薄板弯曲问题分析的解析奇异单元
|
摘要
薄板作为一种重要的构件在结构工程中有着广泛的应用。在实用过程中,板时常由于裂纹以及V型开孔等原因而存在局部应力奇异性问题。有限元等数值方法是非常有效的分析手段,但常规的单元在处理应力奇异性等带有明显局部效应的课题时有刚性问题,都需要在应力奇异点附近采用非常稠密的网格,以保证求解的精度。这不仅降低了求解效率,而且其求解的精度也不是非常令人满意的。因此,提高含局部应力奇异性薄板弯曲问题分析的精度和效率是很有工程实用价值的一个研究课题。
基于辛对偶体系,本博士学位论文开展了薄板弯曲应力奇异性分析和相关问题数值求解方法的研究,构造出系列具有任意高阶精度的薄板弯曲解析奇异单元。论文的主要工作包括：
(1)基于环扇形薄板弯曲问题的通解,利用对偶变量描述的边界条件,给出了单材料以及双材料环扇形薄板弯曲问题在不同边界条件下的辛本征解析解。首先,结合两直边固支以及一直边固支另一直边自由的边界条件,对单材料环扇形薄板弯曲问题进行了解析分析,获得了相关问题的辛本征解,并对相应V型切口问题的应力奇异性进行了讨论。其次,将辛对偶体系的方法论引入到双材料环扇形薄板弯曲问题。在由原变量及其对偶变量组成的辛几何空间中,给出了相关问题的辛对偶方程组以及对偶变量描述的两直边自由的边界条件以及界面协调条件。然后,首次求解出非齐次边界条件下的三组特解,这些解具有特定的物理意义,分别对应于端面作用有集中弯矩、扭矩以及垂直集中力的解。同时,求解出齐次边界条件对应的辛本征解,这些解在端面所形成的力系均是自相平衡的。本文给出了环扇形薄板弯曲问题的一些新的辛本征解,它不仅进一步扩展了应用力学辛对偶体系的应用领域,而且为其后相关问题解析奇异单元的构建奠定了基础。
(2)将单材料以及双材料环扇形薄板弯曲问题的辛本征解析解作为位移模式,构造出三种不同的解析奇异单元,并应用断裂力学的局部-整体分析法分析含有单材料V型切口、双材料界面裂纹以及双材料界面V型切口的薄板弯曲问题。由于在奇异单元内,采用的是解析形式的本征解,因此其位移模式能够准确地描述应力奇点附近奇异应力的特性。解析奇异单元的应用,使得应力奇异点附近不再需要稠密的网格剖分,一个奇异单元替代了几十个甚至更多的常规单元,很好地避免了常规有限单元方法在求解应力奇异性问题时带来的刚性问题,提高了计算精度和效率。同时,反映局部应力奇异性质的应力强度因子等能够被简单、直接地解析给出,而无须借助外推法等其它数值方法二次数值获得。最后,本文还给出了很多数值算例,并与一些基准解做了对比分析,以验证方法的有效性。数值算例结果表明,解析奇异单元的采用明显提高了含应力奇异性问题分析的精度,并具有良好的数值稳定性,本文所提出的薄板弯曲解析奇异单元是分析薄板弯曲应力奇异性问题的一种非常有效的数值方法。
Thin plate, as an important structural element, has been widely used in structrural engineering. Local stress singularities often occur in a plate resulting from a crack or a V-shaped hole in engineering practices. Finite element method (FEM) is one of very efficient numerical methods. However, the application of conventional FEM for numerical analyses of Kirchhoff plate bending problems with local stress singularities from a crack or a V-shaped notch often causes stiffness problem, so dense meshes are required near the singular points in order to ensure the accuracy of solutions. This leads to low computational efficiency and the unsatisfactory precision of solutions. Therefore, it is worthwhile to investigate how to improve the efficiency and precision in analysis of local stress singularities in a thin plate.
In this doctoral dissertation, analysis of the thin plate bending problems with stress singularities and related numerical methods was systematically studied based on symplectic duality system.
(1) Based on the general solution of bending problem of an annular sector thin plate, symplectic eigensolutions of the bending in homogeneous and bi-material annular sector thin plate under various boundary conditions were obtained, using boundary conditions described in dual variables. To begin with, symplectic eigensolutions of thin plates were obtained after the analysis of the bending problem of a homogeneous annular sector using boundary conditions of both straight sides clamped, and one straight side free and the other straight side clamped. Stress singularities around a V-shaped notch under correlative boundary conditions were discussed. Furthermore, methodology of the symplectic duality system was introduced to the solution to bi-material annular sector thin plate bending. In the symplectic geometry space composed of original variables and their dual ones, symplectic dual equations for the related problem and boundary conditions on both straight sides and compatibility conditions along the interface described in dual variables were given. Three particular solutions related to the inhomogeneous boundary conditions were solved for the first time. For a bi-material wedge plate, they are the solutions to a unit bending moment, a unit torsion moment and a unit concentrated vertical force acting at the vertex. Symplectic eigensolutions for the homogeneous boundary conditions were also obtained. For these solutions, the system of forces at the end face is under self-equilibrium. In summary, some new symplectic eigensolutions to the annular sector thin plate bending problems were provided in this dissertation. These results extend the application fields of the symplectic duality system, and lay a foundation for construction of analytical singular finite elements for the related problems.
(2) Using the symplectic eigensolutions of bending problems of the homogeneous and the bi-material annular sector thin plate as the displacement models, three analytical singular finite elements were constructed respectively. Local-global method was applied with these three elements to analyze the thin plate bending problems with V-shaped notches, bi-material interface cracks and bi-material interface V-shaped notches. Due to the use of the analytical eigensolutions within the singular elements, the displacement models can exactly reflect the stress singularities near the singular points. Because of the application of the analytical singular finite elements, dense meshes near the singular points are not required any more. The use of one such singular element instead of tens of or more conventional ones avoids the stiffness problem caused by conventional finite elements in solution to the stress singularity problems. Solution precision and efficiency are improved. Stress intensity factors which reflect the local stress singularities can be determined easily and directly without other numerical methods, such as extrapolation. To verify the validity of these new methods, many numerical examples were presented in this dissertation, comparing with some benchmark solutions. Numerical results show that application of these analytical singular finite elements can improve the analysis precision in problems involving stress singularities and has good numerical stability. The present analytical singular finite element is an effective technique for the analysis of thin plate bending problems with stress singularities.
基于辛对偶体系,本博士学位论文开展了薄板弯曲应力奇异性分析和相关问题数值求解方法的研究,构造出系列具有任意高阶精度的薄板弯曲解析奇异单元。论文的主要工作包括：
(1)基于环扇形薄板弯曲问题的通解,利用对偶变量描述的边界条件,给出了单材料以及双材料环扇形薄板弯曲问题在不同边界条件下的辛本征解析解。首先,结合两直边固支以及一直边固支另一直边自由的边界条件,对单材料环扇形薄板弯曲问题进行了解析分析,获得了相关问题的辛本征解,并对相应V型切口问题的应力奇异性进行了讨论。其次,将辛对偶体系的方法论引入到双材料环扇形薄板弯曲问题。在由原变量及其对偶变量组成的辛几何空间中,给出了相关问题的辛对偶方程组以及对偶变量描述的两直边自由的边界条件以及界面协调条件。然后,首次求解出非齐次边界条件下的三组特解,这些解具有特定的物理意义,分别对应于端面作用有集中弯矩、扭矩以及垂直集中力的解。同时,求解出齐次边界条件对应的辛本征解,这些解在端面所形成的力系均是自相平衡的。本文给出了环扇形薄板弯曲问题的一些新的辛本征解,它不仅进一步扩展了应用力学辛对偶体系的应用领域,而且为其后相关问题解析奇异单元的构建奠定了基础。
(2)将单材料以及双材料环扇形薄板弯曲问题的辛本征解析解作为位移模式,构造出三种不同的解析奇异单元,并应用断裂力学的局部-整体分析法分析含有单材料V型切口、双材料界面裂纹以及双材料界面V型切口的薄板弯曲问题。由于在奇异单元内,采用的是解析形式的本征解,因此其位移模式能够准确地描述应力奇点附近奇异应力的特性。解析奇异单元的应用,使得应力奇异点附近不再需要稠密的网格剖分,一个奇异单元替代了几十个甚至更多的常规单元,很好地避免了常规有限单元方法在求解应力奇异性问题时带来的刚性问题,提高了计算精度和效率。同时,反映局部应力奇异性质的应力强度因子等能够被简单、直接地解析给出,而无须借助外推法等其它数值方法二次数值获得。最后,本文还给出了很多数值算例,并与一些基准解做了对比分析,以验证方法的有效性。数值算例结果表明,解析奇异单元的采用明显提高了含应力奇异性问题分析的精度,并具有良好的数值稳定性,本文所提出的薄板弯曲解析奇异单元是分析薄板弯曲应力奇异性问题的一种非常有效的数值方法。
Thin plate, as an important structural element, has been widely used in structrural engineering. Local stress singularities often occur in a plate resulting from a crack or a V-shaped hole in engineering practices. Finite element method (FEM) is one of very efficient numerical methods. However, the application of conventional FEM for numerical analyses of Kirchhoff plate bending problems with local stress singularities from a crack or a V-shaped notch often causes stiffness problem, so dense meshes are required near the singular points in order to ensure the accuracy of solutions. This leads to low computational efficiency and the unsatisfactory precision of solutions. Therefore, it is worthwhile to investigate how to improve the efficiency and precision in analysis of local stress singularities in a thin plate.
In this doctoral dissertation, analysis of the thin plate bending problems with stress singularities and related numerical methods was systematically studied based on symplectic duality system.
(1) Based on the general solution of bending problem of an annular sector thin plate, symplectic eigensolutions of the bending in homogeneous and bi-material annular sector thin plate under various boundary conditions were obtained, using boundary conditions described in dual variables. To begin with, symplectic eigensolutions of thin plates were obtained after the analysis of the bending problem of a homogeneous annular sector using boundary conditions of both straight sides clamped, and one straight side free and the other straight side clamped. Stress singularities around a V-shaped notch under correlative boundary conditions were discussed. Furthermore, methodology of the symplectic duality system was introduced to the solution to bi-material annular sector thin plate bending. In the symplectic geometry space composed of original variables and their dual ones, symplectic dual equations for the related problem and boundary conditions on both straight sides and compatibility conditions along the interface described in dual variables were given. Three particular solutions related to the inhomogeneous boundary conditions were solved for the first time. For a bi-material wedge plate, they are the solutions to a unit bending moment, a unit torsion moment and a unit concentrated vertical force acting at the vertex. Symplectic eigensolutions for the homogeneous boundary conditions were also obtained. For these solutions, the system of forces at the end face is under self-equilibrium. In summary, some new symplectic eigensolutions to the annular sector thin plate bending problems were provided in this dissertation. These results extend the application fields of the symplectic duality system, and lay a foundation for construction of analytical singular finite elements for the related problems.
