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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Serie: Statistical Physics of Fracture and Breakdown
- Sprache: Englisch
- Veröffentlichungsjahr: 2015
Gathering research from physics, mechanical engineering, and statistics in a single resource for the first time, this text presents the background to the model, its theoretical basis, and applications ranging from materials science to earth science.
The authors start by explaining why disorder is important for fracture and then go on to introduce the fiber bundle model, backed by various different applications. Appendices present the necessary mathematical, computational and statistical background required.
The structure of the book allows the reader to skip some material that is too specialized, making this topic accessible to the engineering, mechanics and materials science communities, in addition to providing further reading for graduate students in statistical physics.
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Table of Contents
Cover
Related Titles
Title Page
Copyright
Series Page
Preface
Chapter 1: The Fiber Bundle Model
1.1 Rivets Versus Welding
1.2 Fracture and Failure: A Short Summary
1.3 The Fiber Bundle Model in Statistics
1.4 The Fiber Bundle Model in Physics
1.5 The Fiber Bundle Model in Materials Science
1.6 Structure of the Book
Chapter 2: Average Properties
2.1 Equal Load Sharing versus Local Load Sharing
2.2 Strain-Controlled versus Force-Controlled Experiments
2.3 The Critical Strength
2.4 Fiber Mixtures
2.5 Non-Hookean Forces
Chapter 3: Fluctuation Effects
3.1 Range of Force Fluctuations
3.2 The Maximum Bundle Strength
3.3 Avalanches
Chapter 4: Local and Intermediate Load Sharing
4.1 The Local-Load-Sharing Model
4.2 Local Load Sharing in Two and More Dimensions
4.3 The Soft Membrane Model
4.4 Intermediate-Load-Sharing Models
4.5 Elastic Medium Anchoring
Chapter 5: Recursive Breaking Dynamics
5.1 Recursion and Fixed Points
5.2 Recursive Dynamics Near the Critical Point
Chapter 6: Predicting Failure
6.1 Crossover Phenomena
6.2 Variation of Average Burst Size
6.3 Failure Dynamics Under Force-Controlled Loading
6.4 Over-Loaded Situations
Chapter 7: Fiber Bundle Model in Material Science
7.1 Repeated Damage and Work Hardening
7.2 Creep Failure
7.3 Viscoelastic Creep
7.4 Fatigue Failure
7.5 Thermally Induced Failure
7.6 Noise-Induced Failure
7.7 Crushing: The Pillar Model
Chapter 8: Snow Avalanches and Landslides
8.1 Snow Avalanches
8.2 Shallow Landslides
Appendix A: Mathematical Toolbox
A.1 Lagrange's Inversion Theorem
A.2 Some Theorems in Combinatorics
A.3 Biased Random Walks
A.4 An Asymmetrical Unbiased Random Walk
A.5 Brownian Motion as a Scaled Random Walk
Appendix B: Statistical Toolbox
B.1 Stochastic Variables, Statistical Distributions
B.2 Order Statistics
B.3 The Joint Probability Distribution
Appendix C: Computational Toolbox
C.1 Generating Random Numbers Following a Specified Probability Distribution
C.2 Fourier Acceleration
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 1: The Fiber Bundle Model
Figure 1.1 The Boeing 737 after the explosive decompression that occurred during flight on April 28, 1988, in Hawaii. (Photo credit: National Transportation Safety Board)
Figure 1.2 The Schenectady after it broke into two on January 16, 1943, in dock in Portland, Oregon. The ship had just been finished and was being outfitted. The failure was sudden and unexpected.
Chapter 2: Average Properties
Figure 2.1 A fiber bundle model stressed by an external force
F
. A bundle is clamped between two rigid supports. The force has displaced one support a distance
x
from its original position (sketched). This has caused some fibers to fail, while other fibers are intact.
Figure 2.2 The behavior of the strain–force relation near the first fiber failure. (a) In a strain-controlled situation, the force drops abruptly. (b) In a force-controlled situation, the strain increases abruptly. Figure (c) is identical to (b) with coordinate axes interchanged. Figure (d) contains both the strain-controlled situation (fully drawn lines) and the force-controlled case with dashed lines.
Figure 2.3 A sketch of how the real elastic force on the bundle may vary with increasing strain
x
for a finite
N
. In a strain-controlled experiment, the bundle follows the solid graph. In a force-controlled experiment, however, the system complies with the non-decreasing graph with the dashed lines.
