We study the rheology of amorphous packings of soft, frictionless particles close to jamming. Implementing a quasistatic simulation method we generate a well defined ensemble of states that directly samples the system at its yield-stress. A continuous jamming transition from a freely-flowing state to a yield stress situation takes place at a well defined packing fraction, where the scaling laws characteristic of isostatic solids are observed. We propose that long-range correlations observed below the transition are dominated by this isostatic point, while those that are observed above the transition are characteristic of dense, disordered elastic media.PACS numbers: 83.80. Fg,83.50.Ax,62.20.de A collection of spherical particles, interacting via a finite-range repulsive (contact) potential, unjams from a solid to a non-rigid state when being decompressed below a critical volume-fraction, φ c [1,2]. This transition, which has been given the name "point J", is accompanied by several interesting and nontrivial scaling relations in the solid phase [1,3]. There, pressure and linear elastic shear modulus vanish as does the ratio of shear to bulk modulus. The average number of particle contacts jumps from a finite value z 0 at point J to zero just below the transition. The value of z 0 is compatible with Maxwell's estimate for the rigidity transition and signals the fact that at point J each particle has just enough contacts for a rigid/solid state to exist (which is called "isostatic" state) [4,5]. Above the transition, additional contacts are generated according to the surprisingly universal law, δz ∼ δφ 1/2 . While the system moves away from its isostatic state the effective size of the remaining isostatic regions, l ⋆ ∼ δz −1 [6,7] has been argued to provide the diverging length-scale that endows point J with a certain "criticality". Based on these findings, jamming has been regarded as a "mixed" transition that shares properties of both, discontinuous (jump in contact number) and continuous phase-transitions (diverging length-scale) [8,9].A different route to approach point J from the fluid phase has been used in the flow simulations of Refs. [10,11,12]. Several scaling relations have been reported, some of which depend on model details. Some others seem to share the universality encountered in the solid phase. In contrast to the linear elastic properties in the solid phase, only little understanding about the flow properties and their relation to nearby point J has been achieved up to now.In this Letter, we employ a quasistatic simulation technique that studies the borderline between fluid and solid state. As we will see, it combines both aspects, the elasticity of a solid and the flow of a fluid, in one simulation. This fact allows us to get insight into how point J, and its isostatic state, affects the flow properties close by. In particular we show that jamming (i.e. the development of a finite shear stress) as probed by shear flow should be viewed as a continuous transition, with isostatic effects showing ...

We present a theory for the elasticity of cross-linked stiff polymer networks. Stiff polymers, unlike their flexible counterparts, are highly anisotropic elastic objects. Similar to mechanical beams, stiff polymers easily deform in bending, while they are much stiffer with respect to tensile forces ͑"stretching"͒. Unlike in previous approaches, where network elasticity is derived from the stretching mode, our theory properly accounts for the soft bending response. A self-consistent effective medium approach is used to calculate the macroscopic elastic moduli starting from a microscopic characterization of the deformation field in terms of "floppy modes"-low-energy bending excitations that retain a high degree of nonaffinity. The length scale characterizing the emergent nonaffinity is given by the "fiber length" l f , defined as the scale over which the polymers remain straight. The calculated scaling properties for the shear modulus are in excellent agreement with the results of recent simulations obtained in two-dimensional model networks. Furthermore, our theory can be applied to rationalize bulk rheological data in reconstituted actin networks.