(2) Using the symplectic eigensolutions of bending problems of the homogeneous and the bi-material annular sector thin plate as the displacement models, three analytical singular finite elements were constructed respectively. Local-global method was applied with these three elements to analyze the thin plate bending problems with V-shaped notches, bi-material interface cracks and bi-material interface V-shaped notches. Due to the use of the analytical eigensolutions within the singular elements, the displacement models can exactly reflect the stress singularities near the singular points. Because of the application of the analytical singular finite elements, dense meshes near the singular points are not required any more. The use of one such singular element instead of tens of or more conventional ones avoids the stiffness problem caused by conventional finite elements in solution to the stress singularity problems. Solution precision and efficiency are improved. Stress intensity factors which reflect the local stress singularities can be determined easily and directly without other numerical methods, such as extrapolation. To verify the validity of these new methods, many numerical examples were presented in this dissertation, comparing with some benchmark solutions. Numerical results show that application of these analytical singular finite elements can improve the analysis precision in problems involving stress singularities and has good numerical stability. The present analytical singular finite element is an effective technique for the analysis of thin plate bending problems with stress singularities.
引文
[1]许金泉.界面力学[M].北京：科学出版社,2006.
[2]钟万勰.弹性力学求解新体系[M].大连：大连理工大学出版社,1995.
[3]Irwin G R. Analysis of stresses and strains near the end of a crack traversing a plate[J]. Journal of Applied Mechanics,1957,24,361-364.
[4]Griffith A A. The phenomena of flow and rupture in solids[J]. Phil Trans Roy Soc,1920, A221,163.
[5]柳春图,蒋持平.板壳断裂力学[M].北京：国防工业出版社,2000.
[6]李洪升,周承芳.工程断裂力学[M].大连：大连理工大学出版社,1990.
[7]周爱细,黄建龙,郎福元.计算断裂力学进展[J].甘肃工业大学学报,1998,24(1),111-116.
[8]Tada H, Paris P C, Irwin G R. The stress analysis of cracks handbook[M]. Hellertown, Pa:Del Reseach Corp,1973.
[9]Sih G C. Handbook of stress intensity factors for researchers and engineers[M]. Bethlehem, Pa:Leheigh University,1973.
[10]中国航空研究院.应力强度因子手册[M].北京：科学出版社,1981.
[11]Murakami Y. Stress intensity factors handbook[M]. Beijing:World Publishing Corporation,1989.
[12]中国航空研究院.应力强度因子手册增订版[M].北京：科学出版社,1993.
[13]Chan S K, Tuba I S, Wilson W K. On the finite element method in linear fracture mechanics[J]. Engineering Fracture Mechanics,1970,2(1),1-17.
[14]Yamamoto Y, Tokuda N. Determination of stress intensity factors in cracked plates by the finite element method[J]. International Journal for Numerical Methods in Engineering,1973,6(3),427-439.
[15]Parks D M. A stiffness derivative finite element technique for determination of crack tip stress intensity factors[J]. International Journal of Fracture,1974,10(4),487-502.
[16]Blandford G E, Ingraffea A R, Liggett J A. Two-dimensional stress intensity factor computations using the boundary element method[J]. International Journal for Numerical Methods in Engineering,1981, 17(3),387-404.
[17]Portela A, Aliabadi M H, Rooke D P. The dual boundary element method:Effective implementation for crack problems[J]. International Journal for Numerical Methods in Engineering,1992,33(6), 1269-1287.
[18]Leung A Y T, Su R K L. Mode I crack problems by fractal two level finite element methods[J]. Engineering Fracture Mechanics,1994,48(6),847-856.
[19]Leung A Y T, Su R K L. Mixed-mode two-dimensional crack problem by fractal two level finite element method[J]. Engineering Fracture Mechanics,1995,51(6),889-895.
[20]Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering,1994,37(2),229-256.
[21]Belytschko T, Lu Y Y, Gu L. Crack propagation by element-free Galerkin methods[J]. Engineering Fracture Mechanics,1995,51(2),295-315.
[22]Awaji H, Yokobori jr A T, Takeo Y. The variation of the stress singularity at the notch tip as notch angle and radius of curvature[J]. Computers & Structures,1986,22(1),25-30.
[23]Sukumar N, Kumosa M. Application of the Finite Element Iterative Method to cracks and sharp notches in orthotropic media[J]. International Journal of Fracture,1992,58(2),177-192.
[24]Yosibash Z, Szabo B. Numerical analysis of singularities in two-dimensions part 1:Computation of eigenpairs[J]. International Journal for Numerical Methods in Engineering,1995,38(12),2055-2082.
[25]Szab6 B A, Yosibash Z. Numerical analysis of singularities in two dimensions. Part 2:computation of generalized flux/stress intensity factors[J]. International Journal for Numerical Methods in Engineering,1996,39,409-434.
[26]Dunn M, Suwito W, Cunningham S, et al. Fracture initiation at sharp notches under mode Ⅰ, mode Ⅱ, and mild mixed mode loading[J]. International Journal of Fracture,1997,84(4),367-381.
[27]Li Z C, Chu P C, Young L J, et al. Models of corner and crack singularity of linear elastostatics and their numerical solutions[J]. Engineering Analysis with Boundary Elements,2010,34(6),533-548.
[28]Lee M G, Young L J, Li Z C, et al. Combined Trefftz methods of particular and fundamental solutions for corner and crack singularity of linear elastostatics[J]. Engineering Analysis with Boundary Elements,2010,34(7),632-654.
[29]Lin K Y, Mar J W. Finite element analysis of stress intensity factors for cracks at a bi-material interface[J]. International Journal of Fracture,1976,12(4),521-531.
[30]Tan C L, Gao Y L. Treatment of bimaterial interface crack problems using the boundary element method[J]. Engineering Fracture Mechanics,1990,36(6),919-932.
[31]马开平,柳春图.双材料界面裂纹平面问题的半权函数法[J].应用数学和力学,2004,25(11),1135-1142.
[32]马开平,柳春图.应用半权函数法计算界面裂纹的应力强度因子[J].应用力学学报,2004,21(1),44-49.
[33]Bazant Z P, Keer L M. Singularities of elastic stresses and of harmonic functions at conical notches or inclusions[J]. International Journal of Solids and Structures,1974,10(9),957-964.
[34]Lin K Y, Hartmann H H. Numerical analysis of stress singularities at a bonded anisotropic wedge[J]. Engineering Fracture Mechanics,1989,32(2),211-224.
[35]Gu L, Belytschko T. A numerical study of stress singularities in a two-material wedge[J]. International Journal of Solids and Structures,1994,31(6),865-889.
[36]Qian Z Q, Akisanya A R. Wedge corner stress behaviour of bonded dissimilar materials[J]. Theoretical and Applied Fracture Mechanics,1999,32(3),209-222.
[37]Xu J Q, Liu Y H, Wang X G. Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients[J]. Engineering Fracture Mechanics,1999,63(6),775-790.
[38]季蓓,王远功,贺鹏飞,等.刚性夹杂角点处奇异场的数值分析及双夹杂间的相互影响[J].力学季刊,1999,20(4),459-464.
[39]许金泉,王效贵,刘一华.振荡应力奇异性及其强度系数的数值分析方法[J].力学季刊,2000,21(2),230-236.
[40]傅列东,许金泉.界面端弹塑性应力奇异性的迭代计算方法[J].浙江大学学报(工学版),2001,35(6),689-694.
[41]王效贵,郭乙木,许金泉.与界面相交的裂纹尖端的应力奇异性分析[J].固体力学学报,2002,23(4),412-418.
[42]宋鸣,李有堂,段红燕.垂直于界面V型缺口尖端的应力奇异性及其应用[J].应用力学学报,2009,26(4),752-756.
[43]Ju S H. Finite element calculation of stress intensity factors for interface notches[J]. Computer Methods in Applied Mechanics and Engineering,2010,199(33-36),2273-2280.
[44]王效贵,王美.考虑尺寸效应的双材料轴对称界面端应力奇异性[J].力学学报,2010,42(3),448-455.
[45]嵇醒.半平面双材料角界面应力奇异性分布规律的探究[J].固体力学学报,2011,32(5),446-458.
[46]程站起,高丹盈,仲政.任意梯度分布功能梯度涂层平面裂纹分析[J].固体力学学报,2011,32(4),426-432.
[47]钟献词,张克实.垂直于双材料非完美界面Ⅱ型裂纹的断裂分析[J].应用数学和力学,2012,33(3),342-352.
[48]Wilson W K, Thompson D G. On the finite element method for calculating stress intensity factors for cracked plates in bending[J]. Engineering Fracture Mechanics,1971,3(2),97-102.
[49]Heming F S. Sixth order analysis of crack closure in bending of an elastic plate[J]. International Journal of Fracture,1980,16(4),289-304.
[50]Alwar R S, Nambissan K N R. Influence of crack closure on the stress intensity factor for plates subjected to bending—A 3-D finite element analysis[J]. Engineering Fracture Mechanics,1983,17(4), 323-333.
[51]Alwar R S, Nambissan K N R. Three-dimensional finite element analysis of cracked thick plates in bending[J]. International Journal for Numerical Methods in Engineering,1983,19,293-303.
[52]Kwon Y W. Fracture analysis of plate bending using the finite element method[J]. International Journal of Fracture,1987,35(4), R79-R81.
[53]Kwon Y W. Finite element analysis of crack closure in plate bending[J]. Computers & Structures, 1989,32(6),1439-1445.
[54]Chen W H, Yang K C, Chang C S. A finite element alternating approach for the bending analysis of thin cracked plates[J]. International Journal of Fracture,1992,56(2),93-110.
[55]Chen W H, Shen C M. A finite element alternating approach to the bending of thin plates containing mixed mode cracks[J]. International Journal of Solids and Structures,1993,30(16),2261-2276.
[56]Leung A Y T, Su R K L. Fractal two-level finite element method for cracked kirchhoffs plates using dkt elements[J]. Engineering Fracture Mechanics,1996,54(5),703-711.
[57]Leung A Y T, Su R K L. Fractal two-level finite element analysis of cracked Reissner's plate[J]. Thin-Walled Structures,1996,24(4),315-334.
[58]Zucchini A, Hui C Y, Zehnder A T. Crack tip stress fields for thin, cracked plates in bending, shear and twisting:A comparison of plate theory and three-dimensional elasticity theory solutions[J]. International Journal of Fracture,2000,104(4),387-407.
[59]Su R K L, Sun H Y. Numerical solution of cracked thin plates subjected to bending, twisting and shear loads[J]. International Journal of Fracture,2002,117(4),323-335.
[60]Wang Y H, Tham L G, Lee P K K, et al. A boundary collocation method for cracked plates[J]. Computers & Structures,2003,81(28-29),2621-2630.
[61]Palani G S, Iyer N R, Dattaguru B. A generalised technique for fracture analysis of cracked plates under combined tensile, bending and shear loads[J]. Computers & Structures,2006,84(29-30), 2050-2064.