Figure 2.4 The solid curve represents for the force per fiber, , as a function of
x
. The dashed lines show when it exceeds . In the limit the parabolic dotted curve is obtained.
Figure 2.5 The uniform distribution (a) and the Weibull distribution (b) with (solid line) and (dotted line).
Figure 2.6 The critical strength per fiber, , for a fiber bundle with thresholds satisfying the Weibull distribution (2.16), as a function of the Weibull index
k
.
Figure 2.7 The average force per fiber,, as a function of
x
for the piecemeal uniform distribution (2.22) with (a) , and (b) .
Figure 2.8 The average force per fiber, , as a function of
x
for the threshold distribution (2.25).
Figure 2.9 The force on fiber
n
at extension
x
. The size of the slack is .
Figure 2.10 The force per fiber, , on fiber
i
as function of its elongation
x
. The elastic regime is , the plastic regime corresponds to . The constant force in the plastic domain equals .
Chapter 3: Fluctuation Effects
Figure 3.1 Two realizations for the force per fiber for a bundle with fibers, as a function of the extension
x
. The uniform fiber strength distribution is assumed. For comparison, a realization with is shown. For such a large number of fibers, fluctuations are tiny, so that the resulting force per fiber deviates little from the parabolic average force, , for this model.
Figure 3.2 The average bundle strength as function of the number
N
of fibers, for bundles with a uniform distribution of the fiber strengths. The dotted line represents a power law with an exponent of . The simulations are based on 10 000 samples for each value of
N
.
Figure 3.3 The average extension beyond criticality, , at which the maximum force occurs, as function of the number of fibers
N
. The bundles are assumed to have a uniform distribution of the fiber strengths. The dotted line represents the power law (3.28). The simulations are based on 10 000 samples for each value of
N
.
Figure 3.4 The Figure shows an example of how the sequence may vary with the fiber number
j
. When the external load compels fiber
k
to fail, the fibers and must necessarily also rupture at the same time. Thus, a burst of size will take place in this example.
Figure 3.5 The probability density for bursts of size , 2, 3, 10, and 20 for a fiber bundle with a uniform threshold distribution, for . The values of are indicated on the graphs.
Figure 3.6 The average burst length as a function of . The fully drawn graph is for the uniform distribution, for , for which . The dashed graph is for the Weibull distribution of index 3, , for which .
Figure 3.7 Simulation results for the normalized avalanche distribution , for strains in the window . Plusses correspond to , crosses correspond to , stars correspond to , squares correspond to , and circles correspond to .
Figure 3.8 Macroscopic force curves are sketched for the value . The values of are 1/3 (upper curve), 1/2 (middle curve), and 2/3 (lower curve). The dashed part of the curve is unstable, and the bundle strength will follow the solid line.
Figure 3.9 The Figure shows an example of a sequence of forces in which both a large burst of size 8 and an smaller internal burst of size 3 are produced.
Figure 3.10 The distribution of inclusive bursts, for the uniform threshold distribution within 0 and 1. The straight line is a plot of Eq. (3.87). The simulation results are based on 1000 bundles with fibers each.
Figure 3.11 The Figure shows an example of a forward burst in a sequence of forces.
Figure 3.12 The distribution of forward bursts, , for the uniform threshold distribution on the unit interval. The straight line shows the asymptotic distribution (3.92). The simulation results are based on 1000 bundles with N=1 000 000 fibers.
Figure 3.13 The probability distribution of the step lengths in the exact one-dimensional random walk.
Figure 3.14 Simulation results of the energy density
g
(
E
) for (a) the uniform distribution and (b) the Weibull distribution of index 2. Open circles represent simulation data, and dashed lines represent the theoretical result (3.130–3.131). In each case, the graphs are based on samples with fibers in each bundle.
Figure 3.15 Simulation results for the energy burst distribution
g
(
E
) in the low-energy regime, for the uniform threshold distribution (circles), the Weibull distribution of index 2 (triangles), and the Weibull distribution of index 5 (squares). The graphs are based on samples with fibers in each bundle.
Figure 3.16 Simulation results for the avalanche size distribution when the load is increased in steps of and (for large avalanche sizes the graph on the right correspond to ). A uniform distribution of fiber strengths is assumed. The dotted lines represent the theoretical asymptotics (3.138) for the two cases, with behavior. The graphs are based on samples with fibers in each bundle.
