction of unidirectional CFRP

Figure 5 Shift factors of the storage modulus for the transverse direction of unidirectional CFRP

261

Figure 6 Master curve of creep compliance for matrix resin calculated from the storage modulus for the transverse direction of unidirectional CFRP 4.2 Master curves of MMF/ATM critical parameters Figures 7 and 8 show the master curves of static and fatigue strengths for longitudinal tension X, longitudinal compression X’, transverse tension Y and transverse compression Y’ for unidirectional CFRP obtained from the strength data at various temperatures by using the time-temperature shift factors a To shown in Figure 5. The MMF/ATM critical parameter master curves Tf, C f, T m and C m are shown in Figures 9 and 10. The critical parameter of Tf was back-calculated fro m the tensile strength X of unidirectional CFRP. This process was repeated to construct other critical parameter master curves, the X’, Y and Y’ yielding the critical parameters C f, Tm and C m, respectively.

Figure 7 Master curves of tensile and bending static and fatigue strengths in the longitudinal direction of unidirectional CFRP

262

Figure 8 Master curves of bending and 20o off-axis tensile static and fatigue strengths in the transverse direction of unidirectional CFRP

Figure 9 Master curves of MMF/ATM critical parameters Tf and C f

Figure 10 Master curves of MMF/ATM critical parameters Tm and C m

263 5. Life determination of CFRP structures As an example of application of MMF/ATM critical parameters master curves, long-term OHC strength for QIL was predicted. Figure 11 shows the initial failure mechanism for static loading by comparing the failure indexes of MMF/ATM parameters under, for example, T=25o C. k T f is failure index of fiber under tension, k Cf is failure index of fiber under compression, and k Tm* is failure index of matrix under tension, and k Cm* is failure index of matrix under compression [7]. When one of these failure indexes reaches to unity, the initial failure of laminate occurs. It is cleared from this figure that the OHC failure of QIL was triggered by fiber compressive failure in 0o layer. Figure 12 shows the initial failure of OHC for QIL under static loading observed from the specimen in which the OHC test was stopped at 98% level of maximum stress under T=25 o C. The microbuckling of fiber in 0o layer as well as the transverse crack in 45o layer are observed. In the -45o and 90o layers, any failures can not be observed. Furthermore, any failures can not be observed for the specimen in which the OHC test was stopped at 95% stress level of maximum stress under all temperatures tested. For all temperature conditions tested, the same failure was observed with the above simulation. Therefore, the predicted master curves of OHC strength for QIL were constructed based on the MMF/ATM parameter of compressive strength of fiber. Figure 13 shows the predicted master curves of OHC strength for QIL with experimental data. The predicted strength agrees well with the experimental data for all region of time to failure t’.

Figure 11 Judgment of initial failure of OHC for QIL under static loading (T=25o C, t’=1 min)

(a) 45°layer

(b) 0°layer

(c) -45°layer

(d) 90°layer

Figure 12 Observation of initial failure of OHC for QIL under static loading (σmax=0.98σs , T=25 o C, V=0.1mm/min)

264

Figure 13 Predicted OHC static and fatigue strengths for QIL 6. Conclusion The time-temperature dependent master curves of MMF/ATM critical parameters were constructed for CFRP laminates MR60H/1053 by tensile and compressive tests under static and fatigue loadings for the longitudinal and transverse directions of unidirectional CFRP under various temperatures based on the time -temperature superposition principle which holds for the viscoelastic behavior of matrix resin. It was confirmed experimentally that the long-term OHC strength of quasi-isotropic CFRP laminates [45/0/-45/90]2s can be predicted using these master curves of MMF/ATM critical parameters. Therefore, the verification of proposed MMF/ATM method was confirmed experimentally. AKNOWLEDGEMENTS The authors thank the Office of Naval Research for supporting this work through an ONR award with Dr. Yapa Rajapakse as the ONR Program Officer. Our award is numbered to N000140611139 and titled “Verification of Accelerated Testing Methodology for Long-Term Durability of CFRP laminates for Marine Use”. The authors thank Professor Richard Christensen, Stanford University as the consultant of this project. References 1. Miyano, Y., Nakada, M., McMurray, M. K. and Muki, R. “Prediction of Flexural Fatigue Strength of CFRP Composites under Arbitrary Frequency, Stress Ratio and Temperature”, Journal of Composite Materials, 31: 619-638, 1997. 2. Miyano, Y., Nakada, M. and Muki, R. “Applicability of Fatigue Life Prediction Method to Polymer Composites”, Mechanics of Time-Dependent Materials, 3: 141-157, 1999. 3. Miyano, Y., Nakada, M., Kudoh, H. and Muki, R., “Prediction of Tensile Fatigue Life under Temperature Environment for Unidirectional CFRP”, Advanced Composite Materials, 8: 235-246, 1999 4. Miyano, Y., Nakada, M. and Sekine, N., “Accelerated Testing for Long-term Durability of FRP Laminates for Marine Use”, Journal of Composite Materials, 39: 5-20, 2005.

265 5. Christensen, R. and Miyano, Y. , “Stress Intensity Controlled Kinetic Crack Growth and Stress History Dependent Life Prediction with Statistical Variability”, International Journal of Fracture, 137: 77-87, 2006. 6. Ha, S. K., Jin, K. K. and Huang Y., “Micro-Mechanics of Failure (MMF) for Continuous Fiber Reinforced Composites”, Journal of Composite Materials, 42: 1873-1895, 2008. 7. Cai, H., Miyano, Y., Nakada, M. and Ha , S. K., “ Long-trem Fatigue Strength Prediction of CFRP Structure Based on Micromechanics of Failure”, Journal of Composite Materials, 42: 825-844, 2008. 8. Iwai, K., Cai, H., Nakada, M. and Miyano, Y., “Prediction of Long-term Fatigue Strength of Quasi-isotropic CFRP Laminates with A Hole Under Compressive Loading”, 8th International Conference on Durability of Composite Systems (DURACOSYS 08), Porto, Portugal, July 2008 9. Miyano, Y., Nakada, M. andCai, H., “Formulation of Long-term Creep and Fatigue Strengths of Polymer Composites Based on Accelerated testing Methodology”, Journal of Composite Materials, 42: 1897-1919, 2008. 10. Hiraoka, M., Iwai, K., Cai, H., Nakada, M. and Miyano, Y., “ Long-term Life Prediction of Quasi-isotropic CFRP Laminates with A Hole Under Compressive Loading”, 17th International Conference of Composite Materials (ICCM-17), Edinburgh, UK, July 2009. 11. Tsai, S. W. and Hahn, H. T. , Introduction to composite materials , Westport, Technomic, 1980.

Non-local Solutions to Direct and Inverse Problems in Mechanics: A Fractional Calculus Approach

C.S. Drapaca, PhD Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802, USA and S. Sivaloganathan, PhD Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada N2J 3G1 Centre for Mathematical Medicine, Fields Institute for Mathematical Sciences, Toronto, ON, Canada M5T 3J1 ABSTRACT In this paper we present a new non-local model of continuum mechanics based on fractional calculus. The model encompasses in a single unified framework both classical continuum mechanics and non-local theories for continua with discontinuities and long range forces. Our theoretical framework can be used to solve both direct and inverse problems of importance in practical applications. We present results to two direct mechanical problems: (1) the deformation of an infinite bar subjected to a self-equilibrated load distribution, and (2) the propagation of longitudinal waves in a thin finite bar. We also show results to the inverse problem of magnetic resonance elastography using our proposed model. 1. INTRODUCTION It is well-known that the classical equations of continuum mechanics are not well-suited to the modeling of many problems of fundamental importance in solid mechanics such as those involving the formation of cracks, phase transitions, or the presence of inclusions or mixtures. The reason why the classical mechanics framework does not work for these problems is that it is a local theory where the displacement fields are represented using partial derivatives which are undefined along discontinuities. In recent years non-local methods have been proposed to address the limitations of classical continuum mechanics [3, 4, 11–17]. More recently, it has been shown that fractional calculus can be related to new non-local constitutive laws of elasticity [7, 8, 18, 23, 25, 26] as well as to the mechanics of fractal media [20–22]. Fractional calculus has emerged lately as a powerful new mathematical method of solution in a variety of problems arising in the mathematical and physical sciences. We refer the reader to [5] for an extensive review of the fractional calculus and its applications. Apart from its use in practical applications, more recent fundamental considerations have led to the introduction of fractional calculus in constitutive laws in an attempt to develop non-local stress-strain relations [7, 12, 13]. On the other hand, attempts have also been made to model the microscopic forces using fractional derivatives [8, 18, 23]. Our approach in this paper differs from previous theoretical work, in that we develop a general framework, defining the laws of motion and the stresses using fractional derivatives. This relaxes the constraint of differentiability imposed on the displacement fields by the classical theory, and extends the class of allowable displacements to continuous but not necessarily differentiable fields. Since fractional derivatives are non-local, this new theory, that we call the fractional model of continuum mechanics, offers an elegant and unified way to study most problems involving continua, including those aforementioned. The model builds upon concepts from the classical theory of continuum mechanics, the peridynamic model [3], and Carpentieri et al. work [20– 22] on the mechanics of fractal media. Our theoretical framework can be used to solve both direct and inverse problems of importance in practical applications. The paper is structured as follows. In section 2 we introduce our new fractional model of continuum mechanics and in section 3 we present applications of the model to two direct mechanical problems: (1) the deformation of an infinite bar subjected to a self-equilibrated load distribution, and (2) the propagation of longitudinal waves in a thin finite bar. We also show results to the inverse problem of magnetic resonance elastography (imaging technique used to estimate the elasticity of biological tissues subject to mechanical stresses in vivo) using our proposed model. The paper ends with a section of conclusions.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_38, © The Society for Experimental Mechanics, Inc. 2011

267

268

2. FRACTIONAL CONTINUUM MECHANICS MODEL Let : and : t , t ! 0 be open subsets of R , t t 0 , where : 0 { : . 3

Definition 1: Let Į be a 3x3 matrix whose elements

D Ii , I , i 1, 2,3 are

continuous functions of

t ! 0 such that

f  D Ii (t ) d 1, t ! 0, I , i 1, 2,3 , and Ȥ (˜; t , Į (t )) : : o Ȥ (:, t , Į (t )) { :t ,Į (t ) be a family of functions in L1 (:) . The motion (deformation) of order Į (t ) of a body is determined by the position x of the material points in space as a function defined for every

x

t ! 0 , and every D Ii (t )  (f,1], I , i 1, 2,3 by:

(t ) ­K 1-Į Ȥ ( X, t ),  f  D Ii (t )  1, I , i 1, 2,3 X , ® Ȥ ( X , t ), D ( t ) 1, I , i 1, 2,3 Ii ¯

(2.1)

In the above definition we denote by: 1 (t ) § K 1-Į 0 0 · ˜ X ¨ ¸ (t ) 2 (t ) K 1-Į K 1-Į 0 ¸, ˜ ¨ 0 ˜ X X 3 (t ) ¸ ¨ 0 K 1-Į ˜¹ X © 0

where

xi

Įi

D1i , D 2i , D3i , i

i (t ) K 1-Į X

1

 x , x , x are given by: 1

2

3

F i (Y 1 , Y 2 , Y 3 , t )dY 1dY 2 dY 3

*(1  D1i (t ))*(1  D 2 i (t ))*(1  D 3i (t )) ³³³H | X 1  Y 1 |D1i ( t ) | X 2  Y 2 |D 2 i ( t ) | X 3  Y 3 |D3i ( t ) f

We denote by

1, 2,3 are the rows of matrix Į , and the components of x

*( s )

³e

.

