Topics on tensor calculus with applications to mechanics (book)

Nikabadze, M.U.

Автор: Nikabadze M.U.
Журнал: Journal of Mathematical Sciences
Том: 225
Номер: 1
Год издания: 2017
Издательство: Plenum Publishers
Местоположение издательства: United States
Первая страница: 1
Последняя страница: 194
DOI: 10.1007/s10958-017-3467-4
Аннотация: There are many fundamental monographs and textbooks on tensor calculus. Remarkable lecture courses on tensor analysis and various areas of deformable solid mechanics and fluid mechanics in tensor calculus were delivered to the author (during his postgraduate and postdoctoral studies) by Professor B.E. Pobedrya (Faculty of Mathematics and Mechanics of the Lomonosov Moscow State University). The said lectures were very important for the author to become an expert in tensor calculus and various areas of deformable solid mechanics (in particular, the theory of thin bodies). For many years, the author delivered the following special lecture courses for students of Mechanics at the Faculty of Mathematics and Mechanics of the Lomonosov Moscow State University: “Tensor Technique of Mechanics”, “Classical Mathematical Theory of Shells”, “Nonclassical Mathematical Theory of Shells”, “Theory of Thin Bodies”, etc. All the above special courses use tensor calculus. Several directions are not sufficiently elucidated in existing sources or are given in books that are not easily accessible. To cover this gap, the books of author “Topics of Tensor Calculus. I, II. Moscow State Univ., (2007) [in Russian]” were issued. The refined content of this book was used to create the initial three chapters and the final chapter of the present monograph. The content of its other two chapters was not published earlier. In the first chapter, we provide basic definitions from linear algebra and functional analysis. In particular, definitions of semigroups, groups, rings, fields, modules, and linear spaces are given, and we formulate the local existence theorem for homomorphisms. Also, we define inner r-products and local scalar products of tensors of ranks not exceeded by r and the local norm of tensors. We provide definitions and prove main assertions related to the linear dependence and independence of a system of tensors of any rank. Also, we provide definitions and prove several theorems related to orthogonal and biorthogonal systems of tensors. Further, we define multiplicative bases (multibases) and study ways to construct module bases by means of bases of modules of lower dimensions: several theorems are proved. Fourth-order tensor and eigenvalue problems for tensors of any even rank are investigated in detail. Canonical representations for any even-rank tensor are provided. Note that a similar problem for the tensor of elasticity modules was considered by Rykhlevski˘ı in 1983-1984, but it was studied for complex tensors of any even rank by Vekua earlier; however, no explicit form of a complete orthonormal system of eigentensors was found that time. In the second chapter, basic information about polynomials with tensor coefficients is provided. A generalized B´ezout theorem is proved. Then it is used to prove the Hamilton–Cayley theorem. Another proof of the last theorem is considered as well. Using relations expressing the associated tensor B(λ) for the tensor binomial λE−A via the tensor A∈C2p(Ω) (complex tensors of rank 2p are elements of that module) and its invariants, we prove several important theorems. Further, we define the minimal polynomials of a tensor of the module C2p(Ω) and of a tensor of the module Cp(Ω) (complex tensors of rank p are elements of the last module) and of a tensor of the module Cp(Ω) with respect to a given tensor of the module C2p(Ω), where Ω is a domain of an n-dimensional Euclidean (Riemannian) space, and prove several theorems for minimal polynomials. Also, we formulate the first, the second, and the third theorems on the decomposition of the module Cp(Ω) into invariant submodules. In particular, we consider theorems on adjoint, normal, Hermitian, and unitary tensors of the modules C2p(Ω) and R2p(Ω) (real tensors of rank 2p are elements of the last module). We prove polar expansion theorems for tensors of the modules C2p(Ω) and R2p(Ω) and existence theorems for complete orthonormal systems of eigentensors for a finite or infinite set of pairwise commuting normal tensors of the modules C2p(Ω) and R2p(Ω). Canonical representations of the above tensors are provided. At the end of the chapter, important exercises are provided. In the third chapter, various ways to construct linearly independent isotropic, gyrotropic, orthotropic, and transversal-isotropic tensors are considered. Theorems and other assertions allowing us to construct the said tensors are formulated. We construct linearly independent tensors of the above kind (from the first to the sixth rank) such that tensor components possess no symmetry even if there are different kinds of symmetries. In the classical linear elasticity theory, mechanical properties of any linearly elastic material are determined by one tensor of the fourth rank of elasticity. In the micropolar linear elasticity theory, it is determined by three tensors (this refers to the general case where the material does not have a symmetry center in the sense of elastic properties). The Hooke law is well known, but tensors of elasticity modules are not sufficiently investigated yet. However, to project and design composite materials possessing a high anisotropy, it is important to know the inner structure of tensors of elasticity modules. In other words, it is important to find eigenvalues and eigentensors for tensors of elasticity modules; if the micropolar material does not have a symmetry center in the sense of elastic properties, then, to investigate the mathematical and mechanical structure of such a material, one has to find eigenvalues and eigentensor-columns of the tensor-block matrix of tensors of elasticity modules. Note that the number of symmetry properties of components of tensors of elasticity modules in the micropolar theory is less than the same number in the classical theory. Moreover, in the micropolar theory, the properties are not the same for all tensors. Therefore, it is harder to find eigenvalues and eigentensors for tensors of elasticity modules in the micropolar theory. In the fourth chapter, we consider eigenvalue problems for even-rank tensors and tensor-block matrices consisting of tensors of the same even rank. We study inner structures of tensors of the module R2p(Ω) and of tensor-block matrices of the module R42p(Ω) (this module consists of tensor-block matrices consisting of four tensors of the module R2p(Ω)). Also, we introduce a tensor-column consisting of two tensors of the module Rp(Ω). The set of such tensor-columns forms a module denoted by