Strain hardening of polymer glasses: Entanglements, energetics, and plasticity

Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While stress-strain curves for a wide range of temperature can be fit to the functional form predicted by entropic network models, many other results are fundamentally inconsistent with the physical picture underlying these models. Stresses are too large to be entropic and have the wrong trend with temperature. The most dramatic hardening at large strains reflects increases in energy as chains are pulled taut between entanglements rather than a change in entropy. A weak entropic stress is only observed in shape recovery of deformed samples when heated above the glass transition. While short chains do not form an entangled network, they exhibit partial shape recovery, orientation, and strain hardening. Stresses for all chain lengths collapse when plotted against a microscopic measure of chain stretching rather than the macroscopic stretch. The thermal contribution to the stress is directly proportional to the rate of plasticity as measured by breaking and reforming of interchain bonds. These observations suggest that the correct microscopic theory of strain hardening should be based on glassy state physics rather than rubber elasticity.

Figure 1

(Color online) Compressive stress $-{\sigma }_z$ as a function of ${\lambda }_z$ for uniaxial compression at $k_{B}T∕u_0=$ 0, 0.1, 0.2, and 0.3 from top to bottom. The chains had $k_{\mathrm{bend}}=1.5u_0$ and $\dot{ϵ}=-{10}^{-4}∕{\tau }_{\mathrm{LJ}}$. Solid lines show a fit to the eight-chain model [Eq. (6)] at $k_{B}T∕u_0=0.2$ and the extrapolation of this fit to other temperatures using the modified model [Eq. (7)]. The initial elastic rise and yield for $1>{\lambda }_z>0.8$ are sensitive to aging and are not fit by the eight-chain model.

Figure 2

(Color online) (a) Compressive stress $-{\sigma }_z$ and (b) perpendicular stress $-{\sigma }_y$ as a function of ${\lambda }_z$ for plane strain compression at $k_{B}T∕u_0=$ 0, 0.1, 0.2, and 0.3 from top to bottom in each panel. The chains had $k_{\mathrm{bend}}=1.5u_0$ and $\dot{ϵ}=-{10}^{-4}∕{\tau }_{\mathrm{LJ}}$. Solid lines show a fit to the eight-chain model [Eq. (6)] at $k_{B}T∕u_0=0.1$ and the extrapolation of this fit to other temperatures using the modified model [Eq. (7)]. The dashed line in (b) shows the variation in ${\sigma }_y$ predicted by Eq. (6) and measured values of ${\sigma }_z$.

Figure 3

(Color online) Total stress (solid lines) and thermal (dashed lines) and potential energy (dot-dashed lines) contributions for (a) uniaxial compression at $k_{B}T∕u_0=0.2$ and (b) plane strain compression at $k_{B}T∕u_0=0$. The systems had $k_{\mathrm{bend}}=2.0$ $(N_e=22)$, $N=350$ and (a) $\dot{ϵ}=-{10}^{-5}∕{\tau }_{\mathrm{LJ}}$ or (b) $\dot{ϵ}=-{10}^{-4}∕{\tau }_{\mathrm{LJ}}$. Dotted lines show best fits of ${\sigma }_z$ to Eq. (7) with (a) $N_e=15.5$ and (b) $N_e=22$. Straight lines are fits to ${\sigma }_z^Q$. Both ${\sigma }_z$ and $g$ are negative under compression.

Figure 4

(Color online) Energetic components of stress ${\sigma }_z^U$ for plane strain compression at $T=0$ with strain rate $\dot{ϵ}=-{10}^{-4}∕{\tau }_{\mathrm{LJ}}$ for $N_e=22$ (solid line), $N_e=26$ (dot-dashed line), $N_e=39$ (dashed line), and $N_e=71$ (dotted line).

Figure 5

(Color online) (a) $R\left(n\right)∕R_{\mathrm{aff}}\left(n\right)$ for the same systems and conditions as Fig. 4 with $n$ scaled by $N_e$. Solid lines indicate $N_e=22$ and dashed lines indicate $N_e=39$. Curves are for $\mid g\mid =$2.5, 5, 7.5, and 10 from top to bottom. (b) ${\sigma }_z^U$ plotted against $R∕R_{\mathrm{aff}}$ evaluated at $n=N_e$. This corresponds to evaluating ${\sigma }_z^U$ along a vertical slice of (a). Solid, dot-dashed, dashed, and dotted lines indicate data for $N_e=$22, 26, 39, and 71, respectively [56].

