We consider a spectral homogenization problem for the linear elasticity system posed in a domain $\varOmega $ of the upper half-space $\mathbb{R}^{3+}$ , a part of its boundary $\varSigma $ being in contact with the plane $\{x_{3}=0\}$ . We assume that the surface $\varSigma $ is traction-free out of small regions $T^{\varepsilon }$ , where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function $M(x)$ and a reaction parameter $\beta (\varepsilon )$ that can be very large when $\varepsilon \to 0$ . The size of the regions $T^{\varepsilon }$ is $O(r_{\varepsilon })$ , where $r_{\varepsilon }\ll \varepsilon $ , and they are placed at a distance $\varepsilon $ between them. We provide all the possible spectral homogenized problems depending on the relations between $\varepsilon $ , $r_{\varepsilon }$ and $\beta (\varepsilon )$ , while we address the convergence, as $\varepsilon \to 0$ , of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on $\varSigma $ . New capacity matrices are introduced to define these strange terms.

中文翻译：

---

过滤温克勒基础上快速交替边界条件谱问题的渐近性

我们考虑在上半空间 $\mathbb{R}^{3+}$ 的域 $\varOmega $ 中提出的线性弹性系统的谱同质化问题，其边界 $\varSigma $ 的一部分接触与飞机 $\{x_{3}=0\}$ 。我们假设表面 $\varSigma $ 在小区域 $T^{\varepsilon }$ 之外是无牵引的，我们在那里施加 Winkler-Robin 边界条件。该条件通过对称和正定矩阵函数 $M(x)$ 和反应参数 $\beta (\varepsilon )$ 将应力和位移联系起来，当 $\varepsilon \to 0$ 时，该参数可能非常大。区域 $T^{\varepsilon }$ 的大小为 $O(r_{\varepsilon })$ ，其中 $r_{\varepsilon }\ll \varepsilon $ ，它们被放置在 $\varepsilon $ 之间的距离他们。我们根据 $\varepsilon $ , $r_{\varepsilon }$ 和 $\beta (\varepsilon )$ 之间的关系提供所有可能的谱同质化问题，同时我们将收敛性处理为 $\varepsilon \to 0$ ，在 $\varSigma $ 上的均质罗宾边界条件上出现一些奇怪项的临界情况下的特征对。引入了新的容量矩阵来定义这些奇怪的术语。