V型切口脆性断裂的研究
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摘要
本文基于弹性力学平面问题的基本方程，引入两个Airy应力函数，推导了
均质材料Ⅴ型切口尖端和裂纹尖端的应力奇异性特征方程及其附近的奇异应力
场和位移场。提出了确定单应力强度因子和双应力强度因子的数值方法以及提
高应力强度因子求解精度的结点选取方法。利用上述数值方法和有限元程序
MSC／Nastran的数值分析结果，对Ⅰ型、Ⅱ型、Ⅰ与Ⅱ复合型平面裂纹以及多
种Ⅴ型切口问题进行了具体计算。计算结果表明，本文所提出的数值方法具有
简单、通用和精度高的特点，便于实际工程应用。
本文基于断裂力学原理，推导出了Ⅴ型切口的临界应力强度因子K_C与材
料的平面应变断裂韧度K_(10C)之间的关系，给出了平面单奇异性和双奇异性Ⅴ型
切口问题的最大周向应力脆性断裂准则，并用有机玻璃板材加工了两种Ⅴ型切
口试样，进行了实验验证。实验表明，本文提出的最大周向应力脆性断裂准则
与实验结果基本吻合。
Based on the basic equations of the elasticity plane problem and the two Airy stress functions in the thesis, stress singularity eigenequations and displacement fields as well as singular stress fields near the V-notch tip and the crack tip for homogeneous materials are obtained. Numerical methods to determine the stress intensity factors for single and double stress singularity problems are presented and a pathway is provided to select nodes for improving the accuracy of the stress intensity factor. By utilizing the above methods and the results of the finite element program named MSC/Nastran, the paper carries through specific calculations corresponding to the single stress singularity and the double stress singularity issues, which consist of the cracks for Mode I , Mode II, the mixed mode crack of mode I and II and many V-notched problems. The calculations show that the numerical methods advanced in the paper have the advantages of simplicity, applicability and high accuracy, so that they are perfectly a
pplied for the projects.
Based on the theory of Fracture Mechanics, the relation between the critical stress intensity factors for V-notch Kc and the plane strain crack toughness K10c and the brittle fracture criterion of the maximum circumferential stress for single and double stress singularity with the V-notch problems are proposed. Experiments were carried out using two kinds of V-notched specimens made out of plexiglas to verify this brittle fracture criterion. It is indicated that the maximum circumferential stress criterion coincides with the experimental results.
均质材料Ⅴ型切口尖端和裂纹尖端的应力奇异性特征方程及其附近的奇异应力
场和位移场。提出了确定单应力强度因子和双应力强度因子的数值方法以及提
高应力强度因子求解精度的结点选取方法。利用上述数值方法和有限元程序
MSC／Nastran的数值分析结果，对Ⅰ型、Ⅱ型、Ⅰ与Ⅱ复合型平面裂纹以及多
种Ⅴ型切口问题进行了具体计算。计算结果表明，本文所提出的数值方法具有
简单、通用和精度高的特点，便于实际工程应用。
本文基于断裂力学原理，推导出了Ⅴ型切口的临界应力强度因子K_C与材
料的平面应变断裂韧度K_(10C)之间的关系，给出了平面单奇异性和双奇异性Ⅴ型
切口问题的最大周向应力脆性断裂准则，并用有机玻璃板材加工了两种Ⅴ型切
口试样，进行了实验验证。实验表明，本文提出的最大周向应力脆性断裂准则
与实验结果基本吻合。
Based on the basic equations of the elasticity plane problem and the two Airy stress functions in the thesis, stress singularity eigenequations and displacement fields as well as singular stress fields near the V-notch tip and the crack tip for homogeneous materials are obtained. Numerical methods to determine the stress intensity factors for single and double stress singularity problems are presented and a pathway is provided to select nodes for improving the accuracy of the stress intensity factor. By utilizing the above methods and the results of the finite element program named MSC/Nastran, the paper carries through specific calculations corresponding to the single stress singularity and the double stress singularity issues, which consist of the cracks for Mode I , Mode II, the mixed mode crack of mode I and II and many V-notched problems. The calculations show that the numerical methods advanced in the paper have the advantages of simplicity, applicability and high accuracy, so that they are perfectly a
pplied for the projects.
