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What-you-prescribe-is-what-you-get orthotropic hyperelasticity. (English) Zbl 1398.74028
Summary: We present a model for incompressible finite strain orthotropic hyperelasticity using logarithmic strains. The model does not have a prescribed shape. Instead, the energy function shape and the material data of the model are obtained solving the equilibrium equations of the different experiments. As a result the model almost exactly replicates the given experimental data for all six tests needed to completely define our nonlinear orthotropic material. We derive the constitutive tensor and demonstrate the efficiency of the finite element implementation for complex loading situations.

74B20 Nonlinear elasticity
74S05 Finite element methods applied to problems in solid mechanics
74L15 Biomechanical solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74E10 Anisotropy in solid mechanics
Full Text: DOI
[1] Bathe KJ (1996) Finite element procedures. Prentice-Hall, New Jersey
[2] Kojic M, Bathe KJ (2005) Inelastic analysis of solids and structures. Springer, New York
[3] Ogden RW (1997) Non-linear elastic deformations. Dover, Mineola
[4] Montáns, FJ; Bathe, KJ; Onate, E (ed.); Owen, R (ed.), Towards a model for large strain anisotropic elasto-plasticity, 13-36, (2007), Netherlands
[5] Caminero, MÁ; Montáns, FJ; Bathe, KJ, Modeling large strain anisotropic elasto-plasticity with logarithmic strain and stress measures, Comp Struct, 89, 826-843, (2011)
[6] Kojic, M; Bathe, KJ, Studies of finite element procedures stress solution of a closed elastic strain path with stretching and shearing using the updated Lagrangian jaumann formulation, Comp Struct, 26, 175-179, (1987) · Zbl 0609.73074
[7] Ogden, RW, Large deformation isotropic elasticity: on the correlation of theory and experiment for incompressible rubberlike solids, Proc R Soc Lond A, 326, 565-584, (1972) · Zbl 0257.73034
[8] Mooney, M, A theory of large elastic deformation, J App Phys, 11, 582-592, (1940) · JFM 66.1021.04
[9] Rivlin, RS, Large elastic deformations of isotropic materials. IV. further developments of the general theory, Phil Trans R Soc Lond A, 241, 379-397, (1948) · Zbl 0031.42602
[10] Blatz, PJ; Ko, WL, Application of finite elasticity theory to the deformation of rubbery materials, Trans Soc Rheol, 6, 223-251, (1962)
[11] Yeoh, OH, Characterization of elastic properties of carbon-black-filled rubber vulcanizates, Rubber Chem Technol, 63, 792-805, (1990)
[12] Arruda, EM; Boyce, MC, A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, J Mech Phys Solids, 41, 389-412, (1993) · Zbl 1355.74020
[13] Twizell, EH; Ogden, RW, Non-linear optimization of the material constants in ogden’s stress-deformation function for incompressible isotropic elastic materials, J Aust Math Soc B, 24, 424-434, (1983) · Zbl 0504.73025
[14] Itskov, M, A generalized orthotropic hyperelastic material model with application to incompressible shells, Int J Number Method Eng, 50, 1777-1799, (2001) · Zbl 0997.74006
[15] Itskov, M; Aksel, N, A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function, Int J Solid Struct, 41, 3833-3848, (2004) · Zbl 1079.74516
[16] Gasser, TC; Ogden, RW; Holzapfel, GA, Hyperelastic modelling of arterial layers with distributed collagen fibre orientations, J R Soc Interface, 3, 15-35, (2006)
[17] Holzapfel, GA; Ogden, RW, Constitutive modelling of passive myocardium: a characterization structurally based framework for material characterization, Phil Trans R Soc A, 367, 3445-3475, (2009) · Zbl 1185.74060
[18] Holzapfel, GA; Gasser, TC, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J Elast, 61, 1-48, (2000) · Zbl 1023.74033
[19] Sussman, T; Bathe, KJ, A model of incompressible isotropic hyperelastic material behavior using spline interpolations of tension-compression test data, Commun Number Method Eng, 25, 53-63, (2009) · Zbl 1156.74008
[20] Kearsley, EA; Zapas, LJ, Some methods of measurement of an elastic strain-energy function of the valanis-landel type, J Rheol, 24, 483-501, (1980)
[21] Latorre, M; Montáns, FJ, Extension of the sussman-bathe spline-based hyperelastic model to incompressible transversely isotropic materials, Comp Struct, 122, 13-26, (2013)
[22] Latorre M, Montáns FJ (2014) On the interpretation of the logarithmic strain tensor in an arbitrary system of representation. Int J Solids Struct (in press) · Zbl 0257.73034
[23] Valanis, KC; Landel, RF, The strain-energy function of a hyperelastic material in terms of the extension ratios, J App Phys, 38, 2997-3002, (1967)
[24] Montáns FJ, Benítez JM, Caminero MÁ (2012) A large strain anisotropic elastoplastic continuum theory for nonlinear kinematic hardening and texture evolution. Mech Res Commun 43: 50-56
[25] Miehe, C; Lambrecht, M, Algorithms for computation of stresses and elasticity moduli in terms of seth-hill’s family of generalized strain tensors, Commun Numer Methods Eng, 17, 337-353, (2001) · Zbl 1049.74011
[26] Anand, L, On H. hencky’s approximate strain-energy function for moderate deformations, J App Mech, 46, 78-82, (1979) · Zbl 0405.73032
[27] Anand, L, Moderate deformations in extension-torsion of incompressible isotropic elastic materials, J Mech Phys Solid, 34, 293-304, (1986)
[28] Morrow, DA; Donahue, TLH; Odegard, GM; Kaufman, KR, Transversely isotropic tensile material properties of skeletal muscle tissue, J Mech Behav Biomed Mater, 3, 124-129, (2010)
[29] Holzapfel, GA; Sommer, G; Gasser, TC; Regitnig, P, Determination of layer-specific mechanical properties of human coronary arteries with nonatherosclerotic intimal thickening and related constitutive modeling, Am J Physiol Heart Circ Physiol, 289, 2048-2058, (2005)
[30] Sacks, MS, A method for planar biaxial mechanical testing that includes in-plane shear, J Biomech Eng, 121, 551-555, (1999)
[31] Dokos S, Smaill BH, Young AA, LeGrice IL (2002) Shear properties of passive ventricular myocardium. Am J Physiol Heart Circ Physiol 283:2650-2659 · Zbl 1051.74539
[32] Diani J, Brieu M, Vacherand JM, Rezgui A (2004) Directional model for isotropic and anisotropic hyperelastic rubber-like materials. Mech Mater 36:313-321
[33] Sussman, T; Bathe, KJ, A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Comp Struct, 26, 357-409, (1987) · Zbl 0609.73073
[34] Hartmann, S; Neff, P, Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility, Int J Solids Struct, 40, 2767-2791, (2003) · Zbl 1051.74539
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