[62]Rabczuk T, Areias P M A. A meshfree thin shell for arbitrary evolving cracks based on an extrinsic basis[J]. Computer Modeling in Engineering and Sciences,2006,16(2),115-130.
[63]Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture[J]. International Journal for Numerical Methods in Engineering,2007,72,524-548.
[64]Marsavina L, Sadowski T. Stress Intensity Factors for an Interface Kinked Crack in a Bi-Material Plate Loaded Normal to the Interface[J]. International Journal of Fracture,2007,145(3),237-243.
[65]杨丽敏,柳春图,曾晓辉.计算Reissner理论各向异性板应力强度因子的半权函数法[J].机械强度,2006,28(1),113-117.
[66]Leung A Y T, Xu X, Zhou Z, et al. Analytic stress intensity factors for finite elastic disk using symplectic expansion[J]. Engineering Fracture Mechanics,2009,76(12),1866-1882.
[67]Maucher R, Hartmann F. Corner singularities of Kirchhoff plates and the boundary element method[J]. Computer Methods in Applied Mechanics and Engineering,1999,173(3-4),273-285.
[68]Li J. Singularity analysis of near-tip fields for notches formed from several anisotropic plates under bending[J]. International Journal of Solids and Structures,2002,39(23),5767-5785.
[69]Treifi M, Oyadiji S O, Tsang D K L. Computations of the stress intensity factors of double-edge and centre V-notched plates under tension and anti-plane shear by the fractal-like finite element method[J]. Engineering Fracture Mechanics,2009,76(13),2091-2108.
[70]Cruse T A, Vanburen W. Three-dimensional elastic stress analysis of a fracture specimen with an edge crack[J]. International Journal of Fracture,1971,7(1),1-15.
[71]Barsoum R S. On the use of isoparametric finite elements in linear fracture mechanics[J]. International Journal for Numerical Methods in Engineering,1976,10(1),25-37.
[72]Ioakimidis N I. Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity[J]. Acta Mechanica,1982,45(1),31-47.
[73]Somaratna N, Ting T C T. Three-dimensional stress singularities in anisotropic materials and composites[J]. International Journal of Engineering Science,1986,24(7),1115-1134.
[74]Shivakumar K N, Tan P W, Newman J C. A virtual crack-closure technique for calculating stress intensity factors for cracked three dimensional bodies[J]. International Journal of Fracture,1988, 36(3), R43-R50.
[75]Nakamura T, Parks D M. Three-Dimensional Stress Field Near the Crack Front of a Thin Elastic Plate[J]. Journal of Applied Mechanics,1988,55(4),805-813.
[76]Barsoum R S, Chen T K. Three-dimensional surface singularity of an interface crack[J]. International Journal of Fracture,1991,50(3),221-237.
[77]Toshio N, Parks D M. Determination of elastic T-stress along three-dimensional crack fronts using an interaction integral[J]. International Journal of Solids and Structures,1992,29(13),1597-1611.
[78]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.
[79]Schmitz H, Volk K, Wendland W. Three-dimensional singularities of elastic fields near vertices [J]. Numerical Methods for Partial Differential Equations,1993,9(3),323-337.
[80]Pageau S S, Biggers S B. Finite element evaluation of free-edge singular stress fields in anisotropic materials[J]. International Journal for Numerical Methods in Engineering,1995,38(13),2225-2239.
[81]Kwon S W, Sun C T. Characteristics of three-dimensional stress fields in plates with a through-the-thickness crack[J]. International Journal of Fracture,2000,104(3),289-314.
[82]Koguchi H, Muramoto T. The order of stress singularity near the vertex in three-dimensional joints [J]. International Journal of Solids and Structures,2000,37(35),4737-4762.
[83]Li R, Tang T, Zhang P W. A Moving Mesh Finite Element Algorithm for Singular Problems in Two and Three Space Dimensions[J]. Journal of Computational Physics,2002,177(2),365-393.
[84]成昌敏,许金泉,王效贵.结合材料界面端的三维应力奇异性[J].力学季刊,2002,23(4),480-486.
[85]吴志学.三维双材料结构的应力奇异性分析[J].计算力学学报,2004,21(5),592-596.
[86]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,2011.
[87]Tracey D M. Finite elements for determination of crack tip elastic stress intensity factors[J]. Engineering Fracture Mechanics,1971,3(3),255-265.
[88]Tong P, Pian T H H, Lasry S J. A hybrid-element approach to crack problems in plane elasticity[J]. International Journal for Numerical Methods in Engineering,1973,7(3),297-308.
[89]Wilson W K. Finite element methods for elastic bodies containing cracks. Mechanics of Fracture 1[M]. Leyden:Noordhoff International Publishing,1973.
[90]Holston A. A mixed mode crack tip finite element[J]. International Journal of Fracture,1976,12(6), 887-899.
[91]蔡承文,刘明杰,黄纯明.计算平面问题应力强度因子的任意高阶奇应变有限单元[J].固体力学学报,1983,4(4),597-605.
[92]Benzley S E. Representation of singularities with isoparametric finite elements[J]. International Journal for Numerical Methods in Engineering,1974,8(3),537-545.
[93]Barsoum R S. Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements[J]. International Journal for Numerical Methods in Engineering,1977,11,85-98.
[94]Lynn P P, Ingraffea A R. Transition elements to be used with quarter-point crack-tip elements[J]. International Journal for Numerical Methods in Engineering,1978,12(6),1031-1036.
[95]Staab G H. A variable power singular element for analysis of fracture mechanics problems[J]. Computers & Structures,1983,17(3),449-457.
[96]徐纪林,吴永礼.计算应力强度因子的奇异等参单元[J].固体力学学报,1983,(2),258-263.
[97]Raju I S. Calculation of strain-energy release rates with higher order and singular finite elements[J]. Engineering Fracture Mechanics,1987,28(3),251-274.
[98]王燕群,张敬宇.一种高精度的裂纹奇异单元’[J].应用数学和力学,1990,11(10),931-936.
[99]陆鸿森.一种计算应力强度因子的奇异元[J].固体力学学报,1992,13(2),182-186.
[100]Tan M A, Meguid S A. Analysis of Bimaterial Wedges Using a New Singular Finite Element[J]. International Journal of Fracture,1997,88(4),373-391.
[101]冯金辉,柳春图.含裂纹有限结构的局部-整体分析[J].应用力学学报,1999,16(2),145-150.
[102]Chen M C, Sze K Y. A novel hybrid finite element analysis of bimaterial wedge problems[J]. Engineering Fracture Mechanics,2001,68(13),1463-1476.
[103]Lim K M, Lee K H, Tay A A O, et al. A new variable-order singular boundary element for two-dimensional stress analysis[J]. International Journal for Numerical Methods in Engineering, 2002,55(3),293-316.
[104]Liu L M, Duan M L, Qin T Y, et al. A Finite Element Method for Cracked Components of Structures[J]. China Ocean Engineering,2003,17(2),177-187.
[105]王海涛,余锦炎.双压电材料界面力电耦合场奇异性研究[J].工程力学,2006,23(1),165-171.
[106]王海涛.分析不同材料界面应力奇异性的一维杂交有限元方法[J].工程力学,2009,26(2),21-26.
[107]平学成,谢基龙,陈梦成,等.夹杂尖端奇异性场的新型杂交元分析[J].北京交通大学学报,2007,31(4),5-9.
[108]平学成,陈梦成,谢基龙.周期分布多边形夹杂奇异性应力干涉问题的研究[J].计算力学学报,2010,27(6),1117-1122.
[109]Chen L, Liu G, Nourbakhsh-Nia N, et al. A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks[J]. Computational Mechanics,2010,45(2),109-125.
[110]Barsoum R S. A degenerate solid element for linear fracture analysis of plate bending and general shells[J]. International Journal for Numerical Methods in Engineering,1976,10,551-564.
[111]Zienkiewicz O C. The finite element method in engineering science[M]. London:McGraw-Hill, 1971.
[112]Ahmad J, Loo F T C. Solution of plate bending problems in fracture mechanics using a specialized finite element technique[J]. Engineering Fracture Mechanics,1979,11(4),661-673.
[113]Williams M L. The bending stress distribution at the base of a stationary crack[J]. Journal of Applied Mechanics,1961,28,78-82.
[114]Yagawa G, Nishioka T. Finite element analysis of stress intensity factors for plane extension and plate bending problems[J]. International Journal for Numerical Methods in Engineering,1979,14, 727-740.
[115]Rhee H C, Atluri S N. Hybrid stress finite element analysis of bending of a plate with a through flaw[J]. International Journal for Numerical Methods in Engineering,1982,18,259-271.
[116]李英治,柳春图Reissner型平板弯曲断裂问题分析[J].力学学报,1983,(4),366-375.
[117]柳春图,李英治Reissner型板裂纹尖端应力应变场及应力强度因子计算[J].力学学报,1984,16(4),351-362.
[118]Chen W H, Chen P Y. A hybrid-displacement finite element model for the bending analysis of thin cracked plates[J]. International Journal of Fracture,1984,24(2),83-106.
[119]Ye T Q, Gallagher R H. A singular finite element for analysis of plate bending problem in fracture mechanics[J]. International Journal of Fracture,1984,24(2),137-147.
[120]Wahba N N. On the use of singular displacement finite elements for cracked plate in bending[J]. International Journal of Fracture,1985,27(1),3-30.
[121]Chen W H, Hwu C B. Hybrid singular element design for the bending analysis of bimaterial thin cracked plates[J]. AIAA J,1986,24(8),1399-1402.
[122]Sih G C, Rice J R. The Bending of Plates of Dissimilar Materials With Cracks[J]. Journal of Applied Mechanics,1964,31(3),477-482.
[123]Viswanath S, Murthy M V V, Murty A V K., et al. A special crack tip element for bending of plates with through cracks from a sixth-order plate theory[J]. Engineering Fracture Mechanics,1989,32(1), 91-109.
[124]Dashevskii E M, Rotter M V. A singular notched finite element for plates in bending[J]. Strength of Materials,1989,21(2),239-244.
[125]Jiang C P, Cheung Y K. A special bending crack tip finite element[J]. International Journal of Fracture,1995,71(1),57-69.
[126]Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering,1999,46,131-150.
[127]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.
[128]孟广伟,陈塑寰,刘寒冰,等.杂交/混合奇异弯曲板单元列式方法及应力强度因子的计算[J].机械强度,2001,23(3),329-331.
[129]Hung N D, Ngoc T T. Analysis of cracked plates and shells using "metis" finite element model[J]. Finite Elements in Analysis and Design,2004,40(8),855-878.
[130]Munaswamy K, Pullela R. Computation of stress intensity factors for through cracks in plates using p-version finite element method[J]. Communications in Numerical Methods in Engineering,2008,24, 1753-1780.
[131]Rethore J, Roux S, Hild F. Hybrid analytical and extended finite element method (HAX-FEM):A new enrichment procedure for cracked solids[J]. International Journal for Numerical Methods in Engineering,2010,81,269-285.
