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Rheological behaviour of partially solidified A356 alloy: Experimental study and constitutive modelling
Journal of Alloys and Compounds 803 (2019) 1141e1154
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Rheological behaviour of partially solidified A356 alloy: Experimental study and constitutive modelling Zhen Ma a, Huarui Zhang a, c, *, Xiaoli Zhang b, Xiaoyan Wu a, Hanwei Fu a, Lina Jia a, Hu Zhang a, c, ** a b c
School of Materials Science and Engineering, Beihang University, Beijing, 100191, China Army Military Transportation University, Tianjin, 300000, China Ningbo Institute of Technology, Beihang University, Ningbo, 315832, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 April 2019 Received in revised form 26 June 2019 Accepted 27 June 2019 Available online 28 June 2019
Rheological behaviour of the mushy zone is dependent on the microstructure and related to parameters such as temperature, shear rate and shear strain/time. However, constitutive modelling of the rheological behaviour is difficult due to the diversity of the parameters. This study focuses on the influence of the microstructure and the above parameters on the rheological behaviour of partially solidified A356 alloy and establishing a constitutive model accordingly. During solidification, the evolution of a-Al phase leads to the changes of the apparent viscosity of mushy zone. Steady-state viscosity and peak viscosity measured by rotational rheometry reveal the thixotropy of the mushy zone. The thixotropic strength is found to increase with increasing shear rate and decreasing temperature. The peak viscosity, as well as the steady-state viscosity, is found to obey a power law model and is modelled using the power law model with the coupling of temperature and shear rate. Furthermore, a constitutive model with temperature, shear rate, shear tress and shear strain/time as parameters which describes the rheological behaviour of the mushy zone of partially solidified A356 alloy accurately and completely is established. Comparing with previously reported experimental results, the FEM simulation employing the constitutive model works well in enabling visualization of the viscoelastic filling process of A356 alloy in mushy zone. The constitutive model provides a solution for characterizing the forming behaviour of other mushy zone/semi-solid aluminium alloys with corresponding rheological parameters. © 2019 Elsevier B.V. All rights reserved.
Keywords: Mushy zone Partially solidified A356 aluminum alloy Microstructural evolution Rheological behaviour Constitutive model
1. Introduction Cast Al-Si alloys are widely used in aerospace, automotive and other fields owing to their excellent mechanical properties, outstanding corrosion resistance and good castability [1e4]. A356 alloy is a type of commercial cast hypoeutectic Al-Si alloy with extensive applications [5]. Cast defects such as segregation, porosity, and hot cracking are easily formed due to the flow of liquid and solid during the filling and feeding processes of casting [6]. This flow is determined by the rheological behaviour of mushy
* Corresponding author. School of Materials Science and Engineering, Beihang University, Beijing, NO.37 Xueyuan Road, 100191, China. ** Corresponding author. School of Materials Science and Engineering, Beihang University, Beijing, 100191, China. E-mail addresses: [email protected] (H. Zhang), [email protected] (H. Zhang). https://doi.org/10.1016/j.jallcom.2019.06.345 0925-8388/© 2019 Elsevier B.V. All rights reserved.
zone i.e. semi-solid state alloy during solidification of the alloy and by the casting parameters [6e9]. In this respect, understanding the rheological behaviour of mushy zone is crucial for obtaining appropriate casting process to eliminate defects. So far, several works have been published on the rheological behaviour of mushy zone or semi-solid state of aluminium alloys. Flemings [10] studied the rheological behaviour of semi-solid Al6.5Si alloys and found that the apparent viscosity decreased with increasing temperature and shear rate. The effect of compositional variation on rheological behaviour of semi-solid Al-Si-Mg alloy at a solid fraction (fs) of 0.45 was reported by Paes et al. [11], showing that increasing silicon and decreasing Mg content lead to lower apparent viscosity. Previous studies indicate that the mushy zone of aluminium alloys is a non-Newtonian fluid whose apparent viscosity varies with shear rate, temperature i.e. solid fraction and composition [10e12]. However, the previous rheological investigations of aluminium alloys are mostly based on the
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experiments of parallel-plate compression [2,13,14], where the material is assumed Newtonian fluid when calculating the rheological parameters [15,16], and the experiments are carried out based on partial melting of the alloys. Therefore, these experiments are unable to accurately reflect the rheological behaviour of the alloys during solidification. Rotational rheometry based on partial solidification, which takes materials as non-Newtonian fluid [17,18], is considered to have high accuracy and good applicability for investigating the rheological behaviour of the alloys during solidification. Using this method, the thixotropic behaviour of aluminium alloys in mushy zone can be measured [19,20]. For instance, Brabazon et al. [19] studied the thixotropic behaviour of Al-4Si alloy in mushy zone