The Hencky strain energy ‖log U‖2 measures the geodesic distance of the deformation gradient to SO(n) in the canonical left‐invariant Riemannian metric on GL(n)

Abstract

AbstractThe well‐known isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set of rigid rotations in a 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. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)