3D-DDM裂纹尖端边界单元形状改良与数值模拟原理
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摘要
裂纹尖端附近应力场具有强奇异性，采用常规的数值方法分析断裂力学的效果往往较差，因此迫切需要另辟蹊径寻找有效方法，而边界元法的基本解也具有这种奇异性，所以通过恰当的单元划分、单元形状改良和数值积分处理技巧等可以明显提高裂纹尖端的计算精度。本文首先提出原常单元(平板单元)位移不连续法由于单元几何形状与裂隙尖端几何形状拟合效果差等原因，计算裂纹尖端附近应力场误差较大，借鉴克劳奇改进二维尖端单元的构想，引入三维抛物线形尖端单元改良原平板单元。如此，既能与边界单元几何形状基本拟合，又能高精度地模拟裂纹尖端附近位移和应力变化规律，并提高计算效率。对改良的抛物线尖端单元影响系数进行了详细推导，给出了计算公式和数值计算方法，特别是自影响系数借助已有的二维抛物线位移不连续法解析解(克劳奇解)求解，无需直接对单元自身积分，解决了奇异积分问题，计算精度提高。由于三维位移不连续法本身具有求解单元不连续位移的特点，因此依此求解裂纹尖端应力强度因子便捷而直接，模拟系统的选择是合理且优越的。
另外，本文还论述了三维位移不连续法求解裂纹问题的具体解题步骤，系统分析了三维边界元数值模拟系统的实现原理。
Having adopted the familiar numerical solution to solve the crack tip issue, but the result generally lead to higher error in evaluated, then solution to nearby crack tip is required. Fortunately, the basis solutions of the boundary element method take on attributive haracter of singularity, so, through fel- icitously element division, shape improvement and dealing with numerical integral ingenious in technique. And else, crack tip precision in calculation can be improved highly. Firstly, demonstrating the original constant element (flat board element) displacement discontinuity method to resolve stress intensity factors and stress field nearby crack tip is inefficient, its arises from the geometrical shape of element is inconsistent with geometrical shape of crack tip, and so on. Use for reference crouch's means to amend two dimensional crack tip, bring forward three dimensional parabolic crack tip element to improve original flat element. So such, both shape consistent with boundary element geometrical shape, and simulate crack tip close-by variety law of stress and strain with higher precision, and enhancing calculative efficiency. Deduce the influence coefficients of the amended parabolic crack tip element detailed, and provided the calculating equations and numerical calculational methods. Especially, by aid of analytical solution of two dimensional parabolic displacem- ent discontinuity methods to solve the self-influence coefficient,then avoiding of element self-integral directly, resolve the singular integral, enhance calcu- lative precision. Because of 3D-DDM is characteristic of solving the discon- tinuity displacement essentially, then adopting it on stress intensity factor close-by crack tip is convenient and directly, moreover, simulative pattern is reasonable and predominantly.
In addition, dissertate the solution steps concretely for three dimensional displacement discontinuity methods, and analyze implementation principium to the three dimensional boundary element numerical simulative system.
另外，本文还论述了三维位移不连续法求解裂纹问题的具体解题步骤，系统分析了三维边界元数值模拟系统的实现原理。
Having adopted the familiar numerical solution to solve the crack tip issue, but the result generally lead to higher error in evaluated, then solution to nearby crack tip is required. Fortunately, the basis solutions of the boundary element method take on attributive haracter of singularity, so, through fel- icitously element division, shape improvement and dealing with numerical integral ingenious in technique. And else, crack tip precision in calculation can be improved highly. Firstly, demonstrating the original constant element (flat board element) displacement discontinuity method to resolve stress intensity factors and stress field nearby crack tip is inefficient, its arises from the geometrical shape of element is inconsistent with geometrical shape of crack tip, and so on. Use for reference crouch's means to amend two dimensional crack tip, bring forward three dimensional parabolic crack tip element to improve original flat element. So such, both shape consistent with boundary element geometrical shape, and simulate crack tip close-by variety law of stress and strain with higher precision, and enhancing calculative efficiency. Deduce the influence coefficients of the amended parabolic crack tip element detailed, and provided the calculating equations and numerical calculational methods. Especially, by aid of analytical solution of two dimensional parabolic displacem- ent discontinuity methods to solve the self-influence coefficient,then avoiding of element self-integral directly, resolve the singular integral, enhance calcu- lative precision. Because of 3D-DDM is characteristic of solving the discon- tinuity displacement essentially, then adopting it on stress intensity factor close-by crack tip is convenient and directly, moreover, simulative pattern is reasonable and predominantly.
In addition, dissertate the solution steps concretely for three dimensional displacement discontinuity methods, and analyze implementation principium to the three dimensional boundary element numerical simulative system.
引文
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12．杜庆华．第一届边界元法在工程中应用全国学术会议文集[J]，重庆：西安交通大学印刷，1985，12．
13．Manfredo P do Carmo．曲线与曲面的微分几何[M]，巴西：机械工业出版社，2005，1．
14．范天佑．断裂力学理论基础[M]，科学出版社，2003，Vol5：50-53．
15. Ioakimidis N I.Acta Mechanical.[J],1982,45:31-47.
16．张兆顺，崔桂香，许春晓．湍流理论与模拟—研究生力学丛书[M]，清华大学出版社，2005，9．Vol(2)：85-96。
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18. Cruse T A. Two dimensional BIE fracture mechanics analysis[J],Appl Math Modeling, 1978,2(3):287-293.
