Analysis of different damage configurations at a macro-crack tip

  • Xu Li
  • Xiaotao Li
  • Shihao Cao
  • Xiaoyu JiangEmail author


The solution of an infinite elastic plane containing a macro-crack and a cluster of micro-cracks is presented based on Muskhelishvili’s complex potential method. A step-by-step sub-problem procedure is used to satisfy the stress boundary conditions on each crack surface. A local damage parameter expressed by stress intensity factor is presented to evaluate the damage magnitude. Three damage configurations as parallel, radiating and arbitrarily oriented micro-cracks have been designed to simulate the damage around the macro-crack tip. The results show the micro-cracks located in front of the macro-crack tip have the amplifying effect on macro-crack growth, and the amplifying effect increases with the number of micro-cracks increase. Moreover, when the macro-crack length decreases or the distance between micro-cracks and macro-crack increases, the damage parameter of macro-crack and micro-cracks will decrease. It is found that the largest damage is the damage configuration of arbitrarily oriented micro-cracks among three damage configurations.


Complex potentials Macro-crack Micro-cracks Damage parameter Damage configuration 



The work was supported by the National Natural Science Foundation of China (11472230) and the National Natural Science Foundation of China Key Project (U1134202/E050303).


  1. 1.
    Bai, Y.L., Wang, H.Y., Xia, M.F., Ke, F.J.: Statistical mesomechanics of solid, linking coupled multiple space and time scales. Adv. Mech. 58(6), 286–305 (2006)Google Scholar
  2. 2.
    Gao, Y., Zhang, C., Xiong, X., Zheng, Z., Zhu, M.: Intergranular corrosion susceptibility of a novel Super304H stainless steel. Eng. Fail. Anal. 24(9), 26–32 (2012)CrossRefGoogle Scholar
  3. 3.
    Hwang, K.C., Lee, S., Hui, C.L.: Effects of alloying elements on microstructure and fracture properties of cast high speed steel rolls: part II. Fracture behavior. Mater. Sci. Eng. A 254(1–2), 296–304 (1998)CrossRefGoogle Scholar
  4. 4.
    Ronevich, J.A., Somerday, B.P., Marchi, C.W.S.: Effects of microstructure banding on hydrogen assisted fatigue crack growth in X65 pipeline steels. Int. J. Fatigue 82, 497–504 (2016)CrossRefGoogle Scholar
  5. 5.
    Bui, H.D.: Fracture Mechanics: Inverse Problems and Solutions. Springer, Berlin (2007)Google Scholar
  6. 6.
    Stepanova, L.: Eigenspectra and orders of stress singularity at a mode I crack tip for a power-law medium. C. R. Méc. 336(1), 232–237 (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Stepanova, L.V.: Eigenvalues of the antiplane-shear crack problem for a power-law material. J. Appl. Mech. Tech. Phys. 49(1), 142–147 (2008)CrossRefzbMATHGoogle Scholar
  8. 8.
    Murakami, S.: Continuum damage mechanics: a continuum mechanics approach to the analysis of damage and fracture. Springer Ebooks 41(B4), 4731–4755 (2012)Google Scholar
  9. 9.
    Stepanova, L.V., Adylina, E.M.: Stress–strain state in the vicinity of a crack tip under mixed loading. J. Appl. Mech. Tech. Phys. 55(5), 885–895 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gong, S.X., Horii, H.: General solution to the problem of microcracks near the tip of a main crack. J. Mech. Phys. Solids 37(1), 27–46 (1989)CrossRefzbMATHGoogle Scholar
  11. 11.
    Gong, S.X., Meguid, S.A.: A general solution to the antiplane problem of an arbitrarily located elliptical hole near the tip of a main crack. Int. J. Solids Struct. 28(2), 249–263 (1991)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gong, S.X., Meguid, S.A.: Microdefect interacting with a main crack: a general treatment. Int. J. Mech. Sci. 34(12), 933–945 (1992)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gross, D.: Stress intensity factors of systems of cracks. Ingenicur-Archiv 51(5), 301–310 (1982)CrossRefGoogle Scholar
  14. 14.
