高对比度弹性复合材料中应力集中的渐近性
Asymptotics of the stress concentration in high-contrast elastic composites
摘要：
材料科学研究的一个长期领域一直是研究复合材料中的静电场、磁场和弹性场，这些复合材料具有与背景材料不同的材料性质的致密包裹体。对于一般的椭圆方程组，当系数是分段Hölder连续且一致有界时，Li和Nirenberg给出了梯度的一个与\cite{ln}无关的界，其中$\varepsilon$表示界面之间的距离.然而，在高对比度复合材料中，当$\varepsilon$趋于零时，应力总是集中在狭窄的区域。为了研究$\varepsilon$在这类集中现象中所起的作用，与Li和Nirenberg的一致有界性结果不同，本文建立了二维和三维具有部分无穷大系数的Lamé系统解的梯度的爆破渐近表达式.我们发现了应力的爆破率与相邻曲面的相对凸性之间的关系，并找到了一族关于边界数据的爆破因子矩阵。因此，本文的工作完全解决了高对比度复合介质中应力集中爆破分析的Babuuska问题。此外，作为这些局部分析的副产品，我们建立了一个扩展的Flaherty-Keller公式的整体有效弹性性能的周期性复合材料与密集的纤维，这是有关的“Vigdergauz微观结构”的形状优化的纤维。
英文摘要：
A long-standing area of materials science research has been the study of electrostatic, magnetic, and elastic fields in composite with densely packed inclusions whose material properties differ from that of the background. For a general elliptic system, when the coefficients are piecewise Hölder continuous and uniformly bounded, an $\varepsilon$-independent bound of the gradient was obtained by Li and Nirenberg \cite{ln}, where $\varepsilon$ represents the distance between the interfacial surfaces. However, in high-contrast composites, when $\varepsilon$ tends to zero, the stress always concentrates in the narrow regions. As a contrast to the uniform boundedness result of Li and Nirenberg, in order to investigate the role of $\varepsilon$ played in such kind of concentration phenomenon, in this paper we establish the blow-up asymptotic expressions of the gradients of solutions to the Lamé system with partially infinite coefficients in dimensions two and three. We discover the relationship between the blow-up rate of the stress and the relative convexity of adjacent surfaces, and find a family of blow-up factor matrices with respect to the boundary data. Therefore, this work completely solves the Babuuska problem on blow-up analysis of stress concentration in high-contrast composite media. Moreover, as a byproduct of these local analysis, we establish an extended Flaherty-Keller formula on the global effective elastic property of a periodic composite with densely packed fibers, which is related to the "Vigdergauz microstructure" in the shape optimization of fibers.
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