Hyperelastic constitutive modelling for transversely isotropic composites and orthotropic biological tissues

Abstract

This work presents an orthotropic hyperelastic strain energy function (SEF) and associated nonlinear constitutive theory that describes the response of transversely isotropic and orthotropic neo-Hookean materials under a range of physical deformations in which the strains are large. The proposed SEF is categorised as invariant-free and is presented as an expression involving quadruple contractions between fourth-order tensors. This form of the SEF retains directional information which can sometimes become lost in typical functions involving scalar invariants. To describe stress-strain relations, the well-known Hookean stiffness tensor for small deformations is decomposed into fourth-order Orthotropic Lamé material tensors separating coupled and non-coupled behaviours. By equating the material parameters in each direction, the model collapses to isotropic Lamé hyperelasticity. The proposed hyperelastic model remains consistent with the linear analysis of anisotropic bodies, when subject to infinitesimal deformations. The proposed SEF compares favourably to large tensile strain and simple shear experimental tests of both isotropic and transversely isotropic materials. Employing an Ogden-type formulation to include additional material terms, the proposed SEF is extended to model highly nonlinear responses such as material stiffening of biological tissues.