A variational critical-state theory of friction

Abstract

Friction plays a fundamental role in many natural processes, including earthquakes, landslides, and volcanic eruptions. Earthquakes occur when highly compressed fault surfaces accumulate large enough shear stresses, causing the faults to move relative to one another, or slip. The slip is accommodated within a thin layer of comminuted granular material -- called fault gouge -- between the fault surfaces. As a result, characterizing the mechanical behavior of fault gouge in response to shear is a major open problem in earthquake source physics. Modeling gouge is complicated by large deformations, inelasticity, rate dependence, and volumetric changes. As such, researchers typically rely on empirical formulations to capture the effective response. Here, we systematically develop a variational, finite-kinematics framework for fault gouge. We first describe a general theory for a rigid-viscoplastic, pressure-sensitive material, where the plasticity evolution follows from the principle of maximum dissipation. Then, we specialize the governing equations for a Cam-Clay material within a shearing and dilating layer. We rely on convexity considerations and experimental observations from consolidation tests of granular layers to calibrate the model and develop explicit solutions for the rate- and state- dependent response of the model to shear tests under constant compressive normal stress and prescribed shearing rate. To validate the model, we select common rate functions and compare numerical material point tests and theoretical solutions to standard laboratory experiments of shearing granular layers. Lastly, we discuss connections of the model to empirical rate-and-state friction laws.

Figures (12)

• Figure 1: Experimental results on the effect of (a) grain diameter and (b) porosity from oedometer tests on sand, reproduced with permission from Brzesowsky2014a. Increasing porosity or grain diameter corresponds to less consolidated granular materials.
• Figure 2: Schematic representation of the effect of consolidation on the (a) shear stress and (b) volumetric response of sheared granular layer. Dense granular assemblies dilate and soften, while loosely compacted layers harden and compact Das2019.
• Figure 3: Response of a granular layer to stepped loading, adapted with permission from Rathbun2013. (a) The shear stress increases (decreases) with step increases (decreases) in the shear displacement rate. (b) The layer dilates (compacts) in response to faster (slower) loading rate. Both the shear stress and layer height approach a rate-dependent steady-state value at constant loading rate.
• Figure 4: Elliptic yield surface in the $(p,q)$ plane. The critical state line (CSL) corresponds to the stress states for which the material response is isochoric.
• Figure 5: Shearing and dilating shear layer of fault gouge with initial height $h_0$ and deformed height $h$.