结构断裂分析的广义参数有限元法
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摘要
本文依据有限元理论和线弹性断裂力学理论等，建立了关于断裂力学的广义参数有限元的计算理论和方法，并利用该方法对一些算例进行了相应的数值试验。
本文的主要工作有：
1．根据等参元理论并利用F90语言编制有限元计算程序，并应用于断裂分析的等参奇异元中。
2．利用1／4边节点法，推导了结构断裂分析的等参数奇异单元，并编制了F90程序。算例表明，这类单元具有较好的计算精度，能正确反映裂纹尖端应力场的奇异性。
3．建立了结构断裂分析的广义参数有限元法的计算模型，推导了奇异域单元和过渡元的刚度方程，利用F90语言编制相应的计算程序，进行大量的数值计算，并和等参奇异元理论的计算结果进行了比较和相互验证。证明了广义参数有限元在断裂问题应用上的优越性。理论分析表明，广义参数有限元法利用整体场和局部单元场相协调的理论导出广义参数位移场，经过单元分析建立奇异区条单元和过渡单元的控制方程，不仅可以避免复杂的单元刚度矩阵转换计算，以及由此带来的过大计算量，同时解决了单元间的位移协调问题。
This dissertation establishes the finite elements with generalized degrees of freedom in fracture mechanics on the basis of finite elements theory and linear elastic fracture mechanics theory. It also gives relevant numerical tests to demonstrate the accuracy and efficiency of this method.
The main contributions of the thesis are as follows:
1. Based on isoparametric element theory and Fortran 90, the finite elements program for normal plane problem is composed;
2. Based on the isoparametric singular element theory a quarter singular element is developed, the finite elements program for fracture analysis of structures is put forward using Fortran 90. Examples show that this element is accurate, and reflects the singular property of the stress field around the crack tip.
3. The finite elements with generalized degrees of freedom developed for fracture analysis of structures. The singular strip element and transition element are constructed based on the theory of globed field assorting with local field. A large quantity of
numerical calculating is undertaken. The results of them are compared with those resulted from isoparametric singular element theory, thus the advantage of the finite elements with generalized degrees of freedom is demenstrated in fracture analysis of structures.
本文的主要工作有：
1．根据等参元理论并利用F90语言编制有限元计算程序，并应用于断裂分析的等参奇异元中。
2．利用1／4边节点法，推导了结构断裂分析的等参数奇异单元，并编制了F90程序。算例表明，这类单元具有较好的计算精度，能正确反映裂纹尖端应力场的奇异性。
3．建立了结构断裂分析的广义参数有限元法的计算模型，推导了奇异域单元和过渡元的刚度方程，利用F90语言编制相应的计算程序，进行大量的数值计算，并和等参奇异元理论的计算结果进行了比较和相互验证。证明了广义参数有限元在断裂问题应用上的优越性。理论分析表明，广义参数有限元法利用整体场和局部单元场相协调的理论导出广义参数位移场，经过单元分析建立奇异区条单元和过渡单元的控制方程，不仅可以避免复杂的单元刚度矩阵转换计算，以及由此带来的过大计算量，同时解决了单元间的位移协调问题。
This dissertation establishes the finite elements with generalized degrees of freedom in fracture mechanics on the basis of finite elements theory and linear elastic fracture mechanics theory. It also gives relevant numerical tests to demonstrate the accuracy and efficiency of this method.
The main contributions of the thesis are as follows:
1. Based on isoparametric element theory and Fortran 90, the finite elements program for normal plane problem is composed;
2. Based on the isoparametric singular element theory a quarter singular element is developed, the finite elements program for fracture analysis of structures is put forward using Fortran 90. Examples show that this element is accurate, and reflects the singular property of the stress field around the crack tip.
3. The finite elements with generalized degrees of freedom developed for fracture analysis of structures. The singular strip element and transition element are constructed based on the theory of globed field assorting with local field. A large quantity of
numerical calculating is undertaken. The results of them are compared with those resulted from isoparametric singular element theory, thus the advantage of the finite elements with generalized degrees of freedom is demenstrated in fracture analysis of structures.
引文
[1] Griffith, A.A., The phenomena of rupture and flow in solids. Phil. Trans. Roy. Soc. of London, (1921) pp. 163-197.
