The Closest Elastic Tensor of Arbitrary Symmetry to an Elasticity Tensor of Lower Symmetry

By Maher Moakher, Andrew N. Norris. 2006

247 citations recorded by the source corpus.

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.

Subfields: Biomedical Engineering, Computational Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/bc79ead906913b851df3887f60b60e2a7553e454

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In Biomedical Engineering

In Mechanics of Materials

Citations in the corpus

Geometry of Logarithmic Strain Measures in Solid Mechanics (2015, background): “Norris and Moakher applied such a distance function in order to find an isotropic elasticity tensor C : Sym(3) → Sym(3) which best approximates a given anisotropic tensor [138, 157].”
Valence electron concentration as an indicator for mechanical properties in rocksalt structure nitrides, carbides and carbonitrides (2018, background): “symmetry, such as ternary systems, the elastic constants were averaged over the directions [69].”
Descriptive tools for the analysis of texture projects with large datasets using MTEX: strength, symmetry and components (2014, background): “Moakher & Norris (2006) discuss other norms suitable for the stiffness and compliance tensors.”