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Numerical simulation of elastoplastic problems by Brezis–Ekeland–Nayroles non-incremental variational principle

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Abstract

This paper is concerned with the energy-dissipation Brezis–Ekeland–Nayroles variational principle (BEN principle) for the numerical study of quasi-static elastoplastic and viscoplastic problems in small strains. This principle is based on the dissipation potential and its Fenchel transform and allows to have a consistent view of the whole evolution by computing the nonlinear response along the whole time history as a solution of a suitable minimization problem. In the present work, the BEN principle is applied to address the elastic perfectly plastic and viscoplastic thick hollow cylinder subjected to internal pressure. It turns out that the BEN variational formulation is based on a two-field functional, that leads naturally to discretize the displacement and stress fields. We present the detailing of the discretization and the numerical implementation of the minimization problem by using the mixed finite element method which is more efficient to enforce the yield condition. Computational accuracy and efficiency of the BEN principle is assessed by comparing the numerical results with the analytical ones and the simulations derived by the classical step-by-step finite element procedure.

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Acknowledgements

The work is funded by the international cooperation project Dissipative Dynamical Systems by Geometrical and Variational Methods and Application to Viscoplastic Structures Subjected to Shock Waves (DDGV) supported by the Agence Nationale de la Recherche (ANR) and the Deutsche Forchungsgemeinschaft (DFG).

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Correspondence to Abdelbacet Oueslati.

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Cao, X., Oueslati, A., Nguyen, A.D. et al. Numerical simulation of elastoplastic problems by Brezis–Ekeland–Nayroles non-incremental variational principle. Comput Mech 65, 1005–1018 (2020). https://doi.org/10.1007/s00466-019-01805-0

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