Approximate closed-form equations are derived for normal contact force between nominally flat rough surfaces in dry contact. The formulation is based on the asperity level interaction in which adhesive forces between two asperities as well as elastic and rate-dependent forces are included. The elastic and time rate-dependent portion of force is derived using a viscoelastic interaction of the two asperities. Statistical consideration of rough surfaces then furnishes the mathematical formulation of total normal force due to adhesion, elastic and rate-dependent properties of the solids in contact. The probabilistic formulation of contact force leads to integral equations. From these are derived approximate closed-form expressions that relate the micron-scale properties of the surfaces to the macro-scale behavior in the form of the total normal contact force between the surfaces. An investigation is carried out to ascertain the relative significance of adhesive, elastic and time-rate dependent forces with respect to material properties of the solids, micron-scale geometry of the surfaces and sliding conditions.

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