Investigation and Modeling of the Preheating E ﬀ ects on Precipitation and Hot Flow Behavior for Forming High Strength AA7075 at Elevated Temperatures

: Preheating is the ﬁrst but critical step for hot stamping high strength precipitate hardened aluminum alloys. To thoroughly understand the e ﬀ ects of preheating conditions—i.e., preheating rate and heating temperature—on the strength and hot deformation of aluminum alloys, a series of thermal–mechanical tests was performed to determine the post-hardness evolution and hot ﬂow behaviors. Typical microstructures with di ﬀ erent preheating conditions were also observed through transmission electron microscopy (TEM), with which a uniﬁed model of both hot ﬂow and strength based on key microstructural variables was developed, enabling the successful prediction of macroscopic properties using di ﬀ erent preheating strategies. The results have shown that for high strength AA7075 at the T6 condition, the dominant mechanism of precipitate evolution with increasing temperature is the coarsening of precipitates ﬁrst, followed by dissolution when they exceed a critical temperature. A higher heating rate results in a slower coarsening and a relatively higher strength level. In addition, the ﬂow stress of hot deformation is also higher using a quick heating rate, with more signiﬁcant softening and reduced ductility.


Introduction
Complex-shaped thin-walled panels of lightweight aluminum alloys are increasingly used in the body structures of automobile, aircraft and aerospace vehicles [1][2][3][4]. It is preferable to form such components by stamping at elevated temperatures to overcome the poor ductility at room temperature and springback [5]. In terms of forming processes at elevated temperatures, warm (T < 0.6 Tm, T and Tm are respectively the forming temperature and melting temperature) and hot (T > 0.6 Tm) stamping are classified depending on whether dynamic recrystallization occurs in the alloy.
For the hot stamping condition, hot form quench (HFQ ® ) is the cutting-edge technology used to manufacture high strength aluminum alloy components, enabling complex shapes and satisfying post-formed strength to be obtained simultaneously [6,7]. The HFQ ® process is independent of the specific temper of the used alloy as forming is performed at solution heat treatment (SHT) temperature, enabling the microstructure to be stored. For the warm forming processes of non-heat-treatable aluminum alloys [8], quick plastic forming [9], patented by GM, has been widely used in AA5xxx panels. The process uses a gas medium to replace the conventional rigid punch; however, warm forming is generally believed to be unsuitable for forming heat-treatable alloys, as the initial precipitates may become coarsened and the post-formed strength is decreased during heating and warm forming [10] and unable to obtain maximum strength. However, the warm forming of high strength heat-treatable aluminum alloys has still been extensively investigated under the premise of compromising strength requirements. Wang et al. [11] have investigated the warm deformation behavior of AA7075, and the best drawability was found to occur at a temperature near 220 • C. However, strength reduced remarkably when the forming temperature was greater than 220 • C. Huo et al. [12] also found that warm forming AA7075-T6 at 250 • C induced strength deterioration, which was caused by the coarsening of matrix precipitates. To address the disadvantage of microstructure deterioration during preheating and even forming, investigations into controlling microstructural evolution during heating by designing a proper combination of heating rate and initial alloy microstructure have been performed. Mori et al. [13] proposed a quick heating process for aircraft alloys, with which satisfactory post-formed strength of the alloys has been obtained. Zhang et al. [14] established a unified model of the complete process of directly heating aluminum alloy at the T4 condition, enabling microstructure, deformation and post-formed strength to be successfully predicted. Essentially, these studies are driven by utilizing heating rate to quickly pass the nose temperature to target forming temperature and accomplish the forming before the temperature of the material drops into sensitive zones. In order to obtain the correlation between heating rate and underlying microstructure, and to understand the subsequent hot flow and post-formed strength evolution comprehensively, quantitative research and modeling are necessary.
In this study, the effects of heating rate on the precipitation and hot behavior of high strength aluminum alloy 7075 were investigated using thermal-mechanical testing and microstructure observation. In addition, a physically based material model considering the precipitation under various heating rates was established for modeling the macro flow behavior for the first time. The mechanism of heating rate effect-i.e., the underlying correlation between micro precipitation and material flow-enables us to provide useful indications for stamping engineers to design process parameters for high strength aluminum alloys, and the developed model also facilities the improvement of simulation accuracy by considering various preheating approaches.

