Non - Classical Mixed Interface Problems for Anisotropic Bodies

The paper deals with the three-dimensional mathematical problems of the elasticity theory of anisotropic piece-wise homogeneous bodies. Non - classical mixed type boundary value problems are studied when on one part (S₁) of the interface the rigid contact conditions (jumps of displacement and stress vectors) are given, while conditions of the non - classical interface Problem H or Problem G are imposed on the remaining part (S₂) of the interface, i. e., in both cases jumps of the normal components of displacement and stress vectors are known on S₂ and, in addition, in the first one (Problem H) the tangent components of the displacement vector and in the second one (Problem G) the tangent components of the stress vector are given from the both sides on S₂. The investigation is carried out by the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

Solving some problems of elasticity theory by the integral equation method for a holomorphic vector

Under study are some problems of elasticity theory with nonclassical boundary value conditions. We assume that the load and displacement vectors are given on a part of the boundary, while on the other parts of the boundary, the load vector or the displacement vector may be given separately, and no conditions are imposed on the remaining part of the surface (of some nonzero measure).We consider the questions of uniqueness for the solutions to these problems. Solving the nonclassical problems is reduced to a system of singular integral equations for a holomorphic vector.     

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.     

On a variational inequality on elasto-hydrodynamic lubrication

We consider the problem of a deformable surface moving over a flat plane. The surfaces are separated by a small gap filled by a lubricant fluid. The mathematical model consists of the Reynolds variational inequality with nonlocal coefficients given by an integral operator which depends on the fluid pressure. The nonlocal operator represents the deformation of the lubricated surfaces. The problem considers the vertical displacement of the elastic surface from its reference configuration. The goal of the paper is to obtain the range of these admissible displacements. We present general results for nonlocal coefficients with applications to particular problems in elasto-hydrodynamic lubrication.     

On the boundary transmission problem of thermoelastostatics in a plane domain with interface corners

This paper deals with bimetal problems of thermoelastostatics. By means of an explicit particular solution a reduction to problems of elastostatics is given. An indirect boundary integral method is applied for solving the traction boundary value problem. The solution is represented by a potential of single layer type having Green's contact tensor as the kernel. Thus, from the first the transmission conditions are satisfied. The Fredholm property of the boundary integral operator as well as the asymptotics of the potential density at an interface corner depend on the symbol of a Mellin convolution operator. The singular functions at corners can be obtained by calculating the potential for terms in the asymptotic expansion of the density.     

Solutions of half-space and half-plane contact problems based on surface elasticity

Analytical solutions for the problems of an elastic half-space and an elastic half-plane subjected to a distributed normal force are derived in a unified manner using the general form of the linearized surface elasticity theory of Gurtin and Murdoch. The Papkovitch–Neuber potential functions, Fourier transforms and Bessel functions are utilized in the formulation. The newly obtained solutions are general and reduce to the solutions for the half-space and half-plane contact problems based on classical linear elasticity when the surface effects are not considered. Also, existing solutions for the half-space and half-plane contact problems based on simplified versions of Gurtin and Murdoch’s surface elasticity theory are recovered as special cases of the current solutions. By applying the new solutions directly, Boussinesq’s flat-ended punch problem, Hertz’s spherical punch problem and a conical punch problem are solved, which lead to depth-dependent hardness formulas different from those based on classical elasticity. The numerical results reveal that smoother elastic fields and smaller displacements are predicted by the current solutions than those given by the classical elasticity-based solutions. Also, it is shown that the out-of-plane displacement and stress components strongly depend on the residual surface stress. In addition, it is found that the new solutions based on the surface elasticity theory predict larger values of the indentation hardness than the solutions based on classical elasticity.     

Axially symmetric problems of stationary heat conduction and thermoelasticity for a body with thermally active or thermally insulated disk inclusion (crack)

We present solutions of axially symmetric problems of stationary heat conduction and thermoelasticity for a body with a thin thermally active disk inclusion (where the temperature or heat flow is given) and also with a thermally insulated inclusion. The heat conduction problems are reduced to integral equations, and exact solutions are obtained in the case where their right-hand sides are polynomials of arbitrary degree. We determine the components of the stress tensor and displacement vector as well as, in the case of cracks, the stress intensity factors.     

