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Modeling of the flow stress behavior and microstructural evolution during hot deformation of Mg–8Al–1.5Ca–0.2Sr magnesium alloy
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ScienceDirect Journal of Magnesium and Alloys 2 (2014) 133e139 www.elsevier.com/journals/journal-of-magnesium-and-alloys/2213-9567
Full length article
Modeling of the flow stress behavior and microstructural evolution during hot deformation of Mge8Ale1.5Cae0.2Sr magnesium alloy Xiao Liu a,b, Luoxing Li a,*, Biwu Zhu a,b a
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, Hunan, China b College of Mechanical and Electrical Engineering, Hunan University Science and Technology, Xiangtan 411002, Hunan, China Received 15 February 2014; revised 8 May 2014; accepted 13 May 2014 Available online 18 June 2014
Abstract Hot compression tests of Mge8Ale1.5Cae0.2Sr magnesium alloy were performed on the Gleeble-1500 machine at temperatures of 300, 350, 400, and 450 C and strain rates of 0.01, 0.1 and 1 s1. The flow stress behavior and microstructural evolution were followed. The work hardening was derived according to the LaasraouieJonas model. An improved approach, which considered the influence of yield stress on flow stress and the effect of grain boundary (GB) migration on the evolution of dislocation density during compression, was used to simulate the microstructural evolution, the flow stress and the volume fraction recrystallized of Mge8Ale1.5Cae0.2Sr magnesium alloy. The simulated results are in good agreement with experimental results. Copyright 2014, National Engineering Research Center for Magnesium Alloys of China, Chongqing University. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Magnesium alloy; Work hardening; Cellular automata; Volume fraction recrystallized; Microstructure
1. Introduction Mge8Ale1.5Cae0.2Sr magnesium alloy with excellent high-temperature creep properties has good application prospect. Microstructure of metallic materials, including grain size, morphology, largely affects the mechanical and functional properties of final products [1]. It is necessary to elucidate the microstructural evolution during deformation. Dynamic recrystallization (DRX) is a common phenomenon in magnesium alloys [2e4]. It is generally accepted that DRX processes have the following characteristics: (1) Critical * Corresponding author. E-mail address: [email protected] (L. Li). Peer review under responsibility of National Engineering Research Center for Magnesium Alloys of China, Chongqing University
Production and hosting by Elsevier
dislocation density and critical strain are required for onset of DRX; (2) The recrystallized grains (R-grains) are equiaxed and the average grain size stays same at a given deformation condition; (3) Preexistent grain boundaries (GBs) are usually nucleation sites [5e8]. It is difficult to study the microstructural evolution characteristics during grain growth by physical experiments. Computer simulations have been used by many researchers to simulate the microstructure evolution during deformation [5,9e11,14,15]. Goetz and Seetharaman [9] first tried to predict DRX and flow stress by cellular automaton (CA) method. This method did not consider the effects of the yield stress, GB migration, and the hot deformation parameters on DRX and the influence of the yield stress on flow stress. Chen et al. [11] predicted the microstructure evolution and flow stress in 30Cr2Ni4MoV rotor steel using deformation parameters coupling the CA method. But the effect of the yield stress on flow stress and the influence of GB migration on dislocation density were also not considered. Yazdipour et al. [12] has taken the yield stress into account and employed the KockseMecking
http://dx.doi.org/10.1016/j.jma.2014.05.001. 2213-9567/Copyright 2014, National Engineering Research Center for Magnesium Alloys of China, Chongqing University. Production and hosting by Elsevier B.V. All rights reserved.
