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

Abstract

Incompressibility is established for three- 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 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 (Proc. R. Soc. London (2002), in press) is employed so that surface wave solutions in the incompressible limit can be obtained. A different formalism, also by Ting (Proc. R. Soc. London A 455 (1999) 69), is employed to study the solutions to two-dimensional elastostatic problems. In the special case of incompressible monoclinic materials 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. Many interesting physical phenomena can be deduced using this property. For instance, there is no interpenetration of the interface crack surfaces in an incompressible bimaterial. When only the inplane deformation is considered, the image force due to a line dislocation in a half-space or in a bimaterial depends on the magnitude, not on the direction, of the Burgers vector.