TURE MODE II CRACK SIMULATION BASED ON A STRUCTURED DEFORMATION DRIVEN NONLOCAL MACRO-MESO-SCALE CONSISTENT DAMAGE MODEL
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摘要:II型载荷作用下裂纹变形模式也为II型的破坏问题称为真II型破坏. 准确定量地把握真II型破坏的全过程是具有挑战性的问题. 本文采用结构化变形驱动的非局部宏−微观损伤模型对真II型破坏问题进行了模拟. 根据结构化变形理论将点偶的非局部应变分解为弹性应变与结构化应变两部分, 进而利用Cauchy-Born准则与结构化应变计算点偶的结构化正伸长量. 在本文中, 结构化应变取为非局部应变的偏量部分. 当点偶的结构化正伸长量超过临界伸长量时, 微细观损伤开始在点偶层次发展. 将微细观损伤在作用域中进行加权求和得到拓扑损伤, 并通过能量退化函数将其嵌入到连续介质−损伤力学框架中进行数值求解. 进一步地, 本文采用Gauss-Lobatto积分格式计算点偶的非局部应变, 将积分点数目降低到4个, 显著降低了前处理和非线性分析的计算成本. 通过对II型加载下裂尖应变场的分析揭示了采用偏应变作为结构化应变的原因. 对两个典型真II型破坏问题的模拟结果表明, 本文方法不仅可以把握II型加载下的真II型裂纹扩展模式, 同时可以定量刻画加载过程中的载荷−变形曲线, 且不具有网格敏感性. 最后指出了需要进一步研究的问题.
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关键词:
- 结构化变形/
- 非局部宏−微观损伤模型/
- 真II型破坏/
- Gauss-Lobatto积分格式
Abstract:The fracture problem in which the crack deformation mode under mode II loading is also mode II is called the true mode II fracture problem. It is challenging to accurately and quantitively capture the whole process of true mode II fracture. In this paper, a structured deformation driven nonlocal macro-meso-scale consistent damage model is adopted to simulate the true mode II fracture problem. The nonlocal strain of a material point pair is decomposed into elastic strain and structured strain based on the theory of structured deformation. Then the structured positive elongation quantity of the material point pair can be evaluated by using the Cauchy-Born rule and the structured strain. In the present paper, the structured strain is taken as the deviatoric part of the nonlocal strain. When the structured positive elongation quantity of a material point pair exceeds the critical elongation quantity, mesoscopic damage starts to emerge at the point-pair level. The topologic damage can be obtained by weighted summing of the mesoscopic damage within the influence domain, then it is embedded into the framework of continuum damage mechanics through the energetic degradation function bridging the geometric damage and energetic damage for numerical solution. Further, the Gauss-Lobatto integration scheme is adopted in this paper to evaluate the nonlocal strain of point pairs, which reduces the number of integral points to 4 and thus considerably reduces the computational cost of preprocessing and nonlinear analysis. The reason for adopting the deviatoric strain as structured strain is revealed based on the analysis of the strain field at the crack tip under mode II loading. Numerical results for two typical true mode II fracture problems indicate that the proposed model can not only well capture the crack deformation pattern of true mode II cracks, but also quantitatively characterize the load-deformation curves without mesh size sensitivity. Problems to be further investigated are also discussed. -
图 2物质点、物质点偶及作用域示意图
Figure 2.Schematic of material point, material point pair and influence domain
图 5紧剪切试件及半结构示意图 (单位: mm)
Figure 5.Schematic of compact shear specimen and semi-structure(unit: mm)
图 8紧剪切试件不同网格的载荷−位移曲线
Figure 8.The load-displacement curves obtained from different meshes for the compact shear specimen
图 10紧剪切试件各载荷步裂纹图(变形放大100倍)
Figure 10.Crack patterns of the compact shear specimen in each load step (deformation magnified by 100 times)
图 11紧剪切试件载荷步III应力云图
Figure 11.Stress contour of compact shear specimen in load step III
图 12两种积分格式计算载荷−位移曲线对比
Figure 12.The load-displacement curves obtained from two integration schemes
图 16长剪切试件不同网格的载荷−位移曲线
Figure 16.The load-displacement curves obtained from different meshes for the long shear specimen
图 20两种积分格式计算载荷−位移曲线对比
Figure 20.The load-displacement curves obtained from two integration schemes
表 14点Gauss-Lobatto格式积分点与积分权重
Table 1.The integral points and weights of 4 point Gauss-Lobatto scheme
Integral points ${s_i}$ Integral weights ${\xi _i}$ $\pm 0.447\;214$ $0.833\;333$ $ \pm 1.0$ $0.166\;667$ 
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表 2紧剪切试件模型参数取值
Table 2.The model parameters for the compact shear specimen
$\ell $/mm ${\lambda _{\text{c}}}$/mm $\gamma $/mm $p$ $q$ $1.5$ $5 \times {10^{ - 4} }$ $\infty$ $11$ $0$ 下载:
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表 3紧剪切试件计算时间
Table 3.The computational time of compact shear specimen
