Microstructure evolution and physical-based diffusion constitutive analysis of Al-Mg-Si alloy during hot deformation

Journal Pre-proof Microstructure evolution and physical-based diffusion constitutive analysis of Al-Mg-Si alloy during hot deformation

Shuhui Liu, Qinglin Pan, Mengjia Li, Xiangdong Wang, Xin He, Xinyu Li, Zhuowei Peng, Jianping Lai PII:

S0264-1275(19)30619-7

DOI:

https://doi.org/10.1016/j.matdes.2019.108181

Reference:

JMADE 108181

To appear in:

Materials & Design

Received date:

14 May 2019

Revised date:

27 August 2019

Accepted date:

2 September 2019

Please cite this article as: S. Liu, Q. Pan, M. Li, et al., Microstructure evolution and physical-based diffusion constitutive analysis of Al-Mg-Si alloy during hot deformation, Materials & Design(2018), https://doi.org/10.1016/j.matdes.2019.108181

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© 2018 Published by Elsevier.

Journal Pre-proof

Microstructure evolution and physical-based diffusion constitutive analysis of Al-Mg-Si alloy during hot deformation Shuhui Liua, Qinglin Pana,b, Mengjia Lib, Xiangdong Wangb, Xin Hea, Xinyu Lia, Zhuowei Pengc, Jianping Laib a

Light Alloy Research Institute, Central South University, Changsha 410083, China

b

School of Materials Science and Engineering, Central South University, Changsha 410083, china

c

Hunan Zhongxing New Materials Co., Ltd. Changsha 410083, China

Abstract: The hot deformation behavior of Al-Mg-Si alloy is studied based on diffusion

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mechanism. The quantity of low angle grain boundaries increases rapidly, accompanied by

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the formation of subgrains and recrystallization grains with the increase of strain. Dynamic recovery (DRV) and continuous dynamic recrystallization (CDRX) are the main softening

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mechanisms, of which the DRV is dominant. The kinetics of dynamic recrystallization (DRX) represented by Avrami relationship shows that the DRX volume fraction increases

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with increasing strain. Physical-based diffusion constitutive model is established to demonstrate the flow behavior of the alloy. The relationship between diffusion activation

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energy and deformation condition is analyzed, and the dependence of creep exponent on temperature and strain is discussed. The result shows that the model describes the flow

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stress accurately. Lattice diffusion is the main diffusion mechanism during hot deformation. The variation of creep exponent can be reflected by dislocation density when the

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deformation mechanism is controlled by dislocation motion. Dynamic precipitation and the impediment of dislocation motion can lead to high value of creep exponent of the alloy. Keywords: Hot deformation; Microstructure evolution; Diffusion model; Creep exponent.

1.Introduction Hot deformation processes are widely applied in manufacture industry for forming and improving properties of metals and alloys. Understanding the flow behavior during hot deformation is important for designers to optimize the forming processes. The coupled effects of strain hardening and thermal softening are evolving with the deformation conditions, i.e. temperature, strain rate and strain, leading to dislocation density variation and microstructure evolution, and finally affect the properties of the products. Hence, it is necessary to study the flow behavior and microstructure evolution during hot deformation 

Corresponding author. E-mail address: [email protected] (M.Li). 1

Journal Pre-proof [1-3]. It is well known that the microstructure of the alloy is sensitive to deformation conditions. The effect of thermal softening is enhanced when increasing deformation temperature or decreasing strain rate [1]. Dynamic recovery (DRV), dynamic recrystallization (DRX) and the coarsening of dynamic precipitating are considered to be the main motivation of thermal softening during thermal-plastic deformation [4]. DRV refers to the dislocation rearrangement and annihilation by dislocation slipping and climbing. The onset of DRX is crucial to control the microstructure evolution and consequently improves the mechanical properties of products. Three types of DRX for

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aluminum alloys were reported [5, 6]: continuous dynamic recrystallization (CDRX),

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discontinuous dynamic recrystallization (DDRX) and geometric dynamic recrystallization (GDRX). The occurrence of CDRX is usually attributed to progressive subgrain rotation in

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high stacking fault energy (SFE) materials. Low angle grain boundaries (LAGBs) can be formed when dislocations accumulate adequately, the misorientation increase due to the

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continuous subgrain boundary rotation, leading to the transformation of LAGBs into high angle grain boundaries (HAGBs) progressively. CDRX is considered to the main softening

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mechanism of Al-Mg-Si alloys [7]. DDRX can be initiated with strain induced grain boundary motion (SIBM), which refers to local grain boundary bugling due to the

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dislocation density difference across the grain boundary [8]. The DDRX progress involves clear nucleation and growth stage, which occurs mainly in low SFE materials. GDRX

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would appear when deformation reaches critical level as the original grain size and deformation conditions are appropriate, which is another common softening mechanism of Al-Mg-Si alloys [9]. Meanwhile, second-phase particles play an important role in recrystallization. Fine second-phase particles slow down grain boundary motion through Zener drag effect, while coarse particles speed up recrystallization by particle stimulated nucleation because of the adequate stored energy in deformed region [10]. Constitutive models are widely used to demonstrate the relationship between flow stress and prevailing deformation conditions. Physical-based models have attracted extensive interest in analyzing the flow behavior in aluminum alloys as they can describe the deformation mechanism with the perspective of physical characteristic [7, 11-14]. Zerilli-Armstrong model (ZA) [15, 16], Avrami model [7], Mechanical threshold stress model (MTS) [17, 18] and Mecking-Bergstrom model (MB) [19] are conductive in describing flow behavior during deformation. However, ZA model is considered to be unsuitable for characterizing the hot behavior at low strain rate, and the reflection of related 2

