Nonlinear Focusing in Dynamic Crack Fronts and the Microbranching Transition

Cracks in brittle materials produce two types of generic surface structures: facets at low velocities and microbranches at higher ones. Here we observe a transition from faceting to microbranching in polyacrylamide gels that is characterized by nonlinear dynamic localization of crack fronts. To better understand this process we derive a first-principles nonlinear equation of motion for crack fronts in the context of scalar elasticity. Its solution shows that nonlinear focusing coupled to rate dependence of dissipation governs the transition to microbranching.

Figure 1

Experimental fracture surface (a) formed by tensile crack fronts (b) moving at $\sim 0.06c_R$ and displayed at 0.13 ms intervals. The surface features both facets (upper and lower right) and microbranches (upper and lower left). A pair of step lines nucleate and diverge forming a facet (black arrows) while a microbranching event ends with cusp formation (red arrows). The black scale bars are $200\mu \mathrm{m}$ long. (c) Geometry of the model. Two line loads $p$ moving with a speed $v$ are driving a planar crack front at a distance $l$. The crack front is perturbed around a straight configuration with amplitude $f(z,t)$. (d) Crack fronts, calculated via Eq. (1), move at velocity $v=0.05c$ and encounter obstacles with increasing toughness; see text following Eq. (3) for details. Fronts are plotted at the same time intervals as in (b) when identifying $c=c_R$.

Figure 2

Change of front dynamics from defocusing to focusing when increasing $a=(d\log {\mathrm{\Gamma }}_0/dv){|}_{v=0}$. Panels show curvature evolution at the obstacle midline for velocity-independent fracture energy (left) and for velocity-dependent fracture energy (right). Here $v=0.3$, $\delta A=1.2$ and ${\kappa }_{\mathrm{disk}}=2/d=80/Z$. The 1st order solution is drawn for comparison (dashed line). Insets depict crack front profiles drawn over the spatial fracture energy distribution.

Figure 3

The growth of step height along the step-line backbone $s$ taken from pair nucleation events. Data are shown from nine step lines taken from five events. For two of the step lines (red and blue points), we performed a high resolution measurement (at $0.5\mu \mathrm{m}/\mathrm{pix}$). (inset) The high resolution surface scan.

Figure 4

Transition from step-line divergence (left) to step-line convergence (right) when increasing the dissipation amplitude $D$. Here $v=0.1$, $a=4$, and $\xi =0.0016Z$. This is the same qualitative behavior as observed experimentally in Fig. 1. See movies in the Supplemental Material [39].