ИСТИНА

The eigenvalue problems tensor-block matrix of any even rank are considered [1]. Formulas are obtained that express the classical invariants of the tensor-block matrix through the first invariants of the degrees of this tensor-block matrix. The inverse relations to these formulas are also given. The complete orthonormal system of eigentensor-columns of the symmetric tensor-block matrix of any even rank is constructed in an explicit form. Using the introduced tensor-columns and the tensor-block matrix the presentations of the elastic deformation energy and the constitutive relations of the micropolar theory [2-4] are given. The definition of a positive definite tensor-block matrix is given and the positive definiteness of the tensor-block matrix of the elastic modulus tensor is shown. The definitions of the eigenvalue and the eigentensor-column of the tensor-block matrix are introduced and the problem of finding the eigenvalues and the eigentensor-column of the tensor-block matrix is considered. The characteristic equation of the positive definite tensor-block matrix has the 18th degree in the micropolar theory and it has 18 positive roots. Each root should be taken as many times as its multiplicity. The complete orthonormal system of tensor columns of the tensor-block matrix consists of 18 tensor columns. Based on the obtained canonical presentation of the tensor-block matrix the canonical forms of the specific strain energy and constitutive relations are given. The concept of the symbol of anisotropy of the tensor-block matrix is introduced. Classification of the tensor-block matrices of the micropolar linear theory of elasticity of anisotropic bodies without a center of symmetry and of some materials is given. All linear anisotropic micropolar elastic materials that do not have a center of symmetry in the sense of elastic properties are divided into 18 classes which equals to the number of different eigenvalues. At the same time these classes, depending on the multiplicities of eigenvalues, are subdivided into subclasses. Using 153 independent parameters the complete orthonormal system of eigentensor-columns of the tensor-block-matrix of the elastic modulus tensors is constructed. As a particular case, we consider a tensor-block-diagonal matrix and tensor of any even rank and fourth-rank. Applying the canonical presentations of the tensor and the tensor-block matrix, we formulated the initial-boundary value problems for the micropolar theory of some anisotropic thin bodies, and also we are considered of decomposition the initial-boundary value problems and the problems of wave propagation in some continuums. Acknowledgement: this work was supported by the ShRNSF (project no. DI-2016-41). 1. M. U. Nikabadze, Topics on tensor calculus with applications to mechanics. J. Math. Sci. 2017. Vol. 225, No. 1. 194 p. DOI: 10.1007/s10958-017-3467-4; 2. A. C. Eringen, Microcontinuum field theories. 1. Foundation and solids. N.Y.: Spr.-Ver., 1999; 3. V. D. Kupradze, T. G. Gegelia and others, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. M., Nauka, 1976; 4. M. U. Nikabadze, Development of the method of orthogonal polynomials in the classical and micropolar mechanics of elastic thin bodies. Moscow: MSU, 2014, 515 p. http://istina.msu.ru/media/publications/book/707/ea1/6738800/Monographiya.pdf