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| A new meshfree-numerical manifold method for solving the fracture problem |
| LI Wei1，2，3，ZHENG Hong4，WANG Hailong1，GUO Hongwei5 |
| (1. School of Civil Engineering and Architecture，Linyi University，Linyi，Shandong 276000，China；2. Linyi City Engineering Technology Research Center of the Sponge City，Linyi，Shandong 276000，China；3. Linyi City Key laboratory of Appraisement and Strengthening in Building Structures，Linyi，Shandong 276000，China；4. College of Architecture and Civil Engineering，
Beijing University of Technology，Beijing 100124，China；5. Gottfried Wilhelm Leibniz Universit?t Hannover
Institute of Continuum Mechanics，Hannover 30167，Germany) |
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Abstract Combined the moving least squares(MLS) based on the approximation of nodes with numerical manifold method(NMM)，the meshfree-numerical manifold method is developed，abbreviated as MLS-NMM. MLS-NMM owns lower mesh-dependency and higher calculate precision than NMM，and naturally solves the continuous and non-continuous problems in a unified way. To increase the adaptability of MLS-NMM about analyzing the linear elastic fracture problems，a new MLS-NMM is proposed. It makes the mathematical patches containing the same crack tip integrate into a complex patch，in which the finite items of Williams’ displacement series are defined as the local approximation in ordering to reproduce the displacement field of the crack tip. And the method of quadtree′s local refinement is adopted to enhance the calculate precision in the vicinity of the crack tip. This method can directly acquire the stress intensity factors(SIFs) using the relationship between SIFs and items of Williams′ displacement series，avoiding post-processing solution such as the interaction integral method. In this paper，the maximum circumferential stress criterion is applied to decide the direction of crack propagation. At last，two examples of SIF and two examples of crack propagation are simulated，the results reveal that the proposed method can effectively deal with the linear elastic fracture problems，and simultaneously suggest that 2 layer local refinement and 9 items of Williams′ displacement series should be used.
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| [1] 张楚汉. 论岩石、混凝土离散-接触-断裂分析[J]. 岩石力学与工程学报，2008，27(2)：217–235.(ZHANG Chuhan. Discrete- contact-fracture analysis of rock and concrete[J]. Chinese Journal of Rock Mechanics and Engineering，2008，27(2)：217–235.(in Chinese))
[2] 徐栋栋，杨永涛，郑 宏，等. 线性无关高阶数值流形法在断裂力学中的应用[J]. 岩石力学与工程学报，2015，34(12)：2 463–2 473. (XU Dongdong，YANG yongtao，ZHENG Hong，et al. Application of the linearly independent high-order numerical manifold method in fracture mechanics[J]. Chinese Journal of Rock Mechanics and Engineering，2015，34(12)：2 463–2 473.(in Chinese))
[3] PRAMOD A L N，ANNABATTULA R K，OOI E T，et al. Adaptive phase-field modeling of brittle fracture using the scaled boundary finite element method[J]. Computer Methods in Applied Mechanics and Engineering，2019，355：284–307.
[4] REN H L，ZHUANG X Y，ANITESCU C，et al. An explicit phase field method for brittle dynamic fracture[J]. Computers and Structures，2019，217：45–56.
[5] SHI G H. Manifold method of material analysis[C]// Transactions of the Ninth Army Conference on Applied Mathematics and Computing. Minneapolish，Minncsoda，USA：[s. n.]，1992：51–76.
[6] ZHENG H，LIU F，LI CG. The MLS-based numerical manifold method with applications to crack analysis[J]. International Journal of Fracture，2014，190(1/2)：147–166.
[7] ZHENG H，LIU F，LI C G. Primal mixed solution to unconfined seepage flow in porous media with numerical manifold method[J]. Applied Mathematical Modelling，2015，39(2)：794–808.
[8] 王水林，葛修润. 流形元方法在模拟裂纹扩展中的应用[J]. 岩石力学与工程学报，1997，16(5)：7–12.(WANG Shuilin，GE Xiurun. Application of manifold method in simulating crack propagation. Chinese Journal of Rock Mechanics and Engineering，1997，16(5)：7–12.(in Chinese))
[9] WU Z J，WONG L N Y. Frictional crack initiation and propagation analysis using the numerical manifold method[J]. Computers and Geotechnics，2012，39：38–53.
[10] 刘 丰，郑 宏，夏开文. 基于 MLS 的数值流形法模拟多裂纹扩展[J]. 岩石力学与工程学报，2016，35(1)：76–86.(LIU Feng，ZHENG Hong，XIA Kaiwen. A MLS-based numerical manifold method for multiple cracks propagation[J]. Chinese Journal of Rock Mechanics and Engineering，2016，35(1)：76–86.(in Chinese))
[11] 李 伟，郑 宏，陈远强，等. 基于 MLS 的增强型数值流形法在动态裂纹扩展中的应用[J]. 岩石力学与工程学报，2018，37(7)： 1 574–1 585.(LI Wei，ZHENG Hong，CHEN Yuanqiang，et al. Application of the MLS based enriched numerical manifold method in dynamic crack propagation[J]. Chinese Journal of Rock Mechanics and Engineering，2018，37(7)：1 574–1 585.(in Chinese))
[12] TSAY R J，CHIOU Y J，CHUANG W L. Crack growth prediction by manifold method[J]. Journal of Engineering Mechanics，1999，125(8)：884–890.
