A model for fibre–matrix interaction in non-linearly elastic incompressible orthotropic materials

Abstract

The strain energy for incompressible anisotropic non-linearly elastic materials is decomposed into an isotropic part representing the mechanical response of an isotropic matrix and an anisotropic part representing the contribution to the mechanical response from the presence of fibres. It is the form of the anisotropic component that is of interest here. We note that the invariants can themselves be divided into two classes: the invariants that are homogeneous functions of degree two and those of degree four in the principal stretches. The approach adopted here is straightforward: assume that there is a linear proportional relationship between terms in the general stress–strain law that are of the same degree in the principal stretches. Setting these constants identically zero recovers many of the simplified strain energies commonly found in the literature. The proportionality constants are interpreted as being a measure of the fibre–matrix interaction and a measure of the interaction between fibres in anisotropic non-linear elasticity. An influential model of fibre dispersion is recovered as a special case. The results are illustrated using the homogeneous deformation of simple shear.