[132]Lasry J, Pommier J, Renard Y, et al. eXtended finite element methods for thin cracked plates with Kirchhoff-Love theory[J]. International Journal for Numerical Methods in Engineering,2010,84, 1115-1138.
[133]Hussain M A, Coffin L F, Zaleski K A. Three dimensional singular element[J]. Computers & Structures,1981,13(5-6),595-599.
[134]Astiz M A. An incompatible singular elastic element for two-and three-dimensional crack problems[J]. International Journal of Fracture,1986,31(2),105-124.
[135]Luo G M, Zhang Y Y. Application of boundary element method with singular and isoparametric elements in three dimensional crack problems[J]. Engineering Fracture Mechanics,1988,29(1), 97-106.
[136]王柏松,贺寅彪.用改进的三维等参奇异元计算应力强度因子[J].核科学与工程,1991,11(2),129-133.
[137]Pageau S S, Biggers Jr S B. A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions[J]. International Journal of Solids and Structures, 1996,33(1),33-47.
[138]Ariza M P, Saez A, Dominguez J. A singular element for three-dimensional fracture mechanics analysis[J]. Engineering Analysis with Boundary Elements,1997,20(4),275-285.
[139]Sukumar N, Moes N, Moran B, et al. Extended finite element method for three-dimensional crack modelling[J]. International Journal for Numerical Methods in Engineering,2000,48(11), 1549-1570.
[140]Ong E T, Lim K M. Three-dimensional singular boundary elements for corner and edge singularities in potential problems[J]. Engineering Analysis with Boundary Elements,2005,29(2),175-189.
[141]赵怀瑞,李强,张向前,等.三维应力奇异单元自动生成的通用算法[J].北京交通大学学报,2009,33(1),62-65.
[142]林广平,梅甫良.12结点三维等参奇异单元的构造和应用[J].计算力学学报,2009,26(2),282-286.
[143]中国科学院力学研究所板壳组.夹层板壳的弯曲、稳定和振动[M].北京：科学出版社,1977.
[144]胡海吕.弹性力学的变分原理及其应用[M].北京：科学出版社,1981.
[145]钱伟长.变分法与有限元[M].北京：科学出版社,1981.
[146]Jiang C P. Finite strip analysis for rectangular plates considering shear deformation[J]. Engineering Analysis,1984,1(4),206-210.
[147]Liu C T, Wu X J, Li Y Z. Study on cracked spherical shells[J]. Scientia Sinaca(Series A),1987,15(9), 944-954.
[148]柳春图,吴犀甲.含轴向裂纹柱壳裂纹尖端应力应变场及应力强度因子计算[J].力学学报,1987,19(2),125-135.
[149]Liu C T, Wu X J. Local stresses and strains inaxially cracked cylindrical shells[J]. Theoretical and Applied Fracture Mechanics,1989,11(1),51-58.
[150]柳春图,韩闻生.含孔边裂纹球壳断裂分析[J].应用力学学报,1993,10(1),1-9.
[151]柳春图,胡互让,吴犀甲.圆柱壳孔边裂纹应力强度因子计算和分析[J].计算结构力学及其应用,1992,9(1),7-15.
[152]钟万腮,钟翔翔.计算结构力学、最优控制及偏微分方程半解析法[J].计算结构力学及其应用,1990,7(1),1-15.
[153]Zhong W X. Plane elasticity in sectorial domain and the Hamiltonian system[J]. Applied Mathematics and Mechanics,1994,15(12),1113-1123.
[154]钟万勰.分离变量法与哈密顿体系[J].计算结构力学及其应用,1991,8(3),229—240.
[155]钟万勰.互等定理与共辘辛正交关系[J].力学学报,1992,24(4),432-437.
[156]张鸿庆.弹性力学的对偶化体系——评《弹性力学求解新体系》[J].大连理工大学学报,1996,36,372.
[157]姚伟岸,钟万勰.辛弹性力学[M].北京：高等教育出版社,2002.
[158]钟万勰.条形域平面弹性问题与哈密尔顿体系[J].大连理工大学学报,1991,3](4),：373-384.
[159]姚伟岸.平面各向异性哈密顿体系及圣维南问题的解析解[J].大连理工大学学报,1999,39(5),612-615.
[160]邹贵平.薄板问题的Hamilion体系和辛儿何方法[J].应用基础与工程科学学报,1996,4(4),335-343.
[161]钟万勰,姚伟岸.板弯曲求解新体系及其应用[J].力学学报,1999,31(2),173-184.
[162]姚伟岸,苏滨,钟万勰.基于相似性原理的正交各向异性板弯曲Hamilton体系[J].中国科学,E辑,2001,31(4),342-347.
[163]Yao W A, Sui Y F. Symplectic solution system for reissner plate bending[J]. Applied Mathematics and Mechanics,2004,25(2),178-185.
[164]鲍四元,邓子辰Mindlin中厚板的辛求解方法[J].固体力学学报,2005,26(1),102-106.
[165]马晨明.固支矩形Mindlin板弯曲问题的辛求解体系[J].力学季刊,2008,29(4),648-655.
[166]Zhong W X, Yao W A. The Saint Venant solutions of multi-layered composite plates[J]. Advances in Structural Engineering,1997,1(2),127-133.
[167]Yao W A, Yang H T. Hamiltonian system based Saint Venant solutions for multi-layered composite plane anisotropic plates[J]. International Journal of Solids and Structures,2001,38(32),5807-5817.
[168]姚伟岸,聂臆瞩,肖峰.基于辛对偶体系的层合板自由边缘效应的分析解[J].应用数学和力学,2011,32(9),1021-1029.
[169]杨有贞,葛修润.基于辛体系的正交各向异性地基梁解析解[J].力学学报,2011,43(2),362-371.
[170]Xu X S, Zhong W X, Zhang H W. The Saint-Venant problem and principle in elasticity[J]. International Journal of Solids and Structures,1997,34(22),2815-2827.
[171]Xu X S, Zhang H W, Zhong W X. A direct method for the problem of the solid cylinder in elasticity[J]. Chinese Quarterly Journal of Mathematics,1998,13(1),44-54.
[172]徐新生,贾宏志,孙发明.横观各向同性弹性柱体中辛本征解方法[J].大连理工大学学报,2005,45(4),617-624.
[173]徐新生,郑新广,张洪武.哈密顿体系与弹性楔体问题[J].应用力学学报,1999,16(2),140-145.
[174]Yao W A, Zhong W X, Su B. New solution system for circular sector plate bending and its application[J]. Acta Mechanica Solida Sinica,1999,12(4),307-315.
[175]鲍四元,邓子辰.环扇形板弯曲问题中环向模拟为时间的辛体系[J].西北工业大学学报,2004,22(6),734-738.
[176]姚伟岸,孙贞.环扇形薄板弯曲问题的环向辛对偶求解方法[J].力学学报,2008,40(4),557-563.
[177]钟万勰,徐新生,张洪武.弹性曲梁问题的直接法[J].工程力学,1996,13(4),1-8.
[178]姚伟岸.极坐标哈密顿体系约当型与弹性楔的佯谬解[J].力学学报,2001,33(1),79-86.
[179]姚伟岸,张兵茹.圆柱型正交各向异性弹性楔的佯谬解[J].固体力学学报,2004,25(2),155-158.
[180]周建方,卓家寿.极坐标下弹性力学的一个新解答[J].力学学报,2001,33(6),839-846.
[181]齐朝晖,唐立民.曲线坐标系下哈密顿体系的建立[J].大连理工大学学报,1997,37(4),376-380.
[182]朱炳麒,周建方,卓家寿.矩形梁受分布荷载的辛解答[J].力学与实践,2007,29(3),15-19.
[183]Lim C W, Yao W A, Cui S. Benchmark symplectic solutions for bending of corner-supported rectangular thin plates[J]. The IES Journal Part A:Civil & Structural Engineering,2008,1(2), 106-115.
[184]谈梅兰,吴光,王鑫伟.矩形薄板面内非线性分布载荷下的辛弹性力学解[J].工程力学,2008,25(10),50-53.
[185]陈伟球,赵莉.功能梯度材料平面问题的辛弹性力学解法[J].力学学报,2009,41(4),588-594.
[186]Xu X S, Zhang W X, Li X, et al. An application of the symplectic system in two-dimensional viscoelasticity[J]. International Journal of Engineering Science,2006,44(13-14),897-914.
[187]徐新生,张维祥,李雪.粘弹性厚壁筒问题的辛本征解方法[J].计算力学学报,2007,24(2),153-158.
[188]张维祥,徐新生,王尕平.辛体系下平面热黏弹性圣维南问题的解析解[J].力学与实践,2008,30(4),76-78.
[189]Zhang W X, Xu X S. A symplectic method in 2D thermo-viscoelasticity[J].中国科学技术大学学报, 2008,38(2),200-206.
[190]Zhang W X, Xu X S, Yuan F. The symplectic system method in the stress analysis of 2D elasto-viscoelastic fiber reinforced composites[J]. Archive of Applied Mechanics,2010,80(8), 829-841.
[191]姚伟岸,杨海天,高强.平面粘弹性问题的辛求解方法[J].计算力学学报,2010,27(1),14-20.
[192]张洪武,李云鹏,钟万勰.双材料楔形结合点的奇性分析[J].大连理工大学学报,1995,35(6),776-782.
[193]张洪武,徐新生,李云鹏,等.多种材料楔形结合点的奇性分析[J].大连理工大学学报,1996,36(4),391-395.
[194]Zhang H W, Zhong W X. Hamiltonian principle based stress singularity analysis near crack corners of multi-material junctions[J]. International Journal of Solids and Structures,2003,40(2),493-510.
[195]王承强,姚伟岸.哈密顿体系在断裂力学Dugdale模型中的应用[J].应用力学学报,2003,20(3),151-155.
[196]郑长良,王承强,基于双线性粘聚力模型混凝土裂纹扩展的半解析有限元法,全国计算力学学术会议，北京,2003.
[197]Wang C Q, Zheng C L. Semi-analytical finite element method for fictitious crack model in fracture mechanics of concrete[J]. Applied Mathematics and Mechanics,2004,25(11),1265-1270.
[198]王承强,郑长良.混凝土双K断裂参数计算的半解析有限元法[J].力学学报,2004,36(4),414-418.
[199]王承强,郑长良.Ⅰ型和Ⅱ型Dugdale模型解析元列式及其半解析有限元法[J].工程力学,2005,22(6),37-40,68.
[200]王承强,郑长良.平面裂纹应力强度因子的半解析有限元法[J].工程力学,2005,22(1)：33-37.
[201]王承强,郑长良.裂纹扩展过程中线性内聚力模型计算的半解析有限元法[J].计算力学学报,2006,23(2),146-151.
[202]姚伟岸,胡小飞.双材料含桥联力Dugdale-Barenblatt模型的解析奇异单元[J].固体火箭技术,2011,34(5),635-638.
[203]周建方,卓家寿,李典庆Hamilton体系下弹性力学半解析法的一个守恒律[J].河海大学学报,2000,28(5),41-43.
