Sliding Contact with Friction at Arbitrary Constant Speeds on a Pre-Stressed Highly Elastic Half-Space

Resisted by Coulomb friction, a rigid indentor slides at a constant arbitrary speed on a generalized neo-Hookean half-space under pre-stress. A dynamic steady-state situation in plane strain is assumed, and is treated as the superposition of contact-triggered infinitesimal deformations upon finite deformations due to pre-stress. Exact solutions are presented for both deformations, and the infinitesimal component exhibits the anisotropy typically induced by pre-stress, and wave speeds that are sensitive to pre-stress. In view of the unilateral constraints of contact, these and other critical speeds define the sliding speed ranges for physically-acceptable solutions. In particular, a Rayleigh speed is the upper bound for subsonic sliding. Solutions are further constrained by the unilateral requirement that contact zone shear must oppose indentor/half-space slip. The generic parabolic indentor is used for illustration, and it is found that traction continuity at the contact zone leading edge is lost for supersonic sliding and at the single sliding speed allowed in the frictionless limit in the trans-sonic range. A range of acceptable pre-stresses is also identified; for pre-stresses that lie out of range, either a negative Poisson effect occurs, or the Rayleigh wave disappears, thereby precluding sliding in the subsonic range. This revised version was published online in August 2006 with corrections to the Cover Date.     

Wave propagation in a pre-stressed highly elastic body: two exact solutions for Boussinesq problems

Two problems of wave propagation induced by surface loads in a compressible neo-Hookean half-space, initially at rest under a uniform pre-stress, are considered. One problem concerns a plane-strain situation, the other, one of axial symmetry. An accepted general procedure, that of superposing infinitesimal deformations upon the possibly large deformations due to pre-stress, is carried out completely in terms of tractable exact solutions for both the surface behavior and the full field, and analytical expressions for all wave speeds.

The results show that for a tensile pre-stress above a critical value, a negative Poisson effect occurs; for compressive pre-stresses, Rayleigh waves disappear at a critical value. Indeed, all wave speeds in the deformed configuration (effective wave speeds), as well as the solutions, for both problems are clearly sensitive to material properties and to both the magnitude and nature (compressive or tensile) of the pre-stress. In particular, the constraint imposed by plane strain appears to enhance this sensitivity.     

Rapid sliding motion of a rigid frictionless indentor with a flat base over a thermoelastic half-space

The three-dimensional, rapid sliding indentation of a deformable half-space by a rigid indentor of a flat elliptical base is treated in this paper. The response of the material that fills the half-space is assumed to be governed by coupled thermoelasticity. The indentor translates without friction on the half-space surface at a constant sub-Rayleigh speed and the problem is treated as a steady-state one. An exact solution is obtained that is based on a Green’s function approach, integral equations, and Galin’s theorem. A closed-form expression for the distributed contact pressure under the elliptical base of the indentor is derived. Representative numerical results are given illustrating the effects of the indentor velocity, indentor geometry, and parameters of the thermoelastic solid on the contact displacement. Since there is an analogy between the steady-state theories of thermoelasticity and poroelasticity, the present results carry over to the latter case directly.     

Multiple-zone sliding contact with friction on an anisotropic thermoelastic half-space

The paper studies a class of multiple-zone sliding contact problems. This class is general enough to include frictional and thermal effects, and anisotropic response of the indented material. In particular, a rigid die (indenter) slides with Coulomb friction and at constant speed over the surface of a deformable and conducting body in the form of a 2D half-space. The body is assumed to behave as a thermoelastic transversely isotropic material. Thermoelasticity of the Green–Lindsay type is assumed to govern. The solution method is based on integral transforms and singular integral equations. First, an exact transform solution for the auxiliary problem of multiple-zone (integer n > 1) surface tractions is obtained. Then, an asymptotic form for this auxiliary problem is extracted. This form can be inverted analytically, and the result applied to sliding contacts with multiple zones. For illustration, detailed calculations are provided for the case of two (n = 2) contact zones. The solution yields the contact zone width and location in terms of sliding speed, friction, die profile, and also the force exerted. Calculations for the hexagonal material zinc illustrate effects of speed, friction and line of action of the die force on relative contact zone size, location of maximal values for the temperature and the compressive stress, and the maximum temperature for a given maximum stress. Finally, from our general results, a single contact zone solution follows as a simple limit.     

