Damageable contact between an elastic body and a rigid foundation

In this work, the contact problem between an elastic body and a rigid obstacle is studied, including the development of material damage which results from internal compression or tension. The variational problem is formulated as a first-kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field. The existence of a unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, three two-dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme.     

Numerical analysis of a bilateral frictional contact problem for linearly elastic materials

We consider a mathematical model which describes the contactbetween a linearly elastic body and an obstacle, the so-calledfoundation. The process is quasistatic and the contact is bilateral,i.e. there is no loss of contact during the process. The frictionis modelled with Tresca's law. The variational formulation ofthe problem is a nonlinear evolutionary inequality for the displacementfield which has a unique solution under certain assumptionson the given data. We study spatially semi-discrete and fullydiscrete schemes for the problem with finite-difference discretizationin time and finite-element discretization in space. The numericalschemes have unique solutions. We show the convergence of thescheme under the basic solution regularity. Under appropriateregularity assumptions on the solution, we derive optimal ordererror estimates. Finally, we present numerical results in thestudy of two-dimensional test problems.     

The conjunction problem for thin elastic and rigid inclusions in an elastic body

Under consideration is the conjunction problem for a thin elastic and a thin rigid inclusions that are in contact at one point and placed in an elastic body. Depending on what kind of conjunction conditions are set at the contact point of inclusions, we consider the two cases: the case of no fracture, where, as the conjunction conditions, we take the matching of displacements at the contact point and preservation of the angle between the inclusions, and the case with a fracture in which only the matching of displacements is assumed. At the point of conjunction, we obtain the boundary conditions for the differential formulation of the problem. On the positive face of the rigid inclusion, there is delamination. On the crack faces, some nonlinear boundary conditions of the type of inequalities are set, that prevent mutual penetration of the faces. The existence and uniqueness theorems for the solution of the equilibrium problem are proved in both cases.     

An approximate solution of the axisymmetric contact problem for an elastic sphere

An axisymmetric, fractionally non-linear contact problem for an elastic sphere with a priori unknown boundary of the contactarea is considered. An integral equation for determining the density of the contact pressures is constructed taking account of the shear displacements of the boundary points of the elastic body. An approximate solution, which refines the equations of Hertz' theory, is constructed in the case of a small contact area.     

A general mathematical model for the collision between a free-fall hammer of a pile-driver and an elastic pile

In this paper we consider a general mathematical model for the collision between the free-fall hammer of a pile-driver and an elastic pile whose ends are furnished with a bearing. When the free-fall hammer collides with the pile, the displacement of a cross-sectional area of the pile is the weak solution of an initial-boundary value problem involving a linear wave equation with memory boundary conditions. We generalize this problem into a nonlinear one with more general boundary conditions. Then we obtain the unique solvability and the regularity of the weak solution of this nonlinear problem. The unique solvability is shortly discussed in regard to the Galerkin method. The regularity result is obtained by a combination of a fixed-point technique and an energy method, and the convenience of this procedure is also pointed out.     

Unilateral contact of a plate with a thin elastic obstacle

We study the problem of contact of an elastic body with a beam. The most attention is paid to describing boundary conditions on the possible contact set. Moreover, we study asymptotic properties of solutions and the energy functional as the rigidity parameters tend to infinity or the length of the beam (or the zone of possible contact) changes.     

The use of the method of boundary states to analyse an elastic medium with cavities and inclusions

The analytical method of boundary states is developed and theoretically substantiated. A corollary of the Weierstrass theorem is proved according to which a function that is harmonic in a bounded, simply connected domain can be approximated by a series of homogeneous harmonic polynomials. A basis of the space of functions that are harmonic outside any neighbourhood of a point is constructed. An algorithm is developed for filling the basis of the space of the states of a multicavity elastic body. The method is used to solve a series of problems of determining of the stress-strain state of an unbounded elastic medium containing spherical cavities or inclusions with different boundary conditions: the boundary of the cavity is free (the Southwell problem), constrained or under conditions of contact with a rigid core. The effect of the width of the intercavity layer on the stress concentration is analysed in a non-axisymmetric problem with two cavities. The form of the relation between the mean-square discrepancy in the boundary conditions of the solution obtained and the number of elements in the basis is indicative of the numerical convergence of the solution of this problem.     

