Generalized law of friction between elastomers and differently shaped rough bodies

In this paper, we study theoretically and experimentally the friction between a rough parabolic or conical profile and a flat elastomer beyond the validity region of Amontons' law. The roughness is assumed to be randomly self-affine with a Hurst exponent H in the range from 0 to 1. We first consider a simple Kelvin body and then generalize the results to media with arbitrary linear rheology. The resulting frictional force as a function of velocity shows the same qualitative behavior as in the case of planar surfaces: it increases monotonically before reaching a plateau. However, the dependencies on normal force, sliding velocity, shear modulus, viscosity, rms roughness, rms surface gradient and the Hurst exponent are different for different macroscopic shapes. We suggest analytical relations describing the coefficient of friction in a wide range of loading conditions and suggest a master curve procedure for the dependence on the normal force. Experimental investigation of friction between a steel ball and a polyurethane rubber for different velocities and normal forces confirms the proposed master curve procedure.