C. Liability Rues Compared: Secondary Considerations
3. Error Cost
Tipo d (nm) h (nm) ν Referencia
(3,3) SWCNT 0.407 0.340 0.298 Popov VN (2000) Physical Review B 61 (4) 3078-84
(6,0) SWCNT 0.470 0.340 0.100 Popov VN (2000) Physical Review B 61 (4) 3078-84
(6,1) SWCNT 0.514 0.340 0.137 Popov VN (2000) Physical Review B 61 (4) 3078-84
(4,4) SWCNT 0.543 0.340 0.273 Popov VN (2000) Physical Review B 61 (4) 3078-84
(7,0) SWCNT 0.549 0.340 0.127 Popov VN (2000) Physical Review B 61 (4) 3078-84
(4,4) SWCNT 0.559 0.34 0.124 Sanchez-Portal D (1999) Phys. Rev. B 59 (19)
12678
(6,2) SWCNT 0.565 0.340 0.186 Popov VN (2000) Physical Review B 61 (4) 3078-84
(7,1) SWCNT 0.592 0.340 0.150 Popov VN (2000) Physical Review B 61 (4) 3078-84
(6,3) SWCNT 0.622 0.340 0.222 Popov VN (2000) Physical Review B 61 (4) 3078-84
(8,0) SWCNT 0.627 0.340 0.145 Popov VN (2000) Physical Review B 61 (4) 3078-84
(5,5) SWCNT 0.679 0.340 0.249 Popov VN (2000) Physical Review B 61 (4) 3078-84
(5,5) SWCNT 0.680 0.34 0.285 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(6,4) SWCNT 0.680 0.34 0.285 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(6,4) SWCNT 0.683 0.340 0.237 Popov VN (2000) Physical Review B 61 (4) 3078-84
(5,5) SWCNT 0.693 0.34 0.140 Sanchez-Portal D (1999) Phys. Rev. B 59 (19)
12678
(7,3) SWCNT 0.700 0.34 0.285 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(9,0) SWCNT 0.705 0.340 0.158 Popov VN (2000) Physical Review B 61 (4) 3078-84
(8,2) SWCNT 0.718 0.340 0.183 Popov VN (2000) Physical Review B 61 (4) 3078-84
(8,2) SWCNT 0.720 0.34 0.285 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(9,1) SWCNT 0.740 0.34 0.285 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(7,4) SWCNT 0.756 0.340 0.225 Popov VN (2000) Physical Review B 61 (4) 3078-84
(10,0) SWCNT 0.780 0.34 0.284 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(10,0) SWCNT 0.784 0.340 0.168 Popov VN (2000) Physical Review B 61 (4) 3078-84
(10,0) SWCNT 0.791 0.34 0.275 Hernandez E (1998) Physical Review Letters 80
(20) 4502-4505
(10,0) SWCNT 0.796 0.34 0.191 Sanchez-Portal D (1999) Phys. Rev. B 59 (19)
12678
(6,6) SWCNT 0.814 0.340 0.237 Popov VN (2000) Physical Review B 61 (4) 3078-84
(6,6) SWCNT 0.820 0.34 0.247 Hernandez E (1998) Physical Review Letters 80
(20) 4502-4505
(10,1) SWCNT 0.826 0.340 0.175 Popov VN (2000) Physical Review B 61 (4) 3078-84
(6,6) SWCNT 0.828 0.34 0.140 Sanchez-Portal D (1999) Phys. Rev. B 59 (19)
Tipo d (nm) h (nm) ν Referencia
(8,4) SWCNT 0.829 0.340 0.217 Popov VN (2000) Physical Review B 61 (4) 3078-84
(8,4) SWCNT 0.842 0.34 0.180 Sanchez-Portal D (1999) Phys. Rev. B 59 (19)
12678
(9,3) SWCNT 0.848 0.340 0.200 Popov VN (2000) Physical Review B 61 (4) 3078-84
(11,0) SWCNT 0.862 0.340 0.175 Popov VN (2000) Physical Review B 61 (4) 3078-84
(8,5) SWCNT 0.890 0.340 0.224 Popov VN (2000) Physical Review B 61 (4) 3078-84
(12,0) SWCNT 0.940 0.340 0.180 Popov VN (2000) Physical Review B 61 (4) 3078-84
SWCNT 0.950 0.066 0.190 Yakobson B I (1996) Phys. Rev. Lett. 76 (14) 2511-
2514
(7,7) SWCNT 0.950 0.340 0.230 Popov VN (2000) Physical Review B 61 (4) 3078-84
(11,2) SWCNT 0.950 0.340 0.189 Popov VN (2000) Physical Review B 61 (4) 3078-84
(10,4) SWCNT 0.979 0.340 0.208 Popov VN (2000) Physical Review B 61 (4) 3078-84
(13,0) SWCNT 1.019 0.340 0.185 Popov VN (2000) Physical Review B 61 (4) 3078-84
(10,5) SWCNT 1.034 0.34 0.265 Hernandez E (1998) Physical Review Letters 80
(20) 4502-4505
(10,5) SWCNT 1.037 0.340 0.215 Popov VN (2000) Physical Review B 61 (4) 3078-84
