The relative motion between two surfaces under a normal load is impeded by friction. Interfacial junctions are formed between surfaces of asperities, and sliding inception occurs when shear tractions in the entire contact area reach the shear strength of the weaker material and junctions are about to be separated. Such a process is known as a static friction mechanism. The numerical contact model of dissimilar materials developed by the authors is extended to evaluate the maximum tangential force (in terms of the static friction coefficient) that can be sustained by a rough surface contact. This model is based on the Boussinesq–Cerruti integral equations, which relate surface tractions to displacements. The materials are assumed to respond elastic perfectly plastically for simplicity, and the localized hardness and shear strength are set as the upper limits of contact pressure and shear traction, respectively. Comparisons of the numerical analysis results with published experimental data provide a validation of this model. Static friction coefficients are predicted for various material pairs in contact first, and then the behaviors of static friction involving rough surfaces are extensively investigated.