Atomistic study of diffusion-mediated plasticity and creep using phase field crystal methods

The nonequilibrium dynamics of diffusion-mediated plasticity and creep in materials subjected to constant load at high homologous temperatures is studied atomistically using phase field crystal (PFC) methods. Creep stress and grain size exponents obtained for nanopolycrystalline systems, $m\simeq 1.02$ and $p\simeq 1.98$, respectively, closely match those expected for idealized diffusional Nabarro-Herring creep. These exponents are observed in the presence of significant stress-assisted diffusive grain boundary migration, indicating that Nabarro-Herring creep and stress-assisted boundary migration contribute in the same manner to the macroscopic constitutive relation. When plastic response is dislocation-mediated, power-law stress exponents inferred from dislocation climb rates are found to increase monotonically from $m\simeq 3$, as expected for generic climb-mediated natural creep, to $m\simeq 5.8$ as the dislocation density ${\rho }_d$ is increased beyond typical experimental values. Stress exponents $m\gtrsim 3$ directly measured from simulations that include dislocation nucleation, climb, glide, and annihilation are attributed primarily to these large ${\rho }_d$ effects. Extrapolation to lower ${\rho }_d$ suggests that $m\simeq 4–4.5$ should be obtained from our PFC description at typical experimental ${\rho }_d$ values, which is consistent with expectations for power-law creep via mixed climb and glide. The anomalously large stress exponents observed in our atomistic simulations at large ${\rho }_d$ may nonetheless be relevant to systems in which comparable densities are obtained locally within heterogeneous defect domains such as dislocation cell walls or tangles.

Figure 1

Diffusional creep in bcc nanopolycrystals with high-angle grain boundaries. (a) Strain normalized system configurations for ${\sigma }_A=0.01$ and $d=50.8a$ are shown at ($0t, \epsilon =0.0$), ($62.5t, \epsilon =0.031$), ($125t, \epsilon =0.060$), and ($187.5t, \epsilon =0.090$). The system thickness is $L_x=1a$. Strain normalization is given by $y\to y(1+\epsilon )$ and $z\to z/(1+\epsilon )$. For analysis and visualization purposes, local peaks in $n\left(\vec{r}\right)$, which represent the most probable atomic positions, are taken to correspond to atomic sites. Sites with bcc coordination (at $t=0$) are shown in pale green, those with irregular coordination (grain boundary atoms) are shown in red $(t=0)$, green $(t=62.5)$, pale blue $(t=125)$, and gold $(t=187.5)$. (b) A representative set of creep curves for the configuration of (a). ${\epsilon }^{\mathrm{pl}}$ vs time is shown at various ${\sigma }_A$ (color legend). The dashed black lines are linear fits.

Figure 2

Morphology of diffusional creep in other bcc nanopolycrystal configurations with high-angle grain boundaries. Strain normalized system configurations are shown for (a) the symmetric six grain cell with rotations of ($0^{\circ }, \pm {30}^{\circ }$) and ${\sigma }_A=0.01, d=76.3a$. Red: ($0t, \epsilon =0.0$), green: ($300t, \epsilon =0.044$), pale blue: ($600t, \epsilon =0.090$), gold: ($900t, \epsilon =0.135$). (b) The symmetric six grain cell with rotations of (${45}^{\circ }, \pm {15}^{\circ }$) and ${\sigma }_A=0.01, d=76.3a$. Red: ($0t, \epsilon =0.0$), green: ($500t, \epsilon =0.108$), pale blue: ($1000t, \epsilon =0.234$), gold: ($1500t, \epsilon =0.384$). (c) The asymmetric 4 grain cell with rotations of ($\pm 12.5^{\circ }, \pm 37.5^{\circ }$) and ${\sigma }_A=0.01, d=101.7a$. Red: ($0t, \epsilon =0.0$), green: ($450t, \epsilon =0.055$), pale blue: ($900t, \epsilon =0.111$), gold: ($1350t, \epsilon =0.166$).