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model [13] to describe the dynamic recovery. But this approach only could be applied when the mean free path for slip was constant and the effect of GB migration on dislocation density was not considered. Our previous work [14,15] has predicted the microstructure evolution in AZ31 magnesium alloy without considering the yield stress and did not predict the volume fraction recrystallized. In the present study, an improved approach, considering the influence of yield stress on flow stress and the effect of GB migration on dislocation density, was used to simulate the microstructural evolution of Mge8Ale1.5Cae0.2Sr magnesium alloy during hot compression. In this approach, the grain structure and the volume fraction recrystallized, which describes the percentage of recrystallized area were simulated. The work hardening based on the LaasraouieJonas model was also first derived. All the simulated results were compared with the experimental results. 2. Experimental The hot compression tests were carried out on Gleeble1500 to obtain the flow curves over four temperatures (300, 350, 400, and 450 C) with three strain rates (0.01, 0.1 and 1 s1). The alloy, used in the present study, was as extruded Mge8Ale1.5Cae0.2Sr with the nominal composition of 8 wt % Al, 1.5 wt% Ca, 0.3 wt% Mn and 0.2 wt% Sr. Cylindrical specimens with a diameter of 10 mm and a height of 15 mm were machined with the compression axis parallel to the extruded solid bar axis. To observe the microstructure, crosssections parallel to compression axis were cut from the deformed specimens. The samples were mounted, polished and etched. Then, the microstructure was observed by optical microcopy with an image analyzer. 3. Mathematical modeling 3.1. Modeling DRX
shown that DRX only occurs beyond critical strain from this equation. The velocity of grain boundary movement is the result of the net pressure on the boundary, which can be written as: ð3Þ Vi ¼ MFi 4pri2 dDb b Qb exp M¼ KT RT
ð4Þ
Fi ¼ 4pri2 tðrm ri Þ 8pri gi
ð5Þ
where Vi is the velocity of GB, M is the grain boundary mobility [17e19], d is the characteristic GB thickness, Db is the boundary self-diffusion coefficient, Qb is the boundary diffusion activation energy, K is Boltzmann's constant, R is the universal gas constant (8.31 J mol1 K1), T is the absolute temperature, Fi is the driving force [20], rm is the dislocation density of the matrix, ri is the dislocation density of ith Rgrain, ri is the radius of the ith R-grain, gi is GB energy. 3.2. Modeling flow stress and work hardening In previous studies [6,7,21], the flow stress is proportional to the square root of dislocation density (see Eq. (6)). To exactly predict flow stress during deformation, the influence of yield stress on flow stress (see Eq. (7)) and the effect of GB migration on dislocation density (see Eq. (1)) were considered to simulate the flow stress and dislocation density evolution. pffiffiffi s ¼ MaGb r ð6Þ pffiffiffi s ¼ so þ MaGb r
ð7Þ
where so is the yield stress, at which point r ¼ ro and ε ¼ εo, a is a constant of 0.5e1, G is the shear modulus, b is the Burger's vector. To predict the work hardening, the following expression is used and the detailed steps involving in this procedure are described in Ref. [22]. 1=2 swh ¼ s2sat s2sat s2o expð rðε εo ÞÞ ð8Þ
To describe the evolution of dislocation density during dynamic recovery, the modified LaasraouieJonas model in Eq. (1) [16] was used and the detailed steps involving in calculation of h and r were shown in our previous researches [10,14,15]. The effect of GB migration on dislocation density evolution was also taken into account in the modified LaasraouieJonas model.
where swh is the work hardening, ssat depending on h and r is the saturate flow stress.
dr ¼ ðh rrÞⅆε rⅆV
3.3. Procedure of simulation
ð1Þ
where r is the dislocation density, dε is the strain increment, h is the thermal work hardening rate, r is the rate of dynamic recovery, and dV is the volume swept by the mobile boundaries. Xiao et al. [5] has proposed the following expression according to experimental and theoretical analysis: p n_ ¼ C1 ε_ Z m ðε εc Þ
ð2Þ
where n_ is the nucleation rate, Z is the ZennereHollomon parameter, and C1, m, and p are the material constant. It is
The CA method is algorithm that represents discrete spatial and temporal evolution of complex systems by applying local or global deterministic or probabilistic transformation rules to the location of a lattice. The system objects are quantified according to generalized state variables [23]. The CA method can easily couple with the physical model to predict DRX during deformation process. In this model, a 2D square lattice and Moore neighboring were employed. In order to simulate the equiaxed growth of R-grain, a probabilistic transformation rule was set to determine which neighbor would switch.
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135
Fig. 1. Flow stress under various temperatures and the following strain rates: (a) 0.01 s1; (b) 0.1 s1; (c) 1 s1.
The initial microstructure was generated via a normal grain growth algorithm. The simulated lattices were set to 150 150, which corresponded to an area of 300 300 mm, and each lattice has three variables: one orientation variable that represents the grain orientation and determines the grain boundary energy; one states variable Sr that indicates which is the R-grains and is set to calculate the volume fraction recrystallized; one dislocation density variable that determines the site energy. The orientation of initial grain and R-grain were set randomly from 1 to 180. When the dislocation density exceeds critical dislocation density, DRX is initiated. Then the nucleation number can be obtained by Eq. (2) in each step. The nucleation sites were selected from the GBs and intersection points. For growth of R-grain, when the driving force for the growth of the ith R-grain is positive, this R-grain can grow continuously until the driving force reaches zero, and the growth velocity in each step is determined by Eq. (3). The
calculation was terminated, while the pre-set strain was attained. The flow stress was determined by the dislocation density and calculated using Eq. (7), suggesting that the evolution of microstructure affects the flow stress during simulation. The initial value of Sr is zero for each lattice. When the lattice changes into R-grain, the value of Sr for the lattice becomes one. The number of lattices for R-grains, can be calculated. Each lattice is corresponding to an area of 2 2 mm. Then the volume fraction recrystallized in each step can be calculated by the following equation: F¼
areanewgrain areatotal
ð9Þ
where F is the volume fraction recrystallized, areanewgrain is the recrystallized area, and areatotal is the simulated area.
