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.
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Other transcriptions of the author name, e.g., Vencel, Ventsel or Wentzell, can also be found in the literature.
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Acknowledgements
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.
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Appendix: Classes of Functions
Appendix: Classes of Functions
For the case of the smooth contour \(L\) with two tips \(c=a\), \(c=b\), the classes of functions considered in the present paper are defined as, see [58]:
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\(H\) class
The function \(\varphi (t )\) is said to belong to \(H\) class on \(L\), if for any two points \(t_{1},\,t_{2}\in L\) it satisfies, for some \(0<\alpha \), the following Hölder condition:
in which \(A\) is an arbitrary positive constant. More specifically, it could be said that the function \(\varphi (t )\) belongs to \(H (\alpha )\) class.
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\(H^{*}\) class
The function \(\varphi (t )\) is said to belong to \(H^{*}\) class on \(L\), if it satisfies Hölder condition with some \(\alpha \) on every closed part of \(L\) that does not contain its tips and near the tips \(c=a\), \(c=b\)
where \(\varphi ^{*} (t )\in H\) class.
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\(h_{0}\) and \(h_{2}\) classes
These classes are particular cases of more general class \(h (c_{1},\ldots ,c_{q} )\) introduced in [58] for the consideration of the Hilbert problems and singular integral equations for a union of smooth contours, see §79, 81, 88 in [58]. The definition of the latter class is more involved and require introduction of the material that is beyond the scope of the present paper. So, instead of giving formal definition of class \(h (c_{1},\ldots ,c_{q} )\), we give its interpretation for the application related to our problem, see also the discussion on page 251 of [58].
In our case, the condition that the function (that belongs to \(H\) class at any internal point of \(L\)) belongs to \(h_{2}\) class simply means that this function is bounded near both tips of \(L\), while the condition that the function belongs to \(h_{0}\) class means that no such restrictions are required on the behavior of the function at any of two tips.
Finally, we reproduce the following result of §20 in [58] that was used in Sect. 5.1: if the function \(\varphi (t )\) belongs to \(H (\alpha )\) class on a smooth contour \(L\), then
also satisfies everywhere on \(L\), except in the neighborhood of those of its tips at which \(\varphi (t )\neq 0\), the \(H (\alpha )\) condition for \(\alpha <1\) and \(H (1-\varepsilon )\) condition for \(\alpha =1\), where \(\varepsilon \) is an arbitrary small parameter.
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Baranova, S., Mogilevskaya, S.G., Mantič, V. et al. Analysis of the Antiplane Problem with an Embedded Zero Thickness Layer Described by the Gurtin-Murdoch Model. J Elast 140, 171–195 (2020). https://doi.org/10.1007/s10659-020-09764-x
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DOI: https://doi.org/10.1007/s10659-020-09764-x