We present a subcritical fracture growth model, coupled with the elastic redistribution of the acting mechanical stress along rugous rupture fronts. We show the ability of this model to quantitatively reproduce the intermittent dynamics of cracks propagating along weak disordered interfaces. To this end, we assume that the fracture energy of such interfaces (in the sense of a critical energy release rate) follows a spatially correlated normal distribution. We compare various statistical features from the obtained fracture dynamics to that from cracks propagating in sintered polymethylmethacrylate (PMMA) interfaces. In previous works, it has been demonstrated that such an approach could reproduce the mean advance of fractures and their local front velocity distribution. Here, we go further by showing that the proposed model also quantitatively accounts for the complex self-affine scaling morphology of crack fronts and their temporal evolution, for the spatial and temporal correlations of the local velocity fields and for the avalanches size distribution of the intermittent growth dynamics. We thus provide new evidence that an Arrhenius-like subcritical growth is particularly suitable for the description of creeping cracks.
Activation Advance Steel 2006 Crack
Different models have successfully described parts of the statistical features of the recorded crack propagation. Originally, continuous line models4,5,20,29 were derived from linear elastic fracture mechanics. While they could reproduce the morphology of slow rugous cracks and the size distribution of their avalanches, they fail to account for their complete dynamics and, in particular, for the distribution of local propagation velocity and for the mean velocity of fronts under different loading conditions. Later on, fiber bundle models were introduced6,30,31, where the fracture plane was discretized in elements that could rupture ahead of the main front line, allowing the crack to propagate by the nucleation and the percolation of damage. The local velocity distribution could then be reproduced, but not the long term mean dynamics of fronts at given loads. One of the most recent models (Cochard et al.8) is a thermally activated model, based on an Arrhenius law, where the fracture energy is exceeded at subcritical stresses due to the molecular agitation. It contrasts to other models that are strictly threshold based (the crack only advances when the stress reaches a local threshold, rather than its propagation being subcritical). A notable advantage of the subcritical framework is that its underlying processes are, physically, well understood, and Arrhenius-like laws have long shown to describe various features of slow fracturing processes26,32,33,34,35,36. In particular, this framework has proven to reproduce both the mean behaviour of experimental fronts37 (i.e., the average front velocity under a given load) and the actual distributions of propagation velocities along these fronts8, whose fat-tail is preserved when observing cracks at different scales38. It has recently been proposed39,40 that the same model might also explain the faster failure of brittle matter, that is, the dramatic propagation of cracks at velocities close to that of mechanical waves, when taking into account the energy dissipated as heat around a progressing crack tip. Indeed, if fronts creep fast enough, their local rise in temperature becomes significant compared to the background one, so that they can avalanche to a very fast phase, in a positive feedback loop39,40.
We consider rugous crack that are characterised by a varying and heterogeneous advancement a(x, t) along their front, x being the coordinate perpendicular to the average crack propagation direction, a the coordinate along it, and t being the time variable (see Fig. 1). At a given time, the velocity profile along the rugous front is modelled to be dictated by an Arrhenius-like growth, as proposed by Cochard et al.8:
where \(V(x,t)=\partial a(x,t)/\partial t\) is the local propagation velocity of the front at a given time and \(V_0\) is a nominal velocity, related to the atomic collision frequency41, which is typically similar to the Rayleigh wave velocity of the medium in which the crack propagates42. The exponential term is a subcritical rupture probability (i.e., between 0 and 1). It is the probability for the rupture activation energy (i.e., the numerator term in the exponential) to be exceeded by the thermal bath energy \(k_B T_0\), that is following a Boltzmann distribution41. The Boltzmann constant is denoted \(k_B\) and the crack temperature is denoted \(T_0\) and is modelled to be equal to a constant room temperature (typically, \(T_0=298\,\) K). Using this constant temperature corresponds to the hypothesis that the crack is propagating slowly enough so that no significant thermal elevation occurs by Joule heating at its tip (i.e., as inferred by Refs.39,40). Such propagation without significant heating is notably believed to take place in the experiments by Tallakstad et al.28 that we here try to numerically reproduce, and whose geometry is shown in Fig. 1. Indeed, their reported local propagation velocities V did not exceed a few millimetres per second, whereas a significant heating in acrylic glass is only believed to arise for fractures faster than a few centimetres per second40,44. See the supplementary information for further discussion on the temperature elevation.
