Effect of time-dependent speed on frictional heat generation and wear in transient axisymmetrical contact of sliding

Summary A transient contact problem with frictional heating and wear for two nonuniform sliding half-spaces is considered. One of the two half-spaces is assumed to be slightly curved to give a Hertzian initial pressure distribution: the other is a rigid nonconductor. Under the assumption that the contact pressure distribution could be described by Hertz formulas during all the process of interaction, the problem is formulated in terms of one integral equation of Volterra type with unknown radius of contact area. A numerical solution of this equation is obtained using a piecewise-constant presentation of an unknown function. The influence of operating parameters on the contact temperature and the radius of the contact area is studied. Accepted for publication 3 November 1996     

The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape

The contact of an indenter of arbitrary shape on an elastically anisotropic half space is considered. It is demonstrated in a theorem that the solution of the contact problem is the one that maximizes the load on the indenter for a given indentation depth. The theorem can be used to derive the best approximate solution in the Rayleigh-Ritz sense if the contact area is a priori assumed to have a certain shape. This approach is used to analyze the contact of a sphere and an axisymmetric cone on an anisotropic half space. The contact area is assumed to be elliptical, which is exact for the sphere and an approximation for the cone. It is further shown that the contact area is exactly elliptical even for conical indenters when a limited class of Green's functions is considered. If only the first term of the surface Green's function Fourier expansion is retained in the solution of the axisymmetric contact problem, a simpler solution is obtained, referred to as the equivalent isotropic solution. For most anisotropic materials, the contact stiffness determined using this approach is very close to the value obtained for both conical and spherical indenters by means of the theorem. Therefore, it is suggested that the equivalent isotropic solution provides a quick and efficient estimate for quantities such as the elastic compliance or stiffness of the contact. The “equivalent indentation modulus”, which depends on material and orientation, is computed for sapphire and diamond single crystals.     

Thermal constriction resistance between two solids for random distribution of contacts

An infinite interface of randomly distributed contacts is modeled as a finite square region with randomly placed contacts inside it. The contacts outside the region are treated as continuum of contacts. The continuum approximation allows for an interaction between the contacts within the square and those outside it. An analytical solution is obtained for the temperature field, and the contact resistance is analyzed for randomness effects. This is the first such analytical model developed to study random distribution of contacts. The result shows an excellent agreement when tested against the the available analytical solution for the case of periodic arrangement of contacts. For the random case, the resistance is observed to be a strong function of the area fraction of contact. Received on 16 March 1998     

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.     

The oblique compression of two elastic spheres

A fundamental problem in the behaviour of the packing of spheres is that of the oblique compression of just two spheres. Here, the solution of this problem is obtained for the case of two identical homogeneous isotropic elastic spheres, since much use can then be made of the existing symmetry. In particular, the normal and shear components of traction on the contact area can be treated separately. Considerations of the normal force show that the contact area is circular and, furthermore, that this part of the solution is precisely that of normal Hertzian contact. To obtain that part of the solution corresponding to shear, two criteria are used. The first is that of no slip between the spheres, and the second is that the energy flux across the contact area must obey the appropriate symmetries of the problem. These symmetries are sufficient to make the solution unique. This solution differs greatly from that obtained when the spheres are first compressed normally and then sheared. In particular, it is shown that if slip does occur, then it will be in the form of sliding; whereas in the latter case, slip occurs only within a circular annulus.     

Steady-state frictional heat generation on axisymmetric sliding contact of a thermosensitive sphere and a fixed, thermally insulated base

An approach to the construction of a calculation scheme for finding the contact area dimensions, pressure, and temperature for a thermosensitive sphere sliding along a fixed, thermally isolated base is suggested. The well-known solution for the constant thermal properties of the sphere material follows as a particular case of the solution obtained. Polytechnical University, Byalystok 15333, Poland. Franko L'vov State University, L'vov 290602. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 174–183, January–February, 1999.     

固定接触界面法向静弹性刚度

基于Hertz接触理论推导了两个微凸体之间互相作用的法向接触静弹性刚度.根据修正后的一个微接触点的平截面积尺寸分布,给出了界面的总法向接触静弹性条件刚度、总条件载荷的解析解.将法向静弹性刚度的解析解嵌入到有限元软件中,获得整机的理论模态.通过实验对解析解进行了定量验证.以机床结合部为研究对象,在理论振型与实验振型一致的...     

