FENG Guoyi, XIAO Junhua, SU Mengyu. Fracture Mechanics Analysis of Mode-Ⅲ Radial Multi Cracks on the Edge of a Hole With Surface Effects[J]. Applied Mathematics and Mechanics, 2020, 41(4): 376-385. doi: 10.21656/1000-0887.400177
 Citation: FENG Guoyi, XIAO Junhua, SU Mengyu. Fracture Mechanics Analysis of Mode-Ⅲ Radial Multi Cracks on the Edge of a Hole With Surface Effects[J]. Applied Mathematics and Mechanics, 2020, 41(4): 376-385.

# Fracture Mechanics Analysis of Mode-Ⅲ Radial Multi Cracks on the Edge of a Hole With Surface Effects

##### doi: 10.21656/1000-0887.400177
Funds:  The National Natural Science Foundation of China(11302186)
• Publish Date: 2020-04-01
• The mode-Ⅲ fracture properties of the radial multi cracks on the edge of a nanoscale circular hole were theoretically investigated. Based on the Gurtin-Murdoch surface elasticity theory and the conformal mapping technique, the analytical solutions of the stress fields around the hole and cracks were given, and the closed form solution of the stress intensity factor at the crack tip was obtained. The size effect of the stress intensity factor was analyzed based on the solution. The effects of the crack number, the ratio of crack/hole radius and the surface defects on the stress intensity factor were discussed. The results indicate that, the dimensionless stress intensity factor has remarkable size-dependent effects when the sizes of the cracked hole are at the nanoscale. The variation of the stress intensity factor with the number of cracks is influenced by the crack/hole size ratio. The effects of the crack/hole size ratio on the stress intensity factors are restricted by the surface defects, and the effects of the surface properties on the stress intensity factor are also limited by the crack/hole size ratio. The surface effects on the stress intensity factors are independent of the number of cracks.
•  [1] 邢帅兵, 王强胜, 生月, 等. 圆形杂质对裂纹扩展的影响[J]. 应用数学和力学, 2019,40(2): 189-199.(XING Shuaibing, WANG Qiangsheng, SHENG Yue, et al. Effect of circular inhomogeneity on crack propagation[J]. Applied Mathematics and Mechanics,2019,40(2): 189-199.(in Chinese)) [2] 余建星, 李修波, 谭玉娜, 等. 管道表面蚀坑-裂纹的应力强度因子分析[J]. 天津大学学报(自然科学与工程技术版), 2019,52(5): 522-528.(YU Jianxing, LI Xiubo, TAN Yuna, et al. Analysis of the stress intensity factor of a pipeline surface with a pit-crack[J]. Journal of Tianjin University(Science and Technology),2019,52(5): 522-528.(in Chinese)) [3] 陈长征, 王琳琳, 周勃, 等. 基于红外热像技术的风力机微裂纹叶片研究[J]. 太阳能学报, 2019,40(2): 417-421.(CHEN Changzheng, WANG Linlin, ZHOU Bo, et al. Study on microcrack of wind turbine blade based on infrared thermography technology[J]. Acta Energiae Solaris Sinica,2019,40(2): 417-421.(in Chinese)) [4] HAJIMOHAMADI M, GHAJAR R. An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks[J]. Applied Mathematics and Mechanics ( English Edition ), 2018,39(8): 1103-1118. [5] 张俊, 靳莹, 熊涛. 集中力作用下多铁性板状复合材料的断裂分析[J]. 应用数学和力学, 2018,39(12): 1390-1399.(ZHANG Jun, JIN Ying, XIONG Tao. Fracture analysis on multiferroic composite plates under concentrated forces[J]. Applied Mathematics and Mechanics,2018,39(12): 1390-1399.