Nonlinear Finite Element Formulation Based on Invariant-Free Hyperelasticity for Orthotropic Materials

The conventional representation of isotropic hyperelastic strain energy densities as functions of scalar invariants of finite deformation tensors does not naturally extend to the field of anisotropic mechanics. Formulating an invariant-free representation of the strain energy function, fourth-order Orthotropic Lamé tensors define the constitutive law whilst naturally collapsing to the transversely isotropic and fully isotropic case where necessary simply as a by-product of known material symmetries. In this study, a simple linear isoparametric hexahedral finite element capable of describing anisotropic invariant-free hyperelasticity is presented. Careful conversion of the fourth-order tensor operations present in the strain energy function to computational arrays then applying to the principle of virtual work generates a weak formulation for finite element analyses. The finite element is then applicable to materials of any degree of anisotropy or compressibility and is particularly useful for predicting highly nonlinear responses such as the stiffening of fibrous biological tissues. A discussion of simple shear experimentation and modelling follows, as well as remarks on modelling nearly-incompressible materials.