Darboux classification of cauchy elastic bodies and some of its consequences

Abstract

The Darboux theorem on the classification and canonical forms of first degree differential forms is used to obtain a classification and canonical representation of Cauchy elastic solids. If the stress is assumed to be determined uniquely by the Green deformation tensor C, the double Piola-Kirchhoff stress tensor is shown to belong to one of six possible classes, each of which possesses a unique representation in terms of linear combinations of gradients with respect to C of scalar-valued functions with scalar-valued coefficients. There are three distinct classes for isotropic materials, each of which has a unique representation. It is shown that these three classes are distinguishable in terms of the existence or nonexistence of cycles with nonzero total work and by the accessibility or inaccessibility of any deformation state from any neighboring deformation state by a path of zero work.