Materials and Methods
The as-received material was high strength aluminum alloy, 7075, provided by Constellium France. The initial heat treatment temper was T6. The sheet thickness was 2 mm. The main chemical composition is given in Table 1.  Figure 1 illustrates the experimentation details. A series of tensile tests under hot forming conditions with different heating rates, temperatures and strain rates were performed. Three heating rates, . T H = 0.5 • C/s, 1 • C/s and 5 • C/s, were selected to simulate the practical heating rates of the industrial heating facilities. The heating temperatures used before forming were T H = 300 • C, 350 • C and 400 • C, ranging from warm to hot forming conditions. Once the specimen was heated to the target temperature, it was soaked for 1 s. After obtaining a uniform temperature distribution, for the heat treatment test (Test A), the specimen was immediately quenched to room temperature using the water quenching module in Gleeble. Then, the hardness was measured within the uniform temperature zone. For Test B, the specimen was hot stretched either to failure or to a certain degree of strain. Water quenching was still used to maintain the microstructure. The objective of such intermediate quenching during hot deformation is to identify the dominant microstructure evolution. For each condition, at least three tests were performed to obtain reliable results. Figure 1b shows the used specimen. A dog-bone specimen was machined from the as-received rolled sheet using electrical discharge machining. The longitudinal direction of the specimen gauge is the rolling direction. The length and width of the parallel zone were 46 mm and 12 mm, respectively, which enables us to guarantee the 10-mm length of the uniform temperature zone using the resistance heating of the Gleeble 3800 thermo-mechanical simulator. In addition, a sufficiently large area for determining mechanical properties and statistical microstructure observations was used to prepare microstructure samples. The conducted mechanical tests are summarized in Table 2 length and width of the parallel zone were 46 mm and 12 mm, respectively, which enables us to guarantee the 10-mm length of the uniform temperature zone using the resistance heating of the Gleeble 3800 thermo-mechanical simulator. In addition, a sufficiently large area for determining mechanical properties and statistical microstructure observations was used to prepare microstructure samples. The conducted mechanical tests are summarized in Table 2.    TEM and Electron backscatter diffraction (EBSD) observations were performed on typical samples under selected conditions to provide microstructure evidence (i.e., precipitates and grain sizes, respectively), aiming to validate the mechanisms in the proposed physically based constitutive equations, where the middle heating rate of 1 • C/s was selected as the typical condition. The detailed conditions for the microstructure observations are given in Table 3. TEM tests of 1 , 2 and 3 were used to examine the temperature effects on the coarsening of the precipitates during heating. EBSD tests of 4 , 5 and 6 were used to examine the grain size evolution after heating and after heating and straining. Table 3. Conditions for microstructure observations.

As-Received
As-Heated As-Heated and Strained

Unified Constitutive Modeling and Calibration
A unified constitutive model is proposed and calibrated in this section, so as to predict the evolutions of both key microstructures (including precipitates and solid solutes) and the mechanical properties (hardness and hot deformation behavior) of AA7075-T6 during and after heating with different heating rate conditions.

Modeling of Precipitation during Heating
It has been widely reported that precipitates are the key microstructures to determine the mechanical properties of age-hardened aluminum alloys during heat treatments. Hence, the evolutions of precipitate related parameters are modeled in this section. Figure 2 summarizes the schematic of the precipitation sequence during heating at a particular heating rate for AA7075 material from previous publications [15], from which the nucleation and dissolution temperature ranges for different precipitates can be observed.

Unified Constitutive Modeling and Calibration
A unified constitutive model is proposed and calibrated in this section, so as to predict the evolutions of both key microstructures (including precipitates and solid solutes) and the mechanical properties (hardness and hot deformation behavior) of AA7075-T6 during and after heating with different heating rate conditions.

