Eigenvalue problem for tensors of even rank and its applications in mechanics

M. U. Nikabadze

Contemporary Mathematics and Its Applications, Tome 98 (2015), pp. 22-52.

Résumé

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 $\mathbb{C}_{2p}(\Omega)$ and $\mathbb{R}_{2p}(\Omega)$, where $p$ is an arbitrary natural number and $\Omega$ is a domain of the $n$-dimensional Riemannian space $\mathbb{R}^n$. 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.