Thermomechanics and Infra-Red Imaging, Volume 7

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

Thermomechanics and Infra-Red Imaging, Volume 7 Proceedings of the 2011 Annual Conference on Experimental and Applied Mechanics

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 978-1-4614-0206-0 e-ISBN 978-1-4614-0207-7 DOI 10.1007/978-1-4614-0207-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011929341 ¤ 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

Thermomechanics and Infra-Red Imaging 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, Mechanics of Time-Dependent Materials and Processes in Conventional and Multifunctional Materials, MEMS and Nanotechnology; Optical Measurements, Modeling and, Metrology; Experimental and Applied Mechanics, and Engineering Applications of Residual Stress. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics. The Thermomechanics and Infra-Red Imaging conference track was organized by: Janice Dulieu-Barton*, University of Southampton, UK, Fabrice Pierron, Arts et Métiers ParisTech, France, Rachel Tomlinson, University of Sheffield, UK and David Backman, National Research Council Canada and sponsored by the Thermomechanics and Infra-Red Imaging Division In recent years the applications of infra-red imaging techniques to the mechanics of materials and structures has grown considerably. The expansion is marked by the increased spatial and temporal resolution of the infra-red detectors, faster processing times and much greater temperature resolution. The improved sensitivity and more reliable temperature calibrations of the devices have meant that more accurate data can be obtained than were previously available. The purpose of the track is to bring together novel work on all aspects of thermomechanics with the focus on the application of infra-red imaging approaches. The main thrust of the session will be on the analysis of thermomechanical behavior of materials and using this behavior to elicit information on material characteristics, stresses and failure. Of particular interest are strong thermomechanical

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couplings that result in nonlinear behavior such as viscoelasticity, diffusivity and material phase changes. An objective is to share experience on how data-rich experimental mechanics can help scientists and engineers to better understand and simulate the behavior of materials and structures. It is also envisaged that papers utilizing other imaging techniques in conjunction with infra-red approaches will be a key part of the track program enabling cross-fertilization over disciplines and applications. The following general technical research areas are included: High speed thermography Multiscale thermodynamic couplings Thermography in fatigue and damage assessment Application to composite materials Thermoelastic stress analysis The track organizers thank the authors, presenters, organizers and session chairs for their participation and contribution to 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. Challenges in Synchronising High Speed Full-field Temperature and Strain Measurement D.A. Crump, J.M. Dulieu-Barton, R.K. Fruehmann, University of Southampton

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2. The use of Infrared Thermography at High Frame Rates R.K. Fruehmann, D.A. Crump, J.M. Dulieu-Barton, University of Southampton

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3. In Situ Heat Generation and Strain Localization of Polycrystalline and Nanocrystalline Nickel T. Chan, University of Toronto; D. Backman, R. Bos, T. Sears, Institute for Aerospace Research, National Research Council; I. Brooks, Integran Technologies Inc.; U. Erb, University of Toronto

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4. Dissipative and Coupling Effects Accompanying the Natural Rubber Elongation B. Wattrisse, R. Caborgan, J.-M. Muracciole, L. Sabatier, A. Chrysochoos, Montpellier University

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5. Experimental Estimation of the Inelastic Heat Fraction From Thermomechanical Observations and Inverse Analysis T. Pottier, F. Toussaint, Université de Savoie; H. Louche, Université Montpellier 2; P. Vacher, Université de Savoie 6. Energy Balance Properties of Steels Subjected to High Cycle Fatigue A. Chrysochoos, A. Blanche, Montpellier University; B. Berthel, École Centrale de Lyon; B. Wattrisse, Montpellier University 7. Contribution of Kinematical and Thermal Full-field Measurements for Identification of High Cycle Fatigue Properties of Steels R. Munier, C. Doudard, S. Calloch, ENSIETA - LBMS; B. Weber, ArcelorMittal Maizières Research & Development

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8. Dissipative Energy: Monitoring Microstructural Evolutions During Mechanical Tests N. Connesson, F. Maquin, F. Pierron, Arts et Metiers ParisTech

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9. Bidirectional Thermo-mechanical Properties of Foam Core Materials Using DIC S.T. Taher, O.T. Thomsen, Aalborg University; J.M. Dulieu-Barton, University of Southampton

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10. Optimization of Transient Thermography Inspection of Carbon Fiber Reinforced Plastics Panels B.G. Bainbridge, Southern Illinois University Carbondale; Y. Pan, University of Akron; T. Chu, Southern Illinois University Carbondale

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11. Experimental Investigation of Thermal Effects in Foam Cored Sandwich Beams R.K. Fruehmann, J.M. Dulieu-Barton, University of Southampton; O.T. Thomsen, Aalborg University; S. Zhang, University of Southampton 12. Intelligent Non-destructive Evaluation Expert System for Carbon Fiber Reinforced Plastics Panel Using Infrared Thermography Y. Pan, University of Akron; T. Chu, Southern Illinois University Carbondale 13. Successful Application of Thermoelasticity to Remote Inspection of Fatigue Cracks T. Sakagami, Kobe University; Y. Izumi, S. Kubo, Osaka University 14. Investigation of Residual Stress Around Cold Expanded Hole Using Thermoelastic Stress Analysis A.F. Robinson, J.M. Dulieu-Barton, S. Quinn, University of Southampton; R.L. Burguete, Airbus Operations Ltd.

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15. TSA Analysis of Vertically- and Incline-loaded Plates Containing Neighboring Holes A.A. Khaja, R.E. Rowlands, University of Wisconsin-Madison

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16. Examination of Crack Tip Plasticity Using Thermoelastic Stress Analysis R.A. Tomlinson, University of Sheffield; E.A. Patterson, Michigan State University

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Challenges in synchronising high speed full-field temperature and strain measurement

Author: D.A. Crump, School of Engineering Sciences, University of Southampton, Highfield, Southampton, SO17 1BJ, UK, [email protected] Co-Author: J.M. Dulieu-Barton, R.K. Fruehmann, University of Southampton ABSTRACT The overall motivation for the research described in the paper is an enhanced understanding of the behaviour of fibre reinforced polymer composites subjected to high velocity loading. In particular, the work described here considers a method that allows the collection of synchronised high speed full-field temperature and strain data to investigate the complex viscoelastic behaviour of fibre reinforced polymer composites material that occurs at high strain rates. The experimental approach uses infra-red thermography (IRT) and digital image correlation (DIC). Because high strain rate events occur rapidly it is necessary to capture the images at high speeds. The paper concentrates on the challenges of the use of IRT and DIC at high speeds to obtain temperature and strain fields from composite materials, and in particular using them in a synchronised manner. In the future such data-rich techniques provide the opportunity for detailed investigation into the viscoelastic behaviour and allow in-depth material characterisation for input to future finite element or numerical models. INTRODUCTION Increasing use of polymer reinforced polymer composites in high performance applications, e.g. military structures, is leading to an increased risk of impact or blast events imparting high velocity loading. Whilst the behaviour of such materials subjected to quasi-static elastic loading is reasonably well understood, the response to high strain rate requires further investigation. To reduce and mitigate the risk of failure it is essential that knowledge of the behaviour of these materials under high velocity deformation is established. Hence the subsequent effects of damage on structural performance can be defined. Therefore the motivation for this work is the need to map the effect of high velocity loading on the overall structural performance. High velocity/strain rate deformations are usually accompanied by a temperature evolution. Therefore the material behaviour is a function of time, strain and temperature, so to fully understand the material structural performance the thermomechanical material constitutive behaviour is required. The overarching aim of the current research is to provide thermomechanical characterisations of glass and carbon fibre polymer composite over a range of strain rates, with the ultimate goal of inputting the constitutive behaviour into a finite element (FE) modelling approach. Hamouda [1] and Sierakowski [2] discuss the range of approaches for high strain rate testing. The preferred technique is the split Hopkinson bar that allows strain rates up to 104 s-1. However in this work a specialised conventional servo-hydraulic test machine (Instron VHS) is used, which allows moderate strain rates up to 102 s-1. Whilst this machine cannot match the strain rates of the split Hopkinson bar, it allows specimens of approximately 25 mm wide by 100 mm long to be used, unlike the much smaller coupons that must be used in conjunction with the split Hopkinson bar. These specimens are of a similar size, and aspect ratio, to those recommended for quasi-static characterisation by testing standards and therefore provides a larger surface for the application of optical measurements techniques. The complex behaviour of fibre reinforced composite materials lends itself to the use of full-field optical measurement techniques, as information from the entire specimen is obtained that allows the identification of failure zones, loading paths etc. Digital image correlation (DIC) is used to measure strain, and infra-red thermography (IRT) to obtain the temperature evolution. One of the primary advantages of techniques such as DIC and IRT is that they are non-contacting, so the measurand does not affect the measurement by, for example, localised reinforcement or heating. However, it is essential that the images are synchronised temporally with any independent load or strain data collected from other sensors used in the experiment. Therefore the relative image capture rates, time delays and thresholding are important considerations. The aim of the current paper is to discuss the application of DIC and IRT to high velocity testing and the corresponding challenges. DIC and IRT are initially applied separately to both metallic and composite specimens, and some initial results are presented demonstrating the approach. The metallic specimens were expected to undergo higher strains and temperature changes, and were therefore chosen as a good starting point for assessing the abilities of DIC and IRT applied to such tests. This is

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_1, © The Society for Experimental Mechanics, Inc. 2011

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2 followed by a discussion of approaches to synchronise data capture from the two optical systems and an independent load measure. DIGITAL IMAGE CORRELATION (DIC) Digital image correlation is a full-field, optical method to measure the deformations and strains in a material or structure. DIC tracks the movement of a random surface pattern to monitor deformation or displacement. The random pattern is usually achieved by covering the surface of a component with a painted speckle pattern. Images of the deformation process are recorded using either one (2D DIC) or two (3D DIC) charge coupled device (CCD) cameras. The images are divided into discrete interrogation windows (or cells) and the displacement is obtained by tracking features within each cell [3]. Strain values are obtained by taking the measured displacement and dividing by the size of the undeformed cell. Strain resolutions are quoted as being as low as 40 µstrain [4], although this is highly dependent upon application and test conditions, such as lighting and alignment. The DIC technique has been successfully used to analyse the strains in heterogeneous engineering materials such as composites [5] and there is some reported work on the use of high speed cameras to collect images for DIC [6, 7]. Tiwari et al [6] described the use of high speed cameras for DIC, and the inherent limitations of such an approach. To apply DIC to high velocity testing, commercially available high speed digital cameras are used to record the images. In this work the images are then imported into the DaVis 7.4 (LaVision) software for analysis. The application of DIC to high speed imaging uses the same speckle analysis algorithm as that applied at quasi-static test speeds. The accuracy of the algorithm is the same as quoted above, but additional sources of error are likely due to the acquisition of images using high speed cameras. To obtain images at the highest possible frame rates it is necessary to reduce the resolution of the sensor, therefore the user must accept a coarse strain map or use smaller cell sizes which give greater uncertainty in the strain result. Secondly it is more difficult to obtain well illuminated images with high contrast at high speed as the integration time must be reduced. Therefore it is important to increase the lighting intensity which may have adverse effects such as specimen heating, or heat haze in the images. Three different high speed cameras are used to capture the images for this work, Photron’s SA-1 and 3 cameras, and Redlake’s MotionPro X3 details of each are described in Table 1. The three cameras have similar maximum resolutions (~ 1 MP), and the X3 and SA-3 are capable of recording this resolution up to 2 kHz whilst the SA-1 extends this to 5.4 kHz. The advantage of the use of the SA-1 to collect images for DIC is clear, allowing full-size images to be used at higher strain rates. The X3 only uses vertical sub-windowing to increase the frame rate, leading to an image with high aspect ratio that lends itself to recording test on tensile strips. The SA-1 and SA-3 sub-window in both directions, producing squarer images. In both cases the compromise is always between spatial and temporal resolution. Table 1 Specification of high speed cameras used in this work Parameter MotionPro X3+ Photron SA-3 Max resolution (pixels) 1280 x 1024 1024 x 1024 Max frame rate at max resolution (kHz) 2 2 Sub-windowing (pixels) Vertical only to 16 Both to 128 x 16 Max frame rate (kHz) 128 120 Storage size (Gb/~full size images) 8/5500 2-8/1500-5500

Photron SA-1 1024 x 1024 5.4 Both to 64 x 16 675 8-32/5500-21500

Steel and unidirectional (UD) GFRP tensile specimens were loaded at both quasi static speed (0.12 m/s, i.e. 2 mm/min), using a standard Instron electro-mechanical test machine, and then at 1 m/s, on a specialised Instron VHS 1000 test machine capable of testing specimens at speeds up to 20 m/s. Steel dog bone specimens with a cross-sectional area in the gauge length of 14.5 mm by 1 mm were used as proof of concept, three specimens at the QS speed and three at 1 m/s. Followed by GFRP specimens manufactured from ACG, MTM28-1\E-glass-200 prepreg with dimensions 200 mm by 20 mm and were 0.4 mm thick. During the tests the load was recorded by the load cell attached to the machine; in the case of the VHS this is a piezoelectric Kistler load cell. For comparison, the strain was separately recorded using Vishay’s CEA-06-240UZ-120 attached to Vishay’s Strainsmart system with a maximum sampling rate of 10 kHz. In addition, Photron’s SA-1 high speed camera was used to capture images during the 1 m/s tests. The specimens were prepared by spraying with black, grey and white paint to provide a speckle pattern. The cameras were recording at 30 kHz with an image size 512 x 256 pixels. The DIC was performed on each image using a cell size of 64 pixels and 50 % cell overlap, therefore providing a strain map with 16 x 8 data points. Figure 1 presents the evolving strain map for a steel specimen tested at 1 m/s, alongside a plot of the average longitudinal and transverse strains across the specimen. The plot highlights the advantages of using DIC. Where the strain gauge has debonded early the DIC measures up to 30 % strain, and allows longitudinal and transverse strain to be measured simultaneously.

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t = 0 ms

2.7 ms 17.2 ms Fig. 1 Full-field strain evolution and average longitudinal and transverse strain for steel at 1 m/s

The load and strain data from each of the three specimens tested at both quasi static and 1 m/s is used to calculate the average Young’s modulus shown in Table 2. There is a distinct difference between the value of Young’s modulus at 1 m/s measured by the strain gauge and DIC. For the steel specimens, the strain gauge data provides a reduction in material stiffness with test speed. The DIC data shows little difference between the quasi static and 1 m/s test. This is expected, as it is likely that 1 m/s is not fast enough to encounter the well-documented stiffening effect. The opposite is true of the data for GFRP specimens. Whilst both strain gauge and DIC show a stiffening effect, the strain gauge has measured almost twice the stiffening to that from DIC. At higher test speeds the strain gauge appears to be providing erroneous results; this can be attributed to sampling rate. Sampling at 10 kHz is not fast enough to accurately measure the strain progression. Further tests are required to increase the confidence in the DIC results, and to ensure all errors from test fixtures, specimen preparation and the processing algorithm are accounted for. This will be the subjected of a future paper. It should also be noted that the Young’s modulus values for the material tested at 1 m/s all have a scatter of around 10 % except for the steel strain gauge value. Further tests are required to assess the sources of this scatter and improve confidence in the results.

Steel GFRP

Table 2 Young’s modulus of steel and GFRP specimens E: Quasi-static (GPa) E: 1 m/s (GPa) strain gauge strain gauge 192.1 ± 2.3 171.0 ± 5.7 42.2 ± 1.4 69.3 ± 5.8

E: 1 m/s (GPa) DIC 192.9 ± 19.5 55.6 ± 6.7

INFRA-RED THERMOGRAPHY (IRT) Infra-red thermography uses an IR detector to monitor the emissions from the surface of a structure, from which the surface temperature is derived. IRT is therefore a full-field non-contacting technique for temperature measurement that has a sensitivity determined by the thermal resolution of the IR detector, and spatial resolution by the number of elements in the detector array. IRT has a large range of commercial uses for non-contact temperature measurement, but has also been used to detect hot-spots in structures that may identify sub-surface damage in non-destructive testing [8]. Commercially available IR detectors, such as those from FLIR systems, are capable of 100s Hz with detector array sizes upwards of 320 x 256. Using these detectors to capture data at higher frame rates has the same limitations as white light high speed cameras used for DIC. The internal electronics that control data transfer from the detector elements into digitised values force a limit on the total number of samples from all elements per second. Therefore to improve the frame rate the number of detector elements utilised must be reduced. As for the white light camera the image is sub-windowed sacrificing spatial for temporal resolution. The necessary reduction in detector integration time is a special case for IR imaging. For white light cameras it is possible to counter the effects of lower integration times by increasing illumination and therefore maintaining an adequate amount of photons striking the CCD array. However IR is dependent on the finite amount of energy emitted from the materials surface due to its temperature. The amount of photons and the sensitivity of the detector elements therefore limit the minimum integration time and hence the maximum frame rate given that the integration time must be equal or less than the reciprocal of the frame rate.

4 There is little research reported in the literature on the use of high speed IR; however the few examples found used specifically designed systems. Noble [9] described the use of a thermal scanning camera to measure the temperature change occurring during high strain rate test on ductile iron at a rate of 1600 s-1 in a split Hopkinson bar rig. The scanner was an AGEMA 880LWB that used a liquid-nitrogen cooled CdHgTe detector with an accuracy of ± 2 K. The camera was only capable of scanning at 2500 Hz, and therefore was not fast enough to record temperature evolution during the test. Instead the camera recorded the temperature change approximately 0.5 ms after specimen fracture. The nature of the material tested, and the high speed applied, produced temperatures up to 573 K at the necking site. Three possible error sources were identified that could account for uncertainty of ~ 100 K; movement of the specimen with respect to the camera, change of specimen orientation and changing emissivity during deformation. Improvements in electronics allowed Zehnder [10, 11], to produce a system capable of 1 MHz with 64 HgCdTe detector elements in an 8 x 8 plane array. Studies of the temperature rise near the tip of a notch in high strength steel sample subjected to an impact showed the system was capable of a temperature resolution of approximately ± 2 K. Finally, more recently, Ranc [12] used a bar of 32 InSb infrared detectors to measure a line of temperature points on a high strain torsion test on titanium. This also sampled at 1 MHz, but was measuring temperatures of the order of 100s K. In the current work infrared data was recorded using a Silver 480M (FLIR systems) detector. The Silver 480M uses a dual layer InSb sensor with 320 x 256 pixels. At maximum resolution it is possible to capture data at 383 Hz, and by windowing down to 48 x 4 pixels it is possible to achieve 20 kHz. It was decided to operate the detector at 15 kHz with a window of 64 x 12 and an integration time of 60 µs as a compromise between spatial, temporal and temperature resolution. When operating the detector outside of its standard configuration it is necessary to perform calibration and non-uniformity correction procedures different to those provided by the manufacturers. These processes set-up the detector and the internal electronics for use at higher speeds, and altered window size to allow calibrated temperatures to be measured. Full details of the calibration and non-uniformity techniques, and their requirement, are discussed in a separate paper [13]. Tests were performed on the Instron VHS machine at an actuator speed of 10 m/s initially on a steel specimen and then a glass fibre chopped strand mat (CSM) specimen. The tests were performed as described in the DIC section, but the surface of the specimen was sprayed only with matt black paint to provide a constant emissivity. Figure 2 is a full window image of the steel specimen before and after the test to demonstrate the location of the subwindowed data, shown by the red box. By varying the stand-off distance it was possible to view the entire gauge length of the specimen, and it is clear from the post test image that the failure section was captured. The temperature evolution with time is plotted in Figure 3 at the six points displayed on the final image from the series show next to the graph. The data is displayed in its raw format as the number of digital levels measured. For reference the load signal from the Kistler load cell on the VHS machine is included shown by the blue line. While it is noisy there is a definite change in signal as the specimen fails at approximately 3 ms into the test time. The temperature at all six points increases steadily at a similar rate as the load is initially applied to the specimen, but towards the end of the test the temperature of the point at the failure location diverges from the rest as it rises rapidly. Therefore there are two distinct stages in the temperature evolution during the high speed test. The first represents temperature change as the specimen is stressed, whilst the second demonstrates large temperature increase due to damage at failure. The apparent cyclic temperature change at point six after failure has been identified as outof-plane movement of the specimen. The temperature evolution during the test on a CSM specimen is plotted in Figure 4. The data is plotted from the same points used for the steel specimen. It should be noted that the timescale of the temperature evolution is far shorter than for the steel specimen. The temperature rise is much sharper, approximately 0.1 ms, whilst the rise for steel was spread across 2 ms. Secondly the data from the composite is a couple of orders of magnitude less than the steel. It is evident that the shorter timescale and lower data values produce noisier IRT. Point six is near to the failure zone, and as for the steel specimen, there is a sudden temperature rise as the specimen fractures. However, point one on the CSM plot also shows a temperature increase lagging slightly behind point six. From the images taken during the test, and from the remains of the specimen post test, it became clear that two failure sites were present. The failure at point six was first to initiate, shortly followed by a fracture at point one. This highlights the more complex nature of tests involving composite materials. CSM specimens are particularly weak, and therefore during failure there is less energy and consequently lower temperature evolutions. It is expected that applying IRT to stronger composites, e.g. UD glass fibre specimens, will provide higher energy failures and higher temperatures.

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Fig. 2 Full window images of steel specimen before and after test

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Fig. 3 Plot of the temperature evolution of the specimen at six points during the test 530 525

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Fig. 4 Plot of the temperature evolution of the specimen at six points during the test

6 SYNCHRONISED DATA CAPTURE The overall goal of the research is to obtain full-field temperature and strain information from composite materials subjected to high velocity loading. Initial tests on the separate use of DIC and IRT at high speeds have shown promise in the two techniques although further work is required to improve consistency and obtain a better understanding of the error sources. The next step is to develop an approach that will allow both systems to capture data concurrently. In the literature there is little mention of the use of white light imaging and IRT together, particularly at high speed. Noble [14], used the IR system mentioned above in the IRT section and a white light high speed camera during a test on iron in a split Hopkinson bar rig. The white light camera was used to obtain deformation information by monitoring the change in shape of the specimen, and did not provide full-field strain. The IR system did not operate at a high enough sampling rate to capture temperature evolution during the test. Instead it was triggered shortly after the specimen failure to measure the maximum temperature at the fracture site. These systems were still effectively used separately to obtain information at different stages of the test. To fully characterise the viscoelastic behaviour of the composite material subjected to high velocity loading it is necessary to measure load, temperature and strain at the same temporal location. Therefore an approach is required to synchronise the three data types together. The first challenge to overcome is to ensure all data capture systems are initiated at the same time; this is achieved using specifically designed LabView code for operation on National Instruments Compact Rio hardware. The Compact Rio monitors the voltage signal from the Kistler load cell and at a user defined threshold triggers the capture of load data and generates a logic signal that is sent to the two camera systems. The white light cameras and IR detector can be triggered from a digital pulse with a known jitter of the order of 100 ns. With a known frame rate it is a simple matter to find a corresponding load value for each white light and IR image. The second challenge is an artefact of the different performances of the white light and IR systems. Whilst the white light cameras can achieve frame rates over 100 kHz (although with much reduced spatial resolution), it has been shown that detector sensitivity will limit the IR system to approximately 15 kHz. Both systems are to be initiated at the same time, but with different frame rates they will quickly lose phase. It is envisaged that using the white light system at a frame rate that is a multiple of the sampling rate of the IR detector would ensure that for each IR image there is also a DIC strain image. It is advantageous to operate the white light camera at higher frame rates to assist the DIC algorithm in obtaining accurate strain information. The final perceived challenge is in the physical collection of the white light and IR images. DIC requires the specimen surface to be sprayed with a speckle pattern, whilst, to obtain the best results, IR requires a surface preparation of matt black paint. Further tests are required to investigate the effect of using a painted speckle pattern on the collection of IR data. The illumination required for DIC will also have an effect on IR data, particularly with specimen heating. ‘Cold’ light LEDs are being investigated to illuminate the white light images. It may also be possible to obtain DIC data from the front of specimen, and IRT data from the rear. Consideration must be taken though that any localised effects from damage may affect the relevance of the two data sets. CONCLUSIONS It has been shown that DIC and IRT with commercially available cameras, detectors and software are feasible techniques to obtain full-field strain and temperature information from high strain rate testing on composite materials. Further work is required to improve consistency and fully understand the errors involved. A system is in place that allows the data capture to be synchronised, even accounting for differences in sampling rates of the DIC and IRT. Consideration is required of the physical aspects of collecting data simultaneously from two techniques that have some conflicting requirements. ACKNOWLEDGEMENTS The work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant number EPG042403/1 and the UK Defence Science and Technology Laboratories (DSTL). The SA3 cameras were provided by Airbus and the SA1 from the UK STFC Equipment Loan Pool. REFERENCES 1. 2. 3. 4.

Hamouda, A.M.S., and Hashmi, M.S.J. Testing of composite materials at high rates of strain: advances and challenges. Journal of Materials Processing Technology, 77, 327-336, 1998. Sierakowski, R.L. Strain rate effects in composites. ASME, 1997. Brillaud, J. and Lagattu, F. Limits and possibilities of laser speckle and white-light image correlation methods: theory and experiments. Applied Optics, 41, 6603-6613, 2002. Anon, Optical deformation and strain field imaging, L. Vision, Editor.

7 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Godara, A. and Raabe, D. Influence of fiber orientation on global mechanical behavior and mesoscale strain localization in a short glass-fiber-reinforced epoxy polymer composite during tensile deformation investigated using digital image correlation. Composites Science and Technology, 67, 2417-2427, 2007. Tiwari, V., Sutton, M., and McNeil, S. Assessment of high speed imaging systems for 2D and 3D deformation measurements: Methodology development and validation. Experimental mechanics, 47, 561-579, 2007. Koerber, H., Xavier, J. And Camanho, P.P. High strain rate characterisation of unidirectional carbon-epoxy IM78552 in transverse compression and in-plane shear using digital image correlation. Mechanics of Materials, 42, 1004-1019, 2010. Maldague, X.P.V. Theory and practice of infrared technology for non-destructive testing. John Wiley and Sons, ed. Chang, K., 2001. Noble, J. P. and J. Harding. Temperature measurement in the tensile Hopkinson bar test. Measurement Science and Technology, 5, 1163-1171, 1994. Zehnder, A.T., Guduru, P.R., et al. Million frames per second infrared imaging system. Review of Scientific Instruments, 71, 3762-3768, 2000. Guduru, P.R., Rosakis, A.J., and Ravichandran, G. Dynamic shear bands: an investigation using high speed optical and infrared diagnostics. Mechanics of Materials, 33, 371-402, 2001. Ranc, N., L. Taravella, et al. Temperature field measurement in titanium alloy during high strain rate loading: Adiabatic shear bands phenomenon. Mechanics of Materials, 40, 255-270, 2008. Fruehmann, R.K., Crump, D.A., and Dulieu-Barton, J.M. The use of infrared thermography at high frame rates. In: International Congress of the Society for Experimental Mechanics (SEM), Uncasville, USA, 2011. Noble, J. P., Goldthorpe, B.D., et al. The use of the Hopkinson bar to validate constitutive relations at high rates of strain. Journal of the Mechanics and Physics of Solids, 47, 1187-1206, 1999.

The use of infrared thermography at high frame rates

R K Fruehmann1, D A Crump1, J M Dulieu-Barton1 1

Faculty of Engineering and the Environment, University of Southampton, University Road, SO17 1BJ, Southampton, UK

ABSTRACT Composite materials are finding increased use in applications where impact and high strain rate loading form a significant part of a component’s service loads. It is therefore imperative to fully characterise the thermomechanical response of composite materials at high strain rates. The work described in the paper forms part of a project investigating the thermomechanical response of composite materials at high strain rates. To obtain the temperature evolutions during the high strain rate event (thermoelastic, viscoelastic and fracture energy), full-field infrared thermography is used. In contrast to visible light photography, the measurand in thermography is the intensity of the emitted radiation from the specimen surface, as opposed to reflected radiation. At increasing recording rates, the emittance available for measurement reduces proportional to the exposure time; the faster the data capture the less the exposure time. Hence, signal noise and detector calibration present a major challenge. This is accompanied by challenges arising from controlling an infrared detector that has not been optimised for the purpose of high speed data acquisition. The present paper investigates the possibility of applying infra-red thermography to high strain rate events and discusses the challenges in obtaining reliable values of the temperature changes that occur over very short time scales during high strain rate events. KEYWORDS: Infrared, thermography, calibration, high speed testing INTRODUCTION High strain rate events, such as those that occur during a collision or impact, are known to invoke a different material response compared with that observed in quasi-static conditions [1]. One of the challenges associated with high strain rate testing is the short duration of the test and the required high data recording frequency. Here an infrared (IR) detector is used to obtain the temperature evolution of a specimen during high strain rate loading. The overall goal is to include the temperature data in constitutive laws that characterise the material behaviour. The present paper describes initial work on the implementation of high speed IR thermography in high strain rate tests using a commercially available system. The paper discusses in detail the challenges associated with obtaining temperature values from an IR detector output at high recording rates. IR detectors are used in a wide variety of applications to measure temperature; the basic physics is described by Planck’s law [2]. At ambient temperature, the radiation band with the strongest emission is the middle IR band (1.5 – 20 μm). In this range two types of detector can be used: bolometers where the detector experiences a rise in temperature due to incident radiation and a temperature sensitive material property (e.g. resistance) is measured or photon detectors where a

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_2, © The Society for Experimental Mechanics, Inc. 2011

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10 semiconductor is excited by an incident photon and releases a charge that is collected in a capacitor. Of the two, photon detectors have a greater sensitivity and faster response rates and are therefore most suited to high speed data capture. To obtain a quantitative temperature measurement it is necessary to characterise the detector response. Calibration procedures are well established in industry and detectors that can be used to obtain quantitative measures of temperature are supplied with a manufacturer calibration. However, no commercial off-the-shelf system is available with a suitable calibration to obtain measurements at the recording frequencies required for high strain rate testing. The equipment used in the this work is a Cedip Silver 480M IR camera with a 320 x 256 element indium / antimonide (InSb) detector array. This is a photon detector, sensitive to radiation with wavelengths from 3 to 5 μm. In standard operation, the detector has a sensitivity of 4.2 mK at 25°C, with a maximum frame rate of 383 Hz at full frame. To capture the temperature evolutions during a high strain rate test however, acquisition frequencies in excess of 10 kHz are required. Therefore the paper describes in detail the characterisation and calibration of an off-the-shelf detector for high speed thermography. DETECTOR CHARACTERISATION Fig. 1 shows a schematic of the data capture for a photon detector such as that used in the 480M system. The photons are focused on the detector array. The electronic shutter controls the exposure time, known as integration time (IT), by controlling the time the switches are left open. To achieve the highest possible frame rate and sensitivity, the system comprises two sets of capacitors in the read out circuit, enabling almost continuous data capture; while data is being read from one bank of capacitors, the second bank is recording. The output from the capacitors is converted in to a 14 bit logic signal using analogue to digital convertors built into the detector device. The digital output is then sent directly to the computer for further processing.

Fig. 1 Schematic of data acquisition Selecting a suitable frame rate is a compromise between three main parameters: the duration of the test, the required thermal sensitivity and the number of detector elements used to collect the data. The frame rate is limited by the (IT), which controls the detector thermal sensitivity and the data handling capacity of the read out circuit which dictates the maximum window size for the image (i.e. number of detector elements) that can be recorded at a given frame rate. Fig. 2 illustrates how the three parameters are related for the system used in this work and shows that a practical limit is reached around 16 kHz above which the sensitivity and image window size fall below a useable threshold.

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Fig. 2 Relationship between a) integration and frame rate and b) window size and frame rate Each detector element has a slightly different sensitivity and similarly each capacitor has its individual characteristics. To obtain quantitative measurements, the first step is to identify this variation so as to be able to correct for it. This process is called the non-uniformity correction (NUC), which comprises exposing the detector to a uniform diffuse emission source (generally a flat plat with a high emissivity coating) and recording several images. The average signal from the whole image (spatial average) is compared with the average signal of each detector element (temporal average). This is done at two different temperatures within the range of temperatures to be measured, as shown in Fig. 3. A linear fit is then assumed between the two points, providing a gain and an offset value for each detector element from which the correction factor can be calculated, to bring it in line with the mean response of the whole array. This linear fit is however only an approximation of the detector response curve. In a typical test setup, the IT is selected to maximise the use of the full range of the detector (typically from 25 to 75% of the detector saturation). The two points for the NUC are chosen at 30 and 70% of the detector saturation to minimise the average error across the full range. The NUC is stored in two tables (one for each array of capacitors) which are loaded directly into the flash memory onboard the camera so that the data output by the camera is already corrected. In the current work however, the IT time is selected to enable a particular frame rate to be achieved. At this IT the detector will only operate between 2 and 10% saturation for room temperature measurements, so the NUC was performed at 30 and 70% of the desired temperature range (i.e. 28 and 51°C for the range 10 to 70°C). Over such a small range, the linear fit assumed in the NUC should be a relatively good approximation.

Fig. 3 Linear fit for NUC

12 CALIBRATION As the necessary IT is so small, the calibration data produced by the manufacturer does not cover this range since it is well outside the usual operating conditions for the IR system. It was therefore necessary to devise a methodology to establish a calibration at small IT. The methodology followed that of the manufacturers approach by taking the detector output in digital level from a high emissivity body with a known temperature. In this work a cavity black body was used [3] with a platinum type thermocouple inserted. The cavity walls were maintained at a uniform temperature by water circulated from a temperature controlled bath with a temperature range from 4 to 75°C. The calibration was conducted twice; the first time a series of videos (20 images each) was collected over a temperature range from 8 – 50°C using an IT of 60 μs and a frame rate of 15 kHz and an image size of 64 x 12 pixels. A NUC was conducted at 10 and 30°C. A calibration curve of detector response against black body temperature was obtained for odd and even numbered frames separately to compare differences that might occur due to the two capacitor arrays. The results showed an almost identical response. It was also possible to obtain the noise in each detector element, calculated as the standard deviation from 10 readings at nominally identical temperature. A standard deviation between 1.5 and 2.5 DL was obtained for most detector elements, with a few pixels having a standard deviation up to a maximum of 3.5 DL at 50°C. This represents a measurement precision of 0.35 to 0.2°C over the range from 15 to 50°C. The calibration procedure was then repeated over the range from 5 to 70°C, this time taking only single images at each temperature to reduce the data processing. A second NUC was performed, this time at 28 and 51°C to account for the extended temperature range. From this data the standard deviation across the image was obtained. This showed a nearly Gaussian distribution at temperatures up to 30°C. Fig. 4 shows that at 50°C, the pixel value distribution across the image shows two spikes, and a greatly increasing spread. Viewed as a percentage standard deviation the noise appears to be quite small, as shown in Fig. 5, but the change in the image noise characteristics seems to display a systematic pattern in the detector response and must therefore be considered significant. This is attributed to an error with the NUC as the pattern in the output has been shown to match the noise pattern of an image obtained with no NUC. The outcome is counter intuitive as the lowest error associated with the NUC process would be expected at the two points at which the NUC was conducted, i.e. at 28 and 51°C. However, it is at 50°C where two spikes start to appear in the image histogram as shown in Fig. 4. The signal noise from an individual pixel, however, remains fairly constant across the temperature rage, increasing from 1.5 to 2.5 DL between 15 and 70°C. Hence, the increase in noise is not a function of the detector elements themselves but of how the NUC process is integrated into the onboard hardware.

Fig. 4 Histogram of image noise at different temperatures

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Fig. 5 Change in image standard deviation with temperature The error with the standard NUC procedure can be avoided by obtaining raw detector data and performing the NUC in a postprocessing stage off-line. The strategy in this work is to conduct a calibration on a pixel-by-pixel basis. This takes into account the responsivity of each individual detector element. The calibration has to be conducted separately for odd and even frames to account for the two arrays of capacitors in the data acquisition hardware. By conducting the calibration directly on the individual pixel response, the two steps are combined. This enables the measurement precision to be assessed for each pixel individually in the resulting image. This calibration was conducted with 2°C intervals across the calibration range from 10 to 70°C. The plot shown in Fig. 6 shows the image average for odd and even frames. The error associated with each measurement is less than 4 DL. Table 1 shows the aggregate detector sensitivity (temperature increment per DL) and precision (detector noise calibrated into °C) over the measurement range.

