ON THE EFFECT OF SPECTRAL CHARACTERISTICS OF ROUGHNESS ON CONTACT PRESSURE DISTIRBUTION

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1 7 Paper present at International Conference on Diagnosis and Prediction in Mechanical Engineering Systems (DIPRE 09) October 2009, Galati, Romania ON THE EFFECT OF SPECTRAL CHARACTERISTICS OF ROUGHNESS ON CONTACT PRESSURE DISTIRBUTION Spiridon CREŢU, Mihaela Rodica BĂLAN, Ana URZICĂ Technical University Gheorghe Asachi of Iasi, ROMANIA ABSTRACT In the first part of the paper, the matrix containing the roughness heights is obtained by a linear transformation of the Gaussian input matrix, the coefficients of the transformation matrix being found as solutions of a non-linear system of algebraic equations in which the free terms are the components of the desired autocorrelation function. In the second part of the paper, a fast computer code is involved to solve the elastic-plastic contact between rough surfaces. The subsurface stress fields were compared using the von Mises equivalent stress and the role played by the autocorrelation length on both pressure distribution and stress state has been pointed out. KEYWORDS: Autocorrelation length, elastic-perfect plastic analysis, von Mises equivalent stress.. INTRODUCTION Every elastic model for the contact of rough surfaces presents very large local asperity contact stresses, able to exceed the yielding limit. The roughness acts as stress concentration sites and induces stresses greater than in an equivalent smooth contact. The real areas of contact and the asperity contact pressures are essential parameters for any wear modeling. These parameters can vary significantly depending on surface topography. A small change in the distribution of heights, wave length and curvature of the surface roughness can have a noticeable effect on the deformation behaviors of the rough surfaces. On machined surfaces the waviness and roughness have comparable magnitudes, typically between 0. and 0 µm, but the corresponding wavelength may differ by two or three orders of magnitude, so that a typical diameter of the roughness bases is 0 to 00 times larger than their heights, Bushan [2], Greenwood [7], Patir [9], Thomas [2]. Greenwood and Wu [7] have shown that the contact is governed by the geometry of the microasperity only in the very early stages. When the asperity has been compressed to the final stage, the microasperity may possible have some influence on the actual area of contact, but the nature of contact will depend on the large scale geometry of the asperity. Any parametric study involving roughness requires surfaces with known statistical proprieties and it is much more convenient to generate them numerically rather to measure manufactured rough surfaces. An essential requirement for any numerical algorithms for roughness simulation is their abilities to generate rough surface which have statistical properties similar to real surfaces. Most of the statistical proprieties of a rough surface can be derived from knowledge of two statistical functions, the frequency density function and the autocorrelation function, Bakolas [], Robbe- Valloire [0], Sayles []. Consequently, a good algorithm should be able to generate surfaces having prescribed frequency density functions and autocorrelation functions.

