Local dynamics of a randomly pinned crack front during creep and forced propagation: An experimental study

We have studied the propagation of a crack front along the heterogeneous weak plane of a transparent poly(methyl methacrylate) (PMMA) block using two different loading conditions: imposed constant velocity and creep relaxation. We have focused on the intermittent local dynamics of the fracture front for a wide range of average crack front propagation velocities spanning over four decades. We computed the local velocity fluctuations along the fracture front. Two regimes are emphasized: a depinning regime of high velocity clusters defined as avalanches and a pinning regime of very low-velocity creeping lines. The scaling properties of the avalanches and pinning lines (size and spatial extent) are found to be independent of the loading conditions and of the average crack front velocity. The distribution of local fluctuations of the crack front velocity are related to the observed avalanche size distribution. Space-time correlations of the local velocities show a simple diffusion growth behavior.

Figure 1

Sketch of the experimental setup. Two PMMA plates with dimensions $(l_1,w_1,h_1)=(30,14,1)$ cm and $(l_1,w_2,h_2)=(30,10,0.4)$ cm are sintered together, creating a weak in-plane layer for the fracture to propagate. Fracture is initiated by lowering a cylindrical press bar, controlled by a step motor, onto the lower plate. The uncracked part of the sample is transparent, whereas the cracked part has lost transparency hence creating a good contrast at the fracture front. The fracture front is imaged from above by a digital camera. The deflection $d$ ($z$ direction) between the plates is indicated in the lower panel. The fracture plane is ($x,y$), where the $x$ direction is transverse to the average direction of fracture propagation whereas the $y$ direction is parallel to the average direction of fracture propagation.

Figure 2

(a) Constant velocity boundary conditions (CVBC). Upper panel shows the force development $F(t)$ on the lower plate as it is bent by the pressbar. The dashed line indicates the onset of fracturing. Lower panel shows the linear increase of the deflection $d(t)$. (b) Creep boundary conditions (CBC). Same as in (a) but $F(t)$ and $d(t)$ are in a semilog scale. The short, solid line in the upper panel indicates the onset of fracturing, whereas the dashed line indicates the time at which the pressbar is stopped and maintained in a constant position according to Eq. (2).

Figure 3

Fracture frontline $h(x,t^{\text{'}})$ at some time $t^{\text{'}}$, superimposed on the corresponding raw image. Direction of propagation is from top to bottom. System size $L$ in the $x$ direction is indicated. The framed raw image corresponds to a tiny central part of the full sample as seen in the upper panel.

Figure 4

Waiting time matrix of a CBC experiment, $\langle v\rangle =1.36$ $\mu$m/s. The map results from the extraction of $24\hspace{0.5em}576$ front lines at a rate of 50 fps. Dark regions correspond to a high waiting time and thus a low velocity, and vice versa for bright regions, as shown in the colorbar indicating the amount of time (in seconds) the front has been fixed at a given position. Black pinning lines are visible, with bright depinning regions in between. The system size $L$ is indicated.

Figure 5

Distribution of local velocities $P(v/\langle v\rangle )$ rescaled by the average propagation velocity for various experimental conditions: A range of roughly four decades in average crack front velocity including both CBC and CVBC experiments. In the legend, symbols labeled with different average velocities are results from Ref. [17]. A fit to all the data for $v>\langle v\rangle$ shows power-law behavior with an exponent $-2.55$. Inset shows the corresponding waiting time distribution $P(w/\langle w\rangle )$. The exponent transforms in this case to $0.55$.

Figure 6

Space correlation functions $G(\Delta x)$. Functions from the same sample are grouped together (sample $2$: filled markers, sample $1$: open markers). A power law with exponential cutoff has been fitted to each group of correlation functions, as indicated by the solid and dashed lines for samples $1$ and $2$, respectively (see text).

Figure 7

(a) Time correlation functions collapsed onto each other according to a power law with an exponential cutoff $G(\Delta t)=A\Delta t^{-{\tau }_t}\exp (-\Delta t/t^{*})$. The exponent is ${\tau }_t\approx 0.43$. Inset shows the scaling between the crossover correlation time and average propagation velocity $t^{*}\approx 7\hspace{0.5em}\mu \text{m}/\langle v\rangle$. (b) Space correlation function $G(\Delta y)$ with logarithmic $\Delta y$ axis. Consistently with (a) and Eq. (7), the local velocities become uncorrelated after only a short distance ($~10-20\hspace{0.5em}\mu$m) in the $y$ direction. Correlation functions from experiments performed on samples $2$ and $1$ have filled and open markers, respectively. To some extent we see also here a grouping of experiments from the same sample. The difference is, however, not as clear as for the spatial correlations along the transverse $x$ axis (subparallel to the fronts), on the inset showing $G(\Delta x)$ with logarithmic $x$ axis. The reason might be that the drop to zero correlation occurs close to the resolution scale for $G(\Delta y)$.

