Boundary Value Problem In Plane Research Articles

On the solution of plane boundary-value problems of elasticity in a rectangle

An algorithm for solving plane boundary-value problems of elasticity for a rectangular domain is expounded. The algorithm is based on a complex-valued representation of the general solution to the differential equations of the plane problem and on the use of Lagrange polynomials to satisfy the boundary conditions. The algorithm can quite easily be implemented in a computer program. This is probably the simplest way of solving boundary-value problems of this class

Scattering of waves by articulated floating elastic plates in water of infinite depth

The wave scattering by an articulated floating elastic plate in water of infinite depth is analyzed in the linearized theory of water waves. Using the geometrical symmetry of the articulated plate, the associated boundary value problem in the half-plane is reduced to two boundary value problems in the quarter plane, whose solutions are derived by the direct application of a mixed-type Fourier transform and the corresponding mode-coupling relation. The articulated plate is modeled as the assembling of two semi-infinite thin elastic plates which are attached by connectors. The hydroelastic behavior of the floating elastic plate is investigated by analyzing the stiffness of the connectors on the reflection and transmission characteristics of the flexural gravity waves. The phase and group velocity, reflection and transmission coefficient and the vertical displacement response of the elastic plate are computed and analyzed to understand the effect of articulation on the wave motion below the plate.

Seismic Wave Scattering By InhomogeneousGeological Deposits With Cracks

In this work, we study seismic wave scattering by cracks in inhomogeneous geological continua with depth-dependent material parameters and under conditions of plane strain. A restricted case of inhomogeneity is considered, with Poisson’s ratio equal to 0.25 and with both shear modulus and density profiles varying proportionally. Also, time-harmonic conditions are assumed to hold. For this type of material, elastic wave speeds remain macroscopically constant and it becomes possible to recover exact Green functions for the crack-free inhomogeneous continuum by using an algebraic transformation proposed in earlier work [1] as a first step. Subsequently, the complete elastodynamic fundamental solution, along with its spatial derivatives and an asymptotic expansion for small argument, are all derived in closed-form using the Radon transform [2]. Next, a non-hypersingular, traction-based boundary element formulation (BEM) is implemented for solving the plane boundary-value problem (BVP) with internal cracks [2]. This formulation is used for computing stress intensity factors (SIF) and scattered wave displacement field amplitudes for the case of an inclined line crack in a continuously inhomogeneous medium swept by either pressure (P) waves or vertically polarized shear (SV) waves at an arbitrary angle of incidence. The numerical results show substantial differences between homogeneous and inhomogeneous materials containing a crack in terms of their dynamic response, with the latter case being a more realistic representation for geological deposits. Finally, these types of results are useful within the context of earthquake engineering, since they can account for the influence of geological cracks in modifying seismically-generated ground motions.

Investigation of the First and the Second Exterior Plane Boundary Value Problems of Steady State Oscillations in the Linear Theory of Elastic Mixtures

Abstract Displacement vectors are represented by combinations of special potentials; singular integral equations of the normal type with zero index are obtained for the first and the second boundary value problem of steady oscillations in the theory of elastic mixtures. It is proved that in the case of positive frequencies the corresponding homogeneous singular integral equations have only a trivial solution.

Electroelastic analysis of a piezoelectric half-plane with finite surface electrodes

In this paper the problem of the electroelastic fields of piezoelectric ceramic in a half-plane with thin collinear surface electrodes is formulated and analyzed. Using the standard analytic continuation method, a mixed boundary value problem in plane is obtained and then reduced to a standard Hilbert problem. The closed form expressions of electroelastic fields are obtained. Some special cases are discussed. The electroelastic filed has singularity with 1/ r at the vicinity of the electrode ends. The numerical examples for two surface electrodes are studied to show the practical distributions of the electroelastic fields graphically.

A variant of Trefftz's method by boundary Fourier expansion for solving regular and singular plane boundary-value problems

The main objective of the present work is to introduce a variant of Trefftz's method for finding approximate solutions to regular or singular two-dimensional boundary-value problems for Laplace's equation. After expressing the solution as a finite linear combination of trial functions, the method foresees the enforcement of the boundary condition by using a boundary Fourier expansion technique, instead of the usual pointwise approach. The procedure ultimately produces a rectangular set of linear algebraic equations. The method is used to find approximate solutions to five prototype problems, mainly in rectangular regions, using polar harmonics as trial functions. More complicated cases could be envisaged. The results are discussed and compared with those obtained by the Boundary Collocation Method for the same set of trial functions. It appears that the proposed method yields better results. In the case of a singular behaviour of the solution at the boundary, for which the singularity could be isolated, we have compared the results obtained by the proposed method, with isolation and without isolation of the singularity. The proposed method may be easily extended to deal with other boundary-value problems involving more complicated differential operators, boundary geometries and boundary conditions in two or in three dimensions.

