Fig. 4: Effects of shear localization on the moment.

a Shear band thickness normalized by the particle diameter as a function of porosity. The data points represent experimental data. The solid line represents the linear fit with a = − 10.7 and b = 125, excluding the cross symbol at the lowest porosity in run #73 (0.15 ≲ ϕ ≲ 0.18), where many particles were not confined to the same monolayer. This exceptional datum is provisionally plotted at ϕ = 0.15. The dashed line indicates the minimum thickness, w/d = 1. The inset indicates the schematic model of the shear band with its outer boundary plane at a radial distance, r = R + w, where each contact patch has a diameter δ. b Mean moment as a function of porosity. The black and white circles represent the mean values from exponential fitting and raw data, respectively. The solid line represents the fitting curve of all the solid black circles with n = 5.0 (Equations (1) and (8)), except for the cross symbol as in (a). Error bars represent standard error based on 4–49 different moment sizes for the exponential fitting, also for (c). c Mean moment rate as a function of porosity. The black circles represent the mean values and the solid line indicates the reference curve of \({\dot{M}}_{0}={M}_{0}/T\) with constant slip velocity (Equations (1) and (2)). The dashed line indicates the maximum value at w/d = 1.