Fluctuations of global energy release and crackling in nominally brittle heterogeneous fracture
The temporal evolution of mechanical energy and spatially-averaged crack speed are both monitored in slowly fracturing artificial rocks. Both signals display an irregular burst-like dynamics, with power-law distributed fluctuations spanning a broad range of scales. Yet, the elastic power released at each time step is proportional to the global velocity all along the process, which enables defining a material-constant fracture energy. We characterize the intermittent dynamics by computing the burst statistics. This latter displays the scale-free features signature of crackling dynamics, in qualitative but not quantitative agreement with the depinning interface models derived for fracture problems. The possible sources of discrepancies are pointed out and discussed.
pacs:46.50.+a, 62.20.M-, 78.55.Qr
Present address: ]Duke University, Durham, North Carolina 27708, USA Present address: ]Univ Paris-Sud, UPMC Univ Paris 6, CNRS, UMR 7608, Laboratoire FAST, Bat 502 – Campus Universitaire, F-91405 Orsay, France
Predicting when and how solids break continues to pose significant fundamental challenges Bonamy and Bouchaud (2011); Alava et al. (2006). This problem is classically addressed within the framework of continuum mechanics, which links deterministically the degradation of a solid to the applied loading. Such an idealization, however, fails in several situations. In heterogeneous solids upon slowly increasing loading for instance, the fracturing processes are sometimes observed to be erratic, with random events of sudden energy release spanning a variety of scales. Such dynamics are e.g. revealed by the acoustic emission accompanying the failure of various materials Petri et al. (1994); Garcimartin et al. (1997); Davidsen et al. (2007); Baro et al. (2013) and, at much larger scale, by the seismic activity going along with earthquakes Bak et al. (2002); Corral (2004); A generic observation in this field is the existence of scale-free statistics for the event energy Bonamy (2009).
These avalanche dynamics Sethna et al. (2001) have attracted much recent attention. They were originally thought to be inherent to quasi-brittle fracture, where the solid starts by accumulating diffuse damage through microfracturing events before collapsing when a macroscopic crack percolates throughout the microcrack cloud van Mier (2012). Phenomenological models such as fiber bundle models (see Pradhan et al. (2010) for review) or random fuse models (see Alava et al. (2006) for review) developed in this case reproduce qualitatively the avalanche dynamics with a minimal set of ingredients. More recently, it has been demonstrated Bonamy et al. (2008) that a situation of nominally brittle fracture, involving the destabilization and propagation of a single crack, can also yield erratic dynamics. Within the linear elastic fracture mechanics (LEFM) framework, the in-plane motion of a crack front was mapped to the problem of a long-range (LR) elastic interface propagating within a two-dimensional (2D) random potential Schmittbuhl et al. (1995); Ramanathan et al. (1997), so that the driving force self-adjusts around the depinning threshold Bonamy et al. (2008). This approach reproduces, in a simplified 2D configuration, the local and irregular avalanches evidenced in the space-time dynamics of an interfacial crack growing along a weak heterogeneous plane Måløy et al. (2006). There exists theoretical arguments to extend this approach to the bulk fracture of real three-dimensional (3D) solids and crackling dynamics at the global (specimen) scale are anticipated Ponson and Bonamy (2010); Barés et al. (2013). Still, fracture experiments are crucially missing to demonstrate this point.
The study reported here aims at filling this gap. Fracture experiments in heterogeneous solids made of sintered polymer beads are found to display irregular burst-like dynamics at the global scale, with large, power-law distributed, fluctuations for the mean failure speed and overall mechanical energy stored in the specimen. Yet and despite their individual giant fluctuations, the ratio between and the power release remains constant and defines a continuum-level scale material-constant fracture energy. The burst statistics displays the scale-free features predicted in elastic interface models. Still, the agreement remains qualitative only, and the scaling exponents are different from those predicted. The possible sources of discrepancies are discussed.
