R. R. Hartley

Logarithmic rate dependence of force networks in sheared granular materials

Many models of slow, dense granular flows assume that the internal stresses are independent of the shearing rate. In contrast, logarithmic rate dependence is found in solid-on-solid friction, geological settings and elsewhere. Here we investigate the rate dependence of stress in a slowly sheared two-dimensional system of photoelastic disks, in which we are able to determine forces on the granular scale. We find that the mean (time-averaged) stress displays a logarithmic dependence on the shear rate for plastic (irreversible) deformations. However, there is no perceivable dependence on the driving rate for elastic (reversible) deformations, such as those that occur under moderate repetitive compression. Increasing the shearing rate leads to an increase in the strength of the force network and stress fluctuations. Qualitatively, this behaviour resembles the changes associated with an increase in density. Increases in the shearing rate also lead to qualitative changes in the distributions of stress build-up and relaxation events. If shearing is suddenly stopped, stress relaxations occur with a logarithmic functional form over long timescales. This slow collective relaxation of the stress network provides a mechanism for rate-dependent strengthening.

Why do granular materials stiffen with shear rate? Test of novel stress-based statistics.

Recent experiments exhibit a rate dependence for granular shear such that the stress grows linearly in the logarithm of the shear rate, gamma. Assuming a generalized activated process mechanism, we show that these observations are consistent with a recent proposal for a stress-based statistical ensemble. By contrast, predictions for rate dependence using conventional energy-based statistical mechanics to describe activated processes, predicts a rate dependence of (ln(gamma))(1/2).

Force networks and elasticity in granular silos

Abstract.We have made experimental observations of the force networks within a two-dimensional granular silo similar to the classical system of Janssen. Models like that of Janssen predict that pressure within a silo saturates with depth as the result of vertical forces being redirected to the walls of the silo where they can then be carried by friction. We use photoelastic particles to obtain information not available in previous silo experiments --the internal force structure. We directly compare various predictions with the results obtained by averaging ensembles of experimentally obtained force networks. We identify several differences between the mean behavior in our system and that predicted by Janssen-like models: We find that the redirection parameter describing how the force network transfers vertical forces to the walls varies with depth. We find that changes in the preparation of the material can cause the pressure within the silo to either saturate or to continue building with depth. Most strikingly, we observe a nonlinear response to overloads applied to the top of the material in the silo. For larger overloads we observe the previously reported “giant overshoot” effect where overload pressure decays only after an initial increase (G. Ovarlez et al., Phys. Rev. E 67, 060302(R) (2003)). For smaller overloads we find that additional pressure propagates to great depth. Analysis of the differences between the inter-grain contact and force networks suggests that, for our system, when the load and the particle weight are comparable, particle elasticity acts to stabilize the force network, allowing deep propagation. For larger loads, the force network rearranges, resulting in the expected, Janssen-like behavior. Thus, a meso-scale network phenomenon results in an observable nonlinearity in the mean pressure profile.

Segregation by friction

Granular materials are known to separate by size under a variety of circumstances. Experiments presented here and elucidated by modeling and MD simulation document a new segregation mechanism, namely segregation by friction. The experiments are carried out by placing steel spheres on a horizontal plane enclosed by rectangular sidewalls, and subjecting them to horizontal shaking. Half the spheres are highly smooth; the remainder are identical to the first half, except that their surfaces have been roughened by chemical etching, giving them higher coefficients of friction. Segregation due to this difference in friction occurs, particularly when the grains have a relatively long mean free path. In the presence of an appropriately chosen small hill in the middle of the container, the grains can be made to completely segregate by friction type.

Statistical Properties of Dense Granular Matter

We review recent work characterizing force fluctuations and transmission in dense granular materials. These forces are carried preferentially on filimentary structures known as force chains. When a system is deformed, these chains tend to resist further deformation; with continued deformation, chains break and rearrange, leading to large spatio-temporal fluctuations. We first consider experiments on force fluctuations, diffusion and mobility under steady-state shear. We then turn to force transmission in static systems as determined by the response to a small point force. These experiments show that the packing structure and friction play important roles in determining the force transmission. Disordered highly frictional packings have responses that are similar to that of an elastic solid. Ordered packings show responses that may be described either by anisotropic elasticity or by a wave-like description.

Logarithmic Strengthening of Granular Materials with Shear Rate

Experiments on sheared granular materials show that the stresses grow as the first power of the log of the shear rate, γ. We suggest that this may be evidence of the stress ensemble recently proposed by Henkes, O’Hern, and Chakraborty. The picture that we propose is that under steady shearing, the local force network builds up over time, and then fails when the force on the network exceeds a characteristic value. In analogy to soft glassy rheology, we assume that this is an activated process, but now, with the Boltzmann factor replaced by the stress ensemble analogue. We assume that the probability that a local part of the network fails is proportional to exp[(σ−σm)/σo], where σ is the local stress, σm is a failure threshold, and σo is related to the generalized temperature, α, of Henkes and Chakraborty. It is then possible to show that these assumptions lead to logarithmic increases in the stress as a function of γ. This contrasts with the SGR result that the stress grows as the square root of log(γ).