3.2 DRX nucleation and grain growth

For a polycrystal with multiple slip systems, when the total dislocation density of grain i reaches a critical value ρ_c, new DRX grains will possibly nucleate. The expectation of DRX grain nucleation can be evaluated by [48]

                           (5)

where N is the number of DRX grain nucleation,  is the nucleation rate, K is the number of grains with ρ_i>ρ_c, C(T) is a function of temperature, S_i is the surface area of grain i, and Δt is the time increment of a simulation step.

The grain boundary velocity V_i is assumed to be proportional to the stored energy difference across the boundary and calculated by [57]

                      (6)

where M(T) is the grain boundary mobility related to temperature,  is the average dislocation density, ρ_i is the dislocation density of grain i, WGT(i) is the volume weight of grain i, and τ is the dislocation line energy [58]

                    (7)

where n^slip is the total number of slop modes.

3.3 GBS strain rate

The total strain rate is composed of strain  rates induced by both dislocation slip and GBS [22,35,59], i.e.

                          (8)

where  and represent strain rates of  grain i due to GBS and dislocation slip, respectively. GBS strain rate can be generally expressed as a function of temperature, grain size and applied stress [6,59-61]. A model originally proposed by CONRAD and NARARYAN [25] and further updated by ZHU et al [35] and WEI and GAO [31] is employed for calculating strain rate of each  grain:

 (9)

where υ_D is a typical lattice vibrational frequency, and υ_D≈10¹³ s^-1; d_i is the grain size; V is the activation volume, taken as  [25,35]; σ_i is the effective stress of grain i; K_B is the Boltzmann constant (1.38×10^-23 J/K); T is the temperature; ΔF is the activation energy for grain boundary diffusion (92 kJ/mol for Mg alloys) [62]; R is the molar gas constant (8.314 J/(mol·K)).

The cavitation damage parameter D_c is introduced to characterize the process of void nucleation, growth and coalescence during high temperature deformation [63]. For grain i, D_c,i is determined by the rupture strain ε_f and GBS strain rate as fallows, considering the difference in tension and compression.

                  (10)

where D_c,i is permanently set to zero in compression and the hydrostatic pressure σ_m,i=1/3(σ_1,i+σ_2,i+σ_3,i).

3.4 Grain rotation mediated by GBS

In the GBS regime [21], initial basal texture of the rolled and annealed magnesium alloys become more random with the increase of deformation, and strain rate and GBS contribution are critical factors for grain rotation [24]. In the present work, a model to introduce GBS induced grain rotation is proposed as follows.

When GBS takes effect, grains affected by GBS are randomly selected to rotate in addition to the rotation induced by dislocation slip. The number of the picked grains is proportional to the ratio of GBS strain rate to total strain rate. Therefore, the expectation of rotated grains z_GBS is determined by

                        (11)

where h_g is the fitting parameter; n_g is the total grain number; ξ_GBS,i is the GBS contribution ratio for grain i.

                           (12)

The extra random rotation degree of each selected grain should be within 45° [6].

Under GBS dominated deformation, the compatible conditions (five slip systems needed for compatible and homogeneous deformation under plastic deformation) are not necessary to be  fulfilled, and grain rotation will randomize the deformation texture [22,24]. In uniaxial tension of the present AZ31B sheet, the c-axis rotates toward the sheet normal by basal slip, while prismatic slip rotates the grain around c-axis to the balance position. After the onset of GBS, the grain rotates about c-axis by a random angle with no specific balance direction, as explained in Section 2.2.

3.5 VPSC-DRX-GBS integrated model

The above DRX and GBS methods are implemented into the VPSC framework. Figure 8 shows the flowchart of the proposed VPSC-DRX- GBS approach. The computational procedures are as follows.

Fig. 8 Flowchart of VPSC-DRX-GBS model

(1) If the strain is less than the critical value for initiating GBS, e_GBS, the status of a given aggregate, including stress, strain and orientation, is updated by the VPSC model, and the total strain rate  for grain i is equal to the plastic strain rate .

(2) When the critical dislocation density for DDRX initiation is achieved, new DRX grains will nucleate, according to Eq. (5), followed by growth through boundary migration described with Eq. (6).