(2.2)

 t s 1

t dt the gamma function. In addition, we assume that Ȥ is zero on the boundary and outside

0

the interval of integration H . The interval of integration H defines the region of influence of the reference position X made of all the material points that contribute to the deformation of X into x . Region H introduces in the model variable length-scales which are not necessarily equal. At the nanoscale level H is determined by the physics of interactions between particles, while at the macroscopic level it may be chosen in such a way that the parameters of the fractional model match the measured mechanical parameters of the material. If domain : is infinite, then we can replace region H by the entire space R 3 . If : is a finite domain, then H is chosen such that the boundary conditions on : are satisfied in the classical way. (We note that although the definition of the region of influence H appears to be similar to that of the horizon used in the peridynamic theory (see, for instance, [3]), their properties are in fact different.) We conjecture that the parameter matrix Į (t ) measures the dynamic deformation of the space scale as a function of time and therefore is related to an intrinsic deformation field present in a body at any given time. This deformation field may be due to electro-magnetic, thermal, and/or chemical processes that may take place sometimes simultaneously and at different time and length scales in objects independently of any existing mechanical stresses. By representing the law of motion as a fractional order integral, we represent, in fact, a multi-scale type of motion in which the microscopic and macroscopic levels are smoothly connected by the presence of the integral. Note that the case Į (t ) 1, t ! 0 could be seen as describing macroscopic motions caused merely by mechanical loading in objects without microstructure (classical continuum mechanics case). Thus, prescribing the region of influence H and the parameter matrix Į (t ) is equivalent to choosing the type of dynamic microstructure in a

269

material of interest. For simplicity, we assume that H does not depend on f  D Ii d 1, I , i 1, 2,3 .

t and Į , and that Į is a constant matrix with

Definition 2: We define the deformation gradient of order Į, f  D Ii d 1, I , i

§ wx j · ¨ J ¸ © wX ¹ j , J

FĮ

1, 2,3 as: (2.3)

1,2,3

We notice that FĮ is a mixture of fractional order derivatives and integrals. In the one-dimensional case, for a particular choice of the region of influence H , (2.3) reduces to the modified Riemann-Liouville derivative introduced by Jumarie in [9, 10]. According to corollary 3.2 in [10], the modified Riemann-Liouville derivative is exactly the local fractional derivative introduced in [24] and used in [20–22] to describe fractal stress fluxes and deformation patterns in materials that have fractal microstructures. Thus the generalization of the geometric interpretation of derivatives in terms of tangents given in [20] remains valid: all the curves with the same order

Į form an equivalence class denoted by x Į .

dP is a contact force acting on the deformed element of area da , then the Į - contact force is by definition K dP (x, t ), and the stress vector of order Į is given by:

Definition 3: If

dPĮ

1-Į x

t Į ( x, t )

lim

da o 0

dPĮ (x, t ) da

K 1-Į x t ( x, t ),

(2.4)

dP is the classical true stress vector. da o 0 da

where t= lim

The Į - contact forces can be seen as interaction forces at long distances: the multi-scaling parameter Į of the motion of a body determines what parts of the body are linked and interact during the deformation process. As before, we assume that on the boundary and outside the region of influence the stress vector is zero. We note that the Į -contact forces differ from the non-local forces introduced in [18, 20–23], but they can be seen as a particular class of the non-local forces of the peridynamic theory [3]. The definition of the Cauchy stress tensor of order Į follows: Definition 4: The Cauchy stress tensor of order

TĮ

Į is given by:

K 1-Į x T,

(2.5)

where T is the classical Cauchy stress tensor. Thus the new local form of the equation of motion of a body of mass density to Į – contact forces, is:

’ ˜ TĮ  b

U

d 2x . dt 2

U (x, t )

subjected to body forces b ( x, t ) and

(2.6)

We note that the conservation of the angular momentum is not required for materials with microstructure in electromagnetic field and thus for such materials TĮ does not need to be a symmetric stress tensor. It can be shown that in order to conserve the angular momentum

Į must be only a vector.

270

3. EXAMPLES 3.1 Self-Equilibrated Loaded Infinite Bar A one-dimensional bar of infinite length and cross-section non-dimensional area A 1 made of a linear elastic material of effective Young modulus E is subjected to a time-independent rescaled body force density function b( x ) that is selff

equilibrated, i.e.

³ b( x)dx

0 , and vanishes outside a loading region  a d x d a, a ! 0. If the pair of self-equilibrated

f

r a , then b( x) G ( x  a )  G ( x  a ) , where G ( x) is the Dirac distribution. In the case when 0  D  1 , we choose the regions of influence H such that the symmetry is conserved and equation (2.6) for the displacement u ( x) in the bar reduces to: concentrated forces, each of unit rescaled magnitude, acts at the points x

d D E D# u r G ( x # a ) dx

0

(3.1)

D

where u (0) 0, E EL with L a characteristic length scale determined by the chosen region H , and the Riemann – Liouville fractional derivatives are: f

1 d f ( x  y )dy , ³ *(1  D ) dx f y#D

D

D# f ( x)

where we denote by

yD

­ 0, y ! 0 , yD ® D ¯| y | , y  0 u ( x)

­| y |D , y ! 0 . The solution to equation (3.1) is: ® 0,  0 y ¯

1 ª| x  a |D  | x  a |D º¼ . EL *(D  1) ¬

(3.2)

D

In figure 1 we show the fractional order displacements (3.2) for

D

0.1, 0.5, 1; E 1, a 1, L / a

0.5, 4.

3.2 Longitudinal Waves in Thin Bars We consider the fractional order generalization of the wave propagation through a finite bar of length section whose ends move simultaneously outward with constant opposite velocities:

d 2u , 0 d x d 2a dt 2 wu u ( x, 0) 0, ( x, 0) 0 wt *( E  1) u (0, t ) k t , u (2a, t ) LD  2E u ED 

with 2 E

u ( x, t )

2a with unit cross

U

(3.3)

k

*( E  1) t LD

D  1, 0  D  1 . Following [17] and [19], the solution to (3.3) is: 1 1 § *( E  1) · § *( E  1) · k / c ¨ c t  (3D 1)/2 x E ¸  k / c ¨ c t  (3D 1)/2 (2a  x) E ¸ , D D L L L L © ¹ © ¹

(3.4)

271

for

td

(2a ) E L1 E , where we denote by  A  c*( E  1)

­ A, A ! 0 and c ® ¯ 0, A d 0

E / U is the classical longitudinal wave speed.

The fractional order displacements (3.4) are shown in figure 2 for a

0.9, c 1, k 1, E

0.6, 1, and

0, 0.5, 1.

3 2

u

1

1

0.1 0.5 1

0 -1

0 -0.5

-2 -3 -5

0.1 0.5 1

0.5

u

t

1, L / a

0 x (a)

5

-1 -5

0 x (b)

Fig.1: The fractional order displacements (3.2) for length scales (a) L / a

5

0.5 , and (b) L / a 4 .

3.3 The Inverse Problem of Magnetic Resonance Elastography Magnetic resonance elastography (MRE) is a non-invasive, in vivo imaging technique used to estimate the elasticity of soft tissues subject to mechanical stresses. The resulting strains are measured using MR imaging and the elastic modulus is computed from viscoelastic models of tissue mechanics. MRE has the potential to become a very powerful non-invasive, in vivo diagnostic tool for tumor detection. We assume that the soft tissues are almost incompressible, linear viscoelastic solids,

b 0 , and U 1 g/cm3 [2]. Under oscillatory loading, equation (2.6) takes the following form in the frequency domain:







/’Į  ’ ˜ U  M ª’ ’Į ˜ U  ’ ˜ ’Į U º -Z2 U ¬ ¼

(3.5)

where U , ȁ, Ȃ are the temporal Fourier transforms of the displacement field, and of the Lamé coefficients Ȝ and ȝ, Į respectively. We denote by Ȧ=2ʌf, where f is the frequency of oscillations, and ’ is the fractional order gradient. We

§ D11 1 ¨ assume that Į ¨ 1 D 22 ¨ 1 1 ©

1 · ¸ 1 ¸ . Since for biological tissues ȁ >> M, we will use the same approximation as in [2] and D33 ¸¹

272

neglect the ȁ-term in equation (3.5). Assuming that the displacement field U is known from the MRE measurements, we want to find M by inverting the following equation:







M ª’ ’ Į ˜ U  ’ ˜ ’ Į U º ¬ ¼

1

0 -0.5 -1 0

(3.6)

0 0.5 1

0.5

u

u

1

0 0.5 1

0.5

-Z2 U

0 -0.5

0.5

1 x (a)

1.5

2

-1

0

0.5

Fig.2: The fractional order displacement (3.4) at time instances 0, 0.5, and 1 for

1 x (b)

E

1.5

2

0.6 (a); the classical displacement (b).

In order to test our new model, we used the displacement field obtained from a two dimensional MRE-type shear test experiment run on a gel phantom containing four cylindrical inclusions of stiffer gel [1]. The outside gel was 1.5% Agar and the cylinders were composed of 10% B-gel (bovine). The cylinders were approximately 5, 10, 16, and 25 mm in diameter. The field of view was 20 cm, and the slice thickness was 5 mm. Eight offsets through time were acquired at a frequency of f = 100 Hz. The expected values of the shear wave speed are of 3-4 kPa in the background and 10-12 kPa in the cylindrical inclusions. In figure 4 we show the values of the shear wave speed with M given by equation (3.6) when D11 0.85, D 22 0.75 (left image) and with M from the classical continuum mechanics theory. Both methods found the expected values of 3-4 kPa in the background and of 10-12 kPa in the inclusions. However, while the classical model was able to find just a few of the correct values in the inclusions, most of the values being around 8 kPa and even lower in the smallest cylinder, our fractional model was able to find all the values in the inclusions in the range of 10 kPa to 12 kPa. Also, as it can be seen in the elastogram profiles shown in figure 4, the classical model is unstable (with values as high as 600 k Pa in one of the inclusions), while our model is very stable. More details on this application are given in [6]. 4. CONCLUSION In this paper, we have presented a new non-local model of continuum mechanics based on fractional calculus. The model encompasses in a single unified framework both classical continuum mechanics and non-local theories for continua with discontinuities and long range forces. The proposed theoretical framework has been applied to two direct problems: (1) the deformation of an infinite bar subjected to a self-equilibrated load distribution, and (2) the propagation of longitudinal waves in a thin finite bar. We have also show promising results to the inverse problem of magnetic resonance elastography using our proposed model.

273

Fig.3: One dimensional profiles: our fractional model (left column) and the classical model (right column) [6]. REFERENCES: 1. MRE/WAVE software and data: http://ndc.mayo.edu/mayo/research/ehman_lab/mrw-wave.cfm 2. Oliphant, T.E., Manduca, A., Ehman, R.L., Greenleaf, J.F., ‘Complex-valued stiffness reconstruction for magnetic resonance elastography by algebraic inversion of the differential equation’, Magn. Reson. Med., 45 (2), 299-310 (2001). 3. Silling, S., ‘Reformulation of elasticity theory for discontinuities and long-range forces’, J. Mech. Physics Solids 48, 175– 209 (2000). 4. Zimmermann, M., A Continuum Theory with Long-Range Forces for Solids, PhD thesis, MIT (2005). 5. Hilfer, R. Applications of Fractional Calculus in Physics, World Scientific (2000). 6. Drapaca, C.S., ‘A novel mechanical model for magnetic resonance elastography’, Rev.Roum.Sci.Techn. - Mec.Appl., 55(1), 3–18 (2010). 7. Lazopoulos, K.A., ‘Non-local continuum mechanics and fractional calculus’, Mech.Researh Comm., 33, 753–757 (2006). 8. Vazquez, L., ‘A fruitful interplay: from nonlocality to fractional calculus’, Nonlinear Waves: Classical and Quantum Aspects, 129–133 (2004). 9. Jumarie, G., ‘Probability calculus of fractional order and fractional Taylor’s series application to Fokker-Plank equation and information of non-random functions’, Chaos Solitons and Fractals, 40, 1428-1448 (2009). 10. Jumarie, G., ‘Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution’, J.Appl.Math.Computing, 24(1-2), 31-48 (2007). 11. Bazant, Z.P., Jirasek, M., ‘Nonlocal integral formulations of plasticity and damage: survey of progress’, Journal of Engineering Mechanics, 128(11), 1119-1149 (2002). 12. Eringen, A.C., Edelen, D.G.B., ‘On non-local elasticity’, Int. J. Eng. Sci., 10, 233-248 (1972). 13. Eringen, A.C., ‘Theory of nonlocal elasticity: some applications’, Research in Mechanics, 21, 313-342 (1987). 14. Kunin, I.A., Elastic Media with Microstructure I: One-Dimenasional Models, Springer (1982). 15. Rogula, D., Nonlocal Theory of Material Media, Springer (1982). 16. Bazant, Z., Belyschko, T., Cheng, T.P., ‘Continuum theory for strain-softening’, J. Eng. Mech., 110, 1666-1692 (1984). 17. Bazant, Z., Belyschko, T., ‘Wave propagation in a strain-softening bar: exact solution’, J. Eng. Mech., 111, 381-389 (1985). 18. Cottone, M., Di Paola, M., Zingales, M., ‘Fractional mechanical model for the dynamics of non-local continuum’, Advances in Numerical Methods, 389-423 (2009). 19. Jumarie, G. ‘Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution’, J.Appl.Math. Comput., 24 (1-2), 31-48 (2007). 20. Carpinteri, A., Cornetti, P. ’A fractional calculus approach to the description of stress and strain localization in fractal media’, Chaos, Solitons and Fractals, 13, 85-94 (2002). 21. Carpinteri, A., Chiaia, B., Cornetti, P., ‘Static-kinematic duality and the principle of virtual work in the mechanics of fractal media’, Comput.Methods Appl.Mech.Engrg., 191, 3-19, (2001).

