DUAN Shujin, FUJII Koju, NAKAGAWA Kenji. Construction of General Analytic Functions With Finite Stress Concentration for Mono-Material Cracks and Bi-Material Interface Cracks[J]. Applied Mathematics and Mechanics, 2018, 39(12): 1364-1376. doi: 10.21656/1000-0887.390030
 Citation: DUAN Shujin, FUJII Koju, NAKAGAWA Kenji. Construction of General Analytic Functions With Finite Stress Concentration for Mono-Material Cracks and Bi-Material Interface Cracks[J]. Applied Mathematics and Mechanics, 2018, 39(12): 1364-1376.

# Construction of General Analytic Functions With Finite Stress Concentration for Mono-Material Cracks and Bi-Material Interface Cracks

##### doi: 10.21656/1000-0887.390030
• Rev Recd Date: 2018-02-03
• Publish Date: 2018-12-01
• The constructing methods for finite stress concentration analysis near the crack tip were summarized. The stress functions for plane problems with cracks were expressed with irrational or exponential functions. For the mono-material crack, with the crack length as the parameter, the direct weighted integration of the irrational-function-type analytic function was conducted to avoid stress singularity at the crack tip, and construct the finite stress concentration functions and the wedge-type opening displacement functions. The indirect integration of the exponential-function-type analytic function was suitable for the interface crack problem, but put the stress distribution within the integral interval into positive-negative inversion and irrational opening displacement shape, which can be improved through combining selection and superposition of different weight functions. The basic solutions for the central cracks and the symmetrical edge cracks were given in 6 stress states of plane stretching, shearing and bending, etc. The reason why the analytic function can avoid the stress singularity at the crack tip was given.
•  [1] WESTERGAARD H M. Bearing pressures and cracks[J]. Journal of Applied Mechanics,1939,6: 49-53. [2] DUGDALE D S. Yielding of steel sheets containing slits[J]. Journal of the Mechanics & Physics of Solids,1960,8(2): 100-104. [3] BARENBLATT G I. The formation of equilibrium cracks during brittle fracture, general ideas and hypotheses, axially-symmetric cracks[J]. Journal of Applied Mathematics & Mechanics,1959,23(3): 434-444. [4] FERDJANI H, ABDELMOULA R. Propagation of a Dugdale crack at the edge of a half plate[J]. Continuum Mechanics & Thermodynamics,2018,30(8): 1-11. [5] TONG D, WU X R. Analysis of crack opening stresses for center- and edge-crack tension specimens[J]. Chinese Journal of Aeronautics,2014,27(2): 291-298. [6] 段树金, 儿岛弘行, 中川建治. 亀裂先端部分で有限な应力集中を与える应力関数[J]. 土木学会論文集, 1986,374: 399-407.(DUAN S J, KOJIAMA H, NAKAGAWA K. Stress functions with finite magnitude of stress concentration at crack tip[J]. Proc JSCE,1986,374: 399-407.(in Japanese)) [7] 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.(WANG Chengqiang, ZHENG Changliang. Semi-analytical finite element method for linear cohesive force model in crack propagation[J]. Chinese Journal of Computational Mechanics,2006,23(2): 146-151.(in Chinese)) [8] WU X, ANTHONY M W. Multiple solutions in cohesive zone models of fracture[J]. Engineering Fracture Mechanics,2017,177: 104-122. [9] HILLERBORG A. Analysis of fracture by means of the fictitious crack model, particularly for fiber reinforced concrete[J]. Endocrine Related Cancer,1980,2(4): 177-184. [10] BAZANT Z P, OHB H. Crack band theory for fracture of concrete[J]. Matériaux et Construction,1983,16(3): 155-177. [11] ELICES M, GUINCA G V, GMEZ I, et al. The cohesive zone model: advantages, limitations and challenges[J]. Engineering Fracture