Incompressible Transversely Isotropic Hyperelastic Materials and Their Linearized Counterparts

Abstract

The strain-energy density $W$ for incompressible transversely isotropic hyperelastic materials depends on four independent invariants of the strain tensor. For consistency with the infinitesimal theory, it is well known that there are three necessary conditions on the derivatives of $W$ (evaluated in the undeformed state) that have to be to be satisfied in terms of the three independent elastic moduli of the linear theory for incompressible transversely isotropic materials. We consider three different sets of these linear elastic moduli and express them in terms of the relevant derivatives of $W $ evaluated in the undeformed state. Necessary and sufficient conditions on the linearized strain-energy to ensure positive-definiteness are given and thus we derive such conditions expressed in terms of derivatives of $W$ evaluated in the undeformed state. These conditions are proposed to be fundamental constitutive inequalities which must be satisfied by any physically realistic strain-energy for incompressible transversely isotropic hyperelastic material, in particular those models proposed for fiber-reinforced soft tissues. Some particular strain-energies that have been proposed in the literature are used as illustrations of the results.