Figure 3.17 Avalanche size distribution for the Weibull threshold distribution of index 5, , with discrete load increase. The load has been increased in steps of . Open circles represent simulation data, the dashed graph is the theoretical result (3.147), while the dotted line represents the asymptotic power law with exponent . The simulation is based on samples of bundles, each with fibers.
Chapter 4: Local and Intermediate Load Sharing
Figure 4.1 An illustration of the equal-load-sharing fiber bundle model in terms of a practical device. When the clamps are moved apart a distance
x
by turning the handle, the fibers will be stretched by the same amount due to the clamps being infinitely stiff.
Figure 4.2 The soft clamp fiber bundle model where the fibers are placed between an infinitely stiff clamp and a soft clamp. The soft clamp responds elastically to the forces carried by the fibers. The distance between the two clamps is
x
as illustrated in the Figure However, the fibers are not extended accordingly as was the case in the equal-load-sharing model, see Figure 4.1.
Figure 4.3 The local-load-sharing fiber bundle model illustrated with the same device as in Figure 4.2 for the soft clamp model. The soft clamp to the right in Figure 4.2 has been substituted for a clamp that reacts as an infinitely stiff clamp for the fibers that have intact neighbors. Where there are missing fibers, the clamp deforms in such a way that the fibers next to the missing ones are stretched further so that the force carried by these equals the force that would have been carried by the missing fibers. We denote this clamp as being “hard/soft.”
Figure 4.4 Inverse of critical load per fiber versus based on samples for up to 2000 samples for . The threshold distribution was uniform on the unit interval.
Figure 4.5 The inverse of the probability to find a hole of size 2 when two fibers have failed as a function of
N
for the threshold distribution for . The data points are based on samples each. The straight line is .
Figure 4.6 The integration area used in calculating in Eq. (4.17).
Figure 4.7 as a function of
N
from Eq. (4.21) compared with numerical simulations based on samples for each
N
. We furthermore compare with the asymptotic expression .
Figure 4.8 for the exponential threshold distribution, Eq. (4.32) compared with numerical calculations for and . The statistics is based on samples for each curve.
Figure 4.9 for the threshold probability given in Eq. (4.34) as a function of the lower cutoff for . We have set .
Figure 4.10 Simulation results for for the equal-load-sharing model for and based on samples (circles). Equation (4.47) is plotted as squares.
Figure 4.11 for the local-load-sharing model for and based on samples. The threshold distribution was uniform on the unit interval.
Figure 4.12 for the local-load-sharing model and equal-load- sharing model for and based on samples. The threshold distribution was uniform on the unit interval.
Figure 4.13 Inverse critical stress versus based on samples for to 2000 samples for (crosses). We also show the predictions of Eqs. (4.82) (broken curve) and (4.83) (dotted curve). The derived approximative solution (4.80) is also shown (black solid line).
Figure 4.14 versus with the values given in Figure 4.10, and . The threshold distribution was uniform on the unit interval. Each curve is based on 2000 samples.
Figure 4.15 versus
k
based on the uniform threshold distribution on the unit interval. The straight line signifies localization and we expect it to follow with . The Figure is based on 2000 samples for each
N
value.
Figure 4.16 versus
k
based on the threshold distribution with . We have set and 1. The Figure is based on 2000 samples with .
Figure 4.17 versus
k
based on the threshold distribution with . We have set and 1. The two straight lines are with () and (), respectively. The Figure is based on 2000 samples with .
Figure 4.18 Burst distribution for the threshold distributions with . The Figure is based on 20 000 samples with for each value of .
Figure 4.19 Burst distribution for the threshold distributions with and , 1 and 2. The Figure is based on 20 000 samples of size for each data set.
Figure 4.20 Inclusive burst distribution in the local-load-sharing model for the threshold distributions with . The curve fits the data very well. The Figure is based on 20 000 samples when .
Figure 4.21 Here we see the two-dimensional local-load-sharing model from “above”. Each intact fiber is shown as a black dot and each failed fiber as a white dot. The system size is . We show the model when fibers have failed. The cumulative threshold distribution was where . In the left panel, we have and in the right panel .