Fig. 6 Calibration curve (image average for odd and even frames)

14 Table 1 Detector Precision Temperature (°C) 15 20 30 50

Sensitivity (°C) 0.14 0.11 0.10 0.07

Precision (°C) 0.35 0.27 0.25 0.20

The final step in obtaining accurate measurements is to evaluate the emissivity of the surface of the test specimen. For this, material coupons were placed in a thermal chamber with a temperature range up to 80°C and a small hole cut into one side to provide optical access to the IR detector. Thermocouples were mounted to the rear of the coupons while the forward facing side was prepared in the same way as for a mechanical test. Images were then collected over a range of temperatures from 15 to 55°C to assess the emissivity of the specimen surfaces over the full range of temperatures expected to occur during the high rate testing. Because the specimen surface is not a perfect emitter, background radiation will be partly reflected from the surface, leading to a measurement error. To assess the effect of a background radiation source, the specimens were also placed at a slight angle to the detector (approximately 20°) and a diffuse emission source (a flat plate with a high emissivity coating) was placed in the reflection path. The emissivity of lightly abraided E-glass / epoxy composite was measured to lie in the range 0.90 – 0.92. This uncertainty is due to some inherent noise in the detectors and variability in the surface preparation. TEST RESULTS Preliminary tensile tests were performed on an E-glass / epoxy specimen made from 4 layers of chopped strand mat to assess the range of temperatures to be expected from a composite specimen in the approach to failure and evaluate the calibration and NUC procedure described above. The specimen was waisted to provide a gauge length that fitted entirely within the field of view of the detector (approximately 10 x 50 mm). The mechanical load was applied using an Instron VHS servo-hydraulic test machine at an actuator velocity of 10 ms-1 and a maximum load capacity of approximately 30 kN. The load was measured using a Kistler piezo electric load cell. However the load data was extremely noisy and could therefore not be used. As a strain gauge was not attached to the specimen directly, the strain rate was estimated using the gauge length, actuator velocity and the assumptions of no slip in the grips and was considered to be between 50 – 100 s-1. The image in Fig. 7 shows the temperature on the specimen surface at the time of fracture. The fracture location is clearly visible on the right hand side of the specimen as a region of significant heating. Temperature measurements taken at five locations on the specimen surface (as marked in Fig. 7) are plotted over time in Fig. 8 a) and b) from 1 ms before fracture to 1 ms after. Fig. 8 b) shows a zoomed in portion of the graph in Fig. 8 a) enabling two distinct regions to be identified: an elastic strain region where there is a uniform decrease in the temperature across the whole specimen due to the thermoelastic effect [4] and a region where the specimen temperature increases all across the specimen surface as the failure strain is reached. The final failure propagates from one location on the specimen (point 5). Here heat is generated by the formation of new surfaces. Post failure inspection indicated manifold failure modes, including fibre pull-out, fibre failure and matrix cracking at the fracture site.

Fig. 7 Image of specimen surface temperature at time of fracture

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Fig. 8 Surface temperature evolution at 5 points on the specimen, a) full range b) zoomed in CONCLUSIONS It has been demonstrated that an off-shelf IR detector can be used to obtain temperature measurements at high speed. In doing so it is necessary to carefully investigate how the detector behaviour changes. Manufacturers’ calibration processes are often integrated into the detector hardware in a manner that may not be transparent and possibly even a trade secret. Apparently straight forward processes such as the NUC may therefore not perform as expected when operating outside the detectors normal range. The work in this paper has highlighted the procedures necessary to obtain quantitative temperature measurements as well as some of the potential pitfalls to be aware of. It is demonstrated that data can be obtained with sufficient precision to make a meaningful addition load and strain measurements during high speed testing. REFERENCES [1]

Hamouda, A.M.S. and Hashmi, M.S.J., “Testing of composite materials at high rates of strain: advances and challenges”, Journal of Materials Processing Technology, vol. 77, pp. 327-336, (1998)

[2]

Bramson, M.A., “Infrared Radiation: A Handbook for applications”, Plenum Press, (1968)

[3]

Irani, K., “Theory and construction of blackbody calibration sources”, Thermosense XXIII - Proceedings of SPIE, vol. 4360, pp. 347-362, (2001)

[4]

Dulieu-Barton, J.M. and Stanley, P., “Development and applications of thermoelastic stress analysis”, Journal of Strain Analysis, vol. 33, pp. 93-104, (1998)

In Situ Heat Generation and Strain Localization of Polycrystalline and Nanocrystalline Nickel

T. Chan1,a, D. Backman2,b, R. Bos2,c, T. Sears2,d, I. Brooks3,e and U. Erb1,f 1

Department of Materials Science and Engineering, University of Toronto, Wallberg Building, 184 College Street, Suite 177, Toronto, ON, Canada M5S 3E4 2

3 a

Institute for Aerospace Research, National Research Council, M-14, 1200 Montreal Road, Ottawa, ON, Canada K1A 0R6

Integran Technologies Inc, 1 Meridian Road, Toronto, ON, Canada M9W 4Z6

[email protected], [email protected], [email protected] d [email protected], [email protected] [email protected]

ABSTRACT Commercially available polycrystalline nickel (Ni200; grain size: 30 µm) and electrodeposited nanocrystalline nickel (grain size: 30 nm) were analyzed for the phenomena of in-situ heat generation and strain localization during plastic deformation at room temperature. Tensile specimens according to ASTM E8 standard dimensions were tested at a strain rate of 10-2/s to record the amount of heat dissipated and the change of localized strain using a high resolution infrared detector and digital image correlation (DIC) camera, respectively. For deformation close to ultimate tensile strength, data recorded for the maximum temperature increase and localized strain for nanocrystalline were 110C and 4.5%, whereas polycrystalline nickel showed 170C and 60%, respectively. The amount of heat generated locally by strain is related by the heat conversion factor (i.e. Taylor Quinney coefficient). Polycrystalline nickel showed a decreasing trend of heat conversion due to lattice distortions or defect formation during deformation. In contrast, nanocrystalline nickel showed an increasing trend, likely due to differences in deformation mechanisms. INTRODUCTION Over the past three decades, major research efforts have been concerned with the study of the mechanical properties of nanocrystalline materials, in particular nanocrystalline metals [e.g.1,2]. For nanocrystalline metals produced by the electrodeposition methods, significant improvements in their yield strength and hardness are observed which are due to HallPetch grain size hardening [3]. The influence of grain size on other mechanical properties such as the Young’s modulus, abrasive and adhesive wear resistance, and coefficient of friction are also well documented [3,4]. However, many issues regarding deformation mechanisms, intrinsic ductility or strain hardening capacity of these materials still require further study. One aspect issue that needs to be addressed is the phenomenon of in-situ heat generation and strain localization during deformation, which could have a considerable effect on the mechanical properties. Temperature increases during deformation could lead to thermal softening, which in turn could affect the failure process. It is well known [5,6] that most of the energy of plastic deformation is dissipated as heat and the remaining balance stored in the material as strain energy (e.g. lattice

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_3, © The Society for Experimental Mechanics, Inc. 2011

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distortions, defect formation). For example, this subject is of interest in the study of stored energy and its effects on the process of recrystallization during deformation. In recent years, infrared imaging technology [7] combined with digital image correlation (DIC) strain measurement [8] during tensile testing have been widely used. Examples include studies on the effect of cold expansion of holes using thermoelasticity and digital correlation [9,10] and the analysis of stress distributions on turbine blades [11]. Effects of heat generation and strain localization on mechanical properties are also well documented. For example, the infrared technique has been applied successfully to study heat generation due to plastic deformation in tensile testing of TRIP steels during deformation at strain rates up to 10-1/s [12]. Results showed a maximum temperature increase of 350C within the strain hardening regime; 580C near the necking regime and 900C just before sample fracture. One particular study on strain localization using DIC acquired full-field necking behaviour of mild steel deformed at a strain rate of 10-3/s [13]. Localized strain gradually developed and built up during the deformation process. The results showed that the highest localized strain was located at the necking zone beyond ultimate tensile strength and the position of highest localized strain corresponded to the area of fracture. In another study, the rapid evolution of shear banding close to the necking region was captured with the DIC method on fully dense nanocrystalline nickel at a strain rate of 10-4/s [14]. The purpose of this study is to correlate the phenomena of in-situ heat dissipation and strain localization for both polycrystalline and nanocrystalline nickel and to analyze the coupling effect to the thermomechanical properties during plastic deformation. EXPERIMENTAL PROCEDURE Tensile coupons of electrodeposited nanocrystalline nickel (grain size: 30 nm) of dimensions according to ASTM E8 standard (12 cm X 2 cm X 0.1 cm) were provided by Integran Technologies Inc, Toronto. Polycrystalline nickel samples with the same size were machined from commercially available Ni 200 (grain size: 30 µm). Prior to tensile testing, tensile samples were spray painted black on one side to equalize emissivity for better infrared detection. On the other side, samples were lightly abraded using 320 grit sandpaper for the measurement of localized strains. The high resolution infrared camera was calibrated before each experiment. Tensile samples were loaded on a MTS servo-hydraulic tensile testing machine. All samples were tested at a strain rate of 10-2/s. Temperature changes and localized tensile strain were measured simultaneously during tensile testing using a high resolution infrared camera (Deltatherm 1410) and a high resolution Digital Image Correlation Camera (Allied Vision Technologies), respectively. Temperature and strain were continuously recorded to sample fracture. However, in this paper the thermomechanical response was only analyzed up to ultimate tensile strength for each material. RESULTS AND DISCUSSION Polycrystalline and nanocrystalline nickel exhibit very distinct engineering stress-strain curves as shown in Fig. 1. Polycrystalline nickel has a yield strength of 208 MPa, ultimate tensile strength of 442 MPa and 50% elongation-to-fracture. On the other hand, nanocrystalline nickel shows a yield strength of 950 MPa, ultimate tensile strength of 1504 MPa and 8% elongation-to-fracture. These results illustrate that yield strength (σy) and ultimate tensile strength (σUTS) improve significantly when the grain size is refined to 30 µm to 30 nm. This can be explained on the basis of the Hall-Petch grain size strengthening mechanism. Both polycrystalline and nanocrystalline show considerable amounts of necking beyond UTS. However, polycrystalline nickel shows much higher ductility compared to nanocrystalline nickel due to the extended region of uniform plastic deformation, which can be attributed to the high intrinsic ductility of fcc materials with large grain size. Localized strain and heat distribution were measured by the DIC camera and the infrared detector, respectively, and the results of strain and heat recorded up to UTS for both polycrystalline and nanocrystalline nickel were also quite distinct (Fig. 2). Polycrystalline nickel with relatively high ductility showed noticeable changes in localized strain along the sample gauge when deformed to 25% and 45% engineering strain. The increase of localized tensile strain is highlighted by the gradual change in colour scale from dark purple and blue to bright green and yellow. At the 45% engineering strain level (i.e. just before UTS), polycrystalline nickel exhibits two regions of highly localized strain (i.e. yellow regions). One of the two regions with the highest strain levels corresponds to the position of final fracture, indicated by the dashed line in Fig. 2. On the other hand, for nanocrystalline nickel with relative high σy and σUTS, and low ductility only one high localized strain region was observed at 5.5% engineering strain (i.e. just before UTS). Such distinct behaviour of polycrystalline and nanocrystalline nickel regarding localized strain distribution is likely due to the nature of plastic flow within the uniform deformation regime. Localized strain of polycrystalline nickel (Fig. 3) showed a transition from homogeneous to inhomogeneous flow from ~ 20% engineering strain onwards. Within the region of inhomogeneous flow, the detected localized strain is higher than the engineering strain. For example, deformation at 40% engineering strain yielded 47% localized tensile strain in the region shown by the dashed line.

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Fig. 1 Engineering stress-strain curves for polycrystalline and nanocrystalline nickel

Fig. 2 Localized tensile strain (top) and heat distribution (bottom) of polycrystalline (left) and nanocrystalline nickel (right) at various engineering strain levels. The dashed lines indicate the position of the final fracture planes

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In contrast, nanocrystalline nickel (Fig. 4) shows that localized tensile strain is consistently lower than the engineering strain within the region of uniform deformation. For example, the deformation at 3% engineering strain in nanocrystalline nickel only yields 2% maximum localized tensile strain. This results in a relative homogeneous strain distribution observed in nanocrystalline nickel and low localized strain hardening capacity. In addition to the localized strain, the heat distribution (Fig. 2) during deformation was also recorded by the infrared camera. Similar to the result in strain localization, polycrystalline nickel also shows noticeable changes in temperature at different engineering strain levels. With the increase of engineering strain to 25% and 45%, an increasing amount of heat is dissipated and a considerable increase of temperature is detected at the center of the gauge (dashed line). The temperature increase reached its highest point of 170C at 45% engineering strain (Fig. 3). On the other hand, nanocrystalline nickel dissipates less heat and heat distribution along the sample gauge is more uniform. At 5.5% engineering strain, the maximum temperature increase reached 110C which is lower than for polycrystalline nickel (Fig. 4). The observed temperature increase correlates with the localized strain. In other words, increasing of localized tensile strain during deformation in both polycrystalline and nanocrystalline nickel induces heat dissipation. The difference in heat dissipation for the two materials could be explained by the heat conversion factor of polycrystalline and nanocrystalline nickel. According to the laws of thermodynamics, plastic work is converted to either stored energy (i.e. lattice distortions, defect formation) or to heat dissipation when metals undergo deformation process [15,16]. The fraction of heat converted from plastic work is known as the Taylor Quinney coefficient (β). 



C p T 



where is the material density; the heat capacity; the heat rate; represents the plastic stress and is the strain rate. In both cases of polycrystalline and nanocrystalline nickel, the heat capacity and density are constant at 480 J/mol K and 8.3g/cm3, respectively [3,17]. Different heat conversion trends were observed for polycrystalline and nanocrystalline nickel. Polycrystalline nickel shows that a decreasing amount of energy is converted to heat for increasing plastic strain to 40% (Fig. 5). In contrast, for nanocrystalline nickel the amount of plastic strain energy converted to heat increases with increasing strain (Fig. 6). The main deformation mechanism during room temperature deformation of polycrystalline nickel is dislocation slip. On the other hand, for nanocrystalline nickel a number of potential mechanisms have been proposed, including diffusional creep, grain boundary sliding and grain rotation [1-3]. We are currently assessing the correlation between heat dissipation/defect formation for the various deformation mechanisms in these materials. 20

Maximum temperature increase Maximum localized tensile strain

80 70

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Fig. 3 The maximum temperature increase and localized tensile strain of polycrystalline nickel

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Fig. 4 Maximum temperature increase and localized tensile strain of nanocrystalline nickel

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Fig. 5 Taylor Quinney coefficient of polycrystalline nickel

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Engineering plastic strain, % Fig. 6 Taylor Quinney coefficient of nanocrystalline nickel CONCLUSIONS In situ heat generation and strain localization during plastic deformation of polycrystalline and nanocrystalline nickel are interdependent. It was shown that localized strain induced localized heat dissipation in both materials. At 45% engineering strain (at UTS) in polycrystalline nickel, 60% localized strain induced a maximum temperature increase of 170C. On the other hand, nanocrystalline nickel at 5.5% engineering strain (at UTS) yielded 4.5% localized strain which induced a maximum temperature increase of 110C. The fractions of energy converted from plastic work to heat during deformation in polycrystalline nickel decreased with increasing deformation, which resulted in 80% of plastic work contributing to lattice distortions or defect formation at UTS. On the other hand, most of the plastic work in nanocrystalline nickel at UTS is converted to heat. ACKNOWLEDGEMENTS The authors would like to acknowledge the financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Ontario Research Fund (ORF). REFERENCES [1] Weertman J.R. in: Nanostructured Materials, 2nd ed., edited by C.C. Koch William Andrew Publishing, Norwich, NY, 2007 [2] Koch C.C., Structural nanocrystalline materials: an overview, Journal of the Materials Science, vol 42, p. 1403-1414, 2007 [3] Erb U., Aust K.T., Palumbo G. in: Nanostructured Materials, 2nd ed., edited by C.C. Koch William Andrew Publishing, Norwich, NY, 2007 [4] Erb U., Size effects in electroformed nanomaterials, Key Engineering Materials, vol 444, p. 163-188, 2010

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[5] Bever M.B., Holt, D.L. Titchener A.L., The stored energy of cold work, Progress in Materials Science, vol 17, p.190, 1973 [6] Taylor G.I., Quinney M.A., The latent energy remaining in a metal after cold working, Proceedings of the Royal Society of London. Series A, vol 143, p. 307-326, 1934 [7] Rogalski A., Infrared detectors: an overview, Infrared Physics & Technology, vol. 43, issue 3-5, p. 187-210, 2002 [8] Chu T.C., Ranson W.F., Sutton M.A., Applications of digital-image-correlation techniques to experimental mechanics, Experimental Mechanics, vol. 25, issue 3, p. 232-244, 1985 [9] Backman D., Liao M., Crichlow L., Yanishevsky M., Patterson E.A., The use of digital image correlation in a parametric study on the effect of edge distance and thickness on residual strains after hole cold expansion, The Journal of Strain Analysis for Engineering Design, vol. 43, issue 8, p. 781-789, 2008 [10] Backman D., Cowal C., Patterson E.A., Analysis of the effects of cold expansion of holes using thermoelasticity and image correlation, Fatigue & Fracture of Engineering Materials & Structures, vol. 33, issue 12, p. 859-870, 2010 [11] Backman D., Greene R.J., Gas turbine blade stress analysis and mode shape determination using thermoelastic methods, Applied Mechanics and Materials, vol. 13-14, p. 281-287, 2008 [12] Rusinek A., Klepaczko J.R., Experiments on heat generated during plastic deformation and stored energy for TRIP steels, Journal of Materials & Design, vol. 30, issue 1, p. 35-48, 2009 [13] Coppieters S., Cooreman S., Sol H., Houtte P.V. and Debruyne D., Identification of the post-necking hardening behaviour of sheet metal by comparison of the internal and external work in the necking zone, Journal of Materials Processing Technology, vol. 211, issue 3, p. 545-552, 2011 [14] Zhu R., Zhou J. Jiang H. Lui Y. Ling X., Multi-scale modeling of shear banding in fully dense nanocrystalline Ni sheet, Materials Science and Engineering: A, vol. 527, issue 7-8, p. 1751-1760, 2010 [15] Zehnder A.T., A model for the heating due to plastic work, Mechanics Research Communications, vol. 18, p. 23-28, 1991 [16] Zehnder A.T., Babinsky E., Palmer T., Hybrid method for determining the fraction of plastic work converted to heat, J. Experimental Mechanics, vol. 38, issue 4, p. 295-302, 1998 [17] Turi T., Erb U., Thermal expansion and heat capacity of porosity-free nanocrystalline materials, Materials Science and Engineering A, vol. 204, issue 1-2, p. 34-38, 1995

Dissipative and coupling effects accompanying the natural rubber elongation

B. Wattrisse, R. Caborgan, J.-M. Muracciole, L. Sabatier, A. Chrysochoos LMGC UMR CNRS5508 Montpellier University, CC048, Place E. Bataillon, 34095 Montpellier, France

ABSTRACT Rubber-like materials can undergo very large strains in a quasi-reversible way. This remarkable behavior is often called hyper (or entropic) elasticity. However, the presence of mechanical loops during a load-unload cycle is not consistent with a purely elastic behavior modeling. Using Digital Image Correlation and Infra-Red Thermography, the present study aims at observing and quantifying dissipative and coupling effects during the deformation of natural rubber at different elongation ratios. For elongation ratios less than 2, the famous thermo-elastic inversion is revisited within the framework of the irreversible processes thermodynamics, and interpreted as a competition between two coupling mechanisms. For elongation of about 3 or 4, the predominance of entropic elasticity is shown and the relevance of the analogy with perfect gases, at the root of its definition, is energetically verified. For very large elongation ratios (about 5), the energy effects associated with stress-induced crystallization-fusion mechanisms are underlined. The current experiments, performed at relatively slow strain rate, did not exhibit any significant dissipation.

Introduction In the literature, the first experimental studies on natural rubber were performed by Gough in the early 19th century [1]. They evidenced the coupled nature of its thermo-mechanical behavior. The experiments were resumed later by Joule [2] who observed that the straining of a vulcanized rubber generated a cooling of the specimen for small elongations, followed by a warming for higher elongations. He also noticed that the thermal expansion coefficient changes its sign – from positive to negative – with the increasing applied stress. This change of sign has since been associated with the so-called thermo-elastic inversion mechanism. This inversion phenomenon has been thoroughly studied by numerous authors (e.g. [3, 4]), and modeled within the framework of finite non-linear thermo-elasticity (e.g. [5,6]). We have recently proposed to override the hypothesis of pure thermo-elasticity to interpret this inversion phenomenon as the result of the competition between two concurrent coupling effects: a standard thermo-elastic effect (associated with the classical thermo-dilatation of materials), and a rubber effect (similar to perfect gas effect). At the micro scale, the macromolecular approach indeed highlights the very high mobility of the rubber molecular chains. Various experiments [2, 7] showed that the elasticity of this material was due to the entropy variation of the molecular chains network. They also suggested that the coupling mechanism was associated with the unfolding of the chains. The macromolecular approach is classically integrated within the statistical thermodynamics framework, and more specifically within the kinetic theory of gases. Using this analogy, which is the basis of “entropic elasticity” or “rubber elasticity”, one can demonstrate that the stress is proportional to the temperature, and that the internal energy depends only of the temperature (as a perfect gas) [4]. These results imply that the deformation energy developed by the material is totally transformed into heat [8, 9]. Numerous models were proposed in order to predict the mechanical behavior of rubber

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_4, © The Society for Experimental Mechanics, Inc. 2011

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26 materials, but very few were able to also take possible dissipative effects (inducing self-heating) and thermo-mechanical coupling (associated with the material thermo-sensitivity) into account. This study aims at observing, understanding and modeling the dissipative and thermo-mechanical coupling effects involved in the deformation of natural rubber during small amplitude cyclic loading around different given elongations. The material behavior analysis is achieved through energy balances performed using complementary imaging techniques, namely InfraRed Thermography (IRT) and Digital Image Correlation (DIC), giving simultaneously temperature and strain fields on the sample surface. The present paper describes the experimental procedure in a first part. The obtained results are shown and discussed in a second part. A heuristic thermo-mechanical model that account for the main characteristics of the observed energy balance is finally proposed.

Experimental setup and tests The experimental setup is presented in Fig.1.a. It involves a testing machine and two cameras set perpendicularly to the sample surface. An infrared “Titanium” camera (Cedip) provides thermal images of the surface, while a “Camelia 8M” camera (Atmel) gives images in the visible spectrum of the sample during the test. The sample is a classical dog bone shaped specimen, as illustrated in Fig.1.b. The dimensions of the gauge part of the sample are the following: initial length L0 (35 mm), width w0 (6 mm), and thickness t0 (2.1 mm).



λ




 






(a)

(b) Fig.1: (a) experimental setup, (b) sample geometry

The in-plane deformation is measured using a mark tracking technique. The initial gauge length between the vertical marks L0GP is 6 mm and the axial elongation λGP is computed as the ratio of the current gauge length LGP obtained by the marks tracking algorithm and the initial one. A specific calibration allows converting the thermal radiation digitized by the infrared camera into temperature. This calibration involves a black body with a uniform high emissivity coating, and guaranteed temperature homogeneity. As the sample undergoes very large deformations (more than 500%), it is not possible to paint or cover it in order to homogenize and increase its emissivity. Un-stretched natural rubber have a high emissivity (>0.9), and we have checked that it does not significantly change with the material elongation. The temperature variations are very small (less than 0.5 K), and it is necessary to take account of the environment thermal fluctuations that modify the heat exchanges with the surroundings.

27 These fluctuations are determined by placing a reference sample (of same material, and same geometry) in the immediate vicinity of the loaded sample. The thermal variations of this sample are used to estimate the thermal fluctuations of the sample environment. The heat sources, responsible for the temperature variations, are deduced from the in-plane thermal measurements, using the integrated form of the heat equation given in Eq. 1 [10]: dθ

ρ 

dt

θ

+ =  ⋅ τ

(1)

where  ⋅ represents the overall heat sources developed within the material, ρ the mass density of the material, C the specific heat, τ a time constant characterizing the linearized heat exchanges by diffusion, convection, and radiation. The particular time derivative of the temperature variation θ in (Eq. 1) is computed using the kinematic and thermal data given by the two cameras. The geometric transformation between the frames of reference of the two cameras is determined using a calibrated target. The whole experimental procedure is described in [11]. Furthermore, the “time constant” τ, characterizing the overall heat losses, depends on the geometry of the sample that changes significantly during the test. A classical heat fin model allowed us to take the evolution of τ with the measured axial elongation λGP into account. Two types of tests were performed. The first ones aimed at characterizing the thermo-elastic inversion phenomenon. The sample was submitted to a constant load (masses varying from 0 g to 730 g), and it was then heated of 25 K. Its deformation was recorded during the thermal return at room temperature. This test is the “dual” of the one performed by Anthony [12] where the stress evolution was measured at constant elongation during the temperature change. The thermal expansion coefficient can be derived by plotting strain vs. temperature. The second type of tests consists in a velocity-controlled ramp (v = 10 mm.s-1) up to a given maximum elongation (λGPM), followed by several loading cycles performed at a given elongation amplitude (∆λGP) with a given loading frequency (fL). Cycles were designed to separate the coupling from the dissipative mechanisms.

Results Fig.2.a illustrates the results obtained on the thermo-elastic inversion test. It represents the evolution of the axial Hencky strain εxx with the temperature variation θ – the initial temperature being higher than the room temperature – for different imposed loads (here the axial Cauchy creep stress σxx). The existence of thermo-mechanical couplings is here obvious as the material deforms when its temperature varies. We can clearly distinguish the manifestations of two opposite couplings since the material tends to contract when temperature decreases for small applied loads, and to expand for higher loads. The inversion stress σinv corresponds to the case where these opposite couplings annihilate one another. Fig.2.b shows the mechanical response of the natural rubber during a cyclic test at λGPM ≈ 1.5, ∆λGP ≈ 0.2 and fL = 0.2Hz. We can observe the quasi-linear response and the non-hysteretic character of the material behavior in this range of elongation. The thermal response is plotted in Fig.2.c that represents the imposed axial strain and the sample temperature with respect to time t. The temperature variations remain small throughout the test (less than 0.05 K), and the existence of a small thermal drift (amplitude smaller than 0.02 K), uncorrected by the experimental protocol described earlier can be noticed. Nevertheless, the coupled nature of the material behavior can be clearly observed as the temperature evolves in phase with the applied load. 

Fig.2.d represents the time course of the deformation energy   =  :  d – D representing the eulerian strain 

rate tensor –, and of the heat   =   ⋅  d. The deformation energy is, in good approximation, equal to the heat

during all the cycles. This experimental result confirms the existence of a thermo-mechanical coupling, and its entropic nature (i.e. similar to perfect gases). Furthermore, the amplitude of the possible dissipative effects seems to be negligible when compared with the amplitude of the thermo-mechanical coupling sources.

28

(a)

(b)

(c) (d) Fig.2: Experimental results for small elongations (λ < 1.5), (a) thermo-elastic inversion test: thermal dilatation response, (b) cyclic test: mechanical response, (c) cyclic test: thermal response, (d) cyclic test: calorimetric response Discussion The preceding results led us to propose a heuristic constitutive model involving two competing coupling mechanism (a classical thermo-elasticity and a rubber elasticity) in series, with no intrinsic dissipation. In a simple one-dimensional approach, three state variables are chosen: the temperature T, the logarithmic strain ε and a rubber strain εr, which play the role of internal state variable. The potentials defining the behavior model are the free energy Ψ,ε,εr , and the dissipation potential !"#, ε$ , %$& . They are defined in Eq. 2 and Eq. 3. The free energy is the combination of a classical thermo-elastic free energy, and the free energy of an affine 1D model. '

(

ρ.

Ψ,ε,εr  = )% * %& * αth  *  - *  (

/0

!"#, ε$ , %$&  =

+ 1α2th :;#⋅:;#

(
34 (

 4εr

+ 5c  

(

9

+ 7 8εr * (

(2) (3)

where T0 is the room temperature (here 293 K), E is the elastic modulus, αth the linear expansion coefficient, Kc a “rubber stiffness” coefficient, "# the heat influx vector, and k denotes the isotropic conduction coefficient. All the coefficients are extracted from the literature (αth ≈ 2.10-4 K-1, ρ ≈ 950 kg.m-3, C ≈ 2150 J.kg-1.K-1), except the time constant τ (τ ≈ 120 s), identified on a thermal return test, E (E ≈ 100 MPa) and Kc (Kc ≈ 1110 Pa.K-1) were identified on the basis of mechanical and thermal responses, namely the previously presented cyclic test. It is important to remark that the elastic modulus given in

29 classical material tables is not E but the equivalent stiffness of the two “elastic moduli” in series, here worth around 1 MPa. The difference in the order of magnitudes of E and Kc insures the smallness of the elastic strain compared to the rubber strain, and consequently the rapid predominance of the entropic coupling, for which the deformation energy-rate is equal to the heat sources.

(a) (b) Fig.3: confrontation model/experiment (a) thermo-elastic inversion test: thermal dilatation response (b) cyclic test (λGPM ≈ 1.5, ∆λGP ≈ 0.2 and fL = 0.2Hz): thermal response The Fig.3 compares the model predictions with the experimental responses. Introducing a competition between two coupling mechanisms leads naturally to a correct prediction of the thermo-elastic inversion phenomenon by the model, as it can be seen in Fig.3.a. The thermal response of the model is also in good agreement with the experimental observations (see Fig.3.b). Naturally, the thermal noise and drift is not accounted for in the model, but the amplitude of the temperature oscillations are properly reproduced.

(a) (b) Fig.4: mechanical energy rate and heat sources evolutions during cyclic tests (fL = 0.1Hz and ∆λGP ≈ 1) at high elongations (a) λGPM = 4, (b) λGPM = 6 Remember that the proposed model is quite voluntarily simple. For instance, the mechanical description of the rubber behavior can be improved by changing the form of the rubber free energy in order to obtain more realistic stress-strain responses (e.g. taking account of the limited elongation of chains). Nevertheless, we have chosen to perform other experiments in order to test the relevance of the thermodynamic hypotheses of this model in a wider range of elongations. Fig.4 illustrates the response of the material to cyclic tests performed at a higher elongations (λGPM = 4 and λGPM = 6). Fig.4.a shows that the entropic coupling remains preponderant up to elongations of 4. For higher elongations (see Fig.4.b), another coupling source appears: exothermic during the loading and endothermic during the unloading. This observation is consistent

30 with microstructural observations of elastomers using X-ray diffraction that show the apparition-disappearance of organized phases during the loading-unloading at similar elongations. These phenomena are interpreted as crystallization-fusion mechanisms [13]. Naturally, the model should be modified to integrate the latent heat of this phase change.

Conclusion Dilatation tests under a constant load were performed on natural rubber samples. They showed the classical effect of the thermo-elastic inversion [3,4]. Cyclic tests around different elongations highlighted the predominance of entropic effects for large elongations, the dissipative effects being negligible for the relatively small loading frequency imposed during the tests. Furthermore, the mechanical response of the natural rubber during cyclic loadings for high elongations underlined the presence of significant hysteresis loops that characterize an irreversible thermo-mechanical behavior. As a matter of fact, the interesting damping properties of this material are directly linked to the amount of energy “lost” in the loop. Let us remind that the mechanisms leading to the development of a hysteresis loop are numerous [14]. They can be entailed by: mechanical energy dissipation (e.g. viscosity effects), internal energy variations induced by microstructural evolutions within the material, thermo-mechanical couplings for non-isothermal or non-adiabatic processes (e.g. thermo-elastic effect). These latter effects seem to be predominant in the case of low frequency loadings of natural rubber. The classical models available in the literature propose to account for the inversion effect by introducing a modified entropic elasticity [5,6] that allows to retrieve the classical thermo-elasticity for the small elongations. We propose here to leave the strict framework of elasticity and to interpret this inversion as the result of the competition between two antagonist coupling effects: a classical thermo-elasticity (induced by the thermo-dilatability of the material, since the material is not purely incompressible), and a rubber effect described by a supplementary state variable called here “rubber strain”. A simplistic constitutive model, written in the framework of the generalized standard material formalism, and integrating the abovementioned properties, was proposed. In its simple “1D” version, the model has been identified using a single cyclic test (λGPM = 1.4) and values from the literature. We have checked that this model was able to predict directly the thermo-elastic inversion effect, and the preponderance of the rubber effects for large deformations (1.05< λGPM < 4) as the deformation energy was totally transformed into heat. For larger deformations (λGPM > 4), another coupling effect appeared. This effect can be correlated with stress-induced crystallization-fusion mechanisms observed in similar materials for similar elongation levels. It is noteworthy that the mechanical dissipation remained negligible for such damping materials. In order to observe the thermal effects of such material dissipation, cyclic tests are now under way at higher loading frequencies (i.e. several tens of Hertz).

References [1] Gough, J., A description of a property of Caoutchouc, or Indian rubber, Mem. Lit. Phil. Soc., 2, 288-295, 1805. [2] Joule, J. P., On some thermo-dynamic properties of solids, Philos. Tr. R. Soc., 149, 91-131, 1859 [3] Chadwick, P., Thermo-mechanics of rubberlike materials, Philos. Tr. R. Soc., 276, 371-403, 1974. [4] Treloar, L. R. G., The physics of rubber elasticity, 3rd ed., Clarendon, Oxford, 1975 [5] Chadwick, P., Creasy, C.F.M., Modified entropic elasticity, J. Mech. Phys. Sol., 32, 337-397, 1984 [6] Ogden, R.W., Aspects of the phenomenological theory of rubber thermoelasticity, Polymer, 28, 379-385, 1987 [7] Meyer, K. H., Ferri, C., Sur l’élasticité du caoutchouc, Helv. Chim. Actu., 18, 570-589, 1935

31 [8] Saurel, J.-L., Thermomecanical study of a thermoplastic elastomer family, PhD thesis Montpellier University France, 1999. [9] Honorat, V., Rubber thermomechanical analysis by field measurements, PhD thesis Montpellier University France, 2006 [10] Chrysochoos, A., Louche, H., An infrared image processing to analyse the calorific effects accompanying strain localisation, Int. J. Eng. Sci., 38, 1759-1788, 2000 [11] Chrysochoos, A.,Wattrisse, B.,Muracciole, J.-M.,El Kaïm, Y., Fields of stored energy associated with localized necking of steel, J. Mech. Mat. Struct., 4, 245-262, 2009 [12] Anthony, R. L., Caston, R. H., Guth, E. Equations of state for natural and synthetic rubber-like materials, J. Phys. Chem., 46, 826-840, 1942 [13] Trabelsi, S., Albouy, P.-A., Rault, J., Crystallization and melting processes in vulcanized stretched Natural Rubber, Macromolecules, 36, 7624-7639, 2003 [14] Chrysochoos, A., Huon, V., Jourdan, F., Muracciole, J.-M., Peyroux R., Wattrisse, B., Use of full-field Digital Image Correlation and Infrared Thermography measurements for the thermomechanical analysis of material behavior, Strain, 46, 117-130, 2010

Experimental estimation of the Inelastic Heat Fraction from thermomechanical observations and inverse analysis Thomas Pottier(1), Franck Toussaint(1), Hervé Louche(2) and Pierre Vacher(1) (1) Laboratoire SYMME - Université de Savoie - Polytech Annecy-Chambéry, BP 80439, 74944 Annecy le Vieux Cedex, France. (2) Laboratoire de Mécanique et de Génie Civil (LMGC), Université Montpellier 2- CC 048 Place Eugène Bataillon, 34095 Montpellier Cedex, France. herve.louche@univ_montp2.fr

ABSTRACT A new method to estimate the inelastic heat fraction (Taylor and Quinney Beta coefficient) during the deformation of a titanium material is proposed. It is based on (1) thermomechanical full field measurements during the loading and on (2) an inverse analysis. First, two cameras (a visible and an infrared) are used to measure the kinematic and the thermal fields on each face of a notched flat sample loaded in tension. Second, two coupled finite element simulations (a mechanical, then a thermal one) of the same tests are conducted. Associated with a Levenberg-Marquardt optimization algorithm, they are able to give, in a first step, optimized values of the anisotropic elastoplastic model parameters. Then, in a second step, parameters of four strain dependent Beta models are identified. Finally, the thermal responses of these models are compared to the experimental values.  

