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- σ_m)/σ_o], where s is the local stress, sm is a failure threshold, and σ_o 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 l og(γ).