FRACTURE BEHAVIOR OF PERIODIC POROUS STRUCTURES BY PHASE FIELD METHOD
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摘要: 周期性多孔结构具有质量轻、比密度低、比强度高、隔音等优良特点, 同时也能很好地满足结构-功能一体化的需求, 在许多领域具有广泛的应用前景. 目前, 对周期性多孔结构在复杂载荷下的力学响应和断裂行为的研究较少. 采用细观力学和相场方法相结合, 基于二维代表性体积单元RVE模型, 施加能实现比例加载的周期性边界条件, 研究周期性多孔结构在复杂多轴比例加载状态下的裂纹萌生位置、断裂模式、承载极限及其变化规律. 本文的数值模拟结果表明: 首先, 周期性多孔结构在竖直方向拉伸载荷作用下, 裂纹均从孔边萌生并沿水平方向同步扩展; 其次, 在双轴载荷作用下, 随着水平载荷的增加, 结构在竖直方向的极限拉伸载荷逐渐增大; 当双轴拉伸载荷等值时, 结构的抗拉强度达到最大, 此时断裂模式呈现为十字正交型开裂; 最后, 面内剪切应力的引入会导致结构的拉伸强度极限降低, 孔边裂纹的萌生位置和扩展路径发生偏移, 裂纹模式从单S型转变为双弧线型, 裂纹向水平位置上相邻的孔洞扩展. 随着水平载荷的增加, 裂纹模式最终转变为斜裂纹, 从孔边对角线位置萌生并沿着45°方向扩展.Abstract: Periodic porous structures have excellent characteristics such as low mass, low density, high specific strength, sound insulation, and they are also well satisfy the needs for structural-functional integration, which have a wide range of applications in many fields. At present, the mechanical response and fracture behavior of periodic porous structures under complex loads have been poorly investigated. In this paper, we use a combination of micro-mechanics method and phase field method to investigate the crack initiation location, crack propagation path, fracture mode and the ultimate strength of periodic porous structures under combined multiaxial loading based on a two-dimensional representative volume element (RVE) model with the periodic boundary condition (PBC) that can implement multiaxial proportional loading. Numerical simulation results in this paper show that all cracks in the periodic porous structure established in this paper initiate from the edge of the holes and propagate consequently along the horizontal direction under the uniaxial tensile loading in the vertical direction. Secondly, under the biaxial loadings in both vertical and horizontal directions, the ultimate strength of the periodic porous structure gradually increases with the increase of the horizontal tensile loading. When the horizontal load is equal to the vertical load, the fracture pattern exhibits as orthogonal cross-type cracking and the ultimate strength reaches the maximum value. Thirdly, the in-plane shear stress simultaneously acted on the RVE model of the periodic porous structure results in a significant decrease of the ultimate strength and the variations of initiation location and propagation trajectory of the hole-edge cracks. Hence, the fracture pattern of periodic porous structure subjected to combined multiaxial loadings changes from the single S-type cracking to the double arc-type cracking, and cracks extend toward adjacent holes in horizontal direction. Finally, with the increase of the horizontal tensile loading, cracks initiate diagonally at the edge of the holes and propagate along the 45-degree direction which lead to the oblique cracking of periodic porous structure.
