动载下缝端应力强度因子计算的扩展有限元法
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摘要
在扩展有限元法(extended finite element methods,XFEM)的理论框架下,重点研究了动荷载作用下稳定裂纹尖端动态应力强度因子(dynamic stress intensity factors,DSIFs)的求解方法。根据XFEM的位移模式,推导了动力XFEM的支配方程,采用Newmark隐式算法进行时间积分。同时,提出一种XFEM质量矩阵的集中策略,给出了求解DSIFs的相互作用积分方法,与静态问题的相互作用积分方法相比,增加了惯性项的贡献。最后,若干典型算例的计算结果表明:XFEM可以准确评价稳定裂纹尖端的DSIFs,建议的质量矩阵集中策略是有效的,为得到正确的DSIFs,惯性项的贡献不可忽略。
In the framework of the extended finite element methods(XFEM),the extraction of dynamic stress intensity factors(DSIFs) for stationary cracks being subjected to dynamic loads was detaily studied.Having constructed the approximation of dynamic XFEM,the derivation of governing equation for dynamic XFEM was presented.The Newmark implicit algorithm was used for time integration.Meanwhile,a mass lumping strategy for XFEM implicit dynamics was proposed.In addition,the interaction integral method was given for evaluating DSIFs.Compared with the interaction integral method for evaluating stress intensity factors(SIFs) of cracks under static conditions,the contribution of inertial effects was added to the interaction integral method for evaluating DSIFs.The numerical illustrations show that the XFEM can evaluate accurately DSIFs and the proposed mass lumping strategy is also quite effective.To obtain DSIFs correctly,the inertial effects on interaction integral cannot be ignored.
In the framework of the extended finite element methods(XFEM),the extraction of dynamic stress intensity factors(DSIFs) for stationary cracks being subjected to dynamic loads was detaily studied.Having constructed the approximation of dynamic XFEM,the derivation of governing equation for dynamic XFEM was presented.The Newmark implicit algorithm was used for time integration.Meanwhile,a mass lumping strategy for XFEM implicit dynamics was proposed.In addition,the interaction integral method was given for evaluating DSIFs.Compared with the interaction integral method for evaluating stress intensity factors(SIFs) of cracks under static conditions,the contribution of inertial effects was added to the interaction integral method for evaluating DSIFs.The numerical illustrations show that the XFEM can evaluate accurately DSIFs and the proposed mass lumping strategy is also quite effective.To obtain DSIFs correctly,the inertial effects on interaction integral cannot be ignored.
引文
[1]章青,刘宽,夏晓舟,杨静.广义扩展有限元法及其在裂纹扩展分析中的应用[J].计算力学学报,2012,29(3):427-432.(ZHANG Qing,LIU Kuan,XIA Xiao-Zhou,YANG Jing.Generalized extended finite element method and its application in crack growth analysis[J].Chinese Journal of Computational Mechanics,2012,29(3):427-432.(in Chinese))
[2]余天堂,万林林.非均质材料热传导问题的扩展有限元法[J].计算力学学报,2011,28(6):884-890.(YU Tian-tang,WAN Lin-lin.Extended finite element method for heat transfer problems in heterogeneous material[J].Chinese Journal of Computational Mechanics,2011,28(6):884-890.(in Chinese))
[3]Belytschko T,Chen H,Xu J,Zi G.Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment[J].International Journal for Numerical Methods in Engineering,2003,58(12):1873-1905.
[4]Chen H.Enriched finite element methods and their applications[D].Evanston,Illinois:Northwestern University,2003.
[5]RéthoréJ,Gravouil A,Combescure A.An energy-conserving scheme for dynamic crack growth using the extended finite element method[J].International Journal for Numerical Methods in Engineering,2005,63(5):631-659.
[6]RéthoréJ,Gravouil A,Combescure A.A combined space-time extended finite element method[J].In-ternational Journal for Numerical Methods in Engineering,2005,64(2):260-284.
[7]Menouillard T,Belytschko T.Dynamic fracture with meshfree enriched XFEM[J].Acta Mechanica,2010,213(1/2):53-69.
