A Riemannian approach to strain measures in nonlinear elasticity

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.

References in corpus (2)

• A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm
• Minimal geodesics on GL(n) for left-invariant, right-O(n)-invariant Riemannian metrics

Cited by in corpus (12)

• Geometry of logarithmic strain measures in solid mechanics
• The exponentiated Hencky-logarithmic strain energy. Part II: Coercivity, planar polyconvexity and existence of minimizers
• Rank-one convexity implies polyconvexity for isotropic, objective and isochoric elastic energies in the two-dimensional case
• The exponentiated Hencky-logarithmic strain energy. Improvement of planar polyconvexity
• A logarithmic minimization property of the unitary polar factor in the spectral norm and the Frobenius matrix norm
• An ellipticity domain for the distortional Hencky-logarithmic strain energy
• A non-ellipticity result, or the impossible taming of the logarithmic strain measure
• Some remarks on the monotonicity of primary matrix functions on the set of symmetric matrices
• The relaxed-polar mechanism of locally optimal Cosserat rotations for an idealized nanoindentation and comparison with 3D-EBSD experiments
• The exponentiated Hencky strain energy in modelling tire derived material for moderately large deformations
• On Grioli's minimum property and its relation to Cauchy's polar decomposition
• Existence theorem for geometrically nonlinear Cosserat micropolar model under uniform convexity requirements