Journal Pre-proof parameters and microstructure characteristic is limited, Avrami model is more appropriate for low SFE alloys, the derivation of deformation parameters in MTS model is complex, and the influence of dynamic precipitation is not considered in MB model, which limit their application. Cabrera et al. [20] developed a physical-based diffusion model with respect to creep exponent to characterize flow behavior during hot deformation. The model considers the combined effects of strain hardening, dynamic restoration and dynamic precipitation, which is appropriate to demonstrate the hot deformation behavior in Al-Zn-Mg-Cu alloys [21, 22]. Besides, the deformation activation energy of the alloy is 174 kJ mol-1 according to the Arrhenius constitutive analysis [15], which is higher than self-diffusion activation

of

energy (142 kJ mol-1 for lattice diffusion and 84kJ mol-1 for grain boundary diffusion [23]). It has been revealed that Young’s modulus is related to the variation of activation energy

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and creep exponent [20, 24]. So, diffusion model with function of Young’s modulus is

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introduced to characterize the relationship of flow stress and deformation conditions. The expression can be given as [21]:

re

 D(T )  B[sinh( E (T ))]n

(1)

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where 𝜀̇ and σ denote the strain rate and flow stress, respectively. B and  are material parameters treated as constants. n represents creep exponent. D(T) and E(T) are

na

self-diffusion coefficient and Young’s modulus taken as functions of temperature, respectively. When deformation mechanism is only controlled by dislocation sliding and climbing, the creep exponent is a theoretical value n = 5 [25]. However, it should be treated

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as variable because the deformation mechanism is affected not only by dislocation motion but also by dynamic precipitation and DDRX [21]. The variation of the creep exponent implies the changing trajectory of deformation characteristic during deformation. Al-Mg-Si alloys have been in common usage due to the excellent properties such as favorite weldability, strong corrosion resistance and good formability. Constitutive relationship considering diffusion mechanism has rarely been reported in Al-Mg-Si alloy. Besides, the understanding on creep exponent and diffusion activation energy of Al-Mg-Si alloy is still inadequate. So it is significant to construct the constitutive model focusing on the diffusion mechanism during hot deformation. In this work, the microstructure evolution and the diffusion constitutive model of a Al-Mg-Si alloy were studied systematically. The flow behavior and dislocation density during hot deformation were discussed. The effect of strain on microstructure evolution was investigated by analyzing the microstructure characteristics during deformation. The DRX kinetic equation was built to describe the 3

Journal Pre-proof relationship of strain and DRX volume fraction. Three types of physical-based diffusion models were introduced to establish the constitutive relationships: the original model with respect to lattice diffusion, the revised model with respect to the coupled effect of lattice diffusion and grain boundary diffusion, the effective model with considering material parameters as variables. 2. Experimental materials and procedures The chemical composition of the studied alloy is presented in Table 1. The ingot was formed by semi-continuous casting method. Pure aluminum (99.9%), iron, Al-Mn, Al-Cr, Al-Si master alloys and magnesium (99.9%) were successively added into the resistance

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furnace with the melt temperature of about 1023 K. After cooling to 993 K, the melt was

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poured into a copper mould. Then the cast ingot was obtained and homogenized at 818 K for 24 h, followed by water quenching. The homogenized sample was machined into

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cylindrical specimens with 10 mm in diameter and 15 mm in length. Isothermal compression test was conducted on a Gleebe 3500 simulation machine. The cylindrical

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specimens were isothermally compressed at 653 K to 803 K with intervals of 50 K and strain rates of 0.01 s-1, 0.1 s-1, 1 s-1, 5 s-1 and 10 s-1, respectively, with height reduction of

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50%. The specimens were heated to the given temperature with heating rate of 5 Ks-1 and then held for 3 minutes before compression to ensure uniform temperature distribution.

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After compression, the specimens were quenching into water immediately to maintain the microstructure. Thermocouple wires were connected to the specimens in the middle of the

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length direction to obtain the real-time temperature data. Graphite foils were placed at both ends of the specimens to reduce friction during compression. The microstructure was observed by transmission electron microscopy (TEM, FEI-F20) technique equipped with energy dispersive spectrometer (EDS) and electron back scattered diffraction (EBSD, ZEISS EVO MA10) equipped with Oxford instruments. The specimens for EBSD analysis were acquired from the central part of the samples along the compression direction. The required EBSD data was obtained from HKL Channel 5 software analysis. In the corresponding EBSD maps, white lines, purple lines, yellow lines and black lines represent boundaries with misorientation angles of 2-5°, 5-10°, 10-15° and >15°, respectively. Table 1 The chemical composition of the alloy (wt. %) Composition

Mg

Si

Mn

Cr

Fe

Al

Content

0.90

1.02

0.55

0.12

0.20

Bal.

3. Results 3.1 Flow behavior and dislocation density 4

Journal Pre-proof A set of flow stress curves of the studied alloys is presented in Fig. 1, and the values of yield strength at different deformation conditions are shown in Table 2. At strain rates of 5 s-1 and 10 s-1, the required strain can be reached in an instant, and the data acquisition is not sensitive enough, which causes data fluctuation at the equilibrium position and leads to the instability in elastic deformation stage of flow stress curves. It can be seen from Fig. 1 that the peak stress and steady state stress increase as deformation temperature decreases or strain rate increases. It is generally known that atomic diffusion and dislocation movement is limited at low temperatures, and the total time for activation energy

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accumulation is relatively short at high strain rates, which leads to the increase of flow

Fig. 1. Flow stress curves of the alloy under different conditions: (a) 0.01 s-1; (b) 0.1 s-1; (c) 1 s-1; (d) 5 s-1; (e) 10 s-1 [15] and (f) 753 K/0.1 s-1. 5

Journal Pre-proof Table 2 Values of yield strengh obtained from the flow stress curves (MPa) Strain rate/s-1