A self-adaptive projection method for contact problems with the BEM

A self-adaptive algorithm, based on the projection and boundary integral methods, is designed and analyzed for frictionless contact problems in linear elasticity. Using the equivalence between the contact problem and a variational formulation with a projection fixed point problem of infinite dimensions, we develop an iterative algorithm that formulates the contact boundary condition into a sequence of Robin boundary conditions. In order to improve the performance of the method, we propose a self-adaptive rule which updates the penalty parameter automatically. As the iteration process is given by the displacement and the stress on the boundary of the domain, the unknowns of the problem are computed explicitly by using the boundary element method. Both theoretical results and numerical experiments show that the method presented is efficient and robust.     

The Behaviour of Elastic Fields and Boundary Integral Mellin Techniques Near Conical Points

For the computation of the local singular behaviour of an homogeneous anisotropic clastic field near the three-dimensional vertex subjected to displacement boundary conditions, one can use a boundary integral equation of the first kind whose unkown is the boundary stress. Mellin transformation yields a one - dimensional integral equation on the intersection curve 7 of the cone with the unit sphere. The Mellin transformed operator defines the singular exponents and Jordan chains, which provide via inverse Mellin transformation a local expansion of the solution near the vertex. Based on Kondratiev's technique which yields a holomorphic operator pencil of elliptic boundary value problems on the cross - sectional interior and exterior intersection of the unit sphere with the conical interior and exterior original cones, respectively, and using results by Maz'ya and Kozlov, it can be shown how the Jordan chains of the one-dimensional boundary integral equation are related to the corresponding Jordan chains of the operator pencil and their jumps across γ. This allows a new and detailed analysis of the asymptotic behaviour of the boundary integral equation solutions near the vertex of the cone. In particular, the integral equation method developed by Schmitz, Volk and Wendland for the special case of the elastic Dirichlet problem in isotropic homogeneous materials could be completed and generalized to the anisotropic case.     

Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

The solvability of a coupled thermoviscoelastic contact problem with small Coulomb friction and linearized growth of frictional heat

A coupled thermoviscoelastic frictional contact problem is investigated. The contact is modelled by the Signorini condition for the displacement velocities and the friction by the Coulomb law. The heat generated by friction is described by a non‐linear boundary condition with at most linear growth. The weak formulation of the problem consists of a variational inequality for the elasticity part and a variational equation for the heat conduction part. In order to prove the existence of a solution to this problem we first use an approximation of the Signorini condition by the penalty method. The existence of a solution for the approximate problem is shown using the fixed‐point theorem of Schauder. This theorem is applied to the composition of the solution operator for the contact problem with given temperature field and the solution operator for the heat equation problem with known displacement field. To obtain this proof, the unique solvability of both problems is necessary. Due to this reason it is necessary to introduce the penalty method. While the penalized contact problem has a unique solution, this is not clear for the original contact problem. The solvability of the original frictional contact problem is verified by an investigation of the limit for vanishing penalty parameter. Copyright © 1999 John Wiley & Sons, Ltd.     

Electro-mechanical sliding frictional contact of a piezoelectric half-plane under a rigid conducting punch

This paper investigates the two-dimensional sliding frictional contact of a piezoelectric half-plane in the plane strain state under the action of a rigid flat or a triangular punch. It is assumed that the punch is a perfect electrical conductor with a constant electric potential. By using the Fourier integral transform technique and the superposition theorem, the problem is reduced to a pair of coupled Cauchy singular integral equations and then is numerically solved to determine the unknown contact pressure and surface electric charge distribution. The effects of the friction coefficient and electro-mechanical loads on the normal contact stress, normal electric displacement, in-plane stress and in-plane electric displacement are discussed in detail. It is found that the friction coefficient has a significant effect on the electro-mechanical sliding frictional contact behaviors of the piezoelectric materials.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

Boussinesq indentation of a transversely isotropic half-space reinforced by a buried inextensible membrane

The normal indentation of a rigid circular disk into the surface of a transversely isotropic half-space reinforced by a buried inextensible thin film is addressed. By virtue of a displacement potential function and the Hankel transform, the governing equations of this axisymmetric mixed boundary value problem are represented as a dual integral equation, which is subsequently reduced to a Fredholm integral equation of the second kind. Two important results of the contact stress distribution beneath the disk region as well as the equivalent stiffness of the system are expressed in terms of the solution of the Fredholm integral equation. When the membrane is located on the surface or at the remote boundary, exact closed-form solutions are presented. For the limiting case of an isotropic half-space the results are verified with those available in the literature. As a special case, the elastic fields of a reinforced transversely isotropic half-space under the action of surface axisymmetric patch loads are also given. The effects of anisotropy, embedment depth of the membrane, and material incompressibility on both the contact stress and the normal stiffness factor are depicted in some plots.     

