FE Analysis of Dynamical Recrystallization during the Seamless Tube Extrusion of Semicontinuous Casting Magnesium Alloy and Experimental Verification

Dynamic recrystallization models

As is shown in the true stress–strain curves, the flow stresses markedly increase as the temperature decreases, and gradually increase with the increase of the strain rate. Compared with the work [21], the curves show that, with the increase in true strain, the flow stress rapidly increases, then decreases slowly after reaching its peak, finally followed by a stabilized condition. The variation of the stress–strain curves is the result of the competition between the workhardening and the dynamic softening caused by dynamic recovery or dynamic recrystallization. In early stage of the flow curve, the effect of workhardening is dominant, although the dynamic recovery occurs so that the flow curve is increased quickly. As strain increases, the stored energy becomes more, and dynamic recrystallization takes place. The effect of the dynamic softening begins to play a leading role so that the flow curve increases slowly, and is up to the peak value. Then, the softening effect becomes predominant obviously. Finally, the flow curve drops until a balanced state between workhardening and dynamic softening has been reached.

The flow stresses markedly increase as the temperature decreases, and gradually increase with the increase of the strain rate. The peak flow stresses are in the true strain of 0.1 to 0.3 and are inclined to delay when the strain rate is increased. As mentioned in the paper, it is sure that at higher temperatures and lower strain rates, flow stress is lower. So that the alloy is subjected to plastic deformation. In the practical extrusion, higher strain rate also can promote the softening process by generating more plastic deform heat. This makes the plastic deformation easy and has a complex effect on dynamic recrystallized fraction and recrystallized grain size (shown in Section 3.2).

The Arrhenius equation (eq. (1)) is frequently used to describe the relationship between the temperature, strain rate, and the flow stress and uncovers the approximate hyperbolic law between Z parameter and flow stress (eq. (2)) [21].

where A (s^–1) are α (MPa^–1) are the material constants, respectively. R is the gas constant (8.314 J· mo^–1· K^–1). The n and Q are stress exponent and activation energy (kJ· mol^–1), indicating deformation difficulty. T is the absolute temperature (K), Z is the Zener–Hollomon parameter, and σ is the flow stress (MPa) for a given strain.

By using the power law (ε˙=A1σn1) and exponential law ε˙=A2exp(βσ), the value of α can be deduced by the following eq. (3) and eq. (4)^[22]:

where β is the material constant. Per eq. (3) and eq. (4), n₁=dln ε˙/dln|σ|, β=dln ε˙/d|σ|, α=β/n₁. The relationships of lnε˙–lnσ and lnε˙–σ were plotted and linearly fitted as shown in Figure 4. The slope rates of the fitted curves in Figure 4(a) and 4(b) represented n₁ and β, respectively, and the mean values of n₁ and β were 8.326 and 0.1778, respectively. Thus, the value of α was calculated as 0.02136 MPa^–1.

Figure 4:

Relationships between lnε˙ and (a) lnσ and (b) σ.

Taking natural logarithms of both sides of eq. (1), it can be rewritten as:

Then, taking partial derivative on both sides of eq. (5), it can be rewritten as:

The linear relationships of ln ε˙-ln[sinh(ασ)] and ln[sinh(ασ)]–1/T were obtained at constant strain rates. The relationships were plotted and linearly fitted as shown in Figure 5. The mean value of the slopes in Figure 5(a) and 5(b) were n' and D, respectively [23]. Thus, the average Q (Q=Rn'D) value was calculated as 254.407 kJ/mol. Generally, the deformation activation energy increases with the increasing temperature and strain rate [24]. For AZ31 magnesium, deformation temperature has a more obvious effect on the activation energy [25]. From 300 to 450 ℃, the calculated values of the activation energy are 238.655, 242.729, 270.537, and 308.325 kJ/mol at strain rate of 0.1 s^-1, respectively. The activation energy increases rapidly as the deformation temperature raises. The increase of the activation energy may be related to the consumption of many dislocations by the DRX during hot deformation. At high temperature, dislocation climb occurs much more rapidly, so that many opposite sign dislocations are annihilated, and the potential source of dislocations is reduced. This will lead to an increase of the activation energy [26]. It is known that as the deformation temperature becomes higher, the dominated deformation mechanism of the magnesium alloy is changed [27]. At lower temperature (≤ 250 ℃), the critical shear stress for nonbasal slip cannot be reached, and basal plane slip and twinning are the dominated deformation mechanisms. Dislocation movement easily encounters obstacles, which leads to stress concentration, so the hardening process is faster. At higher temperature (≥ 300 ℃), the critical shear stress on nonbasal surface decreases rapidly, and more slip system can be started. The cross-slip of dislocations and the dislocation climbs become the leading deformation mechanism in turn, so that the softening process is more prominent, and peak stress and steady-state flow stress decrease [26, 28].