Experiments – The experiments were carried out on artificial rocks made of sintered polystyrene beads: i) a mold filled by monodisperse polystyrene beads was heated to ( of the temperature at glass transition) and compressed (pressure ) between the two jaws of an Instron electromechanical machine while keeping ; ii) the mold was then unloaded and slowly cooled down to ambient temperature and the obtained sample was extracted from it. This sintering process provides heterogeneous solids with homogeneous microstructures, the length-scale of which is set by the bead diameter . In all the experiments reported here, . Large enough heterogeneity scales, indeed, is requested to observe global crackling at finite driving rate Barés et al. (2013) and the fracture of sintered materials with smaller (m, m, m and m) were observed to display continuum-like dynamics.
In the so-obtained materials, stable cracks were driven by means of wedge splitting fracture tests (see Fig. 1A and Refs. Scheibert et al. (2010); Guerra et al. (2012) for details): Parallelepiped samples of size in the (propagation), (loading) and (sample thickness) were loaded in mode I by pushing a wedge at constant speed into a cut out on one the two edges. An initial seed crack (-long) was introduced with a razor blade in the middle of the cut. It prevents dynamic fracture and enables growing slow stable cracks. Two go-between steel blocks were placed between the wedge and the specimen to limit parasitic mechanical dissipation and ensure the damage and failure processes to be the sole dissipation source for mechanical energy in the system (see Bar s (2013) for details).
The wedge speed was varied from to . During each test, the force applied by the wedge was monitored in real time by a S-type Vishay cell force (acquisition rate of , accuracy of ). As soon as the wedge starts to push on the specimen (time origin set at this onset), increases. When gets large enough (), the seed crack starts to propagate. This propagation was imaged at the specimen surface via a camera (USB2 uEye from IDS Imaging Development, space and time accuracy of and , respectively), providing the instantaneous length of the crack edge at the surface. The instantaneous mechanical energy stored in the specimen is given by , and the instantaneous mean crack length (i.e. spatially averaged over specimen thickness) is obtained from the knowledge of the instantaneous specimen stiffness . Indeed, in a linear elastic material, the curve is a continuous decreasing function set by the specimen geometry only, and independent of the other experimental parameters (e.g. , microstructure parameters…). We hence measured the curves in each of our experiments, averaged them over all our experiments, and smoothed the result via a Tikhonov regularization. The so-obtained curve defines the curve for our fracture geometry; it was checked this curve is identical to that obtained using 2D finite element calculations (software Castem 2007) on the exact experimental geometry, assuming plane stress conditions. This reference curve was used to infer from the signal [ denoting the inverse function of ]. Time derivation of finally provides the instantaneous crack speed .
Results – Figure 1(A) presents the time evolution of and in a typical fracture experiment. These profiles exhibit the intermittent features characteristic of crackling dynamics, with random violent bursts (resp. sudden drops) in (resp. in ). The superposition of the two also reveals that the velocity bursts coincide with the energy drops. Beyond this occurrence coincidence, the fluctuation amplitude is proportional to the power released at each moment (fig. 1(B)). This proportionality was observed in all our experiments, irrespectively of the wedge speed. It betrays the characteristics of a nominally brittle fracture: where is the energy release rate. Now, for a stable crack slowly driven in a nominally brittle material, LEFM states that where the fracture energy is a material constant. In other words, a nominally brittle fracture compatible with LEFM assumptions yields at all times , irrespectively of the precise values of and , as observed here. In this scenario, the proportionality constant in fig. 1(B) gives for the considered material: Here .