(3) If the strain reaches e_GBS, the GBS strain rate for each grain i is calculated by using Eq. (9), and the direction of GBS strain rate is assumed to be parallel to the direction of plastic strain rate

(4) Update the macroscopic plastic strain rate  in which the weighted average of the plastic strain rate over the aggregate has to coincide with the macroscopic counterpart [64], i.e.

                    (13)

where the bracket “<>” denotes the average over grains, weighted by associated volume fractions.

(5) Calculate stress and strain by using the macroscopic plastic strain rate in the framework of VPSC, and update effective stress σ_i for grains.

(6) Calculate rotation of grains by Eqs. (11) and (12).

3.6 Modeling of AZ31B sheet

In the present work, the experimental results in the work of ZHOU et al [44] on the same AZ31B sheet subjected to RD tension, RD compression and ND compression at 300 °C and strain rate of 1×10^-3 s^-1 are used for calibration of the modeling parameters, since the same material is studied here. The input parameters of the VPSC-DRX-GBS modeling are as follows.

The initial average grain size at 300 °C is 8 μm from Fig. 3 (Due to the holding at higher temperature, the grain size slightly increases. Therefore, the initial grain size at 300 °C is set to be 8 μm, instead of 7 μm [43,48]), and initial grain number is 300. The initial grain orientations are given according to the initial texture. The shear modulus m at 300 °C is taken as 14000 MPa [62]. The meff homogenization method in framework VPSC with meff=5 is adopted in simulation  process [43,48,55]. According to the EBSD observation, the size of newly generated DDRX grains is assumed to be 1.5 μm. The new grains are assumed to share the same c-axis with their parent grains, and the misorientation angle between each DRX grain and its parent grain, due to the rotation of basal plane, follows the Gaussian distribution with the mean value of 15° and standard deviation of 15° [43].

4 Simulation results and discussion

4.1 Flow stress

Figure 9 shows the comparison between the experimental and fitted stress-strain curves and variation of r value. The calculated results by  the VPSC-DRX-GBS model agree well with the experimental results of RD tension, RD compression and ND compression. For comparison, the stress-strain curves without considering GBS are also calculated and shown in Fig. 9(a), by using the same hardening parameters but turning the GBS module off. Without consideration of GBS, the calculation obviously overestimates the stress at strains higher than 0.2, and cannot reproduce the softening behavior. GBS gives rise to more softening in tension than in compression, because contribution of GBS to deformation is significantly reduced in compression.

Table 1 VPSC-DRX-GBS modeling parameters of AZ31B sheet

Fig. 9 Experimental and fitted results of AZ31B sheets at 300 °C

4.2 Grain size evolution

Figure 10 shows the calculated average grain size varying with true strain under three loading conditions, and comparison with measured results obtained by OM and EBSD observation. The calculated results are consistent with the experimental ones. The average grain size decreases from initial 8 to 7 μm in RD tension, to 6.5 μm in RD compression and to 6.9 μm in ND compression at a strain of 0.2. It decreases gradually at strains less than 0.2 due to DRX, and keeps constant at strains higher than 0.2.

Fig. 10 Evolution of average grain size with change of strain at 300 °C

4.3 Texture

Figure 12 shows the predicted texture evolution with tensile strain. The hexagonal maxima distribution of  planes is gradually developed before at strains higher than 0.2, and due to the occurrence of DRX, the maximum texture intensity slightly fluctuates between 5.2 and 5.3 at strains from 0.05 to 0.2. At strains higher than 0.2, the ring-shaped texture replaces hexagonal maxima distribution due to the grain rotation induced by GBS.

