Persistent homology analysis of craze formation

We apply a persistent homology analysis to investigate the behavior of nanovoids during the crazing process of glassy polymers. We carry out a coarse-grained molecular dynamics simulation of the uniaxial deformation of an amorphous polymer and analyze the results with persistent homology. Persistent homology reveals the void coalescence during craze formation, and the results suggest that the yielding process is regarded as the percolation of nanovoids created by deformation.

Figure 1

Schematic representation of persistent homology. (a) When the radius of each bead $r$ satisfies $r^2<b_1$, all beads are disconnected. At $r^2=b_1$ shown in (b), a loop emerges. In (c), another loop emerges at $r^2=b_2$. The latter loop disappears at $r^2=b_2$ in (d) by meeting balls at the “death position” denoted by the cross mark. In a similar way, the former disappears at $r^2=d_1$ in (e). This persistent homology is represented by the persistent diagram shown in (f).

Figure 2

Stress-strain curve obtained by CG-MD simulation.

Figure 3

Snapshot of the MD simulation (a) before and (b) after yielding. (a) $\epsilon =0.007$ and (b) $\epsilon =0.20$).

Figure 4

$D=2$ persistent diagram obtained from simulation: (a) and (b) before and (c) after yielding. Dashed lines represent diagonal lines. (a) $\epsilon =0.007$, (b) $\epsilon =0.049$, and (c) $\epsilon =0.062$

Figure 5

Plot of the number of voids versus the strain. The dashed line represents the stress-strain curve.

Figure 6

Distribution of the voids along the $z$ axis at $\epsilon =0.0485,0.062$, and 0.076.

Figure 7

Plot of the positions that construct large voids listed in Table 1 in the $xz$ plane.

Figure 8

The forms of large voids listed in Table 1 before and after yielding. Light-gray transparent polygons represent large voids after yielding, and opaque dark-gray polygons (color online) represent voids before yielding. The different colors indicate different voids. (a) Cluster 1, (b) cluster 2, (c) cluster 3, (d) cluster 4, (e) cluster 5, and (f) cluster 6.

Figure 9

Plot of positions of beads that have a Voronoi volume larger than 1.0 in the $xz$ plane.