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.

中文翻译：

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Brezis-Ekeland-Nayroles非增量变分原理对弹塑性问题的数值模拟

本文关注能量耗散 Brezis-Ekeland-Nayroles 变分原理（BEN 原理），用于数值研究小应变中的准静态弹塑性和粘塑性问题。该原理基于耗散势及其 Fenchel 变换，并通过计算沿整个时间历史的非线性响应作为合适的最小化问题的解决方案，允许对整个演化有一致的看法。在目前的工作中，BEN 原理被应用于处理受内压作用的弹性完美塑性和粘塑性厚空心圆柱体。事实证明，BEN 变分公式基于双场泛函，这自然会导致离散化位移场和应力场。我们通过使用更有效地强制屈服条件的混合有限元方法来详细介绍离散化和最小化问题的数值实现。BEN 原理的计算精度和效率是通过将数值结果与解析结果以及经典逐步有限元程序得出的模拟结果进行比较来评估的。