Fig. 2. Microstructure observations at 300 C with a strain of 0.6 and the following strain rates: (a) 0.1 s1; (b) 1 s1.
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Fig. 3. The work hardening rate curves under various temperatures and the following strain rates: (a) 0.01 s1; (b) 0.1 s1; (c) 1 s1.
Fig. 4. The derived work hardening curves under different deformation conditions: (a) 350 C, and a strain rate of 0.01 s1; (b) 350 C and a strain rate of 0.1 s1; (c) 400 C and a strain rate of 0.01 s1; (d) 400 C and a strain rate of 0.1 s1.
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Fig. 5. The comparison between the simulated flow stress and experimental flow stress at a temperature of 400 C with various strain rates: (a) 0.1 s1; (b) 1 s1.
4. Results and discussion The flow stress curves obtained at various temperatures and strain rates are presented in Fig. 1. All the curves display the normal single peak and concave-down appearance associated with DRX. The flow stress increases to a maximum and then decrease to finally attain a steady state in the usual way. Most of the curves at strains from 0.02 to 0.15 are camber line, except the flow curves at 300 C and strain rates of 0.1 s1 and 1 s1. The flow curves of these two conditions at strains from 0.02 to 0.15 are close to a straight line, indicating that twinning occurs during deformation. This can be proved by Fig. 2. Optical microstructure at various strain rates and a temperature of 300 C with a strain of 0.6 are displayed in Fig. 2. Twins are observed inside grains and the number of twins increases with strain rate. It is well known that in order for a polycrystal to accommodate deformation, five independent slip system have to be activated. Basal slip and prismatic slip can only provide two independent slip modes. 1st order pyramidal slip can offer four independent slip modes, but this can not accommodate uniform deformation. Only 2-order pyramidal slip, with its non-basal Burger vectors, will accommodate deformation along the c-axis and meet the requirement for five independent slip modes. But dislocation slip can not meet requirement of uniform plasticity for its low migration velocity at high strain rate and so caused the initiation of twinning. Twinning was not considered during simulation. Therefore, flow stresses, microstructure evolution
and volume fraction recrystallized under these conditions were not simulated. In order to obtain the saturate flow stress and the rate of dynamic recovery, derivate method was applied to the flow stress curves of Fig. 1. The work hardening rates (ds/dε) under various temperatures and strain rates are displayed in Fig. 3. It can be seen that the work hardening increases with increasing temperature. According to the work hardening rate curves, the critical strain εc, the peak strain εp and the saturate stress ssat can be determined [24,25]. The rate of dynamic recovery r can be calculated by the slope of work hardening rate [15]. Then, the derived work hardening curves can be obtained using Eq. (8). The derived work hardening curves under different deformation conditions are displayed in Fig. 4. It can be seen that the derived work hardening is higher than the flow stress from critical stress. The work hardening approached rapidly to the saturate flow stress before a strain of 0.2, due to high work hardening rate. It can be concluded from Fig. 4 that the work hardening increases with decreasing temperature and increasing strain rate. The comparison between the simulated flow curves and the experimental flow curves are illustrated in Fig. 5. Both the simulated flow curves and the experimental flow curves attain a peak, followed by continuous flow softening. At 350 C with a strain rate of 0.1 s1, the simulated flow stress is a little bit higher than experimental flow stress after the critical strain, and then lower than the real flow stress. This discrepancy may be attributed to the possible involvement of additive high
Fig. 6. Simulated microstructure at a strain rate of 1 s1 and a strain of 0.6 with various temperatures: (a) 350 C; (b) 400 C; (c) 450 C.
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Fig. 7. Microstructure observations at a strain rate of 1 s1 and a strain of 0.6 with various temperatures : (a) 350 C; (b) 400 C; (c) 450 C.