Modeling of Precipitation during Heating
It has been widely reported that precipitates are the key microstructures to determine the mechanical properties of age-hardened aluminum alloys during heat treatments. Hence, the evolutions of precipitate related parameters are modeled in this section. Figure 2 summarizes the schematic of the precipitation sequence during heating at a particular heating rate for AA7075 material from previous publications [15], from which the nucleation and dissolution temperature ranges for different precipitates can be observed. For the AA7075-T6 investigated in this study, it is generally believed that η' is the dominant precipitate for strengthening, which starts to dissolve when the temperature is high enough (between 300 to 400 °C ) under different heating rates [16]. Hence, during heating, the initial η' precipitates in the T6 temper material would experience coarsening first, and then start to dissolve when being For the AA7075-T6 investigated in this study, it is generally believed that η' is the dominant precipitate for strengthening, which starts to dissolve when the temperature is high enough (between 300 to 400 • C) under different heating rates [16]. Hence, during heating, the initial η' precipitates in the T6 temper material would experience coarsening first, and then start to dissolve when being heated to above the critical temperature T c . Both the coarsening and dissolution of precipitates are modeled here.
Coarsening of precipitates has been widely reported and modeled before; the radius data is generally used for modeling. The following equation is used to predict the coarsening of precipitates, considering the different temperature effects [17,18] as the following: where r is the normalized precipitate radius, which is defined as r = r/r c (r is the precipitate radius and r c is the critical radius at the peak ageing state) . Q di f f is the activation energy for diffusion, R is the universal constant, Q represents the largest normalized radius of precipitates during heating, and k 1 and n 1 are material constants. When the alloy is heated to a critical temperature T c , the dissolution of coarsened precipitates occurs, and solid solutes will diffuse from precipitates to the matrix of the material, contributing to the strength of the material with solution hardening effects [19,20]. The evolution of solute concentration in the matrix of the material can be modeled as follows [20]: c is the normalized solute concentration in the matrix, Q s is the free energy of the solution of the solute and k 2 is a material constant. The volume fraction of precipitates will change due to dissolution, whose evolution can be modeled according to the solute concentration as the following [21]: in which c e is the equilibrium solute concentration at the previous ageing conditions, defined as follows [21]: When the material is initially at peak aged state, the further coarsening of precipitates would lead to the over-ageing behavior of the material, and, hence, the precipitation hardening mechanism is believed to be controlled by bypassing. The strengthening contribution from the precipitates then can be modeled by the bypassing mechanism, as the following [21]: Meanwhile, solid solution hardening from solid solutes can be modeled by the following [21]: The total yield strength of the material due to precipitation and solid solutes can be modeled according to the mixture law, as the following: In Equations (5) to (7), C a , C ss and σ 0 are temperature-dependent material constants, which are defined as follows:

Modeling of Hot Deformation
The constitutive model for hot deformation behavior after preheating of AA7075 is proposed and introduced in this section. During deformation at high temperatures with different types of precipitates existing in the aluminum alloys, different mechanisms may occur. For example, recrystallization has been reported to occur at high temperatures and high strain levels for aluminum alloys. Meanwhile, the damage evolutions of the alloy may also be affected by the precipitation states. A physical mechanism-based viscoplastic model is proposed in this section for aluminum alloys under hot stamping conditions (heating rate, temperature and strain rate) from [22,23], and the dominant equations are given as follows. . . . .
The viscoplastic behavior of alloys under hot deformation conditions is described by Equation (11), in which . ε p is the plastic strain rate, σ is the flow stress and σ y is the yield stress of the material. The particular yield stress data with different heating conditions can be determined using Equation (7). The dislocation hardening component σ d is modeled according to a normalized dislocation density, ρ, which is defined as ρ = (ρ − ρ i )/ρ max . ρ i and ρ max are the initial and maximum dislocation density of the material, respectively, and ρ then can be set as varying from 0 to 1, when considering that ρ i ρ max . ρ during hot deformation is modeled by Equation (13), in which the accumulation and dynamic recovery of dislocations caused by plastic straining is represented by the first term of the equation, the static recovery effect is then described by the second term [23], and, in addition, the possible recrystallization effect on the dislocation evolution has been modeled by the third term in Equation (13).
Dynamic recrystallization is directly related to the dislocation density, and when ρ reaches a critical level with the proceeding of straining, dynamic recrystallization may occur. The recrystallization model considering the dislocation density and grain size effect proposed by [24,25] is used in this study, as given in Equation (14). The critical level of ρ for recrystallization (ρ c ) is defined in Equation (15). Meanwhile, a kinematic grain size model considering the effects of grain growth and dynamic recrystallization has been used and is given in Equation (16).
A damage factor ω for characterizing the damage evolution during hot deformation is defined in Equation (17). The effect of precipitation of the damage evolution at an over-aged condition has been considered in Equation (17) by the term of 1/ r f m 5 . It is assumed that failure occurs when the damage factor reaches 0.7. Equation (18) is the hook's law, in which the damage factor effect on the decrease in flow stress is introduced.
In the above equations, a number of material parameters are temperature-dependent. The Arrhenius equation is used to model these temperature-dependent parameters, as below:

