Integration algorithm is a key to ensure the better application of the dynamic constitutive model. As for the bounding surface model based on swell–shrink rules in this paper, an explicit integration algorithm based on sub-stepping and error control scheme has be described in detail. For being adaptable to the constitutive model, several modifications have been made. The modified explicit algorithm includes two procedures: the first procedure is the second-order forward modified Euler scheme with sub-stepping and error control; the second procedure is the stress correction. However, due to the shrink rule of the bounding surface, the second procedure is not needed at the unloading phase. Finally, several numerical simulations have been conducted for assessing the performance of the explicit algorithm, which demonstrates that the corresponding algorithm is stable and accurate, and can simulate the mechanical characteristics of the clay pretty well.

中文翻译：

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基于膨胀-收缩规则的黏土循环行为边界曲面模型的显式积分算法

积分算法是保证动态本构模型更好应用的关键。对于本文基于膨胀-收缩规则的边界曲面模型，详细描述了一种基于子步进和误差控制方案的显式积分算法。为了适应本构模型，已经进行了一些修改。改进的显式算法包括两个过程：第一个过程是具有子步进和错误控制的二阶前向修正欧拉方案；第二个程序是应力校正。然而，由于边界表面的收缩规则，在卸载阶段不需要第二个过程。最后，已经进行了几个数值模拟来评估显式算法的性能，