(12,3) SWCNT 1.077 0.340 0.198 Popov VN (2000) Physical Review B 61 (4) 3078-84
(8,8) SWCNT 1.086 0.340 0.225 Popov VN (2000) Physical Review B 61 (4) 3078-84
(14,0) SWCNT 1.097 0.340 0.188 Popov VN (2000) Physical Review B 61 (4) 3078-84
(8,8) SWCNT 1.100 0.34 0.153 Sanchez-Portal D (1999) Phys. Rev. B 59 (19)
12678
(10,7) SWCNT 1.165 0.34 0.266 Hernandez E (1998) Physical Review Letters 80
(20) 4502-4505
(15,0) SWCNT 1.176 0.340 0.193 Popov VN (2000) Physical Review B 61 (4) 3078-84
(14,2) SWCNT 1.183 0.340 0.195 Popov VN (2000) Physical Review B 61 (4) 3078-84
(12,6) SWCNT 1.186 0.340 0.193 Popov VN (2000) Physical Review B 61 (4) 3078-84
(9,9) SWCNT 1.222 0.340 0.222 Popov VN (2000) Physical Review B 61 (4) 3078-84
(16,0) SWCNT 1.254 0.340 0.195 Popov VN (2000) Physical Review B 61 (4) 3078-84
(17,0) SWCNT 1.332 0.340 0.197 Popov VN (2000) Physical Review B 61 (4) 3078-84
(10,10) SWCNT 1.357 0.340 0.220 Popov VN (2000) Physical Review B 61 (4) 3078-84
(10,10) SWCNT 1.360 0.34 0.283 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(10,10) SWCNT 1.360 0.34 0.256 Hernandez E (1998) Physical Review Letters 80
(20) 4502-4505
(10,10) SWCNT 1.373 0.34 0.158 Sanchez-Portal D (1999) Phys. Rev. B 59 (19)
12678
(18,0) SWCNT 1.411 0.340 0.199 Popov VN (2000) Physical Review B 61 (4) 3078-84
(16,4) SWCNT 1.437 0.340 0.204 Popov VN (2000) Physical Review B 61 (4) 3078-84
(14,7) SWCNT 1.451 0.340 0.212 Popov VN (2000) Physical Review B 61 (4) 3078-84
(15,6) SWCNT 1.468 0.340 0.209 Popov VN (2000) Physical Review B 61 (4) 3078-84
(19,0) SWCNT 1.489 0.340 0.201 Popov VN (2000) Physical Review B 61 (4) 3078-84
Tipo d (nm) h (nm) ν Referencia
(20,0) SWCNT 1.567 0.340 0.202 Popov VN (2000) Physical Review B 61 (4) 3078-84
(12,12) SWCNT 1.629 0.340 0.217 Popov VN (2000) Physical Review B 61 (4) 3078-84
(21,0) SWCNT 1.646 0.340 0.205 Popov VN (2000) Physical Review B 61 (4) 3078-84
(22,0) SWCNT 1.724 0.340 0.206 Popov VN (2000) Physical Review B 61 (4) 3078-84
(13,13) SWCNT 1.765 0.340 0.217 Popov VN (2000) Physical Review B 61 (4) 3078-84
(23,0) SWCNT 1.802 0.340 0.206 Popov VN (2000) Physical Review B 61 (4) 3078-84
(24,0) SWCNT 1.881 0.340 0.207 Popov VN (2000) Physical Review B 61 (4) 3078-84
(14,14) SWCNT 1.900 0.340 0.216 Popov VN (2000) Physical Review B 61 (4) 3078-84
(25,0) SWCNT 1.959 0.340 0.208 Popov VN (2000) Physical Review B 61 (4) 3078-84
(15,15) SWCNT 2.034 0.34 0.256 Hernandez E (1998) Physical Review Letters 80
(20) 4502-4505
(20,0) SWCNT 1.571 0.34 0.270 Hernandez E (1998) Physical Review Letters 80
(20) 4502-4505
(50,50) SWCNT 6.780 0.34 0.283 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(100,100) SWCNT 13.560 0.34 0.283 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
1652
(200,200) SWCNT 27.120 0.34 0.283 Lu J P (1997) J. Phys. Chem Solids 58 (11) 1649-
Publicaciones
La rugosidad cinética de interfaces ocurre en una amplia variedad de situaciones físicas que va desde el crecimiento de cristales, invasión de fluidos en medios porosos, movimiento de líneas de flujo en superconductores hasta propagación de frentes de fuego en bosques y los fenómenos de fractura, ya que la textura interna de las aleaciones debido a las estructuras dendríticas desarrolladas durante su solidificación es la principal responsable para la mayoría de las propiedades mecánicas. También se ha utilizado la teoría de la mecánica de fractura fractal para explicar el comportamiento de ruptura en nanotubos, por ello, estudiamos experimentalmente las trayectorias postmorten de grietas en placas de concreto.
