Smooth-to-rough transition in the fracture of fibrous materials

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Physica A 328 (2003) 493 – 504

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Smooth-to-rough transition in the fracture of $brous materials I.L. Menezes-Sobrinho Departamento de F sica, Universidade Federal de Vicosa, Vicosa, MG 36570-000, Brazil Received 19 February 2003

Abstract In this paper, a computational model in (2+1) dimensions which simulates the creep rupture of a $brous material submitted to a constant uniaxial force F is analyzed. This force F produces in the bundle an initial displacement z0 which increases with time. In our model, the breakage of a $ber can provoke a cascade of breaking $bers which describes the propagation of a crack through the $ber bundle. In addition, the crack velocity is not constant, depending on the number of broken $bers in the bundle. Our goal here is to examine the fracture of this type of material as a function of the temperature t and of the initial displacement ratio 0 = z0 =zc (zc is a critical displacement), as well as to investigate the e5ects of these parameters on the fracture roughness and on the roughness exponent . Our results indicate that as 0 approaches the critical initial displacement ratio 0c the roughness W obey the following power law W ∼ (0c − 0 ) , where  0:33 is a critical exponent. These results can be understood in terms of a phase transition from a smooth phase to a rough one. c 2003 Elsevier B.V. All rights reserved.  PACS: 62.20.Mk; 64.60.Fe; 46.50.+a Keywords: Roughness exponent; Crack; Fracture

1. Introduction Fracture process in disordered materials is a subject of intensive research and has attracted much scienti$c and industrial interest [1–3]. Fracture process is extremely sensitive to disorder, that comes into play in many ways during fracture. The e5ect of even small initial disorder can be enormously ampli$ed during the propagation of the crack. Thus, disorder has a strong in?uence on the roughness of the fracture E-mail address: [email protected] (I.L. Menezes-Sobrinho). c 2003 Elsevier B.V. All rights reserved. 0378-4371/$ - see front matter
doi:10.1016/S0378-4371(03)00529-6

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surface. Several computational models have been constructed to study the phenomenon of fracture in disordered materials, such as, the fuse [4] and the well-known $ber bundle models [5–13]. When submitted to a uniaxial force a $brous materials may undergo time-dependent deformation resulting in fracture, called creep rupture. The fracture of the material can occur through a large crack which percolates the sample. In this case, we say that the fracture is catastrophic. It occurs at small temperatures and/or large forces and is similar to a brittle fracture. When the fusion of small cracks cause the fracture of the material, it is called shredding [8], which occurs for large temperatures and/or small forces, and is similar to the ductile fracture. In the catastrophic regime, the fracture pro$le is reasonably smooth, while in the shredding regime it is very rough. A parameter of easy physical interpretation used to characterize fracture surfaces is its roughness, which can be considered as an inheritance of the damage process. It is de$ned as the variance around of the mean height of the fracture pro$le, and measures the complexity of the crack path [14]. Thus, the rougher the fracture pro$le the harder is the crack path. The roughness has a direct relation to the fractal dimension, which characterizes the fractal character of the surface fracture. Therefore, a high fractal dimension indicates a very rough fracture pro$le [15,16]. Experiments have shown that the fracture surface in disordered materials can often be described by self-aEne scaling [15,17,18]. In this case, the roughness (W ) of the fracture surface satis$es the scaling law W () ∼ 

(1)

over the range  of length scales. The roughness exponent  provides a quantitative measurement of the roughness of the fracture surface, and is related to the fractal dimension by the expression [19] =d−D ;

(2)

where d is the Euclidean and D is the fractal dimensionality. Some experimental works have claimed that the roughness exponent  has a universal value of 0.8 [20–22]. However, this universality was questioned by Milman et al. [23], which experimentally found a roughness exponent closer to 0.5. From the theoretical point of view, numerical models have been elaborated in order to evaluate the roughness exponent . Simulations have shown that  ∼ 0:7 in two dimensions [24,25], and that  ranges from 0.4 to 0.5 in three dimensions [26,27]. Nowadays, there is a conjecture relating the smallest and the highest value of the roughness exponent  to the speed of crack propagation through the sample [27]. The greatest value has been associated with a high speed of crack propagation and interpreted as a dynamic regime. In contrast, the smallest value of the exponent was related with a quasi-static regime, where the dynamic e5ects of the propagation are negligible. From the experimental point of view, several works were devoted to characterize the surface fracture of metals submitted to di5erent fracture processes. For example, Daguier et al. [17] investigated the fatigue fracture surfaces of a metallic alloy and the stress corrosion fracture surfaces of a silicate glass. They found that, at large length scales, the roughness exponent is   0:78 and at small

