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Eigenvalue Problems of any even Rank Tensor and Tensor-Block Matrix with some Applications in Mechanicsтезисы доклада
- Автор: Nikabadze M.U.
- Сборник: Generalized Continua as Models for Materials with Multi-Scale-Effects or under Multi-Field-Actions
- Тезисы
- Год издания: 2015
- Место издания: Experimental Factory. Magdeburg, Germany
- Первая страница: 36
- Последняя страница: 37
- Аннотация: Eigenvalue problems of the tensor and the tensor-block matrix of any even rank are considered. Formulas expressing classical invariants of any even rank tensor as with the help of the tensors and the extended tensors of minors as with the help of the tensors and the extended tensors of algebraic cofactors are given. We also obtain formulas for the classical invariants of the any even rank tensor through the first invariants of degrees of this tensor. The inverse formulas to these formulas are also given. Some definitions, statements and theorems are formulated about tensors and tensor-block matrix of any even rank. A complete orthonormal system of eigentensor of symmetric tensor, as well as a complete orthonormal system of eigentensor columns of the symmetric tensor-block matrix are construct. As a special case, we consider the fourth rank tensor and tensor-block matrix, and sixth rank tensor. It is proved that a tensor-block matrix of elastic moduli tensor is positive-definite. In micropolar theory, the characteristic equation of the tensor-block matrix has 18 positive roots counted each root according to its multiplicity. Therefore, the complete orthonormal system of eigentensor columns of the tensor-block matrix consists of 18 tensor-columns. Canonical representation of the tensor-block matrix is given. Using this presentation, we get the canonical form of the elastic strain energy and constitutive relation. We introduce the conception of symbol of structure of the tensor-block matrix and give a classification of the tensor-block matrix of elastic modulus tensor of the micropolar linear elasticity of anisotropic bodies without a center of symmetry. All linear micropolar anisotropic elastic materials which do have not a center of symmetry, are divided into 18 classes according to the number of different eigenvalues, and the classes depending on the multiplicity of the eigenvalues are subdivided into subclasses. All told above is equally true to linear micropolar theory of elasticity of anisotropic bodies with a center of symmetry. In this case, it is enough to study the internal structure of each of the two positive-definite tensors of elastic moduli separately. In the latter case, in contrast to the classical case, the characteristic equation for each elastic modulus tensor has the 9th degree (in the classical theory of elasticity, characteristic equation has the 6th degree). It is shown that, if we make a classification of the set of positive-definite symmetric four rank tensors, then we get the 9 major classes according to the number of different eigenvalues and the classes depending on the multiplicities of the eigenvalues are subdivided into subclasses. In total, we have 256 subclasses (in classical case we have six classes consisting of 32 subclasses). Thus, if each of the anisotropic materials corresponds to the elastic modulus tensors of the same structure, the number of anisotropic materials is 256. If the elastic modulus tensors have the same symbol of structure and belong to the different subclasses, the number of linearly elastic anisotropic materials with a center of symmetry in the sense of the elastic properties is equal to 12870. If tensors have different structures, then the number of materials is 65536. The number of anisotropic materials without a center of symmetry equals 131072. In an explicit form, we have constructed a complete orthonormal system of eigentensor-columns of tensor-block matrix of elastic modulus tensors using 153 independent parameters as well as a complete orthonormal system of eigentensor-columns of tensor-block-diagonal matrix of elastic modulus tensors with the use of 72 independent parameters and a complete orthonormal system of eigentensors for positive-definite symmetric elastic modulus tensor of the micropolar elasticity theory with the help of 36 independent parameters. The orthonormal system of eigentensors for elastic modulus tensor with 15 independent parameters in the classical theory of elasticity was explicitly completed by N.I. Ostrosablin. We also consider the classification of classical anisotropic materials. The eigenvalues and eigentensors for classical materials of crystallographic syngonies different from the forms produced by N.I. Ostrosablin as well as for some micrpolar materials are found.
- Добавил в систему: Никабадзе Михаил Ушангиевич