应力驱动重排不稳定性的统一模型
A unified model for stress-driven rearrangement instabilities
摘要：
介绍了一种同时处理应力驱动重排不稳定性的变分模型，如边界不连续性、内部裂纹、外部细丝、边缘分层、润湿和脆性断裂。该模型的特点是能量显示弹性和表面条款，并允许一个统一的治疗范围广泛的设置，从外延应变薄膜结晶腔，并从毛细作用的问题，断裂模型。采用变分法中的直接法，证明了极小化构形的存在性。给出了一类容许构形中关于适当拓扑的能量等界序列的紧性和能量下半连续性，推广了文献中已有的容许构形.特别是，以前考虑的薄膜和结晶腔的设置的图形约束被更一般的假设所取代，即自由结晶界面是边界，由至多固定的有限数量$m$的连接组件，有限周长的集合。最后，它表明，作为$m\to\infty$，最小容许配置的能量趋于最小能量的一般类的配置没有限制的自由界面上的连接组件的数量。
英文摘要：
A variational model to simultaneously treat Stress-Driven Rearrangement Instabilities, such as boundary discontinuities, internal cracks, external filaments, edge delamination, wetting, and brittle fractures, is introduced. The model is characterized by an energy displaying both elastic and surface terms, and allows for a unified treatment of a wide range of settings, from epitaxially-strained thin films to crystalline cavities, and from capillarity problems to fracture models. Existence of minimizing configurations is established by adopting the direct method of the Calculus of Variations. Compactness of energy-equibounded sequences and energy lower semicontinuity are shown with respect to a proper selected topology in a class of admissible configurations that extends the classes previously considered in the literature. In particular, graph-like constraints previously considered for the setting of thin films and crystalline cavities are substituted by the more general assumption that the free crystalline interface is the boundary, consisting of an at most fixed finite number $m$ of connected components, of sets of finite perimeter. Finally, it is shown that, as $m\to\infty$, the energy of minimal admissible configurations tends to the minimum energy in the general class of configurations without the bound on the number of connected components for the free interface.
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