Convected Stress/Strain Measures in the Physical Frame

Abstract

An oblique, Cartesian coordinate system arises from the geometry affiliated with a QR decomposition of the deformation gradient F, wherein Q is an orthogonal matrix and R is an uppertriangular matrix. We analyzed the deformation of a cube into a parallelepiped, whose convected coordinates are oblique. Components for the metric tensor and its dual evaluated in this convected coordinate system are established for any state of deformation. Strains and strain rates are defined and quantified in terms of these metrics and their rates. Quotient laws are constructed that govern how vector and tensor fields map between the convected coordinate system and the rectangular Cartesian coordinate systems, where boundary value problems are typically solved. We also derived a set of thermodynamically admissible stress-strain pairs that are quantified in terms of physical components from a convected stress and velocity gradient, with elastic models being presented. This model supports two modes of deformation: elongation and shear, which is distinguished by the pure- and simple-shear responses. Then, we start out in an oblique convected coordinate system and finish up in an orthonormal coordinate system. This reversing process results in working with convected tensor fields that are evaluated in a rectangular Cartesian, coordinate system at the current time, which are ‘physical’ by construction. We also compared the classical approach with our proposed approach.