  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.