Adhesive Frictionless Contact Between an Elastic Isotropic Half-Space and a Rigid Axi-Symmetric Punch

In this paper we consider the problem of adhesive frictionless contact of an elastic half-space by an axi-symmetric punch. We obtain integral equations that define the tractions and displacements normal to the surface of the half-space, as well as the size of the contact regions, for the cases of circular and annular contact regions. The novelty of our approach resides in the use of Betti’s reciprocity theorem to impose equilibrium, and of Abel transforms to either solve or substantially simplify the resulting integral equations. Additionally, the radii that define the annular or circular contact region are defined as local minimizers of the function obtained by evaluating the potential energy at the equilibrium solutions for each pair of radii. With this approach, we rather easily recover Sneddon’s formulas (Sneddon, Int. J. Eng. Sci., 3(1):47–57, 1965) for circular contact regions. For the annular contact region, we obtain a new integral equation that defines the inverse Abel transform of the surface normal displacement. We solve this equation numerically for two particular punches: a flat annular punch, and a concave punch.     

Bonded contact of a flexible elliptical disk with a transversely isotropic half-space

The article concerns the problem of bonded contact of a thin, flexible elliptical disk with a transversely isotropic half-space. Three different cases of loading have been considered: (a) the disk is loaded by a transverse force, whose line of action passes through the center of the disk and lies in the plane of the disk; (b) the disk is subjected to a rotation by a torque, whose axis is perpendicular to the surface of the half-space; (c) the half-space with the bonded disk is under uniform stress field at infinity. The problem corresponding to all three cases is reduced, in a unified manner, to a set of coupled two-dimensional integral equations. Closed-form solutions for these equations have been obtained by using Galins theorem.     

Two-dimensional sliding frictional contact of functionally graded materials

A multi-layered model for sliding frictional contact analysis of functionally graded materials (FGMs) with arbitrarily varying shear modulus under plane strain-state deformation has been developed. Based on the fact that an arbitrary curve can be approached by a series of continuous but piecewise linear curves, the FGM is divided into several sub-layers and in each sub-layers the shear modulus is assumed to be a linear function while the Poisson's ratio is assumed to be a constant. In the contact area, it is assumed that the friction is one of Coulomb type. With this model the fundamental solutions for concentrated forces acting perpendicular and parallel to the FGMs layer surface are obtained. Then the sliding frictional contact problem of a functionally graded coated half-space is investigated. The transfer matrix method and Fourier integral transform technique are employed to cast the problem to a Cauchy singular integral equation. The contact stresses and contact area are calculated for various moving stamps by solving the equations numerically. The results show that appropriate gradual variation of the shear modulus can significantly alter the stresses in the contact zone.     

Dynamic Response of a Multilayered Poroelastic Half-Space to Harmonic Surface Tractions

The propagator matrix method is developed to study the dynamic response of a multilayered poroelastic half-space to time-harmonic surface tractions. In a cylindrical coordinate system, a method of displacement potentials is applied first to decouple the Biot’s wave equations into four scalar Helmholtz equations, and then, general solutions to those equations are obtained. After that, the propagator matrix method and the vector surface harmonics are employed to derive the solutions for a multilayered poroelastic half-space subjected to surface tractions. It is known that the original propagator algorithm has the loss-of-precision problem when the waves become evanescent. At present, an orthogonalization procedure is inserted into the matrix propagation loop to avoid the numerical difficulty of the original propagator algorithm. Finally, a high-order adaptive integration method with continued fraction expansions for accelerating the convergence of the truncated integral is adopted to numerically evaluate the integral solutions expressed in terms of semi-infinite Hankel-type integrals with respect to horizontal wavenumber. Furthermore, to validate the present approach, the response of a uniform poroelastic half-space is examined using the formulation proposed in this article. It is shown that the numerical results computed with this approach agree well with those computed with the analytical solution of a uniform half-space.     

A receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate

In this paper, we consider the axisymmetric problem of a frictionless receding contact between an elastic functionally graded layer and a homogeneous half-space, when the two bodies are pressed together. The graded layer is modeled as a nonhomogeneous medium with an isotropic stress–strain law and is subjected over a part of its top surface to normal tractions while the rest of it is free of tractions. Since the contact between the two bodies is assumed to be frictionless, then only compressive normal tractions can be transmitted in the contact area. Using Hankel transform, the axisymmetric elasticity equations are converted analytically into a singular integral equation in which the unknowns are the contact pressure and the receding contact radius. The global equilibrium condition of the layer is supplemented to solve the problem. The singular integral equation is solved numerically using orthogonal Chebychev polynomials and an iterative scheme is employed to obtain the correct receding contact length that satisfies the global equilibrium condition. The main objective of the paper is to study the effect of the material nonhomogeneity parameter and the thickness of the graded layer on the contact pressure and on the length of the receding contact.     

