International Journal of Fracture (2006) 140:39–54 DOI 10.1007/s10704-005-3471-4

© Springer 2006

Scaling properties of mortar fracture surfaces G. MOUROT1 , S. MOREL1,∗ , E. BOUCHAUD2 and G. VALENTIN1 1 Lab. de Rh´eologie du Bois de Bordeaux, UMR 5103 (CNRS/INRA/Univ. Bordeaux 1), Domaine de l’Hermitage, 69 route d’Arcachon, 33612 Cestas Cedex, France 2 C.E.A. Saclay, (D.S.M./D.R.E.C.A.M./S.P.C.S.I.), 91191 Gif-Sur-Yvette Cedex, France ∗ Author for correspondence. (E-mail: [email protected])

Received 30 June 2005; accepted in revised form 22 September 2005 Abstract. Scaling properties of mortar crack surfaces are studied from mode I fracture specimens of six different sizes. Fracture surfaces initiated from a straight notch exhibit an anomalous dynamic scaling which involves two independent roughness indices: the universal local roughness exponent ζloc ≈ 0.8 and the global roughness exponent, estimated to ζ  1.35. We show that there exists a linear relationship between the specimen size and the maximum self-affine correlation length inducing a size effect on the roughness magnitude at saturation and this especially for the smallest length scales. Finally, we argue that anomalous roughening could be an inheritance of the changes in long range elastic interactions which take place in the fracture process zone of quasibrittle materials. Key words: Mortar, quasibrittle fracture, anomalous scaling, size effect.

1. Introduction The failure of quasibrittle materials such as concrete, mortar, rocks or wood, is characterized by the development of a large microcracked fracture process zone ahead of the main crack. The damage development in this process zone is well known to be at the source of the specific fracture properties of quasibrittle materials as the resistance curve behavior and the size effect. Concurrently, the elastic interactions which take place in the fracture process zone between the microcracks and the main crack front have a strong influence on the geometry of this crack front and hence on the resulting morphology of the fracture surface. Quantitative fractography experiments appear to be an interesting way for the understanding of fracture mechanisms in such complex materials. The statistical characterization of fracture surfaces is nowadays a very active field of research. The fracture surfaces obtained in materials as different as metallic alloys (Bouchaud et al.,1990; Dauskardt et al., 1990, Imre et al., 1992, Morel et al., 2004), ceramics (Mecholsky et al., 1989, M˚aløy et al., 1992), glass (Daguier et al., 1997), concrete (Carpinteri et al., 1999), mortar (Mourot et al., 2005), rocks (Schmittbuhl ´ et al., 1995a; Lopez and Schmittbuhl, 1998), sea ice (Weiss, 2001), and wood (Engøy et al., 1994, Morel et al., 1998, 2003) have shown self-affine scaling properties in a large range of length scales (see Bouchaud, 1997, 2003 for reviews). In most cases, and in spite of large differences in the fracture mechanisms, the roughness index measured in the direction perpendicular to crack propagation, shows a surprising robustness. Indeed, this index ζloc called the local roughness exponent is close to 0.8. It is