[204]唐立民,褚致中,邹贵平.混合状态Hamiltonian元的半解析解和叠层板的计算[J].计算结构力学及其应用,1992,9(4),347-360.
[205]邹贵平,唐立民,周哲玮.复合材料条形域问题混合状态Hamiltonian元的半解析解[J].应用力学学报,1995,12(3),1-7.
[206]王治国,唐立民,朱德娥.圆柱坐标系中弹性力学的哈密顿正则方程及其哈密顿单元方法[J].计算结构力学及其应用,1995,12(1),53-59.
[207]周建方,卓家寿,李典庆.基于Hamilton方法的有限元[J].河海大学常州分校学报,2000,14(3),1-6.
[208]卿光辉,邱家俊,刘艳红Hamilton等参元与智能叠层板的半解析法[J].应用数学和力学,2007,28(1),45-52.
[209]胡志强,林皋,王毅,等.基于Hamilton体系的弹性力学问题的比例边界有限元方法[J].计算力学学报,2011,28(4),510-516.
[210]钟万勰.电磁波导的辛体系[J].大连理工大学学报,2001,41(4),379-387.
[211]钟万勰.电磁波导的半解析辛分析[J].力学学报,2003,35(4),401-410.
[212]孙雁,谢军.基于Hamilion体系的辛半解析法在各向异性电磁波导中的应用[J].计算力学学报,2005,22(6),690-693.
[213]陈杰夫,郑长良,钟万勰.电磁波导的辛分析与对偶棱边元[J].物理学报,2006,55(5),2340-2347.
[214]陈杰夫,郑长良,钟万勰.电磁波导的奇异元与对偶有限元分析[J].计算方学学报,2007,24(2),148-152.
[215]张宝善,卢东强,戴世强,等.非线性水波Hamilton系统理论与应用研究进展[J].力学进展,1998,28(4),521-531.
[216]马坚伟,徐新生,杨慧珠,等.平面粘性流体扰动与哈密顿体系[J].应与力学’学报,2001,18(4),82-87.
[217]马坚伟,徐新生,杨慧珠,等.基于哈密顿体系求解空间粘性流体问题[J].工程力学,2002,19(3),1-6.
[218]徐新生,于尕平Stokes流问题中的辛本征解方法[J].力学学报,2006,38(5),682-687.
[219]徐新生,王尕平,孙发明.二维矩形域内Stokes流问题的辛解析和数值方法[J].应用数学和力学,2008,29(6),639-648.
[220]王艳,邓子辰.单板驱动空腔的辛求解方法[J].动力学与控制学报,2007,5(2),105-11 1.
[221]王艳,邓子辰Stokes流问题的环向辛对偶求解方法[J].机械科学与技术,2008,27(3),374-378.
[222]王艳,邓子辰.楔形域中低雷诺数流动问题的辛对偶求解方法[J].西北工业大学学报,2008,26(2),239-243.
[223]Wang G P, Xu X S, Zhang Y X. Influence of inlet radius on Stokes flow in a circular tube via the Hamiltonian systematic method[J]. Physics of Fluids,2009,21(10),103602(103609 pages).
[224]Xu X S, Ma Y, Lim C W, et al. Dynamic buckling of cylindrical shells subject to an axial impact in a symplectic system[J]. International Journal of Solids and Structures,2006,43(13),3905-3919.
[225]刘淼,罗恩,仲政.薄板动力学相空间非传统Hamilton变分原理与辛算法[J].固体力学学报,2007,28(2),207-211.
[226]顾松年,姜节胜,何建军.经典阻尼系统响应的辛解析解[J].西北工业大学学报,2007,25(4),487-491.
[227]胡伟鹏,邓子辰,李文成.膜自由振动的多辛方法[J].应用数学和力学,2007,28(9),1054-1062.
[228]Lim C W, Lu C F, Xiang Y, et al. On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates[J]. International Journal of Engineering Science,2009, 47(1),131-140.
[229]姚征,郑长良.二维非局部线弹性平面问题的辛分析[J].计算方学学报,2009,26(3),395-400.
[230]钟阳,李锐,田斌.矩形中厚板自由振动问题的哈密顿体系与辛儿何解法[J].动方学与控制学报,2009,7(4),302-307.
[231]姚伟岸.电磁弹性固体反平面问题辛求解体系及圣维南原理[J].大连理工大学学报,2004,44(5),630-633.
[232]Gu Q, Xu X S, Leung A Y T. Application of Hamiltonian system for two-dimensional transversely isotropic piezoelectric media[J]. Journal of Zhejiang University SCIENCE,2005,6A(6),915-921.
[233]Xu X S, Gu Q, Leung A Y T, et al. A symplectic eigensolution method in transversely isotropic piezoelectric cylindrical media[J]. Journal of Zhejiang University SCIENCE,2005,6A(9),922-927.
[234]徐新生,段政,马源,等.辛方法和弹性圆柱壳在内外压和轴向冲击下的动态屈曲[J].爆炸与冲击,2007,27(6),509-514.
[235]徐新生,邱文彪,付月,等.辛方法在弹性圆板屈曲问题中的应用[J].应用力学学报,2009,26(3),530-534.
[236]马继光,白象忠,张旭光,等.导电悬臂梁磁弹性弯曲的辛解答[J].燕山大学学报,2009,33(2),147-152.
[237]褚洪杰,徐新生,林志华,等.弹性梁非线性热屈曲行为与辛本征解展开方法[J].大连理工大学学报,2011,51(1),1-6.
[238]Timoshenko S, Woinowsky-Kriege S. Theory of Plates and Shells[M]. New York:MeGraw-Hill Book Company,1959.
[239]Williams M L, Surface stress singularities resulting from various boundary condition in angular corners of plates under bending, First US National Congress on Applied Mechanics,1951, pp. 325-329.
[240]杨春林,郑百林,嵇醒.单材料V型缺口尖端振荡性奇异应力场产生的条件[J].应用力学学报,2001,18(2),65-69.
[2]钟万勰.弹性力学求解新体系[M].大连：大连理工大学出版社,1995.
[3]Irwin G R. Analysis of stresses and strains near the end of a crack traversing a plate[J]. Journal of Applied Mechanics,1957,24,361-364.
[4]Griffith A A. The phenomena of flow and rupture in solids[J]. Phil Trans Roy Soc,1920, A221,163.
[5]柳春图,蒋持平.板壳断裂力学[M].北京：国防工业出版社,2000.
[6]李洪升,周承芳.工程断裂力学[M].大连：大连理工大学出版社,1990.
[7]周爱细,黄建龙,郎福元.计算断裂力学进展[J].甘肃工业大学学报,1998,24(1),111-116.
[8]Tada H, Paris P C, Irwin G R. The stress analysis of cracks handbook[M]. Hellertown, Pa:Del Reseach Corp,1973.
[9]Sih G C. Handbook of stress intensity factors for researchers and engineers[M]. Bethlehem, Pa:Leheigh University,1973.
[10]中国航空研究院.应力强度因子手册[M].北京：科学出版社,1981.
[11]Murakami Y. Stress intensity factors handbook[M]. Beijing:World Publishing Corporation,1989.
[12]中国航空研究院.应力强度因子手册增订版[M].北京：科学出版社,1993.
[13]Chan S K, Tuba I S, Wilson W K. On the finite element method in linear fracture mechanics[J]. Engineering Fracture Mechanics,1970,2(1),1-17.
[14]Yamamoto Y, Tokuda N. Determination of stress intensity factors in cracked plates by the finite element method[J]. International Journal for Numerical Methods in Engineering,1973,6(3),427-439.
[15]Parks D M. A stiffness derivative finite element technique for determination of crack tip stress intensity factors[J]. International Journal of Fracture,1974,10(4),487-502.
[16]Blandford G E, Ingraffea A R, Liggett J A. Two-dimensional stress intensity factor computations using the boundary element method[J]. International Journal for Numerical Methods in Engineering,1981, 17(3),387-404.
[17]Portela A, Aliabadi M H, Rooke D P. The dual boundary element method:Effective implementation for crack problems[J]. International Journal for Numerical Methods in Engineering,1992,33(6), 1269-1287.
[18]Leung A Y T, Su R K L. Mode I crack problems by fractal two level finite element methods[J]. Engineering Fracture Mechanics,1994,48(6),847-856.
[19]Leung A Y T, Su R K L. Mixed-mode two-dimensional crack problem by fractal two level finite element method[J]. Engineering Fracture Mechanics,1995,51(6),889-895.
[20]Belytschko T, Lu Y Y, Gu L. Element-free Galerkin methods[J]. International Journal for Numerical Methods in Engineering,1994,37(2),229-256.
[21]Belytschko T, Lu Y Y, Gu L. Crack propagation by element-free Galerkin methods[J]. Engineering Fracture Mechanics,1995,51(2),295-315.
[22]Awaji H, Yokobori jr A T, Takeo Y. The variation of the stress singularity at the notch tip as notch angle and radius of curvature[J]. Computers & Structures,1986,22(1),25-30.
[23]Sukumar N, Kumosa M. Application of the Finite Element Iterative Method to cracks and sharp notches in orthotropic media[J]. International Journal of Fracture,1992,58(2),177-192.
[24]Yosibash Z, Szabo B. Numerical analysis of singularities in two-dimensions part 1:Computation of eigenpairs[J]. International Journal for Numerical Methods in Engineering,1995,38(12),2055-2082.
[25]Szab6 B A, Yosibash Z. Numerical analysis of singularities in two dimensions. Part 2:computation of generalized flux/stress intensity factors[J]. International Journal for Numerical Methods in Engineering,1996,39,409-434.
[26]Dunn M, Suwito W, Cunningham S, et al. Fracture initiation at sharp notches under mode Ⅰ, mode Ⅱ, and mild mixed mode loading[J]. International Journal of Fracture,1997,84(4),367-381.
[27]Li Z C, Chu P C, Young L J, et al. Models of corner and crack singularity of linear elastostatics and their numerical solutions[J]. Engineering Analysis with Boundary Elements,2010,34(6),533-548.
[28]Lee M G, Young L J, Li Z C, et al. Combined Trefftz methods of particular and fundamental solutions for corner and crack singularity of linear elastostatics[J]. Engineering Analysis with Boundary Elements,2010,34(7),632-654.
[29]Lin K Y, Mar J W. Finite element analysis of stress intensity factors for cracks at a bi-material interface[J]. International Journal of Fracture,1976,12(4),521-531.
[30]Tan C L, Gao Y L. Treatment of bimaterial interface crack problems using the boundary element method[J]. Engineering Fracture Mechanics,1990,36(6),919-932.
[31]马开平,柳春图.双材料界面裂纹平面问题的半权函数法[J].应用数学和力学,2004,25(11),1135-1142.
[32]马开平,柳春图.应用半权函数法计算界面裂纹的应力强度因子[J].应用力学学报,2004,21(1),44-49.
[33]Bazant Z P, Keer L M. Singularities of elastic stresses and of harmonic functions at conical notches or inclusions[J]. International Journal of Solids and Structures,1974,10(9),957-964.
