Fig. 1. A new constitutive model including micro-level internal variables. (a) is the schematic illustration of secondary development of JC model; (b) shows flow chart of VUMAT subroutine; (c) gives schematic of random grain structure modeled by Voronoi diagram; (d) and (e) are illustration of Coupling poly-crystal structure; (f) and (g) are diagrams of Coupling Poly-crystal structure by Cellular Automaton Method.

Fig. 2. The influence of the new parameters associated with the grain size evolution. (a) gives stress - strain curves at different (εc), compared to the Johnson-Cook model. (b) shows grain size distributions with different (εc). (a) and (b) illustrate the effect of the εc on the stress-strain curve and the grain size evolution, respectively (c) presents the stress - strain curves and (d) grain size distributions with different (d0); (c) and (d) reveal the effects of the initial grain size d0 on the stress strain curve and grain size evolution (e) is the stress - strain curves and (f) is grain size distributions with different (d∞). These results reveal that the parameters d∞ and εc have a significant impact on the yielding characteristics of austenite steels as both the stress strain curve and grain size evolution being affected.

Fig. 3. Time evolution of impact velocity and equivalent plastic strain rate. (a) and (b) are predicted grain size distributions after 64 times of random impacts which show the grain size distribution in the interior of SMAT treated sample. It is shown that the SMAT process has induced the gradient gran size distribution from tens of nanometers (in the top surface layer) to several micrometers (in the sub-surface layer) and the uniformity of grain size in the same layer through the large amount of cyclic random impacts. (c) is the monitor of strain rate and velocity during one impact; (d) is the full coverage multiple random impingements model. It is demonstrated that the strain rate predicted by the constitutive model reaches 105 for a 3-mm ball at 10 m/s velocity. Such result agrees with the experiment observation and the theory analysis.

Fig. 4. The predicted spatial variations of grain size distribution during SMAT process. (a, d, g) are the schematic illustrations of local indents produced by random impacts. (b, c, e, f, h and i) clearly reveal the contours of grain size distributions on the impact surface and in the interior. (a, b, c) are the predicted simulated distributions of grain size after five random impacts; (d, e and f) are the assumed distributions of grain size after eight random impacts; (g, h, i) are the predicted distributions of grain size after twelve random impacts. The minimum grain sizes are reduced to 75.94, 51.16 and 43.46 nm after five random impacts, eight random impacts and twelve random impacts, respectively.

Fig. 5. Residual stress and yield stress distributions as a function of position. The residual stress and yield flow stress distributions along depth are calculated. The maximum of compressive residual stress reaches ～900 MPa while the yield stress touches ～1200 MPa. The simulated grain size distributions proved that grain refinement near the top surface contributes largely to the high strength.