2006/08/14 by Maher Moakher, Andrew N. Norris · 2 citations
Engineering · Mathematics · Physics and Astronomy · #Composite Material Mechanics #Elasticity and Material Modeling #Tensor decomposition and applications #cond-mat.mtrl-sci
paper · pdf · doi:10.1007/s10659-006-9082-0
published as J. Elasticity 85(3), 215-263, 2006 · 48 pages, 1 figure
arxiv created 2006/08/14 · openalex publication_date 2006/10/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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.