SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 453-490, January 2021.
A discrete-to-continuum analysis for free-boundary problems related to crystalline films deposited on substrates is performed by $\Gamma$-convergence. The discrete model introduced here is characterized by an energy with two contributions, the surface and the elastic-bulk energy, and it is formally justified starting from atomistic interactions. The surface energy counts missing bonds at the film and substrate boundaries, while the elastic energy models the fact that for film atoms there is a preferred interatomic distance different from the preferred interatomic distance for substrate atoms. In the regime of small mismatches between the film and the substrate optimal lattices, a discrete rigidity estimate is established by regrouping the elastic energy in triangular-cell energies and by locally applying rigidity estimates from the literature. This is crucial to establish precompactness for sequences with equibounded energy and to prove that the limiting deformation is one single rigid motion. By properly matching the convergence scaling of the different terms in the discrete energy, both surface and elastic contributions appear also in the resulting continuum limit in agreement (and in a form consistent) with literature models. Thus, the analysis performed here is a microscopical justification of such models.

中文翻译：

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外延应变结晶膜变化模型的微观验证

SIAM数学分析杂志，第53卷，第1期，第453-490页，2021年1月。
对与沉积在基板上的结晶膜有关的自由边界问题的离散到连续谱分析是通过γ收敛进行的。此处引入的离散模型的特征是具有两个贡献的能量，即表面能量和弹性体能量，并且从原子相互作用开始对其进行正式证明。表面能计数在膜和衬底边界处的缺失键，而弹性能模拟以下事实：对于膜原子，存在一个优选的原子间距离，该距离不同于衬底原子的优选原子间距离。在薄膜和基板最佳晶格之间的较小失配的情况下，通过将弹性能重新组合为三角单元能量并通过局部应用文献中的刚度估算值，可以建立离散的刚度估算值。这对于建立具有相等能量的序列的预紧性以及证明极限变形是一个单一的刚性运动至关重要。通过适当地匹配离散能量中不同项的收敛比例，表面和弹性贡献也都出现在所得的连续谱极限中，与文献模型一致（并且形式一致）。因此，此处执行的分析是此类模型的微观证明。表面和弹性的贡献也都出现在与文献模型一致（且形式一致）的最终连续极限中。因此，此处执行的分析是此类模型的微观证明。表面和弹性的贡献也都出现在与文献模型一致（且形式一致）的最终连续极限中。因此，此处执行的分析是此类模型的微观证明。