Abstract
Background
Mechanical characterization of materials that solely relies on global responses may overlook important local behavior that significantly affects the characterization of material properties. Field displacements such as from digital image correlation (DIC) can provide high-fidelity experimental data, which combined with finite element method (FEM) can form DIC-FEM inverse method that can better account for complex mechanical properties of materials. Despite its capability, the DIC-FEM inverse method has been mainly applied to an elastic-dominant regime even though inelastic deformation is important in many engineering materials. Specifically, the DIC-FEM inverse method has not been fully extended to viscoelastic materials due to the complex representation of the time-dependent modulus.
Objective
This study aimed at establishing a DIC-FEM inverse framework to identify constitutive properties of homogeneous elastic and viscoelastic materials.
Methods
Two example materials (i.e., polyetheretherketone (PEEK) and a viscoelastic fine aggregate matrix (FAM) with a bituminous binder) were selected for the elastic and viscoelastic investigation, respectively. Both were experimentally tested using three-point bending incorporated with DIC. FEM simulated the experiment and the Nelder-Mead nonlinear optimization algorithm was implemented to solve the inverse problem.
Results
The DIC-FEM inverse method successfully identified Young’s modulus of an example linear elastic PEEK and the linear viscoelastic relaxation modulus of FAM.
Conclusions
The resulting DIC-FEM inverse method is applicable to various materials with inelastic deformation and can be extended to localized behavior induced by microstructure heterogeneity and fracture.
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References
Zhao J, Dong J, Liu Z, Xie H (2019) Characterization method of mechanical properties of rubber materials based on in-situ stereo finite-element-model updating. Polym Testing 79:106015
Chu T, Ranson W, Sutton MA (1985) Applications of digital-image-correlation techniques to experimental mechanics. Exp Mech 25(3):232–244. https://doi.org/10.1007/BF02325092
McCormick N, Lord J (2010) Digital image correlation. Mater Today 13(12):52–54. https://doi.org/10.1016/S1369-7021(10)70235-2
Bruck H, McNeill S, Sutton MA, Peters W (1989) Digital image correlation using Newton-Raphson method of partial differential correction. Exp Mech 29(3):261–267
Yoneyama S (2016) Basic principle of digital image correlation for in-plane displacement and strain measurement. Adv Compos Mater 25(2):105–123. https://doi.org/10.1080/09243046.2015.1129681
Shen B, Paulino G (2011) Direct extraction of cohesive fracture properties from digital image correlation: a hybrid inverse technique. Exp Mech 51(2):143–163
Shao X, Dai X, He X (2015) Noise robustness and parallel computation of the inverse compositional Gauss-Newton algorithm in digital image correlation. Opt Lasers Eng 71:9–19. https://doi.org/10.1016/j.optlaseng.2015.03.005
Gajewski T, Garbowski T (2014) Calibration of concrete parameters based on digital image correlation and inverse analysis. Archives of Civil and Mechanical Engineering 14(1):170–180. https://doi.org/10.1016/j.acme.2013.05.012
Genovese K, Casaletto L, Humphrey JD, Lu J (2014) Digital image correlation-based point-wise inverse characterization of heterogeneous material properties of gallbladder in vitro. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470(2167):20140152
Sutton M, Wolters W, Peters W, Ranson W, McNeill S (1983) Determination of displacements using an improved digital correlation method. Image Vis Comput 1(3):133–139
Sutton M, Mingqi C, Peters W, Chao Y, McNeill S (1986) Application of an optimized digital correlation method to planar deformation analysis. Image Vis Comput 4(3):143–150
