One of the most difficult problems in the modeling of deformation processes is the appropriate treatment of frictional boundary condition. The main objectives of this paper are (i) to show that application of the maximum friction law (friction stress is equal to the shear yield stress) leads to solution behavior that may be used to describe different physical phenomena near the friction surface, and (ii) to develop a method to deal with numerical difficulties. In the case of a rigid/perfectly plastic material model, the velocity fields adjacent to surfaces of maximum friction must be describable by nondifferentiable functions where the maximum shear strain rate and the effective strain rate approach infinity. In particular, the equivalent strain rate follows an inverse square root law near the friction surfaces. The coefficient of the singular term in the expansion of the equivalent strain rate near the maximum friction surface may be called, by analogy to the stress intensity factor in linear/elastic fracture mechanics, the strain rate intensity factor. This analogy can be further extended giving a method for predicting material behavior in a thin layer near the friction surface and a method for developing effective numerical codes where the specific approximating functions should be used in elements near this surface. In particular, modifications of the material model may or may not change the singular character of the solutions in the vicinity of maximum friction surfaces. Because of this feature of constitutive equations, several classical problems in plasticity that are often adopted for metal forming applications have no solution. On the other hand, the solutions based on the double shearing model of pressure dependent plasticity, which is applicable for modern aluminum alloys, show the same behavior as those of classical plasticity, and the corresponding generalizations of the aforementioned classical problems are possible.