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length scales   0:5. These two exponents are constant and separated by a crossover length which exhibits a power-law decrease with the measured crack velocity. In the present paper, we present a study of a $ber bundle model in (2+1) dimensions for which it is possible to obtain the fracture pro$le of a $brous material submitted to a constant uniaxial force F, for example, by hanging a weight on it. In this model, the crack velocity is dependent on the number of broken $bers in the bundle. The in?uence of the temperature t on the fracture process was also considered. Our main goal is to use the roughness for investigate the transition between the catastrophic and shredding regime. This paper is organized as follows. Section 2 is devoted to present the model and the way in which computational simulations are carried out. In Section 3, the obtained results are presented and analyzed and, $nally, conclusions are drafted in Section 4. 2. Model Our model consists of a bundle of N0 = L × L parallel $bers, all with the same elastic constant, k, distributed on a triangular lattice. In order to simulate the height of the sample, the $bers are divided in  segments having the same length. A similar procedure was recently used in Ref. [14]. The $ber bundle is $xed at both ends by two parallel plates. One of them is $xed and in the other a constant uniaxial force F is applied. Fig. 1 shows a representation of the model. The force F is equally and completely distributed in the $ber bundle, submitting all the $bers to the same linear

Fig. 1. Schematic representation of the $ber bundle model. A static force F is applied on the $ber bundle provoking a degradation process that inevitably leads to failure.

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displacement z = F=Nk, where N is the number of unbroken $bers. When the constant uniaxial force F is applied to the $ber bundle some may break. The failure probability of a $ber i is given by [8] 
2  ( − 1) ; (3) Pi (; t) = exp (ni + 1) t where ni is the number of unbroken neighboring of the $bers,  = z=zc = F=Nkzc is the displacement ratio of the material, zc is the critical displacement, t = KB T=Ec is the normalized temperature, KB is the Boltzmann constant, T is the absolute temperature and Ec is the critical elastic energy. For an isolated $ber (ni = 0) the failure probability is equal to one, when  = z=zc = 1, i.e., the displacement z reaches a critical value zc . We can relate the critical value zc with the strength of the $ber, the higher the value zc , the higher is the strength of the $ber. In our computational model all $bers have the same critical value zc . In this model, besides $nding the failure probability of a $ber, we have to indicate in which segment it breaks. The probability of a $ber to break in a chosen segment j is given by [14] j (mj ) =

(mj + 1) ; g

(4)

where m j is a variable which indicates how many segments j have broken in the bundle and g= j (mj +1). Eq. (4) simulates a concentration of tension near the region where the $ber bundle is weaker. At the beginning of the simulation, the bundle is submitted to an initial displacement ratio given by 0 =

z0 F = : zc N0 kzc

(5)

At each time step we randomly choose a $ber of a set of Nq = qN0 unbroken $bers. The number q represents a percentage of $bers and allows us to work with any system size. Then, using Eq. (3), we evaluate the $ber failure probability Pi and compare it with a random number r in the interval [0,1). If r ¡ Pi the $ber breaks. At this time, we must select where the $ber will break. A segment j is selected at random. The failure probability for this segment, as well as for its nearest-neighboring segments (j − 1) and (j + 1) is calculated from Eq. (4). So, the segment to break is chosen  with the probabilities: ˜ j ; ˜ j+1 and ˜ j−1 , where ˜ k = k = m m for k and m = j + 1; j; j − 1. Therefore, in our model only those three segments are tested. After breaking the $ber, we begin to test all neighboring unbroken $bers. The $rst segment tested in the neighboring unbroken $bers is the one in which the previous $ber broke. The failure probability Pi of these neighboring $bers increases due to the decreasing of ni and a cascade of breaking $bers may begin. This procedure describes the propagation of a crack through the $ber bundle in the direction perpendicular to the applied force. The process of propagation stops when the test of the probability does not allow the fracture of any other $ber on the border of the crack, or when the crack meets another already formed one. The same cascade propagation is attempted by choosing another $ber of the set Nq . After all the Nq $bers have been tested, the displacement  is