energy sits for nucleation, such as dislocation tangle and dislocation pinning, and the effect of creep behavior on grain boundary migration which has not been taken into account during simulation. Nevertheless, the simulations agree well with the experimental results. The simulated microstructure at various temperatures and a strain rate of 1 s1 with a strain of 0.6 are shown in Fig. 6. Some of regions remain unrecrystallized at 350 C and 400 C. At 450 C, the equiaxed new grains almost replace all of the original microstructure. The volume of new grains increase with temperature. At 350 C and 400 C, initial grains can be observed and the volume of primary grains decreases with increasing temperature. At 450 C, the volume of primary grains dramatically decrease and primary grains are almost replaced by new grains. The experimental grain structures at various temperatures with a strain rate of 1 s1 are shown in Fig. 7. New grains, equiaxed and homogeneously distributed, appear in primary grain boundaries. There are several unrecrystallized grains at 350 C and 400 C, suggesting that DRX is not completely. At 450 C, the original microstructure are almost taken place by R-grains, indicating an approach to steady state flow (see Fig. 1(c)). New grains coarsen with increasing temperature, attributing to the increase of GB energy with temperature. As compared with Fig. 6, it is proved that the improved approach can accurately simulate the microstructure evolution. The simulated microstructural evolution at 400 C and a strain rate of 1 s1 with various strains are exhibited in Fig. 8. At a strain of 0.12, DRX has been initiated and the new
grains are nucleated at the grain boundaries. At a strain of 0.24, substantial new grains can be detected and replace initial grains. The new grains are equiaxed and homogeneously distributed. With the strain reaches to 0.42, the number of unrecrystallized regions has dramatically decreased. The nucleation mechanism obeys DRX characteristic. The simulated volume fraction recrystallized curves are shown in Fig. 9. The volume fraction recrystallized is used to describe the percentage of recrystallized area [3,26,27]. The volume fraction recrystallized increases with strain. The increase of temperature leads to higher grain boundary velocity and lower nucleation rate. Nevertheless, the increment of grain boundary velocity is faster than the decrement of nucleation rate. Thus, the volume fraction recrystallized increases with temperature. The simulated volume fraction recrystallized curves match the actual situation well. 5. Conclusions An improved approach coupling the cellular automaton method and metallurgical principles of dynamic recrystallization has been proposed to simulate the evolution of microstructure, the flow stress and the volume fraction recrystallized during hot compression. This method, considering the influence of the yield stress on flow stress and the effect of GB migration on dislocation density, can accurately predict DRX. The growth variation of each R-grain, including dislocation density and the growth velocity, can be calculated and tracked
Fig. 8. The simulated evolution of grain structure at 400 C and a strain rate of 1 s1 with various strains: (a) 0.12; (b) 0.24; (c) 0.42.
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equipment" (No: 2014ZX04002071), the Changsha Science and Technology Project (No. K1112067-11) and the Supporting Program of the “Twelfth Five-year Plan” for Sci & Tech Research of China (No. 2011BAG03B02). References
Fig. 9. The simulated volume fraction recrystallized at various temperatures with a strain rate of 0.1 s1.
during hot compression. The work hardening was derived by the LaasraourieJonas model. The following conclusions can be obtained: 1. The derived work hardening was successfully obtained and is higher than the flow stress after critical stress. Because of high work hardening rate, work hardening rapidly reaches to the saturate flow stress before a strain of 0.2. 2. The flow stress, determined by the yield stress and dislocation density, was exactly predicted. The equiaxed growth behavior of R-grains was successfully simulated. The simulations agree well with the experimental results, including flow stress and the microstructural evolution. 3. The increase of volume fraction recrystallized with increasing temperature is related to the nucleation rate and the velocity of grain boundaries.
Acknowledgments The authors gratefully acknowledge research support from the National Key Project Of Science and Technology “Highgrade CNC machine tools and basic manufacturing
