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The closest tensors of higher symmetry classes are derived in explicit formfor a given elasticity tensor of arbitrary symmetry. The mathematical problemis to minimize the elastic length or distance between the given tensor and theclosest elasticity tensor of the specified symmetry. Solutions are presentedfor three distance functions, with particular attention to the Riemannian andlog-Euclidean distances. These yield solutions that are invariant underinversion, i.e., the same whether elastic stiffness or compliance areconsidered. The Frobenius distance function, which corresponds to commonnotions of Euclidean length, is not invariant although it is simple to applyusing projection operators. A complete description of the Euclidean projectionmethod is presented. The three metrics are considered at a level of detail fargreater than heretofore, as we develop the general framework to best fit agiven set of moduli onto higher elastic symmetries. The procedures for findingthe closest elasticity tensor are illustrated by application to a set of 21moduli with no underlying symmetry.Comment: 48 pages, 1 figur

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