Numerical solution of the two-dimensional Steigmann–Ogden model of material surface with a boundary

Anna Y. Zemlyanova, Sofia G. Mogilevskaya, Dominik Schillinger

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In this paper, we consider an infinite isotropic elastic matrix that contains a material surface described by the Steigmann–Ogden theory, where the material surface possesses a boundary. In two-dimensional setting, the surface represents a material curve with endpoints. The presence of the surface tension introduces non-linearity in the problem since a superposition with respect to the far-field load is not allowed. We review the governing equations, the jump conditions, and the conditions at the surface tips. For solving the problem in the plane strain setting, the displacements in the matrix are sought in the complex variables form of a single layer elastic potential. The density of the potential represents the jump in complex tractions across the surface. Exact expressions for the elastic fields in the matrix are provided in terms of complex integral representations. The problem of the surface located along a straight segment is considered in detail. Its solution is reduced to that of the system of boundary integral equations using the approximations of the boundary data that involve the series of Chebyshev's polynomials. We present numerical examples to illustrate the influence of dimensionless parameters on the solution.

Original languageEnglish (US)
Article number133531
JournalPhysica D: Nonlinear Phenomena
StatePublished - Jan 2023

Bibliographical note

Funding Information:
The first author (A.Z.) gratefully acknowledges the support from the Simons Collaboration Grant for Mathematicians (2020–2025) , award number 713080 . The second author (S.M.) gratefully acknowledges the support from the National Science Foundation, United States , award number NSF CMMI - 2112894 . This research was also supported through the program “Oberwolfach Research Fellows” by the Mathematisches Forschungsinstitut Oberwolfach in 2022.

Publisher Copyright:
© 2022 Elsevier B.V.


  • Complex variables integral equations
  • Composites with ultra thin reinforcements
  • Series of Chebyshev's polynomials
  • Steigmann–Ogden theory


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