40 G. Mourot et al. now admitted that ζloc has a universal value (Bouchaud et al., 1990), although the origin of this universality is still not fully understood. On the other hand, several recent studies focused on the roughness development ´ of cracks in quasibrittle materials (Lopez and Schmittbuhl, 1998; Morel et al., 1988, 2003; Mourot et al., 2005) have shown the anomalous roughening of fracture surfaces. This anomalous scaling allows to describe scalings of the local and global surface fluctuations by means of an additional roughness index, called the global roughness exponent ζ , different from, and independent of the universal local roughness exponent ζloc . A first consequence of this anomalous roughening is that the magnitude of the roughness related to different length scales is not just a function of the considered length scale but also of the distance to initial notch. A second consequence is the size effect on the roughness magnitude in the stationary regime. Indeed, anomalous scaling leads to an interdependence of the height fluctuations and the specimen size, and this whatever the considered length scale. However, experimental evidence of this size effect is rarely reported in the literature (to our knowledge a size effect evidence is only addressed in Morel et al., 1998) because it is difficult to change the system size in most experimental situations. In this study, the morphology of crack surfaces obtained from geometrically similar mortar fracture specimens of different sizes is studied. We show that the fracture surfaces exhibit self-affine scaling properties, but also that the roughness of these surfaces is anisotropic and must be described on the basis of anomalous scaling. Especially, it is shown that the global roughness seems independent of the specimen size. Moreover, the maximum self-affine correlation length appears proportional to the specimen size leading to a size effect on the maximum roughness magnitude and this whatever the length scale. Finally, we argue that anomalous scaling could be an inheritance of the diffuse damage which progressively develops ahead of the crack front in quasibrittle materials. 2. Experiment The tested material is a mortar composed with a high strength Portland cement (CPA-CEMI 52.5) and a fine sand with a maximum grain size of 2 mm. A superplasticizing agent has been added in order to limit the number and the size of air inclusions. The mortar density is 2.3. Test specimens are notched bending specimens as shown in Figure 1. Geometrically similar specimens of six different sizes characterized by their heights D have been tested. D ranges over a decade, from 20 to 200 mm (Figure 2). Dimensions of the specimens are reported in Table 1. The length of the notch, D/2, has been chosen in order to ensure stable crack propagation. Such a geometry leads to mode I failure. Topographies of the fracture surfaces were recorded with an optical profiler along regular grids. Grid axes are oriented along the x and y directions which are, respectively, perpendicular and parallel to the direction of crack propagation. The first profile, y = 0, is located in the immediate vicinity of the initial straight notch. The number of points in a profile (ranging between 1024 and 8192) is adjusted as a function of the specimen thickness and in order to obtain a step of sampling in the x direction which corresponds approximately to 20  m. The distance between two

Scaling properties of mortar fracture surfaces 41

8/3 D

D 1/2 D

D

8D 10 D Figure 1. Geometry of the notched bending specimens. The height D corresponds to the characteristic specimen size.

Figure 2. Geometrically similar mortar fracture specimens of six different characteristic sizes D ranged between 20 and 200 mm. Table 1. Dimensions of the notched bending specimens. Dimensions are given in mm. Depth

Thickness

Length

Notch length

20 30 50 100 140 200

20 30 50 100 140 200

200 300 500 1000 1400 2000

10 15 25 50 70 100

successive profiles lies between 17 to 60 m, and is a function of D. The vertical resolution (z direction), estimated from the measurements of height differences between two successive samplings along the same line, is close to 5 m. Horizontal resolutions along the x-and y-axis are approximately equal to 2 m. A fracture surface topography obtained on a specimen of characteristic size D = 140 mm (specimen 100-06) is shown in Figure 3.

42 G. Mourot et al.

Figure 3. Topography of the crack surface of a specimen of characteristic size D = 100 mm (specimen 100-06). This map is 82 mm large (x-direction, parallel to initial notch) and 25 mm long (y-direction, parallel to the crack propagation direction).

3. Scaling invariance: self-affine properties Since the 1990s, extensive experimental studies have shown a surprising scaling invariance of fracture surfaces. It is now firmly established that these surfaces are self-affine i.e. the typical h(l), estimated at a length scale l has the following property: h(λl) = λζloc h(l),

(1)

where ζloc is the Hurst or local roughness exponent. ζloc , which characterizes local surface fluctuations, is estimated in this study through three different and independent methods: the ‘variable bandwidth’ method (called RMS in the following), the power spectrum method (PSD) and the average wavelet coefficient method (AWC). The variable band width method consists in computing the root mean square (RMS) h(l) of the fluctuations of height z(x) over a window size l along the x-axis (Bouchaud, 1997; Schmittbuhl et al., 1995b):   l l 1  2 1/2 1 2 h(l) = z(xi ) − z(xi ) , l l i=1

i=1

j

(2)

Scaling properties of mortar fracture surfaces 43 3

2

log10[Δh(l)]

1 ζloc=0.74

1

log10(ξ) 0 1

2

3

4

5

log10(l) Figure 4. Roughness RMS h(l, y) as a function of the length scale l obtained in the case of the specimen 100-06. This scaling is in agreement with Equation (3) and leads to an estimate of the local roughness exponent: ζloc = 0.74.

