J Computer-Aided Mater Des (2007) 14:25–35 DOI 10.1007/s10820-007-9080-y
Fracture in three-dimensional random fuse model: recent advances through high-performance computing Phani K. V. V. Nukala · Srdan Šimunovi´c · Stefano Zapperi · Mikko J. Alava
Received: 21 February 2007 / Accepted: 28 February 2007 / Published online: 15 January 2008 © Springer Science+Business Media B.V. 2008
Abstract The paper presents the state-of-the-art algorithmic developments for simulating the fracture of disordered quasi-brittle materials using discrete lattice systems. Large scale simulations are often required to obtain accurate scaling laws; however, due to computational complexity, the simulations using the traditional algorithms were limited to small system sizes. In our earlier work, we have developed two algorithms: a multiple sparse Cholesky downdating scheme for simulating 2D random fuse model systems, and a block-circulant preconditioner for simulating 3D random fuse model systems. Using these algorithms, we were able to simulate fracture of largest ever lattice system sizes (L = 1024 in 2D, and L = 64 in 3D) with extensive statistical sampling. Our recent massively parallel simulations on 1024 processors of Cray-XT3 and IBM Blue-Gene/L have further enabled us to explore fracture of 3D lattice systems of size L = 128, which is a significant computational achievement. Based on these large-scale simulations, we analyze the scaling of crack surface roughness. Keywords
Random fuse model · Crack roughness
1 Introduction The statistical properties of fracture in disordered media are interesting not only in view of practical applications, but also for purely theoretical reasons [1]. Despite considerable progress, there exist many controversial issues between the theoretically estimated results and the experimentally measured values, and also among various theoretical and numerical
P. K. V. V. Nukala (B) · S. Šimunovi´c Computer Science and Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6164, USA e-mail:
[email protected] S. Zapperi CNR-INFM, Dipartimento di Fisica, Sapienza, Università diRoma, P.le A. Moro 2, 00185 Rome, Italy M. J. Alava Laboratory of Physics, Helsinki University of Technology, Espoo 02105 HUT, Finland
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models used for studying fracture of disordered media. Among these partly still controversial issues, is the scaling of crack geometries; in particular, the origin of both the scaling and the universality of the fracture surface roughness exponent is at the heart of the controversy. Experiments on several materials under different loading conditions have shown that the fracture surface is self-affine [2] and the out of plane roughness exponent displays a universal value of 0.8 irrespective of the material studied [3]. The scaling regime is sometimes quite impressive, spanning five decades in metallic alloys [3]. In particular, experiments have been done in metals, glass, rocks and ceramics (see [3] and the references therein), covering both ductile and brittle materials. Later on, a smaller exponent 0.4–0.6 was observed at smaller length scales. It was conjectured that crack roughness displays a universal value of 0.8 only at larger scales and at higher crack speeds, whereas another roughness exponent in the range of 0.4–0.6 is observed at smaller length scales under quasi-static or slow crack propagation [3]. However, it was recently pointed out that the short-scale value is not present in silica glass, even when cracks move at extremely low velocities [4]. In addition, in glass ceramics and sandstone, one only observes the “small scale” exponent even at high velocities [5]. The current interpretation associates the value ζ 0.75 with rupture processes occurring inside the fracture process zone, where elastic interactions would be screened and the value ζ 0.45 with large scale elastic fracture [6]. The authors of Refs. [4, 6] were also able to show that the fracture surface is anisotropic, with different exponents in parallel and perpendicular directions to the crack propagation. Furthermore, the roughness exponent conventionally measured describes only the local properties, while the fracture surface instead exhibits anomalous scaling [7]: the global exponent describing the scaling of the crack width with the sample size is larger than the local exponent measured on a single sample [7]. It is thus necessary to define two roughness exponents a global one (ζ ) and a local one (ζloc ). Only the local exponent ζloc appears to be universal. The situation that exists today is that there is a large discrepancy between theoretical estimates and experimentally measured values of roughness exponents, and hence numerical methods are in common use. A well established numerical model that deals with explicit description of disorder is based on lattice models, which describe the medium as a discrete set of elastic bonds with randomly distributed failure thresholds [1, 8]. In the last 20 years, the cornerstone of fracture simulations using discrete lattice models has been the Random Fuse Model (RFM), a lattice model of the fracture of solid materials in which as a key simplification vectorial elasticity has been