External Boundary Value Problems Research Articles

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.

External problems of steady vibrations in the theory of elastic materials with a triple porosity structure

AbstractIn this paper, the linear theory of elasticity for materials with a triple porosity (macro, meso and micro levels of porosities) structure based on the volume fraction concept is considered. The external boundary value problems (BVPs) of steady vibrations of this theory are investigated. Namely, the uniqueness theorem for classical solutions of these BVPs is established. Then, the basic properties of the surface (single‐layer and double‐layer) potentials are given. Finally, on the basis of the properties of potentials, the existence theorems for classical solutions of the above mentioned BVPs of steady vibrations are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

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.

Решение задачи о раскрытии метаном трещин в предельно напряженной зоне пласта

The conditions were studied concerning the cracks opening by in-situ methane in the formation marginal zone during its transition to the maximum stressed state. The calculations are based on the joint use of the fundamental methods of deformable solids mechanics, statics of the flowing medium, and the concept of academician S.A. Khristianovich on the distribution of pressure of intra-layer methane during the movement of the mine face at a constant speed. The stress field of a coal-rock mass containing a seam with a weak interlayer and a working through it, was constructed within the framework of an elastic-plastic problem of the stressed state of a coal-rock mass. It is reduced to the integral equation of the second external boundary value problem of elasticity theory and solved by the boundary element method. The stress distribution in the extremely stressed zone of the formation is constructed by the method of characteristics known in the theory of differential equations. In the model of the ultimate stress state of a structurally homogeneous formation, the limiting zones are formed from the edge itself. In the formation with an unstable interlayer, the interlayer passes to the limiting state first, and at some distance from the edge, the formation itself. After that, the formation and the intermediate layer are deformed as integral whole. The cracks in the formation, which are its slip lines, open when the pressure of in-situ methane exceeds the stresses normal to the contour of the slip line. These stresses are related to the values of the principal stresses found from the solution of the boundary value problem. From this condition, the dimensions of the formation edge zone with free methane surfaces are determined.

A Pretreatment Method of Volterra the External Boundary Value Problem of Integral Differential Equations

In the process of traditional methods, the error rate of external boundary value problem is always at a high level, which seriously affects the subsequent calculation and cannot meet the requirements of current Volterra products. To solve this problem, Volterra's preprocessing method for the external boundary value problem of Integro differential equations is studied in this paper. The Sinc function is used to deal with the external value problem of Volterra Integro differential equation, which reduces the error of the external value problem and reduces the error of the external value problem. In order to prove the existence of the solution of the differential equation, when the existence of the solution can be proved, the differential equation is transformed into a Volterra integral equation, the Taylor expansion equation is used, the symplectic function is used to deal with the external value problem of homogeneous boundary conditions, and the uniform effective numerical solution of the external value problem of the equation is obtained by homogeneous transformation according to the non-homogeneous boundary conditions.

Stress analysis of an orthotropic elastic infinite plane with a hole

AbstractA general solution (stress functions) is derived for an orthotropic elastic plane problem using a mapping function to solve arbitrary configurations. Lekhnitskii formulism using complex variable is used. A problem of an infinite orthotropic elastic plane with a hole is solved. The boundary condition is represented by two complex variables. This makes it difficult to solve the orthotropic elastic plane problem. The problem is solved by overcoming this difficulty. Two methods of Cauchy integral and Riemann‐Hilbert problem are applied for the analysis. It is confirmed that the same exact stress functions are finally obtained by these methods and are represented by an irrational mapping function as a closed form. These stress functions are the general solution for the holed problem. The Mittag‐Leffler Theorem is used for the analysis. Therefore, there are two methods to solve an external boundary value problem. Stress components are represented by one complex variable. Therefore, it is not difficult to calculate the stress components. Arbitrarily shaped hole problem can be solved by changing the mapping function in the stress functions. The stress distributions for an infinite plane with a square hole subjected to uniform tension for Case I and III problems are shown. Symmetry of the stress distribution is lost for a Case III problem. A problem of a half plane with an edge crack is solved as Riemann‐Hilbert problem. It is confirmed that the solution coincide with that obtained by Cauchy integral method.

Анализ траектории трещины гидроразрыва около пластовой выработки и условия ее прямолинейного роста

Model of the geomechanical state of a disc hydraulic fracture propagating in the solid rocks near the seam working is based on the methods of solid mechanics and fracture mechanics. Stress field in the coal-rock mass containing in-seam working and growing hydraulic fracture was constructed as a result of solving an elastoplastic problem, in which the area of plasticity is the extremely stressed zones of the edge parts of the seam. The stress field in the edge parts was determined in the course of the numerical solution of three boundary value problems of the seam limiting state. The criteria for the onset of the limiting state are the general criterion of the Coulomb — Mohr limiting state for the formation and a special criterion for the limiting state for its contact with the rock mass. By replacing the extremely stressed zones with the stresses acting on their contact with the rock mass, the elastoplastic problem is reduced to the second external boundary value problem of the theory of elasticity, which is solved by the method of boundary integral equations. At the relatively low fluid pressures in the pumping unit, the trajectory of the hydraulic fracture is a smooth curved line of the small length with a significant deviation of its ends from the direction of the seed crack. With increasing fluid pressure, the crack length increases, and the deviation from the direction of the seed crack decreases. There are fluid pressures at which the crack propagates in a straight line and practically does not change its original direction. The straight-line trajectory of the crack in the vicinity of the working located at different depths corresponds to a point on the graph of the dependence of the relative length of the crack on the fluid relative pressure. This graph is a straight line.