where brackets . . .  denote an average over all possible origins j of the window along the profile at a fixed position y. Figure 4 shows the root mean square h(l, y) of the height fluctuations h(l, y) as a function of the length scale l obtained in the case of a specimen of characteristic size D = 100 mm (specimen 100-06). For length scales l smaller than the self-affine correlation length ξ , the RMS roughness evolves as a power law with exponent ζloc estimated from Figure 4 to 0.74. For larger length scales, the roughness magnitude saturates. The scaling properties of h relative to an isolated profile (Figure 4) can be described as  ζ l loc if l  ξ , (3) h(l, y = const) ∼ const. if l ξ . The power spectrum (PSD) method is based on the computation of the Fourier transform of the self-correlation function z(x + x)z(x) (Schmittbuhl et al., 1995b). In the case of a self-affine profile, the power spectrum is expected to scale as: S(k) ∼ k −(2ζloc +1) ,

(4)

where k is the wave factor. As shown in Figure 5, the scaling obtained from the power spectrum (PSD) for specimen 100-06 is in agreement with Equation (4). The slope of the inclined straight line, i.e. 2ζloc + 1 = 2.58, allows to estimate the local roughness exponent value: ζloc = 0.79. The average wavelet coefficient method uses the Daubechies wavelet transform of z(x) (Simonsen et al., 1998). If the profile z(x) is self-affine, the averaged wavelet coefficients W [z](a) are expected to scale as: W [z](a) ∼ a 1/2+ζloc ,

(5)

44 G. Mourot et al. 6

log10[S(k)]

4

1 2

(2ζloc+1)=2.58

0

-2 -5

-4

-3

-2

-1

log10(k) Figure 5. Log–log plot of the power spectrum S(k) obtained in the case of the specimen 100-06. According to Equation (4), the slope of the inclined line, relative to the self-affine domain, allows to estimate the local roughness exponent value ζloc = 0.79.

where a is the scale factor. Figure 6 shows the average wavelet coefficients W [z](a) as a function of a obtained for specimen 100-06. According to Equation (5), the inclined straight line allows to estimate the local roughness exponent value: ζloc = 0.75. In Table 2, the results obtained for each tested specimen and for the six characteristic sizes are reported. Despite the fact that 20 specimens of the smallest size were tested (D = 20 mm), only one is presented. This is mainly due to the small size of the fracture surfaces, but also to the fact that the roughness development is not clearly visible on small specimens. Note also that a single specimen of size D = 200 mm is presented because of the difficulty to test such large samples. As shown in Table 2, the values of the local roughness exponent ζloc obtained for all the specimens with the three independent methods described above seem independent of the specimen size D. Averages lead to: 0.74 ± 0.04 for the RMS method, 0.78 ± 0.05 for PSD, and 0.84 ± 0.05 for the AWC method. The robustness of these values intimates that the local roughness exponent ζloc has a universal value, close to 0.8, as suggested by Bouchaud et al. (1990). 4. Anomalous scaling 4.1. Anomalous roughness development Since the 1980s, many theoretical and experimental works have been dedicated to the study of self-affine kinetic roughening. The first scaling proposed was the Family–Vicsek ansatz (Family and Vicsek, 1991). In order to illustrate the Family–Vicsek

Scaling properties of mortar fracture surfaces 45

log10[W[z](a)]

4

3 (1/2)+ζloc=1.25 2 1

1

0 0

1

2

3

log10(a) Figure 6. Log–log plot of the averaged wavelet coefficients W [z](a) as a function of the scale factor a obtained in the case of the specimen 100-06. The scaling is in agreement with Equation (5) and the local roughness exponent is estimated to ζloc = 0.75.

