Figure 2: Average yield-stress as function of volume-fraction (left) and pressure (right). The yield-stress is determined as an average over stress- values just before plastic events occur, i.e at the top end of each elastic branch. Figure 1: Stress-strain relation for two different volume-fractions, φ = 0 . 847 (top) and φ = φ$_{c}$ = 0 . 8433 (bottom). At volume-fractions close to φ$_{c}$ the signal is intermittent showing long quiet regions where the system flows without the building up of stress. signal shows a coexistence between jammed and “freely-flowing” states. This is evidence of a distri- bution of jamming thresholds, P ( φ$_{c}$ ), which sharp- ens with increasing the system-size. 16,28 A finite- size scaling analysis of this distribution allows one to extract the critical volume-fraction. To this end we count the number of jamming events that lead from the freely-flowing state to the jammed state and back $^{1}$. We find a maximum number of events at a certain φ$_{c}$ ( L ), which can be extrapolated to L = ∞ to define the critical volume fraction of our simulation. The value we find, φ$_{c}$ = 0 . 8433 is slightly higher than what has been obtained pre- viously, however, evidence is mounting that φ$_{c}$ is non-universal 35 and depends on the details of the ensemble preparation. Scaling properties in the vicinity of a jamming threshold, on the other hand, appear to be universal. 35 In Fig. 2 we display the yield-stress σ$_{y}$ , as a func- tion of volume-fraction, δφ = φ − φ$_{c}$ , and pressure p . Finite-size effects are particularly strong when us- ing φ as control variable. In the intermittent regime the average stress levels off to a system-size depen- dent value. Much better scaling behavior can be obtained, when using pressure as control variable, as this is characterized by the same finite-size effects as the shear-stress. In the following we will therefore use pressure as control variable. Be aware, however, that we do not run pressure-controlled simulations, as for example Peyneau and Roux 25 but use the average pressure, 〈 p 〉 ( φ ), only to plot our simula- tion results. The value σ/p ≈ 0 . 1 obtained from Fig. 2 is consistent with these pressure-controlled simulations. On the other hand, the scaling with volume-fraction, p ∼ δφ 1 . $^{1}$, is slightly stronger than in linear elasticity at zero stress, 16,20 where the pressure simply scales as δφ . In view of the strong finite-size effects, the scaling with volume-fraction should, however, be taken with care. In the following we show results from five dif- ferent volume-fractions, φ = 0 . 846, 0 . 848, 0 . 85, 0 . 86 and φ = 0 . 9, which are all above φ$_{c}$ and outside the intermittent regime (i.e. no freely- flowing zero-stress states occur). For each volume- fraction we study three different system sizes with N = 900 , 1600 and 2500 particles. As reviewed in the introduction a hallmark of the elasticity of solids in the vicinity of point J is the scaling of the linear elastic shear modulus g ∼ p 1 / $^{2}$. Similarly, the number of inter-particle con- tacts scale as z = z$_{0}$ + Ap 1 / $^{2}$. We have analyzed the elastic branches in the steady-state flow to find (Figs. 3 and 4) that the same scaling properties characterize the average nonlinear elastic modulus g$_{avg}$ , which we define as 3.1.1 Elastic properties $^{1}$For a closer illustration of a jamming event see the sup- porting material to our previous paper 34 4  0 1 10 -4 2 10 -4 3 10 -4  1.51  1.55  1.59  1.63  1.67  0 2 10 -5 4 10 -5 6 10 -5  1.51  1.55  1.59  1.63  1.67 10 -5 10 -4 10 -3 10 -2 10 -4 10 -3 10 -2 10 -1 y ~x 1.05 N = 2500 1600 900 400 10 -6 10 -5 10 -4 10 -3 10 -2 10 -5 10 -4 10 -3 10 -2 10 -1 y p ~x 0.95 = 2500 1600 900 400