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On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials. (English) Zbl 07037449
Summary: Quasi-incompressible behavior is a desired feature in several constitutive models within the finite elasticity of solids, such as rubber-like materials and some fiber-reinforced soft biological tissues. The Q1P0 finite element formulation, derived from the three-field Hu-Washizu variational principle, has hitherto been exploited along with the augmented Lagrangian method to enforce incompressibility. This formulation typically uses the unimodular deformation gradient. However, contributions by C. Sansour [Eur. J. Mech. A, Solids 27, No. 1, 28–39 (2008; Zbl 1129.74009)] and J. Helfenstein et al. [Int. J. Solids Struct. 47, No. 16, 2056–2061 (2010; Zbl 1194.74042)] conspicuously demonstrate an alternative concept for analyzing fiber reinforced solids, namely the use of the (unsplit) deformation gradient for the anisotropic contribution, and these authors elaborate on their proposals with analytical evidence. The present study handles the alternative concept from a purely numerical point of view, and addresses systematic comparisons with respect to the classical treatment of the Q1P0 element and its coalescence with the augmented Lagrangian method by means of representative numerical examples. The results corroborate the new concept, show its numerical efficiency and reveal a direct physical interpretation of the fiber stretches.

74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
74E10 Anisotropy in solid mechanics
Full Text: DOI
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