substituted with a scalar field. The RFM represents one of the simplest imaginable pictures of strongly interacting systems that are complicated by the presence of disorder. However, there exists a huge discrepancy between the numerically computed roughness exponents using the three dimensional RFM [9–11] and the experimentally observed universal value of 0.8. The roughness exponent was estimated to be ζ = 0.62 ± 0.05 in [9], whereas a much smaller exponent of ζ = 0.41 ± 0.02 was estimated in [10, 11]. The measured roughness exponents in [10, 11] are similar to the ones describing a minimum energy surface (or a directed polymer in d = 2) which would imply that crack formation occurs by an optimization process, but the issue is still controversial [9, 11]. For the purpose of this paper, it is also worth noting that a roughness exponent of ζ = 0.5 [12] is obtained using threedimensional Born models. It is believed that roughness measurements based on numerical results obtained using large system sizes are necessary to bridge the gap that exists between numerically estimated roughness exponents and the experimentally measured roughness exponents. However,
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large-scale numerical simulation of RFM is computationally expensive. In this paper, we present our recent algorithmic advances as well as high-performance computing efforts in simulating fracture using the RFM. Using our numerical algorithms, for the first time, we can simulate fracture of large 3D cubic lattice systems of sizes up to L = 128. The paper is organized as follows: in Sect. 2, we briefly describe the random fuse model. The state-of-the-art computational algorithms used for simulating fracture in the RFM are presented in Sect. 3. Section 4 discusses further developments in large scale simulation of fracture using high-performance computing. Section 5 discusses the roughness of fracture surfaces.
2 Random fuse model The RFM has been extensively investigated in the literature over the last two decades [1, 8]. In the random thresholds fuse model, the lattice is initially fully intact with bonds having the same conductance, but the bond breaking thresholds, t, are randomly distributed based on a thresholds probability distribution, p(t). The burning of a fuse occurs irreversibly, whenever the electrical current in the fuse exceeds the breaking threshold current value, t, of the fuse. Periodic boundary conditions are imposed in the horizontal direction to simulate an infinite system and a constant voltage difference, V , is applied between the top and the bottom of lattice system bus bars. Numerically, a unit voltage difference, V = 1, is set between the bus bars and the Kirchhoff equations are solved to determine the current flowing in each of the fuses. Subsequently, for each fuse j, the ratio between the current i j and the breaking threshold t j is evaluated, and i the bond jc having the largest value, max j t jj , is irreversibly removed (burnt). The current is redistributed instantaneously after a fuse is burnt implying that the current relaxation in the lattice system is much faster than the breaking of a fuse. Each time a fuse is burnt, it is necessary to re-calculate the current redistribution in the lattice to determine the subsequent breaking of a bond. The process of breaking of a bond, one at a time, is repeated until the lattice system falls apart. In this work, we consider a uniform probability distribution, which is constant between 0 and 1. Numerical simulations on large system sizes are essential to understand the scaling laws of fracture and universality of crack surface roughness exponents. Unfortunately, large scale numerical simulation of these discrete lattice networks has often been hampered for two reasons. –
–
First, a new large set of equations has to be solved everytime a new lattice bond is broken. This becomes especially severe with increasing lattice system size, L d (where L is the linear dimension of the lattice and d is the spatial dimension of the lattice system (d = 2 in 2D and d = 3 in 3D)), since the number of broken bonds at failure, n f , increases with system size L as n f ∼ O(L 1.8 ) in 2D and n f ∼ O(L 2.7 ) in 3D. Second, critical slowing down associated with the iterative solvers close to the macroscopic fracture. That is, as the lattice system gets closer to macroscopic fracture, the condition number of the system of linear equations increases, thereby increasing the number of iterations required to attain a fixed accuracy. This becomes particularly significant for large lattices.
In addition, since the response of the lattice system corresponds to a specific realization of the random breaking thresholds, an ensemble averaging of numerical results over Ncon f ig configurations is necessary to obtain a realistic representation of the lattice system response.
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This further increases the computational time required to perform simulations on large lattice systems. In the following, we describe the state-of-the-art computational algorithms that significantly reduced the computational time thereby enabling the simulation of fracture using large lattice system sizes.
