Figure 1

Left: definition of in-plane loading modes: top—mode-II loading; bottom—mode-I loading. Right: geometry of the experiment reported in 9. An elastic sheet is extended in both $x$ and $y$ directions. The local strain is bigger in the $y$ direction than in the $x$ direction. The crack speed is denoted by $c$. In simulations performed here $W$ was taken equal to 60 and at $y=0$ and $y=W$, $u_x$ was kept constant in time, so that a vertically propagating crack cannot reach $y=W$ or $y=0$.Reuse & Permissions

Figure 2

Left: contour plot of the line $\varphi =0.5$ taken at 1 time unit (t.u.) intervals during the branching of a crack under mode-I loading with ${\Delta }_y=18$ and $\beta =1$. The angle of the two branches at the branching site is approximately $40°$, and the critical speed for which the instability occurs is approximately 0.5 while the shear speed is 1, which is in agreement with predictions in 21. During the evolution of the system, the lower branch will recede while the upper branch will propagate and branch irregularly. The simulations are performed using the undamped model. Right: contour plots of $\varphi =0.5$ taken at 40 t.u. during the propagation of an overdamped crack in a medium where pure mode-II loading is applied. The initial crack is straight and oriented along the $x$ axis. The crack propagates in a direction that nullifies the mode-II stress intensity factor. Inset: same parameters with $\alpha =0$. One can see that, in this case, the crack branches and propagates in two directions symmetrical with respect to the $x$ axis. The second branch propagates in a region where the material is strongly compressed.Reuse & Permissions

Figure 3

Fracture tip trajectories for ${\Delta }_y=10$ and $r=W{\Delta }_x/L=8.0$ during the transient regime (a) and when a stationary state is reached (b). The steady-state tip trajectory matches a simple sinusoid. (c) Tip trajectory for ${\Delta }_y=10$ and $r=W{\Delta }_x/L=7.25$, exhibiting damped oscillations. In (b) the mean tip speed is 0.362 and the mean tip speed along the $x$ axis is 0.354. In (c), the tip speed is 0.347 once straight propagation has resumed. The transverse and longitudinal speeds of sound are, respectively, 1 and 1.22. The period of the oscillations in (b) is about 831 t.u. and the corresponding wavelength is 294 s.u.Reuse & Permissions

Figure 4

Elastic energy landscape (gray scale, dark regions correspond to high density) during the propagation of an oscillating crack. The white region corresponds to the broken region ($\varphi <0.5$). Parameter values are as in Fig. 3a. Between the arches of the sinusoidal line of the crack, the elastic energy density is lower than in the vicinity of the upper and lower boundaries of the slice.Reuse & Permissions

Figure 5

(a) Square of the amplitude of the oscillations as a function of $r=W{\Delta }_x/L$. (b) Frequency $\omega$ ($+$) of the oscillations. The wavelength (not shown) varies like the inverse of $\omega$ since the crack speed is not varying much (about ${10}^{-3}$) over the parameter range presented here.Reuse & Permissions