ИСТИНА

We consider the eigenvalue problems of the tensor and the tensor-block matrix of any even rank. A complete orthonormal system of eigentensors for a symmetric tensor of any even rank, as well as a complete orthonormal system of eigentensor-columns for a symmetric tensor-block matrix of any even rank, are constructed in an explicit form. We represent the elastic energy of deformation and the constitutive relations (Hooke’s law) using the introduced tensor columns and tensor-block matrix. A canonical representation of the tensor-block matrix, as well as the canonical representations of the specific strain energy and the constitutive relations, are given. The symbols of the structure of the tensor and tensor-block matrix are introduced. A classification of some anisotropic materials is given. Using the canonical representations of tensors (tensor-block matrix), dispersion equations and expressions for dispersion tensors are obtained. Formulas for the velocity of the waves for some materials through the eigenvalues of the material tensors (tensor-block matrix) are also obtained.