Figure 1

A fiber bundle model having $N$ parallel fibers. The original position of the rigid platform on which force has been applied is indicated. All the fibers are assumed to have the same elastic constant (normalized to unity here) until they break, while the breaking strengths of these fibers are assumed to be randomly dispersed. As the bundle becomes strained $(x>0)$, some weaker fibers (having lower stress or strain capacity) fail, and the number of intact fibers decreases from its starting value $N$. Consequently, the stress on them (or the strain) increases due to the redistribution or transfer of loads from the failing fibers. Some of these fibers may fail further as they cannot support this extra redistributed load. The process stops if there is no further failure, and the bundle will show nonlinear elastic response (although each fiber has linear elastic behavior until breaking). Otherwise, the bundle fails when the stress concentration on the fibers (due to the dynamic stress redistribution) becomes so high that none of the fibers can withstand that.Reuse & Permissions

Figure 2

Threshold distributions: (a) The uniform distribution and (b) the Weibull distribution with $\rho =5$ (solid line) and 10 (dotted line).Reuse & Permissions

Figure 3

(Color online) Variation of effective stress $\left(x^{*}\right)$, fraction of unbroken fibers $\left(U^{*}\right)$, and the order parameter $\left(O\right)$ with the applied stress $\sigma$ for a bundle with uniform distribution [Eq. (1)] of fiber strengths.Reuse & Permissions

Figure 4

Increasing threshold distributions: (a) The density function $p\left(x\right)$ and (b) the cumulative distribution $P\left(x\right)$ of random fiber strengths $x$ distributed with linearly increasing density in the interval $[C_L,C_R]$. In the particular instance shown $C_L=0.15$ and $C_R=0.75$.Reuse & Permissions

Figure 5

The fixed points of (a) the redistributed stress and (b) the surviving fraction of fibers for the distribution of fiber strengths shown in Fig. 4. In each part the curve for the stable fixed points is shown by a bold solid line and those for the unstable fixed points are shown by bold broken lines. We have $C_L=0.15$ and $C_R=0.75$, so that ${\sigma }_c=0.3375$; the position of the critical point is marked by an arrowhead. For $\sigma ⩽C_L$ the fixed points are trivial: since there are no broken fibers $x^{*}=\sigma$ and $U^{*}=U_0=1$.Reuse & Permissions

Figure 6

Decreasing threshold distribution: (a) The density function $p\left(x\right)$ and (b) the cumulative distribution $P\left(x\right)$ of fiber strengths $x$ distributed with linearly decreasing density in the interval $[C_L,C_R]$. Similar to the cases shown in Fig. 4 we have $C_L=0.15$ and $C_R=0.75$ in this example also.Reuse & Permissions

Figure 7

The fixed points of (a) the redistributed stress and (b) the surviving fraction of fibers for the distribution of fiber strengths shown in Fig. 6. The curve for the stable fixed points is shown by a bold solid line and those for the unstable fixed points are shown by bold broken lines. In this example too we have $C_L=0.15$ and $C_R=0.75$; here ${\sigma }_c=0.173611$, marked by an arrowhead. The critical point is located lower than that in Fig. 5 due to an abundance of fibers of lower strengths compared to the previous case.Reuse & Permissions

Figure 8

Branching ratio as a function of applied stress for three different $\rho$ (Weibull index). From 113.Reuse & Permissions

Figure 9

Number of relaxation steps $t_f\left(\sigma \right)$ for a fiber bundle with a uniform threshold distribution [Eq. (1)]. Here ${\sigma }_c=0.25$. The figure is based on 1000 samples, each with $N={10}^6$ fibers.Reuse & Permissions

Figure 10

Simulation results with postcritical stress for (a) the uniform threshold distribution [Eq. (1)] and (b) the Weibull distribution [Eq. (2)] with index 5. The graphs are based on 10 000 samples with $N={10}^6$ fibers in each bundle. Open circles represent simulation data and dashed lines are the theoretical estimates, Eqs. (82, 83) in (a) and Eq. (92) in (b).Reuse & Permissions

Figure 11

The iteration function $f\left(U\right)$ for the Weibull distribution [Eq. (2)] with index 5. Here $\sigma =0.6$, slightly greater than the critical value ${\sigma }_c=0.5933994$.Reuse & Permissions