[1] S.M. Fatemi-Varzaneh, A. Zarei-Hanzaki, H. Beladi, Mater. Sci. Eng. A 456 (2007) 52e57. [2] J.C. Tan, M.J. Tan, Mater. Sci. Eng. A 339 (2003) 124e132. [3] X.Y. Yang, H. Miura, T. Sakai, Mater. Trans. 44 (2003) 197e203. [4] W. Roberts, B. Ahlblom, Acta Metall. 26 (1978) 801e813. [5] H. Xiao, H.B. Xie, Y.H. Yan, Y.G. Jun, J. Iron Steel Res. Int. 11 (2004) 42e45. [6] R. Ding, Z.X. Guo, Acta Mater. 49 (2001) 3163e3175. [7] R. Ding, Z.X. Guo, Comp. Mater. Sci. 23 (2002) 209e218. [8] R.K. Shelton, D.C. Duand, Acta Mater. 44 (1996) 4571e4585. [9] R.L. Goetz, V. Seetharaman, Scr. Mater. 38 (1998) 405e413. [10] L.X. Li, F.Y. He, X. Liu, Y. Lou, J. Zhou, J. Duczczyk, Key Eng. Mater. 491 (2012) 264e272. [11] F. Chen, Z.S. Cui, J. Liu, X.X. Zhang, W. Chen, Model. Simul. Mater. Sci. Eng. 17 (2009) 1e19. [12] N. Yazdipour, C.H.J. Davies, P.D. Hodgson, Comp. Mater. Sci. 44 (2008) 566e576. [13] Y. Estrin, H. Mecking, Acta Metall. 32 (1984) 57e70. [14] X. Liu, L.X. Li, F.Y. He, J. Zhou, B.W. Zhu, L.Q. Zhang, Trans. Nonferrous Met. Soc. China 23 (2013) 2692e2699. [15] X. Liu, B.W. Zhu, L.X. Li, Chin. J. Nonferrous Metals 23 (2013) 898e904. [16] A. Laasraoui, J.J. Jonas, Metall. Mater. Trans. A 22 (1991) 1545e1558. [17] H.J. Frost, M.F. Ashby, Deformation-mechanism Maps: the Plasticity and Creep of Metal and Ceramics, Pergamon Press, Oxford, 1982. [18] T. Sakai, J. Mater. Process Technol. 53 (1995) 549e561. [19] D. Mclean, Grain Boundaries in Metals, Oxford University Press, Oxford, 1957. [20] W.T. Read, Dislocation in Crystal, McGraw Hill, New York, 1953, p. 181. [21] G. Kugler, R. Turk, Acta Mater. 51 (2004) 4659e4668. [22] X. Quelennec, J.J. Jonas, ISIJ Int. 52 (2012) 1145e1152. [23] M. Qian, Z.X. Guo, Mater. Sci. Eng. A 365 (2004) 180e185. [24] E.I. Poliak, J.J. Jonas, Acta Mater. 44 (1996) 127e136. [25] H.J. McQueen, O.C. Celliers, Can. Metall. Quart. 35 (1996) 305e319. [26] X.Y. Yang, Z.S. Ji, H. Miura, T. Sakai, Trans. Nonferrous Met. Soc. China 19 (2009) 55e60. [27] X. Liu, J.J. Jonas, L.X. Li, B.W. Zhu, Mater. Sci. Eng. A 583 (2013) 242e253.
ScienceDirect Journal of Magnesium and Alloys 2 (2014) 133e139 www.elsevier.com/journals/journal-of-magnesium-and-alloys/2213-9567
Full length article
Modeling of the flow stress behavior and microstructural evolution during hot deformation of Mge8Ale1.5Cae0.2Sr magnesium alloy Xiao Liu a,b, Luoxing Li a,*, Biwu Zhu a,b a
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha 410082, Hunan, China b College of Mechanical and Electrical Engineering, Hunan University Science and Technology, Xiangtan 411002, Hunan, China Received 15 February 2014; revised 8 May 2014; accepted 13 May 2014 Available online 18 June 2014
Abstract Hot compression tests of Mge8Ale1.5Cae0.2Sr magnesium alloy were performed on the Gleeble-1500 machine at temperatures of 300, 350, 400, and 450 C and strain rates of 0.01, 0.1 and 1 s1. The flow stress behavior and microstructural evolution were followed. The work hardening was derived according to the LaasraouieJonas model. An improved approach, which considered the influence of yield stress on flow stress and the effect of grain boundary (GB) migration on the evolution of dislocation density during compression, was used to simulate the microstructural evolution, the flow stress and the volume fraction recrystallized of Mge8Ale1.5Cae0.2Sr magnesium alloy. The simulated results are in good agreement with experimental results. Copyright 2014, National Engineering Research Center for Magnesium Alloys of China, Chongqing University. Production and hosting by Elsevier B.V. All rights reserved.
Keywords: Magnesium alloy; Work hardening; Cellular automata; Volume fraction recrystallized; Microstructure
1. Introduction Mge8Ale1.5Cae0.2Sr magnesium alloy with excellent high-temperature creep properties has good application prospect. Microstructure of metallic materials, including grain size, morphology, largely affects the mechanical and functional properties of final products [1]. It is necessary to elucidate the microstructural evolution during deformation. Dynamic recrystallization (DRX) is a common phenomenon in magnesium alloys [2e4]. It is generally accepted that DRX processes have the following characteristics: (1) Critical * Corresponding author. E-mail address: [email protected] (L. Li). Peer review under responsibility of National Engineering Research Center for Magnesium Alloys of China, Chongqing University
Production and hosting by Elsevier
dislocation density and critical strain are required for onset of DRX; (2) The recrystallized grains (R-grains) are equiaxed and the average grain size stays same at a given deformation condition; (3) Preexistent grain boundaries (GBs) are usually nucleation sites [5e8]. It is difficult to study the microstructural evolution characteristics during grain growth by physical experiments. Computer simulations have been used by many researchers to simulate the microstructure evolution during deformation [5,9e11,14,15]. Goetz and Seetharaman [9] first tried to predict DRX and flow stress by cellular automaton (CA) method. This method did not consider the effects of the yield stress, GB migration, and the hot deformation parameters on DRX and the influence of the yield stress on flow stress. Chen et al. [11] predicted the microstructure evolution and flow stress in 30Cr2Ni4MoV rotor steel using deformation parameters coupling the CA method. But the effect of the yield stress on flow stress and the influence of GB migration on dislocation density were also not considered. Yazdipour et al. [12] has taken the yield stress into account and employed the KockseMecking
http://dx.doi.org/10.1016/j.jma.2014.05.001. 2213-9567/Copyright 2014, National Engineering Research Center for Magnesium Alloys of China, Chongqing University. Production and hosting by Elsevier B.V. All rights reserved.
