基于扩展有限元法的平板模型裂纹扩展研究
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摘要
扩展有限元法(XFEM)是基于单位分解法的思想,在传统有限元框架内,通过在位移场中增加反映裂纹面的不连续函数及反映裂尖局部特性的裂尖渐近位移场函数,体现裂纹的存在,从而使裂纹与有限元模型相互独立,方便了裂纹扩展的模拟。水平集法(Level Set Method, LSM)是一种追踪界面移动的数值技术,本文将水平集法与扩展有限元法相结合,对平面裂纹与板弯曲裂纹进行了系列的模拟和研究,本文做了如下工作：
首先,对断裂力学理论进行了简单的介绍,给出了相互作用能量积分计算应力强度因子的基本公式,给出了等参单元的逆变换算法,并详细的介绍了扩展有限元法的基本原理。在此基础上编制了扩展有限元法计算机计算程序,给出了程序的基本流程,并对程序中的关键部分进行了介绍。
其次,采用自行编制的扩展有限元法程序,对平面裂纹问题进行了系列的数值模拟和研究,研究了J积分路径和网格密度对计算结构的影响,并通过对单边裂纹、双边裂纹、中心裂纹与两条中心裂纹的数值模拟,验证了扩展有限元法具有很高的精确度。同时,采用最大周向应力准则,对裂纹的准静态扩展进行了模拟,并对结果进行了分析；
最后,分别介绍了Kirchhoff板弯曲断裂理论、Reissner板弯曲断裂理论以及能避免剪切闭锁的MITC单元,给出了采用扩展有限元法计算板弯曲裂纹的基本方法以及弯曲应力强度因子的计算。分别将扩展有限元法与Reissner板弯曲断裂理论和MITC单元结合,对板弯曲裂纹进行了数值模拟,并讨论了网格大小、积分区域因子等因素对计算结果的影响。通过对不同厚度板应力强度因子的计算,得到了将扩展有限元法与MITC单元后可以避免剪切闭锁的结论。
文中计算的应力强度因子的结果和变化规律,可以作为工程上相关问题的参考。分析含裂纹板的方法和编制的程序流程,对进一步分析含裂纹平板结构具有重要的意义。
The extended finite element method (XFEM) is based on the concept of partition of unity method and within standard finite element framework. The enrichment functions typically consist of the near-tip asymptotic functions that capture the singularity around the crack tip and a discontinuous function that represents the jump in displacement across the crack surfaces. Thus, the crack surface can be presented and the crack surface is not associated with finite element model. So, the method can simulate crack propagation conveniently. Level set method (LSM) is a numerical method for track the interface propagation paths. This thesis combines extended finite element method and level set method, and simulates a series of plane crack and plate bending crack problems. The major contents are as follows:
Firstly, the concepts of linear elastic fracture mechanics are introduced, and interaction energy integrals are used to compute stress intensity factor. The method for inverse isoparametric elements mapping is introduced. The theory of extend finite element is expounded detailedly. Compiled the XFEM program, the flow of implementing XFEM and the mainly parts of the program is given.
Secondly, the plane crack problems are simulated by using the developed program. The factors of effecting result are discussed, including J-integral and mesh. The precision of extended finite element method is verified by simulating of single edge crack, double edge cracks, a center crack and double center cracks. Meanwhile, the quasi-static crack propagation is simulated using the maximum tangential stress criterion and the results are analyzed.
Finally, Kirchhoff classical fracture theory, Reissner fracture theory and MITC element which can avoid shear locking are introduced. The method for calculate plate bending crack problem and bending stress intensity factors is introduced. Plate bending crack problem is simulated by Reissner fracture theory and MITC element respectively. The factors of effecting result are discussed, including J-integral and mesh. The phenomenon that MITC element can avoid shear locking is certified by simulating different thickness plates.
The computed results of SIFs provide the reference for crack problems in engineering. The method for analytic plates with crack and the flow chart of XFEM are significant for analytic shells with crack.
