Depinning transition in solid mechanics

Stéphane Roux

In statistical physics, the depinning transition describes the onset of motion of an elastic manifold interacting with a random potential. At the critical forcing that separates a pinned regime where the manifold reaches a static equilibrium configuration, and a propagating regime where the manifold acquires a finite velocity, a genuine second order phase transition occurs. Such a transition is encountered in a variety of different contexts, and controls critical currents in superconductors, magnetic walls, wetting contact lines, etc.

In Solid Mechanics also this general phenomenology is at play for different phenomena such as individual dislocation in crystals, or crack propagation in brittle heterogeneous materials, or plasticity of amorphous media. However, depending on the dimensionalities of the problem (elastic manifold and embedding space) and the type of elastic interactions, different universality classes are encountered. The consequences of such depinning transitions will be discussed, focusing mainly on crack arrest and amorphous media plasticity.

One of the most remarkable features is that the random pinning potential does not need to be characterized in full details, but may be reduced to a few characteristic features. Only the mean value and standard deviation of the macroscopic critical forcing at a given reference scale is enough to predict its effective distribution at a different scale. Up to rescaling factors involving the above two moments, the shape of the distribution of the critical forcing (energy release rate for cracks, yield stress for amorphous media plasticity) is universal. Moreover, the scaling of the standard deviation with the system size is shown to display a non-trivial but universal power-law behavior.