scaling, let us consider a fracture surface where the x-and y-axis define the average crack plane as already mentioned in Section 2. The Family–Vicsek scaling expresses the fact that the roughness of the surface, estimated for instance from the root mean square of the height fluctuations h(l, y) (Equation (2)), evolves as a function of the length scale l along the x-axis but also as a function of the position y from the initial notch as:  ζ l loc if l  ξ(y), h(l, y)  A (6) ξ(y)ζloc if l ξ(y), where ξ(y) is the self-affine correlation length measured along the x-axis, i.e. the upper bound of the scaling domain characterized by the local roughness exponent ζloc . ξ(y) evolves as a power law of the distance y to the initial notch: ξ(y) ∼ y 1/z where z is usually called dynamical exponent because position y is generally replaced by time t in surface growth processes1 . On the basis of a Family–Vicsek scaling, local surface fluctuations, i.e. estimated at length scales l smaller than the system size L (l  L) are characterized by the same scaling exponent as the one related to global surface fluctuations. Indeed, according to Equation (6), when the self-affine correlation length ξ has reached the system size L (corresponding to the specimen thickness), the global surface fluctuations scale as: h(L) ∼ Lζloc . However, in several theoretical and experimental surface growth processes, local and global surface fluctuations have different scaling exponents. This singular surface In this study, we consider that there exists a linear relationship between time t and position y because the crack velocity is approximately constant, due the stable character of crack propagation, linked to the R-curve behavior of mortar.

1

46 G. Mourot et al. Table 2. Values of the scaling exponents ζloc , ζ and z and of the maximum self-affine correlation length ξsat . Spec.

Spec.

ζloc

size D

label

RMS

PSD

0.74 0.74 0.76 0.76 0.73 0.72 0.74 0.76 0.76 0.75 0.75 0.76 0.70 0.71 0.73 0.71 0.70 0.74 0.71 0.74 0.76 0.78 0.70 0.75 0.75 0.78 0.78 0.74

0.85 0.85 0.80 0.84 0.86 0.78 0.82 0.77 0.79 0.80 0.81 0.79 0.72 0.81 0.59 0.76 0.75 0.79 0.74 0.74 0.83 0.78 0.80 0.73 0.78 0.79 0.79 0.78

20

20-01 Aver. 30-01 30-02 30 30-03 30-04 Aver. 50-01 50-02 50 50-03 50-04 Aver. 100-01 100-02 100-03 100 100-04 100-05 100-06 Aver. 140-01 140-02 140 140-03 140-04 140-05 Aver. 200 200-01 Aver. Overall average

gA (u) (RMS)

sA (u) (PSD)

ξsat

AWC

ζ

z

ζ

(mm)

0.84 0.84 1.05 1.01 0.99 0.86 0.98 0.83 0.96 0.87 0.85 0.88 0.73 0.77 0.78 0.73 0.77 0.75 0.75 0.82 0.82 0.94 0.70 0.74 0.80 0.85 0.85 0.84

1.1 1.1 1.1 1.2 1.3 1.2 1.2 1.2 1.3 1.4 1.3 1.3 1.1 1.2 1.5 1.3 1.4 1.3 1.3 1.2 1.4 1.3 1.0 1.3 1.2 1.1 1.1 1.2

4.3 4.3 4.1 7.0 6.6 5.4 5.8 4.2 5.2 6.3 8.6 6.1 4.0 5.0 4.8 3.5 6.1 4.8 4.7 5.5 5.2 3.4 2.4 5.6 4.4 2.3 2.3 5.0

z

0.9

3.9

1.9 1.4 1.4 1.4

8.4 6.2 5.3 5.4

0.9 1.2 1.9 1.7 1.9 1.2 1.4 1.9 1.7 1.3 1.4 1.2 1.6 1.2 1.3 1.9 1.9 1.5

8.2 6.3 7.5 4.6 5.8 4.2 6.1 5.7 5.7 5.8 4.1 3.5 6.2 4.2 4.8 4.0 4.0 5.5

2.9 2.9 3.4 4.2 3.3 3.6 3.6 6.0 11.3 4.7 5.4 6.9 8.6 8.3 7.6 15.4 13.9 10.3 10.7 13.6 12.2 10.2 13.2 12.3 12.4 21.1 21.1

The local roughness exponent ζloc is estimated from the ‘variable bandwidth’ method (RMS), the power spectrum (PSD) and the averaged wavelet coefficients method (AWC). The global roughness exponent ζ and the dynamical one z are estimated through the anomalous scaling functions gA (u) (Equation (8)) and sA (u) (Equation (9)) respectively related to the ‘variable bandwidth’ (RMS) and the power spectrum (PSD) methods.

growth leads to the existence of an independent global roughness exponent ζ that characterizes global surface fluctuations and differs from the local roughness exponent ζloc . ´ ´ This particular scaling is called anomalous (Lopez and Rodr´ıguez, 1996; Lopez et al., 1997) by comparison to the Family–Vicsek scaling (Equation (6)). In the case of the RMS roughness (Equation (2)), the anomalous scaling can be expressed as:

Scaling properties of mortar fracture surfaces 47  h(l, y)  A

l ζloc ξ(y)ζ −ζloc if ξ(y)ζ if

l  ξ(y), l ξ(y).

