零温度多边形中的刚度转变
Rigidity transitions in zero-temperature polygons
摘要：
我们研究的刚性过渡的几何线索，由于出现了一个系统的跨越状态的自我应力在约束系统的个别多边形和弹簧网络从这样的多边形。当一个多边形的调和键边和一个面积弹簧约束受到膨胀应变，我们观察到，凸的多边形是一个必要条件，这样的自我应力。证明了多边形的循环构形是自应力的充分条件。这种几何形状和刚度的对应关系类似于将一维弹簧链拉直以使其刚性化。我们使用纯几何方法预测刚度转变的开始。我们还估计过渡应变为一个给定的初始配置近似不规则多边形的规则多边形。这些发现有助于确定一个面积保持多边形的刚度只是看它。由于二维弹簧网络可以被认为是一个网络的多边形，我们寻找类似的几何特征，在各向同性膨胀应变下的约束弹簧网络。特别是，我们观察到，所有的多边形达到凸性的刚性过渡，凸的，但不是循环的，多边形的分数预测发病的刚性过渡。有趣的是，网络中的非循环多边形与较大的张力相关，从而形成有效的力链。
英文摘要：
We study geometrical clues of a rigidity transition due to the emergence of a system-spanning state of self stress in under-constrained systems of individual polygons and spring networks constructed from such polygons. When a polygon with harmonic bond edges and an area spring constraint is subject to an expansive strain, we observe that convexity of the polygon is a necessary condition for such a self stress. We prove that the cyclic configuration of the polygon is a sufficient condition for the self stress. This correspondence of geometry and rigidity is akin to the straightening of a one dimensional chain of springs to rigidify it. We predict the onset of the rigidity transition using a purely geometrical method. We also estimate the transition strain for a given initial configuration by approximating irregular polygons as regular polygons. These findings help determine the rigidity of an area-preserving polygon just by looking at it. Since two-dimensional spring networks can be considered as a network of polygons, we look for similar geometric features in under-constrained spring networks under isotropic expansive strain. In particular, we observe that all polygons attain convexity at the rigidity transition such that the fraction of convex, but not cyclic, polygons predicts the onset of the rigidity transition. Interestingly, acyclic polygons in the network correlate with larger tensions, thus, forming effective force chains.
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