From individual to collective pinning: Effect of long-range elastic interactions

We study the effect of long-range elastic interactions in the dynamical behavior of an elastic chain driven quasistatically in a quenched random pinning potential. This is a generic situation occurring in solid friction, crack propagation, wetting front motion, etc. In the strong pinning limit, the dynamic of the chain is controlled by individual instabilities of each site of the chain. Long-range correlations in the displacement field and in the force field develop progressively. The system self-organizes to a steady state where the propagation of the instabilities is described by scaling laws with characteristic critical exponents. These exponents are numerically estimated through the analysis of the spatio-temporal correlation in the activity map. Tuning the exponent $\alpha$ of the algebraic decay of the elastic interaction with the distance is shown to give rise to three regimes: a mean-field (MF) regime valid for $\alpha <1\mathrm{}$ (very slow decay), a Laplacian regime for $\alpha >3\mathrm{}$ (rapid decay of interactions), and an intermediate regime $1<\mathrm{}\alpha <3\mathrm{}$ where the critical exponents interpolate continuously between the MF and Laplacian limit cases. The latter regime is shown to display, in the range $1<\mathrm{}\alpha <2,$ a mean-field-type character only for time correlations but not for space. The effect of the driving mode on the avalanche statistics is also analyzed.