Logarithmic strain and its material derivative for a QR decomposition of the deformation gradient

Abstract

The well-known polar decomposition of the deformation gradient decomposes it into an orthogonal rotation and a symmetric stretch. We consider a Gram–Schmidt factorization of the deformation gradient, which decomposes it into a different orthogonal rotation with a right-triangular field that we call distortion. Properties of this distortion tensor are discussed, and a work-conjugate stress tensor is derived for this Lagrangian frame. The logarithm of distortion and its material derivative are then introduced, and their components are quantified, resulting in a new logarithmic measure for strain and its rate, distinct from Hencky strain (the logarithm of stretch) and its rate. An eigenprojection analysis and a first-order, differential, matrix equation solved using the technique of back substitution both produce the same matrix components describing the logarithm of distortion. Three homogeneous deformations illustrate similarities and differences between the logarithms of distortion and stretch. They are distinct measures of strain. The new logarithmic strain measure shows monotonic behavior under simple shear as opposed to the non-monotonic behavior of Hencky strain (see Fig. 2).