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Research Papers

Identification of Material Parameters of a Hyper-Elastic Body With Unknown Boundary Conditions

[+] Author and Article Information
M. Hajhashemkhani

Department of Mechanical Engineering,
Shiraz University,
Shiraz 71936, Iran
e-mail: m-hajhashemkhani@shirazu.ac.ir

M. R. Hematiyan

Professor
Department of Mechanical Engineering,
Shiraz University,
Shiraz 71936, Iran
e-mail: mhemat@shirazu.ac.ir

S. Goenezen

Mem. ASME
Department of Mechanical Engineering,
Texas A & M University,
College Station, TX 77843
e-mail: sgoenezen@tamu.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received December 1, 2017; final manuscript received January 27, 2018; published online March 12, 2018. Assoc. Editor: Yihui Zhang.

J. Appl. Mech 85(5), 051006 (Mar 12, 2018) (13 pages) Paper No: JAM-17-1659; doi: 10.1115/1.4039170 History: Received December 01, 2017; Revised January 27, 2018

Identification of material properties of hyper-elastic materials such as soft tissues of the human body or rubber-like materials has been the subject of many works in recent decades. Boundary conditions generally play an important role in solving an inverse problem for material identification, while their knowledge has been taken for granted. In reality, however, boundary conditions may not be available on parts of the problem domain such as for an engineering part, e.g., a polymer that could be modeled as a hyper-elastic material, mounted on a system or an in vivo soft tissue. In these cases, using hypothetical boundary conditions will yield misleading results. In this paper, an inverse algorithm for the characterization of hyper-elastic material properties is developed, which takes into consideration unknown conditions on a part of the boundary. A cost function based on measured and calculated displacements is defined and is minimized using the Gauss–Newton method. A sensitivity analysis is carried out by employing analytic differentiation and using the finite element method (FEM). The effectiveness of the proposed method is demonstrated through numerical and experimental examples. The novel method is tested with a neo–Hookean and a Mooney–Rivlin hyper-elastic material model. In the experimental example, the material parameters of a silicone based specimen with unknown boundary condition are evaluated. In all the examples, the obtained results are verified and it is observed that the results are satisfactory and reliable.

