Eigenvalue Problem for Tensors of Even Rank and its Applications in Mechanics

Abstract

In this paper, we consider the eigenvalue problem for a tensor of arbitrary even rank. In this connection, we state definitions and theorems related to the tensors of moduli ℂ_2p(Ω) and ℝ_2p(Ω), where p is an arbitrary natural number and Ω is a domain of the n-dimensional Riemannian space ℝⁿ. We introduce the notions of minor tensors and extended minor tensors of rank (2ps) and order s, the corresponding notions of cofactor tensors and extended cofactor tensors of rank (2ps) and order (N−s), and also the cofactor tensors and extended cofactor tensors of rank 2p(N−s) and order s for rank-(2p) tensor. We present formulas for calculation of these tensors through their components and prove the Laplace theorem on the expansion of the determinant of a rank-(2p) tensor by using the minor and cofactor tensors. We also obtain formulas for the classical invariants of a rank-(2p) tensor through minor and cofactor tensors and through first invariants of degrees of a rank-(2p) tensor and the inverse formulas. A complete orthonormal system of eigentensors for a rank-(2p) tensor is constructed. Canonical representations for the specific strain energy and determining relations are obtained. A classification of anisotropic linear micropolar media with a symmetry center is proposed. Eigenvalues and eigentensors for tensors of elastic moduli for micropolar isotropic and orthotropic materials are calculated.