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近似不可压软材料动力分析的完全拉格朗日物质点法

章子健 刘振海 张洪武 郑勇刚

章子健, 刘振海, 张洪武, 郑勇刚. 近似不可压软材料动力分析的完全拉格朗日物质点法. 力学学报, 2022, 54(12): 3344-3351 doi: 10.6052/0459-1879-22-471
引用本文: 章子健, 刘振海, 张洪武, 郑勇刚. 近似不可压软材料动力分析的完全拉格朗日物质点法. 力学学报, 2022, 54(12): 3344-3351 doi: 10.6052/0459-1879-22-471
Zhang Zijian, Liu Zhenhai, Zhang Hongwu, Zheng Yonggang. Total Lagrangian material point method for the dynamic analysis of nearly incompressible soft materials. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3344-3351 doi: 10.6052/0459-1879-22-471
Citation: Zhang Zijian, Liu Zhenhai, Zhang Hongwu, Zheng Yonggang. Total Lagrangian material point method for the dynamic analysis of nearly incompressible soft materials. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3344-3351 doi: 10.6052/0459-1879-22-471

近似不可压软材料动力分析的完全拉格朗日物质点法

doi: 10.6052/0459-1879-22-471
基金项目: 国家自然科学基金(12072061, 12072062), 兴辽英才计划(XLYC1807193)和辽宁省重点研发计划(2020JH2/10500003)资助项目
详细信息
    作者简介:

    郑勇刚, 教授, 主要研究方向: 计算力学. E-mail: zhengyg@dlut.edu.cn

  • 中图分类号: O343.5

TOTAL LAGRANGIAN MATERIAL POINT METHOD FOR THE DYNAMIC ANALYSIS OF NEARLY INCOMPRESSIBLE SOFT MATERIALS

  • 摘要: 物质点法(MPM)在模拟非线性动力问题时具有很好的效果, 其已被广泛应用于许多大变形动力问题的分析中. 然而传统的MPM在模拟不可压或近似不可压材料的动力学行为时会产生体积自锁, 极大地影响模拟精度和收敛性. 本文针对近似不可压软材料的大变形动力学行为, 提出一种混合格式的显式完全拉格朗日物质点法(TLMPM). 首先基于近似不可压软材料的体积部分应变能密度, 引入关于静水压力的方程; 之后将该方程与动量方程基于显式物质点法框架进行离散, 并采用完全拉格朗日格式消除物质点跨网格产生的误差, 提升大变形问题的模拟精度; 对位移和压强场采用不同阶次的B样条插值函数并通过引入针对体积变形的重映射技术改进了算法, 提升算法的准确性. 此外, 算法通过实施一种交错求解格式在每个时间步对位移场和压强场依次进行求解. 最后, 给出几个典型数值算例来验证本文所提出的混合格式TLMPM的有效性和准确性, 计算结果表明该方法可以有效处理体积自锁, 准确地模拟近似不可压软材料的大变形动力学行为.

     

  • 图  1  库克膜问题示意图(单位: mm)

    Figure  1.  Schematic plot of Cook’s membrane (unit: m)

    图  2  库克膜: 采用不同数量物质点离散的t = 3 s时右上角竖向位移模拟结果

    Figure  2.  Cook’s membrane: simulation results of the vertical displacement of the top right corner at t = 3 s using different numbers of material points

    图  3  库克膜: t = 3 s时的静水压力分布

    Figure  3.  Cook’s membrane: hydrostatic pressure distribution at t = 3 s

    图  4  二维软梁的弯曲问题示意图

    Figure  4.  Schematic plot of bending of 2D soft beam

    图  5  二维软梁的弯曲: Ax方向位移变化曲线

    Figure  5.  Bending of 2D soft beam: history curves of x-displacement of point A

    图  6  二维软梁的弯曲: 不同时刻的静水压力分布. (a)~(b) 标准格式线性插值; (c)~(d) 标准格式二次插值; (e)~(f) 线性−常数混合格式; (g)~(h) 二次−线性混合

    Figure  6.  Bending of 2D soft beam: hydrostatic distribution at different times. (a)~(b) Standard linear interpolation; (c)~(d) standard quadratic interpolation; (e)~(f) linear-constant mixed formulation and (g)~(h) quadratic-linear mixed formulation

    图  7  三维柱体扭转: 构型及静水压力分布. (a) 几何模型; (b) t = 0.1 s, 标准TLMPM; (c) t = 0.3 s, 标准TLMPM; (d) t = 0.1 s, 混合 TLMPM; (e) t = 0.3 s, 混合 TLMPM

    Figure  7.  Twisting of a 3D column: configurations and distribution of hydrostatic pressure. (a) Geometry; (b) t = 0.1 s, standard TLMPM; (c) t = 0.3 s, standard TLMPM; (d) t = 0.1 s, mixed TLMPM and (e) t = 0.3 s, mixed TLMPM

    图  8  三维柱体扭转: Az方向位移变化曲线(参考解为文献[30]的计算结果)

    Figure  8.  Twisting of a 3D column: history curves of z-displacement of point A (the reference results refer to Ref. [30])

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出版历程
  • 收稿日期:  2022-10-04
  • 录用日期:  2022-10-26
  • 网络出版日期:  2022-10-27
  • 刊出日期:  2022-12-15

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