Geometry and the onset of rigidity in a disordered network

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 ${\gamma }^{★}$, 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, the network acquires a highly anisotropic elastic modulus, resisting further deformation most strongly in the direction of the rigidifying shear. We confirm previously reported critical scaling of the corresponding differential shear modulus. In the subcritical 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 the nascent state of self-stress.

Figure 1

Floppy to rigid. (a) A network, as initially generated with the Mikado method. The periodically repeated simulation area is outlined by the thick black boundary. The average connectivity of this sample network is $z=3.6$. (b) The same network, after a simple shear strain of $\gamma =0.12$ has been applied (using Lees-Edwards boundary conditions [18]). The network is now rigid, betrayed by a single, system-spanning state of self-stress comprising extended (red) and compressed (green) segments. Black segments are unstrained.

Figure 2

Properties of Mikado networks as a function of the number of sticks (fibers) used for stick sizes ranging from 10% of the box size to 19% of the box size in steps of 1% from bottom to top in both panels. (a) The number of nodes in a Mikado network approaches for large numbers of sticks, $N\simeq {(N_{\mathrm{sticks}}{L}_{\mathrm{stick}}/L_{\mathrm{system}})}^2/\pi$, as expected from Eq. (1). (b) The number of floppy modes approaches the number of fibers in the network in the same limit.

Figure 3

The addition of a Mikado stick to an already existing network gives rise to $\mu$ (in this example $\mu =3$) intersections and $2\mu -1$ new springs. Turning the intersection points into nodes before adding the springs that represent the new stick creates a floppy mode at each new node, indicated with the arrows. Adding the $\mu -1$ new springs couples these into a single floppy mode.

Figure 4

Probability density of the critical strain ${\gamma }^{★}$, the strain at which the network acquires a modulus, for Mikado networks at $z=3.6$ with a stick length of 0.57. System sizes are $L=1$ (pluses), $\sqrt{2}$ (circles), 2 (stars), $2\sqrt{2}$ (diamonds) for decreasing distribution width. The inset shows the average ${\gamma }^{★}$ as a function of connectivity $z$ (calculated at $L=\sqrt{2}$).

Figure 5

The scenario for shear-induced rigidification of the Mikado network: As the critical strain ${\gamma }^{★}$ is approached, one singular value drops to zero, spawning a state of self-stress that carries the load, prompting a finite shear modulus. We plot the lowest singular value in black and the second and third lowest singular values in blue and red, respectively. It is evident that even well beyond the transition there remains only one state of self-stress. The lines are guides to the eye.

Figure 6

(a) The differential modulus in the direction of the nonlinear deformation (solid symbols) is much larger than the linear modulus in the other shear direction (open symbols). Circles, diamonds, and stars denote system sizes of 1, $\sqrt{2}$, and 2, respectively. (b) Beyond the critical strain, the shear modulus rises as a power law; $G-\mathrm{\Delta }{G}^{★}\sim {\left[(\gamma -{\gamma }^{★})/{\gamma }^{★}\right]}^f$ with $f=0.82$. Plus symbols, circles, stars, and diamonds denote system sizes of 1, $\sqrt{2}$, 2, and $2\sqrt{2}$, respectively. The line through the shear modulus graph is a fit to Eq. (10).