Boundary-contact Problems Research Articles

Approximation Solution of Mixed Boundary-Contact Problems and their Applications

In this article non-classical diffusion models of theory coupled-elasticity of static systems for isotropic inhomogeneous elastic materials with thermal and diffusion variables has been investigated. Approximate solutions for boundary-contact problems for theory coupled-elasticity in Cauchy hypothesis conditions has been constructed. The tools applied in this development are based on the boundary integral methods and Greens functions applications.

Boundary contact problems with regard to friction of couple-stress viscoelasticity for inhomogeneous anisotropic bodies (quasi-static cases)

Abstract In this paper, quasi-statical boundary contact problems of couple-stress viscoelasticity for inhomogeneous anisotropic bodies with regard to friction are investigated. We prove the uniqueness theorem of weak solutions using the corresponding Green’s formulas and positive definiteness of the potential energy. To analyze the existence of solutions, we equivalently reduce the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter, and study its solvability by the Galerkin approximate method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure, the existence theorem for the original contact problem with friction is proved.

Variable boundary contact problem between pulley and flexible rope

The constraint method (analyzing the constraints between the contacts) has a better computational efficiency than the force method (analyzing the forces between the contacts) in contact dynamics. According to the constraint method, co-position and tangential conditions are proposed and applied to both rigid and flexible body contact problems. A pure rolling problem on the rigid elliptical disk is studied. And a typical slow-release mechanism, including winch, pulley and rope, is also analyzed by decomposing the contact-induced system into several continuous components. The boundary points among the parts can be quickly obtained by the constraint method, and the Kinematic continuity conditions on contact boundary point are deduced.

Solution of some three-dimensional boundary value problems for thermoelastic bodies with voids

The paper considers a three-dimensional system of differential equations describing the thermoelastic static equilibrium of homogeneous isotropic materials with voids. In the Cartesian coordinate system the general solution of the mentioned system of equations is constructed using harmonic and metaharmonic functions. On the basis of the constructed general solution using the method of separation of variables boundary value problems for a semi-infinite prism and a rectangular parallelepiped are analytically solved. The corresponding boundary-contact problems for a multilayer rectangular parallelepiped are also considered.

Stability for a boundary contact problem in thermoelastic Timoshenko’s beam

We demonstrate the existence of solutions to Signorini’s problem for the Timoshenko’s beam by using a hybrid disturbance. This disturbance enables the use of semigroup theory to show the existence and asymptotic stability. We show that stability is exponential, when the waves speed of propagation is equal. When the waves speed is different, we show that the solution decays polynomially. This result is new. We perform numerical experiments to visualize the asymptotic properties.

Finite Element Modeling Method of Centrifugally Rotary Processing

In the present work, the process of interaction of an abrasive particle with the surface of a component is modeled in the framework of dynamic problems of the theory of thermoelasticity, considering plastic deformation, friction and wear of the surface in the contact region. Two boundary contact problems are considered. The first problem deals with the contact interaction of an element of an abrasive particle in the form of a truncated cone and the surface of a part. The circle of smaller diameter of the cone contacts the surface of the part taking into account the friction and plastic deformation of this surface. Kinematic or force boundary conditions are applied to the circle of greater diameter. In the case of kinematic conditions, the normal and tangential displacements of the circle and its rotation are specified. In the case of force conditions, the force and moment are given. In the second task, the hard stamp slides at a constant speed along the flat boundary of the workpiece, the value of the die insertion is set. In the contact area, the sliding friction force is expressed through normal pressure and heating due to the friction and wear. The stress and temperature fields near the contact region are investigated.

Some three-dimensional boundary value and boundary-contact problems for an elastic mixture with double porosity

Some three-dimensional boundary value and boundary-contact problems for an elastic mixture with double porosity

Unilateral contact problems with a friction

The boundary contact problem for a micropolar homogeneous elastic hemitropic medium with a friction is investigated. Here, on a part of the elastic medium surface with a friction, instead of a normal component of force stress there is prescribed the normal component of the displacement vector. We give their mathematical formulation of the Problem in the form of spatial variational inequalities. We consider two cases, the so-called coercive case (when elastic medium is fixed along some part of the boundary) and semi-coercive case (the boundary is not fixed). Based on our variational inequality approach, we prove the existence and uniqueness theorems and show that solutions continuously depend on the data of the original problem. In the semi-coercive case, the necessary condition of solvability of the corresponding contact problem is written out explicitly. This condition under certain restrictions is sufficient, as well.

