Crack-microcrack interactions in dynamical fracture

We address the interaction of fast moving cracks in stressed materials with microcracks on their way, considering it as one possible mechanism for fluctuations in the velocity of the main crack (irrespective whether the microcracks are existing material defects or they form during the crack evolution). We analyze carefully the dynamics (in two space dimensions) of one macrocrack and one microcrack, and demonstrate that their interaction results in a large and rapid velocity fluctuation, in qualitative correspondence with typical velocity fluctuations observed in experiments. In developing the theory of the dynamical interaction we invoke an approximation that affords a reduction in mathematical complexity to a simple set of ordinary differential equations for the positions of the crack tips; we propose that this kind of approximation has a range of usefulness that exceeds the present context.

Figure 1

The geometric configuration of the model problem. The macrocrack and the microcrack extend along the intervals $[-L,0]$ and $[{\ell }_{-},{\ell }_{+}]$, respectively, with $L∕\ell ⪢1$.

Figure 2

The predicted velocity increase as a function of normalized time (where $L_0$ is the crack length at initiation) for a mode I crack under uniform constant load at infinity, Eq. (6).

Figure 3

The normalized static stress intensity factors (SIFs) as a function of $\Delta$. The normalization factor is ${\sigma }^{\infty }\sqrt{\pi L∕2}$, which is the stress intensity factor of the macrocrack in the absence of the microcrack. We fixed ${\ell }_{+}=1$ and varied ${\ell }_{-}$. Note that the stress intensity factors obey $K_{+}<K_{-}<K_{\mathrm{M}}$ and that $K_{\mathrm{M}}\to {\sigma }^{\infty }\sqrt{\pi L∕2}$ as the ratio $\ell ∕\Delta$ decreases, as expected.

Figure 4

The three crack tip velocities for an interaction event when a macrocrack traveling at an initial velocity $v_{\mathrm{M}}=0.62c_R$ interacts with a collinear microcrack of length $\ell =5$ positioned at $\Delta =5$. The figure shows the normalized velocities $v∕c_R$ as a function of the normalized time $c_{R}t∕\ell$.

Figure 5

The normalized experimental velocity $v_{\mathrm{expt}}∕c_R$ as a function of the normalized time $c_{R}t∕\ell$. The experimental velocity was calculated according to Eq. (19) with ${\Delta }_c⪡\ell$.

Figure 6

The simulation data of $L{v}_{\mathrm{M}}∕c_R$ as a function of $L$ are shown by the circles and the linear fit for these data is shown by the solid line. It is clear that the functional form suggested by Eq. (20) indeed describes the single crack dynamics in the lattice simulation. The fit is consistent with the assumptions since the slope $\alpha$ is very close to 1.

Figure 7

The simulation data of $v_{\mathrm{M}}∕c_R$ as a function of $L$ are shown by the circles and the analytic approximation predictions are shown by the solid line. In this simulation we used $L\approx 245$ (the length of the macrocrack in the interaction region) and $\ell =10$. The analytic approximation predicted the simulated data up to an error of 6.5%, even though the ratio $L∕\ell$ was not very large.