(in Chinese)) [6] 彭俊, 蔡明, 荣冠, 等. 裂纹闭合应力及其岩石微裂纹损伤评价[J]. 岩石力学与工程学报, 2015,34(6): 1091-1100.(PENG Jun, CAI Ming, RONG Guan, et al. Stress for crack closure and its application to assessing stress-induced microcrack damage[J]. Chinese Journal of Rock Mechanics and Engineering,2015,34(6): 1091-1100.(in Chinese)) [7] 李政鸿, 徐武, 张晓晶, 等. 多孔多裂纹平板的疲劳裂纹扩展试验与分析方法[J]. 航空学报, 2018,39(7): 149-157.(LI Zhenghong, XU Wu, ZHANG Xiaojing, et al. Experimental and analytical analyses of fatigue crack growth in sheets with multiple holes and cracks[J]. Acta Aeronautica et Astronautica Sinica,2018,39(7): 149-157.(in Chinese)) [8] 郭俊宏, 于静. 多场耦合材料断裂力学[M]. 北京: 科学出版社, 2015.(GUO Junhong, YU Jing. Multi-Field Coupling Material Fracture Mechanics[M]. Beijing: Science Press, 2015.(in Chinese)) [9] GUO J H, LU Z X. Anti-plane analysis of multiple cracks originating from a circular hole in a magnetoelectroelastic solid[J]. International Journal of Solids and Structures,2010,14/15(47): 1847-1856. [10] GUO J H, LU Z X. Exact solution of four cracks originating from an elliptical hole in one-dimensional hexagonal quasicrystals[J]. Applied Mathematics and Computation,2011,217(22): 9397-9403. [11] WANG Y J, GAO C F. The mode Ⅲ cracks originating from the edge of a circular hole in a piezoelectric solid[J]. International Journal of Solids and Structures,2008,45(16): 4590-4599. [12] HASSAN A, SONG T S. Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials[J]. Applied Mathematics and Mechanics ( English Edition ), 2014,35(10): 1261-1270. [13] 宋天舒, 李冬. 压电体中孔边Ⅲ型界面裂纹的动应力强度因子[J]. 力学学报, 2010,42(6): 1219-1224.(SONG Tianshu, LI Dong. Dynamic stress intensity factor for interfacial cracks of mode Ⅲ on a circular cavity in piezoelectric bimaterials[J]. Chinese Journal of Theoretical and Applied Mechanics,2010,42(6): 1219-1224.(in Chinese)) [14] 张乐乐. 纳米复合材料中表面/界面效应对动应力的影响[D]. 硕士学位论文. 石家庄: 石家庄铁道大学, 2011.(ZHANG Lele. Effect of surface/interface on the dynamic stress in nano composites[D]. Master Thesis. Shijiazhuang: Shijiazhuang Tiedao University, 2011.(in Chinese)) [15] 杨娟. 压电效应下一维六方准晶中孔边多裂纹反平面断裂问题研究[D]. 博士学位论文. 宁夏: 宁夏大学, 2015.(YANG Juan. Study on anti-plane fracture problems of multiple cracks emanating from a hole in one-dimensional hexagonal quasicrystals with piezoelectric effects[D]. PhD Thesis. Ningxia: Ningxia University, 2015.(in Chinese)) [16] 肖俊华, 徐耀玲. 纳米夹杂复合材料的有效反平面剪切模量研究[J]. 固体力学学报, 2011,32(3): 287-292.(XIAO Junhua, XU Yaoling. Study on the effective anti-plane shear modulus of nano inhomogeneity composite materials[J]. Chinese Journal of Solid Mechanics,2011,32(3): 287-292.(in Chinese)) [17] 沈海军. 纳米科技概论[M]. 北京: 国防工业出版社, 2007.(SHEN Haijun. Introduction to Nanotechnology [M]. Beijing: National Defense Industry Press, 2007.