653K

703K

753K

803K

0.01 0.1 1 5 10

40.72 46.26 53.43 65 69.58

32.9 37.14 40.21 52.49 66.56

24.62 27.48 36.3 47.36 49.5

19.92 25.72 30.09 44.28 47.66

stress. Meanwhile, The flow stress increases rapidly at the beginning of deformation, then remains stable after peak stress, which is the result of the co-operation of strain hardening and thermal softening during hot deformation. Fig. 2 shows the strain hardening rate

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ddversus temperature at 0.01s-1. The strain hardening rate is relatively high at the initial stage of deformation, then reduces rapidly and remains in stable state with the

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increase of strain, which shows that the influence of strain hardening is much greater than thermal softening at the initial deformation stage, then reaches a dynamic balance as

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deformation continues. The occurrence of strain hardening is attributed to the dislocation

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accumulation and multiplication. Dynamic softening occurs due to the dislocation rearrangement and annihilation, which leads to the decrease of strain hardening rate. When

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is close to 0.

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the two competing processes reach a dynamic equilibrium, the value of strain hardening rate

Fig. 2. Strain hardening rate of the alloy at 0.01 s-1.

The variation of dislocation density is an important characteristic of microstructure evolution. Generally speaking, dislocation density increases because of the dislocation accumulation in strain hardening stage. The DRX occurs when the dislocation accumulation is adequate and the driving force for dynamic softening is large enough, which contribute to the decrease of dislocation density. Then the DRX grains form at grain boundaries and other crystal defect site as the strain reaches the critical level [10]. The critical stress c represents the stress at the onset of DRX [7], which can be defined as the inflection point of the strain 6

Journal Pre-proof hardening ratewith respect to stress line before peak stress [26]. Then the critical strain c can be obtained from the flow stress curve. The -curve at 0.01 s-1 is shown in Fig. 3(a), which illustrates the related stress parameters. The stress of = 0 is peak stress p, and the corresponding strain is peak strain

p, while the saturated stress s can be determined by the extension of the linear portion in

-p

ro

of

-curve to = 0 [27]

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Fig. 3. (a) The plot at 0.01 s-1; (b) The -2plot at 0.01 s-1.

The relationship between dislocation density and strain can be estimated by Estrin and

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Mecking’s DRV model [28, 29], which can be expressed as: (2)

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d  d  h  r 

where  is the dislocation density, h is the athermal strain hardening rate and r denotes the

Eq.(2) [30]:

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rate of DRV at specific deformation condition. Then the flow stress can be obtained from

  [ s2  ( s2   02 ) exp(r )]0.5

(3)

where the σ0 represents the initial stress and can be neglected [31]. Then Eq.(3) can be simplified as:

  0.5r s2  0.5r 2

(4)

The plots of -2 at strain rate of 0.01s-1 are shown in Fig. 3(b). The values of r and

s can be obtained from the slopes and intercepts of the fitting lines, respectively. The calculated result shown in Table 3 implies that r is increased at high deformation temperatures and low strain rates. Generally, the effect of strain hardening and thermal softening reaches a dynamic equilibrium in the steady stage, where the flow stress keeps in stable and = 0. Then the dislocation density should remain unchanged and d/d0. So Eq.(2) transforms into 7

Journal Pre-proof Table 3 Values of the rate of DRV Strain rate/s-1

653K

703K

753K

803K

0.01 0.1 1 5 10

46.05 34.06 20.02 13.78 10.68

58.79 39.97 26.49 20.11 12.13

96.82 53.31 47.61 31.07 15.32

836.34 184.41 55.47 52.38 34.97

Eq.(5) in the steady stage: s  h r

(5)

with

s 2 ) b

of

h  r(

(6)

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where  is the shape factor and equals to 1 [28],  is the shear modulus and b is the

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magnitude of Burger's vector [30]. Then the steady stage dislocation density can be determined. The corresponding critical dislocation density can be obtained according to

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Eq.(2). c  (1  exp(r c )) h r

(7)

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The steady stage dislocation density and critical dislocation density at different deformation conditions are shown in Fig. 4. It can be seen that the critical dislocation

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density is higher than steady stage dislocation density. Meanwhile, dislocation density decreases with increasing temperature and decreasing strain rate. The dislocation motion

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can be accelerated at high temperatures and the time for dislocation motion is longer at low strain rates, which is benefit to promote the dislocation rearrangement and annihilation, leading to the decrease of dislocation density and the decline of flow stress.

Fig.4. Variation of (a) steady state dislocation density and (b) critical dislocation density.

3.2 Microstructure evolution The EBSD images of the alloy deformed at 753 K/0.1 s-1 at various strains are 8

Journal Pre-proof displayed in Fig. 5. It can be seen that the deformed grains become slender with increasing strain. Meanwhile, there are many white and colored lines in the interior of grains, indicating the presence of LAGBs, which may be due to the occurrence of thermal softening during hot deformation. As shown in Fig. 5(a), a slight dynamic softening behavior can be observed. A few of LAGBs scatter in the deformed grains with most locating near the deformed grain boundaries and triple junctions. Simultaneously, individual DRX grains intersperse at deformed grain boundaries as marked with white arrows, indicating that DRV and DRX take place preferentially at deformed grain boundaries. As deformation continues, the number of DRX grains increases as shown in Fig. 5(b). LAGBs occupy the whole

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deformed grains, implying that the quantity of LAGBs increases obviously. The LAGBs

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distribution contrast at 0.2 and 0.4 strains demonstrates that substructures are easier to form near the deformed grain boundaries at low strains, move toward the central of the deformed

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re

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grains when deformation continues. The misorientation between adjacent subgrains at

Fig. 5. EBSD images of the alloy at 753 K/0.1 s-1 with strains of (a) 0.2; (b) 0.4; (c) 0.7 and (d) 1.0. 9