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X. Liu et al. / Journal of Magnesium and Alloys 2 (2014) 133e139
model [13] to describe the dynamic recovery. But this approach only could be applied when the mean free path for slip was constant and the effect of GB migration on dislocation density was not considered. Our previous work [14,15] has predicted the microstructure evolution in AZ31 magnesium alloy without considering the yield stress and did not predict the volume fraction recrystallized. In the present study, an improved approach, considering the influence of yield stress on flow stress and the effect of GB migration on dislocation density, was used to simulate the microstructural evolution of Mge8Ale1.5Cae0.2Sr magnesium alloy during hot compression. In this approach, the grain structure and the volume fraction recrystallized, which describes the percentage of recrystallized area were simulated. The work hardening based on the LaasraouieJonas model was also first derived. All the simulated results were compared with the experimental results. 2. Experimental The hot compression tests were carried out on Gleeble1500 to obtain the flow curves over four temperatures (300, 350, 400, and 450 C) with three strain rates (0.01, 0.1 and 1 s1). The alloy, used in the present study, was as extruded Mge8Ale1.5Cae0.2Sr with the nominal composition of 8 wt % Al, 1.5 wt% Ca, 0.3 wt% Mn and 0.2 wt% Sr. Cylindrical specimens with a diameter of 10 mm and a height of 15 mm were machined with the compression axis parallel to the extruded solid bar axis. To observe the microstructure, crosssections parallel to compression axis were cut from the deformed specimens. The samples were mounted, polished and etched. Then, the microstructure was observed by optical microcopy with an image analyzer. 3. Mathematical modeling 3.1. Modeling DRX
shown that DRX only occurs beyond critical strain from this equation. The velocity of grain boundary movement is the result of the net pressure on the boundary, which can be written as: ð3Þ Vi ¼ MFi 4pri2 dDb b Qb exp M¼ KT RT
ð4Þ
Fi ¼ 4pri2 tðrm ri Þ 8pri gi
ð5Þ
where Vi is the velocity of GB, M is the grain boundary mobility [17e19], d is the characteristic GB thickness, Db is the boundary self-diffusion coefficient, Qb is the boundary diffusion activation energy, K is Boltzmann's constant, R is the universal gas constant (8.31 J mol1 K1), T is the absolute temperature, Fi is the driving force [20], rm is the dislocation density of the matrix, ri is the dislocation density of ith Rgrain, ri is the radius of the ith R-grain, gi is GB energy. 3.2. Modeling flow stress and work hardening In previous studies [6,7,21], the flow stress is proportional to the square root of dislocation density (see Eq. (6)). To exactly predict flow stress during deformation, the influence of yield stress on flow stress (see Eq. (7)) and the effect of GB migration on dislocation density (see Eq. (1)) were considered to simulate the flow stress and dislocation density evolution. pffiffiffi s ¼ MaGb r ð6Þ pffiffiffi s ¼ so þ MaGb r
ð7Þ
where so is the yield stress, at which point r ¼ ro and ε ¼ εo, a is a constant of 0.5e1, G is the shear modulus, b is the Burger's vector. To predict the work hardening, the following expression is used and the detailed steps involving in this procedure are described in Ref. [22]. 1=2 swh ¼ s2sat s2sat s2o expð rðε εo ÞÞ ð8Þ
To describe the evolution of dislocation density during dynamic recovery, the modified LaasraouieJonas model in Eq. (1) [16] was used and the detailed steps involving in calculation of h and r were shown in our previous researches [10,14,15]. The effect of GB migration on dislocation density evolution was also taken into account in the modified LaasraouieJonas model.
where swh is the work hardening, ssat depending on h and r is the saturate flow stress.
dr ¼ ðh rrÞⅆε rⅆV
3.3. Procedure of simulation
ð1Þ
where r is the dislocation density, dε is the strain increment, h is the thermal work hardening rate, r is the rate of dynamic recovery, and dV is the volume swept by the mobile boundaries. Xiao et al. [5] has proposed the following expression according to experimental and theoretical analysis: p n_ ¼ C1 ε_ Z m ðε εc Þ
ð2Þ
where n_ is the nucleation rate, Z is the ZennereHollomon parameter, and C1, m, and p are the material constant. It is
The CA method is algorithm that represents discrete spatial and temporal evolution of complex systems by applying local or global deterministic or probabilistic transformation rules to the location of a lattice. The system objects are quantified according to generalized state variables [23]. The CA method can easily couple with the physical model to predict DRX during deformation process. In this model, a 2D square lattice and Moore neighboring were employed. In order to simulate the equiaxed growth of R-grain, a probabilistic transformation rule was set to determine which neighbor would switch.