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22. Carpinteri, A., Cornetti, P., Sapora, A., ’Static-kinematic fractional operators for fractal and non-local solids’, ZAMM, 89(3), 207-217 (2009). 23. Di Paola, M., Zingales, M., ‘Long-range cohesive interactions of non-local continuum faced by fractional calculus’, Int. J. Solids Struct., 45, 5642-5659, (2008). 24. Kolwankar, K.M., Gangal, A.D., ‘Fractional differentiability of nowhere differentiable functions and dimensions’, Chaos, 6, 505-523 (1996). 25. Di Paola, M., Pirrotta, A., Zingales, M., ‘Mechanically-based approach to non-local elasticity: variational principles’, Int. J. Solids Struct., 47(5), 539-548, (2010). 26. Cottone, G., Di Paola, M., Zingales, M., ‘Elastic waves propagation in 1D fractional non-local continuum’, Physica E, 42(2), 95-103 (2009).

Study on Crystallinity Dependency of Creep Deformation on GFRTP of Polyoxythlene (POM)

S.SOMIYA*, K. YAMADA* and T. SAKAI**

* Keio university; 3-14-1Hiyoshi Kohoku-ku Yokohama Kanagawa 223-8522 **:Metropolitan Tokyo University; 1-1 Minami-Osawa Hachioji Tokyo 192-0397

ABSTRACT It has clarified that the viscoelastic property of some kind of thermoplastic resin is dependent on crystallinity in until now. Concretely, it was confirmed that "Time-temperature superposition principle" was established on the creep deformation in them. And also, in various FRTP, in making fiber volume fraction to be a factor for creep deformation, it was clarified that "Time-temperature-fiber volume fraction superposition principle" was established. In this paper on the effect of crystal on creep deformation, the crystallinity was made to be a factor, and it clarified that "Time - temperature -crystallinity superposition principle" was established in POM resin and GFRPOM. INTRODUCTION Some designers of engineering structure hope to control visco-elastic behavior of thermoplastics and some methods to change them were developed. Famous method was composing of reinforcements [1]-[3] and crystallization [4]-[8]. It is clear that the mechanical property of crystalline polymer depends on the crystallinity. It is well also known that to obtain expecting mechanical property by the adjustment of the crystallinity is difficult, after it becomes a product by molding. The adjustment of the mechanical property by the crystallinity is industrially very interesting, if it is possible in the molding. Today, it is well known that the crystallinity of POM is higher than the other general polymers in which they were about 70%. Because it was very difficult to control the crystallinity, there were few reports for the creep behavior by the control of crystallinity. In this report, the control of creep behavior by crystallinity and fiber mixing were researched. To control the crystallization of POM, the effect of cooling speed from melted solution of resin was focused [9]. Father more, the probability of the calculation of visco-elastic deformation including of the effect of crystallinity was discussed. By confirming “Time, temperature, fiber volume fraction superposition principle”, authors have shown that life design of plastic products and design of durability are possible [10]. EXPERIMENTAL METHOD Polyoxymethylene (POM) and Glass fiber reinforced POM (GFRPOM) were used.  This POM is Tenac C-4510 and GFRPOM is Tenac C-GN455 made by ASAHIKASEI Chemicals Co., Ltd. The fiber fractions were 0%, 5% and 15% in the weight fraction (Wf), and 0%, 2.8% and 8.7% in the volume fraction (Vf). Specimens were cut out to the strip type of 70×10×3(mm). The prepared crystallinity of specimen were 66%,72% and 78% for resin‫ޔ‬68%, 70% and 75% for GFRP with Vf = 2.8% and 62,67 and 73% with Vf=8.6%. Two cooling speeds which were 1C/min and 10C/min were used and they were Annealed specimen made by 1C/min and Quenched material made by 10C/min, which they were called as A-POM and Q-POM. Thermal analysis by DSC of DSC-60 (Simazu Co.,) was carried out under nitro atmosphere. Three points bending creep test was performed, using the modified load deflection temperature instrument S3-MH by TOYOU -SEIKI Co. CRYSTALLIZATION PHENOMENA ON POM RESIN Cooling rate dependence of grain size on POM The process of crystallization of POM from melting state to hard resin was observed under several cooling rate by the polarizing microscope at in-site. The thickness of specimen was about 30ȝm. This means, the crystal size may be different a little even in the same cooling condition in actual molded products.Fig.1 (a) ,(b) shows the crystal progress situation at 10C/min and 50C/min. From these figures, it was confirmed that the crystal growth started from many crystal nuclei, as cooling speed is rapider, and that there is a number of much crystal grain, and that the average grain size also decreases. For grain boundary and cross point of the three grains, the black part regarded as an amorphous part was observed as shown in Fig. 1(b) by the arrow. In the meantime, when cooling speed was10C/min, large crystal was constituted and the crystal grows from small number of crystal nucleus. This means, amorphous part dispersed in the grain of crystals.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_39, © The Society for Experimental Mechanics, Inc. 2011

275

276

100㱘 㱘m

100㱘 㱘m

(b) Cooling speed: 50C/min

(a) Cooling speed: 10C/min

Fig.1 Crystallization observation of POM from 200C to room temperature by two cooling rate The control of crystallinity The speeds on crystallization treatment were 1C/min and 10 C/min and they were called as an annealed (A) and a quenched (Q) treatment. The cooling speed 10C/min was almost the same as cooling rate of practical injection molding. The crystallinity(C=) of A- and Q-material are in C=74% and 66% on each other. POM’s crystallinity is higher than popular engineering polymers. To control the crystallinity of molded specimen the heat treatment time was made to change at 170 C. The prepared crystallinity was C= 66%, 72% and 78% for Q-POM and C=74%, 76% and 78% for A-POM. The size and the shape of the crystal grain completely did not change, even if the crystallinity was improved. It is shown that the growth of folding structure from micelle and the amorphous part was generated on this fact on the rise of the crystallinity. The change of crystal structure caused by crystallization treatment The temperature of the melting point was measured by DSC in order to examine the effect of the change of the cooling speed on the thermal quality. In Q-POM, the peak of melting point was consisted by the two peaks as shown in Fig.2 but in A-POM there is only one peak. The temperature at the single peak of A-POM and the large peak of Q-POM on Fig. 2 were almost the same. This common peak shows the melting of the spherulite which consists of the folded structure, and the sub peak of Q-POM shows the existence of the crystal structure. As this structure, the micelle considers it. As Q-POM, the sub-peak was absorbed into main peak, as  4

Q-material Crystallinity 66% Q-material Crystallinity 72%

Bending modulus (GPa)

3.5

䌅䌸䌯䌴䌨䌥䌲䌭䌩䌣 䌨䌥 䌡䌴

Q-material Crystallinity 䋷䋸% A-material Crystallinity 74% A-material Crystallinity 76% A-material Crystallinity 78%

3

2.5

2

1.5 60

150 160 170 180 190 200

Temperature(㷄 㷄)

Fig.2 Melting peak by DSC analysis of POM

65

70 75 Crystallinity (%)

80

Fig. 4 Strength and Modulus of crystallization of POM and GFRPOM

2

1

(a) Cooling speed : 10㷄 㷄

100㱘 㱘m

(b) Cooling speed : 50㷄 㷄

100㱘 㱘m

Fig.3 Crystallization observation of GFRPOM from 200C to room temperature by two cooling rate Arrow 1: Crystal and Arrow 2: Glass fiber

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crystallinity rose. It is shown that micelle and high polymer in amorphous polymer changed into folded structure. Effect of cooling speed on grain size of GRRP of POM In the crystalline polymer, it is known that the nucleus is made to be an origin, when the crystal is generated, and the material as a nucleus may be mixed, when it wants to raise the crystallinity. There were some reports that fibers have similarly effective. In order to confirm this effect, the resin material which mixed glass fiber was melted, and the cooling was carried out afterwards. Though the example in which the crystal growth started from the surface of fiber was also confirmed as shown in Fig.3 by the arrow 1, as it is shown in the photograph, as well as the case of the resin material, it grew from the resin which is not related to the fiber in much crystal grain. The effect in which the glass fiber in POM increased crystal nucleus was not confirmed. The size of the spherulite is bigger than the size of the fiber. The spherulite grew, while the fiber was taken in. The fiber does not change to alienation factor of the crystal growth from the observation. But it was confirmed that the crystallinity of GFRPOM was deceased by the increase of fiber volume fraction. Dependence of modulus on crystallinity of GFRPOM For both resins, the modulus of A-POM usually was higher than the modulus of Q-POM as shown in Fig.4. With the increase in the crystallinity, the elastic modulus of both materials was improved. It was assumed that this difference depended on the difference in the grain size, because it is both resins were made from the same material. The reason why this difference was possible could not be clarified. It became a similar result on the strength, and it became a result of being completely reverse to the minor rule on metallic material. For FRP, modulus of GFRPOM of A-POM was higher than it’s of Q-POM as same as pure resin as shown in Fig. 4. 1.2

0 50ºC

-1

90ºC

1

110ºC 130ºC

Shift factor aT0(T)

c

Creep compliance D (t,T)(1/GPa)

70ºC

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-2 -3 -4 -5

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0.2

-6

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4 6 Log t'(min)

8

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2.8 2.7 2.6 1/T(˜ ˜10-3/K)

2.5

2.4

                       (a)                 (b) Fig.5 The master curve of creep compliance (a) and time-temperature shift factor (b) of Q-POM of crystallinity 66% THE EFFECT OF CRYSTALLIZATION OF POM AND POM’s GFRP ON CREEP BEHAVIOR The verification of “Time, temperature and crystallinity superposition principle” of Q-POM The creep behavior was measured on the specimen that changed and prepared the crystallinity by the isothermal processing, while the crystal form in the molding was retained. From creep curve group obtained by POM having 66% crystallinity at temperature condition from 50 C and 130 C, the master curve of creep compliance was tried to draw in order to verify whether it establishes “Time-temperature superposition principle” as shown in Fig.5 (a). It is shown that the smooth master curve can be completed by using shift factor shown in Fig.5 (b). It can be confirmed that “Time-temperature superposition principle” is established on the creep of this material. The master curves of POM which have crystallinity C=72% and 78% were obtained too. The value of the compliance lowers so that modulus increased, as the crystallinity increased, when the isothermal crystallinity processing was conducted. It was confirmed that the shape of the master curve of the material of C=72% and 78% was almost same as shown in Fig.6. Two points were researched in order to examine whether it is the factor in which the crystallinity is linearly related to the creep deformation. The first point is the fact of whether it bases the material of the crystallinity variously and can make the gland master curve from the master curve of getting creep compliance in which it is confirmed. The second point is the confirmation of whether there is the linearity between shift factor used for the gland master curve preparation and crystallinity.

