Branching instabilities in rapid fracture: Dynamics and geometry

We propose a theoretical model for branching instabilities in 2-dimensional fracture, offering predictions for when crack branching occurs, how multiple cracks develop, and what is the geometry of multiple branches. The model is based on equations of motion for crack tips which depend only on the time dependent stress intensity factors. The latter are obtained by invoking an approximate relation between static and dynamic stress intensity factors, together with an essentially exact calculation of the static ones. The results of this model are in qualitative agreement with a number of experiments in the literature.

Figure 1

Crack pattern in an araldite tensile sheet [2].

Figure 2

The normalized stress intensity factors $F_{\mathrm{I}}$ and $F_{\mathrm{II}}$ as a function of $\lambda$, where $2\pi \lambda$ is the angle between the branches (see inset). The ratio of branch length and the main crack is $\ell ∕L=0.5\times {10}^{-3}$.

Figure 3

The normalized stress intensity factors $F_{\mathrm{I}}$ and $F_{\mathrm{II}}$ for both tips as a function of $\lambda$, where $\pi \lambda$ is the angle between the branches (see inset). The ratio of branches length and the main crack is $\ell ∕L=0.5\times {10}^{-3}$.

Figure 4

The mode III normalized stress intensity factor for an asymmetric branched configuration. $\pi \lambda$ is the angle between the branches and the ratio of the branches length and the main crack length is $\ell ∕L=0.5\times {10}^{-3}$. The upper (lower) curve corresponds to the forward direction (inclined) branch.

Figure 5

The velocities of the branches as a function of time for $V_b=0.50$ and $V_b=0.55$. The resulting dynamics are such that the unperturbed branch is arrested, while the perturbed one returns after a short time to the original crack path. A representative resulting crack pattern is shown in the inset. See text for details.

Figure 6

The relative change in length $\Delta \ell ∕\ell$ as a function of $V_b$ for $\delta \ell =2\times {10}^{-2}\ell$ (lower curve) and $\delta \ell =0.5\times {10}^{-2}\ell$ (upper curve). Note that the continuation of the lines intersects the $x$ axis at the critical branching velocity $V_b=v_c\approx 0.475$, below which branching is energetically forbidden.

Figure 7

The crack pattern that was formed by three successive branching events. The branching velocity was set to $V_b=0.5$, i.e., slightly more than $v_c$. We introduced a positive small perturbation $\delta \ell =2\times {10}^{-2}\ell$ to the length of the lower branch. Upon reaching the branching velocity again we introduced a symmetric configuration with the same $\ell$ and $\lambda$ with no perturbation added by hand. The bar shows the scale of the process relative to the initial crack length $L$.

Figure 8

A crack pattern in a tensile $260\times 120\times 5\mathrm{mm}$ araldite plate [1]. The plate was notched symmetrically at the edge. Due to minute asymmetries in the production of the notches, only one of them propagates. A second branching event can be observed after some time. In this event almost symmetric branches emerge from the branching point and coexist until one outruns the other and then curves towards the symmetry line. Attempted branching events can be clearly observed before the successful branching event took place. All these features are in qualitative correspondence with our discussion.

Figure 9

Symmetric branching dynamics for different branching length $L$ with $V_b=0.5$. The plot shows the various crack patterns in rescaled coordinates. A power law $y\sim x^{\zeta }$ with $\zeta =0.8$ was added as a guide for the eye.

Figure 10

Two shadow photographs of a symmetric branching event in a $300\times 100\times 9\mathrm{mm}$ glass plate [1]. The left panel shows the early stages of the branching process where the branches are almost straight, while the right panel shows the developed curved branching configuration.

Figure 11

A typical resulting configuration for asymmetric branching. The side branch is almost immediately arrested, while the main branch temporarily deflected from its straight path. The net effect of such an event is a strong fluctuation in the velocity of the main branch. The bar shows the scale of the process relative to the initial crack length $L$.