The implementation of the boundary element method to the Helmholtz equation of acoustics

Introduction: To solve the Helmholtz equation is important for the branches of engineering that require the simulation of wave phenomenon. Numerical methods allow effectiveness’ enhancing of the related computations. Methods: To find a numerical solution of the Helmholtz equation one may apply the boundary element method. Only the surface mesh constructed for the boundary of the three-dimensional domain of interest must be supplied to make the computations possible. This method’s trait makes it possible toconduct numerical experiments in the regions which are external in relation to some Euclidian three-dimensional subdomain bounded in the three-dimensional space. The later also provides the opportunity of not using additional geometric techniques to consider the infinitely distant boundary. However, it’s only possible to use the boundary element methods either for the homogeneous domains or for the domains composed out of adjacent homogeneous subdomains. Results: The implementation of the boundary elementmethod was committed in the program complex named Quasar. The discrepancy between the analytic solution approximation and the numerical results computed through the boundary element method for internal and external boundary value problems was analyzed. The results computed via the finite element method for the model boundary value problems are also provided for the purpose of the comparative analysis done between these two approaches. Practical relevance: The method gives an opportunityto solve the Helmholtz equation in an unbounded region which is a significant advantage over the numerical methods requiring the volume discretization of computational domains in general and over the finite element method in particular. Discussion: It is planned to make a coupling of the two methods for the purpose of providing the opportunity to conduct the computations in the complex regions with unbounded homogeneous subdomain and subdomains with substantial inhomogeneity inside.

Potential method in the coupled theory of elastic double-porosity materials

In the present paper the linear coupled model of elastic double-porosity materials is proposed in which the coupled phenomenon of the concepts of Darcy’s law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions, and its basic properties are presented. The radiation conditions are established, and Green’s identities are obtained. The uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed, and the basic properties of these potentials are given. The determinants of symbolic matrices of the singular integral operators are calculated explicitly. Then, 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 BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

ИССЛЕДОВАНИЕ ПРИБЛИЖЕННОГО РЕШЕНИЯ НЕКОТОРЫХ КЛАССОВ ПОВЕРХНОСТНЫХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ ПЕРВОГО РОДА

A sequence is constructed that converges to an exact solution of a hypersingular integral equation of the first kind of the external Neumann boundary value problem for the Helmholtz equation, which is the boundary value of the solution of the external Neumann boundary value problem on the boundary of the domain. In addition, a sequence is constructed that converges to an exact solution of a weakly singular integral equation of the first kind of the external Dirichlet boundary value problem for the Helmholtz equation, which is the boundary value of the normal derivative of the solution of the external Dirichlet boundary value problem on the boundary of the domain.

Исследование напряженного состояния угольного целика между очистной и подготовительной выработками методами механики деформируемого тела

To ensure safe conditions for mining operations and increase labor productivity, a reliable assessment of the stress state of the coal-rock mass is required. The model is presented concerning the geomechanical state of the massif hosting the coal seam, treatment, and preparatory workings. The model is developed based on the fundamental methods of solid mechanics and ensures a computational experiment and the reliability of the results. Stress distribution in the coal-rock mass in the vicinity of the in-seam workings was calculated in two stages. First, the stress field in the edge zones of the coal seam and in the collapsed rocks was determined by the methods of mechanics of the flowing medium. Distribution of stresses in the extremely stressed zones of the seam and the layer of collapsed rocks behind the working excavation was found by the method of characteristics by solving differential equations of the hyperbolic type. They are obtained based on the of the joint solution of the equilibrium equations, general and special Coulomb — Mohr criteria of the transition of the seam and the collapsed layer, as well as their contacts with the lateral rocks to the limiting state. Then, by replacing extremely stressed zones of the coal seam and the layer of the collapsed rocks with stresses acting at the contacts with the surrounding massif, the problem is reduced to the integral equation of the second external boundary value problem of the theory of elasticity. It is solved by the method of boundary integral equations. Insignificant influence of changes in the angle of internal friction of the collapsed rocks on the parameters of the seam bearing pressure in the vicinity of the treatment and development workings is shown. However, it significantly effects on the bearing pressure in the extremely stressed zone of the collapsed rocks layer.