[8]Menouillard T,Belytschko T.Smoothed nodal forces for improved dynamic crack propagation modeling in XFEM[J].International Journal for Numerical Methods in Engineering,2010,84(1):47-72.
[9]Menouillard T,Song J H,Duan Q L,Belytschko T.Time dependent crack tip enrichment for dynamic crack propagation[J].International Journal of Fracture,2010,162(1/2):33-49.
[10]Menouillard T,RéthoréJ,Combescure A,Bung H.Efficient explicit time stepping for the extended fi-nite element method(X-FEM)[J].International Journal for Numerical Methods in Engineering,2006,68(9):911-939.
[11]Menouillard T,RéthoréJ,Mos N,Combescure A,Bung H.Mass lumping strategies for X-FEM explicit dynamics:application to crack propagation[J].International Journal for Numerical Methods in Engineering,2008,74(3):447-474.
[12]Liu Z L,Menouillard T,Belytschko T.An XFEM/spectral element method for dynamic crack propaga-tion[J].International Journal of Fracture,2011,169(2):183-198.
[13]Motamedi D,Mohammadi S.Dynamic analysis of fixed cracks in composites by the extended finite ele-ment method[J].Engineering Fracture Mechanics,2010,77(17):3373-3393.
[14]方修君,金峰,王进廷.基于扩展有限元法的Koyna重力坝地震开裂过程模拟[J].清华大学学报(自然科学版),2008,48(12):2065-2069.(FANG Xiu-jun,JIN Feng,WANG Jin-ting.Seismic fracture simulation of the Koyna gravity dam using an extended finite element method[J].Journal of Tsinghua University(Science and Technology),2008,48(12):2065-2069.(in Chinese))
[15]王志勇,马力,吴林志,于红军.基于扩展有限元法的颗粒增强复合材料静态及动态断裂行为研究[J].固体力学学报,2011,32(6):566-573.(WANG Zhi-yong,MA Li,WU Lin-zhi,YU Hong-jun.Investigation of static and dynamic fracture behavior of particle-reinforced composite materi-als by the extended finite element method[J].Chinese Journal of Solid Mechanics,2011,32(6):566-573.(in Chinese))
[16]庄茁,柳占立,成斌斌,廖剑晖.扩展有限单元法[M].北京:清华大学出版社,2012.(ZHUANG Zhuo,LIU Zhan-li,CHENG Bin-bin,LIAO Jian-hui.Extended Finite Element Method[M].Beijing:Tsinghua University Press,2012.(in Chinese))
[17]Zhuang Z,Cheng B B.Development of X-FEM methodology and study on mixed-mode crack propaga-tion[J].Acta Mechanica Sinica,2011,27(3):406-415.
[18]范天佑.断裂动力学原理与应用[M].北京:北京理工大学出版社,2006:10-12.(FANG Tian-you.Fracture Dynamics:Principle and Application[M].Beijing:Beijing Institute of Technology Press,2006:10-12.(in Chinese))
[19]Mohammadi S.Extended Finite Element Method[M].Blackwell Publishing Ltd,2008.
[20]Song S H,Paulino G H.Dynamic stress intensity factors for homogeneous and smoothly heterogeneous materials using the interaction integral method[J].International Journal of Solids and Structures,2006,43(16):4830-4866.
[21]Freund L B.Dynamic Fracture Mechanics[M].Cambridge Monographs on Mechanics and Applied Mathematics,1990.
[22]Chen Y M.Numerical computation of dynamic stress intensity factors by a Lagrangian finite-difference method(the HEMP code)[J].Engineering Fracture Mechanics,1975,7(4):653-660.
[2]余天堂,万林林.非均质材料热传导问题的扩展有限元法[J].计算力学学报,2011,28(6):884-890.(YU Tian-tang,WAN Lin-lin.Extended finite element method for heat transfer problems in heterogeneous material[J].Chinese Journal of Computational Mechanics,2011,28(6):884-890.(in Chinese))
[3]Belytschko T,Chen H,Xu J,Zi G.Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment[J].International Journal for Numerical Methods in Engineering,2003,58(12):1873-1905.
[4]Chen H.Enriched finite element methods and their applications[D].Evanston,Illinois:Northwestern University,2003.
