## Shear front propagation and rheology in random elastic networks

A minimal model for studying the mechanical properties of polymer networks, gels, granular media and glasses is a disordered network of point masses connected by harmonic 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. We 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 and highlights the simultaneous breakdown, close to the critical of point, of three common assumptions adopted in elasticity theory.

First, the elastic response becomes increasingly non-affine as manifested by the diverging shear front width, a dynamical analogue of the diverging width of magnetic domain walls near the Curie point. Even in the limit of vanishing microscopic coefficient of dissipation, the fragile solid behaves as if it were over-damped because energy is leaked into the diverging non-affine fluctuations. Second, the quasi-static approximation breaks down leading to the front width growing super-diffusively and the ordinary diffusive regime is only attained after a characteristic time scale that diverges at the transition. Third, the linear response regime vanishes giving rise to strongly non-linear shocks waves even for the tiniest shear strains.