The role of disorder in the yielding transition of amorphous solids

Depending on their preparation, amorphous materials display different types of behavior under shear when passing from an elastic regime to a plastic one. This passage, known as the ”yielding transition” can happen in a smooth way or in a sudden, catastrophic manner. This strongly depends on the preparation of the sample, and in particular on the annealing which it undergoes and which controls the initial disorder. The transition between a smooth (ductile) and a discontinuous (brittle) behavior is reminiscent of the one observed in quasi-statically driven Random Field Ising Model (RFIM) at zero temperature, where the magnetization can either undergo a macroscopic jump or smoothly vary when an applied magnetic field is slowly increased. Both transitions are dependent on the disorder strength. The goal of this thesis has been to investigate what the effective strength of the disorder ∆eff is at the transition of a driven RFIM and at the yielding transition of an EPM, and to make the connection between the two. The quantity we aim at computing is the so-called disconnected susceptibility, which is given by the variance of the sample-to-sample fluctuations of the magnetization or the stress and which characterizes the distribution of the effective disorder. It can indeed be shown (by means of fieldtheoretical arguments) that the disconnected susceptibility is proportional to the square of the so-called connected one, with a coefficient of proportionality precisely given by the effective disorder strength ∆eff.