On the reducibility of solutions for the generalized yield criterion to solutions for Tresca's yield criterion under axial symmetry

Many continuum mechanics models are reduced to simpler models at certain parameter values. However, solutions for the general model may not converge to the corresponding solutions for a simpler model. In the mathematical theory of plasticity, the yield criterion completely determines the material's behavior if the associated plastic flow rule is accepted. In this paper, the reducibility of axisymmetric solutions for the generalized yield criterion to the corresponding solutions for Tresca’s criterion is investigated when the generalized yield condition tends to Tresca’s criterion. It is shown that there is no convergence if the maximum friction law is one of the boundary conditions. In this case, the solutions for both yield criteria are singular. In particular, the quadratic invariant of the strain rate tensor tends to infinity near the friction surface. The strain rate intensity factor controls the magnitude of this invariant in the vicinity of the friction surface. The strain rate intensity factor is involved in some constitutive equations for predicting the evolution of material properties near frictional interfaces in metal forming processes. In this paper, using the solution of a specific boundary value problem, the behavior of this factor is investigated when the generalized yield criterion tends to Tresca’s criterion. It is shown that the strain rate intensity factor continuously changes when the generalized yield criterion deviates from Tresca’s yield criterion. This behavior of the strain rate intensity factor justifies its use in the constitutive equations for the evolution of material properties near friction surfaces.