Figure 12

Simulation results with precritical stress for (a) the uniform threshold distribution [Eq. (1)] and (b) the Weibull distribution [Eq. (2)] with index 5. The graphs are based on 10 000 samples with $N={10}^6$ fibers in each bundle. Open circles represent simulation data and the dotted lines are the theoretical estimates, Eq. (101) in (a) and Eqs. (111, 112) in (b).Reuse & Permissions

Figure 13

The stress-strain curve for an ELS bundle having uniform fiber strength distribution with a low cutoff $C_L=0.2$.Reuse & Permissions

Figure 14

The number of steps of load increase (until final failure) plotted against $C_L$ for an ELS model having 50 000 fibers. The dotted line represents the analytic form [Eq. (136)], the triangles are the simulated data for a strictly uniform strength distribution, and the circles represent the data (averages are taken for 5000 samples) for a uniform on average distribution.Reuse & Permissions

Figure 15

The solid curve indicates the total force $F\left(x\right)$ as a function of $x$. However, when our control parameter is $F$ rather than $x$, the system will follow the dotted line [Eq. (138)]. The bursts are the horizontal parts of $F_{\mathit{ph}}\left(x\right)$. Here $N=100$.Reuse & Permissions

Figure 16

The burst distribution $D\left(\Delta \right)∕N$ for (a) the uniform distribution and (b) the Weibull distribution with index 5. The dotted lines represent the power law with exponent $\xi =5∕2$. Both are based on 20 000 samples of bundles each with $N={10}^6$ fibers.Reuse & Permissions

Figure 17

The macroscopic bundle strength $⟨F⟩\left(x\right)$ for the distribution [Eq. (177)], with $x_0=2x_r$, and for ${\alpha }_0=1∕3$ (upper curve), $1∕2$ (middle curve), and $2∕3$ (lower curve). The broken part of the ${\alpha }_0=2∕3$ curve is unstable and the macroscopic bundle strength will follow the solid line.Reuse & Permissions

Figure 18

The distribution of bursts for thresholds uniformly distributed in an interval $(x_0,x_c)$, with $x_0=0$ and $0.9x_c$. Based on 50 000 samples, each containing $N={10}^6$ fibers. The exponents of the distributions are $\xi =5∕2$ and $3∕2$, respectively.Reuse & Permissions

Figure 19

The distribution of bursts for the uniform threshold distribution for a single fiber bundle with $N={10}^7$ fibers. Results with $x_0=0$, i.e., when all bursts are recorded, are shown as squares and data for bursts near the critical point $(x_0=0.9x_c)$ are shown as circles.Reuse & Permissions

Figure 20

A fuse model of size $100\times 100$. Each bond is a fuse with a burn-out threshold $t$ drawn from a probability distribution $p\left(t\right)$.Reuse & Permissions

Figure 21

The burst distribution based on 300 sample random fuse lattices of size $100\times 100$. The threshold $t$ is uniformly distributed on the unit interval. On average, catastrophic failure sets in after 2097 fuses have blown. The circles denote the burst distribution measured throughout the entire breakdown process. The squares denote the burst distribution based on bursts appearing after the first 1000 fuses have blown. The triangles denote the burst distribution after 2090 fuses have blown. The two straight lines indicate power laws with exponents $\xi =3$ and 2, respectively.Reuse & Permissions

Figure 22

Power dissipation $E$ as a function of the number of broken bonds (upper) and as a function of the total current $I$ flowing in the fuse model (lower).Reuse & Permissions

Figure 23

The power dissipation avalanche histogram $D_d\left(\Delta \right)$ for the fuse model. The slopes of the two straight lines are $-2.7$ and $-1.9$, respectively. The circles show the histogram of avalanches recorded after 1000 fuses have blown, whereas the squares show the histogram recorded after 2090 fuses have blown. This is close to catastrophic failure.Reuse & Permissions

