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Figure 59. Size effects in notched specimens with uniform thresholds disorder. The effective scaling exponent of mean fracture strength depends on the ratio ag/L. Figure 58. Evolution of damage in a notched sample. Although diffuse damage is present, a wandering fracture line starts from the notch breaking the entire lattice. — Figure 59 Size effects in notched specimens with uniform thresholds disorder. The effective scaling exponent of mean fracture strength depends on the ratio ag/L. Figure 58. Evolution of damage in a notched sample. Although diffuse damage is present, a wandering fracture line starts from the notch breaking the entire lattice.

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Figure 1. The setup devised by Leonardo da Vinci to test the strength of metal wires. One end of the wire is suspended and the other end is attached to a basket that was slowly filled with sand. When the wire breaks, one can weigh the sand collected in the basket, determining the strength. From the original notebook [1]. Understanding how materials fracture is a fundamental problem of science and engineering even today, although the effects of geometry, loading conditions, and material characteristics on the strength of materials have been empirically investigated from antique times. Probably the first systematic mathematical analysis of the problem dates back to the pioneering works of Leonardo da Vinci [1] and Galileo Galilei [2]. In Leonardo’s notebooks one finds an inter- esting description of a tension test on metal wires [1]. Attaching a slowly filling sand bag to an end of the wire, Leonardo noticed that longer wires failed ear- lier (see Fig. I). A simple analysis based on continuum mechanics, however, would suggest that wires of the same section should carry the same stress and hence display equal strength. This lead some later commentators to hypothe- size that when Leonardo wrote smaller length, what he really meant was larger diameter [3]. As noted in Ref. [4], there is a simpler explanation: continuum mechanics treats the material as homogeneous, but this is often very far from the truth, especially for metal wires forged in the Renaissance. Disorder has profound effects on material strength, since fracture typically nucleates from weaker spots like preexisting microcracks or voids. The longer the wire, the easier it is to find a weak spot, hence the strength will decrease with length.

Figure 2. An elliptic crack in a two dimensional medium. the crack is in principle unknown, and the shape depends on the stress field acting on it. To overcome this formidable problem, one can prescribe the crack geometry and then compute the stress field. This approximation can provide a useful guidance on the mechanism of crack propagation.

Figure 3. The three modes of fracture.

‘igure 4. One-dimensional example of a crack propagating in a disordered environment. Included are crack arrest due to strong regions, deflection, branching and wandering.

Figure 5. An example of a dilute defect population. These cracks each of some size | are assumed in the simplest case to have no influence on each other, and they have a linear size distribution P(l).

Figure 6. An example of a “first-order” and a critical scenario for damage build-up, via Eets = BD(6)

Figure 7. Examples of trial scaling plots for strength distributions of porous, brittle ceramics (from Ref. [51]). By plotting the distributions in various scales one can try to establish whether the shape resembles the Gumbel or Weibull form the best. In conclusion, the issue of fracture strength from the statistical physics view- point appears to still lack, to a large degree, physics-oriented experiments, which would help to connect scaling theories to reality. It would for instance be of immense interest to have empirical characterizations of the “disorder” (as a microcrack population) from sample to sample, and a concomitant mea- surement of the sample strengths.

Figure 8. The size- dependence of concrete for two different ambient conditions (adapted from [56] * . "> size D (for D sture.

Figure 9. The typical size-effect from various experiments on concrete. It is clear that the data for flexural strength, f;, exhibits a cross-over to a scaling as a function of D, that may (perhaps) follow the Weibull size effect for large D values [74].

Figure 10. The diffuse crack surface in paper (a two-dimensional material) and the thresholded “interface”, shifted down for clarity (courtesy of L. Salminen, HUT). Here, the issue concerns how to define an unique h(x) in the case the damage field is continuous (one can see individual fibers ir the image on the other hand). The conceptually simpler case of crack roughening is in general provided by case (i), where a the final crack path is given by a one-dimensional curve h(x) in two dimensions. Two typical examples are shown in Figures[[Q]and [LJ] from paper fracture. These illustrate the diffusive nature of fracture, such that it is clear that the asymptotic scaling of h(x) can only be attained beyond some microscopic lengthscale €,, in x and h. That is, this might imply that h(x) exhibits rms fluctuations (local width) with w(l) ~ IS only for | >> &. An example is provided of such behavior in Fig.

