Archive of Applied Mechanics

, Volume 89, Issue 11, pp 2321–2334 | Cite as

An accurate and efficient computational method for seeking two equi-tangential stress hole shapes

  • Chonglin Yin
  • Aizhong LuEmail author
  • Xiangtai Zeng


As an infinite elastic plane containing two holes subjected to uniform remote loading, the tangential stress concentration degree around the holes is related to the hole shapes. Conformal transformation in the complex variable method is used to achieve optimal hole shapes, in which equi-tangential stress is produced at the edge of the holes. An effective method and algorithm determining equi-stress hole shapes are proposed. The key of this method is how to map the outer domain of the two holes in physical plane into an annulus in image plane. The mapping function adopted in this article has a general explicit expression, so it is different from the previous ones. Coefficients of the mapping function are design variables in the optimization algorithm of mixed penalty function (Mult-SUMT method) to find optimal holes. The approach presented has a significant computational advantage. The numerical simulations present in detail the influence of hole sizes and external loads on the shape of the optimal holes. Some of the results are verified by ANSYS numerical method.


Hole shape optimization Equi-tangential stress Mapping function Mult-SUMT method 



The study is supported by the National Natural Science Foundation of China (Grant Nos: 11572126, 51704117).


  1. 1.
    Savin, G.N.: Stress Concentrations Around Holes. Pergamon, New York (1961)zbMATHGoogle Scholar
  2. 2.
    Peterson, R.E.: Stress Concentration Design Factors. Wiley, New York (1953)Google Scholar
  3. 3.
    Bjorkman, G.S., Richards, R.: Harmonic holes—an inverse problem in elasticity. J. Appl. Mech. ASME 43, 414–418 (1976)CrossRefGoogle Scholar
  4. 4.
    Bjorkman, G.S., Richards, R.: Harmonic hole for nonconstant field. J. Appl. Mech. ASME 46, 573–576 (1979)CrossRefGoogle Scholar
  5. 5.
    Dai, M., Peter, S., Gao, C.F.: Harmonic holes with surface tension in an elastic plane under uniform remote loading. Math. Mech. Solids 22, 1806–1812 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dhir, S.K.: Optimization in a class of hole shapes in plate structures. J. Appl. Mech. ASME 48, 905–908 (1981)CrossRefGoogle Scholar
  7. 7.
    Sun, H., et al.: Complex variable function method for hole shape optimization in an elastic plane. Appl. Math. Mech. 8, 137–146 (1987)CrossRefGoogle Scholar
  8. 8.
    Lu, A.Z.: Shape optimization based on the criterion of making the absolute maximum tangential stress of the hole be the minimum. Chin. J. Solid Mech. 17, 73–76 (1996) MathSciNetGoogle Scholar
  9. 9.
    Vigdergauz, S.B.: The stress-minimizing hole in an elastic plate under remote shear. J. Mech. Mater. Struct. 1, 387–406 (2006)CrossRefGoogle Scholar
  10. 10.
    Maksymovych, O., Illiushyn, O.: Stress calculation and optimization in composite plates with holes based on the modified integral equation method. Eng. Anal. Bound. Elem. 83, 180–187 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wang, S.J., Lu, A.Z., et al.: Shape optimization of the hole in an orthotropic plate. Mech. Based Des. Struct. Mach. 46, 23–37 (2018)CrossRefGoogle Scholar
  12. 12.
    Sua, Z.X., Xie, C.H., Tang, Y.: Stress distribution analysis and optimization for composite laminate containing hole of different shapes. Aerosp. Sci. Technol. 76, 466–470 (2018)CrossRefGoogle Scholar
  13. 13.
    Cherepanov, G.: Inverse problem of the plane theory of elasticity. J. Appl. Math. Mech. 38, 913–931 (1974)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Vigdergauz, S.B.: Shape optimization in an elastic plate under remote shear: from single to interacting holes. Math. Mech. Solids 3, 1341–1363 (2008)Google Scholar
  15. 15.
    Vigdergauz, S.B.: Energy-minimizing openings around a fixed hole in an elastic plate. Math. Mech. Solids 5, 661–677 (2010)MathSciNetGoogle Scholar
  16. 16.
    Vigdergauz, S.: Equi-stress boundaries in two- and three-dimensional elastostatics: the single-layer potential approach. Math. Mech. Solids 22, 837–851 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Vigdergauz, S.: Simply and doubly periodic arrangements of the equi-stress holes in a perforated elastic plane: the single-layer potential approach. Math. Mech. Solids 23, 805–819 (2018)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Heller, M., Kaye, R., Rose, L.R.: A gradientless finite element procedure for shape optimization. J. Strain Anal. Eng. Des. 34, 323–336 (1999)CrossRefGoogle Scholar
  19. 19.
    Pedersen, P.: Design study of hole positions and hole shapes for crack tip stress releasing. Struct. Multidiscip. Optim. 28, 243–251 (2004)CrossRefGoogle Scholar
  20. 20.
    Pedersen, P.: Suggested benchmarks for shape optimization for minimum stress concentration. Struct. Multidiscip. Optim. 35, 273–283 (2008)CrossRefGoogle Scholar
  21. 21.
    Schmid, F., Hirschen, K., Meynen, S., et al.: An enhanced approach for shape optimization using an adaptive algorithm. Finite Elem. Anal. Des. 41, 521–543 (2005)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Sonmez, F.O.: Shape optimization of 2D structures using simulated annealing. Comput. Methods Appl. Mech. Eng. 196, 3279–3299 (2007)CrossRefGoogle Scholar
  23. 23.
    Miegroet, L.V., Duysinx, P.: Stress concentration minimization of 2D filets using X-FEM and level set description. Struct. Multidiscip. Optim. 33, 425–438 (2007)CrossRefGoogle Scholar
  24. 24.
    Wu, Z.X.: Optimal hole shape for minimum stress concentration using parameterized geometry models. Struct. Multidiscip. Optim. 37, 625–634 (2009)CrossRefGoogle Scholar
  25. 25.
    Lu, A.Z., Xu, Z., Zhang, N.: Stress analytical solution for an infinite plane containing two holes. Int. J. Mech. Sci. 128, 224–234 (2017)CrossRefGoogle Scholar
  26. 26.
    Zeng, X.T., Lu, A.Z., Zhang, N.: Analytical stress solution for an infinite plate containing two oval holes. Eur. J. Mech. A/Solids 67, 291–304 (2018)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Zeng, X.T., Lu, A.Z.: Stress solution for an infinite elastic plate containing two arbitrarily shaped holes. AIAA J. 57, 1691–1701 (2019)CrossRefGoogle Scholar
  28. 28.
    Korn, G.A., Korn, T.M.: Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York (1986)zbMATHGoogle Scholar
  29. 29.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1963)zbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Institute of Hydroelectric and Geotechnical EngineeringNorth China Electric Power UniversityBeijingPeople’s Republic of China

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