Mesh Integration scheme Time preprocessing/ms nonlinear analysis/s mesh A GL4 47.872 50.778 EQ9 141.038 54.955 mesh B GL4 150.087 88.100 EQ9 464.347 126.825 下载:
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表 4长剪切试件模型参数取值
Table 4.The model parameters for the long shear specimen
$\ell $/mm ${\lambda _{\text{c}}}$/μm $\gamma $/mm $p$ $q$ $3$ $3.5$ $7$ $7$ $13$ 下载:
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表 5长剪切试件计算时间
Table 5.The computational time of long shear specimen
Mesh Integration scheme Time/s preprocessing nonlinear analysis mesh A GL4 0.551 107 44.918 EQ9 1.695 55.940 mesh B GL4 1.284 68.400 EQ9 4.137 84.187 下载:
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[1] Gdoutos EE. Fracture Mechanics: An Introduction. Springer Nature, 2020 [2] 嵇醒. 断裂力学判据的评述. 力学学报, 2016, 48(4): 741-753 (Ji Xing. A critical review on criteria of fracture mechanics.Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(4): 741-753 (in Chinese)doi:10.6052/0459-1879-16-069 [3] Rao Q, Sun Z, Stephansson O, et al. Shear fracture (mode II) of brittle rock.International Journal of Rock Mechanics and Mining Sciences, 2003, 40(3): 355-375doi:10.1016/S1365-1609(03)00003-0 [4] 杨卫. 宏−微观断裂力学. 北京: 国防工业出版社, 1995Yang Wei. Macro- and Micro-scale Fracture Mechanics. Beijing: Defense Industrial Press of China, 1995 (in Chinese)) [5] Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear.Journal of Basic Engineering, 1963, 85(4): 519-525doi:10.1115/1.3656897 [6] Bahrami B, Nejati M, Ayatollahi MR, et al. Theory and experiment on true mode II fracturing of rocks.Engineering Fracture Mechanics, 2020, 240: 107314doi:10.1016/j.engfracmech.2020.107314 [7] Bahrami B, Ghouli S, Nejati M, et al. Size effect in true mode II fracturing of rocks: Theory and experiment.European Journal of Mechanics - A/Solids, 2022, 94: 104593 [8] Bobet A, Einstein HH. Fracture coalescence in rock-type materials under uniaxial and biaxial compression.International Journal of Rock Mechanics and Mining Sciences, 1998, 35(7): 863-888doi:10.1016/S0148-9062(98)00005-9 [9] Oda M, Kazama H. Microstructure of shear bands and its relation to the mechanisms of dilatancy and failure of dense granular soils.Géotechnique, 1998, 48(4): 465-481 [10] Kalthoff JF. Modes of dynamic shear failure in solids.International Journal of Fracture, 2000, 101(1): 1-31 [11] Park JW, Song JJ. Numerical simulation of a direct shear test on a rock joint using a bonded-particle model.International Journal of Rock Mechanics and Mining Sciences, 2009, 46(8): 1315-1328doi:10.1016/j.ijrmms.2009.03.007 [12] Zhang L, Thornton C. A numerical examination of the direct shear test.Géotechnique, 2007, 57(4): 343-354 [13] Bourdin B, Francfort GA, Marigo JJ. Numerical experiments in revisited brittle fracture.Journal of the Mechanics and Physics of Solids, 2000, 48(4): 797-826doi:10.1016/S0022-5096(99)00028-9 [14] Silling SA. Reformulation of elasticity theory for discontinuities and long-range forces.Journal of the Mechanics and Physics of Solids, 2000, 48(1): 175-209doi:10.1016/S0022-5096(99)00029-0 [15] Silling SA, Askari E. A meshfree method based on the peridynamic model of solid mechanics.Computers&Structures, 2005, 83(17): 1526-1535 [16] Lancioni G, Royer-Carfagni G. The variational approach to fracture mechanics. A practical application to the French Panthéon in Paris.Journal of Elasticity, 2009, 95(1): 1-30 [17] Freddi F, Royer-Carfagni G. Regularized variational theories of fracture: A unified approach.Journal of the Mechanics and Physics of Solids, 2010, 58(8): 1154-1174doi:10.1016/j.jmps.2010.02.010 [18] Fei F, Choo J. A phase-field model of frictional shear fracture in geologic materials.Computer Methods in Applied Mechanics and Engineering, 2020, 369: 113265doi:10.1016/j.cma.2020.113265 [19] Alessi R, Freddi F, Mingazzi L. Phase-field numerical strategies for deviatoric driven fractures.Computer Methods in Applied Mechanics and Engineering, 2020, 359: 112651doi:10.1016/j.cma.2019.112651 [20] Gavagnin C, Sanavia L, De Lorenzis L. Stabilized mixed formulation for phase-field computation of deviatoric fracture in elastic and poroelastic materials.Computational Mechanics, 2020, 65(6): 1447-1465doi:10.1007/s00466-020-01829-x [21] Kim S, Cervera M, Wu JY, et al. Strain localization of orthotropic elasto-plastic cohesive-frictional materials: Analytical