Approximate closure equations for elastic contact of nominally flat rough surfaces

A hybrid interactive/optimization technique is used to derive in approximate closed form equations relating contact load to mean plane separation. Equations governing Hertz contact for the interaction of surface asperities are considered in which asperity shoulder-to-shoulder contact results in normal and tangential components of force. The normal component of asperity force is summed statistically to find total normal force between the two surfaces. The tangential component of asperity force, summed statistically, yields zero total force due to the symmetry of the distributed tangential force components. The tangential force over a half-plane corresponding to a select direction is found accounting for the directionality of the tangential component of asperity forces. Two sets of approximate equations are found for each of the normal and half-plane tangential force components. The simplest forms of the approximate equations achieve accuracy to within five (5) percent error, while other forms yield approximation error within 0.2 percent.