  INTRODUCTION Heat generation during the deformation can be estimated either by an experimental approach or by a modeling one [1,2]. In these two approaches, the intrinsic dissipation is often computed through In-elastic Heat Fraction (IHF also denoted β). As mentioned by [3], the β factor, introduced and calculated in the pioneer work of Taylor and Quinney [4], was an integrated factor defined by the ratio of the dissipated energy to the plastic work. In many Finite Element codes, the β factor was assumed to be constant and typically about 0.9 for metals (see [5] for example). However, several experimental studies have shown that this assumption does not hold in many material types ([2, 4, 6-9]). In these studies, β was different than 0.9 and was strain (and sometimes strain rate) dependent. A summary of such variations in β values can be found in [8]. Several experimental techniques have been proposed to measure, at a macroscopic scale, the β or the stored energy ratio. In this work, a new method to estimate the inelastic heat fraction (β) during the deformation of a titanium material is proposed. It is based on (1) thermomechanical full field measurements during the loading and on (2) an inverse analysis. After a presentation of the proposed procedure, various models of β evolution with the strain are chosen a priori and identified through an inverse method. The chosen method is the Finite Element Updating (FEU), broadly used in several applications.

  OVERVIEW OF THE PROPOSED PROCEDURE In thermomechanical analysis, the material self-heating is often simulated in the FE calculations as a conversion rate of the plastic power per unit volume !!! into heat: !!! = !!!! , (1) ! where !! is the dissipated power per unit volume generated by irreversibilities (plasticity, damage...), and β is the so called ! Taylor and Quinney factor. In the case of titanium, the thermomechanical coupling power per unit volume !!"#  is only due ! ! to the thermoelastic power per unit volume !!!!" . The heat sources, noted !!! , can then be computed using the following equation: ! ! !!! =   !!! +   !!!!" . (2) In the present work, two FEU inverse methods are successively run (Fig. 1): (i) The first one denoted FEU-M is a mechanical inverse identification and aims to determine the parameters of the mechanical model. Global measurement (reaction force of the sample !!"# (!)) and local displacement data obtained by DIC T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_5, © The Society for Experimental Mechanics, Inc. 2011

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34 are used to identify both Ludwick’s hardening parameters and Hill’s anisotropic parameters. Mechanical parameters are assumed not to depend on the sample self-heating in the observed range of temperature [10].

 

Fig. 1: Mechanical parameters and inelastic heat fraction identification flowchart (ii) The second one denoted FEU-T, identifies the parameters of four different β evolution models: # Model!1:! ! = a % xx xx % Model!2 :! ! (! p ) = b! p + c % xx xx e (3) $ Model!3 :! ! (! p ) = d(! p ) + f , % xx 1"n (! p ) " h % xx % Model!4 :! ! (! p ) = xx (! p )1"n & xx

where ! p is the axial component of the Green Lagrange plastic strain tensor. The first model gives a constant value for β and models 2 to 4 are strain dependent. The three first models exhibit an increasing number of parameters (a to f) and the fourth one is the model proposed by Zenhder [11] with two parameters. ! This identification takes into account the plastic power !!! , the thermo-mechanical coupling power (!!!!" ) both computed in the previous FEU-M analysis and the experimental thermal field !!"# measured by the infrared camera.

  EXPERIMENTAL SETUP

The tested samples are cut from titanium rolled sheets (thickness: 1.6 mm). The rolling and the transverse directions are respectively denoted 0° and 90°. A notched geometry (Fig. 2) is chosen, according to the work of [12], so that the obtained strain fields exhibit a shear band in the central zone of the sample. Digital Image Correlation (DIC) is used for kinematic measurement purpose on one side of the sample. DIC is processed using 7D software [13]. The dimension of the grid is set to 16 × 16 pixels and the dimension of the pattern used to compare sub images is 16 × 16 pixels. The displacement field has a bi-linear form and the gray level interpolation is bi-cubic. The other side of the sample is used for thermal field measurement. A fixed infrared camera (Cedip Jade III MW) captured thermal field at a 10 Hz frequency with a resolution of 320×240 pixels.

 

Fig. 2: Experimental setup with kinematic and thermal full-field measurement devices facing different sides of the sample.

 

35

NUMERICAL MODELLING A 3D Finite Element model is built to duplicate the experimental geometry. The only visible part of the sample is modeled. In other words, the free ends, hidden in the grips, are not considered. The part is meshed using 7056 solids elements (3 layers). Due to the unavailability of through-thickness measurements (whether from a kinematic or a thermal point of view), the surface data are repeated over the 3 layers of elements. Two distinct problems need to be solved successively: the mechanical one and the thermal one. Hence, two cost-functions are defined. The first is needed in the mechanical problem and is then built using strictly mechanical data: the two components of the displacement and the global force. The second, defining the thermal problem, is built from sample temperature. The cost functions are minimized using a Levenberg Marquardt algorithm and sensitivities are computed through a forward finite difference scheme. Mechanical boundary conditions are applied in each node of the sample top and bottom borders according to the corresponding DIC measurement. Thermal boundary conditions are modeled as thermal fluxes on the front/back face of the sample (convection with the ambient air) and on the top/bottom borders (through grip thermal conduction). The parameters of those heat fluxes are experimentally estimated from independent tests.

  RESULTS • Mechanical identification: The six parameters of the model (two from the hardening law and four from the in-plane Hill’s criterion) are identified from two tests led with samples cut at 0◦and 90◦ from the rolling direction. Results show a good agreement between measured and predicted displacement fields (Fig. 3).

 

Fig. 3: Evolution, during loading of the relative error (in %) on longitudinal displacement, after identification, for experiment at 0◦. Hence, the displacement and the stress fields computed with the optimized parameters are used to assess the plastic power !!! ! per unit volume and the thermoelastic coupling power per unit volume !!!!" . Thus, given a value of β, allows the estimations of the heat sources. Fig. 4 shows the time evolution of the volume plastic power in the sample. Higher value of !!! (and so the heat sources) are located in the reduced section of the sample with higher intensities near the edges.

Fig. 4: Evolution, during loading, of the calculated plastic power fields (in W.m−3). • Thermal identification: The thermal problem is solved for a single test led at 0◦. Identified parameters of the four models are gathered in Table 1. The models no.3 and no.4 exhibit the lowest values of the cost function and are thus the most able to predict the measured temperature. Conversely, model no.1 is certainly not sufficient to model the observed phenomenon.

 

36 Table 1: Identified thermal parameters of the four models of β.   Parameter   Initial  value   Final  value   Final  cost  function  value  

Model  no.1   a   1   0.65     2.199  

Model  no.2   b   c   0.3   0.3   0.79   0.43   1.910  

d   0.3   0.99  

Model  no.3   e   1   0.21   1.704  

f   0.3   0.01  

Model  no.4   ℎ   1   4.08   1.810  

  Moreover, the evolution of the β ratio versus strain is plotted in Fig. 5.a. Since the ratio of plastic power converted into heat is xx xx not defined for ! p =0, the plot has been chosen to start at ! p =0,05. Fig. 5.b shows the temperature evolution predicted by the four models and measured at the sample surface for 3 points (P1, P2 and P3) defined in Fig. 4. Fig. 5.b confirms the above considerations among which model 3 is the most able to predict the observed temperature field. Thus, considering the presented results, the evolutionary value of β along strain seems obvious. However, the chosen model to represent this evolution is of the utmost importance. For instance, the model no.2 does not bring major improvements compared to the constant model assumed in many former studies.  

 

Fig. 5: (a) Predicted evolutions of β for the 4 identified models. (b) Comparison between the experimental response and the predicted temperature for the 4 identified models, in the three points P1, P2 and P3 given in Fig. 4.

  CONCLUSIONS In the present paper, a new method to identify the inelastic heat fraction (β) during a quasi static tensile test is proposed. The method is based on an inverse Finite Element Update approach, coupling a Finite Element code (Abaqus) and a LevenbergMarquardt algorithm, and using simultaneous DIC and IR thermal full field measurements. This method is first used to identify the mechanical parameters of material constitutive equations by the means of DIC measurements. Then, the obtained results are used to assess the inelastic heat fraction for the same test. Four strain evolution models for β are tested. Up to the maximum load, results show a good agreement between measured and predicted temperature fields. The identified evolution models of β, on a titanium material, confirm a strain dependency but remain probably too simple to retrieve the whole energetic process. Furthermore, the obtained values of β are very different from the broadly used value (0.9). The best prediction is obtained with a non linear evolution of β with the strain, governed by three parameters (model no.3). The strain evolution is close to the one given by Zehnder model but leads to better temperature estimations at the beginning of the loading. Improvements of β models, in order to obtain non monotonous evolution with the strain, may also be proposed with the same approach.

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REFERENCES [1] Chaboche J-L., Cyclic viscoplastic constitutive equations, Part II: stored energy, comparison between models and experiments, J. Appl. Mech., 60, pp. 822-828, 1993. [2] Dumoulin S., Louche H., Hopperstad O. and Børvik T, Heat sources, energy storage and dissipation in high-strength steels: Experiments and modelling, Europ. J. of Mech - A/Solids, 29, pp. 461-474, 2010. [3] Rittel D., On the conversion of plastic work to heat during high strain rate deformation of glassy polymers. Mech. Mater., 31, 131–139, 1999. [4] Taylor G.I. and Quinney H., The latent heat remaining in a metal after cold working, Proc. Roy. Soc. London A, 163, pp. 157-181, 1937. [5] Hodowany J., Ravichandran G., Rosakis A. J. and Rosakis P., Partition of plastic work into heat and stored energy in metals, Exp. Mech., 40(2), pp. 113-123, 2000. [6] Bever M.B., Holt D.L. and Titchener A.L., The Stored Energy of Cold Work, Third ed. Pergamon Press, Oxford, 1973. [7] Chrysochoos A., Maisonneuve O., Martin G., Caumon H. and Chezeaux, J-C., Plastic and dissipated work and stored energy, Nuclear Eng. and Design, 114, pp. 323-333, 1989. [8] Macdougall D., Determination of the plastic work converted to heat using radiometry, Exp. Mech., 40, pp. 298-306, 2000. [9] Chrysochoos A., Wattrisse B., Muracciole J.-M. and El Kaïm Y., Fields of stored energy associated with localized necking of steel, J. of Mech. of Mater. and Struct., 4, pp. 245-262, 2009. [10] Pottier T., Toussaint F. and Vacher P., Contribution of heterogeneous strain field measurements and boundary conditions modelling in inverse identification of material parameters, Europ. J. Mech. A./Solids., In Press, 2011. [11] Zehnder A.T., A model for the heating due to plastic work, Mech. Research Com., 18, pp. 23-28, 1991. [12] Meuwissen M., Oomens C., Baaijens F., Petterson R. and Janssen, J., Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method, J. of Mater. Process. Tech., 75, pp. 204-211, 1998. [13] Vacher P., Dumoulin S., Morestin F., Mguil-Touchal S., Bidimensional strain measurement using digital images, Proc. Inst. Mech. Eng., 213, pp. 811-817, 1999.

Energy balance properties of steels subjected to high cycle fatigue

A. Chrysochoos1, A. Blanche1, B. Berthel2, B. Wattrisse1, (1) LMGC, UMR CNRS 5508 Montpellier University, CC 048, Place E. Bataillon, 34 095 Montpellier, France (2) LTDS, UMR CNRS 5513, École Centrale de Lyon, 36 Avenue Guy de Collongue, 69 134 Ecully, France

ABSTRACT: This paper presents an experimental protocol developed to locally estimate different energy balance terms associated with the high cycle fatigue (HCF) of steels. Deformation and dissipated energy are respectively derived from displacement and temperature fields obtained using digital image correlation (DIC) and quantitative infrared thermography (QIRT) techniques. The combined processing of visible and infrared images reveals the precocious, gradual and heterogeneous development of fatigue localization zones. It also highlights the plastic character of dissipative heat sources (i.e. proportional to the loading frequencies), and the progress of fatigue dissipation, observing the drift of the mean dissipation per cycle for a given loading. The substantial of internal energy variations during HCF loading are finally underlined. The paper ends with a discussion on the consequences of such energy balance properties in terms of HCF modeling. 1. Introduction Standard characterization of fatigue in materials requires time-consuming and statistical processing of the numerous results generated by expensive mechanical tests. Over the last two decades, alternative experimental approaches have been developed to reliably and rapidly measure fatigue characteristics. These new approaches include thermal methods based on the analysis of self-heating during stepwise loading fatigue tests [1-2]. Although realistic estimates of fatigue characteristics have sometimes been obtained, the thermal approach often leads to questionable results [3]. The direct use of temperature as a fatigue indicator may not always be the most relevant approach since temperature variations are not intrinsic to the material behavior. Energy approaches have been consequently proposed [4-6]. During high cycle fatigue, part of the generated heat comes from irreversible manifestations of micro-structural defects (e.g. intra-granular slip bands), while another part comes from thermoelastic coupling sources linked to reversible thermal expansion of the crystalline network. A combined image processing technique was thus developed to obtain 2D patterns of energy balances. Speckle image correlation techniques, involving a chargecoupled device (CCD) camera, were used to assess surface displacement fields. In parallel, thermal data were provided by an infrared focal plane array (IRFPA) camera. Heat sources were estimated on the basis of partial derivative operators present in a local form of the heat diffusion equation by using a set of approximation functions that locally fits the displacement and temperature field and takes the spectral properties of the sought sources into account [7]. The present paper shows several important properties associated with the fatigue of a dual-phase steel (DP600) commonly used in mechanical industries. This paper laid out as follows: the different terms of the energy balance associated with high cycle fatigue of steel are first recalled. A brief presentation of the experimental set-up is then given, while the main part of the paper presents the energy balance properties. In particular, the precocious, gradual and heterogeneous development of fatigue localization is shown. Then the “plastic” character of dissipation (i.e. proportional to the loading frequency fL) and the evolution of the mean dissipation per cycle during blocks of cycles within a constant stress range Δσ for a given fL and a given stress ratio Rσ are successively

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_6, © The Society for Experimental Mechanics, Inc. 2011

39

40 underlined. Finally, the experiments revealed substential internal energy variations (stored energy) during socalled elastic loading cycles. The paper ends with a discussion the consequences of such energy properties in terms of HCF modeling. 2. Theoretical background a. Heat equation Concepts and results of the thermodynamics of irreversible processes must be used to define the different heat sources induced by HCF processes. In this paper, thermodynamics with internal state variables will be used, whereby the equilibrium state of each volume material element is characterized by a set of n state variables [8]. The chosen state variables are the absolute temperature T, the small strain tensor ε, and the n-2 scalar components (α1, …, αn-2) of the vector α of so-called "internal" variables that model the micro-structural state of the material. By construction, the thermodynamic potential is the Helmholtz free energy ψ. Combining the first and second principles of thermodynamics leads to the local heat diffusion equation:

ρCT − div(k gradT ) = σ : ε − ρψ,ε : ε − ρψ,α .α + ρT ψ,εT : ε + ρT ψ,αT . α + re ,  

 

d1

(1)

sthc

where ρ denotes the mass density, C the specific heat, k the heat conduction tensor, σ the Cauchy stress tensor and ε ƚŚĞstrain rate tensor. The left-hand side consists of a differential operator applied to the temperature, while the right-hand side pools the various types of heat source: the intrinsic dissipation d1, the thermo-mechanical coupling sources sthc, and the possible external heat supply re (e.g. radiation exchanges). The intrinsic dissipation characterizes the material degradation accompanying the irreversible microstructure transformation, while the thermo-mechanical heat sources translate the thermo-sensitivity of the matter, indicating that the mechanical and thermal states are closely coupled. When the couplings are only due to thermo-elasticity, the coupling sources are limited to the famous Lord Kelvin’s term

sthe = ρT ψ,εT : ε 

;ϮͿ

Up to now, the following strong hypotheses have been put forward to compute heat sources: - the mass density and the specific heat are material constants, independent of the thermodynamic state. - the heat conduction tensor remains constant and isotropic during the test (kij = k.δij). - the external heat supply re due to heat exchange by radiation is time-independent, so the equilibrium temperature field T0 verifies –ΔT0 = re. It is then convenient to consider the temperature variation θ defined by θ=T−T0. Taking the latter hypotheses into account, the heat equation can be rewritten in the following compact form:

ρC θ − k Δθ = w h< ,

(3)

where θ is the particular time derivative of the temperature variation, Δ the 3D Laplacian operator, w h< the overall heat source, while wh is the heat locally evolved and where the non standard dot (.)• specifies that the energy rate is path dependent (i.e. such energy is not a state function). These hypotheses are reasonable in many situations and in particular at the beginning of HCF processes. However, they become inadequate when a strong anisotropy pre-exists or develops during the loading, whether a strong strain and/or damage localization occurs or whether the thermo-mechanical loading leads to dynamic instabilities. b. Energy balance Let us consider a load-unload cycle, where A=(TA, εA, αA) and B=(TB, εB, αB) denote the thermo-mechanical states of the material at the extremities of the process. Let us then draw up the energy balance corresponding to the following three situations:

- case 1: A ≠ B, this general situation can be illustrated by the schematic stress-strain diagram proposed in Fig. 1. - case 2: A ≠ B and εA =ҏ εB, a mechanical cycle is then associated with the load-unload test. The stress-strain diagram shows a hysteresis loop. - case 3: A = B, the mechanical cycle is now a thermodynamic cycle.

41 For a quasi-static process and under the small strain hypothesis, the rate of deformation energy is classically defined by: < w def = ı : ε . (4) < , and Following Eq.(1), the intrinsic dissipation d1 is then the difference between the deformation energy rate w def

the sum of the elastic w e< and stored w s< energy rates: < d1 = w d< = w def − w e< − w s< ,

(5)

w <e + w s< = ρψ,ε : ε + ρψ,α .α .

(6)

Both principles of thermodynamics give an alternative expression of the volume deformation energy associated with the load-unload test:

w def =

tB

³t

A

t

t

t

< + w < )dt , d1 dt + ³ B (w e< + w s< )dt = ³ B d1 dt + ³ B (ρe − ρC θ + w the thc tA tA tA

(7

σ

A B T,α

ε

Fig. 1 Schematic stress-strain diagram for a load-unload test; tB-tA is the cycle duration

Equation (7) shows that: -case 1: in the general case, the balance of deformation energy during a load unload test involves, energy dissipation, internal energy variations, variation of heat stored in the material, coupling heat sources. - case 2: for a mechanical cycle, the deformation energy then corresponds to the energy Ah of the area of the hysteresis loop. - case 3: for a thermodynamic cycle, this hysteresis loop is then only due to dissipation and couplings when the heat capacity ρC is assumed to be constant. t

t

< + w < )dt w def = Ah = ³ B d1 dt + ³ B (w the thc tA tA

(8)

The energy balance form then gives the restricted conditions for which the dissipated energy can be estimated by computing the hysteresis area of a uniaxial load-unload cycle. This underlines the necessity of analyzing the thermal effects to verify if a mechanical cycle is also a thermodynamic cycle, and of checking the relative importance of coupling effects. It is worth noting that in the case of HCF of DP 600 steel, the thermoelastic energy remains negligible over a complete cycle. Moreover, after some cycles, the temperature becomes periodic and the mean temperature variation stabilizes. In such cases, the deformation energy over a loading cycle consists of dissipated energy and internal energy variation.

w def =

tB

tB

³tA d1 dt + ³tA ρ e dt = w d + [ρe ]A B

(9)

If the deformation energy is equal to the dissipated energy, the mechanical cycle is a thermodynamic cycle (case 3) and there is no overall microstructure evolution over a cycle. A mechanical illustration of this situation is the perfect elastic-plastic shakedown. If the deformation energy is greater than the dissipated energy (while

42 neglecting coupling heat over a cycle), the material stores some internal energy (case 2). This stored energy is used by the material to modify its microstructure. If the deformation energy becomes declines to below the dissipated energy, part of the stored energy is then transformed into dissipated energy which means that the material is no longer able to adapt its microstructure. Note that dividing the volume energy (resp. energy rate) by the volume heat capacity ρC of the material gives the -1 volume energy (resp. energy rate) expressed in °C ( resp. in °C.s ). This operation is convenient because it associates equivalent heating in adiabatic conditions and then gives the orders of magnitude of this energy (resp. energy rate) in readily understandable units. 3. Experimental arrangement

The experimental set-up, designed to derive local energy balances, involved an MTS hydraulic testing machine (frame: 100 kN, load cell: 25 kN), a Camelia 8M high resolution CCD camera and a Cedip Titanium infrared camera. The optical axis of both cameras was set perpendicularly to the frame of the testing machine, and remained fixed during the test. A simultaneous record of infrared and visible images was performed on each side of the sample surface, and thin flat samples were therefore used.





Fig. 2 Experimental set-up. (Front): CCD camera and cold light sources, (back): IRFPA camera. The grips and columns of the testing machine are roughly wrapped to avoid parasitic reflection

The sample shape corresponded to a standard dog-bone specimen (Fig.3). The sample surface in front of the CCD camera was speckled with white paint on a black background to optimize the contrast of local optical signature, while the surface emissivity of the sample gauge part placed in front of the IR camera was increased 2 and homogenized using mat black paint. Typically the size of the analyzed zone was about 10×10 mm for a -2 -1 space resolution of about 0.1 mm. The conventional strain rates were less than 10 s . This strain rate order of magnitude ensures the quasi-staticity of processes and induces significant (thermal) signal-to-noise ratios.

Fig.3 Shape of the specimen (unit: mm) and frame of reference

43 Each camera was controlled by a separate computer although a specific electronic device was designed to synchronize the frame grabbing of the two cameras. Using this device, the synchronization error between the two cameras is estimated to be less than 0.05 ms. Reference speckle and infrared images were mapped by determining the affine transformation coefficients (rigid body movement and homothetic transformation) of the shape of a calibrated target. The main material constants of the DP600 steel are given in Table 1. Tab. 1 Thermophysical properties of DP600 -1

-3

-1

-1

-1

λth (10 .°C )

ρ (kg.m )

C (J.kg .°C )

k (W.m .°C )

10-11

7800

460

64

-6

-1

For each acquisition time, the thermal data given by the IR camera (measured in the current Eulerian configuration) were linearly interpolated in space and time using the positions of the deformed configuration given by the DIC computation. This operation allowed the material surface element (mse) associated with the DIC mesh to be tracked so that the extent of the volume heat rate induced by matter convection and associated with the particular time derivative of the temperature could subsequently be computed and checked. Readers interested in the many tricky metrology issues and in a detailed presentation of the image processing can refer to [7, 9]. An appropriate image processing method was developed to separately estimate fields of mean intrinsic dissipation per cycle d1 and fields of thermoelastic source amplitude Δsthe. [5, 7]. The local fitting function θfit of the temperature charts is chosen as: θfit ( x, y , τ ) = p1 ( x, y ) τ + p2 ( x, y ) + p3 ( x, y ) cos ( 2πfL τ ) + p4 ( x, y ) sin ( 2πfL τ )

(10)

where the trigonometric time functions describe the periodic part of the thermoelastic effects while the linear time function takes transient effects due to heat losses, dissipative heating and possible drifts in the equilibrium nd temperature into account. Functions pi(x, y), i = 1,…,4, are 2 order polynomials in x and y. These polynomials enabled us to take the possible spatial heterogeneity of the source patterns into account.

4. Experimental results a. Description of tests Tests involved loading blocks, with each block being performed at constant Δσ with Rσ=-1 and fL = 30 Hz. Tests were constructed in stages (Fig. 4), each stage Si, i=1,2, …, including: - a series of 5 “mini” blocks of 3 000 cycles per block with increasing Δσ varying regularly from 175 up to 580 MPa (mi), - a large block of 100 000 cycles performed at Δσ = 580 MPa (pi). 3000 cycles

600

100 000 cycles

400

200

0

1

2 3

mi

4

5

6

pi

7

8

9 10 11 12

mi+1

pi+1

Fig.4 Basic sketch of the fatigue test

44 The stages were repeated until a macro-crack occurred. The series of mini blocks was performed to construct the curve d1 vs. Δσ at a “constant” fatigue state, while the large blocks were defined, close to the fatigue limit so as to speed up the fatigue progress. For the large blocks, camera shots were regularly taken to check the monotonous evolution of dissipative effects.

b. Mechanical response To visualize the hysteretic mechanical response of the material during a loading cycle, and to improve computation of the deformation energy corresponding to the hysteresis area of the stress-strain curve (Fig. 3), the -3 CCD camera data acquisitions were performed at fL= 5.10 Hz to increase the fS/fL ratio, with fS denoting the frame rate of the camera (fS = 3 Hz).

200 σ (MPa) 100

Rσ = − 1 ∆σ = 471 MPa

0 -100 -200

ε

x 10-3

-1 -0.5 0 0.5 1
Fig.5 Mechanical hysteresis loop corresponding to case 2. Note that the the stress state fields were computed assuming a plane stress state and an isochoric transformation. These hypotheses are naturally questionable, particularly at the end of the specimen’s life, where the triaxiality of the stress state is certainly coupled with damage development. Note also that the complete passage from strain to stress field measurements has already been detailed in [10]. The relation used to compute the mean deformation energy rate over a cycle was then: < w def = fL ³

t A +fL−1

tA

(σ xx ε xx + 2 σ xy ε xy + σ yy ε yy )dt

(11)

c. Thermoelastic response Until crack onset, we observed quasi-uniform patterns of thermoelastic nature as expected for a simple tensioncompression test. Besides, the experiments showed that Δsthe increased linearly with stress and the load frequency as predicted by a linear thermoelastic law [5]. An example of the distribution of thermoelastic source amplitudes is given in Fig.6. ∆σ= 580 MPa Rσ = -1

x (pixels)

60

(°C.s -1)

104

∆sthe

40

100

20 10

30 y (pixels)

50

96 th

Fig. 6 2D distribution of mean thermoelastic sources Δs the ( x, y ) averaged over the 5 loading block

45

This 2D distribution is the mean thermoelastic source amplitudes averaged over the fifth loading block. The two

main features to note are the quite good homogeneity of the coupling source field and the order of magnitude of -1 the thermoelastic effects. The mean value of Δs the over the gage part of the specimen is around 101.9 °C.s with -1 a standard deviation of about 2.4 °C.s . Considering the HCF of steel, this heat source amplitude order of magnitude is typically a hundred to a thousand times greater than the dissipation intensities, as shown hereafter (cf. Fig.7). This difference in order of magnitude of both types of sources markedly complicates the image processing. Indeed, the underlying dissipation sources, even those of low intensity, are essential for understanding the fatigue mechanism development, and must be carefully extracted from the overall heat sources.

d. Dissipation field properties Contrary to thermoelastic source fields, heterogeneous distributions of dissipative sources were systematically observed (Fig. 7a). During a fatigue block, this pattern remains spatially constant whatever the set of loading parameters as long as the fatigue effects are minor and do not herald a crack inception [5]. We associated the time independence of the heterogeneous distribution of dissipation with the low development of persistent slip bands (PSB) concentrated in a few grains [11]. We then interpreted the rapid evolution and strong narrowing of the highest dissipation zone as a sign of material damage leading to the crack inception. The experimental method thus enabled us to detect this crack localization several thousands of cycles before its onset.

x (pixels)

80

0.45

d1/ρC

(°C.s -1)

0.25 40

0 hdz 10

ldz

0 30 50 y (pixels)

Fig. 7 Example of a heterogeneous distribution of dissipation averaged over a mini block: fL=30 Hz, Δσ=500 MPa, Rσ=-1. The circles locate low and high dissipation zones, respectively [5] The calorimetric analysis also showed that d1 linearly increases with the loading frequency at constant loading amplitude at every point of the sample gage part, and despite the heterogeneous response of the material. Fig.8 shows 2 examples of mean intrinsic dissipation per block d1 as a function of Δσ and of fL. d1/ρC

d1/ρC 0.2

0.4

°C/s

°C/s 0

0 30 fL (Hz)

9 170

501 ∆σ (MPa)

30

fL (Hz)

9 170

501 ∆σ (MPa)

Fig. 8 Evolution of the mean dissipation per (mini) block as a function of the stress range Δσ and of the loading frequency fL. On the left: “low” dissipation zone, on the right: “high” dissipation zone [5]

46 Two zones were chosen (low/high dissipation zones) to show that, regardless of the zone, the dissipation surfaces were similar, which means that the microstructural mechanisms would probably be the same when they do not have the same intensity. In particular, the linear evolution of dissipation as a function of the loading frequency was compatible with the rate independent formalism of plastic damageable materials. No viscous or yield effects were observable here from a dissipative standpoint. Fig. 9 shows the drift of intrinsic dissipation along with the number of cycles for different stress ranges. The mean dissipation values were computed for the period corresponding to a mini block (i.e. 100 seconds) and averaged over the sample gage part. The curves associated with mini-block series were interpreted as the dissipative signature of the microstructural state at different stress ranges for an approximately constant fatigue state. This signature was associated with the number of active PSBs. In general these PSBs are randomly oriented within a few grains and the greater the stress intensity, the larger the number of activated PSBs. For large blocks, performed close to the fatigue limit (ΔσL≈540 MPa when Rσ=-1), the monotonous increase in dissipation during such blocks was the energy manifestation of the fatigue progress. 4

375k cycles

d1/ρC (°C.s -1)

m3

3

p3

260k cycles 145k cycles

2

m2

30k cycles

1

p2 p1

m1 m0

0

200

300

400

∆σ (MPa)

500

600

Fig. 9 Drift of the mean dissipation




e. Discussion on the size of dissipation zones In Fig. 7, we observe that the order of magnitude of zones that dissipated preferentially was about 10-20 pixels, corresponding to 1-2 mm. This size is naturally surprising considering the size of PSBs, which is about 1 μm. To be convinced by such mesoscopic effects, we considered that dissipative heat sources were induced by activated PSBs. The grain size was chosen to be equal to 10 μM and we supposed that only a few grains had PSBs (see Fig. 10).

10-1 m: specimen gage part source computation

10-3 m: data fitting zone

θµ (x) thermoprofile

µ d1

PSB 10-6 m: PSB size 10-5 m: grain size

xµ, xm

10-4 m :camera resolution

Fig. 10 Characteristic lengths associated with HCF dissipation

48 consuming image processing prevented us from continuing the fatigue test until crack inception. However, it would be interesting to also characterize these potential crack zones in terms of stored energy evolution. x 10-3 °C

16

14 12 x (mm)

10 8 6 4 2

0

0

y (mm)

wdef

6

wd

ws

Fig. 12 Fields of the mean energy balance associated with a cycle [12] 6. Concluding comments In this paper, an infrared image processing method that enables estimates of deformation energy, coupling and dissipative sources accompanying HCF of steel is presented. Applications to Dual Phase 600 steel sheets showed a linear change in the thermoelastic source amplitude as a function of the stress range and the loading frequency, in agreement with the linear thermoelasticity theory. We also observed a heterogeneous dissipation distribution from the beginning of the test. This space distribution remained time independent and did not change with the stress range and/or the loading ratio until fatigue crack onset. The linear variation in the dissipation intensity with the loading frequency was also underlined. In terms of structural design, this method may be useful for detecting fatigue zones. Thermoelasticity is already used to detect potential crack localization due to stress concentrations on structures (see TSA methods [14]). In terms of material analysis, obtaining the distribution of thermoelastic and dissipative sources represents precious information for anyone wishing to model fatigue mechanisms. The coupling sources are indeed related to the constitutive state equations while the dissipation is an indicator of the fatigue kinetics and must be associated with the evolution equations. Moreover, the dissipation fields enabled us to observe fatigue mechanisms at a finer scale insofar as dissipation is mainly due to the development of persistent slip bands. Micro-analyses are currently under way to correlate PSB density fields with dissipation fields. References 1. Luong, M. P., Fatigue limit evaluation of metals using an infrared thermographic technique. Mech. of Mat., 28,155-163, 1998. 2. La Rosa, G. and Risitano, A., Thermographic methodology for rapid determination of the fatigue limit of materials and mechanical components. Int. J. of Fatigue, 22, 1, 65-73, 2002. th 3. Cugy, P. and Galtier, A., Microplasticity and temperature increase in low carbon steels. Proc. 8 Int. Fatigue Conference, Stockholm, 549-556, 2002. 4. Mabru, C. and Chrysochoos, A., Dissipation et couplages accompagnant la fatigue des matériaux métalliques. Proc. Photomécanique’ 01, Poitiers, 375-382, 2001. 5. Boulanger, T., Chrysochoos, A., Mabru, C. and Galtier, A., Calorimetric and thermoelastic effects associated with the fatigue behavior of steels., Int. J. of Fatigue, 26, 221-229, 2004. 6. Morabito, A.E., Chrysochoos, A., Dattoma, V., Galietti, U., Analysis of heat sources accompanying the fatigue of 2024 T3 aluminium alloys, Int. J. of Fatigue, 29, 977-984, 2007. 7. Berthel, B., Chrysochoos, A., Wattrisse, B., Galtier, A., Infrared Image Processing for the Calorimetric Analysis of Fatigue Phenomena, Exp. Mech., 48,1, 79-90, 2008.

49 8. Germain, P., Nguyen, Q. S and Suquet, P., Continuum Thermomechanics, J. Appl. Mech., 50 (4B), 1010-1020, 1983. 9. Honorat, V., Moreau, S., Muracciole, J.-M., Wattrisse, B., Chrysochoos, A., Calorimetric analysis of polymer behaviour using a pixel calibration of an IRFPA camera, Int. J. on Quantitative Infrared Thermography, 2 (2), 153172, 2005. 10. Chrysochoos, A., Wattrisse, B., Muracciole, J.-M., El Kaïm, Y., Fields of stored energy associated with localized necking of steel, J. of Mechanics of Sol. and Struct., 4 (2), 245-262, 2009. 11. Mughrabi, H., Dislocation wall and cell structures and long range internal stresses in deformed metal crystals, Acta Metallurgica, 31, (9), 1367-1379, 1983. 12. Chrysochoos, A., Berthel, B., Latourte, F., Pagano, S., Wattrisse, B., Galtier, A., Local energy approach to fatigue of steel, J. of Strain Anal. for Engngn Design, 43 (6), 411-421, 2008. 13. Kaleta, J., Determination of cold work energy in LCF/HCF region. Proc. of 4th Int. Conf. on LCF and ElastoPlastic Behaviour of Materials, Garmisch-Partenkirchen, 93-98, 1998. 14. Dulieu-Barton, J, Stanley, P, Development and applications of thermoelastic stress analysis, J. Strain Anal. Engng. Design, 33 (2), 93-104, 1998.