Intrinsically anomalous roughness of admissible crack traces in concrete
Alexander S. Balankin, Orlando Susarrey, Rafael García Paredes, Leobardo Morales, Didier Samayoa, and José Alfredo López
Sección de Posgrado e Investigación, ESIME, Instituto Politécnico Nacional, México and Grupo “Mecánica Fractal,” México 07738
共Received 30 July 2005; published 1 December 2005兲
We study the roughness of postmortem cracks in concrete plates of different size. We find that the set of admissible crack paths exhibits an intrinsically anomalous roughness; nevertheless, any individual crack trace in concrete is essentially self-affine. We also find that both the local and the global amplitudes of crack traces are distributed according to a log-logistic distribution characterized by the same scaling exponent, whereas the mean-square width distribution is best fitted by the Pearson distribution, while the log-normal distribution also provides quite good adjustments and cannot be clearly rejected.
DOI:10.1103/PhysRevE.72.065101 PACS number共s兲: 62.20.Mk, 68.35.⫺p, 68.35.Ct One of the most challenging puzzles in statistical physics
and materials science is the fracture phenomena关1,2兴. Frac- ture processes are characterized by the high extent of spa- tiotemporal nonuniformity which results in the complex mor- phology of fracture surfaces and crack shapes 关2,3兴. Since the work of Mandelbrot et al. 关4兴, many works have been dedicated to the characterization of the scaling properties of fracture patterns 关5–13兴. Furthermore, it was shown that crack roughness essentially affects the fracture mechanics
关2,5,12,14,15兴.
Numerous experiments have shown that cracks exhibit self-affine scale invariance within a considerable range of length scales关5–8兴up to five decades 关6兴. Namely, the tra- jectory of a crack, z共x兲, is invariant under an anisotropic scale transformation in the sense that the rescaled trace
−z共x兲 has the same statistical properties as z共x兲, where
⬎0 is the scale factor andis the共local兲roughness expo- nent 关16兴. This implies that the local crack width, w共l兲
=具具关z共x兲−具z典l兴2典L典R
1/2, scales with the apparent crack length, ᐉ, as
w⬀l forl0⬍l⬍x, 共1兲
where具¯典l denotes the average over x in window of size
⌬x=l,具¯典R denotes the average over different realizations,
l0is a microscopic cutoff, andxis the horizontal correlation
length, defined as w共l⬎x兲⬀x
. Furthermore, the structure factor of self-affine trace also exhibits scaling behavior, i.e.,
S共k兲=具Z共k兲Z共−k兲典K⬀k−, where = 2+ 1. Here Z共k兲 is the
Fourier transform ofz共x兲and具¯典Kdenotes the average over
the interval k苸关2/x, 2/l0兴. It should be noted that for
truly self-affine fractals,l0= 0 andx=L, whereLis the sys-
tem size, and so, W共L兲=w共l=L兲⬀L 关16兴. Other important characteristics of crack roughness are the statistical distribu- tions of local width 关17兴 and amplitudes ⌬Zm共l,L兲
= max共0艋x艋l兲苸Lz共x兲− min共0艋x艋l兲苸Lz共x兲 关18兴. It has been ar-
gued that the shape of distribution of the mean-square width can be used to distinguish between different universality classes of kinetic roughening关17,19,20兴. Additionally, crack roughness can be characterized by the moments ofq-order height-height correlation function q共l兲=具兩z共x+l兲−z共x兲兩q典L
1/q
⬀lq关16兴. For self-affine cracks
q=for all q. However, in
some cases the crack roughness is characterized by a non- trivial spectrum of scaling exponentsq=f共q兲, such that 2
=关21兴.