3 State-of-the-art algorithms Algebraically, the process of simulating fracture using discrete lattice systems is equivalent to solving a new set of linear equations An xn = bn ,
n = 0, 1, 2, . . . ,
(1)
every time a new lattice bond is broken. The matrix An refers to the lattice conductance matrix in the case of fuse models and the lattice stiffness matrix in the case of spring and beam models, bn refers to the applied nodal current or force vector, and xn nodal potential or displacement vector. Traditionally, iterative techniques based on preconditioned conjugate gradient (PCG) method have been used to simulate fracture using fuse networks (see [13] for a excellent review of iterative methods; see [14] for a review of multigrid method). However, large-scale numerical simulations using iterative solvers have often been hindered due to the critical slowing down associated with the iterative solvers as the lattice system approaches macroscopic fracture. As a remedy, Fourier accelerated PCG iterative solvers [15] have been suggested to alleviate the critical slowing down. The Fourier acceleration algorithm proposed in [15] ¯ [16] as the preconditioner, where A(i, ¯ chooses an ensemble averaged matrix A j) = r (dw −d f ) , r = |i − j|, the distance between the nodes i and j, and d f and dw refer to the fractal dimension of the current-carrying backbone and the random-walk dimension respectively. However, ¯ is not the optimal circulant preconditioner since it does not the ensemble averaged matrix A minimize the norm I − C−1 A F over all non-singular circulant matrices C. 3.1 Multiple-rank sparse Cholesky downdating algorithm An important feature of fracture simulations using discrete lattice models is that, for each n = 0, 1, 2, . . ., the new matrix An+1 of the lattice system after the (n + 1)th broken bond is equivalent to a rank- p downdate of the matrix An [17]. Mathematically, in the case of the fuse and spring models, the breaking of a bond is equivalent to a rank-one downdate of the matrix An , whereas in the case of beam models, it is equivalent to multiple-rank (rank-3 for 2D, and rank-6 for 3D) downdate of the matrix An . Thus, an updating scheme of some kind is therefore likely to be more efficient than solving the new set of equations formed by Eq. 1 for each n. Consider the Cholesky factorizations PAn Pt = Ln Lnt
(2)
for each n = 0, 1, 2, · · · , where P is a permutation matrix chosen to preserve the sparsity of Ln . Since the breaking of bonds is equivalent to removing the edges in the underlying graph structure of the matrix An , for each n, the sparsity pattern of the Cholesky factorization Ln+1 of the matrix An+1 must be a subset of the sparsity pattern of the Cholesky factorization Ln
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of the matrix An . Hence, for all n, the sparsity pattern of Ln is contained in that of L0 . That is, denoting the sparsity pattern of L by L, we have Lm ⊇ Ln ∀ m < n
(3)
For two-dimensional lattice simulations, in [17], we proposed an efficient algorithm based on multiple-rank sparse Cholesky downdating scheme of Davis and Hager [18] to successively downdate the Cholesky factorizations Ln of An to Ln+1 of An+1 , i.e., Ln → Ln+1 for n = 0, 1, 2, · · · . Since Ln ⊇ Ln+1 , it is necessary to modify only a part of the non-zero entries of Ln in order to obtain Ln → Ln+1 . This results in a significant reduction in the computational time. Once the factorization Ln+1 of An+1 is obtained, the solution vector t xn+1 is obtained from Ln+1 Ln+1 xn+1 = bn+1 by two triangular solves [17]. Using this algorithm [17], the authors have reported numerical simulation results for large 2D lattice systems (e.g., L = 1024), which was so far the largest lattice system used in studying damage evolution using discrete lattice systems. 3.2 Optimal and superoptimal circulant preconditioners Although the sparse direct solver algorithm presented in [17] is superior to iterative solvers in two-dimensional lattice systems, for 3D lattice systems, the memory demands brought about by the amount of fill-in during sparse Cholesky factorization favor iterative solvers. The main observation behind developing preconditioners for the iterative schemes is that the operators on discrete lattice network result in a circulant block structure. Hence, a fast Poisson type solver with a circulant preconditioner can be used to obtain the solution in O(N log N ) operations using FFTs of size N , where N denotes the number of degrees of freedom. However, as the lattice bonds are broken successively, the initial uniform lattice grid becomes a diluted network. Consequently, although the matrix A0 is Toeplitz (also block Toeplitz with Toeplitz blocks) initially, the subsequent matrices An , for each n, are not Toeplitz matrices. However, depending on the pattern of broken bonds, An may still possess block structure with many of the blocks being Toeplitz blocks. The authors have developed an algorithm based on a block-circulant preconditioned conjugate gradient (CG) iterative scheme [19] for simulating 3D random fuse networks. The block-circulant preconditioner was shown to be superior compared with the optimal pointcirculant preconditioner for simulating 3D random fuse networks [19]. Since these blockcirculant and optimal point-circulant preconditioners achieve favorable clustering of eigenvalues, these algorithms significantly reduced the computational time required for solving large lattice systems in comparison with the Fourier accelerated iterative schemes used for modeling lattice breakdown [15]. Using this block circulant preconditioner, we were able to simulate fracture in lattice systems of sizes up to 64 × 64 × 64, which was the largest ever discrete lattice system used in 3D fracture simulation. Table 1 presents a summary of computational times required for simulating the fracture of one such sample configuration of a 3D cubic lattice system. These simulations have been performed on a IBM 1.3 GHz Power4 processor. The CPU times taken for simulating a cubic lattice system of size (L × L × L) scales as O(L 6.5 ), which poses severe computational requirements for simulating fracture of even larger lattice system sizes that are required to obtain accurate scaling laws of fracture in 3D. This is precisely the scenario where massively parallel simulation offers significant advantages. In the following, we present massively parallel simulations of large 3D lattice systems for studying fracture of disordered materials.
