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A minimal model for studying the mechanical properties of amorphous solids isa disordered network of point masses connected by unbreakable springs. At acritical value of its mean connectivity, such a network becomes fragile: itundergoes a rigidity transition signaled by a vanishing shear modulus andtransverse sound speed. We investigate analytically and numerically the linearand non-linear visco-elastic response of these fragile solids by probing howshear fronts propagate through them. Our approach, that we tentatively labelshear front rheology, provides an alternative route to standard oscillatoryrheology. In the linear regime, we observe at late times a diffusive broadeningof the fronts controlled by an effective shear viscosity that diverges at thecritical point. No matter how small the microscopic coefficient of dissipation,strongly disordered networks behave as if they were over-damped because energyis irreversibly leaked into diverging non-affine fluctuations. Close to thetransition, the regime of linear response becomes vanishingly small: thetiniest shear strains generate strongly non-linear shear shock wavesqualitatively different from their compressional counterparts in granularmedia. The inherent non-linearities trigger an energy cascade from low to highfrequency components that keep the network away from attaining the quasi-staticlimit. This mechanism, reminiscent of acoustic turbulence, causes asuper-diffusive broadening of the shock width.

Shear shocks in fragile networks