Crackling noise in three-point bending of heterogeneous materials

We study the crackling noise emerging during single crack propagation in a specimen under three-point bending conditions. Computer simulations are carried out in the framework of a discrete element model where the specimen is discretized in terms of convex polygons and cohesive elements are represented by beams. Computer simulations revealed that fracture proceeds in bursts whose size and waiting-time distributions have a power-law functional form with an exponential cutoff. Controlling the degree of brittleness of the sample by the amount of disorder, we obtain a scaling form for the characteristic quantities of crackling noise of quasibrittle materials. Analyzing the spatial structure of damage we show that ahead of the crack tip a process zone is formed as a random sequence of broken and intact mesoscopic elements. We characterize the statistics of the shrinking and expanding steps of the process zone and determine the damage profile in the vicinity of the crack tip.

Figure 1

(Left) Neighboring polygons of the initial Voronoi tessellation are connected by beams. This way a triangular beam lattice is obtained. (Right) Due to subsequent breaking of beams a crack forms along the edge of polygons.

Figure 2

Three-point bending of a bar composed of polygonal particles. The particles are coupled by elastic beams which are colored according to the longitudinal deformation (yellow: nearly unstressed beams; red and black: elongated beams; blue and green: compressed beams). Beams are allowed to break solely along the center line of the bar. A relatively small sample is presented to have a clear view on the details of the model construction. The two loading plates at the bottom are fixed while the third one on the top moves downward.

Figure 3

Force $F$ normalized by the maximum value $F_{\text{max}}$ as a function of time $t$ during the loading process. Since the deflection of the bar is proportional to $t$ the curve can be considered as the constitutive curve of the sample. Oscillations occur due to elastic waves generated by the loading process. After the peak the decreasing part of $F(t)$ indicates stable crack propagation where our analysis is focused. $t_{\text{tot}}$ denotes the time of the last beam breaking.

Figure 4

Damage accumulated up to the peak of $F(t)$ as a function of the Weibull exponent $m$. The curves can be very well fitted with the functional form Eq. (4). The value of the exponent is $\mu =1.9$ and $\mu =1.5$ for the stretching and bending limits, respectively.

Figure 5

Time series of bursts in a single fracture simulation. The bursts are correlated breaking sequences of beams, which then result in sudden jumps of the extending crack. $N$ denotes the total number of beams along the weak interface where the crack propagates. For all the simulations its value was set to $N=200$. At the beginning of the loading process, for a considerable time no breaking occurs; most of the breaking events appear at larger deflections beyond the peak of the constitutive curve (see Fig. 3). Hence, we magnify the final section of the bending process.

Figure 6

Inset: Avalanche size distributions for the absolute stretching limit varying the value of the Weibull exponent $m$. The main panel presents the excellent data collapse obtained by rescaling the distributions with the average burst size according to Eq. (7). Scaling exponents: ${\alpha }_{\Delta }^s=1.4\pm 0.5,{\beta }_{\Delta }^s=1.8\pm 1$. The parameter values obtained by fitting Eq. (8) are $a_{\Delta }^s=0.55,{\tau }_{\Delta }^s=1.3\pm 0.2,b_{\Delta }^s=2.2,{\delta }_{\Delta }^s=1.5\pm 0.3$.

Figure 7

Inset: avalanche size distributions for the absolute bending limit. The main panel shows that rescaling the distributions according to Eq. (7) an excellent data collapse is obtained. Scaling exponents: ${\alpha }_{\Delta }^b=1.4\pm 0.5,{\beta }_{\Delta }^b=1.8\pm 1$. The fit parameters of the scaling function are $a_{\Delta }^b=0.85,{\tau }_{\Delta }^b=0.8\pm 0.3,b_{\Delta }^b=1.4,{\delta }_{\Delta }^b=1.3\pm 0.3$.

Figure 8

Waiting-time distributions for absolute stretching. The main panel presents curves for low disorder, where the fit was obtained with the exponent ${\tau }_T^s=1.9\pm 0.15$. The inset shows the corresponding curves for high disorder, where a crossover is obtained to a lower exponent ${\tau }_T^s=1.5\pm 0.1$.

Figure 9

Waiting time distributions for the absolute bending limit. The amount of disorder affects only the cutoff, but the exponent is constant ${\tau }_T^b=1.8\pm 0.15$.

Figure 10

Crack tip and the process zone in front of it. Red dots indicate the positions of broken beams. The process zone is identified as a sequence of broken and intact beams starting at the crack tip and ending at the start of the continuous sequence of intact fibers.

Figure 11

Crack tip jump length distributions for the absolute bending limit. The values of the exponent ${\gamma }^b$ of the fitted curves are $1.8$ and $4$.

Figure 12

Crack tip jump length distributions for the absolute stretching limit. The value of the exponent of the power-law regime is ${\gamma }^s=2.2\pm 0.1$; it does not depend on the amount of disorder.

Figure 13

Nucleation length distributions for the absolute bending limit varying the Weibull exponent $m$. The amount of disorder does not have a relevant effect on the distribution. The exponent of the fitted power law is ${\kappa }^b=1.8$.

Figure 14

Nucleation length distributions for the absolute stretching limit. Decreasing disorder leads to localization of damage at the crack tip.

Figure 15

Damage profile for different values of the Weibull exponent $m$. The distance $r$ measured from the crack tip is normalized by the cross section $L_c$ of the specimen. The values of the fitting parameters are $\rho =0.5$, $r_0=0.05$ $(m=2)$, $\rho =1.0$, $r_0=0.035$ $(m=4)$, $\rho =1.5$, $r_0=0.033$ $(m=6)$.