Fractal shear bands at elastic-plastic transitions in random Mohr-Coulomb materials

This paper studies fractal patterns forming at elastic-plastic transitions in soil- and rock-like materials. Taking either friction or cohesion as nonfractal vector random fields with weak noise-to-signal ratios, it is found that the evolving set of plastic grains (i.e., a shear-band system) is always a monotonically growing fractal under increasing macroscopic load in plane strain. Statistical analysis is used to assess the anisotropy of those shear bands. All the macroscopic responses display smooth transitions, but as the randomness vanishes, they turn into a sharp response of an idealized homogeneous material. Parametric study shows that increasing hardening or friction makes the transition more rapid. In addition, randomness in cohesion has a stronger effect than randomness in friction, whereas dilatation has practically no influence. Adapting the concept of scaling functions, the authors find the elastic-plastic transitions in random Mohr-Coulomb media to be similar to phase transitions in condensed-matter physics: the fully plastic state is a critical point, and with three order parameters (reduced Mohr-Coulomb stress, reduced plastic volume fraction, and reduced fractal dimension), three scaling functions are introduced to unify the responses of different materials. The critical exponents are demonstrated to be universal regardless of the randomness in various constitutive properties and their random noise levels.