丝状Kagome晶格的弹性
Elasticity of Filamentous Kagome Lattice
摘要：
在稀释的可果美晶格中，键以概率$1-p$被随机移除，由在最大配位数为4的点处相交的直线组成。如果线被视为半柔性聚合物和交叉点被视为交联，这种晶格提供了一个简单的模型为二维丝状网络。使用基于晶格的有效介质理论和将纤维建模为弹性杆的数值模拟（拉伸模量为$μ$，弯曲模量为$κ$）来研究该晶格的弹性作为$p$和$κ$的函数。在$p=1$处，弹性响应是纯仿射的，并且宏观弹性模量$G$与$κ$无关。当$κ= 0$时，晶格在$p=1$处经历一级刚性逾渗转变。当$κ> 0$时，$G$随着$p$降低到1以下而连续降低，在$p=p_b \approx 0.605$处的连续刚性逾渗转变处达到零，这对于$κ$的所有非零值都是相同的。有效介质理论预测了$G$的标度形式，其在$p=1$和$p=p_b$附近表现出从小$κ/μ$处的弯曲主导响应到大$κ/μ$处的拉伸主导响应的交叉，这与$p=1$附近无可调参数的模拟相匹配。仿射响应作为$p\rightarrow 1$被识别与一个状态的方法与样品交叉直丝处理为弹性杆。
英文摘要：
The diluted kagome lattice, in which bonds are randomly removed with probability $1-p$, consists of straight lines that intersect at points with a maximum coordination number of four. If lines are treated as semi-flexible polymers and crossing points are treated as crosslinks, this lattice provides a simple model for two-dimensional filamentous networks. Lattice-based effective medium theories and numerical simulations for filaments modeled as elastic rods, with stretching modulus $μ$ and bending modulus $κ$, are used to study the elasticity of this lattice as functions of $p$ and $κ$. At $p=1$, elastic response is purely affine, and the macroscopic elastic modulus $G$ is independent of $κ$. When $κ= 0$, the lattice undergoes a first-order rigidity percolation transition at $p=1$. When $κ> 0$, $G$ decreases continuously as $p$ decreases below one, reaching zero at a continuous rigidity percolation transition at $p=p_b \approx 0.605$ that is the same for all non-zero values of $κ$. The effective medium theories predict scaling forms for $G$, which exhibit crossover from bending dominated response at small $κ/μ$ to stretching-dominated response at large $κ/μ$ near both $p=1$ and $p=p_b$, that match simulations with no adjustable parameters near $p=1$. The affine response as $p\rightarrow 1$ is identified with the approach to a state with sample-crossing straight filaments treated as elastic rods.
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