Figure 4.22 The invasion percolation model: a random number – here an integer between 0 and 100 – is assigned to each square in the tiling. We then invade the tiling from below, always choosing the tile with the smallest random number assigned to it, which is next to the already invaded tiles. We illustrate the process after five tiles have been invaded. We have marked the tile with the smallest random number next to the invaded tiles. At the next step, this tile is invaded as shown in the right panel.
Figure 4.23 The size of the largest hole
M
in the two-dimensional Local-load-sharing model as a function of the number of broken fibers,
k
. The threshold distribution was where and the number of fibers . Each data set is based on 5000 samples.
Figure 4.24 The size of the largest hole
M
in the two-dimensional local-load-sharing model as a function of the number of broken fibers,
k
for different values of
N
. The threshold distribution was uniform on the unit interval. Each curve is based on 5000 samples.
Figure 4.25 This is further on in the breakdown process shown in the right panel in Figure 4.21. This breakdown process has been localized – generating a single hole – from the very start. In this figure, 13 568 fibers have failed and those that remain form isolated islands surrounded by the same “sea” of failed fibers. Hence, the remaining fibers all carry the same stress.
Figure 4.26 versus for the two-dimensional local-load-sharing model. The threshold distribution is uniform on the unit interval. The fully drawn graph shows the equal-load-sharing result . The Figure is based on 5000 samples of each size.
Figure 4.27 versus for the two-dimensional local-load-sharing model. The cumulative threshold probability was for . We also show the equal-load-sharing result . The Figure is based on 5000 samples for each curve.
Figure 4.28 Histogram of bursts in the two-dimensional local-load-sharing model. The threshold distribution was where and the number of fibers . Each data set is based on 5000 samples.
Figure 4.29 Comparing the three-dimensional local-load-sharing model with and (upper panel) and the four-dimensional local-load-sharing model with (lower panel) to the equal-load-sharing model containing the same number of fibers. The threshold distribution was uniform on the unit interval. The three-dimensional data set have been averaged over 80 000 samples, and the four dimensional data set has been averaged over 30 000 samples (From [36]). This Figure should be compared to Figure 4.26.
Figure 4.30 as a function of dimensionality
D
. The straight line is . The data are based on those presented in Figure 4.26 and 4.29 (Data from [36]).
Figure 4.31 Apparatus illustrating the soft membrane model.
Figure 4.32 Close-up of the soft membrane model where we define .
Figure 4.33 Critical stress as a function of the number of fibers
N
in the -model. (From Ref. [42].)
Figure 4.34 Density of the largest cluster of failed fibers, , as a function of for and . Here . The vertical bar indicates the percolation critical density 0.59274. Here and 100 samples were generated. (From Ref. [45]).
Figure 4.35 The soft clamp model seen from “above.” Failed fibers are denoted as black. In the left panel, and localization has not yet set in. In the right panel, and the only fibers that fail at this point are those on the border of the growing hole. Here, and the rescaled Young modulus was set to . (From Ref. [45].)
Figure 4.36 Critical versus for the soft clamp model with the rescaled Young modulus of the clamp being either or . In the case of a stiffer clamp, the value of approaches 0.5, which is the equal-load-sharing value. For the soft clamp, critical loading as a function of . There is convergence toward a finite value of as . (Adapted from Ref. [51]).
Figure 4.37 The experimental setup by Schmittbuhl and Måløy [52] and later on used by other authors. (Figure credit: K. J. Måløy.)
Figure 4.38 The fracture front as seen in the experimental setup shown in Figure 4.37. It moves in the positive
y
-direction. (Photo credit: K. J. Måløy.)
Figure 4.39 A graphical representation of the waiting time matrix for a front moving with an average velocity of m s. Each 1/50 s, the position of the front is recorded and added to this Figure by adding to the waiting time matrix. Each pixel is gray colored by the time the front has been sitting at that pixel. The bar to the right shows the relation between time and gray shade. By this rendering, the stick-slip – or jerky – motion of the front is clearly visible. (From Ref. [60].)
Figure 4.40 Distribution of local velocities scaled by the average velocity from the experimental study by Tallakstad
et al.
[60] (left panel) and from the numerical study by Gjerden
et al.
[64] (right panel). In the upper panel, the normalized velocity distribution based on the waiting time matrix technique is shown. The velocity distribution follows a power law with exponent in the depinning regime. The numerical results in the lower panel are based on simulations of sizes and with an elastic constant equal to and , respectively. The threshold gradient was in the range . A fit to the data for yields a power law described by an exponent . The “pinning” and “depinning” regimes refer to how the front moves. The pinning regime is characterized by small incremental position changes of the front, and the associated velocities are small, whereas in the depinning regime, it is dominated by the front sweeping over large areas in avalanches. These events are characterized by large velocities.
