Tribology Letters

, 68:25 | Cite as

Physical Model of Tire-Road Contact Under Wet Conditions

  • J. LöwerEmail author
  • P. Wagner
  • H.-J. Unrau
  • C. Bederna
  • F. Gauterin
Original Paper


A physical model to describe the contact between rubber and a rough surface with water as intermediate medium is presented. The Navier–Stokes equations are simplified and surface properties are approached by the Abbott–Firestone curve to generate an approximative description of the water squeeze out between a visco-elastic rubber block and a macro-rough surface. The model is used to describe the pattern dependent wet grip performance of vehicle tires at moderate water heights between pure wetgrip and full hydroplaning. Influence of surface macro-roughness, water height, tire pattern, and vehicle speed on braking performance is considered in particular. For validation purpose, braking tests on two different surfaces were done at an inner drum test bench. Test results show good agreement with the theory presented.


Wet braking Tribology Friction Road texture Tire-road contact Water height 

List of Symbols


Surface area of tread block

\(A_{\rm F}(h)\)

Area of contact between fluid and rubber


Free area between track and tread block


Free area between track and tread block parallel to yz-plane


Free area between track and tread block parallel to xz-plane

\(A_{\rm R}(h)\)

Area of contact between track and rubber


Width of tread block


Geometric factor


Modulus of elasticity of Kelvin–Voigt element


Viscosity of Kelvin–Voigt element


Water height dependent fluid velocity coefficient in x-direction


Water height dependent fluid velocity coefficient in y-direction


Load on tread block

\(\gamma _{\rm T}\)

Factor for churning losses


Track parameter


Track parameter


Track parameter


Track parameter


Water height


Initial change of water height


Initial water height


Equivalent water height


Track parameter

\(\kappa (z)\)

Correction factor for control volume \(V_{\rm C}\)


Length of tread block

\({\dot{m}}_{{\rm out}}\)

Mass flow density over the control volume boundaries


Dynamic viscosity of fluid

\(p_{\rm F}(t)\)

Mean fluid pressure

\(p_{\rm m}(t)\)

Mean pressure acting on tread block

\(p_{\rm R}(t)\)

Mean contact pressure at interface \(A_{\rm R}(h)\)


Relation between \(f_{L}\) and \(f_{B}\)


Density of water

\(\rho _{\rm R}\)

Density of rubber


Local rubber deformation at interface \(A_{\rm R}(h)\)


Mean rubber deformation at interface \(A_{\rm R}(h)\)


Additional rubber deformation at interface \(A_{\rm F}(h)\)


Time derivative of additional rubber deformation at interface \(A_{\rm F}(h)\)

\(\mathbf {v}_{\rm F}\)

Fluid velocity


Fluid velocity in x-direction


Fluid velocity in y-direction


Fluid velocity in z-direction

\(t_{\rm c}\)

Contact time


Fluid velocities x-direction in equivalent coordinate system


Fluid velocities y-direction in equivalent coordinate system


Fluid velocities z-direction in equivalent coordinate system

\(v_{{\rm out}}(x,y)\)

Fluid velocity at boundary surfaces

\(v_{{\rm out}, A_{q_x}}(y)\)

Fluid velocity at boundary surface \(A_{q_x}\)

\(v_{{\rm out}, A_{q_y}}(x)\)

Fluid velocity at boundary surface \(A_{q_y}\)

\(v_{{\rm out}_{\perp }, A_{q_x}}(y)\)

Perpendicular fluid velocity at boundary surface \(A_{q_x}\)

\(v_{{\rm out}_{\perp }, A_{q_y}}(x)\)

Perpendicular fluid velocity at boundary surface \(A_{q_y}\)

\(V_{\rm C}(h)\)

Control volume

\({\dot{V}}_{{\rm in}}\)

Volume flow in \(V_{{\rm in}}\)

\(V_{\infty }\)

Infinite volume under tread block

\({\dot{V}}_{{\rm out}}\)

Volume flow out \(V_{{\rm out}}\)

\({\dot{W}}_{{\rm in}}\)

Engery flow into \(V_{\rm C}\)

\({\dot{W}}_{{\rm kin}}\)

Change of kinetic energy inside \(V_{\rm C}\)

\(W_{{\rm kin},F}\)

Initial kinetic energy of fluid

\({\dot{W}}_{{\rm out}}\)

Energy flow out of \(V_{\rm C}\)

\({\dot{W}}_{{\rm visc}}\)

Viscous losses inside \(V_{\rm C}\)


State vector


Material share


Coordinate system for equivalent water height

\(\dot{y}_{\perp }\)

Initial vertical velocity of tread block on an undisturbed circular path


Track profile


Track height



Financial support by Continental Reifen Deutschland GmbH is gratefully acknowledged.

Compliance with Ethical Standards

Conflict of interest

No potential conflict of interest was reported by the authors.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Institute of Vehicle System TechnologyKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Continental Reifen Deutschland GmbHHannoverGermany

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