Journal Pre-proof deformed grains is small as marked with white circle, which is the evidence of DRV. The misorientation of subgrain boundaries increases through diffusion and subgrains rotation [10]. When the misorientation angle of boundaries increases to larger than 15°, the subgrain boundaries transformed into grain boundaries, and the new DRX grains can be formed. As can be seen in the white magnified square, the pre-existing new DRX grains are nearly formed with partial LAGBs, some subgrain boundaries with misorientation angles of 5°-10° and 10°-15° can be observed, indicating the transformation process from LAGBs to HAGBs progressively with increasing misorientation. Such a microstructure evolution with continuous rotation of subgrains is the feature of CDRX. With the increase of strain as

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shown in Fig. 5(c-d), more subgrains and DRX grains can be observed, implying dynamic

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softening is enhanced. The subgrains rotate progressively followed by the further increase of misorientation. This matches with the misorientation angle data at LAGBs in Fig. 6, in

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which the mean misorientation angle at LAGBs increases from 4.22 at 0.2 strain to 5.57 at

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1.0 strain.

Fig. 6. Number fractions of subgrain boundaries at misorientation angles between 2° and 15°.

The statistical results of misorientation angles at 753 K/0.1 s-1 are shown Fig. 7. It can be seen that the mean misorientation angle and HAGBs fraction first decrease and then increase with increasing strain. At strain of 0.2, due to the slight dynamic softening, the microstructure of the alloy is consists of large number of deformed grains and a small amount of LAGBs, resulting in the high value of mean misorientation angle and HAGBs fraction. At high strains, i.e. larger than 0.4 strain, dynamic softening is enhanced, the number of LAGBs increases significantly, leading to a decrease in the mean misorientation angle and HAGBs fraction. Meanwhile, it should be pointed out that with the increase of strain from 0.4 to 1.0, the mean misorientation angle increases from 9.51 to 14.96, and the HAGBs fraction increases from 14.16% to 29.54%. The increase tendency under high 10

Journal Pre-proof strains indicates that LAGBs are transformed into HAGBs gradually by subgrains rotate, some subgrains are converted into DRX grains. The main dynamic softening mechanism of the alloy are DRV and CDRX, of which DRV is predominant according to the low DRX

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fraction.

1.0.

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Fig. 7. Statistical results at different misorientation angles with strains of (a) 0.2; (b) 0.4; (c) 0.7 and (d)

The TEM images deformed at 753 K/0.1 s-1 are presented in Fig. 8. At low strains as shown in Fig. 8(a), high density dislocations are pinned by second-phase particles, which reveals that no obvious dynamic softening take place. These second-phase particles are identified as AlMnFeSi and Si precipitates according to EDS analysis. The precipitates act as obstacles to pin the dislocations, resulting in relatively high value of flow stress. Fig. 8(b) shows a cell structures formed at regions filled with dislocations, indicating DRV occurs progressively. Dynamic softening is enhanced at high strains, more subgrains and DRX grains can be formed. The subgrain boundaries of subgrain are not straight as shown in Fig. 8(c). Dislocation walls are built on subgrain boundaries due to the dislocation motion and rearrangement. The dislocation density decreases continuously as a result of the dislocation motion in subgrains and rearrangement/annihilation at subgrain boundaries. Fig. 8(d) shows a DRX grain with regular and clear grain boundaries, which is the evidence of recrystallization. Subgrains boundaries are transformed into grain boundaries due to the 11

Journal Pre-proof sufficient rearrangement of dislocations, and eventually the DRX grains are formed. The

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dislocation density in the DRX grains decreases, leading to the decrease of flow stress.

Fig. 8. TEM images of the alloy at 753 K/0.1 s-1 with strains of (a) 0.2; (b) 0.4; (c) 0.7 and (d) 1.0.

3.3 Diffusion constitutive model

3.3.1 The construction of diffusion constitutive model The expression of Young’s modulus E(T) in the diffusion model shown in Eq.(1) can be described as [24]: E (T )  2(1  )   2(1  ) 0 (1 

TM d  T  300 ) 0 dT TM

(8)

where  is the Poisson’s coefficient and equals to 0.33. is shear modulus and is shear modulus at 300 K, TM is the melting point for aluminum. The term

𝑇𝑀 𝑑𝜇 𝜇0 𝑑𝑇

equals to -0.5

[32]. Another two expressions are introduced to obtain the material parameters through 12

Journal Pre-proof linear regression method.

 D(T )  B1[ E (T )]n

1

(9)

 D(T )  B2 exp(  E (T ))

where B1, B2, n1, are material constants and /n1 Then the diffusion model can be organized according to Eqs.(1) and (9). When considering the lattice diffusion in the original diffusion model, the term D(T) in Eqs.(1) and (9) is represented by DL(T), which can be expressed as [24]: DL (T )  D0 L exp(QL RT )

(10)

of

where D0L is the pre-exponential coefficient of lattice diffusion and equals to 1.7×10-4. QL represents activation energy and equals to 142 kJ mol-1 for lattice diffusion in aluminum. R

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is the ideal gas constant and defined as 8.314J mol-1K-1.

ln( DL (T ))  ln B1  n1 ln( E (T ))

(11)

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na

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ln( DL (T ))  ln B2    E (T )

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ln( DL (T ))  ln B  n ln[sinh( E (T ))]

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Taking natural logarithms to Eqs.(1) and (9) yield to

Fig. 9. Relationships between (a) 𝑙𝑛𝜀̇/𝐷𝐿 (𝑇) and ln(𝜎/𝐸(𝑇)) ; (b) 𝑙𝑛𝜀̇/𝐷𝐿 (𝑇) and 𝜎/𝐸(𝑇) ; (c) ln(𝜀̇/𝐷𝐿 (𝑇)) and ln[sinh(𝛼𝜎/𝐸(𝑇))]. 13

