Contact Mechanics · Friction
Why the coefficient of friction is not a material constant
Start with the first picture: press “Steel in air”, then “Lead film on hard steel”, and watch the friction coefficient fall eight times while the steel stays the same.
1 Adhesion: μ = τ/H 2 Ploughing grooves 3 Stick-slip 4 What changes μ for steel on steel
Where does friction come from? The Bowden-Tabor adhesion model
Two surfaces touch only at the tips of their bumps (asperities). The tips yield until they carry the load, so the real contact area is Ar = W/H (W the load, H the hardness of the softer solid). Sliding must shear these welded tips (junctions) with shear strength τ. So F = τAr and μ = F/W = τ/H. Neither τ nor H belongs to “steel” alone: τ depends on what sits in the interface.
How much friction does ploughing add?
When a hard asperity digs into a softer surface, it must push material aside to move. The front half of the tip carries the load, so for a cone the ploughing part is μp = (2/π) cot θ, where θ is the half-angle of the cone. The attack angle is 90° − θ, the slope of the tip face. The total is roughly μ = μadhesion + μp.
What causes stick-slip?
A block is pulled through a spring by a driver moving at steady speed. If friction falls when the block speeds up (static friction above kinetic), the block can stick, load the spring, then jump. Here friction follows the rate-and-state law (Dieterich, Ruina): μ = μ0 + a ln(v/V0) + b ln(V0θ/Dc), where the state θ is the age of the contacts. Contacts that rest grow stronger, so static friction is higher. At steady sliding μ falls by (b − a) for each factor e in speed. Steady sliding is stable only if the spring is stiffer than kc = W(b − a)/Dc·[1 + mV2/(W a Dc)].
What changes the friction coefficient of steel on steel?
Handbooks list dry steel on steel at 0.74 to 0.80 static and 0.42 to 0.62 kinetic. Change one condition and the number moves by a factor of ten or more. Rows with a bar are published ranges; rows with only an arrow show the direction, because the size depends on the system.
What to take away
More detail: where the formulas come from and where they stop working
Real contact area. Ar = W/H assumes the asperity tips deform plastically at a pressure equal to the indentation hardness. Very smooth or very hard surfaces may touch elastically instead; the area is then still close to proportional to the load, for a random rough surface (Greenwood and Williamson). See the real contact area page.
Junction growth. A junction under a normal pressure p and shear stress s yields when p2 + αs2 = pm2 (Tabor 1959). Under shear it can only yield by growing, so p falls and the area rises. With pm = H and α = 9 (so the shear strength of the solid is k = H/3), sliding starts when s reaches the interface strength τ, giving μ = c/√(α(1 − c2)) with c = τ/k. For c near 1 the junction grows without limit: seizure. The value of α is fitted to experiments; Tabor’s estimates range from about 9 to 25.
Thin films. The simple rule μ = τfilm/Hsubstrate explains why soft metal, MoS2 and PTFE films work. In practice the film’s shear strength under several GPa of contact pressure is much higher than its bulk value, so the lead film preset uses the τ that reproduces the measured 0.085. A film that is too thick lets the load sink into the soft layer (H falls); too thin and the asperities break through.
Ploughing. A cone of half-angle θ supported on the front half of its contact: W = Hπa2/2 and ploughing force Fp = H a2 cot θ, so μp = (2/π)cot θ. For a sphere of radius R with contact radius a: μp = (2/π)[(R/a)2 sin−1(a/R) − √((R/a)2 − 1)]. Adding adhesion and ploughing ignores how they interact; the slip-line field models of Challen and Oxley treat the combination properly.
Rate-and-state stick-slip. The page integrates m·dv/dt = k(Vt − x) − μW with the ageing law dθ/dt = 1 − vθ/Dc and μ0 = 0.6 at V0 = 1 µm/s. kc is the linear stability limit of steady sliding (Rice and Ruina 1983). With little inertia the model depends on speed only through ln V, so stick-slip size barely changes with speed. Real surfaces often change from velocity weakening to strengthening at higher speed, and damping in the machine helps too; this is why speeding up or stiffening a machine often stops squeal.
Classic static/kinetic model. With a constant μs and μk, the sawtooth drop in spring force is about 2(μs − μk)W, because the block overshoots. Rabinowicz showed μs rises with the time the surfaces rest together, which is what the state θ describes.
Questions people ask
Is the coefficient of friction a material property?
No. It is a property of the whole sliding system: both surfaces, any film between them, load, speed, temperature, humidity and the stiffness of the machine. Steel on steel ranges from about 0.001 with a full oil film to above 1 when clean in vacuum.
What is the coefficient of friction of steel on steel?
Dry, in air: about 0.74 to 0.80 static and 0.42 to 0.62 kinetic in handbook tables. Greasy or boundary lubricated: roughly 0.03 to 0.2. With a full oil film: about 0.001 to 0.01. Clean in vacuum: above 1.
Why does friction not depend on the contact area?