We turn now to the statistical characterization of the crack dynamics. We analyzed the temporal evolution in preference to that of since the former is directly obtained from the experimental measurement of applied force , while the latter calls for the addition of the curve. The distribution of instantaneous power released is first analyzed. Note that, in experiments, an "instantaneous" quantity is actually averaged over a finite time scale , the value of which affects the fluctuation amplitude. The distributions of , hence, have been computed for different values of . A Gaussian distribution (centered at zeros, standard deviation decreasing as ) is observed at small scales (plain lines in Figs. 2A and B), and a power-law tail is observed at large scales (Fig. B). The Gaussian part at small is observed throughout the whole experiments, even in the preliminary loading phases where cracks do not propagate. It results from the noise in the measurement of the force signal. This noise yields a -dependent resolution limit below which the true fracture-induced fluctuations of cannot be deconvoluted from the Gaussian noise. Conversely, the probability that a fluctuation of size is due to noise rapidly is insignificant. Keeping only the relevant part above , the distributions all collapse onto a single master curve (Fig. 2C) exhibiting two power-law scaling, a small-scale regime with a scaling exponent and a large-scale regime with . The two scaling regimes, together with the value of the associated crossover (), depend neither on , nor on the loading rate . Conversely, the maximal value (resp. ) of (resp. ) decreases with , as (Inset in Fig. 2C), as expected for independent fluctuation peaks. Note that the large scale power-law exponent observed here at the global scale for fracture experiments in bulk solids is very close to that reported on the local velocity fluctuations in 2D situations of interfacial cracks, both experimentally Måløy et al. (2006); Tallakstad et al. (2013) and numerically Gjerden et al. (2014). This supports the conjecture that at large scales, brittle 3D fracture can be reduced to a 2D elastic interface problem Bonamy et al. (2008); Bonamy (2009); Barés et al. (2013). Conversely, the small scale power-law regime with observed here differs from that observed in the 2D interfacial configuration.
The scale-free statistics observed for the fluctuations (or equivalently for the fluctuations ) is a first hint toward crackling dynamics. We adopt the standard procedure in the field, and identify the underlying avalanches with the bursts where is above a prescribed reference level . Then, the avalanche duration of each pulse is given by the interval between the two intersections of with , and the avalanche size is defined as the energy released during the event, i.e. the integral of between the two intersection points. Note that, in conventional elastic interface formalism, the avalanche size is expressed as the total area swept by the front between two successive pinned configurations. The two definitions are equivalent since, in the nominally brittle situation experienced here, an event releasing an energy creates fracture surfaces of area . As expected for a crackling signal, follows a power-law distribution over nearly three order of magnitude, with event areas up to times the elementary one (Fig. 3:A). [a break in the scaling, around and for and cannot be precluded. Also, the mean avalanche size goes as a power-law with , (Fig. 3:B). The exponents and are independent of the prescribed values for and . Conversely, they both decrease with the loading rate, from at to at .
To complete the dynamics characterization, we computed the average temporal avalanche shape. This observable, indeed, provides an accurate characterization of the considered crackling signal and has been measured in a variety of systems Zapperi et al. (2005); Mehta et al. (2006); Laurson and Alava (2006); Papanikolaou et al. (2011); Danku and Kun (2013); Laurson et al. (2013). The standard procedure was adopted here: First, we identified all the pulses of duration falling within a prescribed interval ); and second, we averaged the shape over all the collected pulses . Figure 4:A shows the resulting shape and its evolution with . Two observations should be noted: i) The shape is parabolic for small and flattens as increases; ii) A small, but clear leftward asymmetry is observed as increases: The bursts start faster than they stop. Quantitatively, the shape evolution (how fast it flattens and how large the asymmetry is) depends on .
A similar shape flattening was observed in Barkhausen experiments Papanikolaou et al. (2011). Therein, it was shown to result from the finite value of the demagnetizing factor . Such a shape flattening is thus expected in the LR interface model for crack growth Bonamy et al. (2008), where the unloading factor plays the same role as Barés et al. (2013). Conversely, the numerical simulation of this LR interface model yielded symmetrical shapes Bar s (2013) (at finite driving rate) or slight rightward asymmetry Laurson et al. (2013) (at vanishing driving rate). In Barkhausen experiments, in contrast, a leftward asymmetry was observed, and attributed to the eddy currents, which provides a negative effective mass to the domain wallsZapperi et al. (2005). We conjecture that the viscoelastic nature of the polymer rocks fractured here acts in a similar way by providing a negative inertia to the crack front (i.e. the addition of a retardation term in the LR interface model of crack Bonamy et al. (2008)).
