Effect of plastic compressibility on the strain rate intensity factor in compression of a material layer between parallel plates

Solutions of boundary value problems for several rigid plastic models may be singular. In particular, some components of the strain rate tensor and the quadratic invariant of the strain rate tensor may approach infinity in the vicinity of certain surfaces. Some models for predicting the high gradient of material properties near frictional surfaces in metal forming processes are based on such behavior of the velocity field. The coefficient of the leading singular term in a series expansion of the quadratic invariant of the strain rate tensor in the vicinity of frictional interfaces is called the strain rate intensity factor. The magnitude of the second invariant of the strain rate tensor is controlled by this factor that depends on geometric parameters of the boundary value problem and parameters involved in the material model. The present paper deals with the effect of plastic compressibility of the material that obeys the double shearing model on the strain rate intensity factor in compression of a plastic layer between two parallel plates. The system of equations comprising the equilibrium equations and constitutive equations is hyperbolic. It is assumed that the surface of the contact between the plates and the deforming material is an envelope of characteristics. An analytic solution of the boundary value problem is found under plane strain deformation. End effects near the free surface of the layer and its center are ignored. The dependence of the strain rate intensity factor on parameters of the boundary value problem including the parameter that controls plastic compressibility is found. In case of the plastically incompressible material, the solution coincides with the available solution.