4.4 Article

Characterization of the symmetry class of an elasticity tensor using polynomial covariants

期刊

MATHEMATICS AND MECHANICS OF SOLIDS
卷 27, 期 1, 页码 144-190

出版社

SAGE PUBLICATIONS LTD
DOI: 10.1177/10812865211010885

关键词

Covariants of tensors; characterization of symmetry classes; fourth-order harmonic tensors; elasticity tensors; covariant algebras

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The study introduces effective conditions to identify the symmetry class of an elasticity tensor and proves that these classes are affine algebraic sets, providing a minimal set of generators. Additionally, an original generalized cross-product is introduced on totally symmetric tensors.
We formulate effective necessary and sufficient conditions to identify the symmetry class of an elasticity tensor, a fourth-order tensor which is the cornerstone of the theory of elasticity and a toy model for linear constitutive laws in physics. The novelty is that these conditions are written using polynomial covariants. As a corollary, we deduce that the symmetry classes are affine algebraic sets, a result which seems to be new. Meanwhile, we have been lead to produce a minimal set of 70 generators for the covariant algebra of a fourth-order harmonic tensor and introduce an original generalized cross-product on totally symmetric tensors. Finally, using these tensorial covariants, we produce a new minimal set of 294 generators for the invariant algebra of the elasticity tensor.

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