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In this Letter, we study the friction between a one-dimensional elastomer and a one-dimensional rigid body having a randomly rough surface. The elastomer is modeled as a simple Kelvin body and the surface as self-affine fractal having a Hurst exponent H in the range from 0 to 1. The resulting frictional force as a function of velocity always shows a typical structure: it first increases linearly, achieves a plateau and finally drops to another constant level. The coefficient of friction on the plateau depends only weakly on the normal force. At lower velocities, the coefficient of friction depends on two dimensionless combinations of normal force, sliding velocity, shear modulus, viscosity, rms roughness, rms surface gradient, the linear size of the system, and the Hurst exponent. We discuss the physical nature of different regions of the law of friction and suggest an analytical relation describing the coefficient of friction in a wide range of loading conditions. An important implication of the analytical result is the extension of the well-known "master curve procedure" to the dependencies on the normal force and the size of the system.

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The penetration of a rigid body with a randomly rough, self-affine surface in a half space filled with a linearly viscous elastomer is studied numerically using the method of boundary elements. Using Radok's principle of functional equations, it is shown analytically that this problem is closely related to the recently investigated problem of contact of self-affine surfaces with an elastic half space. We show that the penetration velocity occurs to be a power function of the applied force and time, the corresponding exponents depending only on the Hurst exponent. For comparison, the same problem is solved using the method of reduction of dimensionality. Both three-dimensional numerical results and the method of reduction of dimensionality support the analytical predictions provided by general scaling arguments.

The present paper contributes to the modeling of contacts with viscoelastic materials. An indenter is pressed into an elastomer and slides on its surface with a constant velocity. Contact regions typically experience periodical loading due to surface roughness with characteristic length scales. Loading within a medium frequency range is studied, in which the viscous properties of the elastomer dominate. After passing a transition zone a stationary state is reached. An onedimensional model is used to determine estimations for the indentation depth and the coefficient of friction. Results for two simply shaped indenters are presented and compared to boundary element simulations.