Interaction of a die with a layered elastic foundation

Plane and axisymmetric contact problems for a three-layer elastic half-space are considered. The plane problem is reduced to a singular integral equation of the first kind whose approximate solution is obtained by a modified Multhopp-Kalandiya method of collocation. The axisymmetric problem is reduced to an integral Fredholm equation of the second kind whose approximate solution is obtained by a specially developed method of collocation over the nodes of the Legendre polynomial. An axisymmetric contact problem for an transversely isotropic layer completely adherent to an elastic isotropic half-space is also considered. Examples of calculating the characteristic integral quantities are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 165–175, May–June, 2006.     

Contact interaction of a rigid heat-conducting punch with an elastic layer involving nonstationary frictional heating

A mathematical formulation is given and a solution is found to the quasistatic contact problem of thermoelasticity for a rigid heat-conducting punch moving over an elastic layer with fixed base. The interaction is accompanied by heating due to frictional forces obeying Amonton’s law. The problem is reduced to a system of integral equations with time-varying limits of integration. The structure of these equations depends on the type of thermal and physical conditions on the contact surface. An algorithm is proposed for the numerical solution of this kind of equations. The variation in the contact pressure and contact area with time is studied __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 35–46, December 2005.     

Asymptotic solution of the problem of the action of a stamp on an elastic layer lying on the surface of a compressible fluid of infinite depth

This paper considers a two-dimensional linear unsteady problem of rigid-stamp indentation on an elastic layer of finite thickness lying on the surface of a compressible fluid of infinite depth. The Lamé equations holds for the elastic layer, and the wave equation for the fluid velocity potential. Using the Laplace and Fourier transforms, the problem is reduced to determining the contact stresses under the stamp from the solution of an integral equation of the first kind, whose kernel has a logarithmic singularity. An asymptotic solution of the problem is constructed for large times of interaction. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 131–142, March–April, 2008.     

Thermoelastic contact problem of two rotating bodies with frictional heat generation in annular region

Summary An axisymmetric contact problem with frictional heating is considered in which a parabolic annular punch is pressed into a plane surface and rotates about its axis of symmetry at constant speed. The problem is formulated in terms of one governing equation with unknown pressure. This equation is solved numerically. The change of the geometry of the contact region and pressure has been investigated. Received 21 May 1997; accepted for publication 23 June 1997     

Approximate Solution of an Axisymmetric Contact Problem with Allowance for Tangential Displacements on the Contact Surface

A structurally nonlinear contact problem of a punch shaped like a paraboloid of revolution is studied. An equation for the contactpressure density is derived with allowance for the radial tangential displacements of the boundary points of an elastic halfspace. A method for constructing a closedform approximate solution is proposed. The effect of the tangential displacements on the main contact parameters is discussed.     

A new formulation for solving 3-D time dependent rolling contact problems of a rigid body on a viscoelastic half-space

The paper deals with a new formulation for solving the rolling contact problem without friction of a rigid body on a viscoelastic half-space in three dimensions. Assuming that the material behavior is independent of time for a sufficiently short time duration, the viscoelastic contact problem is transformed into elastic like problems. Then the contact problem is solved using a direct numerical method at each time step. The convergence of the method in time and space is good for a spherical indenter. The dissymmetry of the contact patch due to hysteresis was found in three dimensions for the spherical indenter and two cylinders of different width. Finally the method was tested for a sinusoidal varying speed and shows a good efficiency.     

Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch

This paper focuses on the study of a frictional sliding contact problem between a homogeneous magneto-electro-elastic material (MEEM) and a perfectly conducting rigid flat punch subjected to magneto-electro-mechanical loads. The problem is formulated under plane strain conditions. Using Fourier transform, the resulting plane magneto-electro-elasticity equations are converted analytically into three coupled singular integral equations in which the main unknowns are the normal contact stress, the electric displacement and the magnetic induction. An analytical closed-form solution is obtained for the normal contact stress, electric displacement and magnetic induction distributions. The main objective of this paper is to study the effect of the friction coefficient and the elastic, electric and magnetic coefficients on the surface contact pressure, electric displacement and magnetic induction distributions for the case of flat stamp profile.     

点群10mm十次对称二维准晶中的两类接触问题

采用复变函数法探讨了在一个刚性压头作用下十次对称二维准晶材料的两类接触问题,即具有有限摩擦的接触问题以及粘结接触问题.特别地对于平底压头,获得了表征声子场和相位子场的全纯函数的显式表达式,以及在压头上的接触应力分布.结果显示,对于具有有限摩擦的接触问题,接触应力在接触区边缘具有实指数奇异性-1／2±β,其中β由准晶体的材料常数及静摩擦系数确定;而对于粘结接触问题,接触应力在接触区边缘具有振荡型奇异性-1／2±iε,其中ε由准晶体的材料常数确定.     