Figure 8

Scaling of the front width as a function of time, rescaled with the average velocity. The dashed line corresponds to $\langle \Delta h(t){}^2\rangle {}^{\frac{1}{2}}\propto t^{0.55}$.

Figure 9

Thresholded matrix $V_C$ ($5600\times 1400$)$\mu$m in the depinning (left) and pinning (right) regime for a CBC experiment with $\langle v\rangle =1.36$ $\mu$m/s. White clusters correspond to velocities $C$ times larger than $\langle v\rangle$ for the depinning case, whereas white clusters or lines correspond to velocities $C$ times less than $\langle v\rangle$ for the pinning case.

Figure 10

Probability distribution function $P(S)$ for all experiments using a threshold $C=3$. A distribution averaged over all experimental conditions is also included for the depinning (dashed line and circular markers) and pinning regime (dashed line and square markers). The pinning size distributions have been shifted along the $y$ axis to enhance visual clarity.

Figure 11

Distributions of pinning clusters $P(S)$ averaged over all different experimental conditions, for a threshold range $C=2-12$. Solid lines show the fits corresponding to a power law with an exponential cutoff.

Figure 12

Collapsed $P(S)$ distributions averaged over all different experimental conditions for both depinning (upper set of data) and pinning (lower set of data). The pinning distributions have been shifted relative to the depinning distributions by a factor $0.01$ for visual clarity. Depinning and pinning thresholds are in the range $C=2-22$ and $C=2-12$, respectively. The dashed and the solid lines both have the slope $\gamma =1.56$. Inset shows the scaling between the cutoff $S^{*}$ and the threshold $C$ for the depinning (solid line ${\sigma }_d=1.77$) and pinning regime (dashed line ${\sigma }_p=2.81$).

Figure 13

Average cluster size $\langle S\rangle$ obtained from the image analysis vs. threshold level $C$. The dashed line shows a power-law fit for $3<C<30$, with the exponent $-{\sigma }_d(2-\gamma )=-0.75$. Inset shows the number of clusters as a function of threshold level for the various experiments. The dashed lines all have the average slope $\chi =0.28$.

Figure 14

(a) Left panel shows a bounding box with sides $l_x$ and $l_y$ embedding a depinning cluster. In this case the bounding box is a good measure of the linear extension of the cluster. Right panel shows a bounding box embedding a pinning cluster. In this case $l_x$ gives a reasonable linear extent measure, however, $l_y$ does not due to the irregular curvature (somewhat exaggerated in the figure) and to the narrow width in the $y$ direction. To characterize this width we use instead the average cross sectional width $l_{\mathit{yw}}$. (b) Upper and lower panels show bounding boxes for depinning and pinning clusters, respectively, from one experiment.

Figure 15

Linear extent of pinning and depinning clusters as a function of cluster size for the full span of threshold levels and averaged over all experimental conditions. The slopes of the different fitted lines (dashed: pinning clusters, solid: depinning clusters) are indicated in the caption. Note that there in all cases are initial transients up to $S\approx 100\hspace{0.5em}\mu$m${}^2$.

Figure 16

(a) Collapsed $P(l_x)$ distributions averaged over all different experimental conditions for both depinning (upper set of data) and pinning (lower set of data). The pinning distributions have been shifted relative to the depinning distributions by a factor of $0.05$ for visual clarity. Depinning and pinning thresholds are in the range $C=2-30$ and $C=2-12$, respectively. The solid and dashed lines both have the slope ${\beta }_x=1.93$. Inset shows for the case of depinning the threshold dependence for the unscaled distributions. (b) Collapsed $P(l_y)$ and $P(l_{\mathit{yw}})$ distributions averaged over all different experimental conditions for the depinning (upper set of data) and pinning (lower set of data) regime, respectively. The pinning distributions have been shifted relative to the depinning distributions by a factor of $0.1$ for visual clarity. Thresholds are in the range $C=2-30$ and $C=2-12$ for depinning and pinning, respectively. The solid line has the slope ${\beta }_y=2.36$. Inset shows for the case of depinning the threshold dependence for the unscaled distributions.

Figure 17

Example of the computation of the waiting time matrix $W(x,y)$ [Eq. (A1)]. All fronts are added to an originally empty matrix in time step unit. Indicated above is the addition of front lines in three timesteps $t_1$ (red), $t_2$ (blue), and $t_3$ (green).

Figure 18

(a) Pixel level zoom in of a front line $h(x^{\text{'}},t)$ at a given position $x^{\text{'}}$ as function of time. Indicated are three waiting times $w_1$, $w_2$, and $w_3$ separated by a one-pixel jump. As an example, note that all captured fronts from $h(x^{\text{'}},t_0)$ to $h(x^{\text{'}},t_0+w_1)$ are given the same constant velocity $v_1\propto 1/w_1$ in making the jump from pixels 127 to 128 along the $y$ axis. This approximation means that the front position increases linearly during this time interval, as indicated. (b) Local velocity fluctuations $v(x,t^{\text{'}})$ along the front line $h(x,t^{\text{'}})$.