Chapter 5 - Conservation laws and stress tensors

Abstract

Conservation laws of mass, linear momentum, angular moment, angular momentum, angular moment, and energy which must be satisfied during a deformation are first delineated in this chapter. Concurrently, the Cauchy stress tensor is defined as the Eulerian tensor. However, the material included in the unit current volume element and its mass changes during the deformation. Then, the stress and the strain rate adopted in constitutive equations for the exact description of the finite deformation must be defined for the specific volume element possessing a fixed mass in the reference configuration leading to the Lagrangian tensors. In addition, the work rate involved in the conservation laws must be the work rate done to the reference volume element, while the pair of stress and strain rate tensors inducing this work rate is called the work-conjugate pair. Then, various work-conjugate Lagrangian stress and strain rate tensors are derived from the Cauchy stress tensor and the Eulerian strain rate tensor in the systematic way by means of the pull-back operations.