正交各向异性地基平面问题动力响应研究

以位移分量为基本未知量,在直角坐标系下建立正交各向异性地基的平面应变问题动力偏微分方程．采用Laplace-Fourier变换和逆变换方法,引入初始条件和边界条件,推导了任意形式表面动荷载作用下正交各向异性地基平面问题在时域内动力反应的积分形式解．基于理论解,编制了相应的计算程序,并对正交各向异性土
体表面作用线性移动谐振荷载进行了算例分析,研究了土体参数、荷载移动速度、荷载频率不同而导致的土体表面各点竖向位移幅值的变化规律,以及荷载速度对竖向应力分量的影响规律．数值分析结果表明：土体的各向异性、荷载频率和移动速度对表面位移幅值有较大影响,土体阻尼比对于荷载中心点附近的位移幅值影响较小;荷载移动速度对于竖向应力分量有较大影响,这对工程实践具有重要指导意义．     

Mechanics of indentation for piezoelectric thin films on elastic substrate

Frictionless normal indentation problem of rigid flat-ended cylindrical, conical and spherical indenters on piezoelectric film, which is either in frictionless contact with or perfectly bonded to an elastic half-space (substrate), is investigated. Both conducting and insulating indenters are considered. With Hankel transform, the general solutions of the homogeneous governing equations for the piezoelectric layer and the elastic half-space are presented. Using the boundary conditions for a vertical point force or a point electric charge, and the boundary conditions on the film/substrate interface, the Green’s functions can be obtained by solving sets of simultaneous linear algebraic equations. The solution of the indentation problem is obtained by integrating these Green’s functions over the contact area with unknown surface tractions or electric charge distribution, which will be determined from the boundary conditions on the contact surface between the indenter and the film. The solution is expressed in terms of dual integral equations that are converted to a Fredholm integral equation of the second kind and solved numerically. Numerical examples are also presented. The comparison between two film/substrate bonding conditions is made. It shows that the indentation rigidity of the film/substrate system is lower when the film is in frictionless contact with the substrate. The effects of the Young’s modulus and Poisson’s ratio of the elastic substrate, indenter electrical condition and indenter prescribed electric potential on the indentation responses are presented.     

Steady-State Response of a Thermoelastic Half-Space to the Rapid Motion of Surface Thermal/Mechanical Loads

Shear and normal tractions and a heat flux are applied to a largely arbitrary area that moves with constant subsonic speed over a half-space surface. The half-space is a coupled thermoelastic solid of the Jeffreys type, so that the governing steady-state equations involve three thermoelastic characteristic lengths and a dimensionless coupling constant. This constant and one of the lengths remain in the limit as the solid reduces to the standard coupled thermoelastic material. The problem is solved exactly in an integral transform space, and asymptotic expressions for the normal displacement and the temperature change induced on the half-space surface are extracted. These are in principle valid for large distances from the loading zone as measured along its line of travel but, because the scaling dimension is of O(10^-14)μ m, they are robust. Exact inversions are performed, and the results show marked dependence on both loading zone speed and thermoelastic parameters. Indeed, the role of the latter is enhanced as the speed is increased. Singular behavior is found, in particular, when the loading zone moves with the effective thermoelastic Rayleigh speed, an exact formula for which is also given. This revised version was published online in August 2006 with corrections to the Cover Date.     

Rapid sliding indentation with friction on a transversely isotropic thermoelastic half-space

A rigid insulated die slides at a constant sub-critical speed on a transversely isotropic half-space in the presence of friction. In a two-dimensional analysis of the dynamic steady-state, the coupled equations of thermoelasticity are invoked. All elements of the Coulomb friction model are strictly enforced, thus giving rise to auxiliary conditions, including two unilateral constraints.Robust asymptotic forms of an exact solution to a related problem with unmixed boundary conditions lead to analytical solutions for the sliding indentation problem. The solution expressions, abetted by calculations for zinc, show the role of frictional heating on the half-space surface. The effects of friction and sliding speed on contact zone size and location and average contact zone temperature are also studied.The analysis is aided by factoring procedures that simplify the complicated forms that arise in anisotropic elasticity. A scheme that renders expressions for roots of certain irrational functions analytic to within a single quadrature also plays a role.     

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.     

A Simplified Model for Calculating Hydrodynamic Lubrication Film Thickness in Elastoplastic Line Contacts

The present paper proposes a simplified model for calculating hydrodynamic lubrication film thickness in elastoplastic line contacts. According to the Saint-Venant’s principle, the pressure in the contact is taken as uniformly distributed, this gives the contact surface elastic deformations in the inlet zone far away from the contact center close to real ones while gives those close to the contact center greater than real ones. This treatment is validated for hydrodynamic lubricated elastic contacts for relatively light loads and high rolling speeds. It gives the film thickness at the contact center a little higher than that calculated based on the real elastic model. The treatment is extended to a hydrodynamic lubricated elastoplastic line contact. The contact surfaces in the inlet zone are assumed as elastic and their deformations are calculated based on the uniform pressure distribution in the elastoplastic contact area. An inlet zone analysis is taken for obtaining the calculating equation of the hydrodynamic film thickness at the contact center. The equation overestimates the central film thickness but gives a satisfactory film thickness prediction for the heavy load which gives significant plastic deformations in the elastoplastic contact. It is found that when the load is lighter than 0.6 w _pc , the contact can be taken as elastic when calculating the central film thickness, while when the load is heavier than 0.6 w _pc , the contact can be taken as fully plastic; Here w _pc is the critical load for the contact fully plastic deformation. The plastic deformation in an elastoplastic line contact is found to reduce the hydrodynamic lubrication film thickness in the contact. This reduction is greater for higher rolling speeds and heavier loads. However, it is significantly dropped with increasing surface hardness.     