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Table 1 Block circulant PCG (3D cubic lattice)
Size
CPU (s)
Iterations
10 16 24 32 48 64
16.54 304.6 2154 12716 130522 1180230
16168 58756 180204 403459 1253331
4 High-performance computing Since iterative solvers exhibit excellent scalability with respect to number of processors, parallel iterative solvers are especially suitable for performing large scale fracture simulations using 3D lattice networks. Using PETSc [20], we were able to start from our existing serial code and rapidly parallelize the most time-consuming (solvers) portions of the computation. At each simulation step, the PETSc KSPSolve() routine is used to solve for the current flowing in each of the fuses using a parallel preconditioned Krylov iterative solver; in the simulations described here, we used conjugate gradient with a block-Jacobi preconditioner in which ILU(0) was applied on each block. The preconditioner we chose is likely not ideal for this system, but has the advantage of being highly parallel. Because the coefficient matrices between steps differ only by a rank-1 update, we do not update the ILU(0) factors at each step; in our simulations we recalculated the factors every 1000 steps. Using PETSc, fracture of a L = 64 cubic lattice system can be simulated within 3 hours (compare this with 14 days of CPU time on a single processor using the block-circulant preconditioner) on 128 Cray-XT3 processors (2.4 GHz). Fracture simulation on a lattice system of size L = 100 in 3D requires a day of computational time on 512 processors of Cray-XT3 (see Fig. 1 for comparison of performance scaling on Cray-XT3 at Oak Ridge National Laboratory and Blue-Gene/L at Argonne National Laboratory). Figure 1 illustrates the performance of our parallel code on a 1003 lattice benchmark for up to 1024 processors. Performance on both the BlueGene/L and Cray XT3 systems is encouraging, especially given
100^3 benchmark Wall-clock time (seconds) per 1000 bonds broken
16384
BlueGene/L (ANL) Cray XT3 (ORNL)
8192 4096 2048 1024 512 256 128 64 8
16
32
64
128
256
512
1024
Number of processors
Fig. 1 Performance comparison of the parallel RFM code in simulating a large 3D cubic lattice system of size L = 100 on Cray-XT3 (Jaguar) at ORNL and on Blue-Gene/L at ANL
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the rapidly-prototyped nature of our code. Our largest RFM simulation in 3D so far has been a cubic lattice system of size L = 128. Along with massive parallelization, it may be worthwhile to look into alternative algorithmic strategies/preconditioners for solving the system of equations that arise in the simulation of large 3D lattice systems. Currently, we are pursuing recycling of Krylov subspaces [21] determined while solving An xn = bn and use it to reduce the cost of solving the subsequent system An+1 xn+1 = bn+1 with minimal effort. Such a recycling process amounts to a reduction of iteration count required to solve the new linear system An+1 xn+1 = bn+1 , and hence increases the overall efficiency of the algorithm.
