In this article, the authors formulate elastoplastic problems as a coupled problem of the equilibrium equation and the yield condition at each material point, and develop a numerical procedure based on the block Newton method to solve the overall structure using the finite element discretization. For the integration of stress, the backward difference scheme is employed. In the conventional return mapping algorithm, the algorithmic tangent moduli are derived analytically so that it is consistent with local iterative calculation to determine internal variables. On the other hand, in the proposed block Newton method, the tangent moduli can be obtained algebraically by eliminating the internal variables, and the internal variables are also updated algebraically without local iterative calculation. The residual of the yield condition is incorporated into the linearized equilibrium equation. The proposed approach enables the errors of the equilibrium equation and the yield condition to decrease simultaneously. Some numerical examples show the validity and effectiveness of the proposed approach by comparing the results of the proposed approach with those of the conventional return mapping algorithm.

中文翻译：

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基于块牛顿法的弹塑性问题同时迭代过程

在本文中，作者将弹塑性问题公式化为平衡方程和屈服条件在每个材料点上的耦合问题，并开发了基于块牛顿法的数值程序，以使用有限元离散化求解整体结构。为了进行应力积分，采用了后向差分方案。在传统的返回映射算法中，算法的切线模量是通过解析得出的，因此它与确定内部变量的局部迭代计算是一致的。另一方面，在提出的块牛顿法中，可以通过消除内部变量代数获得切线模量，并且也可以通过代数方式更新内部变量而无需局部迭代计算。屈服条件的残差被合并到线性平衡方程中。所提出的方法使得平衡方程的误差和屈服条件的误差同时减小。一些数值示例通过将所提出的方法的结果与常规返回映射算法的结果进行比较，证明了所提出的方法的有效性和有效性。