Average crack-front velocity during subcritical fracture propagation in a heterogeneous medium

We study the average velocity of crack fronts during stable interfacial fracture experiments in a heterogeneous quasibrittle material under constant loading rates and during long relaxation tests. The transparency of the material (polymethylmethacrylate) allows continuous tracking of the front position and relation of its evolution to the energy release rate. Despite significant velocity fluctuations at local scales, we show that a model of independent thermally activated sites successfully reproduces the large-scale behavior of the crack front for several loading conditions.

Figure 1

Side view (top) and top view (bottom) of the experimental setup. A stiff aluminum frame is attached to the upper PMMA plate. The lower plate is separated from the upper one by a load $F$ applied by a rod connected to a stepping motor. The rod can freely rotate around its axis. The load causes a deflection $u$ of the bottom plate and the propagation of an interfacial crack. The crack front is located at the average distance $\overline{a}$ from the free end ($\overline{a}$ is obtained as the spatial average of the distance $a$ along the front). Front advance is monitored by a camera set in vertical position, perpendicular to the crack plane. We add glycerol between the upper plate and a thin glass sheet located above the crack front to enhance the optical contrast of the pictures. The narrow plate thickness and width are denoted $h_1$ and $b$, respectively, while the large plate thickness is denoted $h_2$.

Figure 2

Top: Variation of loading point position $u$ as a function of time during an experiment. At time $t=230$ s, the loading point is maintained in a fixed position. Middle: Variation of the average front position $\overline{a}$ as a function of time (black solid line) and evolution of the crack front velocity $d\overline{a}/dt$ (gray solid line). The red solid and short-dashed lines represent a fit to the average front position and velocity, respectively, according to Eq. (19). The blue long-dashed line is a fit to the crack velocity in the relaxation regime following Eq. (26). Bottom: Variation of force as a function of time (black solid line) and force predicted by the beam theory (gray dashed line) using (5) with $E=3.2$ GPa and $b=2.84$ cm, which are the measured properties of the plate. For all figures, the vertical gray line denotes the time at which the loading was stopped and separates the imposed velocity regime from the relaxation regime.

Figure 3

Variation of the crack front position $\overline{a}$ as a function of the loading point displacement $u$. The gray points refer to recorded data from different samples. For each sample we carried out several experiments. The best fit using $\overline{a}\left(u\right)\propto u^{1/2}$ is displayed as a dashed line for each sample. Crack-front positions are shifted vertically for each sample in order to enhance the visibility (by a prefactor corresponding respectively to a factor 0.5 for the lowest and 2 for the highest curves). We see that for each sample and for each experiment the observed behavior is in good agreement with the fitted trend. Notice, however, that the crack initiation does not seem well approximated by this trend.

Figure 4

Evolution of the average front position $\overline{a}\left(t\right)-{\overline{a}}_0$ during several relaxation experiments as a function of time $t-t_{\mathrm{stop}}$. The observed front position is represented by gray dots while the best fits using (1) are displayed as continuous dark lines. Traces have been shifted vertically in order to enhance visibility. The experiment displayed on the top figure lasted more than 18 h (the time representation scale is different from the one in the bottom figure).

Figure 5

Variation of the logarithm of the average crack-front velocity as a function of the average energy release rate during an experiment. The grayscale refers to time since the picture acquisition starts. Both loading regimes (constant loading velocity, diamonds, and relaxation regimes, circles) are included. We see the increase of $\overline{G}$ and $\overline{v}$ during the constant-loading-velocity regime, the stabilization of $\overline{G}$ while the front is propagating at almost constant speed, and the decrease of $\overline{G}$ and $\overline{v}$ during the relaxation process. The linear relationship between $\overline{G}$ and ${\log }_{10}\left(v\right)$ is observed during all these phases.

Figure 6

(a) Variation of the logarithm of the average crack-front velocity as a function of the average energy release rate. All experiments are reported in this figure and are distinguished by gray levels and symbols. Similar gray levels indicate experiments performed in the same region of one specific plate (short separation distance between fracture onsets). The reference velocity $v_{\mathrm{ref}}$ used to estimate ${\overline{G}}^{\mathrm{ref}}$ is shown as a horizontal black line. Upper and lower bounds of ${\overline{G}}^{\mathrm{ref}}$ are indicated with arrows. (b) Superimposition of all experiments when plotted as a function of $\overline{G}-{\overline{G}}^{\mathrm{ref}}$. The dashed line represents the best-fitted model. The slope of this line is related to the value of $\alpha$ and is ${\alpha }^2/k_{B}T$.

Figure 7

Cumulative distribution function of the estimated values of ${\overline{G}}^{\mathrm{ref}}$ for all experiments (gray dots). Values are distributed between 109 and $162\mathrm{J}/{\mathrm{m}}^2$ with a mean of $137\mathrm{J}/{\mathrm{m}}^2$. The dark curve represents the expected cumulative distribution function for a normal distribution with similar mean and standard deviation as inferred from our experimental data.

Figure 8

Variation of force $F$, front position $\overline{a}$, and energy release rate $\overline{G}$ obtained during six experiments (gray points) carried out in the forced regime. The results of the numerical integration of Eq. (14) are drawn as black lines. The model captures most of the features observed in our experiments, such as the peak value of the force and its occurrence time. It also captures the appearance of a Griffith regime, i.e., the fact that $\overline{G}$ is close to constant at large enough crack speed.

Figure 9

Variation of force $F$ and front position $\overline{a}$ obtained during the relaxation regime for the experiment displayed on Fig. 2 and for 5 min [gray dots, (c) and (d)]. Results of the numerical integration are represented as black line. Same representation is made for the experiments monitored for more than 18 h in the relaxation regime and is displayed in the two upper graphs [(a) and (b)]. The dashed line shows the evolution of force deduced from the fit of the front position from Eq. (1).

Figure 10

General configuration of bent plates, adapted from [56] to the configuration used in this work.