有限长滚子线接触热弹流润滑分析

应用多重网格解法 ,求出了有限长滚子线接触热弹流润滑的完全数值解 .结果表明 :在滚子的中部 ,油膜压力、温度和最小膜厚与无限长线接触热弹流润滑的解几乎一致 ;在滚子端部的圆角处 ,油膜压力、温度和最小膜厚与中部均显著不同 ,且最大油膜压力、最大油膜温度和最小油膜厚度均发生在此处 ,端部圆角半径对弹流润滑性能有显著影响 .同时 ,将有限长线接触热解与有限长线接触等温解进行了比较 .     

New,real fundamental solutions to the transient thermal contact problem in a piezoelectric strip under the coupling actions of a rigid punch and a convective heat supply

A thermo-electro-mechanical contact analysis has been performed for a finite piezoelectric strip, which is subjected to the joint actions of a rigid, flat punch and a transient convective heat supply. The Laplace transform and Fourier sine and cosine transforms were applied in solving the governing equations. A detailed analysis of the characteristic roots of the corresponding characteristic equation was made. Real fundamental solutions were derived, which can readily lead to real solutions to the thermo-electro-mechanical quantities. A Cauchy-type singular integral equation was obtained for the stated problem and then solved numerically. Closed form solutions of a special case were obtained. To obtain the accurate solution in the time domain, an effective numerical inversion algorithm of the Laplace transform was applied. Detailed analyses were performed to reveal the variation law of temperature, contact stress beneath the punch, stress intensity factor at the punch edge and strain with time. Parametric studies were performed to discover the effects of the layer thickness on the distribution of temperature, contact stress beneath the punch and stress intensity factor at the punch edge.     

A thermoelastic diffusion interaction in an infinitely long annular cylinder

The present paper is aimed at studying a two-dimensional problem for an infinitely long solid conducting circular cylinder with a permeating substance in contact with its bounding surface. The problem is considered in the context of generalized thermoelastic diffusion theory with one relaxation time. The lateral surface of the solid is traction free and subjected to known temperature and chemical potential as functions of time. The solution is obtained by a transform method and a direct approach without the customary use of potential functions. Numerical inversion of the transformed solution is carried out to obtain the temperature, displacement, stress, and concentration of the diffusive material distributions. Numerical results are represented graphically and discussed. The second sound effect and the asymptotic behavior for the solution are discussed.     

Smooth static and dynamic indentation of a cantilever beam

The static and dynamic indentation of structural elements such as beams and plates continue to be intriguing problems, especially for scenarios where large area contacts are expected to occur. Standard methods of indentation analyses use a beam theory solution to obtain an overall load–displacement relationship and then a Hertzian contact solution to calculate local stresses under the indenter. However, these techniques are only applicable in a fairly limited class of problems: the stress distribution in the contact region will differ significantly from a Hertzian one when the contact length exceeds the thickness of the beam. The indentation models developed herein are improvements over existing GLOBAL/LOCAL models for static and dynamic indentation of cantilever beams. Maximum contact stresses, beam displacements, and contact force time histories are obtained and compared with the predictions of current static and dynamic indentation models. The validity of the solutions presented herein is further assessed by comparing the results obtained to the predictions of modified beam theory solutions.     

On deforming of a linear viscoelastic orthotropic half-space under the turning rigid cylindrical roller

In this paper we consider the problem of rigid cylinder turning on a linear viscoelastic orthotropic half-space with Coulomb's friction acting along the contact area. Results for extents of contact area and pressure under the cylinder are obtained using Volterra's principle. The obtained functions of viscoelastic operators are interpreted by a method based on expansion of such functions in operator continued fractions. A solution is given for the general type of resolvent viscoelastic operators expressing rheological properties of half-space material. Algebra of resolvent Volterrian operators is used to facilitate the calculations. An example is given to illustrate the results for real viscoelastic material with the rheological properties expressed by the operators of Yu.N. Rabotnov.     

