Steady sliding frictional contact problem for a 2d elastic half-space with a discontinuous friction coefficient and related stress singularities

The steady sliding frictional contact problem between a moving rigid indentor of arbitrary shape and an isotropic homogeneous elastic half-space in plane strain is extensively analysed. The case where the friction coefficient is a step function (with respect to the space variable), that is, where there are jumps in the friction coefficient, is considered. The problem is put under the form of a variational inequality which is proved to always have a solution which, in addition, is unique in some cases. The solutions exhibit different kinds of universal singularities that are explicitly given. In particular, it is shown that the nature of the universal stress singularity at a jump of the friction coefficient is different depending on the sign of the jump.     

Stress field at a sliding frictional contact: Experiments and calculations

A MEMS-based sensing device is used to measure the normal and tangential stress fields at the base of a rough elastomer film in contact with a smooth glass cylinder in steady sliding. This geometry allows for a direct comparison between the stress profiles measured along the sliding direction and the predictions of an original exact bidimensional model of friction. The latter assumes Amontons’ friction law, which implies that in steady sliding the interfacial tangential stress is equal to the normal stress times a pressure-independent dynamic friction coefficient μ_d, but makes no further assumption on the normal stress field. Discrepancy between the measured and calculated profiles is less than 14% over the range of loads explored. Comparison with a test model, based on the classical assumption that the normal stress field is unchanged upon tangential loading, shows that the exact model better reproduces the experimental profiles at high loads. However, significant deviations remain that are not accounted for by either calculations. In that regard, the relevance of two other assumptions made in the calculations, namely (i) the smoothness of the interface and (ii) the pressure-independence of μ_d is briefly discussed.     

Thermo-electro-mechanical contact behavior of a finite piezoelectric layer under a sliding punch with frictional heat generation

The thermal contact problem of a piezoelectric strip with heat supply generated by the frictional tangential traction under the action of a rigid sliding punch is investigated. The inertial effects are considered. It is convenient to introduce the Galilean transform. Whole cases of the root distribution of the corresponding characteristic equation are detailed. Appropriate fundamental solutions that can lead to real solutions of the thermo-electro-mechanical quantities are derived for the piezoelectric governing equation. The stated problem is reduced to Cauchy singular integral equation of the second kind finally. Numerical results are also presented. The solutions have a reduced dependence on the material properties. The singular behaviors at the edges of the punch are revealed. The stress distribution and temperature distribution above the punch with the variations of the relative sliding speed, the frictional coefficient and the thickness are plotted. The effects of the material constants on the stress distribution and temperature distribution above the punch are presented.     

Development of a solution method for frictional contact problems with complex loading

In this work, solution methods for frictional contact problems are extended to the case of moving punches and to the external loading history-dependent system states. To solve the frictional contact problems in the contact area, an iterative method is developed and implemented. Solutions of two-dimensional problems are constructed using the boundary element method. Numerical analysis is aimed at the quantitative study of effects such as the interaction of contact pressure and friction forces, estimates of the friction force differences due to the differences in the choice of local basis for the calculation of normal pressure and friction forces, and evaluation of the effects of complex loading (rotation of the rigid punch after its preliminary penetration into the solid). We find that, for the same definition of the friction force, different initial approximations lead to the same solution. At the same time, the friction forces defined either as projections onto the common tangent plane or as projections onto the plane tangent to the punch can differ quite substantially. Similar conclusions are derived for the solutions corresponding to single or multiple loading steps. The work relies on the variational principle for the solution of contact problems and numerical algorithms developed for the problems with one-sided constraints. The variational principle was first applied by Signorini [1] to the determination of the stress-strain state in a linearly deformed body in a rigid smooth shell. The modern view of the problem and its generalizations to the frictional problems and some other problems involving unilateral constraints in given in the monograph [2]. Finite difference and finite element methods in application to the problems with unilateral constraints are described in [3]. Analytical solution methods are developed in the monographs [4–6].     

