A Semi-Analytical Study of the Interaction Between Surface Cracks and Arbitrary Inhomogeneities: Systematic Analysis of Crack Angles, Material Contrasts, and Positional Effects

Yang Bozheng; Song Hengxu; Li Pu; Jin Xiaoqing

doi:10.21656/1000-0887.460204

Volume 47 Issue 8

Aug. 2026

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2026 > 47(8): 978-989

Yang Bozheng, Song Hengxu, Li Pu, Jin Xiaoqing. A Semi-Analytical Study of the Interaction Between Surface Cracks and Arbitrary Inhomogeneities: Systematic Analysis of Crack Angles, Material Contrasts, and Positional Effects[J]. Applied Mathematics and Mechanics, 2026, 47(8): 978-989. doi: 10.21656/1000-0887.460204

Citation:

PDF( 3168 KB)

A Semi-Analytical Study of the Interaction Between Surface Cracks and Arbitrary Inhomogeneities: Systematic Analysis of Crack Angles, Material Contrasts, and Positional Effects

doi: 10.21656/1000-0887.460204

Yang Bozheng^{1, 2
,},
Song Hengxu^{1, 3
,
,},
Li Pu⁴,
Jin Xiaoqing²

1.
State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Science, Beijing 100190, P.R. China
2.
College of Aerospace Engineering, Chongqing University, Chongqing 400030, P.R. China
3.
School of Engineering Science, University of Chinese Academy of Sciences, Beijing 100049, P.R. China
4.
Key Laboratory of Advanced Spatial Mechanism and Intelligent Spacecraft(Ministry of Education), Robotic Satellite Key Laboratory of Sichuan Province, School of Aeronautics and Astronautics, Sichuan University, Chengdu 610065, P.R. China

Received Date: 2025-11-17
Rev Recd Date: 2025-12-08
Publish Date: 2026-08-01

Abstract

Abstract

A unified semi-analytical framework combining the numerical equivalent inclusion method (NEIM) and the distributed dislocation technique (DDT) was employed to investigate the interaction between surface cracks and arbitrarily shaped inhomogeneities in a half-plane. Several representative examples were simulated to systematically examine the effects of crack angles, inhomogeneity stiffnesses, and the inhomogeneity distances from the free surface on the stress intensity factors and local stress fields. The results show that, the crack angle leads to notable changes in the crack-tip field and induces pronounced mode coupling. Stiff inhomogeneities decrease Mode Ⅰ stress intensity factor but increase Mode Ⅱ, whereas soft inhomogeneities exhibit the opposite trend. As the inhomogeneity moves farther from the free surface, its perturbation to the crack-tip field becomes significantly weaker. The semi-analytical results agree well with the finite element solutions, with discrepancies below 4.5%, demonstrating the accuracy and efficiency of the present framework. The numerical examples provide clear insight into the dominant mechanisms governing crack-inhomogeneity interaction under multiple influencing parameters.
- surface crack,
- inhomogeneity,
- semi-analytical method,
- stress intensity factor

(Contributed by Song Hengxu, Member of the Youth Editorial Board of AMM)

FullText(HTML)

References(28)