4.4 GBS contribution

Fig. 11 Comparison of textures between simulation and experiment at strain of 0.4

Fig. 12 Predicted texture evolution under RD tension at different strains

The calculated average GBS contribution ratios  under RD tension, RD compression and ND compression, with the proceeding of deformation at strains higher than 0.2, are presented in Fig. 13. The GBS contribution that in RD tension is obviously higher than ratio in RD compression and ND compression. The evolution trends of GBS contribution ratio vary with the loading modes. For RD tension, the GBS contribution ratio rises steadily, with increasing cavitation damage parameter D_c and strain. For RD and ND compression, the GBS contribution ratio decreases slowly from 0.16 to 0.14 and from 0.18 to 0.165, respectively, as the true strain increases from 0.2 to 0.6, resulting from the stress softening. The significant reduction of the GBS contribution ratio in the compression mode, from the experiments conducted by DIERINGA et al [46] and LI et al [65], is confirmed by the current VPSC-DRX-GBS model. A much higher tension GBS contribution ratio than compression GBS contribution ratio has been reported for the aluminum alloys [65,66] and was attributed to the easier cavity nucleation at grain boundaries under tensile creep condition.

Fig. 13 Evolution of average GBS contribution ratio to overall strain rate at 300 °C

4.5 Slip activities

The relative slip activities of different slip modes in RD tension, RD and ND compression with and without considering the GBS effect are presented in Fig. 14. For each loading condition, when the strain is less than 0.2, the slip activities for different slip modes with consideration of GBS are similar to those without incorporating the GBS module. Thereafter, the slip activities with and without considering GBS mechanism gradually turn to be obviously different.

Basal slip is the dominant slip mode under all the loading conditions, with an increasing trend of activity for RD tension/compression and a decreasing trend for ND compression with deformation. In RD tension, prismatic slip mode accounts for most of other slip activities. After GBS takes effect, basal slip activity increases dramatically, while prismatic slip decreases correspondingly. In RD compression, prismatic slip with a decreasing trend of relative activity acts as a secondary mode at first, and slip takes over the prismatic slip at strains higher than 0.2. GBS mechanism enhances the basal slip and restrains  the prismatic slip. In ND compression, the activity of prismatic slip follows slip, but it vanishes at strains higher than 0.3. The activity of basal slip increases and slip decreases due to GBS.

Fig. 14 Predicted slip activities at 300 °C

According to the above analysis, GBS mainly enhances the basal slip mode. GBS comes into being at strains higher than 0.2, which enhances the activity for basal slip more effectively for RD tension than for RD/ND compression. In order to explain the evidently enhanced basal slip, the relative activities of different slip modes in RD tension are calculated by removing the effect of texture randomization (keeping orientations of DRX grains the same as their parent grains, and setting the GBS-induced grain rotation as zero), as shown in Fig. 15. If excluding the effect of texture randomization, the relative activity of basal slip decreases to the level without GBS or even lower at strains higher than 0.42. The prismatic slip without the effect of texture randomization becomes more obvious than that without GBS because the plastic strain rate is reduced by GBS and the prismatic slip is preferred at higher strain rates [67]. Therefore, the effect of texture randomization induced by DRX and GBS can explain the enhanced basal slip and less activated hard slip modes.

Fig. 15 Comparison of predicted slip activities at 300 °C for RD tension, including with/without considering GBS and removing effect of texture randomization

5 Conclusions

(1) The deformation of the AZ31B sheet experiences DRX at strains higher than 0.2, which leads to refined grains.

(2) A polycrystal plasticity based integration modeling method incorporating both DRX and GBS is developed, and the stress-strain curves, grain size evolution, texture evolution and slip activity can be simultaneously calculated.

(3) By introducing the cavitation damage parameter D_c into GBS model, the polycrystal plasticity model can reproduce the tension- compression difference induced by GBS.

(4) Due to GBS, tension deformation and compression deformation result in significant different texture development. In RD tension, the random grain rotation induced by GBS produces more dispersed distribution textures in the (0002) pole figure, and hexagonal maxima distribution of planes disappears. In the compression modes, GBS effect is much weaker, which hardly influences texture distribution. The proposed model for GBS mediated grain rotation can well reproduce the measured texture development.

(5) GBS enhances the activity of basal slip more obviously for RD tension than for RD and ND compression due to the effect of texture randomization.

Acknowledgments

The authors would like to acknowledge the support from the National Natural Science Foundation of China (51675331 and 51775337), Major Projects of the Ministry of Education (311017). Pei-dong WU would like to acknowledge the support from the Natural Sciences and Engineering Research Council of Canada (NSERC, RGPIN-2016-06464).

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