On anisotropic elasticity and questions concerning its Finite Element implementation

TL;DR

This paper analyzes the compatibility of nonlinear anisotropic hyperelastic material models with classical linear elasticity and reveals limitations in finite element implementations, especially regarding volumetric-deviatoric energy separation.

Contribution

It provides conditions for strain-energy functions to align with linear theories and highlights inaccuracies in current FE codes related to anisotropic material behavior.

Findings

Volumetric-deviatoric separation does not fully capture anisotropic behavior in linear regime.

FE codes incorrectly predict sphere deformation under hydrostatic pressure.

Current models assume fibers can't support compression, leading to unphysical results.

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.