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Crystallinity 78% Crystallinity 66%

0.8

TR=50͠ ͠

0.6

0.4

0.2

0

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2

4 6 Log t'(min)

8

10

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Fig. 6 Master curves of creep compliance of Q-POM which were crystallinity %=66%, 72% and 78%

Crystallinity:72% Crystallinity:66%

0.8

TR=50͠ ͠ C =78% R

0.6

0.4

0.2

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-2

Crystallinity 78%

1

Crystallinity 72%

Creep compliance Dc(t',T)(1/GPa)

Creep compliance Dc(t',T)(1/GPa)

1

-2

0

2

8

10

12

Fig.7 Grand master curve of creep compliance  of Q-POM having %=66%, 72% and 78%

0.05

0 T =50͠ ͠ C =78% R

T =50͠ ͠C =78%

R

R

R

-0.5 Time shift factor aTt'

0.04 Modulus shift factor aDc

4 6 Log t"(min)

0.03

-1

-1.5

0.02

0.01

0

-2

-2.5

78

76

74

72

70

68

Crystallinity (%)

(a)

66

64

-3

78

76

74

72

70

68

66

64

Crystallinity (%)

(b)

Fig.8 (a),(b) Modulus shift factor (a) and time shift factor (b) of Q-POM These curves were perpendicularly and horizontally moved to superpose and the gland master curve of Q-POM is obtained finally as shown in Fig.7. Because, the time shift factor which was shift amount to vertical direction and Modulus shift factor which was shift amount to horizontal direction made straight line to crystallinity, it was confirmed that the crystallinity was linear shift factor as same as temperature as shown in Fig. 8 (a) and (b). From the results, it was confirmed that the crystallinity is an important factor for the superposition principle as same as temperature and time in creep behavior. The verification of “Time-temperature and crystallinity superposition principle” of A-POM Fig. 9 shows the grand curve of creep compliance on A-material which was drawn with three master curves obtained by specimen having crystallinity from %=74% to 78%. This value of creep compliance was usually lower than the grand curve on Q -POM but the curve rapidly rise up and reach to the grand master curve of A- POM according to time. The relationship between sift factor and crystallinity shows smooth line as same as Q-POM. It was confirmed that the crystallinity was a linear factor as same as temperature and time for creep behavior as Q-POM. This means it was clarified that to apply “Time-temperature and crystallinity superposition principle” in the creep deformation was possible. From the result, the creep compliance curve is dependent on not only time and temperature but also fiber volume fraction and crystallinity. To make clear the effect of cooling speed, master curve was compared.  In Fig. 10, the mater curves of Q-POM and A- POM which have the same crystallinity of 78%. Even if the crystallinity of POM is the same, both master curves were clearly different.

279 1

1 Crystallinity 78%

50͠ ͠ 70͠ 90͠ 110͠ 130͠

Creep compliance Dc(t',T)(1/GPa)

Creep compliance Dc(t',T)(1/GPa)

Crystallinity 76% Crystallinity 74%

0.8

TR=50͠ ͠ C =78% R

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4 6 Log t"(min)

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A-material

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4 6 Log t'(min)

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Fig.9 Grand master curve of A-POM having 74%,  Fig.10 Comparison of master curves obtained by 76% and 78%      Q-material and A-POM THE EFFECT OF CRYSTALLIZATION ON CREEP BEHAVIOR ON GFRPOM Discussion of “Time-temperature superposition principle” of Q-GFRPOM Fig.11 shows the three master curves of creep compliance on which crystallinity were 63%, 70% and 75% of GFRPM with Vf=2.8%.It was confirmed that fiber composing decreased not only the compliance value but also rate of creep compliance progress. The shifting factor made straight lines on Arrhenius type figure. It was confirmed that creep behavior can be calculated for GFRPOM having Vf=2.8% by “Time-temperature superposition principle”. Because the three master curves were very smooth and also they have almost same shape, it was discussed the superposition principle for time, temperature and crystallinity. After the curve of C=75% was decided as a standard and other curves were superposed, the grand master curve was finally obtained as shown in Fig 12. Used two shift factor for modulus and time showed straight line. From smooth grand master curve and the two straight shift factors, it was confirmed the verification of “Time-temperature and crystallinity superposition principle”. Fig.13 shows the master curves of creep compliance on GFRPOM which volume fraction was Vf=8.7% of Q-POM and crystallinity were 62%, 67% and 73%. The increase rate of the compliance in the creep time becomes gentle but mixed fiber clearly reduced the creep deformation than master curve of pure resin. As well as the case of GFRP of Vf=2.8%, the gland master curve was drawn using the group of master curves. Fig.14 presented the gland master curve of Q-GFRPOM Vf=8.7 as same as Q-POM which was drawn by the time shift factor and modulus shift factor. From these graphs, it was confirmed that for Q-GFRPOM, crystallinity is a factor for creep behavior. 0.6

0.5 Crystallinity 75%

Crystallinity 70%

0.5

Crystallinity 70%

Crystallinity 63%

Creep compliance Dc(t',T)(1/GPa)

Creep compliance Dc(t',T)(1/GPa)

Crystallinity 75%

TR=50͠ ͠

0.4 0.3 0.2 0.1 0

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4 6 Log t'(min)

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Fig.11 Master curves on V=f2.8% with C=63%, 70% and 75% of Q-GFRPOM

R

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4 6 Log t"(min)

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Fig. 12 Grand master curve of on Vf=2.8% with C=63%, 70%and 75% of Q-GFRPOM

280 0.25

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Crystallinity 73% Crystallinity 67%

Crystallinity 67%

0.25

Creep compliance Dc(t',T)(1/GPa)

Creep compliance Dc(t',T)(1/GPa)

Crystallinity 73% Crystallinity 62% TR=50͠ ͠

0.2 0.15 0.1

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TR=50͠ ͠ C =73% R

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4 6 Log t'(min)

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Fig.13 Master curves of Q-GFRPOM of Vf=8.7% having C=63%, 70% and 75%

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4 6 Log t"(min)

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Fig.14 Grand master curve of Q-GFRPOM of Vf=8.7% having=63%, 70% and 75%

Discussion of “Time-temperature superposition principle” of A-GFRPOM Fig. 15 shows three master curves of creep compliance which were obtained by GFRPOM of Vf=2.8% with crystallinity70%, 73% and 75%. These master curves were very general and they have almost the same shape. And the relationship between shift factors and temperature showed a straight line on Arrhenius type graph on each material. It was found that “Time-temperature superposition principle” can be applied to A-GFRPOM as same as the case of Q-GFRPOM. Increasing crystallinity, because activation energy increased, creep deformation clearly decreased. Fig.16 shows grand curve of the creep compliance for A-GFRPOM of Vf =2.8, which the crystallinity was 65, 67 and 73%. Fig. 17 shows three master curves of creep compliance which were obtained by A-GFRPOM of Vf= 8.7% with crystallinity C=65%, 69% and 73%. It was found that the shift factor shows a linear relationship to time too. And Fig.18 shows the grand curve of the creep compliance for A-GFRPOM of Vf =8.7%, which the crystallinity was 65, 67 and 73% using modulus shift factor and time shift factor. The value of the compliance lowers, as the fiber volume fraction increases. The increase rate of compliance during creep test on A-GFRPOM goes slow down than the case on Q-GFRPOM. And the relationship between shift factors and temperature showed a straight line on Arrhenius type graph on the crystallinity. Finally, it was concluded that that “Time-temperature and crystallinity superposition principle” can be applied to A-GFRPOM as same as Q-GFRPOM. 0.4

0.4

Crystallinity 75%

Crystallinity 75% Crystallinity 73%

Creep compliance Dc(t',T)(1/GPa)

Creep compliance Dc(t',T)(1/GPa)

0.35

0.35

Crystallinity 70%

0.3

TR=50͠ ͠

0.25 0.2 0.15 0.1

Crystallinity 70%

0.3

TR=50͠ ͠ C =75%

0.25

R

0.2 0.15 0.1 0.05

0.05 0

Crystallinity 73%

0

-2

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2

4 6 Log t'(min)

8

10

Fig.15 Master curves of Vf=2.8% having C=70%, 73% and 75% of A-GFRPOM

-2

0

2

4 6 Log t"(min)

8

10

Fig.16 Grand master curve of on Vf=2.8% of A-GFRPOM having C=70%, 73% and 75%

281 0.25 Crystallinity 73%

Crystallinity 73%

Crystallinity 69%

Crystallinity 69%

Creep compliance Dc(t',T)(1/GPa)

Creep compliance Dc(t',T)(1/GPa)

0.25

Crystallinity 65%

0.2

TR=50͠ ͠

0.15

0.1

0.05

0

-2

0

2

4 6 Log t'(min)

Fig.17 Master curves on A-GFRPOM Vf=8.7% .with C=65%, 69% and 73%

TR=50͠ ͠ C =73% R

0.15

0.1

0.05

0

10

8

Crystallinity 65%

0.2

-2

0

2

4 6 Log t"(min)

8

10

Fig.18 Grand curves on A-GFRPOM Vf=8.7% with C=65%, 69% and 73%

THE EFFECT OF THE SHAPE OF GRAIN ON MASTER CURVE OF CREEP COMPLIANCE From previous result, the creep compliance curve is dependent on not only time and temperature but also fiber volume fraction and crystallinity. To make clear the effect of cooling speed in other word the effect of grain size, master curves of Aand Q-GFRPOM having the same crystallinity were compared.  Fig.10 shows the mater curves of Q- and A- POM which have the same crystallinity of 78%. Though the crystallinity of this POM is the same, both master curves were clearly different. The value of compliance on A- POM was smaller than the value of Q-POM because modulus of Q-POM was lower than them of A-POM but the rate of creep progress on A-POM was higher than the speed of Q-POM, and their shape of creep compliance was different. It is not possible that compliance curves are superimposed by moving after all. It is proven that to superimpose two master lines on one, even if the compliance curve is moved, is not possible. It was confirmed that the principle is not established was not possible, when the crystal grain shape is different, though “Time, temperature and crystallinity superposition principle” was established on the materials of the same grain shape. To the make clear difference of the master curves between A- and Q-GFRPOM, specimen having the same crystallinity were prepared. They were C%=75% for GFRPOM Vf=2.8% and C%= 73% for Vf=8.6%. Fig.19 and Fig.20 shows the master curves of for the case of Vf=2.8% and Vf =8.7% on each other. It is shown that both master curves are also similarly 0.2 50͠ ͠ 70͠ 90͠ 110͠ 130͠

0.35 0.3

Creep compliance Dc(t',T)(1/GPa)

Creep compliance Dc(t',T)(1/GPa)

0.4

TR=50͠ ͠

0.25 0.2

Q-material

0.15 0.1

0.15

TR=50͠ ͠

0.1 Q-material

0.05 A-material

A-material

0.05 0

50͠ ͠ 70͠ 90͠ 110͠ 130͠

-2

0

2

4 6 Log t'(min)

8

Fig 19 Comparison of master curves obtained with Q- and A- GFRPOM of Vf=2.8% and C=75%

10

0

-2

0

2

4 6 Log t'(min)

8

10

Fig.20 Comparison of master curves obtained with Q- and A- GFRPOM of Vf=8.7% and C=73%

