two-dimensional and three-dimensional fractures in a

TWO-DIMENSIONAL AND THREE-DIMENSIONAL FRACTURES IN A FACE CENTERED CUBIC ALLOY S. Morel1 and J.-M. Olive2 1 Lab.

de Rh´ eologie du Bois de Bordeaux, UMR 5103 (CNRS/INRA/Universit´ e Bordeaux 1), 69 route d’Arcachon, 33612 Cestas Cedex, France 2 Lab.

de M´ ecanique Physique, UMR 5469 (CNRS/Universit´ e Bordeaux 1), 351 cours de la Lib´ eration, 33605 Talence Cedex, France

ABSTRACT Intergranular and transgranular fracture surfaces obtained in a face centered cubic alloy are studied using 3D maps reconstructed by scanning electron microscopy stereo imaging. The roughness exponents respectively measured in the intergranular and transgranular surfaces, ζ = 0.83±0.05 and ζ = 0.75 ± 0.05, are in agreement with the universal roughness value of 3D fractures. However, the slightly smaller value related to the transgranular surface could be a consequence of crystallographic transgranular zones disseminated on the surface whose the roughness exponent: ζ = 0.65 ± 0.07 is close to the one usually measured on 2D fractures.

1 INTRODUCTION Since the pioneering work of Mandelbrot et al [1], the statistical characterization of fracture surfaces is a very active field of research. The fracture surfaces in brittle materials have shown surprising scaling properties [2] in the sense that they exhibit self-affine scaling properties characterized by a so-called roughness exponent. In the early 1990s, extensive experimental evidence [3] intimate that the roughness exponent ζ has a universal value close to 0.8 in the three-dimensional (3D) brittle fractures (ζ3D ≈ 0.8) while, in the case of two-dimensional (2D) fracture surfaces, the roughness exponent was found equal to ζ ' 0.73 in the case of 2D stacking of collapsible cylinder [4], ζ = 0.68±0.05 for paper tear lines [5], and ζ = 0.68±0.04 for fractures in thin wood plates (where the grain was oriented parallel to the short axis [6]). These values close to 0.7 in 2D fractures were confirmed by the one measured in 2D random fuse networks for a wide class of different disorders [7] ζ = 0.70 ± 0.07, and hence, intimate that ζ2D ≈ 0.7 might be consider as universal value for the 2D fractures. We report here a roughness analysis of fracture surfaces obtained in a face centered cubic alloy for two different fracture modes: intergranular and transgranular. If the intergranular mode exhibits a roughness exponent in agreement with the universal one of the 3D fractures, the roughness exponent measured for the transgranular mode appears slightly smaller than the one relative to intergranular mode and merits thinking about.

2 EXPERIMENT A notched compact tension specimen of a face centered cubic alloy (a type 600 nickel-based alloy) is precracked in fatigue then broken under a corrosion fatigue test in mode I [8]. The fatigue test is performed in air with a constant stress ratio R = σmin /σmax = 0.1 at a frequency f = 30Hz. The corrosion fatigue test is performed at 325◦ C in water with the same stress ratio than the fatigue test (R = 0.1) but at a frequency f = 2.8 10−4 Hz. As illustrated in Fig. 1, the corrosion fatigue test leads to an intergranular fracture mode (Fig. 1a) while the fatigue test induces a transgranular mode (Fig. 1b).

Figure 1: (a) Intergranular fracture mode obtained in corrosion fatigue, (b) transgranular mode obtained in fatigue and (c) crystallographic transgranular mode also obtained in fatigue. The crack propagation direction is oriented from bottom to top of the pictures.

The roughness of fracture surfaces are studied from digital elevation images obtained by three-dimensional reconstruction using a field emission gun of a scanning electron microscope (FEG-SEM, ZEISS Gemini, ONERA Chˆatillon) and stereovision software [9]. The 3D reconstruction using an image matching by dynamic programming [9] is applied from a pair of stereoscopic SEM pictures, in 256 gray levels and 2048×2048 pixels, recorded with tilt angles equal to −8◦ and +8◦ . The digital elevation maps (DEM) in 256 gray levels are constituted by 1850×1850 pixels. For each fracture tests (fatigue and stress corrosion), two DEM corresponding to a small and a large magnification in the same crack surface area, respectively 678 µm (0.366 µm/pixel) and 163 µm (0.088 µm/pixel), are studied. The large magnification DEM are approximately positioned at the center of the small magnification ones. The roughness of the intergranular and transgranular modes are studied from a set of about one hundred profiles of 1850 points oriented along the x-axis (perpendicular to the crack propagation direction) delimited by the dashed lines in Fig. 1(a) and 1(b).