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increased if some $bers have been broken. Since the force is $xed, the greater the number of broken $bers, the larger the displacement on the intact $bers and the higher their failure probability, therefore the higher is the crack velocity. After all the Nq $bers have been tested another set of Nq unbroken $bers is chosen and all fracture process is restarted. The simulation terminates when all the $bers of the bundle have been broken, i.e., when the bundle is divided into two parts. 3. Results We performed simulations considering the elastic constant k = 1, the critical displacement zc = 1 and the number of segments  proportional to the system size. The failure probability equation (3) can be written as !(t; ) Pi (z) = ; (6) (ni + 1) where the parameter !(t; ) is de$ned as   2  −1 : !(t; ) =  exp t

(7)

For a triangular lattice (with coordination number 6) and !(t; ) ¿ 6, the fracture of any $ber induces the failure of the whole bundle, i.e., the bundle breaks with just one crack. Obviously, this crack forms a cluster which percolates through the entire system. We can de$ne the density of the percolation crack as Npc "= ; (8) N0 where Npc is the number of broken $bers belonging to the percolating crack. Thus, when !(t; ) ¿ 6, we have " = 1. Fig. 2 shows the density of the percolation crack " versus the initial displacement ratio 0 , for two di5erent temperatures. Notice that, for high values of 0 ; " = 1 and for low values of 0 the density of the percolating cluster " suddenly jumps to zero. Thus, we may assume that there is a critical value 0c (=F=N0 kzc ) which depends on the temperature t. For the temperatures t = 0:5 and 2.0 used in our simulation the obtained values for 0c are 1.11 and 1.27, respectively. Above 0c there is a percolation crack and below it no single crack percolates the $ber bundle. The critical value, 0c , represents the transition between catastrophic and the slowly shredding regime. In the catastrophic regime, there are cracks that percolate the $ber bundle and in the slowly shredding none percolates the bundle. In Ref. [7], we have shown that these two regimes are separated by a second-order phase transition and determined the critical line separating these two failure regimes in the plane t × 0 . Since in the present model each $ber of the bundle can break at di5erent parts, a fracture surface is produced. Fig. 3 shows the fracture surfaces obtained for t = 2:0 and three di5erent initial displacement ratios 0 . In Fig. 3(a) the fracture surface is very rough, a pro$le that is characteristic of a shredding regime. Here a small force

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1.0

ρ (δ0)

0.8

0.6

0.4

0.2

0.0 1.00

1.09

1.18 δ0

1.27

1.36

Fig. 2. Density of the percolating cluster " versus the initial displacement ratio 0 for two di5erent temperatures: t = 0:5 (circles) and t = 2:0 (squares). The data were averaged over 1000 statistically independent samples.

was applied on the bundle provoking a slow process of successive ruptures of $bers in the material, which weakened the bundle causing its failure. In Fig. 3(c), the fracture surface exhibits a small roughness. In this case, the bundle was submitted to a large force causing its catastrophic fracture where the breakage of a single $ber induced the rupture of the whole bundle. For 0 = 1:27 (Fig. 3(b)) the fracture occurs at the boundary between the two failure regimes. It can be seen from Fig. 3 that the rupture of the sample begins at di5erent segments, since we did not use a deterministic starting notch in our simulations. In order to evaluate the roughness exponent , various one-dimensional cuts in the fracture surface were considered. The roughness W of each cut was found by the method of the best linear least-squares $tting described in Ref. [28]. In this method, the roughness W () in the scale  is given by M

W () =

1 wi () M

(9)

i=1

and the local roughness wi () is de$ned as wi2 () =

i+  1 [hj − (ai ()xj + bi ())]2 : (2 + 1) j=i−

(10)

ai () and bi () are the linear $tting coeEcients to the displacement ratio data on the interval [i − ; i + ] centered on the $ber i. Fig. 4(a) shows, for t = 2:0, how the roughness W decreases with the increase of the initial displacement ratio 0 for several lattice sizes L (remember that 0 is the

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Fig. 3. Fracture surfaces for three di5erent initial displacement ratios: (a) 0 = 0:4, (b) 0 = 1:27 and (c) 0 = 1:4. In these simulations a total of N0 = 4 × 104 $bers are involved.