Analysis of FRP composites under frictional contact conditions

This work studies numerically the tribological behavior of fiber-reinforced plastics (FRP) under different frictional contact conditions, using a boundary element methodology. The formulation uses the Boundary Element Method (BEM) with an explicit approach for fundamental solutions evaluation, for computing the elastic influence coefficients. To enforce the contact constraints on the potential contact zone: Signorini’s contact conditions and an orthotropic law of friction, contact operators over the augmented Lagrangian are considered in the formulation. The methodology and the proposed algorithm are applied to study two types of glass FRP and two types of carbon FRP, with the same fiber volume fraction, under frictional contact. In these studies, it can be observed how the fiber orientation and sliding orientation affect the normal and tangential contact compliance, as well as the contact traction distribution. Furthermore, the formulation considers a micromechanics model for FRP that allows also to study the influence of fiber volume fraction on normal and tangential contact compliances.     

Slip-shakedown analysis of a system of circular beams in frictional contact

In automotive components, the cumulative microslip phenomenon is often observed for engine assemblies. This phenomenon results in an accumulation of the relative slips in a preferred tangential direction on the contact interface of two solids under cyclic loadings. A significant relative displacement may occur and leads to the assembly failure. In particular, a global rotation of the bearing shell may result from this mechanism of cumulated slips in conrod big end systems. To discuss this rotation problem, a model of two circular beams in frictional contact and submitted to a periodical rotating load is considered here. The aim is to give some simplified estimates of the critical rotation load based on a slip-shakedown analysis. The discussion holds for Tresca friction and can be extended to Coulomb friction under the assumption of small coupling. The static and kinematic slip-shakedown approaches are discussed. The obtained analytical results are shown to be in agreement with the finite element computations.     

An embedded crack in a functionally graded orthotropic coating bonded to a homogeneous substrate under a frictional Hertzian contact

In this paper, we consider the elasto-static problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subject to statically applied normal and tangential surface loading. The crack direction is parallel to the free surface. The coating is graded in the thickness direction and is orthogonal to the crack direction. This coating is modelled as a non-homogeneous medium with an orthotropic stress–strain law. The equivalent crack surface stresses are first obtained and substituted in the plane elasticity equations. Using integral transforms, the governing equations are converted into singular integral equations which are solved numerically to yield the displacement field as well as the crack-tip stress intensity factors. This study presents a complete theoretical formulation for the problem in the static case. A numerical predictive capability for solving the singular integral equations and computing the crack-tip stress intensity factors is proposed. Since the loading is compressive, a previously developed crack-closure algorithm is applied to avoid interpenetration of the crack faces. The main objective of the paper is to investigate the effects of the material orthotropy and non-homogeneity of the graded coating on the crack-tip stress intensity factors, with and without using the crack-closure algorithm, for the purpose of gaining better understanding on the behavior and design of graded coatings.     

The contact problem of a rubber half-space dented by a rigid cone apex

Summary  The smooth contact of a rubber half-space dented by a rigid cone apex is analyzed based on the large deformation theory. The problem is treated as an axially symmetric case, and the material is assumed to be incompressible. The asymptotic equations for the domain near the apex are derived. They are solved analytically for the shrinking domain, while a numerical solution is found for the expanding domain in the vicinity of the stress singularity. The purpose of this paper is not only to solve a typical problem but also to provide an analytical method to solve a large-strain problem with a singular point. Received 10 July 2001; accepted for publication 24 January 2002     

The frictional sliding response of elasto-plastic materials in contact with a conical indenter

Over the past decade, many computational studies have explored the mechanics of normal indentation. Quantitative relationships have been well established between the load–displacement hysteresis response and material properties. By contrast, very few studies have investigated broad quantitative aspects of the effects of material properties, especially plastic deformation characteristics, on the frictional sliding response of metals and alloys. The response to instrumented, depth-sensing frictional sliding, hereafter referred to as a scratch test, could potentially be used for material characterization. In addition, it could reproduce a basic tribological event, such as asperity contact and deformation, at different length scales for the multi-scale modeling of wear processes. For these reasons, a comprehensive study was undertaken to investigate the effect of elasto-plastic properties, such as flow strength and strain hardening, on the response to steady-state frictional sliding. Dimensional analysis was used to define scaling variables and universal functions. The dependence of these functions on material properties was assessed through a detailed parametric study using the finite element method. The strain hardening exponent was found to have a greater influence on the scratch hardness and the pile-up height during frictional sliding than observed in frictionless normal indentation. When normalized by the penetration depth, the pile-up height can be up to three times larger in frictional sliding than in normal indentation. Furthermore, in contrast to normal indentation, sink-in is not observed during frictional sliding over the wide range of material properties examined. Finally, friction between indenter and indented material was introduced in the finite element model, and quantitative relationships were also established for the limited effects of plastic strain hardening and yield strength on the overall friction coefficient. Aspects of the predictions of computational simulations were compared with experiments on carefully selected metallic systems in which the plastic properties were systematically controlled. The level of accuracy of the predicted frictional response is also assessed by recourse to the finite element method and by comparison with experiment.     

