A Crystal Structure-Based Eigentransformation and its Work-Conjugate Material Stress

Abstract

In the abstract of his 1970 paper, Eshelby stated: “The force on a dislocation or point defect, as understood in solid-state physics, and the crack extension force of fracture mechanics are examples of quantities which measure the rate at which the total energy of a physical system varies as some kind of departure from uniformity within it changes its configuration.” He then went on to demonstrate that the elastic energy-momentum tensor proves to be a useful tool in calculating such forces. The ‘forces’ turn out to be the appropriate traction vectors associated with the energy-momentum (stress) tensor. It is therefore natural and perhaps even fundamental to look for the ’strain tensor’ that can be paired with the ’stress tensor’ to form work. The ’strain rate’ would then be that some kind of departure from uniformity within a physical system. In this paper, we examine the configurational changes brought about by atomic diffusion in a nonuniform alloy crystal. The transformation from a reference, single-parameter simple cubic cell to a six-parameter alloy crystal cell, called the eigentransformation, is identified as the needed kinematic tensor.