Figure 5:

Relationships between ln[sinh(ασ)] and (a) lnε˙ and (b) 1/T.

According to eq. (2), taking natural logarithms of both sides, it can be written as:

Substituting Q into eq. (2), the value of Z can be acquired at the certain strain and temperature. Thus, the linear relationship between lnZ and ln[sinh(ασ)] was obtained as shown in Figure 6. The slope rate was the value of n, n=8.78. The intercept of the fitted curve was the value of lnA, lnA=43.1, so A was equal to 5.19×10¹⁸. The correlation factor of the fitted line was 0.9851.

Figure 6:

Relationship between lnZ and ln[sinh(ασ)].

Thus, substituting A, α, n, and Q into eq. (1), the expression can be:

During hot deformation, DRX evolution is mainly dependent on the density and distribution of dislocation. Dislocations continually increase and accumulate to such an extent that DRX nucleus will form and grow up near grain boundaries, twin boundaries, and deformation bands. The stress–strain curves can reflect the DRX evolution. From the curves, some key parameters are used to describe the DRX process: ε_c (corresponding to σ_c) is the critical strain for the initiation of the DRX; ε^* (corresponding to σ^*) is the strain for maximum softening rate, σ_p is the peak stress, σ_sat is the saturate stress, σ_ss is the steady-state stress [6, 29]. These parameters were determined by the plot of θ (θ=∂σ/∂ε, strain hardening rate) versus σ.

For example, Figure 7 shows the relationship between θ (θ=∂σ/∂ε) versus σ at the deformation temperature of 400 ℃ with the strain rate of 0.1 s^–1. From the curve, the inflections in the plot of θ–σ up to the peak point indicated the occurrence of the DRX. That is, the inflection point (σ_c) was determined by the lowest point in the −∂θ/∂σ curve [23, 30], and then return to the corresponding true stress–strain (σ–ε) curve; the ε_c was identified by the value of σ_c. The value of σ_sat was determined by the extrapolation of the θ–σ plot to θ=0 using only the linear portion of the curve relating to σ values just below σ_c. The value of σ_p was obtained at the point at θ=0. The value of σ^* is the valley point of θ–σ plot and obtained as the value of θ reaches the negative peak after the peak point. The stress σ_ss is the steady-state stress at the point at θ=0 after the peak stress.

Figure 7:

Relationship between θ (θ=∂σ/∂ε) versus σ under the deformation temperature of 400 ℃ with strain rate of 0.1 s^–1.

The value of ε_c and ε^* were identified from the stress–strain curve along with the value σ_c and σ^*. They can be described as a function of dimensionless parameter, Z/A, as shown in Figure 8(a) and 8(b), viz:

Figure 8:

Relationships between the ln(Z/A) and (a) ln|ε_c|; (b) ln|ε^*|.

The Avarami equation was used for describing the kinetics of DRX evolution [31]. The volume fraction was expressed as a function of strain at constant strain rate, viz:

where, k and n are the material constants. By identifying the deformation conditions as X_DRX equal to 100 %, the material constant k and n were calculated. In the work, n=1.1854, k=1.3165. Thus, the volume fraction of DRX for the AZ31 magnesium alloy was expressed as the following:

Based on the grain size measurements under different deformation conditions, the relation between the dynamic recrystallized grain size (D_DRX) and Z was obtained [32]. The model was built as:

FE analysis for the hot extrusion of seamless tube by the dynamic recrystallization models

Based on the DRX models and the extrusion parameters, FE analysis was performed for the microstructure evolution of the AZ31 magnesium alloy during the seamless tube extrusion. The extrusion temperature and ram speed are two key parameters as they directly affect the DRX volume fraction and grain size in the extruded part.

