The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower SymmetryReport as inadecuate

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Journal of Elasticity

, Volume 85, Issue 3, pp 215–263

First Online: 05 October 2006Received: 21 February 2006Accepted: 22 August 2006


The closest tensors of higher symmetry classes are derived in explicit form for a given elasticity tensor of arbitrary symmetry. The mathematical problem is to minimize the elastic length or distance between the given tensor and the closest elasticity tensor of the specified symmetry. Solutions are presented for three distance functions, with particular attention to the Riemannian and log-Euclidean distances. These yield solutions that are invariant under inversion, i.e., the same whether elastic stiffness or compliance are considered. The Frobenius distance function, which corresponds to common notions of Euclidean length, is not invariant although it is simple to apply using projection operators. A complete description of the Euclidean projection method is presented. The three metrics are considered at a level of detail far greater than heretofore, as we develop the general framework to best fit a given set of moduli onto higher elastic symmetries. The procedures for finding the closest elasticity tensor are illustrated by application to a set of 21 moduli with no underlying symmetry.

Key wordselastic symmetry anisotropy closest moduli Reimannian metric log-Euclidean Mathematics Subject Classifications 200073C30 74B05 15A48 15A69  Download to read the full article text

Author: Maher Moakher - Andrew N. Norris



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