3 ROUGHNESS ANALYSIS In order to evaluate the roughness exponent ζ of the different crack surfaces, three independent methods have been used. The first method is the ”variable band width method” in which the root mean square (RMS) w(l) of the height fluctuations z(x) is computed over a window size l along the x-axis and averaged over all possible origins j of the window belonging the profile [12]. For a self-affine profile characterized by a roughness exponent ζ, the root mean square is expected to scale as ¿ X l l ³1 X ´2 À1/2 1 2 w(l) = z(xi ) − z(xi ) ∼ lζ l i=1 l i=1

(1)

j

The second method is the power spectrum density (PSD), that is, the Fourier transform of the autocorrelation function hz(x + ∆x)z(x)i. For a self-affine profile, the power spectrum scales as S(k) ∼ k −(1+2ζ) (2) where k is the wave factor [12]. The third independent method used in this study is the averaged wavelet coefficient method (AWC) which consists to transform the data into the wavelet domain and to averaged the wavelet coefficients W [h](a) as a function of the scale parameters a [13]. The averaged wavelet coefficients are expected to scale as 1

W [h](a) ∼ a 2 +ζ

(3)

The roughness development being non-existent in the direction of the crack propagation in the sense that neither Family-Vicsek [10] nor anomalous scaling [11] is observable and this for both modes, solely the averages obtained from each statistical method are presented in the following.

3.1 Intergranular fracture mode For the intergranular fracture mode, the RMS method (Eq. 1) is applied and respectively plotted in Fig. 2a and 2b for the large (L = 678µm) and the small (L = 163µm) DEM. The roughness exponents directly measured for large and small DEM are respectively 0.81 ± 0.04 and 0.77 ± 0.06 (Tab. 1). The PSD (Eq. 2) and AWC (Eq. 3) methods (not represented in this paper) give evaluations of the roughness exponent (Tab. 1) which are close to the one obtained from the RMS method. Average of the ζ values obtained from the three independent methods gives the roughness exponent: ζ = 0.79 ± 0.05 which is perfectly compatible with the universal roughness exponent ζ3D ≈ 0.8 of the 3D fractures. A correction taking into account the analysis biases (see [12] for more details) is systematically carried out on all roughness exponents reported in Table 1 and the average of the corrected values leads to ζ = 0.83 ± 0.05 which is also in very good agreement with the universal exponent ζ3D . 3.2 Transgranular and crystallographic transgranular fracture modes In the case of the transgranular fracture mode (Fig. 1b), the three independent methods are applied (see Fig. 2 for RMS) and the averages of the directly measured and corrected roughness exponents are respectively 0.71 ± 0.05 and 0.75 ± 0.05 (Tab. 1). Hence, the roughness exponent relative to the transgranular mode can be considered in agreement with the universal value ζ3D , 0.8 being within the error bar of the corrected exponent. Nevertheless, the relative weakness of the roughness exponent measured for the transgranular mode compared to the one related to the intergranular mode merits thinking about. An accurate observation of the transgranular surface (Fig. 1b) allows to distinguish several small areas which are symptomatic of an intergranular fracture mode and others linked to a crystallographic transgranular fracture mode, as shown respectively by white and black dashed circles in Fig. 1b. A large crystallographic transgranular zone, located immediately above the region delimited by the dashed line in Fig. 1b, can be seen in Fig. 1c. In this figure, two grains have been completely broken in this mode and the long and relatively

Table 1: Roughness exponents ζ estimated from the ”variable band width” (RMS), the power spectrum (PSD) and the averaged wavelet coefficient (AWC) methods. The values in brackets are the corrected values of ζ taking into account the errors due to analysis biases. Fracture Mode Intergranular Transgranular Crystallographic

L (µm) 163 678 163 678 375

RMS 0.81 0.77 0.73 0.70 0.58

(0.90) (0.86) (0.80) (0.77) (0.64)

PSD 0.77 (0.78) 0.78 (0.81) 0.57 ( ) 0.68 (0.70) 0.63 (0.65)

AWC 0.83 0.80 0.73 0.72 0.64

(0.84) (0.81) (0.74) (0.73) (0.66)

