On the use of convected coordinate systems in the mechanics of continuous media derived from a QR factorization of F

Abstract

An oblique, Cartesian, coordinate system arises from the geometry affiliated with a Gram–Schmidt (QR) factorization of the deformation gradient F, wherein Q is a proper orthogonal matrix and R is an upper-triangular matrix. Here a cube deforms into a parallelepiped whose edges are oblique and serve as the base vectors for a convected coordinate system. Components for the metric tensor, its dual, and their rates, 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 and their affiliated Jacobians are constructed that govern how vector and tensor fields transform between this oblique coordinate system, where constitutive equations are ideally cast, and the reference, rectangular, Cartesian, coordinate system described in terms of Lagrangian variables, where boundary value problems are solved.