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Key words:
- periodic porous structure /
- phase field /
- fracture /
- RVE model /
- multiaxial loading
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图 3 弥散裂纹与弹性体受力情况, 左侧为尖裂纹, 右侧为弥散裂纹
Figure 3. Diffused crack and elastomeric stresses, with sharp cracks on the left and diffused crack on the right
图 5 周期性多孔结构RVE模型的几何构型(单位: mm)
Figure 5. Geometric configuration of the RVE model for periodic porous structures (unit: mm)
图 6
$ 8 \times 8 $ 多孔模型的单轴拉伸边界条件Figure 6. Boundary conditions for uniaxial stretch of
$ 8 \times 8 $ periodic porous model
图 7 单轴拉伸载荷下单元数分别为80000, 115200和156800的周期性多孔结构的2×2 RVE模型名义应力-应变曲线的比较
Figure 7. Comparison of nominal stress-strain curves of the RVE model with 80000, 115200 and 156800 elements under tensile loading condition
图 8 单轴拉伸载荷下周期性多孔结构的
$ 2 \times 2 $ RVE模型与$ 8 \times 8 $ 模型的名义应力-应变曲线的比较Figure 8. Comparison of nominal stress-strain curves of the
$ 2 \times 2 $ RVE model and the$ 8 \times 8 $ periodic porous model under uniaxial tensile loading condition
图 9 单轴拉伸载荷下的周期性多孔结构的
$ 2 \times 2 $ RVE模型与$ 8 \times 8 $ 模型的断裂相场云图Figure 9. Phase-field fracture contours of the
$ 2 \times 2 $ RVE model and$ 8 \times 8 $ model for periodic porous structure under uniaxial tensile loading
图 11 周期性多孔结构在竖直方向的承载极限
${P_{22 \text{-} \max }}$ 随应力比$ {\rho _1} $ 和$ {\rho _2} $ 变化的曲线: (a)保持应力比$ {\rho _2} $ 不变, (b)保持应力比$ {\rho _1} $ 不变, (c)局部放大图Figure 11. Changing curves of 2-directional extreme load
${P_{22 \text{-} \max }}$ with$ {\rho _1} $ and$ {\rho _2} $ for periodic porous structures: (a) keep$ {\rho _2} $ unchanged, (b) keep$ {\rho _1} $ unchanged and (c) partial enlarged graph
图 12 周期性多孔结构在不同多轴加载条件下的裂纹扩展模式相图
Figure 12. Phase diagram of crack propagation mode for periodic porous structure under different multiaxial proportional loading
表 1 双轴比例加载下的周期性多孔结构的承载极限
${P_{22 \text{-} \max }}$ 与断裂相场云图Table 1. Extreme load
${P_{22 \text{-} \max }}$ and phase-field fracture contours for the RVE model under biaxial proportional loading condition$ {\rho _1} $ −1 −0.5 0 0.5 1 P22-max/MPa 702.26 782.25 867.91 947.08 971.39 Phase-
field fracture contours
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表 2 多轴比例加载下的周期性多孔结构的断裂相场云图
Table 2. Phase-field fracture contours of periodic porous structure model under multiaxial proportional loading
$ {\rho _1} $, $ {\rho _2} $ −1 −0.5 0 0.5 1.0 0 
0.25 
0.5 
0.75 
1.0 
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表 3 5种不同裂纹扩展模式的表征符号和渐进破坏的断裂相场云图
Table 3. Representative symbols and progressive phase-field fracture contours for the five distinguished crack propagation modes
Representative symbol 
crack propagation modes horizontal
typeS-type double arc-type 45°-type orthogonal cross-type progressive phase field
fracture contours
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[1] 张洪武, 王鲲鹏. 材料非线性微-宏观分析的多尺度方法研究. 力学学报, 2004, 36(3): 359-363 (Zhang Hongwu, Wang Kunpeng. Multiscale methods for nonlinear analysis of composite materials. Chinese Journal of Theoretical and Applied Mechanics, 2004, 36(3): 359-363 (in Chinese) [2] Liu QH, Liu XW, Zhang CZ, et al. A novel multiscale porous composite structure for sound absorption enhancement. Composite Structures, 2021, 276(22): 114456 [3] 樊宣青, 刑誉锋. 周期多孔材料力学性能分析中的多尺度均匀化方法的精度// 第十二届全国振动理论及应用学术会议论文集. 