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dependent on the grain size as same as the case of resin and that the shape of the curve is different, and that it does not complete the superposition in the simplicity. The creep behavior of materials of different grain size in the equal crystallinity was compared. As the result, though the creep compliance of A-POM in which grain size is large was lower than creep compliance of Q-POM in which grain size is small, it was confirmed that the gap between two master curves decreased with the progress in the time rapidly. The reason why the compliance lowers is dependent on the increase in the elastic modulus. This phenomenon is dependent on the increase of the proportion of long folded structure oriented to tensile direction, when the grain size is large. And, the proportion in amorphous division which disperses in the crystal grain increases, when the grain size increases, because amorphous division of the meeting division and boundary of grain decreases. The grain boundary with deformation blocked effect decreases, when grain size increases, and degree of freedom of the viscoelastic flow movement in the grain increases. As the result, in POM that the crystal grain was big, it was assumed that the rate of the increase of the creep compliance increased with the progress in the time rapidly. This means creep deformation depended on the not only crystallinity but also the shape of crystal grain. For this case, it means that the creep behavior can not be estimated by “Time-temperature and crystallinity superposition principle” CONCLUSION Based on “Time-temperature superposition principle”, the creep deformation on several POM resin and GFRPOM having some kind of crystallinity clarified calculation and that it can estimate it. In this paper, it was found that the decrease of the creep deformation quantity was possible by increase of crystallinity as same as the fiber mixing and the effect of crystallinity is predictable by the calculation with the theory of “Time-temperature and crystallinity superposition principle”. And it was found that “Time-temperature and crystallinity superposition principle” was not established if the crystal structure differs, even if the crystallinity is same. Finally, it was clarified that that the creep deformation quantity decreases was possible and that it is predictable, if temperature- time, crystallinity and fiber volume fraction superposition principle is applied, when the shape of grain did not change by heat treatment. Reference (1)Ming Chen㧘Shiou-Chang Chao㧘“Thermal Stability and Nonisothermal Crystallization of Short Fiber-Reinforced Poly (ether ether ketone) Composites”㧘Journal of Polymer Science: Part B: Polymer physics, 36, 12, pp. 2225-2235, 1998 (2)Biswas K. K.㧘Somiya S.㧘Endo J.㧘“Creep Behavior of Metal Fiber-PPE Composites and Effect of Test Surroundings”㧘 Mechanics of Time-Dependent Materials, 3, 1, pp. 85-101, 1999. (3) Somiya S., Yamada K. and Sakai T.,”Bending creep Behavior of Glass Fiber Reinforced Polyoxymethylene”, Innovative Developments Characterilizations and Applications of Composites, pp239-249, 2006. (4) Ming Chen㧘Chia-Ting Chung㧘“Crystallinity of Isothermally and Nonisothermally Crystallized Poly (Ether Ether Ketone ) Composites”㧘Polymer Composite, pp. 689-697, 1998. (5) Srinivas S㧘Wilkes G L㧘“Structural and relaxation studies during crystallization of New TPI polyimide”㧘Polymer㧘39㧘 23㧘pp. 5839-5851, 1998. (6) Biswas K. K.㧘Somiya S.㧘“Study of the Effect of Aging Progression on Creep Behavior of PPE Composites”㧘Mechanics of Time-Dependent Materials㧘3㧘4㧘pp. 335-350, 1999. (7) Somiya S., Creep Behavior of a Carbon-Fiber Reinforced Thermoplastic Resin, Journal of Thermoplastic Composite Materials, 7, 2 , pp.91-99, 1994. (8) Keating M. Y., Malone L. B. and Saunders W. D. “Annealing effect on semi-crystalline materials in creep behavior”, Journal of Thermal Analysis and Calorimeter, 69, pp37-52, 2002 (9) F. Y. C. Boey, T. H. Lee and K. A. Khor, Polymer crystallinity and its effect on the non-linear bending creep rate for polyphenylene sulphide themoplastic composite, Polymer Testing, 14, pp.425-438 2002. (10) Sakai T. and Somiya S.., “Estimating Creep Deformation of Glass-Fiber-reinforced Polycarbonate”, Mechanics of Time Dependent Materials, Vol. 10, No. 3, pp185-199. 2006.

Numerical simulation of hot imprint process of periodical lamellar microstructure into polycarbonate

Rimvydas Gaidys, Assoc. Prof., Kaunas University of Technology, Studentǐ str. 50, Kaunas, LT-51368, Lithuania Birutơ Narijauskaitơ, Phd. Student, Kaunas University of Technology, A. Mickeviþiaus str. 37, Kaunas, LT-44244, Lithuania Arvydas Paleviþius, Prof., Kaunas University of Technology, A. Mickeviþiaus str. 37, Kaunas, LT44244, Lithuania Giedrius Janušas, Assoc. Prof., Kaunas University of Technology, A. Mickeviþiaus str. 37, Kaunas, LT44244, Lithuania

ABSTRACT Thermoplastic polymers are frequently used in the industry. They represent the most important group of polymeric materials. Most of all, injection molding, extrusion, spinning and hot embossing are used for plastic processing. Today polymers are used in precision systems. So it is necessary to create new powerful modeling and analyzing tools. A finite element model for hot imprint process of periodical microstructure into polycarbonate has been developed. In the finite element model polycarbonate is assumed to be a nonlinear elasto-plastic material. The model covers the main three steps of hot imprint process: polycarbonate heating, imprinting and demolding. Periodical lamellar microstructure was chosen as die in the hot imprint process, because it is common structure in the practice. The model is solved using the heat transfer and the solid stress-strain application modes with thermal contact problem between die and polycarbonate. This multiphysics polycarbonate hot imprint model includes the heat transport, structural mechanical stresses and deformations resulting from the temperature distribution. Finite-element simulation of the hot imprint process has been performed using COMSOL Multiphysics. Nonlinear elasto-plastic model was created. It allows evaluation of temperature distributions and stresses in the polycarbonate during hot imprint process. Obtained theoretical results were compared with experimental. INTRODUCTION The fabrication of micro structures (using hot imprint technology) is an important technology due to its advantages as low cost, high efficiency and parallel operation [1,2]. On the other hand, typical problems still remain, such as low pattern fidelity, long hot imprinting time and structure defects especially when high aspect ratio patterns are fabricated [3,4,5]. Increasing of the imprinting area and decreasing of the structure size down into nanoscale requires of new and adapted hot imprint technologies. Mostly viscoelastic, viscoplastic, visco-elastic-plastic, hyperelastic and elasto-plastic material models are used to model the hot imprint process. In this paper, the finite element method (FEM) was used to analyze the heating, imprinting and demolding steps. Polycarbonate was modeled as elasto-plastic material, because it was analyzed below glassing temperature. In general, the simulation model and numerical results provide a useful understanding of the fundamental formation mechanism during the thermal imprint process and serve as a useful guide for specifying the optimal processing conditions for variety of thermal imprint applications. Hot imprint is a state-of-art thermal forging methods which is able to maintain at a low constant working displacement on the pattern of template, to form the complex shape onto the polymer substrate with heat. The thermal energy is used to melt the polymer layer so that it can flow and fill up the gaps in the template. When this energy is drawn away, the polymer substrate cools down and hardens.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_40, © The Society for Experimental Mechanics, Inc. 2011

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Materials properties Coupled time-depend thermo-mechanical analysis is essential with all necessary material parameters such as thermal conductivity (k in W/mK), density (U in kg/m3), heat capacity (cp in J/kgK), Young’s modulus (E in N/m2), Poisson’s ratio (Q) ant the thermal expansion coefficient (D in K-1) defined as function of temperature [6,7]. A transient analysis of heat transfer conduction is used for the thermal analysis. The main polymer parameters such as thermal conductivity, density and heat capacity in connection with temperature were taken from Comsol Multiphysics material library. Coupled thermo-mechanical contact problem of a polymer into nanometer-scale cavities during hot imprint with a temperature 421 K creating FE model was analyzed in this paper. Polycarbonate (PC) was chosen as the material for replication of microstructure. HOT IMPRINT PROCESS MODEL Hot imprint process of polycarbonate consists of three steps: heating, imprinting and demolding. In the model, the periodic microstructure of the mold was simplified and analyzed as a two-dimensional plain strain model of a unit cavity of the microstructure [8, 9]. Boundary conditions of the model according to its regularity and symmetry are presented in the Fig. 1. pressure symmetric in x-dir

master

T=f(T,t) MOLD

hm

symmetric in x-dir

slave

symmetric T

symmetric T

POLYCARBONATE hp symmetric in x-dir

symmetric in x-dir

fixed in x and y dir, y = 0 T = 293 K

Fig. 1 Schematic diagram of imprint process with boundary conditions. In the Fig. 1 hm corresponds to the depth of the cavity of the periodic microstructure and h p is the depth of the polycarbonate. The initial temperature of the mold and polycarbonate is 293 K. T f (T , t ) corresponds to the mold depending on heating temperature function on time (T – temperature (Kelvin) and t – time (second)). The model demonstrates nonlinear thermal-stress analysis with thermal contact, large deformations, and the use of elastoplastic material model [10]. Typical hot imprint process can be divided into three steps: heating, imprinting and demolding [11]. In our case: 1) Heating. The initial temperature of the mold and polycarbonate is 293 K, the same as the temperature of environment. In this step, when the stamp touches polycarbonate, during their initial contact, the heating of the mold begins up to chosen 421 K temperature. During the heating process of the mold, the heat is carried to the polycarbonate and it starts to deform against of the effect of the heat. The heating step takes very short time about t 2 ˜10 7 s. 2) Imprinting. During this process, the mold goes down and presses polycarbonate, at the same time the contact force between the mold and polycarbonate increases. Polycarbonate keeps deformation and plastic deformation appears. 3) Demolding. In this step, the hot mold ( T 421 K) is demolded and finally polycarbonate is cooled. Polycarbonate takes the form of the mold periodic microstructure.

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RESULTS Numerical simulation tool for hot imprint process of polycarbonate has been developed. COMSOL Multiphysics FEA was used to simulate three steps of hot imprint process: heating, imprint and demolding. Temperature field, displacement field, and stress field were analyzed in each step. Simulation results provided the insights of how the large deformation occurred during imprint process and the stress distribution on the polycarbonate. As example of modeling of the heating step, is presented in Fig. 2.

a) b) Fig. 2 Example of simulation: a - The mesh of the FEM model; b - Von Mises stress distribution and temperature fields in the polycarbonate during heating process after 1.1 ˜10 7 s. The imprint and demolding simulations were produced according to below presented material properties and boundary conditions. CONCLUSIONS 1.Usage of COMSOL Multiphysics FEA allowed to simulate three steps of hot imprint process: heating, imprint and demolding and the temperature field, displacement field, and stress field were got in each step. 2. The imprint step the maximum stresses were observed in empty cavities of the mold and in the intersection field with the polycarbonate (about 30 MPa) in heating simulation step. 3. In terms of absolute magnitude maximum displacements in the x direction reaches 77 nm. After imprint process some areas appeared to be unfilled due to polycarbonate features. 4. While demolding the mold, deformations, which mostly exceed (5 times) allowed strain limit, were observed in the area of empty cavity of the mold in contact with polycarbonate. REFERENCES [1] Heckele M., Schomburg W. K. Review on micro molding of thermoplastic polymers. Journal of Micromechanics and Microengineering, vol. 14, no. 3, p. R1-R14, 2004. [2] Quake S. R., Scherer A. From micro- to nanofabrication with soft materials. Science 290, p. 1536-40. [3] Yoshihiko H., Yoshida S., Nobuyuki T. Defect analysis in thermal nanoimprint lithography. J. Vac. Sci. Technol. B 21 2765–70, 2003. [4] Yoshihiko H., Takaaki K., Takashi Y. Simulation and experimental study of polymer deformation in nanoimprint lithography. Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures, B22, p. 3288-93, 2004. [5] Worgull M., Heckele M., Hetu J. F., Kabanemi K. K. Modeling and optimization of the hot embossing process for microand nanocomponent fabrication, J. Microlith., Microfab., Microsyst. 5, 011005, 2006. [6] Reiter J., Pierer R. Thermo-mechanical simulation of a laboratory test to determine mechanical properties of steel near the solidus temperature, Excerpt from the Proceedings of the COMSOL Multiphysics User's Conference. – Frankfurt, 2005. [7] Jin P., Gao Y., Liu T., Tan J., Wang Z., Zhou H. Simulation and experimental study on recovery of polymer during hot embossing, The Japan Society of Applied Physics, p. 06FH10-1 – 06FH10-4, 2009. [8] Lan S., Lee H. J., Lee S. H., Ni J., Lai X., Lee H. W., Song J. H., Lee M. G. Experimental and numerical study on the viscoelastic property of polycarbonate near glass transition temperature for micro thermal imprint process, Materials and design, vol. 30, p. 3879-3884, 2009. [9] Yao D., Virupaksha V. L., Kim B. Study on Squeezing Flow During Nonisothermal Embossing of Polymer Microstructures, Polymer engineering and science, vol. 45, no. 5, p. 652-660, 2005. [10] Hung C., Chen R. H., Lin C. R. The Characterization and Finite-Element Analysis of a Polymer under Hot Pressing, Advanced Manufacturing Technology, vol. 20, no. 3, p. 230-235, 2002. [11] Becker H., Heim U. Hot embossing as a method for the fabrication of polymer high aspect ratio structures, Sensors and Actuators, vol. 83, p. 130-135, 2000.