Self-affine roughness is commonly associated with the ki- netic roughening obeying the Family–Vicsek dynamic scal- ing ansatz
W共L,t兲=L␣f共t1/z/L兲, 共2兲
where the scaling function f共y兲behaves as f⬀y␣ with␣=, when yⰆ1 and f共y兲 is a constant foryⰇ1. Accordingly, at the early times, tⰆLz, the horizontal correlation length and
global width of growing interface increase with time as x
⬀t1/zandW⬀t␣/z, respectively, wherezis the dynamic expo- nent and=␣/zis the so-called growth exponent关16兴. The roughness exponent␣=and the dynamic exponentzchar- acterize the universality class of the model under study关16兴. However, in many cases the interface roughening displays an anomalous scaling behavior, characterized by different roughness exponents in the local and global scales; namely
␣⬎ 关10–13,22兴. Accordingly, the local fluctuations of the growing interface behave as
w共lⰆx,tⰆLz兲⬀lt共␣−兲/z andw共tⰇLz,lx兲⬀L␣−lx
;
共3兲
nevertheless the global width,W共L,t兲, exhibits the Family– Vicsek scaling共2兲. Moreover, in the case of so-called super- roughening, the structure factor also satisfies the Family– Vicsek ansatz with scaling exponent= 2␣+ 1, nevertheless, generally,
S⬀L共␣−␣s兲k−共2␣s+1兲, 共4兲
where␣s is the spectral exponent 关22兴. Specifically, the in- trinsically anomalous kinetic roughening is characterized by
␣s=⬍␣ 关22,23兴. It is pertinent to note that in this case the
saturated interface exhibits self-affine invariance, so its “anomalous nature” can be detected only through the study of roughening dynamics 共3兲 or by using test specimens of different sizes关24兴.
Unfortunately, in most experimental works devoted to the scaling analysis of fractures, only the local roughness of cracks was measured using the “postmortem” fractures in test specimens of standard dimensions关4–7兴. So the experi- 1539-3755/2005/72共6兲/065101共4兲/$23.00 065101-1 ©2005 The American Physical Society
mental data reported in these works do not permit to distin- guish between the self-affine and the intrinsically anomalous nature of crack roughening. More recently, the intrinsically anomalous roughening of cracks in quasibrittle materials was observed in experiments with woods, some kinds of paper, and mortar关10–13兴. Intrinsically anomalous roughening was also observed in some numerical simulations of crack growth
关25兴. It was found that the global crack roughness exponent is material dependent 关11–13兴. With respect to the local roughness, in a number of works the local roughness expo- nent was conjectured to be universal, i.e., independent of the material, mechanism of fracture, and of the fracture mode
关5兴. Namely, it is suggested that one-dimensional共1D兲cracks are characterized by⬵0.6, whereas two-dimensional 共2D兲
cracks are characterized by⬵0.8关5,6兴. The universality of
, however, is still controversial共see Refs.关7–9,26兴, and ref- erences therein兲. There are strong experimental evidence and theoretical reasons that, at least in materials with long-range correlations in microstructure, the value of is determined by the scaling properties of the material structure 关8,27兴. Moreover, the authors of关9兴have detected a dependence of
on the mechanism of fracture. The dependence of on the crack orientation in anisotropic materials was observed in
关26兴. In this respect, the authors of关28兴noted thatdepends on the strength of disorder in fractured media and the uni- versal exponent emerges as disorder is increased关28兴.
In this work we studied the statistical properties of the postmortem cracks in fractured concrete slabs covering pe- destrian paths in our university. The pavement was installed about three years ago. A sampling analysis of concrete has shown that the aggregate/cement ratio共by weight兲is within the range of 2 ÷ 4; the grain size of gravel and sands varies from 2 to 16 mm, the density of concrete is 2300± 500 kg/ m3, and the compressive strength is 45± 15 MPa.
