Anisotropic Elastic Materials that Uncouple All Three Displacement Components, and Existence of One-Displacement Green's Function

Abstract

It was shown in an earlier paper that, under a two-dimensional deformation, there are anisotropic elastic materials for which the antiplane displacement u3 and the inplane displacements u1, u2 are uncoupled but the antiplane stresses σ31, σ32 and the inplane stresses σ11, σ12, σ22 remain coupled. The conditions for this to be possible were derived, but they have a complicated expression. In this paper new and simpler conditions are obtained, and a general anisotropic elastic material that satisfies the conditions is presented. For this material, and for certain monoclinic materials with the symmetry plane at x3 = 0, we show that the unnormalized Stroh eigenvectors ak for k = 1, 2, 3 are all real. The matrix A =[a1, a2, a3] is a unit matrix when the material has a symmetry plane at x2 = 0. Thus any one of the u1, u2, u3 can be the only nonzero displacement, and the solution is a one-displacement field. Application to the Green's function due to a line of concentrated force f and a line dislocation with Burgers vector v in the infinite space, the half-space with a rigid boundary, and the infinite space with an elliptic rigid inclusion shows that one can indeed have a one-displacement field u1, u2 or u3. One can also have a two-displacement field polarized on a plane other than the (x1, x2)-plane. The material that uncouples u1, u2, u3 is not as restrictive as one might have thought. It can be triclinic, monoclinic, orthotropic, tetragonal, transversely isotropic, or cubic. However, it cannot be isotropic.