Rapid Characterization of Visco-elastic Properties of Polymeric Materials

Yejin Kim and Bongtae Han University of Maryland, College Park 2181 Glenn L. Martin Hall, University of Maryland, College Park, MD 20742

ABSTRACT: As polymers are widely used in electronic packaging, their visco-elastic behavior must be characterized in order to predict the behavior of package assemblies during manufacturing and operation. Current known testing methods for visco-elastic properties are often too time-consuming or too complex to be implemented as the specimen preparation and the testing conditions are critical to reliability and repeatability of measurements.

In this work, a method is proposed to characterize the visco-elastic behavior of polymeric materials utilizing a fiber Bragg grating (FBG) sensor. The specimen is formed by curing a cylindrical shape of polymer around a FBG. An instantaneous mechanical load is then applied to the specimen while equilibrated at a temperature inside an environmental chamber. As constant stress is applied to the polymer substrate, strain is applied to the fiber, and the Bragg wavelength (BW) shift is documented as a function of time. The BW shift data can be converted into the creep compliance directly from the theoretical behavior of the FBG. The creep compliance obtained from each temperature can be converted into the time dependent relaxation modulus using a de-convolution process [1], which results in relaxation modulus data and the initial modulus at each temperature. The individual relaxation modulus curves can be shifted along the logarithmic time axis to produce the “master carve” for a given reference temperature based on the study of linear viscoelastic materials [2]. With the Williams-Landel-Ferry (WLF) shift function, the shift factors can be fit to approximate any arbitrary temperature values within the measurement range so that the master curve can be described by continuous functions. The total behavior of the master curve can then be approximated using a Prony series. Finally, the visco-elastic properties are then defined using numerical modeling with the WLF function and the Prony-series.

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_41, © The Society for Experimental Mechanics, Inc. 2011

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Estimation of Fatigue life of Cortical Bone Considering Viscoelastic Properties and Damage Mechanics Takenobu Sakaia, Keita Yasuia and Shuichi Wakayamaa a Tokyo Metropolitan University, 1-1, Minami-Osawa, Hachioji, Tokyo, 192-0397, Japan ABSTRACT Recently, the Osteoporosis victims increase in the senior citizen. Therefore, the danger of the stress fracture due to the decrease in bone strength is pointed out. Especially, the damage accumulation behavior of the bone in the cyclic load becomes a problem for the fatigue of the bone. Moreover, it is necessary to consider the viscoelastic property for the prediction of the fatigue behavior, because the bone is a viscoelastic material. In this study, our objective is to estimate the fatigue behavior that considers the viscoelasticity behavior and the damage accumulation behavior as damage mechanics. The viscoelastic properties were revealed with the strain rate tests, and we calculated the 3-mode Maxwell parameters with these results. The damage accumulation behaviors were measured with the Acoustic Emission method. Using these results, we could estimate the fatigue behavior of cortical bone with the viscoelastic parameter and cumulative AE energy. Key words : Cortical bone, Fatigue, Viscoelasticity, Acoustic emission, Strain rate dependency, 3-mode Maxwell model Introduction Recently, the Osteoporosis victims increase in the senior citizen. Therefore, the danger of the stress fracture due to the decrease in bone strength is pointed out. Especially, the damage accumulation behavior of the bone in the cyclic load becomes a problem for the fatigue of the bone. Moreover, it is necessary to consider the viscoelastic property for the prediction of the fatigue behavior, because the bone is a viscoelastic material. The bone is a specifically shaped and structured organ, which is made of bone tissue. The bone tissue, in turn, is mainly made of bone material (comprising mineral, collagen, and water) and bone cells. The cells provide the bone with the ability to respond to mechanical loading, but in the end, it is the predominant loading that determines the bone shape and macroscopic structure – not the cells. In recent years, many researchers seek to find how to give the effective mechanical stimulation to the bone at low amplitude and broad frequency strain. Flieger et al. (1998), Tanaka et al. (2003) reported the results of 0 Hz ~ 50 Hz[1, 2]. Furthermore, Hsieh et al. (1999)[3] tried to characterize the mechanical strain in the rat tiba or ulna during applied loading at frequencies of 1, 2, 5, 10 and 20 Hz. And other researchers investigated the mechanical character [4 – 11]. However, few researchers could estimate with viscoelastic parameter and damage mechanics depended on test time. In this study, our objective is to estimate the fatigue behavior that considers the viscoelasticity behavior and the damage accumulation behavior as damage mechanics. Calculation of the viscoelastic parameters by static compression testing On the static strain rate test, we can describe strain İ(t) and stress ı(t) as follows; ߳ሺ‫ݐ‬ሻ ൌ ߳௧ ሺ‫ݐ‬ሻ ൌ ܴ‫ݐ‬ሺͳሻ ߪሺ‫ݐ‬ሻ ൌ ߪ௧ ሺ‫ݐ‬ሻሺʹሻ where, the subscript t shows the static strain rate testing and R is the strain rate. Then, these two equations were Laplace transformed, following equations were obtained. ܴ ሺ͵ሻ ‫ݏ‬ଶ ɐ ഥሺ•ሻ ൌ ߪഥ௧ ሺ‫ݏ‬ሻሺͶሻ

Ԗതሺ•ሻ ൌ ߳ഥ௧ ሺ‫ݏ‬ሻ ൌ

Equations (3) and (4) were substituted for the following eq. (5). ɐ ഥሺ•ሻ ൌ ‫ܧ‬ሺ‫ݏ‬ሻ߳ҧሺ‫ݏ‬ሻሺͷሻ Equation (5) showed the pseudo-elastic. Then, we got eq. (6).

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_42, © The Society for Experimental Mechanics, Inc. 2011

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ɐ ഥ௧ ሺ•ሻ ൌ ‫ܧ‬ሺ‫ݏ‬ሻ

ܴ ሺ͸ሻ ‫ݏ‬ଶ

When eq. (7) was substituted for eq. (6), eq. (8) was obtained. തതത௥ ሺ‫ݏ‬ሻሺ͹ሻ ‫ܧ‬ത ሺ•ሻ ൌ ‫ܧݏ‬ ͳ തതത ‫ܧ‬௥ ሺ•ሻ ൌ ‫ߪݏ‬ഥ௧ ሺ‫ݏ‬ሻሺͺሻ ܴ where, the Er shows the relaxation modulus. Equation (8) was inverse Laplace transformed, then eq. (9) was obtained. ‫ܧ‬௥ ሺ‫ݐ‬ሻ ൌ

ͳ †ɐ୲ ሺ–ሻ †ɐ୲ ሺ–ሻ ൌ ሺͻሻ  †– †Ԗ୲ ሺ‫ݐ‬ሻ

Using the strain and stress of the static compression testing, we can calculate the relaxation modulus. To calculate the viscoelastic parameter, we use the 3-mode Maxwell model. The 3-mode Maxwell model was described as follows; ߪሺ‫ݐ‬ሻ ൌ ቎ ୣ ൅

݇‫ܦ‬ ‫ܦ‬൅

ͳ ߬

቏ Ԗሺ–ሻሺͳͲሻ

where, ke and k are elastic constants and IJ is the relaxation time. Then, eq. (10) was Laplace transformed and shown in eq. (11). ߪതሺ‫ݏ‬ሻ ൌ ቎ ୣ ൅

݇‫ݏ‬ ‫ݏ‬൅

ͳ ߬

቏ Ԗതሺ•ሻሺͳͳሻ

By the boundary conditions of relaxation testing, the equations as follows were obtained. ߳ሺ‫ݐ‬ሻ ൌ ߳଴ ሺͳʹሻ ߪሺ‫ݐ‬ሻ ൌ ߪ௥ ሺ‫ݐ‬ሻሺͳ͵ሻ The Laplace transformed equations were eq. (14) and eq. (15). ߳଴ ሺͳͶሻ • ߪതሺ‫ݏ‬ሻ ൌ തതതሺ‫ݏ‬ሻሺͳͷሻ ߪ௥

߳ҧሺ‫ݏ‬ሻ ൌ

Equations (14) and (15) were substituted for eq. (11), we got eq. (16) തതത௥ ሺ‫ݏ‬ሻ ൌ ‫ܧ‬

ɐഥ୰ ሺ‫ݏ‬ሻ ݇௘ ݇ ൌ ൅ ሺͳ͸ሻ ߳଴ ‫ݏ ݏ‬൅ͳ ߬

Then, eq. (16) was inverse Laplace transformed, eq. (17) was obtained. ௧

‫ܧ‬௥ ሺ–ሻ ൌ ݇௘ ൅ ݇݁ ିఛ ሺͳ͹ሻ In this study, the 3 types of strain rate testing were carried out. We can calculate the relaxation modulus from these results by using eq. (9), and the viscoelastic parameters were calculated using eq. (17). Experimental Procedure Bone Specimens Bovine cortical bone was used as a material investigated in this study. There are two types of cortical bones with

291 d different micrrostructure; i.e. Haversian n and plexiform bones. Pllexiform bone e was used in n this study. Figure 1 shows a microstruccture of bovvine plexiform m bone and d a schema atic geometrry of dyaphisis. It is ob bserved in the t m microstructurre perpendiccular to the longitudinal l axis (Fig. 1 (a)) that lam mellar structu ure is develo oped along tthe ccircumferential axis. A nu umber of lacu unae were allso observed d among lame ellae, which w would contrib bute to fractu ure b behavior as flaws f under loading. Sincce lamellae a are aligned along a the long gitudinal axiss, elastic properties of bo one a are orthogon nal. Specimen ns with the size of 5 mm x 5 mm x 12.5 mm were cut c from the diaphysis of bovine femo oral b bone, which has the tub bular structurre as shown in Fig. 1(b). The symbo ols ‘x1’, ‘x2’ a and ‘x3’ refe er to the radial, ttangential an nd longitudinal directions, respectivelyy. In the follo owing tests, compressive c load was ap pplied along x3 a axis as living g bodies. The e specimenss were kept frreezing at -2 20ºC, then cu ut under flow wing water an nd kept wet ju ust b before the mechanical m tessts.

(a) Optical m micrograph o of cross sectio on (b) Schematic representatio on of dyaphissis Fig. 1 C Cross section of bovine bo one (Plexiform m bone) and geometry off dyaphisis S Static Comprression Test B Before the cyyclic compre ession test, sttatic compresssion tests of o bovine corttical bone (plexiform bone e) were carriied o out in the airr at room tem mperature to calculate the e viscoelasticc parameter. Uniaxial com mpressive loa ad was appliied a along the lon ngitudinal (x3) axis of the b bone specime ens under 3 types consta ant crosshead d speed of 1.0, 0.1 and 0..01 m mm/min. unttil the applied d load reached ~10MPa. Testing systtems of static compressio on test and fatigue f test a are sshown in Fig g. 2, schema atically. Durin ng the tests, longitudinal strain was measured m ussing a strain gage attach hed d directly on th he specimen. To discuss the fatigue b behavior afte er the static ccompression tests, the sp pecimens mu ust n not have dam mages. To de etect the materials’ dama age, we used d Acoustic Em mission (AE)) Systems. A wide band AE A ssensor (NF; AE-900M) was w attached on the speccimen. The surfaces s of sp pecimens we ere wiped be efore gluing tthe sstrain gage and AE sen nsors with cyyano-acrylate e adhesives and moiste ened immediately after th he attachme ent. Detected AE E signals were e amplified b by a pre-amp plifier (Gain; 60 6 dB) throug gh the band pass filter with a terminall of tthe a pre-am mplifier). Amplified AE sign nals, load and strain were e then record ded by an AE E analyzer (AMSY-5; Valle en). T The mechanical noise be etween the sp pecimen and loading plate was restrained using Te eflon sheets on the platess.

