从二维刃位错系统导出线性化多晶体
Derivation of Linearised Polycrystals from a 2D system of edge dislocations
AI总结：
本文建立二维刃位错系统的变分模型，通过核心半径方法推导晶格间距趋于零时弹性能的 Γ-极限，揭示塑性与弹性项解耦，并说明极小元对应晶粒微小互旋的线性化多晶体。
摘要：
本文证明多晶结构可作为弹性能最小化的结果而涌现。为此，我们在所谓的核心半径方法框架内，引入了一个针对二维刃位错系统的变分模型，并在晶格间距趋于零时推导了弹性能泛函的 Γ-极限。在所研究的能量尺度下，应变的对称部分与反对称部分在极限中解耦，位错测度即为应变反对称部分的旋度。极限能量由两部分之和给出：作用于位错密度的塑性项，以及依赖于对称应变的弹性项。在适当边界条件下，极小元为分段常数的反对称应变场，在我们的模型中表示晶粒之间以无穷小角度相互旋转的多晶体。
英文摘要：
In this paper we show the emergence of polycrystalline structures as a result of elastic energy minimisation. For this purpose, we introduce a variational model for two-dimensional systems of edge dislocations, within the so-called core radius approach, and we derive the $Γ$-limit of the elastic energy functional as the lattice space tends to zero. In the energy regime under investigation, the symmetric and skew part of the strain become decoupled in the limit, the dislocation measure being the curl of the skew part of the strain. The limit energy is given by the sum of a plastic term, acting on the dislocation density, and an elastic term, which depends on the symmetric strains. Minimisers under suitable boundary conditions are piece-wise constant antisymmetric strain fields, representing in our model a polycrystal whose grains are mutually rotated by infinitesimal angles.
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