Cracks and Crazes: On Calculating the Macroscopic Fracture Energy of Glassy Polymers from Molecular Simulations

We combine molecular dynamics simulations of deformation at the submicron scale with a simple continuum fracture mechanics model for the onset of crack propagation to calculate the macroscopic fracture energy of amorphous glassy polymers. Key ingredients in this multiscale approach are the elastic properties of polymer crazes and the stress at which craze fibrils fail through chain pullout or scission. Our results are in quantitative agreement with dimensionless ratios that describe experimental polymers and their variation with temperature, polymer length, and polymer rigidity.

Figure 1

Some length scales involved in the fracture of glassy polymers. A $\sim 1\mathrm{m}\mathrm{m}$ long crack penetrates the material. A craze of typical width $d=10–50\mu \mathrm{m}$ and length $l\sim 10d$ forms in the “process zone” in front of the crack tip (top left). Craze formation (A), deformation (B), and failure (C) are studied with molecular dynamics simulations of $\sim 100\mathrm{n}\mathrm{m}$-sized volume elements. A: craze fibrils emerge from the dense polymer glass (top) within a narrow “active zone”; B: fully developed craze structure; C: failure of the craze immediately in front of the crack tip. Individual beads of the polymer chains are shown as small ellipsoids. Chains carrying the largest tension are colored blue and broken bonds in C are colored red. The width of each image is 128 bead diameters. The fibril spacing $D_0$ is $\sim 10\mathrm{\text{bead}}$ diameters for this figure, but depends on temperature, chain rigidity and other parameters.

Figure 2

(a) Normalized stress ${\sigma }_3/S$ during craze growth at $T=0.3u_0/k_B$ for flexible (solid lines) and semiflexible (dotted lines) chains of the indicated length in systems composed of 32 768 beads. Here we take $N_e=32$ for the semiflexible and $N_e=64$ for the flexible chains. The value of $S_{\max }/S$ is obtained from the maximum height of the curves. Panel (b) summarizes $S_{\max }/S$ for the flexible ($\circ$) as well as semiflexible ($▵$) chains at $T=0.3u_0/k_B$ (open symbols) and $T=0.1u_0/k_B$ (solid symbols) as a function of chain length.