2015/09/08 by Yakov Itin · 7 citations
Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Analysis Techniques #Binary quadratic form #Cartesian tensor #Composite Material Mechanics #Elasticity (physics) #Elasticity and Material Modeling #Exact solutions in general relativity #Geometry #Isotropic quadratic form #Isotropy #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quadratic equation #Quadratic function #Quantum mechanics #Symmetric tensor #Tensor (intrinsic definition) #Tensor contraction #Tensor density #Tensor field #Tensor product #cond-mat.other #math-ph #math.MP
paper · pdf · doi:10.1007/s10659-016-9569-2
published in Journal of Elasticity 125(1), 39-62 (Springer Science+Business Media)
arxiv created 2015/09/08 · openalex publication_date 2016/01/15 · arxiv updated 2017/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We study the quadratic invariants of the elasticity tensor in the framework of its unique irreducible decomposition. The key point is that this decomposition generates the direct sum reduction of the elasticity tensor space. The corresponding subspaces are completely independent and even orthogonal relative to the Euclidean (Frobenius) scalar product. We construct a basis set of seven quadratic invariants that emerge in a natural and systematic way. Moreover, the completeness of this basis and the independence of the basis tensors follow immediately from the direct sum representation of the elasticity tensor space. We define the Cauchy factor of an anisotropic material as a dimensionless measure of a closeness to a pure Cauchy material and a similar isotropic factor is as a measure for a closeness of an anisotropic material to its isotropic prototype. For cubic crystals, these factors are explicitly displayed and cubic crystal average of an arbitrary elastic material is derived.