# Two-dimensional scaling properties of fracture surfaces

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## Failure of disordered materials: The physical approach

Position of the problem : Linear Elastic Fracture Mechanics provides a frame-work to predict the motion and trajectory of a crack in an ideal homogeneous elastic medium. In real materials, one would expect that this framework remains valid but at a scale larger than both the process zone size and the microstructure scale. This is not the case, unfortunately for engineers and thankfully for physicists. Microscopic mechanisms, even very localized in a small zone in the vicinity of the crack tip, can have macroscopic consequences on the motion of the crack and its trajectory. This eﬀect is characteristic of failure problems: Because of the stress concentration, the crack enhances catastrophically the eﬀects of mechanisms localized at its tip. In other words, rare specific processes can have a giant eﬀect on the averaged macroscopic be-havior. This is true in the case of an ideal elastic homogeneous material where the crack motion is determined by the quantities KI and KI I that are rigorously defined √ √ at the tip by KI = limr→0 2πrσyy (r, θ = 0) and KI I = limr→0 2πrσxy (r, θ = 0) (see Eq. (1.2) and (1.3)). And this is also true for more realistic cases. Therefore, theoret-ical approaches – such as homogenization technique – that would amount to neglect small-scale eﬀects to predict macroscopic behavior must be considered carefully.
The central point of this work is the transition from microscopic processes to macroscopic behavior in failure problems for real material: What are the eﬀects of the microstructural disorder of the material on the macroscopic behavior of the crack? The classical approach defines an ”eﬀective” equivalent homogeneous medium using ”eﬀective” quantities. The macroscopic behavior of the material at length scales larger than the disorder would then coincide with that of this so-defined ”equivalent” homogeneous medium. For example, it is natural to define the eﬀective Young’s mod-ulus Eeﬀ either (i) experimentally by measuring its response to a perturbation with a wavelength larger than the typical length scale ξd of the disorder/microstructure (see Annexe A for the experimental setup) or (ii) theoretically by using homogenization techniques that gives the relation between the eﬀective Young’s modulus Eeﬀ and the ones of each component of the disordered material (see for example Ref. ). With regard to its elastic response, the macroscopic (at scales larger than ξd) behav-ior of the disordered material is strictly identical to that of the so-defined eﬀective homogeneous medium with Young’s modulus Eeﬀ.
For failure problems, this approach is not valid. For example, let us consider a material characterized by a one-dimensional disorder as represented in Fig. 1.6. It is made of two compounds a and b characterized by their toughness KIa and KIb, respectively (KIa < KIb). To make the crack propagate through the whole material, one must apply a stress intensity factor KIapplied at least equal to KIb. Indeed, if KIa < KIapplied < KIb, the crack is ”pinned” in the grey regions. Therefore, the macroscopic stress intensity factor of this model heterogeneous material is equal to KIb, irrespective of the properties of the phase a. We can now consider a material made of N diﬀerent sections of toughness KIi . The macroscopic toughness is then given by the maximum value of the KIi so that a relatively small part of the disordered material rules its whole behavior.

### Statistics of fracture surfaces

Scaling of the 1D correlation function : Since the 80’s and the pioneering work of Mandelbrot et al. , the roughness of fracture surface has been widely studied. These works have been motivated by the puzzling scaling invariance properties of these surfaces. Extensive experimental investigations have lead to the conclusion that fracture surfaces are self-aﬃne, characterized by a universal roughness exponent ζ ≃ 0.8 [26, 27]. A key consequence of this scale invariance is that the height−height correlation function defined as the standard deviation σΔh(Δr) = (ΔhΔr )2 1r/2 =(h(r + Δr) − h(r))2 r1/2 of the distribution {ΔhΔr } of height variation scales as l =l ζ (2.1) σΔh Δr. Here ζ is the roughness or Hurst exponent and l the topothesy, i.e. the length scale at which σΔh is equal to Δr. Many other methods have been proposed to study the scaling properties of signals [57, 48, 49]. The choice of the height−height correlation function made here is motivated by the fact that this method is most eﬃcient3 when the signal is characterized by two self-aﬃne regimes – or one regime and one saturation – which is the usual case for experimental signals.