Figure 11. An example of a 2D crack-line from a paper web, of width 6500 mm [78]. The samp has an effective exponent of about 0.6 as shown by various measures, but only above a scale of about 2-3 mm (x 30 pixels in the lateral direction).

Figure 12. Examples of crack fronts in a quasi-two-dimensional setup where a rough interface between two elastic plates is used as the substrate for the crack line. Notice the avalanche-like localized advancement of the line between subsequent frames. These can be used to measure locally the roughness exponent ¢ via the relation Ah ~ 1S where Ah is the average displacement, and | is the lateral extent of the region which moved [87].

Figure 13. The avalanche size distribution P(A) in the Plexiglas experiment of Ref. [87]. Notice the independence of the scaling exponent of the actual average crack velocity.

Figure 14. Roughness scaling measured in a metallic alloy broken in tension. Zmazx(r), the maximum h-value difference inside a window of size r shows scaling over five decades with an exponent of ¢ ~ 0.78. The inset shows the overall between two different measuring techniques (from Ref. [77]). attempted via elaborate experiments by stopping the experiment and cleaving the sample ( [77]). Very recent research has brought new light into these issues. Ponson, Bouchaud, and co-workers have tested the line propagation scenario on four materials with very different microstructural and brittleness/ductility properties [89,92]: silica glass, an aluminum alloy, mortar, and wood. Fig. [16] shows an example of the correlation functions and the appropriate exponents, that seem to be valid for all these cases, and independent of the detailed mode of fracture (tensile, two-point bending, fatigue etc.). The parallel scaling is in agreement with earlier ¢-results. It is noteworthy that in both cases one has the same dynamical exponent z, so that the two-dimensional correlation functions (to use the notation of Ponson et al., where x is the direction of propagation) would follow the scaling Ansatz. Ponson et al. find that z ~ 1.25 which together with the Family-Vicsek relation then sets the three exponents (see Fig. 77)

Figure 15. Zmax(r)/WEc, the maximum h-value difference inside a window of size r/&- for two materials. Here €¢ is a cross-over scale that can be determined separately and may be thought of as the FPZ scale. As can be seen for both cases and for various velocities of crack growth, there seems to be two different regimes of roughness with ¢ ~ 0.5 at small scales and ¢ ~ 0.8 at larger scales ( [91]).

Figure 16. 1D height-height correlation functions for hj and h_, in the notation of Sec. 2-6-1] for an aluminum alloy fracture surface [89]. The apparent scaling exponents are ¢ = ¢y 0.75 and B=. ~ 0.58.

Figure 17. 2D height-height correlation functions Aha,(Az) for various Ax, Az for a fracture surface of an aluminum alloy ( [89]). The collapse uses the form suggested by Ponson et al., Eq. 82]

Figure 18. Size effect on critical energy release rate Gro for two kinds of samples: pine (a), and spruce (b). The expected slopes are using the associated roughness exponents ¢ — Cige = 0.47 + 0.17 for pine (a), and 0.73 + 0.17 for spruce (b) (from [47]).

Figure 19. A typical example of the crackling noise of acoustic emission (courtesy of L. Salminen, HUT). The two events that are detected are often well-separated in time and are characterized by duration intervals 7; and energy E;. Note the presence of noise in the setup, which means one has to threshold the data to define events.

Figure 20. Two examples of acoustic data series (crosses and circles) for two strain rates of paper samples in tensile testing. The stress-strain curves change with the strain rate (from Ref. [125]).

Figure 21. Three different energy distributions from a tensile AE test for paper (é = 1.0 [%/min]). The data is separated in addition to the total distribution for the post- and pre-maximum stress parts (from Ref. [125]).

Figure 22. “b-values” or the corresponding amplitude distributions for acoustic emission from rock samples. It is noteworthy how the exponent of the cumulative distribution becomes smaller (from “stage 1”) as the maximum point along the stress-strain curve is approached [127].

