XU Zhi, FAN Xue-ming. The Explicit Expression of Erdogan’s Fundamental Solution for Plane Problems With Cracks[J]. Applied Mathematics and Mechanics, 2017, 38(9): 1009-1020. doi: 10.21656/1000-0887.370253
 Citation: XU Zhi, FAN Xue-ming. The Explicit Expression of Erdogan’s Fundamental Solution for Plane Problems With Cracks[J]. Applied Mathematics and Mechanics, 2017, 38(9): 1009-1020.

# The Explicit Expression of Erdogan’s Fundamental Solution for Plane Problems With Cracks

##### doi: 10.21656/1000-0887.370253
Funds:  The National Natural Science Foundation of China(General Program)(51378009)
• Rev Recd Date: 2017-05-11
• Publish Date: 2017-09-15
• The fundamental solution plays an essential role in many numerical methods such as the boundary element method (BEM), the method of fundamental solution and the boundarytype meshless methods. For fracture problems, application of fundamental solutions in domains with cracks can avoid deeming the crack surface as boundary condition and simplify the problems significantly. Based on the complex variable expression of Erdogan’s fundamental solution (EFS) for plane problems, the application conditions for EFS were noted, some faults in EFS were corrected and the explicit expression of the displacement in EFS was obtained. The computer program of the spline fictitious boundary element method (SFBEM) based on EFS was compiled to calculate the numerical solutions to mixed boundary condition problems. Numerical examples prove the correctness and accuracy of the derived explicit expression of EFS.
•  [1] 郦正能. 应用断裂力学[M]. 北京: 北京航空航天大学出版社, 2012.(LI Zheng-neng. Applied Fracture Mechanics [M]. Beijing: Beihang University Press, 2012.(in Chinese)) [2] Zehnder A T. Fracture Mechanics [M]. Netherlands: Springer, 2012. [3] 付宇明, 田振国, 郑丽娟. 轴对称金属模具电磁热裂纹止裂中热应力场的分析[J]. 应用数学和力学, 2006,27(3): 331-336.(FU Yu-ming, TIAN Zhen-guo, ZHENG Li-juan. Thermal stress field when crack arrest in an axial symmetry metal die using electromagnetic heating[J]. Applied Mathematics and Mechanics,2006,27(3): 331-336.(in Chinese)) [4] Benzley S E. Representation of singularities with isoparametric finite elements[J]. Clinical Endocrinology,1974,8(8): 537-545. [5] 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. [6] MENG Qing-hua, WANG Zhen-qing. Extended finite element method for power-law creep crack growth[J]. Engineering Fracture Mechanics,2014,127: 148-160. [7] Mos N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing[J]. International Journal for Numerical Methods in Engineering,1999,46(1): 131-150. [8] Portela A, Aliabadi M H, Rooke D P. Dual boundary element incremental analysis of crack propagation[J]. Computers & Structures,1993,46(2): 237-247. [9] Leonel E D, Venturini W S. Dual boundary element formulation applied to analysis of multi-fractured domains[J]. Engineering Analysis With Boundary Elements,2010,34(12): 1092-1099. [10] Erdogan. On the stress distribution in a plate with collinear cuts under arbitrary loads[C]// Proceedings of the 〖STBX〗4th US National Congress of Applied Meehanies . 1962: 547-553. [11] Tan P W, Raju I S, Newman I, et al. Boundary force method for analyzing two-dimensional cracked bodies[R]. Hampton, Virginia: NASA Langley Research Center, 1986. [12] 苏成, 郑淳. 基于Erdogan基本解边界元法计算应力强度因子[J]. 力学学报, 2007,39(1): 93-99.(SU Cheng, ZHENG Chun. Calculation of stress intensity factors by boundary element method based on Erdogan fundamental solutions[J]. Chinese Journal of Theoretical and Applied Mechanics,2007,39(1): 93-99.(in Chinese)) [13] SU Cheng, ZHENG Chun. Probabilistic fracture mechanics analysis of linear-elastic cracked structures by spline fictitious boundary element method[J]. Engineering Analysis With Boundary Elements,2012,36(12): 1828-1837. [14] 吴家龙. 弹性力学[M]. 上海: 同济大学出版社, 2010.(WU Jia-long. Elasticity Mechanics[M]. Shanghai: Tongji University Press, 2010.(in Chinese)) [15] Mogilevskaya S G, Linkov A M. Complex fundamental solutions and complex variables boundary element method in elasticity[J]. Computational Mechanics,1998,22(1): 88-92. [16] Banerjee P K, Butterifeld R. Boundary Element Method in Engineerng Seienee [M]. London: McGraw Hill, 1981. [17] SU Cheng, HAN Da-jian. Multidomain SFBEM and its application in elastic plane problems[J]. Journal of Engineering Mechanics,2000,126(10): 1057-1063.

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