Figure 6

(Color online) Time dependent relaxation at $T=0.4u_0∕k_B$ of (a) true strain and (b) chain orientation parameter $P_2$ for entangled $N=350$ (solid lines) and unentangled $N=16$ (dashed lines) systems. Systems were prepared by loading uniaxially to ${ϵ}_z=1.5$ at $T=0.2u_0∕k_B$, unloading to zero stress, and then heating to $0.4u_0∕k_B$ over $100{\tau }_{\mathrm{LJ}}$.

Figure 7

(Color online) Dependence of (a) stress and (b), (c) microscopic chain orientation on $\mid g\mid$ for flexible chains under plane strain compression at $T=0.2u_0∕k_B$ with $\dot{ϵ}=-{10}^{-5}∕{\tau }_{\mathrm{LJ}}$. (d) Ratio of changes in chain volume to macroscopic volume. The chains have lengths $N=500$ (solid lines), $N=107$ (dotted lines), $N=36$ (dash-dotted lines), $N=18$ (dashed lines), and $N=12$ (dash-dot-dotted lines). The lower dashed line in (c) shows the macroscopic stretch ${\lambda }_z$.

Figure 8

(Color online) Compressive stress as a function of microscopic orientation function $g_{\mathrm{eff}}\equiv ({\lambda }_z^c{)}^2-({\lambda }_x^c{)}^2$ at $T=0.2u_0{k}_B$ with $\dot{ϵ}=-{10}^{-5}∕{\tau }_{\mathrm{LJ}}$. (a) Plane strain compression of flexible chains with lengths $N=500$ (solid line), 107 (dotted line), 36 (dash-dotted line), 18 (dashed line), or 12 (dot-dash-dot line). (b) Uniaxial compression of semiflexible chains $(N_e=39)$ with $N=350$ $(\star )$, 175 $(-)$, 70 $(◻)$, 40 $(\cdots )$, 25 $(▵)$, 16 (- - -), and 10 $(◇)$.

Figure 9

(Color online) Stress difference ${\sigma }_z-{\sigma }_y$ as a function of (a) the macroscopic $g$ and (b) the microscopic orientation function $g_{\mathrm{eff}}\equiv ({\lambda }_z^c{)}^2-({\lambda }_y^c{)}^2$ at $T=0.2u_0{k}_B$ with $\dot{ϵ}=-{10}^{-5}∕{\tau }_{\mathrm{LJ}}$. Chains have length $N=500$ (solid line), 107 (dotted line), 36 (dash-dotted line), 18 (dashed line), or 12 (dot-dash-dot line).

Figure 10

(Color online) (a) Stress as a function of $g$ during uniaxial compression of $k_{\mathrm{bend}}=2.0u_0$ chains with $N=350$ (dotted), 88 (solid), 44 (dash-dot-dotted), 22 (dash-dotted), and 11 (dashed) at $T=0.2u_0∕k_B$ and $\dot{ϵ}=-{10}^{-5}∕{\tau }_{\mathrm{LJ}}$. (b) Stress replotted against $g_{\mathrm{eff}}$. (c) Energetic stress ${\sigma }_z^U$ plotted against $g_{\mathrm{eff}}$.

Figure 11

(Color online) Rate of plastic deformation $R_P$ (solid lines) and normalized dissipative stress (dashed lines) for uniaxial compression of $k_{\mathrm{bend}}=0.75u_0$ chains with $N=350$ (upper curves) and $N=4$ (lower curves). Here $T=0$, $\dot{ϵ}=-{10}^{-4}∕{\tau }_{\mathrm{LJ}}$, and ${\sigma }^{*}=1.02u_0∕a^3$ for $N=350$ and $1.1u_0∕a^3$ for $N=4$.

Figure 12

(Color online) Rate of plasticity (solid lines) and dissipative stress (dashed lines) for plane strain compression of flexible $(k_{\mathrm{bend}}=0)$ $N=500$ chains (lower curves) and semiflexible $(k_{\mathrm{bend}}=1.5u_0)$ $N=350$ chains (upper curves). Here $T=0$, $\dot{ϵ}=-{10}^{-4}∕{\tau }_{\mathrm{LJ}}$, and ${\sigma }^{*}=1.05u_0∕a^3$ for $k_{\mathrm{bend}}=1.5u_0$ and $0.98u_0∕a^3$ for $k_{\mathrm{bend}}=0u_0$.