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A modified Laurent series for hole/inclusion problems in plane elasticity

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Abstract

We propose a modified Laurent series for investigating the elastic field around holes/inclusions under plane deformation by dividing the classical Laurent series over the first derivative of the conformal mapping, which is justified from a mathematical point of view. Through a group of numerical examples concerning the stress concentration around holes of common shapes, the accuracy of the modified Laurent series is verified. It is also demonstrated that the use of the modified Laurent series as compared with the classical one generally leads to significantly more accurate results with even much fewer terms of the series. Utilizing the modified Laurent series, we revisit the stress field around an irregularly shaped elastic inclusion embedded in an infinite matrix and effectively eliminate the severe interfacial stress fluctuation reported earlier in the literature. More importantly, the modified Laurent series may be very useful for a highly accurate prediction of the physical field in particulate composites with densely packed inclusions.

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References

  1. Savin, G.N.: Stress Concentration Around Holes. Pergamon, London (1961)

    MATH  Google Scholar 

  2. Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Springer, Dordrecht (1977)

    Book  Google Scholar 

  3. Kang, H., Kim, E., Milton, G.W.: Inclusion pairs satisfying Eshelby’s uniformity property. SIAM J. Appl. Math. 69(2), 577–595 (2008)

    Article  MathSciNet  Google Scholar 

  4. Wang, X.: Uniform fields inside two non-elliptical inclusions. Math. Mech. Solids 17(7), 736–761 (2012)

    Article  MathSciNet  Google Scholar 

  5. Dai, M., Gao, C.F., Ru, C.Q.: Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation. Proc. R. Soc. A 471(2177), 20140933 (2015)

    Article  Google Scholar 

  6. Dai, M., Schiavone, P.: Need the uniform stress field inside multiple interacting inclusions be hydrostatic? J. Elast. 143, 195–207 (2021)

    Article  MathSciNet  Google Scholar 

  7. Gao, C.F., Fan, W.X.: Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack. Int. J. Solids Struct. 36(17), 2527–2540 (1999)

    Article  MathSciNet  Google Scholar 

  8. Gao, C.F., Zhang, T.Y.: Fracture behaviors of piezoelectric materials. Theor. Appl. Fract. Mech. 41(1–3), 339–379 (2004)

    Google Scholar 

  9. Li, M., Gao, C.F.: Electro-elastic fields in an elliptic piezoelectric plane with an elliptic hole or a crack of arbitrary location. Meccanica 53(1), 347–357 (2018)

    Article  MathSciNet  Google Scholar 

  10. Wang, G.F., Wang, T.J.: Deformation around a nanosized elliptical hole with surface effect. Appl. Phys. Lett. 89(16), 161901 (2006)

    Article  Google Scholar 

  11. Wang, S., Chen, Z.T., Gao, C.F.: Analytic solution for a circular nano-inhomogeneity in a finite matrix. Nano Mater. Sci. 1(2), 116–120 (2019)

    Article  Google Scholar 

  12. Dai, M., Yang, H.B., Schiavone, P.: Stress concentration around an elliptical hole with surface tension based on the original Gurtin–Murdoch model. Mech. Mater. 135, 144–148 (2019)

    Article  Google Scholar 

  13. Yu, C.B., Wang, S., Gao, C.F., Chen, Z.T.: Thermal stress analysis of current-carrying media containing an inclusion with arbitrarily-given shape. Appl. Math. Modell. 79, 753–767 (2020)

    Article  MathSciNet  Google Scholar 

  14. Song, K., Schiavone, P.: Thermal conduction around a circular nanoinhomogeneity. Int. J. Heat Mass Trans 150, 119297 (2020)

    Article  Google Scholar 

  15. Yang, H.B., Yu, C.B., Tang, J.Y., Qiu, J., Zhang, X.Q.: Electric-current-induced thermal stress around a non-circular rigid inclusion in a two-dimensional nonlinear thermoelectric material. Acta Mech. 231(11), 4603–4619 (2020)