Chapter 5: Recursive Breaking Dynamics
Figure 5.1 The order parameter as function of the stress , for the uniform threshold distribution.
Figure 5.2 The increasing fiber strength distribution (5.33).
Figure 5.3 The average total force per fiber, , for the increasing strength distribution (5.33).
Figure 5.4 The relation between the fixed-point value and the applied stress for the increasing strength distribution (5.33). The maximum and minimum values are and occur at .
Figure 5.5 Simulation for the number of iterations until every fiber is broken, for a bundle with the uniform threshold distribution on the unit interval. The simulation results, marked with asterisks, are averaged over 100 000 samples with fibers in each bundle. The solid curve is the upper bound, Eq. (5.74), and the dashed curve is the lower bound, Eq. (5.75).
Figure 5.6 Iterations for the slightly supercritical uniform fiber strength distribution model (). The path of the iteration moves to and fro between the diagonal and the iteration function .
Figure 5.7 The iteration function (5.81) for the Weibull threshold distribution with index 5, together with the start of the iteration path. Here , slightly larger than the critical value .
Figure 5.8 Simulation results for the number of iterations necessary to reach equilibrium for the uniform threshold distribution (5.16). The graph is based on samples with fibers in each bundle. The dotted line is the theoretical estimate (5.109).
Figure 5.9 Simulation results for the number of iterations necessary to reach equilibrium for the Weibull distribution with index 5. The graph is based on 1000 samples with fibers in each bundle. The dashed line is the theoretical estimate (5.117).
Chapter 6: Predicting Failure
Figure 6.1 The distributions of bursts in the fiber bundle model with and . The Figure is based on samples with fibers with uniformly distributed fiber thresholds between 0 and 1.
Figure 6.2 The distribution of bursts for the uniform threshold distribution (a) with and for a Weibull distribution of index 10 (b) with (square) and (circle). Both the Figure are based on samples with fibers each. The straight lines represent two different power laws, and the arrows indicate the crossover points and , respectively.
Figure 6.3 Burst distributions: all bursts (squares) and bursts within an interval (circles). The Figure is based on a single bundle containing fibers with uniformly distributed fiber thresholds between 0 and 1.
Figure 6.4 Distribution of first bursts (squares) and total bursts (circles) for the critical strength distribution with . The simulation results are based on samples with fibers. The “star” symbol stands for the analytic result (6.14).
Figure 6.5 Distribution of energy bursts (circles) for the uniform strength distribution with . The simulation results are based on samples with fibers. The “arrow” indicates the crossover energy magnitude , which follows the analytic result (6.18).
Figure 6.6 Crossover signature in the local magnitude (
m
) distributions of earthquakes in Japan. The exponent of the distribution during 100 days before a main shock is about , much smaller than the average value [74].
Figure 6.7 Each bond in this network is a fuse. A voltage
V
is applied across the network, resulting in a corresponding current
I
. As
V
is increased, the fuses will burn out one by one.
Figure 6.8 The burst distribution in the fuse model based on 300 fuse networks of size using a uniform threshold distribution on the unit interval. 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.
Figure 6.9 The power dissipation avalanche histogram in the fuse model. The slopes of the two straight lines are and , respectively. The squares show the histogram of avalanches recorded through the entire process, whereas the circles show the histogram recorded only after 2090 fuses have blown. The system size and the number of samples are the same as in Figure 6.8.
Figure 6.10 Simulation results for the average burst size versus elongation
x
for a single bundle of fibers with thresholds uniformly distributed in the unit interval. The solid line is the theoretical result (6.31).
Figure 6.11 Inverse of average burst size is plotted against
x
for the same data set as in Figure 6.10. The solid line is the theoretical expression for the inverse of burst size versus elongation
x
.
Figure 6.12 Variation of and with applied stress for a bundle with averaging over 10 000 samples. The dotted straight lines are the best linear fits near the critical point.
Figure 6.13 Variation of and with applied stress for a single bundle of fibers with uniform distribution of fiber strengths. The straight line is the best linear fit near the critical point.