References

[1]	Guan J, Wang L, Zhang C, et al. Effects of non-metallic inclusions on the crack propagation in bearing steel[J]. Tribology International, 2017, 106: 123-131. doi: 10.1016/j.triboint.2016.10.030
[2]	Tamate O. The effect of a circular inclusion on the stresses around a line crack in a sheet under tension[J]. International Journal of Fracture Mechanics, 1968, 4(3): 257-266. doi: 10.1007/BF00185261
[3]	Muskhelishvili N I. Some Basic Problems of the Mathematical Theory of Elasticity[M]. Groningen: P. Noordhoff, 1963.
[4]	Gdoutos E E. Interaction effects between a crack and a circular inclusion[J]. Fibre Science and Technology, 1981, 15(3): 173-185. doi: 10.1016/0015-0568(81)90002-6
[5]	Eshelby J D. The determination of the elastic field of an ellipsoidal inclusion, and related problems[J]. Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences, 1957, 241(1226): 376-396.
[6]	Jin X, Wang Z, Zhou Q, et al. On the solution of an elliptical inhomogeneity in plane elasticity by the equivalent inclusion method[J]. Journal of Elasticity, 2014, 114(1): 9423.
[7]	Li P, Lyu D, Soewardiman H, et al. Analytical and numerical evaluation of the interaction energy between screw dislocation and inhomogeneous inclusion[J]. Mechanics of Materials, 2021, 156: 103788. doi: 10.1016/j.mechmat.2021.103788
[8]	Jin X, Zhang X, Li P, et al. On the displacement of a two-dimensional Eshelby inclusion of elliptic cylindrical shape[J]. Journal of Applied Mechanics, 2017, 84(7): 074501. doi: 10.1115/1.4036820
[9]	Jin X, Keer L M, Wang Q. A closed-form solution for the Eshelby tensor and the elastic field outside an elliptic cylindrical inclusion[J]. Journal of Applied Mechanics, 2011, 78(3): 031009. doi: 10.1115/1.4003238
[10]	Li Z, Chen Q. Crack-inclusion interaction for mode Ⅰ crack analyzed by Eshelby equivalent inclusion method[J]. International Journal of Fracture, 2002, 118(1): 29-40. doi: 10.1023/A:1022652725943
[11]	Yang L, Chen Q, Li Z. Crack-inclusion interaction for mode Ⅱ crack analyzed by Eshelby equivalent inclusion method[J]. Engineering Fracture Mechanics, 2004, 71(9/10): 1421-1433.
[12]	Lal A, Vaghela M B, Mishra K. Numerical analysis of an edge crack isotropic plate with void/inclusions under different loading by implementing XFEM[J]. Journal of Applied Computational Mechanics, 2019, 7(3): 1362-1382.
[13]	Li R, Wu S, Ivanova E, et al. Finite element model and experimental analysis of crack-inclusion interaction[J]. Journal of Applied Polymer Science, 1993, 50(7): 1233-1238. doi: 10.1002/app.1993.070500714
[14]	Lipetzky P, Schmauder S. Crack-particle interaction in two-phase composites, part Ⅰ: particle shape effects[J]. International Journal of Fracture, 1994, 65(4): 345-358. doi: 10.1007/BF00012373
[15]	Nguyen T T, Yvonnet J, Zhu Q Z, et al. A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography[J]. Computer Methods in Applied Mechanics and Engineering, 2016, 312: 567-595. doi: 10.1016/j.cma.2015.10.007
[16]	Kumar A, Sain T. A unified thermo-viscoelastic phase-field fracture model for fiber-reinforced polymer composites[J]. Journal of the Mechanics and Physics of Solids, 2026, 206: 106378. doi: 10.1016/j.jmps.2025.106378
[17]	Bian P L, Liu Q, Zhang H, et al. Adaptive phase-field cohesive-zone model for simulation of mixed-mode interfacial and bulk fracture in heterogeneous materials with directional energy decomposition[J]. Computer Methods in Applied Mechanics and Engineering, 2025, 443: 118062. doi: 10.1016/j.cma.2025.118062
[18]	Hills D A, Kelly P, Dai D, et al. Solution of Crack Problems: the Distributed Dislocation Technique[M]. Springer Science & Business Media, 1996.
[19]	Yang B, Li P, Liu K, et al. Analysis of kinked cracks interacting with multiple inhomogeneities[J]. International Journal of Mechanical Sciences, 2025, 304: 110703. doi: 10.1016/j.ijmecsci.2025.110703
[20]	Yang B, Li P, Liu K, et al. Semi-analytical modeling of coating-crack-defect interactions using a combined distributed dislocation technique and numerical equivalent inclusion method[J]. Tribology International, 2026, 214: 111199. doi: 10.1016/j.triboint.2025.111199
[21]	Eshelby J D. The elastic field outside an ellipsoidal inclusion[J]. Proceedings of the Royal Society of London Series A: Mathematical and Physical Sciences, 1959, 252(1271): 561-569.
[22]	Zhou Q, Jin X, Wang Z, et al. Numerical implementation of the equivalent inclusion method for 2D arbitrarily shaped inhomogeneities[J]. Journal of Elasticity, 2015, 118(1): 39-61. doi: 10.1007/s10659-014-9477-2
[23]	Zhou Q, Jin X, Wang Z, et al. Numerical EIM with 3D FFT for the contact with a smooth or rough surface involving complicated and distributed inhomogeneities[J]. Tribology International, 2016, 93: 91-103. doi: 10.1016/j.triboint.2015.09.001
[24]	金晓清, 牛飞飞, 张睿, 等. 均布激励基本单元解析解的一种记号方法[J]. 上海交通大学学报, 2016, 50(8): 1221-1227. Jin Xiaoqing, Niu Feifei, Zhang Rui, et al. A notation for elementary solution to uniformly distributed excitation over a rectangular/cuboidal domain[J]. Journal of Shanghai Jiao Tong University, 2016, 50(8): 1221-1227. (in Chinese)
[25]	谢东东, 金晓清, 蒋志桢, 等. 弹性半平面矩形夹杂基本单元解及其应用[J]. 重庆大学学报, 2022, 45(12): 26-35. Xie Dongdong, Jin Xiaoqing, Jiang Zhizhen, et al. Elementary solution of the elastic half-plane containing a rectangular inclusion: theory and applications[J]. Journal of Chongqing University, 2022, 45(12): 26-35. (in Chinese)
[26]	Liu K, Li P, Yang B, et al. Cuboidal inclusion problem revisited: unified expressions for the complete elastic fields and numerical implementation based on FFT[J]. Tribology International, 2026, 213: 111099. doi: 10.1016/j.triboint.2025.111099
[27]	Liu K, Li P, Yang B, et al. A versatile three-dimensional contact analysis for heterogeneous materials[J]. International Journal of Solids and Structures, 2025, 318: 113440. doi: 10.1016/j.ijsolstr.2025.113440
[28]	Zhou K, Keer L M, Wang Q J. Semi-analytic solution for multiple interacting three-dimensional inhomogeneous inclusions of arbitrary shape in an infinite space[J]. International Journal for Numerical Methods in Engineering, 2011, 87(7): 617-638. doi: 10.1002/nme.3117