Contribution of Kinematical and Thermal Full-field Measurements for Identification of High Cycle Fatigue Properties of Steels

R. Muniera,b, C. Doudarda, S. Callocha, B. Weberb a LBMS EA4325, ENSTA Bretagne / UBO / ENIB, 2 rue F. Verny, 29806 Brest Cedex 9, France b ArcelorMittal Maizières Research & Development BP 30320, 57283 Maizières-les-Metz Cedex, France

ABSTRACT Using kinematical and thermal full-field measurements for identification of mechanical parameters has become a very promising area of experimental mechanics. The purpose of this work is to extend the use of non-conventional tests and full field measurements (kinematical and thermal) to the identification of the fatigue properties of a dual-phase steel. A particular attention is paid to the influence of plastic pre-strain on the fatigue limit. Indeed, an analytical approach is proposed to define the geometry of the specimen permitting to obtain a constant gradient of plastic strain within the zone of interest after a monotonic pre-strain. Then, a self-heating test under cyclic loading is carried out on the pre-strained specimen. During this cyclic test, the thermal field is measured using an infrared camera. Finally, a suitable numerical strategy is proposed to identify a given thermal source model taking into account the influence of a plastic pre-strain. The results show that, with the non-conventional test and the procedure developed in this work, the influence of a range of plastic pre-strain on fatigue properties can be identified by using only one specimen. It is worth noting that a great number of specimens is required to determine this effect by using classical fatigue campaign. INTRODUCTION The measurement and analysis of kinematical and thermal full-field during mechanical tests are more and more used in the field of mechanics of solid materials and structures. In the past, full-field measurements were essentially applied as means to qualitatively identify the heterogeneity of a field (e.g., kinematical or thermal field [1-3]). Today, the cumulated experience and the available information about the metrological performances of full-field measurements open up new horizons [4, 5]. Indeed, full-field measurements may be used as a starting point for identification procedures providing access to parameters of constitutive laws and mechanical properties. It may provide very rich experimental data when applied to tests conducted under non-homogeneous conditions (i.e., for which strain and stress or temperature are not uniform in the zone of interest of the specimen). So, two of the main challenges about identification tests in mechanics of material concern, on the one hand, the design of the experiments (i.e., geometry of the specimen and an associated loading and boundary conditions) to controlled the heterogeneous conditions and, on the other hand, the development of suitable numerical strategies to identify the material parameters. The purpose of this work is to extend the use of non-conventional tests and full-field measurements (i.e., strain field and temperature field) to the identification of the fatigue properties of a dual-phase steel. A particular attention is paid to the influence of plastic pre-strain on the fatigue limit [6-9]. This is particularly important, for example, in the context of the fatigue design of structures and chassis frame metal parts produced by metal forming operations (e.g., stamping, ...). During these drawing operations, the sheets are severely strained, including thickness variations, cumulated plastic strain, residual stress and evolution of mechanical properties (e.g., yield stress, fatigue limit, ...). As an example, the figure 1 shows the S-N

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_7, © The Society for Experimental Mechanics, Inc. 2011

51

52 curves (i.e., stress amplitude of the cyclic loading relating to the number of cycles to failure) of a dual-phase steel in two different states (i.e., undeformed state and 20% plastic pre-strained state) [8-9]. One can see that the fatigue properties are considerably changed. The fatigue limit (i.e. endurance limit), which is initially equal to 250 MPa goes up to 320 MPa after a pre-straining of 20%. This evolution of fatigue properties is often not taken into account in conventional fatigue design approaches. One of the major reasons of this lack is the prohibitive time needed to characterize this effect by traditional fatigue tests campaigns.

material: dual-phase steel

450

material: dual-phase steel

450

p

Σ = 0, ε = 0%

p

Σ = 0, ε = 20% m

400

Stress Amplitude, Σ0(MPa)

Stress Amplitude, Σ0(MPa)

m

350 300

250

400 350 300 unbroken

250

unbroken

200 4 10

5

6

10 10 Number of cycles to failure, N -a-

10

7

200 4 10

5

6

10 10 Number of cycles to failure, N -b-

10

7

Fig. 1 Effect of a plastic pre-strain on the S-N curve of a dual-phase steel: a) unstrained state; b) 20% plastic pre-strained state [24] In order to reduce the time dedicated to fatigue properties characterization, several authors have worked on an estimation of high cycle fatigue properties based upon self-heating measurements under cyclic loading [3,8-16]. The proposed test, hereafter called ‘self-heating test’, consists in observing thermal effects during cyclic loadings and is performed on a ~ specimen with a constant cross-section. For each stress amplitude, the change of the temperature variation, θ = T-T0 (where T is the mean temperature of the specimen and T0 the surrounding value) is recorded. It is generally observed that the mean temperature becomes stable after a fixed number of cycles depending on the stress level and the loading frequency. Figure 2 shows the evolution of the steady-state temperature relating to the stress amplitude. One can see that beyond a given stress amplitude that is close to the fatigue limit, the steady-state temperature starts to increase significantly. Recently, a model [15] allows us, on the one hand, to describe thermal effects and, on the other hand, to relate the thermal effects to the fatigue properties. In a previous work, a self-heating curve per plastic pre-strain level has been determined (i.e., one specimen is needed per plastic pre-strain level). This set of self-heating curves permits to quantitatively characterize the influence of plastic pre-strain on the fatigue limit. As shown in this work and in the considered pre-strain range (10% - 20%), a linear evolution of the mean endurance limit with respect to the plastic pre-strain can be assumed. The slope of this linear evolution is denoted by β [8]. In this paper, an alternative method based on the use of only one 1-D non-conventional specimen and a suitable numerical strategy is proposed to identify the influence of a plastic pre-strain range on the fatigue limit. The paper is divided into main sections. In the first one, the design of a non-conventional specimen is presented. The main goal is to define the geometry of the specimen permitting to obtain a controlled heterogeneous plastic strain field on the zone of interest after a monotonic prestrain test. The monotonic pre-strain test is performed on the specimen and the plastic strain field is checked using kinematical field measurements with digital image correlation. In the second section, a self-heating test under cyclic loading is performed on the pre-strained specimen. The heterogeneous thermal field is measured using an infrared camera and analyzed by using a suitable numerical strategy permitting to identify a given thermal source model taking into account the influence of plastic pre-strain.

material: dual-phase steel DP600 Σ = 0, f = 10Hz

4

m

r

3

2

1

Σ =250 MPa

0

8

∼ Steady state temperature elevation, θ (K)

53

0

50

100 150 200 250 300 350 Stress amplitude, Σ (MPa) 0

Fig. 2 Empirical identification of the mean fatigue limit from self-heating measurements for a dual-phase steel [16] 1. Preliminary plastic strain of a non-conventional specimen In this section, the design of a non-conventional specimen is described. An analytical approach is proposed to define the geometry of the specimen permitting to obtain a constant gradient of plastic strain within the zone of interest after a monotonic pre-strain. Then the pre-strained specimen is obtained by performing a uniaxial tension test under displacement control, followed by a complete unload. The plastic strain field is determined using a Digital Image Correlation technique and a suitable numerical strategy. 1.1 Design of the geometry of a non-conventional specimen In so far as the specimen is machined in a constant thickness sheet, only the width, 2b0, is a free parameter (i.e., depending on the abscissa x1). One assume standard plastic compressibility of the steel during the pre-strain test, so that the ratio between the initial section, S0(x1)=2 e0b0(x1), and the strained section, S(x1), is given by

S0 (x1) du = 1+ (x1) , S(x1 ) dx1 where

(1)

du (x1) is the 1D gradient of displacement, u(x1). The maximum true stress, σ(x1), is then written as dx1

  F F du 1 + = (x1)  , (2) S(x1) 2e0 b0 (x1)  dx1  where F is the maximum load reached during the monotonic tensile test. σ(x1) depends on the plastic strain level, εp(x1), reached at the abscissa x1. Consequently, the half-width of the specimen is given by F b0 (x1 ) = exp(ε p (x1 )) , (3) 2e 0σ(ε p (x1 )) σ(x1 ) =

where εp(x1) is the logarithmic plastic strain. In order to obtain a constant plastic pre-strain gradient, K, the plastic strain is related to the displacement u(x1) by the following relation

  du ε p (x1) = ln 1 + (x1 )  := K (x1 + u(x1 ) ) .  dx1  The previous equation is solved by using the boundary condition u(x1=0)=0, so that

(4)

54 u(x1 ) = -x1 − and

1 ln(1 − Kx1 ) , K

 1 ε p (x1) = ln  1 − Kx1

(5)

 .  

(6)

material: dual phase steel DP600 800 700

Stress, σ (MPa)

600 500 400 300 200 100 0

0

5

10 15 20 25 Logarithmic strain, ε (%)

30

Fig. 3 Monotonic tensile test of a dual-phase steel According to the monotonic tensile curve of the studied steel (Fig. 3), K is chosen to reach a maximum value of 17% of plastic pre-strain at the center of the specimen. On Figure 4, the geometry of the specimen is given. It is worth noting that the plane (x1=30mm) is a plane of symmetry. 5.2 5.1

Width of the "specimen", 2b (mm)

20

5 4.9

15

4.8 20

10

25

30

35

40

5 0 -5 -10 -15 -20

0

10

20

30 40 Abscissa, x1 (mm)

Fig. 4 Geometry of the non-conventional specimen on the zone of interest 1.2 Identification of the gradient of plastic pre-strain field, Kid

50

60




55 The plastic deformation is performed on a servo-hydraulic testing machine. During this test, successive images are taken by using a CCD camera. Thus the displacement map is computed with a correlation technique between an initial picture and a subsequent one [17]. Figure 5a shows the experimental displacement field, Uexp(x1,y), of the zone of interest at the last stage of the test (i.e., at the end of the unloading). One can see that this displacement field is constant by bands. The theoretical expression of the 1D displacement is given by Equation 5. The value of Kid (i.e., the gradient of plastic pre-strain field) is identified by squares minimization between the experimental field and its analytical expression (i.e. Kid = 0.0053). Figures 5b-d show different fields at the last stage of the test. Figures 5b shows the identified displacement field, Uid(x1). An error map representing the relative error, ErrorU, defined by U exp (x1, y) − Uid (x1) ErrorU = , (7) U exp (x1, y) is showed on Figure 5c. The maximum relative error is less than 3.5%. Finally, Figure 5d shows the identified plastic prestrain field given by equation (6) with K= Kid. A maximum value of 17% is reached at the center of the specimen, as expected.

Displacement, u (mm) exp

a

Displacement, u (mm) id

b

Error, Error u (%)

c

Plastic pre-strain, ε (%) p

d

Fig. 5 Kinematical field measurements: a) Experimental displacement field obtained with Digital Image Correlation; b) Identified displacement field; c) Displacement error map; d) Calculated plastic pre-strain field 2. Self-heating test on the plastic pre-strained specimen In this section, the self-heating test carried out on the plastic pre-strained specimen is described. Then the identification of the influence of the plastic pre-strain on the fatigue limit from the results of the self-heating test is presented. This identification consists in solving the heat conduction equation. In the particular case of the designed specimen, the 1D heat conduction equation is considered. 2.1 Experimental result The self-heating test consists in applying a uniaxial cyclic test under load control conditions with constant load amplitude, (without mean stress) during 6000 cycles (i.e., the time to reach a thermal equilibrium). The load frequency, fr, is 30Hz. During the self-heating test, the temperature field is measured using an infrared camera. In order to consider only the temperature elevation, θ(M), the difference between the temperature T(M) and the initial temperature at the same point M, Tt=0(M), is considered. Figure 6a shows the steady-state experimental temperature elevation field, θexp(x1,y), of the zone of

56 interest. One can see that this temperature field is constant by bands, so that the 1D heat conduction equation is considered in the following of this paper. 2.2 Principle of the 1D heat conduction equation solving By considering Fourier's law including conduction losses [18] but neglecting any convection threw the surrounding air, the 1D heat conduction equation can be written

∂θ(x1 , t) ∂ 2 θ(x1 , t) ∂θ(x 1 , t) ∂S(x 1 ) 1 - λ' - λ' = S t (x1 , t) , (8) 2 ∂t ∂x 1 ∂x 1 S(x1 ) ∂x1 with λ’ the isotropic thermal conductivity, S(x1) the section of the non-conventional specimen after the pre-strained test, ρ the mass density, c the specific heat, St(x1,t) the 1D thermal sources field and θ(x1,t) the 1D temperature elevation field. The determination of heat source field from the measurement of temperature fields is still a difficult task. Due to the signal-tonoise ratio and the regularizing effects of heat diffusion, regularization method is required to handle this problem [1,19]. In this work, another approach is proposed, based on the determination of a theoretical temperature field from an a priori considered expression of the heat source field given by ρc

(Σ 0 (x1 )) m + 2

S t (x1 ) = δf r

, (9) (1 + βε p (x1 )) m where δ is a parameter, fr is the load frequency, Σ0(x1) and εp(x1) are the stress amplitude and the plastic strain, respectively, m is a material parameter (for this steel grade, m=12.5 is identified from a classical self-heating test [8]) and β is the material parameter that characterizes the influence of the plastic pre-strain viewed as linear. The main goal of the proposed approach is to identify the value of the parameter β from the field measurements results of the self-heating test previously exposed. From the analytical expression of the thermal source field, an analytical temperature is obtained by solving the heat conduction equation on a particular basis [18], namely, the Eigen basis regardless of the boundary conditions, generally used to solve the heat conduction equation, completed with a second order polynomial basis. The first one is used to describe the local variations of the steady-state temperature field. The second one describes the boundary conditions at both ends. The following expression of the 1D steady state temperature field variation is then considered

θ(x1 ) = ax 1 ² + bx 1 + c + ∑ θ' ak cos(w k x1 ) + θ' bk sin(w k x1 ) ,

(10)

k >0

where a, b, c are three parameters, k is the number of components of the Fourier basis, θ’ak and θ’bk are functions of Fourier coefficients with wk=2kπ/L, L representing the length of the zone of interest of the pre-strained specimen. The heat conduction equation can be then projected on the proposed particular basis and be written as

h bk h ak    S t0  − 2(1 + g 0 ) w k 2L − w k 2L   a  S  = λ'  − 2g   θ'  , Ak Dk (11) ak  tak     ak  S tbk   − 2g bk  θ' bk  Ck Bk   with St0 the constant value of the Fourier basis, Stak and Stbk the successive harmonics of this basis. The terms Ak, Bk, Ck and Dk of the previous equation are defined as follows wj A k = w 2k + ∑ h bj-k + h bj+ k − h bk - j , (12) j>0 2L

(

Bk = − ∑

(h aj-k + h aj+k − h ak- j ),

(13)

(h aj-k − h aj+k + h ak- j ) ,

(14)

wj

j>0 2L

Ck =

∑

wj

j>0 2L

D k = w 2k − ∑

wj

j>0 2L




)

(- h bj-k + h bj+k − h bk - j ),

(15)

57 and gi and hi the components of the Fourier basis of the functions, defined by ∂S(x 1 ) L ∂S(x 1 ) x 1 h(x 1 ) = and g(x 1 ) = . ∂x1 S(x 1 ) ∂x 1 S(x 1 )

(16)

By using the proposed approach, it is not necessary to identify the heat exchange parameters beforehand. Thus it is possible to define a region of interest that is independent of the length of the specimen. 2.3 Identification of the parameter β The value of β, that characterizes the influence of the plastic pre-strain on fatigue limit, is identified by squares minimization between the experimental field and its theoretical expression calculated by solving the previous linear system (i.e. β = 1.32). It is worth noting that the identified value using the considered approach in this paper is close to the one determined from specimens with a constant cross-section (i.e. β = 1.33). Figure 6b shows the identified temperature elevation field, θid(x1). An error map representing the relative error, Errorθ, defined by

Errorθ =

θ exp (x1 , y) − θ id (x1 ) θ exp (x 1 , y)

,

(14)

is showed on Figure 6c. The maximum relative error is less than 3%. Finally, Figure 5d shows the identified thermal source field given by equation (9). Experimental temperature, θ

a

(K)

m rature t ,θ Calculated t empe

(K)

m rat ra ure error, Εrror (%) Te mpe θ

6

-3

Thermal sources, S (.10 W.m )

c

Fig. 6 Thermal field measurements: a) Experimental temperature field obtained with infrared camera; b) Identified temperature field; c) Temperature error map; d) Calculated thermal source Conclusion In this paper, a new methodology has been proposed, combining the use of kinematical and thermal full-field measurements, to identify the influence of a wide range of plastic pre-strain on the fatigue properties of a dual-phase steel. This new methodology is based on the two main following steps: 1. the carrying out of a monotonic pre-strain test on a non conventional specimen coupled with kinematical full-field measurements permitting to check the plastic strain field; 2. the carrying out of a self-heating test under cyclic loadings on the pre-strained non-conventional specimen coupled with thermal full-field measurements analyzed by using a suitable numerical strategy permitting to identify a given thermal source field depending on plastic pre-strain.

58 It has been shown that tests conducted under controlled non-homogeneous conditions associated to full-field measurements permit to extract information from a small number of tests. It opens a lot of perspectives concerning the mechanical parameters identification methods. References 1. A. Chrysochoos, H. Louche, Infrared image processing to analyse the calorific effects accompanying strain localization, Int J Eng Sci 38 16, 1759-1788, 2000. 2. J. Medgenberg and T. Ummenhofer, Detection of localized fatigue damage in steel by thermography, In Knettel, K. M., Vavilov, V. P., et Miles, J. J., editors, Proceedings of Thermosense XXIX, volume 6541, 17.117.11, 2007. 3. M. Poncelet, C. Doudard, S. Calloch, F. Hild, B. Weber, Dissipation measurements in steel sheets under cyclic loading by use of infrared microthermography, Strain 46, 101-116, 2010. 4. M. Bornert, F. Brémand, P. Doumalin, J.-C. Dupré and M. Fazzini, et al., Assessment of Digital Image Correlation Measurement Errors: Methodology and Results, Experimental Mechanics 49 3, 353-370, 2009. 5. S. Roux, F. Hild, Digital Image Mechanical Identification (DIMI), Experimental Mechanics 48 4, 495-508, 2008. 6. A. Gustavsson and A. Melander, Variable-amplitude fatigue of a dual-phase sheet steel subjected to prestrain, Int. J. Fat. 16, 503-509, 1995. 7. K. Nakajima, S. Kamiishi, M. Yokoe, et T. Miyata. The influence of microstructural morphology and prestrain on fatigue crack propagation of dual-phase steels in the near-threshold region, ISIJ International 39(5), 486-492, 1999. 8. R. Munier, C. Doudard, S. Calloch, B. Weber, Towards a faster determination of high cycle fatigue properties taking into account the influence of a plastic pre-strain from self-heating measurements, Procedia Engineering 2 1, 1741-1750, 2010. 9. C. Doudard, Détermination rapide des propriétés en fatigue à grand nombre de cycles à partir d'essais d'auto-échauffement, PhD ENS Cachan, 2004. 10. C. E. Stromeyer, The determination of fatigue limits under alternating stress conditions, Proc. Roy. Soc. London, A90, 411-425, 1914. 11. HF. Moore, JB. Kommers, Fatigue of metals under repeated stress, Chem Metall Eng 25, 1141-1144, 1921. 12. M.P. Luong (1998), Fatigue limit evaluation of metals using an infrared thermographic technique. Mech Mater 28(1-4), 155-163 13. G. La Rosa and A. Risitano, Thermographic Methodology for Rapid Determination of the Fatigue Limit of Materials and Mechanical Components, Int. J. Fat. 22 [1], 65-73, 2000. 14. P.K. Liaw, H. Wang, L. Jiang, B. Yang, J. Y. Huang, R. C. Kuo and J. C. Huang, Thermographic detection of fatigue damage of pressure vessel at 1 Hz and 20 Hz, Scripta Materialia 42 [4], 389-395, 2000. 15. C. Doudard, S. Calloch, F. Hild, P. Cugy and A. Galtier, Identification of the scatter in high cycle fatigue from temperature measurements, C.R. Mcanique 332 [10], 795-801, 2004. 16. C. Doudard, S. Calloch, P. Cugy, A. Galtier and F. Hild, A probabilistic two-scale model for high-cycle fatigue life predictions, Fat. Fract. Eng. Mat. Struct. 28, 279-288, 2005. 17. L. Chevalier, S. Calloch, F. Hild, Y. Marco, Digital image correlation used to analyse the multiaxial behavior of rubberlike materials, Eur J Mech A Solid 20 2, 168-187, 2001. 18. C. Doudard, S. Calloch, F. Hild and S. Roux, Identification of heat source fields from infra-red thermography: Determination of 'self-heating' in a dual-phase steel by using a dog bone sample, Mechanics of Materials 42 1, 55-62, 2010. 19. N. Connesson, F. Maquin, F. Pierron, Experimental energy balance during the first cycles of cyclically loaded specimens under the conventional yield stress, Experimental Mechanics online first, 2010.

Dissipative energy : monitoring microstructural evolutions during mechanical tests

N. Connesson (Ph.D), [email protected] F. Maquin (Ph.D), [email protected] F. Pierron (Prof), [email protected] Laboratoire de mécanique et procédés de fabrication (LMPF), Arts et Metiers ParisTech, rue St Dominique, BP 508, 51006 Châlons-en-Champagne cedex, France ABSTRACT

Fatigue characterization is an expensive operation commonly undertaken in industry. Some authors thus developed experimental measurement methods based on the materials thermomechanical behaviour to provide faster fatigue limit estimations. Yet, the physical ground of these methods needs to be understood. In this work, it has been assumed that heat dissipation phenomena are related to dislocation movements in the material lattice (internal friction); changes in the dislocation characteristics (through plastic straining for example) will affect the material dissipative behaviour. The dissipative energy characteristics of a Dual-Phase 600 grade (DP600) have been experimentally estimated during traction-traction cyclic loadings on thin sheet specimens. The specimens surface temperature variations have been recorded using an infrared camera and analysed using the heat balance equation. Each dissipative energy measurement has been performed for a specific microstructural state of the material (no macroscopic plasticity occurs during the measurement). The effect of different loading sequences on the material dissipative behaviour has been tested and interpreted using the commonly used specific damping capacity. The dissipative energy (the dislocation mobility) has been proved to increase with the macroscopic plastic strain and to be affected by aging periods at ambient temperature.

INTRODUCTION Fatigue characterization is a time consuming and expensive operation commonly undertaken in industria. The development of many theories and experimental measurement techniques is thus of first interest to provide faster fatigue limit estimation methods. Thus, with the development of infrared cameras, some authors proposed experimental methods to estimate the fatigue limits based on the material temperature increase under cyclic loading [1,2]. Yet, the physical explanation of such methods is not fully understood, and heat dissipation phenomena needs to be more thoroughly studied. The thermomechanical sources of materials can be dissociated into two main phenomena, such as the thermomechanical coupling and the dissipative energy. The dissipative energy, responsible for the mean temperature increase of the specimen under harmonic loading, can be experimentally measured by solving a heat balance equation [3]. An experimental method to precisely measure the dissipative energy is proposed in [4] and is used in this study. The dissipative energy phenomena is related to internal friction mechanisms [5]: under loading, the material defects (inclusion, dislocations, etc) are moving in the material lattice and dissipate energy, the main part of dissipated energy being attributed to dislocation movements. It seems thus that the dissipative energy should depend on the material microstructure such as the mobile dislocation density, their spatial repartition and interactions, the lattice friction coefficient etc. In fewer words, the dissipative energy should be a good experimental criterion to monitor the material microstructural evolution. Moreover, it is well known that loading history has a direct impact on the material microstructure. Depending on the applied loading, the dislocations move, extend, multiply [6,7], or create specific spatial structures [8–11] such as cells, persistent slip bands etc. The dislocation density usually increases with the plastic strain. In this study, it is thus chosen to observe the correlation between the dissipative energy, the plastic strain and from a more general point of view, with the loading history.

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_8, © The Society for Experimental Mechanics, Inc. 2011

59

60 Few works have been made to analyse the dissipative energy evolution with the cold work. The temperature versus alternate stress curve has been studied in [12,13] on different cold worked specimens. Yet, as the materials were loaded with Rσ= σmin/σmax =-1 (where σmin and σmax are respectively the minimum and maximal stress), heat dissipation was probably due both to internal friction and macroscopic plasticity. In this work, every dissipative energy measurements have been performed during traction-traction tests (Rσ= σmin/σmax =0.1) and when no macroscopic plastic strain (ratcheting effect) occurs. Thanks to these loading conditions, internal friction and macroscopic plasticity have been dissociated [14] and attention has been focussed only on the heat dissipation due to internal friction. In the rest of this study, the heat dissipation due to internal friction will be simply referred as “heat dissipation” or “dissipative energy”. This paper presents thus an experimental measurement of the dissipative energy variation along with the uniaxial plastic strain of a dual phase steel (DP600). The idea is that such experimental data may help to better understand heat dissipation phenomena and provide some data to temperature increase predictive models.

EXPERIMENTAL PROCEDURE AND RESULTS This study has been performed on a specimen machined in a 2 mm thick steel plate of DP600 steel which geometry is presented in Figure 1.

Fig. 1 Specimen geometry The dissipative energy and plastic strain in the observed area Ωsp have been monitored with the experimental setup presented hereafter. Plastic strain measurement. The strain has been measured with a 350 strain gauge (Vishay, CEA-06-250UW-350) connected to a quarter Wheatstone Bridge. This strain gage was bonded on one side of the specimen in the loading direction. In all this study, plastic strain has been measured while no loading was applied to the specimen. Dissipative energy measurement. The dissipative energy is measured using an infrared CEDIP Jade camera by applying the experimental procedure proposed in [4,14]: during a dissipative energy measurement, the specimen is cyclically loaded and the temperature field of the observed area Ωsp is acquired. The dissipative sources are estimated using a local energy balance which general shape is:

 ∂θ

  ∂θ   − k∆ 2θ + ρC θ2 D = d1 + Sth −   τ th  ∂t  ∂t  (t = 0 − ) 

ρC 

(1)

where ρ is the material density, C the calorific capacity, k the thermal conductivity and ∆2 the laplacien operator. θ is here the local temperature variation measured with the infrared camera. This equation gives thus the local dissipative sources d1 and the sources due to the thermoelastic coupling Sth by estimating :    • the energy stored by temperature increase  ρC  ∂θ −  ∂θ   ,   ∂t  ∂t  (t =0 − )      • energy exchanges by conduction ( k∆ 2θ ) • and energy exchanges by convection and radiation ( ρC θ ). τ th2 D

τ th2 D is

a time constant characterizing the convection and radiation exchanges. Equation 1 is then integrated over the

observed area Ωsp and over each cycle which provides the dissipative energy per cycle E i = d1

t i +1 / f L

∫

ti

1 Ω sp

∫d

1

dx dy dt

Ω sp

where fL is the loading frequency and ti the beginning of the cycle i. In this study, the specimen have been harmonically loaded with Rσ= σmin/σmax =0.1. For each stress levels, a high enough number of cycles have been applied so that the material reached a steady state (the mean strain remains constant). As the loading ratio is greater than zero, no

61 macroscopic plastic strain occurred during the dissipative energy measurements. The dissipative energies measured in these conditions have been attributed to internal friction. Moreover, the dissipative energy is constant during each measurement. Only the temporal mean of the dissipative energy has thus been used in this paper. This temporal mean will be noted E dm in the rest of this paper. A noise level analysis showed that the measurement incertitude of a mean 1

-1

dissipative energy E dm (±2 Standard deviations) is of ±100 J.m-3.cycle for a loading frequency of fL=14 Hz [14]. 1

To summary, this experimental setup provides the plastic strain εp while the specimen is not loaded and the dissipative energy E dm under cyclic loading in steady state conditions. Measurements sequences have been organized in this work to 1

strudy the correlation between dissipative energy and plastic strain. Six measurement sequences at different loading levels (Phase I to VI) have successively been applied to the specimen, each sequence being designed to reach a different goal: the material initial state behaviour is analyzed during Phase I. The material is then plastically strained and its evolution is analyzed during Phase II. Phase III and IV are then performed to monitor the changes in the material dissipative energy due to the previous two phases. Phase V and VI are applied after a rest period to analyse recovery period effect on the material dissipative behaviour. The tests sequences are represented graphically in the upper graph of Figure 2. In this representation, each point represents three successive tests: • a cyclic loading at the maximum stress σmax (represented by the point ordinate) applied until a steady state is reached, • then a dissipative energy measurement in the same loading conditions, • and eventually a plastic strain measurement εp while the loading is zero. The cumulated plastic strain εp at the end of each test is presented in the bottom graph of Figure 2.

a)

350

Strain gage failure during loading

B

D

F

(MPa)

300

σmax

250

A

200

D0

D1 D3 D5 D7 D9

C

E

150

Rest : 1 month

400

100 50 Phase 0 2500

Phase II

Phase III

Phase IV

Phase V

Phase VI

b)

1500 1000

εcp

(10−6)

2000

I

500 0

Test sequence

Fig. 2 DP600: test sequence and associated cumulated plastic strain εp measurements

62 Phase I: the dissipative behaviour of the material in its initial state has been analyzed during this phase. Dissipative energy measurements have been performed for increasing maximal stress σmax from 100 to 240 MPa by steps of 20 MPa (figure 2, upper graph): the dissipated energy per cycle E dm increases with the alternate stress (figure 3, Phase I). The yield stress 1

at 0.02% being of 295 MPa for the DP600, the cumulated plastic strain εp remained small during these tests (figure 2, bottom graph). Phase II: the reproducibility of the results of Phase I has been checked during the 1rst levels of Phase II: as in Phase I, dissipated energy measurements have been performed while increasing the maximal stress σmax from 100 to 240 MPa by steps of 20 MPa. As expected, the plastic strain did not change during these 1rst steps and the dissipative energies per cycle Edm overlap; the measurements are well reproducible on the same specimen. 1

Then, the maximal stress σmax has been increased from 260 to 440 MPa by steps of 20 MPa (figure 2, Phase II). The material microstructural state after each loading step have been characterized both with a cumulated plastic strain measurement εp (figure 2, bottom graph) and a dissipated energy measurement under identical loading conditions (σmax =240 MPa, dots D2..D10 in figures 2 and 3). The measured dissipated energies show a progressive increase after each load. Phase III : at the end of Phase II (point B, figure 2), the material has been plastically strained at σmax =440 MPa and reached an elastic behaviour. The material microstructural state in B (referred to as "state B") is thus different from its initial state and should have a different thermomechanical behaviour. The thermomechanical behaviour of the material microstructure in state B has been tested by increasing the maximal stress σmax from 100 to 440 MPa by steps of 20 MPa. As expected, the measured dissipated energy (Figure 3, Phase III) is always greater than the dissipated energy of the material in its initial state (Phase I) and for lower plastic strains (Phase II): Phase III is a thermomechanical characterization of the material in state B while Phase II is a step by step characterization of the material progressive evolution up to state B. Moreover, the measurements are reproducible for the stress level σmax =440 MPa. Phase IV: a reproducibility test of the dissipated energy measurements during Phase III has been performed immediately after Phase III. Once again, the dissipated energy measurements prove to be well reproducible on the same specimen (Figure 3, Phase IV). Phase V and VI: In [15], Lazan reported that materials damping could be affected by a recovery period. Phases V and VI have thus been performed on the same specimen after a one month recovery period at ambient temperature. The dissipative energy results will be described later in this paper. The dissipated energy per cycle Edm depends here on the alternate stress σa and thus on the input mechanical work per cycle. 1 These results can be further analyzed by computing the specific damping capacity ψ . As the tests have been performed in the elastic hysteretic domain), the specific damping capacityψ can be written as: m ∆W Ed1 2 E m Ψ= = = Ed W W σa 1

(2)

where E is the Young’s modulus. During traction-compression test (Rσ=−1), W represents the maximum work given to the material during one cycle. This definition has been kept in this work even if the loading ratio is here Rσ = 0.1. The results presented as specific damping capacity ψ in figure 4 are thus the same data as presented in figure 3 but on a scale which emphasizes the data for low alternate stress σa. Moreover, as Edm is known at ±100 J.m-3.cycle-1, equation (2) has also 1

been used to compute the measurement error-bars depending on the alternate stress amplitude σa. The specific damping capacity results (figure 4) underline that whichever phase is considered, the specific damping capacity ψ increases with the alternate stress amplitude σa and is not constant: the dissipated energy per cycle E dm1 is not simply proportional to σa2. The fact that the proportion of dissipated energy per received mechanical work increases with the alternate stress can be interpreted as an increase of the dislocation mobility, either through an improved mobility per dislocation or an increase of mobile dislocation density [6]. The variations of the specific damping capacity ψ with the alternate stress could be used as a non destructive indicator of the material microstructure evolution.

63 σ

(MPa)

max

−3

−1

(J.m .cycle )

m

10000

Ed1

(J.m−3.cycle−1)

12000

8000

4000

6000

350

400

10

3500 7

3000 2500

0.05

D8

D

D6

D5

D0,D1

A

σ

0.04

D4

D2,D3

a

0.03

105

(MPa)

0.02

C,E

2000

Phase I Phase II Phase II int 0.01 Phase III Phase IV 0 160 180

A,D0 20

0.07

0.06

4000

0 0

300

D

D9

2000 100

d1

Em

250

40

60

80

100

σ

120

(MPa)

a

−1

C E

4500

14000

200

B,D,F

18000 16000

150

(K.s )

100

1

50

d /ρC

0

140

Fig. 3 DP600: dissipated energy per cycle E (left scale) and equivalent heat rate d1/ρC (right scale) versus alternate stress m d1

(bottom scale) or maximal stress (top scale). Rσ= 0.1

σmax 0 0.2 0.18

50

100

150

(MPa)

200

250

350

400

B,D,F

Phase I Phase II Phase II int Phase III Phase IV

C E

0.16

D10 D

0.14

ψ

300

D

0.12

D

8

7

D6

D

0.1

5

D ,D ,D ,D 0

0.08

1

2

9

D

4

3

Error bars ±2σ for σ =40 MPa and σ =190 MPa

A

a

a

0.06 0.04 0.02 0 0

20

40

60

80

σa

100

120

(MPa)

140

160

180

200

Fig. 4 Specific damping capacity ψ versus alternate stress (bottom scale) or maximal stress (top scale). Rσ= 0.1

64 It is also possible to analyze the variations of dissipative energy per cycle Edm measured during Phase II (D2..D8) with the 1

cumulated plastic strain εp (figure 5). The dissipative energies per cycle E dm of this graph have all been measured with the 1

same experimental settings (σa = 104.6 ±0.25 MPa, Rσ=0.1, etc.). The only difference from one test to another is thus the material microstructure modification due to cold work. Figure 5 presents thus in a different way that the dissipative energy E dm increases with the plastic strain. It is yet worth noting that the increase rate of the dissipative energy is greater for 1

small plastic cumulated strain. 3500

Phase I A Phase II D0

3400

Phase II D ..D 1

Em d1

(J.m−3.cycle−1)

3200

8

3000 Error bars ±2σ

2800 2600 2400

σa = 104,6±0.25* MPa

2200

* ±2Std

2000 0

500

1000

1500

Déformation plastique (µdef)

2000

2300

Fig. 5 Dissipative energies per cycle Edm versus cumulated plastic strain εp. σmax =240 MPa, Rσ= 0.1 1

Eventually, the influence of ambient temperature recovery period on the dissipated energy behaviour has been tested during Phases V and VI. The experimental results have been plotted directly as specific damping capacity ψ in Figure 6. σ (MPa) max 0 0.2 0.18

50

100

150

200

250

300

350

400

Error bars ±2σ σ = 40 MPa and σ =200 MPa a

a

0.16

ψ

0.14 0.12 0.1 0.08 0.06 0.04

Phase II Phase III Phase V Phase VI

0.02 0 0

20

40

60

80

σ

a

100

120

(MPa)

140

160

180

200

Fig. 6 Specific damping capacity ψ versus alternate stress (bottom scale) or maximal stress (top scale). Rσ= 0.1

65 Phase V : this test sequence has been performed on the same specimen after a one month recovery period at ambient temperature (figure 2, upper graph). The specific damping capacity ψ during the first levels of Phase V is comparable to the phase-shift of the material in its initial state at the first levels of Phase II (figure 6) and is lower than the specific damping capacity ψ measured one month before during Phase III (state B, figure 2). This specific damping capacity ψ then quickly increases with the alternate stress amplitude σa to reach its previous value during Phase III. The one month recovery period thus affected the dissipated energy behaviour but did not totally erase the effect of the previous test sequences. Phase VI: this test is performed just after Phase V. The measured specific damping capacity ψ overlaps with the results obtained during Phase III; Phase V totally erased the recovery period effect. These results underlines thus that the thermomechanical behaviour of the DP600 depends both on the loading sequences and on the recovery periods. The proposed experimental method to measure the dissipated energy is sensitive enough to observe such effects.

CONCLUSION The dissipative energy due to internal friction of a dual phase steel evolution with the plastic strain has been measured. First, it clearly appeared that the dissipative energy due to internal friction increases with the plastic strain. When no plastic strain occurs between two tests, the dissipative energy measurement is well reproducible on the same specimen. Thus, the dissipative energy variations versus the alternate stress in the viscoelastic domain could be used as a “signature” of the material. Eventually, it has been underlined that the dissipative energy measurement method used in this work is sensitive enough to observe small changes in the material microstructure due, for example, to recovering periods. The dissipative energy measurement is thus a sensitive tool that could be used to monitor the material microstructure variations due to the loading history. The variations of the DP600 dissipative energy signature during fatigue tests will be studied in future works.