Concluding discussion – The experiments reported here demonstrate that crackling at global scale can be observed in nominally brittle fractures (due to the propagation of a single crack) and is not restricted to quasi-brittle (multi-fracturing) situation: Three main observations emerge: i) Despite their individual giant fluctuations, the ratio between spatially-averaged velocity and power release remains fairly constant and defines a continuum-level scale material-constant fracture energy; ii) The event size, defined either as the increase of crack length or as the energy release during the event, is power-law distributed, and scales as a power-law with the event duration; iii) the associated exponents depend on the crack loading rate.
These observations are in qualitative agreement with what is predicted by a recent model Bonamy et al. (2008) identifying stable crack growth with a LR elastic interface driven in a random potential so that the driving force self-adjust around the depinning threshold. Still, the agreement is qualitative only: i) In the universality class of the LR depinning transition, the scaling exponents are predicted to be Dobrinevski (2013), Bonamy (2009), and Bonamy (2009), significantly different from the experimental values measured here; ii) These predicted exponents are independent of the driving rate Bares et al. (2014), contrary to what is observed here. This discrepancy is thought to result from the finite width of the fracture specimens; the LR elastic kernel in the interface model Bonamy et al. (2008) arises from Rice’s perturbative analysis of the elastic problem of a corrugated crack front embedded in a sample of infinite width Rice (1985). Conversely, it is interesting to note that the variations of and observed in our experiments are compatible with those expected in the mean-field ABBM model Alessandro et al. (1990): In the ABBM model, i) decreases from 3/2 to 1 as the driving rate increases; ii) exhibits two values: for short pulses, for large ones, with a crossover decreasing with driving rate Papanikolaou et al. (2011). In our experiments, the mean-field approximation may be relevant since the separation between the microstructure scale (bead size ) and the continuum-level scale (specimen width: ) is quite small. The simplicity of the ABBM model has allowed the derivation of exact analytical solutions for the avalanche distribution and shape Papanikolaou et al. (2011); Colaiori (2008); Dobrinevski et al. (2012, 2013). A very interesting future extension of this study is to accurately characterize how the avalanche statistics evolves with the driving rate and to confront those against the ABBM solutions. Work in this direction is under progress.
Acknowledgements.Support through ANR project MEPHYSTAR (ANR-09-SYSC-006-01) is gratefully acknowledged. Special thanks to Thierry Bernard for technical support.
- Bonamy and Bouchaud (2011) D. Bonamy and E. Bouchaud, Physics Report 498, 1 (2011).
- Alava et al. (2006) M. J. Alava, P. K. V. V. Nukala, and S. Zapperi, Advances in Physics 55, 349 (2006).
- Petri et al. (1994) A. Petri, G. Paparo, A. Vespignani, A. Alippi, and M. Costantini, Physical Review Letters 73, 3423 (1994).
- Garcimartin et al. (1997) A. Garcimartin, A. Guarino, L. Bellon, and S. Ciliberto, Physical Review Letters 79, 3202 (1997).
- Davidsen et al. (2007) J. Davidsen, S. Stanchits, and G. Dresen, Phys. Rev. Lett. 98, 125502 (2007).
- Baro et al. (2013) J. Baro, A. Corral, X. Illa, A. Planes, E. K. H. Salje, W. Schranz, D. E. Soto-Parra, and E. Vives, Physical Review Letters 110, 088702 (2013).
- Bak et al. (2002) P. Bak, K. Christensen, L. Danon, and T. Scanlon, Physsical Review Letter 88, 178501 (2002).
- Corral (2004) A. Corral, Phys. Rev. Lett. 92, 108501 (2004).
- Bonamy (2009) D. Bonamy, Journal of Physics D: Applied Physics 42, 214014 (2009).
- Sethna et al. (2001) J. P. Sethna, K. A. Dahmen, and C. R. Myers, Nature 410, 242 (2001).
