The material point method is a versatile technique which can be used to solve various types of solid mechanics problems, especially those involving large deformations. However, the capability of the material point method to track a load-displacement response can deteriorate once a limit point, such as snap-through or snap-back, in the response is encountered. One way of overcoming this is to use path following techniques, such as an arc-length method. This technique is well established in finite element analysis but not within any material point method formulation. This paper provides for the first time an arc-length controlled implicit, quasi-static material point method. The modifications to the standard arc-length scheme to allow for the stable execution of an arc-length solver within the material point method are detailed. The capability of the material point-based arc-length method is demonstrated through a number of problems, which include linear elastic, non-linear elastic, linear elastic-perfectly plastic and linear elastic-plastic softening material behaviour under large deformations. The techniques presented in this paper are essential for arc-length techniques to be applied effectively to the material point method and the combination of these techniques makes the method suitable for new problems that cannot be solved with existing implicit material point approaches.

中文翻译：

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基于质点的弧长方法的实现

质点法是一种通用技术，可用于解决各种类型的固体力学问题，特别是涉及大变形的固体力学问题。然而，一旦遇到响应中的极限点（例如突跳或回弹），材料点方法跟踪载荷-位移响应的能力就会下降。克服这个问题的一种方法是使用路径跟踪技术，例如弧长方法。该技术在有限元分析中得到了很好的应用，但并未在任何质点方法公式中得到应用。本文首次提出了一种弧长控制的隐式准静态材料点方法。详细介绍了对标准弧长方案的修改，以允许在质点方法中稳定执行弧长求解器。基于材料点的弧长方法的能力通过一系列问题得到证明，包括大变形下的线弹性、非线性弹性、线弹-完全塑性和线弹-塑性软化材料行为。本文提出的技术对于将弧长技术有效地应用于质点方法至关重要，并且这些技术的组合使得该方法适用于现有隐式质点方法无法解决的新问题。