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y^K 539.62
On the dependence of the static friction force between a rigid, randomly rough fractal surface and a viscoelastic body on the normal force
J. Benad
Berlin University of Technology, Berlin, 10623, Germany
The present investigation is based on the assumption that the static coefficient of friction can be characterized roughly by the rms value of the surface gradient in the contact region. We numerically investigate the contact between a rigid, randomly rough surface and an elastic half-space and determine the dependence of the rms slope in the contact region on the normal force. For fractal surfaces with a Hurst exponent of H * 1, the rms slope can be approximated very well by a logarithmic function of the normal force. Parameters of the approximation have been determined as functions of the Hurst exponent. The rms value of the slope always increases with the normal force, which indicates that the coefficient of friction should be an increasing function of the normal force.
Keywords: coefficient of friction, surface roughness, contact stiffness, method of reduction of dimensionality
1. Introduction
Surface roughness plays an important role in many contact properties, such as contact stiffness, electrical conductivity, adhesion, friction and wear. The importance of surface roughness was first recognized by Bowden and Tabor [1]. The pioneering works on contact of rough surfaces were carried out by Archard [2] and Greenwood and Williamson [3]. In the last years the problem of contact of bodies with rough surfaces has once again become a hot topic [4-6]. The main attention has been paid to the real contact area and, quite recently, to the contact stiffness [7]. The real contact area is a very intuitively clear property. However, there are no macroscopic physical properties directly associated with this quantity. The “contact length”, roughly defined as a sum of diameters of the microcontacts, has more direct physical applications. Thus, the contact stiffness as well as electric and thermal conductance are associated with the contact length [8]. Another interesting geometrical property of real contacts, which has not been studied until now, is the rms value of the surface gradient. This property can be associated roughly with the coefficient of friction. Imagine, for example, a viscoelastic body which is characterized by some elasticity and viscosity. If a rigid rough body is pressed onto such a body and is held under this force for a long time, then the contact configuration will only depend on the elastic modulus of the medium. If the body is
now moved rapidly in the tangential direction, then it will react as a viscous body and the instantaneous coefficient of friction will be just of the order of magnitude of the rms value of the surface slope [8]. This consideration justifies our interest in the dependence of the rms slope in the real contact area on the normal force: it will give us, at least qualitatively, the dependence of the static friction force between a rigid body and an elastomer on the normal force. This question is of very high scientific and technical importance. As many technical surfaces of interest are self-affine fractal surfaces [5], we will investigate this class of rough surfaces in this work. The contact problem of fractal surfaces is very complicated and time consuming. This is the reason why we use the so-called method of reduction of dimensionality in this paper. This method has been proposed for normal contact problems in [9], and was extended to contact of rough surfaces in [10, 11]. In the meantime, many theorems of the method have been proven exactly for arbitrary bodies of revolution in [12] and have been applied to the simulation of friction [13]. The idea of the method is that the original contact of three-dimensional bodies is replaced by an equivalent contact of one-dimensional elastic foundations. In order to be equivalent to the three-dimensional problem, the stiffness of the single springs of the elastic foundation must be chosen as
c = E *A x, (1)
© Benad J., 2012
Fig. 1. Fractal line in contact with discretized elastic foundation
where E * =-
1 -v2
(2)
where E is the elastic modulus and v is the Poisson ratio.
2. Numerical model
We start with the generation of randomly rough lines with the Hurst exponent H, according to the rule [5, 8] z(x) = Z B1D( g)exp(i( qx + M q))), (3)
q
where 0(q) is a random phase,
1
B1D(q) = ^“^C1D(q) = BlD(-q) (4)
and C1D (q) is the one-dimensional spectral density of the form
C1D(q) = const • q ~2 H-1. (5)
Note that (z(x)) = 0. Summation in (3) is over the wave vectors in the interval
nii 1 n — < q <-,
L 11 10 Ax
(6)
where L = NAx is the size of the system and N is the total number of discretization points. The relation (6) means that there is no cut-off wave vector at the lower limit of the interval apart from the natural cut-off due to the finite size of the system. On the other side, at the upper limit there is a cutoff wave vector which guarantees that the line is smooth enough even at the smallest scale.