Figure 24

Mixed uniform distribution.Reuse & Permissions

Figure 25

Total avalanche size distribution $D\left(\Delta \right)$, $D_1\left(\Delta \right)$, and $D_2\left(\Delta \right)$ obtained by numerical integration of Eqs. (200, 201). (a) corresponds to $x_1=0.08,x_2=0.28$, and $f=0.1$ and (b) corresponds to $x_1=0.25$, $x_2=0.42$, and $f=0.3$. As $x_2\to 0.5$, the nonuniversal region increases in the small $\Delta$ region, whereas $\Delta \propto {\Delta }^{-5∕2}$ for large $\Delta$. The dotted line has a slope of $-5∕2$. From 41.Reuse & Permissions

Figure 26

Avalanche size distribution for the uniform threshold distribution [Eq. (1)] when the load is increased in steps of $\delta =10$ and 50 (upper curve). The dotted lines show the theoretical asymptotics [Eq. (208)] for $\delta =10$ and 50. Based on 10 000 samples with $N={10}^6$ fibers in the bundle.Reuse & Permissions

Figure 27

Avalanche size distribution for the Weibull distribution [Eq. (2)] with index 5. Open circles represent simulation data, dashed lines are analytic expressions [Eq. (213)], and the dotted line is the asymptotic power law with exponent $-3$. The load is increased in steps of $\delta =20$. Based on 10 000 samples of bundles with $N={10}^6$ fibers.Reuse & Permissions

Figure 28

Simulation results for $g\left(E\right)$ characterizing energy bursts in fiber bundles with (a) the uniform threshold distribution [Eq. (1)] and (b) the Weibull distribution [Eq. (2)] of index 2. The graphs are based on 1000 samples with $N={10}^6$ fibers in each bundle. Open circles represent simulation data and dashed lines are the theoretical results [Eqs. (226, 227)] for the asymptotics.Reuse & Permissions

Figure 29

Simulation results for the burst distribution $g\left(E\right)$, in the low-energy regime, for the uniform threshold distribution (circles), the Weibull distribution with $\rho =2$ (triangles), and Weibull distribution with $\rho =5$ (squares). The graphs are based on 1000 samples with $N={10}^6$ fibers in each bundle.Reuse & Permissions

Figure 30

A fiber bundle with periodic boundary conditions. The externally applied force $F$ is the control parameter.Reuse & Permissions

Figure 31

Burst distribution in local model as found numerically for 4 000 000 samples with $N=20000$ fibers (crosses) and calculated from Eq. (233) (boxes). The straight line shows the power law ${\Delta }^{-5}$ and the broken curve the function $\exp (-\Delta ∕{\Delta }_0)$ with ${\Delta }_0=1.1$. Note the small value of ${\Delta }_0$.Reuse & Permissions

Figure 32

Numerical estimate of the upper bound of $C_L$ in the LLS model: for $C_L⩾0.5$ the average step values go below 1.5, i.e., the bundle fails at one step in most of the realizations.Reuse & Permissions

Figure 33

The strength $-C_L$ is plotted against $1∕N$ for different $C_L$ values: 0.3 (square), 0.35 (circle), 0.4 (up triangle), 0.45 (down triangle), and 0.5 (star). All the straight lines approach 0 as $N\to \infty$.Reuse & Permissions

Figure 34

Space-time diagram of breaking sequences of the interpolating fiber bundle model of 136. Black lines represent broken fibers and white regions are unbroken parts of the bundle. As the interpolation parameter $g$ is increased, there is increasing localization of the failing fibers.Reuse & Permissions

Figure 35

Strength at failure ${\sigma }_c$ as a function of $\gamma$ in the variable range fiber bundle of 74.Reuse & Permissions

Figure 36

Strength at failure ${\sigma }_c$ as a function of the number of fibers $N$ for different range exponents $\gamma$ in the variable range fiber bundle of 74.Reuse & Permissions

Figure 37

Burst size distribution for different range exponents $\gamma$ in the variable range fiber bundle of 74. The data points where $\gamma ⩽2.0$ can be fitted to a power law with exponent $\xi =5∕2$. For larger $\gamma$, the slope becomes steeper and resembles the behavior seen in the local load-sharing model (see Fig. 31).Reuse & Permissions

Figure 38

Size distribution of clusters of broken bonds at collapse in the variable range fiber bundle of 74. There is no evident power-law behavior.Reuse & Permissions