Mechanics,2002,69(2): 137-163. [12] REDDY K C, SUBRAMANIAMK V L. Analysis for multi-linear stress-crack opening cohesiverelationship: application to macro-synthetic fiber reinforcedconcrete[J]. Engineering Fracture Mechanics,2017,169: 128-145. [13] ISSA S, ELIASSON S, LUNDBERG A, et al. Cohesive zone modeling of crack propagation influenced by martensiticphase transformation[J]. Materials Science & Engineering A,2018,712: 564-573. [14] DO B C, LIU W, YANG Q D, et al. Improved cohesive stress integration schemes for cohesive zoneelements[J]. Engineering Fracture Mechanics,2013,107(7): 14-28. [15] TRYDING J, RISTINMAA M. Normalization of cohesive laws for quasi-brittle materials[J]. Engineering Fracture Mechanics,2017,178: 333-345. [16] OTTOSEN N S, RISTINMAA M. Thermodynamically based fictitious crack/interface model for generalnormal and shear loading[J]. International Journal of Solids and Structures,2013,50(22/23): 3555-3561. [17] 管俊峰, 卿龙邦, 赵顺波. 混凝土三点弯曲梁裂缝断裂全过程数值模拟研究[J]. 计算力学学报, 2013,30(1): 143-148.(GUAN Junfeng, QING Longbang, ZHAO Shunbo. Research on numerical simulation on the whole cracking processes of three-point bending notch concrete beams[J]. Chinese Journal of Computational Mechanics,2013,30(1): 143-148.(in Chinese)) [18] 张鹏, 胡小飞, 姚伟岸. 内聚力模型裂纹问题分析的解析奇异单元[J]. 固体力学学报, 2017,38(2): 157-164.(ZHANG Peng, HU Xiaofei, YAO Weian. An analytical singular element to study the cohesive zone model for cracks[J]. Chinese Journal of Solid Mechanics,2017,38(2): 157-164.( in Chinese)) [19] 黄才政, 郑丹. 无限大平板黏聚裂纹模型的数值解[J]. 水利学报, 2014,45(S1): 137-142.(HUANG Caizheng, ZHENG Dan. Numerical solution of the cohesive crack model in finite plate[J]. Journal of Hydraulic Engineering,2014,45(S1): 137-142.(in Chinese)) [20] DUAN S J, NAKAGAWA K. Stress functions with finite stress concentration at the crack tips for a central cracked panel[J]. Engineering Fracture Mechanics,1988,29(5): 517-526. [21] DUAN S J, FUJII K, NAKAGAWA K. Finite stress concentrations and J-integrals from normal loads on a penny-shaped crack[J]. Engineering Fracture Mechanics,1989,32(2): 167-176. [22] DUAN S J, NAKAGAWA K, SAKAIDA T. A mathematical model to approach the fracture process of overconsolidatedclay[J]. Engineering Fracture Mechanics,1991,38(6): 361-369. [23] FUJII K, DUAN S J, NAKAGAWA K. A mathematical model for fracture process of four point bending concrete beam[J]. Engineering Fracture Mechanics,1991,40(1): 37-41. [24] 段树金, 中川建治. せん断を受ける円盤状亀裂の周边で有限な应力集中を与える弹性解について[J]. 土質工学論文報告集, 1988,28(1): 153-160.(DUAN S J, NAKAGAWA K. Finite stress concentrations around a penny-shaped crack in three-dimensional body under shear loading[J]. Proc JSSC,1988,28(1): 153-160.(in Japanese)) [25] 段树金, 中川建治. 含有直线状裂纹正交异性板的平面问题的应力函数[J]. 应用数学和力学, 1988,9(6): 491-498.(DUAN Shujin, NAKAGAWA K. Stress functions for central straight cracked anisotropic plates[J]. Applied Mathematics and Mechanics,1988,9(6): 491-498.(in Chinese)) [26] 胡启平. Ⅲ型静态与运动Griffith裂纹的有限应力集中解[J]. 工程力学, 1995,12(S1): 474-478.(HU Qiping. Finite stress concentration solutions of mode-Ⅲ static and moving Griffith cracks[J]. Engineering Mechanics,1995,12(S1): 474-478.(in Chinese)) [27] ZHU M, CHANG W V. An unsymmetrical fracture process zone model and its application to the problem of radical crack with an inclusion in longitudinal shear deformation[C]// Fracture Mechanics of Concrete Structures, Proceedings FRAMCOS-3.Freiburg, Germany, 1997. [28] 段树金, 前田春和, 藤井康寿, 等. 沿直线有多条裂纹的薄板弯曲问题[J]. 工程力学, 1999,16(3): 21-29.(DUAN Shujin, MAEDA H, FUJII K, et al. On the bending of an elastic plate containing multi-cracks[J]. Engineering Mechanics,1999,16(3): 21-29.(in Chinese)) [29] 栖原秀郎, 中川建治. 亀裂先端で有限な応力集中を構成する応力関数に関する研究[J]. 