X. Liu et al. / Journal of Magnesium and Alloys 2 (2014) 133e139
135
Fig. 1. Flow stress under various temperatures and the following strain rates: (a) 0.01 s1; (b) 0.1 s1; (c) 1 s1.
The initial microstructure was generated via a normal grain growth algorithm. The simulated lattices were set to 150 150, which corresponded to an area of 300 300 mm, and each lattice has three variables: one orientation variable that represents the grain orientation and determines the grain boundary energy; one states variable Sr that indicates which is the R-grains and is set to calculate the volume fraction recrystallized; one dislocation density variable that determines the site energy. The orientation of initial grain and R-grain were set randomly from 1 to 180. When the dislocation density exceeds critical dislocation density, DRX is initiated. Then the nucleation number can be obtained by Eq. (2) in each step. The nucleation sites were selected from the GBs and intersection points. For growth of R-grain, when the driving force for the growth of the ith R-grain is positive, this R-grain can grow continuously until the driving force reaches zero, and the growth velocity in each step is determined by Eq. (3). The
calculation was terminated, while the pre-set strain was attained. The flow stress was determined by the dislocation density and calculated using Eq. (7), suggesting that the evolution of microstructure affects the flow stress during simulation. The initial value of Sr is zero for each lattice. When the lattice changes into R-grain, the value of Sr for the lattice becomes one. The number of lattices for R-grains, can be calculated. Each lattice is corresponding to an area of 2 2 mm. Then the volume fraction recrystallized in each step can be calculated by the following equation: F¼
areanewgrain areatotal
ð9Þ
where F is the volume fraction recrystallized, areanewgrain is the recrystallized area, and areatotal is the simulated area.
Fig. 2. Microstructure observations at 300 C with a strain of 0.6 and the following strain rates: (a) 0.1 s1; (b) 1 s1.
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Fig. 3. The work hardening rate curves under various temperatures and the following strain rates: (a) 0.01 s1; (b) 0.1 s1; (c) 1 s1.
Fig. 4. The derived work hardening curves under different deformation conditions: (a) 350 C, and a strain rate of 0.01 s1; (b) 350 C and a strain rate of 0.1 s1; (c) 400 C and a strain rate of 0.01 s1; (d) 400 C and a strain rate of 0.1 s1.
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Fig. 5. The comparison between the simulated flow stress and experimental flow stress at a temperature of 400 C with various strain rates: (a) 0.1 s1; (b) 1 s1.
4. Results and discussion The flow stress curves obtained at various temperatures and strain rates are presented in Fig. 1. All the curves display the normal single peak and concave-down appearance associated with DRX. The flow stress increases to a maximum and then decrease to finally attain a steady state in the usual way. Most of the curves at strains from 0.02 to 0.15 are camber line, except the flow curves at 300 C and strain rates of 0.1 s1 and 1 s1. The flow curves of these two conditions at strains from 0.02 to 0.15 are close to a straight line, indicating that twinning occurs during deformation. This can be proved by Fig. 2. Optical microstructure at various strain rates and a temperature of 300 C with a strain of 0.6 are displayed in Fig. 2. Twins are observed inside grains and the number of twins increases with strain rate. It is well known that in order for a polycrystal to accommodate deformation, five independent slip system have to be activated. Basal slip and prismatic slip can only provide two independent slip modes. 1st order pyramidal slip can offer four independent slip modes, but this can not accommodate uniform deformation. Only 2-order pyramidal slip, with its non-basal Burger vectors, will accommodate deformation along the c-axis and meet the requirement for five independent slip modes. But dislocation slip can not meet requirement of uniform plasticity for its low migration velocity at high strain rate and so caused the initiation of twinning. Twinning was not considered during simulation. Therefore, flow stresses, microstructure evolution
and volume fraction recrystallized under these conditions were not simulated. In order to obtain the saturate flow stress and the rate of dynamic recovery, derivate method was applied to the flow stress curves of Fig. 1. The work hardening rates (ds/dε) under various temperatures and strain rates are displayed in Fig. 3. It can be seen that the work hardening increases with increasing temperature. According to the work hardening rate curves, the critical strain εc, the peak strain εp and the saturate stress ssat can be determined [24,25]. The rate of dynamic recovery r can be calculated by the slope of work hardening rate [15]. Then, the derived work hardening curves can be obtained using Eq. (8). The derived work hardening curves under different deformation conditions are displayed in Fig. 4. It can be seen that the derived work hardening is higher than the flow stress from critical stress. The work hardening approached rapidly to the saturate flow stress before a strain of 0.2, due to high work hardening rate. It can be concluded from Fig. 4 that the work hardening increases with decreasing temperature and increasing strain rate. The comparison between the simulated flow curves and the experimental flow curves are illustrated in Fig. 5. Both the simulated flow curves and the experimental flow curves attain a peak, followed by continuous flow softening. At 350 C with a strain rate of 0.1 s1, the simulated flow stress is a little bit higher than experimental flow stress after the critical strain, and then lower than the real flow stress. This discrepancy may be attributed to the possible involvement of additive high
Fig. 6. Simulated microstructure at a strain rate of 1 s1 and a strain of 0.6 with various temperatures: (a) 350 C; (b) 400 C; (c) 450 C.
