OFFPRINT Critical jamming of frictional grains in the generalized isostaticity picture

While frictionless spheres at jamming are isostatic, frictional spheres at jamming are not. As a result, frictional spheres near jamming do not necessarily exhibit an excess of soft modes. However, a generalized form of isostaticity can be introduced if fully mobilized contacts at the Coulomb friction threshold are considered as slipping contacts. We show here that, in this framework, the vibrational density of states (DOS) of frictional discs exhibits a plateau when the generalized isostaticity line is approached. The crossover frequency ω∗ scales linearly with the distance from this line. Moreover, we show that the frictionless limit, which appears singular when fully mobilized contacts are treated elastically, becomes smooth when fully mobilized contacts are allowed to slip. Finally, we elucidate the nature of the vibrational modes, both for slipping and for non-slipping fully mobilized contacts. Copyright c © EPLA, 2010 The jamming transition in disordered media has been the focus of intensive research efforts in recent years. Most work has focussed on the simplest case of purely repulsive frictionless spheres that jam when the pressure becomes positive. The corresponding jamming point, known as point J, acts as a critical point: close to it one observes non-trivial powerlaw scaling of the elastic constants, an abundance of soft modes, a diverging length-scale and singularly non-affine behavior [1–4]. These phenomena are deeply rooted in the emergence of isostaticity at the jamming point: for frictionless spheres, the packings reached for vanishingly small pressure are marginally stable and the contact number z at point J reaches z iso = 2d, where d is the number of spatial dimensions [1]. What is becoming clear for the last two years is that isostaticity and jamming do not go hand in hand for soft particles with different interactions [5–8]. In particular, friction has to be introduced to make connection with the granular systems studied in experiments [9], in engineering [10] and in the geosciences [11]. For frictional spheres, the contact number at the jamming transition, zc, lies between the frictional isostatic value z μ iso = d+1 and the frictionless isostatic value z iso = 2d, depending (a)E-mail: [email protected] on the preparation method and the friction coefficient μ of the material. Hence, isostaticity and jamming do not coincide for frictional spheres, and the central question is to understand how friction modifies the critical jamming scenario developed for frictionless spheres. The density of states (DOS) has played a crucial role in understanding the jamming scenario, and for frictionless spheres the spectrum of vibrational modes shows an excess density of states at low frequencies, with a finite number of zero-energy modes at the jamming transition and a crossover frequency ω ∼ (z− z iso) [2,3]. Calculating the DOS for frictional discs, Somfai et al. studied the cross-over frequency ω in the presence of friction and concluded that ω scales with the distance to the frictional isostatic limit (z− z iso), but not with distance to the frictional jamming transition (z− zc): ω ∼ (z− z iso) [6]. We will show here that so-called fully mobilized contacts, contacts where the tangential force reaches the maximum allowed by the Coulomb criterium, play a crucial role for jamming of frictional spheres. In the work of Somfai et al. [6], all contacts where assumed to be non-sliding under small vibrations. Since it is far from obvious how fully mobilized contacts react to tangential perturbations, the question is raised how to deal with vibrations in the presence of such contacts. Here we