REFERENCES

1. H.F. Moore, J.B. Kommers, Fatigue of metals under repeated stresses, Chemical and Metallurgical Engineering 25, 11411144, (1921) 2. G. Fargione, A. Geraci, G. La Rosa, A. Risitano, Rapid determination of the fatigue curve by the thermographic method, International Journal of Fatigue 24, 11-19, (2002) 3. A. Chrysochoos, H. Louche, An infrared image processing to analyse the calorific effects accompanying strain localisation, International Journal of Engineering Science 38, 16, 1759-1788, (2000) 4. F. Maquin, F. Pierron, Heat dissipation measurements in low stress cyclic loading of metallic materials : from internal friction to micro-plasticity, Mechanics of Materials 41, 928-942, (2009) 5. D. Caillard, J.L. Martin, Thermally activated mechanisms in crystal plasticity, Pergamon, Amsterdam London 85-123, (2003) 6. A. Granato, K. Lücke, Theory of mechanical damping due to dislocations, Journal of Applied Physics, 27:6, 583-593, (1956) 7. T. Tanaka, S. Hattori, Initial behavior of hysteresis loop of low carbon steels under repeated load (theoretical treatment on the multiplying model of dislocations), Bulletin of the JSME, 21:161, 1557-1564, (1978) 8. C. Déprés, M. Fivel, L. Tabourot, A dislocation-based model for low-amplitude fatigue behaviour of face-centred cubic single crystals., Scripta Materialia, 58:12, 1086-1089, (2008) 9. T. Fujii, C. Watanabe, Y. Nomura, N. Tanaka, M. Kato, Microstructural evolution during low cycle fatique of a 3003 aluminum alloy, Materials Science and Engineering A, 319-321, 592-596, (2001) 10. P. Lukáš, L. Kunz, Cyclic plasticity and substructure of metals, Materials Science and Engineering A, 322:1-2, 217-227, (2002) 11. H. Mughrabi, F. Ackermann, K. Herz, Persistent slipbands in fatigued Face-Centered and Body-Centered Cubic metals, ASTM Special Technical Publication, 675, 69-105, (1979) 12. C. Doudard, Détermination rapide des propriétées en fatigue à grand nombre de cycles à partir d'essais d'échauffement. PHD Thesis, ENS Cachan, (2004), in French 13. C. Mareau, Modélisation micromécanique de l'échauffement et de la microplasticité des aciers sous solliciations cycliques, PHD Thesis, ENSAM, (2007), in French 14. N. Connesson, F. Maquin, F. Pierron, Experimental Energy Balance During the First Cycles of Cyclically Loaded Specimens Under the Conventional Yield Stress, Experimental Mechanics, 51 :1, 23-44, (2010) 15. B. Lazan, A study with new equipment of the e®ects of fatigue stress on the damping capacity and elasticity of mild steel, Transactions of the ASM, 42:499-558, (1950)

Bidirectional Thermo-Mechanical Properties of Foam Core Materials Using DIC

1

S T Taher1, O T Thomsen1, J M Dulieu-Barton2 Department of Mechanical and Manufacturing Engineering, Aalborg University, Denmark 2 School of Engineering Sciences, University of Southampton, UK

KEYWORDS: PVC foam, Modified Arcan fixture, Digital image correlation, Thermal degradation, Finite element analysis ABSTRACT Polymer foam cored sandwich structures are often subjected to aggressive service conditions which may include elevated temperatures. A modified Arcan fixture (MAF) has been developed to characterize polymer foam materials with respect to their tensile, compressive, shear and bidirectional mechanical properties at room and at elevated temperatures. The MAF enables the realization of pure compression or high compression to shear bidirectional loading conditions that is not possible with conventional Arcan fixtures. The MAF is attached to a standard universal test machine equiped with an environmental chamber using specially designed grips that allow the specimen to rotate, and hence reduces paristic effects due to misalignment. The objective is to measure the unidirectional and bidirectional mechanical properties of PVC foam materials at elevated tempreature using digital image correlation (DIC), including the elastic constants and the stress-strain response to failure. To account for nonhomogeneity of the strain field across the specimen cross sections, a “correction factor” for the measured surface strain is determined using nonlinear finite element analysis (FEA). The final outcome is a set of validated mechanical properties that will form the basis input into a detailed finite element analysis (FEA) study of the nonlinear thermo-mechanical response of foam cored sandwich panels. 1. INTRODUCTION Polymer foam cored sandwich structures are often subjected to aggressive service conditions which may include elevated temperatures. The mechanical properties of polymer foam cores degrade significantly with elevated temperatures, and significant changes in the properties may occur well within the operating range of temperatures. The material properties of foam cored sandwich structures depend on the temperature field imposed, and this is usually ignored in engineering analysis and design. As an example, the thermal degradation problem for wind turbine blades is especially associated with the use of polymer foam cores in the wing shells when these are exposed to high temperatures. This occurs most severely under hot climate conditions, but can also occur in temperate climates. An example would be very high gusting winds increasing on a warm/hot summer day, for instance due to the development of a thunder storm. This scenario may generate large structural loads in combination with hot conditions in the outer sandwich face sheet (70-80°C or more even under temperate climate conditions [1]). At such temperatures a significant stiffness reduction occurs in typical PVC or PET foam cores (40-50% stiffness reduction) [2, 3], and when high mechanical loads act at the same time, nonlinear interaction effects may occur and subsequently lead to a loss of structural integrity [4-6]. Furthermore sandwich core materials may experience multidirectional mechanical stress states. In a conventional sandwich panel the in-plane and bending loads are carried by the face sheets, while the core resists the transverse shear loads. A well known failure mode of such sandwich panels is „core shear failure‟ in which the core fails due shear stress overloading. However, although the shear stress is often the main core stress, there are conditions in which the normal stresses in the core are of comparable size or even higher than the shear stresses. Such conditions may occur in the vicinity of concentrated loads or supports and also in the vicinity of geometrical and material discontinuities. Under such condition a material element in the core is subjected to a multidirectional state of stress. Therefore, proper design of sandwich structures requires the characterization of the core material response under multi-directional stress states. Previously, the Arcan test rig has been used to measure mechanical properties of polymer foams and other light materials used for sandwich core, especially in the bidirectional tensile-shear stress region [7, 8]. In this work a modified Arcan fixture (MAF) has been developed to characterize polymer foam materials with respect to their tensile, compressive, shear and bidirectional mechanical properties at room and at elevated temperatures, and including the elastic coefficients and the full stress-strain response to failure. The MAF enables the realization of pure compression or high compression to shear bidirectional loading conditions that are not possible with conventional Arcan fixtures. The MAF is T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_9, © The Society for Experimental Mechanics, Inc. 2011

67

68 attached to a standard universal test machine equipped with an environmental chamber using specially designed grips using universal joints that do not constrain the specimen rotation, and hence reduces parasitic effects due to misalignment. The present paper focuses on characterizing the orthotropic material behaviour of Divinycell cross linked PVC foam at room temperature and to describe the design of the test setup to be used for testing at elevated temperatures. Eventually, the overall outcome will be a set of validated mechanical properties that will form the basis input into a detailed FE analysis study of the nonlinear thermo-mechanical response of foam cored sandwich structures that will in turn be compared with full scale experimental results obtained for sandwich beams subjected to thermo-mechanical loading. 2. EXPERIMENTAL SETUP 2.1 MATERIAL Fully cross-linked PVC closed-cell cellular foam manufactured by DIAB (Laholm, Sweden) was investigated. In particular the core grade Divinycell H100 with a nominal density of 100 kg/m3 (PVC foam relative density ≈ 8%) was investigated. Other PVC foam densities will be investigated in future work. A PVC foam sheet of 60 mm thickness and with the rise direction of the cells orientated in the thickness direction was used for the manufacturing of the specimens. 2.2 MODIFIED ARCAN FIXTURE (MAF) The standard Arcan testing apparatus can be used to apply bidirectional loading to a butterfly shaped (BS) specimen. Fig.1 (a) shows a standard Arcan rig with circular distribution of griping holes, which is limited to apply only combinations of tensile and shear loadings. A novel modified Arcan Fixture (MAF) has been designed to enable the application of any combination of axial (tension or compression) and shear loadings (Fig.1 (b)) with a quasi-spiral distribution of griping holes. The MAF provides an S-shape fixture that consists of two boomerang shaped arms and a two specimen tabs bonded to the test specimen in the centre of the fixture. The new apparatus appear as a simple fixture that may be simply attached to a test machine, capable of only imparting a tension load, to provide biaxial deformation at different shear to axial deformation ratios. The different shear to axial deformation ratios are provided by selecting different attachment points on the boomerang shaped arms. A load applying double sided fork-lug is connected to each boomerang shaped arm at one end, while at the other end each arm is connected to a universal joint to compensate for any misalignment effect from the loading machine as shown in Fig. 1(c). (a)

(c)

(b)

Fig. 1 Bidirectional test rigs: (a) classic Arcan rig with only tension-shear deformation envelope coverage; (b) Modified Arcan fixture (MAF) with full range coverage of the axial-shear deformation envelope; (c) perspective view of MAF

69 The foam specimens were fabricated with high precision using a 3-axis CNC milling machine to cut the foam specimens symmetrically relative to centre plane of the PVC foam sheets. Since PVC H100 has a higher stiffness in the rise direction of the foam cells, both in-plane and through-thickness specimens were prepared for the tests as shown in Fig. 2(a). After machining the specimens were bonded to “dove” tailed aluminium tabs using Araldite epoxy adhesive as shown in Fig 2(b). Three different specimen shapes were used for different loading conditions. Fig. 2 (c) shows the three different shapes of test specimens designed for the MAF test rig: a butterfly shaped (BS) specimen geometry for shear and bidirectional loading, a short dogbone (SD) specimen geometry for tensile loading, and a block specimen geometry for compression loading. The BS type specimens were initially machined with three different notch radii 6.67, 4.5 and 2.5 mm. All specimen types were manufactured with a constant thickness of 15 mm. (b)

(a) In-plane specimen

Through-thickness specimen Foam sheet thickness

Bonded tab

Fig. 2 Specimen preparation and geometries: (a) manufacturing of in-plane and through-thickness short dogbone (SD) specimens; (b) perspective view of a butterfly shaped (BS) specimen bonded to “dove” tailed tabs; (c) dimensions of butterfly shaped (BS), short dogbone (SD) and block specimen shapes. 2.3 DIGITAL IMAGE CORRELATION (DIC) SETUP The non contact optical full field Digital Image Correlation (DIC) method was used to obtain the strain field over the gauge area of the test specimens. An ARAMIS 4 M system (from GOM GmbH) was used for the measurements and the further processing of the recorded displacement field. Two cameras (1 inch, CMOS chip) with a resolution of 2048 × 2048 pixels were placed perpendicular to and on both sides of the specimen to provide simultaneous 2D strain measurements of the specimen surfaces. Both cameras and the load cell were synchronized and connected to the computer interface as indicated in Fig. 3(a). Fig. 3(b) shows the MAF rig in a pure shear test setup using double sided DIC measurement. Low heat lights provided the required light without heating the specimen. To generate an appropriate surface pattern for the DIC black ink was smeared onto the surface of specimen after which zinc oxide powder (white) was spread on the surface. After this the top of the surface was cleaned carefully to visualize the foam cell walls. A facet size of 60 × 60 pixels and a step size of 30 pixels in the X and Y directions were chosen. Each camera recorded a first image before the loading sequence was started, and about 150-300 images were recorded during loading up to the fracture point.

70

(b)

Fig. 3 DIC setup in room temperature configuration: (a) Schematic of synchronized control system; (b) Photograph of MAF rig and DIC setup for a pure shear test

cameras and load cell connected to

Load cell

Polymer isolator (Delrin) Heat exchanger

Connection rod Universal joint Environmental chamber Fixture arm

Base Fig. 4 Test setup for elevated temperatures

71 2.4 ELEVATED TEMPERATURE TEST SETUP Extensive experimental core characterisation is being planned at elevated temperatures. The elevated temperature tests will be carried out using an Instron environmental chamber. The specimens will be allowed to equilibrate inside the chamber before testing. The environmental chamber includes a window in the access door, and DIC measurements will be conducted through the window on the front side of specimen. It has been established recently that DIC through the window is indeed feasible [9]. A double 2D DIC setup as shown in Fig. 3(a) will be applied for each specimen prior to the elevated temperature testing, and a small mechanical load will be applied to verify the symmetry of the strain response on both sides of the specimen. After this initial mechanical test the back side camera will be removed and the environmental chamber will be inserted around the MAF rig and the front camera will be used to acquire images through the environmental chamber window. To acquire accurate load data from the load cell of the test machine, any heat transfer into the load cell should be restricted. To ensure temperate operation of the load cell, an intermediate isolating connection that can operate up to 200°C has been designed and manufactured (see Fig. 4); it includes an air cooled heat exchanger and a heat isolator made of Delrin polymer. The connecting rod, heat exchanger and polymer isolator have standard connection pins that are compatible with the Instron test machine connection. Fig. 5(a) shows a photograph of the isolating connector including the aluminium connection rod, the steel heat exchanger and the Delrin polymer isolator. A finite element analysis (FEA) has been conducted to analyse the heat transfer of isolating connector using the commercial FEA package ANSYS 12.1. The FEA steady state simulation included conduction, radiation and convection over the constituent components. As shown in Fig. 5(b) the FEA results predict a reduction of temperature from 200°C in the connection rod to 150°C in the Delrin isolator corresponding to the maximum operation temperature for the Delrin polymer material. The final temperature at end of the isolator is predicted to be about 26°C which is sufficiently low for safe and accurate operation of the load cell.

Fig. 5 Thermal isolating connection for separation of MAF rig and load cell: (a) photograph of thermal isolator including heat exchanger and polymer isolator; (b) temperature contour map over thermal isolator predicted using FEA 3. EXPERIMENTAL RESULTS OBTAINED AT ROOM TEMPERATURE Fig. 6 shows examples of the full stress vs. strain curves for uniaxial tension and shear loading recorded at a strain rate of 6×10-4 s-1. Each data point on the stress-strain curves is computed from an average of the measured strain on the specimen surface gauge line (DIC measurements) that is multiplied by a “correction factor” derived from nonlinear FEA to compensate for the inhomogeneity of the strain field across the specimen cross sections (to be further described ahead). A specially developed MATLAB code reads each image DIC result data file to extract the all the data points on the stress vs. strain curves, and a robust local polynomial regression approach is used to generate a smoothed stress vs. strain curve for each tested specimen. Fig. 6(a) shows an in-plane (core plate direction) tensile test response up to the fracture obtained for a short dogbone (SD) test specimen. The curve displays a characteristic nonlinear behaviour up to fracture, where after an initially linear region, the curve show a substantial nonlinear softening response until fracture occurs. A series of preliminary shear tests were carried out on butterfly shape (BS) shaped specimens with three different notch radii (6.67, 4.5 and 2.5mm). Only the specimen with minimum radius displayed fracture initiation at the (shear) gauge section, and accordingly only specimens with a notch radius of R=2.5mm were subsequently considered for the shear tests. The shear stress vs. shear strain curves displayed a distinctly nonlinear behaviour as shown in Fig. 6(b).

72

Fig. 6 Tensile and shear material behaviour of H100 foam: (a) SD tensile normal stress vs. normal strain response up to fracture (in-plane core material direction); (b) BS shear stress vs. shear strain behaviour of material (transverse core material direction)

Fig. 7 Bilinear material approximation for nonlinear FEA modelling 4. NONLINEAR FINITE ELEMENT MODELLING 3D nonlinear finite FE analyses, including both material and geometric nonlinearity, have been conducted using the FE code ANSYS 12.1 to estimate the “correction factors” that are used to compensate for the difference between the measured surface field and the inhomogeneous strain field over the specimen cross section. Higher order solid twenty nodded hexahedral elements (solid186) were used for both the PVC foam and the aluminium fixture materials. Different FE meshes, ranging from 5,000 to 35,000 elements, were used. A bilinear approximation of the experimentally obtained nonlinear shear stressstrain curves (see Fig. 7) has been implemented in the nonlinear FEA model for each specimen type (SD and BS specimens),

73 and an iterative solution procedure is used to “correct” the material model in the FE analyses until convergence of the derived “correction factor” is achieved after typically 2-4 iterations. Fig. 8(a) shows the shear stress field in a butterfly shaped (BS) shear test specimen as predicted by nonlinear FE analysis. The distribution of the shear strain on the specimen gauge cross section and the specimen surface gauge line are shown in Fig. 8(b) and Fig.8(c), respectively. The “correction factor” is then used to “correct” the stress-strain response measured on the surface gauge line to obtain the average shear strain on the whole gauge cross section. Fig 8(a) displays the variation of the calculated strain “correction factor” as a function of the average strain on the gauge cross section. As expected, the strain “correction factor” displays it highest values in the linear (elastic) region of the PVC foam material. However, it decreases with increasing gauge section strains where the specimen gauge section undergoes increasing plastic strains that will smooth the strain distribution, becomes almost uniform before specimen fracture occurs. The “correction factor” appears to increase again just before fracture (see data point to the outermost right in Fig. 9(a)), but this behaviour is believed to be nonphysical and caused by numerical instability in the nonlinear numerical solution. Original data obtained from the shear testing of Divinycell grade H100 PVS foam at room temperature and the corrected shear stress vs. shear strain curve (core through-thickness direction) are shown in Fig. 9(b).

Fig. 8 Nonlinear FEA results: (a) shear strain distribution; (b) gauge section shear strain distribution; (c) defined gauge line on external specimen surface of FEA model (a)

(b)

Fig. 9 Nonlinear FEA correction results for shear test: (a) shear strain “correction factor” computed as a function of total shear strain; (b) shear data and corrected shear stress vs. shear strain curve for Divinycell H100 (through-thickness direction)

74 5. DERIVATION OF MATERIAL PROPERTIES A set of representative tensile and shear material properties measured for Divinycell grade H100 PVC foam at room temperature is shown in Table 1. The Young´s moduli are 130 MPa and 67 MPa in the through-thickness and in-plane directions, respectively. The in-plane modulus is approximately 55% lower than the through-thickness modulus, and the inplane strength is about 36% lower than the through-thickness strength. Table 1 shows that the properties obtained using the MAF rig compares well with recent measurements conducted at the University of Southampton, UK [9], as well as with the data sheet valued provided by the manufacturer of the foam core material [2]. It should be noted that the MAF data includes both of the elastic material coefficients as well as the strengths and strains to failure. Also it should be emphasized that all properties have been measured using the same MAF rig. Recent work has shown that the standard rigs used in [9] have limited performance beyond the elastic limit. Table 1 Divinycell grade H100 PVC foam orthotropic material properties measured using the MAF rig and comparison with measurements using other test methods E1 1 E2 G12 σ1max σ2max τ12max ε1max ε2max ν12 ε12max (MPa) (MPa) (MPa) (MPa) (MPa) (MPa) (%) (%) MAF

130.51 ±1.38

58.90 ±1.53

0.40 ±0.02

32.53 ±0.58

4.0 ±0.15

2.55 ±0.55

1.44 ±0.02

0.073 ±0.01

0.177 ±0.02

0.15 ±0.01

UoS2

132.78 ±0.88

58.70 ±0.89

0.41 ±0.01

30.12 ±0.18

N.A.

N.A.

N.A.

N.A.

N.A.

N.A.

DIAB3

130

N.A.

N.A.

35

3.5

N.A.

1.6

N.A.

N.A.

20

1

Indices 1 and 2 represent the through-thickness and in-plane directions, respectively. Linear elastic properties measured using DIC on the tensile and lap shear fixtures at the University of Southampton, UK [9]. 3 Material data sheet data provided by DIAB [2]. 2

6. ONGOING AND FUTURE WORK Work is presently ongoing to characterize the stress vs. strain curves for PVC foam core materials loaded in compression and at elevated temperatures for tensile, shear and compressive loads. In a further continuation of the work the bidirectional properties of PVC foams at both room and elevated temperatures will be also be investigated. ACKNOWLEDGEMENT The work presented was co-sponsored by the Danish Council for Independent Research | Technology and Production Sciences (FTP), Grant Agreement 274-08-0488, “Thermal Degradation of Polymer Foam Cored Sandwich Structures”, and by the US Navy, Office of Naval Research (ONR), Grant Award N000140710227. The ONR programme manager was Dr. Yapa D. S. Rajapakse. The financial support received is gratefully acknowledged. REFERENCES [1] Alcan Composites Newsletter, 2006, “Temperature of a Sandwich Panel when Exposed to Sunlight”, www.alcanairex.com. [2] Data sheets for cross-linked PVC foams, DIAB, www.diabgroup.com. [3] Data sheets for linear PVC foams, Alcan Airex, www.alcanairex.com. [4] Frostig, Y. and Thomsen, O.T., Thermal Buckling and Post-Buckling of Sandwich Panels with a Transversely Flexible Core, AIAA Journal, Vol. 46, No. 8, pp. 1976-1989, 2008. [5] Frostig, Y. and Thomsen, O.T., Buckling and Non-Linear Response of Sandwich Panels with a Compliant Core and Temperature-Dependent Mechanical Properties, Journal of Mechanics of Materials and Structures, Vol. 2. No. 7, 2007. [6] Frostig, Y. and Thomsen, O.T., Non-linear Thermal Response of Sandwich Panels with a Flexible Core and Temperature Dependent Mechanical Properties, Composites Part B: Engineering, Vol. 39, Issue 1, pp. 165-184, 2008. [7] Deshpande, V. S. and Fleck, N. A., Multi-Axial Yield Behaviour of Polymer Foams, Acta Materialia, Vol. 49, pp. 1859– 1866, 2001. [8] Gdoutos, E. E., Daniel, I.M., Failure of cellular foams under multiaxial loading, Composites Part A: Applied Science and Manufacturing, 33(2): 163-176, 2002. [9] Zhang, S., Dulieu-Barton, J.M., Fruehmann, R. and Thomsen, O.T., A methodology for obtaining material properties of polymeric foam at elevated temperatures. Submitted to Experimental Mechanics, 2011.

Optimization of Transient Thermography Inspection of Carbon Fiber Reinforced Plastics Panels Bradley G. Bainbridge1, Yicheng “Peter” Pan2, and Tsuchin “Philip” Chu3 1Master student, Southern Illinois University Carbondale, 2Research Associate, the University of Akron, Professor, Southern Illinois University Carbondale, 3 Department of Mechanical Engineering and Energy Process, 1230 Lincoln Drive Carbondale, Illinois 62901 Tel:(618)453-7039 ; Fax (618) 354-7658; Email [email protected]

ABSTARCT Transient thermography non-destructive evaluation methods are being used in aerospace industry to inspect flaws and damages for various composite materials. The purpose of this paper is to establish a set of guidelines for testing Carbon Fiber Reinforced Panels (CFRP) panels using infrared thermography. These guidelines insure that the inspection process is efficient and effective. Samples with simulated defects were made and modeled using a finite element program. Heat will be applied to the models and the temperature profiles analyzed. Along with changing the heat and time, different post-processing techniques were used to improve the method in determining defects in the sample. Once this has been optimized, actual CFRP panels with the same simulated defects were experimentally tested using the conditions from the analytical model. The analytical and experimental data was compared to insure that the testing process has been optimized. A standardized process was developed for evaluating the CFRP panels using infrared thermography. 1. Introduction CFRP have been around for many years and their material properties have been improving significantly. These plastics are made up of one or more layers depending on the application requirements. There are two main types of carbon fiber patterns that use rovings, unidirectional and woven. The unidirectional CFRP, lines a layer of fibers parallel to each other, then the next layer will be the same but will have a different orientation compared to the first layer and the layers will continue to change orientation as more layers are added. In general, a simple woven pattern would have fibers going in one direction and then another set of fibers perpendicular, woven in between the first set of fibers in the same layer. Depending on the needed parameters, many layers are put on top of each other making a panel or some other desired shape. An epoxy or thermoplastic based resin will be used to bond the fibers together, and is referred to as the matrix. A tighter weave will add stiffness and strength to the panel but will be more brittle where a loose weave will be inversely proportional. Also the type of matrix used will dictate how the panel acts in different loading conditions [1]. The manufacturing process is very complicated and there is a chance that defects shall develop internally within the CFRP. Visually the panel may look sufficient but it is critical to know that the internal structure was formed correctly. Infrared thermography can be used to help insure this quality. When a material is heated up from an external source, the heat will penetrate the sample at a constant rate. If there is a defect in the sample the rate that the heat is dispersed through the sample will be deferent at this spot affecting the surface temperature compared to the rest of the material. When the applied heat reaches the defect it will either build up at the defect or diffuse through the material faster creating a cold spot. When using an infrared camera during this process, the defect shall be shown visually on the camera’s display. However, sometimes the defects are deep compared to their size and can be hard to detect, because of this, post-processing computer programs are used to help find these defects. Optimizing this inspection process can help insure the quality of the product. The main goal of this paper is to establish a set of guidelines for testing CFRPs using infrared thermography. This will insure that the inspection process is efficient and effective. The main objective in reaching this goal will be optimizing the parameters at which the heat applied to the sample and the most efficient way to look for defects. These are the two main parameters that can be controlled by the user with the current set-up. The supplied samples of CFRPs were experimentally tested to find values of heat and time that will identify defect areas. This gave a baseline of input data for FEA. Simulations were run to see how different material properties affected the temperature vs. time profile of the models. These models were compared with the experimental results to verify that the FEA models were correct. The experimental data was also analyzed

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_10, © The Society for Experimental Mechanics, Inc. 2011

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76 using a post processing technique called the subtract function. This is used to refine the infrared images. 2. FEA Analysis In engineering problems one simple equation is not typically adequate in solving a problem. Many times multiple sets of the same equation must be used on the same part to see how different areas of the part in question react to the applied conditions. Finite element analysis (FEA) is a method that clearly organizes multiple equations representing a modeled object and shows how the different areas interact with one another. FEA can be applied to many different conditions including: stress analysis, heat transfer, electro magnetism, and fluid flow [2]. The methods used to solve these equations can be done by hand but this would be very time consuming so a computer based program is typically used. Computer based programs like ANSYS not only save a large amount of time but can also display the results in a visual manner that is easy to understand. Many researchers have used these programs to determine the potential reactions materials will have under different conditions. Badghaish et al. used analytical models to evaluate flatbottom and embedded defects of glassed reinforced plastics. There models used a constant heat flux from the top surface and assumed that any heat transfer from other surfaces were negligible. The thermal resistance was the main parameter that was studied, which is how well one can detect the resistance of the flow of heat. This is based upon the thickness of the material which is inversely proportional to the materials thermal conductivity [3]. To completely model a composite structure would be extremely complex. Not only would each fiber have to be modeled individually but also the epoxy matrix around the fibers. Then there would be the different contact resistances for the different materials. If modeling a defect then the delamination (or other type of defect) would have to be simulated. This would require the delamination to have its own set of boundary conditions as well as the general conditions the panel has. Fortunately, this does not have to be done and a homogeneous solid can be made to represent the complex structure as long as the proper thermal properties are applied to the material [4]. If the material properties are known for the individual components but not for the entire panel, they can be combined together using C p   f C pf   mCmf (1   ) (1) where φ is the volume fraction of the fibers, ρ is the density, Cp is the heat capacity and the subscripts f and m note the fibers and matrix respectively To verify the results of the experimental testing, analytical models were made and inserted into an environment to mimic that of the experimental tests. The models were created using AutoDesk Inventor 2010 and were based off of the specifications that the actual panels were made from. Since the panels can be considered uniform though the thickness of the material the individual layers were not modeled individually. The inserts were of known thickness and size and were modeled accordingly, then inserted at depths that would represent them being between layers. Meshing was developed by using ANSYS’s automatic mesh generator which uses an automatic patch conforming sweep.

Figure 1: (a) The white region represents the eighth model compared to the full model with the defect represented by the blue middle square. (b) Eighth model used in ANSYS These models started out as full size models but time and space limitations prevented these models to be used. The models were cut down to smaller one defect pieces and from there to quarter and eighth size models. The eighth model is shown in Figure 1(a). The Autodesk Inventor model is based off an actual panel that is 0.49mm thick with a 6mm x 6mm x 0.2mm defect inserted 0.21mm from the bottom. The CFRP was made in 3 pieces, the part with no defect in it was one piece and then the part above and below the defect were their own separate pieces. Then the defect was modeled by itself and later all four pieces were put together into an assembly. From here the model was cut down to a 6.5mm x 6.5mm right triangle. Figure 2 shows that the scaled models reproduce the same result as the quarter scale models do within 0.01%. A panel with an inserted disk defect will transfer heat through the thickness of the material and radically outward from the center of the

77 disk. Therefore the model can be scaled down to an eighth of its original size due to symmetry. The sides of the eighth model were also modeled to be perfectly insulated to insure accuracy.

Figure 2: Difference between using an eighth model compared to a quarter model yields no difference. Therefore an eighth model may be used to save computing time. A convergence study on the size of the mesh was studied. ANSYS has a built in mesh generator that was used where the relevance center and element size among other things can be adjusted. The default coarse mesh was used with a default element size, resulting in 12086nodes and 2455 elements; this was compared to a fine mesh with 212025nodes and 47934elements shown in Figure 3 and Figure 4 respectively. Figure 5 shows that there is little difference between the two mesh sizes, the percent difference was calculated and although the time step was initially slightly different the largest error was found to be 0.23% occurring at 0.4097sec. This is within an acceptable range allowing the coarse mesh to be used saving evaluation time.

Figure 3: Top view of an eighth model’s mesh set to the default coarse relevance.

Figure 4: Top view of an eighth model’s mesh set to the fine relevance.

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Figure 5: Difference between using the coarse and fine mesh. There is no perceivable difference. Boundary conditions were reviewed to see what effect they had on the temperature of the model. There are three types of heat transfer that could be used, conduction, convection, and radiation. The model was created to mimic the actual test specimens, to do this convection and radiation effects were studied. Figure 6 is a plot of actual data compared to simulations with different types of heat transfer. When only heat flux is applied the temperature rise is very similar to the actual data but after the heating period the heat uniformly distributes throughout the sample and remains constant since there is no source for it to escape. The temperature curves of using only radiation as a source of heat loss and another curve with only convection as a source of heat loss are also plotted. Individually the amount of heat loss increases but the cooling curve is not sufficient to the actual data. However, when both these conditions are applied to the model the temperature vs. time curve of the model and of the actual experimental data is very similar. The radiation emissivity value was set to 0.97; this is the value that the experimental data is set to record as well. The temperature for this was correlated to the ambient temperature which was set to range from 22°C up to 51°C then back down to 30°C, these are estimated temperatures based on thermal couple readings of the air temperature under the hood after 5sec of heating. The convection coefficient was also based off of these air temperatures coupled with an excel sheet that calculated the natural convection of the air based on an approximate panel temperature and ambient air temperatures. The rear face of the panel was elevated from the top of the resting surface so convection was used on the rear face with ambient temperature at 22°C. The air temperature under the panel was considered to remain the same since the panel would block any direct heating, also since warm air rises the part under the panel is the coolest part.

Figure 6: Comparison of different boundary conditions compared to experimental data The amount of heat flux used was determined by curve fitting the Ansys result temperatures to the experimental temperatures found on the panel the computer model was built to emulate. The heat flux was not considered completely constant as it is known that the halogen bulbs take a short time to heat up, so the heat flux was ramped to mimic this. The material properties of the CFRP were based on values close to the values referenced, the thermal properties of CFRP change depending on the manufacturing process and the materials used so no two panels will necessarily have the same properties so values taken were in reason. 3. Experiments Set-up and Test Knowing the type of materials being tested is important to obtaining accurate results from the experiments. Also the more known about a material, more conclusions can be made from experimental data. With this it is important to know the equipment being used and how to set it up properly. Combining these two things will help lead to greater success in finding defects and also will help insure the accuracy of the project.

79 3.1 Materials The sample materials in the lab are of woven carbon fiber reinforced by a polymer, where the carbon fibers add strength and rigidity. The panels are made of different layers, the number of layers range from 2 to 32 layers of carbon fiber weave. Since CFRP are made layer by layer there is possibility that defects will occur in between the layers. The most common defects that are seen in CFRP are delaminations, voids, inclusions, porosity, and regions having non uniform distribution of fibers [4]. When the panels were laid out, Teflon inserts were placed in between some layers at known locations and sizes to represent defects, mainly delaminations. To find the range of defects that can be found, each panel had the Teflon parts inserted at different depths and sizes, each panel has different shaped defects. Figure 7 is schematic of the CFRP panels that were used in testing. The actual material properties of any fiber reinforced plastic depend on several parameters such as fiber density, layer thickness, matrix material, shape of the fibers, and quality. The fiber density is how tight the weave of the CFRP is. Since carbon fibers have a much higher thermal conductivity along the length of the fibers as opposed to normal direction, a different fiber density will change the thermal conductivity along the length and width of the panel. Continuing with this would be the layer thickness, how large are the fibers that are being used and what the spacing between the layers is, and the amount of epoxy between the layers. The type of matrix (epoxy) can be different as well. Then the shape of the fiber section, it was briefly mentioned that fibers could be unidirectional or woven but even the woven fibers can have different patterns depending on the application.

Figure 19: (a) Schematic used in experimental testing. The Teflon inserts are equal in size throughout the pane, and are show in there approximates locations. (b) Schematic of a panel used in testing, showing the locations of the Teflon inserts. 3.2 Experimental Set-up The panels were tested experimentally using the equipment provided by the Intelligence Measurement and Evaluation Laboratory (IMEL) and the Center for Advanced Friction Studies (CAFS). Figure 21 is a schematic showing the basic testing set up for the CFRP panels. Figure 22 shows a picture of a typical test set up, where the infrared camera is sitting on top of the hood which encloses the four 1000W heat lamps aimed downward towards the surface of the panel. The heat lamps are computer controlled using a microcontroller that was developed in house, where the camera is controlled using the MikroSpec R/T software that was provided with the camera. The infrared camera being used is a MikroScan 7600PRO manufactured by MIKRON INFRARED. The camera can be used as a hand held camera, and has a built-in visual camera. For infrared cameras, this is considered high resolution at 320x240pixels and uses an uncooled focal plane array microbolometer. This detector can measure temperature differences up to 0.06°C at 60Hz at 30°C, and is accurate within 2% or ±2°C. The temperature range can be set between -40°C-120°C, 0°C-500°C, or 200°- 2000°C, and can focus on an object as close as 0.3m. With an emissivity range that can be adjusted from 0.1 to 1.0 in .01 increments [5]. In general CFRPs have a relatively shiny surface because of the epoxy which results in low emissivity. Although the infrared camera can adjust for different emissivity values there is a greater likely hood of the panel reflecting other light sources resulting in a false temperature reading by the camera. To help reduce this error the surface of the objects are typically painted with high emissivity paint. When the paint is used it is considered as a uniform thin layer that does not affect the result of the temperature profile [6]. The testing done in the IMEL used a temporary black paint manufactured by Dupli-Color. This paint provided a flat black surface reducing the reflectivity effectively raising the emissivity to approximately 0.97. It is easy to apply, takes very little time to dry and can be easily removed without affecting the material itself, making it ideal for non-destructive testing [7].

