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Stress analysis of anti-plane finite elastic solids with hole by the method of fundamental solutions using conformal mapping technique

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Abstract

Stress fields of anti-plane finite elastic solids containing hole subjected to the traction or displacement boundaries are studied by using the method of fundamental solutions (MFS) in this paper. The conformal mapping technique is applied to achieve the robustness in computation for the anti-plane elastic problem. The performances of the MFS utilizing the direct MFS method and the MFS based on the conformal mapping technique are studied in solving multi-connected anti-plane elastic problem. Based on the complex analysis, the approximate solution of the complex analytic function is derived for the MFS. To avoid the derivatives on the traction boundaries, a modified boundary condition utilizing the value of the analytic function is given to construct the interpolation equations. Furthermore, the interpolation equations are given in the mapped plane by the conformal mapping technique. The accuracy of the solutions of stress with the MFS is compared between the direct MFS method and the MFS based on the conformal mapping technique in three numerical examples of the traction boundary and the mixed boundary conditions. It is illustrated that stress fields obtained by the MFS based on the conformal mapping technique can achieve good accuracy for the multi-connected anti-plane problems, whereas the direct MFS can not. The proposed method is an expansion of the traditional MFS in solving the elastic problems and can be applied for the heat transfer and the electrostatics problems with the simple concept and the easy numerical implementation.

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References

  1. Chen, W., Fu, Z.J., Chen, C.S.: Recent Advances in Radial Basis Function Collocation Methods. Springer, New York (2013)

    Google Scholar 

  2. Alves, C., Karageorghis, A., Leitão, V., et al.: Advances in Trefftz methods and their applications. SEMA SIMAI Springer Series 23. Springer, Berlin (2020)

  3. Cruse, T.: Recent advances in boundary element analysis method. Comput. Methods Appl. Mech. Eng. 62(3), 227–244 (1987)

    Article  Google Scholar 

  4. Mukhtar, F.: Relative performance of three mesh-reduction methods in predicting mode III crack-tip singularity. Latin Am. J. Solids Struct. 14(7), 1226–1250 (2017)

    Article  Google Scholar 

  5. Aleksidze, M.A.: On approximate solutions of a certain mixed boundary value problem in the theory of harmonic functions. Differ. Equ. 2(2), 515–518 (1966)

    MathSciNet  MATH  Google Scholar 

  6. Fairweather, G., Karageorghis, A.: The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9(1–2), 69–95 (1998)

    Article  MathSciNet  Google Scholar 

  7. Karageorghis, A., Fairweather, G.: Simple layer potential method of fundamental solutions for certain biharmonic problems. Int. J. Numer. Methods Fluids 9(10), 1221–1234 (1989)

    Article  MathSciNet  Google Scholar 

  8. Karageorghis, A., Fairweather, G.: The method of fundamental solutions for the numerical solution of the biharmonic equation. J. Comput. Phys. 69(2), 434–459 (1987)

    Article  MathSciNet  Google Scholar 

  9. Cheng, A.H.D., Hong, Y.: An overview of the method of fundamental solutions—solvability, uniqueness, convergence, and stability. Eng. Anal. Bound. Elem. 120(5), 118–152 (2020)

    Article  MathSciNet  Google Scholar 

  10. Zhang, L.P., Li, Z., Chen, Z., et al.: The Laplace equation in three dimensions by the method of fundamental solutions and the method of particular solutions. Appl. Numer. Math. 154(1), 47–69 (2020)

    Article  MathSciNet  Google Scholar 

  11. Li, M., Chen, C.S., Karageorghis, A.: The MFS for the solution of harmonic boundary value problems with non-harmonic boundary conditions. Comput. Math. Appl. 66(11), 2400–2424 (2013)

    Article  MathSciNet  Google Scholar 

  12. Lin, J., Chen, C.S., Liu, C.S.: Fast solution of three-dimensional modified Helmholtz equations by the method of fundamental solutions. Commun. Comput. Phys. 20(2), 512–533 (2016)

    Article  MathSciNet  Google Scholar 

  13. Fan, C.M., Huang, Y., Chen, C., et al.: Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations. Eng. Anal. Bound. Elem. 101, 188–197 (2019)

    Article  MathSciNet  Google Scholar 

  14. Karageorghis, A., Lesnic, D., Marin, L.: A survey of applications of the MFS to inverse problems. Inverse Probl. Sci. Eng. 19(3), 309–336 (2011)

    Article  MathSciNet  Google Scholar 

  15. Karageorghis, A., Lesnic, D., Marin, L.: The MFS for inverse geometric inverse problems. Inverse Problems and Computational Mechanics, Chapter: 8. Vol. 1. The Publishing House of the Romanian Academy (2011)

  16. Karageorghis, A., Lesnic, D., Marin, L.: The method of fundamental solutions for an inverse boundary value problem in static thermo-elasticity. Comput. Struct. 135, 32–39 (2014)