Modeling Calibration
Most of the equations developed above are non-linear ordinary differential equations, which cannot be solved analytically. Some material constants (such as Q di f f and Q s ) have particular physical meanings and have been obtained in previous studies, while other material constants need to be calibrated according to the experimental results obtained in Section 2. An evolutionary algorithm optimization method to obtain the minimum sum of the squares of errors between experiments and numerical data, which has been detailed in [26,27], has been used for the calibration process in order to achieve the optimized material constants in this study. The experimental data of hardness results were used to calibrate the material constants in the yield strength equations, while the hot flow behavior results from Section 2 were used to calibrate the material constants in the hot deformation equations above. It should be noted that the determined hot uniaxial tensile results were used, which has been extensively used for model calibration [5,7,28] and is believed to be able to investigate the effects of hot stamping conditions (heating rate, temperature and strain rate). The objective function of the evolutionary optimization method is defined by experimental and numerical differences of hardness or flow behavior at the same testing condition. The more detailed functions and algorithms can be referred to in previous studies [26,27].
The material constants in the developed equations for the heating and hot deformation stages are calibrated according to the results shown in Section 4 and are listed in Tables 4 and 5, respectively.

Results and Discussions
In this section, to facilitate the understanding of precipitation and grain evolution during heating and hot straining, and to obtain quantitative data for modeling, TEM and EBSD observations were performed in order to identify the key microstructures, including grains and precipitates. Figure 3 shows the TEM photos of the precipitate evolutions of AA7075 heated to different temperatures with a rate of 1 • C/s. Compared to the fine and dense distributed precipitates, η', in the as-received T6 condition, an obvious coarsening of precipitates can be observed once the material is heated to 300 • C. such coarsened precipitates is believed to indicate the over-aged state. With further heating of the material, as illustrated in Figure 2, once the temperature becomes greater than the critical temperature, T > T c , the previously coarsened precipitates begin to dissolve into the matrix again. The size of precipitates is reduced for the heated temperature of 350 • C, as shown in the Figure 3c. The results correspond well to the proposed mechanism in Figure 2. The critical temperature depends on various factors: for the preheating conditions investigated in this study, both initial temper and heating rate could affect the critical temperature. In general, a higher heating rate would lead to a higher T c [19]. The obtained precipitates under different conditions would result in different hot flow behavior and post-strength if the final forming temperature is lower than the SHT temperature. The coarsened precipitates inevitably occupied useful elements for post artificial ageing, resulting in sacrificed strength under warm forming conditions. In terms of the advanced HFQ ® process, the effect of heating rate is believed to be weakened as the material experiences a full solution heat treatment. For different heating rates, the precipitate behavior is expected to be similar, with precipitates coarsened first and then dissolved depending on the critical dissolution temperatures, which has been reported using the differential scanning calorimetry (DSC) measures in quite a few studies [29,30]. Hence, we only performed TEM for a single heat rate in order to provide indications of this behavior in this study and support the development of the microstructure-based constitutive model in the following sections. Figure 4 shows the EBSD maps of the samples before heating, after heating and after heating and hot straining to demonstrate the grain evolution during the investigated process. Misorientations > 15 • are identified as grain boundaries (black lines in Figure 4). As seen in Figure 4a, coarse grains, ranging from 50 to 100 µm, were presented in the as-received sample, where some small grains with a size of 20 µm were also observed. Heating the sample to 400 • C using a rate of 1 • C/s did not significantly affect the grain sizes, where coarse grains were still present, as shown in Figure 4b. However, after the hot deformation to a true strain level of 0.3, an apparent grain size reduction was observed, as in Figure 4c. This is evidenced by the increased amount of small grains in the deformed samples, indicating the occurrence of dynamic recrystallization. Such microstructure observation is also consistent with the findings in [31], confirming the reliability of our results.  Figure 4 shows the EBSD maps of the samples before heating, after heating and after heating and hot straining to demonstrate the grain evolution during the investigated process. Misorientations > 15° are identified as grain boundaries (black lines in Figure 4). As seen in Figure 4a, coarse grains, ranging from 50 to 100 µ m, were presented in the as-received sample, where some small grains with a size of ~20 µ m were also observed. Heating the sample to 400 °C using a rate of 1 °C /s did not significantly affect the grain sizes, where coarse grains were still present, as shown in Figure 4b. However, after the hot deformation to a true strain level of 0.3, an apparent grain size reduction was observed, as in Figure 4c. This is evidenced by the increased amount of small grains in the deformed samples, indicating the occurrence of dynamic recrystallization. Such microstructure observation is also consistent with the findings in [31], confirming the reliability of our results.