The indentation of a punch in the form of an elliptic paraboloid into the plane boundary of an elastic body

The three-dimensional contact problem for an elastic body of arbitrary geometry with a single plane face, into which a punch in the shape of an elliptic paraboloid is indented, is considered. The curvilinear boundary of the body is partially clamped, and the remaining boundary (outside the contact region) is stress-free. It is assumed that the dimensions of the contact area are small compared with the characteristic dimension of the body. Using the method of matched asymptotic expansions a model problem of unilateral contact without friction is derived for the boundary layer, which is solved using the apparatus of Hertz's theory. Asymptotic models of the contact interaction of different degrees of accuracy are constructed, including corrections to the geometry and clamping conditions of the elastic body. The sensitivity of the parameters of the elliptic region of the contact to these factors is investigated.     

Regularity of solutions to a contact problem

We consider a variational inequality for the Lamé system which models an elastic body in contact with a rigid foundation. We give conditions on the domain and the contact set which allow us to prove regularity of solutions to the variational inequality. In particular, we show that the gradient of the solution is a square integrable function on the boundary.

     

Dynamic contact between a spherical inclusion and a matrix upon incidence of an elastic wave

Using the boundary element method, we have studied the dynamic displacements and stresses in an infinite elastic matrix with a spherical elastic inclusion, caused by the propagation of an elastic wave. The original problem has been reduced to a system of boundary integral equations for the contact displacements and tractions on the interface between the inclusion and matrix. Based on the numerical solution of these equations, we have analyzed the influence of the direction of wave propagation and frequency on the important physical parameters, depending on the elastic characteristics of composite constituents.     

The variational contact problem for elastic objects of different dimensions

We consider the variational free boundary problem describing the contact of an elastic plate with a thin elastic obstacle. The contact domain is unknown a priori and should be determined. The problem is described by a variational inequality for a fourth-order operator. The constraint on the displacement is given on a set of dimension less than that of the solution domain. We find the boundary conditions on the set of the possible contact and their exact statement. We justify the mixed statement of the problem and analyze the limit cases corresponding to the unbounded increase of the elasticity coefficients of the contacting bodies.     

The dynamics of the motion of a flat punch on the boundary of an elastic half-plane

The dynamic contact problem of the motion of a flat punch on the boundary of an elastic half-plane is considered. During motion, the punch deforms the elastic half-plane, penetrating it in such a manner that its base remains parallel to the boundary of the half-plane at each instant of time. In movable coordinates connected to the moving punch, the contact problem reduces to solving a two-dimensional integral equation, whose two-dimensional kernel depends on the difference between the arguments for each of the variables. An approximate solution of the integral equation of the problem is constructed in the form of a Neumann series, whose zeroth term is represented in the form of the superposition of the solutions of two-dimensional integral equations on the coordinate semiaxis minus the solution of the integral equation on the entire axis. This approach provides a way to construct the solution of the two-dimensional integral equation of the problem in four velocity ranges of motion of the punch, which cover the entire spectrum of its velocities, as well as to perform a detailed analysis of the special features of the contact stresses and vertical displacements of the free surface on the boundary of the contract area. An approximate method for solving the integral equation, which is based on a special approximation of the integrand of the kernel of the integral equation in the complex plane, is proposed for obtaining effective solutions of the problem that do not contain singular quadratures.     

The axisymmetric contact problem taking into account transient heat production due to sliding friction

The contact problem of the sliding of a solid heat insulator with a plane surface along the boundary of an axisymmetric elastic body is considered, taking into account heat release and the thermal distortion of the boundary of the deformable body due to friction. It is assumed that the shear stresses have no effect on the value of the contact pressures, which enables the problem to be investigated in an axisymmetric formulation. The solution is constructed in two stages: first the form of the thermally distorted surface is determined using known expressions, obtained by Carslaw and Jaeger and also by Barber, and then the contact condition is considered taking into account the elastic displacements and distortion of the form of the surface due to heating, and the integral equation of the problem for determining the unknown contact pressures is derived. The latter equation is solved numerically by approximating the unknown contact pressures by a piecewise-constant function.     

