Three-dimensional molecular dynamics simulations of void coalescence during dynamic fracture of ductile metals

Void coalescence and interaction in dynamic fracture of ductile metals have been investigated using three-dimensional strain-controlled multimillion atom molecular dynamics simulations of copper. The correlated growth of two voids during the coalescence process leading to fracture is investigated, both in terms of its onset and the ensuing dynamical interactions. Void interactions are quantified through the rate of reduction of the distance between the voids, through the correlated directional growth of the voids, and through correlated shape evolution of the voids. The critical intervoid ligament distance marking the onset of coalescence is shown to be approximately one void radius based on the quantification measurements used, independent of the initial separation distance between the voids and the strain rate of the expansion of the system. The interaction of the voids is not reflected in the volumetric asymptotic growth rate of the voids, as demonstrated here. Finally, the practice of using a single void and periodic boundary conditions to study coalescence is examined critically and shown to produce results markedly different than the coalescence of a pair of isolated voids.

Figure 1

A sketch of the simulation configuration. The simulation box includes $120\times 120\times 120$ fcc unit cells with periodic boundary conditions for a total of 6 912 000 atoms initially. From this two equal-size voids are created by removing approximately 3620 atoms for each void. The thin arrow shows the intervoid ligament distance (ILD), the shortest surface-to-surface distance between the voids, identified as $A$ and $B$. The initial ILD ranges between $1∕2$ and 5 times the initial void radius. The bold arrows denote triaxial, hydrostatic, expansion of the box. The strain-controlled expansion is applied with constant strain-rates of $\dot{\epsilon }={10}^8∕\sec$ and ${10}^9∕\sec$.

Figure 2

(Gray scale) A three-dimensional snapshot of the two voids with the prismatic dislocation loops forming from the voids. Only those atoms belonging to the void surfaces or to dislocation cores, stacking faults or other defects are shown (and a small number of extraneous atoms due to thermal fluctuations). The stacking fault ribbons are the broad plates between leading and trailing partial dislocations. The snapshot is from the simulation with the initial intervoid ligament distance ${\mathrm{ILD}}_0=1.81\text{diameters}$ and the strain rate $\dot{\epsilon }={10}^9∕\sec$. The strain at the snapshot is $\epsilon =2.93\%$.

Figure 3

Dislocation activity at six instants of time shown on one particular slice through the system in order to expose the plasticity near the void surfaces. These snapshots are from the simulation with $\dot{\epsilon }={10}^9∕\sec$ and again only those atoms in dislocation cores, stacking faults, void surfaces, or other defects are shown (see text for more details and Fig. 2 for a full three-dimensional figure). The dashed loop in panel (c) is drawn around a slice of a prismatic dislocation loop. The plane shown passes through the centers of both voids with normal [0.145, 0.145, $-0.979$]. The snapshots show the initial plasticity (a),(b), interacting plastic zones (c),(d), and the final coalescence (e),(f). The frames correspond to the following values of strain $\epsilon$: 1.72%, 2.42%, 3.47%, 3.89%, 4.52%, and 5.21%, respectively.

Figure 4

(a) Mean stress ${\sigma }_m$ versus strain $\epsilon$ from the simulations with the initial intervoid ligament distance ${\mathrm{ILD}}_0=1.81$ and the strain rates $\dot{\epsilon }={10}^8∕\sec$ (dashed line) and ${10}^9∕\sec$ (solid line). (b) Linear mean void size ${\overline{f}}^{1∕3}$ (see the text for its definition) versus the strain from the same simulations as the data in (a). The void sizes are calculated until the coalescence of the voids. The inset shows the stress from (a) versus linear mean void size from (b). Note the strain scales are different.