(in Chinese)) [18] GURTIN M E, MURDOCH A I. A continuum theory of elastic material surfaces[J]. Archive for Rational Mechanics and Analysis,1975,57(4): 291-323. [19] GURTIN M E, MURDOCH A I. Surface stress in solids[J]. International Journal of Solids and Structures,1978,14(6): 431-440. [20] GURTIN M E, WEISSMULLER J, LARCHE F. A general theory of curved deformable interfaces in solids at equilibrium[J]. Philosophical Magazine A: Physics of Condensed Matter Structure Defects and Mechanical Properties,1998,78(5): 1093-1109. [21] LUO J, WANG X. On the anti-plane shear of an elliptic nano inhomogeneity[J]. European Journal of Mechanics A: Solids,2009,28(5): 926-934. [22] PINYOCHOTIWONG Y, RUNGAMORNART J, SENJUNTICHAI T. Analysis of rigid frictionless indentation on half-space with surface elasticity[J]. Procedia Engineering,2011,14: 2403-2410. [23] KIM C I, SCHIAVONE P, RU C Q. The effects of surface elasticity on an elastic solid with mode-Ⅲ crack: complete solution[J]. Journal of Applied Mechanics,2010,77(2): 021011-1-7. DOI: 10.1115/1.3177000. [24] 肖俊华, 韩彬, 徐耀玲, 等. 考虑表面弹性效应时正三角形孔边裂纹反平面剪切问题的断裂力学分析[J]. 固体力学学报, 2017,38(6): 530-536.(XIAO Junhua, HAN Bin, XU Yaoling, et al. Fracture analysis of cracked equilateral triangular hole with surface elasticity effect under antiplane shear[J]. Chinese Journal of Solid Mechanics,2017,38(6): 530-536.(in Chinese)) [25] XIAO J H, SHI C F, XU Y L, et al. Interface stress of orthotropic materials with a nanodefect under antiplane shear loading[J]. Journal of Mechanics of Materials and Structures,2016,11(5): 491-504. [26] WANG X, ZHOU K. Interface cracks with surface elasticity in anisotropic bimaterials[J]. International Journal of Solids and Structures,2015,59: 110-120. [27] KIM C I, SCHIAVONE P, RU C Q. Effect of surface elasticity on an interface crack in plane deformations[J]. Proceedings Mathematical Physical & Engineering Science,2011,467(2136): 3530-3549. [28] KIM C I, SCHIAVONE P, RU C Q. The effects of surface elasticity on a mode-Ⅲ interface crack[J]. Archives of Mechanics,2011,63(3): 267-286. [29] XU J Y, DONG C Y. Surface and interface stress effects on the interaction of nano-inclusions and nano-cracks in an infinite domain under anti-plane shear[J]. International Journal of Mechanical Sciences,2016,111/112: 12-23. [30] 肖俊华, 徐耀玲, 张福成. 基于表面弹性理论的开裂椭圆孔断裂力学研究[J]. 固体力学学报, 2019,40(1): 82-89.(XIAO Junhua, XU Yaoling, ZHANG Fucheng. Fracture mechanics of cracked elliptical hole based on surface elasticity theory[J]. Chinese Journal of Solid Mechanics,2019,40(1): 82-89.(in Chinese)) [31] GUO J H, LU Z X. The fracture behavior of multiple cracks emanating from a circular hole in piezoelectric materials[J]. Acta Mechanica,2010,215(1/4): 119-134. [32] MUSKHELISHVILI N I. Some Basic Problems of the Mathematical Theory of Elasticity [M]. Groningen: Noordhoff, 1953. [33] 肖俊华, 崔友强, 徐耀玲, 等. 纳米尺度孔边裂纹裂尖Ⅲ型应力强度因子研究[J]. 中国机械工程, 2018,29(19): 2347-2352.(XIAO Junhua, CUI Youqiang, XU Yaoling, et al. Study on mode Ⅲ stress intensity factor at tip of nano cracks emanating from a circular hole[J]. China Mechanical Engineering,2018,29(19): 2347-2352.(in Chinese)) [34] XIAO J H, XU Y L, ZHANG F C. An analytic solution for problem of two symmetrical edge cracks emanating from a circular hole with surface effect under antiplane shear[J]. Acta Mechanica,2018,229(12): 4915-4925.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142