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W

1000

500

0 0.3

0.8

(a)

δ0

0.4

1.3

1.8

4

10

3

10

W

0.3

2

10

1

W/L

ζ

10

0

0.2

10

10

1

10

2

10

3

4

10

L 0.1 0.0 0.3

(b)

0.8

δ0

1.3

1.8

Fig. 4. (a) Roughness W versus the initial displacement ratio 0 for t = 2:0. (b) W=L versus 0 for t = 0:5 (long dashed line: L = 2000 and solid line: L = 4000) and t = 2:0 (symbols). The insert shows the roughness W as a function of the system size L for three initial displacement ratios: circle 0 = 0:4, square 0 = 0:8 and triangle up 0 = 1:2. The full lines, with slope 1.0, are guides to the eyes. The data were averaged over 500 statistically independent samples.

displacement immediately after the application of the force F on the bundle). As can be readily seen from Fig. 4(a), increasing 0 decreases continuously the values of W up to a critical initial displacement ratio 0c (=Fc =L2 ) beyond which W $nally continuously vanishes. This fact can be seen as a phase transition from a smooth phase (0 ¿ 0c ) to a rough one (0 ¡ 0c ), a behavior called smooth-to-rough transition. Using 0 =F=L2 , we assume that W for systems of di5erent sizes (L) can be expressed in terms of the following scaling form:   F ; (11) W (L; F) = L + L2 where +(x) is a scaling function, and  is a scaling exponent. Plotting on the horizontal axis F=L2 = 0 , and on the vertical W (L; F)=L , the di5erent curves in Fig. 4(a) fall on a common curve, as shown in Fig. 4(b). This almost perfect collapse, con$rms the simple scaling assumption expressed in Eq. (11). There are two di5erent scaling regimes depending on the value of F of the scaling function +(x). For F ¡ Fc , the scaling function +(x) does not depend on the system size. In this regime, the roughness W scales with the system size as W (L) ∼ L , where the value of the exponent  can be obtained from the slope of the log–log plot of W versus L for 0 ¡ 0c (see inset

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398

W

251

158

100

−1

10

|1−δ0/δ0c|

0

10

Fig. 5. Log–log plot of the roughness W versus |1 − 0 =0c |, for t = 0:5 (open circles) and t = 2:0 ($lled squares). The data were obtained near the critical initial displacement ratio 0c .

in Fig. 4(b)). For F ¿ Fc the scaling function +(x) tends to zero. Notice that (see Fig. 4(b)) the change of the temperature t a5ects only the value of 0c , the behavior of the roughness W as a function of the initial displacement ratio 0 remaining the same. For 0 ¿ 0c , i.e., for an external force F higher than a critical force Fc , the rupture of the material occurs due to one large crack that percolates the $ber bundle. In this regime, the fracture surface is smooth (see Fig. 3(c)) and the fracture process is relatively rapid. Technologically speaking, a fracture with these characteristic is called brittle fracture. For 0 ¡ 0c the $rst displacement produces small cracks in the bundle and with time evolution new small crack formed. The coalescence of individual microcracks produces the rupture of the $ber bundle. In this regime, the fracture surface is rough (see Fig. 3(a)) similar to the ductile regime. Our results indicate that decreasing the temperature t the critical initial displacement 0c also decreases, therefore, the tendency to the brittle fracture increases. This behavior agrees with the observations obtained experimentally in fracture mechanics [1], which shows that the brittle to ductile transitions happen with increase of the temperature or with decrease of the speed of the deformation, i.e., small external force. In analogy with the general theory of critical phenomena, the roughness W can be thought as the order parameter characterizing this smooth-to-rough transition, while the initial relative displacement 0 is plays the role of the tuning parameter. Near 0c ; W follows a power law W ∼ (0c − 0 ) ;

(12)

where is a critical exponent. The log–log plot of W versus |1 − 0 =0c | is a straight line, where the slop is the exponent  0:35, which does not depend on the temperature t (see Fig. 5). The exponent shows how the fracture surface becomes smooth as 0 approaches 0c . Using Eqs. (1) and (9) the roughness exponent  was calculated along the x and y directions. In both directions, the same value for the roughness exponent was found.

