Acta Mechanica

, Volume 230, Issue 6, pp 2105–2110 | Cite as

On the numerical implementation of special solutions to the homogeneous Riemann–Hilbert problem in two-dimensional elasticity

  • Ming DaiEmail author
Original Paper


The special solutions proposed by Muskhelishvili for a particular kind of homogeneous Riemann–Hilbert problems are very important in the mechanical analysis of cracked materials. The numerical implementation of these special solutions relies on specific choices of the arguments of relevant parameters. We establish here a unified principle to specify the admissible arguments of relevant parameters in the numerical implementation of these special solutions. We show that the use of the conventional argument branch (such as \(\left( {-\pi ,\pi } \right] \) or \(\left[ {0,2\pi } \right) )\) may lead to erroneous evaluation of these special solutions when the corresponding crack is arc-shaped. In the context of our principle, we provide also the consistent asymptotic expressions of these special solutions in the neighborhood of a certain point or in the remote region.


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The author appreciates a start-up grant of the Nanjing University of Aeronautics and Astronautics and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).


  1. 1.
    Muskhelishvili, N.I.: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen (1975)zbMATHGoogle Scholar
  2. 2.
    England, A.H.: An arc crack around a circular elastic inclusion. ASME J. Appl. Mech. 33(3), 637–640 (1966)CrossRefGoogle Scholar
  3. 3.
    Hwu, C.: Explicit solutions for collinear interface crack problems. Int. J. Solids Struct. 30(3), 301–312 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Gao, C.F., Kessler, H., Balke, H.: Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks. Int. J. Eng. Sci. 41(9), 983–994 (2003)CrossRefzbMATHGoogle Scholar
  5. 5.
    Dai, M., Schiavone, P., Gao, C.F.: Periodic cracks in an infinite electrostrictive plane under the influence of a uniform remote electric field. Eng. Fract. Mech. 157, 1–10 (2016)CrossRefGoogle Scholar
  6. 6.
    Dai, M., Gao, C.F., Schiavone, P.: Arc-shaped permeable interface crack in an electrostrictive fibrous composite under uniform remote electric loadings. Int. J. Mech. Sci. 115, 616–623 (2016)CrossRefGoogle Scholar
  7. 7.
    Wang, X., Yang, M., Schiavone, P.: Debonding of an elastic inhomogeneity of arbitrary shape in anti-plane shear. Z. Angew. Math. Phys. 67(4), 82 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Wang, X., Schiavone, P.: Debonded arc-shaped interface conducting rigid line inclusions in piezoelectric composites. C. R. Mec. 345(10), 724–731 (2017)CrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.State Key Laboratory of Mechanics and Control of Mechanical StructuresNanjing University of Aeronautics and AstronauticsNanjingChina

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