Effects of Preheating on Strength
As reasoned in Figure 2, preheating-i.e., heating rate and heated temperature-would result in varied precipitation in the matrix, which exhibits the varied post-formed strength of the material. Figure 5 shows the effects of heating rate on the strength of AA7075 for a series of heated temperatures.
In order to suppress the natural ageing effect, hardness was used instead of yield strength for a quick manual manipulation. Two heating rates, 1 °C /s and 5 °C /s, were compared. As can be seen in this figure, for both heating rates, the hardness drops initially, which is believed to be caused by the coarsening of precipitates (η'). Compared to the as-received T6 condition, the microstructure is similar to an over-aged state, resulting in the reduced strength. When the temperature is higher than 400 °C , the hardness level starts to increase as the dissolution of coarsened precipitates occurs, and solid solution hardening contributes to this. Compared the hardness levels of both heating rates, initially, hardness using 1 °C /s is lower than that of 5 °C /s. The reason is believed to be that, for a higher heating rate, the relative soaking time is short and the coarsening of precipitates is not severe. Meanwhile, when the temperature is higher than 400 °C , a slower heating rate would result in a longer heating time, which contributes to the dissolution of more precipitates, and the resulted solid solution hardening is enhanced, which results in the reversion of hardness levels. The comparisons between experimental results (symbols) and model predictions (solid lines) are also shown in Figure  5, where symbols represent the experimental determined hardness and the solid lines represent the model predictions. Both the above-mentioned hardness evolutions can be predicted excellently using the established model.