(7)

Note that the Family–Vicsek scaling (Equation (6)) is recovered from the anomalous scaling (Equation (7)) when the global roughness exponent ζ is equal to the local exponent ζloc . According to Equation (7), when the self-affine correlation length ξ reaches the system size L, the scaling of global surface fluctuations h(L) ∼ Lζ differs from the local one h(l  L) ∼ l ζloc . Anomalous roughening has been observed in several numerical models such as molecular-beam epitaxial growth (Schroeder et al., 1996; Krug, 1994; Das Sarma et al., ´ ´ 1996; Lopez et al., 1997), some growth models in the presence of disorder (Lopez and Rodr´ıguez, 1996), electrochemical deposition models (Castro et al., 1998), etc. It has also been reported in many experimental studies, especially in molecular beam-epitaxy (Yang et al., 1994), in Cu electrodeposition process (Huo and Schwarzacher, 2001) in sputter-deposition growth of Pt on glass (Jeffries et al., 1996), in the interface roughness of Fe–Cr superlattices (Santamaria et al., 2002), in the roughening of Hele-Shaw flows (Soriano et al., 2003) and also in the roughness development of fracture surfaces ´ (Lopez and Schmittbuhl, 1998; Morel et al., 1998, 2003; Mourot et al., 2005). In order to illustrate anomalous roughening in propagating cracks, the RMS roughness of the mortar specimen 100-06 is plotted in Figure 7 as a function of the distance y to initial notch for several length scales. In Figure 7, roughening can be observed between positions ymin and ysat . For distances y < ymin ≈ 1 mm, the roughness magnitude seems approximately constant. This could be due to the non–zero thickness of the initial straight notch (0.4 mm) imposing a non zero roughness at the onset of crack propagation, as previously mentioned in Mourot et al. (2005). For distances y > ysat , the roughness magnitude saturates for all length scales. This global saturation is due to the fact that the self-affine correlation length ξ has reached its maximum value: ξ(y > ysat ) = ξsat . Indeed, according to Equation (7), if the self-affine correlation length becomes constant for distances y > ysat , the roughness magnitude is also expected to be independent of y. The roughness exponent observed for the smallest length scale, determined from the slope of the lower line in Figure 7, is characteristic of the anomalous scaling. According to Equation (7), the slope of this lower line corresponds to the scaling exponent (ζ − ζloc )/z, while the slope of the upper line, related to a large length scale, i.e. l ξ(y), allows to estimate the scaling exponent: ζ /z. More details about the roughness development of mortar crack surfaces can be found in Mourot et al. (2005). In order to obtain a more accurate estimate of the scaling exponents ζ and z, it is useful to express the anomalous scaling (Equation (7)) as h(l, y) = l ζ gA (l/y 1/z ) ´ where gA (u) is the anomalous scaling function (Lopez et al., 1997) which has the following asymptotic behaviors:  gA (u) 

u−(ζ −ζloc ) if u−ζ if

u  1, u 1.

(8)

Two examples of experimentally determined gA (u) (Equation (8)) obtained in the case of specimens 050-01 and 140-03 are plotted in Figure 8. In fact, Figure 8 shows the best data collapses of all profiles ranging between positions ymin and ysat . For

48 G. Mourot et al. 3

log10[Δh(l,y)]

0.31

1

2

1 1 log10(ymin) 0 2,5

0.16 log10(ysat)

3

3,5

4

log10(y) Figure 7. Log–log plot of the roughness RMS as a function of the distance y to the initial notch (specimen 100-06). The part between the position ymin and ysat corresponds to the roughness growth domain.