Figure 39

(Color online) Burst size distributions in the variable-range fiber bundle model for different $\gamma$ and ${ϵ}_L$ values: $\gamma =\left(\mathrm{a}\right)$ 2.0, (b) 2.5, (c) 3.0, and (d) 6.0. From 145.Reuse & Permissions

Figure 40

Force-displacement curve, $128\times 128$ systems with $e=10$.Reuse & Permissions

Figure 41

Force-displacement curve, $128\times 128$ systems with $e=100$.Reuse & Permissions

Figure 42

Burst distribution for $128\times 128$, $e=10$. The slope of the straight line is $-2.5$.Reuse & Permissions

Figure 43

Burst distribution for $128\times 128$, $e=100$. The slope of the straight line is $-2.5$.Reuse & Permissions

Figure 44

Area distribution of zones where glue has failed for systems of size $128\times 128$ and elastic constant $e=10$. The straight line is a least-squares fit and indicates a power law with exponent $-1.6$.Reuse & Permissions

Figure 45

Failure time ${\tau }_f$ of a homogeneous bundle $(k{T}_d=0)$ in a creep test. (a) ${\tau }_f$ as a function of the normalized force ${(1-x_0)}^2$ for several values of thermal noise variance $kT$: $kT=0.0045$ (cross); 0.006 (circle); and 0.01 (triangle). (b) ${\tau }_f$ as a function of $1∕kT$ for several values of $x_0$: $x_0=0.45$ (box); 0.54 (circle); and 0.7 (triangle). Continuous lines in (a) and (b) are the fits with Eqs. (259, 260), respectively. Adapted from 26.Reuse & Permissions

Figure 46

Failure time ${\tau }_f$ of a heterogeneous bundle $(k{T}_d\ne 0)$ in creep test. The lifetime ${\tau }_f$ is plotted as a function of $1∕kT$ at $x_0=0.45$. Different symbols correspond to different values of $k{T}_d$: $k{T}_d=0$ (circle); 0.02 (triangle); and 0.04 (box). Adapted from 26.Reuse & Permissions

Figure 47

The simulation results showing variation of average failure time ${\tau }_f$ against (a) external stress $\sigma$ and (b) noise $T$ ($k=1$ here) for a homogeneous bundle containing $N={10}^5$ fibers. The insets show the variation of the fraction $U$ of unbroken fibers with time $t$ for different $T$ values [1.2 (cross) and 1.0 (plus)] in (a) and different $\sigma$ values [0.15 (cross) and 0.12 (plus)] in (b). The dotted and dashed lines represent the theoretical result [Eq. (277)].Reuse & Permissions

Figure 48

Simulation results of fatigue behavior: (a) average failure time ${\tau }_f$ vs noise $T$ ($k=1$ here) for three different stress values and (b) ${\tau }_f$ vs $\sigma$ for three different noise values. The bundle contains $N={10}^5$ fibers with uniformly distributed strength thresholds. The time variations of a fraction of surviving fibers are shown in the insets. The dotted lines in (a) and (b) correspond to the fit with expression [Eq. (278)] where ${\sigma }_c\simeq 0.245$ [exact $\text{value}=1∕4$ (130; 17)].Reuse & Permissions

Figure 49

The viscoelastic fiber bundle: intact fibers are modeled by Kelvin-Voigt elements which consist of a spring and a dashpot in parallel. From 91.Reuse & Permissions

Figure 50

The behavior of the time to failure $t_f$ for uniform and Weibull distributions with two different Weibull indices for the ELS case. All three curves are parallel to each other on a double logarithmic plot with an exponent close to 0.5, in agreement with the general result [Eq. (288)]. From 72.Reuse & Permissions

Figure 51

The fibers are modeled by Maxwell elements. From 91.Reuse & Permissions

Figure 52

Lifetime $t_f$ as a function of the distance from the critical point, ${\sigma }_o-{\sigma }_c$, for two different values of the parameter $m$. The number of fibers in the bundle is $N={10}^7$. From 91.Reuse & Permissions

Figure 53

(Color online) Asphalt samples set up for experimental testing of fatigue failure. (a) shows how fatigue failure under these experimental conditions is modeled using a fiber bundle model and (b) shows a postfailure sample. From 90.Reuse & Permissions

