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Application of eigenvalue problems for tensor and tensor-block matrices for mathematical modeling of micropolar thin bodiesтезисы доклада
- Автор: Nikabadze M.
- Сборник: 7th International conference «Problems of Mathematical Physics and Mathematical Modelling»: Book of abstracts (Moscow, NRNU MEPhI, 25–27 June) / M.B. Kochanov
- Тезисы
- Год издания: 2018
- Место издания: M Moscow
- Первая страница: 128
- Последняя страница: 129
- Аннотация: The eigenvalue problems of any even rank tensor and tensor-block matrix which consists of the identical even rank tensors are considered. Formulas are obtained that express the classical invariants of the tensor of arbitrary even rank through the first invariants of the degrees of this tensor. The inverse relations to these formulas are also given. The complete orthonormal system of eigentensors of the symmetric tensor of arbitrary even rank, as well as the complete orthonormal system of eigentensor-columns of the symmetric tensor-block matrix of arbitrary even rank are constructed in an explicit form. Some applications to mechanics are given. In particular, the representations of the elastic deformation energy and the constitutive equations (Hooke’s law) of the micropolar theory are given using the introduced tensor columns and the tensor-block matrix. 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 tensor-block matrix has the 18th degree in the micropolar theory and according to the positive definiteness of the tensor-block matrix it has 18 positive roots. Each root should be taken as many times as its multiplicity. Consequently, each eigenvalue has the corresponding eigentensor-column. The complete orthonormal system of tensor columns of the tensor-block matrix consists of 18 tensor columns. A canonical representation of the tensor-block matrix is given. Based on this representation the canonical forms of the specific strain energy and constitutive relations are given. The concept of the structure symbols (the symbol of anisotropy) of the tensor and the tensor-block matrix are introduced. Classification of the tensor-block matrices of the elastic modulus tensor of the micropolar linear theory of elasticity of anisotropic bodies without a center of symmetry 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. The complete orthonormal system of eigentensor-column of the tensor-block-matrix of the elastic modulus tensor using 153 independent parameters, and the complete orthonormal system of eigentensor-columns of the tensor-block-diagonal matrix of the elastic modulus tensor using 72 independent parameters as well as the complete orthonormal system of eigentensors for the positive definite symmetric elastic modulus tensor of the micropolar elasticity theory by means of 36 independent parameters were constructed in an explicit form. Applying the canonical representations 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 considered the problems of waves propagation in some continuum. Acknowledgements: this work was supported by the Shota Rustaveli National Science Foundation (project no. DI-2016-41).
- Добавил в систему: Никабадзе Михаил Ушангиевич