A frictional contact problem with wear and damage for electro-viscoelastic materials

We consider a quasistatic contact problem for an electro-viscoelastic body. The contact is frictional and bilateral with a moving rigid foundation which results in the wear of the contacting surface. The damage of the material caused by elastic deformation is taken into account, its evolution is described by an inclusion of parabolic type. We present a weak formulation for the model and establish existence and uniqueness results. The proofs are based on classical results for elliptic variational inequalities, parabolic inequalities and fixed point arguments.     

The calculation of the energy losses in a sliding elastic contact with friction and wear (the plane problem)

The plane problem of the sliding contact of a punch with an elastic foundation when there is friction and wear is considered. Assuming the existence of a steady solution in a moving system of coordinates, relations are derived between the sliding velocity, the wear, the contact stresses and the displacements for an arbitrary dependence of the wear rate on the contact pressure. Taking into account the presence of a deformation component of the friction force, an equation is written for the balance of the mechanical energy for the punch - elastic base system considered. It is shown that the equality of the work of the external force in displacing the punch to the losses due to friction and the change in the shape of the foundation due to wear is satisfied when the work done by the contact stresses on the increments of the boundary displacements is equal to zero, and the frictional losses must be determined taking into account the non-uniformity of the distributions of the shear contact stresses and the sliding velocity in the contact area. Two special cases of the foundation in the form of a wide and narrow strip are considered, for which the total coefficient of friction is calculated, taking into account the deformation component of the friction force.     

The Apollonius contact problem and Lie contact geometry

A simple classification of triples of Lie cycles is given. The class of each triad determines the number of solutions to the associated oriented Apollonius contact problem. The classification is derived via 2-dimensional Lie contact geometry in the form of two of its subgeometries—Laguerre geometry and oriented M?bius geometry. The method of proof illustrates interactions between the two subgeometries of Lie geometry. Two models of Laguerre geometry are used: the classic model and the 3-dimensional affine Minkowski space model.     

The spatial contact problem for an elastic layer and wavy punch when there is friction and wear

The spatial (three-dimensional) problem of the wear of a wavy punch sliding over an elastic layer bonded to a rigid base, assuming there is complete contact between the punch and the layer, is considered. It is assumed that there is Coulomb friction and wear of the punch. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the harmonic functions in which are represented in the form of double Fourier integrals, after which the problem reduces to a linear system of differential equations. It is established that the harmonics constituting the shape of the punch and the contact pressure are shifted with respect to one another in time along the sliding line of the punch. The velocity of this shift depends on the longitudinal and transverse frequencies of the harmonic, that is, dispersion of the waves is observed.     

The contact problem for an elastic strip and a wavy punch when there is friction and wear

The plane problem of the mutual wear of a wavy punch and an elastic strip, bonded to an undeformable foundation under the condition of complete contact between the punch and the strip is considered. An analytical expression for the contact pressure is constructed using the general Papkovich–Neuber solution, the two harmonic functions in which are represented in the form of Fourier integrals after which the problem reduces to a non-linear system of differential equations. In the case of a small degree of wear of the strip, this system becomes linear and admits of a solution in explicit form. The harmonics, constituting the profile of the punch and the contact pressure, move along the strip with respect to one another and are shifted in time. Conditions are obtained that ensure the hermetic nature of the contact between the wavy punch and the strip when there is friction and wear.     

Elastic antiplane contact problem with adhesion

We study a mechanical problem modeling the antiplane shear deformation of a linearly elastic body in adhesive contact with a foundation. The material is assumed to be homogeneous and isotropic and the process is quasistatic. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding is included. We establish the existence of a unique weak solution for the problem, by construction of an appropriate mapping which is shown to be a contraction on a Banach space.     

The contact problem for a rectangle with stress-free side faces

The plane contact problem for an elastic rectangle into which two symmetrically positioned punches are impressed is considered. Homogeneous solutions are constructed that leave the side faces of the rectangle stress-free. When the modified boundary conditions using generalized orthogonality of the homogeneous solutions are satisfied, the problem reduces to a Friedholm integral equation of the first kind in the function describing the displacement of the surface of the rectangle outside the contact area. This function is sought in the form of the sum of a trigonometric series and a power function with a root singularity. The ill-posed infinite system of algebraic equations thereby obtained is regularized by introducing a small positive parameter (Ref. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978), and, after reduction, has a stable regularized solution. Since the matrix elements of the system are determined by a poorly converging number series, an effective method was developed for calculating the residues of the series. Formulae are found for the contact pressure distribution function and dimensionless indentation force. Since the first formula contains a third-order derivative of the functional series, when it is used, a numerical differentiation procedure is employed (Refs. Kalitkin NN. Numerical Methods. Moscow: Nauka; 1978; Danilina NI, Dubrovskaya NS, Kvasha OP et al. Numerical Methods. Textbook for Special Colleges. Moscow: Vysshaya Shkola; 1976). Examples of a calculation for a plane punch are given.     

The three-dimensional contact problem with friction for an elastic wedge

Solutions of three-dimensional boundary-value problems of the theory of elasticity are given for a wedge, on one face of which a concentrated shearing force is applied, parallel to its edge, while the other face is stress-free or is in a state of rigid or sliding clamping. The solutions are obtained using the method of integral transformations and the technique of reducing the boundary-value problem of the theory of elasticity to a Hilbert problem, as generalized by Vekua (functional equations with a shift of the argument when there are integral terms). Using these and previously obtained equations, quasi-static contact problems of the motion of a punch with friction at an arbitrary angle to the edge of the wedge are considered. In a similar way the contact area can move to the edge of a tooth in Novikov toothed gears. The method of non-linear boundary integral equations is used to investigate contact problems with an unknown contact area.     