Jiang Y, Li G-Y, Qian L-X, Hu X-D, Liu D, Liang S, Cao Y (2015) Characterization of the nonlinear elastic properties of soft tissues using the supersonic shear imaging (SSI) technique: inverse method, ex vivo and in vivo experiments. Med Image Anal 20(1):97–111
He W, Goudeau P, Le Bourhis E, Renault P-O, Dupré JC, Doumalin P, Wang S (2016) Study on Young’s modulus of thin films on Kapton by microtensile testing combined with dual DIC system. Surf Coat Technol 308:273–279
He W, Han M, Goudeau P, Le Bourhis E, Renault P-O, Wang S, Li L-A (2018) Strain transfer through film-substrate interface and surface curvature evolution during a tensile test. Appl Surf Sci 434:771–780
He W, Duan Q, Shi W, Xie H (2019) Elastic property characterization of soft substrate-supported thin films using multiscale digital image correlation. Opt Lasers Eng 121:112–119
Tuninetti V, Gilles G, Péron-Lührs V, Habraken A (2012) Compression test for metal characterization using digital image correlation and inverse modeling. Procedia IUTAM 4:206–214
Laurin F, Charrier J-S, Lévêque D, Maire J-F, Mavel A, Nuñez P (2012) Determination of the properties of composite materials thanks to digital image correlation measurements. Procedia IUTAM 4:106–115
Caminero MA, Lopez-Pedrosa M, Pinna C, Soutis C (2013) Damage monitoring and analysis of composite laminates with an open hole and adhesively bonded repairs using digital image correlation. Compos B Eng 53:76–91
Ghiassi B, Xavier J, Oliveira DV, Lourenço PB (2013) Application of digital image correlation in investigating the bond between FRP and masonry. Compos Struct 106:340–349
Kubo S (1988) Inverse problems related to the mechanics and fracture of solids and structures. JSME international journal. Ser. 1. Solid Mechanics, Strength Mater 31(2):157–166
Lin L, Li H, Fok AS, Joyce M, Marrow J (2008) Characterization of heterogeneity and nonlinearity in material properties of nuclear graphite using an inverse method. J Nucl Mater 381(1–2):158–164. https://doi.org/10.1016/j.jnucmat.2008.07.042
Lin L, Li H, Fok AS, Joyce M, Marrow TJ (2006) Characterization of material properties using an inverse method. Vol. 5. Trans Tech Publ
Nguyen T, Boyce B (2011) An inverse finite element method for determining the anisotropic properties of the cornea. Biomech Model Mechanobiol 10(3):323–337. https://doi.org/10.1007/s10237-010-0237-3
Kim SK, Jung BS, Kim HJ, Lee WI (2003) Inverse estimation of thermophysical properties for anisotropic composite. Exp Thermal Fluid Sci 27(6):697–704. https://doi.org/10.1016/S0894-1777(02)00309-6
Schnur DS, Zabaras N (1992) An inverse method for determining elastic material properties and a material interface. Int J Numer Meth Eng 33(10):2039–2057. https://doi.org/10.1002/nme.1620331004
Mathieu F, Leclerc H, Hild F, Roux S (2015) Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC. Exp Mech 55(1):105–119
Pan B (2018) Digital image correlation for surface deformation measurement: historical developments, recent advances and future goals. Meas Sci Technol 29(8):082001
Grédiac M (1989) Principe des travaux virtuels et identification. Comptes rendus de l'Académie des sciences. Série 2, Mécanique, Physique, Chimie. Sciences de l'univers, Sciences de la Terre 309(1):1–5
Pierron F, Grédiac M (2012) The virtual fields method: extracting constitutive mechanical parameters from full-field deformation measurements. Springer Science & Business Media
Pierron F, Forquin P (2012) Ultra-high-speed full-field deformation measurements on concrete spalling specimens and stiffness identification with the virtual fields method. Strain 48(5):388–405
Brigham J, Aquino W, Mitri F, Greenleaf JF, Fatemi M (2007) Inverse estimation of viscoelastic material properties for solids immersed in fluids using vibroacoustic techniques. J Appl Phys 101(2):023509. https://doi.org/10.1063/1.2423227
Lutif JE, Souza FV, Kim YR, Soares JB and Allen DH (2010) Multiscale modeling to predict mechanical behavior of asphalt mixtures. Trans Res Rec 2181(1):28–35. https://doi.org/10.3141/2181-04
