Frictional granular matter is shown to be fundamentally different in its plastic responses to external strains from generic glasses and amorphous solids without friction. While regular glasses exhibit plastic instabilities due to a vanishing of a real eigenvalue of the Hessian matrix, frictional granular materials can exhibit a previously unnoticed additional mechanism for instabilities, i.e. the appearance of a pair of complex eigenvalues leading to oscillatory exponential growth of perturbations which are tamed by dynamical nonlinearities. This fundamental difference appears crucial for the understanding of plasticity and failure in frictional granular materials. The possible relevance to earthquake physics is discussed.It is often stressed that the mechanical properties of frictional granular matter and of glassy amorphous solids share many similarities [1][2][3][4][5], although the effective forces in frictional solids are not derivable from a Hamiltonian. Here we show that the lack of a Hamiltonian description is responsible for previously unreported oscillatory instabilities in frictional granular matter. These oscillatory instabilities furnish a micromechanical mechanism for a giant amplification of small perturbations that can lead eventually to major events of mechanical failure. We will demonstrate this physics in the context of amorphous assemblies of frictional disks, but will make the point that the mechanism discussed here is generic for systems with friction. To motivate the new ideas recall that the understanding of plastic instabilities, shear banding and mechanical failure in athermal amorphous solids with an underlying Hamiltonian description had progressed significantly in the last twenty years. Beginning with the seminal papers of Malandro and Lacks [6,7] it became clear that an object that controls the mechanical responses of athermal glasses is the Hessian matrix. In an athermal (T=0) system of N particles at positions (r 1 , r 2 · · · r N ) we define the Hamiltonian U (r 1 , r 2 , · · · r N ). The Hessian matrix isHere F i is the total force on the ith particle, and in systems with binary interactions we can write F i ≡ j F ij with the sum running on all the particles j interacting with particle i. Being real and symmetric, the Hessian matrix has real eigenvalues which are all positive as long as the material is mechanically stable. Under strain, the system may display a saddle node bifurcation in which an eigenvalue goes to zero, accompanied by a localization of an eigenfunction, signalling a plastic instability that is accompanied by a drop in stress and energy [8]. Significant amount of work was dedicated to understanding the density of states of the Hessian matrix which differs in amorphous solids from the classical Debye density of purely elastic materials [3,9,10]. The well known "Boson peak" was explained by the prevalence of "plastic modes" that can go unstable and do not exist in pure elastic systems. The system size dependence of the eigenvalues of the Hessian [9], their role in ...
