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Comptes Rendus Physique
Volume 18, n° 5-6
pages 297-313 (mai 2017)
Doi : 10.1016/j.crhy.2017.09.012
Dynamics of cracks in disordered materials
Dynamique des fissures dans les matériaux désordonnés
 

Fig. 1




Fig. 1 : 

The different stages underlying the advent of continuum fracture mechanics. Panel A: crude atomistic view of a flawless solid loaded by a constant tensile stress, σ ext . The atoms are placed on a square lattice with an interatomic distance . They are connected by bonds whose energy, U bond , varies with as depicted in panel A. 0 denotes the interatomic distance at rest, and γ b the associated bond energy. The “atomistic” stress–strain curve presented in panel A can be deduced. The slope at the origin gives the Young modulus, E , the area below is  , and the maximum defines the strength, σ . Panel B: continuum-level scale view of the solid, which now includes in its center an elliptical hole of semi-minor axis b (along the loading direction) and of semi-major axis a (perpendicular to loading). As shown by Inglis, the tensile stress is maximum at the apex (point M) and given by σ max =σ ext ×(1+2a /b ). Panel C: Griffith's view of the crack problem: the semi-minor axis b goes to zero so that the elliptic defect reduces to a slit crack of length a . Its presence in the stressed plate leads to the release of the stress in a roughly circular zone centered on the crack with a diameter close to a (dark gray zone). Panel C: the onset of crack growth is given from the comparison between two energies: The potential energy Πpot (dash green) decaying as a 2 (i.e. as the area of the released zone) and the energy to create new fracture surfaces Πsurf (dot blue), increasing linearly with a . For small a , the total energy, Πtot =Πpot +Πsurf increases with a and the crack does not move. For large a , Πtot decreases with increasing a and the crack extends. The motion onset is at a , so that dΠtot /da (a )=dΠpot /da (a )+dΠsurf /da (a )=0. Panel D: notations used to describe the stress field near the tip of a slit crack.


Fig. 2




Fig. 2 : 

Crackling dynamics of a slowly driven crack in an artificial rock. Panel A: sketch of the experimental setup. Panel B: microscope image of the fracture surfaces. Note the facet-like structure illustrating the intergranular fracture mode and the absence of visible porosity. The diameter of the beads used to synthesis this rock was 583 μm. Panel C: zoomed view of the crack speed v (t ) (black) and the potential elastic energy Πpot (t ) stored in the specimen (red) as a function of time in a typical fracture experiment. Panel D: instantaneous released power   as a function of v (t ) for all t . The proportionality constant (slope of straight line) sets the fracture energy Γ=100±10 J/m2. Panel E: standard procedure to extract the avalanche size and duration from such a crackling signal within the depinning interface framework (see also Fig. 3). A threshold is prescribed and the avalanches are identified as the individual bursts above this threshold. The avalanche duration is defined from the two successive times the curve crosses this threshold. The avalanche size, S , is defined as the integral of the burst above the threshold. Panel F: distribution of S , expressed either as the energy released during the event (bottom x -axis) or as the area swept during the events made dimensionless by the bead diameter d (top x -axis). The various symbols correspond to various coarsening times δt and different values for the prescribed threshold Cv 〉, where 〈v 〉 is the speed averaged over the whole experiment: 〈v=2.7 μm/s (empty symbol) and 〈v=40 μm/s (filled symbol); the latter has been shifted vertically for the sake of clarity. (Adapted from [34].)


Fig. 3




Fig. 3 : 

Elastic depinning approach applied to stable crack propagation. Panel A: sketch and notations used to derive Eqs. (15) and (16). The crack propagates by a series of jumps (avalanches) between successive pinned configurations. Panel B: for each avalanche, the duration, T , is defined by the duration of the jump and the times t 1 and t 2 coincide with the start and the end of the jump. The avalanche size S is set by the area A swept over this jump. Panel C: as a result, the time evolution of the spatially-averaged crack length   exhibits a step-like form where the pinned regions coincide with the horizontal portions, and the avalanches coincide with the stiff portions. Panel D: the spatially-averaged crack speed,   (resp. the instantaneous power released,  ) exhibits a crackling dynamics made of successive bursts, the duration of which are set by T . Moreover, the integral below the curve is given by S /L , where L is specimen thickness (resp.  , where   is the material fracture energy). (Adapted from [28].)


