In this paper we introduce Peierls–Nabarro type models for edge dislocations at semi-coherent interfaces between two heterogeneous crystals, and prove the optimality of uniformly distributed edge dislocations. Specifically, we show that the elastic energy \(\Gamma \)-converges to a limit functional comprised of two contributions: one is given by a constant \(c_\infty >0\) gauging the minimal energy induced by dislocations at the interface, and corresponding to a uniform distribution of edge dislocations; the other one accounts for the far field elastic energy induced by the presence of further, possibly not uniformly distributed, dislocations. After assuming periodic boundary conditions and formally considering the limit from semi-coherent to coherent interfaces, we show that \(c_\infty \) is reached when dislocations are evenly-spaced on the one dimensional circle.

在本文中，我们介绍了Peierls-Nabarro型模型用于两个异质晶体之间半相干界面处的位错，并证明了均匀分布的位错的最优性。具体来说，我们表明弹性能\（\ Gamma \）-收敛到一个包含两个贡献的极限函数：一个由常数\（c_ \ infty> 0 \）给出。测量由界面处的位错引起的最小能量，并对应于边缘位错的均匀分布；另一种是由于存在进一步的，可能不是均匀分布的位错而引起的远场弹性能。在假定周期性边界条件并正式考虑了从半相干到相干界面的限制之后，我们表明，当位错在一维圆上均匀分布时，就达到了\（c_ \ infty \）。