超越经典柯西-玻恩（Cauchy-Born）法则
Beyond the classical Cauchy-Born rule
AI总结：
本研究针对含非凸能量的离散变分问题中的几何阻挫现象，探究非凸性、非局域性与离散性的相互作用，提出符合广义柯西-玻恩（GCB）法则的新几何阻挫问题类，并建立混合行为问题的通用求解方法。
摘要：
由物理驱动的含非凸能量的变分问题通常在离散框架下表述，且包含边界条件。此类问题中的长程相互作用，结合晶格离散性带来的约束，即使在一维情形下也会引发几何阻挫现象。非凸性会导致微结构的形成，而不同尺度下相互作用的不相容性则会产生非平凡的混合效应；当最优微结构与底层晶格尺度之间存在非公度性时，这种效应会进一步加剧。阐明非凸性、非局域性与离散性之间相互作用的复杂机制，是本研究的核心目标。虽然一般而言，我们无法期望这类问题的基态具备周期性等全局性质，但在某些情形下，恰当定义的全局解是存在的，且足以描述相应的连续（均匀化）极限。我们将这类情形解释为符合广义柯西-玻恩（Generalized Cauchy-Born, GCB）法则，并提出了一类新的存在几何阻挫的问题：这类问题在某一（加载）参数范围内符合GCB法则，而在互补参数范围内则完全不属于该类别。本文还针对此类具有混合行为的问题，开发了一种通用方法。
英文摘要：
Physically motivated variational problems involving non-convex energies are often formulated in a discrete setting and contain boundary conditions. The long-range interactions in such problems, combined with constraints imposed by lattice discreteness, can give rise to the phenomenon of geometric frustration even in a one-dimensional setting. While non-convexity entails the formation of microstructures, incompatibility between interactions operating at different scales can produce nontrivial mixing effects which are exacerbated in the case of incommensuration between the optimal microstructures and the scale of the underlying lattice. Unraveling the intricacies of the underlying interplay between non-convexity, non-locality and discreteness, represents the main goal of this study. While in general one cannot expect that ground states in such problems possess global properties, such as periodicity, in some cases the appropriately defined global solutions exist, and are sufficient to describe the corresponding continuum (homogenized) limits. We interpret those cases as complying with a Generalized Cauchy-Born (GCB) rule, and present a new class of problems with geometrical frustration which comply with GCB rule in one range of (loading) parameters while being strictly outside this class in a complementary range. A general approach to problems with such mixed behavior is developed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