[2] Griffith, A.A., The theory of rupture, Proc. 1st Int. Congress Appl. Mech., (1924) pp. 55-63. Biezecn ed. Waltman. (1925).
[3] Shank, M. E. (ed.), "A Critical Survey of Brittle Fracture in CarbonSteel Structures other than Ships", Weld. Res. Coun. Bull., 17 (1954).
[4] Parker, E. R., Brittle Behavior of Engineering Structures, John Wiley & Sons, Inc., New York, 1957.
[5] Orowan, E., Physical Soc. Lond., 12(1949) 185.
[6] Irwin, G. R., J. Appl. Mech., 24, 4 (1957) 361.
[7] 杨绿峰，李桂青，二次样条元，广西科学，1(2)，1994，7-11
[8] 杨绿峰等，有限样条元法在薄板弯曲分析中的应用，广西大学学20(3)，1995，228-232．
[9] L.F. Yang (杨绿峰), Y.L. Zhao, and G.Q. Li, Finite element with generalized coefficients for analysis of beams and plates. Proceeding of 6th International Conference on Education and Practice of Computational Methods in Engineering and Science. Guangzhou, China, Press of Huanan University of Science and Technology. 1997. pp: 560-565
[10] 《数字弹性力学的几个基本问题》，赵惠元译，科学出版社．1958．
[11] Westergaard, H.M., J. Appl. Mech., 6(1939)A49.
[12] Irons B.M. Engineering application of numerical integration in stiffness methods. AIAA. J. 11, 2035-2037, 1966.
[13] Barsoum, R.S., "On the Use of Isoparametric Finite Elements in Linear Elastic Fracture Mechanics", International Journal of Numerical Methods in Engineering, 10, pp.25-37(1976).
[14] Henshell, R.D. & Shaw, K.G., "Crack-Tip Finite Elements are unnecessary", International Journal for Numerical Method in Engineering, 9, pp, 495-507(1975).
[15] 朱伯芳．有限单元法原理与应用．北京：水利电力出版社，1979．
[16] 杨绿峰．模糊随机有限元法及其应用．博士论文．武汉工业大学．1998．
[17] 杨绿峰．断裂力学讲义．广西大学．2004．
[18] 黎在良，王元汉，李廷芥．断裂力学中的边界数值方法．地震出版社．1996．
[19] 甘舜仙编著．有限元技术与程序．北京理工大学出版社．1988．
[20] 宋天霞编著．有限元法理论及应用基础教程．华中工学院出版社．1987．
[21] R．D．库克著．有限元分析的概念和应用．科学出版社．1989．
[22] 徐次达，华佰浩．固体力学有限元理论、方法及程序．水利水电出版社．1983．
[23] 范天佑编著．断裂力学基础．江苏科学技术出版社．1978．
[24] 张允真，曹富新编著．弹性力学及其有限元法．中国铁道出版社．1983．北京．
[25] 王元汉，李丽娟，李银平编著．有限元法基础与程序设计．华南理工大学出版社．2001．广州．
[26] 有限元法新论：原理、程序、进展．龙志飞，岑松编著．中国水利水电出版社．2001．北京．
[27] 刘正兴，孙雁，王国庆编著．计算固体力学．上海交通大学出版社．
[28] 范天佑．断裂力学引论．北京理工大学出版社．1990．北京．
[29] 张彦秋．关于用有限元计算缝端应力强度因子的“过渡单元”．水利学报．1982．
[30] J.F.Xie, S.L.Fok & A.Y.T.Leung. A parametric study on the fractal finite element method for two-dimensional crack problems. Int. J. Numer. Meth. Engng. 2003; 58: 631-642.
[31] A.Y.T.Leung & R.K.L.Su. Fractal Two-Level Finite-Element Method for Two-Dimensional Cracks. Microcomputers in Civil Engineering. 11(1996)249-257.
[32] 张建民，秦荣．等参奇异裂纹单元的研究．广西科学．7(4)：257-259．
[33] L.J.Gray., A.-V. Phan, Glaucio H. Paulino, T. Kaplan. Improved quarter-point crack tip element. Engineering Fracture Mechanics. 70(2003)269-283.
