Relation between driving energy, crack shape, and speed in brittle dynamic fracture

We report results on the interrelation between driving force, roughness exponent, branching, and crack speed in a finite element model. We show that for low applied loadings the crack speed reaches the values measured in the experiments, and the crack surface roughness is compatible with logarithmic scaling. At higher loadings, the crack speed increases, and the crack roughness exponent approaches the value measured at short length scales in experiments. In the case of high anisotropy, the crack speed is fully compatible with the values measured in experiments on anisotropic materials, and we are able to interpret explicitly the results in terms of the efficiency function introduced by us in our previous work [A. Parisi and R. C. Ball, Phys. Rev. B 66, 165432 (2002)]. The mechanism which leads to the decrease of crack speed and the appearence of the logarithmic scaling is attempted branching, while power law roughness develops when branches succeed in growing to macroscopic size.

Figure 1

The efficiency for planar crack propagation in our fcc lattice model. The overall increase of $E\left(v\right)$ with the crack speed still remains to be quantitatively understood, but the drop of the efficiency at the Rayleigh speed as well as the fine structure are well understood in terms of vibrational resonances.

Figure 2

The histogram method for measuring the crack speed. The histogram represents the number of broken tetrahedra $N\left(l\right)$ at distance $l$ from the front side of the sample. As time advances, the histogram grows: we can build a set of histograms corresponding to a set of intermediate positions of the crack. The approximate position of the crack front is then obtained as the intercept with the dashed threshold.

Figure 3

Comparison between the results for free-running planar cracks (filled circles) and the efficiency description (continuous line). Higher values of the driving energy $ϵ$ correspond to lower values of $E\left(v\right)$ as described by Eq. (2.3). The figure shows how this also corresponds to higher crack speeds. The dashed line refers to the special case of $ϵ=1$.

Figure 4

(Color online) (Top) Crack in a sample $300\times 60\times 60$ tetrahedra wide (projection), for $ϵ=1$. Shading (and color online) refers to the depth in the third dimension. The crack appears “fat,” thicker than one layer of tetrahedra. A single-layer wide section of the sample (bottom) shows that the crack itself is driven slightly off planar. Microbranching is visible, suggesting attempted branching as the mechanism which leads to thicker cracks and slows the crack speed.

Figure 5

(Color online) (Top) Crack in a sample $300\times 60\times 60$ tetrahedra wide (projection), $ϵ=5$: branches reach the sample’s boundaries. (Bottom) Crack in a larger sample, $500\times 120\times 120$ tetrahedra wide (projection), for $ϵ=5$. In this case branches do not reach the sample boundaries. Shading (and color online) refers to the depth in the third dimension.

Figure 6

Sequence of frames for a typical crack growing in a sample $500\times 120\times 120$ tetrahedra wide. The gray shading refers to the height with respect to the crack notch plane. In this case $ϵ=3$.

Figure 7

(Color online) Slices of the fracture of Fig. 6 viewed from the side. Each slide corresponds to $1∕8$ of the sample. Shadings (and colors online) refer to the depth in the third dimension.

Figure 8

(Color online) Final fracture surface corresponding to a driving energy $ϵ=3$.

Figure 9

Log-log plots of the height-height correlation functions along the $\widehat{x}$ direction (on the top), and the $\widehat{z}$ direction (on the bottom) for the $300\times 60\times 60$ samples. For comparison, also one of the correlation functions for the $500\times 120\times 120$ samples is shown. The above slope corresponds to roughness exponent $\zeta =0.45$.

Figure 10

Log-log plots of the height power spectra for cuts along the $\widehat{x}$ direction (on the top) and the $\widehat{z}$ direction (on the bottom) for the $300\times 60\times 60$ samples. The above slope corresponds to $\zeta =0.45$.

Figure 11

Change of the roughness exponent from the correlation functions, with the driving factor $ϵ$ for cuts along the $\widehat{x}$ direction (on the top) and the $\widehat{z}$ direction (on the bottom). Results for the $300\times 60\times 60$ samples are in black, those for the $500\times 120\times 120$ samples are in light gray.

Figure 12

Change of the roughness exponent from the power spectra, with the driving factor $ϵ$ for cuts along the $\widehat{x}$ direction (on the top) and the $\widehat{z}$ direction (on the bottom). Results for the $300\times 60\times 60$ samples are in black, those for the $500\times 120\times 120$ samples are in light gray.

Figure 13

Linear-log plots of the height-height correlation functions along the $\widehat{x}$ direction (on the top) and the $\widehat{z}$ direction (on the bottom) for the $300\times 60\times 60$ samples. In both plots, the lowest curves corresponding to the lowest $ϵ$ are straight, corresponding to a logarithmic scaling.

Figure 14

(Color online) Shape of cracks in a sample $300\times 60\times 60$ tetrahedra wide (projections) for increasing values of $ϵ$. Values shown are from the top: $ϵ=0.8$, 1.0, 1.4, 1.6, 2.0, 3.0, 4.0, and 5.0. Shading refers to the depth in the third dimension.

Figure 15

(Color online) Sideview of the final fracture in the case of a nonconnected fracture as discussed in Sec. 6, for a $500\times 120\times 120$ tetrahedra wide sample and $ϵ=5$. Shading refers to the depth in the third dimension.