5 Crack roughness
z
Once the complete fracture of the lattice system has occurred, we identify the final crack by removing the dangling ends and overhangs as shown in Fig. 2. The crack surface is represented by a single valued height function z(x, y) in the x − y reference plane, where x ∈ [0, L] and y ∈ [0, L]. We estimate the crack surface roughness in the x and y directions by considering number of slices along each direction. The local width using the variable bandwidth method is computed as wx () ≡ x (z x − (1/) X z X )2 1/2 , where the sums are restricted to windows of length along the x direction and the average . . . is taken over all possible origins of the windows along the profile, and over different realizations. The local width w y () is calculated similarly. The self-affine scaling properties of crack surfaces results in a scaling law of form w() ∼ ζ for L that saturates to a value W = w(L) ∼ L ζ corresponding to the global width. A more precise value of the exponents is obtained from the power spectrum, which is expected to yield more precise estimates [22]. The power spectrum or the structure factor is computed as S(ω) ≡ ˆz ω zˆ −ω /L, where zˆ ω ≡ x z x exp(2πi xω/L), and decays as S(ω) ∼ ω−(2ζ +1) . Figure 3 presents the scaling of local crack width in the x and y directions for different lattice system sizes. The results presented in Fig. 3a, b indicate that the crack width scaling in the x and y directions is identical. A reference line, obtained by fitting the scaling of
90 80 70 0
100 90 10
20
80 30
70 40
x
60 50
50 60
40 70
30 80
y
20 10 90 100 0
Fig. 2 (Color online) Crack surface in a typical cubic lattice system of size L = 100. The color coding is such that the z value of the surface is lowest for blue, and increases as the color changes to cyan, magenta and red, respectively
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L = 10 L = 16 L = 24 L = 32 L = 48 L = 64 L = 100 ζ = 0.52
0.1
10
wx( l )
Fig. 3 (Color online) The local width w(l) of the crack in the x and y directions for different lattice sizes in log-log scale. The crack width scaling in x and y directions is identical. The global width yields a scaling exponent ζ = 0.52
P. K. V. V. Nukala et al.
−0.1
10
−0.3
10
−0.5
10
0
1
10
10
2
10
l L = 10 L = 16 L = 24 L = 32 L = 48 L = 64 L = 100 ζ = 0.52
0.1
wy( l )
10
−0.1
10
−0.3
10
−0.5
10
0
10
1
10
2
10
l
the global width with L, indicates a roughness exponent ζ = 0.52. We have obtained the same scaling exponents in both x and y directions. It should be noted that the exponent ζ = 0.52 differs considerably from the exponent ζ = 0.62 ± 0.05 proposed in [9] and that of ζ = 0.41 ± 0.02 proposed in [11]. The difference may however be due to the fewer number of statistical samples considered in these earlier studies. A direct fit of the local width based on Fig. 3a, b would not be reliable due to the very small scaling regime. Thus in Fig. 4, we present the collapse of the local crack width data using the scaling form w() ∼ ζ f (/L) and the result is not very good. In two-dimensional RFM, our data with improved statistics and large system sizes [23] indicated that the crack roughness scaling is anomalous [7]. In previous measurements on two-dimensional RFM, anomalous scaling could not be detected since the data was available only for smaller system sizes. Anomalous scaling has been observed not only in various growth models [7] but also in fracture surfaces in granite [24] and wood samples [25]. Anomalous scaling implies that the exponent describing the system size dependence of the surface differs from the local exponent measured for a fixed system size L. In particular, the local width scales as w() ∼ ζloc L ζ −ζloc , so that the global roughness W scales as L ζ with
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Fracture in three-dimensional random fuse model Fig. 4 (Color online) The collapse of the local crack widths in the x direction with ζ = 0.52. The collapse of the data however is not very good. The scenario for the collapse of local crack widths in y direction is similar
33 L = 10 L = 16 L = 24 L = 32 L = 48 L = 64
−1
w( l )/Lζ
10
−1
0
10
10
l/L 3
10
L=10 L=16 L=24 L=32 L=48 L=64 −1.82 ω
2
0.2
10
S(ω)/L
Fig. 5 The power spectrum of the crack surface of the three dimensional RFM. The spectrum exponent 2ζ + 1 = 1.82, corresponds to ζloc = 0.41. The global width scales with ζ = 0.52 (see inset), so that the power spectrum should be normalized by 2(ζ − ζloc ) 0.2
10
1
10
ζ=0.52
W
1 0
10
0
10
100
−1
10
−2
10
−1
10
0
10
ω/L
ζ > ζloc . Consequently, the power spectrum scales as S(ω) ∼ ω−(2ζloc +1) L 2(ζ −ζloc ) . The upward shift of the w() presented in Fig. 3a, b for different system sizes is a fingerprint of anomalous scaling since such a shift indicates that w() grows with L. This is a key feature of anomalous scaling that can be spotted by just looking at the local roughness amplitudes. In addition, an analysis of the power spectrum data results in a value of ζ 0.52, with possible anomalous scaling corresponding to a local value of ζloc 0.41 (see Fig. 5). Although the value of ζ − ζloc as estimated using the power spectrum method is small, it is significantly larger than zero and it may even correspond to a logarithmic growth. However, based on our numerical results, which included lattice system sizes up to L = 64 with extensive statistical sampling, it is not possible to conclude with certainty whether anomalous scaling of crack surface roughness is present in three-dimensional RFM or not. Simulations on much larger lattice systems such as those presented in here (sizes L = 100 and higher), but with extensive statistical sampling are anticipated to clarify this. But, the important thing to note is that these roughness exponents are considerably lower than those observed in experiments.
