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arXiv · 1305.2393

A Riemannian approach to strain measures in nonlinear elasticity

Patrizio Neff·Bernhard Eidel·Frank Osterbrink·Robert Martin
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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.

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Patrizio Neff, Bernhard Eidel, Frank Osterbrink, Robert Martin. 2013-05-10. A Riemannian approach to strain measures in nonlinear elasticity. https://doi.org/10.1016/j.crme.2013.12.005

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