首先,对断裂力学理论进行了简单的介绍,给出了相互作用能量积分计算应力强度因子的基本公式,给出了等参单元的逆变换算法,并详细的介绍了扩展有限元法的基本原理。在此基础上编制了扩展有限元法计算机计算程序,给出了程序的基本流程,并对程序中的关键部分进行了介绍。
其次,采用自行编制的扩展有限元法程序,对平面裂纹问题进行了系列的数值模拟和研究,研究了J积分路径和网格密度对计算结构的影响,并通过对单边裂纹、双边裂纹、中心裂纹与两条中心裂纹的数值模拟,验证了扩展有限元法具有很高的精确度。同时,采用最大周向应力准则,对裂纹的准静态扩展进行了模拟,并对结果进行了分析；
最后,分别介绍了Kirchhoff板弯曲断裂理论、Reissner板弯曲断裂理论以及能避免剪切闭锁的MITC单元,给出了采用扩展有限元法计算板弯曲裂纹的基本方法以及弯曲应力强度因子的计算。分别将扩展有限元法与Reissner板弯曲断裂理论和MITC单元结合,对板弯曲裂纹进行了数值模拟,并讨论了网格大小、积分区域因子等因素对计算结果的影响。通过对不同厚度板应力强度因子的计算,得到了将扩展有限元法与MITC单元后可以避免剪切闭锁的结论。
文中计算的应力强度因子的结果和变化规律,可以作为工程上相关问题的参考。分析含裂纹板的方法和编制的程序流程,对进一步分析含裂纹平板结构具有重要的意义。
The extended finite element method (XFEM) is based on the concept of partition of unity method and within standard finite element framework. The enrichment functions typically consist of the near-tip asymptotic functions that capture the singularity around the crack tip and a discontinuous function that represents the jump in displacement across the crack surfaces. Thus, the crack surface can be presented and the crack surface is not associated with finite element model. So, the method can simulate crack propagation conveniently. Level set method (LSM) is a numerical method for track the interface propagation paths. This thesis combines extended finite element method and level set method, and simulates a series of plane crack and plate bending crack problems. The major contents are as follows:
Firstly, the concepts of linear elastic fracture mechanics are introduced, and interaction energy integrals are used to compute stress intensity factor. The method for inverse isoparametric elements mapping is introduced. The theory of extend finite element is expounded detailedly. Compiled the XFEM program, the flow of implementing XFEM and the mainly parts of the program is given.
Secondly, the plane crack problems are simulated by using the developed program. The factors of effecting result are discussed, including J-integral and mesh. The precision of extended finite element method is verified by simulating of single edge crack, double edge cracks, a center crack and double center cracks. Meanwhile, the quasi-static crack propagation is simulated using the maximum tangential stress criterion and the results are analyzed.
Finally, Kirchhoff classical fracture theory, Reissner fracture theory and MITC element which can avoid shear locking are introduced. The method for calculate plate bending crack problem and bending stress intensity factors is introduced. Plate bending crack problem is simulated by Reissner fracture theory and MITC element respectively. The factors of effecting result are discussed, including J-integral and mesh. The phenomenon that MITC element can avoid shear locking is certified by simulating different thickness plates.
The computed results of SIFs provide the reference for crack problems in engineering. The method for analytic plates with crack and the flow chart of XFEM are significant for analytic shells with crack.
引文
[1]陈景杰,黄一,刘刚.基于奇异元计算分析裂纹尖端应力强度因子[J].中国造船,2010,51(3)：56-64.
[2]孟玮.边界元法在结构断裂分析中的应用[D].广西：广西大学,2005.
[3]孙和成,张永元.边界积分方程-边界元法在三维断裂力学中的应用[J].上海交通大学学报,1984,18(3)：99-108.
[4]刘佳莉.无网格法在断裂力学中的应用研究[D].北京：中国石油大学,2008.
[5]刘凯远.无网格局部Petrov-Galerkin方法在断裂力学中的应用[D].湖南：湖南大学,2007.
[6]Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing[J]. International Journal for numerical Methods in Engineering,1999,45(5):601-620.
[7]Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering, 1999,46(1):131-150.
[8]Daux C, Moes N, Dolbow J, et al. Arbitrary branched and intersecting cracks with the extended finite element method [J]. International Journal for Numerical Methods in Engineering,2000,48:1741-1760.
[9]Dolbow J, Mo S N, Belytschko T. Modeling fracture in Mindlin-Reissner plates with the extended finite element method[J]. International Journal of Solids and Structures, 2000,37(48-50):7161-7183.
[10]Sukumar N, Moes N, Moran B, et al. Extended finite element method for three-dimensional crack model ing[J]. International Journal for Numerical Methods in Engineering, 2000,48(11):1549-1570.
