Antiplane Shear Deformations for Homogeneous and Inhomogeneous AnisotropicLinearly Elastic Solids

摘要

Antiplane shear deformations of a cylindrical body, with a single displacementfield parallel to the generators of the cylinder and independent of the axial coordinate, are one of the simplest classes of deformations that solids can undergo. They may be viewed as complementary to the more familiar plane deformations. Antiplane (or longitudinal) shear deformations have been the subject of the considerable recent interest in nonlinear elasticity theory for homogeneous isotropic solids. In contrast, for the linear theory of isotropic elasticity, such deformations are usually not extensively discussed. The purpose of the present paper is to demonstrate that for inhomogeneous anisotropic linearly elastic solids the antiplane shear problem does provide a particularly tractable and illuminating setting within which effects of an isotropy and inhomogeneity may be examined. We consider infinitesimal antiplane shear deformations of an inhomogeneous an isotropic linearly elastic cylinder subject to prescribed surface tractions on its lateral boundary whose only nonzero component is axial and which does not vary in the axial direction. In the absence of body forces, not all arbitrary anisotropic cylinders will sustain an antiplane shear deformation under such tract ions. Necessary and sufficient conditions on the elastic moduli are obtained which do allow an antiplane shear. The resulting boundary value problems governing the axial displacement are formulated. The most general elastic symmetry consistent with an an antiplane lane shear is described. There are at most 15 independent elastic coefficients associated with such a material. In general, there is a normal axial stress present, which can be written as a linear combination of the two dominant shear stresses.