Disordered spring networks that are undercoordinated may abruptly rigidify when sufficient strain is applied. Since the deformation in response to applied strain does not change the generic quantifiers of network architecture - the number of nodes and the number of bonds between them - this rigidity transition must have a geometric origin. Naive, degree-of-freedom based mechanical analyses such as the Maxwell-Calladine count or the pebble game algorithm overlook such geometric rigidity transitions and offer no means of predicting or characterizing them. We apply tools that were developed for the topological analysis of zero modes and states of self-stress on regular lattices to two-dimensional random spring networks, and demonstrate that the onset of rigidity, at a finite simple shear strain $γ^\star$, coincides with the appearance of a single state of self stress, accompanied by a single floppy mode. The process conserves the topologically invariant difference between the number of zero modes and the number of states of self stress, but imparts a finite shear modulus to the spring network. Beyond the critical shear, we confirm previously reported critical scaling of the modulus. In the sub-critical regime, a singular value decomposition of the network's compatibility matrix foreshadows the onset of rigidity by way of a continuously vanishing singular value corresponding to nascent state of self stress.
Rigidity percolation control of the brittle-ductile transition in disordered networks
在普通固体中，已知材料无序会增加应力集中在裂纹尖端的过程区的尺寸，从而导致从局部失效到扩散失效的转变。在这里，我们报告的实验无序的二维晶格，来自摩擦粒子包装，其中的平均配位数$\langle z \rangle$的底层网络提供了类似的控制。我们的实验表明，调整网络的连通性提供了一系列的行为，从脆性到韧性破坏。我们阐明了合作的起源，这种过渡使用摩擦卵石游戏算法的原始，完整的晶格。我们发现，过渡对应于均衡值$\langle z \rangle = 3$在大摩擦的限制，与脆性破坏发生的结构垂直跨越刚性集群，和韧性失败的软盘网络包含nonspanning刚性集群。此外，我们发现，个别故障事件通常发生在由刚性集群分开的软盘区域内。
