Fibre stretch models of transversely isotropic incompressible hyperelasticity

A new strain–energy function for incompressible transversely isotropic non-linearly hyperelastic materials is obtained by assuming that the deviatoric stress in the preferred direction is a function only of the stretch in that direction. The model is an additive combination of two arbitrary functions of the strain invariants, one of which is an arbitrary function of the fibre stretch. The behaviour of the model for the canonical deformations of simple tension and shear is analysed. It is shown that the stress–stretch response in simple tension along and perpendicular to the fibres is determined by the fibre stretch function alone. It is also shown that a linearisation of the general model containing only three arbitrary constants is an excellent fit to a comprehensive experimental data set of other authors for simple tension along and perpendicular to the fibres and simple shear along the fibres.