Figure 6.14 The breaking rate versus step
t
(upper plot) versus the rescaled step variable (lower plot) for the uniform threshold distribution for a bundle of fibers. Different symbols are used for different excess stress levels : 0.001 (circles), 0.003 (triangles), 0.005 (squares), and 0.007 (crosses).
Figure 6.15 Simulation result for the breaking rate versus the rescaled step variable for a bundle of fibers having a Weibull threshold distribution of index 5. Different symbols are used for different excess stress levels : 0.001 (circles), 0.003 (triangles), 0.005 (squares), and 0.007 (crosses).
Figure 6.16 Simulation results for the energy emission versus step
t
(a) and versus the rescaled step variable (b) for the uniform threshold distribution for a bundle of fibers. Different symbols are used for different excess stress levels : 0.001 (circles), 0.003 (triangles), 0.007 (squares).
Figure 6.17 Simulation results for the energy emission versus the scaled step variable for a bundle of fibers obeying the Weibull threshold distribution (6.71). Different symbols are used for different excess stress levels : 0.001 (circles), 0.003 (triangles), 0.007 (squares).
Figure 6.18 Position of the emission minimum in terms of versus excess stress for the uniform fiber strength distribution, computed using the continuous time approximation. The dashed line marks the maximum stress with an emission minimum.
Figure 6.19 Energies of AE signals recorded during a rock fracturing test on Castlegate Sandstone sample at SINTEF lab.
Figure 6.20 Energy emission versus step number
t
in the load redistribution process in a bundle with fibers having uniform strength distribution within the unit interval. Different symbols indicate different stress levels: critical stress (circles); (triangles), and (squares).
Figure 6.21 Distribution of energy emissions in the same fiber bundle as in Figure 6.20 for sub-critical stresses: (circles), (triangles). The straight line has a slope .
Figure 6.22 Distribution of energy emissions in the same fiber bundle as in Figure 6.20 for stresses above the critical value: (circles), (triangles). The straight line has a slope .
Chapter 7: Fiber Bundle Model in Material Science
Figure 7.1 Load curves for Weibull fibers that can be damaged just once. The dotted curve is for the case of fibers with Weibull index , when a damaged fiber has its strength reduced by 50%. The fully drawn curve corresponds to Weibull index , and a strength reduction factor . Note that in a force-controlled experiment, the minimum is not realized, the system will follow the least monotonic function of , as indicated.
Figure 7.2 Load curves for a large bundle with thresholds following a Weibull distribution of index , with a strength reduction factor , and for different values of the maximum number
m
of damages.
Figure 7.3 Load curves for a large bundle with thresholds following a Weibull distribution of index 2, with a strength reduction factor , and for different values of the maximum number
m
of damages before collapse.
Figure 7.4 Deformation
x
as a function of time
t
due to creep.
Figure 7.5 Elongation
x
(
t
) as a function of
t
for different values of . The Figure is based on 500 samples for and one sample for and 100. .
Figure 7.6 Elongation
x
(
t
) as a function of for different values of . The straight line is . The Figure is based on the data from Figure 7.5.
Figure 7.7 Lifetime versus for different threshold distributions using the equal-load-sharing model. Here and 5 samples were generated for each . We used a threshold distribution that was flat on the unit interval.
Figure 7.8 Histogram of the number of fibers that simultaneously fail – avalanches – throughout the failure process. The data are based on and 500 samples. The threshold distribution was flat on the unit interval.
Figure 7.9 Lifetimes as function of the external load for fiber bundles with uniform and with Weibull, index , strength distributions. The slope of the graphs at low load equals the damage accumulation parameter .
Figure 7.10 Extension of fibers as function of time, for different values of the load . The fiber thresholds are uniformly distributed on the unit interval.
Figure 7.11 The general behavior of the strain change versus the strain
x
, according to Eq. (7.42). The dashed graph corresponds to the critical stress , and the intersection of a graph with the ordinate axis gives the stress value. The arrows indicate the flow direction along the
x
axis. The fixed points are marked as circles. The graphs in the Figure correspond to a Weibull threshold distribution of index , .
Figure 7.12 The asphalt sample modeled as a fiber bundle.
Figure 7.13 Experimental deformation
x
(
t
) as a function of the number of cycles . The continuous line is a fit with theoretical fiber bundle results, assuming a uniform distribution of thresholds . From Ref. [88].
