A constitutive model for fibre-matrix interaction in fibre-reinforced hyperelastic materials

Abstract The strain energy for incompressible transversely isotropic nonlinearly elastic materials reinforced by a single family of fibres is decomposed into an isotropic part representing the mechanical response of the isotropic matrix and an anisotropic part representing the contribution to the mechanical response from the fibres. It is the form of the anisotropic component that is of interest here. We note that the classical deformation invariants can be divided into two classes: the invariants that are of degree two and those of degree four in the principal stretches. The approach adopted here is straightforward: assume that there is a proportional relationship linking derivatives of the strain energy with respect to invariants of the same degree in the principal stretches in the general stress-strain law. The proportionality constants are interpreted as being measures of the fibre-matrix interaction in transversely isotropic non-linear elasticity. Setting these constants identically zero recovers many of the simplified strain energies commonly found in the literature. It is shown that inclusion of a matrix-fibre interaction can have a significant effect on the mechanical response. An influential method of incorporating fibre dispersion into the modelling procedure is shown to be a special case of the approach introduced here.