A Finite Element (FE) procedure based on a fully implicit backward Euler predictor/corrector scheme for the Cosserat continuum is here presented. The integration algorithm is suitable for yield and plastic potential surfaces with general shape in the deviatoric plane. The key element of the integration scheme is the spectral decomposition of the stress tensor, which is achieved, despite the lack of symmetry, because of the mathematical structure of the yield function and the set of invariants chosen as independent variables. It is also shown that the choice of invariants enables considerable mathematical simplifications, which result in the reduction of the system of equations and unknowns of the elasto-plastic problem from 19 to 1, and to rigorously handle the discontinuity at the apex of the surfaces. The algorithm has been implemented in a proprietary FE programme, and used for the constitutive model recently proposed by the same authors in this journal for the Cosserat continuum, which allows to set various classical failure criteria as yield and plastic potential surfaces. Numerical analyses have been conducted to simulate a biaxial compression test and a shallow strip footing resting on a Tresca, Mohr–Coulomb, Matsuoka–Nakai and Lade–Duncan soil. The benefits of the Cosserat continuum over the Cauchy/Maxwell medium are discussed considering mesh refinement, non-associated flow and softening behaviour.

中文翻译：

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Cosserat连续体各向同性弹塑性模型基于不变量的隐式积分算法

此处介绍了基于 Cosserat 连续统的完全隐式后向欧拉预测器/校正器方案的有限元 (FE) 过程。该积分算法适用于偏平面内具有一般形状的屈服面和塑性势面。积分方案的关键要素是应力张量的谱分解，尽管缺乏对称性，但由于屈服函数的数学结构和选择作为自变量的一组不变量，它得以实现。它还表明，不变量的选择可以实现相当大的数学简化，从而将弹塑性问题的方程组和未知数从 19 减少到 1，并严格处理曲面顶点处的不连续性。该算法已在专有的 FE 程序中实施，并用于同一作者最近在本期刊中为 Cosserat 连续体提出的本构模型，该模型允许将各种经典失效标准设置为屈服和塑性势表面。已经进行了数值分析来模拟双轴压缩试验和搁在 Tresca、Mohr-Coulomb、Matsuoka-Nakai 和 Lade-Duncan 土壤上的浅条形基础。考虑到网格细化、非关联流动和软化行为，讨论了 Cosserat 连续体相对于 Cauchy/Maxwell 介质的好处。已经进行了数值分析来模拟双轴压缩试验和搁在 Tresca、Mohr-Coulomb、Matsuoka-Nakai 和 Lade-Duncan 土壤上的浅条形基础。考虑到网格细化、非关联流动和软化行为，讨论了 Cosserat 连续体相对于 Cauchy/Maxwell 介质的好处。已经进行了数值分析来模拟双轴压缩试验和搁在 Tresca、Mohr-Coulomb、Matsuoka-Nakai 和 Lade-Duncan 土壤上的浅条形基础。考虑到网格细化、非关联流动和软化行为，讨论了 Cosserat 连续体相对于 Cauchy/Maxwell 介质的好处。