Journal Pre-proof Taken the steady state stress (=0.4) as an example to build the constitutive relationship. The values of n1, and B1, B2 are obtained from the slopes and the intercepts of the fitting lines ln(𝜀̇/𝐷𝐿 (𝑇)) -ln(𝜎/𝐸(𝑇)) and ln(𝜀̇/𝐷𝐿 (𝑇)) -𝜎/𝐸(𝑇) as shown in Figs.9(a) and (b), respectively. Then the value of can be determined. Similarity, the values of n and B can be gained from the slope and the intercept of the fitting lines ln(𝜀̇/𝐷𝐿 (𝑇)) - ln[sinh(𝛼𝜎/𝐸(𝑇))], respectively, as shown in Fig.9(c). It can be seen that n is a variable and evolving with deformation temperature. The values of the related material parameters in original model are presented in Table 4. It is necessary to take the compensation of strain into consideration when accurately

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characterizing the flow behavior of the alloy. The material parameters under different strains

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can be obtained by repeating the solution procedure discussed above at strains of 0.05 to 0.7 with interval of 0.05. A third-order polynomial function is introduced to display the

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relationship between material parameters and strain, the expression can be described as:

re

Y  Y0  Y1  Y2 2  Y3 3

(12)

where Y represents the material parameters (, n, lnB). Fig. 10 displays the polynomial

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fitting results. The adjusted R-squares of the three material parameters are 0.981, 0.995 and

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na

0.965, respectively. Then the original diffusion model can be established.

Fig. 10. Relationships between (a) α; (b) n; (c) lnB and true strain by polynomial fitting. 14

Journal Pre-proof Table 4 Values of material parameters in original model and revised model at strain of 0.4 Model



n

lnB

k1B(×1012)

k2B(×1016)

Original model

1199.48

7.03

30.10

/

/

Revised model

1254.36

6.89

/

9.63

7.84

During hot deformation process, the second-phase particles induce pinning effect on dislocation, which results in stress concentrations at the interface between particles and matrix. It was pointed out that both lattice diffusion and grain boundary diffusion were effective for relaxing the stress during deformation [33]. So, the coupled effect of lattice diffusion and grain boundary diffusion should be involved in diffusion model to better

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understand the intrinsic atomic mechanisms. When considering the coupled effect of lattice diffusion and grain boundary diffusion

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in the revised model, the term D (T ) in Eqs.(1) and (9) is represented by the effective

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diffusion coefficient Deff (T ) , which stands for the self-diffusion including lattice diffusion and grain boundary diffusion. The expression of Deff (T ) can be described as:

DGB (T )  D0GB exp(QGB RT )

lP

with

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Deff (T )  k1DL (T )  k2 DGB (T )

(13)

(14)

na

where DGB(T) is the self-diffusion coefficient of grain boundary diffusion. D0GB and QGB are the pre-exponential coefficient (5.0 × 10-14) and activation energy of grain boundary

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diffusion (84 kJ mol-1), respectively. k1, k2 are material constants [21]. The material parameters in the revised model can be obtained as described in original model. The values of k1 and k2 can be determined by binary linear equation. After strain compensation by a third-order polynomial fitting, the revised model is constructed and the related material parameters are confirmed as shown in Table 4. In the two types of diffusion models discussed above, material parameters such as creep exponent n and diffusion activation energy QL/Qeff are always considered as material constants, and the mean values are used in constitutive relationship derivation process. In fact, n varies with deformation temperatures, and is a temperature related parameter. Similarly, QL/DL (Qeff/Deff) are both temperature and strain rate dependent. So, a effective model is built with considering the material parameters as variables during hot deformation. The creep exponent n', material parameter B', effective activation energy Q'eff and effective self-diffusion coefficient D'eff in the effective model are treated as material variables. Then the effective model can be proposed as: 15

Journal Pre-proof   Deff' (T ,  ) B' (T ,  )[sinh(  E (T ))]n(T )

(15)

where the material variables Deff' (T ,  ) , B(T ,  ) are functions of temperature and strain rate, n'(T) is function of temperature. Taking the nature logarithm of Eq.(15) and substituting the expression of Deff' (T ,  ) , the following equation can be obtained: ln   n(T )[ln(sinh(  E(T )))]  ln B(T ,  )  ln D0eff  Qeff' (T ,  ) RT

(16)

The values of creep exponent at various temperatures can be derived from the revised model. The relationship of creep exponent and temperature at strain of 0.4 is obtained as

of

shown in Fig. 11(a) and expressed in Eq.(17). It is obvious that the value of creep exponent

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decreases with the increase of temperature. n(T )  0.0115T  15.2862

(17) The expression of Qeff (T ,  ) can be obtained by partial differentiation of Eq.(16) with

-p

'

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respect to temperature under specific strain rate, and simplified as:

Qeff' (T ,  )  Rn(T )[[ln(sinh(  E(T )))] (1 T )]  Rn(T )S ( )

(18)

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Then S ( ) can be obtained from the plot of ln(sinh(  / E (T ))) -(1/T) at different

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strain rates, as shown in Fig. 11(b). The relationship of S ( ) and ln  can be determined

Fig. 11. Relationships between (a) n' and T; (b) ln[sinh(𝛼′𝜎/𝐸(𝑇))] and1/T; (c) S and ln𝜀̇. 16

Journal Pre-proof by regression fitting (Fig. 11(c)), which is expressed in Eq.(19).

S ( )  2579.12  171.703ln   21.504(ln  )2

(19)

Substituting Eqs.(17) and (19) into Eq.(18), the expression of Q' (T ,  ) at different eff

temperatures and strain rates is determined.

