Semi-analytical finite element analysis of waveguides in initially-stressed elastic rods

Abstract

The analysis of waveguides in initially-stressed elastic media is of interest in many realms of the physical sciences and engineering. It is, therefore, necessary to understand how the solids behave under the impact of initial stress. A body may still hold a certain amount of internal stress called the residual stress, even though all loads are removed. In manufacturing, residual stress has long been noted to be harmful to, or enhance material performance. For biological tissues, residual stress is used to self-adjust stress and strain, which results in preserving ideal mechanical conditions for the tissue, whilst in geophysics, due to gravity and active tectonics, the Earth has developed extremely high initial stresses below its surface, which greatly affects the propagation of seismic waves. The term initial stress, here, is used to broadly mean the internal stress of some reference configuration, irrespective of how the stress was formed. In this sense, residual stress is a type of initial stress. The initial stress sensed by any material region is owing to the push and pull of the surrounding regions. Provided that any region were to be cut out from the material, the stress on its newly formed boundary would be equal to zero, and thus reducing the potential energy in the bulk. Based on this concept, Hoger, therefore, developed constitutive laws for residually stressed materials [9–12]. An ideal way to account for the initial stress would be to write explicitly a strain-energy function W = W (F, τ) in terms of the deformation gradient tensor F and the initial stress τ without any a priori restrictions. From this, constitutive laws are developed to study the effect of initial stress on infinitesimal wave propagation (M. Shams et al. [5] and W. Ogden [6])