How Fast Cracks in Brittle Solids Choose Their Path

While we fundamentally understand the dynamics of simple cracks propagating in brittle solids within perfect (homogeneous) materials, we do not understand how paths of moving cracks are determined. We experimentally study strongly perturbed cracks that propagate between 10% and 95% of their limiting velocity within a brittle material. These cracks are deflected by either interaction with sparsely implanted defects or via an intrinsic oscillatory instability in defect-free media. Dense high-speed measurements of the strain fields surrounding the crack tips reveal that crack paths are governed by the direction of maximal strain energy density, even when the near-tip singular fields are highly disrupted. This fundamentally important result may be utilized to either direct or guide running cracks.

Figure 1

Experimental system. (a) Brittle polyacrylamide gel sheets are loaded in tension, and dynamic cracks are initiated at sheet edges. Collimated light-emitting diode (LED) beams produced shadowgraph images of shallow grids embossed on gel surfaces. Grid deformations provide instantaneous displacement and strain fields near crack tips. (b) Top: a crack ($v\sim 0.25c_R$) within a pure gel sample with no embedded particles. Bottom: the measured ${ϵ}_{yy}$ strain component has the $r^{-1/2}$ scaling (inset) predicted by LEFM of ${ϵ}_{yy}$ along $y=0$. (c) Top: a crack ($v\sim 0.3c_R$) approaches embedded polyamide spheres (highlighted by diamonds). Bottom: nearby particles strongly affect the ${ϵ}_{yy}$ field (see also Supplemental Material [24]). Inset: ${ϵ}_{yy}$ along the path selected by the crack (blue) strongly deviates from LEFM predictions (red).

Figure 2

Crack tip path deviation and acceleration upon encountering particles. (a) Interactions with embedded particles cause crack tip opening displacements (CTODs) to become nonparabolic and force cracks to deviate significantly from straight paths. We define a crack tip as the leading point in the CTODs, as denoted by the red dot. (b) A time series of images separated by 0.125 ms, as a crack rapidly evolves while interacting with embedded particles (yellow diamonds). Its resulting path, denoted by the red line connecting successive crack tips (red dots), becomes tortuous. (c) Corresponding local tip velocities, normalized by $c_R=5.3\mathrm{m}/\mathrm{s}$, as a function of time. Such particles may even induce crack arrest (see Supplemental Material [24]).

Figure 3

The SED-based prediction agrees well with observed propagation directions. (a) A typical crack ($\approx 0.5c_R$) whose trajectory is governed by crack-particle interactions. The trajectory is indicated by the red line. Particles are outlined by yellow diamonds. (b) The SED (in $\mathrm{J}/{\mathrm{m}}^3$) field ahead of the crack within the dashed red box in (a). Magenta diamonds denote the maximal value for the SED for each value of $x$ in the reference frame. Current and future tip locations are indicated by red dots. Linear fits (magenta arrow) to the local SED maxima ahead of the tip agree well with the future crack tip location. Yellow diamond: a particle location.

Figure 4

The maximum SED criterion predicts the instantaneous direction of oscillatory cracks. (a) The trajectory of a spontaneously oscillating crack propagating at approximately $0.95c_R\sim 0.9c_s$ in a homogeneous gel at the onset of the oscillatory instability [16, 23]. The trajectory of successive crack tips leading to this instant is denoted in red. (b),(c) At left, snapshots of the crack tip in the laboratory frame that are indicated by yellow dots in (a). Center: the respective SED fields, in $\mathrm{J}/{\mathrm{m}}^3$, in the material area ahead of each crack tip. As in Fig. 3, diamonds denote the locations of the maximal SED value for each $x$. The line fitted to these maxima agrees well with the propagation direction denoted by the red line connecting the sequential tips (red dots). Predicted (measured) angles were 30° (31°) in (b) and $-40°$ ($-36°$) in (c). Right: ${ϵ}_{yy}$ (in blue) along the trajectory direction compared to the LEFM prediction (red). Singular fields are highly perturbed by large amplitude waves excited by crack oscillations.

Figure 5

All deflected crack directions are accurately predicted by the maximum SED condition. Filled circles: predicted angles for 21 crack-particle interactions with differing $v/c_R$ (colors) and particle ensembles ($2–6\mathrm{particles}/{\mathrm{mm}}^2$). Open symbols are due to the oscillatory instability. Open (closed) triangles are sequential data from Fig. 4 (Fig. 2). Horizontal (vertical) error bars are the ($1/3$ of the) point sizes. Dashed line represents perfect agreement.