Stress-intensity factors for 3-D dynamic loading of a cracked half-space

A half-space containing a surface-breaking crack of uniform depth is subjected to three-dimensional dynamic loading. The elastodynamic stress-analysis problem has been decomposed into two problems, which are symmetric and antisymmetric, respectively, relative to the plane of the crack. The formulation of each problem has been reduced to a system of singular integral equations of the first kind. The symmetric problem is governed by a single integral equation for the opening-mode dislocation density. A pair of coupled integral equations for the two sliding-mode dislocation densities govern the antisymmetric problem. The systems of integral equations are solved numerically. The stress-intensity factors are obtained directly from the dislocation densities. The formulation is valid for arbitrary 3-D loading of the half-space. As an example, an applied stress field corresponding to an incident Rayleigh surface wave has been considered. The dependence of the stress-intensity factors on the frequency, and on the angle of incidence, is displayed in a set of figures.     

Vertical vibrations of elastic foundation resting on saturated half-space

This paper is mainly concerned with the dynamic response of an elastic foun- dation of finite height bounded to the surface of a saturated half-space.The foundation is subjected to time-harmonic vertical loadings.First,the transform solutions for the governing equations of the saturated media are obtained.Then,based on the assumption that the contact between the foundation and the half-space is fully relaxed and the half- space is completely pervious or impervious,this dynamic mixed boundary-value problem can lead to dual integral equations,which can be further reduced to the Fredholm integral equations of the second kind and solved by numerical procedures.In the numerical exam- ples,the dynamic compliances,displacements and pore pressure are developed for a wide range of frequencies and material/geometrical properties of the saturated soil-foundation system.In most of the cases,the dynamic behavior of an elastic foundation resting on the saturated media significantly differs from that of a rigid disc on the saturated half-space. The solutions obtained can be used to study a variety of wave propagation problems and dynamic soil-structure interactions.     

Mechanics of non-slipping adhesive contact on a power-law graded elastic half-space

In previous work about axisymmetric adhesive contact on power-law graded elastic materials, the contact interface was often assumed to be frictionless, which is, however, not always the case in practical applications. In order to elucidate the effect of friction and the coupling between normal and tangential deformations, in the present paper, the problem of a rigid punch with a parabolic shape in non-slipping adhesive contact with a power-law graded half-space is studied analytically via singular integral equation method. A series of closed-form analytical solutions, which include the frictionless and homogeneous solutions as special cases, are obtained. Our results show that, compared with the frictionless case, the interfacial friction tends to reduce the contact area and the indentation depth during adhesion. The magnitude of the coupling effect depends on both the Poisson ratio and the gradient exponent of the half-space. This effect vanishes for homogeneous incompressible as well as for linearly graded materials but becomes significant for auxetic materials with negative Poisson’s ratio. Furthermore, influence of mode mixity on the adhesive behavior of power-law graded materials, which was seldom touched in literature, is discussed in details.     

Uniform indentation of the elastic half-space by a rigid rectangular punch

The problem considered is that of a rigid flat-ended punch with rectangular contact area pressed into a linear elastic half-space to a uniform depth. Both the lubricated and adhesive cases are treated. The problem reduces to solving an integral equation (or equations) for the contact stresses. These stresses have a singular nature which is dealt with explicitly by a singularity-incorporating finite-element method. Values for the stiffness of the lubricated punch and the adhesive punch are determined: the effect of adhesion on the stiffness is found to be small, producing an increase of the order of 3%.     

Analytical solution of the spherical indentation problem for a half-space with gradients with the depth elastic properties

The contact problem for the impression of spherical indenter into a non-homogeneous (both layered and functionally graded) elastic half-space is considered. Analytical methods for solving this problem have been developed. It is assumed that the Lame coefficients vary arbitrarily with the half-space depth. The problem is reduced to dual integral equations. The developed methods make it possible to find the analytical asymptotically exact problem solution, suitable for a PC. The influence of the Lame coefficients variation upon the contact stresses and size of the contact zone with different radius of indenter as well as values of the impressing forces are studied. The effect of the non-homogeneity is examined. The developed method allows to construct analytical solutions with presupposed accuracy and gives the opportunity to do multiparametric and qualitative investigations of the problem with minimal computation time expenditure.