Pinning of crack fronts by hard and soft inclusions: A phase field study

Through tridimensonal numerical simulations of cracks propagating in material with an elastic moduli heterogeneity, it is shown that the presence of a simple inclusion can dramatically affect the propagation of the crack. Both the presence of soft and hard inclusions can lead to the arrest of a crack front. Here the mechanism leading to the arrest of the crack are described and shown to depend on the nature of the inclusion. This is also the case in regimes where the presence of the inclusion leads to a slowdown of the crack.

Figure 1

Perspective view of the simulation setup. The crack is propagating from left to right and the inclusion is the sphere seen on the right-hand side of the simulation domain.

Figure 2

Plots of the crack tip trajectories together with the inclusion boundary for four different values of the loading in 2D. In both cases, the radius of the obstacle is 13.5, and the value of $Y_{\text{off}}$ is 19.5 in (a) and 22.5 in (b). The crack is propagating from left to right along the $x$ axis. In the inset, a larger part of the simulation domain is represented. It corresponds to its total extension along the $y$ axis and the to its rightmost half along the $x$ axis.

Figure 3

(a1) Perspective view of the crack front and inclusion at different times during the pinning of the crack. The crack is propagating from left to right and the color of the line is changed according to the time. (b1) Top view of the crack front. Only half of the front is shown. The side of the sample is at the bottom of the graph. (c1) Side view of the position of the crack tip in different $xy$ planes (center of the sample, side of the sample, and an intermediate plane). Graphs use the actual aspect ratio. (a2), (b2), (c2) Same views in a case where the crack front is pinned but does not intersect the inclusion as it is evidenced in (c2).

Figure 4

Top: (a) Perspective view of the crack front and inclusion at different times during the slowdown and unpinning of the crack. The crack is propagating from left to right and the color of the line is changed according to the time. (b) Top view of the crack front. Only half of the front is shown. The side of the sample is at the bottom of the graph. (c) Side view of the position of the crack tip in different $xy$ planes (center of the sample, side of the sample, and an intermediate plane). Bottom: Sequence of images of the crack (iso $\mathrm{surface}\varphi =0.5$ of the phase field) when it goes around the inclusion. The images are labeled by a dimensionless time variable. One can see that between images 2 and 8, the crack moves very little. This contrasts with the advance of the crack between images 1 and 2 or 8 and 9. It is worth mentioning that the crack isosurface may intersect the sphere while the crack tip is less than a regularization length away from the obstacle.

Figure 5

Phase diagram in the $R, Y$ plane for simulations for soft inclusions [2D in (a) and 3D in (b)] and hard inclusions [2D in (c) and 3D in (d)]. $G$ is set to $1.5G_c$ for soft inclusions and to $1.25G_c$ for hard inclusions. The criteria chosen to differentiate between freely moving is arbitrary. Here it is chosen to consider that the latter was observed when the minimal crack front velocity is smaller than 1/2 its free propagation speed. Obviously, changing the criteria would lead to different shapes of the partial pinning region.

Figure 6

Snapshots of the crack going around a hard inclusion at different times of propagation. The crack surface (isosurface $\varphi =0.5$) is represented in blue while the spherical inclusion is in gray. At $t=4$ and $t=5$, one can see that the crack partly breaks the inclusion. The lower rightmost image corresponds to a crack stopped by a hard inclusion. In all cases, only a portion of the simulation domain is represented.