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y^K 539.62
Dependence of the kinetic force of friction between a randomly rough surface and simple elastomer on the normal force
Q. Li
Berlin Technical University, Berlin, D-10623, Germany
The force of friction between a self-affine fractal rough surface and an elastomer with the simplest linear rheology is simulated with the method of reduction of dimensionality. The coefficient of friction increases with normal force approximately according to logarithmic law.
Keywords: method of reduction of dimensionality, friction, roughness, elastomer
1. Introduction
Since classical works by Bowden and Tabor [1], it is widely accepted that the roughness plays a central role in friction processes. In particular, roughness is one of the main reasons for the validity of Coulomb’s law of friction: the frictional force is proportional to the normal force and is — in first approximation — independent from the apparent contact area and sliding velocity [2]. However, it is well known that this law is only a very rough empirical law and that the friction coefficient, even between the same material pairing, can change by a factor of about four depending on the geometry of the tribological system as a whole and loading conditions [3-5]. In the present paper, friction between a randomly rough, self-affine rigid body (for definitions, see [6]) and an elastomer with the simplest linear rheo-logical material behavior equivalent to that of a linearly viscous fluid (however, with slipping boundary conditions) is considered. Complete three-dimensional simulations, even for a case as simple as this, do not exist until now. To investigate this problem, we use the so-called method of reduction of dimensionality first proposed in [7] and developed for randomly rough surfaces in [8, 9]. Recently, Hess [10] has rigorously proven some theorems of the method of reduction of dimensionality and Pohrt and Popov [11] have studied the contact stiffness of randomly rough surfaces and have shown the equivalence of results of the true three-dimensional calculations with those obtained with the method of reduction of dimensionality [12]. The method of reduction of dimensionality has already been applied to the in-
vestigation of the friction between a rigid body and an elastomer, but the normal force acted on the surface was in the medium range, where the friction coefficient was approximately considered to be constant [13, 14]. In the present paper we study the friction coefficient in a wider range of forces using the method of reduction of dimensionality. The results show that the friction coefficient increases with normal force.
2. Model
In the method of reduction of dimensionality [8], a threedimensional rough profile with a spectral density C2D (q) is replaced by a one-dimensional “rough line” with the spectral density C1D(q) = nqC2D(q) in contact with viscoelastic foundation consisting of springs with the normal stiffness
kz = E A x. (1)
Here, E is the effective elastic modulus
E* = E/ (1 -v2), (2)
E is the Young’s modulus, v is the Poisson ratio, and Ax is the discretization step of the foundation. The profile of the rough line is generated according to the rule qi
z( x) = X Bid (q) exP(X qx + q))) (3)
qo
f2n —
with B1D(q) = . — C1D(q) = B1D(-q), where L is the
length of the system and §(q) is the phase distributed ran-
© Li Q., 2012
domly on the interval [0, 2n]. Many technically important surfaces are self-affine fractal surfaces [6]. Their power spectra can be written as C2D( q) = const • q-2 H -2, where H is the Hurst exponent, which can vary from 0 to 1 (in this paper H will be from 0.7 to 1, which corresponds to most practical cases). It follows that C1D(q) = const • q—H-1.
In the present paper we would like to consider friction with an elastomer having the simplest rheological law — that of a linearly viscous fluid with viscosity n. According to the ideas of Radok [15], the contact with an elastomer can be considered as equivalent to that with an elastic continuum provided the corresponding force operators are used instead of elastic constants. This idea has been used in [2] to formulate and generalize the method of reduction of dimensionality to viscoelastic bodies. The force rule for each “spring” in the foundation must be defined as
fn( Xi) = 4nA xz(Xi). (4)
Thus, we have a “viscous foundation” consisting of a series of “dampers” (Fig. 1).
The system is initiated by bringing the rough line and the fluid first into contact. Then, under the normal force Fn, the rough line starts to ‘sink’ while it is driven to move in the direction of the x axis at velocity v. At every step the contact condition and the force balance are checked so that the contact points are determined and the equilibrium equation Fn = ^ f (xcont) is satisfied. Meanwhile, the tangential forces fx (xcont) in the contact range are determined by fn (xcont), multiplying with the slope of rough line at the contact points. The calculation is performed until stationary sliding is achieved. This means that the coefficient of friction approaches a constant.
3. Results
In the simulation, the lines were generated with 105 and 106 points. All values shown below were obtained by averaging of 100 realizations for each normal force and Hurst exponent. For interpretation of results, we used the following analytical considerations. It is shown in [2] that the order of magnitude of the coefficient of friction u should be approximately equal to the rms value of the slope of the
Fig. 1. One-dimensional “damper” model for contact between a rough surface and a simple elastomer
surface profile Vz in this case. It is, therefore, sensible to use the quantity Z = u/Vz instead of u. At the given contact configuration, the normal force is strictly proportional to n and v. If we assume that the only further surface parameter determining the force of friction is the rms value of the height distribution h, then we can only calculate the following dimensioneless quantity proportional to the force and having the necessary scaling properties f = Fn / (n v h).
The dependences of the value Z on the dimensionless force for the Hurst exponents in the range of 0.7 to 1.0 are shown in Fig. 2(a). As shown, the coefficient Z increases with the dimensionless force. The shapes of curves for different Hurst exponents are similar and the values are also close at the same dimensionless force. Figure 2(6) shows the fitted curves of the dependence of the coefficient of friction on the normal force at the small force range for H= = 0.7, where the result is also plotted with more points (N= = 106). We can see that in the range of small forces, Z increases linearly with the natural logarithm off According to the fitting, the functions of the dependences are the following:
Fn
-^ = 0.22ln
Vz
_y_
Vz
= 0.22ln
nv h
nv h
+ 1.48 for N = 105
+1.79 for N = 106
(5)
(6)
One can see easily that the frictional force depends approximately logarithmically on the normal force. The result that the frictional force depends on the length of the system means that the hypothesis about the dependence of
1.6- | a 1.6 Lb Ay A
1.2 - 1.2 - y*' P A X X X 9,'
i i i CO d d ZA/rl s/a/ —+—H = 0.7 i i i CO d d ZA/rl yy* ^ y'A y' O N = 105 x' ° A N = 106
- —£*— H = 0.9 A ys6 N = 105
o.o- ■ ■■■I 0.0 - V' N = 106
10-
10~2 10~1 F n/(r|vh)
10°
101
IQ-
10-2 10~1 F n/(r|vh)
10°
Fig. 2. Dependence of Z on the dimensionless force with N = 105 points for a high Hurst exponent (a) and with N = 105 and N = 106 for H = 0.7 (b)
the coefficient of friction only on the dimensionless variable f = Fn/(nvh) is not correct. The dependence on the system length will be studied and reported in another paper.
In conclusion, the method of reduction of dimensionality was used effectively to simulate the contact process between a rough surface and an elastomer. The result obtained shows that Z is not constant but increases logarithmically with the normal force.
The author acknowledges valuable discussions and critical comments by V.L. Popov, R. Pohrt and A.E. Filippov.
References
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Поступила в редакцию 10.06.2012 г.
Сведения об авторе
Li Qiang, MSc, Berlin Technical University, Germany, llqq0108@mailbox.tu-berlin.de