土木学会論文集, 1994,501: 65-74.(SUHARA H, NAKAGAWA K. Studies on the stress function with finite stress concentration at crack tip[J]. Proc JSCE,1994,501: 65-74.(in Japanese)) [30] 中川建治, 国富康志, 藤井康寿. クラック先端で有限で滑らかな応力集中を与える応力関数の構成法[J]. 名城大学理工学部研究報告, 2002,42: 62-69.(NAKAGAWA K, KUNITOMI Y, FUJII K. Introductions of stress functions with finite magnitude of stress concentrations at the crack tips in a infinite plate[J]. Research Reports of the Faculty of Science and Technology Meijou Univ,2002,42: 62-69.(in Japanese)) [31] 段树金, 张彦龙, 安蕊梅. 基于裂纹尖端二阶弹性解的断裂过程区尺度[J]. 应用数学和力学, 2013,34(6): 598-605.(DUAN Shujin, ZHANG Yanlong, AN Ruimei. Fracture process zone size based on the secondary elastic crack tip stress solution[J]. Applied Mathematics and Mechanics,2013,34(6): 598-605.(in Chinese)) [32] AN Ruimei, DUAN Shujin, GUO Quanmin. A new method to determine tensile strain softening curve of quasi-brittle materials[C]// Sustainable Solutions in Structural Engineering and Construction, Proc of ASEA-SEC-2.Bangkok, Thailand, 2014: 177-182. [33] DUAN S J, YAZAKI H, NAKAGAWA K. A crack at the interface of an elastic half plane and a rigid body[J]. Engineering Fracture Mechanics,1989,32(4): 573-580. [34] FUJII K, KATO Y, DUAN S J, et al. Stress analysis around a partially bonded rigid cylinder in an elastic medium with process zones[J]. Engineering Fracture Mechanics,1993,45(1): 31-38. [35] FUJII K, NAKAGAWA K, DUAN S, et al. Stress function with finite magnitude of stress concentration around an interface crack[J]. Engineering Fracture Mechanics,1994,47(6): 881-891. [36] 藤井康寿, 中川建治. 面内引張りを受ける境界面き裂問題の応力関数[J]. 土木学会論文集, 1994,502: 23-32.(FUJII K, NAKAGAWA K. Stress function around an interface crack under uniform tension at infinity[J]. Proc of JSCE,1994,502: 23-32.(in Japanese)) [37] MURASE Y, DUAN S J, NAKAGAWA K. Stress analysis around a circular interface crack between dissimilar media loaded by uniform tension at infinity[J]. Engineering Fracture Mechanics,1994,48(3): 325-337. [38] MURASE Y, NAKAGAWA K, DUAN S. Introduction of stress functions around a circular interface crack between dissimilar materials[J]. Engineering Fracture Mechanics,1996,53(4): 661-673. [39] 国富康志, 藤井康寿, 中川建治. 面内力問題における等方性き裂と異材界面き裂に対して適用可能な応力関数[J]. 材料,2005,54(9): 946-951.(KUNITOMI Y, FUJII K, NAKAGAWA K. Some applicable stress functions for an isotropic crack and an interface crack in in-plane problems[J]. J Soc Mat Japan,2005,54(9): 946-951.(in Japanese)) [40] 中川建治, 藤井康寿. 異材界面き裂問題の解析法に関する基礎的な研究[J]. 名城大学理工学部研究報告, 2008,48: 32-35.(NAKAGAWA K, FUJII K. A Study of analytical method of solution of interface crack problems in an infinite plate[J]. Research Reports of the Faculty of Science and Technology Meijou University,2008,48: 32-35.(in Japanese)) [41] ERDOGAN F. Stress distribution in a nonhomogeneous elastic plane with cracks[J]. Trans ASME Series E, Journal of Applied Mechanics,1963,30(2): 232-236. [42] ENGLAND A H. A crack between dissimilar media[J]. ASME Journal of Applied Mechanics,1965,32(2): 400-402. [43] MONFARED M M, AYATOLLAHI M, BAGHERI R. In-plane stress analysis of dissimilar materials with multiple interface cracks[J]. Applied Mathematical Modelling,2016,40: 8464-8474. [44] COMNINOU M. The interface crack[J]. Transaction of the ASME Journal of Applied Mechanics,1977,44(4): 631-636. [45] 安蕊梅, 段树金. 低体积含量的钢纤维混凝土三折线拉应变软化曲线的确定[J]. 石家庄铁道大学学报(自然科学版), 2017,30(1): 14-18.(AN Ruimei, DUAN Shujin. Determination of trilinear tension softening curve of reinforced concrete with steel fiber in low volume-ratio[J]. Journal of Shijiazhuang Tiedao University(Natural Science Edition),2017,30(1): 14-18.(in Chinese))

### 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