Basic Boundary Value Problems Research Articles

Potential method in the coupled theory of thermoelastic triple-porosity nanomaterials

In this paper, the coupled linear theory of thermoelasticity for nanomaterials with triple porosity is considered in which the combination of Darcy’s law and the volume fraction concept for three levels of pores (macro-, meso- and micropores) is provided. The 3D basic boundary value problems (BVPs) of steady vibrations of this theory are formulated and these BVPs are investigated using the potential method (boundary integral equation method) and the theory of singular integral equations. Namely, the formula of integral representation of regular vectors is obtained. The surface (single-layer and double-layer) and volume potentials are introduced and their basic properties are given. Some useful singular integral operators are defined for which Noether’s theorems are valid. The symbolic determinants and indexes of these operators are calculated. The BVPs of steady vibrations are reduced to the equivalent singular integral equations. Finally, with the help of the potential method, we prove the existence theorems for classical solutions of the aforementioned BVPs.

SOLVING BASIC BOUNDARY VALUE PROBLEMS FOR AN ORTHOTROPIC LAYER

We consider the first, second basic and mixed boundary value problems of the linear theory of elasticity for a homogeneous orthotropic layer of constant thickness. The layer is subjected to a surface load. There are no internal forces. It is required to determine the stresses and displacements at an arbitrary point of the layer, assuming plane deformation. The solutions to the problem are constructed. The solution to the problem is sought in the space of transformants of the one-dimensional integral Fourier transform, where the relationship between the transformants of the stress function and the transformants of displacements and stresses is established. The desired values are expressed through four auxiliary functions that are associated with the stress and displacement transforms at the upper boundary. To obtain the true values of stresses and displacements at the points of the orthotropic layer, the inverse integral Fourier transform is applied to the found stress and displacement transforms. Numerical solutions for specific cases are obtained and the results are analyzed.

BASIC BOUNDARY VALUE PROBLEM WITH RETARDED ARGUMENT CONTAINING AN EIGENPARAMETER IN THE TRANSMISSION CONDITION

In this paper basic boundary value problem with retarded argument that has a discontinuity point inside the interval will be studied. At the discontinuity point transmission conditions contain eigenparameter. Existence of eigenvalues and eigenfunctions will be studied. Asmyptotic properties of eigenvalues and eigenfunctions will be obtained.

On the coupled linear theory of thermoelasticity for nanomaterials which triple porosity

In this paper, the linear mathematical model of thermoelastic nanomaterials with triple porosity is proposed in which the coupled phenomenon of Darcy’s law and the concept of the volume fractions of three levels of pores (macro-, meso- and micropores) is considered. Then, the 3D internal and external basic boundary value problems (BVPs) of steady vibrations of this model are formulated. The radiation conditions are established and Green’s identities are obtained in the considered theory. Finally, the uniqueness theorems for classical solutions of the above mentioned BVPs are proved.

COMPLEX BOUNDARY VALUE PROBLEMS FOR THE CAUCHY–RIEMANN OPERATOR ON A TRIANGLE

There are three over-determined boundary value problems for the inhomogeneous Cauchy–Riemann equation, the Dirichlet problem, the Neumann problem and the Robin problem. These problems have found their significant importance and applications in diverse fields of science, for example, applied mathematics, physics, engineering and medicine. In this paper, we explicitly investigate the solvability of these three basic boundary value problems for the Cauchy–Riemann operator on an isosceles orthogonal triangle. The solvability conditions are explicitly obtained. The plane parqueting reflection principle is used. Cauchy–Pompeiu-type formulae for the triangle are established. The boundary behavior of a related integral operator of the Schwarz type is studied in detail. Then, the boundary values of solutions at the corner points are found.

On the coupled theory of thermoelastic double-porosity materials

In this paper, the linear model of thermoelastic double-porosity materials is proposed in which the coupled phenomenon of the concepts of Darcy’s law and the volume fractions is considered. The basic systems of governing equations of motion and steady vibrations are obtained. Then, the boundary value problems of steady vibrations are investigated by means of the potential method (boundary integral equation method) and the theory of singular integral equations. Namely, on the basis of Green’s identities and the surface and volume potentials the existence and uniqueness theorems for the classical solutions of the internal and external basic boundary value problems of the considered theory are proved.

