From macroscopic yield criteria to atomic stresses in polymer glasses

The relationship between macroscopic shear yield criteria and local stress distributions in deformed polymer glasses is investigated via molecular dynamics simulations on different scales of coarse-graining. Macroscopic shear stresses at the yield point obey a pressure-modified von Mises (pmvM) criterion for many different loading conditions and strain rates. Average local stresses in small volume elements obey the same yield criterion for volumes containing approx. 100 atoms or more. Qualitatively different behavior is observed on smaller scales: the average octahedral atomic shear stress has a simple linear relationship to hydrostatic pressure regardless of macroscopic stress state and failure mode. Local plastic events are identified through a threshold in the mean-squared nonaffine displacement and compared to the local stress state. We find that the pmvM criterion only predicts local yield events when stress and displacements are averaged over at least 100 atoms. By contrast, macroscopic shear yield criteria appear to lose their ability to predict plastic activity on the atomic scale.

Figure 1

(Color online) (a) Stress-strain curves of the polymer glass for loading conditions #1 (◻), #2 (○), #15 (△), and #5 (◇) as indicated in Table . (b) Octahedral stress vs pressure at yield for all loading conditions indicated in Table as well as additional combinations with strain rates of order ${10}^{-5}/{\tau }_{\text{LJ}}$ (●). Also shown are yield stresses from similar loading conditions with higher strain rates on order ${10}^{-4}/{\tau }_{\text{LJ}}$ (○). Straight lines show fits to Eq. (1) with constants ${\tau }_0=0.41ϵ/{\sigma }^3$, $\alpha =0.086$ (red, lower curve) and ${\tau }_0=0.54ϵ/{\sigma }^3$, $\alpha =0.092$ (blue, upper curve).

Figure 2

(Color online) Distribution of Voronoi volumes at the yield point for loading conditions #7 (◼, pure cavitational failure), #4 (○), #8 (●), #9 (△), and #11 (▲, pure shear failure) as indicated in Table .

Figure 3

(Color online) Distributions of pressure $P\left(p\right)$ and octahedral stress $P\left({\tau }_{\text{oct}}\right)$ at yield at the atomic level (a,b) as well as in bins containing approximately 10 (c,d) and 78 atoms (e,f), respectively. Symbols correspond to loading states as indicated in Fig. 2.

Figure 4

(Color online) Average octahedral stress $⟨{\tau }_{\text{oct}}⟩$ at the macroscopic yield point for the atomic stress (▲) and averaged within bins containing 10 (△), 78 (●), 625 (○), and 5000 (◼) atoms and the macroscopic yield stress (◻). Solid lines are fits to Eq. (1) and the coefficients ${\tau }_0^{\text{av}}$ and ${\alpha }_{\text{av}}$ are reported in Table .

Figure 5

(Color online) Scatter plots of octahedral stresses at yield vs pressure. The bins contained approximately (a) 5000, (b) 625, (c) 78, and (d) 10 atoms, and (e) shows the atomic stress. Solid lines are fits to Eq. (1) and the coefficients ${\tau }_0$ and $\alpha$ are reported in Table .

Figure 6

(Color online) Parametric plot of $⟨{\tau }_{\text{oct}}⟩$ against pressure for strains between zero and the yield strain. The bins contained approximately (a) 5000, (b) 625, (c) 78, and (d) 10 atoms, and (e) shows the average atomic stress. Solid lines are the fit lines obtained in Fig. 4.

Figure 7

(Color online) Average atomic nonaffine displacements $⟨{\left(\Delta r^{\text{na}}\right)}^2⟩$ as a function of effective strain for several different deformations. The yield stresses for these runs are ${\tau }_{\text{oct}}^y=0.11$ (◼), 0.24 (○), 0.31 (●), 0.37 (△), and 0.41 (▲). The average was taken over all atoms in the simulation box.

Figure 8

(Color online) (a) $⟨{\left(\Delta r^{\text{na}}\right)}^2⟩$ at yield as a function of pressure at yield for different deformations. (b) Strain for $⟨{\left(\Delta r^{\text{na}}\right)}^2⟩$ to reach a threshold on 0.1 ${\sigma }^2$ vs strain to reach macroscopic yield.

Figure 9

(Color online) Yield rates and fraction of bins that have undergone a yield event for (a,c) caviation and (b,d) shear yielding. The bins contained 1 (◆), 10 (◇), 78 (●), 625 (○), 5000 (◼), and 40000 atoms (◻).

Figure 10

(Color online) Scatter plot of octahedral shear stress vs pressure in bins that have reached the yield threshold ${⟨{\left(\Delta r^{\text{na}}\right)}^2⟩}_{\text{bin}}=0.1{\sigma }^2$ (red, solid symbols). The bins contained approximately (a) 40000, (b) 5000, (c) 625, (d) 78, and (e) 10 atoms, and (f) shows the atomic stress. Also shown (blue, open symbols) in (a–c) are the maximum octahedral stresses and corresponding pressures. Solid lines are fits to Eq. (1) and the coefficients ${\tau }_0^{\text{na}}$ and ${\alpha }^{\text{na}}$ are reported in Table .

Figure 11

(Color online) Scatter plot of octahedral shear stress vs pressure in bins at random times (red, solid symbols). The bins contained approx. (a) 40000 (b) 5000, (c) 625, (d) 78, and (e) 10 atoms, and (f) shows the atomic stress. Also shown (blue, open symbols) in (a)–(c) are the maximum octahedral stresses and corresponding pressures. Solid lines are fits to Eq. (1) and the coefficients ${\tau }_0^{\text{rand}}$ and ${\alpha }^{\text{rand}}$ are reported in Table .