results and numerical verification.Materials, 2021, 14(8): 2040doi:10.3390/ma14082040 [22] Ren H, Zhuang X, Rabczuk T. A new peridynamic formulation with shear deformation for elastic solid.Journal of Micromechanics and Molecular Physics, 2016, 1(2): 1650009doi:10.1142/S2424913016500090 [23] Zhang Y, Qiao P. A new bond failure criterion for ordinary state-based peridynamic mode II fracture analysis.International Journal of Fracture, 2019, 215(1): 105-128 [24] Madenci E, Barut A, Phan N. Bond-based peridynamics with stretch and rotation kinematics for opening and shearing modes of fracture.Journal of Peridynamics and Nonlocal Modeling, 2021, 3(3): 211-254. [25] Lu GD, Chen JB. A new nonlocal macro-meso-scale consistent damage model for crack modeling of quasi-brittle materials.Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112802doi:10.1016/j.cma.2019.112802 [26] 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): 1-14 (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): 1-14 (in Chinese) [27] Wu JY. A unified phase-field theory for the mechanics of damage and quasi-brittle failure.Journal of the Mechanics and Physics of Solids, 2017, 103: 72-99doi:10.1016/j.jmps.2017.03.015 [28] 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 theoretical and numerical aspects of the unified phase-field theoryfor damage and failure in solids and structures.Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(2): 301-329 (in Chinese)doi:10.6052/0459-1879-20-295 [29] Lu GD, Chen JB. Dynamic cracking simulation by the nonlocal macro-meso-scale damage model for isotropic materials.International Journal for Numerical Methods in Engineering, 2021, 122(12): 3070-3099doi:10.1002/nme.6654 [30] Chen JB, Ren YD, Lu GD. Meso-scale physical modeling of energetic degradation function in the nonlocal macro-meso-scale consistent damage model for quasi-brittle materials.Computer Methods in Applied Mechanics and Engineering, 2021, 374: 113588doi:10.1016/j.cma.2020.113588 [31] 任宇东, 陈建兵. 基于一类非局部宏−微观损伤模型的混凝土典型试件力学行为模拟. 力学学报, 2021, 53(4): 1196-1121Ren Yyudong, Chen Jianbing. Simulation of behaviour of typical concrete specimens based on a nonlocal macro-meso-scale consistent damage model.Chinese Journal of Theoretical and Applied Mechanics. 2021, 53(4): 1196-1121 (in Chinese)) [32] 杜成斌, 黄文仓, 江守燕. SBFEM与非局部宏-微观损伤模型相结合的准脆性材料开裂模拟. 力学学报, 2022, 54(4): 1-14 (Du Chengbin, Huangwencang, Jiang Shouyan. Cracking simulation of quasi-brittle materials by combining SBFEM with nonlocal macro-micro damage model.Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(4): 1-14 (in Chinese) [33] Ren YD, Chen JB, Lu GD. A structured deformation driven nonlocal macro-meso-scale consistent damage model for the compression/shear dominate failure simulation of quasi-brittle materials//Computer Methods in Applied Mechanics and Engineering, 2022 [34] Del Piero G, Owen DR. Structured deformations of continua.Archive for Rational Mechanics and Analysis, 1993, 124(2): 99-155doi:10.1007/BF00375133 [35] Del Piero G, Owen DR. Multiscale Modeling in Continuum Mechanics and Structured Deformations. Springer, 2004 [36] Bažant ZP, Jirásek M. Nonlocal integral formulations of plasticity and damage: Survey of progress.Journal of Engineering Mechanics, 2002, 128(11): 1119-1149doi:10.1061/(ASCE)0733-9399(2002)128:11(1119) [37] 李杰, 吴建营, 陈建兵. 混凝土随机损伤力学. 北京: 科学出版社, 2014Li Jie, Wu Jianying, Chen Jianbing. Stochastic Damage Mechanics of Concrete Structures. Beijing: Science Press, 2014 (in Chinese)) [38] Bathe KJ. Finite Element Procedures. Prentice Hall, 2006 [39] De Borst R. Computation of post-bifurcation and post-failure behavior of strain-softening solids.Computers&Structures, 1987, 25(2): 211-224 [40] May IM, Duan Y. A local arc-length procedure for strain softening.Computers&Structures, 1997, 64(1): 297-303 [41] Zehnder AT. Fracture Mechanics. Netherlands: Springer, 2012 [42] Chisholm DB, Jones DL. An analytical and experimental stress analysis of a practical mode II fracture test specimen.Experimental Mechanics, 1977, 17(1): 7-13doi:10.1007/BF02324265 [43] Miehe C, Hofacker M, Welschinger F. A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits.Computer Methods in Applied Mechanics and Engineering, 2010, 199(45): 2765-2778 -
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