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Fig. 7. Microstructure observations at a strain rate of 1 s1 and a strain of 0.6 with various temperatures : (a) 350 C; (b) 400 C; (c) 450 C.
energy sits for nucleation, such as dislocation tangle and dislocation pinning, and the effect of creep behavior on grain boundary migration which has not been taken into account during simulation. Nevertheless, the simulations agree well with the experimental results. The simulated microstructure at various temperatures and a strain rate of 1 s1 with a strain of 0.6 are shown in Fig. 6. Some of regions remain unrecrystallized at 350 C and 400 C. At 450 C, the equiaxed new grains almost replace all of the original microstructure. The volume of new grains increase with temperature. At 350 C and 400 C, initial grains can be observed and the volume of primary grains decreases with increasing temperature. At 450 C, the volume of primary grains dramatically decrease and primary grains are almost replaced by new grains. The experimental grain structures at various temperatures with a strain rate of 1 s1 are shown in Fig. 7. New grains, equiaxed and homogeneously distributed, appear in primary grain boundaries. There are several unrecrystallized grains at 350 C and 400 C, suggesting that DRX is not completely. At 450 C, the original microstructure are almost taken place by R-grains, indicating an approach to steady state flow (see Fig. 1(c)). New grains coarsen with increasing temperature, attributing to the increase of GB energy with temperature. As compared with Fig. 6, it is proved that the improved approach can accurately simulate the microstructure evolution. The simulated microstructural evolution at 400 C and a strain rate of 1 s1 with various strains are exhibited in Fig. 8. At a strain of 0.12, DRX has been initiated and the new
grains are nucleated at the grain boundaries. At a strain of 0.24, substantial new grains can be detected and replace initial grains. The new grains are equiaxed and homogeneously distributed. With the strain reaches to 0.42, the number of unrecrystallized regions has dramatically decreased. The nucleation mechanism obeys DRX characteristic. The simulated volume fraction recrystallized curves are shown in Fig. 9. The volume fraction recrystallized is used to describe the percentage of recrystallized area [3,26,27]. The volume fraction recrystallized increases with strain. The increase of temperature leads to higher grain boundary velocity and lower nucleation rate. Nevertheless, the increment of grain boundary velocity is faster than the decrement of nucleation rate. Thus, the volume fraction recrystallized increases with temperature. The simulated volume fraction recrystallized curves match the actual situation well. 5. Conclusions An improved approach coupling the cellular automaton method and metallurgical principles of dynamic recrystallization has been proposed to simulate the evolution of microstructure, the flow stress and the volume fraction recrystallized during hot compression. This method, considering the influence of the yield stress on flow stress and the effect of GB migration on dislocation density, can accurately predict DRX. The growth variation of each R-grain, including dislocation density and the growth velocity, can be calculated and tracked
Fig. 8. The simulated evolution of grain structure at 400 C and a strain rate of 1 s1 with various strains: (a) 0.12; (b) 0.24; (c) 0.42.
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equipment" (No: 2014ZX04002071), the Changsha Science and Technology Project (No. K1112067-11) and the Supporting Program of the “Twelfth Five-year Plan” for Sci & Tech Research of China (No. 2011BAG03B02). References
Fig. 9. The simulated volume fraction recrystallized at various temperatures with a strain rate of 0.1 s1.
during hot compression. The work hardening was derived by the LaasraourieJonas model. The following conclusions can be obtained: 1. The derived work hardening was successfully obtained and is higher than the flow stress after critical stress. Because of high work hardening rate, work hardening rapidly reaches to the saturate flow stress before a strain of 0.2. 2. The flow stress, determined by the yield stress and dislocation density, was exactly predicted. The equiaxed growth behavior of R-grains was successfully simulated. The simulations agree well with the experimental results, including flow stress and the microstructural evolution. 3. The increase of volume fraction recrystallized with increasing temperature is related to the nucleation rate and the velocity of grain boundaries.
Acknowledgments The authors gratefully acknowledge research support from the National Key Project Of Science and Technology “Highgrade CNC machine tools and basic manufacturing
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