A non-incremental numerical method for dynamic elastoplastic problems by the symplectic Brezis–Ekeland–Nayroles principle

Abstract The goal of the present work is the demonstration of the ability of the Symplectic Brezis–Ekeland–Nayroles (SBEN) principle for the numerical simulation of dynamic elastoplastic problems in small strains. This non-incremental approach is an alternative method to conventional step-by-step techniques, as it allows the computation of the whole evolution curve over the entire loading history. We show that the SBEN variational formulation yields to a constrained time–space minimization problem . The cost function depends on stress, displacements, and plastic multiplier fields, which, naturally, leads to a mixed finite element discretization . The details of the solution algorithm are illustrated through the numerical study of the elastoplastic response of the thin and thick pressurized tubes, including inertia effects. Moreover, the balance of momentum is handled using two approaches. Firstly, the Schaefer superposition technique is considered, where the balance of momentum is exactly satisfied. Secondly, the balance of momentum is considered as an optimization constraint and satisfied only at the Gauss integration point level of the finite element problem. The accuracy and efficiency of the SBEN principle are assessed by comparing the numerical results with the analytical solutions (for the thin tube) and the predictions derived by the conventional incremental finite element procedure.