Non-affine response: jammed packings versus spring networks

We compare the elastic response of spring networks whose contact geometry is derived from real packings of frictionless discs, to networks obtained by randomly cutting bonds in a highly connected network derived from a well-compressed packing. We find that the shear response of packing-derived networks, and both the shear and compression response of randomly cut networks, are all similar: the elastic moduli vanish linearly near jamming, and distributions characterizing the local geometry of the response scale with distance to jamming. Compression of packing-derived networks is exceptional: the elastic modulus remains constant and the geometrical distributions do not exhibit simple scaling. We conclude that the compression response of jammed packings is anomalous, rather than the shear response. The jamming transition governs the onset of rigidity in disordered media as diverse as foams, colloidal suspensions, granular media and glasses [1]. While jamming in general is controlled by a combination of density, shear stress and temperature, most progress has been made for frictionless soft spheres that interact through purely repulsive contact forces, and that are at zero temperature and zero load [2–7]. This simple model applies to static foams or emulsions [8, 9], and represents a simplified version of granular media, if one ignores friction [10, 11] and nontrivial grain shapes [12–15]. From a theoretical point of view, this model is ideal for two reasons. First, it exhibits a well defined jamming point, “point J”, which in the limit of large system sizes, occurs at a well-defined density φ = φc [2]. Here the system is a disordered packing of frictionless undeformed spheres, which is marginally stable and isostatic, i.e., its contact number (average number of contacts per particle) z equals ziso = 2d in d dimensions [2, 16]. Second, in recent years it has been uncovered that the mechanical and geometric properties of such jammed packings exhibit a number of non-trivial power law scalings as a function of the distance to the jamming point: (1) The excess contact number ∆z := z−ziso scales as (φ−φc) [2,6,9,10]; (2) The ratio of shear (G) and bulk (K) elastic moduli vanishes at point J as G/K ∼ ∆z [2]. The latter behavior — a shear rigidity which becomes much smaller than the compression modulus as the jamming point is approached – is in many ways surprising. It also differs markedly from what is found in two simplified models of jammed systems, effective medium theory (EMT) and random elastic networks, as is illustrated schematically in fig. 1 for the simple case of harmonic particles. EMT predicts that the elastic moduli vary smoothly through the isostatic point where ∆z = 0 and that the moduli are of order of the local spring constant k. This is because effective medium theory is essentially “blind” to local packing considerations and isostaticity. Thus, besides failing to capture the vanishing of G near jamming, its prediction for the bulk modulus fails spectacularly as well: it predicts finite rigidity below isostaticity. The failure of EMT to describe elasticity near jamming motivated earlier suggestions that elasticity of jammed packings might be captured by random networks of springs — this problem is known as rigidity percolation [8,17–19]. However, in such random spring networks, both G and K are expected to go to zero as k∆z, as fig. 1c illustrates [17]. Thus, while from the point of view of effective medium theory the shear rigidity of jammed packings behaves anomalously, from the point of view of rigidity percolation, the compression modulus behaves unexpectedly. What sets jammed packings apart from either of these two limiting models? How to understand the difference in terms of the local packing or response? Is the difference with