高温下钢管自密实混凝土界面接触热阻试验研究

钢管混凝土结构的火灾性能研究通常要考虑钢管和混凝土的界面接触热阻问题,本文对钢管自密实混凝土柱的界面接触热阻进行了试验研究。制作了8个钢管自密实混凝土柱,依据Fourier定律和Newton冷却定律得到了钢管混凝土界面接触热阻的求解方法。利用有限元和多项式拟合外推得到界面温度,得出钢管混凝土接触热阻随着钢管界面温度的变化规律。试验结果表明,未受载的钢管自密实混凝土界面接触热阻平均值范围在0.002~0.01m2·K/W,与其他文献结果相比有一定可靠性。     

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.     

Parametric study of thermal constriction resistance

Thermal contact resistance is due to imperfect contact of two bodies at the interface. It plays an important role in the dissipation of heat from electronic devices. The concept of individual heat flux tubes consisting of a single contact area and the corresponding gap which extends far in either solid was used in this study. The three dimensional conduction equation in the contact region was solved numerically for different shapes of gap and contact area and various thermal boundary conditions at locations far from the contact area. Constriction resistance defined as the ratio of the temperature difference across the contact surface to the rate of heat transfer through a heat flux channel was calculated for each case. The results have indicated that constriction resistance is strongly affected by the gap geometry, shape of contact area and certain end surface boundary conditions. The geometry dependence becomes more significant as the ratio of contact to total area becomes smaller. Given the fact that the shape of the contact region is highly unpredictable, the heat flux tube approach can hardly provide a reasonable estimate of the thermal contact resistance, unless the geometry of the contact region is properly modeled. Received on 5 January 1999     

Influence of the temporal profile of the contact pressure on the maximal temperature in pad/disc tribosystem

The solution to a thermal problem of friction during braking for a pad/disc (strip/semi-space) tribosystem with a time-dependence upon friction power is obtained. The convective cooling on an upper surface of a strip and the heat transfer through a contact surface are considered, too. The influence of duration of increase in pressure from zero (at the initial moment of time) to nominal value (at the moment of a stop) on the maximal temperature for a friction pair metal-ceramic pad/cast iron disc is studied.     

Calculation of contact temperatures generated by the rotation of a shaft in a bearing

The problem of the distribution of contact stresses resulting from the interaction between a journal and its bearing was considered in [1]. This paper deals with the problem of temperature distribution in the area of contact of a rotating cylindrical shaft and a bearing. The process is assumed to be stabilized.The problem reduces to an integral equation with respect to the contact temperature at the shaft surface.An approximate method is proposed for solving the integral equation which had permitted the derivation of a simple approximate formula for the contact temperature within any range of variation of the parameters of this problem.     

Effect of friction in wedging of elastic solids

In this paper the contact problem for an elastic wedge of arbitrary angle is considered. It is assumed that the external load is applied to the medium through a rigid wedge and the coefficient of friction between the loading wedge and the elastic solid is constant. The problem is reduced to a singular integral equation of the second kind with the contact pressure as the unknown function. An effective numerical solution of the integral equation is described and the results of three examples are presented. The comparison of these results with those obtained from the frictionless wedge problem indicates that generally friction has the tendency of reducing the peak values of the stress intensity factors calculated at the wedge apex and at the end points of the contact area.This work was supported by NASA-Laugley under the Grant NGR 39-007-011 and by NSF under the Grant GK-42771X.     

Slow nonstationary vertical motions of a die on the surface of an elastic half-space

We consider the dynamic contact problem on vertical motions of an absolutely rigid body on an elastic half-space. We assume that the contact region does not vary during the motion and there is no friction under the die bottom. We construct an approximate solution of the problem under the assumption that the variation in the contact pressure under the die bottom on the time interval in which the Rayleigh wave runs the distance equal to the contact area diameter is small. Computational formulas are obtained for the cases of circular and elliptic dies.     

Temperature distribution in a liquid spherical layer around a sphere falling under gravity

An examination is made of the stationary problem of temperature distribution in a liquid of known mass covering a solid sphere falling under gravity, the temperatures of both media with which the liquid is in contact being given. A solution has been obtained under the assumption of constant coefficient of thermal expansion of the liquid. A number of particular cases is examined, the question of stability of the liquid under the temperature distribution obtained is solved, and the total heat content of the liquid is found.