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Figure 8: Schematic of the testing equipment used

Figure 9: Actual photo of the testing equipment and set up used for experiments 4. Results and Discussion There are many different types of post processing methods, and they have been shown to help improve the detection of defects. Depending on the type of inspection process will dictate some of the post processing methods that can be used. An image is made up of tiny squares and each square has a number associated to it, typically between 0 and 255, the higher the number the whiter the color (on a gray scale). Many post processing methods find a unique way to make the pixel value of a defect further from the pixel value of the non-defect region. The subtract function is another popular method used and is built into the MikroSpec software. The subtract function lets the user pick a frame that they would like to subtract from the rest of the images. The software takes the frame that is selected and finds the temperature value for each pixel, then it subtracts that value from the temperature value of all the other frames. For example if the image to be subtracted was a uniform color with every pixel value equal to 100°C, and the frame to be analyzed had pixel values ranging from 10-255°C. The new range of pixel values would be from -90-155°C. There are many options when it comes to using the subtract function since any frame can be chosen, however certain frames should prove to be better than others. Using ANSYS the ideal points for using the subtract function are evaluated. Figure 10(a) shows a typical temperature vs. time graph with the COP at around 10.5sec. In this figure there are three ovals labeled as 1, 2, 3, these are the three different type of frames that could be used in the subtract function, one before the COP, at the COP and one after the COP. Figure 10(b) shows the temperature vs. time graph using region one, it can be seen that the temperature contrast is 13.9°C this is 11.4°C less than the original maximum temperature contrast of 25.3°C. However, the amount of contrast over the entire experiment above 1°C (about the amount of contrasted need to identify a defect) is about 6% higher than the original. Figure 10(c) is the new graph when the frame at which both the defect and non-defect region temperatures are the same, the temperature contrast for this has not changed significantly and the number of frames above 1°C of contrast did not change either. Choice three, shown in Figure 10 (d), has an increase in temperature contrast by 5.3°C, but the number of frames that the contrast is over 1°C has dropped by 4.2%.

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Figure 10: (a)Three locations that can be chosen to be used for the subtract function.(b) Temperature difference when Frame 1 is subtracted from the peak temperature, Frame 0. (c) Temperature difference when Frame 2 is subtracted from the peak temperature, Frame 0. (d) Temperature difference when Frame 3 is subtracted from the peak temperature, Frame 0. To show the change in thermal contrast another way line profiles have been taken of the original frame and the three manipulated frames. A line profile finds the temperature value of each pixel along the length of a line (length determined by user). For this case a line was drawn across an area with no defect while also crossing though the defect region as shown in Figure 11 (a) by the red line. MikroSpec R/T was used to do this evaluation and since the camera is mounted to the hood and all the subtract function images are from the same experiment there is no error in the line profile location. When the subtract function is used the temperature values change from image to image, so to help better understand the results the temperatures were adjusted so that they all started off as the same temperature. Since this is done by addition there it has no effect on the overall temperature contrast. Using a line profile shown in Figure 11(a), Figure 11(b) shows the adjusted values, and the thick blue line represents the original image without any post processing.

Figure 11: (a) Location of where the line profile is taken so the thermal contrast can be Evaluated (b)Line profiles of a defect and non-defect regions for different subtract function images In this case it is not an issue but in cases where there is a much smaller temperature difference the extra 1.4°C could lead to finding a defect not previously seen. Another set of experiments were done on a different CFRP panel. The sample was heated for 5sec with four 1000W bulbs and allowed to cool. When the raw data was analyzed only 4 out of the 9 defects could be identified in the data collected, and one frame is shown in Figure 12(a). A frame before the maximum temperature was captured and subtracted from the rest of the images. This frame is Figure 12(b), and only shows 3 out of the 4 defects. When subtracted from the rest of the images more defects are revealed, frame 99 shows the best contrast and is shown in Figure 12(c), where an additional 3 defects are revealed and shown inside of the black circles. The subtract function is not

82 always repeatable but can be very beneficial. There can be some noise and perceived temperature difference in CFRP because of the weave and the inherent nature of thermography. This noise will cause information to be lost and or masked, because of this sometimes some frames work better than others. In theory if a point is taken after the temperature regions cross over there is a better chance of increasing the temperature contrast. However, due to noise and reflections this is not always the case as seen below and will many times come down to the operator’s experience.

Figure 40: (a) Original image with no post processing, showing four defects.(b)Image of the frame that was subtracted from Figure 40 (c)Result of subtracting Figure 41 from Figure 40, 3 more defects can be seen 5. Conclusion It is concluded that the amount of heat and the duration is important to the temperature contrast of the material. If there is not enough energy put into the material then the intensity will be decreased, also if the amount of time is too short the change in temperature will be too small for the camera to detect. Increasing both of these will increase the contrast however, if too much energy is input into the material it will start to burn and change the material properties. Since this is nondestructive testing this needs to be avoided. The maximum temperature will depend on the epoxies used in the CFRPs so will depend on a case to case basis but around 120°C will typically be the limit. This limit should not be pushed too far, this is because if the known amount of heat to raise a non defect panel to 120°C is used and the defect creates a hot spot on the surface then this temperature limit compromised. A smaller amount of energy over a longer time period increases the chances of finding deeper defects in thicker materials. Where shorter higher amounts of energy will increase the contrast and allow for faster results in thinner panels. This research has started a broad foundation of information that can be expanded. It has reviewed the temperature trends in the defect material allowing one to better understand the materials being used in the experiment. Future work should look at different post processing methods and combine them experimentally with different types of defects to see if there are other large temperature contrasts that can be detected. Also a study of the COP and its relationship between time and material properties could serve as useful information for processing image data and knowing the ideal amount of time a material should be evaluated. In addition, the variables for the FEA model should be refined so that more accurate predictions can be made between the models and the experimental data. This additional information will help refine CFRP testing even more and the concepts would likely be transferable to other materials as well. Acknowledgements The Authors thank the Center for Advanced Friction Studies (CAFS) of Southern Illinois University Carbondale and all CAFS’s industrial sponsors for their support. Reference [1] Soutis, C., 2005, “Carbon Fiber Reinforced Plastics in Aircraft Construction,” Material Science and Engineering: A, 412 (1-2), pp. 171-176. [2] Moaveni, S., 2008, “Finite Element Analysis Theory and Application with ANSYS”, Prentice Hall, Upper Saddle River, pp. 1, Chap 1. [3] Badghaish, A. A. and Fleming, D. C., 2008 “Non-destructive inspection of composites using step heating thermography” Journal of Composite Materials 42 (13), pp. 1337-1357. [4] Marinetti S., Musicio, A., Bison, P. G., and Grinzato, E., 2000, “Modeling of thermal non-destructive evaluation techniques for composite materials,” Proceedings of SPIE - The International Society for Optical Engineering, 4020, pp.164-173. [5] MikroScan Operators Manual, Version 15.4F. [6] Pan, Y.P., Miller, R.A., Chu, T.C., and Filip, P., 2009, “Comparative Study of Thermography Systems For C/C Composite Disk Brakes,” Proceedings of the 2009 SEM Annual Conference Albuquerque, New Mexico, Session 11, No. 369. [7] Pan, Y., 2010, “Intelligent Non-Destructive Evaluation Expert System for Aircraft Carbon/Carbon Composite Brakes by Using Thermography and Air-Coupled Ultrasonic” Southern Illinois University, Carbondale.

Experimental investigation of thermal effects in foam cored sandwich beams

R K Fruehmann1, J M Dulieu-Barton1, O T Thomsen2, S Zhang1 1

Faculty of Engineering and the Environment, University of Southampton, University Road, SO17 1BJ, Southampton, UK 2

Department of Mechanical and Manufacturing Engineering, Aalborg University, Fibigerstræde 16, DK-9220 Aalborg East, Denmark

ABSTRACT Polymer foam cored sandwich structures are commonly used in applications where mechanical loads and elevated temperatures form the normal service conditions. The temperature sensitivity of the mechanical properties of the polymer foam cores leads to compromised mechanical performance of the overall sandwich structure at elevated temperatures. So far this phenomenon has primarily been investigated using analytical techniques. The present paper provides a basis for experimental studies of the temperature sensitivity of sandwich structures through the design of a test rig that can simulate the mechanical and thermal conditions experienced in service. The sandwich structure is modelled as a simple beam specimen with the mechanical load introduced using a standard servo-hydraulic test machine. A fixture has been specially designed that can apply a variety of constraints. A through thickness temperature gradient is introduced to the beam via an infrared (IR) radiator applied to one face sheet. The rig is designed to accommodate non-contact full-field techniques. An infrared detector is used to obtain the temperature field and high resolution white light cameras to capture the displacement using digital image correlation (DIC). An arrangement of mirrors enables both the face sheet and through-thickness surfaces to be viewed. The paper presents the design and evaluation of the rig, together with initial data obtained from a PVC foam cored sandwich specimen with aluminium face sheets. KEYWORDS: Sandwich structures, thermal degradation, model validation INTRODUCTION Sandwich structures with polymer foam cores have been used extensively in the marine industry, and are finding increased use in wind energy and civil engineering applications where the low weight to stiffness ratio and cost effectiveness for manufacturing large structures with complex shapes is advantageous. In many cases the service environment includes a large range of temperatures and often temperature gradients. Surfaces exposed to direct sunlight can reach temperatures in excess of 60°C [1], leading to large thermal gradients through the thickness. Many polymeric foams start to loose stiffness at such temperatures; in the case of a cross linked PVC foam this loss in stiffness could be as much as 15% of its ambient temperature stiffness, as shown in Fig. 1 for an Divinycell H100 PVC foam by DIAB [2]. Due to the sensitivity of polymeric

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_11, © The Society for Experimental Mechanics, Inc. 2011

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84 foams to changes in temperature, such service conditions present a challenge to designers when predicting the load response of a structure over the full range of service conditions to be expected. The loss in stiffness of the foam core changes the overall nature of the mechanical response of the sandwich structure. Simulations of a beam in three point bending with a temperature induced through thickness stiffness gradient using an analytical model developed in references [3, 4], predict a change in the mechanical behaviour from linear to nonlinear when the stiffness gradient in the core exceeds a threshold value. This change in behaviour is not currently considered in any design codes but it may lead to a significant change in the failure mode of a structure and possibly unexpected catastrophic failures in service. However, before the model predictions can be applied with confidence in practice, experimental validation is required.

Fig. 1 Degradation of moduli with temperature for Divinycell H100 PVC foam from DIAB [2] The present paper describes the evaluation of a test rig designed to validate the analytical model from references [3, 4] (a model based on higher order sandwich panel theory (HSAPT)). The HSAPT model has shown that sandwich beam response is very sensitive to the applied mechanical and thermal boundary conditions. Three specific mechanical boundary conditions have been identified that show an especially interesting load / deformation response, two of which are presented in the current work. The rig has been designed to accommodate the use of full-field optical measurement techniques to monitor the temperature and displacement / strain fields on the specimen surface. Infrared thermography (IRT) is used to obtain the temperature field and digital image correlation (DIC) to provide displacements and strains. Both techniques are also used to assess the validity of the thermal and mechanical boundary conditions. MECHANICAL BOUNDARY CONDITIONS The three mechanical boundary conditions that have been defined for this study are illustrated in Fig. 2 and comprise simply supported lower face sheet with horizontal movement allowed (Fig. 2 a) – referred to as SS-1, simply supported lower face sheet with movement restraints in both the vertical and horizontal directions, while being able to rotate about its mid-plane (Fig. 2 b) – referred to as SS-2, and simply supported lower face sheet as in SS-2 but with the addition of a horizontal constraint to the upper face sheet, allowing vertical displacement and rotation about its mid-plane (Fig. 2c) – referred to as SS-3. Nominal specimen dimensions of 500 x 50 x 30 mm (length x width x thickness) were specified, with a variety of face-sheet materials and thickness from 0.5 to 2 mm. An end fixture was designed that could be used to apply the mechanical boundary conditions whilst allowing optical access to the specimen. The end fixture is mounted in a 200 kN Denison Mayes servo-hydraulic test machine as shown in Fig. 3. Two large beams with a T-section groove are bolted, one to the actuator and one to the test bed, and provide the interface with the end fixture shown in Fig. 4. The beam specimens are mounted on a platform that is free to rotate about a horizontal axis, across the width of the beam as shown in Fig. 4. The height of the platform relative to the axis of rotation is adjustable so that the rotation axis always passes

85 through the mid-plane of the lower face sheet, thereby enabling the rig to accommodate face sheets of different thickness. For SS-1, the specimen rests freely on the platform. Lubricant is used to avoid frictional effects constraining the specimen in the in-plane direction. For SS-2, the lower face sheet of the specimen is extended by 25 mm on either side, allowing the lower face sheet to be clamped to the platform as shown in Fig. 4 a), thereby constraining the beam in the vertical and horizontal directions but allowing in plane rotation. The edge of the clamp has a radius to provide a gradual introduction of the clamping loads into the face-sheet and minimises stress concentrations resulting from the clamping force.

Fig. 2 Schematic of boundary conditions to be applied; a) SS-1, b) SS-2 and c) SS-3 For SS-3, the upper face sheet is also extended either side. Two vertical supports are mounted on bearings on the shaft of the lower platform (as in Fig. 4 b)). A platform to which the upper face sheet can be clamped is mounted on bearings in a similar manner to the lower platform. These bearings, however, are set in long holes to enable vertical displacement while constraining the face sheet in the in-plane direction. A combination of shims is used to adjust the height of the clamping surface relative to the axis of rotation to accommodate different face sheet thicknesses. A counterbalance is fitted to both platforms to ensure that the centre of mass is vertically aligned with the rotational axes so that no bending moment is applied to the face sheets by the fixture. In the configuration for SS-1 and SS-2, the whole length of the beam can be viewed for DIC and thereby the surface strain field can be evaluated right into the specimen corners enabling effects of clamping to be assessed. In the configuration for SS-3, the ends of the beam are slightly obscured, however, because the geometry of the clamping surfaces is the same as for SS-2 the clamping effect does not need to be investigated in this configuration. Furthermore, all components of the rig can be imaged and therefore the rig itself can be assessed to exactly quantify the mechanical boundary conditions. ASSESSMENT OF THERMAL BOUNDARY CONDITIONS The HSAPT model is a two dimensional model in the formulation presented in [3, 4]. As such, it is assumed that the temperature across the width of the specimen is uniform and the steady state temperature distribution through the thickness of the specimen is therefore linear. Experimentally, the temperature gradient is achieved by heating the upper surface of the specimen using an IR radiator, while maintaining the lower face sheet at ambient temperature by means of forced convection. The finite width of the specimen, however, leads to a non-uniform temperature distribution across the specimen width. This was shown by initial measurements of the through thickness temperature profile, obtained from the side of the specimen using IR thermography which revealed a strongly nonlinear temperature gradient, as shown in Fig. 5. This is due to heat losses at the specimen sides.

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Fig. 3 Experimental arrangement, a) test machine and b) end fixture

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Position (mm) Fig. 5 Surface temperature profile of a specimen with a 30 mm thick Divinycell H100 foam core and 1 mm thick aluminium face sheets (data from IR thermography – Expt and FEA surface) To assess the internal temperature field of the specimen, a two dimensional finite element model of the specimen crosssection was created using PLANE55 (4 node rectangular elements) in ANSYS 12. Material conductivity values were obtained from the manufacturer’s data sheet for the foam [5] and from reference [6] for the aluminium. The heat flux through the top surface was adjusted to control the top surface temperature. It was found that to achieve a good correlation with the experimental data, the conductivity in the foam core needed to be reduced from 0.032 to 0.028 Wm-1K-1. This provided good correlation with the experimental data over a range of temperature gradients, as shown in Fig. 5, although the model consistently under-predicted the lower surface temperature. In this configuration, a temperature difference of approximately 20°C exists between the centre of the specimen and the free edges (sides) of the specimen. Considering the stiffness degradation curve of the Divinycell H100 foam shown in Fig. 1, this would result in a 35% variation in stiffness across the width relative to the specimen mid-plane stiffness. To reduce the width-wise temperature gradient, insulation in the form of additional PVC foam was fixed to both sides of the specimen during the heating process. Due to the space available the practical limit of the thickness of the insulation was approximately 25 mm. The addition of insulation to the sides enabled a higher temperature to be achieved and provided a through thickness profile much closer to the desired linear distribution, as shown in Fig. 5 where the surface measurement is compared with the FEA. Fig. 6 shows the temperature distribution across the specimen width. Again, the temperatures derived from the model were compared with temperatures obtained experimentally. To obtain the internal temperature through the width of the beam thermocouples were installed within the specimen as shown in the insert in Fig. 6. To facilitate this, 2 mm diameter holes were drilled to a controlled depth in the specimen. K-type thermocouples were pushed into the holes until the hot junction made contact with the end of the hole. The agreement between the FEA model and the thermocouple measurements is good, although the internal temperature measurements were slightly lower than the model predictions. This could be attributed to the conduction of heat out of the foam block due to the presence of the thermocouple wires which have a significantly higher conductivity than the foam.

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Fig. 6 Comparison between thermocouple and FEA data across the specimen width, taken 3 mm below the specimen surface; the insert shows the thermocouple placement in the specimen To observing the foam core behaviour, the foam insulation must be removed from one side of the specimen to enable images to be collected from the specimen side for DIC. The rate of convective cooling at the surface was measured using IR thermography. This indicated that 10 to 15 seconds were available (depending on the initial temperature condition) in which conduct the test. Hence a relatively high displacement rate of 20 mm / min is required for a test to failure. INITIAL TEST RESULTS Initial tests were conducted for the SS-1 and SS-2 condition at four temperatures between 25 and 90 °C. In both cases specimens of 450 x 50 x 27 mm (span x width x thickness) were used with a 25 mm thick PVC H100 foam core and 1 mm thick aluminium face sheets. A constant displacement rate of 20 mm / min was applied at the mid-span. The load was applied via a 12 mm diameter steel roller across the width of the specimen and the temperature of the top face sheet was monitored during the heating phase via a mirror attached to the actuator as shown in Fig. 3. When a steady temperature was reached (this took approximately 40 minutes depending on the temperature condition) the IR detector was moved to image the specimen side, to monitor the rate of cooling during the test. Images of the central portion of the beam were captured using a LaVision Imager E-lite (5 MPx 12 bit CCD camera) from which the displacements of both top and bottom face sheets were calculated using DIC. Fig. 7 shows the force vs. displacement curves for the SS-1 condition to failure. A clear loss in overall stiffness is observed, most marked in the 90°C case, as would be expected based on the thermal degradation curve shown in Fig. 1. A difference can also be observed in the gradient of the curves for the upper and lower face sheets for all temperature conditions indicating a compressive strain in the foam core. Interestingly, the difference is most marked for the 25°C case and least observable in the 90°C case. Failure was defined as the point where the load decreases. Two types of failure mode were observed. At room temperature and 50°C failure occurred as a combination of indentation of the foam core and interfacial failure between the foam core and the face sheet in the region immediately adjacent to the load application. At 70°C and 90°C the failure mode changed to indentation of the foam core and plastic deformation of the aluminium under the roller only. No debonding was observed at these temperatures. At failure, the 90°C case shows the largest difference between the upper and lower face sheet displacements, indicating the largest core compression.

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Fig. 7 Force vs. top and bottom face sheet displacement for SS-1 The load vs. displacement curve for SS-2 is shown in Fig. 8 for the lower face sheet only. The specimen shows a slightly higher stiffness and failure load than the SS-1 case. The same trend of loss of stiffness can be observed as for the SS-1 case. The most significant difference between the two boundary conditions was the failure mode. For SS-2 all specimens failed by failure of the interface between the face sheet and the foam core. Only minimal indentation was observed, mostly of the face sheet although at the higher temperatures a slight impression was also left in the foam core. CONCLUSIONS Initial work has been conducted to experimentally validate the HSAPT model using full-field measurement techniques. The challenge has been addressed to meet specific therrmal and mechanical boundary conditions through the design of a bespoke end-fixture that enables full optical access to the specimen throughout the loading. A compromise was required in the form of obscuring one side of the specimen during heating in order to achieve the very demanding thermal boundary conditions resulting from the limitations of a two dimensional model. However, it has been demonstrated that this does not present an obstruction to obtaining the images so that full-field displacement data can be obtained. Obtaining accurate predictions of the mechanical and thermal performance of such a test setup using, for example, FEA techniques, is prohibitively complex due to the large number of surfaces in contact and the inherent nonlinearity and potential for stick-slip phenomena this implies. Instead, the full-field optical techniques used to measure the specimen deformations have been employed to assess the performance and provide information that can be fed back into the HASPT. Thereby it is possible to perform a valid comparison between calculated and measured data, despite the physical limitations of the experiment and the conceptual limitations of the model.

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Fig. 8 Force vers. bottom face sheet displacement for SS-2 Preliminary data has been obtained for SS-1 and SS-2 that indicates a distinct progressive softening that resembles the predicted behaviour found using HSAPT. Observation of the full development of a nonlinear load-displacement curve has not been possible in these initial specimens due to the face / core interface failure that defines the maximum load. Optimisation of specimen geometry and materials combination is currently in progress using the experimental techniques laid out in this work. Ongoing work is focusing on conducting a systematic comparison of SS-1, SS-2 and SS-3 measurements with HSAPT and nonlinear FEA model predictions. REFERENCES [1] Feng, Y., “Thermal design standards for energy efficiency of residential buildings in hot summer/cold winter zones”, Energy and Buildings, 36, pp. 1309-1312, 2004 [2] Zhang, S., Dulieu-Barton, J.M., Fruehmann, R.K. and Thomsen, O.T., “Thermomechanical Interaction Effects in Composite Sandwich Structures – Initial Experimental Analysis and Derivation of Mechanical Properties”, 7th Asian Conference on Composite Materials, Taipai, 4 pages on CD, 2010 [3] Frostig, Y. and Thomsen, O.T., “Buckling and non-linear response of sandwich panels with a compliant core and temperature-dependent mechanical properties”, J. Mech. of Matls. and Structures, 2, pp. 1355-1380, 2007 [4] Frostig, Y. and Thomsen, O.T., “Thermal buckling and post-buckling of sandwich panels with a transversely flexible core”, AIAA Journal, 46, pp. 1976-1989, 2008 [5]

DIAB, “Divinycell PVC datasheet”, http://www.diabgroup.com, November 2010

[6]

Automation Creations Inc., “Matweb.com - online material datasheet”, http://www.matweb.com, January 2011

Intelligent Non-Destructive Evaluation Expert System for Carbon Fiber Reinforced Plastics Panel Using Infrared Thermography Yicheng “Peter” Pan1 and Tsuchin “Philip” Chu2 1Research Associate, the University of Akron, 2Professor, Southern Illinois University Carbondale, Department of Mechanical Engineering and Energy Process, 1230 Lincoln Drive Carbondale, Illinois 62901 Tel: (618) 453-7039; Fax (618) 354-7658; Email [email protected]

ABSTARCT This research developed a reliable intelligent non-destructive evaluation (NDE) expert system for Carbon Fiber Reinforced Panels (CFRP) panels based on infrared thermography testing (IRT) and post processing by means of fuzzy expert system technique. Data features and NDE expert knowledge are seamlessly combined in the intelligent system to provide the best possible diagnosis of the potential defects and problems. As a result, this research help ensure CFRP panels’ integrity and reliability. Specimens with simulated defects were evaluated to demonstrate the usefulness of the intelligent IRT NDE expert system in NDE inspection. The testing data pattern corresponding to feature and quantification of defects were found. This fuzzy expert system not only eliminates human errors in defect detection but also functions as NDE experts. In addition, fuzzy expert system improves the defect detection by incorporating fuzzy expert rules to remove noises and to measure defect size more accurately. In the future, the expert system model could be continuously updated and modified to quantify the size and distribution of defects. The system developed here can be adapted and applied to build an intelligent NDE expert system for better quality control as well as automatic defect and porosity detection in CFRP production process. 1. Introduction Carbon Fiber Reinforced Plastics (CFRP) are becoming more popular every day. They are currently being used in a variety of applications in aerospace and researchers are looking into using it in building structures due to their high strength to weight ratio [1]. Currently the manufacturing process to make CFRP is very complex so it is crucial to make sure that the final product is flawless. Infrared Thermography (IRT) is an ideal non-destructive evaluation method for looking at CFRP because it is non-invasive, non-contacting and can analyze large areas at one time [2-8]. However, being able to find these defects is based on the operator’s experience, and improper testing can result in defects not being found. Therefore, implementing and operating intelligent IRT NDE expert system for CFRP would be reliable, economical, and easy to the industry. In summary, the ability to evaluate CFRP by IRT NDE methods is important to many industries which produce and use CFRP. Hence, the goal of this study is to develop intelligent NDE expert systems which provide expert knowledge and experiences to help the NDE technicians during inspection. Although IRT NDE techniques have been tested and proven for CFRP, an intelligent IRT NDE system for CFRP has not been demonstrated. Therefore, this research contributed to advancing intelligent IRT NDE system technologies in CFRP. 2. IRT Method The experiments were conducted at Southern Illinois University Carbondale’s (SIUC) Intelligent Measurements and Evaluation Laboratory (IMEL). Experiments were conducted using four 500 Watt halogen linear tubes as the heat source. The equipment used in these experiments was the SIUC thermography system. The infrared camera that was used in these experiments to record the thermal images was a MikroSpecRT thermal imaging camera with a resolution of 0.06°C at 30°C, a measurement accuracy of + 2°C of reading, and 320 x 240 dpi. The error of IR camera rounds the 0.06°C up to 1°C, since there is noise associated with the camera and the material properties such as thermal diffusivity and emissivity. The CFRP is an orthotropic material and the thermal diffusivity changes throughout the sample which can cause different temperature readings on the surface. To make sure that and actual defect is being detected and not just noise due to either the camera or the material, the analyzed thermal changes should not be smaller than 0.2°C. The thermal image infrared camera unit incorporated a black and white or (grayscale) image viewing screen as well as a colored viewing screen for a more easily

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_12, © The Society for Experimental Mechanics, Inc. 2011

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92 seen thermal contrast imaging. The infrared camera was linked to a Dell Latitude D820 computer, which used the MikroSpecRT software that came with the IR camera. This software allowed for even further rendering and processing of the images. The software could record in real time and create video recordings or take snap shots as the sample was heated. 2.1 Transient IRT Experimental Setup The schematic of a typical test setup is shown in Figure 1. Long pulse thermography was used to inspect composite materials with the use of SIUC’s IMEL thermography equipment. The analyses were performed on the CFRP samples. The long pulse thermography method subjected one face of a specimen to a constant and uniform heat flux, and recorded the surface temperature response on the same face. The heat source was applied for approximately 5 sec, and then the sample was allowed to cool down. The IR camera recorded the data from the start of the heat flux application and up to 5 sec after the heat flux was removed. The distance between the lamps and the sample was adjusted to approximately 18in or 460mm from the front of the specimen to the front of the IR camera. The lamps were positioned so that they would be exactly at the same distance from the sample for each experiment. Temperature data was taken every 0.03 second when the experiments were performed. All thermal response images and frame rate set-up were collected by a computer.

Figure 1: A Typical Transient Thermography System 2.2 IRT NDE Software Current IRT system enables the capturing and processing of sixty of digital images per second. The thermal response images are typically analyzed for temperature differentiation which may indicate the suspected defects. “Noise of multiple nature and optical distortions produce difficulties for the interpretation of the recorded data. Post-processing is a powerful tool for the determination of the shapes and sizes of sub surface defects in inspected subjects. Regular thermal image analysis combined with special methods based on heat transfer theory can be applied to each IRT inspection” [9]. Several thermal data processing algorithms are currently used for enhancement of defect detection such as Thermal Signal Reconstruction (TSR), contract computations, pulse phase imaging, derivative function, and slope function. Therefore, this research developed IR NDE evaluation software with algorithms for thermal data processing, analysis of temperature-time history data, and qualitative visualization of thermal response data. The software was coded by MATLAB. Some of built-in functions used in the programs are contained in the image analysis toolbox. Approximately fifteen MATLAB subroutines were developed as part of this research for pre-processing, processing, and post-processing analysis of IRT images. The goal of this software is to detect the subsurface anomalies and provide quantitative information about defects with minimal input from the user. The screen of software as in Figure 2:

Figure 2: screen of transient infrared thermography NDE system software

93 2.3 Materials The CFRP sample is shown in Figure 3(a). The CFRP has seven embedded defects (films of Teflon) in different layers ranging from 2 plies, 4 plies, 8 plies, 12 plies, 16 plies, 22 plies, to 28 plies deep. The panel is made of different layers, the number of layers range from 2 to 32 layers of carbon fiber weave. Since CFRP are made layer by layer there is possibility that defects will occur in between the layers. The most common defects that are seen in CFRP are delaminations, voids, inclusions, porosity, and regions having non uniform distribution of fibers [10]. When the panels were laid out, Teflon inserts were placed in between some layers at known locations and sizes to represent defects, mainly delaminations. To find the range of defects that can be found, each panel had the Teflon parts inserted at different depths and sizes, each panel has different shaped defects. Figure 3(b) is schematics of the CFRP panel that were used in testing.

Figure 3: Images of commercial CFRP 3. Fuzzy Expert System Architecture of IRT After the IRT inspection, testing data is processed by the intelligent IRT NDE expert system, which is implemented in MATLAB. The fuzzy logic expert system consists of two main components: The first is a set of fuzzy inference systems (FISs) (MathWorks, 2005a) and the second is an m-file (i.e. a MATLAB program (Math-Works, 2005b)). The software was coded by MATLAB because it integrates mathematical matrix with visualization capabilities and fuzzy inference functions. Some built-in functions of MATLAB executed in the programs include the image analysis and fuzzy logic tool boxes. 3.1 Input Data The data that is inputted to the expert system consist of the IRT thermal response serious images measured by the long pulsed thermography system. Thermal response serious images are then converted to a three dimensional array (I). The data array I contains three dimensions, an area of image and time frame number with the value of temperature. For example, the value of I (100, 150, 1) means that the temperature value of the pixel of in the X-direction is 100 and the Y-direction is 150 at time frame number 1. The IRT analyzed the presence of hot spots which may specify the existence of subsurface defects. Surface emissivity effects, heterogeneous heat sources and optical distortions produce difficulties in interpretating the recorded data. In order to avoid these problems this research had to develop algorithms for determining the defects of CFRP materials. Based on the subtract function algorithm, temperature gradient data at five certain time frames were created by the IRT NDE system. The temperature gradient data at a certain time frame, data (I), is directly used as membership of the fuzzy clustering algorithm. The temperature gradient data is normalized in standard score to satisfy the normalizations in statistics conditions in Eq. (1) and (2)

Z( i, j ) 

I ( i ,J )  u



I ( i ,J )

(1)

Where is a raw temperature gradient value to be standardized in the i and j position; μ is the mean of the temperature gradient image array (I); σ is the standard deviation of the temperature gradient image array (I);.

94 U Z( i, j ) 

Z ( i , j )  min Z max Z  min Z

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U Z( i, j )

is a normalized temperature gradient standard score in the i and j position; max Z is the maximum U value of the standard score array; min Z is the minimum value score of the standard score array; Z normalized standard score which will be used as fuzzy memberships. where

“In statistics, a standard score indicates how many standard deviations an observation or datum is above or below the mean. It is a dimensionless quantity derived by subtracting the population mean from an individual raw score and then dividing the difference by the population standard deviation. This conversion process is called standardizing or normalizing” [11]. In the fuzzy model for IRT fuzzy expert system, two variables were used in each time frame data set. These input variables sent to the FISs include (1) temperature gradient standard score (TGSC) value of a pixel and (2) the mean value of the temperature gradient standard score in a 3x3 pixel window. The value of the mean and standard score of the pixel is allocated to central pixel of each window. 3.2 Knowledge Base The knowledge base of fuzzy expert system includes fuzzy set and fuzzy rules. At first, the two input variables and output variable are modeled by fuzzy sets in their respective ranges. The range of normal TGSC levels is considered to be from 0.25 to 0.75, from 0.5 to 0 it is considered low, between 0.5 and 1 it is high, a level below 0.25 is very low , and a level above 0.75 is very high. Thus, the following linguistic labels were used for the variable standard score level and the mean TGSC level: very low (VL), low (L), medium (N), high (H), very high (VH). The output variable (defect confidence level) was attributed in the following five labels: very low (VL), low (L), medium (M), high (H), very high (VH). The inference method proposed by mamdani-style inference was utilized .The output is a constant value in the range of [0, 1] for presenting the defect confidence level. The membership functions of the fuzzy sets assumed by input variables and output variable of the system are shown in Figure 4. Fuzzifier is then depicted into triangles. To increase the effect of signal amplitude and the size of the defect, this study used a number of fuzzy it-then rules. The rule-base is shown in Figure 5 and it is composed of 25 rules, like this “If standard score level is VH and mean density level is VH then defect confidence level is very high.” The system output is obtained by utilizing the fuzzy rules and the fuzzy inference method. This output is a real number with a linguistic label. Complex relationships between all variables used in the fuzzy system can be represented best by the hierarchical structure shown in Figure 6.

Figure 4: Membership functions of fuzzy sets assumed by input variables and an output variable representation.

Figure 5:25 rules of the rule-base elaborated of IRT fuzzy expert system

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Figure 6: hierarchical fuzzy model for IRT fuzzy expert system 4. Results and Discussion 4.1 Experimental Result of IRT The IRT inspection was conducted on CFRP sample. Simulated defects were detected in all three samples with the use of the long pulse thermography method. The temperature profiles and defects detected are shown in Figures 7 which were captured at frame 200 and 240.An advance data processing algorithm (subtract function method, SF) was used for enhancement of signal-noise ratio of thermal images. The line profile plots are shown in Figures 8which captured frames 200. This allowed for a temperature variation to be seen within the defects

Figure 7: Thermal image on CFRP sample

96 Figure 7 shows defects within the CFRP sample using long pulse thermography. This sample had 1mm by 1mm square simulated defects at 2, 4, 8, 12, 16, 22, and 28 plies deep from the surface. Seven simulated defects were easily detected on two sections. Figure 8 is the thermal line profiles for the CFRP simulated defects. The graph in Figure 8 shows seven distinctive temperature changes; (1) is the square simulated defect at a depth of 2 plies, (2) is at 4 plies deep, (3) is at 8 plies deep, and (4) is at 12 plies deep, (5) is at 16 plies deep, (6) is at 22 plies deep, and (7) is at 28 plies deep.

Figure 8: Line Profile of CFRP sample Thermography was able to detect seven embedded simulated defects sealed by resin in CFRP sample to a depth of 28 plies deep with reasonable accuracy. The distinctive temperature change for depth of 28 plies deep defect with a maximum temperature change is about 4ºC ~ 5 ºC. Some of the defects were not displayed clearly by a bright white spot on the IR camera, again instead it was revealed by using the thermal profile over the area. Overall, being able to find these defects is based on the operator’s experience, and improper testing and skill can result in defects not being found. 4.2 Defects Detection by IRT Fuzzy Expert System In the IRT fuzzy expert system, the specimen was used to determine the capability and performance of the system for identifying the defect’s size and shape in particular samples. This research used two fuzzy logic inference methods in the IRT fuzzy expert system which are Mamdani-style and Sugeno-style methods.

Figure 9: Fuzzy expert detected defects in CFRP sample (8 plies)

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Figure 10: Detected Defects comparison in CFRP sample (28 plies) Figure 9 shows fuzzy expert defect detection input and output images of the embedded defect in 8plies deep within the CFRP sample. Figure 10 shows the fuzzy expert defect detection input and output images of the embedded defect in 28plies deep within the CFRP sample. The spatial resolution of the sample was found to be 0.66mm/pixel. With this information the defect size can be determined for the samples tested. According to the output of IRT expert system, the dimension of the defects is around 5.94mm by 5.94 mm in square shape which are shown in Figure 9 and 10. 5. Conclusion This IRT NDE logic fuzzy expert system gave plausible results for the porosity location and distribution which match well with the result of performance testing in IRT NDE method by IRT expert. This fuzzy logic expert system similar to the human eye and can be used in applications to eliminate the human errors during inspection. Also, fuzzy logic expert systems improved the inspection capability by inferring some fuzzy logic expert rules for better defect detection and noise elimination. It is recommended that further work be done on the quantitative study of the particular defect size, type, and depth in order to determine the effect of these parameters on the results using the intelligent IRT NDE expert system. Also, the accuracy and the range of this system for any specified defect condition are necessary to be investigated. The intelligent IRT NDE expert system employed in this study can be applied to building an expert system for automated detection of defects and porosity levels in commercial aerospace CFPR materials production process. Acknowledgements The Authors thank the Center for Advanced Friction Studies (CAFS) of Southern Illinois University Carbondale and all CAFS’s industrial sponsors for their support and for acting as technical monitor of this research. Reference [1] Soutis, C., 2005, “Carbon Fiber Reinforced Plastics in Aircraft Construction,” Material Science and Engineering: A, 412 (1-2), pp. 171-176. [2] Chu, T.C., 1998, “Finite Element Modeling of Transient Thermography Inspection of Composite Materials,” NASA Contractor Reports – 1997 NASA/ASEE SFFP, pp. VII.1–VII.4. [3] Chu, T.C., DiGregorio, A.M., and Russell, S. S., 1999, “Determination of Experimental Parameters for Transient Thermography,” Proceedings of the SEM Annual Conference on Theoretical, Experimental, and Computational Mechanics, pp. 318-321. [4] Chu, T.C., Mahajan, A., DiGregorio, A., and Russell, S., 2005, "Determination of Optimal Experimental Parameters for Transient Thermography Imaging using Finite Element Models," The Imaging Science Journal, 53, pp. 20-26. [5] Shepard, S. M., 2007, “Therography of Composites” Materials Evaluation, pp. 690-696. [6] Zalameda, J. N., 1999, “Measured Through-the-Thickness Thermal Diffusivity of Carbond Fiber Reinforced Composite Materials,” Journal of Composite Technology & Research, JCTRER, v 21, n 2, pp.98-102. [7] Maldague, X., 2000 “Applications of Infrared Thermography in Nondestructive Evaluation” Trends in Optical Nondestructive Testing, pp. 591–609.

