Basic Boundary Value Problems Research Articles

The nullfield method for the ellipsoidal Stokes problem

The ellipsoidal Stokes problem is one of the basic boundary-value problems for the Laplace equation which arises in physical geodesy. Up to now, geodecists have treated this and related problems with high-order series expansions of spherical and spheroidal (ellipsoidal) harmonics. In view of increasing computational power and modern numerical techniques, boundary element methods have become more and more popular in the last decade. This article demonstrates and investigates the nullfield method for a class of Robin boundary-value problems. The ellipsoidal Stokes problem belongs to this class. An integral equation formulation is achieved, and existence and uniqueness conditions are attained in view of the Fredholm alternative. Explicit expressions for the eigenvalues and eigenfunctions for the boundary integral operator are provided.

On Laplace and Stokes potentials

Integral equations associated with the basic boundary value problems for the Laplace and Stokes equations are considered. The integral operators for these integral equations are interpreted as the pseudodifferential operators, and their principal symbols are calculated. The symbols are obtained in terms of the principal curvatures and the coefficients of the first quadratic form of the boundary. As a consequence, the initial approximation is suggested for the iterative methods solving the integral equations. © 1998 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

Torsion of Functionally Graded Isotropic Linearly Elastic Bars

The purpose of this research is to investigate the effects of material inhomogeneity on the torsional response of linearly elastic isotropic bars. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The classic approach to the torsion problem for a homogenous isotropic bar of arbitrary simply-connected cross-section in terms of the Prandtl stress function is generalized to the inhomogeneous case. The special case of a circular rod with shear modulus depending on the radial coordinate only is examined. It is shown that the maximum shear stress does not, in general, occur on the boundary of the rod, in contrast to the situation for the homogeneous problem. It is shown that the material inhomogeneity may increase or decrease the torsional rigidity compared to that for the homogeneous rod. Optimal upper and lower bounds for the torsional rigidity for nonhomogeneous bars of arbitrary cross-section are established. A new formulation of the basic boundary-value problem is given. The results are illustrated using specific material models used in the literature on functionally graded elastic materials.

A circular inclusion with circumferentially inhomogeneous interface in antiplane shear

A general method is developed for the rigorous solution of a problem associated with a circular inclusion embedded within an infinite matrix in antiplane shear. The bonding at the inclusion–matrix interface is assumed to be imperfect. Most significant is the fact that the imperfection in the interface is assumed to be circumferentially inhomogeneous. Using analytic continuation, the basic boundary–value problem for two analytic functions is reduced to a first‐order differential equation for a single analytic function and the closed‐form solution is obtained. The method is illustrated using several specific examples. The results from these examples are compared to the corresponding results when the imperfection in the interface is homogeneous. These comparisons illustrate how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the stresses induced within the inclusion.

Similarity analysis of inelastic contact

Analysis of mechanical contact of solids is of interest not only regarding a variety of mechanical assemblies but also on a smaller scale such as roughness properties of surfaces and compaction of powder particles. Indentation testing is another prominent problem in the context. To analyse the phenomena involved is inherently difficult at application essentially due to the presence of large strains, nonlinear material behaviour, time dependence and moving contact boundaries. Recently, progress has been made, however, to explicitly solve basic boundary value problems especially due to advances in computational techniques. A substantial ingredient which facilitates solution procedures is self-similarity and it is the present purpose to explore in detail the advantages in a general setting when this feature prevails. A viscoplastic framework is laid down for a wide class of constitutive properties where strain-hardening plasticity, creep and also nonlinear elasticity arise as special cases. It is then shown that when surface shapes and material properties are modelled by homogeneous functions, associated boundary value problems posed may be reduced to stationary ones. As a consequence, within Hertzian kinematics, relations between contact impression and regions become independent of loading and time and the connection to loading characteristics does not usually require a full solution of the problem. In particular it is shown that for general head-shapes it proves efficient to use an approach where an intermediate flat die solution serves as a basic tool also for hereditary materials. An invariant computational procedure based on the intermediate problem is arrived at and decisive results shown to be found by simple cumulative superposition. Illustrations are given analytically for ellipsoidal contact of Newtonian fluids and by detailed computations for spherical indentation of viscoplastic solids for which also universal hardness formulae are proposed. For several bodies in contact it is shown how general results may be extracted from fundamental solutions for a half-space.

Steady unsaturated flow in two-dimensional scale-heterogeneous porous media

Unsaturated flow in scale heterogeneous porous media is described by a class of nonlinear partial differential equations involving three general coefficient functions. It is shown that among these, there is a large class of integrable models, being transformable to linear equations even when the spatially varying geometric dilation scale factor is an arbitrary twice differentiable function. Explicit solutions are constructed for basic boundary value problems that relate to field and laboratory measurements. In one of these solutions, the flow pattern is unaffected by the heterogeneity even though the water concentration is modified. From another solution, we predict the bulk conductivity of a heterogeneous soil sample with a constant hydraulic potential difference maintained between the ends.

Three-Dimensional Boundary Value Problems of Elastothermodiffusion with Mixed Boundary Conditions

Abstract We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [Kupradze, Gegelia, Basheleishvili, and Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Publishing Company, 1979, Russian original, 1976–Mikhlin, Multi-dimensional singular integrals and integral equations, 1962], we prove theorems on the existence and uniqueness of the classical solutions of these problems.

Boundary Integral Equation Method in the Steady State Oscillation Problems for Anisotropic Bodies

The three-dimensional steady state oscillation problems of the elasticity theory for homogeneous anisotropic bodies are studied. By means of the limiting absortion principle the fundamental matrices maximally decaying at infinity are constructed and the generalized Sommerfeld-Kupradze type radiation conditions are formulated. Special functional spaces are introduced in which the basic and mixed exterior boundary value problems of the steady state oscillation theory have unique solutions for arbitrary values of the oscillation parameter. Existence theorems are proved by reduction of the original boundary value problems to equivalent boundary integral (pseudodifferential) equations.