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References

Basdogan, C. , Ho, C. H. , and Srinivasan, M. A. , 2001, “Virtual Environments for Medical Training: Graphical and Haptic Simulation of Laparoscopic Common Bile Duct Exploration,” IEEE/ASME Trans. Mechatronics, 6(3), pp. 269–285. [CrossRef]
Satava, R. M. , and Jones, S. B. , 1997, “Virtual Environments for Medical Training and Education,” Presence: Teleoperators Virtual Environ., 6(2), pp. 139–146. [CrossRef]
Tendick, F. , Downes, M. , Goktekin, T. , Cavusoglu, M. C. , Feygin, D. , Wu, X. , Eyal, R. , Hegarty, M. , and Way, L. W. , 2000, “A Virtual Environment Testbed for Training Laparoscopic Surgical Skills,” Presence: Teleoperators Virtual Environ., 9(3), pp. 236–255. [CrossRef]
Avis, N. J. , 2000, “Virtual Environment Technologies,” Minimally Invasive Ther. Allied Technol., 9(5), pp. 333–339. [CrossRef]
Ayache, N. , Cotin, S. , Delingette, H. , Clement, J. M. , Russier, Y. , and Marescaux, J. , 1998, “Simulation of Endoscopic Surgery,” Minimally Invasive Ther. Allied Technol., 7(2), pp. 71–77. [CrossRef]
Cotin, S. , Delingette, H. , and Ayache, N. , 2000, “A Hybrid Elastic Model for Real-Time Cutting, Deformations, and Force Feedback for Surgery Training and Simulation,” Visual Comput., 16(8), pp. 437–452. [CrossRef]
Picinbono, G. , Lombardo, J. C. , Delingette, H. , and Ayache, N. , 2002, “Improving Realism of a Surgery Simulator: Linear Anisotropic Elasticity, Complex Interactions and Force Extrapolation,” Comput. Animation Virtual Worlds, 13(3), pp. 147–167.
Snedeker, J. G. , Bajka, M. , Hug, J. M. , Szekely, G. , and Niederer, P. , 2002, “The Creation of a High‐Fidelity Finite Element Model of the Kidney for Use in Trauma Research,” Comput. Animation Virtual Worlds, 13(1), pp. 53–64.
Jiang, Y. , Li, G. Y. , Qian, L. X. , Hu, X. D. , Liu, D. , Liang, S. , and 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), pp. 97–111. [CrossRef] [PubMed]
Goenezen, S. , Dord, J. F. , Sink, Z. , Barbone, P. E. , Jiang, J. , Hall, T. J. , and Oberai, A. A. , 2012, “Linear and Nonlinear Elastic Modulus Imaging: An Application to Breast Cancer Diagnosis,” IEEE Trans. Med. Imaging, 31(8), pp. 1628–1637. [CrossRef] [PubMed]
Mazurkiewicz, D. , 2010, “Problems of Identification of Strength Properties of Rubber Materials for Purposes of Numerical Analysis: A Review,” Arch. Civ. Mech. Eng., 10(1), pp. 69–84. [CrossRef]
Le Saux, V. , Marco, Y. , Bles, G. , Calloch, S. , Moyne, S. , Plessis, S. , and Charrier, P. , 2011, “Identification of Constitutive Model for Rubber Elasticity From Micro-Indentation Tests on Natural Rubber and Validation by Macroscopic Tests,” Mech. Mater., 43(12), pp. 775–786. [CrossRef]
Samani, A. , and Plewes, D. , 2004, “A Method to Measure the Hyperelastic Parameters of Ex Vivo Breast Tissue Samples,” Phys. Med. Biol., 49(18), p. 4395. [CrossRef] [PubMed]
O'Hagan, J. J. , and Samani, A. , 2009, “Measurement of the Hyperelastic Properties of 44 Pathological Ex Vivo Breast Tissue Samples,” Phys. Med. Biol., 54(8), p. 2557. [CrossRef] [PubMed]
Tran, H. V. , Charleux, F. , Rachik, M. , Ehrlacher, A. , and Ho Ba Tho, M. C. , 2007, “In Vivo Characterization of the Mechanical Properties of Human Skin Derived From MRI and Indentation Techniques,” Comput. Methods Biomech. Biomed. Eng., 10(6), pp. 401–407. [CrossRef]
Ruggiero, L. , Sol, H. , Sahli, H. , Adriaenssens, S. , and Adriaenssens, N. , 2011, “An Inverse Method to Determine Material Properties of Soft Tissues,” Mechanics of Biological Systems and Materials, Vol. 2, pp. 19–32. [CrossRef]
Chen, Z. , Scheffer, T. , Seibert, H. , and Diebels, S. , 2013, “Macroindentation of a Soft Polymer: Identification of Hyperelasticity and Validation by Uni/Biaxial Tensile Tests,” Mech. Mater., 64, pp. 111–127. [CrossRef]
Liu, H. , Sangpradit, K. , Li, M. , Dasgupta, P. , Althoefer, K. , and Seneviratne, L. D. , 2014, “Inverse Finite-Element Modeling for Tissue Parameter Identification Using a Rolling Indentation Probe,” Med. Biol. Eng. Comput., 52(1), pp. 17–28. [CrossRef] [PubMed]