R2p(Ω). Further, for any tensor A∈ R2p(Ω), we introduce the tensor and the extended tensor of minors of any rank and order as well as the tensor and the extended tensor of cofactors corresponding to the said minors. Using those tensors, we generalize the Laplace expansion theorem for the determinant of the tensor A∈R2p(Ω). The classical invariants of any tensor of the module R2p(Ω), i.e., the invariants included into the characteristic equation, are expressed both via tensors and extended tensors of minors of the said tensor and via tensors and extended tensors of its cofactors. Also, we express classical invariants of any tensor A∈R2p(Ω) via first invariants of powers of that tensor and provide relations inverse to those expressions. Further, we formulate definitions, theorems, and other assertions regarding tensor-block matrices of the module R42p(Ω).For any symmetric tensor of the module R2p(Ω), we explicitly construct the complete orthonormal system of eigentensors and the complete orthonormal system of eigentensor-columns of the symmetric tensor-block matrix of the module R42p(Ω). The said eigentensor-columns belong to the module R2p(Ω). The fifth chapter contains applications to mechanics. In particular, for the linear micropolar elasticity theory of anisotropic bodies without symmetry centers in the sense of elastic properties, we introduce (similarly to the general case) tensor-columns of stress tensors, tensors of moment stresses, deformation tensors, and bend-torsion tensors belonging to the module R22(Ω). Also, we introduce a tensor-block matrix of tensors of elasticity modules; it consists of four fourth-rank tensors and belongs to the module R44(Ω).The introduced tensor-columns and tensor-block matrix are used to represent the elastic deformation energy and the Hooke law. We define positive definite tensor-block matrices and prove that the tensorblock matrix of tensors of elasticity modules is positive definite. For tensor-block matrices, we define eigenvalues and eigentensor-columns. In the micropolar theory, the characteristic equation of a tensorblock matrix has the 18th power. Therefore, it has 18 roots (the roots are counted according to their multiplicity) because the said matrix is positive definite. Therefore, an eigentensor-column corresponds to any eigenvalue. The complete orthonormal system of tensor-columns of a tensor-block matrix consists of 18 tensor-columns. We provide a canonical representation for tensor-block matrices. Then we use it to find canonical representations of the specific energy of deformation and definiens. We introduce a structure symbol of a tensor-block matrix and classify tensor-block matrices for elasticity modules of the micropolar linear elasticity theory for anisotropic bodies without symmetry centers (micropolar linearly elastic anisotropic materials without symmetry centers). All linear anisotropic micropolar elastic materials without symmetry centers in the sense of elastic properties are distributed into 18 classes depending on the number of different eigenvalues. The classes are decomposed into subclasses according to the multiplicities of the eigenvalues. The same is valid for the linear micropolar elasticity theory of anisotropic bodies possessing symmetry centers. In such a case, the tensor-block matrix is a tensor-block-diagonal matrix; this simplifies the investigation of its inner structure. In particular, it suffices to study the inner structure of each of the two positive definite tensors of elasticity modules separately. Then, unlike the classical case, the characteristic equation for each tensor of elasticity modules is of the ninth power (in the classical elasticity theory, the tensor of elasticity modules is unique and the characteristic equation is of the sixth power because components of the tensor of elasticity modules have symmetries). Direct computations show that, classifying the set of positive definite symmetric tensors of the fourth rank (tensors of elasticity modules in the linear micropolar elasticity theory of anisotropic bodies possessing symmetry centers belong to that set), we obtain nine main classes depending on the number of different eigenvalues. The classes are decomposed into subclasses according to the multiplicities of the eigenvalues. There are 256 subclasses (six classes exist in the classical case and they are decomposed in 32 subclasses). If tensors of elasticity modules of the same structure (this means that they have the same structure symbol and belong to the same subclass) correspond to each anisotropic material, then the number of anisotropic materials is equal to 256; if tensors of elasticity modules have the same structure symbol but belong to different subclasses, then the number of linearly elastic anisotropic materials possessing symmetry centers in the sense of elastic properties is equal to 12870. Hence, the number of anisotropic materials without symmetry centers in the sense of elastic properties substantially exceeds the number of anisotropic materials possessing symmetry centers. In particular, the number of all anisotropic micropolar linearly elastic materials without symmetry centers is equal to 131072 (see below). Using 153 independent parameters, we explicitly construct a complete orthonormal system of eigentensor-columns of the tensor-block matrix of tensors of elasticity modules. Using 72 independent parameters, we explicitly construct a complete orthonormal system of eigentensor-columns of the tensor-block-diagonal matrix of tensors of elasticity modules. Using 36 independent parameters, we explicitly construct a complete orthonormal system of eigentensors for the positive definite symmetric tensor of elasticity modules of the micropolar elasticity theory (in the classical elasticity theory, Ostrosablin explicitly constructed eigentensors for the tensor of elasticity modules using 15 independent parameters). We consider the classification of classical anisotropic materials as well. We find eigenvalues and eigentensors for classical materials of crystallographic syngonies in a form different from the Ostrosablin one and for other micropolar materials. In the sixth chapter, we explain the tensor calculus under a new parametrization of domains of thin bodies: several base surfaces are used to construct theories of one-layer and multilayer thin bodies as well as theories of plates, envelopes, and multilayer constructions. The chapter is completed by exercises. The author expresses his profound gratitude to Professors Pobedrya, Dimitrienko, and Georgievskii for their useful advice and considerations and to PhD Senior Lecturer Ulukhanyan for the great aid in the preparation of the manuscript. This work is financially supported by RFBR grants No. 14-01-00317-a and 15-01-00848-a.
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