The incompressible limit in linear anisotropic elasticity, with application to surface waves and elastostatics

TL;DR

This paper establishes conditions for incompressibility in anisotropic linear elasticity, derives explicit surface wave equations, and explores elastostatic solutions, revealing unique properties like the absence of crack interpenetration in incompressible bimaterials.

Contribution

It introduces a formalism for analyzing incompressible anisotropic elasticity using compliances, enabling direct derivation of surface wave and elastostatic solutions from compressible cases.

Findings

Explicit secular equation for surface waves in incompressible monoclinic materials.

Incompressible monoclinic materials exhibit specific tensor properties, such as vanishing and inverse relations.

Dislocation image force depends only on Burgers vector magnitude in in-plane deformation.

Abstract

Incompressibility is established for three-dimensional and two-dimensional deformations of an anisotropic linearly elastic material, as conditions to be satisfied by the elastic compliances. These conditions make it straightforward to derive results for incompressible materials from those established for the compressible materials. As an illustration, the explicit secular equation is obtained for surface waves in incompressible monoclinic materials with the symmetry plane at x_3=0. This equation also covers the case of incompressible orthotropic materials. The displacements and stresses for surface waves are often expressed in terms of the elastic stiffnesses, which can be unbounded in the incompressible limit. An alternative formalism in terms of the elastic compliances presented recently by Ting is employed so that surface wave solutions in the incompressible limit can be obtained. A different formalism, also by Ting, is employed to study the solutions to two-dimensional elastostatic problems. In the special case of incompressible monoclinic material with the symmetry plane at x_3=0, one of the three Barnett-Lothe tensors S vanishes while the other two tensors H and L are the inverse of each other. Moreover, H and L are diagonal with the first two diagonal elements being identical. An interesting physical phenomenon deduced from this property is that there is no interpenetration of the interface crack surface in an incompressible bimaterial. When only the inplane deformation is considered, it is shown that the image force due to a line dislocation in a half-space or in a bimaterial depends only on the magnitude, not on the direction, of the Burgers vector.