[34] 姚敬之．奇应变圆单元与等参数单元结合的有限元法．水利学报．1981．
[35] 徐纪林，吴永礼．计算应力强度因子的奇异等参单元．固体力学学报．1983．
[36] 吴永礼．用自然三角形单元计算应力强度因子．固体力学学报．1984．
[37] 岗村弘之著，李顺林译．线性断裂力学入门．江苏科学技术出版社．1980．
[38] 李灏，陈树坚编著．断裂理论基础．四川人民出版社．1983．成都．
[39] H．L．Ewalds R．J．H．Wanhill著．朱永昌，浦素云等译．断裂力学．北京航空航天大学出版社．1988．
[40] MEL VIN F．KANNINEN & CARL H．POPELAR著．洪其麟，郑光华，郑祺选译．高等断裂力学．北京航空学院出版社．1987．
[41] 郦正能，何庆之编．工程断裂力学．北京航空航天大学出版社．1992．
[42] 王勖成，邵敏编著．有限单元法基本原理和数值方法．清华大学出版社．
[43] 彭国伦编著．Fortran95程序设计．中国电力出版社．2002．
[44] 殷家驹，张元冲．计算力学教程．西安交通大学出版社．1992．
[45] 王秀喜主编．计算力学及在工程中的应用．中国科学技术大学出版社．1992．
[46] Andrzej Seweryn. Modeling of singular stress fields using finite element method. International Journal of Solids and Structures 39 (2002) 4787-4804.
[47] B.L.Karihaloo, Q.Z.Xiao. Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review. Computer and structures 81 (2003) 119-129.
[48] Q.S. Li, L.F.Yang (杨绿峰), G.Q. Li. The quadratic finite element and strip with generalized degrees of freedom and their applications, Finite Elements in Analysis and Design, Vol.34, No.4,
pp. 325-339, 2001.
[49] Q.S. Li, L.F. Yang (杨绿峰), X.D. Ou, G.O. Li, D.E. Liu, The quintic finite element and finite strip with generalized degrees of freedom in structural analysis, International Journal of Solids and Structure & Vol.38, No.30-31, pp.5355-5372, 2001.
[50] 徐芝纶．弹性力学简明教程．高等教育出版社．
[2] Griffith, A.A., The theory of rupture, Proc. 1st Int. Congress Appl. Mech., (1924) pp. 55-63. Biezecn ed. Waltman. (1925).
[3] Shank, M. E. (ed.), "A Critical Survey of Brittle Fracture in CarbonSteel Structures other than Ships", Weld. Res. Coun. Bull., 17 (1954).
[4] Parker, E. R., Brittle Behavior of Engineering Structures, John Wiley & Sons, Inc., New York, 1957.
[5] Orowan, E., Physical Soc. Lond., 12(1949) 185.
[6] Irwin, G. R., J. Appl. Mech., 24, 4 (1957) 361.
[7] 杨绿峰，李桂青，二次样条元，广西科学，1(2)，1994，7-11
[8] 杨绿峰等，有限样条元法在薄板弯曲分析中的应用，广西大学学20(3)，1995，228-232．
[9] L.F. Yang (杨绿峰), Y.L. Zhao, and G.Q. Li, Finite element with generalized coefficients for analysis of beams and plates. Proceeding of 6th International Conference on Education and Practice of Computational Methods in Engineering and Science. Guangzhou, China, Press of Huanan University of Science and Technology. 1997. pp: 560-565
[10] 《数字弹性力学的几个基本问题》，赵惠元译，科学出版社．1958．
[11] Westergaard, H.M., J. Appl. Mech., 6(1939)A49.
[12] Irons B.M. Engineering application of numerical integration in stiffness methods. AIAA. J. 11, 2035-2037, 1966.
[13] Barsoum, R.S., "On the Use of Isoparametric Finite Elements in Linear Elastic Fracture Mechanics", International Journal of Numerical Methods in Engineering, 10, pp.25-37(1976).
[14] Henshell, R.D. & Shaw, K.G., "Crack-Tip Finite Elements are unnecessary", International Journal for Numerical Method in Engineering, 9, pp, 495-507(1975).
