Modelling slight compressibility for hyperelastic anisotropic materials

In order to avoid the numerical difficulties in locally enforcing the incompressibility constraint using the displacement formulation of the Finite Element Method, slight compressibility is typically assumed when simulating the mechanical response of nonlinearly elastic materials. The current standard method of accounting for slight compressibility of hyperelastic materials assumes an additive decomposition of the strain-energy function into a volumetric and an isochoric part. A new proof is given to show that this is equivalent to assuming that the hydrostatic stress is a function of the the volume change only and that uniform dilatation is a possible solution to the hydrostatic stress boundary value problem, with therefore no anisotropic contribution to the mechanical response. An alternative formulation of slight compressibility is proposed, one that does not suffer from this defect. This new model generalises the standard model by including a mixed term in the volume change and isochoric response. Specific models of slight compressibility are given for isotropic, transversely isotropic and orthotropic materials