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We give conditions on the strain-energy function of nonlinear anisotropichyperelastic materials that ensure compatibility with the classical lineartheories of anisotropic elasticity. We uncover the limitations associated withthe volumetric deviatoric separation of the strain energy used, for example, inmany Finite Element (FE) codes in that it does not fully represent the behaviorof anisotropic materials in the linear regime. This limitation has importantconsequences. We show that, in the small deformation regime, a FE code based onthe volumetric-deviatoric separation assumption predicts that a sphere made ofa compressible anisotropic material deforms into another sphere underhydrostatic pressure loading, instead of the expected ellipsoid. For finitedeformations, the commonly adopted assumption that fibres cannot supportcompression is incorrectly implemented in current FE codes and leads to theunphysical result that under hydrostatic tension a sphere of compressibleanisotropic material deforms into a larger sphere.

On anisotropic elasticity and questions concerning its Finite Element implementation