Boundary Value Problems of Thermoelasticity for Porous Sphere and for A Space with Spherical Cavity

This article is concerned with the coupled linear quasi-static theory of thermoelasticity for porous materials under local thermal equilibrium. The system of equations is based on the constitutive equations, Darcy's law of the flow of a fluid through a porous medium, Fourier's law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of governing equations is expressed in terms of displacement vector field, the change of volume fraction of pores, the change of fluid pressure in pore network and the variation of temperature of porous material. The present paper is devoted to construct explicit solutions of the quasi-static boundary value problems (BVPs) of coupled theory of thermoelasticity for a porous elastic sphere and for a space with a spherical cavity. In this research the regular solution of the system of equations for an isotropic porous material is constructed by means of the elementary (harmonic, bi-harmonic and meta-harmonic) functions. The basic boundary value problems (the Dirichlet type boundary value problem for a sphere and the Neumann type boundary value problem for a space with a spherical cavity) are solved explicitly. The obtained solutions are given by means of the harmonic, bi-harmonic and meta-harmonic functions. For the harmonic functions the Poisson type formulas are obtained. The bi-harmonic and meta-harmonic functions are presented as absolutely and uniformly convergent series.

Problems of steady vibrations in the coupled linear theory of double-porosity viscoelastic materials

In the present paper the coupled linear theory of double-porosity viscoelastic materials is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Indeed, in the beginning, the systems of equations of motion and steady vibrations are presented. Then, Green’s identities are established and the uniqueness theorems for classical solutions of the BVPs of steady vibrations are proved. The fundamental solution of the system of steady vibration equations is constructed and the basic properties of the potentials (surface and volume) are given. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (the boundary integral equations method) and the theory of singular integral equations.

Polyanalytic boundary value problems for planar domains with harmonic Green function

There are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

Oblique bearing capacity of shallow foundations placed near slopes determined by the method of rigorous characteristics

The ultimate bearing capacity of shallow foundations placed near slopes has long been a critical issue, especially for the obliquely loaded case. However, it is too complicated to find a statically and kinematically admissible field and then present a rigorous solution. Herein, the method of rigorous characteristics is applied to establish a slip line field that satisfies the stress and velocity boundary conditions simultaneously. Meanwhile, five one-sided failure modes combined with basic boundary value problems are presented to consider a variation in the failure mechanism. The application of boundary value problems avoids an arbitrary assumption of the failure shape and identifies the failure region automatically. The analytical solutions are located in the narrow range determined by the upper bound and lower bound solutions found in the literature, and hence they are rigorous. The ultimate bearing capacity is estimated by accounting for the load inclination, soil properties and the geometry model. The variation in the ultimate bearing capacity and the failure mechanism of obliquely loaded cases shows great non-linearity, due to the complicated boundary conditions.

A method for solving basic boundary value problems for a second-order self-adjoint differential operator with constant coefficients in domains of arbitrary shape

В данной работе метод опорных функций модифицируется для решения основных краевых задач для самосопряжённого дифференциального оператора в областях произвольной формы. Метод основывается на методе функций Грина и кратных преобразованиях Фурье

Inversion Symmetry of the Solutions of Basic Boundary-Value Problems of Two-Dimensional Elasticity Theory for a Wedge

By using the Mellin integral transformation, we obtain the solutions of basic two-dimensional problems for an elastic wedge. It is shown that, in the case of boundary conditions symmetric under the inversion transformation, some components of the solutions of plane strain problems and the solutions of antiplane strain problems for a wedge also have the inversion symmetry.

On the classification of boundary value problems for elastic homogeneous and piecewise homogeneous spaces with a penny-shaped crack

The classification of the basic axisymmetric boundary value problems of the mathematical theory of elasticity for homogeneous and piecewise homogeneous elastic spaces with a penny-shaped crack is given. Using the Hankel integral transform, governing integral equations (GIEs) of these problems are derived and methods for finding their exact solutions are indicated.