Fig. 2 Sch hematic diagrram of static compression n test system m

292 Cyclic Compression After the static compression test for calculating the viscoelastic parameter, cyclic compression tests were performed in the air at room temperature. The bone specimen was subjected to sinusoidal load of 3 Hz along the longitudinal (x3) axis of the bone. Stress ratio (minimum stress divided by maximum stress) was 0.05, and mean stress was 50MPa. The damage accumulation in bone under cyclic loading was monitored by AE measurement. The system was almost same as the static compression test (Fig. 2). In order to keep the specimens as wet as possible, surfaces of the specimen were moistened by the wet cotton. Experimental Results Static Compression Test To calculate the viscoelastic parameter, the static compression tests were carried out with 3 types of strain rate. Figure 3 shows a typical result of stress-strain relationship for the static compression test. As shown in this figure, the stress increased with the strain rate increase. This is because the viscoelastic behavior made these change between different strain rates. Using these results, (9) and (17), we calculated the viscoelastic parameters, Ee, E1 and IJ, and the results are shown in table 1. During static compression tests, AE measurement was carried out to detect the mechanical damage. However, we could not detect the damage and microdamage by AE measurement. Therefore, we use same specimens for the fatigue tests.

1mm/min 0.1mm/min 0.01mm/min

10

㻌

Stress (MPa)

15

5

0 0.0000

0.0005

0.0010

Strain

0.0015

Fig. 3 The results of static compression tests with 3 types strain rate test Table 1 Viscoelastic parameters Ee (MPa) E1 (MPa) IJ

7000 800 100

Cyclic Compression Test Figure 4 shows a typical result of strain-time relationship and the behavior of cumulative AE energy for the fatigue compression test. This strain shows the mean strain during fatigue test. As shown in this figure, the mean strain decreased little by little, and they vibrated from 2000 sec. to 3500 sec. It assumed that this vibration was caused by some environmental factors. In this paper, we ignored this vibration; therefore, we calculated the smooth curve of mean strain. Figure 5 shows the smoothed mean strain. As shown in this result, smoothed mean strain decreased little by little. It was considered that this decrease had two affected factors. One is damage accumulation, and the other is viscoelastic behavior as creep deformation. In the condition of fatigue test, there is constant mean stress about 80MPa, and the stress worked as creep load. To divide the effect of two factors on fatigue tests, we estimate the creep behavior under the mean stress. The estimated data was shown in fig. 5. This curve shows only the viscoelastic behavior, that is, this curve shows the result of fatigue test without the damage accumulation. On the other hand, cumulative AE energy shows only the damage accumulation, since the AE sounds are generated by

293 some cracking. Therefore, we tried to make the curve that shows the difference between estimated and experimental data, and it is shown in fig. 6. As shown in fig. 6, the curves, which indicate the difference data, and smoothed cumulative AE energy show good agreement without initial stage. When we use this relationship between difference and cumulative AE energy, we can estimate the mean strain during fatigue test.

0.16

0.14

150000 100000

0.12

Strain Cumulative AE Energy 0.10

0

1000

2000

3000

4000

50000 0

Cumulative AE Energy (10

Strain (PH)

200000

-14

2

V s)

250000

Time (s) Fig. 4 Mean strain and cumulative AE energy during fatigue test

Strain (PH)

㻜㻚㻝㻠

㻝㻜㻜㻜㻜㻜 㻜㻚㻝㻞 Strain Calculated Strain Cumulative AE Energy Smooth of strain Smooth of Cumulative AE Energy

㻜

㻜㻚㻝㻜 㻜

㻞㻜㻜㻜 Time (s)

Cumulative AE Energy (10

㻞㻜㻜㻜㻜㻜

-14

2

V s)

㻜㻚㻝㻢

㻠㻜㻜㻜

Fig. 5 Smoothed mean strain and cumulative AE energy and estimated strain curve during fatigue test

300000

2 -14

0.003 0.002

200000

0.001 0.000 -0.001

100000

-0.002 -0.003 -0.004

Difference of strain Smooth of Cumulative AE Energy

-0.005 -0.006

0

1000

2000

Time (s)

3000

4000

0

Smoothed cumulative AE energy (10

Difference of strain (PH)

0.004

V s)

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Fig. 6 Difference between experimental and estimated strain data and smoothed cumulative AE energy Conclusions The static and cyclic compression tests of bovine cortical (plexiform) bone were carried out in this study. The viscoelastic parameters (3-mode Maxwell model and Boltzmann’s superposition principles) of bovine cortical bone were calculated with the results of the static compression tests with 3-types of strain rate. Damage accumulation during the fatigue test was monitored by AE measurement. The difference between experimental data and estimated data without damage information and smoothed cumulative AE energy were shown good agreement. Using the viscoelastic parameters and cumulative AE energy, we can estimate the mean strain during fatigue test. Reference 1) Flieger, J., Karachalios, Th., Khaldi, L., Raptou, P., Lyritis, G., 1998. Mechanical stimulation in the form of vibration prevents postmenopausal bone loss in Oveariectomized rats. Calcif. Tissue Int. 63, 510. 2) Tanaka, S.M., Li, J., Duncan, R.L., Yokota, H., Burr, D.B., Turner, C.H., 2003. Effects of broad frequency vibration on cultured osteoblasts. J. Biomech. 36, 73. 3) Hsieh, Y.F., Wang, T., Turner, C.H., 1999. Viscoelastic response of the rat loading model: implications for studies of strain-adaptive bone formation. Bone 379, 25–33. 4) The Japan Society of Mechanical Engineers ed., Biomechanical Engineering: A First Course, (1997), p.101, The Japan Society of Mechanical Engineers. 5) Mori, S., Fatigue in Bone (in Japanese), J. Soc. Biomechanisms Japan, Vol. 21, No. 2, (1997) pp.81-85. 6) Lee, S. C., Coan, B. S. and Bouxsein, M. L., Tibial Ultrasonic Velocity Measured In Situ Predicts the Material Properties of Tibial Cortical Bone, Bone, Vol. 21, No. 1, (1997) pp.119-125. 7) Caler, W. E. and Carter, D. R., Bone Creep-Fatigue Damage Accumulation, J. Biomechanics, Vol. 22, No. 6, (1989) pp.625-635 8) Pattin, C. A., Caler, W. E. and Carter, D. R., Cyclic Mechanical Property Degradation During Fatigue Loading of Cortical Bone, J. Biomechanics, Vol. 29, No. 1, (1996) pp.69-79. 9) Hirai, T., Katayama, T., Inoue, N. and Yamamoto, H., Nonlinear Deformation of Cortical Bone under Statical Tensile Load, Trans. JSME, Ser. A, Vol. 58, No. 551, (1992) pp.34-39. 10) Niimoto, M., Akahori, T., Kim, J., Tajima, H. and Kodama, T., Fracture Characteristics and Microstructures at Different Positions of Bovine Femoral Compact Bone, Trans. JSME, Ser. A, Vol. 69, No. 687, (2003) pp.121-128. 11) Lasaygues, P. and Pithioux, M., Ultrasonic Characterization of Orthogonal Elastic Bovine Bones, Ultrasonics, Vol. 39, (2002) pp.567-573.

Crack initiation and viscoplasticity in polyethylene joint replacement components Jevan Furmanski*, P. Abhiram Sirimamilla+, Clare M. Rimnac+ *MST-8, Mail Stop G755, Los Alamos National Laboratory, Los Alamos, NM, 87545 +Department of Mechanical and Aerospace Engineering, Case Western Reserve University, Cleveland, Ohio, 44121 Abstract Ultrahigh molecular weight polyethylene (UHMWPE) is an abrasion resistant and bioinert polymer widely used as a bearing material in total joint replacements. Recent reports of fracture and crack initiation in these systems make the prediction of crack initiation a primary concern. Past work in assessing the resistance to crack propagation in UHMWPE has typically ignored the creeping (quasi-static) constitutive contribution to the process of failure. We conducted constant load experiments on pre-notched fracture specimens and observed the elapsed time to crack initiation and subsequent crack velocity as a function of the applied load. A hyperelastic-viscoplastic constitutive model was calibrated to three uniaxial tensile experiments; one at a constant crosshead velocity, and two delayed yield creep (viscoplastic) experiments at an engineering stress either slightly or moderately below the short-term yield strength, until the strain saturated or failure. The crack initiation phase of the fracture experiment was modeled in ABAQUS, predicting the time-dependent J-integral and average molecular chain stretch to be single-valued at the experimentally obtained crack initiation times for the three tested boundary conditions. The crack initiation time and propagation velocity were also found to scale with the applied load in agreement with an analytical power-law viscous fracture model. Introduction Cross-linked ultrahigh molecular weight polyethylene (UHMWPE) has seen increasing use as a compliant bearing material in total joint replacements (e.g., artificial knees and hips). While a highly cross-linked UHMWPE can experience volumetric wear less than 20% of that for conventional (uncross-linked) formulations in a wear simulator [1], the resistance to crack propagation concomitantly is deleteriously reduced by 50% [2, 3]. This increased wear resistance is of great importance to the clinical community, as microscopic UHMWPE wear debris in the joint capsule have been linked to a runaway immune response termed wear-mediated osteolysis [3], which ultimately can result in large defects in the surrounding bone and failure of the implant. Thus, cross-linked UHMWPE is widely used for its wear resistant properties, despite it being more susceptible to crack propagation and fracture. While all of the structural implications of this trade-off have not yet been elucidated, recent reports of the clinical fracture of cross-linked UHMWPE total hip replacement components demonstrate their susceptibility to crack initiation and propagation near a notch under typical service conditions (Fig.1) [4].

Fig.1 Two cross-linked UHMWPE total hip replacement bearings that fractured in patients under ordinary service conditions. Adapted from [4].

T. Proulx (ed.), Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 99999, DOI 10.1007/978-1-4614-0213-8_43, © The Society for Experimental Mechanics, Inc. 2011

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Thus, it is clear that the material resistance of various grades of UHMWPE to crack initiation and propagation in a notchaffected region is of particular interest. Unfortunately, the fatigue and fracture behavior of UHMWPE and remains only marginally understood; the purpose of this work is to elucidate the intrinsic failure of these materials and provide a clear failure criterion for evaluating the structural suitability of a given formulation of UHMWPE in a particular component design. The crack propagation behavior of UHMWPE has been shown to behave in a static (non-cyclic) mode, i.e., predictable from quasi-static experiments, with no true cyclic damage mechanism [5, 6], i.e., the average crack propagation velocity in a variety of cyclically loaded correlates to the maximum of the waveform, independent of mean stress, stress range, waveformspecific effects, or frequency effects. Static mode fatigue behavior is expected for a viscous solid, where relaxation processes ahead of a crack tip dominate the deformation and energy dissipation phenomena in the fracture process zone. A mathematical model due to J.G. Williams predicts crack propagation velocity (da/dt) and time to crack initiation (ti), related to the rate of energy accumulation near the crack tip under static loading (Eqs.1-3) [7].

(Eq.1)

(Eq. 2)

(Eq. 3) This model employs a power-law increasing J with exponent n (often, but not necessarily, derived from a time power-law increasing material compliance), time constant IJ0, and instantaneous J-integral J0 (Eq.1), and an exponent n which roughly is seen to govern the overall severity of the viscous response. The behavior described by this model for a notched component is schematized in Fig.2 for a typical ramp-and-hold load controlled experiment.

Fig.2 Schematic of Williams time-dependent J on log-log axes. An initial load ramp applied to a notch increases J to J0, after which the load is held and J increases according to a power-law with exponent n. Ultimately J reaches Jc, the threshold for crack initiation, and the crack begins to propagate at an approximately constant J.