[34]Lin K Y, Hartmann H H. Numerical analysis of stress singularities at a bonded anisotropic wedge[J]. Engineering Fracture Mechanics,1989,32(2),211-224.
[35]Gu L, Belytschko T. A numerical study of stress singularities in a two-material wedge[J]. International Journal of Solids and Structures,1994,31(6),865-889.
[36]Qian Z Q, Akisanya A R. Wedge corner stress behaviour of bonded dissimilar materials[J]. Theoretical and Applied Fracture Mechanics,1999,32(3),209-222.
[37]Xu J Q, Liu Y H, Wang X G. Numerical methods for the determination of multiple stress singularities and related stress intensity coefficients[J]. Engineering Fracture Mechanics,1999,63(6),775-790.
[38]季蓓,王远功,贺鹏飞,等.刚性夹杂角点处奇异场的数值分析及双夹杂间的相互影响[J].力学季刊,1999,20(4),459-464.
[39]许金泉,王效贵,刘一华.振荡应力奇异性及其强度系数的数值分析方法[J].力学季刊,2000,21(2),230-236.
[40]傅列东,许金泉.界面端弹塑性应力奇异性的迭代计算方法[J].浙江大学学报(工学版),2001,35(6),689-694.
[41]王效贵,郭乙木,许金泉.与界面相交的裂纹尖端的应力奇异性分析[J].固体力学学报,2002,23(4),412-418.
[42]宋鸣,李有堂,段红燕.垂直于界面V型缺口尖端的应力奇异性及其应用[J].应用力学学报,2009,26(4),752-756.
[43]Ju S H. Finite element calculation of stress intensity factors for interface notches[J]. Computer Methods in Applied Mechanics and Engineering,2010,199(33-36),2273-2280.
[44]王效贵,王美.考虑尺寸效应的双材料轴对称界面端应力奇异性[J].力学学报,2010,42(3),448-455.
[45]嵇醒.半平面双材料角界面应力奇异性分布规律的探究[J].固体力学学报,2011,32(5),446-458.
[46]程站起,高丹盈,仲政.任意梯度分布功能梯度涂层平面裂纹分析[J].固体力学学报,2011,32(4),426-432.
[47]钟献词,张克实.垂直于双材料非完美界面Ⅱ型裂纹的断裂分析[J].应用数学和力学,2012,33(3),342-352.
[48]Wilson W K, Thompson D G. On the finite element method for calculating stress intensity factors for cracked plates in bending[J]. Engineering Fracture Mechanics,1971,3(2),97-102.
[49]Heming F S. Sixth order analysis of crack closure in bending of an elastic plate[J]. International Journal of Fracture,1980,16(4),289-304.
[50]Alwar R S, Nambissan K N R. Influence of crack closure on the stress intensity factor for plates subjected to bending—A 3-D finite element analysis[J]. Engineering Fracture Mechanics,1983,17(4), 323-333.
[51]Alwar R S, Nambissan K N R. Three-dimensional finite element analysis of cracked thick plates in bending[J]. International Journal for Numerical Methods in Engineering,1983,19,293-303.
[52]Kwon Y W. Fracture analysis of plate bending using the finite element method[J]. International Journal of Fracture,1987,35(4), R79-R81.
[53]Kwon Y W. Finite element analysis of crack closure in plate bending[J]. Computers & Structures, 1989,32(6),1439-1445.
[54]Chen W H, Yang K C, Chang C S. A finite element alternating approach for the bending analysis of thin cracked plates[J]. International Journal of Fracture,1992,56(2),93-110.
[55]Chen W H, Shen C M. A finite element alternating approach to the bending of thin plates containing mixed mode cracks[J]. International Journal of Solids and Structures,1993,30(16),2261-2276.
[56]Leung A Y T, Su R K L. Fractal two-level finite element method for cracked kirchhoffs plates using dkt elements[J]. Engineering Fracture Mechanics,1996,54(5),703-711.
[57]Leung A Y T, Su R K L. Fractal two-level finite element analysis of cracked Reissner's plate[J]. Thin-Walled Structures,1996,24(4),315-334.
[58]Zucchini A, Hui C Y, Zehnder A T. Crack tip stress fields for thin, cracked plates in bending, shear and twisting:A comparison of plate theory and three-dimensional elasticity theory solutions[J]. International Journal of Fracture,2000,104(4),387-407.
[59]Su R K L, Sun H Y. Numerical solution of cracked thin plates subjected to bending, twisting and shear loads[J]. International Journal of Fracture,2002,117(4),323-335.
[60]Wang Y H, Tham L G, Lee P K K, et al. A boundary collocation method for cracked plates[J]. Computers & Structures,2003,81(28-29),2621-2630.
[61]Palani G S, Iyer N R, Dattaguru B. A generalised technique for fracture analysis of cracked plates under combined tensile, bending and shear loads[J]. Computers & Structures,2006,84(29-30), 2050-2064.
[62]Rabczuk T, Areias P M A. A meshfree thin shell for arbitrary evolving cracks based on an extrinsic basis[J]. Computer Modeling in Engineering and Sciences,2006,16(2),115-130.
[63]Rabczuk T, Areias P M A, Belytschko T. A meshfree thin shell method for non-linear dynamic fracture[J]. International Journal for Numerical Methods in Engineering,2007,72,524-548.
[64]Marsavina L, Sadowski T. Stress Intensity Factors for an Interface Kinked Crack in a Bi-Material Plate Loaded Normal to the Interface[J]. International Journal of Fracture,2007,145(3),237-243.
[65]杨丽敏,柳春图,曾晓辉.计算Reissner理论各向异性板应力强度因子的半权函数法[J].机械强度,2006,28(1),113-117.
[66]Leung A Y T, Xu X, Zhou Z, et al. Analytic stress intensity factors for finite elastic disk using symplectic expansion[J]. Engineering Fracture Mechanics,2009,76(12),1866-1882.
[67]Maucher R, Hartmann F. Corner singularities of Kirchhoff plates and the boundary element method[J]. Computer Methods in Applied Mechanics and Engineering,1999,173(3-4),273-285.
[68]Li J. Singularity analysis of near-tip fields for notches formed from several anisotropic plates under bending[J]. International Journal of Solids and Structures,2002,39(23),5767-5785.
[69]Treifi M, Oyadiji S O, Tsang D K L. Computations of the stress intensity factors of double-edge and centre V-notched plates under tension and anti-plane shear by the fractal-like finite element method[J]. Engineering Fracture Mechanics,2009,76(13),2091-2108.
[70]Cruse T A, Vanburen W. Three-dimensional elastic stress analysis of a fracture specimen with an edge crack[J]. International Journal of Fracture,1971,7(1),1-15.
[71]Barsoum R S. On the use of isoparametric finite elements in linear fracture mechanics[J]. International Journal for Numerical Methods in Engineering,1976,10(1),25-37.
[72]Ioakimidis N I. Application of finite-part integrals to the singular integral equations of crack problems in plane and three-dimensional elasticity[J]. Acta Mechanica,1982,45(1),31-47.
[73]Somaratna N, Ting T C T. Three-dimensional stress singularities in anisotropic materials and composites[J]. International Journal of Engineering Science,1986,24(7),1115-1134.
[74]Shivakumar K N, Tan P W, Newman J C. A virtual crack-closure technique for calculating stress intensity factors for cracked three dimensional bodies[J]. International Journal of Fracture,1988, 36(3), R43-R50.
[75]Nakamura T, Parks D M. Three-Dimensional Stress Field Near the Crack Front of a Thin Elastic Plate[J]. Journal of Applied Mechanics,1988,55(4),805-813.
[76]Barsoum R S, Chen T K. Three-dimensional surface singularity of an interface crack[J]. International Journal of Fracture,1991,50(3),221-237.
[77]Toshio N, Parks D M. Determination of elastic T-stress along three-dimensional crack fronts using an interaction integral[J]. International Journal of Solids and Structures,1992,29(13),1597-1611.
[78]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.
[79]Schmitz H, Volk K, Wendland W. Three-dimensional singularities of elastic fields near vertices [J]. Numerical Methods for Partial Differential Equations,1993,9(3),323-337.
[80]Pageau S S, Biggers S B. Finite element evaluation of free-edge singular stress fields in anisotropic materials[J]. International Journal for Numerical Methods in Engineering,1995,38(13),2225-2239.
[81]Kwon S W, Sun C T. Characteristics of three-dimensional stress fields in plates with a through-the-thickness crack[J]. International Journal of Fracture,2000,104(3),289-314.
[82]Koguchi H, Muramoto T. The order of stress singularity near the vertex in three-dimensional joints [J]. International Journal of Solids and Structures,2000,37(35),4737-4762.
[83]Li R, Tang T, Zhang P W. A Moving Mesh Finite Element Algorithm for Singular Problems in Two and Three Space Dimensions[J]. Journal of Computational Physics,2002,177(2),365-393.
[84]成昌敏,许金泉,王效贵.结合材料界面端的三维应力奇异性[J].力学季刊,2002,23(4),480-486.
[85]吴志学.三维双材料结构的应力奇异性分析[J].计算力学学报,2004,21(5),592-596.
[86]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,2011.
[87]Tracey D M. Finite elements for determination of crack tip elastic stress intensity factors[J]. Engineering Fracture Mechanics,1971,3(3),255-265.
[88]Tong P, Pian T H H, Lasry S J. A hybrid-element approach to crack problems in plane elasticity[J]. International Journal for Numerical Methods in Engineering,1973,7(3),297-308.
[89]Wilson W K. Finite element methods for elastic bodies containing cracks. Mechanics of Fracture 1[M]. Leyden:Noordhoff International Publishing,1973.
[90]Holston A. A mixed mode crack tip finite element[J]. International Journal of Fracture,1976,12(6), 887-899.
[91]蔡承文,刘明杰,黄纯明.计算平面问题应力强度因子的任意高阶奇应变有限单元[J].固体力学学报,1983,4(4),597-605.
[92]Benzley S E. Representation of singularities with isoparametric finite elements[J]. International Journal for Numerical Methods in Engineering,1974,8(3),537-545.
[93]Barsoum R S. Triangular quarter-point elements as elastic and perfectly-plastic crack tip elements[J]. International Journal for Numerical Methods in Engineering,1977,11,85-98.
[94]Lynn P P, Ingraffea A R. Transition elements to be used with quarter-point crack-tip elements[J]. International Journal for Numerical Methods in Engineering,1978,12(6),1031-1036.
[95]Staab G H. A variable power singular element for analysis of fracture mechanics problems[J]. Computers & Structures,1983,17(3),449-457.
[96]徐纪林,吴永礼.计算应力强度因子的奇异等参单元[J].固体力学学报,1983,(2),258-263.
[97]Raju I S. Calculation of strain-energy release rates with higher order and singular finite elements[J]. Engineering Fracture Mechanics,1987,28(3),251-274.
[98]王燕群,张敬宇.一种高精度的裂纹奇异单元’[J].应用数学和力学,1990,11(10),931-936.
[99]陆鸿森.一种计算应力强度因子的奇异元[J].固体力学学报,1992,13(2),182-186.
