Session D22: Focus Session: Fracture

Failure of heterogeneous materials: Scaling properties of fracture surfaces and implications on models of cracks in disordered media.

While there exists a unified theoretical framework - Linear Elastic Fracture Mechanics (LEFM) - to describe the failure of homogeneous materials, understanding and modelling the mechanical properties of heterogeneous media continue to raise significant fundamental challenges. These mechanical properties, observed at the macroscopic scale, result from microscopic processes occurring at the scale of the material. To include these local processes into a statistical description constitutes then a crucial step toward the setup of predictive macroscopic models. Crack surface roughness is a consequence of these local processes. Consequently, many fractography experiments have focussed on their analysis. In this context, it was recently evidenced that, in many materials, fracture surfaces exhibit anisotropic scaling properties reminiscent to interface growth problems, fully characterized by two couples of parameters: The roughness exponents and the characteristic length-scales measured along and perpendicular to the direction of crack growth respectively. While the characteristic length-scales do depend on the considered material, the exponents are surprisingly universal: Two \textit{distinct} sets of critical exponents are observed whether the surfaces are examined at scales below or above the size of the damaged zone at the crack front. Models of crack growth in disordered media are discussed at the light of these experimental observations. In particular, one can derive a model from LEFM which describe the development of crack roughness as an ``elastic'' manifold creeping in a random media. This approach captures quantitatively the experimental observations performed at length-scales above the size of the process zone. In this approach, the onset of crack propagation can be interpreted as a dynamic phase transition while sub-critical crack growth can be assimilated to thermally-assisted depinning.   

Low self-affine exponents of fractured glass ceramics surfaces

The morphology of fracture surfaces encodes the various complex damage and fracture processes occurring at the microstructure scale during crack propagation. It is now well established that fracture surfaces are self-affine characterized by a roughness exponent usually found close to $\zeta \quad \approx $ 0.75 for a wide range of materials. Recently, fracture surfaces of sandstone were found to be also self-affine but with a lower roughness exponent $\zeta \quad \approx $ 0.4-0.5. To investigate its origin, we studied fracture surfaces of glassy ceramics which are obtained by sintering glass beads. Such a material mimics the structure of sandstone with the advantage that their porosity may be tuned. They are also found to be self-affine, characterized by a roughness exponent $\zeta \quad \approx $ 0.40 $\pm $ 0.04 significantly lower than the ``universal'' roughness exponent $\zeta \quad \approx $ 0.75 widely reported in the literature. Its value is found to depend very slightly on the crack growth velocity and the microstructure (grain diameter, porosity) in the range studied. This suggests the existence of a second universality class in failure problems. Its physical origin is then discussed and a model proposed.   

Rapid and slow self-affine fracture in glass

We discuss the self-affine properties of the fracture surfaces of soda-lime glass obtained in conditions of both rapid and slow fracture in bending. The fracture surfaces were studied by SEM and AFM. The analysis of the mirror and mist-hackle zones for the two conditions suggest the existence of two well defined self-affine regimes governed by universal or attractor values. At low-speed/fine-scales the roughness exponent$\zeta =0.5 \quad \zeta $ dominates whereas the value$\zeta =0.8$ is recovered for high-speed/large scales regimes. These values are subjected to significant deviations that give rise to a possible transitional regime at intermediate scales and speeds, where both attractor values may coexist, particularly in the case of slow fracture. In this context the transitional regime can thus be regarded as the result of the competition of these attractors at intermediate scales and velocities.   

Roughness Exponent Measurements for the Central Force Model

We study the roughness properties of fracture profiles from the two-dimensional central force lattice model for a wide range of disorders. The intrinsic and the extrinsic roughness exponent have been measured together with the step size distribution $p(|\Delta h|)$ and the height difference distribution $p(\Delta h,l)$. We find that the profiles are self-affine for systems with narrow disorders and that broader disorders introduces overhangs in the fracture surface leading to deviation from self-affinity for small length scales and to non-trivial finite size scaling.   

