A Riemannian approach to strain measures in nonlinear elasticity

作者：Neff Patrizio^*; Eidel Bernhard; Osterbrink Frank; Martin Robert

来源：Comptes Rendus Mecanique, 2014, 342(4): 254-257.

DOI：10.1016/j.crme.2013.12.005

• Nonlinear elasticity
• Geodesic distance
• Hencky strain
• Polar decomposition

The isotropic Hendry 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 only 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 parameterized and a novel minimization problem, involving the matrix logarithm for non-symmetric arguments, has to be solved.

• 出版日期2014-4