A Riemannian approach to strain measures in nonlinear elasticity

TL;DR

This paper introduces a Riemannian geometric framework for strain measures in nonlinear elasticity, showing that isotropic Hencky strain energy can be viewed as a distance in a specific Riemannian metric on GL(n).

Contribution

It develops a family of Riemannian metrics on GL(n) satisfying objectivity and isotropy, revealing their near-independence and deriving explicit geodesic and minimization formulas involving the matrix logarithm.

Findings

Hencky strain energy as a Riemannian distance measure

Family of metrics satisfying invariance properties

Explicit geodesic and minimization formulas involving matrix logarithm

Abstract

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n) of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n). Objectivity requires the Riemannian metric to be left-GL(n)-invariant, isotropy requires the Riemannian metric to be right-O(n)-invariant. The latter two conditions are satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n). Surprisingly, the final result is basically independent of the chosen parameters. In deriving the result, geodesics on GL(n) have to be parametrized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved.