Abstract. 1 Introduction

Transcription

1 Elasto-plastic contact of rough surfaces K. Willner Institute A of Mechanics, University of Stuttgart, D Stuttgart, Germany willner@mecha. uni-stuttgart. de Abstract If two rough surfaces are brought into contact, asperities will deform either elastically or plastically. While the mode of initial deformation will depend on geometrical and material properties at any single asperity, the deformation upon unloading and reloading will be in the elastic regime. As it is not feasible to model the behaviour of each single asperity in a structural mechanics application, global contact laws are needed. In the present paper such contact laws are derived for several deformation models. 1 Introduction The derivation of contact laws for rough surfaces is a two step task. First, a description of the rough surface is needed. Possible approaches are statistical [3, 4, 8] or fractal models [7, 9]. The second step is the description of the mechanical behaviour of single summits. Early investigations assumed either a rigid, perfectly plastic or a pure elastic contact. These simplified models werefirstextended by GREENWOOD and WiLLiAMSON[4], who introduced the plasticity index as a measure of the elastic fraction. Since no analytical solution to the elasto-plastic contact problem is known, approximate models like the cavity model by JOHNSON [5] and the volume conservation model by CHANG ET AL. [1] are necessary. More recently, WARREN and KRAJCINOVIC [9] used a volume conservation model in conjunction with a fractal surface description. Another approach was taken by KUCHARSKI ET AL. [6] who used afiniteelement model to determine the asperity behaviour. Introducing the single asperity model into the surface description leads to the desired global law. In the following a statistical model is used and frictionless contact is assumed.

2 14 Contact Mechanics III 2 Surface model GREENWOOD[S] introduced a statistical model of a rough surface using finite difference approximations of the slope and curvature of summits. Figure 1: Finite differences grid, slopes On a regular grid with even spacing h 4 the heights z with respect to a refer- ence plane can be measured (Fig. 1). 1 Assuming a Gaussian distribution of heights the reference plane is chosen such that a mean value of zero is obtained. Now, slopes and curvatures at a point 0 can be approximated by fidifferences as - 23 curvatures mean curvatures k --(k^ -f ky) *, = (2) = --(&z ky). (3) As these values depend strongly on the spacing /i, the influence of h will be discussed in some detail later on. The conditions that 0 is a summit are (4) - (5) Assuming Gaussian distribution of each of the variables the joint probability distribution of a set of random variables Zi is given by where M^ is the inverse of the co-variance matrix dj and A is the determinant of Cij. Here the Gaussian random variables are z,sx,sy,k,kd as introduced above. No further details on the probability function will be given, which can be found elsewhere [3]. After some lengthy calculations a joint probability distribution P(z,k) can be obtained. This function gives the probability of a summit with given height z and given curvature &, depending on the standard deviation of heights c^, the standard deviation of curvature a^ and a mixed term o^k- (6)

3 Contact Mechanics III 15 3 Contact laws To obtain a global contact law the local relations have to be distributed on the apparent area of contact AQ. This is done by the surface model. Since the resolution of surface measurement is given by ft, the total number of points is no = Ao/h?. With the probability P(z,k) of a summit with curvature k and height z, the number of all summits above a given height g can be calculated as n(2) = no /V P(z, &) d& dz. (7) 9 0 Introducing the normal force of a single summit Ni(z, k) into the probability distribution leads to the apparent normal pressure upon contact with a plane at a distance g oo oo 9 0 In the same way the real area of contact can be determined using the area of contact of a single summit Ai(z, k) Ar(9) oo oo = l/y^(z,6)p(z,6)d6dz. (9) 9 o If both surfaces are rough, the contact can be described as an equivalent rough surface in contact with a plane using equivalent parameters,6,2 <?t = ^,i + <2 (10) The next step is to choose the deformation model 7V% and A% of a single summit in contact. The material and geometric parameters of a summit are described by the Hertz modulus and the equivalent curvature and k k\ -\- 62, (11) respectively. Additionally, if elasto-plastic behaviour is considered, an elastic, perfectly-plastic material with a yield stress of &y is assumed.

4 16 Contact Mechanics III l/k elastic 3.1 Elastic contact Figure 2: Cavity model The elastic deformation mode of a single summit is described by Hertz theory. Given the compression d = z - g of a summit with initial height z, the normal force N and contact radius a are and (12) The limit of elastic deformation is given by the Tresca yield condition 2TVnax > <V The maximum shear stress of an elastic contact occurs below the surface and its value is about Tmax 0.31po [5]. With po = fj the compression atfirstyield can be calculated as, 2 4 = G.42^g (13) 3.2 Cavity model Johnson [5] developed a model for the indentation of an elastic, perfectlyplastic half space by a rigid sphere. If the compression exceeds the limiting value dy, it is assumed that an incompressible core with radius a develops and the stress state within the core has a hydrostatic pressure component p. Outside the core it is assumed that the stresses and displacements have radial symmetry and are the same as in an infinite elastic, perfectly-plastic body which contains a cavity under a pressure p. Volume conservation of the core requires 27ra*du(a) = -KO?dd =?ra^ tan /3da, (14)

5 Contact Mechanics HI 17 with tan/3 ak for a spherical indentor. Integrating the second part of (14), with d dy and a = dy/k at the start of integration, leads to <»> The radial displacement at the boundary between core and plastic zone assuming incompressible material is Using the first part of (14) and integrating again leads to 4 (Jy for a > -\fdyk. Equilibrium requires that p = Or at r = a, with the radial stress component 0y given by Using eq. (17) leads finally to - (18) a 3 As the mean contact pressure at the onset of yielding is about pm = l.lo\/, an approximation of Pm is given by 1 ^0.71. (20) j The force-displacement relation is given by Ni(d) = Pm^c? with a(d) given by (15). The limit of elasto-plastic deformation is reached at a mean pressure of Pm ~ 3<7y, the limiting value given by rigid, perfectly plastic theory. The corresponding indentation is dp w 160^. In the fully plastic range d > dp the normal force and contact radius are simply and o(d) % 2. (21)

6 18 Contact Mechanics III 2a Figure 3: Volume conservation model 3.3 Volume conservation model CHANG ET AL. [1] presented an alternative model. They use a control space with constant volume after the yield limit is reached. The geometry of the control volume is given by L2 _ "" k ' (22) with an open constant C. Conservation of volume requires that From this equation the radius of contact a is obtained as (24) As an upper limit the fully plastic state requires a? «2^ and therefore (25) Choosing a sufficient large C, equation (24) simplifies to &\ which is the same relationship as for the cavity model (15). d (26)

7 Contact Mechanics HI 19 Assuming the mean contact pressure in the elasto-plastic range to be equal to the maximum elastic pressure at the yield limit leads to Pm =PQ,y «1.61(7%, (27) Comparing with equation (12) shows that at the transition from elastic to elasto-plastic contact, d = dy, the normal force is discontinuous. This discrepancy was removed by EVSEEV ET AL. [2], who assumed the contact area to consist of an plastic part with constant pressure Pm = Po,y and a surrounding annulus with a Hertzian pressure distribution. Nevertheless, CHANG ET AL.[1] pointed out that for a surface with many summits in contact equation (27) gives a good approximation in sense of a mean value. This is true as long as only a few summits are in the plastic range. At higher loadings eq. (27) will be an estimate significantly to low, as more summits approach the limit value Pm 3cjy. 3.4 Simple Elastic-plastic model A much simpler model of deformation is obtained by ignoring the elastoplastic transition and to switch directly from elastic (12) to fully plastic contact (21) at d > dy. Although this deformation model is discontinuous at the point of first yield, it will be a fairly good approximation, if a rough surface with many plastically deforming asperities is considered. As only very few asperities will be in the transition phase, only a slight overestimation of the mean apparent contact pressure will occur. 3.5 Elastic unloading Upon unloading any deformation will be purely elastic. If the contact was loaded over the plastic limit a decrease in height and an increase in curvature occurred. The new values can be calculated by elastic theory. Given the contact radius amax and normal force Nmax at the maximum compression dmax during any loading, the new curvature k' is given by ^ (29) With the elastic restitution d' = a^k' the effective compression d^ is d^=d-(dm_-d'), (30) with dmax ~ d' being the plastic deformation of the summit. The force and contact radius upon unloading are given by eq.(12), if k' and d<,g are substituted.

