Abstract
The statement of the eigenvalue problem for a tensor–block matrix (TBM) of any orderand of any even rank is formulated, and also some of its special cases are considered. In particular,using the canonical presentation of the TBM of the tensor of elastic modules of the micropolartheory, in the canonical form the specific deformation energy and the constitutive relations arewritten. With the help of the introduced TBM operator, the equations of motion of a micropolararbitrarily anisotropic medium are written, and also the boundary conditions are written down bymeans of the introduced TBM operator of the stress and the couple stress vectors. The formulationsof initial-boundary value problems in these terms for an arbitrary anisotropic medium are given.The questions on the decomposition of initial-boundary value problems of elasticity and thin bodytheory for some anisotropic media are considered. In particular, the initial-boundary problems of themicropolar (classical) theory of elasticity are presented with the help of the introduced TBM operators(tensors–operators). In the case of an isotropic micropolar elastic medium (isotropic and transverselyisotropic classical media), the TBM operator (tensors–operators) of cofactors to TBM operators(tensors–tensors) of the initial-boundary value problems are constructed that allow decomposinginitial-boundary value problems. We also find the determinant and the tensor of cofactors to the sumof six tensors used for decomposition of initial-boundary value problems. From three-dimensionaldecomposed initial-boundary value problems, the corresponding decomposed initial-boundary valueproblems for the theories of thin bodies are obtained.
Highlights
For isotropic materials, eigenmodules and eigenstates are known since monograph [1]
These questions are described in some detail in [27,28], and in this work, special attention is paid to the canonical representations of equations and boundary conditions
If kinematic boundary conditions are given on one part of the body boundary S1, and on the remaining part of it S2 are given the static boundary conditions, where S1 ∪ S2 = S, S1 ∩ S2 = ∅, such boundary conditions are called mixed boundary conditions, and the problem of micropolar solids mechanics (SM), using them and initial conditions is called the mixed initial-boundary value problem
Summary
Eigenmodules (eigenvalues) and eigenstates (eigentensors) are known since monograph [1]. Representations of general solutions of the Lamé’s equations were made by many scientists (see, for example, [19,20,21,22,23]), and representations of general solutions of equations in displacements and rotations of the micropolar theory of elasticity can be found, for example, in [22,24,25,26] Note that in this case the equations of the classical and micropolar theory of elasticity are decomposed, but for decomposing the equations, as well as static boundary conditions, the algebraic method turned out to be more efficient. The above results and eigenvalue problems for tensor and tensor–block matrices (see [29]) are used in this work for mathematical modeling of the micropolar thin bodies
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