Potential method in the linear theory of viscoelastic porous mixtures

In the present paper, the linear theory of viscoelasticity for binary porous mixtures is considered. The fundamental solution of the system of steady vibration equations is constructed, and its basic properties are established. Green’s identities of this theory are obtained. The uniqueness theorems for classical solutions of the internal and external basic boundary value problems (BVPs) of steady vibrations are proved. The surface and volume potentials are introduced, and their basic properties are established. The determinants of symbolic matrices of the singular integral operators are calculated explicitly, and the BVPs are reduced to the always solvable singular integral equations for which Fredholm’s theorems are valid. Finally, the existence theorems for classical solutions of the internal and external BVPs of steady vibrations are proved by means of the potential method and the theory of singular integral equations.

Modeling triple porosity under local thermal non-equilibrium

In this paper, we present a model describing the evolutionary behavior of materials with triple porosity under local thermal non-equilibrium. The constitutive equations are presented for anisotropic and inhomogeneous materials and so they take into account the terms representing connectivity between the pressures and temperatures in the various pore scales. Further, the basic initial boundary value problems are investigated by using the Lagrange identity and the logarithmic convexity methods. We introduce a functional representing an appropriate measure of the solution to the initial boundary value problem for the model in concern and by means of the Lagrange identity method we are able to establish a uniqueness result. We also establish some estimates describing the continuous data dependence of solution with respect to the initial data and with respect to the various supply terms. Finally, we show how the above functional can be used in combination with the logarithmic convexity method in order to prove again that the solution is unique.

თერმოდრეკადობის თეორიის სტატიკის ძირითადი სასაზღვრო ამოცანები სხვადასხვა ველის გათვალისწინებით

თერმოდრეკადობის თეორიის სტატიკის ძირითადი სასაზღვრო ამოცანები სხვადასხვა ველის გათვალისწინებით

Simulation of Harmonic Temperature and Magnetic Fields Based on the Method of Separation of Variables in Regions of Complex Toroidal Forms

Basic boundary-value problems of calculating harmonic physical fields for toroidal forms (including the cases of complex sectors and nonlinear boundary-value conditions) are considered on the basis of exact analytical representation of the solution in the form of a series in toroidal harmonics. The results obtained can be used in calculating harmonic fields in problems of heat and mass transfer and of electromagnetism, coefficients of the medium of specific technical objects of toroidal form, as well as a test problem when employing other methods.

On the linear theory of double porosity thermoelasticity under local thermal non-equilibrium

In the present paper, the linear theory of thermoelasticity of double porosity materials under local thermal non-equilibrium is considered and the basic boundary value problems are investigated by using the boundary integral equation method (potential method). Indeed, the fundamental solution of system of steady vibrations equations of the considered theory is constructed explicitly by means of elementary functions and its basic properties are established. A Galerkin-type solution to this system of equations is presented and the completeness of this solution is proved. The basic internal and external boundary value problems of steady vibrations are formulated and the uniqueness theorems for classical solutions of these problems are proved. The basic properties of the surface and volume potentials and singular integral operators are established. Finally, the existence theorems for classical solutions of the above-mentioned boundary value problems are proved by using the potential method and the theory of singular integral equations.

External boundary value problems in the quasi static theory of thermoelasticity for materials with triple voids

AbstractIn this paper the quasi static linear theory of thermoelasticity for materials with triple voids is considered. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established.

On the exact solutions of the biharmonic problem of the theory of elasticity in a half-strip

We have constructed the solution of the first basic odd-symmetric boundary value problem in the theory of elasticity in a half-strip with free longitudinal sides. The solution is represented as series in Papkovich–Fadle eigenfunctions whose coefficients are found in an explicit form by using functions biorthogonal to the Papkovich–Fadle eigenfunctions. It is shown that the obtained solution describes residual stresses in an infinite strip with zero boundary conditions on its sides and the displacements that arise when the residual stresses are released as a consequence of the formation of a discontinuity. The same formulas give the exact solution of a boundary value problem for the half-strip with stresses specified at its end in the traditional statement, and only the displacements should now be taken with the opposite sign. In the constructed solutions, the angular points have the properties of infinitesimal elements, where, for the uniqueness of the solution, the boundary functions must be specified together with all their derivatives. In this, they are different from an angular point in an infinite wedge. The final formulas are simple and can easily be used in engineering.