A general duality principle in elasticity

Abstract

Anisotropic elasticity problems are considered, where the field variables depend on only two spatial coordinate variables, say, the rectangular Cartesian coordinates, x1 and x1. The equilibrium equations are then identically satisfied (no body forces), if the stress components, σij, are such that σj1 = ∂φj/∂x2 and σj2 = - ∂φj/∂x1, j = 1, 2, 3, where φ with the components φj is the stress vector potential. In terms of the six-dimensional vector field η = [u, φ], the basic elasticity equations reduce to {∂/∂x2}[u, φ]T = N{∂/∂x1}[u, φ]T, where u is the displacement field and N is called the fundamental elasticity matrix, given by the elastic moduli of the solid. This formulation and the corresponding duality remove the distinction between the displacement vector field u, and the stress vector-potential field φ. Indeed, these two vector fields are, in many respects, each other's dual, in the sense that, for a given solid with given geometry and elasticity, the solution to a class of prescribed mixed displacement-traction boundary-value problems also provides the solution of the corresponding dual mixed traction-displacement boundary-value problems, without a change in geometry or elasticity of the solid. This duality principle is first proved rigorously. Then the results are applied to a general mixed boundary-value problem of a multiply connected heterogeneous finite solid with arbitrary piecewise constant anisotropic elasticity. A number of interesting illustrations involving dislocations and concentrated forces, cracks and anticracks (rigid line inclusions), interface problems, and the half-space Green functions, are also presented as special cases of the general duality principle. In addition to the general duality principle which provides a two-way relation between u and φ, a number of other one-way relations between the displacement field u and the stress vector potential φ of the same problem, as well as u of one problem and u of its dual, and φ of one problem and φ of its dual, are also presented.