98 [8] Badghaish, A. A. and Fleming, D. C., 2008 “Non-destructive inspection of composites using step heating thermography” Journal of Composite Materials 42 (13), pp. 1337-1357. [9] Plotnikov. Y. A., Winfree, W. P., 1998, “Advanced image processing for defect visualization in infrared thermography” SPIE Vol.3361, pp. 331-338. [10] Marinetti S., Musicio, A., Bison, P. G., and Grinzato, E., 2000, “Modeling of thermal non-destructive evaluation techniques for composite materials,” Proceedings of SPIE - The International Society for Optical Engineering, 4020, pp.164-173. [11] Moore, D.S., Notz, W.I., 2006, “statistics: concepts and controversies” 6th Edition New York: W. H. Freeman

Successful Application of Thermoelasticity to Remote Inspection of Fatigue Cracks

Takahide Sakagami, Professor, Graduate School of Engineering, Kobe University 1-1 Rokkodai, Nada, Kobe 657-8501 Japan Yui Izumi, Doctoral Student, Graduate School of Engineering, Osaka University 2-1 Yamadaoka, Suita, Osaka 565-0871 Japan Shiro Kubo, Professor, Graduate School of Engineering, Osaka University 2-1 Yamadaoka, Suita, Osaka 565-0871 Japan

ABSTRACT A new remote nondestructive inspection technique based on thermoelastic temperature measurement by infrared thermography was developed for the detection of fatigue cracks in steel bridges. Fatigue cracks were detected from localized thermoelastic temperature changes at crack tips due to stress singularities generated by wheel loading from traffic on a bridge. Self-reference lock-in data-processing technique and motion compensating technique were developed to improve the thermal images obtained in the crack detection process. Advantages and limitations of the proposed nondestructive evaluation technique were discussed based on results of field experiments for highway bridges. Thermoelastic stress analyses in the vicinity of crack tips were also carried out after the crack detection process by self-reference lock-in thermography. The stress distribution under wheel loading by traffic was measured by infrared thermography. Stress intensity factors were evaluated from measured stress distribution. It was found that these fracture mechanics parameters can be evaluated with reasonable accuracy by the proposed technique, enabling the assessment of structural integrity based on the evaluated fracture mechanics parameters. 1. Introduction Recently, crack propagation in aged structures has become a serious problem that can lead to their catastrophic failure. In large-scale steel structures of critical importance, such as highway bridges, the nondestructive inspection for deterioration and damage is necessary to ensure safety and to estimate the remaining life of these structures. As conventional nondestructive testing (NDT) techniques for steel bridges, visual testing, magnetic particle testing and ultrasonic testing have been commonly employed. However, these techniques are time- and labor- consuming techniques, because special equipment is required for inspection, such as scaffolding or a truck mount aerial work platform. Furthermore they can only be employed for crack detection, and can not be used to directly measure physical quantities for evaluating the remaining strength based on fracture mechanics. Thermoelastic stress analysis (TSA) by infrared thermography has been widely used as an effective full-field experimental stress measurement technique [1, 2]. TSA has been gaining increasing attention as a nondestructive testing and evaluation method for fatigue cracks in steel structures. When TSA measurement is performed on a cracked structure, cracks can be detected from singular stress fields due to the cracks, since significant thermoelastic temperature changes can be observed due to stress concentrations around crack tips. Steel bridges being concerned with fatigue crack propagations are always subjected to frequent and heavy wheel loadings from traffics on the bridge. Therefore fatigue cracks can be identified from localized thermoelastic temperature change due to stress singularity at crack tips under variable wheel loadings. Lock-in infrared thermography using reference signal synchronized with stress is commonly employed to improve the precision of stress measurements, since thermoelastic temperature change is very small. A load signal from an external source, such as load-cell or strain gage, is usually employed as a reference signal in the conventional TSA technique.

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However, it is difficult to obtain a reference signal from steel bridges in service. Furthermore the observed load signal does not have a clear sinusoidal waveform, because it contains the random waveform components due to the in-service wheel loading by the vehicles. These facts indicate that the conventional lock-in infrared thermography is not applicable for TSA of steel bridges. The present authors developed a self-reference lock-in thermography [3] that does not require any external reference signals and can be employed even under random loading. Nondestructive inspections of fatigue cracks in steel bridges were conducted by the proposed self-reference lock-in thermography in our previous papers [4, 5]. TSA is extremely beneficial not only for crack detection but also for the on-site measurement of stress distributions around crack tips, which is important for crack propagation analysis. The present authors [6] carried out thermoelastic stress measurement under random wheel loading in the vicinity of fatigue cracks. The stress intensity factors KI and KII for mixed- mode cracks were obtained from measured stress distributions in actual steel bridges in service. In this paper, experimental results for fatigue crack detection by self-reference lock-in thermography and subsequent on-site stress evaluation based on thermoelastic temperature measurement are reviewed. 2. Thermoelastic stress measurement Dynamic stress change causes very small temperature change under the adiabatic condition in a solid. This phenomenon is known as the thermoelastic effect and is described by Lord Kelvin’s equation relating the temperature change (ΔT) to a change in the sum of the principal stresses (Δσ) under cyclic variable loading as follows. (1) α : Coefficient of thermal expansion ρ : Mass density Cp : Specific heat at constant pressure T : Absolute temperature A change in the sum of the principal stresses (Δσ) is obtained by measuring the temperature change (ΔT) using infrared thermography. 3. Self-reference lock-in thermography In self-reference lock-in thermography, a reference signal is constructed from a reference region arbitrarily set on the same sequential infrared images as those showing the thermoelastic temperature change. The distribution of the relative intensity of the thermoelastic temperature change against that in the reference region can be obtained by the following least-squares approach even under random loading, provided that the temperature change in the reference region has a similar and in-phase waveform to that in the objective area under measurement. Assume that a body is subjected to random loading whose waveform is expressed as fn. The infrared signal in an objective region can be approximated as follows: (2) where a is the DC offset, b is an influence coefficient of the reference, and n is the frame number. To calculate b, the sum of the squares of the deviations between Yn and the infrared signal yn obtained from the region, defined as follows, is minimized. (3) Here, N is the total frame number. Then, b is obtained from the following equation.

(4)

When this calculation is performed on all the pixels of infrared thermography, it is possible to obtain the correlation between the infrared signal in the reference region and that in any region. The value of b indicates the intensity of the thermoelastic

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temperature change relative to that in the reference region. The self-reference lock-in method does not require any external reference signals and can be applied even under random loading. The obtained values of b are effectively employed for the detection of areas of stress concentration around notches or cracks. 4. Motion compensation technique for sequential infrared images Displacement in infrared image is caused by the vibration of the bridge under vehicle loading. This displacement sometimes causes an error called as edge effect in infrared measurement when time-series processing is conducted for sequential infrared images. Displacement in the infrared image was corrected by motion compensation by the post data processing based on the movements of characteristic points in infrared images. Determination of image shift is conducted by the two-dimensional SSD (Square Sum of Differences) parabola fitting method [7]. Pixel values of the subset, i.e., the region of interest of pattern matching, in the shifted image are compared with those in the basic image and square sum of the differences is calculated between two images. The value of image shift which minimizes SSD is determined as the most plausible value of the image shift. SSD is given by the following equation. (5) Fig. 1 Schematics of 2-D SSD parabola fitting method where I1 and I2 are pixel values in the basic image and shifted image, respectively. (x,y) is a coordinate in the subset region W. dx and dy are values of image shift in x-direction and y-direction, respectively. The values of (dx,dy) which minimizes SSD is obtained by the following procedure. (1) The combination of (dx,dy) which minimizes SSD is obtained as the combination of integer pixel unit values; (dx=i, dy=j). (2) It is assumed that distribution of SSD close to its minimum value can be expressed by the following quadratic function. (6) (3) Paraboloid which involves five points, SSD(i,j), SSD(i-1,j), SSD(i+1,j), SSD(i,j-1) and SSD(i,j+1) is defined as shown in Fig. 1. (4) The coefficients ak (k=1,2,3,4,5) is calculated from the following equation.

(7)

(5) (dx,dy) which minimizes SSD is obtained as the coordinate of the apex of the paraboloid. The coordinate of the apex can be obtained as (-a2/a1, -a4/a3). 5. Detection of fatigue cracks by self-reference lock-in thermography A schematic illustration of part of a steel bridge reinforced by trough ribs is shown in Fig. 2. There are two different types of fatigue crack that propagate from a weld root. Crack A initiates at the weld root and propagates through the weld throat. This type of fatigue crack is known as a “weld-bead-penetrant-type” fatigue crack. On the other hand, Crack B initiates at the weld root and propagates

Fig. 2 Fatigue cracks at rib-to-deck joint

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through the deck plate. This type is known as a “through-deck-type” fatigue crack. The through-deck-type fatigue crack is not open at the inspection surface; therefore it is very difficult to detect. 5.1 Detection of weld-bead penetrant type fatigue cracks in actual steel deck Detection of weld-bead penetrant type fatigue cracks in a steel deck of the bridge was conducted by the self-reference lock-in thermography as shown in Fig. 3. Result of magnetic particle testing for the objective fatigue crack is shown in Fig. 4. This crack propagated in the weld-bead and deflected into the trough rib. Distribution of thermoelastic temperature change near the crack tip was sequentially measured under the variable wheel loading caused by the traffics on the bridge. Motion compensation by 2-D SSD parabola fitting method was applied for obtained sequential infrared images. Infrared image captured at the maximum loading is shown in Fig. 5. Three different reference points for generating reference signals for lock-in processing were set at points A, B and C in the figure. Waveform generated at each reference point is shown in Fig. 6. It is found that clear signal was obtained at point A located near the crack tip. The experimental results obtained by the self-reference lock-in processing were shown in Fig. 7. It is found that significant contrast change can be observed at the crack tip due to the singular stress field in the vicinity of the crack tip and the location of the fatigue crack tip is clearly detected by the present technique. It is also found that improved contrast image can be obtained when clear reference signal is employed for self-reference lock-in processing.

Fig. 3 Objective steel deck and experimental setup

Fig. 5 Infrared image and points for generating reference signal

(a) Reference signal at point A

(b) Reference signal at point B

Fig. 4 Objective fatigue crack

Fig. 6 Waveform of reference signal

(c) Reference signal at point C

Fig. 7 Experimental results obtained by the self-reference lock-in processing

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5.2 Remote detection of fatigue crack Remote detection of fatigue cracks in steel deck in actual steel bridge in-service was conducted by the self-reference lock-in thermography with telescope lens as shown in Fig. 8. Infrared measurement was conducted from distant places in distances of 8m and 12m. Thermoelastic temperature change near the crack tip under variable wheel loading by the traffics on the bridge was measured by infrared camera. The experimental results obtained by the self-reference lock-in thermography are shown in Fig. 9. The region for generating the reference signal of lock-in processing was indicated by “Ref.” in the figure. The location of the fatigue crack tip was indicated by arrows in Fig. 9. It is found that significant contrast change can be observed at the crack tip due to the singular stress field in the vicinity of the crack tip. It is found that the location of the fatigue crack tip is clearly detected from remote place by the present technique. It is also found that the crack could be detected in spite of degradation due to infrared attenuation by optical system of telescopic lens.

Fig. 8 Remote measurement with telescopic lens

(a) From 8m (b) From 12m Fig. 9 Result of crack detection in actual steel deck from distant place

5.3 Detection of through-deck type fatigue cracks Detection of through-deck type fatigue cracks by the self-reference lock-in thermography was conducted for a steel bridge specimen. The steel bridge specimen which simulated a part of the actual steel deck of the bridge is shown in Fig. 10. The thickness of the deck plate was 19 mm. Dimensions of cross section of the trough rib were indicated as U320×240×6; i.e., top width, height and thickness were 320 mm, 240 mm and 6 mm, respectively. Fatigue test was conducted for the steel deck specimen under the cyclic bending load of 10 - 110 kN as shown in Fig. 11. Load frequency was set to be 9 Hz. Distribution of thermoelastic temperature change was measured by the infrared thermography and obtained sequential infrared data were processed by self-reference lock-in technique. A fatigue crack was initiated from the back surface of the weld bead and propagated to the deck plate; however it was not open to the inspection surface. The result of the self-reference lock-in processing was shown in Fig. 12. The reference region for generating the reference signal of lock-in processing was set at “Reference signal” in the figure. Characteristic stress concentration in the deck-to-rib weld bead was not shown in the early stage of fatigue testing as shown in Fig. 12(a). On the other hand, significant stress concentration zone in the weld bead were found in Fig. 12(b) and (c), and it moved from cross rib to the left in the figure. It was found that significant stress concentration zone can be observed near the crack front, which enabled us to detect through deck type fatigue cracks. It was also found that half length of the fatigue crack can be estimated from the distance between the center of stress concentration zone and the cross rib. Comparison of crack length measurement between the self-reference lock-in thermography and the ultrasonic inspection is shown in Table 1. It was found that crack length can be accurately determined by the present technique.

Fig. 10 Schematics of steel deck plate specimen

Fig. 11 Testing apparatus and steel deck specimen

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(a) 4x104 cycles (b) 1x106 cycles (c) 2x106 cycles Fig.12 Results of self lock-in thermography for crack detection in crack propagation test Table 1 Comparison of crack measurement between self-reference lock-in thermography and ultrasonic inspection Number of Load cycles 500,000 1,000,000 2,000,000

Distance from cross rib to stress concentration zone obtained by the self-reference lock-in thermography Difficult to determine 35 mm 40 mm

Distance from cross rib to semielliptical crack tip obtained by ultrasonic inspection 17 mm 34 mm 40 mm

6. Stress intensity factor evaluation based on on-site thermoelastic stress measurement From the quantitative analyses of the obtained infrared waveform, thermoelastic temperature data can be obtained with a good signal/noise ratio under the relatively high wheel loading applied by heavy vehicles. Thermoelastic temperature data can be employed not only for crack detection by the self-reference technique but also for near-tip stress analysis under in-service loading conditions. This is highly beneficial for conducting remaining-strength or remaining-life assessments based on fracture mechanics. 6.1 Stress Intensity Factor Analysis As the change in the sum of the principal stresses Δσx+Δσy can be measured by thermoelastic stress measurement, the corresponding changes in the stress intensity, ΔKI, ΔKII, can be assessed by the following procedure. The relationships between the stresses near the crack tip and the K-values can be expressed by Eq. (8) for mode I cracks and by Eq. (9) for mode II cracks using the coordinate system shown in Fig. 13.

(8)

(9) Fig. 13 Coordinate system near the crack tip For mixed-mode cracks, the equation (10)

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can be derived from Eqs. (8) and (9). Changes in the K-value, ΔKI and ΔKII, can be obtained from the measurement of Δσx+Δσy for two different values of θ as shown in the following equations. (11) (12) To determine appropriate values for ΔK, the stress extrapolation method given by the following equations can be used. (13) (14)

Stress intensity factor evaluation based on thermoelastic stress measurement was performed for a weld-bead-penetrant-type fatigue crack. The thermoelastic temperature change near the fatigue crack tip under wheel loading by traffic on the bridge was measured by infrared camera. Temperature values were obtained from a calibration relation between infrared values and temperatures obtained by a blackbody furnace. Stress values were calculated from the thermoelastic coefficient, which can be calculated from handbook values for physical quantities of mild steel. After motion compensation of the infrared images affected by the vibration of the bridge under vehicle loading by the SSD parabola fitting method, stress distribution was evaluated. Figure 14 shows the waveform of the change in the sum of the principal stresses during the passing of a vehicle, which was obtained at point A located near the crack tip and point B located at a distance from the crack tip. It was found that a significant stress change appeared near the crack tip under variable wheel loading by the vehicle. The distributions of the change in the sum of the principal stresses in the inspection area at 0.47 sec and 0.66 sec are shown in Fig. 15. It was found from the figures that the singular stress field in the vicinity of the crack tip is clearly obtained. From these stress distributions, the values of KI and KII were determined by the stress extrapolation technique. The time series variation of the K-value was calculated from the measured stress distribution during the passage of a vehicle. The obtained results are shown in Fig. 16. In the thermoelastic stress measurement, the measured stress values mean difference from the fundamental stress values including the mean stress or residual stress. Therefore the K-value is given by the difference from the fundamental value K0. It was found from the figure that stress intensity factors can be evaluated with good accuracy by the proposed technique and that changes in KII values are dominant in this case. Such results for the near-tip stress intensity provide direct information on the driving force of fatigue crack propagation. This information can be used in assessing of the urgency of repair or the evaluation of the remaining life based on the crack propagation rate.

Change in the sum of the principal stress Δσ (MPa)

6.2 Measurement of Fatigue Cracks in Actual Steel Bridge 150 A (107,134)

100

B (240,124)

50 0 ‐50 ‐100 ‐150 0.0 

0.2 

0.4 

0.6 

0.8 

1.0 

Time (sec)

Fig. 14 Waveform of change in the sum of the principal stresses

(a) at 0.47 sec

(b) at 0.66 sec Fig. 15 Distributions of change in the sum of the principal stresses

Difference of stress intensity factor from fundamental value K0 Diffenrent of stress intensity factor KI-KI0, KII-KII0

106 15 10 5 0 ‐5 ‐10 ‐15 0

0.2

0.4

0.6

0.8

1

Time (sec)

Fig. 16 Time series variation of K-value 7. Conclusions The practicability of a nondestructive evaluation technique based on the thermoelastic stress analysis of fatigue cracks in aging steel bridges was examined. Crack detection by the self-reference lock-in thermography was conducted for fatigue cracks in steel bridges. It was found that self-reference lock-in thermography was practicable for the detection of different types of fatigue cracks, such as weld-bead-penetrant-type cracks and through-deck-type cracks. Measurement of the stress intensity factor was performed for weld-bead-penetrant-type fatigue cracks in the steel deck of a bridge in service. The stress intensity factors KI and KII were evaluated from the measured stress distribution under variable wheel loading by traffic. It was found that these fracture mechanics parameters can be evaluated with good accuracy by the proposed technique. The evaluated values can be used in assessing of the urgency of repair or the evaluation of the remaining strength based on the crack propagation rate. Acknowledgements This research was partly supported by a Grant-in Aid for Scientific Research from Japan Society for the Promotion of Science and a grant from the National Institute for Land Infrastructure Management. References (1) Barton, J. M. D., and Stanley, P., Development and applications of thermoelastic stress analysis, The Journal of Strain Analysis for Engineering Design, Vol. 33, No.2, pp.93-104, 1998. (2) Pitarresi, G. and Patterson, E. A., A review of the general theory of thermoelastic stress analysis, The Journal of Strain Analysis for Engineering Design, Vol. 38, No.5, pp.405-417, 2003. (3) Sakagami, T., Nishimura, T., Kubo, S., et al., Development of a self-reference lock-in thermography for remote nondestructive testing of fatigue crack: 1st report, fundamental study using welded steel samples, Trans. JSME, Vol.72-A, No.724, pp.1860-1867, 2006. (4) Izumi, Y., Sakagami, T., Kubo, S., et al., Nondestructive evaluation of fatigue cracks in steel bridges by infrared thermography, Proc. of ASCE 2008 International Orthotropic Bridge Conference, pp.502-513 (CD-ROM), 2008. (5) Sakagami, T., Izumi, Y., Kubo, S., Application of infrared thermography to structural integrity evaluation of steel bridges, Journal of Modern Optics, Volume 57, Issue 18, pp. 1738-1746, 2010. (6) Izumi, Y., Sakagami, T., Kubo, S., Stress intensity factor measurements for fatigue cracks in steel bridges based on themoelastic stress analysis, Proc. of 12th International Conference on Fracture, pp.1-8, (CD-ROM, Paper No.T30-003), 2009. (7) Sakagami, T., Matsumoto, T., Kubo, S., Nondestructive testing by super-resolution infrared thermography, SPIE Proceedings Vol.7299, pp.72990V-1 - 72990V8, 2009.

Investigation of residual stress around cold expanded holes using thermoelastic stress analysis

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A. F. Robinson1, Prof. J. M. Dulieu-Barton1, Dr. S. Quinn1, Dr. R. L. Burguete2 University of Southampton, School of Engineering Sciences, Highfield, Southampton, SO17 1BJ, UK 2 Airbus Operations Ltd., New Filton House, Filton, Bristol, BS99 7AR, UK

ABSTRACT Thermoelastic stress analysis (TSA) is a well established tool for non-destructive full-field experimental stress analysis. In TSA the change in the sum of the principal stresses is derived and it is generally accepted that the TSA relationship does not allow the evaluation of residual stress, which is essentially a mean stress. However, modifications to the linear form of the thermoelastic equation that incorporate the mean stress have enabled estimations of residual stresses. It has also been shown that the application of plastic strain modifies the thermoelastic constant, K, in some materials, causing a change in thermoelastic response, which can also be related to the residual stress. The changes in response due to plastic strain and mean stress are of the order of a few mK, and are significantly less than those expected to be resolved in standard TSA. Recent developments in infra-red detector technology have enabled these small variations in the thermoelastic response to be identified, leading to renewed interest in the use of TSA for residual stress analysis. The residual stress distribution around cold expanded holes is relatively well defined, and hence provides an opportunity to examine any changes in thermoelastic response caused by residual stress. In the present paper, the thermoelastic response around cold expanded holes in aluminium plates is investigated, and the feasibility of applying a TSA based approach for residual stress analysis to components containing realistic residual stress levels is assessed. Key Words: Thermoelastic stress analysis, Residual stress, Cold expanded hole, Non-destructive evaluation INTRODUCTION Residual stresses may be introduced into a component throughout the entire manufacturing process, so it is highly unlikely that an in-service component is entirely free of residual stresses. Since residual stresses are an almost unavoidable biproduct of manufacture, it is important to understand how residual stresses are distributed in a component; at present, there are several techniques available for measuring residual stresses. Destructive methods are not always practical for an inservice industrial environment, while the non-destructive methods are typically expensive and time consuming. Therefore, demand for a cheaper and quicker non-destructive, non-contact, full-field residual stress evaluation technique is increasing. Thermoelastic stress analysis (TSA) [1] has been identified as a possible solution for a robust and portable means of nondestructive residual stress assessment. TSA is a well established non-contacting analysis technique that provides full-field stress data over the surface of a cyclically loaded component. It is based on the small temperature changes that occur when a material is subject to a change in elastic strain, generally referred to as the ‘thermoelastic effect’. When a material is subjected to a cyclic load, the induced strain produces a cyclic variation in temperature. The temperature change ( T ) can be related to the change in the ‘first stress invariant’,  ( 1   2 ) , or the sum of the principal stresses [1]. An infra-red detector is used to measure the small temperature change, which can then be related to the stress using the following equation: T  K T0 ( 1   2 )

(1)

where T0 is the absolute temperature and K is the thermoelastic constant, K = α / (ρCp), where α, ρ, Cp are the material constants of the coefficient of thermal expansion, mass density and the specific heat at constant pressure, of the material respectively.

T. Proulx, Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_14, © The Society for Experimental Mechanics, Inc. 2011

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As residual stress is essentially a mean stress, it is accepted that the linear form of the TSA relationship given in Eqn. 1 does not allow its evaluation. However, there are situations where this linear relationship is not valid so that a means of establishing the residual stresses from the thermoelastic response has been developed. The changes in the thermoelastic response resulting from the inclusion of residual stress produce temperature change differences of a few mK, which are significantly less than those expected to be resolved in standard TSA. At present, three approaches have been investigated as potential candidates for residual stress measurement using the thermoelastic response [2]. Two are based on the mean stress effect and the revised higher order thermoelastic effect [3]. One utilises the thermoelastic response at the second harmonic of the loading frequency [4], and the other directly relates the change in the thermoelastic response to the principal stresses [5]. The major limitation of these two approaches is that they are not suitable for steel components since the temperature dependence of the elastic properties of steel are negligible at room temperature. The third approach [6] is based on Eqn. 1 and the change in the thermoelastic constant, K, resulting from plastic deformation during manufacture or assembly. In the third approach the main disadvantage is that plastic deformation must have taken place, but it has the advantage that it may be valid for a larger range of materials, not just those with temperature dependent elastic properties. A significant disadvantage common to all three approaches is that any change in the thermoelastic response resulting from either the mean stress, m, or from the modification of K will be small. In actual components the changes in the response are around the noise threshold of the detectors. Success in detecting these changes has been achieved by applying very large residual stress or plastic strain, by using materials that are very sensitive to the mean stress effect, or from investigation of specifically designed specimens. Recently the sensitivity of infra-red detectors has improved to the extent where it may be possible to accurately measure changes representative of those in actual components, hence leading to a renewed interest using TSA for residual stress analysis. Since the variations in thermoelastic response are small, it is important to minimise sources of signal attenuation and understand the possible sources of error within the measurement. The major factors known to influence the change in thermoelastic response (and subsequently K) are the high emissivity coating, background temperature, applied stress and the infra-red detector settings; these effects must always be considered in parallel to any evaluations of the significance of changes in response due to mean stress or from the modification to K. Typical manufacturing residual stresses may be detrimental to the components performance; however it is possible to enhance a component by introducing beneficial residual stresses. An example widely used in the aerospace industry is the cold expansion of holes to prevent fatigue crack growth and initiation. The residual stress distribution around cold expanded holes is relatively well defined, providing an opportunity to investigate the variations in thermoelastic response due to residual stress, and assess the feasibility of applying a TSA based approach for residual stress assessment to actual components containing realistic levels of plastic strain. The present paper focuses on examining small changes in thermoelastic response obtained from the region around cold expanded holes where residual stress is present. TSA is conducted on aluminium plate (AA2024-T351 and AA7085-T7651) containing holes with different levels of cold expansion, which are similar to the levels found in service. Direct point by point comparison of the thermoelastic response is made between specimens to attempt to identify the areas affected by the cold expansion process. The effect of plastic strain on the thermoelastic constant for each alloy is investigated using tensile dogbone-type specimens loaded uniaxially; similarly, the effects of material directionality (due to cold rolling) and strain hardening on the thermoelastic response is also examined. BACKGROUND AND APPROACH The cold expansion process is an established means of generating a beneficial compressive residual stress distribution around a hole, preventing fatigue crack growth and initiation; it has also been shown to inhibit growth in existing cracks. The cold expansion process involves plastically deforming the material directly adjacent to the hole to develop compressive residual stress [7]. The split-sleeve expansion method is most commonly used, and involves pulling an oversized tapered mandrel through an internally lubricated split-sleeve that has been positioned inside the hole, as shown in Fig. 1. The combined thickness of the sleeve and mandrel relative to the size of the hole controls the amount of expansion. The hole is expanded to cause plastic deformation, and as the mandrel is removed the material undergoes partial, but not full elastic recovery. The material further from the hole, which is only elastically deformed springs back from the expanded state and forces the plastically deformed material, closer to the hole, into compression. Consequently a large compressive residual stress is formed close to the hole, and reduces with distance from the hole. Fig. 2 shows the typical residual stress field following cold expansion; the maximum compressive stress is typically similar in magnitude to the yield stress of the material. As the mandrel is removed from the hole, the material on the entry side of the hole begins to relax, however the material on the exit face does not, since it is still constrained by the presence of the mandrel. As a result, the final residual stress distribution is three-dimensional with a different magnitude of residual stress on the entry and exit sides of the hole.

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Fig. 1 Schematic of the split sleeve cold expansion method

Fig. 2 Typical tangential residual stress profile around a cold expanded hole

There are many features that influence the residual stress field, and although the shape is well defined, it can be difficult to quantify the final distribution; some of the influencing factors include: method of expansion, level of expansion, material, plate thickness, and the holes’ position relative to other holes. In this study, each specimen consists of a 10 mm thick aluminium plate containing a single central hole that has been cold expanded using the split-sleeve method. Holes containing three different levels of cold expansion are inspected, and both the exit and entry sides of the plate are examined due to the three-dimensional residual stress profile. It is estimated that the region of compressive residual stress extends approximately 5 mm from the hole edge, and the reverse yielding zone approximately 1 – 2 mm from the hole edge. The variations in thermoelastic response are investigated, and the possible causes discussed, which include the mean stress effect and the effects of plastic strain on the thermoelastic constant. The mean stress effect was first observed by Belgen [8] and later confirmed by Machin et al [9]; the revised higher order theory of the thermoelastic effect was proposed by Wong et al [3], which accounted for the temperature dependence of elastic properties, and it was shown that the temperature response is dependent on the stress state, i.e. the mean stress, as well as the change in stress, i.e. the stress amplitude. Since the presence of residual stress within a component would form a contribution to the mean stress, there would be a small difference in the thermoelastic response from that measured in the same component with zero residual stress, subjected to the same loading conditions. In the simplest case of a uniaxial stress field, if (a) the change in thermoelastic response between a specimen containing unknown residual stress and a specimen with zero residual stress was measured, (b) the effect of changing the mean stress on the thermoelastic response was known for the given material, and (c) known adiabatic loading conditions were prescribed, there is potential for the amount of residual stress within the specimen to be estimated [4]. However, this process of residual stress assessment becomes considerably more difficult in specimens containing non-uniform stress fields, i.e. that seen around a cold expanded hole, or in situations of multi-axial loading. While the mean stress effect is very small, and often considered negligible at room temperature in many engineering materials [10], is has been shown to be measurable in materials with temperature dependent elastic properties, including titanium (Ti-6Al-4V), Inconel 718, and some grades of aluminium [11]. Thus, it is possible that any variation in thermoelastic response between holes that have experienced different amounts of expansion could be a result of a change in the effective mean stress, due to the presence of residual stress. The other potential cause of variations in thermoelastic response around cold expanded holes is the effect of plastic deformation on the thermoelastic constant. It has been shown [6], that the introduction of plastic deformation caused a modification of the thermoelastic constant in some metals due to a change of the material properties contained in K. Rosenholtz et al [12], and Rosenfield et al [13] have both demonstrated that in steel and aluminium, an application of plastic strain will cause a change in the material property, α, the coefficient of linear thermal expansion. Rosenfield et al [13] also noted that this change in α increases significantly when subjected to compressive strains, and less with tensile plastic straining. It has also been suggested that the change in α is affected by the strain hardening capability of the material [6], i.e. the change in K for a material that does strain harden will be different than for a material that does not. Since the area immediately adjacent to the hole has undergone significant plastic deformation, is it possible that this effect will contribute to a change in thermoelastic response. Plates were made from two aluminium alloys (AA2024-T351 or AA7085-T7651) that have different strain hardening characteristics; enabling any contribution to variations in thermoelastic response due to the effect of plastic deformation on K to be established.

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To supplement the investigation of cold expanded holes, it is necessary to perform calibration tests to characterise the materials under investigation; this involves loading dog-bone type specimens in uniaxial tension to: (a) determine the effect of plastic deformation on the thermoelastic constant, K, for each alloy, (b) establish the different strain hardening characteristics and the potential affect on the thermoelastic response around the hole, and (c) examine any material directionality that is present due to cold rolling and investigate what affect this has on the thermoelastic response. In a dog-bone type specimen loaded in uniaxial tension, the stress can be calculated in a straightforward manner, given that the loading conditions are defined, thus allowing the thermoelastic constant, K, to be computed from the applied stress and measured ∆T and T values using Eqn. 1. This methodology has been used for determining the effect of plastic strain on the thermoelastic constant for the following three reasons: (i) If this type of specimen is loaded beyond the material’s yield point and then unloaded, it will result in a residual strain; however, there will be no residual stress as the stress can be fully relaxed by the elastic unloading. Without a residual stress that would result in an increase in σm when loaded, any change in the thermoelastic response would be due to a change in one of the material properties, α, ρ or Cp, and not due to the mean stress effect. (ii) In a dynamically loaded tensile specimen, non-adiabatic conditions cannot occur because there is no stress gradient, and therefore there is no heat transfer within the specimen. (iii) Since the stress in a specimen loaded uniaxially can be calculated, the change in thermoelastic constant due to plastic deformation can be obtained by calculating thermoelastic constant, KP, for a specimen containing a known amount of plastic strain, and comparing it to K0, which is calculated from a reference specimen containing zero plastic strain. Similarly, by comparing tensile specimens loaded in uniaxial tension, any variations in the measured thermoelastic response due to strain hardening, material directionality or the mean stress effect will become apparent, assuming prescribed loading conditions are used and the background temperature does not change significantly. Having ascertained the mechanical and thermoelastic behaviour from tensile specimens, a catalogue of information is available that characterises the materials under consideration. Subsequently, the variations in thermoelastic response around the cold expanded holes can be exhaustively investigated to provide a complete view of the material behaviour. Hence revealing the potential for TSA to be used to detect or measure residual stresses in real components, and provide a focus for further work on the topic. EXPERIMENTAL PROCEDURE The plate specimens used in this work have been chosen to be representative of in-service applications, whilst at the same time realising the limitations associated with the load capacity of test machines and loading jigs (shown in Fig. 3 and Table 1). Aluminium plate specimens were manufactured with dimensions of 300 mm by 150 mm by 10 mm containing holes of nominal diameter 5/8” (15.875 mm). These dimensions were chosen for two reasons: firstly, the distance from the edge of the plate relative to the diameter of the hole (e/D ratio) is sufficiently large (i.e. greater than 3) for edge effects to not cause any unusual variation of the residual stress distribution during the cold expansion process. Secondly, it was shown in [14] that infinite plate conditions (i.e. no edge effects during cyclic loading for TSA) can be assumed if the distance from the hole to the plate edges (d1 and d2) are greater than 3.3D and 6.7D in the x and y directions respectively. It has been noted on several occasions that the level of strain hardening a material experiences may affect the change in thermoelastic constant that can be expected from plastic deformation. Subsequently, two grades of aluminium are investigated; one which experiences a high level of strain hardening (2024-T351) and one which does not strain harden significantly (7085-T7651). Finally, since the optimum level of cold expansion is 4% using the split sleeve method, four plates of each material have been manufactured containing holes with 0%, 2% and 4% cold expansion. Typically holes are reamed to a final dimension after expansion, for this reason one plate contains a 4% cold expanded hole that has not been reamed. It should also be noted that the mandrel split was aligned with the top of the hole, and that prior to cold expansion, the plates had been coldrolled (in the 300 mm direction). In total, 8 plates were manufactured from each aluminium alloy; 4 containing cold expanded holes, and 4 additional plates from which the dog-bone type specimens were manufactured. Specimens had a nominal cross sectional area of 15 mm by 5 mm and were cut at 0º, 45º and 90º to the rolling direction; these were used to obtain the material properties, to assess the effect of material directionality, to obtain the baseline thermoelastic constant, and to determine the effect of plastic deformation on the thermoelastic constant for the two grades of aluminium. A Cedip Silver 480M infra-red detector was used to obtain thermoelastic data. A thin layer of RS matt black paint was applied to the surface of all specimens to provide a high emissivity surface in accordance with established preparation guidelines [15]. An Instron 8500 servo-hydraulic test machine was used to statically apply load to induce plastic deformation in the tensile specimens, as well as cyclically load both the tensile and plate specimens for TSA. To assess the effect of plastic deformation, dog-bone type specimens were strained quasi-statically in three loading steps: (i) 0.5 mm min-1 extension until yield, (ii) 0.5 mm min-1 until an additional 2% strain (or 4%), (iii) –0.5 mm min-1 until initial

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load (i.e. the pre-load when gripping the specimen) prior to applying the cyclic load for TSA. Linear strain gauges were applied to one side and an Imetrum video-extensometer was used on the opposite side of the specimen to measure the strain during the plastic deformation procedure. A calibration specimen was used to obtain the thermoelastic constant, K0, for 0% strain for each material. Specimens were cyclically loaded at a mean load of 8 kN (107 MPa), with a load amplitude of 4 kN (54 MPa) to obtain ∆T and T, allowing KP for each level of plastic strain to be calculated. Tests were conducted at 5 Hz, 7.5 Hz and 10 Hz to assess the effect of any non-adiabatic behaviour caused by the paint coating. Plate specimens were cyclically loaded (in the rolling direction) at a mean load of 30 kN, with a load amplitude of 25 kN. The loading jig shown in Fig. 3 was used to facilitate loading of the specimens and to minimise bending during each test. Thermoelastic data was recorded from both the entry and exit faces of the plate, at two viewing distances: (i) at a stand off distance of 385 mm, showing the hole as well as areas of uniform stress away from the hole, and (ii) at a stand off distance of 200 mm, showing the areas immediately adjacent to the hole. Once again to assess any possible non adiabatic behaviour resulting from the paint coating, tests were conducted at 5 Hz, 7.5 Hz and 10 Hz.