Analysis of a mixed boundary value problem for an orthotropic elasticity using a mapping function

A solution is derived for a mixed boundary orthotropic elastic plane problem using a mapping function to solve arbitrary configurations. No stress function using a mapping function seems to have been derived. The problem is solved as a Riemann–Hilbert problem. The final exact stress functions are represented by an irrational mapping function as a closed form. Stress components are represented by one complex variable. Arbitrarily shaped hole problems can be solved by changing the mapping function. Once the analytical solution has been derived, calculating the stress components is simpler than for an isotropic problem. It is easier to use an irrational mapping function than to form a rational mapping function for an isotropic problem. As an example, an infinite plane with a square hole subjected to uniform tension is analyzed. The stress distributions are shown for Cases I and III problems corresponding to two characteristic roots of the fundamental equation. In a Case III problem, the symmetry of stress distributions is lost. A solution of an external force boundary value problem can be derived from that of the mixed boundary value problem.

Quasi-static problems in the coupled linear theory of thermoporoelasticity

This article is concerned with the coupled linear quasi-static theory of thermoporoelasticity under local thermal equilibrium. The system of equations of this theory is based on the constitutive equations, Darcy’s law of the flow of a fluid through a porous medium, Fourie’s law of heat conduction, the equations of equilibrium, fluid mass conservation and heat transfer. The system of general governing equations is expressed in terms of the displacement vector field, the changes of the volume fraction of pores, the fluid pressure in pore network and temperature. The fundamental solution of the system of quasi-static equations in the considered theory is constructed and its basic properties are established. On the basis of Green’s formulas the uniqueness theorems for classical solutions of the internal and external boundary value problems (BVPs) are proved. Then, the surface and volume potentials are presented and their basic properties are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

Regularization of an external boundary value problem for open axisymmetric magnetic systems

Important characteristic of magnetic-field with the high measure of intensity and heterogeneity in the swept volume of the electromagnetic systems is a size of tension of the field Н. Calculation of tension in the field H in these systems, characterized by relatively large air interval, given its three-dimensionalty, presents a very difficult task. Analytical decision of equalization of Laplace that describes distribution of magnetic-field in the interpolar volume of the axisymmetrical magnetic systems, difficultly from difficult geometry of bodies that is included in a calculation area, that is why for research of distribution of tension the expedient use of numeral methods of calculation - to the method of eventual differences, method of eventual elements, method of maximum integral equalizations. The calculation of distribution of scalar magnetic potential in open magnetic systems with the use of numeral methods of calculation causes the difficulties related to limitation of calculation  area in which the calculation is conducted. The static magnetic fields, analysed in this research, obey one of basic equalizations of mathematical physics, and namely, to equalization of Laplace in partial differential. In case of calculation of magnetic field of the axisymmetrical magnetic systems it is necessary to resolve the Cauchy problem for equalization of Laplace. It is known, that this task does not have characteristic of steadyness, and thus does not obey Hadamars third condition of correctness and, therefore, is considered incorrectly defined. For the number of unsteady tasks of mathematical physics of R.Lattes and ZH.Lions developed the quasi-random method, that can be applied both for evolutional tasks, as well as for constant. The basic idea of quasi-random method lays in proper update of differential operators that are part of the task. This change is done by introducing additional differential elements. Application of this method allows to use effectively the numeral methods of calculation of regional task for  open axisymmetrical systems.

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.

Quasi-static problems in the coupled linear theory of elasticity for porous materials

In this paper, the coupled linear quasi-static theory of elasticity for porous materials is considered. The system of equations of this theory is based on the constitutive equations, Darcy’s law, the equations of equilibrium, and fluid mass conservation. The system of general governing equations is expressed in terms of the displacement vector field, the volume fraction of pores, and the fluid pressure in pore network. The fundamental solution of the system of steady vibration equations in the considered theory is constructed, and its basic properties are established. Green’s formulas are obtained, and the uniqueness theorems of the internal and external boundary value problems (BVPs) are proved. Then, the surface and volume potentials are constructed, and their basic properties are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

Steady vibration problems in the coupled linear theory of porous elastic solids

This paper concerns the coupled linear theory of elasticity for isotropic porous materials. In this theory the coupled phenomena of the concepts of Darcy’s law and the volume fraction is considered. The basic internal and external boundary value problems (BVPs) of steady vibrations are investigated. Indeed, the fundamental solution of the system of steady vibration equations is constructed explicitly by means of elementary functions, and its basic properties are presented. The radiation conditions are established and Green’s identities are obtained. The uniqueness theorems for the regular (classical) solutions of the BVPs are proved. The surface (single layer and double layer) and volume potentials are constructed and the basic properties of these potentials are given. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method (boundary integral equation method) and the theory of singular integral equations.

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.

Potential method in the theory of thermoelasticity for materials with triple voids

In the present paper the linear theory of thermoelasticity for isotropic and homogeneous solids with macro-, meso- and microporosity is considered. In this theory the independent variables are the displacement vector field, the changes of the volume fractions of pore networks and the variation of temperature. The fundamental solution of the system of steady vibrations equations is constructed explicitly by means of elementary functions. The basic internal and external boundary value problems (BVPs) are formulated and the uniqueness theorems of these problems are proved. The basic properties of the surface (single-layer and double-layer) and volume potentials are established and finally, the existence theorems for regular (classical) solutions of the internal and external BVPs of steady vibrations are proved by using the potential method (boundary integral equation 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.