One-sided contact problems with friction arising along the normal

We study a boundary contact problem for a micropolar homogeneous elastic hemitropic medium with regard of friction; in the considered case, friction forces do not arise in the tangential displacement but correspond to a normal displacement of the medium. We consider two cases: the coercive case (in which the elastic body has a fixed part of the boundary) and the noncoercive case (without fixed parts). By using the Steklov–Poincare operator, we reduce this problem to an equivalent boundary variational inequality. Existence and uniqueness theorems are proved for the weak solution on the basis of properties of general variational inequalities. In the coercive case, the problem is unconditionally solvable, and the solution depends continuously on the data of the original problem. In the noncoercive case, we present closed-form necessary conditions for the existence of a solution of the contact problem. Under additional assumptions, these conditions are also sufficient for the existence of a solution.

The boundary contact problem of electroelasticity and related integral differential equations

The problem of finding mechanical and electrical fields in a homogeneous piezoelectric plate supported by a thin wedged-shaped cover plate is considered. Using the methods of the theory of analytic functions, the problem is reduced to a system of singular integro-differential equations. With the help of integral transformation, the exact solution of the posed by us problem is obtained.

The Analysis of Pipeline’s Penetration Laid on the Elastic-Plastic Seabed Based on the ABAQUS Software

The pipeline’s stability is the key problem of submarine pipeline’s design. In order to supply the theory basis for the pipeline’s design, the ABAQUS software has been adopted to analyze the pipe/soil system. The Ramberg-Osgood model has been adopted to simulate the seabed; the contact pair also adopted to simulate the pipe/soil system, the dynamic boundary contact problem of pipe embedded into the seabed has been analyzed. The computation has been operated according to change pipe’s subweight, environment load, pipe’s diameter, yielding stress and so on. The computation results have shown that these parameters have effects on the pipe’s penetration.

On the numerical solution of some two-dimensional boundary-contact delocalization problems

The elastic equilibrium of a multi-layer confocal elliptic ring is studied. The ring consists of steel, rubber and celluloid layers which differ in thickness and in the order in which they are placed relative to one another. Using the solutions of the considered problems, the following delocalization problem is solved: for a three-layer elliptic body, the external elliptic boundary of which is loaded by normal point force and the internal boundary is stress-free and the layers of which are in rigid or sliding contact with one another, by an appropriate choice of layer thickness and arrangement of the layers relative to one another we can obtain a sufficiently uniform distribution of normal displacements on the internal elliptic boundary. Numerical solutions are obtained by the boundary element method and the related graphs are constructed. For the two-layer ellipse, exact and approximate solutions of the same problem are obtained respectively by the method of separation of variables and by the boundary element method. The results obtained by both methods are compared and the conclusion as to the reliability of the numerical boundary element method is made.

Electromagnetic scattering by cylindrical orthotropic waveguide irises

Abstract The paper is devoted to the mathematical analysis of scattered time-harmonic electromagnetic waves by an infinitely long cylindrical orthotropic waveguide iris. This is modeled by an orthotropic Maxwell system in a cylindrical waveguide iris for plane waves propagating in the x 3-direction, imbedded in an isotropic infinite medium. The problem is equivalently reduced to a 2-dimensional boundary-contact problem with the operator div M grad+k 2 inside the domain and the (Helmholtz) operator Δ+k 2 = div grad+k 2 outside the domain. Here M is a 2 × 2 positive definite, symmetric matrix with constant, real valued entries. The unique solvability of the appropriate boundary value problems is proved and the regularity of solutions is established in Bessel potential spaces.

Energy release rate in a prestressed sandwich plate-strip containing interface cracks

Within the framework of a piecewise homogeneous body model, by utilizing the three-dimensional linearized theory of elasticity, the problem of opening of interface cracks in an initially stressed sandwich plate-strip is studied. To solve the corresponding boundary-contact problem, the FEM is employed. Numerical results for the influence of the initial stresses on the energy release rate (ERR) at various values of problem parameters are presented and discussed. In particular, it is established that the ERR in creases (de creases) with growing initial tension (compression) of the plate-strip.