8 20 Contact Mechanics HI 4 Determination of surface parameters The necessary parameters of the surface model have to be determined by measuring the height distribution of a real surface and the grid space h becomes a crucial parameter. In the following example the same surface is measured with two different values of h and the resulting rms values of height GZ and curvature cr& are compared: , Also this example is rather crude, the main effect is clearly seen. The grid spacing influences mainly the curvature, while the heights are nearly unaffected. So which values should be used? As continuum mechanics has been used to derive the single summit models, each summit should be treated as a continuum and effects resulting from anisotropy of single grains etc. should be homogenised be considering a sufficiently large wavelength. Assuming a grain size of about 1/xm, the shortest meaningful wavelength is about 10 //ra. The maximum wavelength is given the sampling interval / 1cm. Using these data as a guideline, the smallest possible grid space h = 0.25^ra was used and the resulting spectrum was filtered accordingly. This leads to values of a^ 3.92^ra, a^ = 0.12 and a*. = 0.031//^ra, with the significantly lower value of 0% resulting from the filtering of long wavelength. 5 Results As Fig.4 shows, all of the elasto-plastic models give the same fraction of real area of contact, which is about twice the area of the elastic theory as expected. As the models assume that no interaction between the summits occurs, the limit of validity is given at a value of Ar/A$ «0.1 and a gap distance ofg/&z ~ 1.5, respectively. The corresponding pressure relation is shown in Fig.5. While the simple elastic-plastic model and the cavity model gives nearly the same result, the volume conservation model predicts lower pressures at the same gap distance. The former observation indicates that here the transition state between pure elastic and fully plastic has no influence. This is no surprise as the surface parameters indicate a rather rough surface, which will deform mainly plastically on initial contact. The latter observation is due to the fact that the volume conservation model assumes a maximum pressure significantly to low, as discussed in section 3.3. Using the higher value of Pm, = 3<r/,it would give the same result as the simple elastic-plastic model.

9 Contact Mechanics III 21 elastic model 0- elastic-plastic model * cavity model + volume conservation model I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I Figure 4: Fraction of real area of contact vs. normalised gap distance 4-y elastic model o elastic-plastic model 3- * - + cavity model volume conservation model I 0.1 I 0.2 I Figure 5: Normalised apparent pressure vs. normalised gap distance 4-T Figure 6: Normalised apparent pressure vs. normalised gap distance, unloading of cavity model

10 22 Contact Mechanics HI Fig.6 shows the unloading relations for different maximum loadings. Only the cavity model is shown as the modified volume conservation model [2] will lead to a similar result, while the standard volume conservation model and the simple elastic-plastic model will give acceptable results only as long as the majority of summits is deformed into the fully plastic state on initial contact. For mainly elastically deforming surfaces a lot of summits will be in the transition phase where both models are discontinuous, as discussed in section 3.3 and 3.4. Here, only little restitution occurs, again indicating a rather rough surface with only a small elastic fraction. Therefore, one can conclude that the contact stiffness during unloading and reloading of initially plastically deformed surfaces is much higher than those which deform mainly elastically. References [1] Chang, W.R., Etsion,!. & Bogy, D.P. An elastic-plastic model for the contact of rough surfaces, Journal of Tribology, 1987, 109, [2] Evseev, D.G., Medvedev, B.M. & Grigoriyan, G.G. Modification of the elastic-plastic model for the contact of rough surfaces Wear, 1991, [3] Greenwood, J.A. A unified theory of surface roughness, Proc. R. Soc. Lond. A, 1984, 393, [4] Greenwood, J.A. & Williamson,J.B.P. Contact of nominallyflatsurfaces, Proc. R. Soc. Lond. A, 1966, 295, [5] Johnson, K.L. Contact Mechanics, Cambridge University Press, [6] Kucharski, S., Klimczak, T., Polijaniuk, A.& Kaczmarek J. Finiteelements model for the contact of rough surfaces, Wear, 1994, [7] Majumdar,A. & Bhushan, B. Role of fractal geometry in roughness characterization and contact mechanics of surfaces, Journal of Tribology, 1990, 112, [8] Nayak, P.R. Random process model of rough surfaces, Journal of Lubrication Technology, 1971, 93, [9] Warren,T.L. & Krajcinovic, D. Random cantor set models for the elastic-perfectly plastic contact of rough surfaces, Wear, 1996, 196, 1-15.