Average 0.79 (0.83) ± 0.05 0.71 (0.75) ± 0.05 0.62 (0.65) ± 0.07

flat surfaces separated by quasi vertical steps correspond respectively to the primary facets and ligaments. It is interesting to note that such a surface has very much in common with the crack surfaces obtained in thin wood plates [6] and qualified as quasi-two-dimensional surfaces, i.e., surfaces which are almost invariant in one direction and whose the roughness exponent is consistent with the one of the 2D surfaces. Such an analogy is intriguing and a roughness analysis of the crystallographic transgranular surface appears necessary. A set of about one hundred profiles of 1024 points oriented along the x-axis (0.366 µm/pixel) are extracted from Fig. 1c and studied. The three independent methods (Eq. 1, 2 and 3) are applied (see Fig. 2 for RMS) and the average value of the corrected roughness exponents gives: ζ = 0.65 ± 0.07 (Tab. 1). This unexpected value disagrees with the universal value of the 3D fractures but appears consistent with the one related to the 2D fractures (ζ2D ≈ 0.7).

log10 w(l)

1.5

0.5

−0.5

inter. trans. cryst.

(a)

inter. trans.

(b)

log10 w(l)

−1.5 1.0

0.0

−1.0

−2.0 −1.5

−0.5

0.5

1.5

2.5

log10 l Figure 2: Roughness w(l) (RMS) versus l for the large (a) and small (b) elevation maps. According to Eq.(1), the straight lines correspond to the power law w(l) ∼ lζ .

4 CONCLUSION The weak roughness exponent observed on the transgranular mode (0.75 ± 0.05) could be seen as a consequence of the existence of crystallographic transgranular zones. Indeed, if the crystallographic zones are characterized by a roughness exponent ζ ≈ 0.65 while the intergranular zones are linked to an exponent ζ ≈ 0.83, the exponent characterizing the surface as a whole should be bounded by both latter values. But, how explain that the intergranular surfaces are characterized by the 3D universal exponent while the crystallographic transgranular surfaces exhibit the 2D one ? The notion of 2D or 3D interactions with the crack front during the fracture process could be at the source of such a difference. Indeed, the crystallographic transgranular surface arises from an extremely localized fracture process while the intergranular surface is obtained through a branching process of the crack. Thus, in the intergranular surface, the crack front morphology results from more or less long range 3D interactions which take place in the multi-cracked zone ahead of the main crack while, in the crystallographic surface, the morphology arises from very short range interactions and quasi invariant in the crack propagation direction (i.e. quasi two dimensional) due to the coincidence of crystallographic planes with the average plane of the main crack. The authors thank J.-L. Pouchou for the SEM pictures and 3D reconstructions.

References [1] B.B. Mandelbrot, D.E. Passoja, and A.J. Paullay Nature 308, 721 (1984). [2] E. Bouchaud, J. Phys. Cond. Mat. 9, 4319 (1997). aløy, [3] E. Bouchaud, G. Lapasset, and J. Plan´es, Europhys. Lett. 13, 73 (1990). K.J. M˚ A. Hansen, E.L. Hinrichsen and S. Roux, Phys. Rev. Lett. 68, 213 (1992). J. Schmittbuhl, F. Schmitt, and C. Scholz, J. Geophys. Res. 100, 5953 (1995). [4] C. Poirier, M. Ammi, D. Bideau, and J.P. Troadec, Phys. Rev. Lett. 68, 216 (1992). [5] J. Kertesz, V. K. Horvath, and F. Weber, Fractals 1, 67 (1993). [6] T. Engøy, K.J. M˚ aløy, A. Hansen, and S. Roux, Phys. Rev. Lett. 73, 834 (1994). [7] Statistical Models for the Fracture of Disordered Materials, edited by H.J. Herrmann and S. Roux (Elsevier, Amsterdam, 1990). A. Hansen, E. L. Hinrichsen, and S. Roux, Phys. Rev. Lett. 66, 2476 (1991). [8] C. Bosch, M. Puiggali, J.-M. Olive, and F. Vaillant, in Proceedings of the Conference EUROMAT 2000 on Advances in Mechanical Behaviour, Plasticity and Damage, Tours (France), November 7-9 2000. [9] J.-L. Pouchou et al., Mikrochim. Acta 139, 135 (2002). [10] J. Schmittbuhl, S. Roux, and Y. Berthaud, Europhys. Lett. 28, 585 (1994). [11] S. Morel, J. Schmittbuhl, J.M. L´opez, and G. Valentin, Phys. Rev. E 58 (6), 6999 (1998). S. Morel, G. Mourot, and J. Schmittbuhl, Int. J. Fract. 121 (1), 23 (2003). [12] J. Schmittbuhl, J.P. Vilotte, and S. Roux, Phys. Rev. E 51, 131 (1995). [13] I. Simonsen, A. Hansen, and O.-M. Nes, Phys. Rev. E 58, 2779 (1998).