Dynamic analysis of a disc brake under frictional and thermomechanical internal loading

The study of the vibratory response of a mechanical system as complex as a disc brake needs to consider the complexity of the problem induced by the coupling of tribological, thermomechanical and dynamical effects. Experimental consideration are discussed here for two set ups at the full scale of the disc brake and at a local scale focussed on the third body interface. A numerical model with thermomechanical and dynamical couplings is then presented, followed by a substantial discussion.     

Modeling the deformation of rock with rough surfaces of block contact under quasistatic and dynamic loading conditions

The effect of layers consisting of small particles or asperities on contact surfaces on the deformation of a block medium as a whole and the failure of the blocks constituting the medium was studied experimentally. The samples were subjected to quasistatic uniaxial compression perpendicular to the contact surfaces. Numerical modeling was performed of wave propagation during pulse loading of a pair of blocks having rough surfaces of contact and made of a material with elastic characteristics close to the characteristics of marble and limestone. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 3, pp. 173–178, May–June, 2007.     

Investigation of changes in the spherical shell shape under the action of pulsed loading due to contact interaction with a rigid block

An axisymmetric problem of high strains in a spherical lead shell enclosed into an aluminum “spacesuit” under the action of pulsed loading is considered. The shell straining is described with the use of equations of mechanics of elastoviscoplastic media in Lagrangian variables, and the kinematic relations are determined in the current state metrics. Equations of state are taken in the form of equations of the flow theory with isotropic hardening. The problem is solved numerically by using the variational difference method and the “cross” explicit scheme of integration with respect to time. The influence of the yield stress as a function of the strain rate on changes in the shell shape is studied for different values of loading. The calculated final shape and residual strains are demonstrated to be in good agreement with experimental data.     

Propagation of harmonic waves through micro gap with consideration of frictional contact

Transmission of elastic waves through a micro gap between two solids with consideration of frictional contact is investigated.By using the Fourier analysis technique and the corrective solution method,the nonlinear boundary problem is reduced to a set of algebraic equations.Numerical results exhibit the locations and extents of separation, slip,and stick zones,the interface tractions,and the energy partition.The effects of gap width,frictional coefficients,and the incident angle on the wave transmission ...     

Non-linear response of laminated composite plates under thermomechanical loading

The non-linear response of laminated composite plates under thermomechanical loading is studied using the third-order shear deformation theory (TSDT) that includes classical and first-order shear deformation theories (CLPT and FSDT) as special cases. Geometric non-linearity in the von Kármán sense is considered. The temperature field is assumed to be uniform in the plate. Layers of magnetostrictive material, Terfenol-D, are used to actively control the center deflection. The negative velocity feedback control is used with the constant gain value. The effects of lamination scheme, magnitude of loading, layer material properties, and boundary conditions are studied under thermomechanical loading.     

Response of a finite beam in contact with a tensionless foundation under symmetric and asymmetric loading

In this work a general analytical model is developed for the static response of a beam resting on a tensionless elastic foundation subjected to a lateral point load. This load may either be located at the center of the beam or may be offset. An analytical/numerical solution is obtained to the governing equations; this solution makes no assumption about either the contact area or the kinematics associated with the transverse deflection of the beam. This is in contrast to previous work in which, for an infinite beam (where the load is symmetric by definition), implicit assumptions about the contact area and the response kinematics were made. Because these assumptions are dropped, the contact behavior differs in several fundamental ways from its infinite counterpart. Specifically, it is shown that (i) the contact area is a sensitive function of the beam length and that this function may change nonmonotonically, (ii) the contact area may depend on the magnitude of the load, (iii) asymmetric loads, which cannot exist in the infinite problem, have a dramatic influence the contact area for the finite system. These features are demonstrated with specific examples and explained in terms of the fundamental physics of the system. The implications for these behaviors are also discussed.     