Shen B, Stanciulescu I, Paulino GH (2010) Inverse computation of cohesive fracture properties from displacement fields. Inverse Probl Sci Eng 18(8):1103–1128
Shen B, Paulino GH (2011) Identification of cohesive zone model and elastic parameters of fiber-reinforced cementitious composites using digital image correlation and a hybrid inverse technique. Cement Concr Compos 33(5):572–585
Shen B, Functionally graded fiber-reinforced cementitious composites—Manufacturing and extraction of cohesive fracture properties using finite elements and digital image correlation. Civil Engineering. Vol. Doctor of Philosophy. (2009) Urbana. University of Illinois at Urbana-Champaign, Illinois
Geymonat G, Hild F, Pagano S (2002) Identification of elastic parameters by displacement field measurement. CR Mec 330(6):403–408. https://doi.org/10.1016/S1631-0721(02)01476-6
Kiełczyński P, Szalewski M (2011) An inverse method for determining the elastic properties of thin layers using Love surface waves. Inverse Problems in Science and Engineering 19(1):31–43. https://doi.org/10.1080/17415977.2010.531472
Ruggiero L, Sol H, Sahli H, Adriaenssens S, Adriaenssens N (2011) An inverse method to determine material properties of soft tissues. Mechanics of Biological Systems and Materials 2:19–32. https://doi.org/10.1007/978-1-4614-0219-0_3
Lin L (2009) Characterization of Material Properties Using Inverse Method Ann Arbor: The University of Manchester (United Kingdom). 209. https://scholar.uwindsor.ca/etd/4825
Mathieu F, Leclerc H, Hild F and Roux S (2015) Estimation of elastoplastic parameters via weighted FEMU and integrated-DIC. Experimen Mech 55:105–119
de-Carvalho R, Valente RAF, Andrade-Campos A (2011) Optimization strategies for non-linear material parameters identification in metal forming problems. Comp and Struct 89:246–255
Gajewski M, Kowalewski L (2016) Inverse analysis and DIC as tools to determine material parameters in isotropic metal plasticity with isotropic strain hardening. Mater Test 58(10). https://doi.org/10.3139/120.110925
Im S, Kim Y-R, Ban H (2013) Rate-and Temperature-Dependent Fracture Characteristics of Asphaltic Paving Mixtures. J Test Eval 41(2):257–268
Kim YR, Teixeira JE, Kommidi SR, Little DN, Aragao FT, Manrique‐Sanchez L, Souza FV (2021) Rate‐dependent fracture modeling of bituminous media using nonlinear viscoelastic cohesive zone with Gaussian damage function. Comput -Aided Civ Infrastruct Eng
Abaqus V (2014) 6.14 Documentation, in Dassault Systemes Simulia Corporation 6.2
Kowalewski Ł, Gajewski M (2019) Assessment of Optimization Methods Used to Determine Plasticity Parameters Based on DIC and back Calculation Methods. Exp Tech 43(4):385–396
Sutton MA, Orteu JJ, Schreier H (2009) Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications. Springer Science & Business Media
Negahban M (2012) The mechanical and thermodynamical theory of plasticity. Crc New York, NY
Vendroux G, Knauss W (1998) Submicron deformation field measurements: Part 2. Improved digital image correlation Experimental Mechanics 38(2):86–92
Lu H, Cary P (2000) Deformation measurements by digital image correlation: implementation of a second-order displacement gradient. Exp Mech 40(4):393–400
Lagarias JC, Reeds JA, Wright MH, Wright PE (1998) Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J Optim 9(1):112–147
Singer S, Nelder J (2009) Nelder-mead algorithm Scholarpedia 4(7):2928. https://doi.org/10.4249/scholarpedia.2928
Barton RR, Ivey JS Jr (1991). Modifications of the Nelder-Mead simplex method for stochastic simulation response optimization. https://doi.org/10.1109/WSC.1991.185709
Nocedal J and Wright S (2006) Numerical optimization. Springer Science & Business Media
Anderson TL (2017) Fracture mechanics: fundamentals and applications. CRC press