中国振动工程学会, 2017Fan Xuanqing, Xing Yufeng. The accuracy of multi-scale homogenization method in the analysis of mechanical properties of porous materials//Proceedings of the 12th National Conference on Vibration Theory and Applications. Chinese Society for Vibration Engineering, 2017 (in Chinese) [4] Hayes AM, Wang A, Dempsey BM, et al. Mechanics of linear cellular alloys. Mechanics of Materials, 2004, 36(8): 691-713 doi: 10.1016/j.mechmat.2003.06.001 [5] 杜映洪. 层级多孔结构的力学及断裂性能研究. [硕士论文]. 天津: 天津大学, 2016Du Yinghong. On the mechanical and fracture properties of the hierarchical cellular structure. [Master Thesis]. Tianjin: Tianjin University, 2016 (in Chinese) [6] Jelitto H, Schneider GA. Extended cubic fracture model for porous materials and the dependence of the fracture toughness on the pore size. Materialia, 2020, 12(4): 100761 [7] 吴建营. 固体结构损伤破坏统一相场理论, 算法和应用. 力学学报, 2021, 53(2): 301-329 (Wu Jianying. On the unified phase-field theory for damage and failure in solids and structures: theoretical and numerical aspects. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 301-329 (in Chinese) [8] 卢广达, 陈建兵. 基于一类非局部宏-微观损伤模型的裂纹模拟. 力学学报, 2020, 52(3): 749-762 (Lu Guangda, Chen Jianbing. Cracking simulation based on a nonlocal macro-meso-scale damage model. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(3): 749-762 (in Chinese) [9] Fan R, Fish J. The rs-method for material failure simulations. International Journal for Numerical Methods in Engineering, 2008, 73(11): 1607-1623 doi: 10.1002/nme.2134 [10] Xu XP, Needleman A. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids, 1994, 42(9): 1397-1434 doi: 10.1016/0022-5096(94)90003-5 [11] 李录贤, 王铁军. 扩展有限元法(XFEM)及其应用. 力学进展, 2005, 35(1): 5-21 (Li Luxian, Wang Tiejun. The extended finite method and its application. Advances in Mechanics, 2005, 35(1): 5-21 (in Chinese) doi: 10.3321/j.issn:1000-0992.2005.01.002 [12] 王理想, 文龙飞, 肖桂仲等. 非连续问题中单元分割的模板方法. 力学学报, 2021, 53(3): 823-836 (Wang Lixiang, Wen Longfei, Xiao Guizhong, et al. A templated method for partitioning of solid elements in discontinuous problems. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(3): 823-836 (in Chinese) [13] Wu JY, Qiu JF, Nguyen VP, et al. Computational modeling of localized failure in solids: XFEM vs PF-CZM. Computer Methods in Applied Mechanics and Engineering, 2019, 345(3): 618-643 [14] Griffith AA. The phenomena of rupture and flow in solids. Philosophical Transactions of The Royal Society A: Mathematical Physical and Engineering Sciences, 1920, A221(4): 163-198 [15] Francfort GA, Marigo JJ. Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 1998, 46(8): 1319-1342 doi: 10.1016/S0022-5096(98)00034-9 [16] Bourdin B, Francfort GA, Marigo JJ. Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 2000, 48(4): 797-826 doi: 10.1016/S0022-5096(99)00028-9 [17] Molnár G, Gravouil A. 2D and 3D Abaqus implementation of a robust staggered phase-field solution for modeling brittle fracture. Finite Elements in Analysis and Design, 2017, 130(5): 27-38 [18] Ambati M, Gerasimov T, Delorenzis L. Phase-field modeling of ductile fracture. Computational Mechanics, 2015, 55(5): 1017-1040 doi: 10.1007/s00466-015-1151-4 [19] Fang JG, Wu CQ, Li J, et al. Phase field fracture in elasto-plastic solids: Variational formulation for multi-surface plasticity and effects of plastic yield surfaces and hardening. International Journal of Mechanical Sciences, 2019, 156(6): 382-396 [20] Geelen JM, Liu Y, Hu T, et al. A phase-field formulation for dynamic cohesive fracture. Computer Methods in Applied Mechanics and Engineering, 2019, 348(6): 680-711 [21] Lo YS, Borden MJ, Ravi CK, et al. A phase-field model for fatigue crack growth. Journal of the Mechanics and Physics of Solids, 2019, 132(11): 103684 [22] Clayton JD, Knap J. Phase field modeling of directional fracture in anisotropic polycrystals. Computational Materials Science, 2015, 98(3): 158-169 [23] 吴建营, 陈万昕, 黄羽立. 