Action of an elliptic punch on a transversally isotropic half-space

The 3D contact problem on the action of a punch elliptic in horizontal projection on a transversally isotropic elastic half-space is considered for the case in which the isotropy planes are perpendicular to the boundary of the half-space. The elliptic contact region is assumed to be given (the punch has sharp edges). The integral equation of the contact problem is obtained. The elastic rigidity of the half-space boundary characterized by the normal displacement under the action of a given lumped force significantly depends on the chosen direction on this boundary. In this connection, the following two cases of location of the ellipse of contact are considered: it can be elongated along the first or the second axis of Cartesian coordinate system on the body boundary. Exact solutions are obtained for a punch with base shaped as an elliptic paraboloid, and these solutions are used to carry out the computations for various versions of the five elastic constants. The structure of the exact solution is found for a punch with polynomial base, and a method for determining the solution is proposed.     

The contact problem of electroelasticity for a flat punch with parabolic cross-section

The solution of the problem of a rigid punch with a parabolic cross-section and flat base that is forced into an elastic piezoelectric ceramic half-space is derived in explicit form. The punch is somewhat displaced, being parallel to the isotropy plane that coincides with the boundary surface of the half-space. The symmetry axis coincides with the direction of the force lines of the field with the previous polarization. Formulas are derived to determine the stresses on the surface of the half-space under the punch and the components of the conjugate electric field for certain boundary conditions on the contact area. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 11, pp. 20–26, November, 1999.     

Contact with friction of a rigid cylinder with an elastic half-space

An exact solution to the problem of indentation with friction of a rigid cylinder into an elastic half-space is presented. The corresponding boundary-value problem is formulated in planar bipolar coordinates, and reduced to a singular integral equation with respect to the unknown normal stress in the slip zones. An exact analytical solution of this equation is constructed using the Wiener-Hopf technique, which allowed for a detailed analysis of the contact stresses, strain, displacement, and relative slip zone sizes. Also, a simple analytical solution is furnished in the limiting case of full stick between the cylinder and half-space.     

Self-similar problems of elastic contact for non-convex punches

Self-similar problems of contact for non-convex punches are considered. The non-convexity of the punch shapes introduces differences from the traditional self-similar contact problems when punch profiles are convex and their shapes are described by homogeneous functions. First, three-dimensional Hertz type contact problems are considered for non-convex punches whose shapes are described by parametric-homogeneous functions. Examples of such functions are numerous including both fractal Weierstrass type functions and smooth log-periodic sine functions. It is shown that the region of contact in the problems is discrete and the solutions obey a non-classical self-similar law. Then the solution to a particular case of the contact problem for an isotropic linear elastic half-space when the surface roughness is described by a log-periodic function, is studied numerically, i.e. the contact problem for rough punches is studied as a Hertz type contact problem without employing additional assumptions of the multi-asperity approach. To obtain the solution, the method of non-linear boundary integral equations is developed. The problem is solved only on the fundamental domain for the parameter of self-similarity because solutions for other values of the parameter can be obtained by renormalization of this solution. It is shown that the problem has some features of chaotic systems, namely the global character of the solution is independent of fine distinctions between parametric-homogeneous functions describing roughness, while the stress field of the problem is sensitive to small perturbations of the punch shape.     

Determination of optimal shape of a moving punch with friction taken into account

The optimization problem for the contact interaction between a rigid punch and an elasticmediumis considered. It is assumed that that the punch is under the action of some prescribed forces and momenta and moves along a surface bounding a half-space filled with an elastic medium. It is also assumed that themotion is quasistatic and the friction forces arising in the region of contact are taken into account. The punch shape is considered as the desired design variable, and the integral functional characterizing the discrepancy between the pressure distribution in the region of contact that corresponds to the optimized shape of the punch and a given goal distribution of pressure is taken as the minimizing criterion. The optimal shape can be determined efficiently by solving the following two problems: first, to obtain the optimal pressure distribution and then to solve a boundary value problemfor the elastic half-space under the action of the obtained normal pressure and friction forces. By way of example, the optimal shape is analytically determined for a punch of rectangular shape in horizontal projection.     

Boundary film shear elastic modulus effect in hydrodynamic contact. Part II: influence of operational parameters

The present paper is the subsequent research of the first part (Theor Comput Fluid Dyn, 2009). It investigates the boundary film shear elastic modulus effect in a hydrodynamic contact in different operating conditions. The hydrodynamic contact is one-dimensional, composed of two parallel plane surfaces, which are respectively rough rigid with rectangular micro projections in profile periodically distributed on the surface and ideally smooth rigid. The whole contact consists of cavitated area and hydrodynamic area. The hydrodynamic area consists of many micro Raleigh bearings which are discontinuously and periodically distributed in the contact. The hydrodynamic contact in a micro Raleigh bearing consists of boundary film area and fluid film area which, respectively, occur in the outlet and inlet zones. In boundary film area, the film slips at the upper contact surface due to the limited shear stress capacity of the film–contact interface, while the film does not slip at the lower contact surface due to the shear stress capacity of the film–contact interface large enough. In boundary film area, the viscosity, density, and shear elastic modulus of the film are varied across the film thickness due to the film–contact interactions, and their effective values are used in modeling which depends on the film thickness. In fluid film area, the film does not slip at either of the contact surfaces, and the shear elastic modulus of the film is neglected. It is found from the simulation results that the boundary film shear elastic modulus influences are normally negligible on the mass flow through the contact, the carried load of the contact and the overall film thickness of the contact, and the boundary film shear elastic modulus would normally influence the local film thickness in an elastic contact when the local film thickness is on the film molecule diameter scale. It is also found that the boundary film shear elastic modulus effect has the tendency of being increased with the reduction of the width of a micro contact. It is increased with the reduction of the boundary film–contact interfacial shear strength or with the increase of the critical boundary film thickness, while it is strongest at certain values of the contact surface roughness, the width ratio of fluid film area to boundary film area, and the lubricant film shear elastic modulus.

