Micromechanics of rock friction 2: Quantitative modeling of initial friction with contact theory

Abstract

A constitutive model is developed which predicts the mechanical properties of two rough surfaces in contact under shear load during the early stages of the development of frictional sliding. The model includes the development of slip at the contacts, a phenomenon which begins immediately upon shear loading. Upon initial application of the normal load, the model predicts that the joint consists of a finite number of contacts which are subject to a wide variety of local normal loads. As the shear load is increased, sliding of the contacts develops progressively, with the contacts under low local normal load sliding first. This gradual development of sliding is the cause of the experimentally observed nonlinear force‐displacement relation for deformation of the joint in shear. Two asperity scale strength laws are examined, one based on the adhesion theory of friction and the other based on observations of frictional strength for brittle elastic solids. The model is tested with experiments on lapped surfaces of Westerly Granite with a variety of surface roughnesses and under a range of normal loads from 10 to 35 MPa. Geometric parameters used in the model are constrained by direct measurement of surface profiles. For both strength laws, the model quantitatively predicts the shear compliance and development of slip for the first few microns of shear displacement, successfully describing the effect of surface roughness and normal load. This model helps explain the initial yield in the friction curve which corresponds to the gradual transition from the elastic deformation and partial slip of asperity contacts to a condition of fully sliding contacts. When a large population of contacts are fully sliding, the model under‐estimates the frictional strength, indicating that displacement strengthening mechanisms are important to the ultimate frictional strength of rocks.

Rate and state dependent friction and the stability of sliding between elastically deformable solids

“…That is consistent with an estimate by Boitnott et al (1992) and with microhardness indentation strengths measured independently by Dieterich and Kilgore. (Since typically c = c = = ≈ 0:6 for rocks such as granite, that implies that c is of order 0:1 in such systems; 0:1 is a standard estimate of the theoretical shear strength, and that suggests an unusual type of local plasticity at the contacts.)…”

On the Relationship Between Ultrasonic and Micro-Structural Properties of Imperfect Interfaces in Layered Solids

“…As pointed out by Boitnott et al [12], the initial transverse stiffness, that is, the value of K T at w = 0, is independent of the constitutive law used to describe the shear interaction between the contacts. This observation is relevant to this work insofar as it implies that the model's ability to reproduce the experimental results depends strongly on the assumptions made about the probability density function <p(z).…”

“…The quantity do is related to the maximum height of the surface, Zmax, and depends on the degree of conformity of the two surfaces. The normal stiffness constant, KN, is found by differentiating P with respect to 0, Adopting a similar approach, Boitnott et al [12] derived the following relationship between the applied shear stress, t , and the relative displacement between the two rough surfaces, w,…”

“…Quite different approaches have been developed to model the elasto-plastic behavior of rough surfaces in contact [1,2,5,11,12,13,14,15,16]. In the most sophisticated models the contacts between the rough surfaces are assumed to occur at the surface asperities, and in many cases their interaction is assumed to be purely elastic [1,11,12,13].…”

“…The models were developed by Brown and Scholz [11] and by Boitnott et al [12] to describe the closure, 8, and the sliding of two rough surfaces against each other when they are subjected to an external normal or transverse load, respectively. The models account for the topography ofthe two rough surfaces through a fictitious composite surface.…”