Temporal force ﬂuctuations measured by tracking individual particles in granular materials under shear Eric I. Corwin, Eric T. Hoke, Heinrich M. Jaeger, and Sidney R. Nagel James Franck Institute, Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA Received 7 August 2007; revised manuscript received 2 May 2008; published 27 June 2008 The network of forces between contacts in a granular material determines the response of the material to an external perturbation. As the shear stress is increased, a static, rigid material can start to yield and ﬂow as it becomes unjammed. The time- and ensemble-averaged distribution of forces shows a marked change in character at this jamming-unjamming transition. Here, we investigate the temporal ﬂuctuations of the forces between individual particles and the external boundary in a dense, slowly sheared system. We ﬁnd that the ﬂuctuations occur over a very broad range of time scales, resulting in power spectra characterized by a 1 / f -type shape and a rollover to 1 / f 2 behavior above a turnover frequency f  . At very low frequencies additional enhancements over the 1 / f contributions are observed in regions of signiﬁcant in-plane shear. Moreover, the magnitude of the force ﬂuctuation spectra produced by one particle is different from those produced by its neighbors. Such heterogeneities persist over periods of time exceeding our measurement interval 60 h. In this respect they are reminiscent of the dynamic heterogeneities that occur in liquids supercooled toward the glass transition temperature. DOI: 10.1103/PhysRevE.77.061308 PACS numbers: 45.70.-n, 74.40.+k, 71.55.Jv, 47.57.Gc Granular material is precariously perched on the threshold between two very different kinds of behavior—depending on the nature of the force network between its constituent par- ticles, a granular material can either maintain its rigidity and withstand shearing forces like a solid or yield to the shear and ﬂow as if it were a ﬂuid 1. What are the differences in the force networks between the rigid and the ﬂowing states that allow such a dramatic change in material properties as the system unjams 2,3? We have previously shown that in the ﬂowing, or sheared, regime the distribution of contact forces between particles and the wall can be modeled as an equilibrium system with an effective temperature, whereas in the static, rigid, solid the force-magnitude distributions are those of an out-of-equilibrium system 4. These measure- ments considered ensemble- and time-averaged distributions. In the present paper, we study how the contact force of a single particle ﬂuctuates over time as the granular material is sheared. We ﬁnd that these local ﬂuctuations have an ex- tremely broad distribution of time scales. Moreover, when we measure the ﬂuctuations of seemingly identical neighbor- ing particles, we ﬁnd large differences in the intensity of the ﬂuctuations over the course of our longest measurement in- terval 60 h. This result is surprising, since the particles are all in the same environment undergoing shear, and is remi- niscent of the dynamic heterogeneities that appear in glasses near the glass transition. The time- and ensemble-averaged distribution of contact forces has been investigated extensively for static, un- sheared, granular media 5–18. By contrast, the temporal variations of contact forces and impulses in sheared material have been studied in only a few situations by experiment 18–20 and simulation 21. Experiments in two- dimensional 2D systems tracked individual particles. Ex- periments in 3D systems used a ﬁxed force transducer mounted ﬂush with the container wall; when beads traveled over the transducer surface the measurements recorded the net force or impulse of all particles in contact with it at a given time which, depending on the transducer area and bead size, could be hundreds of beads. The resultant data therefore were a convolution of the force ﬂuctuations of many beads with the step function of each bead entering and exiting the measurement surface. Our experiments use a different ap- proach that tracks the motion of single particles at the bound- ing surface of 3D systems and records the individual contact force variations locally. In order to measure force ﬂuctuations occurring on a single particle, we use the same shear cell, shown schemati- cally in Fig. 1, as was used to measure the time-averaged distribution of contact forces between particles and the bot- tom plate 4. Particles are sheared by rotating the roughened plunger at the top surface at ﬁxed angular velocity  =2 f drive . This plunger is maintained at constant vertical pressure by loading it with a stiff spring monitored by an- other force transducer. Because the bottom and sidewalls of this shear cell are stationary, the rotating plunger results in a complex shear ﬂow proﬁle where particles are retarded by friction with the side and bottom but nevertheless can slip along those surfaces. Granular material, consisting of 3.06  0.04 mm diameter soda lime glass beads, is conﬁned in a cylinder or radius R =6.5 cm and ﬁlled to a height of approximately H =4.5 cm 15 bead diameters. For the ex- periments reported here, the plunger at the top surface rotates at a frequency f drive = 0.18 Hz and applies a compressive force of 1380 N, chosen to bring the average force on each bead into a range appropriate for easy detection and mea- surement. The contact forces in the normal direction applied by each individual bead on the bottom plate are detected using a photoelastic force transducer covering the whole bottom sur- face of the cylinder. This force transducer consists of a thin layer of photoelastic polymer Vishay Micro-Measurements, Raleigh, NC, thickness 0.25 mm, hardness 80 Durometer, mirrored on its top surface and bonded to a 9.5-mm-thick clear glass plate for mechanical support. The position and PHYSICAL REVIEW E 77, 061308 2008 1539-3755/2008/776/0613086 ©2008 The American Physical Society 061308-1