Figure 5

Evolution of the ILD and the critical ILD. (a) Dynamical ILD, the distance between the surfaces of the voids along the line connecting the original center positions for various initial ${\mathrm{ILD}}_0=1.81$, 1.50, 1.20, 1.00, plotted versus strain. For ${\mathrm{ILD}}_0=1.81$ the thick solid line and thick medium dashed line denote $\dot{\epsilon }={10}^9∕\sec$ and ${10}^8∕\sec$, respectively. The thin solid lines show the hypothetical ILD for spherical voids with the same ${\mathrm{ILD}}_0$ impinging freely on each other (see text). The short dashed line shows the hypothetical ILD computed by duplicating a single void (in the same box size as the two void simulations) at fixed centers (here the duplication is to the position with ${\mathrm{ILD}}_0=1.81$). (b) The same as in (a), but now plotted versus the linear average void size ${\overline{f}}^{1∕3}$ and the distances are given in the units of the current average diameter $d$ of the voids, calculated from their volumes assuming that they are spherical (see text). The horizontal line is at $\mathrm{ILD}=0.5\text{diameters}$, the value we identify as the ${\mathrm{ILD}}_c$. [Panel (b) is after Ref. 27.]

Figure 6

Void center-of-mass displacement. The distance $d_{\mathit{cv}}$ from the original center of void to the instantaneous void center, projected onto the line connecting the original void centers, is plotted versus the average void size to the point of coalescence $(\mathrm{ILD}\simeq 0)$. The sign of the displacement $d_{\mathit{cv}}$ is positive for movement toward the other void. Solid and long dashed lines are for voids $A$ and $B$, respectively. Strain rates are $\dot{\epsilon }={10}^9∕\sec$ in (a) and $\dot{\epsilon }={10}^8∕\sec$ in (b). The thin solid line is for a single void in the same size of the box and with the same radius and strain rate projected to the same line. Here the distance $d_{\mathit{cv}}$ is given in the units of the original void diameter $d_0$. ${\mathrm{ILD}}_0=1.81$ in both (a) and (b).

Figure 7

(a) Quadrupole moments (2) of the surface of void $A$ from the start of the simulation until coalescence. The coordinate axes have been rotated using Eq. (3) so that the $z$ axis is aligned with the center of void $B$. The initial intervoid ligament distance is ${\mathrm{ILD}}_0=1.00$ and the strain rate is $\dot{\epsilon }={10}^9∕\sec$. (b) Quadrupole moments for void $B$, located at $[0.1778,0.0829,0.0387]L$ from void $A$. Now Eq. (3) has been used so that the positive $z$ axis is toward the void $A$. (c) Quadrupole moments (not rotated) for a single void in a simulation with same box size as in (a) and (b) and the strain rate $\dot{\epsilon }={10}^9∕\sec$.

Figure 8

(a) Growth of the void $A$ until coalescence presented by void fraction $f=V_{\mathit{\text{void}}}∕V$ versus strain for ${\mathrm{ILD}}_0=0.50$, 1.00, 1.20, 1.50, 1.81, and $4.62\text{diameters}$ as well as in a single-void case (in the same box size) at $\dot{\epsilon }={10}^9∕\sec$. The asymptotic behavior (before finite-size effects) is exponential growth with $\exp \left(200ϵ\right)$ as seen from the line drawn as a guide to the eye. (b) The void size $f$ presented in (a) has been scaled with the exponential and plotted versus linear void size $f^{1∕3}$. The circles point where the dynamical ILD’s cross the horizontal line $\mathrm{ILD}=0.5d$ in Fig. 5b.

Figure 9

Growth of a single void with varying initial box size. The initial radius of the void is kept constant $r_0=2.17\mathrm{nm}$, while the initial side length of the cube is varied as $L=6.50$, 9.75, 13.0, 17.3, 21.7, and $43.3\mathrm{nm}$. The void fraction $f$ has been divided by ${(r_0∕L)}^3$ in order to take into account different initial volumes $V_0=L^3$ of the box. The vertical axis has been divided also with the exponential of the strain in order to show the asymptotic behavior as in Fig. 8 for the largest box size. The strain rate is for all the simulations $\dot{\epsilon }={10}^9∕\sec$.