Figure 3

Creep exponents of the configuration shown in Fig. 1. (a) Compilation of steady-state creep rates vs ${\sigma }_A$ at various grain sizes $d$. The dashed lines are fits to Eq. (1), the solid lines are fits to Eq. (1) with ${\sigma }_A$ replaced by ${\sigma }_A-{\sigma }_{-}$. The resulting creep rate stress exponents $m$ indicated in the figure are those of the solid lines, while the corresponding values for the dashed lines are $m=1.11$, 1.10, 1.14, and 1.13, respectively. The inset shows the same data plotted vs ${\sigma }_A-{\sigma }_{-}$ with the same solid line fits. (b) Dependence of ${\dot{\epsilon }}^{\mathrm{ss}}$ on grain size $d$ at various ${\sigma }_A$. Solid lines are fits to Eq. (1) with variable grain size exponent $p$. See Ref. [37] for animations of the four simulations at ${\sigma }_A=0.01$.

Figure 4

(a) Effect of stochastic temperature $M\sim k_{B}T$ on grain boundary structure. Colors are as in Fig. 1. (b) Variation of the grain size exponent $p$ with $M$.

Figure 5

Diffusional creep in bcc nanopolycrystals with only low-angle grain boundaries. (a) Strain normalized system configurations for ${\sigma }_A=0.03$ and $d=101.7a$ are shown at ($0t, \epsilon =0.0$), ($40t, \epsilon =0.0211$), ($100t, \epsilon =0.0342$), and ($180t, \epsilon =0.0499$). The system thickness is $L_x=1a$. Sites with bcc coordination (at $t=0$) are shown in pale green, those with irregular coordination (dislocation core atoms) are shown in red $(t=0)$, green $(t=40)$, pale blue $(t=100)$, and gold $(t=180)$. Burgers vectors of dislocations along each boundary are indicated with white $\perp$ symbols, along with the subsequent climb directions expected upon deformation. (b) Steady state creep rates vs ${\sigma }_A$ at $d=101.7a$. The dashed line is a fit to Eq. (1) with $m=1.19$. (c) (Left) Creep curves for the data shown in (b). ${\epsilon }^{\mathrm{pl}}$ vs time is shown at various ${\sigma }_A$ (color legend). The dashed black lines are linear fits. (Right) Dependence of ${\dot{\epsilon }}^{\mathrm{ss}}$ on grain size $d$ at ${\sigma }_A=0.01$ and 0.03. Solid lines are fits to Eq. (1) with grain size exponent $p=1.68$.

Figure 6

Dislocation nucleation, climb, and pile-up in a simple dislocation creep scenario at ${\sigma }_A=0.09$ and $L_y=L_z=264a$. (a) $50t, \epsilon =0.0358$, (b) $130t, \epsilon =0.0397$, (c) $360t, \epsilon =0.0544$, and (d) $480t, \epsilon =0.0751$. Interior atoms with bcc coordination are shown in pale green, exterior hard wall atoms (also bcc coordination) in pale blue, those with irregular coordination (dislocation core atoms) in red, and those with fcc coordination in bright green. Red atoms are displayed with larger radii to highlight the dislocation positions.

Figure 7

(a) Stress exponent $m$ for simple dislocation creep vs dislocation density ${\rho }_d$. (Inset) Steady-state strain rate data from which $m$ values were determined $(L=L_y=L_z)$. (b) Steady-state dislocation velocity $v_{\mathrm{ss}}$ vs applied stress ${\sigma }_A$ for the first six dislocation pairs in the lowest ${\rho }_d$ system examined. The red lines are fits to the first pair of nucleated dislocations that climb along $\pm \vec{z}$ [see Fig. 6].

Figure 8

Combined dislocation and $22.5^{\circ }/{45}^{\circ }$ grain boundary creep at $d=152.5a$ with 1 central dislocation source per grain. (a) System representation at ${\sigma }_A=0.04$ and $t=160$, with colors as in Fig. 6. (b) Steady state strain rate data and stress exponents $m$. (Inset) The corresponding creep curves at various ${\sigma }_A$ (color legend).

Figure 9

Dislocation creep at $L_y=L_z=352a$ with 16 randomly positioned sources and four hard wall boundaries. (a) System representation at ${\sigma }_A=0.08$ and $t=65$, with colors as in Fig. 6. (b) ${\dot{\epsilon }}^{\mathrm{ss}}$ data from which $m$ was determined. (Inset) Creep curves at various ${\sigma }_A$ for the configuration shown in (a).

Figure 10

Comparison of dislocation pile-up spacing with predictions of continuum theory. Simulations are at ${\sigma }_A=0.045$ and various $L_y=L_z$.