Local waiting time fluctuations along a randomly pinned crack front

The propagation of an interfacial crack along a heterogeneous weak plane of a transparent Plexiglas block is followed using a high resolution fast camera. We show that the fracture front dynamics is governed by local and irregular avalanches with very large size and velocity fluctuations. We characterize the intermittent dynamics observed, i.e. the local pinnings and depinnings of the crack front which trigger a rich burst activity, by measuring the local waiting time fluctuations along the crack front during its propagation. The local front line velocity distribution deduced from the waiting time analysis exhibits a power law behavior, $P(v) \propto v^{-\eta}$ with $\eta = 2.55 \pm 0.15$, for velocities $v$ larger than the average front speed $\langle v \rangle$. The burst size distribution is also a power law, $P(S)\propto S^{-\gamma}$ with $\gamma=1.7 \pm 0.1$. Above a characteristic length scale of disorder $L_d \sim 20 \mu m$, the avalanche clusters become anisotropic, and the scaling of the anisotropy ratio provides an estimate of a local roughness exponent, $H=0.6$.   

A Dissipative Particle Dynamics Model of Fracture

The role of thermal fluctuations and dissipative physical processes in fracture initiation and propagation has not been systematically studied due to the absence of appropriate simulation models. In order to investigate this issue, we have developed a dissipative particle dynamics (DPD) model, in which the elastic interactions between adjacent nodes in a two-dimensional spring network model are supplemented by dissipative interactions and random forces related through the fluctuation-dissipation theorem. With this newly developed model, we have simulated two different scenarios: One is self-initiated spontaneous fracturing, and the other is externally forced fracturing. Our preliminary results show that the fluctuating and dissipative forces have an important influence on the propagation mode, and propagation path. Both qualitative analysis and quantitative results will be presented and discussed.   

Sub-critical crack growth in a sheet of paper

We present experiments on the slow growth of a single crack in a fax paper sheet submitted to a constant force $F$. The non-averaged crack growth curves present a stepwise growth dynamics. Modelling the material as a lattice where the crack is pinned by elastic traps and grows due to thermal noise, we find that, in agreement with experiments, the distribution of step sizes follows subcritical point statistics with a power law (exponent 3$/$2) and a stress-dependent exponential cutoff diverging at the critical rupture threshold [1]. Taking into account the microstructure of cellulose fibers, the model is able to reproduce the shape of the statistically averaged crack growth curves, the dependence of the characteristic growth length on $F$ as well as the effect of temperature on the rupture time. Finally, roughness of the crack interface is shown to depend on whether the crack grows in the subcritical regime, or in the rapid regime, over the critical rupture threshold. We analyze this roughness difference using a new approach based on the cumulants of the statistical distribution of the crack front height variations. \newline [1] S. Santucci, L. Vanel and S. Ciliberto, Phys. Rev. Lett. 93, 095505 (2004).   

Stability and roughness of crack paths in 2D heterogeneous brittle materials

We present a recent study on the stability of propagating cracks in heterogeneous two-dimensional brittle materials and on the roughness of the surfaces created by this irreversible process. We introduce a stochastic model describing the propagation of the crack tip based on an elastostatic description of crack growth in the framework of linear elastic fracture mechanics. The model recovers the stability of straight cracks and allows for the study of the roughening of fracture surfaces. We show that in a certain limit, the problem becomes exactly solvable and yields analytic predictions for the power spectrum of the paths. This result suggests a surprising alternative to the conventional power law analysis often used in the analysis of experimental data and thus calls for a revised interpretation of the experimental results.   

Statistical properties of microcracking in polyurethane foams under tensile and creep tests: influence of temperature and density.

