Hysteresis and elastic interactions of microasperities in dry friction

Velocity independent dry friction of a slider upon a base is due to an hysteretic response of relative displacement rho to a tangential driving force F. We show that the purely elastic model for multistability considered in a previous publication is in no way essential: multistability arises just as well from adhesion. We emphasize the physical consequences of multistability for dynamic/static, a.c./d.c. friction. When the slider is moved from rest by an amount rho the transition from the zero force static configuration to dynamic behaviour is progressive, spreading on a range equal to the width of the hysteresis cycle. When rho is small, an elastic restoring force ensues; in agreement with observations. The competition of that elastic pinning with bulk elasticity generates a screening length which we believe is the natural size of Burridge Knopoff blocks. We then study the effect of elastic interactions between asperities: it is weak for dilute asperities, but its long range makes it important. In lowest order the interaction mediated displacement of a given asperity has logarithmically divergent fluctuations: they become comparable to the asperity radius when the slider sire reaches another characteristic "Larkin length" lambda(L), which for dilute micronic asperities is exponentially large. Wie give arguments suggesting that individually monostable asperities display collective multistability on scales larger than lambda(L). For individually multistable sites we show that elastic interactions give rise to cascade processes in which the spinodal jump of a given asperity triggers the jump of others. We estimate the size of these cascades that should show up in the noise spectrum.