A B C D d1 d2 h

Table. 1 Plate Dimensions Plate length 300 mm Plate width 150 mm Attachment holes 6 mm Hole diameter 15.875 mm (5/8”) Edge distance (x) 9.4 D Edge distance (y) 4.7 D Plate thickness 10 mm

Fig. 3 Plate specimen dimensions and loading mechanism

RESULTS AND DISCUSSION Table 2 shows the relevant mechanical and thermoelastic properties of the two aluminium alloys obtained from tensile specimens. Five specimens of each alloy were used to obtain the data, and the values shown are the average over all five specimens; numbers in brackets represent the largest variation seen within each set of tests. AA2024 strain hardened by 32% and AA7085 by approximately 9%, confirming the different strain hardening characteristics of the two alloys (where the ability to strain harden is defined by the percentage increase of the ultimate tensile strength in comparison to the yield strength).

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Table. 2 Mechanical and thermoelastic properties of AA2024-T351 and AA7085-T7651 Yield Stress Ultimate Tensile % Strain Thermoelastic Material [MPa] Stress [MPa] Hardening Constant, K [Pa-1] AA2024-T351 352 (± 5) 464 (± 3) 31.8 (± 1.8) 9.8 x 10-12 (± 0.05) AA7085-T7651 494 (± 6) 538 (± 9) 8.9 (± 1.2) 9.4 x 10-12 (± 0.06) Fig. 4 and Fig. 5 show the variation of KP/K0 for AA2024 and AA7085 specimens aligned with the 0°, 90° and 45° rolling directions. It can be seen that the change in KP/K0 is different for each alignment direction, and that the effect of plastic strain is greatest for the 45° specimen; the smallest change was measured for the 0° direction. While the temperature changes associated with this effect are very small (approximately 5 – 15 mK for the data shown), it can be seen that KP/K0 increases with increasing levels of plastic strain for each material. For specimens aligned with the 0° direction there was no visible effect of strain hardening on the change in KP/K0 which is in contrast to previous work [6], however the material directionality was not previously considered. It was seen that for the 90° and 45° specimens, the change in KP/K0 is larger for the AA2024 material, which does exhibit more strain hardening.

Fig. 4 Variation of KP/K0 for different specimen orientations for AA2024 at 10 Hz

Fig. 5 Variation of KP/K0 for different specimen orientations for AA7085 at 10 Hz

Some analysis of the microstructure is required to understand the physical processes causing the change in KP/K0 due to the introduction of plastic strain and how material directionality influences this change. Finally, due to the differences in KP/K0 for different alignment directions, it is now apparent that knowledge of material directionality would be required to enable an evaluation of plastic strain based on a change in thermoelastic constant from a reference specimen. While the plates containing cold expanded holes have been cold rolled and the direction is known, the stress distribution is more complex such that a simple calculation of KP is not necessarily possible; however, the fact that plastic deformation has occurred is likely to change the measured thermoelastic response, even though for cold expansion the relaxation is constrained, and thus residual stress is present. The 0% cold expanded hole should contain no significant residual stress, since it has not been cold expanded, and any residual stress present due to cold rolling would be somewhat relieved during the subsequent hole drilling and reaming. In comparison, the 2% and 4% cold expanded holes would contain large compressive residual stress very close to the hole (within a few mm) reducing to small tensile residual stress further from the hole. Therefore any differences in thermoelastic response due to residual stress should be identifiable from comparison of the thermoelastic data recorded from the specimens. Every effort was made to ensure the infra-red detector was positioned such that the location of the hole remained the same in each test, thus allowing a pixel by pixel comparison of the thermoelastic data to be performed for each plate. However, due to the high dynamic loading, motion was noticeable in the TSA video data and this was particularly apparent at the specimen edges. Motion compensation was performed on the TSA data to remove this motion and the resulting errors than can be associated with it. It was achieved by identifying two significant and contrasting features in the first frame of the recorded data, which are then tracked through subsequent frames allowing them to be rotated, distorted or translated

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such that any motion appears stationary. Fig. 6 shows TSA data observed from around 0%, 2% and 4% cold expanded holes on the entry side of AA7085 plates; the stress distribution that would normally be expected around a hole is clearly identifiable.

Fig. 6 Thermoelastic response around cold expanded holes in AA7085-T7651 plates, Entry side loaded at 10 Hz; (Top) 4%, (Middle) 2%, (Bottom) 0%. The circular area (1) was applied to each image to check the hole was in the same position and that motion compensation was successful; during each test, the hole itself was filled with Blue-Tac, and therefore shows little or no thermoelastic response. Two line plots have been taken from the data; the first (2) shows the thermoelastic response directly across the hole, the second (3) shows the response further from the hole, in a region of more uniform stress. It can be seen that the background temperature change is approximately 0.10ºC and the maximum temperature occurring at the hole is approximately 0.25ºC. Using Eqn. 1 and the thermoelastic constant for AA7085 from Table 2, the applied stresses can be approximated as 35 MPa in the plate and 90 MPa at the hole. In an ideal situation, increasing the applied load would increase the stress, and thus increase the differences in the temperature changes that are of interest, however, further increasing the applied load could result in some areas containing tensile residual stress entering plasticity, as well as approaching the load capacity of the test machine and other deleterious effects arising from bending or distortion of the loading rig. To examine differences in thermoelastic response, the ∆T data around the 0% cold expanded holes was subtracted from the data around the corresponding 4% cold expanded holes on a pixel by pixel basis; this formed new data sets revealing areas with a different thermoelastic response (Fig. 7); note, the data from within the holes has been removed, along with the immediate the edge data that may be erroneous. Fig. 7(a) shows clear differences in the thermoelastic response from around the 4% and 0% cold expanded holes can be seen in the new data sets; however, the data is relatively noisy. The marks seen in the top left and top right corners of each image correspond to paint that was scratched from the surface to provide a reference point for motion compensation. In an attempt to remove the noise, a line of data from an area of uniform stress in each 0% plate was subtracted from the same area on the corresponding 4% plate; this provided an estimation of the background noise variation caused by the subtraction process. A threshold filter was then applied to each new data set to remove any data that fell within the calculated noise limits, providing an image where departures in the thermoelastic response between identically loaded plates of the same material

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can be clearly identified. Fig. 7(b) shows the ∆T4% - ∆T0% data sets for the entry and exit sides of the AA2024 and AA7085 plates with the threshold noise filter applied. The variations in ∆T are small, but it is encouraging that measurable differences exist that are larger than the proposed noise levels, and far enough from the hole edge to be a result of erroneous measurement or edge effects.

(a)

(b)

Fig. 7 Difference in thermoelastic response around the 4% and 0% cold expanded holes, (∆T4 - ∆T0) (a) Without threshold noise filter applied, (b) With threshold noise filter applied. (Top left) AA7085 entry side, (Bottom left) AA7085 exit side, (Top right) AA2045 entry side, (Bottom right) AA2024 exit side. Since the loading conditions are the same for each plate, the difference in the thermoelastic response between the specimens must be a result of the residual stress around the hole caused by the cold expansion process. The expected residual stress profile (Fig. 2) would take the form of a ring of compressive residual stress close to the hole, with a very small region of tensile residual stress directly above the hole corresponding to the location of the split in the sleeve. The data for the AA7085 entry side and the AA2024 exit side show variations in ∆T all around the hole which correlates with the expected areas of residual stress; however a different distribution is seen for the AA7085 exit side and the AA2024 entry side. At this point, it is not clear what the causes of the differences are, and since there is no consistency between the shape and magnitude of the variation, it is difficult to ascertain if this is a result of the mean stress effect, plastic deformation causing a change in K or a combination of the two. CONCLUSIONS The residual stress distribution around cold expanded holes presented an opportunity to investigate changes in thermoelastic response that occur as a result of residual stress, and assess the feasibility of using TSA as a potential residual stress assessment tool. It was known that the presence of residual stress may influence the effective mean stress, possibly providing an additional component of the thermoelastic signal; it was also shown that the occurrence of plastic strain can modify the thermoelastic constant K, for some materials, again possibly resulting in a change in thermoelastic response. The loading conditions were rigorously maintained during experimental work and the position and settings of the infra-red detector remained constant. Differences in the thermoelastic response were observed from around the cold expanded holes in regions that would be expected to contain residual stress, and a basic noise filter applied to identify areas where significant departures in thermoelastic response were measured. Experimental work using tensile specimens (utilised to eliminate the mean stress effect) revealed that plastic deformation did affect the thermoelastic constant for both materials analysed in this work, confirming that the changes in thermoelastic response around the holes could be attributed to the deformation experienced during cold expansion. However, since the purpose of cold expansion is to create compressive residual stress around the hole, the 4% hole contained residual stress from cold expansion, and had experienced plastic deformation, whereas the 0% hole did not contain residual stress from cold expansion and had not experienced plastic deformation; therefore there is also the possibility that the change in response or part of, was a result of an increase in effective mean stress due to the residual stress. Nevertheless, small but measurable differences in the thermoelastic response from areas that were known to contain different levels of residual

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stress were detected, suggesting that there may be potential for TSA to be used for assessing residual stress at some point. Currently, the extent of its practical application is limited and significant further work is required to identify the cause of the change in thermoelastic response, and to develop procedures that could possibly relate this change to residual stress. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Dulieu-Barton, J. M. and Stanley, P., Development and applications of thermoelastic stress analysis, Journal of Strain Analysis for Engineering Design, 33, 93-104, (1998). Robinson, A. F., Dulieu-Barton, J. M., Quinn, S. and Burguete, R. L., A Review of Residual Stress Analysis using Thermoelastic Techniques, Proceedings of 7th International Conference on Modern Practices in Stress and Vibration Analysis, Journal of Physics: Conference Series 181 012029, (2009). doi:10.1088/1742-6596/181/1/012029. Wong, A. K., Sparrow, J. G. and Dunn, S., On the revised theory of the thermoelastic effect, Journal of Physics and Chemistry of Solids, 49, 395-400, (1988). Wong, A. K., Dunn, S. A. and Sparrow, J. G., Residual stress measurement by means of the thermoelastic effect, Nature, 332, 613-615, (1988). Patterson, E. A., The potential for quantifying residual stress using thermoelastic stress analysis, Proceedings of SEM Conference, Springfield, MA, (2007). Quinn, S., Dulieu-Barton, J.M., and Langlands, J.M., Progress in thermoelastic residual stress measurement. Strain, 40, 127-133, (2004). Özdemir, A. and Edwards, L., Measurement of the three-dimensional residual stress distribution around split-sleeve cold-expanded holes, The Journal of Strain Analysis for Engineering Design, 31, 6, 413-421, (1996). Belgen, M. H., Infrared Radiometric Stress Instrumentation Application Range Study, NASA Report CR-1067, 142, (1967). Machin, A. S., Sparrow, J. G. and Stimson, M. G., Mean stress dependence of the thermoelastic constant, Strain, 23, 1, 27-30, (1987). Eaton-Evans, J. M., Dulieu-Barton, J. M., Little, E. G. and Brown I. A., Thermoelastic Studies on Nitinol Stents, Journal of Strain Analysis for Engineering Design, 41, 481-495, (2006). Gyekenyesi, A. L. and Baaklini, G. Y., Thermoelastic stress analysis: The mean stress effect in metallic alloys, NASA, NASA-TM-1999-209376, (1999). Rosenholtz, J. and Smith, D., The effect of compressive stresses on the linear thermal expansion of magnesium and steel, Journal of Applied Physics, 21, 396-399, (1950). Rosenfield, A. R. and Averbach, B. L., Effect of stress on the expansion coefficient, Journal of Applied Physics, 27, 154-156, (1956). Stanley, P. and Day, B., Photoelastic investigation of stresses at an oblique hole in a thick flat plate under uniform uniaxial tension, The Journal of Strain Analysis for Engineering Design, 25, 3, 157-175, (1990). Robinson, A.F., Dulieu-Barton, J. M., Quinn, S. and Burguete, R. L., Paint coating characterisation for Thermoelastic stress analysis of metallic materials, Measurement, Science and Technology, 21, (2010). doi:10.1088/09570233/21/8/085502.

TSA Analysis of Vertically- and Incline-loaded Plates containing Neighboring Holes

1

A. A Khaja and R. E. Rowlands

2

Department of Mechanical Engineering, University of Wisconsin, Madison, WI 53706 1

[email protected]

2

[email protected]

ABSTRACT This paper presents an effective way to determine the individual stresses in arbitrarily-loaded multiply-perforated finite plates whose various size cutouts are randomly distributed. Thermoelastic stress analysis (TSA) is used such that the recorded temperature data are processed utilizing an Airy’s stress function. Advantages of TSA include full-field, non-contacting, nondestructive, no surface preparation other than a paint is needed, unnecessary to differentiate the measured information and has a resolution comparable to that of commercial foil strain gages. The present analysis was inspired by previous studies involving the TSA stress analysis of an incline-loaded clamped plate containing a single cutout [1] and a tensile plate containing multiple holes which were located collinearly to the external load [2]. The individual stresses in vertically- and incline-loaded aluminum plates containing two side-by-side circular holes are determined here. The holes are sufficiently close together that their stress fields interact with each other. The absence of any universal loading fixturing causes the inclineloaded plate to be subjected to some unknown in-plane bending as well as the tension. This unknown loading of the inclined plate hampers stress analyzing the plate numerically, and theoretical solutions to finite geometries are extremely difficult. Notwithstanding the aforementioned statement, TSA results agree well with those from commercial strain gages and ANSYS. Load equilibrium is also satisfied. 1. Introduction Many engineering structures involve multiple perforated finite plates and it is imperative to be able to evaluate the stresses reliably for cases where theoretical or numerical approaches are not available. Although illustrated here for the particular situation of vertically- and incline-loaded plates containing neighboring holes, the general approach described is applicable for stress analyzing arbitrarily-loaded, multiply-perforated or -notched finite plates whose various-size cutouts are randomly distributed. For example, locating auxiliary holes about an original hole can influence the stresses at the latter compared with those in the absence of the auxiliary holes. Unlike theoretical or FEM analyses, the beauty of this technique includes it requires neither accurately knowing the boundary/loading conditions, figure 1(b), nor the constitutive properties, nor differentiating the measured data. In addition to those from strain gages, the present TSA results are compared with those from an approximate FEM prediction. Recognizing that the thickness of the plate exceeds the diameter of the small hole, a 3-D FEA validates the plane-stress assumptions. The two cases of figures 1 actually involve the same plate, the plate being vertically loaded in figure 1(a) but inclined in figure 1(b).

T. Proulx (ed.), Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_15, © The Society for Experimental Mechanics, Inc. 2011

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118

Plate thickness = 0.25” (6.35mm)

(a) (b) o Figure 1: Schematic of ( a) symmetrically-loaded (b) unsymmetrically-loaded (~15 w.r.t. loading) plates The circular holes in the plate of figure 1(a) are horizontal with respect to the vertical direction of loading whereas o in figure 1(b) they are inclined at -15 to the horizontal axis. In each case a two dimensional thermoelastic stress analysis was conducted to determine stress distribution around the neighboring holes, the latter being sufficiently close together that their respective stress fields interact. 2. Relevant Stress Functions Equations (1) and (2) are two specific Airy stress functions for the symmetrically-and unsymmetrically-loaded plate of figures 1. Derivatives of the stress function provide the individual components of stress. These are omitted here due to lack of space. Having derived the individual stress components, the traction-free conditions on the boundary of the hole were imposed analytically. This reduced the number of independent Airy coefficients and consequently simplified the stress functions from the original more complicated form. Reference [1] provides the detail derivation and expressions for individual components of stresses and isopachic stress.

 sym

'  '  c1 ' 3   a0  b0 ln r  c0 r  a1 r   d1 r  cos    r   2



N

 a r

n  2 , 3...

' n

n

 bn' r ( n  2)  cn' r  n  d n' r ( n 2 )

cos(n ) (1)

119

unsym  a0  b0  ln r  c0  r 2  A0    '  c1' c1  3   a1  r   d1  r   sin    a1  r   d 1'  r 3   cos    r r      

N



 an  r n  bn  r  n  2  cn  r  n  d n  r  n  2   sin  n      n  2 ,3 ,4... N



 a'  r n  b'  r  n  2  c'  r  n  d '  r  n  2   cos  n    n n n  n  n  2 ,3 ,4...

(2)

where r is the radius measured from the center of a hole, angle  is measured counter-clockwise from the horizontalx-axis (figures 1) and N is the terminating value of the above series (N can be any positive integer greater than one). An individual coordinate system is employed for analyzing each hole of each plate of figures 1. TSA-wise, the doubly perforated plates of figures 1 are stress analyzed by employing a separate stress function associated with individual coordinate systems having their origins at the center of the respective holes. 3. Experimental Details, Analyses and Results The aluminum (E = 68.95 GPa and ν = 0.33) plate of figures 1 was sprayed with Krylon Ultra-Flat black paint, loaded in a MTS testing machine using hydraulic grips and the data recorded using a DeltaTherm DT1410 camera as shown in figures 2. The symmetrical plate of figure 1(a) was clamped vertically making the plate o symmetrical about x-axis, figure 2(a). The unsymmetrical plate of figure 1(b) was inclined at an angle of 15 with respect to the vertical hydraulic loading grips, figure 2(b). These plates were sinusoidally loaded between 3560 N (800lb) ± 2224 N (500lb) at a frequency of 20 Hz. Figures 3 show an actual TSA image (256x256 = 65,536 data values) for each of the symmetrical and unsymmetrical loadings. Since the plate of figure 1(a) is symmetrical about the line through the holes (x-axis), the recorded temperature data were averaged about that axis. A separate uniaxial tension calibration specimen was used to determine the thermomechanical coefficient, K = 406 U/MPa (2.8 U/psi). Some vertical strain gages were bonded along lines CD and C’D’ of figures 1. The large and the small holes of the symmetrically-loaded plate of figure 1(a), were analyzed individually i.e., when analyzing the large hole, the temperature data near that large hole were used along with the corresponding stress function, equation 1, and similarly for the small hole. A total of m1 = 4,378 (small hole) and m2 = 2,031 (large hole) input values were used to evaluate the corresponding unknown Airy coefficients (equation 1) for the large and the small hole, respectively. Their respective source locations are shown in the figures 4. For the

(a) (b) Figure 2: Specimen loading and TSA recording: (a) symmetrically-loaded (b) unsymmetrically-loaded plate.

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(a) (b) Figure 3: Actual recorded TSA images, S*: (a) symmetrically-loaded (b) unsymmetrically-loaded plate. symmetrically-loaded plate, k1 = 9 was found to be an appropriate number of Airy coefficients for each of the small and large holes. The large and the small holes of the unsymmetrically-loaded plate of figure 1(b) were similarly analyzed individually. In this case a total of m3 = 9,372 (small hole) and m4 = 4,094 (large hole) input values were used to evaluate thermoelastically the corresponding unknown Airy coefficients for the large and the small hole, respectively. Their source locations are illustrated in the figures 5. For unsymmetrically-loaded plate, k2 = 17 was found to be an appropriate number of Airy coefficients for each of the small and large holes. Using the TSA input information, S*, all the unknown Airy coefficients of equations 1 and 2 were evaluated for each hole of the symmetrically- and unsymmetrically-loaded plate and thereby provided the individual components of stress i.e., the expressions for individual components of stress involve the now-known Airy coefficients. TSA results are compared with those from finite element analysis (ANSYS). For the symmetrically-loaded plate of figure 1(a), the symmetrical boundary condition along the horizontal line of symmetry and a uniform end tensile stress of 9.19 MPa (1333.33 psi) were applied. The end loading conditions of the unsymmetrically-loaded plate are not well known, which challenges the modeling of finite element analysis. Nevertheless an approximate finite element model was made for the unsymmetrically-loaded plate in which, in addition to applying a far-field stress of 9.19 MPa (1333.33 psi) at the ends of the plate, the inside ends of the clamped plate were restrained horizontally, compatible with the physical situation that the ends of the plate do not move in the x-direction (horizontally) when loaded.

y

y x

x

(a) Large hole, m1 = 4,378 input values. (b) Small hole, m2 = 2,031 input values. Figure 4: TSA source locations for symmetrically-loaded plate.

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y

y x

x

(a) Large hole, m3 = 9,372 input values (b) Small hole, m4 = 4,094 input values. Figure 5: TSA source locations for unsymmetrically-loaded plate. Tangential stresses, σθθ, normalized with respect to the far field stress, σ0 (= 9.2 MPa (1333.33 psi)), are plotted in figures 6 around the boundary of each of the holes for the symmetrically-loaded plate and similarly for unsymmetrically-loaded plate in figures 7. Figures 8 compare the strains along the line CD (symmetrical-loading) and C’D’ (unsymmetrical-loading) extending from the large hole (figures 1) from TSA, strain gages and finite element analysis (ANSYS).

(a) Large hole (m1 = 4,378 TSA input values) (b) Small hole (m 2 = 2,031 input TSA values) Figure 6: σθθ/σ0 on the edge of the holes of symmetrically-loaded plate from TSA (k1 = 9 coefficients) and ANSYS.

(a) Large hole (m3 = 9,372 TSA input values) (b) Small hole (m4 = 4,094 input TSA values) Figure 7: σθθ/σ0 on the edge of the holes for unsymmetrically-loaded plate from TSA (k2 = 9 coefficients) and ANSYS

Figure 8: Strain εyy along CD of figure 1(a) (left) and εθθ along C’D’ of figure 1(b) (right) obtained from TSA (k1 = 9 and k2 = 17 coefficients; m1 = 4,378 andm4 = 9,372TSA values), strain gages and ANSYS

122 4. Summary and Conclusions A major objective of the present research was to evaluate the individual stress components associated with two neighboring holes which are either perpendicular or inclined to the direction of the loading and whose respective stresses interact and so as to influence the stress concentration factor at the boundary of the original larger hole alone, figures 1. The TSA technique utilized is not confined/restricted to specific geometry/loading conditions, location/shape of holes, specific arrangement of multiple holes, hole spacing, finite/infinite geometries or number of holes. Here, two different cases were conducted: one in which a finite plate is symmetrical about horizontal xaxis, and the other does not enjoy such symmetry.The stress functions are based on the geometry and tractionfree conditions, and are irrespective of far-field loading conditions. TSA results agree with those from strain gages and as predicted by an approximate FEA. The uncertainty of the ANSYS model (results) for the unsymmetrically-loaded plate of figure 1(b) raises the question of the validity or usefulness of comparing TSA and ANSYS results for that case. However, the excellent agreement between TSA and measured strains for the unsymmetrical situation substantiates the reliability of TSA for stress analyzing practical engineering members. Load equilibrium from TSA-based stresses agrees with the applied load (= 4715 N (1000 lbs)) within 6%, further supporting the reliability of the TSA results. Although the thickness of the plate exceeds the diameter of the small hole, a separate 3-D FEA justifies the present plane stress assumptions. Other results illustrate that whether or not compatibility of the respective individual stress components is enforced between the holes has little consequence. While demonstrated here for neighboring circular holes, the general approach is applicable to non-circular holes or notches, or combinations thereof. 5. References [1] Lin, S-J (Katherine), “Two- and Three-Dimensional Hybrid Photomechanical-Numerical Stress Analysis”, PhD Thesis, University of Wisconsin-Madison (2007). [2] A. A. Khaja, “Thermoelastically Determined Stresses around Neighboring Holes in a Finite Plate whose Individual Stress Fields Interact”,MS Thesis, University of Wisconsin-Madison (2010).

Examination of Crack Tip Plasticity Using Thermoelastic Stress Analysis Rachel A Tomlinson1 and Eann A Patterson2 Department of Mechanical Engineering, The University of Sheffield, Sheffield, S1 3JD, ENGLAND; email: [email protected] 2 Michigan State University, CVRC, 2727 Alliance Drive, East Lansing, MI 48910, USA 1

INTRODUCTION Thermoelastic Stress Analysis (TSA) is based on the principle that under adiabatic and reversible conditions, a cyclically loaded structure experiences temperature variations that are proportional to the sum of the principal stresses. These temperature variations may be measured using a sensitive infra-red detector and thus the cyclic stress field on the surface of the structure may be obtained. In order to achieve the adiabatic and reversible conditions, the test specimen must be cyclically loaded at a high enough frequency to prevent heat transfer, which is not only dependent of the loading frequency, but also stress gradients in the specimen and the thermal conductivity of the material. TSA has been found to be an ideal method to study crack tip strain fields, due to its non-contacting, full-field, data collection capabilities, however the adiabatic assumption breaks down close to the crack tip due to the steep stress gradients. Hence the majority of crack tip studies have focussed on the determination of linear elastic parameters, where data are recorded from regions surrounding the crack tip where the adiabatic and reversible assumptions are valid. More recently the non-adiabatic region close to the crack tip has been explored, with the aim of gaining a greater understanding of crack-tip plasticity. Infrared thermography has been used to correlate the energy dissipation at the crack tip to the plastic zone [1,2], whereas others [3,4,5] have explored using the TSA phase data to quantify the size and shape of the crack-tip plastic zone. The latter method allows the elastic and plastic strain field information to be recorded simultaneously, and thus has the potential for near real-time studies of fatigue crack growth. Thus far, the phase method has only been applied at fairly low frequencies on aluminium alloys. Since the method considers non-adiabatic effects at the crack tip and heat transfer is known to be dependent on material and frequency, it was decided to investigate the extent of plasticity at the tip of propagating fatigue cracks in two different aerospace materials at a range of frequencies. CRACK TIP PLASTIC ZONE SIZE USING TSA In order to make thermoelastic measurements, the specimen is subjected to a cyclic load and a sensitive infra-red (IR) camera records the temperature variation from the surface of the specimen in the form of a photon flux. The signal recorded by the camera is correlated with a reference signal from the loading device in order to filter out any background IR radiation and extract the TSA signal from the specimen. Each point in the data field may be represented as a vector where the magnitude is the TSA signal and the angle is the phase shift between the thermoelastic and reference signals. If adiabatic and reversible conditions apply, then for a component under uniaxial load, the phase shift will be uniform over the whole specimen. However if stress gradients are steep, there will be a relative phase difference due to non-adiabatic effects. Figure 1 shows the TSA output around the tip of a crack under fatigue loading. The phase in Figure 1(b) has been shifted so that the far field is zero, so the map is really the phase difference relative to the far field. A line plot through the crack tip of the phase difference is shown in Figure 1(d). This phase difference indicates the non-adiabatic region caused by heat generation associated with plastic work and may be used as an estimate of the plastic zone size as indicated in Figure 1(d) [3].The region with a phase of opposite sign behind the crack tip may be attributed to contact of the crack faces, therefore it has been suggested [4] that the transition between positive and negative phase is an indication of the location of the crack tip. This hypothesis may be tested by comparing the map of phase difference with the visible crack in Figure 1(c). In order to enhance the line of the crack a thermal image of the crack under a static load was recorded and the crack tip can be seen clearly. The visible crack tip location coincides with the crack tip location indicated on the phase map, therefore the method of indentifying the crack tip location using the phase map will be taken throughout the remainder of this research. The dimensions of the plastic zone will be found using the method employed by Patki and Patterson [3] who used a binary filter on the map of phase difference to identify the shape and size of the plastic zone from the negative phase difference ahead of the crack tip. One of the T. Proulx (ed.), Thermomechanics and Infra-Red Imaging, Volume 7, Conference Proceedings of the Society for Experimental, Mechanics Series 9999999, DOI 10.1007/978-1-4614-0207-7_16, © The Society for Experimental Mechanics, Inc. 2011

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124 dimensions which they used was the distance from the crack tip to the extremity of the negative phase zone in the direction of the crack, Rcrack (Figure 1(d)).

Fig. 1 TSA data at a fatigue crack tip, showing how the phase difference may be used to size the plastic zone. PLASTIC ZONE DETERMINATION AT INCREASING FREQUENCY Two aerospace materials were used to explore the measurement of the crack-tip plastic zone using the maps of phase difference: an aluminium alloy 2024 and a commercially pure titanium (CP Ti). The dimensions and material properties are given in Table 1. A compact tension specimen design (BS ISO 12108 (2002))was chosen, but since the material supplied was of different thicknesses, in order to prevent buckling of the thinner titanium, the specimen sizes had to be scaled accordingly (see Figure 2). Table 1 Dimensions and material properties of the CT specimens All dimensions in mm Thickness, B Height, H Notch length, a Width from loading holes, W Yield Stress, MPa Modulus of elasticity, GPa KIc, MPa√m

Aluminium alloy 2024

Commercially pure titanium

2 67 20 72 344 73 32

0.7 24 3 20 366 99 44

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Fig. 2 CT specimens: AL2024 (left); CP titanium (right) Each specimen was sprayed with a thin coat of matt black paint using an airbrush in order to increase the emissivity of the surface. The aluminium specimen was loaded in a 50kN MTS servo-hydraulic test machine at a frequency of 10Hz with a minimum load of 500N and a maximum load of 1500N, and a fatigue crack was grown until it was 32.75 mm long. The load was then reduced to a mean of 1000N and an amplitude of 100N in order to prevent the crack from growing during the following test. The specimen was loaded at frequencies of 5, 10, 15, 20, 25 and 30Hz, and at each frequency at set of TSA data was captured. Each phase map was shifted so that that far field phase was zero and the plastic zone size was determined using the method proposed by Patki and Patterson [3]. The test was repeated. A similar procedure was followed for the titanium specimen. The titanium specimen was loaded at a frequency of 20Hz with a minimum load of 150N and a maximum load of 450N, and a fatigue crack was grown until the crack length was 7.25 mm. The load was then reduced to a mean of 464 N and an amplitude of 46N. These loads were selected so that (K/KIc)Ti = (K/KIc)Al = 0.1. The frequency experiment was then repeated twice at this load.

Fig. 3 Line plot of the phase difference through the crack tip of the AL2024 CT specimen at increasing frequency

126 RESULTS AND DISCUSSION Figure 3 shows a line plot at the same location through the map of phase difference for the AL2024 specimen at each frequency. Figure 4 shows the plastic zone size, Rcrack (see Figure 1(d)), for both materials. On a second axis the thermal diffusion length (TDL) for each material is plotted as a solid line. From Figures 3 and 4 it may be deduced that although the actual plastic zone sizes did not change during these experiments, the measured plastic zone size is dependent on frequency. The measured plastic zone size has the same rate of decrease with frequency as the thermal diffusion length. By increasing the frequency of the test, thermal diffusion length is decreased, thus reducing the heat conduction effects. At a high enough frequency, the rate of change of thermal diffusion length with frequency is negligible, and thus above this frequency, quantitative measurements of the plastic zone may be made. Because heat conduction is material dependant, this limiting frequency will be different for different materials, as can be seen in Figure 4; the limiting frequency is about 30Hz for the AL2024 and 20 Hz for the titanium.

Fig. 4 Measured plastic zone size (left axis) and thermal diffusion length (right axis) for AL2024 and CT Ti at increasing frequency FURTHER EXPLORATION OF THE PLASTIC ZONE Further experiments were performed to investigate how the plastic zone determined by TSA relates to a theoretical model of crack tip plasticity. Fatigue cracks were grown in additional specimens and TSA data recorded at intervals of crack growth. Data maps and line plots through the crack tip for phase difference and the uncalibrated TSA signal are shown in Figure 5 for the AL2024 and Figure 6 for CP Ti. Data for the longest cracks are presented. The shape of the plastic zone found using the phase difference map is superimposed on the line plot. In his model for crack tip plasticity, Irwin [6] proposed a correction which assumes that the crack is behaving as if it were longer than its physical size. He proposed that the crack has a notional tip ahead of the real one and that the plastic zone centres around this notional crack tip. The elastic stress field is essentially pushed forward ahead of the visible crack tip, as shown in Figure 7, and the plastic zone is approximated to a circle of radius ry, where:

1 K  2ry   I     y 

2

(1)

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Fig. 5 Maps of (a)TSA signal and (b) phase difference, and (c) line plots through the crack tip of an Aluminium 2024 CT specimen of fatigue crack length = 31.2 mm with mean load = 1000N; load amplitude = 500N; frequency 30Hz. (d) The shape of the plastic zone found using the phase difference map is superimposed on the line plot.

Fig. 6 Maps of (a)TSA signal and (b) phase difference, and (c) line plots through the crack tip of a CP Titanium CT specimen of fatigue crack length = 7.13 mm; mean load = 300N; load amplitude = 150N; frequency 20Hz. (d) The shape of the plastic zone found using the phase difference map is superimposed on the line plot.

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Fig. 7 Showing Irwin’s notional crack tip and the estimation of the plastic zone size, 2ry The elastic stress field equations proposed by Williams [7] were surface fit to the thermoelastic data in Figures 5 and 6 using a Downhill-Simplex method proposed by Diaz et al [4], where the crack tip coordinates (the origin of the stress field equations) were unknowns in the surface fitting routine. It was found that the elastic stress field which fits the TSA data has its origin at a point ahead of the crack, rather than at the visible crack tip. This origin of the elastic stress field exactly coincided with the minimum of the phase difference, and is shown in Figures 5 and 6 as the dotted vertical line. So the minimum of the phase difference could be used as a way to identify the origin of the elastic stress field and that this is the location of the “notional” crack tip. Irwin’s correction assumes the origin of the elastic field to be in the centre of a circle of radius ry from the visible crack tip. However the origin found from TSA is not in the centre of the plastic zone “size”, Rcrack (see Figure 1). Also the plastic zone shape found from this method is not a circle as suggested by Irwin, but a kidney shape, as can be seen in Figures 5 and 6. By inspecting Figure 2, it may be seen that, with the exception of the data recorded at 5 Hz, the minimum of the phase difference does not change location with respect to the phase map. However, the determined location of the visible crack tip, and also the plastic zone size, Rcrack, change with increasing frequency. So, the determined location of the notional crack tip does not appear to be affected by the adiabatic assumption, whereas the plastic zone size must be determined at a high enough frequency as to ensure adiabatic conditions. Further work is needed to explore these ideas but the results appear to partially support Irwin’s theory. CONCLUSIONS The evidence presented here supports the idea proposed by Irwin of a notional crack tip ahead of the visible crack tip, but not its location relative to the plastic zone size and the visible crack tip. It is suggested that the minimum of the phase difference could be used as a way to identify the origin of the elastic stress field and that this is the location of the “notional” crack tip. It is also found that the method of using the phase difference to determine the size and shape of the crack-tip plastic zone is highly dependent on frequency and material. ACKOWLEDGEMENTS These experiments were conducted at Michigan State University and Tomlinson’s visit was funded by an Overseas Travel Grant from the Engineering and Physical Science Research Council.

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