Geng, J; Behringer, RP, *Slow drag in two-dimensional granular media*,
Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, vol. 71 no. 1
(2005) [doi] .
**Abstract:**

*We study the drag force experienced by an object slowly moving at constant velocity through a two-dimensional granular material consisting of bidisperse disks. The drag force is dominated by force chain structures in the bulk of the system, thus showing strong fluctuations. We consider the effect of three important control parameters for the system: the packing fraction, the drag velocity and the size of the tracer particle. We find that the mean drag force increases as a power law (exponent of 1.5) in the reduced packing fraction, (γ-γ c)/γ c as γ passes through a critical packing fraction, γ c. By comparison, the mean drag grows slowly (basically logarithmic) with the drag velocity, showing a weak rate dependence. We also find that the mean drag force depends nonlinearly on the diameter, a of the tracer particle when a is comparable to the surrounding particles' size. However, the system nevertheless exhibits strong statistical invariance in the sense that many physical quantities collapse onto a single curve under appropriate scaling: force distributions P(f) collapse with appropriate scaling by the mean force, the power spectra P(ω) collapse when scaled by the drag velocity, and the avalanche size and duration distributions collapse when scaled by the mean avalanche size and duration. We also show that the system can be understood using simple failure models, which reproduce many experimental observations. These observations include the following: a power law variation of the spectrum with frequency characterized by an exponent α=-2, exponential distributions for both the avalanche size and duration, and an exponential fall-off at large forces for the force distributions. These experimental data and simulations indicate that fluctuations in the drag force seem to be associated with the force chain formation and breaking in the system. Moreover, our simulations suggest that the logarithmic increase of the mean drag force with rate can be accounted for if slow relaxation of the force chain networks is included. © 2005 The American Physical Society.*