Eigenvalue Problems for Tensor-Block Matrices and Their Applications to Mechanics

In this paper, we state and examine the eigenvalue problem for symmetric tensor-block matrices of arbitrary even rank and arbitrary size $m\times m$, $m\geq 1$. We present certain definitions and theorems of the theory of tensor-block matrices. We obtain formulas that express classical invariants (that are involved in the characteristic equation) of a tensor-block matrix of arbitrary even rank and size $2\times2$ through the first invariants of powers of the same tensor-block matrix and also inverse formulas. A complete orthonormal system of tensor eigencolumns for a tensor-block matrix of arbitrary even rank and size $2\times2$ is constructed. The generalized eigenvalue problem for a tensor-block matrix is stated. As a particular case, the tensor-block matrix of tensors of elasticity moduli is considered. We also present canonical representations of the specific energy of deformation and defining relations. We propose a classification of anisotropic micropolar linearly elastic media that do not possess a symmetry center.