Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8

River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8 Antonio Baldi John M. Considine Simon Quinn Xavier Balandraud Proceedings of the 2017 Annual Conference on Experimental and Applied Mechanics River Publishers

Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Kristin B. Zimmerman, Ph.D. Society for Experimental Mechanics, Inc., Bethel, CT, USA

River Publishers Antonio Baldi • John M. Considine • Simon Quinn • Xavier Balandraud Editors Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8 Proceedings of the 2017 Annual Conference on Experimental and Applied Mechanics

Published, sold and distributed by: River Publishers Broagervej 10 9260 Gistrup Denmark www.riverpublishers.com ISBN 978-87-7004-963-4 (eBook) Conference Proceedings of the Society for Experimental Mechanics An imprint of River Publishers © The Society for Experimental Mechanics, Inc. 2018 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, or reproduction in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Preface Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems represents one of nine volumes of technical papers presented at the 2017 SEM Annual Conference & Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics and held in Indianapolis, IN, June 12–15, 2017. The complete Proceedings also includes volumes on: Dynamic Behavior of Materials; Challenges In Mechanics of TimeDependent Materials; Advancement of Optical Methods in Experimental Mechanics; Mechanics of Biological Systems, Materials and other topics in Experimental and Applied Mechanics; Micro-and Nanomechanics; Mechanics of Composite, Hybrid & Multifunctional Materials; Fracture, Fatigue, Failure and Damage Evolution; and Mechanics of Additive and Advanced Manufacturing. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics; Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems being three of these areas. Residual stresses are self-balanced stress fields induced during most materials processing procedures, for example, welding/joining, casting, thermal conditioning, and forming. Their hidden character often causes them to be underrated or overlooked. However, they profoundly influence structural design and substantially affect strength, fatigue life, and dimensional stability. Thus, they must be taken seriously and included in practical applications. In recent years the applications of infrared 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 infrared 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. Advances in inverse identification have been coupled with optical methods that provide surface deformation measurements and volumetric measurements of materials. In particular, inverse methodology was developed to more fully use the dense spatial data provided by optical methods to identify mechanical constitutive parameters of materials. Since its beginnings during the 1980s, creativity in inverse methods has led to applications in a wide range of materials, with many different constitutive relationships, across material heterogeneous interfaces. Complex test fixtures have been implemented to produce the necessary strain fields for identification. Force reconstruction has been developed for high strain rate testing. As developments in optical methods improve for both very large and very small length scales, applications of inverse identification have expanded to include geological and atomistic events. Researchers have used in-situ 3D imaging to examine microscale expansion and contraction and used inverse methodologies to quantify constitutive property changes in biological materials. Sardinia, Italy Antonio Baldi Madison, WI, USA John M. Considine Southampton, UK Simon Quinn Sigma Clermont, France Xavier Balandraud v

Contents 1 Residual Stresses in Bovine Femurs ................................................................................... 1 Yongbo Zhang and Drew Nelson 2 Experimental Stress Analysis of Unsymmetrical, Irregularly-Shaped Structure Containing an Arbitrarily-Shaped Hole ............................................................................................ 9 B. Kalayciogli, A. Alshaya, and R. Rowlands 3 Quantitative Calorimetry and TSA in Case of Low Thermal Signal and Strong Spatial Gradients: Application to Glass Materials ......................................................................................... 13 Guillaume Corvec, Eric Robin, Jean-Benoît Le Cam, Pierre Lucas, Jean-Christophe Sangleboeuf, and Frédéric Canevet 4 A New Denoising Methodology to Keep the Spatial Resolution of IR Images Equal to 1 Pixel................. 21 Guillaume Corvec, Eric Robin, Jean-Benoît Le Cam, Jean-Christophe Sangleboeuf, and Pierre Lucas 5 Calorific Signature of PLC Bands Under Biaxial Loading Conditions in Al-Mg Alloys ........................ 29 Jean-Benoît Le Cam, Eric Robin, Lionel Leotoing, and Dominique Guines 6 How Does Cristallizable Rubber Use Mechanical Energy to Deform?............................................ 37 Jean-Benoît Le Cam 7 Use of Bulge Test Geometry for Material Property Identification ................................................. 43 John M. Considine and X. Tang 8 Crystal Plasticity Parameter Identification by Integrated DIC on Microscopic Topographies................. 47 J. P. M. Hoefnagels, M. Bertin, C. Du, and F. Hild 9 Comparison of Residual Stress Characterization Techniques Using an Interference Fit Sample.............. 51 Jun-Sang Park, John Okasinski, Jonathan Almer, Paul Shade, and T.J. Turner 10 Influence of Thermographic Image Filtering on Hybrid TSA...................................................... 57 W. A. Samad and X. Balandraud 11 Optical Analysis of Residual Stress with Minimum Invasion....................................................... 65 Sanichiro Yoshida, Fumiya Miura, Tomohiro Sasaki, Daniel Didie, and Shahab Rouhi 12 Determination of Constitutive Properties in Inverse Problem Using Airy Stress Function..................... 73 A. Alshaya, John M. Considine, and R. Rowlands 13 High-Speed Infrared Imaging for Material Characterization in Experimental Mechanic Experiments ...... 83 Marc-André Gagnon, Frédérick Marcotte, Philippe Lagueux, and Vince Morton 14 A Spatio-Temporal Approach for iDIC-Residual Stress Measurement ........................................... 91 Antonio Baldi 15 Detection of Early Stage Material Damage Using Thermophysical Properties................................... 95 Mulugeta A. Haile, Natasha C. Bradly, Michael D. Coatney, and Asha J. Hall vii

