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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
  • [1] 王勖成, 邵敏. 有限单元法基本原理和数值方法. 北京: 清华大学出版社有限公司, 1997

    Wang Xucheng, Shao Min. Fundamental Principles and Mumerical Approaches of FEM. Beijing: Tsinghua Univeristy Press, 1997. (in Chinese))
    [2] 周旭. 导弹毁伤效能试验与评估技术. 北京: 国防工业出版社, 2014

    Zhou Xu. Test and Assessment Technology of the Damage Efficiency of Missiles. Beijing: National Defense Industry Press, 2014 (in Chinese))
    [3] Elices M, Guinea GV, Gomez J, et al. The cohesive zone model: advantages, limitations and challenges. Engineering Fracture Mechanics, 2002, 69(2): 137-163 doi: 10.1016/S0013-7944(01)00083-2
    [4] Camacho GT, Ortiz M. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures, 1996, 33(20-22): 2899-2938 doi: 10.1016/0020-7683(95)00255-3
    [5] Cervera M, Barbat GB, Chiumenti M, et al. A comparative review of XFEM, mixed FEM and phase-field models for quasi-brittle cracking. Archives of Computational Methods in Engineering, 2022, 29(2): 1009-1083 doi: 10.1007/s11831-021-09604-8
    [6] Benson DJ. Computational methods in Lagrangian and Eulerian hydrocodes. Computer Methods in Applied Mechanics and Engineering, 1992, 99(2-3): 235-394 doi: 10.1016/0045-7825(92)90042-I
    [7] Anderson Jr CE. An overview of the theory of hydrocodes. International Journal of Impact Engineering, 1987, 5(1-4): 33-59 doi: 10.1016/0734-743X(87)90029-7
    [8] Mair HU. Hydrocodes for structural response to underwater explosions. Shock and Vibration, 1999, 6(2): 81-96 doi: 10.1155/1999/587105
    [9] Belytschko T, Liu WK, Moran B, et al. Nonlinear Finite Elements for Continua and Structures. John Wiley & Sons, 2014
    [10] 龙述尧. 无网格方法及其在固体力学中的应用. 北京: 科学出版社, 2014

    Long Shurao. Meshless Method and Its Application in Solid Mechanics. Beijing: Science Press, 2014 (in Chinese))
    [11] Lucy LB. A numerical approach to the testing of the fission hypothesis. The Astronomical Journal, 1977, 82: 1013-1024 doi: 10.1086/112164
    [12] Gingold RA, Monaghan JJ. Smoothed particle hydrodynamics: theory and application to non-spherical stars. Monthly Notices of The Royal Astronomical Society, 1977, 181(3): 375-389 doi: 10.1093/mnras/181.3.375
    [13] Liu GR, Liu MB. Smoothed Particle Hydrodynamics: A Meshfree Particle Method. World Scientific, 2003
    [14] 孙晓旺. 高效率, 高精度耦合算法及对材料冲击响应特性应用的研究. [博士论文]. 中国科学技术大学, 2017

    Sun Xiaowang. A study on coupled arithmetic of high efficiency and high precision and its application on material impact response. [PhD Thesis]. HeFei: University of Science and Technology of China, 2017 (in Chinese))
    [15] Sulsky D, Chen Z, Schreyer HL. A particle method for history-dependent materials. Computer Methods in Applied Mechanics and Engineering, 1994, 118(1-2): 179-196 doi: 10.1016/0045-7825(94)90112-0
    [16] 马上, 张雄, 邱信明. 超高速碰撞问题的三维物质点法. 爆炸与冲击, 2006, 26(3): 273-278 (Ma Shang, Zhang Xiong, Qiu Xinming. Three-dimensional material point method for hypervelocity impact. Explosion and Shock Waves, 2006, 26(3): 273-278 (in Chinese) doi: 10.3321/j.issn:1001-1455.2006.03.014
    [17] Zhang F, Zhang X, Liu Y. An augmented incompressible material point method for modeling liquid sloshing problems. International Journal of Mechanics and Materials in Design, 2018, 14(1): 141-155 doi: 10.1007/s10999-017-9366-5
    [18] 廉艳平, 张帆, 刘岩等. 物质点法的理论和应用. 力学进展, 2013, 43(2): 237-264 (Lian Yanping, Zhang Fan, Liu Yan, et al. Material point method and its applications. Advances in Mechanics, 2013, 43(2): 237-264 (in Chinese) doi: 10.6052/1000-0992-12-122