Homogenization for contact problems with friction on rough interface

AbstractWe consider a contact problem between a macroscopic solid with a smooth boundary and a technical textile, while the textile has a periodic microscopic structure and microscopically rough surface. Two–scale homogenization approach is applied to the problem. The microscopic solution is approximated in terms of macroscopic solution and some concentration factor, given as a solution of auxiliary boundary value or contact problems of elasticity on the periodicity cell. Local friction condition is represented as a continuous non–linear functional over the stress field. Two–scale convergence is used to prove the convergence of friction functional. The macroscopic initial frictional limit is found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the expansion of the resolvent for elliptic boundary contact problems

Let A be an elliptic operator on a compact manifold with boundary \({\overline{M}}\) , and let \({\wp:\partial\overline{M} \to Y}\) be a covering map, where Y is a closed manifold. Let A C be a realization of A subject to a coupling condition C that is elliptic with parameter in the sector Λ. By a coupling condition we mean a nonlocal boundary condition that respects the covering structure of the boundary. We prove that the resolvent trace \({Tr_{L^2} (A_C-\lambda)^{-N}}\) for N sufficiently large has a complete asymptotic expansion as \({|\lambda|\to \infty}, \lambda \in \Lambda\) . In particular, the heat trace \({Tr_{L^2}e^{-tA_C}}\) has a complete asymptotic expansion as \({t \to 0^+}\) , and the \({\zeta}\) -function has a meromorphic extension to \({\mathbb {C}}\).

Potential fields caused by point sources in regions with apertures and foreign inclusions

A semi-analytic approach is developed for obtaining potential fields generated by point sources in regions embedded with foreign inclusions and containing holes of various shape. This study promotes an extension of the range of effective implementation of the Green's function version of the boundary integral equation method in mechanics of contemporary materials. Equivalents of Green's functions are obtained to boundary-contact value problems posed for Laplace equation on piecewise homogeneous multiply connected regions. Dirichlet, Neumann and Robin boundary conditions can be imposed on the outer boundary of the region and on contours of the apertures, while the ideal contact conditions are assumed on the interfacial contours. Source points could be located either outside or inside the inclusion. A Green's function modification of the method of functional equations is applied. A number of illustrative examples included to show the potential of the approach.

Boundary-contact problems for domains with edge singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have edge singularities. Such problems represent continuous operators between weighted edge spaces and subspaces with asymptotics. Ellipticity is formulated in terms of a principal symbolic hierarchy, containing interior, transmission, and edge symbols. We construct parametrices, show regularity with asymptotics of solutions in weighted edge spaces and illustrate the results by boundary-contact problems for the Laplacian with jumping coefficients.

Dynamical (time-harmonic) axisymmetric interface stress field in the finite pre-strained half-space covered with the finite pre-stretched layer

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), the dynamical (time-harmonic) stress field in the initially finite strained half-space covered with an initially and finitely stretched layer is investigated. It is assumed that on the upper free face of the covering layer the point located force which acts is harmonic with respect to time. The corresponding boundary contact problem is solved by employing the Hankel integral transformation. Moreover, it is assumed that the material of the layer and half-space are incompressible and elastic relations for those are given through the Treloar’s potential. In the case where the initial strains are absent in the layer and half-space the considered problem formulation and solution to that coincide with the corresponding ones of the classical linear theory of elasticity for an incompressible body. The algorithm for obtaining numerical results is proposed. The numerical results regarding the stresses acting on the interface plane are presented. These results are obtained for the case where the stiffness (distortion wave velocity) of the covering layer material is less (greater) than that for the half-space material. In this case, the main attention is focused on the dependencies between the values of the stresses and frequency of the external force and also the influence of the initial strains on these dependencies. In particular, it is established that the “resonance” values of the frequency of the external force increase, but the absolute maximum values of the stresses decrease significantly with the amount of the initial tension of the covering layer.

Boundary-contact problems for domains with conical singularities

We study boundary-contact problems for elliptic equations (and systems) with interfaces that have conical singularities. Such problems represent continuous operators between weighted Sobolev spaces and subspaces with asymptotics. Ellipticity is formulated in terms of extra transmission conditions along the interfaces with a control of the conormal symbolic structure near conical singularities. We show regularity and asymptotics of solutions in weighted spaces, and we construct parametrices. The result will be illustrated by a number of explicit examples.