viii Contents 16 Repeatability of Contour Method Residual Stress Measurements for a Range of Material, Process, and Geometry............................................................................................................ 101 Mitchell D. Olson, Adrian T. DeWald, and Michael R. Hill 17 System Identification of Structures with Modal Interference ...................................................... 115 Chang-Sheng Lin 18 Influence of Printing Constraints on Residual Stresses of FDM Parts ............................................ 121 C. Casavola, A. Cazzato, V. Moramarco, and G. Pappalettera

Chapter1 Residual Stresses in Bovine Femurs Yongbo Zhang and Drew Nelson Abstract The slitting method has become well-established for determining residual stresses in engineering materials. This study develops and applies a version of that method using a small slot to find residual stresses vs. depth in layers near the surface of bovine femurs. Results are obtained for the central region (diaphysis) of hydrated femurs from both mature and young cows. The magnitude of residual stresses was found to be greatest in thin layers near the surface, typically 100– 200 m deep. Residual stresses in those layers were compressive in mature femurs at the circumferential location tested, but tensile in hydrated young femurs. Keywords Residual stress • Bone • Femur • Slitting method • X-ray diffraction 1.1 Introduction The presence of residual stresses in components made of engineering materials is well known. Residual stresses and strains also exist in arteries [1–6], the esophagus [7–10], intestines [11, 12], brain [13], skin [14], etc. and may play an important role in the mechanical behavior of biological structures. For instance, at the inner diameter of arteries, compressive circumferential residual strains are believed to significantly reduce tensile stresses from blood pressure [4, 15–17] and enhance resistance to failure [18]. (The distinction between residual stresses and strains is made here because residual strains can have a different influence on the nonlinear stress-strain behavior of soft tissue than residual stresses). The existence and possible role of residual stresses in bone does not appear to be well-understood. X-ray diffraction (XRD) has been widely applied to find residual stresses in engineering materials with crystalline structures [19]. More recently, it has been applied to bone, a major constituent of which is the mineral hydroxyapatite (HAP) [20]. In the following summary, residual stresses refer to values determined by XRD with HAP crystals serving as “miniature strain sensors.” Residual stresses have been measured in specimens taken mainly from bovine femurs [21, 22–29], but also from canine fibula [30, 31] and the extremities of rabbits [32, 33]. The size and condition of specimens prior to and during experiments has varied considerably from study-to-study. A number of studies [21, 23–27] have used sizeable specimens removed from the central portion (diaphysis) of bovine femurs as depicted in Fig. 1.1 and measured residual stresses at the surface or at various depths. All but one of those studies used air dried specimens, and reported longitudinal tensile stresses, in some cases exceeding 100 MPa at the surface, with smaller values of compressive stress ( 10 MPa) at depths of 1 mm. Residual stresses were found to vary considerably with position around the circumference of specimens. Other studies used much smaller specimens (dimensions on the order a few mm) of bovine femurs [22, 28, 29] or canine fibula [31] that were kept hydrated. Compressive residual stresses and strains were measured through the thickness with synchrotron X-rays, with values as large as 150MPa and 2500 ©, respectively. For perspective on the magnitude of residual stress and strain values mentioned above, the longitudinal tensile yield stress of bovine femurs is approximately 100–130 MPa [20, 34]. The yield strain is estimated to be about 6500 © [35]. The interpretation of results from the XRD studies can be complicated by the following factors. Using 1 1 10mm specimens taken from bovine femurs, Tung et al. [29] found that initial compressive residual stresses (exceeding 100MPa) became tensile after about 20 min without hydration, climbing to approximately C75 MPa after an hour. Measured values of residual stresses may thus be altered if dehydration occurs during XRD experiments, although the extent of that influence Y. Zhang Institute of Solid Mechanics, Beihang University, Beijing 100191, China D. Nelson ( ) Mechanical Engineering Department, Stanford University, Stanford, CA 94305, USA e-mail: dnelson@stanford.edu © The Society for Experimental Mechanics, Inc. 2018 A. Baldi et al. (eds.), Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62899-8_1 1

