Theory of rubber friction and contact mechanics
Summary (3 min read)
Introduction
- Theory of rubber friction and contact mechanics B. N. J. Persson Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany ~Received 12 March 2001; accepted 7 June 2001!.
- The nature of the friction when rubber slides on a hard substrate is a topic of considerable practical importance, e.g., for the construction of tires,1 wiper blades,1 and in the cosmetic industry.
- Surface roughness, but explicit results are presented for self affine fractal surface profiles.
- It is clear that even if the rubber is able to make direct contact with the substrate in the large cavities, the pressure acting on the rubber at the bottom of a large cavity will be much smaller than the pressure at the top of a large asperity.
- Note that the longest roughness wavelength possible are of order ;L .
II. SELF-AFFINE FRACTAL SURFACES AND CONTACT THEORIES
- It has been found that many ‘‘natural’’ surfaces, e.g., surfaces of many materials generated by fracture, can be approximately described as self-affine surfaces over a rather Downloaded 21 Dec 2006 to 134.94.122.39.
- A self-affine fractal surface has the property that if the authors make a scale change that is different for each direction, then the surface does not change its morphology,10,11 see Fig.
- Since H,1 ~H51 correspond to D f52!, these theories predict that the area of real contact increases faster than linear with the load.
- Roberts20 has shown that polar substances like soaps prevent direct contact between track and rubber ~see Sec. VII!; this explain why the friction is slightly lower for the wet 15% detergent case, compared to the wet, clean carborundum surface.
III. AREA OF REAL CONTACT: QUALITATIVE DISCUSSION
- I have already emphasized the importance of knowing the nature of the area of real contact when discussing rubber friction.
- This is at least one order of magnitude smaller than the ~static or lowfrequency!.
- Thus, the deformations induced by the largest asperities are relatively well described by using the low frequency elastic modulus E(v)'E(0).
- In Sec. V and Appendices B and C, I develop a new contact theory for surfaces with roughness on many different length scales.
IV. SLIDING FRICTION
- Using the theory of elasticity ~assuming an isotropic elastic medium for simplicity!, one can calculate the displacement field ui on the surface z50 in response to the surface stress distributions s i5s3i .
- Note that, in accordance with the discussion in Sec.
- The friction coefficient m can be obtained by dividing the frictional shear stress ~21! with the pressure s0 , m5 1 2 E d2q q2 cos f C~q !.
- This has a simple but important physical origin: Consider two cosine-surface corrugations, where the ‘‘wave vector’’ points ~a!.
V. CONTACT THEORY FOR RANDOMLY ROUGH SURFACES
- If A0 denotes the nominal contact area, the load FN5s0A0 .
- Now, assume that the macroscopic pressure s0 depends on the lateral position x in the nominal contact region, as would be the case if, e.g., a rubber ball is squeezed against a nominally flat substrate @where s0(x) is given by the Hertz expression#.
- Elastic modulus E , the rubber friction coefficient is ~nearly!.
- Numerical evaluation of Eq. ~31! shows that Eq. ~34! is an accurate representation of P(q) for all q ~or, equivalently, all G!.
- This will, of course, not occur in real systems, where there always exist an upper cutoff zmax5q1 /q0 in the integral over z.
VI. NUMERICAL RESULTS
- This model is, in fact, not a very good description of real rubbers, since the transition with increasing frequency from the rubbery region to the glassy region is much too abrupt, leading to a much too narrow ~and too high! m(v) peak.
- Later the authors will use experimental data for E(v) for two different rubbers, illustrating how the results based on the present model ~Fig. 10! are quantitatively modified.
- When s0 becomes of order E(0) the friction coefficient is no longer independent of s0 , but decreases with increasing s0 ~see 10 MPa curve in Fig. 16!.
- This nonlinearity is associated with the breakdown of the filler network, which occurs in the range of a few % strain amplitude.
- Is about half as large as when the calculation is based on the 8% stain amplitude data.
VII. DISCUSSION
- The theory developed above can be used to estimate the kinetic friction coefficient for rubber sliding on a rough hard substrate.
- In a typical case an equilibrium film of liquid some 200 Å thick becomes established between the surfaces.
- The generation of repulsive forces between rubber and glass surfaces means that the pair will make a microconforming contact with a uniform thin film of liquid between them ~see Fig. 18!.
- For this reason it is plausible that a liquidlike water layer may exist at the rubber–ice interface but not when ice is in contact with a hard, high energy solid surface, such as glass or metal oxides.
- 38 Finally, for practical applications it is necessary to study the heating of the rubber during sliding.
