On the Cauchy Strain Tensor, Compatibility Conditions, and Defining Equations of an Elastic Medium

Abstract

Using the example of four-dimensional equilibrium equations for kinetic stresses in Eulerian rectangular coordinates, it is shown that the operator of the four-dimensional Cauchy strain tensor is conjugate (transposed) to the operator of the equilibrium equations. The same connection between the operators of the equilibrium equations and the Cauchy strain tensor also holds in the three-dimensional case. Three variants of the derivation of the conditions for the compatibility of Cauchy deformations are given. In the four-dimensional case, there are 21 compatibility conditions, and in the three-dimensional case, there are six Saint-Venant compatibility conditions. It is shown that the Cauchy strain tensor, both in Eulerian and Lagrangian variables, completely determines the deformed state of a continuous medium. At the same time, no restrictions on the amount of displacements, deformations or rotations are required. The Lagrange-Green and Euler-Almancy tensors, the so-called large or finite deformations, and the displacements are expressed using Cesaro formulas in terms of the Cauchy strain tensor. The defining equations of an elastic continuous medium relate the Cauchy true stress tensor and the Cauchy strain tensor one to another. Using proper bases in the spaces of symmetric stress and strain tensors, the de ning relations can be written as six separate independent equations containing functions of only one argument. For continuous media with crystallographic symmetries, we can use the bases obtained on the basis of the generalized Hooke’s law.