A variant of Trefftz's method by boundary Fourier expansion for solving regular and singular plane boundary-value problems

The main objective of the present work is to introduce a variant of Trefftz's method for finding approximate solutions to regular or singular two-dimensional boundary-value problems for Laplace's equation. After expressing the solution as a finite linear combination of trial functions, the method foresees the enforcement of the boundary condition by using a boundary Fourier expansion technique, instead of the usual pointwise approach. The procedure ultimately produces a rectangular set of linear algebraic equations. The method is used to find approximate solutions to five prototype problems, mainly in rectangular regions, using polar harmonics as trial functions. More complicated cases could be envisaged. The results are discussed and compared with those obtained by the Boundary Collocation Method for the same set of trial functions. It appears that the proposed method yields better results. In the case of a singular behaviour of the solution at the boundary, for which the singularity could be isolated, we have compared the results obtained by the proposed method, with isolation and without isolation of the singularity. The proposed method may be easily extended to deal with other boundary-value problems involving more complicated differential operators, boundary geometries and boundary conditions in two or in three dimensions.

A Representation Formula for the Robin Function

The Robin function R is the fundamental solution of a mixed boundary value problem in the complex plane, where Dirichlet conditions are posed on one part of the boundary and Neumann conditions on the remaining part. We prove a representation formula for R in a multiply connected domain under rather weak assumptions on the structure of the Dirichlet and the Neumann boundary. The key to this formula is to relate the Robin function to the Green function of a certain Riemann surface. This surface is embedded in a compact surface that arises from the action of a classical Schottky group in its domain of discontinuity.

GREEN'S FUNCTION FOR THERMAL STRESS MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ARBITRARY HOLE UNDER A POINT HEAT SOURCE

Green's function of a point heat source is derived for a mechanical mixed boundary value problem of an infinite plane with an arbitrary hole, for which zero-displacement and traction-free boundary conditions are prescribed to its boundary. As the thermal boundary condition on the hole, either an adiabatic or isothermal condition is considered. By employing the mapping technique and complex variable method, an explicit solution including a hypergeometrical function is obtained. Stress distributions are shown in illustrative examples for a square hole.

Displacement Boundary Value Problem of Piezoelectric Material Plane With an Elliptic Hole

In this paper, the displacement boundary value problem of a piezoelectric material plane with an elliptic hole is studied. As the permittivity of the medium in the hole is far less than that of the piezoelectric material, the electric induction of the medium is negligible. An exact solution is obtained. Its application to the inclusion problem has been given. The components of the stress tensor are discussed.

On the justification of approximate unbounded solutions of mixed plane boundary value problems

The approximate unbounded solution of a class of singular integral equations is fully justified and the error is estimated. This is the class to which lot of mixed plane boundary value problems are reducible. An example is given in which the procedures were carried out right to the numerical results.

On the solutions of mixed plane problems that are unbounded where the boundary conditions change

In this paper, we apply a modification to the method of the integral equation formulation of mixed plane boundary value problems so that it enables us to obtain the solutions unbounded at the points where the boundary conditions change. Such solutions are of great physical interest. The modification is illustrated by means of a typical problem. As is it the case in the original method proposed by Cherskii [1], the problem is reduced to an infinite system of algebraic equations. The justification of the truncation of such systems has been established.

GREEN'S FUNCTION FOR A THERMOMECHANICAL MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ELLIPTIC HOLE

This article derives the Green's function for a thermomechanical mixed boundary value problem of an infinite plane with an elliptic hole under a pair of heat source and sink. To derive the Green's function in closed form, the Cauchy integral method and a basic Green's function for an external force boundary value problem with a pair of heat source and sink are employed. Illustrative numerical results for temperature, heat flux, and stress along the hole edge and stress intensity factors when the hole collapses into a crack are presented graphically.