The periodic contact problem of the plane theory of elasticity. Taking friction,wear and adhesion into account

A solution of the plane problem of the contact interaction of a periodic system of convex punches with an elastic half-plane is given for two forms of boundary conditions: 1) sliding of the punches when there is friction and wear, and 2) the indentation of the punches when there is adhesion. The problem is reduced to a canonical singular integral equation on the arc of a circle in the complex plane. The solution of this equation is expressed in terms of simple algebraic functions of a complex variable, which considerably simplifies its analysis. Asymptotic expressions are obtained for the solution of the problem in the case when the size of the contact area is small compared with the distance between the punches.     

The contact problem of elasticity theory for bodies with cracks

The plane contact problem of a stamp impressed into an elastic half-plane containing arbitrarily arranged rectilinear subsurface cracks is formulated and investigated by asymptotic methods. Partial or total overlapping of the crack edges is assumed. The problem reduces to a system of linear singular integrodifferential equations with side conditions in the form of equalities and inequalities. An analytic solution of the problem is obtained in the form of asymptotic power series /1/ in the relative dimension of the greatest crack. Dependences of the first terms of the asymptotic expansions of the desired functions on the mutual location of the cracks and the contact domains, the pressure and friction stress distributions, and the crack size and orientation are determined. Numerical results are presented.

Analysis of the influence of the stress-free boundary of the half-plane on the state of stress and strain of the elastic material near the cracks is presented in /2, 3/. The problem of a crack in an elastic plane whose edges overlap partially is also examined in /3/ by numerical methods.     

The completion problem for partial packings

Packings (resolutions) of designs have been of interest to combinatorialists in recent years as a way of creating new designs from old ones. Line packings of projective 3-space were the first packings studied, but it is still unknown when a partial packing can be completed to a packing in this case. In this paper we show that there is no guarantee from a combinatorial point of view of completing such a partial packing even when the deficiency is 2. In particular, we construct for every odd prime power q a set of 2(q ²+1) lines which doubly cover the points of PG(3,q) and yet cannot be partitioned into two spreads (resolution classes). The method is based on manipulations of primitive elements of finite fields.This work was supported in part by the National Research Council of Canada.     

The completeness problem in partial hyperclones

The composition-closed sets of partial multi-valued operations, called partial hyperclones, defined on the finite set are investigated. It is shown that the lattice of all partial hyperclones is dually atomic, i.e., any non-full partial hyperclone is contained in a maximal partial hyperclone. Based on it some completeness criteria in the full partial hyperclone are established. Next the total list of maximal restriction-closed partial hyperclones is obtained and, thus, the completeness problem with respect to compositions and restrictions of partial hyperoperations is solved.     

The applicability of a hereditary model of wear with an exponential kernel in the one-dimensional contact problem taking frictional heat generation into account

An analytic solution is obtained for the contact problem for a stiff thermally insulated plate and an elastic heat-conducting layer, subject to the conditions of wear and frictional heating, when the contacting bodies are not drawn nearer. The evolution of the contact pressure, the temperature and the wear are traced. Conditions for the occurrence of thermoelastic instability are established. The conditions under which the wear model considered is applicable are given.     

Completion problem with partial correlation vines

This paper extends the results in [D. Kurowicka, R.M. Cooke, A parametrization of positive definite matrices in terms of partial correlation vines, Linear Algebra Appl. 372 (2003) 225-251]. We show that a partial correlation vine represents a factorization of the determinant of the correlation matrix. We show that the graph of an incompletely specified correlation matrix is chordal if and only if it can be represented as an m-saturated incomplete vine, that is, an incomplete vine for which all edges corresponding to membership-descendents (m-descendents for short) of a specified edge are specified. This enables us to find the set of completions, and also the completion with maximal determinant for matrices corresponding to chordal graphs.     

Sturm-LiouVille算子部分谱信息的逆问题

研究了Sturm-Liouville算子Aq,h,Hj,j=1,2,…中势函数q(x)的确定性问题,即已知部分区间[a,1](a∈(0,1))上的势函数q(x),则无限组部分谱信息可唯一确定整个区间[0,1]上的势函数.推广了Simon的方法,且将选择条件的范围从一组谱扩展到了无限组.     

The Yamabe problem on quaternionic contact manifolds

By constructing normal coordinates on a quaternionic contact manifold M, we can osculate the quaternionic contact structure at each point by the standard quaternionic contact structure on the quaternionic Heisenberg group. By using this property, we can do harmonic analysis on general quaternionic contact manifolds, and solve the quaternionic contact Yamabe problem on M if its Yamabe invariant satisfies λ(M) < λ(ℍ ⁿ ). Mathematics Subject Classification (2000) 53C17, 53D10, 35J70     

A viscoelastic frictionless contact problem with adhesion

We consider a model for the quasistatic, bilateral, adhesive and frictionless contact between a viscoelastic body and a rigid foundation. The adhesion process on the contact surface is modeled by a surface internal variable, the bonding field, and the tangential shear due to the bonding field is included. The problem is formulated as a coupled system of a variational equality for the displacements and an integro-differential equation for the bonding field. The existence of a unique weak solution for the problem is established by construction of an appropriate mapping which is shown to be a contraction on a Hilbert space. We also consider the problem describing the bilateral contact between two viscoelastic bodies, and establish similar results.