Dynamic Buckling and Fragmentation in Brittle Rods

We present experiments on the dynamic buckling and fragmentation of slender rods axially impacted by a projectile. By combining the results of Saint-Venant and elastic beam theory, we derive a preferred wavelength $\lambda$ for the buckling instability, and experimentally verify the resulting scaling law for a range of materials including teflon, dry pasta, glass, and steel. For brittle materials, buckling leads to the fragmentation of the rod. Measured fragment length distributions show two peaks near $\lambda /2$ and $\lambda /4$. The nonmonotonic nature of the distributions reflect the influence of the deterministic buckling process on the more random fragmentation processes.

Figure 1

The dynamic buckling and fragmentation of dry pasta ($d=1.9\mathrm{m}\mathrm{m}$, $L=24\mathrm{c}\mathrm{m}$) just after the impact of an aluminum projectile at velocity $U_0=3.5\mathrm{m}/\mathrm{s}$ (interval between pictures: $236\mu \mathrm{s}$).

Figure 2

The measured dynamic buckling wavelength for various materials versus (a) smallest dimension and (b) impact speed. The line in (a) corresponds to the scaling $\lambda \sim d$ and in (b) corresponds to the scaling $\lambda \sim U_0^{-1/2}$. Note that the legend applies to both plots.

Figure 3

Buckled shape of a cylindrical PTFE teflon rod ($d=2.0\mathrm{m}\mathrm{m}$, $L=14\mathrm{c}\mathrm{m}$), shown $\sim 250\mu \mathrm{s}$ after impact by a steel projectile with speeds (a) $0.7\mathrm{m}/\mathrm{s}$, (b) $4.6\mathrm{m}/\mathrm{s}$, (c) $11.2\mathrm{m}/\mathrm{s}$, and (d) $26.0\mathrm{m}/\mathrm{s}$.

Figure 4

The nondimensional buckling wavelength $\lambda /d$ versus the normalized impact speed. The $\gamma$ factor accounts for the reflection of the initial impulse (see text). The lines correspond to the scaling law in Eq. (5), plotted for $\gamma =1$ and 2.

Figure 5

Size distribution normalized by the experimental buckling wavelength $\lambda$ for impact fragmentation of pasta at $U_0=3.5\mathrm{m}/\mathrm{s}$ ($c\simeq 1400\mathrm{m}/\mathrm{s}$): $d=1.1\mathrm{m}\mathrm{m}$, $\lambda =48\mathrm{m}\mathrm{m}$ (solid symbols) and $d=1.9\mathrm{m}\mathrm{m}$, $\lambda =70\mathrm{m}\mathrm{m}$ (open symbols), vertically offset for clarity. The smooth lines are meant to guide the eye. The dashed lines indicate the fragment sizes $\lambda /4$ and $\lambda /2$. The inset shows a breaking event (captured $76\mu \mathrm{s}$ after impact) contributing to both the $\lambda /2$ and $\lambda /4$ peaks.