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 die 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.
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