    [19] Atluri SN, Zhu T. A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Computational Mechanics, 1998, 22(2): 117-127 doi: 10.1007/s004660050346
    [20] Atluri SN, Zhu T. New concepts in meshless methods. International Journal for Numerical Methods in Engineering, 2000, 47(1-3): 537-556
    [21] Belytschko T, Lu YY, Gu L. Element‐free Galerkin methods. International Journal for Numerical Methods in Engineering, 1994, 37(2): 229-256 doi: 10.1002/nme.1620370205
    [22] Lu YY, Belytschko T, Gu L. A new implementation of the element free Galerkin method. Computer Methods in Applied Mechanics and Engineering, 1994, 113(3-4): 397-414 doi: 10.1016/0045-7825(94)90056-6
    [23] Belytschko T, Tabbara M. Dynamic fracture using element‐free Galerkin methods. International Journal for Numerical Methods in Engineering, 1996, 39(6): 923-938 doi: 10.1002/(SICI)1097-0207(19960330)39:6<923::AID-NME887>3.0.CO;2-W
    [24] 吴俊超, 邓俊俊, 王家睿等. 伽辽金型无网格法的数值积分方法. 固体力学学报, 2016, 37(3): 208-233 (Junchao Wu, Junjun Deng, Jiarui Wang, et al. A review of numerical integration approaches for Galerkin meshfree methods. Chinese Journal of Solid Mechanics, 2016, 37(3): 208-233 (in Chinese)
    [25] Wang D, Wang J, Wu J. Super convergent gradient smoothing meshfree collocation method. Computer Methods in Applied Mechanics and Engineering, 2018, 340: 728-766 doi: 10.1016/j.cma.2018.06.021
    [26] Hillman M, Chen JS. An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics. International Journal for Numerical Methods in Engineering, 2016, 107(7): 603-630 doi: 10.1002/nme.5183
    [27] 苏伯阳, 李书欣, 刘立胜等. 湿热环境下复合材料冲击损伤的近场动力学模拟. 科学技术与工程, 2018, 18(1): 201-206 (Su Boyang, LI Shuxin, Liu Lisheng, et al. Peridynamic simulation of impact damage of composite material under hygrothermal environment. Science Technology and Engineering, 2018, 18(1): 201-206 (in Chinese) doi: 10.3969/j.issn.1671-1815.2018.01.035
    [28] 郁杨天, 章青, 顾鑫. 含单边缺口混凝土梁冲击破坏的近场动力学模拟. 工程力学, 2016, 33(12): 80-85 (Yu Yangtian, Zhang Qing, Gu Xin. Impact failure simulation of a single-edged notched concrete beam based on peridynamics. Engineering Mechanics, 2016, 33(12): 80-85 (in Chinese) doi: 10.6052/j.issn.1000-4750.2015.05.0396
    [29] 黄丹, 章青, 乔丕忠等. 近场动力学方法及其应用. 力学进展, 2010, 40(4): 448-459 (Huang Dan, Zhang Qing, Qiao Pizhong, et al. A review on peridynamics(pd)method and its applications. Advances in Mechanics, 2010, 40(4): 448-459 (in Chinese) doi: 10.6052/1000-0992-2010-4-J2010-002
    [30] Han F, Lubineau G, Azdoud Y. Adaptive coupling between damage mechanics and peridynamics: A route for objective simulation of material degradation up to complete failure. Journal of the Mechanics and Physics of Solids, 2016, 94: 453-472 doi: 10.1016/j.jmps.2016.05.017
    [31] 高效伟, 郑保敬, 刘健. 功能梯度材料动态断裂力学的径向积分边界元法. 力学学报, 2015, 47(5): 868-873 (Gao Xiaowei, Zheng Baojing, Liu Jian. Dynamic fracture analysis of functionally graded materials by radial integration BEM. Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(5): 868-873 (in Chinese) doi: 10.6052/0459-1879-15-150
    [32] 于福临, 郭君, 姚熊亮等. 基于 RKDG 方法的船体板架爆炸冲击响应数值模拟. 振动与冲击, 2015, 34(13): 60-65 (Yu Fulin, Guo Jun, Yao Xiongliang, et al. Numerical simulation of hull grillage responses under blast loading based on RKDG method. Chinese Journal of Theoretical and Applied Mechanics, 2015, 34(13): 60-65 (in Chinese)