2 Y. Zhang and D. Nelson Fig. 1.1 Schematic of a specimen cut from the central portion (diaphysis) of a bovine femur Fig. 1.2 Slitting geometry is unknown for larger specimens. Dissecting a bone into successively smaller pieces changed values of residual stresses measured by XRD [32]. Recent studies [22, 29] have also found that compressive residual stresses in small specimens, as measured via HAP crystals, dropped significantly with radiation dose. Doses are not reported in most of the XRD studies of bone and may or may not have influenced results. Residual stresses can also be measured in objects by releasing residual stresses, measuring resulting strains or deflections, and then using a computational model that relates the strains or deflections to the residual stresses. Stanwyck et al. [36] applied a strain gage in the longitudinal direction of a bovine metatarsal bone and sawed a 2 mm deep cut in the transverse direction of the bone, near the gage. A compressive strain of 180 © was reported. When the cut was deepened to 3 mm, the strain increased to 280 ©. The area surrounding the cut was irrigated with saline solution during the sawing. This experiment could be considered an early form of the slitting method for residual stress determination. Residual stresses were not computed from the measured strains, which is understandable since the methodology to do so was in its infancy when the experiment was conducted in the early 1980s. This paper will explore a version of the slitting method adapted to find residual stress vs. depth in bovine femurs, using a refined experimental approach and a finite element model. 1.2 Slitting Method As background, key features of the slitting method will be summarized. Suppose that a slit is introduced incrementally in depth into an object containing residual stresses normal to the slit and varying in an unknown manner with depth x, as depicted in Fig. 1.2. The slit releases residual stresses, causing the surface to develop strains © normal to the slit, which are typically measured with a strain gage near the slit location (and/or on the opposite side of the object if desired). Measured strain vs. depth data can be used with a computational model to determine the variation of with depth [37–39]. Assuming that residual stresses are constant in the z-direction, residual stresses can be related to strains by [40]: .ai/ DZ ai 0 G.x; ai/ .x/dx (1.1) where (ai) is the measured strain when a slit is at depthai. The functionG(x, ai) gives the strain response from a unit stress at depthx for a slit of depthai. Residual stresses vs. depth can approximated by .x/ DXn jD1 Aj Uj.x/ (1.2)

1 Residual Stresses in Bovine Femurs 3 Fig. 1.3 Illustration of unit stresses applied to different increments of depth to find compliance matrix Cij. where Aj are coefficients to be found andUj (x) are unit pulses withUj (x) D1 for a depth increment aj – 1 x aj and zero for x aj – 1, x aj. Substituting Eq. 1.2 into Eq. 1.1 gives .ai/ DX n jD1 AjCij (1.3) with a compliance matrix given by Cij D 1 E Z aj 1 aj G.x; ai/Uj.x/dx (1.4) andEDmodulus of elasticity. From Eqs. 1.1 and 1.4, the matrix elements Cij represent strains at the surface from unit stresses applied to various increments of depth aj –1 x aj. Values of Cij can be found by a creating a finite element model of a slit and applying stresses as in Fig. 1.3. Expressing Eq. 1.3 as f gD[C] fAg leads to a solution for the coefficients Aj in Eq. 1.2 in terms of measured strains: fAgD ŒC T ŒC 1 ŒC T f "measg (1.5) The determination of residual stresses vs. depth using the “unit pulse” method can be improved by Tikhonov regularization [41] to reduce effects of experimental uncertainties. 1.3 Slotting Model The slitting method was adapted for application to the central portion of femurs by the use of a small slot as depicted in Fig. 1.4 at regions that were flat over the length of a slot. Prior to performing experiments on bone, a finite element model was developed to determine compliance coefficients Cij for incremental slotting. The finite element code ABAQUS was with the model shown in Fig. 1.5, which employed eight-node linear brick elements. The nodes on the bottom of the model were fixed. The slot length D was 2.6 mm and the width 2R was 0.8 mm. The width of the slit was based on the smallest diameter end mill that would not break when making a slot. The slot was extended to a depth of 0.61 mm in ten steps. Each of the first six steps was 0.051 mm, followed by four steps of 0.076 mm each. Compliance Cij values were computed by applying a unit stress step-by-step as illustrated in Fig. 1.3. Slot extension in depth was simulated by deleting elements. Displacement data were used to compute strains using the method in Ref. 42. Orthotropic material behavior was used in the model, with longitudinal and transverse (tangential) moduli of elasticity EL andET values as described shortly, plus values for Poisson’s ratios and shear moduli available for bovine femur [43]. As might be anticipated, the value of EL governed strains from slotting. Experiments utilized hydrated femurs from mature (20–24 months old) and young (3–4 months old) cows. Rather than assuming that published values of EL and ET for bovine femurs would be applicable over the depth of slotting used here, tests were performed to find EL, ET values relevant to that depth. Specimens with a length of 9.0 mm and a rectangular cross-section with a 1.0 mm width and 0.4–0.5 mm thickness were carefully milled from surface layers of different femurs and stored in PBS at room temperature. The specimens were tested in a miniature three point bending fixture with a span of 8.0 mm. The mid-point deflection of each specimen was monitored using a 100 power microscope. Modulus of elasticity