Two-phase boundary dynamics near a moving contact line over a solid

Consideration is given to the nonlinear problem on a shape of boundary between two viscous liquids, of which one displaces the other one from a solid surface, the Reynolds number being rather low. An asymptotic theory of wetting dynamics is developed that is of the second order with respect to small capillary numbers and valid for any ratio of viscosity coefficients of the media. A formula describing the dynamic contact angle (i.e. the inclination angle of the tangent to the interface) as a function of a distance to the solid is derived. Limitations on the angles for which the second-order theory is valid are shown. If the phase 2 viscosity is zero, the asymptotic second-order theory is valid for angles below 128.7°. A theory applicability domain depends on the ratio of viscosity coefficients. The applicability domain is not limited if the viscosity coefficients differ by a factor of less than four.     

Boundary element method for moving and rolling contact of 2D elastic bodies with defects

A scheme of boundary element method for moving contact of two-dimensional elastic bodies using conforming discretization is presented. Both the displacement and the traction boundary conditions are satisfied on the contacting region in the sense of discretization. An algorithm to deal with the moving of the contact boundary on a larger possible contact region is presented. The algorithm is generalized to rolling contact problem as well. Some numerical examples of moving and rolling contact of 2D elastic bodies with or without friction, including the bodies with a hole-type defect, are given to show the effectiveness and the accuracy of the presented schemes. The project supported by the National Natural Science Foundation of China (19772025)     

Three-Dimensional Dynamic Contact Problem for an Elliptic Crack Interacting with a Normally Incident Harmonic Compression–Expansion Wave

The paper deals with the three-dimensional dynamic problem for an elliptic crack interacting with a normally incident harmonic compression–expansion wave, considering the contact interaction of the crack faces. An asymmetric solution is obtained using an iteration algorithm developed earlier. Numerical results are presented     

The unilateral contact buckling problem of continuous beams in the presence of initial geometric imperfections: An analytical approach based on the theory of elastic stability

An analytical method for the treatment of the elastic buckling problem of continuous beams with intermediate unilateral constraints is presented, which is based on the fundamental theory of elastic stability. The study focuses on the unilateral contact buckling problem of beams in the presence of initial geometric imperfections. The mathematical Euler approach, based on the fundamental solution of the boundary value problem of the buckling of continuous beams, is appropriately modified in order to take into account the unilateral contact conditions. Furthermore, in order the obtained analytical solutions to be applicable for practical design cases, the actual strength of the cross-section of the beam under combined compression and bending is considered. The implementation of the proposed method is demonstrated through a characteristic example.     

Invariant Integrals in the Problem of a Crack on the Interface between Two Media

The equilibrium problem for an elastic body containing a crack on the interface between two media is considered. It is proved that there exist invariant (independent of the integration surface) integrals in this problem. The existence of invariant integrals is also established in the problem of a contact between an elastic body and a rigid stamp. Nonlinear boundary conditions of mutual non-penetration are prescribed on the contact boundaries. The physical meaning of invariant integrals is established.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 5, pp. 123–137, September–October, 2005.     

Axisymmetric contact problem of cubic quasicrystalline materials

The axisymmetric elasticity theory of cubic quasicrystal was developed in Ref. [1]. The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function, based on which, the exact analytic solutions for the elastic field of an axisymmetric contact problem of cubic quasicrystalline materials are obtained for universal contact stress or contact displacement. The result shows that if the contact stress has order −1/2 singularity on the edge of the contact domain, the contact displacement is a constant in the contact domain. Conversely, if the contact displacement is a constant, the contact stress must have order −1/2 singularity on the edge of the contact domain. Project supported by the National Natural Science Foundation of China (No. 19972011).     

Deformation of an elastic helix in contact with a rigid cylinder

Summary The deformation of a short helix in contact with a rigid cylinder is investigated. Deformations occur due to bending, torsion and longitudinal elasticity of the helix. Shear deformation is neglected. Some of the equations describing the problem have been given already in Love's Treatise on the Mathematical Theory of Elasticity, in terms of curvature changes. Nevertheless, the equations for small deformations have to be reformulated in terms of displacements and rotations, because contact constraints cannot be expressed in terms of curvature. Friction is neglected, thus the problem is symmetric, and it is sufficient to determine its solution for one half of the helix.Without friction between the cylinder and the helix, the contact problem arises only for a helix longer than one length of twist. For a shorter helix, the global equilibrium conditions cannot be satisfied for nonvanishing contact forces. For the minimum length, there are two noncontact zones, and the helix is in contact with the cylinder only at three points: at the ends and in the middle. For a slightly longer helix, four contact points and three noncontact regions are found. The dependence of the noncontact zones and the contact forces, which are of practical interest, can be calculated as a function of the length of the helix and its geometrical parameters. The case of a very long helix with more than four contact points remains unsolved.     