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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%.     

New analytical and numerical results for two-dimensional contact profiles

Recently, a generalized Coulomb law for elastic bodies in contact has been developed by the author, which assumes that the tangential traction is the difference of the slip stress of the contact and the stick area, whereby each stick area corresponds to a smaller contact area. It holds for multiple contact regions also. Several applications for elastic half planes, half spaces, thin and thick layers and impact problems have been published. For plane contact of equal bodies with friction, it provides exact solutions, and the interior stress field can be expressed with analytical results in closed form. In this article, a singular superposition of flat punch solutions is outlined, in which the punches are aligned with an edge of the contact area. It is shown that this superposition satisfies Coulomb's inequalities directly, and new results for the Muskhelishvili potentials of several profiles are presented. It is illustrated how problems of singularity and multi-valuedness of complex functions can be solved in closed form, and the Chebyshev approximation used by earlier authors can be avoided. For comparison, some previous solutions for symmetric profiles are appended. Some results for the interior stress field, the pressure, the frictional traction and the surface displacements are compared with FEM solutions of an equivalent problem. The small differences between both methods show characteristic features of the FEM model and the theoretical assumptions, and are shortly explained. Further, this example can be used as benchmark test for FEM and BEM programs.     

Effect of friction on subsurface stresses in sliding line contact of multilayered elastic solids

A numerical integral scheme based on Fourier transformation approach is employed to investigate the effect of friction on subsurface stresses arising from the two-dimensional sliding contact of two multilayered elastic solids. The analysis incorporates bonded and unbonded interface boundary conditions between the coating layers. Two line contact problems are presented. The first one is the contact problem between a rigid cylinder and a two-layer half space and the second one is the indentation of a multilayered elastic half-space by a flat rigid punch. The effects of the surface coating on the contact pressure distribution and subsurface stress field are presented and discussed.     

Wave induced sliding at the interface between a layered elastic medium and a half-space

The sliding interface between an unrestrained elastic half-space and a grounded layered half-space excited by an incident harmonic wave is investigated. The problem is formulated considering various possible boundary conditions and boundary inequalities at the sliding interface. The Coulomb friction model without distinction between the static and kinetic coefficients of friction is considered to govern the sliding condition. Three possible bands at the interface, namely slip, stick, and separation, are considered. The interface is assumed to be preloaded under normal and shear stresses. The solution is developed by modifying the problem of welded interface, which then is reduced to a set of algebraic equations. The effects of the incident angle, layer thickness, friction coefficient and externally applied stresses on the drift velocity of the unrestrained half-space are studied numerically for a pair of materials. It is shown that the sliding interface, and hence the drift velocity of unrestrained half-space is noticeably influenced by the layered medium. These results are expected to be useful for the development of a new kind of ultrasonic drive in future.     

Analytical solutions for inherently incremental,similar elastic contact problems with bulk stress

Analytical solutions are obtained for a class of inherently incremental, similar elastic half-plane frictional contact problems which involve simultaneous application of a changing, remote bulk-stress during oblique loading. The method is based on Abel Integral Equations. The spatial gradient of the slip function is obtained uniquely by the conditions determining the existence of a bounded solution to the shear traction Cauchy SIE and global shear equilibrium, provided there is stick at all points in the contact throughout the loading. Solutions are obtained for both the cylindrical indenter and the flat punch with rounded edges. The method also lends itself easily to generalization to non-linear load paths and a number of such problems are considered. Problems involving the development of slip zones within the contact may be solved in some special cases.     

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

This paper studies the contact vibration problem of an elastic half-space coated with functionally graded materials (FGMs) subject to a rigid spherical punch. A static force superimposing a dynamic time-harmonic force acts on the rigid spherical punch. Firstly, we give the static contact problem of FGMs by a least-square fitting approach. Next, the dynamic contact pressure is solved by employing the perturbation method. Lastly, the dynamic contact stiffness with different dynamic contact displacement conditions is derived for the FGM coated half-space. The effects of the gradient index, coating thickness, internal friction, and punch radius on the dynamic contact stiffness factor are discussed in detail.     