The asymmetrical mixed temperature problem for a transversely isotropic elastic layer

The problem of the uniform heating of a two-layer plate is solved. The transversely isotropic elastic layer (soft plate) investigated is in ideal contact with an absolutely rigid layer, deformable only by thermal expansion. The generalized plane temperature problem reduces to determining the stress-strain state of the soft anisotropic layer investigated using the equations of the mixed problem of elasticity theory. At the ends of the boundary layer of the soft plate (a thin contact layer), no conditions are imposed. On the remaining part of the ends of the soft plate, the boundary conditions correspond to a free boundary. The problem has a bounded smooth solution. Unlike the approach described earlier [1], it is proposed to seek an accurate solution in the form of ordinary Fourier series with respect to a single longitudinal coordinate. Solutions in polynomials are also used. It is shown that the existence of these solutions in polynomials enables the convergence of the Fourier series to be improved considerably.     

A rigid inclusion in the contact problem for elastic plates

A family of problems under consideration describes the contact of elastic plates situated at a given angle to each other and, in the natural condition, touching along a line. The plates are subjected only to bending. The limiting process from the elastic inclusion to the rigid one is studied. It is demonstrated that the limit problems precisely describe the contact of an elastic plate with a rigid beam and the problem of the equilibrium of an elastic plate with a rigid inclusion. The solvability of the problems is established; the boundary conditions holding on the possible contact set are found as well as their precise interpretation.     

Spatial contact problems for rough elastic bodies under elastoplastic deformations of the unevenness

On the basis of /1, 2/, a model is constructed for the contact between a rigid stamp and a rough body taking elastoplastic deformations of the unevenness into account. The contact model for rough bodies with elastic deformations of the unevenness is a special case. A classical approach utilizing boundary integral equations is applied in the mathematical formulation of the contact problem. Under quite general assumptions (for instance, the multiconnectedness of the contact domain desired), the uniqueness and existence of the solution are investigated. A method is developed to determine the contact pressure, the closure of the bodies, and also the contact area which consists of two parts in the general case, a zone of elastoplastic deformation of the unevenness and a zone of their elastic deformation. The efficiency of the method is shown in examples of new contact problems. The solution is represented in a convenient form for analysing the influence of the roughness. This is of considerable value for material testing by a contact method. A fairly complete survey of research on contact problems for rough bodies can be found in /1–4/.     

The interaction of punches on the face of an elastic wedge

The interaction of two punches, which are elliptic in plan, on the face of an elastic wedge is investigated in a three-dimensional formulation for different types of boundary conditions on the other face. The wedge material is assumed to be incompressible. An asymptotic solution is obtained for punches which are relatively distant from one another and from the edge of the wedge. For the case when the punches are arranged relatively close to the edge of the wedge (or reach the edge, the contact area is unknown) the numerical method of boundary integral equations is used. The mutual effect of the punches is estimated by means of calculations. The asymptotic solution of the generalized Galin problem, concerning the effect of a concentrated force applied on the edge of the three-dimensional wedge on the contact pressure distribution under a circular punch relatively far from the edge, is obtained.     

Mixed impedance transmission problems for vibrating layered elastic bodies

Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady‐state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so‐called nonlocal approach and reduce the mixed transmission problem to an equivalent functional‐variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.     

The impact of a plane punch on an elastic half-plane

The transient dynamic contact problem of the impact of a plane absolutely rigid punch on an elastic half-plane is considered. The solution of the integral equation of this problem in terms of the unknown Laplace transform of the contact stresses at the punch base is constructed by a special method of successive approximations. The solution of the transient dynamic contact problem is obtained after applying an inverse Laplace transformation to the solution of the integral equation over the whole time range of the impact process, and the law of the penetration of the punch into the elastic medium is determined from a Volterra-type integrodifferential equation. The conditions for the punch to begin to separate from the elastic half-plane are formulated from the solution obtained, and all the stages of the separation process are investigated in detail. The law of the punch motion on the elastic half-plane and the width of the contact area, which varies during the separation, are then determined from the solution of the Volterra-type integrodifferential equation when an additional condition is satisfied.     

Solving the problem of contact equilibrium of a punch and an elastic body

The variational problem of contact equilibrium of a punch and an elastic body is considered. An equivalent formulation of the problem is given in variational inequality form. Existence and uniqueness of the solution is investigated in a particular case. A penalty method is proposed for approximate solution of the problem.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 97–103, 1985.