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The present work is devoted to an analysis of the contribution to friction of different scales for one class of materials — elastomers with linear rheology. Since Greenwood and Tabor 3 , and especially after the classic work of Grosch 4 , it has been widely accepted that elastomer friction is mostly due to internal dissipative losses in the material that are caused by deformation through surface asperities of the counter body. In the present paper, we follow the above paradigm of Greenwood-Tabor-Grosch and do not discuss the adhesive contribution to friction.

The force of friction typically increases with velocity, reaches a plateau and decreases again, as schematically shown in Fig. The plateau is normally of most practical interest. Physically, the behavior of an elastomer in this range is dominated by the loss modulus of the elastomer 5 , and the elastomer behaves roughly speaking as a simple fluid.

Elastomere Friction

In particular, the relaxation of the elastomer after indentation or ploughing by an asperity is very slow. Practically all micro contacts therefore will be in one-sided contacts as shown in Fig. It is easy to understand that the coefficient of friction is then roughly equal to the one-sided average of the local slope of the surface profile in the contact region. This simplified picture is valid if the contact between elastomer and the rigid body is friction free on scales smaller than that of the asperities.

Despite the apparent simplicity of this physical picture, already at this point an interesting and non-trivial question arises: What is the average slope of the profile? Let us start the discussion of this question with an estimation of the rms value of the gradient of the surface profile over the whole surface. In this limited range, the spectral power density of typical fractal surfaces is known to be a power-function of the wave-vector q : where H is the Hurst-Exponent and q 0 is some reference wave-vector 8. For Hurst exponents smaller than one, the resulting integral diverges at the upper limit of integration.

This means that for a true fractal surface without an upper cut-off wave vector , the surface gradient would be infinitely large. In practice, of course, there is always some upper cut-off wave vector q max , and the surface gradient is determined by one or two orders of magnitude of wave vectors at and below q max.

In other words, for typical fractal rough surfaces, the friction force is determined by the roughness components with the largest wave-vectors or the smallest scale of the system. One can say that understanding friction is equivalent to understanding the nature of this smallest relevant scale.

The conclusion that the surface gradient is mainly determined by the smallest space scales follows from very general scaling considerations of self-affine fractals and is not limited to the spectral representation of rough surfaces. However, for the sake of simplicity and transparency of argumentation, we will confine ourselves to consideration of randomly rough surfaces according to definition 6.

Of course, the above estimation is oversimplified in the sense that it is the surface slope in the contact region and not over the whole surface which is determining the coefficient of friction at the plateau. However, as has been shown already by Archard in th 9 , the main effect of changing normal force is the number of asperities coming into contact, while the local conditions in the real contact area, including the surface gradient, depend only weakly on the normal force. Thus, the average gradient over the whole surface is already a good estimation for the gradient in the real contact area.

However, the mentioned relatively weak dependence on the normal force is exactly what we would like to discuss in more detail. The actual surface gradient is a function of the current contact configuration e. In the next section we will argue that the governing parameter for the contact configuration is the indentation depth d.

The indentation depth, in turn is connected with the normal force over the contact stiffness, which is dependent practically only on the large wavelength components of roughness or on the macroscopic form of the indenter We will thus come to the following hypothesis: while the friction force is almost entirely dependent on the smallest-scale roughness, its weak dependence on the normal force is related only to the large scale roughness.

We then will substantiate this two-scale hypothesis with a numerical simulation of the force of friction between an elastomer and a randomly rough fractal surface using a simplified one-dimensional model.

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If a rigid body of an arbitrary shape is pressed against a homogeneous elastic half-space then the resulting contact configuration is only a function of the indentation depth d. At a given indentation depth, the contact configuration does not depend on the elastic properties of the medium, and will be the same even for indentation of a viscous fluid or of any linearly viscoelastic material. This general behavior was recognized by Lee 12 and Radok 13 and was verified numerically for fractal rough surfaces Further, the contact configuration at a given depth remains approximately invariant for media with thin coatings 15 and for multi-layered systems, provided the difference of elastic properties of the different layers is not too large In 17 , it was argued that this is equally valid for media which are heterogeneous in the lateral direction along the contact plane.

Along with the contact configuration, all contact properties including the real contact area, the contact length, the contact stiffness, as well as the rms value of the surface gradient in the contact area will be unambiguous functions of the indentation depth. Note, that this is equally valid for tangential contact. This result does not depend on the form of the body and is valid for arbitrary bodies of revolution and even for randomly rough fractal surfaces In practice, however, the controlled and measured quantity is normally not the indentation depth but the normal force.

The latter is connected with the indentation depth through the contact stiffness.


