Thermodynamic theory of dislocation-enabled plasticity

The thermodynamic theory of dislocation-enabled plasticity is based on two unconventional hypotheses. The first of these is that a system of dislocations, driven by external forces and irreversibly exchanging heat with its environment, must be characterized by a thermodynamically defined effective temperature that is not the same as the ordinary temperature. The second hypothesis is that the overwhelmingly dominant mechanism controlling plastic deformation is thermally activated depinning of entangled pairs of dislocations. This paper consists of a systematic reformulation of this theory followed by examples of its use in analyses of experimentally observed phenomena including strain hardening, grain-size (Hall-Petch) effects, yielding transitions, and adiabatic shear banding.

Figure 1

Log-log plot of the steady-state stresses as functions of strain rate for copper at temperatures $T=1173\text{K}$ (lower curve) and $300\text{K}$ (upper curve). Parameters are $T_P=40800K$, ${\tau }_0={10}^{-12}{\text{s}}^{-1}$, and ${\chi }_0/e_D\cong 0.25$. The values of ${\mu }_T$ are $1343\text{MPa}$ and $1600\text{MPa}$ for the higher and lower temperatures, respectively. The four data points are taken from the stress-strain curves shown in Figs. 2 and 3.

Figure 2

Experimental data and theoretical stress-strain curves for copper at $T=1173\text{K}$, for strain rates $0.066{\text{s}}^{-1}$ (lower blue curve) and $980{\text{s}}^{-1}$ (upper red curve). The parameters used for computing both theoretical curves are ${\kappa }_1=3.1$, ${\kappa }_2=120$, ${\chi }_0=0.25$, and ${\mu }_T=1343\text{MPa}$. The initial values of $\tilde{\rho }$ are ${10}^{-6}$ for both cases; but the initial value of $\tilde{\chi }$ is 0.22 for the lower curve and 0.18 for the upper one.

Figure 3

Experimental data and theoretical stress-strain curves for copper at $T=298\text{K}$, for strain rates $0.002{\text{s}}^{-1}$ (lower blue curve) and $2000{\text{s}}^{-1}$ (upper red curve). The parameters used for computing both theoretical curves are ${\kappa }_1=3.1$, ${\kappa }_2=11.2$, ${\chi }_0=0.25$, and ${\mu }_T=1600\text{MPa}$. The initial values of $\tilde{\rho }$ and $\tilde{\chi }$ are ${10}^{-6}$ and 0.18, respectively.

Figure 4

Experimental data and theoretical stress-strain curves for aluminum at $T=573\text{K}$, for strain rates $0.25{\text{s}}^{-1}$ (lower blue curve), $2.5{\text{s}}^{-1}$ (middle black curve), and $25{\text{s}}^{-1}$ (upper red curve). The parameters used for computing all theoretical curves are ${\kappa }_1=0.97$, ${\kappa }_2=12$, ${\chi }_0=0.249$, $T_P=24000\text{K}$, and ${\mu }_T/\mu =0.040$. The initial values of $\tilde{\rho }$ and $\tilde{\chi }$ are 0.0035 and 0.224, respectively, for all cases.

Figure 5

Magnified stress-strain curves for aluminum showing the yielding transitions seen in Fig. 4 at $\epsilon \cong 0$.

Figure 6

Rate-hardening anomaly. The four curves, from bottom to top, show stresses as functions of strain rate for copper, for four different strains, $\epsilon =0.05,0.10,0.15,\mathrm{and}0.20$. The dashed curve at the top is the theoretical steady-state prediction.

Figure 7

Stress-strain curves for adiabatic shear banding in steel. The experimental data are from Fig. 8 of Marchand and Duffy [22]. The upper curve (red) is for strain rate $\dot{\epsilon }=3300{\text{s}}^{-1}$, the lower (blue) is for ${10}^{-4}{\text{s}}^{-1}$. The system parameters for both curves are: $T_P=6\times {10}^5\text{K},T=300\text{K},\mu =5\times {10}^4\text{MPa},{\mu }_T=1200\text{MPa},{\tilde{\chi }}_0=0.25,{\kappa }_1=4,K_0=2\times {10}^{-6},\mathrm{and}{K}_1=K_2=0$. For the upper curve, ${\kappa }_2=16$, and the initial values of the parameters are ${\tilde{\rho }}_i=0.008,{\tilde{\chi }}_i=0.19$. The inscribed line defect is given by Eq. (7.3) with $\delta =0.017,y_0=0.05$. For the lower curve, ${\kappa }_2=5,{\tilde{\rho }}_i=0.007,{\tilde{\chi }}_i=0.22$.

Figure 8

Relative plastic strain rates $q(\epsilon ,y)/Q$ at strains $\epsilon =0.45,0.47,0.49,$ and 0.497, for the top stress-strain curve shown in Fig. 7. For increasing $\epsilon$, these shear flows are increasingly concentrated in a narrowing band centered at $y=0$.

Figure 9

Temperature in degrees K as a function of strain $\epsilon$ at the center of the band, $y=0$, for the top stress-strain curve shown in Fig. 7.