For the first time, strict analytical solutions for the elastic fields and the strain energies of an infinitely thin dilatational disk (DD) and a dilatational cylindrical inclusion (CyI) of finite length coaxially embedded in an infinite elastically isotropic cylinder with free surface are given and analyzed in detail. The solutions are represented in an integral form that is suitable for further analytical use and numerical study. The screening effect of the cylinder free surface on the elastic fields and the strain energies of the DD and the CyI is discussed. It is shown that this effect is significant for both the axial displacement and stress fields when the DD and CyI radii are comparable with the cylinder radius, however it is rather weak for the DD strain energy (the energy release does not exceed ∼10%). In contrast, for the CyI strain energy, the screening effect can be very strong. It is also shown that the hydrostatic stress is inhomogeneous and exists not only inside the CyI, as is the case with this stress for inclusions in an infinite medium, but also outside it. This stress is concentrated on the free surface at the points that are closest to the CyI boundary. Inside the CyI, the hydrostatic stress is much higher in magnitude than outside it.

中文翻译：

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弹性纳米线中的膨胀盘和有限圆柱形夹杂物


首次给出了无限薄的膨胀盘 （DD） 和有限长度的膨胀圆柱体 （CyI） 的弹性场和应变能的严格解析解，这些夹杂物同轴嵌入具有自由表面的无限弹性各向同性圆柱体中，并进行了详细分析。解以积分形式表示，适合进一步的分析使用和数值研究。讨论了圆柱体自由表面对弹性场以及 DD 和 CyI 的应变能的屏蔽作用。结果表明，当 DD 和 CyI 半径与圆柱半径相当时，这种影响对轴向位移和应力场都很重要，但对于 DD 应变能来说，这种影响相当弱（能量释放不超过 ∼10%）。相反，对于 CyI 应变能，屏蔽效果可能非常强。研究还表明，静水应力是不均匀的，不仅存在于 CyI 内部（如无限介质中夹杂物的应力），还存在于 CyI 外部。该应力集中在最靠近 CyI 边界的自由表面上。在 CyI 内部，静水应力的大小比外部高得多。