Solution of the Basic Boundary Value Problems of Stationary Thermoelastic Oscillations for Domains Bounded by Spherical Surfaces

The boundary value problems of stationary thermoelastic oscillations are investigated for the entire space with a spherical cavity, when the limit values of a displacement vector and temperature or of a stress vector and heat flow are given on the boundary. Also, consideration is given to the boundary-contact problems when a nonhomogeneous medium fills up the entire space and consists of several homogeneous parts with spherical interface surfaces. Given on an interface surface are differences of the limit values of displacement and stress vectors, also of temperature and heat flow, while given on a free boundary are the limit values of a displacement vector and temperature or of a stress vector and heat flow. Solutions of the considered problems are represented as absolutely and uniformly convergent series.

THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS OF ELASTOTHERMODIFFUSION WITH MIXED BOUNDARY CONDITIONS

We investigate the basic boundary value problems of the connected theory of elastothermodiusion for three-dimensional do- mains bounded by several closed surfaces when the same boundary conditions are fulÞlled on every separate boundary surface, but these conditions dier on dierent groups of surfaces. Using the results of papers (1-8), we prove theorems on the existence and uniqueness of the classical solutions of these problems.

Solution of the Basic Boundary Value Problems of Stationary Thermoelastic Oscillations for Domains Bounded by Spherical Surfaces

The boundary value problems of stationary thermoelastic oscillations are investigated for the entire space with a spherical cavity, when the limit values of a displacement vector and temperature or of a stress vector and heat flow are given on the boundary. Also, consideration is given to the boundary-contact problems when a nonhomogeneous medium fills up the entire space and consists of several homogeneous parts with spherical interface surfaces. Given on an interface surface are differences of the limit values of displacement and stress vectors, also of temperature and heat flow, while given on a free boundary are the limit values of a displacement vector and temperature or of a stress vector and heat flow. Solutions of the considered problems are represented as absolutely and uniformly convergent series.

Two-dimensional steady-state oscillation problems of anisotropic elasticity

The paper deals with the two-dimensional exterior bound- ary value problems of the steady-state oscillation theory for anisotro- pic elastic bodies. By means of the limiting absorption principle the fundamental matrix of the oscillation equations is constructed and the generalized radiation conditions of Sommerfeld-Kupradze type are established. Uniqueness theorems of the basic and mixed type boundary value problems are proved. In the paper we treat the uniqueness theorems of basic and mixed type exterior boundary value problems (BVPs) for equations of two-dimensional steady-state elastic oscillations of anisotropic bodies. In this case questions regarding the correctness of BVPs have not been investigated so far. Here any analogy with the isotropic case is completely violated, since the ge- ometry of the characteristic surface becomes highly complicated and the fundamental matrix cannot be written explicitly in terms of elementary functions. This in turn creates another diculty in obtaining asymptotic estimates. As is well known, even for the metaharmonic equation v(x) + k 2 v(x) = 0; k 2 > 0; x 2R 2 ;

Two-Dimensional Steady-State Oscillation Problems of Anisotropic Elasticity

The paper deals with the two-dimensional exterior boundary value problems of the steady-state oscillation theory for anisotropic elastic bodies. By means of the limiting absorption principle the fundamental matrix of the oscillation equations is constructed and the generalized radiation conditions of Sommerfeld–Kupradze type are established. Uniqueness theorems of the basic and mixed type boundary value problems are proved.

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

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.

Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. II

In the first part [1] 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 $$(\mathbb{B}_{p, q}^s )$$ and Bessel-potential (ℍ ) 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.

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

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.

Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. I

Basic boundary value problems of thermoelasticity for anisotropic bodies with cuts. I

Formulation of boundary-value problems of statics for thin elastic asymmetrically-laminated anisotropic plates and solution using functions of a complex variable

The internal stress-strain state (SSS) of a plate under coupled bending and extension-compression-shear is determined. It is proved that the two-dimensional equilibrium equations for the materials of layers with anisotropy of general form are elliptic, and natural boundary conditions are derived. The displacements are expressed in terms of functions of a complex variable, which make it possible to reduce the basic boundary-value problems to a determination of these functions from the values of the real part of linear combinations of the functions on the lateral surface of the plate. Conditions are established for the functions to be single-valued; these conditions are related to the self-balance of the boundary load. Exact solutions of the boundary-value problems are presented for a finite ellipse, including the case where polynomial loads are applied to the plate faces. Solutions are presented for a plate with an elliptic cut-out, including the case in which the boundary loads are not self-balanced. The singular problem of a plate with a loaded finite cut through the material or a rigid insertion is also considered. Exact solutions are constructed, using Schwartz's formula, for simply-connected regions that can be mapped conformally onto a disk. The method proposed is a new variation of the Kolosov-Muskhelishvili-Lekhnitskii method for non-classical laminated plates and has the same advantages and disadvantages. The main difference compared with the standard two-dimensional problem or bending problem is the higher dimension.

Existence and continuous dependence results in the theory of microstretch elastic bodies

This paper is concerned with the linear theory of microstretch elastic solids. First, we establish existence and uniqueness results for the basic boundary-value problems of elastostatics. Then, we use some results of the semigroups theory to prove an existence theorem in the dynamic theory.

The basic boundary value problems ofstatic elasticity theory and their Cosserat spectrum

The basic boundary value problems ofstatic elasticity theory and their Cosserat spectrum