[11]Stolarska M, Chopp D L, Moes N, et al. Modeling crack growth by level sets in the extended finite element method [J]. International Journal for Numerical Methods in Engineering, 2001,51 (8):943-960.
[12]Sukumar N, Chopp D L, Moes N, et al. Modeling holes and inclusions by level sets in the extended finite element method[J]. Computer Methods in Applied Mechanics and Engineering,2001,190:6183-6200
[13]Moes N, Gravouil A, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets-Part I:Computational Procedure [J]. International Journal for Numerical Methods in Engineering,2002,53:2549-2568.
[14]Sukumar N, Chopp D L, Moran B. Extended finite element method and fast marching method for three-dimensional fatigue crack propagation[J]. Engineering Fracture Mechanics, 2003,70:29-48.
[15]Belytschko T, Parimi C, Moes N, et al. Structured extended finite element methods for solids defined by implicit surfaces[J]. International Journal for Numerical Methods in Engineering,2003,56(4):609-635.
[16]Mariani S, Perego U. Extended finite element method for quasi-brittle fracture[J]. International Journal for Numerical Methods in Engineering,2003,58(1):103-126.
[17]Ventura G, Budyn E, Belytschko T. Vector level sets for description of propagating cracks in finite elements[J]. International Journal for Numerical Methods in Engineering,2003,58 (10):1571-1592.
[18]Toshio N, Youhei 0, Shuichi T. Stress intensity factor analysis of interface cracks using X-FEM. Int. J. Numer. Meth. Engng,2003,56:1151-1173.
[19]X. Y. Liu, Q. Z. Xiao, B. L. Karihaloo. XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials[J]. International Journal for Numerical Methods in Engineering,2004,59:1103-1118.
[20]Dolbow J E, Devan A. Enrichment of enhanced assumed strain approximations for representing strong discontinuities:addressing volumetric incompressibility and the discontinuous patch test[J]. International Journal for Numerical Methods in Engineering,2004,59(1):47"67.
[21]Ventura G, Moran B, Belytschko T. Dislocations by partition of unity[J]. International Journal for Numerical Methods in Engineering,2005,62(11):1463-1487.
[22]陈胜宏,汪卫明,徐明毅,等.小湾高拱坝坝踵开裂的有限单元法分析[J].水利学报,2003,1(1)：66-71.
[23]李录贤,王铁军.扩展有限元法(XFEM)及其应用[J].力学进展,2005,35(1)：5-20.
[24]余天堂.含裂纹体的数值模拟[J].岩石力学与工程学报,2005,24(24)：4434-4439.
[25]方修君,金峰.基于ABAQUS平台的扩展有限元法[J].工程力学,2007,24(7)：6-10.
[26]方修君,金峰,王进廷.基于扩展有限元法的粘聚裂纹模型[J].清华大学学报,2007,47(3)：344-347.
[27]董玉文,余天堂,任青文.直接计算应力强度因子的扩展有限元法[J].计算力学学报,2008,25(1)：72-77.
[28]余天堂.摩擦接触裂纹问题的扩展有限元法[J].工程力学,2010,,27(4)：84-89.
[29]余天堂.模拟三维裂纹问题的扩展有限元[J].岩土力学,2010,31(10)：3280-3285.
[30]Irwin G. R. Fracture Dynamics [M]. In:Fracturing of Metal. Clevlad:Am. Soc. Metals, 1948:147-166.
[31]解德,钱勤,李长安.断裂力学中的数值计算方法及工程应用[M].北京：科学出版社,2009.
[32]Irwin G R. One set of fast crack propagation in high strength steel and aluminum alloys[J]. Sagamore Rearch Conference Proceedings,1956,2:289-305.
[33]Rybicki E F, Kanninen M F. A finite element calculation of stress intensity factors by a modified crack closure integral[J]. Engineering Fracture Mechanics, 1977,9:931-938.
[34]Raju I S. Calculation of strain-energy release rates with hig-order and singular finite-elements[J]. Engineering Fracture Mechanics,1987,28:251-274.
[35]Rice J R. A path independent integral and the approximate analysis of strain concentration by notches and cracks[J]. Journal of Applied Mechanics, 1968,35:379-386.
[36]Li F Z, Shih C F, Needleman. A comparison of methods for calculating energy release rates[J]. Engineering Fracture Mechanics,1985,21(2),405-421.
