Eigenvalue Problems for Tensor-Block Matrices and Their Applications to Mechanics
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EIGENVALUE PROBLEMS FOR TENSOR-BLOCK MATRICES AND THEIR APPLICATIONS TO MECHANICS UDC 512.64; 517.958
M. U. Nikabadze
Abstract. In this paper, we state and examine the eigenvalue problem for symmetric tensor-block matrices of arbitrary even rank and arbitrary size m × m, m ≥ 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 2×2 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 × 2 is constructed. The generalized eigenvalue problem for a tensor-block matrix is stated. As a particular case, the tensor-block matrix of tensors of elastic 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. Keywords and phrases: eigenvalue problem for a tensor-block matrix, tensor column, eigentensor, anisotropy symbol of a tensor-block matrix, anisotropy symbol of a material. AMS Subject Classification: 74B05
CONTENTS 1. Tensors of the Module R2p (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Eigenvalue Problem and Construction of a Complete System of Eigentensor Columns of Symmetric Tensor-Block Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Applications to Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Eigenvalue Problem and Construction of a Complete System of Eigentensors for a Symmetric Tensor of the Fourth Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Classification of Micropolar Linearly Elastic Anisotropic Materials without Centers of Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 896 . 898 . 916 . 926 . 927 . 929
Mechanical properties of linearly elastic material in the classical elasticity theory are described by a fourth-rank tensor (the tensor of elastic moduli) and in the micropolar linear elasticity theory—in the general case where materials do not possess centers of symmetry in the sense of elastic properties—by four tensors. In the latter case, the defining relations can be represented by a tensor-block matrix consisting of four tensors of elastic moduli. Therefore, the question on the study of the intrinsic structure of tensor-block matrices appears. Despite the fact that the defining relations (generalized Hooke’s law) were formulated long ago, tensors of elastic moduli and especially tensor-block matrices have not yet been sufficiently studied, whereas for developing and creating composite materials that possess higher-degree an
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