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The contact stiffness, however, is known to be determined almost entirely by the long wave-length part of the spectral density of the surface profile or the macroscopic form of the contacting bodies and does not depend on the detailed topography on the micro-scale This supports the hypothesis that frictional force is determined mainly by the smallest spatial scale while its weak dependence on the normal force is governed almost exclusively by the largest spatial scale of the system.

Despite the simplicity and robustness of the arguments for the two-scale picture of friction and contact of rough surfaces, they are not exact. It is therefore of interest to test by direct simulation if the arguments are valid and with what accuracy. We therefore performed numerical simulation of the contact of a rigid fractal surface using the full fractal spectrum and a spectrum with a truncated middle part.

Note that we use the fractal approach as described e. This is done for illustrative purposes, as the randomly rough surfaces can be easily handled numerically in the frame of this approach. However, we beleve that the main qualitative arguments of the paper remain valid also for other types of fractal surfaces. We considered the simplest contact of rigid rough surfaces with a viscoelastic Kelvin counter-body. According to the general logic presented in the introductory sections, we wanted to check the following properties:.

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The indentation depth is the governing parameter of the contact. Thus, the length of the system under consideration which determines the minimal wave-vector of the fractal surface does not influence the coefficient of friction, provided the indentation depth is kept constant.

Under conditions of constant normal force, the length of the system and, thus, the minimal wave-vector does influence the coefficient of friction.

The coefficient of friction is influenced mostly by the short wavelength part of power spectrum and is insensitive to changes in the middle part of the power spectrum. To test the first property, we calculated the coefficient of friction for systems of different total length and thus different long wavelength cut-off wave-vector, while the upper part of the spectrum remained unchanged.

The results are presented in Fig. They show that under conditions of constant indentation depth the coefficient of friction practically does not depend on the length of the system. However, under conditions of controlled normal force, there is distinct dependency on the size of the system, as it should be, Fig. The reason is that the size of the system influences the contact stiffness and thus the indentation depth for a given normal force.

Dependencies of the coefficient of friction on indentation depth a and normal load b for the surfaces of different length L. The central point of the two-scale picture of rough contacts is the insensitivity of contact properties to the middle part of the power spectrum.

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Elastomere Friction - Theory, Experiment and Simulation | Dieter Besdo | Springer

To show this, simulations were carried out first with the complete fractal power spectrum given by equation 2 as shown with a dashed line in Fig. The spectrum then was modified by setting equal to zero the middle part of the spectrum as shown in Fig.

Hyperelastic Simulations

The coefficients k 1 and k 2 determine the size of the inner truncated region of the spectral density. The red solid line shows a modified spectral density. By changing the coefficients k 1 and k 2 one can change the size of the truncated region. This allowed studying the influence of the coefficients k 1 and k 2 in a wide range of their values. Only in this limiting case, the friction is truly multi-scale, and all parts of power spectrum contribute essentially to the coefficient of friction.

The results shown in Fig. However, such high Hurst exponents are not typical for frictional surfaces. We have shown that the multi-scale view of friction adds very little to the accuracy of friction models — at least for the case of elastomer friction — while making them significantly more complicated. In practice, of course, there is always some upper cut-off wave vector q max , and the surface gradient is determined by one or two orders of magnitude of wave vectors at and below q max. In other words, for typical fractal rough surfaces, the friction force is determined by the roughness components with the largest wave-vectors or the smallest scale of the system.

One can say that understanding friction is equivalent to understanding the nature of this smallest relevant scale. The conclusion that the surface gradient is mainly determined by the smallest space scales follows from very general scaling considerations of self-affine fractals and is not limited to the spectral representation of rough surfaces.

However, for the sake of simplicity and transparency of argumentation, we will confine ourselves to consideration of randomly rough surfaces according to definition 6. Of course, the above estimation is oversimplified in the sense that it is the surface slope in the contact region and not over the whole surface which is determining the coefficient of friction at the plateau. However, as has been shown already by Archard in th 9 , the main effect of changing normal force is the number of asperities coming into contact, while the local conditions in the real contact area, including the surface gradient, depend only weakly on the normal force.

Thus, the average gradient over the whole surface is already a good estimation for the gradient in the real contact area. However, the mentioned relatively weak dependence on the normal force is exactly what we would like to discuss in more detail. The actual surface gradient is a function of the current contact configuration e.

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In the next section we will argue that the governing parameter for the contact configuration is the indentation depth d. The indentation depth, in turn is connected with the normal force over the contact stiffness, which is dependent practically only on the large wavelength components of roughness or on the macroscopic form of the indenter We will thus come to the following hypothesis: while the friction force is almost entirely dependent on the smallest-scale roughness, its weak dependence on the normal force is related only to the large scale roughness. We then will substantiate this two-scale hypothesis with a numerical simulation of the force of friction between an elastomer and a randomly rough fractal surface using a simplified one-dimensional model.