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δ0 = 0.6 δ0 = 0.9 δ0 = 1.0 δ0 = 1.11

10

W(ε)

t = 0.5

t = 2.0

2

10

3

10

2

10

10

10

10

1

10

0

10

1

10

3

2

10

2

1

10

0

10

1

10

2

1

10 0 10

1

10

2

10

ε

3

10

4

10

0

10

1

10

2

10

ε

3

10

4

10

Fig. 6. Log–log plots of the roughness W as a function of the scale  for two di5erent temperatures: t = 0:5 and 2.0. The data were averaged over 500 statistically independent samples with lattice size of 5000 × 5000. The inset shows that all the $ts near 0c have the same slope, corresponding to a value of   0:42 over at least one decade.

This behavior is expected for a large variety of materials in which the two directions have similar scaling properties [29,30]. Fig. 6 shows log–log plots of the roughness W as a function of the scale  for four initial displacement ratios 0 6 0c and two temperatures t. Notice that each curve in Fig. 6 has a characteristic crossover c separating the large and short length scales regimes. At small length scales the asymptotic relationship corresponds to a straight line, i.e., W ∼ loc . In the neighborhood of the critical initial displacement ratio 0c ; loc  0:42 ± 0:02. This value is remarkably close to the local roughness exponent loc obtained in three-dimensional simulations [26,27]. Decreasing 0 the crossover c becomes diEcult to be de$ned and the slope of the curve log(W )×log() tends to zero (loc → 0). This means that, for smaller values of 0 , the fracture surface becomes very rough and $lls the space (see Fig. 3(a)). In this region, the fractal dimension D of the surface tends to the Euclidean dimension d. Then, from Eq. (2) we have loc → 0. This result supports the idea that the fracture roughness exponent is related to the di5erent dynamics of the crack. For 0 ¿ 0c the rupture is catastrophic and the material breaks in the $rst time step. In this regime, it was not possible to $nd the roughness exponent loc , since in this region the fracture surface becomes ?at and the fracture roughness W tends to zero (see Fig. 3(c)). The behavior of the crossover c as a function of the initial relative displacement 0 is shown in Fig. 7 for two temperatures, t = 0:5 and 2.0. Our results suggest a power law behavior between the crossover c and the initial relative displacement 0 c ˙ 0 ; where the exponent  depends on the temperature t.

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εc(δ0)

100

36

10 1

δ0

Fig. 7. Log–log plot of the crossover c as a function of the initial displacement ratio 0 for two di5erent temperatures: t =0:5 (circles) and t =2:0 (squares). The data were averaged over 500 statistically independent samples with lattice size of 5000 × 5000.

4. Conclusions In conclusion, we studied the process of creep rupture of a $brous material submitted to a constant uniaxial force F, in which is possible to obtain the fracture surface of the sample. In our model, the crack velocity depends on the number of broken $bers in the bundle and changes with the time step. We have analyzed the roughness of the fracture surface as a function of the $rst displacement produced in the $ber bundle due to the application of an external force. A decrease of the roughness W with the increase of the initial displacement 0 , up to the critical initial displacement ratio 0c was observed. This decrease of the roughness can be seen as a phase transition of the second-order type and can be understood in terms of a roughening transition, i.e., a smooth-to-rough transition. This behavior is similar to the brittle-to-ductile transition observed in metals. The brittle fracture occurs when a relatively high external force (which is proportional to the initial displacement ratio 0 ) is applied in the material. In this regime, the fracture process is rapid and the fracture surface is smooth. The ductile fracture happens when an external force relatively lower is applied in the material. In this case, the rupture of the material is slow and the fracture surface is very rough. Near the critical point 0c , which depends on the temperature t, the roughness W behaves as W ∼(0c −0 ) , where is a critical exponent which does not depend on the temperature t. Ours results also support the idea of dependency of the local roughness exponent loc upon the initial displacement ratio 0 . For short length scales and close to the critical value 0c the local roughness exponent is loc  0:42 ± 0:02. This value is the same as the one obtained in other simulations in three dimensions [26,27] that related the exponent loc with the quasi-static regime.