Qeff' (T ,  )  246.59T  16.42T ln   2.06T (ln  )2  327779.46  21821.69ln   2732.88(ln  )2 (20) Then the term Deff' (T ,  ) can be determined. ln B(T ,  ) can be derived according to

ln  - [ln(sinh(  E(T )))] plots and expressed in Eq.(21). Finally, the effective diffusion

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model can be established.

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ln B(T ,  )  0.037T  50.25  0.97ln   0.25(ln  )2  39425.00 T  2624.69ln  T  328.71(ln  )2 T (21)

Fig. 12. Comparisons between predicted flow stress and experimental flow stress of three types of 17

Journal Pre-proof models under different conditions: (a) 0.01 s-1; (b) 0.1 s-1; (c) 1 s-1; (d) 5 s-1 and (e) 10 s-1.

3.3.2 Validity of prediction ability The predicted flow stress data and experimental flow stress data based on the three types of diffusion models are shown in Fig. 12. It can be seen that the predicted data coincides with the experimental data well, which means that the diffusion model can well reflect the flow behavior of the alloy. Meanwhile, the effective model displays the best prediction ability to track the flow stress among the three models. Validity of prediction ability of the three types of models is conducted by (AARE) and correlation coefficient (R), the expressions are shown as follows:



q i 1

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

( Ei  E )( Pi  P)

( Ei  E )2  i 1 ( Pi  P) 2 i 1 q

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R

1 q Ei  Pi  100%  n i 1 Ei

q

(22)

(23)

-p

AARE 

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where Ei and Pi are the experimental and predicted flow stresses, while E and P are the mean values of Ei and Pi, respectively. q represents the number of data included in the

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prediction. The correlation of the experimental and predicted data is illustrated in Fig. 13. It

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can be seen that most of the data points in the three types of models are close to the best

Fig. 13. Correlation between experimental and predicted results under three types of models: (a) original 18

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fitting line, which implies that the three types of diffusion models describe the flow stress accurately. The R values and AARE values of original model, revised model and effective model are 0.993, 0.992, 0.995 and 4.03%, 3.52%, 3.00%, respectively, implying that the effective model has the best accurate prediction ability among the three models. 4. Discussion 4.1 The analysis of DRX kinetics DRX is a crucial phenomenon during hot deformation, which reflects the microstructure evolution and affects the final properties of the alloy. Understanding the

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DRX kinetics is helpful to optimize the deformation conditions and obtain the desired microstructure characteristic. DRX kinetics can be applied in aluminum alloys [2, 7]. Based

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on the approach of the kinetics of DRX, a general form of Avrami equation is employed as

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follows [34]: X  1  exp(kt m )

(24)

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where X is the volume fraction of DRX grains, which can be obtained from EBSD data. t is

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the transformation time and a strain-function (   c )  p is used instead at fixed strain rate [27]. m and k are material constants. Rearranging the equation and taking the nature

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logarithm of both sides, the following expression can be obtained:

ln(ln1 (1  X ))  ln k  m ln t

(25)

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Then the values of m and k can be obtained from the slope and intercept of the fitting line ln(ln1 (1  X )) - lnt, respectively, as shown in Fig. 14(a). The fitting line shows favorable linear correlation. The predicted volume fraction of DRX curve along with the measured values of the alloy deformed at 753 K/0.1 s-1 is shown in Fig. 14(b). It can be seen that the predicted value matches well with the measured value. The volume fraction of DRX equals to zero unless strain exceeds the critical value, then increases with increasing strain. Besides, unlike classic S-shaped curves, the DRX kinetics curve of the alloy exhibits a rapid increase of DRX volume fraction at the initial stage, implying high DRX driving force at early stage of deformation. With increasing strain, the DRX fraction increases slowly with a nearly steady rate. Besides, the rate of DRX can be analyzed based on the theory from Liu [35]. The expression of the rate of DRX can be obtained by taking partial differentiation of DRX volume fraction with strain. 19

Journal Pre-proof 

X  kmt m1 exp(kt m ) 

(26)

where  represents the rate of DRX. Fig. 14(c) illustrates the rate of DRX with respect to strain at 753 K/0.1 s-1. It can be seen that DRX starts at a rapid rate at the initial deformation stage when strain exceeds critical value, followed by a quick decrease with the increase of strain, then maintain in a slow rate. After the incubation period, the rate of DRX is high at beginning because of the DRX driving force. With the increase of strain, the influence of DRX on unrecrystallized grains tends to be saturated and the deformation continues in the

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DRX grains, resulting in the low rate of DRX [36].

Fig. 14. (a) The relationship of ln(ln1 (1  X )) and lnt; (b) The predicted and measured results of DRX fraction; (c) The rate of DRX for deformation at 753K/0.1 s-1.

4.2 The analysis of diffusion mechanism and diffusion parameters Diffusion plays a considerable role during hot deformation. The transition information between lattice diffusion and grain boundary diffusion during deformation can be obtained from the diffusion model. The main diffusion mechanism is lattice diffusion if k1DL (T )  k2 DGB (T ) , otherwise is controlled by grain boundary diffusion under particular

deformation condition. The calculated result shows that the influence of lattice diffusion is stronger than grain boundary diffusion except at temperatures of 653 K and 703 K below the strain of 0.05, implying that the main diffusion mechanism is controlled by lattice 20

Journal Pre-proof diffusion. Besides, the diffusion activation energy map is demonstrated in Fig. 15(a), where the contour numbers stand for the values of effective diffusion activation energy. The values of diffusion activation energy of the alloy are between 104.14 to 188.39 kJ mol -1 at present deformation condition, and increase with decreasing temperature and strain rate. The diffusion activation energy higher than 142 kJ mol-1 at low temperature and strain rate can be attributed to the dislocation pinning effect caused by the second-phase particles [37]. In the case of lower than 142 kJ mol-1, it may be related to the dislocation motion and

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short-circuit diffusion along the grain boundary [38].

