Depinning of a discrete elastic string from a two dimensional random array of weak pinning points

The present work is essentially concerned with the development of statistical theory for the low temperature dislocation glide in concentrated solid solutions where atom-sized obstacles impede plastic flow. In connection with such a problem, we compute analytically the external force required to drag an elastic string along a discrete two-dimensional square lattice, where some obstacles have been randomly distributed. The corresponding numerical simulations allow us to demonstrate a remarkable agreement between simulations and theory for an obstacle density ranging from 1 to 50 % and for lattices with different aspect ratios. The theory proves efficient on the condition that the obstacle-chain interaction remains sufficiently weak compared to the string stiffness.