[100]Tan M A, Meguid S A. Analysis of Bimaterial Wedges Using a New Singular Finite Element[J]. International Journal of Fracture,1997,88(4),373-391.
[101]冯金辉,柳春图.含裂纹有限结构的局部-整体分析[J].应用力学学报,1999,16(2),145-150.
[102]Chen M C, Sze K Y. A novel hybrid finite element analysis of bimaterial wedge problems[J]. Engineering Fracture Mechanics,2001,68(13),1463-1476.
[103]Lim K M, Lee K H, Tay A A O, et al. A new variable-order singular boundary element for two-dimensional stress analysis[J]. International Journal for Numerical Methods in Engineering, 2002,55(3),293-316.
[104]Liu L M, Duan M L, Qin T Y, et al. A Finite Element Method for Cracked Components of Structures[J]. China Ocean Engineering,2003,17(2),177-187.
[105]王海涛,余锦炎.双压电材料界面力电耦合场奇异性研究[J].工程力学,2006,23(1),165-171.
[106]王海涛.分析不同材料界面应力奇异性的一维杂交有限元方法[J].工程力学,2009,26(2),21-26.
[107]平学成,谢基龙,陈梦成,等.夹杂尖端奇异性场的新型杂交元分析[J].北京交通大学学报,2007,31(4),5-9.
[108]平学成,陈梦成,谢基龙.周期分布多边形夹杂奇异性应力干涉问题的研究[J].计算力学学报,2010,27(6),1117-1122.
[109]Chen L, Liu G, Nourbakhsh-Nia N, et al. A singular edge-based smoothed finite element method (ES-FEM) for bimaterial interface cracks[J]. Computational Mechanics,2010,45(2),109-125.
[110]Barsoum R S. A degenerate solid element for linear fracture analysis of plate bending and general shells[J]. International Journal for Numerical Methods in Engineering,1976,10,551-564.
[111]Zienkiewicz O C. The finite element method in engineering science[M]. London:McGraw-Hill, 1971.
[112]Ahmad J, Loo F T C. Solution of plate bending problems in fracture mechanics using a specialized finite element technique[J]. Engineering Fracture Mechanics,1979,11(4),661-673.
[113]Williams M L. The bending stress distribution at the base of a stationary crack[J]. Journal of Applied Mechanics,1961,28,78-82.
[114]Yagawa G, Nishioka T. Finite element analysis of stress intensity factors for plane extension and plate bending problems[J]. International Journal for Numerical Methods in Engineering,1979,14, 727-740.
[115]Rhee H C, Atluri S N. Hybrid stress finite element analysis of bending of a plate with a through flaw[J]. International Journal for Numerical Methods in Engineering,1982,18,259-271.
[116]李英治,柳春图Reissner型平板弯曲断裂问题分析[J].力学学报,1983,(4),366-375.
[117]柳春图,李英治Reissner型板裂纹尖端应力应变场及应力强度因子计算[J].力学学报,1984,16(4),351-362.
[118]Chen W H, Chen P Y. A hybrid-displacement finite element model for the bending analysis of thin cracked plates[J]. International Journal of Fracture,1984,24(2),83-106.
[119]Ye T Q, Gallagher R H. A singular finite element for analysis of plate bending problem in fracture mechanics[J]. International Journal of Fracture,1984,24(2),137-147.
[120]Wahba N N. On the use of singular displacement finite elements for cracked plate in bending[J]. International Journal of Fracture,1985,27(1),3-30.
[121]Chen W H, Hwu C B. Hybrid singular element design for the bending analysis of bimaterial thin cracked plates[J]. AIAA J,1986,24(8),1399-1402.
[122]Sih G C, Rice J R. The Bending of Plates of Dissimilar Materials With Cracks[J]. Journal of Applied Mechanics,1964,31(3),477-482.
[123]Viswanath S, Murthy M V V, Murty A V K., et al. A special crack tip element for bending of plates with through cracks from a sixth-order plate theory[J]. Engineering Fracture Mechanics,1989,32(1), 91-109.
[124]Dashevskii E M, Rotter M V. A singular notched finite element for plates in bending[J]. Strength of Materials,1989,21(2),239-244.
[125]Jiang C P, Cheung Y K. A special bending crack tip finite element[J]. International Journal of Fracture,1995,71(1),57-69.
[126]Moes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering,1999,46,131-150.
[127]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.
[128]孟广伟,陈塑寰,刘寒冰,等.杂交/混合奇异弯曲板单元列式方法及应力强度因子的计算[J].机械强度,2001,23(3),329-331.
[129]Hung N D, Ngoc T T. Analysis of cracked plates and shells using "metis" finite element model[J]. Finite Elements in Analysis and Design,2004,40(8),855-878.
[130]Munaswamy K, Pullela R. Computation of stress intensity factors for through cracks in plates using p-version finite element method[J]. Communications in Numerical Methods in Engineering,2008,24, 1753-1780.
[131]Rethore J, Roux S, Hild F. Hybrid analytical and extended finite element method (HAX-FEM):A new enrichment procedure for cracked solids[J]. International Journal for Numerical Methods in Engineering,2010,81,269-285.
[132]Lasry J, Pommier J, Renard Y, et al. eXtended finite element methods for thin cracked plates with Kirchhoff-Love theory[J]. International Journal for Numerical Methods in Engineering,2010,84, 1115-1138.
[133]Hussain M A, Coffin L F, Zaleski K A. Three dimensional singular element[J]. Computers & Structures,1981,13(5-6),595-599.
[134]Astiz M A. An incompatible singular elastic element for two-and three-dimensional crack problems[J]. International Journal of Fracture,1986,31(2),105-124.
[135]Luo G M, Zhang Y Y. Application of boundary element method with singular and isoparametric elements in three dimensional crack problems[J]. Engineering Fracture Mechanics,1988,29(1), 97-106.
[136]王柏松,贺寅彪.用改进的三维等参奇异元计算应力强度因子[J].核科学与工程,1991,11(2),129-133.
[137]Pageau S S, Biggers Jr S B. A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions[J]. International Journal of Solids and Structures, 1996,33(1),33-47.
[138]Ariza M P, Saez A, Dominguez J. A singular element for three-dimensional fracture mechanics analysis[J]. Engineering Analysis with Boundary Elements,1997,20(4),275-285.
[139]Sukumar N, Moes N, Moran B, et al. Extended finite element method for three-dimensional crack modelling[J]. International Journal for Numerical Methods in Engineering,2000,48(11), 1549-1570.
[140]Ong E T, Lim K M. Three-dimensional singular boundary elements for corner and edge singularities in potential problems[J]. Engineering Analysis with Boundary Elements,2005,29(2),175-189.
[141]赵怀瑞,李强,张向前,等.三维应力奇异单元自动生成的通用算法[J].北京交通大学学报,2009,33(1),62-65.
[142]林广平,梅甫良.12结点三维等参奇异单元的构造和应用[J].计算力学学报,2009,26(2),282-286.
[143]中国科学院力学研究所板壳组.夹层板壳的弯曲、稳定和振动[M].北京：科学出版社,1977.
[144]胡海吕.弹性力学的变分原理及其应用[M].北京：科学出版社,1981.
[145]钱伟长.变分法与有限元[M].北京：科学出版社,1981.
[146]Jiang C P. Finite strip analysis for rectangular plates considering shear deformation[J]. Engineering Analysis,1984,1(4),206-210.
[147]Liu C T, Wu X J, Li Y Z. Study on cracked spherical shells[J]. Scientia Sinaca(Series A),1987,15(9), 944-954.
[148]柳春图,吴犀甲.含轴向裂纹柱壳裂纹尖端应力应变场及应力强度因子计算[J].力学学报,1987,19(2),125-135.
[149]Liu C T, Wu X J. Local stresses and strains inaxially cracked cylindrical shells[J]. Theoretical and Applied Fracture Mechanics,1989,11(1),51-58.
[150]柳春图,韩闻生.含孔边裂纹球壳断裂分析[J].应用力学学报,1993,10(1),1-9.
[151]柳春图,胡互让,吴犀甲.圆柱壳孔边裂纹应力强度因子计算和分析[J].计算结构力学及其应用,1992,9(1),7-15.
[152]钟万腮,钟翔翔.计算结构力学、最优控制及偏微分方程半解析法[J].计算结构力学及其应用,1990,7(1),1-15.
[153]Zhong W X. Plane elasticity in sectorial domain and the Hamiltonian system[J]. Applied Mathematics and Mechanics,1994,15(12),1113-1123.
[154]钟万勰.分离变量法与哈密顿体系[J].计算结构力学及其应用,1991,8(3),229—240.
[155]钟万勰.互等定理与共辘辛正交关系[J].力学学报,1992,24(4),432-437.
[156]张鸿庆.弹性力学的对偶化体系——评《弹性力学求解新体系》[J].大连理工大学学报,1996,36,372.
[157]姚伟岸,钟万勰.辛弹性力学[M].北京：高等教育出版社,2002.
[158]钟万勰.条形域平面弹性问题与哈密尔顿体系[J].大连理工大学学报,1991,3](4),：373-384.
[159]姚伟岸.平面各向异性哈密顿体系及圣维南问题的解析解[J].大连理工大学学报,1999,39(5),612-615.
[160]邹贵平.薄板问题的Hamilion体系和辛儿何方法[J].应用基础与工程科学学报,1996,4(4),335-343.
[161]钟万勰,姚伟岸.板弯曲求解新体系及其应用[J].力学学报,1999,31(2),173-184.
[162]姚伟岸,苏滨,钟万勰.基于相似性原理的正交各向异性板弯曲Hamilton体系[J].中国科学,E辑,2001,31(4),342-347.
[163]Yao W A, Sui Y F. Symplectic solution system for reissner plate bending[J]. Applied Mathematics and Mechanics,2004,25(2),178-185.
[164]鲍四元,邓子辰Mindlin中厚板的辛求解方法[J].固体力学学报,2005,26(1),102-106.
[165]马晨明.固支矩形Mindlin板弯曲问题的辛求解体系[J].力学季刊,2008,29(4),648-655.
[166]Zhong W X, Yao W A. The Saint Venant solutions of multi-layered composite plates[J]. Advances in Structural Engineering,1997,1(2),127-133.
[167]Yao W A, Yang H T. Hamiltonian system based Saint Venant solutions for multi-layered composite plane anisotropic plates[J]. International Journal of Solids and Structures,2001,38(32),5807-5817.
[168]姚伟岸,聂臆瞩,肖峰.基于辛对偶体系的层合板自由边缘效应的分析解[J].应用数学和力学,2011,32(9),1021-1029.
[169]杨有贞,葛修润.基于辛体系的正交各向异性地基梁解析解[J].力学学报,2011,43(2),362-371.
[170]Xu X S, Zhong W X, Zhang H W. The Saint-Venant problem and principle in elasticity[J]. International Journal of Solids and Structures,1997,34(22),2815-2827.