Based on the theory of Fracture Mechanics, the relation between the critical stress intensity factors for V-notch Kc and the plane strain crack toughness K10c and the brittle fracture criterion of the maximum circumferential stress for single and double stress singularity with the V-notch problems are proposed. Experiments were carried out using two kinds of V-notched specimens made out of plexiglas to verify this brittle fracture criterion. It is indicated that the maximum circumferential stress criterion coincides with the experimental results.
引文
[1] 沈成康．断裂力学，上海：同济大学出版社，1996
[2] 陆毅中．工程断裂力学．西安：西安交通大学，1989
[3] 许金泉，丁浩江．现代固体力学理论及应用．浙江：浙江大出版社，1997
[4] 黄克智，徐秉业．固体力学发展趋势．北京：北京理工大学出版社，1995
[5] 罗祖道，王震鸣．复合材料力学进展．北京：北京大学出版社，1988
[6] Lu H,Chiang F P. Photo-elastic determination of stress intensity factor of interface crack in a bi-materials. [J]. Journal Of Applied Mechanics 1991,60(3):607-613
[7] 柳兆涛，刘一华，王炯华．平面V型切口双应力强度因子的弹性实验研究．合肥工业大学学报2002，25(1)：59-62
[8] Inglis. Stress in a plate due to the presence of cracks and sharp comers. Trans.Inst. Naval Architects, 1913, 55:219-241
[9] Griffith A A. The phenomena of rupture and flow in solids. Phil. Trans. Roy.Soc. of London, 1921,A221:163-197
[10] IrwinG A.Analysis of stress and strains near the end of a crack transversing a plate, J. Appl. Mech. 1957, 24(4): 361-384
[11] Orowan E.Energy criteria of fracture, Welding Journal. 1955, 34:1575-1605
[12] Wells A A. Unstable crack propagation in metals-cleavage and fast fracture.Proc. Crack Propagation Symposium 1961:210-230
[13] Rice J R. A path independent integral and approximate of strain concentration by notches and cracks, J. Appl. Mech., 1968:379-386
[14] Hutchinson J W. Singular behavior at the end of a tensile crack in a hardening Material. J. Mech. Phys. Solids, 1968, 16:13-31
[15] Rice J R and Rosengren G R. Plate strain deformation near a crack tip in a power law hardening material, J. Mech. Phys. Solids, 1968, 16:1-12
[16] N.Haseba and Y. Kutana. Calculation of stress intensity factor from stress concentration factor. [J]. Engineering Fracture Mechanics, 1976,10:215-221
[17] D.J.Hart .A note on the plate width and notch angle correction factors for single edge V-notched plates in tension. [J]. Engineering Fracture Mechanics, 1976, 8:755-761
[18] N.I.Ioakimidis and P.S.Theocaris,A note on stess intensity factor for single edge V-notched plates in tension. [J]. Engineering Fracture Mechanics,
1978,10:685-686
[19] B.Zhao and F. Lang,Determining k_1 of a V-notch from the existing results ora linear crack. [J]. Engineering Fracture Mechanics, 1988,31:923-929
[20] M.L.Williams. Stress singularities resulting from various boundary conditions in angular coners of plates in tension. [J].,Journal Of Applied Mechanics 1952,19:526-528
[21] M,L.Williams. On the stress distribution at the base of a stationary crack.[J].dournal Of Applied Mechanics. 1957, 24:109-114
[22] Hartranft R.J,Sih G.C. The use of eigenfunction expansions in the general solution of three-dimension crack problems. [J]. Journal of Mathematics and Mechanics 1969,19(2):123-138