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When the J-integral (under a static load) eventually reaches the value necessary for crack initiation, Jc, Eq.1 can be manipulated to give the time to crack initiation, ti. It can also be shown (Eq.3) that the subsequent crack propagation velocity depends on the initially applied J in a similar power law, with a scaling coefficient Q which is itself a function of the compliance, cohesive stress, and toughness. The exponential factor n unifies Eqs. 1-3 and ties them to the viscous J response, providing a means of predicting the fatigue performance of a material based on its dynamic constitutive behavior. Mechanistically-motivated constitutive modeling of UHMWPE has produced numerical results that reproduce its hyperelastic-viscoplastic material behavior in wide variety of loading and boundary configurations [8-10]. The Hybrid Model [9] and Three Network Model (TNM) [11] are two very promising constitutive models for UHMWPE that combine hyperelastic rubber-elastic responses with power law viscoplastic behavior representing amorphous chain disentanglement and crystal plasticity, in addition to a hardening response for the crystal plasticity. Both models predict the average molecular chain strain (İch) in the amorphous polymer phase, which is related to the average of the trace of the right stretch tensor (Eq.4), (Eq.4) and has been shown to predict tensile rupture in UHMWPE better and more generally than typical engineering failure criteria [10]. This factor does not appear in the Williams analytical model, but serves as an additional quantity of interest for predicting the local failure of material near a notch root or crack tip. In this work we conducted constant load crack initiation and propagation experiments on pre-notched compact tension specimens, and simulated the same experiment using FEA with the TNM constitutive model calibrated to three different uniaxial tensile experiments. We hypothesized that, in a notched UHMWPE specimen, the initiation time (ti) and crack propagation velocity (da/dt) under static loading are driven by the viscoplastic dynamic response of the material, and thus UHMWPE crack behavior can be predicted by an appropriately calibrated viscoplastic FEA model. It was also hypothesized that a critical value of J and İch each would correlate with crack initiation in UHMWPE. Materials and Methods Ram extruded rods of GUR 1050 UHMWPE resin were gamma irradiated with a dose of 65 kGy and subsequently remelted (Orthoplastics, Lancashire, UK), and were machined into notched round compact tension (CT) and waisted cylindrical tensile test specimens. The CT specimens had a crack length a=17 mm, length W=40 mm, thickness B=20 mm, and notch root radius R=0.25 mm. Three tests were run to failure under a static load of either 900 N, 800 N, or 750 N on an Instron servohydraulic frame. Crack propagation was recorded with a travelling microscope. Crack advance was reported as loss of ligament length (remaining material ahead of crack tip; a-W) to account for gross motion of the crack tip during the test. Ligament reduction was assumed to be due to both gross specimen creep and true crack growth. Tensile specimens had a gage diameter of D=7.67 mm and a gage length L=14 mm. One specimen was taken to failure at a monotonically increasing displacement of 30 mm/min. Two other specimens were tested in a creep mode with a constant engineering stress of either 14 or 20 MPa, which were respectively chosen to be moderately or marginally below the instantaneous yield stress of approximately 22 MPa. A video extensometer (Instron, Norwood, MA) recorded true strain throughout the tests. CT specimen deformation was numerically modeled with two-dimensional plane stress finite element analyses (FEA) in ABAQUS CAE 6.8 (Simulia, Providence, RI). The model consisted of 2450 8-node quad elements (CPS8R), with a strongly biased mesh (crack tip element 4x4 μm, Fig.3). No element failure was modeled, and so the FEA analysis was undertaken to understand the material behavior in the specimen prior to the instant of material failure at the crack tip.

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(a)

(b)

Fig.3 Deformed FEA mesh; whole model (a) and detail view of crack tip (b) The load was applied as a distributed pressure, linearly ramped over 0.1 s, on the horizontally constrained specimen pin holes. We employed the TNM constitutive model as a user material subroutine (Veryst, Cambridge, MA). The model parameters were iteratively fit simultaneously to the three sets of experimental monotonic and creep data until agreement with each test was achieved. The FEA was used to predict the creep-related ligament reduction, J-integral, and molecular chain strain (İch) at the notch root. J showed moderate path-dependence near the crack tip, so J was defined as the mean of the first 10 integration contours. Results The 800 N and 750 N tests both clearly showed logarithmic initial ligament reduction prior to crack initiation, consistent with FEA creep predictions (below). The ligament reduction data then transitioned to an approximately linear stable growth region followed by instability and failure (Fig.4a). True crack growth was determined by regression fitting the initial logarithmic specimen creep component of the ligament reduction and subtracting it from the total ligament reduction at all times (Fig.1B). Constant velocity crack growth initiated at a time ti, before which true crack growth was negligible (Fig.4b, Table 1). The ti for 907 N was difficult to confidently extract; hence, related values in Table 1 are estimates.

Fig.4 a) Total ligament reduction in CT tests, showing an initial specimen creep-dominated logarithmic response, followed by approximately 2 mm of stable growth, then instability. b) True crack growth, i.e., logarithmic creep contribution subtracted ligament reduction, with constant velocity regression shown. Note ti for each, before which true crack growth is negligible. The TNM model was successfully calibrated simultaneously to the monotonic and creep experiments (Fig.5). The creep results transition from small-strain creep, through a viscoplastic yield transition up to an asymptotic strain plateau (without failure). Similar large strain yield and creep behavior has been reported elsewhere for UHMWPE [12]. From a constitutive behavior perspective, accurately modeling the stress-dependent rate of yield and the subsequent strain asymptote are critical for an accurate representation of the viscous deformation in the highly stressed region near a notch or crack tip. Short times were not calibrated in the creep tests due to minimum time step limitations, but short time responses were accurately probed

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in the monotonic experiment. Thus, multiple orders of magnitude of time, strain rate, and strain were calibrated between the three tensile experiments.

Fig.5 Experimental and FEA predicted behavior using calibrated TNM for a) monotonic uniaxial tension to failure, and b) creep at 20 and 14 MPa (engr. stress). Note the distributed yield transition in the creep results. FEA predicted a J-integral with a power law time-dependence, satisfying Eq.1 with exponent n: nFEA=0.15, and a logarithmically increasing İch at the notch root (Fig.6). The model also predicted a logarithmic gross ligament reduction prior to crack initiation, thus supporting the subtraction of that portion of the experimental results to obtain true crack growth (i.e., Fig.4b). The experimentally determined ti’s (Fig.6) can be plotted on the FEA results, and these map to for each predicted response to a single FEA predicted value of J and İch at crack initiation (respectively denoted, J(ti) and İch(ti), Table 1). Thus, J(ti) and İch(ti) both appear to represent single-valid failure criteria for crack initiation, and also appear to be in one-to-one correspondence.

Fig.6 FEA predictions of (a) J-integral and (b) average molecular chain strain. Superimposed on the numerical predictions are the experimentally observed crack initiation times (ti), which in each case map to a single value of the predicted parameter for all three tests, supporting the notion of J and İch as failure criteria. Table 1. Experimental, analytical, and numerical FEA results

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Discussion The experimental crack initiation and propagation results match the expected response, i.e., a delayed initiation event followed by a constant crack velocity for a constant applied load, and ultimately instability and failure. The delayed initiation time was extracted by fitting subtracting the initial logarithmic response for all time, which the FEA predicted was due to whole specimen creep prior to crack initiation. An estimate of the experimental value of the exponential factor n can be obtained by taking the ratio of either Eq.2 or Eq.3 from the experimental values of ti or da/dt, while assuming the applied J-integral to be proportional to the square of the applied load (this is expected from fracture mechanics and was confirmed with FEA). When this is done, the prefactor Q drops out, since it is related to material properties that are constant in this case, as does the time constant. The resulting values of the exponent are termed nEQ2 and nEQ3, and are listed in Table 1. These experimental estimates of the exponent n agree well with that predicted by the FEA, which indicates that the viscoplasticity effects in the mathematical and numerical analysis match with one another. Agreement in the exponent n links the analytical, experimental, and numerical components of this work, and strongly motivates the contention that the viscoplastic material behavior drives crack initiation and propagation phenomena in UHMWPE. The results for J are generally applicable to design of arbitrarily shaped components with stress concentrations, via traditional arguments of similarity employed in fracture analyses. Likewise, İch is a locally calculated invariant parameter, and has no direct dependence on the boundary conditions far from the crack tip, and therefore can be abstracted and used in the analysis of engineered components. However, both J and İch have limitations to their interpretation; J is path-dependent and does not correspond to an intrinsic local material state, İch is an empirical simplification of the stretch tensor and does not have a rigorous link to the local terminal behavior of the molecules in the amorphous phase of the polymer. The fact that they both correlate to crack initiation, while nevertheless representing such different quantities in fracture mechanics, strengthens the case for each as a failure criterion for prediction of crack initiation from notch roots. In this regard, İch may have an advantage in future analyses, where a critical value can be used to trigger element deletion and model crack growth. In such a scenario, J can be used as a check to ensure that crack initiation triggers at the appropriate state and proceeds at the approximately constant value predicted by the analytical model. It is worth mentioning that Eq.2 contains not just intrinsic material information relevant to incipient crack formation (through Jc, n, and IJ0), but also extrinsic information related to the boundary conditions and design (through J0). When designing components, engineers often have to play extrinsic constraints on the design against material selection decisions, and to balance these trade-offs to produce a safe and efficacious final product. When no clear link can be made between failure phenomena and such extrinsic factors, it can be difficult to predict the performance of the final design with confidence. A clear case in point is the failures shown in Fig.1, where the designed geometry was inherited from a previous design that had employed an uncross-linked UHMWPE, and cross-linked UHMWPE was substituted in the absence of a clear design rule regarding the danger of substituting a material with inferior fracture resistance in designs with stress-concentrating features [4]. This work was motivated in large part by the lack of such a design rule guiding the trade-off of design and material features, and Eq.2 represents an easily applicable method to achieve this in fracture mitigation. Furthermore, having a direct way to accommodate these trade-off decisions allows design engineers the freedom to achieve a given level of performance without restriction on whether it is accomplished by material substitution, redesign to reduce stress concentration or far-field stress (J0), or some combination of these tactics. References 1. McKellop, H., et al., Development of an extremely wear-resistant ultra high molecular weight polyethylene for total hip replacements. Journal of Orthopaedic Research, 1999. 17(2): p. 157-167. 2. Baker, D.A., A. Bellare, and L. Pruitt, The effects of degree of crosslinking on the fatigue crack initiation and propagation resistance of orthopedic-grade polyethylene. Journal of Biomedical Materials Research - Part A, 2003. 66(1): p. 146-154. 3. Pruitt, L.A., Deformation, yielding, fracture and fatigue behavior of conventional and highly cross-linked ultra high molecular weight polyethylene. Biomaterials, 2005. 26(8): p. 905-915. 4. Furmanski, J., et al., Clinical fracture of cross-linked UHMWPE acetabular liners. Biomaterials, 2009. 30(29): p. 5572-5582. 5. Furmanski, J., and Pruitt, L.A., Peak stress intensity dictates fatigue crack propagation in UHMWPE. Polymer, 2007. doi:10.1016/j.polymer.2007.04.006.

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Furmanski, J., E. Feest, and L.A. Pruitt. Static mode fatigue of UHMWPE. in International Congress on Mechanics of Biomaterials and Tissues. 2007. Kauai, HI, USA: Elsevier. Williams, J.G., Fracture Mechanics of Polymers. 1984, Chichester: Ellis Horwood Ltd. Bergstrom, J. and J.E. Bischoff, An Advanced Thermomechanical Constitutive Model for UHMWPE. Submitted to Int J Struct. Changes Solids, 2010. Bergstrom, J.S., C.M. Rimnac, and S.M. Kurtz, An augmented hybrid constitutive model for simulation of unloading and cyclic loading behavior of conventional and highly crosslinked UHMWPE. Biomaterials, 2004. 25(11): p. 21712178. Bergstrom, J.S., C.M. Rimnac, and S.M. Kurtz, Molecular chain stretch is a multiaxial failure criterion for conventional and highly crosslinked UHMWPE. Journal of Orthopaedic Research, 2005. 23(2): p. 367-375. Bergstrom, J. and J.E. Bischoff, An Advanced Thermomechanical Constitutive Model for UHMWPE. Submitted to Int J Structural Changes in Solids, 2010. Crissman, J.M. and L.J. Zapas, Creep Failure and Fracture of Polyethylene in Uniaxial Extension. Polymer Engineering and Science, 1979. 19(2): p. 99-103.