    Article  MathSciNet  Google Scholar 

  16. Style, R.W., Wettlaufer, J.S., Dufresne, E.R.: Surface tension and the mechanics of liquid inclusions in compliant solids. Soft Matter. 11(4), 672–679 (2015)

    Article  Google Scholar 

  17. Mancarella, F., Style, R.W., Wettlaufer, J.S.: Surface tension and the Mori–Tanaka theory of non-dilute soft composite solids. Proc. R. Soc. A 472(2189), 20150853 (2016)

    Article  MathSciNet  Google Scholar 

  18. Wu, J., Ru, C.Q., Zhang, L.: An elliptical liquid inclusion in an infinite elastic plane. Proc. R. Soc. A 474(2215), 20170813 (2018)

    Article  MathSciNet  Google Scholar 

  19. Dai, M., Li, M., Schiavone, P.: Plane deformations of an inhomogeneity-matrix system incorporating a compressible liquid inhomogeneity and complete Gurtin-Murdoch interface model. J. Appl. Mech. 85(12), 12101085 (2018)

    Article  Google Scholar 

  20. Dai, M., Hua, J., Schiavone, P.: Compressible liquid/gas inclusion with high initial pressure in plane deformation: modified boundary conditions and related analytical solutions. Eur. J. Mech. A Solids 82, 104000 (2020)

    Article  MathSciNet  Google Scholar 

  21. Dai, M., Schiavone, P.: Deformation-induced change in the geometry of a general material surface and its relation to the Gurtin–Murdoch model. J. Appl. Mech. 87(6), 061005 (2020)

    Article  Google Scholar 

  22. Luo, J.C., Gao, C.F.: Faber series method for plane problems of an arbitrarily shaped inclusion. Acta Mech. 208(3), 133–145 (2009)

    Article  MathSciNet  Google Scholar 

  23. Luo, J.C., Gao, C.F.: Stress field of a coated arbitrary shape inclusion. Meccanica 46(5), 1055–1071 (2011)

    Article  MathSciNet  Google Scholar 

  24. Chen, D., Nisitani, H.: Singular stress field near the corner of jointed dissimilar materials. J. Appl. Mech. 60(3), 607–613 (1993)

    Article  Google Scholar 

  25. Dai, M., Sun, H.Y.: Thermo-elastic analysis of a finite plate containing multiple elliptical inclusions. Int. J. Mech. Sci. 75, 337–344 (2013)

    Article  Google Scholar 

  26. Faber, G.: Über polynomische Entwickelungen. Math. Ann. 57(3), 389–408 (1903)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Li and Cai acknowledge the partial supports from the National Natural Science Foundation of China (Grant No. 52005256), the Natural Science Foundation of Jiangsu Province (Grant No. BK20190394), the Jiangsu Post-doctoral Research Funding Program (Grant No. 2020Z437) and the Priority Academic Program Development of Jiangsu Higher Education Institutions. Huang appreciates the support from the National Natural Science Foundation of China (Grant No. 11802040) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 18KJB130001). Wang thanks the support from the National Natural Science Foundation of China (Grant No. 12002004).

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Author notes

  1. Chao Li and Cheng Huang have contributed equally to this work.

Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Chao Li & Deng’an Cai

  2. Nanjing Fiberglass Research and Design Institute, Nanjing, 210012, China

    Chao Li

  3. School of Aviation and Mechanical Engineering, Changzhou Institute of Technology, Changzhou, 213032, China

    Cheng Huang

  4. State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, BIC-ESAT, College of Engineering, Peking University, Beijing, 100871, China

    Shuang Wang

Authors
  1. Chao Li
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  2. Cheng Huang
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  3. Shuang Wang
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  4. Deng’an Cai
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Correspondence to Shuang Wang or Deng’an Cai.

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Li, C., Huang, C., Wang, S. et al. A modified Laurent series for hole/inclusion problems in plane elasticity. Z. Angew. Math. Phys. 72, 124 (2021). https://doi.org/10.1007/s00033-021-01552-4

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  • DOI: https://doi.org/10.1007/s00033-021-01552-4

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