Figure 23. The cumulative energy E, normalized to Emax, as a function of the reduced control parameter (7 — t)/7, plotted in the proximity of the failure in the case of a constant applied load (pressure). The circles are the averages, for 9 wood samples. The solid line is a trial fit E= Eo (= ty 7. According to Ref. [124], -~ = 0.26, is independent of the “load”. Note the range of FE in which the scaling seems to apply. In the case of pure dislocation plasticity, outside of “fracture” there are col lective effects (see the review of Zaiser for an outline [155]), which often neec to be treated with the tools of statistical mechanics. Examples of collective be havior are the Portevin-LeChatelier effect [156], and the temporal and spatia fluctuations in dislocation dynamics [138,157]. One would like to understanc the laws of damage accumulation and how to define a RVE or correlatio1 length [134, 158-162]. The plastic deformation develops in non-crystalline ma terials patterns, whose statistical structure is largely not understood (see e.g. Ref. [163]) though simple models have been devised [164]. These question: could also be studied again via statistical mechanics ideas and models.

Figure 24. a): the microfractures localize (in 2D) as the control parameter is increased. The final panel has all the located events from the five equidistant intervals in five subplots (from Ref. [122]). Note that any localization experiment typically misses some events that can not be placed with certainty. b): The entropy, which decays roughly linearly from one (corresponding to random locations) as the final fracture is approached for the pressure (control parameter) value of P/P, = 1.

Figure 25. Random thresholds fuse network with triangular lattice topology. Each of the bonds i1 the network is a fuse with unit electrical conductance and a breaking threshold randomly assigned based on a probability distribution (for example, a uniform distribution). The behavior of the fuse is linear up to the breaking threshold. Periodic boundary conditions are applied in the horizontal direction and a unit voltage difference, V = 1, is applied between the top and bottom of lattice system bus bars. As the current J flowing through the lattice network is increased, the fuses will burn out one by one.

Figure 26. Examples of perfectly brittle and plastic fuses. A fuse undergoes an irreversible transformation from (V,I)¢ to (Vc,0) in the case of a brittle fuse or keeps its current at the yield current (I. or I) in the case of a plastic fuse. In the brittle fuse case the response is linear up to (V, I)c. On the other hand, in the plastic fuse case, the unloading tangent modulus is same as the original tangent modulus, and in the fully unloaded state (zero current) there is a remnant “strain” or voltage in the bond.

Figure 27. The schematic of a current-voltage curve using RFM. Two alternatives, current- and voltage-driven, are outlined. The schematic corresponds to a failure process in which one bond at a time fails. when the path energy is given by

Figure 28. An example of a local amplitude at a site 7. Clearly, individual events can be distinguished with an envelope that decays as a function of time. The lowest panel presents the integrated wave energy as a function of time. With linearly increasing applied strain, the increas in local amplitude as the failure point is approached is also visible in the time-series of “events” presented in the topmost panels. (see Ref. [227])

Figure 29. Void initiation and subsequent growth in a binary LJ mixture. Each of the three rows represents five cross-cuts across a 3D system, while the rows themselves correspond to snapshots taken beyond the maximum stress (from Ref. [245]).

Figure 30. The constitutive law for the ELS model Eq. (I) (solid line) is compared with the local stress transfer model. Increasing the system size L, the failure stress decreases in the LLS model (from Ref. [269]).

Figure 31. The time evolution of the strain of a viscoelastic fiber bundle for different values of the load f. For f < Fy the strain accelerates up to a time to failure ty (from Ref. [268]).

Figure 32. A free energy representation of nucleation. In the absence of a field the system has two equivalent stable states (left). In the presence of a weak positive field one of the two minima becomes metastable and thermally activated nucleation will eventually occur (middle). When the field reaches the spinodal value, the negative minimum becomes unstable (right).

Figure 33. The phase diagram of the non-equilibrium RFIM. The dashed line represents the location of the spinodal, which is obtained from the mean-field theory. When the interactions are short-ranged the reversal occurs on the solid line through a first-order transition. The first-order line ends at a critical point, above which the reversal is continuous.

Figure 35. The backbone of a bond percolation problem in 2d: only the bonds belonging to it are depicted. The “red bonds” are marked, naturally, with red. So-called busbar boundary conditions are used, which makes the backbone more dense close to the boundaries.