4 Y. Zhang and D. Nelson Fig. 1.4 Schematic of femur, typical specimen and a slot with adjacent strain gage (D D2.6, R D0.4, H D1.0, P D0.55, GL D3.0, GWD1.5mm) Fig. 1.5 Finite element model to simulate slotting with unit stresses applied along the straight portion of the slot was computed from a relation between mid-point deflection and bending moment. For tests of femurs from two mature cows, average EL andET were 22.0 and 9.1 GPa, respectively. The range of EL for specimens from previous studies [34, 43–45], which used specimens roughly an order of magnitude larger than those here, was 19.3–22.6 GPa, and 12.4 to 14.6 GPa for ET. Tests using specimens from two young cows gave average EL and ET of 14.6 and 8.5 GPa. Values of EL and ET of 6.6 and 5.3 GPa have been reported in a recent study involving young bovine femurs [45]. 1.4 Slotting Experiments with Bone Specimens Each refrigerated bovine femur (with ends removed as illustrated in Fig. 1.4) was obtained from a butcher within 24 h of slaughter. Soft tissue was removed and the resulting specimen placed in phosphate buffered saline (PBS) for 48 h. Specimens were 125–150 mm long, with cross sectional widths between 50 and 75 mm. Next, each specimen was placed in a fixture that held it steady for slotting and enabled fine adjustments of tilt in two directions. A specimen and its holding fixture were then submerged in a water tank at room temperature, and a slotting setup illustrated in Fig. 1.6 installed over the specimen.

1 Residual Stresses in Bovine Femurs 5 Fig. 1.6 Schematic of setup used to perform slotting experiments, adjustable in x,y,z directions The setup enabled translation in x,y and z directions. Next, a displacement probe was slid into the guide tube and used to map the flatness of the region. Through adjustment of the location and tilt of a specimen, it was possible to identify surface regions that were on average flat to within 0.013 mm (0.0005 in.) over a prospective slot length. Those regions were marked with waterproof ink. With a specimen temporarily removed from the water tank, a strain gage was attached to the surface adjacent to a prospective slot using cyanoacrylate adhesive, which cures well in the presence of moisture. Care was taken to ensure that no adhesive extended into the region of a prospective slot. A strain gage and its terminal pad were covered with a polyurethane coating for protection from water, with the slot area masked temporarily to prevent it from being covered. After allowing 20 min for the coating to dry, a specimen was returned to the water tank and the orientation of an intended slot adjusted to ensure it was horizontal. The specimen was submerged in the water tank for 24 h for additional hydration and to allow the temperature of the specimen to equilibrate with that of the water surrounding it. Prior to making a slot, the thermal stability of strain readings was checked. Strains did not fluctuate by more than 2 © over the anticipated duration of an experiment. Slotting was performed by sliding a boring bar with an end mill into the tubular guide in Fig. 1.6. Slot depth was set using a precision translation stage (y-direction). Each slot was made by manually rotating the boring bar and gradually translating the end mill in the z direction using a second translation stage. Powered drilling has been found to damage bone tissue by heating [46] and thus gentle manual rotation (less than 10 RPM) was used in an effort to avoid such damage and its unknown effect on results of the experiments. After each slotting step, actual depth was measured at four locations along the length of a slot, and the average of those values used in expressing strain vs. depth. The actual depth after each step during slotting experiments was not exactly the same as in the finite element model. Depths were measured after each step to a resolution of 0.0025 mm (0.0001 in.). Values of strain corresponding to the depths used in the finite element model were found by interpolation from the measured strains vs. depth. Each specimen, including slot and strain gage, was submerged during the entirety of a slotting experiment. Final slot depths were close to 0.61 mm. 1.5 Initial Results Residual stresses vs. depth as determined by slotting are shown in Fig. 1.7. Magnitudes are most significant in thin layers just beneath the surface. Compressive residual stresses near the surface were found for femurs from mature cows, while those from young cows had tensile residual stresses. The magnitudes of residual stresses found by slotting may seem insignificant at first glance. However, the magnitudes may be of more significance when noting again that the tensile yield stress of mature bovine femurs is on the order 100–130 MPa [20, 34] (and perhaps somewhat lower for young femurs). Experiments to provide residual stress data for other circumferential locations on young and mature femurs are planned.

6 Y. Zhang and D. Nelson Fig. 1.7 Residual stresses vs. depth for (a) mature and (b) young femurs at location A 1.6 Discussion Determination of residual stresses in layers near the surface may be of interest since fracture [47], fatigue [48, 49] and bone growth [50] mechanisms are prominent there. The observation in this study of tensile residual stresses in surface layers of hydrated young femurs was not anticipated. An XRD study of young but air dried femurs [25] found minimal residual stresses at the surface (between about 0 and 10MPa) and stresses that alternated between tension and compression (approx. C10 to 10 MPa depending on circumferential location) at depths between 0.5 and 3 mm. The unknown effect of air drying makes comparison with results observed here difficult. Maintaining hydration of larger bone specimens during XRD experiments can be challenging. Bone is a microstructurally complex material [20]. An example is shown in Fig. 1.8, where the outer layers of a bovine femur have a lamellar structure like layers of bricks, while deeper layers have cylindrical osteons (Haversian structure). Residual stresses in this study represent values averaged over the volume of material removed by each step of slotting and may differ from values of residual stresses in smaller volumes.