Effects of Preheating on Strength
As reasoned in Figure 2, preheating-i.e., heating rate and heated temperature-would result in varied precipitation in the matrix, which exhibits the varied post-formed strength of the material. Figure 5 shows the effects of heating rate on the strength of AA7075 for a series of heated temperatures.  Figure 6 shows the modeling results of the microstructures during heating of the alloy using heating rates of 0.1, 1 and 5 °C /s. In general, before the temperature of 400 °C , the coarsening of precipitates occurs during heating, and a higher coarsening speed is observed with a lower heating rate. Dissolution occurs when the heated temperatures reach a critical value, which has been predicted in Figure 6b (Tc is defined as 350 °C in this study [19]). Solute concentration starts to increase due to the dissolution that occurs with the decreasing volume fraction of precipitates. In addition, a comparatively higher critical temperature is predicted with higher heating rates, as shown in Figure   Figure 5. Effect of heating rate on hardness of AA7075-T6 heated to different temperatures, where the symbols represent the experimental determined hardness and the solid lines represent the model predictions.
In order to suppress the natural ageing effect, hardness was used instead of yield strength for a quick manual manipulation. Two heating rates, 1 • C/s and 5 • C/s, were compared. As can be seen in this figure, for both heating rates, the hardness drops initially, which is believed to be caused by the coarsening of precipitates (η'). Compared to the as-received T6 condition, the microstructure is similar to an over-aged state, resulting in the reduced strength. When the temperature is higher than 400 • C, the hardness level starts to increase as the dissolution of coarsened precipitates occurs, and solid solution hardening contributes to this. Compared the hardness levels of both heating rates, initially, hardness using 1 • C/s is lower than that of 5 • C/s. The reason is believed to be that, for a higher heating rate, the relative soaking time is short and the coarsening of precipitates is not severe. Meanwhile, when the temperature is higher than 400 • C, a slower heating rate would result in a longer heating time, which contributes to the dissolution of more precipitates, and the resulted solid solution hardening is enhanced, which results in the reversion of hardness levels. The comparisons between experimental results (symbols) and model predictions (solid lines) are also shown in Figure 5, where symbols represent the experimental determined hardness and the solid lines represent the model predictions. Both the above-mentioned hardness evolutions can be predicted excellently using the established model. Figure 6 shows the modeling results of the microstructures during heating of the alloy using heating rates of 0.1, 1 and 5 • C/s. In general, before the temperature of 400 • C, the coarsening of precipitates occurs during heating, and a higher coarsening speed is observed with a lower heating rate. Dissolution occurs when the heated temperatures reach a critical value, which has been predicted in Figure 6b (T c is defined as 350 • C in this study [19]). Solute concentration starts to increase due to the dissolution that occurs with the decreasing volume fraction of precipitates. In addition, a comparatively higher critical temperature is predicted with higher heating rates, as shown in Figure 6b, which corresponds well with the findings in previous publications.  Figure 6 shows the modeling results of the microstructures during heating of the alloy using heating rates of 0.1, 1 and 5 °C /s. In general, before the temperature of 400 °C , the coarsening of precipitates occurs during heating, and a higher coarsening speed is observed with a lower heating rate. Dissolution occurs when the heated temperatures reach a critical value, which has been predicted in Figure 6b (Tc is defined as 350 °C in this study [19]). Solute concentration starts to increase due to the dissolution that occurs with the decreasing volume fraction of precipitates. In addition, a comparatively higher critical temperature is predicted with higher heating rates, as shown in Figure  6b, which corresponds well with the findings in previous publications.  Figure 7 shows the effects of preheating on the hot flow behavior for different temperatures using three different heating rates, 0.5, 1, and 5 °C /s, where the solid symbols represent the experimental results and the solid lines are the model predictions. The used strain rate was 1/s. Good predictions were achieved as both the flow stress level and ductility could be precisely predicted. For the stress-strain curves at 350 °C shown in Figure 7a, the flow stress level of 5 °C /s is the highest among the three heating rates, while those of the other two rates are similar. Such a finding is consistent with that of hardness variation in Figure 5. During heating to 350 °C , the coarsening of precipitates occurs, in which that of 5 °C /s is the smallest, resulting in the highest flow stress level. In addition, for all three rates, an obvious softening of flow stress can be observed, which is believed to be caused by the occurrence of dynamic recrystallisation and microdamage around precipitates. During stretching, precipitates are hard particles, and microcracks are easy to initiate at the  Figure 7 shows the effects of preheating on the hot flow behavior for different temperatures using three different heating rates, 0.5, 1, and 5 • C/s, where the solid symbols represent the experimental results and the solid lines are the model predictions. The used strain rate was 1/s. Good predictions were achieved as both the flow stress level and ductility could be precisely predicted. For the stress-strain curves at 350 • C shown in Figure 7a, the flow stress level of 5 • C/s is the highest among the three heating rates, while those of the other two rates are similar. Such a finding is consistent with that of hardness variation in Figure 5. During heating to 350 • C, the coarsening of precipitates occurs, in which that of 5 • C/s is the smallest, resulting in the highest flow stress level. In addition, for all three rates, an obvious softening of flow stress can be observed, which is believed to be caused by the occurrence of dynamic recrystallisation and microdamage around precipitates. During stretching, precipitates are hard particles, and microcracks are easy to initiate at the boundaries due to the incompatible deformation between precipitates and matrix. For the strain to failure, the slower the heating rate, the greater the ductility; the reason for this is that microcracks may initiate around the precipitates during stretching. The denser the precipitates distributed, the higher the probability of damage, and, subsequently, the ductility becomes lower. With further heating of the material to 400 • C, the previously coarsened precipitates begin to dissolve; a quicker heating rate induces a relatively shorter soaking time. Therefore, the flow stress level of 5 • C/s is still the highest, as the solid solution hardening is limited. The flow stress levels of 1 and 0.5 • C/s are similar as the coarsened precipitates might be dissolved to a similar extent at such a high temperature.