Figure 36. The scaling of both conductivity and strength in 2d RFM as a function of the bonc probability p (from Ref. [316]). exponential or Gumbel-form, of extremal statistics [318]. One can write for the integrated distribution

Figure 37. Example of a 2d fuse network for p = 0.75. What is the largest linear defect in such a system? separated from an empty region by a corrugated front. The front occurs around the region where p(x) ~ p, and displays a self-similar geometry at short scales, crossing over to a flat profile at larger scale. The crossover scale €, is set. by the value of concentration gradient Vp around p,. Using scaling arguments it is possible to show that

Figure 38. A fractal front in gradient percolation. We also display the solid-on-solid projection of the front which is single valued but not self-affine. Courtesy of A. Baldassarri, see also [323].

Figure 39. The high order correlation function for the solid-on-solid projection of a gradient percolation front. Since the original front is self-similar, the projection displays an artificial multi-affinity with effective exponents ¢qg = 1/q for g > 1. Courtesy of A. Baldassarri, see also [323].

Figure 40. Typical I — V response of a 2D RFM triangular lattice system of size L = 64. The thin line is the evolution that follows a quasistatic dynamics in which bonds are broken one at a time. The envelope of the J — V response (thick line) represents the voltage controlled response. the current becomes zero for large enough V: in that limit @ vanishes and is thus not monotonic. For small x the function has to scale as power-law of its argument, which directly implies that the FV collapse is not applicable unless the peak load has a power-law scaling form (Imax ~ LY, for some y), and the same holds also for the corresponding voltage. In addition, the scaling of ¢ couples the exponents 92; and Qe: since for small V, I « V, it follows that o~ «x and Q) = Q2. While this discussion is valid in general, for arbitrary models and experiments, we consider as an example the numerical data for the RFM.

arises from an attempt to correctly take into account the strength size effect. For such a scaling to hold the scaling function ¢2 would also need to contain logarithmic terms so that the small-voltage limit is independent of L. Their origin would be unknown. Notice that the analysis presented here is also valid for experiments, and for instance questions the application of FV scaling to the experiment of Otomar et al. [177]. Notice that for the data in Figs. 41] ensemble averaging is performed us- ing the accumulated damage as a control parameter as originally done in Refs. [326, 328-330]. This produces an unphysical hysteresis in the curves, which corresponds to unstable crack growth in voltage or current controlled cases. This effect was not apparent for the small lattices sizes at hand in Refs. [326, 328-330], but becomes clear for larger lattices employed here. To avoid this problem, it is more natural to average the envelopes of the I — V curves, using the voltage as a control parameter. The total current J flowing through the lattice in the RFM is given by J = © V where © is the total conductance of the lattice system expressed as

Figure 42. Normalized ensemble averaged I — V response based on Eq. (92) for different 2D RFM triangular lattice systems of sizes L = {4, 8, 16, 24, 32, 64, 128, 256,512}. the effective medium prediction, with strong sample-to-sample and lattice size dependent fluctuations. This brings back the question of ensemble averaging: by averaging together several curves as the one reported in Fig. 44] the abrupt character of the discontinuity is partly lost and one obtains smooth curves as the ones reported in Fig. [45{a). This effect becomes less and less important as the sample size is increased, but has lead to some confusion in the early literature, when only small samples were available. Alternatively, the averaging of the conductivity degradation responses may be performed by first shifting the data by the respective peak load locations (pp, U(pp)). In this way, one preserves the main character of the curves as shown in Fig. [Z5{b).

Figure 43. Ensemble averaged damage variable D(V) for different 2D RFM triangular lattice systems of sizes L = {4, 8, 16, 24,32, 64, 128, 256,512}. After peak load the damage variable is not intensive. Also, the nonlinearity in the plots of D(V) close to 1.0 is an artifact of J — V averaging close to fracture.

Figure 44. Typical conductivity (stiffness) degradation in various 2D RFM samples: triangular lattices of sizes L = 256 (dotted), 512 (solid), 1024 (thin dashed). The thick dashed line corresponds to the degradation of the conductivity predicted by effective medium theory.

Figure 45. Average conductivity scaling in a 2D RFM triangular lattice system. (a) Averaging performed at constant mn, number of broken bonds. (b) Averaging by first shifting each of the sample responses by their respective peak load positions.

