Abstract

The equilibrium problem for an elastic body having an inhomogeneous inclusion with curvilinear boundaries is considered within the framework of antiplane shear. We assume that there is a power-law dependence of the shear modulus of the inclusion on a small parameter characterizing its width. We justify passage to the limit as the parameter vanishes and construct an asymptotic model of an elastic body containing a thin inclusion. We also show that, depending on the exponent of the parameter, there are the five types of thin inclusions: crack, rigid inclusion, ideal contact, elastic inclusion, and a crack with adhesive interaction of the faces. The strong convergence is established of the family of solutions of the original problem to the solution of the limiting one.

中文翻译：

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反平面剪切问题中弹性体中薄夹杂物模型的渐近合理性

摘要

在反平面剪切力的框架内考虑了弹性体的平衡问题，该弹性体的曲线边界不均匀地包含在内。我们假设夹杂物的剪切模量与幂律之间的相关性取决于表征其宽度的小参数。当参数消失时，我们证明传递到极限是合理的，并构造了包含薄夹杂物的弹性体的渐近模型。我们还表明，根据参数的指数，有五种类型的薄夹杂物：裂纹，刚性夹杂物，理想接触，弹性夹杂物以及带有表面粘合作用的裂纹。原始问题的解系列与极限问题的解的强收敛性得以建立。