Effects of Preheating on Hot Flow Behavior
boundaries due to the incompatible deformation between precipitates and matrix. For the strain to failure, the slower the heating rate, the greater the ductility; the reason for this is that microcracks may initiate around the precipitates during stretching. The denser the precipitates distributed, the higher the probability of damage, and, subsequently, the ductility becomes lower. With further heating of the material to 400 °C, the previously coarsened precipitates begin to dissolve; a quicker heating rate induces a relatively shorter soaking time. Therefore, the flow stress level of 5 °C /s is still the highest, as the solid solution hardening is limited. The flow stress levels of 1 and 0.5 °C /s are similar as the coarsened precipitates might be dissolved to a similar extent at such a high temperature.

Model Verification
To further verify the robustness of this model, hot flow behaviors at a variety of temperatures and strain rates were predicted and compared to the experimental results (solid symbols). As shown in Figure 8a, the flow stress level, ductility and softening can be accurately predicted. The severest softening is observed for the curve at the temperature of 300 °C; this is assumed to be caused by damage around precipitates, as dynamic recrystallisation is unable to occur at this temperature. For different strain rates at the temperature of 400 °C, obvious hardening is observed, as both strain and strain rate hardening cooperatively functioned. The quick heating rate results in higher density of accumulated dislocation and a shorter time of recovery. The flow behavior is believed to be the mutual effect of dislocation accumulation, recovery and recrystallisation. Meanwhile, for the lowest strain rate, 0.1/s, a gentle increase in flow stress is observed, which might be caused by the grain growth. For the middle strain rate, 1/s, the severest softening is found as the dynamic recrystallisation dominates.

Model Verification
To further verify the robustness of this model, hot flow behaviors at a variety of temperatures and strain rates were predicted and compared to the experimental results (solid symbols). As shown in Figure 8a, the flow stress level, ductility and softening can be accurately predicted. The severest softening is observed for the curve at the temperature of 300 • C; this is assumed to be caused by damage around precipitates, as dynamic recrystallisation is unable to occur at this temperature. For different strain rates at the temperature of 400 • C, obvious hardening is observed, as both strain and strain rate hardening cooperatively functioned. The quick heating rate results in higher density of accumulated dislocation and a shorter time of recovery. The flow behavior is believed to be the mutual effect of dislocation accumulation, recovery and recrystallisation. Meanwhile, for the lowest strain rate, 0.1/s, a gentle increase in flow stress is observed, which might be caused by the grain growth. For the middle strain rate, 1/s, the severest softening is found as the dynamic recrystallisation dominates.
boundaries due to the incompatible deformation between precipitates and matrix. For the strain to failure, the slower the heating rate, the greater the ductility; the reason for this is that microcracks may initiate around the precipitates during stretching. The denser the precipitates distributed, the higher the probability of damage, and, subsequently, the ductility becomes lower. With further heating of the material to 400 °C, the previously coarsened precipitates begin to dissolve; a quicker heating rate induces a relatively shorter soaking time. Therefore, the flow stress level of 5 °C /s is still the highest, as the solid solution hardening is limited. The flow stress levels of 1 and 0.5 °C /s are similar as the coarsened precipitates might be dissolved to a similar extent at such a high temperature.

Model Verification
To further verify the robustness of this model, hot flow behaviors at a variety of temperatures and strain rates were predicted and compared to the experimental results (solid symbols). As shown in Figure 8a, the flow stress level, ductility and softening can be accurately predicted. The severest softening is observed for the curve at the temperature of 300 °C; this is assumed to be caused by damage around precipitates, as dynamic recrystallisation is unable to occur at this temperature. For different strain rates at the temperature of 400 °C, obvious hardening is observed, as both strain and strain rate hardening cooperatively functioned. The quick heating rate results in higher density of accumulated dislocation and a shorter time of recovery. The flow behavior is believed to be the mutual effect of dislocation accumulation, recovery and recrystallisation. Meanwhile, for the lowest strain rate, 0.1/s, a gentle increase in flow stress is observed, which might be caused by the grain growth. For the middle strain rate, 1/s, the severest softening is found as the dynamic recrystallisation dominates.