Figure 46. Snapshots of damage in a typical 2D triangular lattice system of size L = 512. Number of broken bonds at the peak load and at failure are 83995 and 89100, respectively. (a)-(i) represent the snapshots of damage after breaking ny, number of bonds. (a) np = 25000 (b) ny = 50000 (c) np = 75000 (d) np = 80000 (e) np = 83995 (peak load) (f) np = 86000 (g) np = 87000 (h) np = 88000 (i) nm» = 89100 (failure)

Figure 47. Scaling of number of broken bonds at peak load for two-dimensional triangular and diamond random fuse lattices, and triangular spring lattices. In all these models disorder is uniformly distributed in [0,1] and the scaling exponents are very close to each other. The last curve represents data obtained using a power low disorder distribution: the exponent changes to b = 1.

Figure 48. 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. The normal distribution fit in the re-parameterized form is presented in the inset. In the RFM case, 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. [325].

Figure 49. 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. Data are for RFM simulations with uniform thresholds disorder.

Figure 50. Data collapse of the localization profiles for the damage accumulated between peak load and failure. The profiles show exponential tails. Data are for RFM simulations with uniform thresholds disorder.

Figure 51. Localization of damage (nj — na) in a 2D RFM simulation of lattice system size L = 512 averaged over 200 samples. Localization of damage occurs abruptly at the peak load (np/np = 1).

Figure 52. Damage cluster distribution at the peak load in cubic lattices of system sizes L = {10, 16, 24, 32, 48} in log-log and log-linear scales (inset). The cluster distribution data for different lattice system sizes is neither a power-law or an exponential distribution. exponential representations are not adequate, and a similar result is obtained in 2D as well [341]. For completeness, one should note that the distribution of the largest crack cluster at the peak load seems to follow a Lognormal dis- tribution as demonstrated in Fig. [3] for various system sizes. It is interesting to note that the fracture strength data presented in the following subsection indicate that a Lognormal distribution represents an adequate fit for the frac- ture strength distribution in broadly disordered materials. At first glance, this result appears to be trivial: assuming that the stress concentration around the largest crack cluster is given by a power law, the fracture strength follows a Lognormal distribution since the power of a Lognormal distribution is also a Lognormal distribution. However, as noted in Ref. [341], a close examination of the largest crack cluster size data at peak load and the fracture strength data reveals that the largest crack cluster size and the fracture strength are uncorrelated [341]. Indeed, quite often, the final spanning crack is formed not due to the propagation of the largest crack at the peak load, but instead due to coalescence of smaller cracks (see Figs. 6 and 7 of Ref. [341]).

Figure 53. Re-parameterized form of Lognormal fit for the cumulative probability distribution of largest crack cluster size at the peak load in cubic lattices of system sizes L = {10, 16, 24, 32, 48} LN al refer to the mean and the standard deviation of the logarithm of largest cluster sizes, Sp, at peak load.

Figure 54. Entropy S versus the damaged stage index D; for a triangular lattice of size L = 512 with uniform threshold distribution (A = 1) threshold distribution. Symbols are By = (0,1,--+, Noow — 1) = (*,2,4,9, , ©, <1). The inset presents the entropy at the peak load versus the box size index for uniform and power law threshold distributions for L = (64, 128, 256, 512, 1024). The data for smaller systems is shifted such that the points correspond to the same relative box size. The slope of the graph in the inset is equal to —log(2) = —0.693 corresponding to randomly distributed damage. the inset equals —log(2) = —0.693, which matches exactly with the slope of the maximum entropy Smazx versus By since Sparx = log(Mpor) = log(L/Bsize) = log(L/2?") = log(L) — Bylog(2).

Figure 55. Examples of how the Gumbel and Weibull forms fit into the percolation disorder RFM data (from Ref. [316]).