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#### Anisotropy of fracture surfaces

Context and motivation : In the previous part, the statistical properties of the surface height parallel to the z-axis, perpendicular to the crack growth direction, have been shown to be fully described by a single scaling exponent, the roughness exponent ζ . From now on, we will focus on the standard deviation σΔh (noted Δh for sake of simplicity) which is suﬃcient to estimate this roughness exponent and therefore the whole statistical properties of the profiles. In this section, we go beyond the analysis of profiles parallel to the z-direction and we study the statistics of surface height along other directions.
The scaling properties of fracture surfaces are usually believed to be isotropic [48, 35]. However, for surfaces obtained by shear fracture (mode II), it was reported in Ref.  that the scaling exponent measured on profiles parallel to the crack propagation was slightly smaller than for profiles along the perpendicular direction. The analysis of such an anisotropy on samples obtained under tensile failure (mode I) is the central point of this paragraph. This point is crucial because, as we will see in Section 2.5, it will determine the kind of models developed to describe crack front propagation in heterogeneous materials. As reviewed in Section 1.2, the various competing theoretical approaches for failure of disordered materials lead to conflicting conclusions about the isotropy of fracture surfaces. Scaling behavior parallel and perpendicular to the crack growth direction: In order to investigate the anisotropy of the experimental fracture surfaces, the 1D height−height correlation functions Δh(Δz) = (h(z + Δz, x) − h(z, x))2 1/2 along the z direction, and Δh(Δx) = (h(z, x + Δx) − h(z, x))2 1/2 along the x direction were computed for each material. They are represented in Figure 2.5.

Two-dimensional scaling properties of fracture surfaces

Motivation : The observation of the anisotropy of fracture surfaces raises a crucial question debated in  and : Is this anisotropy a universal property of the fracture surface due to a physically relevant underlying phenomenon? Or is it a simple perturbation of the isotropic case and thus an experimental bias due to the choice of a particular fracture test configuration?
In this last case, the fracture surface might be described as an isotropic object with an additional anisotropic perturbation: this would lead to a roughness exponent that would vary continuously with the direction of analysis without any remarkable structure. In the former case, the whole anisotropic geometry of the fracture surface is expected to display universal properties that would be a signature of an underlying physical phenomenon. The analysis of profiles extracted along a random directions suggests that they are not self-aﬃne, but display two distinct scaling behaviors (cf. Fig. 2.6). In order to study in detail the two-dimensional structure of the fracture surface roughness, the computation of the 1D correlation function only is not enough. A new approach based on the analysis of the 2D height−height correlation function Δh(Δr ) = (hr( + Δr ) −hr( ))2 r1/2 has therefore appeared to be necessary. This anal-ysis should confirm the anisotropy measured by 1D technique and provides additional information on the 2D properties of fracture surfaces.
2D height−height correlation function : The observation of two pure scaling behaviors along the two diﬀerent directions z and x of the fracture surface (see Section 2.3) suggests that the 2D height−height correlation function defined in the Cartesian frame e( ze, x) could be the appropriate quantity. This function Δh is defined as: 2 1/2 (2.4) Δh(Δz, Δx) = [h(z + Δz, x + Δx) − h(x, z)] z,x

Acknowledgments
Collaborations
Introduction
1 Context and motivation
1.1 At the continuum scale: The Linear Elastic Fracture mechanics
1.2 Failure of disordered materials: The physical approach
2 Morphology of fracture surfaces revisited
2.1 Materials and methods
2.2 Statistics of fracture surfaces
2.3 Anisotropy of fracture surfaces
2.4 Two-dimensional scaling properties of fracture surfaces
2.5 Physical interpretation
2.6 Concluding remarks
3 Low roughness exponents of fractured porous material surfaces
3.1 Materials and methods
3.2 Fracture surfaces of glass ceramics
3.3 Roughness amplitude
3.4 Morphology of fractured sandstone surfaces
3.5 Concluding remarks
4 Fracture surfaces for model linear elastic disordered materials
4.1 Model of crack propagation in ideal linear elastic disordered materials
4.2 Fracture surface of porous materials: Interpretation
4.3 Fracture surfaces of other brittle materials
4.4 Fracture surfaces of ductile materials : Interpretation
Conclusion
Bibliography

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