VIII. SUMMARY AND CONCLUSION
- There is at present a strong drive by tire companies to design new rubber compounds with lower rolling resistance, higher sliding friction, and reduced wear.
- At present these attempts are mainly based on a few empirical rules and on very costly trial-and-error procedures.
- I have shown that for stationary surfaces ~or low sliding velocity!, and for typical pressures in the contact area between a tire and a road, the rubber will only make ~apparent!.
- Contact with about 5% of the road surface.
- I have developed a contact theory which describes how the ~apparent!.
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Frequently Asked Questions (17)
Q2. What have the authors contributed in "Theory of rubber friction and contact mechanics" ?
In this paper, a general theory of the hysteretic contribution to rubber friction has been presented for rubber sliding on selfaffine fractal surfaces, e.g., a tire on a road surface.
Q3. How can one calculate the displacement field ui on the surface z50?
Using the theory of elasticity ~assuming an isotropic elastic medium for simplicity!, one can calculate the displacement field ui on the surface z50 in response to the surface stress distributions s i5s3i .
Q4. What is the reason why the rubber surface is likely to be microscopically rough?
the rubber surface is likely to be microscopically rough and the rubber molecules undergoes large thermal movement which may tend to break up any icelike structure at the interface.
Q5. How does the temperature field T(x,t) be determined?
The temperature field T(x,t) must be determined by solving the heat diffusion equation with the heat source Q(x,t), and with the appropriate boundary conditions which depend on the external conditions ~e.g., road temperature!.
Q6. What is the effect of a surface active agent on the film?
however, a surface active agent ~e.g., sodium dodecyl sulphate!, is included in the electrolyte solution, monolayer protection prevents surfaces from coming into intimate contact at points where the separating liquid film is locally punctured.
Q7. What is the problem with applying the present theory to filled rubbers?
One problem with applying the present theory to filled rubbers is the strongly nonlinear relation between the shear stress and the shear strainDownloaded 21 Dec 2006 to 134.94.122.39.
Q8. What is the elasticity of the filler network?
The complex elastic modulus E(v) used in the calculation was measured at 8% strain amplitude, which is so large that a complete break down of the filler network has occurred.
Q9. What is the current drive by tire companies to design new rubber compounds?
There is at present a strong drive by tire companies to design new rubber compounds with lower rolling resistance, higher sliding friction, and reduced wear.
Q10. What is the pressure in the contact area of the large surface asperities?
Since for a road surface the authors expect D'R it is clear that the local pressure in the contact area of the large surface asperities will be of order of E , i.e., just large enough in order for the rubber to deform and fill out at least some of the smaller sized surface cavities.
Q11. What is the reason for the low distance cut-off in the sum over length scales?
The authorbelieve that because of surface contamination there will be a low distance cut-off in the sum over length scales which is larger than 0.1 mm, and for this reason, in the context of the tire-road friction, The authordo not believe that the adhesion rubber–substrate interaction is important.
Q12. What is the first step in studying the contact mechanics of a fractal surface?
In order to study rubber friction on a hard self-affine fractal surface, it is first necessary to be able to describe the contact mechanics.
Q13. How can the authors study the asperity of the contact areas of thin fluid films?
27By using a transparent substrate, it should be possible to study the asperity contact areas during squeezing and shearing of thin fluid films.
Q14. What is the average pressure in the contact area of the largest asperities?
according to the contact theory of Greenwood,14,17 the average pressure which acts in the rubber–substrate contact area at the largest asperities is of order '(D/R)1/2E , where D is the rms surface roughness amplitude and R the ~average!
Q15. What is the friction coefficient for a tire?
In Fig. 17 the authors only show m(v) up to the velocity where the friction coefficient is maximal; higher velocities are probably of no direct interest for tires since the region where m(v) decreases with increasing v should be avoided, as interfacial stick-slip may occur when m8(v) ,0, which may result in an enhanced wear rate, and a loud noise.
Q16. What is the kinetic friction coefficient for a flat rubber surface?
The steady state kinetic friction coefficient for a flat rubber surface sliding on a nominally flat substrate is in the most general case is given byAIP license or copyright, see http://jcp.aip.org/jcp/copyright.jspm5 1 2 EqL q1 dq q3 C~q !
Q17. How many surfaces can be described as self-affine?
It has been found that many ‘‘natural’’ surfaces, e.g., surfaces of many materials generated by fracture, can be approximately described as self-affine surfaces over a ratherDownloaded 21 Dec 2006 to 134.94.122.39.