2 8 2. ROUGHNESS PARAMETERS 2.. Probability Density Function If for convenience z was measured from the mean plane of the surface, then the height z(x, y) of a rough surface may be considered as a twodimensional random variable. The spatial characterristics can be adequately described with the use of probability function p(z) which denotes the probability that a point on the surface has a height equal to z. Most of the parameters used to describe the microtopography of a rough surface are related to the probability distribution function. It has been found, Bakolas [], Bhushan [2], Patir [9], that many real surfaces, notably freshly grounded surfaces, reveal a height distribution which is close to the normal Gaussian probability function: σ 2 2 p( z ) = exp( z / ( 2 σ )) () 2π where σ is the standard (R.M.S.) deviation from the mean height. The shape of the probability function can give useful information about the nature of the roughness profile. A mathematical presentation of this shape is provided by the moments of the probability density function about the mean. The moment of order k is denoted by m k, while the central moment of order k and is denoted by c k m, and are given by the equations: k mk = ( z p(z) ) dz, c m ( ) k k = z m p(z)dz (2) c m = m, m = 0, Ra = z m p(z) dz. (3) 2 2 m2 = ( z p(z) ) dz = Rq ; m c ( ) 2 2 = z m p(z) dz = σ 2 = Rq 2 m 2 (4) 3 z m) ( m 3 = Sk = p{ z} dz (5) 3 σ m ( ) 4 4 = K = z m p(z) dz 4 σ (6) 2 The second moment is the variance R q of the roughness heights, meaning the standard deviation R, or the root mean square (R.M.S.) σ, of the q surface heights. The third normalized central moment is called skewness and represents a measure of the symmetry of the statistical distribution. Symmetrical distributions have skewness equal to 0, which means that they have evenly distributed peaks and valleys of specific height. The fourth normalized central moment is called kurtosis and represents the spikiness of the statistical distribution being a measure of the degree of pointdness or bluntness. Symmetric Gauss distribution has a kurtosis of The Autocorrelation Function (ACF) The autocorrelation function R(x, y) is the expected value of the product z( x, y ) z( x + λx, y + λy ) (7) of the surface height at position (x, y) and at the position (x + λx,y + λy ), where λx, λ y are the delay length. Under the stationarity assumption this expectation is independent of (x, y): λ λ = + λ + λ, R( x, y ) E z( x, y ) z( x x, y y ) 2 R( 0,0 ) = σ (8) The ACF provides information about the spectral proprieties of a rough surface in the same sense that the probability density function describes its amplitude statistical proprieties. It has been found that for most of the manufacturing processes the ACF follows a negative exponential function. In this assumption the necessary variables to prescribe an ACF are the decay length in two directions that are perpendicular to each other. These lengths are known as the autocorrelation * * lengths, λx, λ y and are widely defined as the length in the x and y directions where the autocorrelation length drops to 0% from its original value. Using the values of the autocorrelation lengths, the ellipticity ratio γ is defined as a measure of the shape of the asperity heights, as well as of the * * anisotropy of the rough surface: γ = λx / λy. The profile of a random surface is regarded as a random signal represented by a height distribution and an autocorrelation function [2]. It is proved that all features of a surface with Gaussian distribution of heights and a negative exponential autocorrelation function could be represented by two parameters: σ and λ *. 4. RANDOM ROUGHNESS SIMULATION To generate a rough surface having a Gaussian density distribution function and a prescribed autocorrelation function, the following steps should be carried out:. Develop the desired autocorrelation matrix. 2. Generate a standardized Gaussian random sequence with zero mean and unit variance. 3. Solve the non-linear system of equations to obtain the coefficients of the transformation matrix. 4. Transform the random sequence to the input sequence.