1 Residual Stresses in Bovine Femurs 7 Fig. 1.8 Example showing (a) Haversian (osteons) and (b) lamellar microstructures (in circumferential sheath) of bovine femur, with micrographs from[44] 1.7 Conclusions 1. Development and application of a slotting method to find longitudinal residual stresses vs. depth in specimens of bone is feasible. 2. In hydrated bovine femurs from both mature and young animals, residual stresses were found to be greatest in thin layers near the surface, typically 100–200 mdeep. 3. In those layers, residual stresses in hydrated mature femurs were compressive at the circumferential location tested, but tensile in young femurs. References 1. Fung, Y.: Biodynamics: Circulation, pp. 54–60. Springer, New York (1984) 2. Vishnav, R., Vossoughi, J.: Estimation of residual strains in aortic segments. In: Hall, C (ed.) Biomedical Engineering II, Recent Developments, pp 330–333. Pergamon Press, Elmsford, New York (1983) 3. Choung, C., Fung, Y.: On residual stresses in arteries. J. Biomech. Eng. 108, 189–192 (1986) 4. Rachev, A., Greenwald, S.: Residual strains in conduit arteries. J. Biomech. 36, 661–670 (2003) 5. Humphrey, J.: Cardiovascular Solid Mechanics: Cells, Tissues and Organs. Springer, New York (2002) 6. Holzapfel, G., et al.: Layer-specific 3D residual deformation of human aortas with non-atherosclerotic intimal thickening. Ann. Biomed. Eng. 35, 530–545 (2007) 7. Gregersen, H., Lee, T., Chien, S., Skalak, R., Fung, Y.: Strain distribution in the layered wall of the esophagus. J. Biomech. Eng. 121, 442–448 (1999) 8. Laio, F.Y., Zeng, G., Gregersen, H.: Stress distribution in the layered wall of the rat oesophagus. Med. Eng. Phys. 25, 731–738 (2003) 9. Zhao, J., et al.: Opening angle and residual strain in a three-layered model of pig oesophagus. J. Biomech. 40, 3187–3192 (2007) 10. Sokolis, D.: Strain-energy function and three-dimensional stress distribution in esophageal biomechanics. J. Biomech. 43, 2753–2764 (2010) 11. Gao, C., Gregersen, H.: Biomechanical and morphological properties in rat large intestine. J. Biomech. 33, 1089–1097 (2000) 12. Dou, Y., et al.: Longitudinal residual strain and stress-strain relationship in rat small intestine. Biomed. Eng. Online. 5, 37 (2006). doi:10.1186/1475-925-5-37 13. Xu, G., Bayly, P., Taber, L.: Residual stress in the adult mouse brain. Biomech. Model. Mechanobiol. 8, 253–262 (2009)

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Chapter2 Experimental Stress Analysis of Unsymmetrical, Irregularly-Shaped Structure Containing an Arbitrarily-Shaped Hole B. Kalayciogli, A. Alshaya, and R. Rowlands Abstract This paper describes the ability to process load induced temperature information with an Airy stress function in real polar coordinates and some local known boundary conditions to determine the stresses experimentally in an isotropic linear elastic finite arbitrarily-shaped structure containing an irregularly-shaped hole. The proposed method simultaneously smooths the measured data, separates the stress components, and evaluates the individual stress components full-field, including at the boundary of the hole (location of highest tensile stress). Keywords Thermoelastic stress analysis • Irregularly-shaped holes • Stress concertation • Airy stress function • Hybrid method 2.1 Introduction References [1–3] treat non-circular cutouts using complex variables and conformal mapping but are restricted to infinitely large members having relatively simple and known external boundary conditions. References [4–16] combine thermoelastic stress data with a series solution of Airy stress function in either real or complex variables along with imposed traction-free boundary conditions on the edge of a hole to determine full-field individual components of stress, strain and displacements. Unlike with Fig. 2.1, all previous situations enjoyed a simple eternal shape, symmetry and/or necessitated imposing the boundary condition only on the internal boundary. Although a few cases imposed the internal boundary conditions discretely, most enabled imposing them analytically. The predominance of asymmetrically-loaded arbitrarily-shaped mechanical structures having irregularly-shaped holes necessitates the present technical extension. Thermoelastic stress analysis (TSA) is a non-contacting, nondestructive experimental method for determining the fullfield stresses in loaded members [17]. The technique enables the stress analysis of actual structures in their operating environment with a sensitivity comparable to that of strain gages. No surface preparation is required other than perhaps a flat black paint to provide an enhanced and uniform emissivity. For proportional loading under adiabatic, reversible elastic conditions, the stress- induce temperature information, S*, is proportional to the changes in the sum of the normal stresses, S, S DK S DK . 1 C 2/ DK xx C yy DK . rr C / DK C (2.1) where Kis experimentally-determined thermoelastic material coefficient, and 1, 2, rr, , xx, yy, , and are the normal stress components in the principal, polar, Cartesian rectangular coordinates, and normal and tangent to the edges of the structure, respectively. Combining experimental information with analytical and numerical tools enables one to separate stress components. The plate was sinusoidally compressed at 1334.46˙667.23 N at a rate of 20 Hz in a 20 kips capacity MTS hydraulic testing machine. The corresponding load-induced TSA data were recorded using a DeltaTherm model DT1410 system (Stress Photonics, Madison, WI) having a sensor array of 256 horizontal by 256 vertical pixels. The camera is cooled with liquid nitrogen to maintain the sensor at a very low temperature necessary for the accurate readings. The thermoelastic B. Kalayciogli Kirikkale University, Kirikkale, Turkey A. Alshaya Kuwait University, Kuwait, Kuwait R. Rowlands ( ) University of Wisconsin-Madison, Madison, WI, USA e-mail: rowlands@engr.wisc.edu © The Society for Experimental Mechanics, Inc. 2018 A. Baldi et al. (eds.), Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62899-8_2 9