Figure 7.14 The number of cycles causing total failure as a function of the load amplitude . Both experimental and numerical fiber bundle results are shown. The fibers have Weibull distributed strengths, damage accumulation parameters , and healing range . From Ref. [88].
Figure 7.15 The phase boundary in the plane for three different types of fiber strength distributions. The data points show simulation results for a homogeneous bundle with (circles), for a uniform threshold distribution (triangles), and for a Weibull distribution of index (squares). The lines are curves of the form (7.83). The number of fibers in each bundle was .
Figure 7.16 Simulation results for the waiting time distribution for three different types of fiber strength distributions. The graphs can be fitted with gamma functions of the type (7.84), where for a homogeneous bundle, and for uniform and Weibull distributions. Each bundle contains fibers.
Figure 7.17 The pillar model. The fibers have been substituted by pillars that fail under stress. The pillars are loaded under compression, and they fail according to their failure thresholds.
Figure 7.18 Photo of sandstone after being loaded uniaxially along the long axis. (Photo credit: SINTEF.)
Figure 7.19 Stress–strain relation for Red Wildmoor sandstone as determined in a triaxial test. (Curve taken from Ref. [96].)
Chapter 8: Snow Avalanches and Landslides
Figure 8.1 The upper photo shows a buried weak layer above a hoarfrost sheet. Photo from Jamieson and Schweizer [99]. In the lower picture, a schematic drawing of fibers representing the partly fractured weak layer is overlying the photo. The snow bonds are intact on the right, fractured on the left [97, 98].
Figure 8.2 Schematic representation of the model under shear. The upper plate represents the snow slab, the lower plate the hoarfrost layer. A displacement of the upper plate, orthogonal to the initial fiber direction, implies stretching of the fibers.
Figure 8.3 Roots stretched across a tension crack. (Picture credit: belop GmbH, Switzerland.)
Figure 8.4 A landslide triggering model. The hill slope is viewed as a collection of soil columns with hexagonal cross sections. In analogy with the sandpile cellular automaton [102], each column can accommodate a certain amount of earth, in units pictured as spheres. Here, loads with more than four units induce a chain reaction moving material in the downslope direction. The stabilizing mechanism is modeled as intercolumn fiber bundles. (From Ref. [100].)
Figure 8.5 Stress–strain relation for a fiber mixture representing soil samples containing roots.
List of Tables
Chapter 3: Fluctuation Effects
Table 3.1 Asymptotic behavior of the avalanche size distribution for different parameter sets in the threshold distribution (3.73)
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Book Series: Statistical Physics of Fracture and Breakdown
Editors: Bikas K. Chakrabarti and Purusattam Ray
Why does a bridge collapse, an aircraft or a ship break apart? When does a dielectric insulation fail or a circuit fuse, even in microelectronic systems? How does an earthquake occur? Are there precursors to these failures? These remain important questions, even more so as our civilization depends increasingly on structures and services where such failure can be catastrophic. How can we predict and prevent such failures? Can we analyze the precursory signals sufficiently in advance to take appropriate measures, such as the timely evacuation of structures or localities, or the shutdown of facilities such as nuclear power plants?
Whilst these questions have long been the subject of research, the study of fracture and breakdown processes has now gone beyond simply designing safe and reliable machines, vehicles and structures. From the fracture of a wood block or the tearing of a sheet of paper in the laboratory, the breakdown of an electrical network on an engineering scale, to an earthquake on a geological scale, one finds common threads and universal features in failure processes. The ideas and observations of material scientists, engineers, technologists, geologists, chemists and physicists have all played a pivotal role in the development of modern fracture science.
Over the last three decades, considerable progress has been made in modeling and analyzing failure and fracture processes. The physics of nonlinear dynamic, many-bodied and non-equilibrium statistical mechanical systems, the exact solutions of fibre bundle models, solutions of earthquake models, numerical studies of random resistor and random spring networks, and laboratory-scale innovative experimental verifications have all opened up broad vistas of the processes underlying fracture. These have provided a unifying picture of failure over a wide range of length, energy and time scales.
This series of books introduces readers – in particular, graduate students and researchers in mechanical and electrical engineering, earth sciences, material science, and statistical physics – to these exciting recent developments in our understanding of the dynamics of fracture, breakdown and earthquakes.
Preface
One of the authors, Alex Hansen, first came across the fiber bundle model in 1989 in a paper entitled Elasticity and failure of a set of elements loaded in parallel1