3 9 5. Transform the output sequence to roughness heights. 3.. Autocorrelation Matrix If the sampling intervals in the x and y directions were denoted by x, y then the coordinates x and y of a current point on the mean plane can be written as: x = i x,y = j y. The roughness amplitude in the considered point is denoted z ij and the autocorrelation function is defined as: Rpq = R( p x,q y) = E zij zi+ p,j+ q (9) Let n and m be two integers such that: R pq =0 if p n or q m that yields to an (n x m) autocorrelation matrix. Defining the correlation lengths as the length at which the autocorrelation function equals zero, the corresponding correlation lengths of the x and y profiles are: 0 x = n x, 0 y = m y. λ λ The relevant parameters are chosen to be: λ * ( ) x = n x, λ * ( ) y = m y, σ =, (0) The amplitude of the generated roughness will be comparable with that of a real surface measured x = / n and at sampled intervals ( ) y = / ( m ) of the respective 0.0 correlation length. In order to obtain a finite order autocorrelation matrix, the exponential autocorrelation function is assumed to drop to zero after 0.0 correlation length, and consequently the numeric form of the autocorrelation function is: 2 2 / 2 p q Rpq = exp n m if < (a) p < n and q m; Rpq = 0 if p 0 or q 0. (b) 3.2. Transformation Matrix The first step is to generate a ((N+n)x(M+m)) matrix η ij whose components are identically distributed Gaussian numbers with zero mean, (m=0) and unit standard deviation, (σ=). Patir N. [9] and later Bakolas V. [] have shown that by using a linear transformation on random matrices it is possible to generate a (NxM) matrix of roughness amplitudes [z ij ] that has a Gaussian distribution of heights with an imposed (n x m) autocorrelation matrix: n m i =,2,...,N; zij = ( akl η i + k, j + l ) (2) j =,2,...,M k= l= where a kl are the coefficients to be determined in order to provide the needed autocorrelation function. Since η ij are independent and have unit variance, the following equations are valid:, if i = k, j = l E ( ηη ij kl ) = (3) 0, if i k, j l Using these equations along with the definition of the autocorrelation function, the matrix representation of the ACF is obtained as: n p m q p = 0,,2..., n Rpq = ( akl ak+ p,l+ q), q = 0,,2,..., m k= l= (4) Equation (4) represents a system of ( n m) non-linear equations with the coefficients a ij as unknowns. The iterative technique of Non-Linear Conjugate Gradients Method has been used to solve the system (4), Cretu S. [4, 6] The Roughness Matrix If the solutions of the system (4) were obtained, they are used further in system (2) to provide the roughness heights z ij with the desired autocorrelation function. Choosing a unit value for the standard deviation, (σ=) will produce normalized roughness amplitudes. If we needed to obtain roughness with a certain value for σ, all we have to do is to multiply the roughness amplitudes by the prescribed value of the standard deviation. To compare the autocorrelation function (ACF) of the generated surfaces with the expected ACF, the x and y profile ACF_s of the generated surfaces are calculated by using the definition equations: N p M R p0 = E zij zi+ p, j = ( ) zij z (5) i+ p, j ( N p) M i= j= N M ( ) q R0q = E zij zi, j + q = N M q i= j = ( z ) ij zi, j + q (6) Two cases have been considered which correspond to 0% autocorrelation lengths λ x =7, λ y =4, and respectively to λ x = λ y =3. In figure and figure 2 are presented as follows: - the 3-D representation of the independent identically distributed Gaussian random numbers, (figures a and 2a); - the 3-D representation of the transformed random matrix numbers to have the desired ACF, (figures b and 2b) ; - the ACF of the generated surfaces versus the desired negative exponential ACF, (figures c and 2c). Since the autocorrelation function is fundamentally a random function, one cannot expect that the