Zisis, T. , Zafiropoulou, V. I. , and Giannakopoulos, A. E. , 2015, “Evaluation of Material Properties of Incompressible Hyperelastic Materials Based on Instrumented Indentation of an Equal-Biaxial Prestretched Substrate,” Int. J. Solids Struct., 64–65, pp. 132–144. [CrossRef]
MacManus, D. B. , Pierrat, B. , Murphy, J. G. , and Gilchrist, M. D. , 2016, “Mechanical Characterization of the P56 Mouse Brain Under Large-Deformation Dynamic Indentation,” Sci. Rep., 6(1), p. 21569. [CrossRef] [PubMed]
Hendriks, F. M. , Brokken, D. V. , Van Eemeren, J. T. W. M. , Oomens, C. W. J. , Baaijens, F. P. T. , and Horsten, J. B. A. M. , 2003, “A Numerical‐Experimental Method to Characterize the Non‐Linear Mechanical Behaviour of Human Skin,” Skin Res. Technol., 9(3), pp. 274–283. [CrossRef] [PubMed]
Nava, A. , Mazza, E. , Kleinermann, F. , Avis, N. J. , and McClure, J. , 2003, “Determination of the Mechanical Properties of Soft Human Tissues Through Aspiration Experiments,” International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI), Montreal, QC, Canada, Nov. 15–18, pp. 222–229.
Delalleau, A. , Josse, G. , Lagarde, J. M. , Zahouani, H. , and Bergheau, J. M. , 2008, “A Nonlinear Elastic Behavior to Identify the Mechanical Parameters of Human Skin In Vivo,” Skin Res. Technol., 14(2), pp. 152–164. [CrossRef] [PubMed]
Nguyen, T. D. , and Boyce, B. L. , 2011, “An Inverse Finite Element Method for Determining the Anisotropic Properties of the Cornea,” Biomech. Model. Mechanobiol., 10(3), pp. 323–337. [CrossRef] [PubMed]
Delgadillo, J. O. V. , Delorme, S. , Thibault, F. , DiRaddo, R. , and Hatzikiriakos, S. G. , 2015, “Large Deformation Characterization of Porcine Thoracic Aortas: Inverse Modeling Fitting of Uniaxial and Biaxial Tests,” J. Biomed. Sci. Eng., 8(10), p. 717. [CrossRef]
Mei, Y. , Fulmer, R. , Raja, V. , Wang, S. , and Goenezen, S. , 2016, “Estimating the Non-Homogeneous Elastic Modulus Distribution From Surface Deformations,” Int. J. Solids Struct., 83, pp. 73–80. [CrossRef]
Mei, Y. , Wang, S. , Shen, X. , Rabke, S. , and Goenezen, S. , 2017, “Mechanics Based Tomography: A Preliminary Feasibility Study,” Sensors, 17(5), p. 1075. [CrossRef]
Gambarotta, L. , Massabo, R. , Morbiducci, R. , Raposio, E. , and Santi, P. , 2005, “In Vivo Experimental Testing and Model Identification of Human Scalp Skin,” J. Biomech., 38(11), pp. 2237–2247. [CrossRef] [PubMed]
Wang, Z. G. , Liu, Y. , Wang, G. , and Sun, L. Z. , 2011, “Nonlinear Elasto-Mammography for Characterization of Breast Tissue Properties,” Int. J. Biomed. Imaging, 2011, p. 5.
Lago, M. A. , Rupérez, M. J. , Martínez-Martínez, F. , Monserrat, C. , Larra, E. , Güell, J. L. , and Peris-Martínez, C. , 2015, “A New Methodology for the In Vivo Estimation of the Elastic Constants That Characterize the Patient-Specific Biomechanical Behavior of the Human Cornea,” J. Biomech., 48(1), pp. 38–43. [CrossRef] [PubMed]
Simón-Allué, R. , Calvo, B. , Oberai, A. A. , and Barbone, P. E. , 2017, “Towards the Mechanical Characterization of Abdominal Wall by Inverse Analysis,” J. Mech. Behav. Biomed. Mater., 66, pp. 127–137. [CrossRef] [PubMed]
Roan, E. , and Vemaganti, K. , 2007, “The Nonlinear Material Properties of Liver Tissue Determined From No-Slip Uniaxial Compression Experiments,” ASME J. Biomech. Eng., 129(3), pp. 450–456. [CrossRef]
Zhang, C. , Wu, J. , Hwang, K. C. , and Huang, Y. , 2016, “Postbuckling of Hyperelastic Plates,” ASME J. Appl. Mech., 83(5), p. 051012. [CrossRef]
Breslavsky, I. D. , Amabili, M. , and Legrand, M. , 2016, “Static and Dynamic Behavior of Circular Cylindrical Shell Made of Hyperelastic Arterial Material,” ASME J. Appl. Mech., 83(5), p. 051002. [CrossRef]
Zhang, C. , Wu, J. , and Hwang, K. C. , 2015, “Hyperelastic Thin Shells: Equilibrium Equations and Boundary Conditions,” ASME J. Appl. Mech., 82(9), p. 094502. [CrossRef]