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In the literature, there are several experimental works that characterize the fracture surfaces of di5erent metals submitted to di5erent fracture processes. We hope that the results presented in this paper stimulate further experiments on $brous materials in order to con$rm the validity of the present results. Acknowledgements The author thanks M.L. Martins, M.S. Couto and J.A. Redniz for a careful reading of the manuscript, and the kind hospitality of the Departamento de FMNsica, UFMG. The author also acknowledges FAPEMIG and CNPq (Brazilian agencies) for $nancial support. References [1] H.J. Herrmann, S. Roux (Eds.), Statistical Models for the Fracture of Disordered Media, North-Holland, Amsterdam, 1990. [2] S. Zapperi, P. Ray, H.E. Stanley, A. Vespignani, Phys. Rev. Lett. 78 (1997) 1408. [3] C. Maes, A. van Mo5aert, H. Frederick, H. Strauven, Phys. Rev. B 57 (1998) 4987. [4] L. de Arcangelis, H.J. Herrmann, J. Phys. Lett. (Paris) 46 (1985) L585. [5] S.D. Zhang, E.J. Ding, Phys. Lett. A 193 (1994) 425. [6] P.M. Duxbury, P.L. Leath, Phys. Rev. B 49 (1994) 12 676. [7] I.L. Menezes-Sobrinho, J.G. Moreira, A.T. Bernardes, Int. J. Mod. Phys. C 9 (1998) 851. [8] I.L. Menezes-Sobrinho, J.G. Moreira, A.T. Bernardes, Eur. Phys. J. B 13 (2000) 313. [9] I.L. Menezes-Sobrinho, Phys. Rev. E 65 (2001) 011 502. [10] A.T. Bernardes, J.G. Moreira, J. Phys. I France 5 (1995) 1135. [11] R.C. Hidalgo, F. Kun, H.J. Herrmann, Phys. Rev. E 64 (2001) 066 122. [12] E.L. Bonnaud, J.M. Neumeister, Phys. Lett. A 290 (2001) 261. [13] H.E. Daniels, Proc. R. Soc. A 183 (1945) 404. [14] I.L. Menezes-Sobrinho, J.G. Moreira, A.T. Bernardes, Phys. Rev. E 63 (2001) 025 104(R). [15] B.B. Mandelbrot, D.E. Passoja, A.J. Paullay, Nature (London) 308 (1984) 721. [16] J.A. Rodrigues, V.C. Pandolfelli, Mater. Res. 1 (1998) 47. [17] P. Daguier, B. Nghiem, E. Bouchaud, F. Creuzet, Phys. Rev. Lett. 78 (1997) 1062. [18] J.M. Lopez, J. Schmittbuhl, Phys. Rev. E 57 (1998) 6405. [19] A.L. BarabMasi, H.E. Stanley (Eds.), Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, 1995. [20] E. Bouchaud, G. Lapasset, J. PlanMes, Europhys. Lett. 13 (1990) 73. [21] K.J. MaR lHy, A. Hansen, K.L. Hinrichsen, S. Roux, Phys. Rev. Lett. 68 (1992) 213. [22] J. Schmittbuhl, S. Roux, Y. Berthaud, Europhys. Lett. 28 (1994) 585. [23] V.Y. Milman, R. Blumenfeld, N.A. Stelmashenko, R.C. Ball, Phys. Rev. Lett. 71 (1993) 204. [24] A. Hansen, E.L. Hinrichsen, S. Roux, Phys. Rev. Lett. 66 (1991) 2476. [25] G. Caldarelli, R. Ca$ero, A. Gabrielli, Phys. Rev. E 57 (1998) 3878. [26] V.I. Raisanen, E.T. Seppala, M.J. Alava, P.M. Duxbury, Phys. Rev. Lett. 80 (1998) 329. [27] G.G. Batrouni, A. Hansen, Phys. Rev. Lett. 80 (1998) 325. [28] J.G. Moreira, J. Kamphorst Leal da Silva, S. Oli5son Kamphorst, J. Phys. A 27 (1994) 8079. [29] F. PlourabouMe, P. Kurowski, J.P. Hulin, S. Roux, J. Schmittbuhl, Phys. Rev. E 51 (1995) 1675. [30] A. Parisi, G. Caldarelli, Physica A 280 (2000) 161.