Concrete is a quasibrittle material, and the nature of its fracture behavior continues to be the subject of intensive research 关13,29,30兴. A common feature of fracture in con- crete is the presence of damage or process zone, which grows near the crack tip. The size of fracture process zone
共its width and length兲depends on both the concrete structure and the stress field, and commonly it is much larger than the typical grain size关31兴. It was found that crack trajectories in concrete possess a self-affine invariance over a wide range of length scales关29,30兴. Furthermore, recently it was found that the front of slowly growing crack in notched mortar beam subjected to four points bending displays intrinsically anomalous dynamic scaling关13兴.
To detect an anomalous roughness, in this work we ana- lyzed the statistical properties of rupture lines in slabs of different sizes L⫻L 共see Fig. 1兲 from different pedestrian paths. Specifically, we studied the slabs with linear sizesL
= 10, 25, 50, 100, 200, and 300 cm. The ratio of thickness共d兲
to width共L兲for all slabs is in the range of 0.02艋d/L⬍0.1. Furthermore, we note that in all cases the size of damage zone is larger thand. So, we assumed that the cracks may be treated as 1D traces. At leastNL= 50 specimens of each size
were analyzed. We are not able to define specific fracture modes and loads associated with each crack. The postmor- tem analysis shows that cracks mainly propagate at the grain- matrix interface. The photoimage of each crack trace was
digitized with the resolution of 1 mm/ pixel关see Fig. 1共a兲兴. Figures 2 and 3 show the scaling behavior of the crack width, amplitude, and structure factor for specimens of dif- ferent size, from which follows that the ensemble of post- mortem cracks in concrete exhibit an intrinsically anomalous roughness characterized by
q==␣S= 0.75 ± 0.02⬍␣= 1.35 ± 0.02 共5兲
for 1艋q艋5; e.g., multiscaling was not detected. The micro- scopic cutoff of observed scaling is of the order of the typical grain size,l0⬵1 cm andx⬵L 共see Figs. 2 and 3兲.
It should be pointed out that the values of local roughness exponents obtained for 320 crack traces are normally distrib- uted with the standard deviation of 0.03; nevertheless, the large variations in loads and environmental conditions asso- ciated with studied cracks 关32兴. Furthermore, we note that the values of both roughness exponents coincide with those obtained for crack fronts in three-dimensional 共3D兲 mortar beams 关13兴, nevertheless the differences in the material structure and the fracture mode. At first glance, this result may be interpreted as support for the hypothesis of scaling universality, e.g.,= 0.8± 0.05 for 2D cracks关5,13兴, although the coincidence may be accidental关33兴. With respect to this point, we note that experimental value= 0.75± 0.02 agrees with the result of simulations of the two-dimensional fuse model关34兴, as well as with the predictions of the correlated percolation theory for 1D cracks关35兴; whereas in three di- mensions the fuse关36兴, as well as the beam lattice关37兴mod- els and the correlated percolation theory 关35兴, predict
⬵0.6.
The scale-free nature of the crack roughness implies that crack growth is essentially probabilistic in nature关15兴. This means that in the test specimen, under given conditions, ex- FIG. 1. 共a兲Photographs of cracks in the concrete slabs of dif- ferent sizeL= 2共1兲, 1共2兲, 0.5共3兲, and 0.25 m共4兲; and共b兲digitized graph of crack trace in the plate of sizeL= 3 m.
ists a set of admissible crack paths characterized by the same local roughness exponent共see关15,38兴兲. The statistical prop- erties of this set can be determined from the analysis of crack traces in macroscopically identical specimens 关15,18兴. The ensemble of postmortem crack traces can be characterized by statistical distributions of the mean-square widthW共L兲 关17兴
and the global amplitudes⌬ZM共L兲=⌬Zm共l=L兲 关18兴.
Accordingly, we found that the distribution of the mean- square crack width is best fitted关39兴by the Pearson distri- bution关see Fig. 4共a兲兴,
f共wL2兲= exp共−k/y兲 k⌫共k+ 1兲具wL2典
冉
y k冊
k+2 , wherey= wL 2 具wL2典 共6兲and⌫共¯兲is the␥function. However, the log-normal distri- bution also provides quite good adjustments 共pvalue = 0.3946兲 and cannot be clearly rejected in terms of2 and
Kolmogorov-Smirnov statistics. Notice that both distribu- tions have a tail heavier than exponential, but lighter than power-law distributions, as expected for self-affine traces
关17,19兴.
