Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint

TL;DR

This paper derives exact equations for surface waves in prestrained isotropic hyperelastic materials with internal constraints, providing stability criteria and analyzing conditions for wave existence and uniqueness.

Contribution

It introduces a direct method to derive the secular equation for surface waves in constrained hyperelastic materials and establishes universal bifurcation criteria based on principal stretches.

Findings

Derived the exact secular equation for surface waves in constrained hyperelastic materials.

Established explicit bifurcation/stability criteria depending only on principal stretches.

Analyzed surface wave conditions for various isotropic constraints, including incompressibility and Bell constraints.

Abstract

An isotropic elastic half space is prestrained so that two of the principal axes of strain lie in the bounding plane, which itself remains free of traction. The material is subject to an isotropic constraint of arbitrary nature. A surface wave is propagated sinusoidally along the bounding surface in the direction of a principal axis of strain and decays away from the surface. The exact secular equation is derived by a direct method for such a principal surface wave; it is cubic in a quantity whose square is linearly related to the squared wave speed. For the prestrained material, replacing the squared wave speed by zero gives an explicit bifurcation, or stability, criterion. Conditions on the existence and uniqueness of surface waves are given. The bifurcation criterion is derived for specific strain energies in the case of four isotropic constraints: those of incompressibility, Bell, constant area, and Ericksen. In each case investigated, the bifurcation criterion is found to be of a universal nature in that it depends only on the principal stretches, not on the material constants. Some results related to the surface stability of arterial wall mechanics are also presented.