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基于碎点法的动态断裂分析

沈宝莹 王松 李明净 董雷霆

沈宝莹, 王松, 李明净, 董雷霆. 基于碎点法的动态断裂分析. 力学学报, 2022, 54(12): 3383-3397 doi: 10.6052/0459-1879-22-498
引用本文: 沈宝莹, 王松, 李明净, 董雷霆. 基于碎点法的动态断裂分析. 力学学报, 2022, 54(12): 3383-3397 doi: 10.6052/0459-1879-22-498
Shen Baoying, Wang Song, Li Mingjing, Dong Leiting. Dynamic fracture analysis with the fragile points method. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3383-3397 doi: 10.6052/0459-1879-22-498
Citation: Shen Baoying, Wang Song, Li Mingjing, Dong Leiting. Dynamic fracture analysis with the fragile points method. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(12): 3383-3397 doi: 10.6052/0459-1879-22-498

基于碎点法的动态断裂分析

doi: 10.6052/0459-1879-22-498
基金项目: 国家自然科学基金资助项目(12102023, 12072011)
详细信息
    作者简介:

    李明净, 助理教授, 主要研究方向: 飞机结构冲击防护和生存力设计技术、航空复合材料结构强度预测和损伤分析技术、飞机结构多尺度分析技术以及相应的计算力学分析方法/软件 . E-mail: limingjing@buaa.edu.cn

  • 中图分类号: O302

DYNAMIC FRACTURE ANALYSIS WITH THE FRAGILE POINTS METHOD

  • 摘要: 工程中的冲击防护结构在撞击、爆炸等冲击载荷下可能发生动态断裂并最终破坏, 抑制结构的动态断裂是提升结构防护能力的重要手段, 为此需要准确预测结构在动态载荷下的断裂行为. 数值仿真是预测动态断裂的重要手段, 然而当前工程中常用的有限元法在模拟断裂方面仍存在网格畸变和难以显式引入裂纹等问题. 碎点法是近年来提出的一种不连续型伽辽金弱形式无网格方法, 适合模拟断裂问题, 本文提出一种显式动力学格式的碎点法并将该方法应用于动态断裂分析. 一方面, 碎点法参考弱形式无网格类方法, 将求解域离散为空间中的节点和子域, 并基于支持域内的节点群构造子域的位移试函数, 因此该方法的子域具有抵抗畸变的能力. 另一方面, 碎点法参考间断伽辽金有限元法, 使用分片连续的位移试函数, 并引入内部界面数值通量修正保证方法的一致性和稳定性, 因此该方法易于在结构中显式引入裂纹. 本文首先介绍碎点法的核心思想和离散形式, 接着推导了动力学碎点法弱形式动量方程, 然后建立了碎点法的显式动力学求解格式, 最后通过算例验证动力学碎点法预测应力波传播和动态断裂行为的能力.

     

  • 图  1  (a) 碎点法模型的离散和(b) 支持域及其节点群的定义

    Figure  1.  (a) Discretization of FPM model and (b) definition of support range and its point cloud

    图  2  (a) 碎点法形函数示意图 (b) 位移场为${{\rm{e}}^{ - 10{{\left( {{x_1} - 0.5} \right)}^2} - 10{{\left( {{x_2} - 0.5} \right)}^2}}}$的碎点法位移试函数[34]

    Figure  2.  (a) Shape function of FPM (b) FPM’s trial function of a field of displacement :${{\rm{e}}^{ - 10{{\left( {{x_1} - 0.5} \right)}^2} - 10{{\left( {{x_2} - 0.5} \right)}^2}}}$

    图  3  碎点法裂纹的引入

    Figure  3.  Steps of introducing crack in FPM

    图  4  中心差分法时间轴示意图

    Figure  4.  Time axis of the Verlet method

    图  5  拉力作用下的矩形平板模型

    Figure  5.  The model for rectangular plate under tension

    图  6  载荷$P\left( t \right)$随时间的关系

    Figure  6.  The curve for the load-time correlation

    图  7  拉力作用下平板应力波传播问题精确解及FPM结果

    Figure  7.  Exact solutions and FPM’s results of the displacement and stress at different points

    7  拉力作用下平板应力波传播问题精确解及FPM结果(续)

    7.  Exact solutions and FPM’s results of the displacement and stress at different points (continued)

    图  8  (a)悬臂梁示意图和(b)计算模型节点分布

    Figure  8.  (a) Configuration of the beam and (b) the scatter of subdomains

    图  9  自由端中点竖直方向位移响应曲线

    Figure  9.  Vertical displacement curve of the mid-point at the free end of the beam

    图  11  悬臂梁A点处应力瞬态响应曲线

    Figure  11.  Stress-time curve of point A

    图  10  固定端中点应力响应曲线

    Figure  10.  Curves of stress components of the mid-point at the fixed end of the beam

    图  12  (a)含裂纹金属板冲击试验装置示意图和(b)试验观测到的裂纹扩展路径[43]

    Figure  12.  (a) Experimental setup for the Kalthoff plate impact test and (b) the experimentally-observed crack path[43]

    图  13  计算模型: (a) 6516个子域节点和(b) 11995个子域节点

    Figure  13.  Domain discretization with (a) 6516 FPM subdomains and (b)11995 FPM subdomains

    图  14  静水压云图

    Figure  14.  Configuration with hydrostatic pressure

    14  静水压云图(续)

    14.  Configuration with hydrostatic pressure (continued)

    图  15  FPM仿真裂纹扩展路径

    Figure  15.  Crack propagation from FPM simulation

    图  16  裂纹扩展速度−时间曲线

    Figure  16.  Curve of crack propagation speed versus time

    表  1  有限元、碎点法的位移峰值与理论解的比较

    Table  1.   Relative error of maximum displacement obtained from FEM and FPM

    Maximum ${u_1}$ at AMaximum ${u_1}$ at B
    ExactFEMFPMExactFEMFPM
    results0.01820.01820.01810.01000.01000.0100
    error−0.12%−0.58%−0.01%0.18%
    下载: 导出CSV

    表  2  有限元、碎点法的应力峰值与理论解的比较

    Table  2.   Relative error of maximum stress obtained from FEM and FPM

    Peak ${\sigma _{11}}$ at BPeak ${\sigma _{11}}$ at C
    ExactFEMFPMExactFEMFPM
    results399.9910399.5224399.1767399.9936400.2177399.8739
    error−0.12%−0.20%−0.10%0.13%
    下载: 导出CSV
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  • 收稿日期:  2022-10-16
  • 录用日期:  2022-11-10
  • 网络出版日期:  2022-11-11
  • 刊出日期:  2022-12-15

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