Figure 54

Deformation $\epsilon$ as a function of number of cycles $N_{\text{cycle}}$, showing both experimental data and theoretical curves based on the fiber bundle model. From 90.Reuse & Permissions

Figure 55

Number of cycles at catastrophic failure $N_f$ as a function of load amplitude ${\sigma }_0∕{\sigma }_c$. The two curves show experimental and fiber bundle model data. From 90.Reuse & Permissions

Figure 56

Variation of ${\chi }^{-2}$ and ${\tau }^{-2}$ with applied stress for a bundle having $N=50000$ fibers. Uniform distribution of fiber thresholds is considered and averages are taken over 1000 samples. The dotted straight lines are the best linear fits near the critical point.Reuse & Permissions

Figure 57

Variation of ${\chi }^{-2}$ and ${\tau }^{-2}$ with applied stress for a single bundle having $N=10000000$ fibers with uniform distribution of fiber thresholds. Straight lines represent the best linear fits near the critical point.Reuse & Permissions

Figure 58

Log-log plot of breaking rate with step of load redistribution for seven different stress values. Circles are for stresses below the critical stress and triangles are for stresses above the critical stress. The simulation has been performed for a single bundle with $N=10000000$ fibers having uniform distribution of fiber thresholds. The dotted straight line has a slope $-2$.Reuse & Permissions

Figure 59

The breaking rate $R\left(t\right)$ vs step $t$ (upper plot) and vs the rescaled step variable $t∕t_f$ (lower plot) for the uniform threshold distribution for a bundle of $N={10}^7$ fibers. Different symbols are used for different excess stress levels $\sigma -{\sigma }_c$: 0.001 (circles), 0.003 (triangles), 0.005 (squares), and 0.007 (crosses).Reuse & Permissions

Figure 60

The avalanche size distributions for different values of $x_0$ in the ELS model with uniform fiber strength distribution. Here bundle size $N=50000$ and averages are taken over 10 000 samples. Two power laws (dotted lines) have been drawn as reference lines to compare the numerical results.Reuse & Permissions

Figure 61

(Color online) Crossover signature in the local magnitude distributions of earthquakes in Japan. The exponent of the distribution during $100\text{days}$ before a mainshock is about 0.60; much smaller than the average value of 0.88. From 86.Reuse & Permissions

Figure 62

(Color online) This dental bridge is made from a fiber-reinforced composite with braided fibers made from polyethylene. Due to the braided structure, this composite is four times tougher than a composite made from the same materials but without the braiding (85). Courtesy of H. Strassler, University of Maryland Dental School and V. Karbhari, University of Alabama, Huntsville.Reuse & Permissions

Figure 63

Postmortem micrograph showing a fiber-reinforced composite where the matrix has undergone brittle failure followed by failure of the bridging fibers through debonding. From 85.Reuse & Permissions

Figure 64

The discretization of a three-dimensional unidirectional fiber-reinforced composite used by 184.Reuse & Permissions

Figure 65

Fiber bundle model on networks: (a) The system size $\left(N\right)$ dependence of critical points $\left({\sigma }_c\right)$ for various networks with $N=2^8,2^9,\dots ,2^{15}$ vertices. (b) The susceptibility for the networks with $N=2^{14}$. $p$ and $\gamma$ are the rewiring probability in the WS networks and the exponent of degree distribution $P\left(k\right)\sim k^{-\gamma }$, respectively. The data points are obtained from the averages over ${10}^4$ (${10}^3$ for $N=2^{15}$) ensembles. From 88.Reuse & Permissions

Figure 66

The highway traffic current $I_O$ gets fragmented into uniform currents $i$ in each of the narrower roads and the roads having threshold current $i_{c_n}⩽i$ get congested or blocked. This results in extra load for the uncongested roads. We assume that this extra load per uncongested roads gets equally redistributed and gets added to the existing load, causing further blocking of some more roads.Reuse & Permissions

Figure 67

The transport current $I$ in the one-dimensional lane or road is possible if, say, the $l\text{th}$ site is occupied and the $(l+1)\text{th}$ site is vacant. The intersite hopping probability is indicated by $J$.Reuse & Permissions