[37]Matthew C, Walters G, Paulino R, et al. Interaction integual procedures for 3-D curved cracks including surface tractions[J]. Engineering Fracture Mechanics, 2005,72:1635-1663.
[38]Soheil M. Extended finite element method for Fracture Analysis of Structures[M]. UK:Blackwell Pu-blishing Ltd,2008.
[39]丁遂栋,孙利民.断裂力学[M].北京：机械工业出版社,1997.
[40]李志安,金志浩,宫殿民.压力容器断裂理论与缺陷评定[M].大连：大连理工大学出版社,1994.
[41]钱向东,任青文,赵引.一种高效的等参有限元逆变换算法[J].计算力学学报,1998,15(4)：437-441.
[42]Osher S, Sethian J A. Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations[J]. Journal of Computational Physics, 1988,79(1):12-49.
[43]中国航空研究院.应力强度因子手册[M].北京：科学出版社,1981.
[44]Bathe K. Finite Element Procedures[M]. New Jersey:Prentice-Hall,1996.
[45]Bathe K, Bucalem M, Brezze F. Displacement and stress convergence of our MITC plate bending elements[J].1990,7:291-302.
[46]柳春图,蒋持平.板壳断裂力学[M].北京：国防工业出版社,2000.
[47]Jiang C P, Cheung Y K. A special bending crack tip finite element[J]. International Journal of Fracture,1995,1:55-69.
[48]Sih G C, Paris P C, Erdogan F. Crack tip stress-intensity factors for plane extension and plate bending problems[J]. Journal of Applied Mechanics,1962,29:306-310.
[2]孟玮.边界元法在结构断裂分析中的应用[D].广西：广西大学,2005.
[3]孙和成,张永元.边界积分方程-边界元法在三维断裂力学中的应用[J].上海交通大学学报,1984,18(3)：99-108.
[4]刘佳莉.无网格法在断裂力学中的应用研究[D].北京：中国石油大学,2008.
[5]刘凯远.无网格局部Petrov-Galerkin方法在断裂力学中的应用[D].湖南：湖南大学,2007.
[6]Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing[J]. International Journal for numerical Methods in Engineering,1999,45(5):601-620.
[7]Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering, 1999,46(1):131-150.
[8]Daux C, Moes N, Dolbow J, et al. Arbitrary branched and intersecting cracks with the extended finite element method [J]. International Journal for Numerical Methods in Engineering,2000,48:1741-1760.
[9]Dolbow J, Mo S N, Belytschko T. Modeling fracture in Mindlin-Reissner plates with the extended finite element method[J]. International Journal of Solids and Structures, 2000,37(48-50):7161-7183.
[10]Sukumar N, Moes N, Moran B, et al. Extended finite element method for three-dimensional crack model ing[J]. International Journal for Numerical Methods in Engineering, 2000,48(11):1549-1570.
[11]Stolarska M, Chopp D L, Moes N, et al. Modeling crack growth by level sets in the extended finite element method [J]. International Journal for Numerical Methods in Engineering, 2001,51 (8):943-960.
[12]Sukumar N, Chopp D L, Moes N, et al. Modeling holes and inclusions by level sets in the extended finite element method[J]. Computer Methods in Applied Mechanics and Engineering,2001,190:6183-6200
[13]Moes N, Gravouil A, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets-Part I:Computational Procedure [J]. International Journal for Numerical Methods in Engineering,2002,53:2549-2568.
[14]Sukumar N, Chopp D L, Moran B. Extended finite element method and fast marching method for three-dimensional fatigue crack propagation[J]. Engineering Fracture Mechanics, 2003,70:29-48.
[15]Belytschko T, Parimi C, Moes N, et al. Structured extended finite element methods for solids defined by implicit surfaces[J]. International Journal for Numerical Methods in Engineering,2003,56(4):609-635.
[16]Mariani S, Perego U. Extended finite element method for quasi-brittle fracture[J]. International Journal for Numerical Methods in Engineering,2003,58(1):103-126.
[17]Ventura G, Budyn E, Belytschko T. Vector level sets for description of propagating cracks in finite elements[J]. International Journal for Numerical Methods in Engineering,2003,58 (10):1571-1592.
[18]Toshio N, Youhei 0, Shuichi T. Stress intensity factor analysis of interface cracks using X-FEM. Int. J. Numer. Meth. Engng,2003,56:1151-1173.
[19]X. Y. Liu, Q. Z. Xiao, B. L. Karihaloo. XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials[J]. International Journal for Numerical Methods in Engineering,2004,59:1103-1118.
