Journal of Advances in Applied Mathematics

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DOI: 10.22606/jaam.2020.52002

• Fuzhang Zhao^*
  APD Optima Study, Lake Forest, CA 92630, USA

Constitutive models and finite element implementations of compressible finite deformation are straightforwardly formulated by the general isotropic continuum stored energy (CSE) functional without the isochoric–volumetric split. Coupled stress and elasticity tensors in reference and current configurations are derived. Modeling and predicting capabilities of the general CSE functional are exhibited through multiaxial experimental tests of compressible NR and SBR rubbers. Characterization of kinematic relation, rather than pressure–volume relation, is emphasized in experimental tests of compressibility. The isochoric–volumetric split does not hold based on either theoretical analyses or experimental validations.

Compressible finite deformation, finite element implementation, isochoric–volumetric split, Poisson function, work-conjugacy.

[1] I. Fried, “Reflections on the computational approximation of elastic incompressibility,” Computers & Structures, vol. 17, no. 2, pp. 161–168, 1983.

[2] T. Sussman and K.-J. Bathe, “A finite element formulation for nonlinear incompressible elastic and inelastic analysis,” Computers & Structures, vol. 26, no. 1-2, pp. 357–409, 1987.

[3] H. Richter, “Das isotrope elastizit¨atsgesetz,” Zeitschrift f¨ur angewandte Mathematik und Mechanik, vol. 28, no. 7-8, pp. 205–209, 1948.

[4] P. J. Flory, “Thermodynamic relations for high elastic materials,” Transactions of the Faraday Society, vol. 57, pp. 829–838, 1961.

[5] S. H. C. Lu and K. S. Pister, “Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids,” International Journal of Solids and Structures, vol. 11, no. 7-8, pp. 927–934, 1975.

[6] C. Sansour, “On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy,” European Journal of Mechanics A/Solids, vol. 27, no. 1, pp. 28–39, 2008.

[7] J. Lubliner, “A model of rubber viscoelasticity,” Mechanics Research Communications, vol. 12, no. 2, pp. 93–99, 1985.

[8] J. C. Simo and R. L. Taylor, “Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms,” Computer Methods in Applied Mechanics and Engineering, vol. 85, no. 3, pp. 273–310, 1991.

[9] R. W. Ogden, Non-linear Elastic Deformations, 1st ed. New York: Dover, 1997.

[10] G. A. Holzapfel, Nonlinear Solid Mechanics: A Continuum Approach for Engineering, 1st ed. Chichester: John Wiley & Sons, 2000.

[11] D. W. Nicholson, “Tangent modulus matrix for finite element analysis of hyperelastic materials,” Acta Mechanica, vol. 112, no. 1-4, pp. 187–201, 1995.

[12] J. A. Weiss, B. N. Maker, and S. Govindjee, “Finite element implementation of incompressible, transversely isotropic hyperelasticity,” Computer Methods in Applied Mechanics and Engineering, vol. 135, no. 1-2, pp. 107–128, 1996.

[13] M. Itskov, “On the theory of fourth-order tensors and their applications in computational mechanics,” Computer Methods in Applied Mechanics and Engineering, vol. 189, no. 2, pp. 419–438, 2000.

[14] C. Suchocki, “A finite element implementation of Knowles stored-energy function: Theory, coding and applications,” The Archive of Mechanical Engineering, vol. LVIII, no. 3, pp. 319–346, 2011.

[15] J. Cheng and L. T. Zhang, “A general approach to derive stress and elasticity tensors for hyperelastic isotropic and anisotropic biomaterials,” International Journal of Computational Methods, vol. 15, no. 1, pp. 1 850 028 (1–33), 2018.

[16] M. C. Boyce and E. M. Arruda, “Constitutive models of rubber elasticity: A review,” Rubber Chemistry and Technology, vol. 73, no. 3, pp. 504–523, 2000.

[17] P. J. Blatz and W. L. Ko, “Application of finite elastic theory to the deformation of rubbery materials,” Transactions of the Society of Rheology, vol. 6, no. 8, pp. 223–251, 1962.

[18] R. J. Ogden, “Volume changes associated with the deformation of rubber-like solids,” Journal of the Mechanics and Physics of Solids, vol. 24, no. 6, pp. 323–251, 1976.

[19] I. Fried and A. R. Johnson, “A note on elastic energy density functions for largely deformed compressible rubber solids,” Computer Methods in Applied Mechanics and Engineering, vol. 69, no. 1, pp. 53–64, 1988.

[20] L. Anand, “A constitutive model for compressible elastomeric solids,” Computational Mechanics, vol. 18, no. 5, pp. 339–355, 1996.