A New Method of Solving the Basic Plane Boundary Value Problems of Statics of the Elastic Mixture Theory

The basic plane boundary value problems of statics of the elastic mixture theory are considered when on the boundary are given: a displacement vector (the first problem), a stress vector (the second problem); differences of partial displacements and the sum of stress vector components (the third problem). A simple method of deriving Fredholm type integral equations of second order for these problems is given. The properties of the new operators are established. Using these operators and generalized Green formulas we investigate the above-mentioned integral equations and prove the existence and uniqueness of a solution of all the boundary value problems in a finite and an infinite domain.

Explanation of locking of four‐node plane element by considering it as elastic Dirichlet‐type boundary value problem

The aim of this study is to arrive at a better understanding of the phenomenon of locking of low-order compatible displacement type of finite elements in particular for the hour-glass mode of the plane four-node element and dilative materials. To this end the properties of finite elements are investigated in an analytical way, where a finite element is considered as a plane boundary value problem with prescribed boundary displacement (Dirichlet problem). In this paper for the sake of simplicity the simplest possible linear comparison solid, namely isotropic linear elasticity, is applied, although recognizing fully that for a dilative material elasto-plasticity would be more realistic. From the study described in this paper it is concluded that locking of the four-node element is not due to any particular numerical formulation of this compatible finite element since, even the analytical solution suffers from this problem. The locking of this element is not related to incompressibility of the material either as the analytical solution shows locking to occur at a parameter set which differs significantly from the one in case of incompressibility. It is shown that locking is a consequence of the combination of the dilative material behaviour and the compatible displacement type of boundary conditions, which leads to infinite isotropic stresses in the element. These infinite isotropic stresses occur at the limit of uniqueness of the solution, which for this element is shown to occur outside the parameter range of the sufficiency of uniqueness.

The Basic Mixed Plane Boundary Value Problem of Statics in the Elastic Mixture Theory

Abstract The basic mixed plane boundary value problem of equations of statics of the elastic mixture theory is considered in a simply connected domain when the displacement vector is given on one part of the boundary and the stress vector on the remaining part. The problem is investigated using the general displacement vector and stress vector representations obtained in [Basheleishvili, Georgian Math. J. 4: 223–242, 1997]. These representations enable us to reduce the considered problem to a system of singular integral equations with discontinuous coefficients of special kind. The solvability of this system in a certain class is proved, which implies that the basic plane boundary value problem has a solution and this solution is unique.

A generalization of D. A. Grave's method for plane boundary-value problems in harmonic potential theory

The article describes and proves D. A. Grave's method that solves classical plane boundary-value problems for the Green's function of the Laplace equation in regions whose boundaries are smooth analytical curves defined by finite-order irreducible polynomials. The proposed method has certain advantages compared with the method that constructs the Green's function by conformally mapping the original region onto the unit disk. A class of regions are identified for which Grave's methods produces an explicit analytical solution in convergent-series form. This is a natural generalization of the conformal mapping method for simplest regions.

Friction of solid bodies with formation of a melt layer

Some laws governing friction of solid bodies with formation of a developed layer of melt are studied. In contrast to [1], we study a case when a finite body moves along a melting substrate (half-plane) with the heat required for melting being liberated by frictional heat release or from a heat flux from the body. To determine the outflow of heat from the melt layer to the half-plane and the zone of melt in front of the moving body we considered two plane boundary-value problems of the equation of heat conduction.

Application of Analogues of General Kolosov–Muskhelishvili Representations in the Theory of Elastic Mixtures

Abstract The existence and uniqueness of a solution of the first, the second and the third plane boundary value problem are considered for the basic homogeneous equations of statics in the theory of elastic mixtures. Applying the general Kolosov–Muskhelishvili representations from [Basheleishvili, Georgian Math. J. 4: 223–242, 1997], these problems can be splitted and reduced to the first and the second boundary value problem for an elliptic equation which structurally coincides with the equation of statics of an isotropic elastic body.

Application of Analogues of General Kolosov–Muskhelishvili Representations in the Theory of Elastic Mixtures

The existence and uniqueness of a solution of the Þrst, the second and the third plane boundary value problem are consid- ered for the basic homogeneous equations of statics in the theory of elastic mixtures. Applying the general Kolosov-Muskhelishvili rep- resentations from (1), these problems can be splitted and reduced to the Þrst and the second boundary value problem for an elliptic equa- tion which structurally coincides with the equation of statics of an isotropic elastic body.