Dhavale, N. N. , Tamadapu, G. , and DasGupta, A. , 2014, “Finite Inflation Analysis of Two Circumferentially Bonded Hyperelastic Circular Flat Membranes,” ASME J. Appl. Mech., 81(9), p. 091012. [CrossRef]
Boyce, M. C. , and Arruda, E. M. , 2000, “Constitutive Models of Rubber Elasticity: A Review,” Rubber Chem. Technol., 73(3), pp. 504–523. [CrossRef]
Vahapoğlu, V. , and Karadeniz, S. , 2006, “Constitutive Equations for Isotropic Rubber-Like Materials Using Phenomenological Approach: A Bibliography (1930–2003),” Rubber Chem. Technol., 79(3), pp. 489–499. [CrossRef]
Payan, Y. , and Ohayon, J. , 2017, Biomechanics of Living Organs: Hyperelastic Constitutive Laws for Finite Element Modeling, World Bank Publications, London.
Demiray, H. , Weizsäcker, H. W. , Pascale, K. , and Erbay, H. , 1988, “A Stress-Strain Relation for a Rat Abdominal Aorta,” J. Biomech., 21(5), pp. 369–374. [CrossRef] [PubMed]
Holmes, M. H. , and Mow, V. C. , 1990, “The Nonlinear Characteristics of Soft Gels and Hydrated Connective Tissues in Ultrafiltration,” J. Biomech., 23(11), pp. 1145–1156. [CrossRef] [PubMed]
Holzapfel, A. G. , 2000, Nonlinear Solid Mechanics II, Wiley, West Sussex, UK.
Bower, A. F. , 2009, Applied Mechanics of Solids, CRC Press, Boca Raton, FL.
Björck, Å. , 1996, Numerical Methods for Least Squares Problems, Society for Industrial and Applied Mathematics, North-Holland, The Netherlands. [CrossRef]
Ortega, J. M. , and Rheinboldt, W. C. , 2000, Iterative Solution of Nonlinear Equations in Several Variables, Society for Industrial and Applied Mathematics, Philadelphia, PA. [CrossRef]
Hematiyan, M. R. , Khosravifard, A. , Shiah, Y. C. , and Tan, C. L. , 2012, “Identification of Material Parameters of Two-Dimensional Anisotropic Bodies Using an Inverse Multi-Loading Boundary Element Technique,” Comput. Model. Eng. Sci. (CMES), 87(1), pp. 55–76.
Hematiyan, M. R. , Khosravifard, A. , and Shiah, Y. C. , 2015, “A Novel Inverse Method for Identification of 3D Thermal Conductivity Coefficients of Anisotropic Media by the Boundary Element Analysis,” Int. J. Heat Mass Transfer, 89, pp. 685–693. [CrossRef]
Hematiyan, M. R. , Khosravifard, A. , and Shiah, Y. C. , 2017, “A New Stable Inverse Method for Identification of the Elastic Constants of a Three-Dimensional Generally Anisotropic Solid,” Int. J. Solids Struct., 106–107, pp. 240–250. [CrossRef]
Avril, S. , Bouten, L. , Dubuis, L. , Drapier, S. , and Pouget, J. F. , 2010, “Mixed Experimental and Numerical Approach for Characterizing the Biomechanical Response of the Human Leg Under Elastic Compression,” ASME J. Biomech. Eng., 132(3), p. 031006. [CrossRef]
Franquet, A. , Avril, S. , Le Riche, R. , and Badel, P. , 2012, “Identification of Heterogeneous Elastic Properties in Stenosed Arteries: A Numerical Plane Strain Study,” Comput. Methods Biomech. Biomed. Eng., 15(1), pp. 49–58. [CrossRef]
Affagard, J. S. , Feissel, P. , and Bensamoun, S. F. , 2015, “Identification of Hyperelastic Properties of Passive Thigh Muscle Under Compression With an Inverse Method From a Displacement Field Measurement,” J. Biomech., 48(15), pp. 4081–4086. [CrossRef] [PubMed]
Rauchs, G. , Bardon, J. , and Georges, D. , 2010, “Identification of the Material Parameters of a Viscous Hyperelastic Constitutive Law From Spherical Indentation Tests of Rubber and Validation by Tensile Tests,” Mech. Mater., 42(11), pp. 961–973. [CrossRef]
Moerman, K. M. , Holt, C. A. , Evans, S. L. , and Simms, C. K. , 2009, “Digital Image Correlation and Finite Element Modelling as a Method to Determine Mechanical Properties of Human Soft Tissue In Vivo,” J. Biomech., 42(8), pp. 1150–1153. [CrossRef] [PubMed]
Genovese, K. , Casaletto, L. , Humphrey, J. D. , and Lu, J. , 2014, “Digital Image Correlation-Based Point-Wise Inverse Characterization of Heterogeneous Material Properties of Gallbladder In Vitro,” Proc. R. Soc. A, 470(2167), p. 20140152. [CrossRef]