[171]Xu X S, Zhang H W, Zhong W X. A direct method for the problem of the solid cylinder in elasticity[J]. Chinese Quarterly Journal of Mathematics,1998,13(1),44-54.
[172]徐新生,贾宏志,孙发明.横观各向同性弹性柱体中辛本征解方法[J].大连理工大学学报,2005,45(4),617-624.
[173]徐新生,郑新广,张洪武.哈密顿体系与弹性楔体问题[J].应用力学学报,1999,16(2),140-145.
[174]Yao W A, Zhong W X, Su B. New solution system for circular sector plate bending and its application[J]. Acta Mechanica Solida Sinica,1999,12(4),307-315.
[175]鲍四元,邓子辰.环扇形板弯曲问题中环向模拟为时间的辛体系[J].西北工业大学学报,2004,22(6),734-738.
[176]姚伟岸,孙贞.环扇形薄板弯曲问题的环向辛对偶求解方法[J].力学学报,2008,40(4),557-563.
[177]钟万勰,徐新生,张洪武.弹性曲梁问题的直接法[J].工程力学,1996,13(4),1-8.
[178]姚伟岸.极坐标哈密顿体系约当型与弹性楔的佯谬解[J].力学学报,2001,33(1),79-86.
[179]姚伟岸,张兵茹.圆柱型正交各向异性弹性楔的佯谬解[J].固体力学学报,2004,25(2),155-158.
[180]周建方,卓家寿.极坐标下弹性力学的一个新解答[J].力学学报,2001,33(6),839-846.
[181]齐朝晖,唐立民.曲线坐标系下哈密顿体系的建立[J].大连理工大学学报,1997,37(4),376-380.
[182]朱炳麒,周建方,卓家寿.矩形梁受分布荷载的辛解答[J].力学与实践,2007,29(3),15-19.
[183]Lim C W, Yao W A, Cui S. Benchmark symplectic solutions for bending of corner-supported rectangular thin plates[J]. The IES Journal Part A:Civil & Structural Engineering,2008,1(2), 106-115.
[184]谈梅兰,吴光,王鑫伟.矩形薄板面内非线性分布载荷下的辛弹性力学解[J].工程力学,2008,25(10),50-53.
[185]陈伟球,赵莉.功能梯度材料平面问题的辛弹性力学解法[J].力学学报,2009,41(4),588-594.
[186]Xu X S, Zhang W X, Li X, et al. An application of the symplectic system in two-dimensional viscoelasticity[J]. International Journal of Engineering Science,2006,44(13-14),897-914.
[187]徐新生,张维祥,李雪.粘弹性厚壁筒问题的辛本征解方法[J].计算力学学报,2007,24(2),153-158.
[188]张维祥,徐新生,王尕平.辛体系下平面热黏弹性圣维南问题的解析解[J].力学与实践,2008,30(4),76-78.
[189]Zhang W X, Xu X S. A symplectic method in 2D thermo-viscoelasticity[J].中国科学技术大学学报, 2008,38(2),200-206.
[190]Zhang W X, Xu X S, Yuan F. The symplectic system method in the stress analysis of 2D elasto-viscoelastic fiber reinforced composites[J]. Archive of Applied Mechanics,2010,80(8), 829-841.
[191]姚伟岸,杨海天,高强.平面粘弹性问题的辛求解方法[J].计算力学学报,2010,27(1),14-20.
[192]张洪武,李云鹏,钟万勰.双材料楔形结合点的奇性分析[J].大连理工大学学报,1995,35(6),776-782.
[193]张洪武,徐新生,李云鹏,等.多种材料楔形结合点的奇性分析[J].大连理工大学学报,1996,36(4),391-395.
[194]Zhang H W, Zhong W X. Hamiltonian principle based stress singularity analysis near crack corners of multi-material junctions[J]. International Journal of Solids and Structures,2003,40(2),493-510.
[195]王承强,姚伟岸.哈密顿体系在断裂力学Dugdale模型中的应用[J].应用力学学报,2003,20(3),151-155.
[196]郑长良,王承强,基于双线性粘聚力模型混凝土裂纹扩展的半解析有限元法,全国计算力学学术会议，北京,2003.
[197]Wang C Q, Zheng C L. Semi-analytical finite element method for fictitious crack model in fracture mechanics of concrete[J]. Applied Mathematics and Mechanics,2004,25(11),1265-1270.
[198]王承强,郑长良.混凝土双K断裂参数计算的半解析有限元法[J].力学学报,2004,36(4),414-418.
[199]王承强,郑长良.Ⅰ型和Ⅱ型Dugdale模型解析元列式及其半解析有限元法[J].工程力学,2005,22(6),37-40,68.
[200]王承强,郑长良.平面裂纹应力强度因子的半解析有限元法[J].工程力学,2005,22(1)：33-37.
[201]王承强,郑长良.裂纹扩展过程中线性内聚力模型计算的半解析有限元法[J].计算力学学报,2006,23(2),146-151.
[202]姚伟岸,胡小飞.双材料含桥联力Dugdale-Barenblatt模型的解析奇异单元[J].固体火箭技术,2011,34(5),635-638.
[203]周建方,卓家寿,李典庆Hamilton体系下弹性力学半解析法的一个守恒律[J].河海大学学报,2000,28(5),41-43.
[204]唐立民,褚致中,邹贵平.混合状态Hamiltonian元的半解析解和叠层板的计算[J].计算结构力学及其应用,1992,9(4),347-360.
[205]邹贵平,唐立民,周哲玮.复合材料条形域问题混合状态Hamiltonian元的半解析解[J].应用力学学报,1995,12(3),1-7.
[206]王治国,唐立民,朱德娥.圆柱坐标系中弹性力学的哈密顿正则方程及其哈密顿单元方法[J].计算结构力学及其应用,1995,12(1),53-59.
[207]周建方,卓家寿,李典庆.基于Hamilton方法的有限元[J].河海大学常州分校学报,2000,14(3),1-6.
[208]卿光辉,邱家俊,刘艳红Hamilton等参元与智能叠层板的半解析法[J].应用数学和力学,2007,28(1),45-52.
[209]胡志强,林皋,王毅,等.基于Hamilton体系的弹性力学问题的比例边界有限元方法[J].计算力学学报,2011,28(4),510-516.
[210]钟万勰.电磁波导的辛体系[J].大连理工大学学报,2001,41(4),379-387.
[211]钟万勰.电磁波导的半解析辛分析[J].力学学报,2003,35(4),401-410.
[212]孙雁,谢军.基于Hamilion体系的辛半解析法在各向异性电磁波导中的应用[J].计算力学学报,2005,22(6),690-693.
[213]陈杰夫,郑长良,钟万勰.电磁波导的辛分析与对偶棱边元[J].物理学报,2006,55(5),2340-2347.
[214]陈杰夫,郑长良,钟万勰.电磁波导的奇异元与对偶有限元分析[J].计算方学学报,2007,24(2),148-152.
[215]张宝善,卢东强,戴世强,等.非线性水波Hamilton系统理论与应用研究进展[J].力学进展,1998,28(4),521-531.
[216]马坚伟,徐新生,杨慧珠,等.平面粘性流体扰动与哈密顿体系[J].应与力学’学报,2001,18(4),82-87.
[217]马坚伟,徐新生,杨慧珠,等.基于哈密顿体系求解空间粘性流体问题[J].工程力学,2002,19(3),1-6.
[218]徐新生,于尕平Stokes流问题中的辛本征解方法[J].力学学报,2006,38(5),682-687.
[219]徐新生,王尕平,孙发明.二维矩形域内Stokes流问题的辛解析和数值方法[J].应用数学和力学,2008,29(6),639-648.
[220]王艳,邓子辰.单板驱动空腔的辛求解方法[J].动力学与控制学报,2007,5(2),105-11 1.
[221]王艳,邓子辰Stokes流问题的环向辛对偶求解方法[J].机械科学与技术,2008,27(3),374-378.
[222]王艳,邓子辰.楔形域中低雷诺数流动问题的辛对偶求解方法[J].西北工业大学学报,2008,26(2),239-243.
[223]Wang G P, Xu X S, Zhang Y X. Influence of inlet radius on Stokes flow in a circular tube via the Hamiltonian systematic method[J]. Physics of Fluids,2009,21(10),103602(103609 pages).
[224]Xu X S, Ma Y, Lim C W, et al. Dynamic buckling of cylindrical shells subject to an axial impact in a symplectic system[J]. International Journal of Solids and Structures,2006,43(13),3905-3919.
[225]刘淼,罗恩,仲政.薄板动力学相空间非传统Hamilton变分原理与辛算法[J].固体力学学报,2007,28(2),207-211.
[226]顾松年,姜节胜,何建军.经典阻尼系统响应的辛解析解[J].西北工业大学学报,2007,25(4),487-491.
[227]胡伟鹏,邓子辰,李文成.膜自由振动的多辛方法[J].应用数学和力学,2007,28(9),1054-1062.
[228]Lim C W, Lu C F, Xiang Y, et al. On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates[J]. International Journal of Engineering Science,2009, 47(1),131-140.
[229]姚征,郑长良.二维非局部线弹性平面问题的辛分析[J].计算方学学报,2009,26(3),395-400.
[230]钟阳,李锐,田斌.矩形中厚板自由振动问题的哈密顿体系与辛儿何解法[J].动方学与控制学报,2009,7(4),302-307.
[231]姚伟岸.电磁弹性固体反平面问题辛求解体系及圣维南原理[J].大连理工大学学报,2004,44(5),630-633.
[232]Gu Q, Xu X S, Leung A Y T. Application of Hamiltonian system for two-dimensional transversely isotropic piezoelectric media[J]. Journal of Zhejiang University SCIENCE,2005,6A(6),915-921.
[233]Xu X S, Gu Q, Leung A Y T, et al. A symplectic eigensolution method in transversely isotropic piezoelectric cylindrical media[J]. Journal of Zhejiang University SCIENCE,2005,6A(9),922-927.
[234]徐新生,段政,马源,等.辛方法和弹性圆柱壳在内外压和轴向冲击下的动态屈曲[J].爆炸与冲击,2007,27(6),509-514.
[235]徐新生,邱文彪,付月,等.辛方法在弹性圆板屈曲问题中的应用[J].应用力学学报,2009,26(3),530-534.
[236]马继光,白象忠,张旭光,等.导电悬臂梁磁弹性弯曲的辛解答[J].燕山大学学报,2009,33(2),147-152.
[237]褚洪杰,徐新生,林志华,等.弹性梁非线性热屈曲行为与辛本征解展开方法[J].大连理工大学学报,2011,51(1),1-6.
[238]Timoshenko S, Woinowsky-Kriege S. Theory of Plates and Shells[M]. New York:MeGraw-Hill Book Company,1959.
[239]Williams M L, Surface stress singularities resulting from various boundary condition in angular corners of plates under bending, First US National Congress on Applied Mechanics,1951, pp. 325-329.
[240]杨春林,郑百林,嵇醒.单材料V型缺口尖端振荡性奇异应力场产生的条件[J].应用力学学报,2001,18(2),65-69.