The contact problem when there are friction forces in the contact area for a three-component cylindrical base

The plane contact problem of elasticity theory on the interaction when there are friction forces in the contact area of an absolutely rigid cylinder (punch) with an internal surface of a cylindrical base, consisting of two circular cylindrical layers rigidly connected to one another and with an elastic space, is considered. The layers and space have different elastic constants. A vertical force and a counterclockwise torque, act on the punch, and the punch – base system is in a state of limiting equilibrium,. An exact integral equation of the first kind with a kernel represented in an explicit analytical form, is obtained for the first time for this problem using analytical calculation programs. The main properties of the kernel of the integral equation are investigated, and it is shown that the numerator and denominator of the kernel symbols can be represented in the form of polynomials in products of the powers of the moduli of the displacement of the layers and the half-space. A solution of the integral equation is constructed by the direct collocation method, which enables the solution of the problem to be obtained for practically any values of the initial parameters. The contact stress distributions, the dimensions of the contact area, the interconnection between the punch displacement and the forces and torques acting on it are calculated as a function of the geometrical and mechanical parameters of the layers and the space. The results of the calculations in special cases are compared with previously known results.     

Oscillatory integral operators in scattering theory

We consider a particular Fourier integral operator with folding canonical relations, which arises in scattering theory: the Radon Transform of Melrose and Taylor. We obtain the regularity properties of this operator when the obstacle admits tangent planes with contact of precise order k (Theorem 1.1 and its Corollary).

For these purposes, we derive asymptotic estimates for oscillatory integral operators in Wn with folding canonical relations (Theorem 2.2). Asymptotics correspond to vanishing principal curvature of a fold of one of the projections from the canonical relation, and to small support of the localization of oscillatory integral operator.     

Mixed interface problems for anisotropic elastic bodies

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where, on one part of the interface, rigid contact conditions are give (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

A strip cut in a transversely isotropic elastic solid

The three-dimensional problems of a strip cut in a transversely isotropic elastic space, when the isotropy planes are perpendicular to the plane of the cut, are investigated using the asymptotic methods developed by Aleksandrov and his coauthors. Two cases of the location of the strip cut are considered: along the first axis of a Cartesian system of coordinates (Problem A) or along the second axis (Problem B). Assuming that the normal load, applied to the sides of the cut (normal separation friction) can be represented by a Fourier series, one-dimensional integral equations of problems A and B are obtained, the symbols of the kernels of which are independent of the number of the term of the Fourier series. A closed solution of the problem is derived for a special approximation of the kernel symbol. Regular and singular asymptotic methods are also used to solve the integral equations by introducing a dimensionless geometrical parameter, representing the ratio of the period of the applied wavy normal load to the thickness of the cut strip. The normal stress intensity factor on the strip boundary is calculated using the three methods of solving the integral equations indicated.     

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lamé functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.     

An approximation theory for conjugate surfaces and solutions of elliptic multiple integral problems: Application to numerical solutions of generalized Laplace's equation

An approximation theory is given for a class of elliptic quadratic forms which include the study of conjugate surfaces for elliptic multiple integral problems. These ideas follow from the quadratic form theory of Hestenes, applied to multiple integral problems by Dennemeyer, and extended with applications for approximation problems by Gregory.The application of this theory to a variety of approximation problem areas in this setting is given. These include conjugate surfaces and conjugate solutions in the calculus of variations, oscillation problems for elliptic partial differential equations, eigenvalue problems for compact operators, numerical approximation problems, and, finally, the intersection of these problem areas.In the final part of this paper the ideas are specifically applied to the construction and counting of negative vectors in order to obtain new numerical methods for solving Laplace-type equations and to obtain the “Euler-Lagrange equations” for symmetric-banded tridiagonal matrices. In this new result (which will allow the reexamination of both the theory and applications of symmetricbanded matrices) one can construct, in a meaningful way, negative vectors, oscillation vectors, eigenvectors, and extremal solutions of classical problems as well as efficient algorithms for the numerical solution of partial differential equations. Numerical examples (test runs) are given.