A duality principle and correspondence relations in elasticity

Abstract

The equilibrium equations for elasticity problems with field variables depending on two rectangular Cartesian coordinates, χ 1 and χ 2, reduce to σ j1, 1 + σ j2, 2 = 0, j = 1, 2, 3. These equations are satisfied if a vector potentialis introduced for the stress components σ ij, such that σ j1 = ∂ j/∂χ 2 and σ j2 = − ∂ j/∂χ 1. In terms of the six-dimensional vector field η = [u, ] T, the basic elasticity equations reduce to ∂ ∂ x 2 [ u , φ ] T = N ∂ ∂ x 1 [ 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 removes the distinction between the displacement vector field u, and the stress potential vector field . Indeed, these two vector fields are, in many respects, each other's dual, in the sense that the solution to a class of “displacement problems” also provides the solution of the corresponding dual “force problems”. This duality principle is examined in terms of the known solutions of an edge dislocation and a concentrated line force. Then several correspondence relations are also pointed out and discussed. It is shown that the displacement (stress potential) field of a displacement (stress) boundary-value problem can be reduced to the corresponding stress potential (displacement) field by a simple parameter manipulation.