each specimen, the couple of scaling exponents (ζ, z) leading to the best data collapse are reported in Table 2. A similar method based on the power spectrum can also be used. Indeed, in the case of anomalous scaling, the power spectrum (Equation (4)) can be expressed as ´ S(k, y) ∼ k −(2ζ +1) sA (ky 1/z ) where the anomalous scaling function sA (u) (Lopez et al., 1997; Soriano et al., 2003) scales as:  2ζ +1 u if u  1, sA (u) ∼ 2(ζ −ζloc ) (9) u if u 1, where u = ky 1/z . As for the RMS roughness, the experimental values sA (u) obtained in the case of the mortar specimens 050-01 and 140-03 are shown in Figure 9. For each specimen, the couple of values (ζ, z) leading to the best data collapse are reported in Table 2. As shown in Table 2, the values of the scaling exponents ζ and z estimated from gA (u) (Equation (8)) are consistent with those obtained from sA (u) (Equation (9)). As observed in previous studies (Morel et al., 1998, 2003; Mourot et al., 2005), if the values of the global roughness exponent are quite robust, the values of the dynamical exponent z are more scattered. However, despite this scattering, the values of this exponent and of the global roughness exponent ζ seem independent of the specimen size. The average scaling exponents estimated from the RMS and the PSD anomalous scaling functions are equal to: ζ ≈ 1.35 and z ≈ 5.25. ´ Thus, as previously observed for other quasibrittle materials, granite (Lopez and Schmittbuhl, 1998) and various species of wood (Morel et al., 1998, 2003), mortar fracture surfaces also exhibit anomalous scaling properties. If the measured values of

Scaling properties of mortar fracture surfaces 49 0

-ζ

log10[Δh(l,y)l ]

1 ζ−ζloc=0.44

-1

1 ζ=1.2

-2 Size D=50 mm -3 0

1

2

3

4

5

0

-ζ

log10[Δh(l,y)l ]

1 ζ−ζloc=0.52

-1

-2

1 ζ=1.3

-3 Size D=140 mm -4

0

1

2

3

4

5

1/z

log10[l/y ] Figure 8. Anomalous scaling function gA (u) [Equation (8)] corresponding to the roughness RMS in the case of two specimens of characteristic sizes D = 50 and 140 mm (specimens 50-01 and 140-03).

the dynamical exponent z are quite scattered, the global roughness exponent ζ could be a material dependent parameter. But, what could be the source of the anomalous roughening? As previously mentioned in the introduction, the morphology of the fracture surface of a quasibrittle material is strongly influenced by the more or less long range elastic interactions which take place in the fracture process zone. ´ Moreover, it has been shown recently by Lopez et al., (2005) from dynamic renormalization group arguments that the intrinsic anomalous roughening cannot occur in local growth models and that, as a consequence, disorder and/or nonlocal effects must be responsible for the occurrence of intrinsic anomalous roughening in experimental systems. Concurrently, Hansen et al. (2005) have recently shown, through a wavelet analysis of numerical self-affine traces, that anomalous scaling is linked to intrinsic non-stationary averaging effects of the signal. Moreover, the fact that this property is not recovered in experimental anomalous roughening suggests that the physical mechanism inducing the singular small scale behavior of anomalous scaling is linked to changes in the large scale correlation length (Hansen et al., 2005). On ´ the basis of these recent numerical studies (Lopez et al., 2005; Hansen et al., 2005),

50 G. Mourot et al. -4

log10[S(k,y)k

(2ζ+1)

]

1 -6

2(ζ−ζloc)=1.26

-8 1

-10

2ζ+1=3.8

-12 -14

Size D=50 mm

-16 -6

-5

-4

-3

-2

-1

0

log10[S(k,y)k

(2ζ+1)

]

-4 1

-6

2(ζ−ζloc)=0.84

-8 1 -10

2ζ+1=3.4

-12 -14 -16

Size D=140 mm -6

-5

-4

-3

-2

-1

0

1/z

log10[ky ] Figure 9. Anomalous scaling function sA (u) (Equation (9)) related to the power spectrum obtained in the case of two specimens of characteristic sizes D = 50 and 140 mm (specimens 50-01 and 140-03).

