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.

中文翻译：

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由 Gurtin-Murdoch 模型描述的具有嵌入式零厚度层的反平面问题分析

考虑了由 Gurtin-Murdoch 模型描述的无限各向同性弹性介质在远场载荷下包含任意形状的零厚度层的反平面问题。结果表明，在反平面假设下，完整的 Gurtin-Murdoch 模型的控制方程对于非零表面张力是不一致的。对于表面张力消失的情况，引入了弹性场的解析积分表示和控制问题的无量纲参数。问题的解被简化为用层的弹性应力写成的超奇异积分方程的解。对于沿直线段的层的情况，对超奇异方程进行了理论分析，并研究了尖端附近弹性场的渐近行为。讨论了适当的数值求解技术，并给出了几个数值结果。此外，还证明了所研究的问题与嵌入均匀弹性介质中的薄而硬的弹性不均匀性这一众所周知的问题的具体情况密切相关。