Interplay between Process Zone and Material Heterogeneities for Dynamic Cracks

Using an elastodynamic boundary integral formulation coupled with a cohesive model, we study the problem of a dynamic rupture front propagating along an heterogeneous plane. We show that small-scale heterogeneities facilitate the supershear transition of a mode-II crack. The elastic pulses radiated during front accelerations explain how microscopic variations of fracture toughness change the macroscopic rupture dynamics. Perturbations of dynamic fronts are then systematically studied with different microstructures and loading conditions. The process zone size is the intrinsic length scale controlling heterogeneous dynamic rupture. The ratio of this length scale to asperity size is proposed as an indicator to transition from quasihomogeneous to heterogeneous fracture. Moreover, we discuss how the shortening of the process zone size with increasing crack speed brings the front to interact with smaller details of the microstructure. This study shines new light on recent experiments reporting perturbations of dynamic rupture fronts, which intensify with crack propagation speed.

Figure 1

Geometry of the in-plane heterogeneous fracture problem. A crack of length $a$ is inserted along an interface with constant macroscopic toughness $G_c^H$ at rest under a uniform shear loading ${\tau }_0$. The interface is made of a homogeneous portion $L_{\mathrm{hom}}$ and a region with a heterogeneous toughness $L_{\mathrm{het}}$ in form of alternately weaker (yellow)/tougher (orange) stripes of constant width $w$.

Figure 2

Space-time diagrams of two macroscopically equivalent dynamic fracture events where the normalized slip velocity ${\dot{\delta }}_x/c_s$ is shown using the same color scale while the crack tip position at $a_1=5L_c$ and $a_2=20L_c$ is highlighted with cyan stars. In (a) the crack grows on a perfectly homogeneous interface, while in (b) the rupture front interacts with smaller-scale heterogeneities.

Figure 3

Effect of heterogeneities on the supershear transition. Color curves trace the observed boundary between sub-Rayleigh and supershear regimes for different loading conditions (seismic ratio) and toughness distribution. The dashed arrow draws the trajectory of ruptures of Fig. 2 where the crack is initially in the sub-Rayleigh regime (cyan star at $a_1=5L_c$) and grows toward a size (cyan star at $a_2=20L_c$) where it either crosses the boundary toward supershear regime [as in the interface of Fig. 2 with $w=0.6L_c$], or not [as in the homogeneous interface of Fig. 2]. The dark blue star shows the maximum seismic ratio allowing supershear crack in homogeneous plane strains interface [25, 34].

Figure 4

The process zone size is the length scale controlling crack front interaction with heterogeneities. For $v=0.5c_s$ and a toughness ratio of 3.5, colors in plots (b)–(d) divide broken surface (sky blue), cohesive zone (blue), and intact interface which is either dark blue (average properties), yellow (weaker properties), or orange (tougher properties). (a) Normalized increase of slip velocity as a function of interface heterogeneity, namely, asperity size and toughness ratio for $v=0.5c_s$. (b) Asperities are much smaller than $l_{\text{coh}}$ leading to quasihomogeneous dynamics. (c) Collective interaction between depinning events when $w$ is in the range of $l_{\text{coh}}$ leading to a significant impact on rupture dynamics. (d) When the asperities are much larger than $l_{\text{coh}}$, the material is macroscopically heterogeneous.

Figure 5

Faster crack fronts interact with smaller heterogeneities. The main plot details how the process zone size contracts as the crack accelerates toward $c_R$. Cyan dots show simulation data compared with the theoretical prediction in yellow (details are provided in [23]). The inset shows the increase of slip velocity as function of asperity size for $G_c^{\text{strong}}/G_c^{\mathrm{weak}}=3.5$ and different crack speeds using $l_{\text{coh}}(v)$ for normalization to collapse data.