Figure 9 shows the effects of the extrusion temperature and the ram speed on the recrystallized volume fraction. The DRX was not fully realized at low extrusion temperature of 300 ℃ with ram speed of 0.5 mm/s (Figure 9(a)), the recrystallized fraction was 96 % in the extruded tube, while about 100 % at the other the extrusion conditions. Magnesium alloys have fewer slip systems due to magnesium's hcp crystal structure which promotes twinning as a main mechanism for deformation. The increase in the twinning phenomenon would appear to hinder the occurrence of DRX, particularly at low deformation temperature. When the deformation temperatures are increased along with the extrusion temperature, DRX was easily achieved. As is shown in Figure 9(a) and 9(b) (ram speeds of 0.5 mm/s and 1 mm/s, respectively), when the ram speed was increased, the recrystallized fraction also increased. This is because more heat was generated in the deformed tube with the increase of the ram speed along with the deformation temperature becoming higher. As is shown in Figure 10(a) and 10(b), the ram speed was changed from 0.5 to 1 mm/s at extrusion temperature of 300 ℃ and the deformation temperature at the exit of the die apparently was 407 ℃ and increased by 47 ℃ from 360 ℃.

Figure 9:

Distribution of dynamic recrystallization fraction on the cross-section, under different extrusion conditions: (a) 300 ℃, 0.5 mm/s; (b) 300 ℃, 1 mm/s; (c) 400 ℃, 0.5 mm/s; (d) 400 ℃, 1 mm/s; and (e) volume fraction.

Figure 10:

Distribution of temperature under different extrusion conditions: (a) 300 ℃, 0.5 mm/s; (b) 300 ℃, 1 mm/s; (c) 400 ℃, 0.5 mm/s; and (d) 400 ℃, 1 mm/s.

Figure 11 shows the effect of the extrusion temperature and the ram speed on the dynamic recrystallized grain size. When the extrusion temperature was increased from 300 ℃ to 400 ℃, the dynamic recrystallized grain size became larger. The grain size was increased from 5.3 μm to 7.8 μm at ram speed of 0.5 mm/s, and from 5.9 μm to 8.4 μm at ram speed of 1 mm/s. The increase in grain size was due to the rise of the extrusion temperature which led to the decrease of the Zener–Hollomon parameter. When the Zener–Hollomon parameter decreases, the coarsening of the recrystallized grain will occur [33]. When the ram speed was increased, the recrystallized grain in the extruded tube also became coarse, but to a relatively smaller extent because of the conflicting effect of the deformation temperature and strain rate on the parameter. The increased ram speed led to the rises of both deformation temperature and the strain rate at the exit of die. The former rise reduced the parameter and enlarged the recrystallized grain size, but the latter rise enlarged the parameter and refined the recrystallized grain size [34].

Figure 11:

Dynamic recrystallization grain size on the cross-section, under different extrusion conditions: (a) 300 ℃, 0.5 mm/s; (b) 300 ℃, 1 mm/s; (c) 400 ℃, 0.5 mm/s; and (d) 400 ℃, 1 mm/s.

Interestingly, when the workpieces were extruded at temperature of 300 ℃ with ram speed of 1 mm/s and at temperature of 400 ℃ with ram speed of 0.5 mm/s, the deformation temperature distributions at the die exit were very close (shown in Figure 10(b) and 10(c)). The difference of the temperature value was less than 3 ℃. However, the dynamic recrystallized grain sizes were different because of the distinct strain rates at the exit of die due to the varied ram speeds. Larger ram speed gave rise to higher strain rate and smaller dynamic recrystallized grain size. The results mean that the grain size in the tube extruded at a temperature of 300 ℃ with a ram speed of 1 mm/s was smaller than at a temperature of 400 ℃ with a ram speed of 0.5 mm/s

Figure 12 shows the microstructures in the extruded tubes under different extrusion conditions, and a comparison between the experimental and FEM results is performed, as shown in Figure 13. In Figures 12 and 13, the grain size was fine to several micrometer orders. The grain sizes were about 7.4 ~ 8.6 μm at extrusion temperature of 400 ℃ and complete DRX occurred, while they were 8.5 ~ 12.3 μm at temperature of 300 ℃. It can be seen that the predicted average grain sizes well agree with the experimental ones. So that it can be concluded that the derived models for the AZ31 magnesium alloy were valid in this work. The average grain size was bigger under extrusion temperature of 300 ℃ with a ram speed of 0.5 mm/s than under the other extruding conditions, even though the dynamic recrystallized grain size was the smallest. The case of bigger overall average grain size was because the DRX was not fully achieved under the given condition. Except for this condition, the effect of the deformation conditions on the average grain size was similar to that of the dynamic recrystallized size.

Figure 12:

Microstructures in the extruded tubes under different extrusion conditions: (a) 300 ℃, 0.5 mm/s; (b) 300 ℃, 1 mm/s; (c) 400 ℃, 0.5 mm/s; (d) 400 ℃, 1 mm/s.

Figure 13:

Comparisons between the experimental and simulated results (average grain sizes).