[23] 柳春图．承受弯曲的板在裂纹顶端附近的应力和变形．固体力学学报，1983，3：441-448
[24] 龙驭球，支秉深，匡文起等．分区混合有限元法计算应力强度因子．力学学报，1982，4：341-353
[25] 徐永君，袁驷．余能积分提取法计算应力强度因子．工程力学，1996(增刊)：196-201
[26] 袁驷，张亿果．常微分特征值问题的求解器解法，地震工程与工程震动，1993，13(2)：94-102
[27] B.Gross. Some plane problems elastostatic solutions for plates having a V-notch. Ph.D. dissertation, Case Western Reserve University (1970)
[28] K.Y. Lin and Pin Tong. International Jounal for Numerical Methods in Engineering 1980,15:1343-1354
[29] Stem M,Becker E B and Dunham R S. A contour integral computation of mixed-mode stress intensity factors. International Journal of Fracture,1976, 12:359-368
[30] Carpenter W C. Calculation of fracture mechanics parameters for a general corner. International Journal of Fracture, 1984, 24:45-48
[31] Carpenter W C. A collocation of procedure for determining fracture mechanics parameters at a corner. International Journal of Fracture, 1984,24:255-266
[32] 杨晓翔，毛银洁，匡震邦．V型切口的断裂研究．固体力学学报，1996，17：353-359
[33] 苏世清，匡震邦．V型切口试样脆断准则的研究．上海交通大学学报，1990，24：89-96
[34] Andrzej Seweryn. Solution for the stress and displacement fields in the
vicinity of a V-notch of negative wedge angle in plate problems of elasticity. [J]. Engineering Fracture Mechanics. 1993,44(2):275-281
[35] 徐芝纶．弹性力学．北京：高等教育出版社，1986
[36] 高庆．工程断裂力学．重庆：重庆大学出版社，1986
[37] Tan C L,Gao Y L and Afagh F F. Boudary element analysis of interface cracks between dissimilar anisotropic materials. Int. J. Solids Struct. 1992,29(24):3201-3220
[38] Wu Y L. A new method for evaluation of stress intensities for interface cracks. [J]. Engineering Fracture Mechanics. 1994, 48(6):755-761
[39] Chen D H. A normal to termination at a bi-material interface. Eng.Fracture Mech. 1994, 49:517-533
[40] Dong D H. Analysis of singular stress field around the inclusion comer tip. Eng. Fracture Mech. 1994,49(4):533-544
[41] Z.ZHAO,H.G. HAHN. Determing the sifofa V-notch from the results of a mixed-mode crack. Engineering Fracture Mechanics 43:511-518(1992)
[42] Chen D H. Stress intensity factors for V-notched strip under tension or in-plane bending. [J]. International Journal of Fracture, 1995, 70:81-97
[43] 刘一华．结合材料的空间轴对称界面端和界面角点的应力奇异性研究浙江大学博士论文，1999
[44] 王海军．确定多重应力强度因子的数值方法的研究合肥工业大学硕士论文，2002
[45] 王能超．数值分析简明教程．武汉：高等教育出版社，1997
[46] 李庆扬，王能超．数值分析．武汉：华中理工大学出版社，1999
[47] 丁浩江，谢贻权．弹性和塑性力学中的有限单元法．浙江：机械工业出版社，1988
[48] 王勖成，邵敏．有限单元法基本原理和数值方法．北京：清华大学出版社，1997
[49] Morten Strandberg. Upper bounds for the notch intensity factor for some geometries and their use in general interpolation formulate. [J].Engineering Fracture Mechanics, 2001,68:577-585
[50] Kuang Zhenbang ,Xu Xiaoping. Singular behavior of a sharp V-notch tip in power hardening material, in "Role of fracture mechanics in modem technology", edit by Sih G C,Nisitani H,Ishihara T. Norh-holland 1986
[51] Andrzej Seweryn. Brittle fracture criterion for structures with sharp notches. [J]. Engineering Fracture Mechanics, 1994, 47(5): 673-681
[52] V.V. Novozhilov. On necessary and sufficient criterion of brittle fracture.