This profile is brought into contact with the above defined elastic foundation and pressed with the total normal force F (see Fig. 1).
f, 10-9
Fig. 2. Dependence of the dimensionless mean slope in contact on the dimensionless normal force. Curves are shown for N = 105, L = 105, H = 1, E = 1 and number of realizations t = 300. With 400 steps, F goes from 1 to 20. For F = 1, there are 4 points in contact on average. For F = 20, this number increases to 12. The simulation returns a = 1.8847 and b = 0.0805 for the natural logarithmic fit (gray line)
Fig. 3. Dependence and input values as shown in Fig. 2, however F goes from 1 to 500. For F = 500, the average number of points in contact is 40. The simulation returns a = 1.5289 and b = 0.0614 for the natural logarithmic fit (gray line)
After the penetration depth has been acquired numerically for a given normal force, the mean slope of the single line in places of contact can be calculated:
Vz =
zcont
z(xt+1) - z(xt) A x
(7)
only for points in contact.
The rms gradient found in this way was averaged over 300 realizations of the rough line with the same spectral density.
To characterize the surface, we further introduce the rms value of the height distribution
h = J (z (x)2) (8)
and the rms value of the surface gradient over the whole system
Vz = ^((dz/ dx )2Sj. (9)
3. Simulation results
We now search for the relation between Vzcont and F. There are several strict scaling relations which must be fulfilled independent of the particular form of the spectral density. If the rms value of the height distribution as well as the indentation depth is increased by some factor, both gradients (7) and (9) as well as the normal force will increase by the same factor. For the given contact configuration, the force must be proportional to the elastic modulus. We fulfill both requirements if we introduce the dimensionless variables
Fig. 4. Dependence and input values as shown in Fig. 2, H = 0.5. The simulation returns a = 2.6252 and b = 0.1341 for the natural logarithmic fit (gray line)
Fig. 5. Dependencies of the parameters a and b on the Hurst exponent H. Points are shown for N = 105, L = 105, E = 1 and t = 300. With 400 steps, F goes from 1 to 20 for each point
Vz
£ _ COnt
Vz : F
f-
(10)
(11)
E hL
and search for a function £(f).
Examples of the numerical simulation are shown in Figs. 2-4. For Hurst exponents of H * 1 they can be approximated very accurately by logarithmic function of the form
£ = a + b ln(f). (12)
For greater normal forces, one can also observe a fairly close fit (see Fig. 3). Another interesting relation might be the dependence of a and b on H (Fig. 5).
However, it seems that for smaller Hurst exponents, the natural logarithmic fit in the form of relation (12) does not fit the curve as well as for higher exponents. See Fig. 4 for a fit with H = 0.5.
4. Conclusion
The numerical experiment has shown that the dependence of the rms slope of the rough stochastically generated fractal line, in places of contact with the elastic foundation, on the normal force acting on this line is very likely to be given by a natural logarithmic relation of the form £ = a + b ln( f). Apparently, the best fits can be obtained for Hurst exponents near 1. For the most typical values of H * 0.7, the following relation has been derived:
£ = a +b ln(f) (13)
with a * 2.9 and b * 0.14. Thus, the dependence of the rms gradient value on the force can be written as
F
Vz r
—CO^ _ 2.9 + 0.14ln Vz
E hL
(14)
Increasing the force by a factor of 500 changes the rms gradient (and, therefore, the coefficient of friction) by a factor of 3 to 4. This is in accordance with the strong dependence of the coefficient of friction on the normal force observed in many experiments (see, e.g., [14]). According to the method of reduction of dimensionality, these results should be valid for three-dimensional systems as well, pro-
vided the length of the system is replaced by L = 2 V^/ Vrc, where A is the apparent area of contact:
Vz
Vz
= 2.9 + 0.14ln
УЛF
2E *h4A
(15)
The author is thankful to V.L. Popov for suggesting the topic of this investigation as well as for many valuable discussions and critical comments to the manuscript. The author further thanks A.E. Filippov, R. Pohrt and B. Grzemba for providing the initial version of the program generating fractal lines and support in the initial stages of this work.
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Поступила в редакцию 10.06.2012 г.
Сведения об авторе
Benad Justus, Berlin University of Technology, Germany, justusbenad@mailbox.tu-berlin.de
