Tension in the fibres of anisotropic non-linearly hyperelastic materials. Some stability results and constitutive restrictions

A new notion of stability for compressible, transversely isotropic hyperelastic, nonlinear materials is introduced whereby an anisotropic material is considered stable if the stress in the direction of anisotropy is both tensile and is a monotonically increasing function of the fibre stretch. The physical motivation for these criteria is that the fibres which typically induce anisotropy are stable only in tension and that increasing fibre stretch should result in greater fibre tension. These criteria are then used as the basis of a constitutive assumption for fibre tension: specifically, it is assumed that the fibres are a monotonically increasing function of the fibre stretch and the strain-energy function consistent with this assumption is derived. Such materials automatically satisfy the stability criteria proposed here. The stability criteria are then reformulated in terms of the strain-energy function commonly used to account for slight compressibility. The stability of a specific slightly compressible material is then studied for illustrative purposes. The corresponding stability criteria, and the practical difficulties in enforcing them, for perfectly incompressible materials are briefly discussed.