[5]RéthoréJ,Gravouil A,Combescure A.An energy-conserving scheme for dynamic crack growth using the extended finite element method[J].International Journal for Numerical Methods in Engineering,2005,63(5):631-659.
[6]RéthoréJ,Gravouil A,Combescure A.A combined space-time extended finite element method[J].In-ternational Journal for Numerical Methods in Engineering,2005,64(2):260-284.
[7]Menouillard T,Belytschko T.Dynamic fracture with meshfree enriched XFEM[J].Acta Mechanica,2010,213(1/2):53-69.
[8]Menouillard T,Belytschko T.Smoothed nodal forces for improved dynamic crack propagation modeling in XFEM[J].International Journal for Numerical Methods in Engineering,2010,84(1):47-72.
[9]Menouillard T,Song J H,Duan Q L,Belytschko T.Time dependent crack tip enrichment for dynamic crack propagation[J].International Journal of Fracture,2010,162(1/2):33-49.
[10]Menouillard T,RéthoréJ,Combescure A,Bung H.Efficient explicit time stepping for the extended fi-nite element method(X-FEM)[J].International Journal for Numerical Methods in Engineering,2006,68(9):911-939.
[11]Menouillard T,RéthoréJ,Mos N,Combescure A,Bung H.Mass lumping strategies for X-FEM explicit dynamics:application to crack propagation[J].International Journal for Numerical Methods in Engineering,2008,74(3):447-474.
[12]Liu Z L,Menouillard T,Belytschko T.An XFEM/spectral element method for dynamic crack propaga-tion[J].International Journal of Fracture,2011,169(2):183-198.
[13]Motamedi D,Mohammadi S.Dynamic analysis of fixed cracks in composites by the extended finite ele-ment method[J].Engineering Fracture Mechanics,2010,77(17):3373-3393.
[14]方修君,金峰,王进廷.基于扩展有限元法的Koyna重力坝地震开裂过程模拟[J].清华大学学报(自然科学版),2008,48(12):2065-2069.(FANG Xiu-jun,JIN Feng,WANG Jin-ting.Seismic fracture simulation of the Koyna gravity dam using an extended finite element method[J].Journal of Tsinghua University(Science and Technology),2008,48(12):2065-2069.(in Chinese))
[15]王志勇,马力,吴林志,于红军.基于扩展有限元法的颗粒增强复合材料静态及动态断裂行为研究[J].固体力学学报,2011,32(6):566-573.(WANG Zhi-yong,MA Li,WU Lin-zhi,YU Hong-jun.Investigation of static and dynamic fracture behavior of particle-reinforced composite materi-als by the extended finite element method[J].Chinese Journal of Solid Mechanics,2011,32(6):566-573.(in Chinese))
[16]庄茁,柳占立,成斌斌,廖剑晖.扩展有限单元法[M].北京:清华大学出版社,2012.(ZHUANG Zhuo,LIU Zhan-li,CHENG Bin-bin,LIAO Jian-hui.Extended Finite Element Method[M].Beijing:Tsinghua University Press,2012.(in Chinese))
[17]Zhuang Z,Cheng B B.Development of X-FEM methodology and study on mixed-mode crack propaga-tion[J].Acta Mechanica Sinica,2011,27(3):406-415.
[18]范天佑.断裂动力学原理与应用[M].北京:北京理工大学出版社,2006:10-12.(FANG Tian-you.Fracture Dynamics:Principle and Application[M].Beijing:Beijing Institute of Technology Press,2006:10-12.(in Chinese))
[19]Mohammadi S.Extended Finite Element Method[M].Blackwell Publishing Ltd,2008.
[20]Song S H,Paulino G H.Dynamic stress intensity factors for homogeneous and smoothly heterogeneous materials using the interaction integral method[J].International Journal of Solids and Structures,2006,43(16):4830-4866.
[21]Freund L B.Dynamic Fracture Mechanics[M].Cambridge Monographs on Mechanics and Applied Mathematics,1990.
[22]Chen Y M.Numerical computation of dynamic stress intensity factors by a Lagrangian finite-difference method(the HEMP code)[J].Engineering Fracture Mechanics,1975,7(4):653-660.
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