    Stepanova, L., Roslyakov, P.: Complete Williams asymptotic expansion near the crack tips of collinear cracks of equal lengths in an infinite plane medium. Solid State Phenom. 258, 209–212 (2016)CrossRefGoogle Scholar
  15. 15.
    Stepanova, L., Roslyakov, P.: Multi-parameter description of the crack-tip stress field: analytic determination of coefficients of crack-tip stress expansions in the vicinity of the crack tips of two finite cracks in an infinite plane medium. Int. J. Solids Struct. 100–101, 11–28 (2016)CrossRefGoogle Scholar
  16. 16.
    Hello, G., Tahar, M.B., Roelandt, J.M.: Analytical determination of coefficients in crack-tip stress expansions for a finite crack in an infinite plane medium. Int. J. Solids Struct. 49(3), 556–566 (2012)CrossRefGoogle Scholar
  17. 17.
    Chudnovsky, A., Kachanov, M.: Interaction of a crack with a field of microcracks. Int. J. Eng. Sci. 21(8), 1009–1018 (1983)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kachanov, M.: A simple technique of stress analysis in elastic solids with many cracks. Int. J. Fract. 28(1), R11–R19 (1985)Google Scholar
  19. 19.
    Kachanov, M.: Elastic solids with many cracks: a simple method of analysis. Int. J. Solids Struct. 23(1), 23–43 (1987)CrossRefzbMATHGoogle Scholar
  20. 20.
    Tamuzs, V., Romalis, N., Petrova, V.: Fracture of Solids with Microdefects. Nova Science Publishers, New York (2000)Google Scholar
  21. 21.
    Petrova, V., Schmauder, S.: Thermal fracture of a functionally graded/homogeneous bimaterial with system of cracks. Theor. Appl. Fract. Mech. 55(2), 148–157 (2011)CrossRefGoogle Scholar
  22. 22.
    Petrova, V., Schmauder, S.: Mathematical modelling and thermal stress intensity factors evaluation for an interface crack in the presence of a system of cracks in functionally graded/homogeneous bimaterials. Comput. Mater. Sci. 52(1), 171–177 (2012)CrossRefGoogle Scholar
  23. 23.
    Petrova, V., Tamuzs, V., Romalis, N.: A survey of macro–microcrack interaction problems. Appl. Mech. Rev. 53(5), 117–146 (2000)CrossRefGoogle Scholar
  24. 24.
    Wang, H., Liu, Z., Xu, D., Zeng, Q., Zhuang, Z.: Extended finite element method analysis for shielding and amplification effect of a main crack interacted with a group of nearby parallel microcracks. Int. J. Damage Mech. 25(1), 4–25 (2014)CrossRefGoogle Scholar
  25. 25.
    Loehnert, S., Belytschko, T.: Crack shielding and amplification due to multiple microcracks interacting with a macrocrack. Int. J. Fract. 145(1), 1–8 (2007)CrossRefzbMATHGoogle Scholar
  26. 26.
    Loehnert, S., Belytschko, T.: A multiscale projection method for macro/microcrack simulations. Int. J. Numer. Methods Eng. 71(12), 1466–1482 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ouinas, D., Bouiadjra, B.B., Benderdouche, N., Saadi, B.A., Vina, J.: Numerical modelling of the interaction macro-multimicrocracks in a plate under tensile stress. J. Comput. Sci. 2(2), 153–164 (2011)CrossRefGoogle Scholar
  28. 28.
    Soh, A.K., Yang, C.H.: Numerical modeling of interactions between a macro-crack and a cluster of micro-defects. Eng. Fract. Mech. 71(2), 193–217 (2004)CrossRefGoogle Scholar
  29. 29.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Dordrecht (2009)Google Scholar
  30. 30.
    Tanaka, K.: Fatigue crack propagation from a crack inclined to the cyclic tensile axis. Eng. Fract. Mech. 6(3), 493–507 (1974)CrossRefGoogle Scholar
  31. 31.
    Murakami, Y.E.: The stress intensity factors handbook. J. Appl. Mech. 1(4), 1063 (1987)CrossRefGoogle Scholar
  32. 32.
    Kachanov, M.: Elastic solids with many cracks and related problems. Adv. Appl. Mech. 30(114), 259–445 (1993)CrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and EngineeringSouthwest Jiaotong UniversityChengduPeople’s Republic of China
  2. 2.School of Civil EngineeringSouthwest Jiaotong UniversityChengduPeople’s Republic of China

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