Stabilized finite element method for viscoplastic flow: formulation with state variable evolution

A stabilized, mixed nite element formulation for modelling viscoplastic ow, which can be used to model approximately steady-state metal-forming processes, is presented. The mixed formulation is expressed in terms of the velocity, pressure and state variable elds, where the state variable is used to describe the evolution of the material’s resistance to plastic ow. The resulting system of equations has two sources of well-known instabilities, one due to the incompressibility constraint and one due to the convection-type state variable equation. Both of these instabilities are handled by adding mesh-dependent stabilization terms, which are functions of the Euler–Lagrange equations, to the usual Galerkin method. Linearization of the weak form is derived to enable a Newton–Raphson implementation into an objectoriented nite element framework. A progressive solution strategy is used for improving convergence for highly non-linear material behaviour, typical for metals. Numerical experiments using the stabilization method with hierarchic shape functions for the velocity, pressure and state variable elds in viscoplastic ow and metal-forming problems show that the stabilized nite element method is e ective and e cient for non-linear steady forming problems. Finally, the results are discussed and conclusions are inferred.