Modeling the mechanics of rate and state friction with linear viscoelasticity

In slide-hold-slide friction experiments, rock surfaces loaded by normal stress sigma slide against one another at velocity V until steady-state conditions prevail; the applied shear stress tau required for steady-state sliding is f sigma, where f is the friction coefficient. Sliding is then stopped abruptly for a specified hold time t(H), and then resumed at velocity V. Shear stress tau and the separation Delta w between the rough surfaces are measured throughout the test. During the hold period, both applied shear stress and separation are found to decay with time from their steady-state values. We show that the constitutive behavior in shear during the hold period is in effect a relaxation test, and likewise, the closure measurements describe a creep test. Analyzing the data in detail, we find that they can be modeled as the response of a standard linear solid, with a characteristic time T-1 for creep of 1900 s and a characteristic time T-2 for relaxation of 1400 s. The shear stress required to initiate sliding at the end of the hold period is found to peak Delta f sigma above the steady-state stress f sigma, then decay over a distance D-c to the steady-state value. In our experiments, measurements of separation and shear load were made for hold periods as long as 10(6) s, i.e., one hundred times longer than in similar previous studies. We find that our measured values of Delta f increase gradually with increasing hold times - in agreement with previous measurements up to about 10(4) s - but then approach a limiting value asymptotically. In our analysis we show that Delta f is equal to (1 + f(2)) Delta delta(H)/f lambda, where Delta delta(H) is the total decrease in separation during the hold period and lambda is a length parameter that describes the roughness of the surfaces. The parameter D-c is found to be given by Delta H-delta/Delta f. Using measurements from our experiments, we find that both relationships are valid to a reasonable degree, considering the uncertainties in the data and the simplicity of the analysis.