Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model

S. Baranova, S. G. Mogilevskaya, V. Mantič, S. Jiménez-Alfaro

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6 Scopus citations

Abstract

The antiplane problem of an infinite isotropic elastic medium subjected to a far-field load and containing a zero thickness layer of arbitrary shape described by the Gurtin-Murdoch model is considered. It is shown that, under the antiplane assumptions, the governing equations of the complete Gurtin-Murdoch model are inconsistent for non-zero surface tension. For the case of vanishing surface tension, the analytical integral representations for the elastic fields and the dimensionless parameter that governs the problem are introduced. The solution of the problem is reduced to the solution of the hypersingular integral equation written in terms of elastic stress of the layer. For the case of a layer along a straight segment, theoretical analysis of the hypersingular equation is performed and asymptotic behavior of the elastic fields near the tips is studied. The appropriate numerical solution techniques are discussed and several numerical results are presented. Additionally, it is demonstrated that the problem under study is closely related to the specific case of the well-known problem of a thin and stiff elastic inhomogeneity embedded into a homogeneous elastic medium.

Original languageEnglish (US)
Pages (from-to)171-195
Number of pages25
JournalJournal of Elasticity
Volume140
Issue number2
DOIs
StatePublished - Aug 1 2020

Bibliographical note

Funding Information:
The first author (S.B.) gratefully acknowledges the support provided by the International Student Work Opportunity Program (ISWOP), University of Minnesota. The second author (S.M.) gratefully acknowledges the support from the Theodore W. Bennett Chair, University of Minnesota and the Isaac Newton Institute for Mathematical Sciences (INI) for support and hospitality during the programme CAT, when work on a part of this paper was undertaken. This part of work was supported by: EPSRC grant number EP/R014604/1. The support of the Simons Foundation through Simons INI Fellowship is also gratefully acknowledged. The research by V.M. and S.J.-A. was conducted with the support of the Spanish Ministry of Science, Innovation and Universities and European Regional Development Fund (Project PGC2018-099197-B-I00).The authors are also grateful to Prof. Eric Bonnetier (Institut Fourier, Université Grenoble-Alpes) for pointing out the connections between the boundary conditions for the present problem and the Ventcel boundary condition.

Funding Information:
The first author (S.B.) gratefully acknowledges the support provided by the International Student Work Opportunity Program (ISWOP), University of Minnesota. The second author (S.M.) gratefully acknowledges the support from the Theodore W. Bennett Chair, University of Minnesota and the Isaac Newton Institute for Mathematical Sciences (INI) for support and hospitality during the programme CAT, when work on a part of this paper was undertaken. This part of work was supported by: EPSRC grant number EP/R014604/1. The support of the Simons Foundation through Simons INI Fellowship is also gratefully acknowledged. The research by V.M. and S.J.-A. was conducted with the support of the Spanish Ministry of Science, Innovation and Universities and European Regional Development Fund (Project PGC2018-099197-B-I00).The authors are also grateful to Prof. Eric Bonnetier (Institut Fourier, Universit? Grenoble-Alpes) for pointing out the connections between the boundary conditions for the present problem and the Ventcel boundary condition.

Keywords

  • Antiplane elasticity
  • Gurtin-Murdoch model
  • Nanocomposites
  • Thin and stiff elastic inhomogeneity
  • Tip/edge asymptotics
  • Zero thickness layer

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