At the same time, the local and the global amplitudes of
cracks are more likely关39兴to follow a log-logistic distribu- tion关see Figs. 4共b兲and 4共c兲兴,
f共y兲=mym−1/Z*关1 +ym兴2, wherey=共⌬Z−z*兲/Z*; 共7兲
⌬Z is ⌬Zm共l兲 or ⌬ZM共L兲 and Z*= median兵⌬Z其−z*, z*
= min兵⌬Z其, andmis the distribution tail exponent, which in the case of self-affine cracks depends on⬅␣ 关15,18兴. For intrinsically anomalous cracks in concrete we find that both normalized amplitudes ⌬zl=⌬Zm/L␣−l and ⌬zg=⌬ZM/L␣
exhibit a log-logistic distribution共p value 0.6838关40兴兲char- acterized by the same tail exponentm= 4 ± 0.1共see Fig. 4共d兲
and Ref.关41兴兲.
Intrinsically anomalous crack roughness implies that the FIG. 3. Data collapse for共a兲 crack width,w=w共l,L兲/L0.6, and
共b兲power spectrum,s共k兲=S共k,L兲/L0.6, for crack traces in concrete plates of different size. Straight line slopes = 0.75 共1兲 and −
= −2␣s− 1 = −2.5共2兲.
FIG. 4. 共a兲–共c兲 Conditional probability distributions of共a兲 the mean-square crack width and共b兲the global and共c兲the local am- plitudes of crack traces. Bins—experimental data, lines—data fit- ting by共a兲Pearson distribution withk= 0.76共pvalue= 0.4846兲and
共c–d兲log-logistic distribution共pvalue= 0.6838兲with共c兲m= 4.053,
=z*/Z*= 0.0139 and共d兲m= 3.963,= 0.0048.共d兲Log-log plot of =F共y兲/关1 −F共y兲兴 vsy, whereF共y兲 is the cumulative distribution of the normalized global, y=⌬ZM共L兲/L␣ 共full circles兲, and local,
y=⌬Zm共l,L兲/lL␣−共circles兲, crack width共straight line corresponds
to the log-logistic distribution withm= 4兲. FIG. 2. 共a兲Log-log plots of the global共1兲 and the local共2–6兲
crack width vsl共cm兲in concrete plates of sizeL= 10共2兲, 25共3兲, 50
共4兲, 100共5兲, and 300 cm共6兲; solid line slope␣= 1.35, dotted lines slope= 0.75.共b兲 Log-log plots of␦L共l兲=具具⌬Zm共l,L兲典L典Nvsl共cm兲
in plates of sizeL= 300 cm共1兲and⌬共L兲=具⌬ZM共L兲典Nvs specimen
size共2兲; solid line slope␣= 1.35, dotted line slope= 0.76.
statistical properties of the set of admissible crack paths in concrete are dependent on the system size; nevertheless, any individual crack possesses a self-affine invariance. This means that a crack has information about the specimen size before it fails. Surprisingly, the local and the global ampli- tudes of cracks exhibit the same fat-tailed statistical distribu- tion. These observations provide a unique insight into the
physics of fracture and the nature of intrinsically anomalous roughening.
The authors would like to thank J. M. López, M. A. Ro- dríguez, and R. Cuerno for useful discussions. This work has been supported by CONACyT of the Mexican Government under Project No. 44722.
关1兴M. Adda-Bedia, Phys. Rev. Lett. 93, 185502 共2004兲; E. Bouchbinder, D. Kessler, and I. Procaccia, Phys. Rev. E 70, 046107共2004兲; S. Bohnet al.,ibid. 71, 046214 共2005兲; V. I. Marconi and E. A. Jagla,ibid. 71, 036110共2005兲.
关2兴A. S. Balankin, Eng. Fract. Mech. 57, 135共1997兲; Z. P. Bazant and A. Yavari,ibid. 72, 1共2005兲, and references therein.
关3兴G. P. Cherepanovet al., Eng. Fract. Mech. 51, 997共1995兲; E. Bouchaud, J. Phys.: Condens. Matter 9, 4319 共1997兲; F. Bo- rodich, Int. J. Fract. 95, 239 共1999兲; A. Carpinteri and A. Spagnoli, Int. J. Fatigue, 26, 125 共2004兲, and references therein.