[20]Dolbow J E, Devan A. Enrichment of enhanced assumed strain approximations for representing strong discontinuities:addressing volumetric incompressibility and the discontinuous patch test[J]. International Journal for Numerical Methods in Engineering,2004,59(1):47"67.
[21]Ventura G, Moran B, Belytschko T. Dislocations by partition of unity[J]. International Journal for Numerical Methods in Engineering,2005,62(11):1463-1487.
[22]陈胜宏,汪卫明,徐明毅,等.小湾高拱坝坝踵开裂的有限单元法分析[J].水利学报,2003,1(1)：66-71.
[23]李录贤,王铁军.扩展有限元法(XFEM)及其应用[J].力学进展,2005,35(1)：5-20.
[24]余天堂.含裂纹体的数值模拟[J].岩石力学与工程学报,2005,24(24)：4434-4439.
[25]方修君,金峰.基于ABAQUS平台的扩展有限元法[J].工程力学,2007,24(7)：6-10.
[26]方修君,金峰,王进廷.基于扩展有限元法的粘聚裂纹模型[J].清华大学学报,2007,47(3)：344-347.
[27]董玉文,余天堂,任青文.直接计算应力强度因子的扩展有限元法[J].计算力学学报,2008,25(1)：72-77.
[28]余天堂.摩擦接触裂纹问题的扩展有限元法[J].工程力学,2010,,27(4)：84-89.
[29]余天堂.模拟三维裂纹问题的扩展有限元[J].岩土力学,2010,31(10)：3280-3285.
[30]Irwin G. R. Fracture Dynamics [M]. In:Fracturing of Metal. Clevlad:Am. Soc. Metals, 1948:147-166.
[31]解德,钱勤,李长安.断裂力学中的数值计算方法及工程应用[M].北京：科学出版社,2009.
[32]Irwin G R. One set of fast crack propagation in high strength steel and aluminum alloys[J]. Sagamore Rearch Conference Proceedings,1956,2:289-305.
[33]Rybicki E F, Kanninen M F. A finite element calculation of stress intensity factors by a modified crack closure integral[J]. Engineering Fracture Mechanics, 1977,9:931-938.
[34]Raju I S. Calculation of strain-energy release rates with hig-order and singular finite-elements[J]. Engineering Fracture Mechanics,1987,28:251-274.
[35]Rice J R. A path independent integral and the approximate analysis of strain concentration by notches and cracks[J]. Journal of Applied Mechanics, 1968,35:379-386.
[36]Li F Z, Shih C F, Needleman. A comparison of methods for calculating energy release rates[J]. Engineering Fracture Mechanics,1985,21(2),405-421.
[37]Matthew C, Walters G, Paulino R, et al. Interaction integual procedures for 3-D curved cracks including surface tractions[J]. Engineering Fracture Mechanics, 2005,72:1635-1663.
[38]Soheil M. Extended finite element method for Fracture Analysis of Structures[M]. UK:Blackwell Pu-blishing Ltd,2008.
[39]丁遂栋,孙利民.断裂力学[M].北京：机械工业出版社,1997.
[40]李志安,金志浩,宫殿民.压力容器断裂理论与缺陷评定[M].大连：大连理工大学出版社,1994.
[41]钱向东,任青文,赵引.一种高效的等参有限元逆变换算法[J].计算力学学报,1998,15(4)：437-441.
[42]Osher S, Sethian J A. Fronts propagating with curvature-dependent speed:algorithms based on Hamilton-Jacobi formulations[J]. Journal of Computational Physics, 1988,79(1):12-49.
[43]中国航空研究院.应力强度因子手册[M].北京：科学出版社,1981.
[44]Bathe K. Finite Element Procedures[M]. New Jersey:Prentice-Hall,1996.
[45]Bathe K, Bucalem M, Brezze F. Displacement and stress convergence of our MITC plate bending elements[J].1990,7:291-302.
[46]柳春图,蒋持平.板壳断裂力学[M].北京：国防工业出版社,2000.
[47]Jiang C P, Cheung Y K. A special bending crack tip finite element[J]. International Journal of Fracture,1995,1:55-69.
[48]Sih G C, Paris P C, Erdogan F. Crack tip stress-intensity factors for plane extension and plate bending problems[J]. Journal of Applied Mechanics,1962,29:306-310.