(in Russian).Prikl.Mat.Mek.1969,33:212-216
[53]张如一，陆耀桢．实验应力分析．北京：机械工业出版社，1981
[54]刘宝琛．实验断裂、损伤力学测试技术．北京：机械工业出版社，1994
[2] 陆毅中．工程断裂力学．西安：西安交通大学，1989
[3] 许金泉，丁浩江．现代固体力学理论及应用．浙江：浙江大出版社，1997
[4] 黄克智，徐秉业．固体力学发展趋势．北京：北京理工大学出版社，1995
[5] 罗祖道，王震鸣．复合材料力学进展．北京：北京大学出版社，1988
[6] Lu H,Chiang F P. Photo-elastic determination of stress intensity factor of interface crack in a bi-materials. [J]. Journal Of Applied Mechanics 1991,60(3):607-613
[7] 柳兆涛，刘一华，王炯华．平面V型切口双应力强度因子的弹性实验研究．合肥工业大学学报2002，25(1)：59-62
[8] Inglis. Stress in a plate due to the presence of cracks and sharp comers. Trans.Inst. Naval Architects, 1913, 55:219-241
[9] Griffith A A. The phenomena of rupture and flow in solids. Phil. Trans. Roy.Soc. of London, 1921,A221:163-197
[10] IrwinG A.Analysis of stress and strains near the end of a crack transversing a plate, J. Appl. Mech. 1957, 24(4): 361-384
[11] Orowan E.Energy criteria of fracture, Welding Journal. 1955, 34:1575-1605
[12] Wells A A. Unstable crack propagation in metals-cleavage and fast fracture.Proc. Crack Propagation Symposium 1961:210-230
[13] Rice J R. A path independent integral and approximate of strain concentration by notches and cracks, J. Appl. Mech., 1968:379-386
[14] Hutchinson J W. Singular behavior at the end of a tensile crack in a hardening Material. J. Mech. Phys. Solids, 1968, 16:13-31
[15] Rice J R and Rosengren G R. Plate strain deformation near a crack tip in a power law hardening material, J. Mech. Phys. Solids, 1968, 16:1-12
[16] N.Haseba and Y. Kutana. Calculation of stress intensity factor from stress concentration factor. [J]. Engineering Fracture Mechanics, 1976,10:215-221
[17] D.J.Hart .A note on the plate width and notch angle correction factors for single edge V-notched plates in tension. [J]. Engineering Fracture Mechanics, 1976, 8:755-761
[18] N.I.Ioakimidis and P.S.Theocaris,A note on stess intensity factor for single edge V-notched plates in tension. [J]. Engineering Fracture Mechanics,
1978,10:685-686
[19] B.Zhao and F. Lang,Determining k_1 of a V-notch from the existing results ora linear crack. [J]. Engineering Fracture Mechanics, 1988,31:923-929
[20] M.L.Williams. Stress singularities resulting from various boundary conditions in angular coners of plates in tension. [J].,Journal Of Applied Mechanics 1952,19:526-528
[21] M,L.Williams. On the stress distribution at the base of a stationary crack.[J].dournal Of Applied Mechanics. 1957, 24:109-114
[22] Hartranft R.J,Sih G.C. The use of eigenfunction expansions in the general solution of three-dimension crack problems. [J]. Journal of Mathematics and Mechanics 1969,19(2):123-138
[23] 柳春图．承受弯曲的板在裂纹顶端附近的应力和变形．固体力学学报，1983，3：441-448
[24] 龙驭球，支秉深，匡文起等．分区混合有限元法计算应力强度因子．力学学报，1982，4：341-353
[25] 徐永君，袁驷．余能积分提取法计算应力强度因子．工程力学，1996(增刊)：196-201
[26] 袁驷，张亿果．常微分特征值问题的求解器解法，地震工程与工程震动，1993，13(2)：94-102
