Pitchfork bifurcations in simple hyperelastic orthotropic arterial models and their constitutive implications

Abstract

This paper is concerned with the response of orthotropic hyperelastic incompressible materials in the homogeneous deformation of simple tension. The problem of out-of-plane simple tension of a cuboid reinforced with two in-plane families of mechanically equivalent initially straight fibres is considered. For this deformation, the material characterisation test where the normal in-plane stresses are equal is examined. Analytical results are obtained for the special case of orthonormal fibres that is where the fibres are initially perpendicular to one another in the undeformed state. It is shown that in this case, there are two distinct solution branches namely the symmetric solution in which the in-plane stretches are equal and an asymmetric solution where this is not the case. The results are illustrated for two specific strain energy densities one of which has been used to model the mechanical response of arteries. For these two models, the asymmetric solution is shown to be energetically favourable at a sufficiently large critical out-of-plane stretch. For small enough out-of-plane stretch, for weakly anisotropic materials, the symmetric solution branch is unique and stable while beyond this critical stretch, this solution is unstable and a pitchfork bifurcation into two stable asymmetric branches is demonstrated. For slight departures from orthonormality of the fibres, a numerical approach is used for one of these models to demonstrate that the response undergoes a significant change. A constitutive restriction is suggested that eliminates the general non-uniqueness demonstrated in this work.