    [33] Dong L, Yang T, Wang K, et al. A new Fragile Points Method (FPM) in computational mechanics, based on the concepts of Point Stiffnesses and Numerical Flux Corrections. Engineering Analysis with Boundary Elements, 2019, 107: 124-133 doi: 10.1016/j.enganabound.2019.07.009
    [34] Yang T, Dong L, Atluri SN. A simple Galerkin meshless method, the fragile points method (FPM) using point stiffness matrices, for 2D linear elastic problems in complex domains with crack and rupture propagation. International Journal for Numerical Methods in Engineering, 2021, 122(2): 348-385 doi: 10.1002/nme.6540
    [35] Wang K, Shen B, Li M, et al. A Fragile Points Method, with an interface debonding model, to simulate damage and fracture of U‐notched structures. International Journal for Numerical Methods in Engineering, 2022, 123(8): 1736-1759 doi: 10.1002/nme.6914
    [36] Guan Y, Dong L, Atluri SN. A new meshless Fragile Points Method (FPM) with minimum unknowns at each point, for flexoelectric analysis under two theories with crack propagation, I: Theory and implementation. Journal of Mechanics of Materials and Structures, 2021, 16(2): 159-195 doi: 10.2140/jomms.2021.16.159
    [37] Guan Y, Dong L, Atluri SN. A new meshless fragile points method (FPM) with minimum unknowns at each point, for flexoelectric analysis under two theories with crack propagation, II: Validation and discussion. Journal of Mechanics of Materials and Structures, 2021, 16(2): 197-223 doi: 10.2140/jomms.2021.16.197
    [38] Guan Y, Grujicic R, Wang X, et al. A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part I: Theory and implementation. Numerical Heat Transfer, Part B:Fundamentals, 2020, 78(2): 71-85 doi: 10.1080/10407790.2020.1747278
    [39] Guan Y, Grujicic R, Wang X, et al. A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part II: Validation and discussion. Numerical Heat Transfer, Part B: Fundamentals, 2020, 78(2): 86-109 doi: 10.1080/10407790.2020.1747283
    [40] 曾亿山, 卢德唐, 曾清红. 无单元伽辽金法的并行计算. 计算力学学报, 2008, 25(3): 385-391 (Zeng Yishan, Lu Detang, Zeng Qinghong. Parallel computing of element-free galerkin method for elasto-dynamics. Chinese Journal of Computational Mechanics, 2008, 25(3): 385-391 (in Chinese)
    [41] Kalthoff JF, Winkler S. Failure mode transition at high rates of shear loading. Impact Loading and Dynamic Behavior of Materials, 1988, 1: 185-195
    [42] Kalthoff JF. Modes of dynamic shear failure in solids. International Journal of Fracture, 2000, 101(1): 1-31
    [43] Wang S, Li D, Li Z, et al. A rate-dependent model and its user subroutine for cohesive element method to investigate propagation and branching behavior of dynamic brittle crack. Computers and Geotechnics, 2021, 136: 104233 doi: 10.1016/j.compgeo.2021.104233
    [44] Song JH, Belytschko T. Cracking node method for dynamic fracture with finite elements. International Journal for Numerical Methods in Engineering, 2009, 77(3): 360-385 doi: 10.1002/nme.2415
    [45] Park K, Paulino GH, Celes W, et al. Adaptive mesh refinement and coarsening for cohesive zone modeling of dynamic fracture. International Journal for Numerical Methods in Engineering, 2012, 92(1): 1-35 doi: 10.1002/nme.3163
    [46] Lee YJ, Freund LB. Fracture initiation due to asymmetric impact loading of an edge cracked plate. Applied Mechanics, 1990, 57(1): 104-111 doi: 10.1115/1.2888289
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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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