LETTER doi:10.1038/nature10667 Jamming by shear Dapeng Bi 1 , Jie Zhang 2,3,4 , Bulbul Chakraborty 1 & R. P. Behringer 4 A broad class of disordered materials including foams, glassy molecular systems, colloids and granular materials can form jammed states. A jammed system can resist small stresses without deforming irreversibly, whereas unjammed systems flow under any applied stresses. The broad applicability of the Liu–Nagel jamming concept 1,2 has attracted intensive theoretical and modelling interest but has prompted less experimental effort 1–6 . In the Liu–Nagel framework, jammed states of athermal systems exist only above a certain critical density. Although numerical simulations for particles that do not experience friction broadly support this idea 7–13 , the nature of the jamming transition for frictional grains is less clear 14–17 . Here we show that jamming of frictional, disk-shaped grains can be induced by the application of shear stress at densities lower than the critical value, at which isotropic (shear-free) jamming occurs. These jammed states have a much richer phenomenology than the isotropic jammed states: for small applied shear stresses, the states are fragile, with a strong force network that percolates only in one direction. A minimum shear stress is needed to create robust, shear-jammed states with a strong force network percolating in all directions. The transitions from unjammed to fragile states and from fragile to shear-jammed states are controlled by the fraction of force-bearing grains. The fractions at which these transitions occur are statistically independent of the density. Jammed states with densities lower than the critical value have an anisotropic fabric (contact network). The minimum anisotropy of shear-jammed states vanishes as the density approaches the critical value from below, in a manner reminiscent of an order–disorder transition. Cohesionless granular materials form jammed states only under external stress, as explored extensively in the soil mechanics literature 18 . In the zero-temperature (T 5 0) plane of the Liu–Nagel jamming diagram 1–3 (Fig. 1a), increased packing fraction (w) induces jamming and positive pressure, and shear stress (t) induces irreversible flow at the yield stress line (Fig. 1a, black line). Simulations of frictionless grains typically probe jamming near the critical point at T 5 0, t 5 0 and w 5 w J , through isotropic compression or decompression 7–9 , or along the yield stress line 11–13,19,20 . The numerical value of w J depends on the protocol for preparing the jammed states. However, the characteristics of the transition are robust 21 . Only a few experi- ments 4–6,22 have investigated the Liu–Nagel jamming model 1,2 for physical systems consisting of particles with friction. For example, by using isotropically confined frictional disks it was found 4 that fric- tion only weakly affects certain aspects of jamming, such as the pack- ing fraction (w<0:842), but that other aspects, such as average number of contacts at jamming, are more strongly dependent on friction, as expected 15 . We report stable static states that jam only under a minimum shear stress. These states are outside the jammed region of Fig. 1a and alter the jamming diagram as illustrated in Fig. 1b. Of special note is the line separating two qualitatively different classes of states: the fragile states and the shear-jammed states. As we show below, this line is the locus of shear stresses marking a percolation transition. Shear-jammed states have not been reported in typical (that is, frictionless) models of jamming, which involve isotropically compressed particles and where additional relaxation may be induced to find a lower-energy state 7,15 . Our systems differ from these models in several key aspects: they consist of frictional (photoelastic) disks (Supplementary Fig. 1) pre- pared at densities below (and above) w J ; they are subjected to pure (volume-preserving) shear applied in small strain steps, allowing the system spontaneously to relax between steps; and they rest on a weakly frictional substrate, with forces that are an order of magnitude smaller than typical interparticle forces at jamming. We obtained stress data, that is, values of t, pressure (P) and the fabric tensor ( ^ R). The eigenvalues, R 1 and R 2 , of ^ R yield the mean contact number, Z 5 R 1 1 R 2 , and a measure of contact anisotropy, r 5 R 2 2 R 1 . In addition, we analysed the spatial organization of contact forces. Experimental details can be found in Methods. Ascertaining that states are macroscopically jammed is non-trivial. Necessary requirements are non-zero P and t and the ability to resist any small incremental stress. For frictionless grains, these conditions are met if Z exceeds Z iso (refs 7, 11), where the number of mechanical equilibrium constraints equals the number of degrees of freedom. For large N, isotropic jammed states exist for w $ w J , where w J depends only on the spatial dimension for a given protocol 7 . For frictional grains, a minimal parameter set for jamming has not been clearly determined. Although Z is a key parameter for mechanical stability, the minimum number of contacts needed for jamming can span a range of values, depending on friction and preparation 14,15,23 . For the disks used here, a reasonable criterion for isotropic jamming was found to be Z $ 3.0 to ,3.1 (ref. 4), for which w J <0:842 (Supplementary Fig. 3). A distinguishing property of granular jammed states is the existence of force networks. To characterize the states obtained by shearing at w , w J , we consider the contact force and fabric networks, and their correlation with the properties of the stress tensor 24 and the fraction of non-rattler grains, f NR (Supplementary Figs 2 and 4). It is known from earlier studies that force networks in jammed packings of dry grains 1 Martin Fisher School of Physics, Brandeis University, Waltham, Massachusetts 02454, USA. 2 Institute of Natural Sciences and Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China. 3 Department of Physics, Indiana University Purdue University Fort Wayne, Fort Wayne, Indiana 46805, USA. 4 Department of Physics, Duke University, Durham, North Carolina 27708, USA. Jammed Unjammed Jammed Unjammed F a b SJ τ τ φ φ J φ J φ S φ Figure 1 | Jamming phase diagrams in the T 5 0 plane. a, Original Liu– Nagel jamming phase diagram 1 . The boundary between unjammed and jammed regions is the yield stress line. Unjamming can be induced by decreasing the packing fraction or increasing the shear stress. b, Generalized jamming diagram including the shear-jammed (SJ) and fragile (F) states. Along the w axis, there are two special densities: w S , below which there is no shear jamming, and w J , above which isotropically jammed states exist. For w S # w # w J , jamming can occur with application of shear stress. 15 DECEMBER 2011 | VOL 480 | NATURE | 355 Macmillan Publishers Limited. All rights reserved ©2011