More on stress invariance conditions for the traction boundary value problem of plane linear elasticity

Abstract

Cherchaev (1992) gave a sufficient condition for a change in the elastic tensor field to preserve the stress state of a plane linearly elastic solid subject to prescribed traction forces on its boundary. This condition turned out to have a number of fruitful applications in the mechanics of composite materials and was later extended by Dundurs and Markenstoff (1993). The present work addresses and answers the question whether there is a yet more general condition. We show that: (i) if the change in the elastic tensor field is required to be hyperelastic, the extended condition of Dundurs and Markenscoff is not only sufficient but also necessary; (ii) if the change in the elastic tensor field is relaxed so as to be elastic, a more general necessary and sufficient condition exists. In proving these two conclusions, several orthogonal decompositions are constructed for third- and fourth-order tensors presenting index permutation symmetries. Such decompositions are probably also useful for solving other problems in mechanics.