Thermalized formulation of soft glassy rheology

We present a version of soft glassy rheology that includes thermalized strain degrees of freedom. It fully specifies systems' strain-history-dependent positions on their energy landscapes and therefore allows for quantitative analysis of their heterogeneous yielding dynamics and nonequilibrium deformation thermodynamics. As a demonstration of the method, we illustrate the very different characteristics of fully thermal and nearly athermal plasticity by comparing results for thermalized and nonthermalized plastic flow.

Figure 1

Dependence of nonlinear mechanics and thermodynamics on $\epsilon$ and $T$ for thermalized SGR. Panel (a): Elastic stress-strain curves $[{\sigma }^{\text{el}}\left(\epsilon \right)$ [Eq. (15)]]. Panels (b) and (c): average zone relaxation time $〈\tau 〉/{\tau }_0$ and its dispersion $\mathrm{\Delta }\tau /〈\tau 〉$. Panels (d)–(f): free energy $F\left(\epsilon \right)$, energy $E\left(\epsilon \right)$, and $\text{temperature}\times \text{entropy}$ $TS\left(\epsilon \right)$ [Eqs. (7) and (8)]. Blue, green, and red lines respectively indicate $T/T_g=1/2$, $3/4$, and $39/40$. Systems have $\alpha =\sqrt{8}$ and are deformed at constant strain rate $\dot{\epsilon }={\tau }_0^{-1}$. All energies are scaled by the maximum zone activation energy ${\alpha }^2{k}_B{T}_g$, and stresses are further scaled by ${\alpha }^2/v_0$.

Figure 2

Energy dissipation in the same systems depicted in Fig. 1. Dotted, dashed, and solid lines respectively show $\mathrm{\Delta }F\left(\epsilon \right)$, $W\left(\epsilon \right)$, and $Q\left(\epsilon \right)$. Note that the stress $\sigma \left(\epsilon \right)$ appearing in the definition of $W\left(\epsilon \right)={\int }_0^{\epsilon }\sigma \left({\epsilon }^{\prime }\right)d{\epsilon }^{\prime }$ is $\sigma \left(\epsilon \right)={\sigma }^{\text{el}}\left(\epsilon \right)+{\sigma }^{\text{entr}}\left(\epsilon \right)$, where the entropic term ${\sigma }^{\text{entr}}\left(\epsilon \right)=-T(\partial S/\partial \epsilon )$.

Figure 3

Strain-history-dependent position of the thermalized $T/T_g=1/2$ system on its energy landscape. Panel (a): $w(u,{\epsilon }^{\text{el}})$ for unstrained systems ($\epsilon =0$). Panel (b): $w(u,{\epsilon }^{\text{el}})$ at the yield strain ($\epsilon ={\epsilon }^y=0.0468$). Panel (c): $w(u,{\epsilon }^{\text{el}})$ at the post-yield stress minimum ($\epsilon ={\epsilon }^{pysm}=0.0730$). Because this $\alpha =\sqrt{8}$ system remains low on its energy landscape, $w(u,{\epsilon }^{\text{el}})$ is shown only for $5/8\le {\alpha }^{-2}u\le 1$; values for ${\alpha }^{-2}u<5/8$ are finite but remain small.

Figure 4

Strain-history-dependent probability distributions of zones' relaxation times $\left[P\right(\tau ;\epsilon )$ from Eq. (16): panel (a)] and elastic stresses $[P({\sigma }^{\text{el}};\epsilon )$ from Eq. (17): panel (b)]. Systems are the same and line colors indicate temperatures as in Figs. 1 and 2. Dotted, solid, and dashed curves respectively indicate data for $\epsilon =0$, $\epsilon ={\epsilon }^y$, and ${\epsilon }^{pysm}$ (Table 2).

Figure 5

Dependence of nonlinear mechanics and thermodynamics on $\epsilon$ and $T$ for nonthermalized SGR. Panel (a): elastic stress-strain curves $[{\sigma }^{\text{el}}\left(\epsilon \right)$ [Eq. (15)]]. Panels (b) and (c): average zone relaxation time $〈\tau 〉/{\tau }_0$ and its dispersion $\mathrm{\Delta }\tau /〈\tau 〉$. Panels (d)–(f): free energy $F\left(\epsilon \right)$, energy $E\left(\epsilon \right)$, and $\text{temperature}\times \text{entropy}$ $TS\left(\epsilon \right)$ [Eqs. (7) and (8)]. Blue, green, and red lines respectively indicate $T/T_g=1/2$, $3/4$, and $39/40$. Systems have $\alpha =\sqrt{8}$ and are deformed at constant strain rate $\dot{\epsilon }={\tau }_0^{-1}$. All energies are scaled by the maximum zone activation energy ${\alpha }^2{k}_B{T}_g$, and stresses are further scaled by ${\alpha }^2/v_0$. Note that replacing the $\delta \left({\epsilon }^{\text{el}}\right)$ proportionality in Eq. (18) with a $\theta (u,{\epsilon }^{\text{el}})$ proportionality eliminates the post-softening ($\epsilon >{\epsilon }^{pysm}$) stress increases shown in panel (a); instead, systems exhibit perfect-plastic flow at a constant stress ${\sigma }_{\text{flow}}\left(T\right)$, which in turn affects other measures of response like those shown in panels (b)–(f).

Figure 6

Rate-dependent mechanical response for thermalized flow. Panel (a): stress-strain curves for $T/T_g=3/4$, for strain rates ${10}^{-5}\le \dot{\epsilon }{\tau }_0\le {10}^3$; brighter green indicates higher rate. Panel (b): scaled yield stresses ${\sigma }^y\left(\dot{\epsilon }\right)$ (solid curves) and yield strains ${\epsilon }^y\left(\dot{\epsilon }\right)$ (dashed curves). Systems are the same and line colors indicate temperatures as in Figs. 1 and 2.