Fig. 15. Maps of (a) diffusion activation energy and (b) creep exponent at various deformation conditions.

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The creep exponent map is presented in Fig. 15(b), where the contour numbers stand for the values of creep exponent. It can be seen that the creep exponent value is larger than

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5 at present deformation condition and decreases with the increase of temperature and strain. The creep exponent value equals to 5 represents the softening mechanism is only controlled by dislocation gliding and climbing [25]. The reason for creep exponent value larger than 5 is related to the dynamic precipitates during hot deformation at low temperatures [39], and the grain boundary sliding may result in creep exponent value smaller than 5 at high temperatures and low strain rates [21]. In this work, the high value of creep exponent means that the deformation mechanism is mainly controlled by dislocation gliding and climbing. Dynamic precipitation and the impediment of dislocation motion could be the reasons for the value of creep exponent larger than 5 at present deformation condition. The fine Si and AlMnFeSi second-phase particles induce impediment effect on dislocation movement during deformation. Meanwhile, the effective time for dynamic softening is limited at high strain rates, the inadequate process of dislocation movement and reorganization makes contribution to the high value of creep exponent. Besides, the value of creep exponent varies with the deformation condition. The second-phase particles at different temperatures 21

Journal Pre-proof are presented in Fig. 16. It can be seen that numerous fine precipitates are surrounded by high density of dislocation at temperature of 653 K, implying the dislocation motion is impeded. As deformation temperature increases to 753 K, the size of second-phase particles increases and the number of hindered dislocations decreases, indicating a weak pinning effect on dislocation motion. The decreased dislocation density at high temperatures is consistent with the variation of steady state dislocation density as discussed in section 3.1. The pinning effect of second-phase particles on dislocations decreases when increasing temperature, which is benefit to the decrease of dislocation density and the decline of creep exponent. Dislocation density can reflect the variation of creep exponent when the

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deformation mechanism is controlled by dislocation motion. Meanwhile, DRX softening

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somehow reduces the value of creep exponent [25]. The phenomenon that DRX volume fraction increases with strain was discussed in section 4.1, which implies the high DRX

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fraction in large strains contributes to the creep exponent reduction.

Fig. 16. TEM images of the alloy at conditions of: (a) 653 K/10 s-1; (b)753 K/10 s-1.

5. Conclusions The hot deformation behavior of Al-Mg-Si alloy was studied through diffusion constitutive model. The microstructure evolution was analyzed by EBSD and TEM observations. The main results are presented as follows: (1) Diffusion models are suitable for predicting the flow behavior of the alloy. The effective model has the best prediction ability with R value of 0.995 and AARE value of 3.00%. (2) Critical dislocation density, steady state dislocation density and the rate of DRV of the alloy decrease with increasing temperature and decreasing strain rate. (3) High density of dislocations are pinned by the second-phase particles at the initial stage 22

Journal Pre-proof of deformation. The quantity of LAGBs increases obviously and the substructure network is developed with the increase of strain. DRV and CDRX operate together during hot deformation, between which DRV is predominant. (4) The volume fraction of DRX increases rapidly with strain at the initial deformation stage when it exceeds the critical value, then maintain in a slow increasing rate. The rate of DRX is high at beginning, then decreases rapidly with the increase of strain. (5) Lattice diffusion is the main diffusion mechanism at present deformation conditions. The diffusion activation energy is sensitive to deformation temperature and strain rate, the value of which are between 104.14 to 188.39 kJ mol-1. The impediment of precipitates on

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dislocation motion result in the value of creep exponent is larger than 5. Dislocation density

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can reflect the change of creep exponent when the deformation mechanism is controlled by

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dislocation motion.

Acknowledgements

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This work was supported by the Science and Technology Major Project of Changsha City, Hunan Province [grant number kc1703017]; and the Fundamental Research Funds for

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the Central Universities of Central South University [grant number 2018zzts153].

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Compliance with ethical standards

References

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Declarations of interest: None

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combined with weight optimization method of Al-7.82Zn-1.96Mg-2.35Cu-0.11Zr alloy during hot deformation, J. Alloys Compd. 732 (2018) 902-914. [7] C. Zhang, C. Wang, R. Guo, et al., Investigation of dynamic recrystallization and modeling of microstructure evolution of an Al-Mg-Si aluminum alloy during high-temperature deformation, J. Alloys Compd. 773 (2019) 59-70. [8] A. M. Wusatowska-Sarnek, H. Miura, T. Sakai, Nucleation and microtexture development under dynamic recrystallization of copper, Mater. Sci. Eng. A 323 (2002) 177-186. [9] C. Poletti, M. Rodriguez-Hortalá, M. Hauser, et al., Microstructure development in hot deformed AA6082, Mater. Sci. Eng. A 528 (2011) 2423-2430. [10] K. Huang, R. E. Logé, A review of dynamic recrystallization phenomena in metallic materials, Mater. Des. 111 (2016) 548-574. [11] C. Q. Huang, J. Deng, S. X. Wang, et al., A physical-based constitutive model to describe the strain-hardening and dynamic recovery behaviors of 5754 aluminum alloy, Mater. Sci. Eng. A 699 (2017) 106-113. [12] Y. C. Lin, W.Y. Dong, M. Zhou, et al., A unified constitutive model based on dislocation density for an Al-Zn-Mg-Cu alloy at time-variant hot deformation conditions, Mater. Sci. Eng. A 718 (2018) 165-172. [13] S. Acharya, R. K. Gupta, J. Ghosh, et al., High strain rate dynamic compressive behaviour of Al6061-T6 alloys, Mater. Charact. 127 (2017) 185-197. [14] W. S. Lee, Z. C. Tang, Relationship between mechanical properties and microstructural response of 6061-T6 aluminum alloy impacted at elevated temperatures, Mater. Des. 58 (2014) 116-124. [15] S. Liu, Q. Pan, H. Li, et al., Characterization of hot deformation behavior and constitutive modeling of Al–Mg–Si–Mn–Cr alloy, J. Mater. Sci. 54 (2018) 4366-4383. [16] D. Samantaray, S. Mandal, A. K. Bhaduri, et al., Analysis and mathematical modelling of elevated temperature flow behaviour of austenitic stainless steels, Mater. Sci. Eng. A 528 (2011) 1937-1943. [17] P. S. Follansbee, U. F. Kocks, A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable, Acta Metall. 36 (1988) 81-93. [18] B. Banerjee, The Mechanical Threshold Stress model for various tempers of AISI 4340 steel, Int. J. Solids Struct. 44 (2007) 834-859. [19] Lin, C. Y, Chen, et al., A physically-based constitutive model for a typical nickel-based superalloy, Comput. Mater. Sci. 83 (2014) 282-289. [20] J. M. Cabrera, A. A. Omar, J. M. Prado, et al., Modeling the flow behavior of a medium carbon microalloyed steel under hot working conditions, Metall. Mater. Trans. A 28 (1997) 2233-2244. [21] S. Wang, J. R. Luo, L. G. Hou, et al., Physically based constitutive analysis and microstructural evolution of AA7050 aluminum alloy during hot compression, Mater. Des. 107 (2016) 277-289. [22] X. Wang, Q. Pan, S. Xiong, et al., Prediction on hot deformation behavior of spray formed ultra-high strength aluminum alloy—A comparative study using constitutive models, J. Alloys Compd. 735 (2018) 1931-1942. 24