Fracture-mechanical assessment of electrically permeable interface cracks in piezoelectric bimaterials by consideration of various contact zone models

Summary An interface crack with an artificial contact zone at the right-hand side crack tip between two piezoelectric semi-infinite half-planes is considered under remote mixed-mode loading. Assuming the stresses, strains and displacements are independent of the coordinate x ₂, the expression for the displacement jumps and stresses along the interface are found via a sectionally holomorphic vector function. For piezoceramics of the symmetry class 6 mm and for electrically permeable crack faces, the problem is reduced to a combined Dirichlet-Riemann boundary value problem which can be solved analytically. Further, analytical expressions for the stresses, electrical displacements, derivatives of elastic displacement jumps, stress and electrical intensity factors are found at the interface. Real contact zone lengths and the well-known oscillating solution are derived from the obtained solution as well. Analytical relationships between the fracture-mechanical parameters of various models are found, and recommendations are suggested concerning the application of numerical methods to the problem of an interface crack in the discontinuity area of a piezoelectric bimaterial. Received 16 March 1999; accepted for publication 31 May 1999     

Critical equilibrium state of a bushing in a contact pair with cracks possessing plastic tip zones

The problem of mechanics of contact fracture is considered for a bushing in a friction pair. It is assumed that multiple reciprocating motion of the plunger leads to fracture of the bushing material owing to friction caused by contact interaction and accompanied by the joint effect of loading and temperature. It is assumed that there are several arbitrarily located straight-line cracks with tip zones near the contact surface of the bushing. The stress state of the bushing is examined in the presence of regions where the crack faces (or some part of them) come into contact. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 145–156, September–October, 2006.     

Thermoelastic Contact of a Shroud Ring and a Cylinder under Unsteady Friction-Induced Heat Release

Mathematical formulation is performed and a solution is found for a quasi-static thermoelastic problem of contact interaction of an elastic shroud ring and a hollow circular cylinder inserted into this ring, which are compressed by a load varied along the axis of the system, under the condition of an unloaded contact over the ring surface or over the circumference contour. The radial displacements of the contact surface of the shroud ring are approximated by displacements of the surface of a long circular hollow cylinder. Unsteady friction-induced heat release caused by the action of friction forces owing to shroud ring rotation over the cylinder with a time-dependent low angular velocity is taken into account. The problem is reduced to a system of integral equations whose structure is determined by the form of thermophysical contact conditions. A numerical algorithm of the solution is proposed, and the influence of the problem parameters on the contact pressure and temperature distributions is considered. Based on an analysis of results, a conclusion is made that the character of axial variation of the compressing load has a significant effect on the distribution of contact pressure in describing the kinematic condition of interaction of bodies in accordance with Hertz’s theory.__________Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 4, pp. 161–178, July– August, 2005.     

The equilibrium state of a structure subject to frictional contact

A structure in frictional contact subject to static loads has not, in general, a unique static equilibrium state. This is because the state, displacements and contact forces, depend on the load history of the structure.In cases where the exact load history is not known it would be of interest to find a state that is in some sense likely and define this as the equilibrium state. In this paper, it is assumed that the state with the smallest potential energy is the most likely one. The implication of this definition of likely state is analysed and shows that the resulting problem basically can be seen as a generalization of the frictionless contact problem to structures where no frictionless state is possible, i.e. structures where non-zero friction forces are necessary to satisfy force equilibrium.The results of several numerical experiments are given. The structures in the experiments are trusses and structures modelled by the finite element method. Both a sequential quadratic programming method and an enumeration method are used to solve the likely-state problem.     

Dynamique multi-contact d'un système de solides tridimensionnels rigidesMulti-contact dynamics of a set of three-dimensional rigid bodies

This study is in keeping with the general pattern of dynamical simulations of a set of rigid three-dimensional bodies submitted to unilateral contact constraints with dry friction. An exact formulation (respecting the contact and friction laws) of the problem of predicting the system accelerations and the contact status, in further evolution is proposed. A numerical treatment of this kind of nonlinear problem is presented. This approach is applied to a simple multi-contact example, and yields results in agreement with those of analytical and numerical type, known for this example. To cite this article: C. Le Saux et al., C. R. Mecanique 331 (2003).