El-Qoubaa Z, Othman R (2015) Characterization and modeling of the strain rate sensitivity of polyetheretherketone’s compressive yield stress. Mater and Des (1980–2015) 66:336–345
Rivard CH, Rhalmi S, Coillard C (2002) In vivo biocompatibility testing of peek polymer for a spinal implant system: a study in rabbits. Journal of Biomedical Materials Research: An Official Journal of The Society for Biomaterials, The Japanese Society for Biomaterials, and The Australian Society for Biomaterials and the Korean Society for Biomaterials 62(4):488–498
Chen F, Ou H, Lu B, Long H (2016) A constitutive model of polyether-ether-ketone (PEEK). J Mech Behav Biomed Mater 53:427–433
Chen F, Gatea S, Ou H, Lu B, Long H (2016) Fracture characteristics of PEEK at various stress triaxialities. J Mech Behav Biomed Mater 64:173–186
Aragão FTS, Kim YR (2012) Mode I Fracture Characterization of Bituminous Paving Mixtures at Intermediate Service Temperatures. Exp Mech 52(9):1423–1434. https://doi.org/10.1007/s11340-012-9594-4
Al-Rub RKA, Darabi MK, You T, Masad EA, Little DN (2011) A unified continuum damage mechanics model for predicting the mechanical response of asphalt mixtures and pavements. Int J Roads Airports 1(1):68–84
Im S, You T, Ban H, Kim Y-R (2017) Multiscale testing-analysis of asphaltic materials considering viscoelastic and viscoplastic deformation. Int J Pavement Eng 18(9):783–797
Aragão FTS, Badilla-Vargas GA, Hartmann DA, de Oliveira AD, Kim Y-R (2017) Characterization of temperature-and rate-dependent fracture properties of fine aggregate bituminous mixtures using an integrated numerical-experimental approach. Eng Fract Mech 180:195–212
Kim Y-R, Aragao FT, Allen DH, Little DN (2010) Damage modeling of bituminous mixtures considering mixture microstructure, viscoelasticity, and cohesive zone fracture. Can J Civ Eng 37(8):1125–1136
Ban H, Im S, Kim Y-R (2015) Mixed-mode fracture characterization of fine aggregate mixtures using semicircular bend fracture test and extended finite element modeling. Constr Build Mater 101:721–729
Park S and Schapery R (1999) Methods of interconversion between linear viscoelastic material functions. Part I—A numerical method based on Prony series. Int J Solid Struct 36(11):1653–1675
Newcomb D, Martin AE, Yin F, Arambula E, Park ES, Chowdhury A, Brown R, Rodezno C, Tran N, Coleri E (2015) Short-term laboratory conditioning of asphalt mixtures
Wineman AS and Rajagopal KR (2000) Mechanical response of polymers: an introduction. Cambridge University Press
Haque A (2019) Timoshenko Beam Theory. Independently published
Gutierrez-Lemini D (2014) Engineering viscoelasticity. Springer
Cowper G (1966) The shear coefficient in Timoshenko’s beam theory
Sobieraj MC, Kurtz SM, Rimnac CM (2009) Notch sensitivity of PEEK in monotonic tension. Biomaterials 30(33):6485–6494
Kurtz SM (2019) PEEK biomaterials handbook. William Andrew
Jaekel DJ, MacDonald DW, Kurtz SM (2011) Characterization of PEEK biomaterials using the small punch test. J Mech Behav Biomed Mater 4(7):1275–1282
Lesiuk G, Sawicka A, Correia J, Frątczak R (2017) Fracture resistance analysis of PEEK-polymer. Engineering Structures and Technologies 9(4):207–213
Tschoegl NW, Knauss WG, Emri I (2002) Poisson’s ratio in linear viscoelasticity–a critical review. Mechanics of Time-Dependent Materials 6(1):3–51
Lu H, Zhang X, Knauss W (1997) Uniaxial, shear, and Poisson relaxation and their conversion to bulk relaxation: studies on poly (methyl methacrylate). Polym Compos 18(2):211–222
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Partial financial support was received from the Texas A&M Engineering Experiment Station.
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Nsengiyumva, G., Kim, YR. Field Displacement-Based Inverse Method for Elastic and Viscoelastic Constitutive Properties. Exp Mech 62, 1553–1568 (2022). https://doi.org/10.1007/s11340-022-00876-0
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DOI: https://doi.org/10.1007/s11340-022-00876-0