the source of experimental anomalous scaling should be found in the nonlocal effects or in other terms in the changes of a large length scales physical mechanism. Thus, as intuitively expected by Mourot et al. (2005), the anomalous scaling could be an inheritance of the diffuse damage which characterizes quasibrittle fracture. Indeed, the existence of more or less long range elastic interactions and the fact the intensity of these interactions evolves with the crack advance could explain the singular roughness growth at small length scales characteristic of anomalous roughening. Finally, on the basis of what has been mentioned before, the global roughness exponent ζ could be dependent of the damage development at large length scales and of the material rather than only material dependent. 4.2. Size effect As mentioned in the introduction, one of the main expected consequence of anomalous roughening is the size effect on the roughness magnitude in the stationary regime (or saturation regime).

Scaling properties of mortar fracture surfaces 51 25

ξsat [mm]

20

15

10

5

0 0

50

100

150

200

D [mm] Figure 10. Maximum self-affine correlation length ξsat = ξ(y > ysat ) as a function of the characteristic size D. ξsat appears proportional to D: ξsat /D ≈ 9%.

In order to show if there exists a size effect on roughening, the maximum selfaffine correlation length ξsat , i.e. estimated for distances y to initial notch larger than ysat , is first given in Table 2 and plotted in Figure 10 as a function of the characteristic size D. As shown in Figure 10, the maximum self-affine correlation length seems proportional to the characteristic size D, the ratio ξsat /D being around 9%. Then, according to Equation (7), for distances y > ysat , i.e. when the self affine correlation length has reached its maximum value ξsat = ξ(y > ysat ), the ratio h(l  ξsat , y > ysat )/ l ζloc must evolve as a power law of the maximum self-affine correlation length ξsat (D) with exponent is ζ − ζloc . Moreover, because of the linear relationship between ξsat and D, it is expected that this ratio is a power law of the characteristic size D:   h(l  ξsat , y > ysat ) ∼ D ζ −ζloc , (10) l ζloc l where brackets . . .  denote the average over all length scales l  ξsat . Ratios defined in Equation (10) have been computed for each specimen and are plotted in Figure 11 as a function of the corresponding specimen size D. It can be seen in Figure 11 that, as expected from Equation (10), the ratios seem to evolve as a power law of the size D. Nevertheless, if a size effect is effectively observed in Figure 11, the estimate of the scaling exponent leads to the value 0.20 which is different from the expected value deduced from the average results of the RMS roughness (Table 2), i.e. ζ − ζloc ≈ 0.46. However, despite the fact that the experimental scaling exponent is not exactly the same than the expected one, anomalous scaling observed on mortar crack surfaces leads to a size effect on the height fluctuations estimated for all length scales smaller than the maximum self-affine correlation length ξsat in the saturation regime of the roughness. The fact that this size effect takes especially place from the smallest length

52 G. Mourot et al.

log10[Δh(l<<ξsat,y>>ysat)/l

ζloc

]l

0,1

0,0

-0,1

-0,2

-0,3

-0,4 1,2

1,4

1,6

1,8

2,0

2,2

2,4

log10(D) Figure 11. Log–log plot of the ratios h(l  ξsat , y > ysat )l −ζloc l (Equation (10)) as a function of the specimen size D. The slope of the straight line is estimated to 0.2.

scales suggest that the fracture process (which results into the crack surface morphology) is size dependent. Note that the size effect shown in Figure 11 is characteristic of the anomalous scaling and differs from what happens in the case of Family–Vicsek scaling where a ratio such as the one defined in Equation (10) is expected to lead to a constant, i.e. no size effect, according to Equation (6). Thus, the source of anomalous scaling may reside in the evolution of the process zone (size, structure, induced stress, ability to screen out elastic interactions, etc.) but also in the interactions between this process zone and the specimen boundaries. Indeed, in a quasibrittle material such as mortar, the process zone is very rapidly macroscopic in size and, as a consequence, its evolution is actually influenced all along by boundary conditions and hence by the specimen size. On the contrary, in materials like glasses, where the process zone size is always small compared to the specimen dimensions (the process zone does not exceed a few hundreds of nanometers as shown by Prades et al. (2004)), this evolution is unlikely to be driven by boundary conditions imposed by the finite sample size. However, a process zone develops as soon as a crack is introduced into the sane material and submitted to an external stress. It would be instructive to analyze the roughening of the crack and see if it can be described by any anomalous scaling. Unfortunately, growing a crack in glass from a straight notch is difficult to perform, and no experimental observation of this kind is available, to our knowledge. 5. Conclusion We have shown in this study that the roughness development of mortar crack surfaces obtained from geometrically similar fracture specimens of six different sizes is anisotropic and follows an anomalous scaling law. In the direction perpendicular