A friction contact stiffness model of fractal geometry in forced response analysis of a shrouded blade

To investigate the nonlinear vibration behavior of a shrouded blade with friction dynamic contact interface, a friction contact stiffness model is proposed to describe the friction force at different rough interfaces and different normal loads. In the proposed model, the friction contact interface is discretized to a series of friction contact pairs and each of them can experience stick, slip, or separate states. Fractal geometry is used to simulate the topography of contact surfaces. The contact stiffness is calculated using the Hertz contact theory and fractal geometry, which is related to contact interfaces parameters including normal load, roughness, Young??s modulus, and Poisson??s ratio. The trajectory tracking method is used to predict the friction force and it is not necessary to judge the transition condition among stick, slip, and separate states. It is suitable for complicated periodic motion of the contact interfaces. The forced response of a real shrouded blade is predicted using the proposed model and the multi-harmonic balance method. The effect of surface roughness, initial normal load, and contact area on the forced response of a shrouded blade is studied. It is shown that contact stiffness increases with normal load and fractal dimension. The resonant amplitude is sensitive to the initial normal load and contact surface roughness. The response can be influenced by the contact area, which is an important parameter for blade designers.     

Effective macroscopic adhesive contact behavior induced by small surface roughness

In this paper we study a model contact problem involving adhesive elastic frictionless contact between rough surfaces. The problem's most notable feature is that it captures the phenomenon of depth-dependent-hysteresis (DDH) (e.g., see Kesari et al., 2010), which refers to the observation of different contact forces during the loading and unloading stages of a contact experiment. We specifically study contact between a rigid axi-symmetric punch and an elastic half-space. The roughness is represented as arbitrary periodic undulations in the punch's radial profile. These undulations induce multiple equilibrium contact regions between the bodies at each indentation-depth. Assuming that the system evolves so as to minimize its potential energy, we show that different equilibrium contact regions are selected during the loading and unloading stages at each indentation-depth, giving rise to DDH. When the period and amplitude of our model's roughness is reduced, we show that the evolution of the contact force and radius with the indentation-depth can be approximated with simpler curves, the effective macroscopic behavior, which we compute. Remarkably, the effective behavior depends solely on the amplitude and period of the model's roughness. The effective behavior is useful for estimating material properties from contact experiments displaying DDH. We show one such example here. Using the effective behavior for a particular roughness model (sinusoidal) we analyze the energy loss during a loading/unloading cycle, finding that roughness can toughen the interface. We also estimate the energy barriers between the different equilibrium contact regions at a fixed indentation-depth, which can be used to assess the importance of ambient energy fluctuations on DDH.     

The stress on an elastic half-space due to sectionally smooth-ended punch

The axisymmetric contact problem for an elastic half-space and a rigid punch is considered using integral transform methods. The end of the punch is sectionally smooth and there is incomplete penetration. The normal stress under the punch is calculated and found to have an elliptic integral type of singularity.     

Plane contact problem for periodic laminated composite involving frictional heating

Summary The stationary problem of a rigid thermally insulated punch sliding over the boundary surface of a periodic two-layered thermoelastic half-space is considered. The heat generated in the contact area is assumed to be caused by frictional forces. The problem is formulated within the framework of thermoelasticity with microlocal parameters, and it is reduced to a system of two integral equations, which is solved numerically. The effects connected with the composite structure are analyzed.     

Impression of a punch with a flat square base into an elastic half-space

The three-dimensional contact problem on impression of a square punch into an elastic isotropic half-space is considered. An explicit formula for determining the normal stress on the contact area is obtained. International University of Civil Aviation, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 21–26, October, 1999.     

On the Analytical Approach to Galin’s Stick-Slip Problem. A Survey

Galin’s classical work (PMM J Appl Math Mech 9:413–424, 1945) on the contact of a rigid flat-ended indenter with an elastic half-plane with partial slip was the first successful attempt to take into account friction in the problem of normal contact. As Galin was unable to find an exact solution of the formulated problem, the problem of contact with partial slip of a rigid punch with an elastic half-plane was challenged by many researchers. At the same time Galin’s seminal work stimulated development of solutions for other contact problems with friction that feature different punch geometries and different material responses. This paper presents an overview of the developments in the area of elastic contact with partial slip. In the spirit of Galin’s work the focus is placed on contributions with substantial analytical merit.      

磁电弹复合材料和刚性导电导磁圆柱压头的二维微动接触分析

本文求解平面应变状态下磁电弹复合材料半平面和刚性导电导磁圆柱压头的二维微动接触问题。假设压头具有良好的导电导磁性,且表面电势和磁势是常数。微动接触依赖载荷的加载历史,所以首先求解单独的法向加载问题,然后在法向加载问题的基础上求解循环变化的切向加载问题。整个接触区可以分为内部的中心粘着区和两个外部的滑移区,其中滑移区满足Coulomb摩擦法则。利用Fourier积分变换,磁电弹半平面的微动接触问题将简化为耦合的Cauchy奇异积分方程组,然后数值离散为线性代数方程组,利用迭代法求解未知的粘着/滑移区尺寸、电荷分布、磁感应强度、法向接触压力和切向接触力。数值算例给出了摩擦系数、总电荷和总磁感应强度对各加载阶段的表面接触应力、电位移和磁感应强度的影响。     

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 ...