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[23] M. F. A. H.J. Frost, Deformation-mechanism maps: The Plasticity and Creep of Metals and Ceramics, Pergamon Press, Oxford (1982). [24] A. Thomas, M. El-Wahabi, J. M. Cabrera, et al., High temperature deformation of Inconel 718, J. Mater. Process. Technol. 177 (2006) 469-472. [25] H. Mirzadeh, J. M. Cabrera, A. Najafizadeh, Constitutive relationships for hot deformation of austenite, Acta Mater. 59 (2011) 6441-6448. [26] E. I. Poliak, J. J. Jonas, A one-parameter approach to determining the critical conditions for the initiation of dynamic recrystallization, Acta Mater. 44 (1996) 127-136. [27] J. J. Jonas, X. Quelennec, L. Jiang, et al., The Avrami kinetics of dynamic recrystallization, Acta Mater. 57 (2009) 2748-2756. [28] Y. Estrin, H. Mecking, A unified phenomenological description of work hardening and creep based on one-parameter models, Acta Metall. 32 (1984) 57-70. [29] K. D. A. Momeni, G.R. Ebrahimi, Modeling the initiation of dynamic recrystallization using a dynamic recovery model.pdf, J. Alloys Compd. 509 (2011) 9387-9393. [30] X. Q. John J. Jonas, Lan Jiang., The Avrami kinetics of dynamic recrystallization, Acta Mater. 57 (2009) 2748-2756. [31] A. Momeni, K. Dehghani, G. R. Ebrahimi, Modeling the initiation of dynamic recrystallization using a dynamic recovery model, J. Alloys Compd. 509 (2011) 9387-9393. [32] A. G. Atkins, Deformation-mechanism maps (the plasticity and creep of metals and ceramics), Pergamon Press, 1982. [33] F. J. Humphreys, P. N. Kalu, Dislocation-particle interactions during high temperature deformation of two-phase aluminium alloys, Acta Metall. 35 (1987) 2815-2829. [34] M. Avrami, Granulation, Phase Change, and Microstructure Kinetics of Phase Change. III, J. Chem. Phys. 9 (1941) 177-184. [35] J. Liu, Z. Cui, L. Ruan, A new kinetics model of dynamic recrystallization for magnesium alloy AZ31B, Mater. Sci. Eng. A 529 (2011) 300-310. [36] A. Hadadzadeh, F. Mokdad, M. A. Wells, et al., Modeling dynamic recrystallization during hot deformation of a cast-homogenized Mg-Zn-Zr alloy, Mater. Sci. Eng. A 720 (2018) 180-188. [37] S. Chen, K. Chen, G. Peng, et al., Effect of heat treatment on hot deformation behavior and microstructure evolution of 7085 aluminum alloy, J. Alloys Compd. 537 (2012) 338-345. [38] G. Meng, B. Li, H. Li, et al., Hot deformation and processing maps of an Al–5.7wt.%Mg alloy with erbium, Mater. Sci. Eng. A 517 (2009) 132-137. [39] M. Malu, J. K. Tien, The elastic modulus correction term in creep activation energies: applied to oxide dispersion strengthened superalloy, Scripta Metallurgica 9 (1975) 0-1120.

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Author statement Manuscript title: Microstructure evolution and physical-based diffusion constitutive analysis of Al-Mg-Si alloy during hot deformation

Authors’ contributions:

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Shuhui Liu Formal analysis, Writing - original draft; Writing - review & editing Qinglin Pan Funding acquisition and Validation Mengjia Li Conceptualization, Formal analysis, Writing - review & editing Xiangdong Wang Data curation Xin He Methodology Xinyu Li Investigation Zhuowei Peng Investigation Jianping Lai Investigation

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All authors have read and commented on the revised manuscript.

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Highlights 

Three types of physical-based diffusion models are established to demonstrate the flow behavior of the alloy.



The main diffusion mechanism during hot deformation is determined by constitutive analyses.



The relationships between creep exponent, diffusion activation energy and deformation

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The microstructure evolution and the dynamic softening mechanisms are investigated

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during hot deformation.

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

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condition are discussed.

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