[15] 朱伯芳．有限单元法原理与应用．北京：水利电力出版社，1979．
[16] 杨绿峰．模糊随机有限元法及其应用．博士论文．武汉工业大学．1998．
[17] 杨绿峰．断裂力学讲义．广西大学．2004．
[18] 黎在良，王元汉，李廷芥．断裂力学中的边界数值方法．地震出版社．1996．
[19] 甘舜仙编著．有限元技术与程序．北京理工大学出版社．1988．
[20] 宋天霞编著．有限元法理论及应用基础教程．华中工学院出版社．1987．
[21] R．D．库克著．有限元分析的概念和应用．科学出版社．1989．
[22] 徐次达，华佰浩．固体力学有限元理论、方法及程序．水利水电出版社．1983．
[23] 范天佑编著．断裂力学基础．江苏科学技术出版社．1978．
[24] 张允真，曹富新编著．弹性力学及其有限元法．中国铁道出版社．1983．北京．
[25] 王元汉，李丽娟，李银平编著．有限元法基础与程序设计．华南理工大学出版社．2001．广州．
[26] 有限元法新论：原理、程序、进展．龙志飞，岑松编著．中国水利水电出版社．2001．北京．
[27] 刘正兴，孙雁，王国庆编著．计算固体力学．上海交通大学出版社．
[28] 范天佑．断裂力学引论．北京理工大学出版社．1990．北京．
[29] 张彦秋．关于用有限元计算缝端应力强度因子的“过渡单元”．水利学报．1982．
[30] J.F.Xie, S.L.Fok & A.Y.T.Leung. A parametric study on the fractal finite element method for two-dimensional crack problems. Int. J. Numer. Meth. Engng. 2003; 58: 631-642.
[31] A.Y.T.Leung & R.K.L.Su. Fractal Two-Level Finite-Element Method for Two-Dimensional Cracks. Microcomputers in Civil Engineering. 11(1996)249-257.
[32] 张建民，秦荣．等参奇异裂纹单元的研究．广西科学．7(4)：257-259．
[33] L.J.Gray., A.-V. Phan, Glaucio H. Paulino, T. Kaplan. Improved quarter-point crack tip element. Engineering Fracture Mechanics. 70(2003)269-283.
[34] 姚敬之．奇应变圆单元与等参数单元结合的有限元法．水利学报．1981．
[35] 徐纪林，吴永礼．计算应力强度因子的奇异等参单元．固体力学学报．1983．
[36] 吴永礼．用自然三角形单元计算应力强度因子．固体力学学报．1984．
[37] 岗村弘之著，李顺林译．线性断裂力学入门．江苏科学技术出版社．1980．
[38] 李灏，陈树坚编著．断裂理论基础．四川人民出版社．1983．成都．
[39] H．L．Ewalds R．J．H．Wanhill著．朱永昌，浦素云等译．断裂力学．北京航空航天大学出版社．1988．
[40] MEL VIN F．KANNINEN & CARL H．POPELAR著．洪其麟，郑光华，郑祺选译．高等断裂力学．北京航空学院出版社．1987．
[41] 郦正能，何庆之编．工程断裂力学．北京航空航天大学出版社．1992．
[42] 王勖成，邵敏编著．有限单元法基本原理和数值方法．清华大学出版社．
[43] 彭国伦编著．Fortran95程序设计．中国电力出版社．2002．
[44] 殷家驹，张元冲．计算力学教程．西安交通大学出版社．1992．
[45] 王秀喜主编．计算力学及在工程中的应用．中国科学技术大学出版社．1992．
[46] Andrzej Seweryn. Modeling of singular stress fields using finite element method. International Journal of Solids and Structures 39 (2002) 4787-4804.
[47] B.L.Karihaloo, Q.Z.Xiao. Modelling of stationary and growing cracks in FE framework without remeshing: a state-of-the-art review. Computer and structures 81 (2003) 119-129.
[48] Q.S. Li, L.F.Yang (杨绿峰), G.Q. Li. The quadratic finite element and strip with generalized degrees of freedom and their applications, Finite Elements in Analysis and Design, Vol.34, No.4,
pp. 325-339, 2001.
[49] Q.S. Li, L.F. Yang (杨绿峰), X.D. Ou, G.O. Li, D.E. Liu, The quintic finite element and finite strip with generalized degrees of freedom in structural analysis, International Journal of Solids and Structure & Vol.38, No.30-31, pp.5355-5372, 2001.
[50] 徐芝纶．弹性力学简明教程．高等教育出版社．