Figure 1

(Color online) Envelope of a typical force-displacement response obtained using the RSM.Reuse & Permissions

Figure 2

(Color online) Snapshots of damage evolution in a typical simulation of size $L=256$. Number of broken bonds at the peak load and at failure is 13 864 and 16 695, respectively. (1)–(9) represent the snapshots of damage after $n_b$ bonds are broken. (1) $n_b=5000$ (2) $n_b=10000$, (3) $n_b=12000$, (4) $n_b=13000$, (5) $n_b=14000$ (just after peak load), (6) $n_b=15000$, (7) $n_b=15500$, (8) $n_b=16000$, and (9) $n_b=16500$ (close to failure).Reuse & Permissions

Figure 3

(Color online) Snapshots of prepeak damage profiles of a typical RSM simulation with uniform threshold distribution on a triangular lattice of size $L=256$. The damage is uniform in the prepeak regime. In each of the subplots, the abscissa refers to the $y$ coordinate of the lattice section and the ordinate is the number of broken bonds in the section.Reuse & Permissions

Figure 4

(Color online) Snapshots of post-peak damage profiles of a typical RSM simulation with uniform threshold distribution on a triangular lattice of size $L=256$. The damage is clearly localized in the post-peak regime. In each of the subplots, the abscissa refers to the $y$ coordinate of the lattice section and the ordinate is the number of broken bonds in the section.Reuse & Permissions

Figure 5

(Color online) Average damage profiles at peak load obtained by first centering the data around the center of mass of the damage and then averaging over different samples. For each of the samples, the damage profile is evaluated as $p\left(y\right)=n_b\left(y\right)∕(3L+1)$, where $n_b\left(y\right)$ denotes the number of broken bonds in the $y\text{th}$ section.Reuse & Permissions

Figure 6

(Color online) Data collapse of the average profiles for the damage accumulated between peak load and failure using a power-law scaling. We have considered the damage profiles for $L=\{16,24,32,64,128,256\}$ system sizes. The average has been performed after shifting by the center of mass. The profiles show exponential tails.Reuse & Permissions

Figure 7

(Color online) Scaling of the number of broken bonds at peak load for triangular random threshold spring lattices. The scaling exponent $b=0.92$ is very close to the exponent obtained for a random threshold fuse network using triangular $(b=0.93)$ and diamond $(b=0.91)$ lattices. The difference could be attributed to finite-size effects. The number of broken bonds at peak load can also be fit by a linear function times a logarithmic correction by plotting $n_p∕N_{\mathrm{el}}$ as a function of $N_{\mathrm{el}}$ in a log-linear plot (inset).Reuse & Permissions

Figure 8

(Color online) The cumulative probability distribution for the fraction of broken bonds at the peak load for triangular spring lattices of different system sizes.Reuse & Permissions

Figure 9

(Color online) The collapsed cumulative probability distribution for the fraction of broken bonds at the peak load in the random threshold spring model (RSM) using triangular lattices of different system sizes $(L=8,16,24,32,64,128,256)$ with uniform disorder when plotted as a function of the reduced variable ${\overline{p}}_p=(p-p_p)∕{\Delta }_p$. In the inset, a comparison between the cumulative probability distributions of the fraction of broken bonds at the peak load is presented for the RSM and RFM. For the RSM, triangular lattices of sizes $(L=8,16,24,32,64,128,256)$ and for the RFM, triangular lattices of sizes $(L=16,24,32,64,128,256,512)$ are plotted. In the RFM case, the collapse of cumulative probability distributions at the peak load for different lattice topologies (triangular and diamond) and different disorder distributions (uniform and power law) is presented in Ref. 22.Reuse & Permissions

Figure 10

(Color online) Normal distribution fit for the cumulative probability distributions of the fraction of broken bonds at the peak load for triangular spring lattices of different system sizes $L=\{8,16,24,32,64,128,256\}$.Reuse & Permissions

Figure 11

(Color online) Universality of fracture strength distribution in the random threshold spring and fuse models. (a) Fracture strength density distributions for triangular spring lattices $(L=\{8,16,24,32,64,128\})$ and triangular fuse lattices $(L=\{4,8,16,24,32,64,128\})$ with uniform disorder. (b) Cumulative fracture strength distribution for triangular spring lattices $(L=\{8,16,24,32,64,128,256\})$ and triangular fuse lattices $(L=\{4,8,16,24,32,64,128,256,512\})$ with uniform disorder. The collapse of the data in random spring and fuse models suggests universality of fracture strength distribution. In the RFM case, the universality of fracture strength distributions with respect to different lattice topologies is presented in Ref. 17.Reuse & Permissions

Figure 12

(Color online) Probability distribution fits for fracture strengths at the peak load in a triangular spring lattice network for different lattice system sizes $L=\{8,16,24,32,64,128,256\}$. (a) Modified Gumbel distribution (top). (b) Weibull distribution (middle). (c) Log-normal distribution fit for all the 16 curves [see Fig. 11b] (bottom). Since the data for different lattice system sizes do not collapse onto a single curve, Weibull distribution may not be an adequate fit for representing fracture strengths in the RSM. On the other hand, the collapse of the data in the reparametrized log-normal distribution fit suggests that the log-normal distribution describes the fracture strength distribution adequately.Reuse & Permissions

Figure 13

(Color online) Proposed scaling law for the mean fracture strength $(F_{\text{peak}}=C_0{L}^{\overline{\alpha }}+C_1)$. (1) Triangular spring network (symbol: -$+$-): $\overline{\alpha }=0.97$. (2) Triangular fuse network (symbol: ${\text{−}}^{*}\text{−}$): $\overline{\alpha }=0.956$; the corresponding Weibull fit for the mean fracture strength is shown in the inset. Nonlinearity of the plots in the inset suggests that mean fracture strength does not follow a power-law scaling consistent with the Weibull distribution.Reuse & Permissions

Figure 14

(Color online) The distribution of avalanche sizes (without the last catastrophic event) for triangular spring lattices of different sizes.Reuse & Permissions

Figure 15

(Color online) Data collapse of the avalanche size distributions excluding the final catastrophic event. The exponents used for the collapse are $\tau =2.5$ (the reference line has this slope) and $D=1.1$. The distributions have been logarithmically binned to reduce fluctuations.Reuse & Permissions

Figure 16

(Color online) The mean avalanche size of the last catastrophic event $[s_m=(n_f-n_p)]$ scales as a power law of $N_{\mathrm{el}}$. Once again, the scaling exponent $b=0.68$ for RSM is similar to the scaling exponent $b\simeq 0.7$ obtained for RFM using triangular and diamond lattices (see Fig. 14 of Ref. 22).Reuse & Permissions

Figure 17

(Color online) The collapsed cumulative distribution of the last avalanche. (a) RSM, (b) RFM. The insets in each of these figures show how well the data can be represented by a normal distribution fit. The presence of significant nonlinearity of the data in these insets suggests that normal distribution may not be an adequate fit for representing the distribution of the last avalanche size for the RSM model, whereas it may be an adequate fit for the RFM model.Reuse & Permissions