Only the tips of the bumps touch. Their total area is W/H, set by the load and the hardness, not by the size of the block. A bigger block spreads the same load over more, smaller contacts with the same total area.
How do solid lubricants like MoS2, PTFE or lead films work?
A thin layer that is easy to shear sits on a hard base. The hard base keeps the contact area small (large H) while sliding happens in the weak layer (small τ), so μ = τ/H is low. The same soft material in bulk gives high friction, because then H is small too.
Why is static friction higher than kinetic friction?
Contacts at rest slowly grow and strengthen with time (creep of the asperities, chemical bonding). Once sliding, each contact lives only for a short time, so it is weaker. Static friction grows roughly with the logarithm of the resting time.
What causes stick-slip and squeal?
Friction that falls as speed rises, plus a soft enough spring in the system. The block sticks, the spring loads, the block jumps and sticks again. A stiffer machine, a friction pair whose friction rises with speed, or more damping stops it.
Why do clean metals stick together in vacuum?
Oxide and adsorbed films normally keep the metals apart. Remove them and the contacts form metal-to-metal bonds as strong as the metal. Under a sideways force the junctions grow instead of shearing, so friction rises above 1 and the parts can seize (cold welding).
Does a rougher surface always have more friction?
No. Very rough surfaces add ploughing and interlocking, but very smooth, clean surfaces can also give high friction, because more of the area comes into close contact. For many metal pairs friction is lowest at an intermediate roughness.
Related: Asperities and the real area of contact · Lubrication regimes and the Stribeck curve · Flash temperature at sliding contacts · Wear mechanisms and wear depth (Archard) · Wear rate calculator · Scratch test critical loads (when a coating breaks through) · Scratch hardness and ploughing regimes · Nanoindentation hardness H · Strengthening mechanisms (what sets H)
References
Show the 11 references
- F. P. Bowden and D. Tabor, The Friction and Lubrication of Solids, Part I, Clarendon Press, Oxford (1950; reissued 2001): real contact area, μ = s/p, thin metal films on hard substrates, clean metals in vacuum.
- D. Tabor, Junction growth in metallic friction: the role of combined stresses and surface contamination, Proceedings of the Royal Society A 251, 378 to 393 (1959). doi:10.1098/rspa.1959.0114
- E. Rabinowicz, Friction and Wear of Materials, 2nd ed., Wiley, New York (1995): chapter 4 (friction, roughness, static friction rising with rest time, stick-slip).
- I. M. Hutchings and P. Shipway, Tribology: Friction and Wear of Engineering Materials, 2nd ed., Butterworth-Heinemann (2017): chapter 3 (adhesion and ploughing, μp = (2/π)cot θ for a cone and the sphere formula, junction growth, stick-slip) and chapter 4 (lubrication regimes).
- B. N. J. Persson, Sliding Friction: Physical Principles and Applications, 2nd ed., Springer, Berlin (2000): stick-slip, rate-and-state friction, contact ageing.
- P. J. Blau, The significance and use of the friction coefficient, Tribology International 34(9), 585 to 591 (2001). doi:10.1016/S0301-679X(01)00050-0
- E. A. Avallone, T. Baumeister and A. M. Sadegh (eds.), Marks’ Standard Handbook for Mechanical Engineers, 11th ed., McGraw-Hill (2007), section 3.2, friction coefficient table: steel on steel dry 0.74 to 0.80 static, 0.42 to 0.62 kinetic; greasy or lubricated about 0.03 to 0.23. The same values appear in P. J. Blau, Appendix: Static and kinetic friction coefficients for selected materials, ASM Handbook Vol. 18, ASM International (1992).
- J. R. Rice and A. L. Ruina, Stability of steady frictional slipping, Journal of Applied Mechanics 50, 343 to 349 (1983). doi:10.1115/1.3167042 (critical stiffness, with the inertia term mV2/(W a Dc)).
- T. Spalvins and B. Buzek, Frictional and morphological characteristics of ion-plated soft metallic films, NASA TM-81723 (1983), also Thin Solid Films 84, 267 to 272 (1981): lead films about 0.2 µm thick on 440C steel, μ = 0.085, in vacuum.
- S. Philippon, G. Sutter and A. Molinari, An experimental study of friction at high sliding velocities, Wear 257, 777 to 784 (2004). doi:10.1016/j.wear.2004.03.017 (dry metal pair, 0.01 to 60 m/s; μ falls with speed above 14 m/s).
- F. H. Stott, The role of oxidation in the wear of alloys, Tribology International 31, 61 to 71 (1998) (compacted oxide glaze layers at high temperature). M. R. Vazirisereshk, A. Martini, D. A. Strubbe and M. Z. Baykara, Solid lubrication with MoS2: a review, Lubricants 7, 57 (2019), doi:10.3390/lubricants7070057 (humidity raises MoS2 friction).
BibTeX
@misc{tripathy2026frictionconstant,
author = {Tripathy, Manisha},
title = {Why the Coefficient of Friction Is Not a Material Constant},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tribology/why-friction-is-not-a-material-constant}},
note = {Interactive web tool}
}