- van Mier (2012) J. G. M. van Mier, Concrete Fracture (CRC Press, 2012).
- Pradhan et al. (2010) S. Pradhan, A. Hansen, and B. K. Chakrabarti, Review of Modern Physics 82, 499 (2010).
- Bonamy et al. (2008) D. Bonamy, S. Santucci, and L. Ponson, Phys. Rev. Lett. 101, 045501 (2008).
- Schmittbuhl et al. (1995) J. Schmittbuhl, S. Roux, J. P. Vilotte, and K. J. Måløy, Physical Review Letters 74, 1787 (1995).
- Ramanathan et al. (1997) S. Ramanathan, D. Ertas, and D. S. Fisher, Physical Review Letters 79, 873 (1997).
- Måløy et al. (2006) K. J. Måløy, S. Santucci, J. Schmittbuhl, and R. Toussaint, Physical Review Letters 96, 045501 (2006).
- Ponson and Bonamy (2010) L. Ponson and D. Bonamy, International Journal of Fracture 162, 21 (2010).
- Barés et al. (2013) J. Barés, L. Barbier, and D. Bonamy, Physical Review Letters 111, 054301 (2013).
- Scheibert et al. (2010) J. Scheibert, C. Guerra, F. Célarié, D. Dalmas, and D. Bonamy, Physical Review Letters 104, 045501 (2010).
- Guerra et al. (2012) C. Guerra, J. Scheibert, D. Bonamy, and D. Dalmas, Proceedings of the National Academy of Sciences USA 109, 390 (2012).
- Bar s (2013) J. Bar s, Failure in brittle heterogeneous materials: Intermittency, crackling and seismicity, Ph.D. thesis, Ecole Polytechnique (2013).
- Tallakstad et al. (2013) K. T. Tallakstad, R. Toussaint, S. Santucci, and K. J. Måløy, Physical Review Letter 110, 145501 (2013).
- Gjerden et al. (2014) K. J. Gjerden, A. Stormo, and A. Hansen, Frontiers in Physics 2, 66 (2014).
- Zapperi et al. (2005) S. Zapperi, C. Castellano, F. Colaiori, and G. Durin, Nature Physics 1, 46 (2005).
- Mehta et al. (2006) A. P. Mehta, K. A. Dahmen, and Y. Ben-Zion, Physical Review E 73, 056104 (2006).
- Laurson and Alava (2006) L. Laurson and M. J. Alava, Physical Review E 74, 066106 (2006).
- Papanikolaou et al. (2011) S. Papanikolaou, F. Bohn, R. L. Sommer, G. Durin, S. Zapperi, and J. P. Sethna, Nature Physics 7, 1745 (2011).
- Danku and Kun (2013) Z. Danku and F. Kun, Physical Review Letters 111, 084302 (2013).
- Laurson et al. (2013) L. Laurson, X. Illa, S. Santucci, K. T. Tallakstad, K. J. Måloy, and M. J. Alava, Nature Communications 4, 2927 (2013).
- Bares et al. (2014) J. Barés, M. Barlet, C. L. Rountree, L. Barbier, and D. Bonamy, Frontiers in Physics 2, 70 (2014).
- Dobrinevski (2013) A. Dobrinevski, Dynamics of elastic interfaces in random media, Ph.D. thesis, Ecole Normale Sup rieure PAris (2013).
- Rice (1985) J. R. Rice, Journal of Applied Mechanics 52, 571 (1985).
- Alessandro et al. (1990) B. Alessandro, C. Beatrice, G. Bertotti, and A. Montorsi, Journal of Applied Physics 68, 2901 2908 (1990).
- Colaiori (2008) F. Colaiori, Advances in Physics 57, 287 (2008).
- Dobrinevski et al. (2012) A. Dobrinevski, P. Le Doussal, and K. J. Wiese, Physical Review E 85, 031105 (2012).
- Dobrinevski et al. (2013) A. Dobrinevski, P. Le Doussal, and K. J. Wiese, Physical Review E 88, 032106 (2013).