Fig. 4




Fig. 4 : 

Crackling vs. continuum dynamics in heterogeneous fracture. Panel A: time evolution of the spatially-averaged velocity   predicted by Eq. (15) and (16) for increasing values of G : G =4.75×10−5 (A1), G =2×10−4 (A2), G =5.5×10−3 (A3). The other parameters are kept constant:  ,  , μ =1, L =1024, and η (x ,z ) is an uncorrelated random landscape of zero average and unit variance. At low G ,   wanders around the value  , as predicted within the LEFM framework. When G increases, the dynamics becomes jerky and switches to crackling dynamics made of separate pulses the duration of which decreases with increasing G . Panel B: phase diagram of the crack dynamics predicted within the depinning interface framework (Eqs. (15) and (16)). This diagram is fully defined by two reduced variables mingling all the parameters involved in Eqs. (15) and (16). Panel C: Fourier spectrum of   at increasing G (value indicated in the right-handed color bar), keeping all the other parameters constant (same value as in panel A1→A3). Note the qualitative change as the transition line in panel B is crossed (i.e. as G crosses  ). Note also that only the lowest frequencies of the spectra evolve with G below  . Note finally the power law, characteristic of a scale-free dynamics above   (adapted from [59]).


Fig. 5




Fig. 5 : 

Signature of microcracking onset on the dynamic fracture of PMMA. Panel A: fracture energy Γ as a function of the (macroscale) crack speed v , for five different experiments with different potential energy values at crack initiation, U 0 . The horizontal dotted line indicates the quasi-static value, Γ(v =0)=420 J/m2. The vertical dotted line points out the kink occurring at the microcracking onset,  . Panel B: sequences of microscope images (1×1.4 mm2) showing the evolution of the fracture surfaces as v increases. Beyond v microcrack , conics marks are visible and their number increases with v . They sign the existence of microcracks forming ahead of the propagating main front. Panel C: density of conic marks as a function of v . In both panels (A) and (C), the vertical dashed line indicates v microcrack and the error bars indicate a 95% confident interval (adapted from [61,62]).


Fig. 6




Fig. 6 : 

From local front speed within FPZ to apparent speed at the continuum scale. Data are for PMMA. Panel A : reconstruction scheme of the microscale damage dynamics from the post-mortem fracture surfaces. The bright white regions provide the nucleation centers (red ×). Red dots sketch the successive positions of two growing microcracks, denoted by (1) and (2). The crossing points give rise to the green branch of the conic mark. The fit of this branch permits to infer both the ratio c 2 /c 1 of the microcrack speeds and the time interval t 2 t 1 between the two nucleation events. From the nucleation positions, the speed ratio c j /c i and the inter-nucleation times t j t i , it is possible to reconstruct the time-space dynamics of microcracking events, within nanosecond and micrometer resolution. Panels BB show such a reconstructed sequence. The blue part is the uncracked material, and the grey one is the cracked part. The different gray levels illustrate the fact that the fracture surface does not result from the propagation of the main crack front, but is the sum of the surfaces created by each microcrack. A different gray level has been randomly assigned to each of them. The analysis of these reconstructions has shown that all microcracks grow with the same velocity c m . Panel C: evolution of the mean crack front as a function of c m ×t for different values of microcrack density. The slope of these curves provides the ratio between the apparent macroscale crack speed v and the true local speed c m of the propagating (micro)crack front. This boosting factor is plotted as a function of microcrack density in panel D. Panel E: deduced variation of c m with ρ . The horizontal red line indicated the mean value   (adapted from [62,71]).

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