Evaluation of the thermal phonon emission in dynamic fracture of brittle crystals

We describe in detail how to quantify, computationally, the contribution of the anisotropic and velocity-dependent thermal phonon emission energy release rate during dynamic crack propagation in brittle crystals. The calculations were performed using a procedure based on combined continuum elastodynamics Freund equation of motion and molecular dynamics computer calculations. We used a precracked strip-like specimen subjected to prescribed displacement on the boundaries, commonly used in atomistic calculations of crack dynamics. Various cleavage planes and directions of silicon-like crystal for a wide range of crack speeds were investigated. It is shown that in addition to being strongly dependant on crack speed and atomistic arrangement, the relative phonon emission energy release rate is size dependent, hence governing the size-dependent terminal crack speed. This speed, however, is not influenced by the specimen's initial temperature, ranging between 10K and 300K.

Figure 1

Schematic presentation of the normalized crack speed vs the normalized strain ERR by the Freund equation of motion (red/dark gray line) and MD calculations (blue/medium gray line); the difference is attributed to the energy dissipated by phonon emission ERR, ${\mathcal{G}}_{\mathrm{ph}}$. Star denotes normalization by $C_{\mathrm{R}}$, bar by $2{\gamma }_s$.

Figure 2

Flowchart of the dynamic fracture computer “experiments” performed in this study.

Figure 3

A schematic presentation of the strip-like specimen dimensions and boundary conditions.

Figure 4

Visualization of the initial atomic configuration.

Figure 5

Visualization of the atomic arrangements near the crack tips for seven cleavage systems considered in this study.

Figure 6

The deformation of the strip-like specimen of the (110)$\left[1\overline{1}0\right]$ cleavage system: applying an additional strain increment beyond a critical value caused the specimen to split into two relaxed parts. The lattice trapping strain for this case is 6.23$\%$.

Figure 7

Visualization of the atomic configuration after crack initiation stage. The colors/shades of gray represent instantaneous local temperature.

Figure 8

Typical surface instabilities observed in a highly strained specimen.

Figure 9

Crack propagation on the (110)[001] cleavage system of silicon is unstable; the crack tends to deflect to the (111)$\left[11\overline{2}\right]$ cleavage system.

Figure 10

Crack tip position vs time for different strain ERR values of a crack propagating on the (110)$\left[1\overline{1}0\right]$ cleavage system of a single silicon crystal.

Figure 11

The instantaneous average kinetic energy per atom vs time in the (110)$\left[1\overline{1}0\right]$ system for increased ${\mathcal{G}}_0$.

Figure 12

The normalized crack tip speed, $V$/$C_{\mathrm{R}}$, vs the normalized ${\mathcal{G}}_0/2{\gamma }_s$ for the six cleavage systems of silicon. The full black line presents the Freund equation of motion.

Figure 13

The normalized crack tip speed, $V/C_{\mathrm{R}}$, with normalized kinetic ERR, ${\mathcal{G}}_{\mathrm{K}}^{\mathrm{MD}}/2{\gamma }_s$, for the six cleavage systems of silicon. The black line presents the Freund equation of motion.

Figure 14

Variations of the normalized kinetic ERR, ${\mathcal{G}}_{\mathrm{K}}^{\mathrm{MD}}/2{\gamma }_s$, with the normalized strain ERR, ${\mathcal{G}}_0/2{\gamma }_s$. This relationship suggests that no other energy dissipation mechanisms exist during crack propagation in silicon.

Figure 15

The normalized phonon ERR, ${\mathcal{G}}_{\mathrm{ph}}/2{\gamma }_s$, vs the normalized crack speed, $V/C_{\mathrm{R}}$, for the six different cleavage systems of silicon crystal.

Figure 16

The maximum normalized crack speed vs specimen height, $H$, corresponds to higher specimen volume and relative phonon energy with respect to strain ERR (black and blue/medium gray full squares) and that estimated for experimental results (black and blue/medium gray full circles).

Figure 17

Variations of the terminal crack speed with specimen initial temperature. The crack propagation takes place on the (110)$\left[1\overline{1}0\right]$ cleavage system.