Scaling properties of mortar fracture surfaces 53 to the crack propagation one, the mortar fracture surfaces exhibit self-affine scaling properties characterized by the universal local roughness exponent ζloc ≈ 0.8. Anomalous scaling implies that the roughness growth exhibits different scalings between the local and the global surface fluctuations as a function of the crack advance. This difference leads to the existence of an independent global roughness exponent ζ ≈ 1.35, that characterizes the global surface fluctuations, and which is appeared independent of the specimen size. Moreover, it has been shown that there exists a size effect on the roughness magnitude at saturation: the roughness magnitude is not only function of the scale of observations but also of the specimen size. The fact that, on one hand, the fluctuations heights estimated at small length scales exhibit a size ´ effect, and that, on the other hand, recent studies (Lopez et al., 2005; Hansen et al., 2005) have shown that intrinsic anomalous scaling in experimental systems is due to changes in non-local effects, seem to indicate that anomalous roughening is linked to the development of a diffuse microcracking damage ahead of the crack front in quasibrittle materials, but also to the interactions between this process zone evolution and the boundary conditions imposed by the finite specimen size. Thus, the global roughness exponent appears dependent on the material but could be also dependent on the damage development at large length scales. Acknowledgements We wish to thank A. Hansen, L. Ponson and D. Bonamy for very stimulating and fruitful discussions. References Bouchaud, E., Lapasset, G. and Plan´es, J. (1990). Fractal dimension of fractured surfaces: a universal value? Europhysics Letters 13, 73–79. Bouchaud, E. (1997). Scaling properties of cracks. Journal of Physics Condensed Matter 9(21), 4319–4344. Bouchaud, E. (2003). The morphology of fracture surfaces: a tool for understanding crack propagation in complex materials. Surface Review and Letters 10(5), 797–814. Carpinteri, A., Chiaia, B. and Invernizzi, S. (1999). Three-dimensional fractal analysis of concrete fracture at the meso-level. Theoretical and Applied Fracture Mechanics 31(3), 163–172. Castro, M., Cuerno, R., S´anchez, A. and Dom´ınguez-Adame, F. (1998). Anomalous scaling in a nonlocal growth model in the Kardar–Parisi–Zhang universality class. Physical Review E 57, R2491–R2494. Daguier, P., Nghiem, B., Bouchaud, E. and Creuzet, F. (1997). Pinning and depinning of crack fronts in heterogeneous materials. Physical Review Letters 78, 1062–1065. Das Sarma, S., Lanczycki, C.J., Kotlyar, R. and Ghaisas, S.V. (1996). Scale invariance and dynamical correlations in growth models of molecular beam epitaxy. Physical Review E 53, 359–389. Dauskardt, R.H., Haubensk, F. and Ritchie, R.O. (1990). On the interpretation of the fractal character of the fracture surfaces. Acta Metallurgica Materialia 38(2), 143–159. Engøy, T., M˚aløy, K.J., Hansen, A. and Roux, S. (1994). Roughness of two-dimensional crack in wood. Physical Review Letters 73(6), 834–837. Family, F. and Vicsek, T. (1991). Dynamics of fractal surfaces. World Scientific, Singapore. Hansen, A., Kalda, J. and M˚aløy, K.J. (2005). Wavelet analysis of anomalous scaling in self–affine traces. preprint. Huo, S. and Schwarzacher, W. (2001). Anomalous scaling of the surface width during Cu electrodeposition. Physical Review Letters 86, 256–259. Imre, A., Pajkossy, T. and Nyikos, L. (1992). Electrochemical determination of the fractal dimension of fractured surfaces. Acta Metallurgica Materialia 40(8), 1819–1826.

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