Figures

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Fig. 1

Reference and deformed configuration for a deformable solid

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Fig. 2

(a) target material attached to an unknown material and (b) separate illustration of deformed target material after applying external forces for inverse analysis

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Fig. 3

The geometry of the direct problem and its boundary conditions

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Fig. 4

The geometry of the inverse problem with some controlling and sampling points

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Fig. 5

The deformed shape of the interface AB obtained by the inverse analysis for the neo–Hookean model with one controlling point and six sampling points under plane strain conditions

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Fig. 6

The deformed shape of the interface AB obtained by the inverse analysis for the neo–Hookean model with one controlling point and eight sampling points under plane strain conditions

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Fig. 7

The inverse problem under plane strain conditions with two controlling points and eight sampling points

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Fig. 8

The deformed shape of the interface AB obtained by the inverse analysis for the neo–Hookean model with two controlling points and eight sampling points under plane strain conditions

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Fig. 9

The inverse problem with three controlling points and ten sampling points for the plane stress condition

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Fig. 10

Nominal stress–stretch curves for the Mooney–Rivlin hyper-elastic material with constants μ1 = 80 Pa, μ2 = 20 Pa, and K1 = 2000 and the material parameters reported in Table 8

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Fig. 11

A part of the speckle pattern on the sample surface and a unique feature in a facet

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Fig. 12

(a) Dimensions of the rectangular sample and (b) the sample with sandpaper at both ends and random gray pattern on its surface

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Fig. 13

Force-stretch curve of the rectangular silicone sample

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Fig. 14

(a) Dimensions of the silicone sample and (b) the silicone sample molded in the customized wooden mold

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Fig. 15

The silicone sample under tension with INSTRON 5567

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Fig. 16

The part of the sample used for inverse analysis

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Fig. 17

Sampling points used for inverse analysis

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Fig. 18

Force-stretch curve of the neo–Hookean model obtained from the inverse analysis in comparison with the experimental curve

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Fig. 19

(a) Displacement contour in the member obtained from ISTRA 4D software in mm and (b) displacement contour corresponding to the results of the inverse analysis in m

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Fig. 20

Comparison of the deformed shape of the interface AB in the experiment and reconstructed from the inverse analysis

Tables

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