[27] B.Gross. Some plane problems elastostatic solutions for plates having a V-notch. Ph.D. dissertation, Case Western Reserve University (1970)
[28] K.Y. Lin and Pin Tong. International Jounal for Numerical Methods in Engineering 1980,15:1343-1354
[29] Stem M,Becker E B and Dunham R S. A contour integral computation of mixed-mode stress intensity factors. International Journal of Fracture,1976, 12:359-368
[30] Carpenter W C. Calculation of fracture mechanics parameters for a general corner. International Journal of Fracture, 1984, 24:45-48
[31] Carpenter W C. A collocation of procedure for determining fracture mechanics parameters at a corner. International Journal of Fracture, 1984,24:255-266
[32] 杨晓翔，毛银洁，匡震邦．V型切口的断裂研究．固体力学学报，1996，17：353-359
[33] 苏世清，匡震邦．V型切口试样脆断准则的研究．上海交通大学学报，1990，24：89-96
[34] Andrzej Seweryn. Solution for the stress and displacement fields in the
vicinity of a V-notch of negative wedge angle in plate problems of elasticity. [J]. Engineering Fracture Mechanics. 1993,44(2):275-281
[35] 徐芝纶．弹性力学．北京：高等教育出版社，1986
[36] 高庆．工程断裂力学．重庆：重庆大学出版社，1986
[37] Tan C L,Gao Y L and Afagh F F. Boudary element analysis of interface cracks between dissimilar anisotropic materials. Int. J. Solids Struct. 1992,29(24):3201-3220
[38] Wu Y L. A new method for evaluation of stress intensities for interface cracks. [J]. Engineering Fracture Mechanics. 1994, 48(6):755-761
[39] Chen D H. A normal to termination at a bi-material interface. Eng.Fracture Mech. 1994, 49:517-533
[40] Dong D H. Analysis of singular stress field around the inclusion comer tip. Eng. Fracture Mech. 1994,49(4):533-544
[41] Z.ZHAO,H.G. HAHN. Determing the sifofa V-notch from the results of a mixed-mode crack. Engineering Fracture Mechanics 43:511-518(1992)
[42] Chen D H. Stress intensity factors for V-notched strip under tension or in-plane bending. [J]. International Journal of Fracture, 1995, 70:81-97
[43] 刘一华．结合材料的空间轴对称界面端和界面角点的应力奇异性研究浙江大学博士论文，1999
[44] 王海军．确定多重应力强度因子的数值方法的研究合肥工业大学硕士论文，2002
[45] 王能超．数值分析简明教程．武汉：高等教育出版社，1997
[46] 李庆扬，王能超．数值分析．武汉：华中理工大学出版社，1999
[47] 丁浩江，谢贻权．弹性和塑性力学中的有限单元法．浙江：机械工业出版社，1988
[48] 王勖成，邵敏．有限单元法基本原理和数值方法．北京：清华大学出版社，1997
[49] Morten Strandberg. Upper bounds for the notch intensity factor for some geometries and their use in general interpolation formulate. [J].Engineering Fracture Mechanics, 2001,68:577-585
[50] Kuang Zhenbang ,Xu Xiaoping. Singular behavior of a sharp V-notch tip in power hardening material, in "Role of fracture mechanics in modem technology", edit by Sih G C,Nisitani H,Ishihara T. Norh-holland 1986
[51] Andrzej Seweryn. Brittle fracture criterion for structures with sharp notches. [J]. Engineering Fracture Mechanics, 1994, 47(5): 673-681
[52] V.V. Novozhilov. On necessary and sufficient criterion of brittle fracture.
(in Russian).Prikl.Mat.Mek.1969,33:212-216
[53]张如一，陆耀桢．实验应力分析．北京：机械工业出版社，1981
[54]刘宝琛．实验断裂、损伤力学测试技术．北京：机械工业出版社，1994