10 B. Kalayciogli et al. Fig. 2.1 CAD model of finite plate with arbitrary-shaped aluminum plate containing irregularly-shaped hole signal, S*, was recorded by the data acquisition system which is equipped with Delta Vision Software and TSA images were captured and averaged over two minute durations. Since TSA data typically are unreliable on and near an edge, no recorded TSA information was used within at least two pixel positions (0.92 mm) of the boundary. 2.2 Relevant Equations For elasto-static plane isotropic problems in the absence of the body forces, the Airy stress function, ˆ, satisfying stress equilibrium and strain compatibility conditions gives the biharmonic equationr4ˆD0where r2 is the Laplacian operator andr2 D @ @r2 C 1 r @ @r C 1 r2 @ @ 2 . The general solution tor4ˆD0 in polar coordinates for the plate in Fig. 2.1 is [18] ˆDa0 Cb0 lnr Cc0r 2 CA0 C a1r C c1 r Cd1r 3 sin C a ’ 1r C c’ 1 r Cd ’ 1r 3 cos CP1n D2;3;:: anr n Cbnr nC2 Cc nr n Cdnr .n 2/ sinn CP1n D2;3;:: a’ nr n Cb’ nr nC2 Cc’ nr n Cd’ nr .n 2/ cosn (2.2) wherer is the radial coordinate measured from the center of the notch, is measured counter-clockwise from the horizontal xaxis as shown in Fig. 2.1, andNis the terminating index of the series which is any positive integer. The individual components of stresses in polar coordinate can be evaluated from rr D 1 r @ˆ @r C 1 r2 @2ˆ @ 2 ; D @2ˆ @r2 ; r D @ @r 1 r @ˆ @ (2.3) Determination of individual stresses, strains, or displacements necessitates evaluating the unknown Airy coefficients of Eq. 2.2. To help evaluate all of the Airy coefficients, boundary conditions in terms of stresses as shown in Fig. 2.2 were imposed discretely at multiple locations on the interior and exterior boundaries of the structure. 2.3 Results From the mD8564 recorded thermoelastic values, and, h D2 1192D2384 and t D2 (2731C250) D5962 boundary conditions on the internal and external boundaries, the Airy coefficients can be solved by forming a linear system of equation represented in matrix form as [A](mChCt) kfcgk 1 Dfdg(mChCt) 1. The number of equations, mCh Ct D16,910 will exceed the number of coefficients, k. The resulting overdetermined system of equations with which to evaluate the unknown

2 Experimental Stress Analysis of Unsymmetrical, Irregularly-Shaped Structure Containing an Arbitrarily-Shaped Hole 11 Fig. 2.2 Locations of TSA data and imposed traction-free boundary conditions -8 -6 -4 -2 0 2 4 6 8 -90 -70 -50 -30 -10 10 30 50 70 90 Normalized tangential stress Angel, θ, in degree FEM Fig. 2.3 Variations of tangential stress, ¢ , on the edge of the irregularly-shaped hole from FEA and hybrid-TSA (k D103 and m Ch C t D16,910) Table 2.1 Strains E at locations on the internal boundary by each of hybrid-TSA, FEA (ANSYS), and strain gages for a static load of 1,334.46 N on structure of Fig. 2.1 Location on the inner surface of the hole Strain value ( ") Average strain value ( ") Hybrid-TSA method FEA (ANSYS) D58 ı 890 910 890 870 D232 ı 930 D158 ı 420 410 480 440 D338 ı 400 coefficients was solved using least-squares. The number of Airy coefficients to retain, k D103, was assessed by monitoring the condition number of the respective Airy matrix, computing the Root Mean Square (RMS) for a series of different number of Airy coefficients and by comparing the reconstructed thermoelastic data with the actual measured thermoelastic signals. With ŸŸ and Ÿ in the -coordinate system (tangential-normal) being numerically zero as dictated by imposing the traction-free conditions on the edge of the of the irregularly-shaped hole, the normalized tangential stress, ¢ / 0, along the edge from FEA and hybrid-TSA is shown in Fig. 2.3. The results of this hybrid experiment based on discrete input values of S agree with ANSYS, including along edges where no input values were employed. The stresses are normalized with respect to a far-field stress 0 D1334.46/99 9.53D1.415MPa. Table 2.1 compares the strains from hybrid-TSA, FEA, and the strain gages. The good agreement between the current hybrid-TSA results and those from FEA and the strain gages provides strong confidence in the presently developed ability to obtain reliable stresses from recorded thermal information.