[13] LIU Z J，ZHENG H. Two-dimensional numerical manifold method with multilayer covers[J]. Science China-Technological Sciences，2016，59(4)：515–530.
[14] LIU F，XIA K W. Structured mesh refinement in MLS-based numerical manifold method and its application to crack problems[J]. Engineering Analysis with Boundary Elements，2017，84：42–51.
[15] 李树忱，程玉民. 考虑裂纹尖端场的数值流形方法[J]. 土木工程学报，2005，38(7)：96–101.(LI Shuchen，CHENG Yumin. Numerical manifold method for crack tip fields[J]. China Civil Engineering Journal，2005，38(7)：96–101.(in Chinese))
[16] MA G W，AN X M，ZHANG H H，et al. Modeling complex crack problems using the numerical manifold method[J]. International Journal of Fracture，2009，156(1)：21–35.
[17] WU Z J，WONG L N Y，FAN L F. Dynamic study on fracture problems in viscoelastic sedimentary rocks using the numerical manifold method [J]. Rock Mechanics and Rock Engineering，2013，46(6)：1 415–1 427.
[18] WU Z J，WONG L N Y. Elastic-plastic cracking analysis for brittle-ductile rocks using manifold method [J]. International Journal of Fracture，2013，180(1)：71–91.
[19] WU Z J，WONG L N Y. Modeling cracking behavior of rock mass containing inclusions using the enriched numerical manifold method[J]. Engineering Geology，2013，162：1–13.
[20] ZHENG H，XU D D. New strategies for some issues of numerical manifold method in simulation of crack propagation[J]. International Journal for Numerical Methods in Engineering，2014，97(13)：986–1 010.
[21] ZHENG H，LIU F，DU X L. Complementarity problem arising from static growth of multiple cracks and MLS-based numerical manifold method[J]. Computer Methods in Applied Mechanics and Engineering，2015，295：150–171.
[22] YANG Y T，TANG X H，ZHENG H，et al. Three-dimensional fracture propagation with numerical manifold method [J]. Engineering Analysis with Boundary Elements，2016，72：65–77.
[23] YANG Y T，TANG X H，ZHENG H，et al. Hydraulic fracturing modeling using the enriched numerical manifold method [J]. Applied Mathematical Modelling，2018，53(462–486.
[24] ZHANG H H，MA G W，FAN L F. Thermal shock analysis of 2D cracked solids using the numerical manifold method and precise time integration[J]. Engineering Analysis with Boundary Elements，2017，75：46–56.
[25] 苏海东，祁勇峰，龚亚琦. 裂纹尖端解析解与周边数值解联合求解应力强度因子[J]. 长江科学院院报，2013，30(6)：83–89.(SU Haidong，QI Yongfeng，GONG Yaqi. Compute stress intensity factors via combining analytical solutions around crack tips with surrounding numerical solutions[J]. Journal of Yangtze River Scientific Research Institute，2013，30(6)：83–89.(in Chinese))
[26] LIU F，ZHENG H，DU X L. Hybrid analytical and MLS-based NMM for the determination of generalized stress intensity factors[J]. Mathematical Problems in Engineering，2015，1–9.
[27] LIU Z J，ZHENG H，SUN C. A domain decomposition based method for two-dimensional linear elastic fractures[J]. Engineering Analysis with Boundary Elements，2016，66：34–48.
[28] CHIOU Y J，LEE Y M，TSAY R J. Mixed mode fracture propagation by manifold method[J]. International Journal of Fracture，2002，114(4)：327–347.
[29] FAN H，ZHENG H，LI C G，et al. A decomposition technique of generalized degrees of freedom for mixed mode crack problems[J]. International Journal for Numerical Methods in Engineering，2017，112(7)：803–831.
[30] WILLIAMS M L. On the stress distribution at the base of a stationary crack[J]. Journal of Applied Mechanics，1957，24(1)：109–114.
[31] TADA H，PARIS P C，IRWIN G R. The stress analysis of cracks handbook[M]. 3rd ed. New York：ASME Press，2000：52–53.
[32] MURAKAMI Y. Stress Intensity factors handbook[M]. New York：Pergamon，1987：909.
[33] TANG X，WU S，ZHENG C，et al. A novel virtual node method for polygonal elements[J]. Applied Mathematics and Mechanics，2009，30(10)：1 233.
[34] YANG Y，SUN G，ZHENG H，et al. A four-node quadrilateral element fitted to numerical manifold method with continuous nodal stress for crack analysis[J]. Computers and Structures，2016，177：69–82. |
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2020, Vol. 39 