    Article  Google Scholar 

  17. Alves, C.J.S., Antunes, P.R.S.: Determination of elastic resonance frequencies and eigenmodes using the method of fundamental solutions. Eng. Anal. Bound. Elem. 101, 330–342 (2019)

    Article  MathSciNet  Google Scholar 

  18. Alves, C.J.S., Martins, N.F.M., Valtchev, S.S.: Extending the method of fundamental solutions to non-homogeneous elastic wave problems. Appl. Numer. Math. 115, 299–313 (2017)

    Article  MathSciNet  Google Scholar 

  19. Askour, O., Mesmoudi, S., Tri, A., et al.: Method of fundamental solutions and a high order continuation for bifurcation analysis within Föppl-von Karman plate theory. Eng. Anal. Bound. Elem. 120, 67–72 (2020)

    Article  MathSciNet  Google Scholar 

  20. Askour, O., Tri, A., Braikat, B., et al.: Method of fundamental solutions and high order algorithm to solve nonlinear elastic problems. Eng. Anal. Bound. Elem. 89, 25–35 (2018)

    Article  MathSciNet  Google Scholar 

  21. Buryachenko, V.A.: Method of fundamental solutions in micromechanics of elastic random structure composites. Int. J. Solids Struct. 124, 135–150 (2017)

    Article  Google Scholar 

  22. Buchukuri, T., Chkadua, O., Natroshvili, D.: Method of fundamental solutions for mixed and crack type problems in the classical theory of elasticity. Transactions of A. Razmadze Math. Inst. 171(3), 264–292 (2017)

    Article  MathSciNet  Google Scholar 

  23. Guimaraes, S., Telles, J.C.F.: The method of fundamental solutions for fracture mechanics—Reissner’s plate application. Eng. Anal. Bound. Elem. 33(10), 1152–1160 (2009)

    Article  MathSciNet  Google Scholar 

  24. Ma, J., Chen, W., Zhang, C., et al.: Meshless simulation of anti-plane crack problems by the method of fundamental solutions using the crack Green’s function. Comput. Math. Appl. 79(5), 1543–1560 (2019)

    Article  MathSciNet  Google Scholar 

  25. Karageorghis, A., Poullikkas, A., Berger, J.: Stress intensity factor computation using the method of fundamental solutions. Comput. Mech. 37(5), 445–454 (2006)

    Article  Google Scholar 

  26. Berger, J., Karageorghis, A., Martin, P.: Stress intensity factor computation using the method of fundamental solutions: Mixed-mode problems. Int. J. Numer. Meth. Eng. 69(3), 469–483 (2007)

    Article  MathSciNet  Google Scholar 

  27. Liu, Q.G., Šarler, B.: Method of fundamental solutions without fictitious boundary for three dimensional elasticity problems based on force-balance desingularization. Eng. Anal. Bound. Elem. 108, 244–253 (2019)

    Article  MathSciNet  Google Scholar 

  28. Chen, W., Wang, F.: A method of fundamental solution without fictitious boundary. Eng. Anal. Bound. Elem. 34(5), 530–532 (2010)

    Article  Google Scholar 

  29. Lavrentiev, M.A., Shabat, B.V.: Methods of Functions of a Complex Variable (Chinese Edition). Higher Education Press, Beijing (2006)

    Google Scholar 

  30. Chen, R.L.: A notch problem of finite bodies containing elliptic hole in the condition of anti-plane deformation. Comput. Struct. Mech. Appl. 4(1), 89–96 (1984). ((In Chinese))

    Google Scholar 

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Acknowledgements

We would like to thank the Editor and Reviewers for giving valuable improvements to the paper. This work was supported by the National Natural Science Foundation of China under Grant No. 11802145 and Jiangsu Provincial Natural Science Foundation of China under Grant No. BK20191450. We also thank Mr. Jicheng Jiang for his patience and help on our work.

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Authors and Affiliations

  1. School of Transportation and Civil Engineering, Nantong University, Nantong, 226019, China

    Xiaoguang Yuan & Quan Jiang

  2. School of Science, Nantong University, Nantong, 226019, China

    Quan Jiang

  3. College of Materials, Xiamen University, Xiamen, 361005, China

    Zhidong Zhou

  4. School of Naval Architecture, Ocean & Civil Engineering, Shanghai Jiaotong University, Shanghai, 200240, China

    Fengpeng Yang

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  1. Xiaoguang Yuan
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  4. Fengpeng Yang
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Correspondence to Quan Jiang.

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Yuan, X., Jiang, Q., Zhou, Z. et al. Stress analysis of anti-plane finite elastic solids with hole by the method of fundamental solutions using conformal mapping technique. Arch Appl Mech 92, 1823–1839 (2022). https://doi.org/10.1007/s00419-022-02150-0

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