4 0 autocorrelation function of every generated surface to be identical. Typical 2-D profiles of the rough surfaces corresponding to the input matrices, as well as for generated surfaces, are presented in figure CONTACT MODELLING 4.. Contact Geometry Two identical steel spheres loaded with a normal force Q=5000 N have been considered, the corresponding Hertzian values being: - the maximum pressure, σ 0 = 2724N ; - the radius of the circular contact area, a=0.936 mm. The previous 3D roughness, with zero mean and σ=0.4 µm, has been superimposed on the surface of one sphere, the mate surface being maintained smooth, figure Elastic-Plastic Contact Solver The algorithm developed for pressure distribution considers an elastic-perfect plastic behavior of the material, so that the maximum values of the contact pressure are limited to py = 3σ Y, where σ Y is the yielding limit of the material subjected to pure traction. A discrete formulation of the elastic-perfect plastic asperity deformation has been involved to obtain the pressure distribution and real contact area, Creţu [3, 5]. The subsurface elastic stresses state was further solved as a Neumann type problem, Creţu [3]. 5. RESULTS Both the contact areas and pressure distributions obtained with the generated roughness having the 0% autocorrelation lengths λ x =7, λ y =4, are presented in figure 5. The results obtained in similar conditions but with the value of the 0% autocorrelation length modified to λ x = λ y =3, are presented in figure 6. The corresponding stresses distributions are presented in figure 7 and figure 8, respectively. In the detailed pictures only the von Mises stresses greater than 0.6 have been maintained. The results show near-surface stresses much greater than those due to smooth (Hertzian) contact which means that this roughness gives rise to stationary concentrations of stresses near the surface. Even when the statistical values of the roughness amplitude were the same, the changing of the spectral characteristics, in the studied case the autocorrelation length, were able to induce severe changes of the stress state in the vicinity of the contact surfaces. 7. CONCLUSIONS. The real areas of contact and the asperity contact pressures are essential parameters for any wear modeling. Any parametric study involving roughness requires surfaces with known statistical proprieties and it is much more convenient to generate them numerically rather than to measure manufactured rough surfaces. An essential requirement for any numerical algorithms for roughness simulation is their abilities to generate rough surface which have statistical proprieties similar to real surfaces. 2. By using a random number generator, an input matrix is formed as a first representation of a Gaussian roughness with zero mean, (m=0), and unit standard deviation, (σ=). The autocorrelation function was assumed to have an exponential form. To fulfill this requirement, the matrix containing the roughness heights was obtained by a linear transformation of the input matrix. 3. A discrete formulation of the elastic-perfect plastic asperity deformation solver has been involved to obtain the pressure distribution in concentrated rough contacts. 4. The stress tensor components, obtained by convolution products, pointed out that roughness acts as stress concentration sites and induces stresses greater than in an equivalent smooth contact. 5. The change of the spectral characteristics of micro-topography proved to be able to induce severe changes of the stress state in the vicinity of the contact surfaces. REFERENCES. Bakolas V., 2003, Numerical Generation of Arbitrarily Oriented non-gaussian Three-Dimensional Rough Surfaces. Wear, 254, pp Bhushan B., 998, Contact Mechanics of Rough Surfaces in Tribology: Multiple Asperity Contact, Tribology Letters, 4, pp Cretu S.Sp., 2009, Contactul Concentrat Elastic-Plastic, Ed. Polytehnium, Iasi, Romania. 4. Cretu S. Sp, 2006, Random Simulation of Gaussian Rough Surfaces. Part. Theoretical Formulations, Bul. IPI, LII (LVI), -2, pp Cretu S.Sp., Antaluca E., 2003, The Study of Non-Hertzian Concentrated Contacts by a GC-DFFT Technique, The Annals of Dunarea de Jos University of Galati, Fascicle VIII, Tribology, pp Creţu S. Sp., 2006, The Correlation Length and the Stresses State in Elastic-Perfect Plastic Rough Contacts, Proc. of the STLE/ ASME IJTC_2006, paper 2399, Oct , San Antonio, TX, USA. 7. Greenwood J.A., Wu J.J., 2002, Surface Roughness and Contact: An Apology. Meccanica, vol. 36, pp Nelias D., Antaluca E., Boucly J., Cretu S.Sp., 2007, A 3D Semi-Analytical Model for Elastic-Plastic Sliding Contacts, Trans. ASME, J. of Tribology, 29, pp Patir N., 978, A Numerical Procedure for Random Generation of Rough Surfaces. Wear, pp Robbe-Valloire F., 2000, Statistical Analysis of Asperities on a Rough Surface. Wear, vol. 249, pp Sayles R. S., 966, Basic Principles of Rough Surface Contact Analysis Using Numerical Methods, Tribology International, vol. 29, no. 8, pp Thomas T.R., 982, Rough Surfaces, Longman, London.

5 FASCICLE VIII, 2009 (XV), ISSN Fig.. Autocorrelation functions (n=7, m=4, N=M=28). Fig. 2. Autocorrelation functions (n=3, m=3, N=M=28).

6 2 Fig. 3. Typical x-x profiles. Fig. 4. The separation between a rough and a smooth sphere.

7 3 Fig. 5. The pressure distribution and contact area for roughness with λx = 7, λy = 4. Fig. 6. The pressure distribution and contact area for roughness with λx = λy = 3.

8 4 Fig.7. The pressure distribution and von Mises stresses for roughness with λx = 7, λy = 4. Fig. 8. The pressure distribution and von Mises stresses for roughness with λ x = λ y = 3.