On anisotropic elasticity and questions concerning its Finite Element implementation

Abstract

We give conditions on the strain-energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the volumetric deviatoric separation of the strain energy used, for example, in many Finite Element (FE) codes in that it does not fully represent the behavior of anisotropic materials in the linear regime. This limitation has important consequences. We show that, in the small deformation regime, a FE code based on the volumetric-deviatoric separation assumption predicts that a sphere made of a compressible anisotropic material deforms into another sphere under hydrostatic pressure loading, instead of the expected ellipsoid. For finite deformations, the commonly adopted assumption that fibres cannot support compression is incorrectly implemented in current FE codes and leads to the unphysical result that under hydrostatic tension a sphere of compressible anisotropic material deforms into a larger sphere.

Finite element implementation of a new model of slight compressibility for transversely isotropic materials

“…This is best illustrated by simulating the hydrostatic tension of a transversely isotropic cube, which unrealistically deforms into another larger cube instead of a rectangular cuboid because of the additive split of the SEF (Ní Annaidh et al 2013;Gilchrist et al 2014). This deficiency has also been shown for a transversely isotropic sphere deforming into another sphere (Vergori et al 2013). It also results in an incompatibility with the linear theory, which gives correct anisotropic results.…”

Constitutive modelling of fibre reinforced nonhomogenous hyperelastic materials

“…In tasks i) and ii) there is solution for the orthotropic circular non-homogeneous disk. In the paper [21], the boundary problem of hydrostatic tension test for sphere made of transversally isotropic hyper-elastic material was considered. For this type of material available in ABAQUS [9], non-physical results were obtained.…”

“…In particular, the proposed SE functions are simplifications of the SE functions applied in our earlier works [10,12,13]. These models do not lead to non-realistic results, as shown in [21], where the transversally isotropic material models available in the ABAQUS program [9] were used. The heterogeneity of the material result from the different arrangement of the fibres in the matrix material.…”

Hyperelastic behaviour of auxetic material in tension and compression tests

“…Anisotropic hyper-elastic materials form a narrow group among constitutive models which describe elastic behaviour. An energetic approach to the analysis of anisotropic hyper-elastic materials is proposed by Vegori et al, [18]. According to work by Dłużewski [6], the explicit form of a constitutive equation for anisotropic nonlinear materials for Cauchy stress has not yet been formulated.…”

“…According to work by Dłużewski [6], the explicit form of a constitutive equation for anisotropic nonlinear materials for Cauchy stress has not yet been formulated. When the material undergoes finite deformation, the material properties are strongly dependent on the deformation and the effective material properties have to be calculated for each deformation state (Vegori,[18]). Therefore, it is possible for instantaneous stiffness in uniaxial strain test.…”