19．Dennis G, Zill Michael R, Culln．微分方程与边界值问题(原书第5版)中文[M]，机械工业出版社，2005，11，Vol(2)：40-45．
20. Atkinson C. Fracture Mechanics Stress Analysis,Progress in Boundary Element Methods[J],Edited by C.A.Brebbia, Pentech Press,1983,Vol(2):53-100.
21. Cruse T A.Two-Dimensional BIE Fracture Mechanics Analysis.Applied Mathematical Modelling[M],1978,2:287-293.
22. Blandford G E,Ingraffea A R,Liggett A.Two-dimensional stress intensity factor computations using the boundary element method[J],Intl.Jour.Num.Meth.Eng,1981,17(4):387-404.
23. Crouch S L, Starfield A M.Boundary Element Method in Solid Machanics[M],London:Geore Allon&Unwin, 1983,79-109.
24. Portela A, Aliabadi M H, Rock D P.The dual boundary element method:effective implemention for crack problems[J]. Int.J.Numer Methods Engrg, 1992,88(2): 1269-1287.
25. Mi Y, Aliabadi M H.Dual-boundary element method for three dimensional fracture mechanics analysis[J], Engineering Analysis With Boundary Elements,1992,10(2):161-171.
26. Tanaka M, Itoh H.New crack elements for boundary element analysis of elastostatics considering arbitrary stress singularities[J], Appl Math Modelling, I987,11(4):357-363.
27．杜庆华．边界积分方程方法——边界元法之力学基础与工程应用[M]，高等教育出版社．1989，10：52-63．
28．朱合华，陈清军，杨林德．边界元法及其在岩土工程中的应用[M]，上海：同济大学出版社，1996，10：38．
29. Crouch S L. Solution of Plane Elastictity Problems by the Displacement Discontinuity Method[J].Int,Num,Meth,Engng, 1976,10:218-315.
30. Brebbia C A.Boundary Elements Seminar in BEM[J],Proc.7-12th.Int.Berlin:SpringerBerlag, 1985-1990.
31．方企勤．复变函数教程[M]，北京大学出版社，2003，8：120-131．
32．徐仲，张凯院，陆全，冷国伟．矩阵论简明教程(第二版)研究生数学教学系列(工科类)[M]，科学出版社，2005，6：130-138．
33. Williams M L.The stress around a fault or crack in dissimilar media[J],Bull Seismol.Soc.Ameriea,1959,Vol(49): 199-204.
34．李增福，杨桂长．弹性力学中的基本解及在断裂力学中的应用[M]，第五届全国断裂学术林，1988，6(4)：427-443．
35．徐汉忠．不连续位移基本解的简单推导法[J]．应用力学学报，1991，8(2)：105-108．
36. Nengquan Liu, Altiero N.An Integral Equation Method Applied to Mode I Crack Problems[J],Engng.Frac.Mech., 1992,41 (4).
37．文丕华．计算反平面裂纹应力强度因子的K_I的位移间断法[J]．上海力学，1992，13(2)．
38．马维青，汤任基．带裂纹圆柱体扭转的强奇异积分方程解法[J]．应用数学和力学，1993，14(10)：853-859．
39．[英]P．K．班努杰，R．白脱费尔德编著．工程科学中的边界元法冯振兴等译[M]，北京：国防工业出版社，1988．
40. J Larmor.The influence of flaws and air cavities on the strength materials[M],Phil. Mag., 1892,Vol.331:70-80.
41. M L Williams.On the stress distribution at the base of a stationary crack[J],.Appl. Mech.,1957,Vol24,109-114.
42. England A H.A crack between dissimilar media. ASME[J],Appl,Mech.1965,Vol(32):400-402.
43. Comninous M.Interface crack with ftiction in the contact zone[J],ASMEJ.Appl.Mech. 1977,Vol44:780-782.
44. Rice J R, Sih G C. Plane problems of cracks in dissimilar media[J],ASMEJ,Appl,Mech, 1965,Vol(32): 114-128.
45. Comninous M.The interface crack[J], ASMEJ.Appl.Mech,1977,Vol(44):631-636.
46. J R Rice. Mathematical analysis in the mechanics of fracture, in Fracture:An Advanced Treatise[M], Academic Press, New York, 1968,Vol.Ⅱ(ed.by H.Liebowitz): 191-371.
47. F Erdogan.On the stress distribution on plates with collinear cuts under arbitrary loads[M], Proc.of 4th U.S.National Congress of Applied Mechanics,1962.,547-553.
48. Shih C F, Asaro R J. Elastic-plastic analysis of cracks on biomaterial interfaces,Part Ⅰ-small scale yielding[M],ASMEJ.Appl.Mech,1988,Vol:55:299-316.
49．沈养中．工程力学(第二分册)(第二版)[M]，北京：高等教育出版社，1988．
50. Banks-sills L.Weight functions for interface cracks[J], Int.Fract,1993,Vol(60):89-95.
51. Symington M F.Eigenvalues for interface cracks in linear elasticity[M],ASMEJ.Appl. Mech,1987,Vol(54):973-974.
52. Chen Y Z, Hasebe N. Eigenfunction expansion and higher order weight functionof in terface cracks[M],ASMEJ.Appl. mech. 1994,Vol61:843-849.
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