Contact Mechanics · Rolling contact
Rolling contact traction: creep, stick and slip, and rolling resistance
A driven wheel grips the rail only because part of its contact patch slips.
Where does a driven wheel stick and where does it slip?
Push a driving or braking force Q through the wheel.
Try it: raise Q / μP from 0.6 to 0.95: the stick zone shrinks from 63% to 22% and the creepage doubles; past 1.0 the wheel spins.
Why does rolling cost any force at all?
Roll a cylinder on a flat that loses energy when squeezed.
Try it: switch to steel: the contact narrows, the pressure evens out, and μR falls from 0.0038 to 0.000011.
What to take away
Grip needs creep
A driven wheel turns a little faster than it rolls; a braked one, slower. That difference is the creepage.
Stick at the front
Slip starts at the trailing edge and spreads forward as Q grows. At Q = μP the wheel spins or skids.
Lossy and soft rolls hard
Hysteresis puts more pressure at the front. The resistance grows with α and with a/R.
More detail: the equations and the limits of these models
Carter's solution (1926)
A cylinder of radius R rolls on a flat (plane strain, line contact). Hertz gives the half-width a = (4P'R / πE*)1/2 and peak pressure p0 = 2P' / πa, with P' the load per unit length. The traction is the full friction limit μp(x) minus a smaller elliptical term over the stick zone. The stick zone touches the leading edge and has half-width c = a (1 − Q/μP)1/2.
In the stick zone the two surfaces must stretch by the same amount, which fixes the creepage: ξ = (μa/R)[1 − (1 − Q/μP)1/2] (Johnson 1985, section 8.3). Turned round, Q/μP = 1 − (1 − ξ/ξs)2 with ξs = μa/R: this is the curve on the right. The page uses a 920 mm wheel (R = 460 mm) and steel on steel, E = 210 GPa, ν = 0.3, so E* = 115 GPa.
Scope: two dimensional, same elastic constants in both bodies (so normal and tangential problems do not couple), one constant μ, steady rolling, no spin. A real wheel on a rail has an elliptical patch with lateral creep and spin; Kalker's theory (1990) handles that. Measured traction curves also fall after the peak and depend on surface films and roughness.
Rolling resistance from hysteresis
Tabor (1955) argued that the work done on the front half of the contact is partly lost; call the lost fraction α. For a cylinder with the Hertz pressure, the front half produces a moment 2P'a / 3π about the centre. Losing α of it gives μR = (2α / 3π)(a/R) for a cylinder, and μR = (3α / 16)(a/R) for a sphere (Johnson 1985, section 9.1). The push then acts ahead of the centre by e = μR R.
The drawn pressure is the Hertz shape tilted so that its centre sits at e; it shows the idea, not an exact viscoelastic solution. α is a measured property, not predicted here (Greenwood, Minshall and Tabor 1961 measured it for rubbers). The preset α values are rough orders of magnitude for illustration: 0.005 for hard steel, 0.1 for a lossy rubber, 0.05 for nylon. Moduli: steel 210 GPa (ν 0.3), rubber 3 MPa (ν 0.5), nylon 3 GPa (ν 0.4).
Scope: elastic and small strain. Above about a/R = 0.2 Hertz is only a rough guide, and a pneumatic tyre is a shell, not a solid rubber roller. For hard steel, hysteresis is not the main loss in real bearings: microslip, roughness, lubricant and seals add more.
On this site: Subsurface contact stresses · Fretting and partial slip · Why friction is not a material constant · Wear mechanisms and wear depth
Questions people ask
What is creepage in rolling contact?
Creepage is the difference between the surface speeds of the wheel and the rail, divided by the rolling speed. A driven wheel's surface moves slightly faster than the rail; a braked one slightly slower. In this model it is a fraction of a percent until full slip.
Why is the stick zone at the leading edge?
Material enters the patch at the leading edge with no strain difference between the two bodies, so it can stick. As it moves back, the strain difference builds up until the traction needed to hold it reaches μp. From there to the trailing edge it slips.
Why do trains lose grip on wet or leafy rails?
The largest traction a wheel can carry is μP. Water and crushed leaves lower μ, so the same driving or braking force now asks for more than μP. The stick zone vanishes and the wheel spins or slides.
What causes rolling resistance?
In the model here, energy lost inside the material each time it is squeezed and released (hysteresis). The material at the back pushes back less than it was pushed at the front, so the pressure is higher at the front. Microslip, roughness and plastic flow add to it in real contacts.
Why does a steel wheel roll so much more easily than a tyre?
Steel is stiff, so the contact is narrow (small a/R), and it loses little energy per squeeze (small α). Rubber is soft and lossy, so both factors are larger. At the presets here the rubber roller has about 340 times the rolling resistance of the steel one.
How does rolling cause fatigue?
Each pass loads the metal below the surface, with the largest shear stress a little below the surface for pure rolling. Millions of passes can start cracks there. Traction moves the highest stress up towards the surface.
References
Show the 7 references
- F. W. Carter, On the action of a locomotive driving wheel, Proceedings of the Royal Society of London A 112, 151 to 157 (1926). doi:10.1098/rspa.1926.0100
- K. L. Johnson, Contact Mechanics, Cambridge University Press (1985): section 8.3, tractive rolling of elastic cylinders; section 9.1, elastic hysteresis.
- J. J. Kalker, Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer Academic Publishers, Dordrecht (1990).
- D. Tabor, The mechanism of rolling friction. II. The elastic range, Proceedings of the Royal Society of London A 229, 198 to 220 (1955). doi:10.1098/rspa.1955.0082
- J. A. Greenwood, H. Minshall and D. Tabor, Hysteresis losses in rolling and sliding friction, Proceedings of the Royal Society of London A 259, 480 to 507 (1961). doi:10.1098/rspa.1961.0004
- O. Reynolds, On rolling friction, Philosophical Transactions of the Royal Society of London 166, 155 to 174 (1876).
- M. Shahzamanian Sichani, Wheel-rail contact modelling in vehicle dynamics simulation, Licentiate thesis, KTH Royal Institute of Technology, Stockholm (2013): Carter's theory (used here to check the traction curve).
BibTeX
@misc{tripathy2026rollingcontacttraction,
author = {Tripathy, Manisha},
title = {Rolling Contact Traction Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tribology/rolling-contact-traction}},
note = {Interactive web tool}
}