On Lagrangian mechanics and the implicit material point method for large deformation elasto-plasticity
On Lagrangian mechanics and the implicit material point method for large deformation elasto-plasticity
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DOI：
10.1016/j.cma.2019.112622
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发表时间：
2020-01
影响因子：
7.2
通讯作者：
W. Coombs;C. Augarde;A. Brennan;Michael Brown;T. J. Charlton;J. Knappett;Yousef Ghaffari Motlagh-Yousef-Ghaffari Motlagh-2128299607;Lei Wang
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文献类型：
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作者：
W. Coombs;C. Augarde;A. Brennan;Michael Brown;T. J. Charlton;J. Knappett;Yousef Ghaffari Motlagh-Yousef-Ghaffari Motlagh-2128299607;Lei Wang
材料点法非常适合于传统的基于网格的方法难以解决的涉及大变形的建模问题。然而，总的拉格朗日方法和更新的拉格朗日方法分别不适合和不理想地表述该方法的平衡。这是由于材料点方法的基函数，特别是基函数的导数，通常是在未形成的，有时是规则的背景网格上定义的。利用材料点处的变形映射基函数空间导数是可能的，但这引入了额外的算法复杂性和计算费用。本文提出了一种新的拉格朗日平衡表述，它满足加载步骤开始时未变形背景网格上的平衡，是理想的质点法。该公式采用拟静态隐式算法实现，该算法包含一致切线的推导，以实现全局平衡迭代的最优收敛。该方法适用于许多大变形弹塑性问题，并特别关注该方法的收敛性，以标准，广义插值和CPDI2材料点法的解析解。对于广义插值方法，研究了不同的区域更新方法，结果表明，在一定的简单变形场下，现有的方法都是退化的。针对这些问题，提出了一种新的领域更新方法。提出的物质点法框架可以适用于所有现有的物质点法，并可用于隐式和显式分析，但其优势主要与前者相关。
The material point method is ideally suited to modelling problems involving large deformations where conventional mesh-based methods would struggle. However, total and updated Lagrangian approaches are unsuitable and non-ideal, respectively, in terms of formulating equilibrium for the method. This is due to the basis functions, and particularly the derivatives of the basis functions, of material point methods normally being defined on an unformed, and sometimes regular, background mesh. It is possible to map the basis function spatial derivatives using the deformation at a material point but this introduces additional algorithm complexity and computational expense. This paper presents a new Lagrangian statement of equilibrium which is ideal for material point methods as it satisfies equilibrium on the undeformed background mesh at the start of a load step. The formulation is implemented using a quasi-static implicit algorithm which includes the derivation of the consistent tangent to achieve optimum convergence of the global equilibrium iterations. The method is applied to a number of large deformation elasto-plastic problems, with a specific focus of the convergence of the method towards analytical solutions with the standard, generalised interpolation and CPDI2 material point methods. For the generalised interpolation method, different domain updating methods are investigated and it is shown that all of the current methods are degenerative under certain simple deformation fields. A new domain updating approach is proposed that overcomes these issues. The proposed material point method framework can be applied to all existing material point methods and adopted for implicit and explicit analysis, however its advantages are mainly associated with the former.