On the performance of two tangent operators for finite element analysis of large deformation inelastic problems

An important step in the solution of non‐linear deformation problems using a Newton‐Raphson type of iterative scheme is the calculation of a tangent operator (the so‐called Jacobian) by linearizing the involved field equations. In this paper, full linearization of the virtual work equation is performed in an updated Lagrangian framework together with a calculation of the consistent linearized material moduli. To update the material state, a material frame indifferent two parameter integration scheme introduced earlier by the authors is used. In general, the reference configuration is updated after each iteration to coincide instantaneously with the present guess of the unknown equilibrium configuration. Another approach is to use the previous equilibrium state (at the beginning of a time step) as the reference configuration until the new equilibrium configuration at the end of the time step is found. The present paper explores the performance of two different Jacobians based on the above two approaches, with emphasis on their accuracy and convergence characteristics when large incremental steps are used. Finally, some details of their finite element implementation are given and results for several numerical tests including upset forging and extrusion of an axisymmetric billet are presented and discussed.