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6 Conclusions We have presented two state-of-the-art algorithms for simulating fracture of disordered quasibrittle materials using 2D and 3D discrete lattice systems. These algorithms have enabled us to simulate fracture of largest ever lattice systems, which are necessary to obtain accurate scaling laws of fracture. We have further expanded this finite size scaling regime using massively parallel simulations on thousands of processors of Cray-XT3 (Jaguar) and IBM Blue-Gene/L. Using these parallel algorithms, we can currently simulate fracture of a 3D lattice system of size L = 64 within 3 hours on Cray-XT3 (within 9 hours on IBM BlueGene/L), whereas it requires 14 days of computational time on a single processor. We can currently simulate fracture of 3D lattice systems of sizes up to L = 128, which is a significant computational achievement considering that traditionally fracture of lattice systems of sizes up to L = 48 were simulated, and that computational complexity increases as O(L 6.5 ). Based on these large scale simulations, we have analyzed the roughness of the final crack and found a local exponent ζloc = 0.41, which is slightly different from the global roughness exponent ζ = 0.52. A similar difference between local and global exponents was also found in two dimensional simulations for both triangular and diamond lattices [23], suggesting that anomalous scaling is a generic feature of the fracture of disordered media as already found in fracture experiments [7]. The numerical value of the exponents is, however, quite far from the three dimensional experimental results indicating that additional physical mechanisms should probably be taken into account. In summary, the present results seem to indicate that the RFM provides only a qualitative description of the fracture surface morphology found in experiments. The quantitative differences may be attributed to the strong simplifications present in the model such as its scalar nature, the quasistatic dynamics and the absence of a preexisting notch; although a roughness exponent of ζ = 0.5 [12] is obtained using three-dimensional Born model as well. On the other hand, the initial phase of damage accumulation leading to localization and failure in a strongly disordered sample are well captured by the RFM model, but further work is needed to clarify these issues. Acknowledgements PKVVN is sponsored by the Mathematical, Information and Computational Sciences Division, Office of Advanced Scientific Computing Research, U.S. Department of Energy under contract number DE-AC05-00OR22725 with UT-Battelle, LLC. PKVVN also acknowledges the use of “BGL”, a 1024-node BG/L machine operated by the Mathematics and Computer Science Division at Argonne National Laboratory through his INCITE Award. In addition, the research used resources of the Center for Computational Sciences at Oak Ridge National Laboratory.
References 1. Herrmann, H.J., Roux, S. (eds.): Statistical Models for the Fracture of Disordered Media. North-Holland, Amsterdam, (1990); Hansen, A., Roux, S.: Statistical toolbox for damage and fracture. In: Krajcinovic, D., van Mier, J.G.M. (eds.) Book Damage and Fracture of Disordered Materials, pp. 17–101. Springer Verlag, New York (2000); Alava, M.J., Nukala, P.K.V.V., Zapperi, S.: Statistical models of fracture. Adv. Phys. 55, 349–476 (2006) 2. Mandelbrot, B.B., Passoja, D.E., Paullay, A.J.: Fractal character of fracture surfaces of metals. Nature (London) 308, 721–722 (1984) 3. For a review see Bouchaud, E.: Scaling properties of cracks. J. Phys. Condens. Matter 9, 4319–4344 (1997); Bouchaud, E., The morphology of fracture surfaces, a tool to understand crack propagation in complex materials. Surf. Sci. Review & Lett. 10:794–814(2003) 4. Ponson, L., Bonamy, D., Bouchaud, E.: Two-dimensional scaling properties of experimental fracture surfaces Phys. Rev. Lett. 96, 035506(4pages) (2006)
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