[21] J. E. Bischoff, E. M. Arruda, and K. Grosh, “A new constitutive model for the compressibility of elastomers at finite deformations,” Rubber Chemistry and Technology, vol. 74, no. 4, pp. 541–559, 2001.

[22] J.-B. Le Cam, “A review of volume changes in rubbers: The effect of stretching,” Rubber Chemistry and Technology, vol. 83, no. 3, pp. 247–269, 2010.

[23] R. W. Penn, “Volume changes accompanying the extension of rubber,” Transactions of the Society of Rheology, vol. 14, no. 4, pp. 509–517, 1970.

[24] W. Ehlers and G. Eipper, “The simple tension problem at large volumetric strains computed from finite hyperelastic material laws,” Acta Mechanica, vol. 130, no. 1-2, pp. 17–27, 1998.

[25] F. Zhao, “Continuum constitutive modeling for isotropic hyperelastic materials,” Advances in Pure Mathematics, vol. 6, no. 9, pp. 571–582, 2016.

[26] T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, 1st ed. Englewood Cliffs, NJ: Prentice-Hall, 1987.

[27] O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu, The Finite Element Method: Its Basis and Fundamentals, 6th ed. New York: Elsevier Butterworth-Heinemann, 2005.

[28] K.-J. Bathe, Finite Element Procedures, 2nd ed. Watertown, MA: Klaus-Jürgen Bathe, 2014.

[29] F. Zhao, “Anisotropic continuum stored energy functional solved by Lie group and differential geometry,” Advances in Pure Mathematics, vol. 8, no. 7, pp. 631–651, 2018.

[30] O. Kintzel and Y. Basar, “Fourth-order tensors–tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis,” Zeitschrift f¨ur angewandte Mathematik und Mechanik, vol. 86, no. 4, pp. 291–311, 2006.

[31] O. Kintzel, “Fourth-order tensors–tensor differentiation with applications to continuum mechanics. Part II: Tensor analysis on manifolds,” Zeitschrift f¨ur angewandte Mathematik und Mechanik, vol. 86, no. 4, pp. 312–334, 2006.

[32] B. Storåkers, “On material representation and constitutive branching in finite compressible elasticity,” Journal of the Mechanics and Physics of Solids, vol. 34, no. 2, pp. 125–145, 1986.

[33] F. Zhao, “On constitutive modeling of arteries,” Journal of Advances in Applied Mathematics, vol. 4, no. 2, pp. 54–68, 2019.

[34] O. Starkova and A. Aniskevich, “Poisson’s ratio and the incompressibility relation for various strain measures with the example of a silica-filled SBR rubber in uniaxial tension tests,” Polymer Testing, vol. 29, no. 3, pp. 310–318, 2010.

[35] C. O. Horgan and J. G. Murphy, “Constitutive modeling for moderate deformations of slightly compressible rubber,” Journal of Rheology, vol. 53, no. 1, pp. 153–168, 2009.

[36] A. F. Amin, M. S. Alam, and Y. Okui, “Measurement of lateral deformation in natural and high damping rubbers in large deformation uniaxial tests,” Journal of Testing and Evaluation, vol. 31, no. 6, pp. 524–532, 2003.

[37] P. Charrier, B. Dacorohna, B. Hanouzet, and P. Laborde, “An existence theorem for slightly compressible materials in nonlinear elasticity,” SIAM Journal on Mathematical Analysis, vol. 19, no. 1, pp. 70–85, 1988.

[38] J. G. Murphy and G. A. Rogerson, “Modelling slight compressibility for hyperelastic anisotropic materials,” Journal of Elasticity, vol. 131, no. 2, pp. 171–181, 2018.

[39] F. Zhao, “The continuum stored energy for constitutive modeling finite deformations of polymeric materials,” Advances in Pure Mathematics, vol. 7, no. 10, pp. 597–613, 2017.

[40] Z. P. Baˇzant, M. Gattu, and J. Vorel, “Work conjugacy error in commercial finite-element codes: its magnitude and how to compensate for it,” Proceedings of the Royal Society A, vol. 468, pp. 3047–3058, 2012.

[41] W. Ji, A. M. Waas, and Z. P. Baˇzant, “Errors caused by non-work-conjugate stress and strain measures and necessary corrections in finite element programs,” Journal of Applied Mechanics T-ASME, vol. 77, no. 4, pp. 044 504 (1–5), 2010.

[42] ——, “On the importance of work-conjugacy and objective stress rates in finite deformation incremental finite element analysis,” Journal of Applied Mechanics T-ASME, vol. 80, no. 4, pp. 041 024 (1–9), 2013.