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Eigenvalue problem for tensors of even rank and its applications in mechanics
M. U. Nikabadze Lomonosov Moscow State University
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
$\mathbb{C}_{2p}(\Omega)$ 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.
UDC:
539.8