Dissipative interface waves and the transient response of a three-dimensional sliding interface with Coulomb friction

We investigate the linearized response of two elastic half-spaces sliding past one another with constant Coulomb friction to small three-dimensional perturbations. Starting with the assumption that friction always opposes slip velocity, we derive a set of linearized boundary conditions relating perturbations of shear traction to slip velocity. Friction introduces an effective viscosity transverse to the direction of the original sliding, but offers no additional resistance to slip aligned with the original sliding direction. The amplitude of transverse slip depends on a nondimensional parameter η=c_sτ₀/μv₀, where τ₀ is the initial shear stress, 2v₀ is the initial slip velocity, μ is the shear modulus, and c_s is the shear wave speed. As η→0, the transverse shear traction becomes negligible, and we find an azimuthally symmetric Rayleigh wave trapped along the interface. As η→∞, the inplane and antiplane wavesystems frictionally couple into an interface wave with a velocity that is directionally dependent, increasing from the Rayleigh speed in the direction of initial sliding up to the shear wave speed in the transverse direction. Except in these frictional limits and the specialization to two-dimensional inplane geometry, the interface waves are dissipative. In addition to forward and backward propagating interface waves, we find that for η>1, a third solution to the dispersion relation appears, corresponding to a damped standing wave mode. For large-amplitude perturbations, the interface becomes isotropically dissipative. The behavior resembles the frictionless response in the extremely strong perturbation limit, except that the waves are damped. We extend the linearized analysis by presenting analytical solutions for the transient response of the medium to both line and point sources on the interface. The resulting self-similar slip pulses consist of the interface waves and head waves, and help explain the transmission of forces across fracture surfaces. Furthermore, we suggest that the η→∞ limit describes the sliding interface behind the crack edge for shear fracture problems in which the absolute level of sliding friction is much larger than any interfacial stress changes.     

A multi-scale constitutive model for the sintering of an air-plasma-sprayed thermal barrier coating, and its response under hot isostatic pressing

A micromechanical model is developed for the sintering of an air-plasma-sprayed, thermal barrier coating, and is used to make predictions of microstructure evolution under free sintering and under hot isostatic pressing. It is assumed that the splats of the coating are separated by penny-shaped cracks; the faces of these cracks progressively sinter together at contacting asperities, initially by the mechanism of plastic yield and subsequently by interfacial diffusion. Diffusion is driven by the reduction in interfacial energy at the developing contacts of the cracks and also by the local contact stress at asperities. The contact stress arises from the remote applied stress and from mechanical wedging of the rough crack surfaces. Sintering of the cracks leads to an elevation in both the macroscopic Young's modulus and thermal conductivity of the coating, and thereby leads to a degradation in thermal performance and durability. An assessment is made of the relative roles of surface energy, applied stress and crack face roughness upon the sintering response and upon the evolution of the pertinent mechanical and physical properties. The evolution in microstructure is predicted for free sintering and for hot isostatic pressing in order to provide guidance for experimental validation of the micromechanical model.     

高冲击载荷作用下弹载记录仪防护系统动力学响应特性

为了给弹载记录仪的防护设计提供依据,从机械振动的角度揭示了高冲击载荷作用下弹载记录仪防护系统的动力学响应机理。在分析弹载记录仪内部载荷传递关系的基础上,基于双自由度弹簧-质量-阻尼系统建立了一种简化的防护系统动力学响应模型,并开展了数值模拟,通过脉冲响应分析和谐响应分析辨识了模型参数。理论计算与数值模拟的对比分析结果表明：建立的动力学响应模型能较准确地预测高冲击载荷作用下弹载记录仪防护系统的动力学响应特性。在此基础上,以模型的幅频响应特性为依据,分析了防护系统动力学响应特性随各种参数的变化规律。研究结果可为更有效地指导弹载记录仪的防护设计提供依据。

     

Stress Analysis for an Elastic Semispace with Surface and Graded Layer Coatings under Induced Torsion

In this paper, the axisymmetric torsional problem of a coating structure consisting of a surface coating, a functionally graded layer and a substrate under a rigid cylindrical punch is investigated. The coating and substrate are homogeneous materials with distinct physical properties while the intermediate layer is inhomogeneous with its shear modulus changing exponentially along the thickness direction. The Hankel integral transform technique is employed to reduce the torsional problem to a singular integral equation with a Cauchy kernel. The circumferential shear stress and displacement fields in the coating structure are calculated by solving the integral equation numerically. The results show that the stiffness ratio has significant effect on the distribution of the circumferential stress and displacement at the interface.