Figure 56. Universality of fracture strength distribution in the random thresholds fuse and springs models. Data are for different lattice system sizes, L, corresponding to triangular fuse lattice with uniform disorder (L = {4, 8, 16, 24, 32, 64, 128}), diamond fuse lattice with uniform disorder (L = {4, 8, 16, 24, 32, 64, 128}), triangular fuse lattice with power law disorder (L = {4, 8, 16, 24, 32, 64, 128}), and triangular spring lattice with uniform disorder (L = {8, 16, 24, 32, 64, 128}). The inset reports a Lognormal fit for the corresponding cumulative distributions. UL UID 1LACUULYS SvUlLelebil UCitolby Ulovurtouvlgoii. In addition, the collapse of the data in Fig. B6] suggests that P(o < of) = W(E), where P(o < of) refers to the cumulative probability of fracture strength o <of, WV is a universal function such that 0 < UW < 1, and E= Eaton is the standard Lognormal variable. The inset of Fig. [56] presents a Lognorma! fit for fracture strengths, tested by plotting the inverse of the cumulative probability, 6~'(P(cf)), against the standard Lognormal variable, €. In the above description, ®(-) denotes the standard normal probability function. As discussed in Ref. [334], a Lognormal distribution is an adequate fit for 3D random fuse models as well. This further confirms the notion of universality of fracture strength distribution in broadly disordered materials.

Figure 57. Proposed scaling law for the mean fracture strength (Fpean = CoL® + C1). (1) Triangular spring network (symbol: -L1): a = 0.97. (2) Triangular fuse network (symbol: -*-): a = 0.956; (3) Diamond fuse network: (symbol: -+-): a = 0.959; The corresponding Weibull fit for the mean fracture strength is shown in the inset.

Figure 60. (a) The local width w(l) of the crack surface for different lattice sizes of 2D RFM triangular lattice systems plotted in log-log scale. A line with the local exponent Cjo- = 0.7 is plotted for reference. The global width displays an exponent ¢ > Cigc.(b) The corresponding power spectrum in log-log scale. The spectra for all of the different lattice sizes can be collapsed indicating anomalous scaling.

Figure 61. The distribution of the crack width is universal for diamond and triangular random fuse lattices. A fit with a log normal distribution is shown by a dashed line.

Figure 62. The power spectrum of the crack surface of the three dimensional RFM. The spectrum exponent 2¢ + 1 = 1.82, corresponds to Cig = 0.41. The global width scales with ¢ = 0.52 (see inset), so that the power spectrum should be normalized by 2(¢ — ioe) & 0.2

Figure 63. A summary of two-dimensional results for avalanche distributions integrated along the entire loading curve. The results are for the RFM on triangular and diamond lattices [350] and for the RSM [182]. Data for different lattice sizes have been collapsed according to Eq. an A line with exponent + = 5/2 is reported for reference.

Figure 64. Data collapse of the integrated avalanche size distributions for the three dimensional RFM. The exponents used for the collapse are 7 = 2.5 and D = 1.5. A line with a slope rT = 5/2 is reported in Ref. [350].

Figure 65. The avalanche size distributions sampled over a small bin of the reduced current [* for a diamond lattice of size L = 128. The dashed line represents a fit according to Eq. with 7 = 1.9 (from Ref. [350]).

Table 1. A summary of the values of the roughness exponent measured in experiments, in the RFM and in the depinning transition (DT) scenario, both with long-range (LR) and (SR) interactions. Refer to Sec. [2-6-T]for the definition of the exponents, to Sec.[5-2]for details about experiments, to Sec. [6-4] for the RFM results and to Sec. [53] for the depinning transition.

Table A3. Block Circulant PCG (2D Triangular Lattice) [able A4. Optimal Circulant PCG (2D Triangular Lattice)

circulant preconditioner and the naive sparse direct solution technique based on factorizing each of the A,, matrices.

Table A2. Solver Type B (2D Triangular Lattice)

Table A6. Optimal Circulant PCG (3D Cubic Lattice)

Table A8. Incomplete Cholesky PCG (3D Cubic Lattice) ilar to that of block-circulant preconditioner (see Table A5), even though the performance of incomplete Cholesky preconditioner is far more superior in the case of 2D lattice simulations [187]. Indeed, for 3D lattice systems of sizes L = 48 and L = 64, the CPU time taken by the sparse direct solver types A and B is much higher than that of block-circulant PCG solver.

Table A7. Un-preconditioned CG (3D Cubic Lattice)