GSSI Lecture on Applied Mathematics and Mechanics
A minimal model for studying the mechanical properties of amorphous solids is a disordered network of point masses connected by springs. At a critical value of its mean connectivity, such a network becomes fragile: it undergoes a rigidity transition signaled by a vanishing shear modulus and transverse sound speed. We first investigate analytically and numerically the linear and non-linear visco-elastic response of these fragile solids by probing how shear fronts propagate through them.
Our approach, that we tentatively label shear front rheology, provides an alternative route to standard oscillatory rheology. In the linear regime, we observe at late times a diffusive broadening of the fronts controlled by an effective shear viscosity that diverges at the critical point. No matter how small the microscopic coefficient of dissipation, strongly disordered networks behave as if they were over-damped because energy is irreversibly leaked into diverging non-affine fluctuations. Close to the transition, the regime of linear response becomes vanishingly small: the tiniest shear strains generate strongly non-linear shear shock waves. The inherent non-linearities trigger an energy cascade from low to high frequency components that keep the network away from attaining the quasi-static limit. This mechanism, reminiscent of acoustic turbulence, causes a super-diffusive broadening of the shock width.
Finally, we show that the mechanism of failure of such networks, consists of meandering cracks whose width diverges at the transition. Thus, upon approaching the critical point, we can effectively zoom inside the fracture process zone.