In Eq. (1), the rupture activation energy is proportional to the difference between an intrinsic material surface fracture Energy \(G_c\) (in J m\(^-2\)) and the energy release rate G at which the crack is mechanically loaded, which corresponds to the amount of energy that the crack dissipates to progress by a given fracture area. As the front growth is considered subcritical, we have \(G
Equations (1) and (2) thus define a system of differential equations for the crack advancement a, which we have then solved with an adaptive time step Runge-Kutta algorithm46, as implemented by Hairer et al.47. The complete code for the crack simulation is available as a Software Heritage archive48. Further details on the code can be obtained by contacting the authors.
Experiments in two different regimes were run28: a forced one where the deflection of the lower plate (see Fig. 1) was driven at a constant speed, and a relaxation regime, where the deflection was maintained constant while the crack still advances. In both scenarii, the long term evolution of the average load \(\overlineG(t)\) and front position \(\overlinea(t)\) was shown8,37 to be reproduced by Eq. (1). In the case of the experiments of Tallakstad et al.28, the intermittent dynamics measured in the two loading regimes were virtually identical. Such similarity likely arises from the fact that the avearge load \(\overlineG\) was, in both cases, computed to be almost constant over time, in regard to the spatial variation in G, described by Eq. (2) (see the supplementary information). Here, we will then consider that the crack is, in average along the front, always loaded with the same intensity (i.e., \(\overlineG(t)=\overlineG\)).
Among the various statistical features studied by Tallakstad et al.28, was notably quantified the temporal evolution of their fracture fronts morphology. It was interestingly inferred that the standard deviation of the width evolution of a crack front h scales with the crack mean advancement:
\(\overlinea\) being the average crack advancement at a given time. To mitigate the effect of the limited resolution of the experiments and obtain a better characterization of the scaling of the interfacial fluctuations on the shorter times, we computed the subtracted width,
(a) Standard deviation of the width evolution of the crack front as a function of the mean crack advancement, as defined by Eqs. (3) to (5) for the chosen simulation (plain points) and for the experiments28 (hollow stars) (out of Fig. 8, Expt. 5 of the experimental paper). The continuous line has a slope 0.6, close to that of the experimental points: \(\beta _G\sim 0.55\). The numerical \(\beta _G\), obtained with a linear root mean square fit of the growth of W, is estimated as \(\beta _G=0.60\pm 0.05\). The dashed lines mark the observation scale \(\Delta x\), corresponding to the experimental camera pixel size, and the chosen correlation length for the simulation \(l_c=50\,\upmu\)m. (b) The same width function for simulations with different correlation lengths \(l_c\). The rest of the parameters are as defined in Table 1. The slope and plateau of the experimental data (shown in (a)) is marked by the dashed line for comparison.
While the crack advances at an average velocity \(\overlineV\), the local velocities along the front, described by Eq. (1), are, naturally, highly dependent on the material disorder: the more diverse the met values of \(G_c\) the more distributed shall these velocities be.
Maløy et al.26 and Tallakstad et al.28 inferred the local velocities of their cracks with the use of a so-called waiting time matrix. That is, they counted the number of discrete time steps a crack would stay on a given camera pixel before advancing to the next one. They then deduced an average velocity for this pixel by inverting this number and multiplying it by the ratio between the pixel size and the time between two pictures: \(\Delta a/\Delta t\). Such a method, that provides a spatial map V(x, a), was applied to our simulated fronts, and we show this V(x, a) map in Fig. 3c. As to any time t corresponds a front advancement a(x, t) (recorded with a resolution \(\Delta a\)), an equivalent space-time map V(x, t) can also be computed, and it is shown in Fig. 3b. The experimental report28 presented the probability density function of this latter (space-time) map \(P(V/\overlineV)\), and it was inferred that, for high values of V, the velocity distribution scaled with a particular exponent \(\eta = 2.6 \pm 0.15\)28,38 (see Fig. 5a). That is, it was observed that 2ff7e9595c
Comments