For some heterogeneous materials fracture can be described as a clustering of microcracks: global rupture being not controlled by a single event. We focus on polyurethane foams whose heterogeneities (pores) constitute the termination points where microcracks can stop. We record both the spatial and time distributions of acoustic emission emitted by a sample during mechanical tests: each microcrack nucleation corresponds to a burst of energy that can be localized on the widest face of the specimen. The probability distributions of the energy released is power-law distributed, independently of the material density, the loading mode or the mechanical behavior. On the other hand, the agreement of a power law for the time intervals between two damaging events seems to require a quasi constant stress during damaging. Moreover, we notice a behavior difference of the cumulative number of events and the cumulative energy of the localized events with temperature in the case of tensile tests and not any more for creep tests. The occurrence of a unique behavior and a power law in a restricted time interval for the cumulative number of events and the cumulative energy in creep allow us to apprehend interesting later studies of materials' lifetime prediction.   

Do Plastic Zones form at Crack Tips in Silicate Glasses?

In a number of recent studies, the claim has been made that silicate glasses fracture by the formation, growth and coalescence of cavities, in the same way as in metals but at a much smaller scale. Evidence for this premise is based on the examination of side surfaces of fracture mechanics specimens, at the point where the crack intersects the free surface. Such measurements were carried out with an atomic force microscope, which demonstrated finite depressions in the regions around and in front of crack tips in silicate glasses. The height profile around crack tips supposedly differed from that obtained from a simple linear elastic fracture mechanics analysis; while, in front of the crack tip small depressions were observed which were interpreted as cavities. We used a three-dimensional finite element analysis to show that the calculated depression around the crack tip is in excellent agreement with that obtained by atomic force microscopy. In addition, we used AFM measurements on the fracture surfaces themselves to demonstrate the absence of the kind of residual damage that should be present on fracture surfaces if cavitation occurred at crack tips in glass. Our results are proof that cracks in glass propagate by brittle fracture; glass is elastic and bond snapping is the dominant feature of crack growth.   

Fragmentation in brittle rods

When a rod made of brittle material is axially impacted it breaks into fragments of various sizes. Before the first breaking event, an axial compression wave propagates along the rod, triggering a buckling instability. The instability selects a transverse mode with a well defined wavelength. Recently, Gladden et al. have shown that the fragment size distribution exhibits two peaks corresponding to the length selected by the buckling instability. In the present work we explore in more details the dynamics of elastic waves in the rod and the different phenomena that may explain the broad distribution of fragment sizes. In particular, we will discuss the coupling between the longitudinal and the transverse displacement in the post buckling dynamics.   

Mechanisms for Fragment Formation in Brittle Solids

The fracture process is usually analyzed in terms the fractal dimension of a crack, the crack surface roughness, or fragment size distributions. It is established that relatively simple scaling laws exist for the crack surface roughness in mode I fracture and for the power law distribution for fragment sizes in fracture by impact. These two types of fracture are usually studied separately. Consequently, much less is known about the relationship between crack roughness and fragment size distribution. In this work, we study this relationship by developing a simple model of mode I fracture, which nevertheless produces sufficiently rich behavior in terms of crack roughness and fragment formation. Using this model, we show that different roughness in local regions of the crack path leads to different mechanisms for the subsequent fracture of those regions. We observe two robust power laws for the size distribution of smaller and larger fragments. We connect measurements in fragment size distribution with the local fractal dimension of cracks in the region of fragment formation.   

Competition between Diffusion and Fragmentation: Evolution of polycrystalline materials under stress

We propose a dynamical model for the grain evolution in polycrystalline materials. The model is based on the competition of the common physical processes diffusion and fragmentation. Specifically, we formulate a rate equation in terms of the distribution N(x, t) of grains or crystallites of linear size x at time t. The grains either grow by boundary diffusion or shrink by deformation and subsequent fragmentation. The equation leads to a third order differential equation which we solve exactly in terms of Bessel functions. The stationary state is a universal Bessel distribution described by one parameter. Our model perfectly fits experimental data on grain evolution in sheets of ice.