12 B. Kalayciogli et al. 2.4 Summary, Discussion and Conclusions A major contribution of this paper is the demonstrated ability to combine experimental (TSA), numerical (least-squares) and analytical (Airy stress) techniques for the full-field determination of the separate components of stresses at and in the neighborhood of the irregularly-shaped hole in a loaded finite arbitrarily-shaped structure, Fig. 2.1, and to do so without having to model the situation, know the external loading or constitutive properties, or differentiate the recorded data. This paper deals with an irregularly-shaped internal boundary and arbitrarily-shaped external boundary. The authors are unaware of prior utilization of Airy stress function to evaluate experimentally the stresses for such complex geometry of the plate in Fig. 2.1 using recorded load-induced thermal information. References 1. Muskhelishvili, N.: Some basic problems of the mathematical theory of elasticity. Springer, Leyden (1977) 2. Lekhnitskii, S.G.: Anisotropic plates. Gordon & Breach Scientific Publishers, New York (1968) 3. Timoshenko, S., Goodier, J.: Theory of elasticity. McGraw-Hill Publishing Company, New York (1970) 4. Foust, B.E., Rowlands, R.E.: Thermoelastic determination of individual stresses in a diametrally loaded disk. Strain. 47(2), 146–153 (2011) 5. Lin, S.-J., Matthys, D.R., Rowlands, R.E.: Separating stresses thermoelastically in a central circularly perforated plate using an airy stress function. Strain. 45(6), 516–526 (2009) 6. Samad, W.A., Rowlands, R.E.: Full-field thermoelastic stress analysis of a finite structure containing an irregularly-shaped hole. Exp. Mech. 54(3), 457–469 (2014) 7. Samad, W.A., Khaja, A.A., Kaliyanda, A.R., Rowlands, R.E.: Hybrid thermoelastic stress analysis of a pinned joint. Exp. Mech. 54(4), 515–525 (2014) 8. Lin, S.-J., Quinn, S., Matthys, D.R., New, A.M., Kincaid, I.M., Boyce, B.R., Khaja, A.A., Rowlands, R.E.: Thermoelastic determination of individual stresses in vicinity of a near-edge hole beneath a concentrated load. Exp. Mech. 51(6), 797–814 (2011) 9. Lin, S.J., Matthys, D.R., Samad, W.A., Khaja, A.A., Boyce, B.R., Rowlands, R.E.: Infrared stress analysis of unsymmetrically-loaded perforated member, ISEM-ACEM-SEM-7th ISEM’12, Taipei (2012) 10. Samad, W.A., Rowlands, R.E.: Hybrid thermoelastic analysis of an unsymmetrically-loaded structure containing an arbitrarily-shaped cutout. In: Residual stress, thermomechanics & infrared imaging, hybrid techniques and inverse problems, vol 8, pp. 51–57. Springer (2014) 11. Abdel Samad, W.: Hybrid full-field stress analysis of structures containing irregularly-shaped cutouts, PhD Thesis, University of WisconsinMadison (2013) 12. Lin, S.J., Matthys, D.R., Quinn, S., Davidson, J.P., Boyce, B.R., Khaja, A.A., Rowlands, R.E.: Stresses at and in the neighborhood of a near-edge hole in a plate subjected to an offset load from measured temperatures. Eur. J. Mech. A Solids. 39, 209–217 (2013) 13. Kurunthottikkal Philip, S.: Stress analysis of a finite structure containing an asymmetrical, arbitrarily-shaped cutout based on recorded temperature data, Master Thesis, University of Wisconsin – Madison (2015) 14. Lin, S.T., Rowlands, R.E.: Thermoelastic stress analysis of orthotropic composites. Exp. Mech. 35(3), 257–265 (1995) 15. Alshaya, A., Shuai, X., Rowlands, R.: Thermoelastic stress analysis of a finite orthotropic composite containing an elliptical hole. Exp. Mech. 56(8), 1373–1384 (2016) 16. Alshaya, A.A.: Experimental, analytical and numerical analyses of orthotropic materials and biomechanics application, PhD Thesis, University of Wisconsin-Madison (2017) 17. Greene, R., Patterson, E., Rowlands, R.: Thermoelastic stress analysis. In: Sharpe, J., William, N. (eds.) Springer handbook of experimental solid mechanics, pp. 743–768. Springer, New York (2008) 18. Soutas-Little, R.W.: Elasticity. Dover Publications, Mineola (1999)

