We present a gentle approach to the justification of effective media approximations, for PDE’s set outside the union of n≫1 spheres with low volume fraction. To illustrate our approach, we consider three classical examples: the derivation of the so-called strange term, made popular by Cioranescu and Murat, the derivation of the Brinkman term in the Stokes equation, and a scalar analogue of the effective viscosity problem. Under some separation assumption on the spheres, valid for periodic and random distributions of the centers, we recover effective models as n→+∞ by simple arguments.

中文翻译：

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带有稀疏球形夹杂物的传导或流体介质的有效模型的简单证明

对于PDE在n≫1球体的体积分数较小的并集之外的集合，我们提出了一种有效的介质近似的合理证明方法。为了说明我们的方法，我们考虑三个经典示例：由Cioranescu和Murat流行的所谓奇数项的推导，Stokes方程中Brinkman项的推导以及有效粘度问题的标量类似物。在球体上的一些分离假设下，对中心的周期和随机分布有效，我们通过简单的论点将有效模型恢复为n→+∞。