关4兴B. B. Mandelbrotet al., Nature共London兲 308, 721共1984兲.
关5兴E. Bouchaud, G. Lapasset, and J. Planés, Europhys. Lett. 13, 73共1990兲; K. J. Maloyet al., Phys. Rev. Lett. 68, 213共1992兲; J. Schmittbuhl and K. J. Maloy, ibid. 78, 3888 共1997兲; A. Delaplaceet al., Phys. Rev. E 60, 1337共1999兲.
关6兴P. Daguieret al., Phys. Rev. E 60, 1337共1999兲.
关7兴J. Kertszet al., Fractals 1, 67共1993兲; A. S. Balankin and O. Susarrey, Int. J. Fract. 81, R27共1996兲; A. S. Balankinet al.,
ibid. 87, L37共1997兲; 90, L57共1998兲; 106, L21共2000兲; Phys. Lett. A 297, 376共2002兲; Phys. Rev. E 64, 066131共2001兲.
关8兴A. S. Balankinet al., Phys. Rev. Lett. 90, 096101共2003兲.
关9兴S. Morelet al., Phys. Rev. Lett. 93, 065504共2004兲.
关10兴J. M. López and J. Shmittbuhl, Phys. Rev. E 57, 6405共1998兲.
关11兴S. Morelet al., Phys. Rev. E 58, 6999共1998兲; A. S. Balankin
et al., Philos. Mag. Lett. 80, 165共2000兲.
关12兴S. Morelet al., Phys. Rev. Lett. 85, 1678共2000兲.
关13兴G. Mourotet al., Phys. Rev. E 71, 016136共2005兲.
关14兴A. B. Mosolov, Europhys. Lett. 24, 673共1993兲; F. M. Borod- ich, J. Mech. Phys. Solids 45, 239共1997兲; G. Palasantzas, J. Appl. Phys. 83, 5212共1998兲; A. Yavari et al., Int. J. Fract.
101, 365共2000兲; J. Weiss,ibid. 109, 365共2001兲.
关15兴A. S. Balankin, Int. J. Fract. 79, R63 共1996兲; Philos. Mag. Lett. 74, 415 共1996兲; A. S. Balankin et al., Proc. R. Soc. London, Ser. A 455, 2565共1999兲; I. Campos and A. S. Bal- ankin, Theor. Appl. Fract. Mech. 44, 187共2005兲.
关16兴A.-L. Barabási and H. E. Stanley,Fractal Concepts in Surface Growth共Cambridge University Press, Cambridge, 1995兲.
关17兴Z. Rácz and M. Plischeke, Phys. Rev. E 50, 3530共1994兲; M. Plischeke and Z. Rácz,ibid. 50, 3589共1994兲; T. Antal and Z. Rácz, ibid. 54, 2256 共1996兲; E. Marinari et al., ibid. 65, 026136共2002兲.
关18兴A. S. Balankin and O. Susarrey, Philos. Mag. Lett. 79, 629
共1999兲.
关19兴G. Tripathy and W. van Saarloos, Phys. Rev. Lett. 85, 3556
共2000兲; T. Antal et al., ibid. 87, 240601 共2001兲; S. Ray- chaudhuriet al.,ibid. 87, 136101共2001兲.
关20兴A. Rossoet al., Phys. Rev. E 68, 036128共2003兲; S. Moulinet
et al.,ibid. 69, 035103共2004兲.
关21兴J. Schmittbuhlet al., J. Geophys. Res. 100, 5953共1995兲; E. Bouchbinderet al., e-print cond-mat/0508183.
关22兴J. J. Ramascoet al., Phys. Rev. Lett. 84, 2199共2000兲.
关23兴Intrinsically anomalous multifractal roughness is characterized byq⬍␣and␣S=2关see M.-P. Kuittuet al., Phys. Rev. E 59,
2677共1999兲; J. Asikainenet al.,ibid. 65, 052104共2002兲兴.
关24兴We note that both graphs presented in Fig. 4 of Ref.关10兴are essentially self-affine and so, if we do not know the models to which they correspond, we cannot determine which of them is related to intrinsically anomalous kinetic roughening process.
关25兴S. Zapperiet al., e-print cond-mat/0407568.
关26兴I. L. Menezes-Sobrinho et al., Phys. Rev. E 71, 066121