Chapter3 Quantitative Calorimetry and TSA in Case of Low Thermal Signal and Strong Spatial Gradients: Application to Glass Materials Guillaume Corvec, Eric Robin, Jean-Benoît Le Cam, Pierre Lucas, Jean-Christophe Sangleboeuf, and Frédéric Canevet Abstract In the present paper, the thermo-mechanical characterization of a holed glass sample under cyclic loading is carried out. Due to the low thermoelastic response obtained for such a material, the thermal movie has been preliminary filtered. The experimental stress field obtained from the Thermoelastic Stress Analysis (TSA) is well correlated to the finite element model. It validates both the use of this experimental technique to study the thermoelastic response of brittle materials and the filtering methodology. Finally, the corresponding calorimetric response has been determined by using a simplified formulation of the heat diffusion equation. This permits to quantify heat sources and to carry out energy balances. Keywords Infrared thermography • Denoising methodology • Inorganic oxide glass • Thermoelastic stress analysis • Quantitative calorimetry 3.1 Introduction The Thermoelastic Stress Analysis (TSA) [1, 2] and the quantitative calorimetry are non-contact techniques, which have experienced an impressive expansion since the 1980s with the development of thermal cameras. They are used to access to the thermoelastic and the calorimetric effects accompanying the deformation of materials in order to better understand and model their mechanical behavior. Most materials have already benefited from these techniques including smart memory alloys [3], aluminum alloys [4], polymers [5], composites [6–10] and elastomers [11, 12]. These materials exhibit temperature variations in the range of one degree or more. In these conditions the experimental noise does not extensively affect the measurement and basic filters can be used to detect and to quantify temperature variations. Concerning inorganic glasses, although these materials are used in a wide range of applications due to their transparency, heat resistance, pressure resistance, and chemical resistance, their fragility and low fracture toughness prevent them from use in most mechanical components. To understand or improve their mechanical behavior, most of the studies have been carried out on the crack tip movement [13], the mechanical properties [14] or fracture [15, 16], but rarely on their thermo-mechanical properties. A possible cause of this state is that the low strain level supported by glasses, combined with their low thermal conductivity, lead to very low temperature variations during the deformation process. Hence, to the best of authors knowledge, only two studies have been dedicated to the thermal and thermo-mechanical response of inorganic glasses [17, 18]. The aim of this paper is to present strong thermal gradients measurement at the surface of a holed disc in case of low temperature variation conditions, without altering the spatial resolution of the infrared images after the filtering process. The stress fields obtained by TSA are compared to a Finite Element Method model. As the mechanical behavior is well known under such tests conditions, this allows us to validate the presented denoising methodology. Then, quantitative calorimetry analysis is carried out by computing heat sources produced and absorbed from the temperature field measured at thespecimen G. Corvec ( ) • E. Robin • J.-B. Le Cam • J.-C. Sangleboeuf Universitée de Rennes 1, Institut de Physique UMR 6251, CNRS/Université de Rennes 1, Campus de Beaulieu, Bât. 10B, 35042 Rennes Cedex, France e-mail: guillaume.corvec@gmail.com P. Lucas Arizona Materials Laboratory, 4715 East Fort Lowell Rd, Tucson, AZ 85712, USA F. Canevet Cooper Standard France, 194 route de Lorient, 35043 Rennes, France © The Society for Experimental Mechanics, Inc. 2018 A. Baldi et al. (eds.), Residual Stress, Thermomechanics & Infrared Imaging, Hybrid Techniques and Inverse Problems, Volume 8, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-62899-8_3 13

14 G. Corvec et al. surface. The methodology to identify intrinsic dissipation is given. The validation of this methodology at ambient temperature is a first step towards the use of thermal field measurements on glassy materials at temperatures close to glass temperature transition. 3.2 Experimental Setup 3.2.1 Specimen Geometry and Testing Conditions The material considered here is a soda lime glass. The sample, presented in Fig. 3.1 corresponds to a disc of 2.1 mm in thickness and 29.7 mm in diameter with three elliptical holes. The major and minor axis lengths are respectively 6 and 3 mm. The holes are oriented in relation to each other according to the major axis with an angle of 120ı. The centre of the holes are 6 mm far from the disc center. The holes were cut with a water jet cutting machine. During the mechanical test, one of the holes was oriented with an angle of 27.91ı according to its major axis and the loading axis. The disc was submitted to cyclic compressive loading by means of a 5543 Instron testing machine. An overview of the experimental setup is given in Fig. 3.1. The test was conducted under a prescribed periodic triangular signal. The minimum and maximum values of the compression force are 5 N and 520 N, respectively. The sample was submitted to ten cycles at a frequency of 2.9 Hz. Fig. 3.1 Overview of the experimental setup

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