Phase Field Modeling of Fast Crack Propagation

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase-field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.

Figure 1

Steady state growth of a crack in a strip, as obtained from phase-field simulations. A constant displacement is prescribed at the upper and lower boundary of the system. The parameters used here are $\Delta =2.03$ and $D/\xi v_R=6.18$; the system size is $600\times 400$. The resulting velocity is $v/v_R=0.71$ and the radius $r_0=0.69D/v_R=4.26\xi$ are dynamically selected.

Figure 2

Steady state velocity versus dimensionless driving force $\Delta$; $\Delta =1$ is the Griffith point.

Figure 3

Tip radius $r_0$ as a function of the kinetic coefficient $D$ for different driving forces $\Delta$. In the intermediate linear regime the length scales are well separated.

Figure 4

The quantity $v{r}_0/D$ as a function of the kinetic coefficient for different driving forces $\Delta$.

Figure 5

$v{r}_0/D$ as a function of the driving forces $\Delta$; the curves for different kinetic coefficients do not differ significantly, in agreement with the theoretical prediction.

Figure 6

Irregular tip splitting scenario. We used $D/\xi v_R=1.85$ and $\Delta =3.6$; the system size is $600\times 400$. Time is given in units $D/v_R^2$. Minor numerical noise breaks the symmetry of the competing sidebranches.