基于统一相场理论的早龄期混凝土化-热-力多场耦合裂缝模拟与抗裂性能预测. 力学学报, 2021, 53(5): 1367-1382 (Wu Jianying, Chen Wanxin, Huang Yuli. Computational modeling of shrinkage induced cracking in early-age concrete based on the unified phase-field theory. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(5): 1367-1382 (in Chinese) [24] Noll T, Kuhn C, Olesch D, et al. 3D phase field simulations of ductile fracture. GAMM-Mitteilungen, 2020, 43(2): 1-16 [25] 王博臣, 侯玉亮, 夏凉等. 基于子结构法与损伤识别的周期性结构脆性断裂相场模拟. 航空学报, 2022, 43(3): 293-304 (Wang Bochen, Hou Yuliang, Xia Liang, et al. Phase field modeling of the brittle fracture of periodic structures based on substructuring and damage identification. Journal of Aeronautics, 2022, 43(3): 293-304 (in Chinese) [26] 郑晓霞, 郑锡涛, 缑林虎. 多尺度方法在复合材料力学分析中的研究进展. 力学进展, 2010, 40(1): 42-52 (Zheng Xiaoxia, Zheng Xitao, Gou Linhu. The Research progress on multiscale method foe the mechanical analysis of composites. Advance in Mechanics, 2010, 40(1): 42-52 (in Chinese) doi: 10.6052/1000-0992-2010-1-J2008-104 [27] Tyrus JM, Gosz M, Desantiago E. A local finite element implementation for imposing periodic boundary conditions on composite micromechanical models. International Journal of Solids & Structures, 2007, 44(9): 2972-2989 [28] 卫宇璇. 周期性复合材料结构力学性能的多尺度分析. [硕士论文]. 吉林: 吉林大学, 2018Wei Yuxuan. Multi-scale analysis of mechanical properties of periodic composite structures. [Master Thesis]. Jilin: Jilin University, 2018 (in Chinese) [29] Liang X, Crosby AJ. Uniaxial stretching mechanics of cellular flexible metamaterials. Extreme Mechanics Letters, 2020, 35(2): 100637 [30] Sun Q, Zhou G, Meng Z, et al. Failure criteria of unidirectional carbon fiber reinforced polymer composites informed by a computational micromechanics model. Composites Science and Technology, 2019, 172(4): 81-95 [31] Msekh MA, Sargado JM, Jamshidian M, et al. Abaqus implementation of phase-field model for brittle fracture. Computational Materials Science, 2015, 96(1B): 472-484 [32] Miehe C, Welschinger F, Hofacker M. Thermodynamically consistent phase‐field models of fracture: Variational principles and multi‐field FE implementations. International Journal for Numerical Methods in Engineering, 2010, 83(10): 1274-1311 [33] Fang JG, Wu CQ, Rabczuk T, et al. Phase field fracture in elasto-plastic solids: Abaqus implementation and case studies. Theoretical and Applied Fracture Mechanics, 2019, 103(5): 102252 [34] Guo YD, Huang W, Ma YE. Buckling pattern transition of periodic porous elastomers induced by proportional loading conditions. International Journal of Applied Mechanics, 2021, 13(6): 2150067 doi: 10.1142/S1758825121500678 [35] 刘国威, 李庆斌, 左正. 相场断裂模型分步算法在ABAQUS中的实现. 岩石力学与工程学报, 2016, 35(5): 1020-1029 (Liu Guowei, Li Qingbin, Zuo Zheng. Implementation of phase field fracture model step algorithm in ABAQUS. Journal of Rock Mechanics and Engineering, 2016, 35(5): 1020-1029 (in Chinese) doi: 10.13722/j.cnki.jrme.2015.1264 [36] Sarac B, Schroers J. Designing tensile ductility in metallic glasses. Nature Communications, 2013, 4: 2158 doi: 10.1038/ncomms3158 -
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