Miniversal Deformations of Pairs of Symmetric Second-order Tensors in the Context of Solid Mechanics

Abstract

In order to study the perturbations of stress‐strain tensors by means of versal deformations techniques, we consider their representations as pairs of symmetric matrices, partitioned in equivalent classes corresponding to change of bases. Both when only orthonormal bases are considered or for general ones, these equivalence classes are differentiable submanifolds (in fact, orbits under the action of suitable groups), and specific expressions of their normal subspaces (with regard to a natural scalar product) are obtained. The normal subspace gives a miniversal deformation of any pair and it allows to compute the dimension of its equivalence class.