Phonon Emission Induced Dynamic Fracture Phenomena

We use molecular dynamics simulations to calculate the phonon energy emitted during rapid crack propagation in brittle crystals. We show that this energy is different for different crack planes and propagation directions and that it is responsible for various phenomena at several length scales: energetically preferred crack systems and crack deflection at the atomic scale, reduced maximum crack speed with volume at the micrometer scale, and the inability of a crack to attain the theoretical limiting speed at the macroscale. We propose to include the contribution of this energy in the Freund equation of motion of a dynamically propagating crack.

Figure 1

(a) The normalized phonon emission ERR, $G_{\mathrm{ph}}/2\gamma$, as a function of the normalized crack speed, $V/C_R$, for six cleavage systems (first brackets denote the cleavage plane, the second the crack propagation direction) obtained by Eq. (4). The energy dissipated by phonon emission increases rapidly around $V/C_R>0.5$, which may describe surface phenomena such as in-plane anisotropy of the silicon (111) cleavage plane as shown in the (b) AFM image of the fracture surface of cleaved silicon specimen under tension, at speed larger than $0.5C_R$. The subnanoscale ridges indicate that the crack is propagating in the $[21\overline{3}]$ direction. The anisotropic phonon ERR also explains the deflection phenomenon in silicon under bending [11, 13] schematically shown in (c) crack deflected primarily from$(110)[1\overline{1}2]$ to $(111)[11\overline{2}]$ at $V>0.45C_R$. The dashed lines in (a) were calculated by Eq. (6).

Figure 2

$V/C_R$ vs $G_0/2\gamma$ for the six cleavage systems obtained by our MD calculations. The solid line is the Freund equation of motion [1] [Eq. (1)]. The extended Freund equation of motion containing the phonon ERR by the line heat source model, Eq. (5), is shown by the red dot-dashed line and by our MD calculations, Eq. (6), by dashed lines, with two limiting values of the material parameter, $\xi =2.5$ and 4.5. The inset shows that $G_0=G_K+2\gamma$, and, presumably, no other energy dissipation mechanisms exist.

Figure 3

The maximum normalized crack speed, $V_{\max }/C_R$, and the normalized phonon ERR, $G_{\mathrm{ph}}/G_0$, as a function of the specimen height $H$ obtained by our MD calculations (full squares) for the $(110)[1\overline{1}0]$ cleavage system and by experiments (full circle) [13]. The dashed lines show the trend. The radius of the hot zone and the maximum temperature as a function of crack speed, obtained by the temperature profile equation and the material parameters given in 25, are shown in the inset.