scholarly journals Stability of mathematical models of the main problems of the anisotropic theory of elasticity

2020 ◽  
Vol 30 (1) ◽  
pp. 112-124
Author(s):  
A.V. Yudenkov ◽  
A.M. Volodchenkov
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Research Result ◽  
Theory Of Elasticity ◽  
Boundary Problems ◽  
Main Research ◽  
Contour Shape ◽  
Variable Function ◽  
Anisotropic Bodies

The boundary problems of the complex-variable function theory are effectively used while investigating equilibrium of homogeneous elastic mediums. The most complicated systems of the boundary value problems correspond to the case when an elastic body exhibits anisotropic properties. Anisotropy of the medium results in the drift of boundary conditions of the function that in general disrupts analyticity of the functions of interest. The paper studies systems of the boundary value problems with drift for analytic vectors corresponding to the primal elastic problems (first, second and mixed problems). Systems of analytic vectors with drift are reduced to equivalent systems of Hilbert boundary value problems for analytic functions with weak singularity integrators. The obtained general solution of the primal boundary value problems for the anisotropic theory of elasticity allows us to check the above problems for stability with respect to perturbations of boundary value conditions and contour shape. The research is relevant as there is necessity to apply approximate numerical methods to the boundary value problems with drift. The main research result comes to be a proof of stability of the systems of the vector boundary value problems with drift for analytic functions on the Hölder space corresponding to the primal problems of the elastic theory for anisotropic bodies in the case of change in the boundary value conditions and contour shape.

Basic Boundary Value Problems of Thermoelasticity for Anisotropic Bodies with Cuts. II

1995 ◽  
Vol 2 (3) ◽  
pp. 259-276
Author(s):  
R. Duduchava ◽  
D. Natroshvili ◽  
E. Shargorodsky
Keyword(s):  
Boundary Value Problems ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Arbitrary Configuration ◽  
Anisotropic Bodies ◽  
Present Part ◽  
Classical Potential Theory ◽  
Basic Boundary

Abstract In the first part [Duduchava, Natroshvili and Shargorodsky, Georgian Math. J. 2: 123–140, 1985] of the paper the basic boundary value problems of the mathematical theory of elasticity for three-dimensional anisotropic bodies with cuts were formulated. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems were formulated in the Besov () and Bessel-potential () spaces. In the present part we give the proofs of the main results (Theorems 7 and 8) using the classical potential theory and the nonclassical theory of pseudodifferential equations on manifolds with a boundary.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Basic Boundary Value Problems of Thermoelasticity for Anisotropic Bodies with Cuts. I

1995 ◽  
Vol 2 (2) ◽  
pp. 123-140
Author(s):  
R. Duduchava ◽  
D. Natroshvili ◽  
E. Shargorodsky
Keyword(s):  
Boundary Value Problems ◽  
Displacement Vector ◽  
Boundary Value ◽  
Three Dimensional ◽  
Manifolds With Boundary ◽  
Hölder Continuous ◽  
Arbitrary Configuration ◽  
Anisotropic Bodies ◽  
Potential Methods ◽  
Basic Boundary

Abstract The three-dimensional problems of the mathematical theory of thermoelasticity are considered for homogeneous anisotropic bodies with cuts. It is assumed that the two-dimensional surface of a cut is a smooth manifold of an arbitrary configuration with a smooth boundary. The existence and uniqueness theorems for boundary value problems of statics and pseudo-oscillations are proved in the Besov and Bessel-potential spaces by means of the classical potential methods and the theory of pseudodifferential equations on manifolds with boundary. Using the embedding theorems, it is proved that the solutions of the considered problems are Hölder continuous. It is shown that the displacement vector and the temperature distribution function are Cα -regular with any exponent α < 1/2. This paper consists of two parts. In this part all the principal results are formulated. The forthcoming second part will deal with the auxiliary results and proofs.

scholarly journals Some Applications of Eigenvalue Problems for Tensor and Tensor–Block Matrices for Mathematical Modeling of Micropolar Thin Bodies

2019 ◽  
Vol 24 (1) ◽  
pp. 33 ◽  
Author(s):  
Mikhail Nikabadze ◽  
Armine Ulukhanyan
Keyword(s):  
Boundary Value Problems ◽  
Anisotropic Medium ◽  
Eigenvalue Problems ◽  
Constitutive Relations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Theory Of Elasticity ◽  
Initial Boundary Value Problems ◽  
Special Cases ◽  
Initial Boundary Value

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.

scholarly journals Generalized Analytic Functions in Higher Dimensions

2007 ◽  
Vol 14 (3) ◽  
pp. 581-595
Author(s):  
Wolfgang Tutschke
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Higher Dimensions ◽  
Partial Differential ◽  
Generalized Analytic Functions ◽  
Linear Partial Differential Equations ◽  
Weakly Singular

Abstract Originally I. N. Vekua's theory of generalized analytic functions dealt only with linear systems of partial differential equations in the plane. The present paper shows why I. N. Vekua's ideas are also fruitful for the solution of linear and non-linear partial differential equations in higher dimensions. One of the highlights of the theory of generalized analytic functions in the plane is the reduction of boundary value problems for general (linear or nonlinear) equations to boundary value problems for holomorphic functions using the well-known weakly singular and strongly singular 𝑇- and П-operators, respectively. The present paper is mainly aimed at reducing boundary value problems in higher dimensions to boundary value problems for monogenic functions.

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