at a solid fraction of 0.36 and found that the thixotropic behaviour could be evaluated by peak and steady-state viscosities. The peak and steady-state viscosities are thought to result from the collapse, agglomeration and disagglomeration of solid particles during continuous shearing [16,20e22]. Quaak's study [23] on A356 alloy confirmed that the relationship between steady-state viscosity and shear rate obey a power law model, which is widely accepted and confirmed by other studies [19,20,24]. However, the mathematic relationship between the peak viscosity and shear rate is still unclear. Moreover, the thixotropic behaviour of mushy zone aluminium alloys indicates that the rheological behaviour of mushy zone is not only dependent on shear rate, temperature and composition, but also on shear strain/time. Therefore, to accurately describe the rheological behaviour, constitutive modelling of mushy zone taking into consideration temperature, shear rate and shear strain/time as parameters is of great importance. Some models have been established to describe the rheological behaviour of mushy zone, which are fundamental for the numerical simulation of casting processes [16,25e28]. Hu et al. [16] established a time-dependent power law model for 319s alloy considering the influence of geometry based on compression tests. Ludwig et al. [27] obtained a compressible constitutive model of AA5182 alloy considering the state of cohesion of mushy zone based on shear and tensile tests. However, previous rheological models are mostly incomplete due to the neglect of thixotropic behaviour. Given the diversity of parameters, a constitutive model that is capable of describing the rheological behaviour of mushy zone accurately and completely needs to be developed. In the present study, a rheological investigation of the mushy zone of partially solidified A356 is performed in a wide temperature range (580e610 C) and shear rate range (0.1e1000 s1) using rotational rheometry. The relationship between rheological behaviour and temperature, shear rate, shear strain/time and microstructure of the mushy zone of partially solidified A356 alloy was investigated. A power law model for the peak and steady-state viscosities are obtained with the coupling of temperature and shear rate. Furthermore, a constitutive model of the mushy zone of partially solidified A356 alloy is established and applied to simulate the filling process of the alloy in mushy zone.
chemical composition listed in Table 1. Partial solidification and quenching experiment was conducted along with differential scanning calorimetry (DSC) analysis using STA-449F3 to study the microstructural evolution of A356 alloy during solidification. During the partial solidification and quenching experiment, samples with a volume of 14 10 5 mm were first cut from A356 alloy ingots and placed in crucibles after polishing and ultrasonic cleaning; the samples were then heated to 700 C at a heating rate of 5 C/min in a tube furnace and isothermally treated for 5 min. The samples were finally cooled to 610 - 550 C at a cooling rate of 5 C/min followed by rapid quenching in water. Rheological experiments were performed using a hightemperature rotational rheometer MCR 502 with a concentriccylinder device (Fig. 1). During the experiment, the external cup remained static, and the internal bob rotated at a speed corresponding to the set shear rate in order to shear the samples inside the cup. The concentric-cylinder device was located in a heating chamber equipped with a hot air circulation device to provide a high-temperature environment with an accuracy of ±1 C. Inert atmosphere was maintained within the heating chamber using nitrogen gas to prevent sample oxidation. The rheological tests were based on the partial solidification of A356 alloy. Samples cut from A356 alloy ingots were polished and ultrasonically cleaned. Fig. 2 shows the procedure of the rheological tests. Two different types of rheological tests, continuous cooling tests and shear time tests, were performed to characterize the rheological behaviour of partially solidified A356 alloy. The continuous cooling tests were adopted to study the effect of temperature on rheological behaviour. The shear time tests were performed to reveal the thixotropic behaviour of the alloy. Three specimens were prepared for each condition in the shear time tests to ensure repeatability and accuracy of the results. During the continuous cooling tests, three different cooling rates were used with the shear rate being kept constant, whilst the shear time tests were performed with 6 combinations of temperature and shear rate. The shear time was 200 s for each shear time test. Table 2 summarizes the parameters of the rheological tests.
2. Experimental procedure The material studied in this research is A356 alloy with the
Fig. 1. Concentric-cylinder device of high-temperature rotational rheometer.
Table 1 A356 alloy composition. Element
Si
Mg
Fe
Cu
Ti
Sr
Zn
Ni
Al
Wt. %
7.128
0.346
0.111
0.089
0.145
0.0144
0.006
0.005
Balance
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phases before quenching. As shown in Fig. 3(h), the larger a-Al dendrites are considered to be the resultant solid phase transformed from the mushy zone, while other microstructures (such as extremely fine a-Al and eutectic microstructures) are considered to be liquid phase of the mushy zone. At the beginning of solidification (Fig. 3(a)), the alloy was in liquid state. With decreasing quenching temperature (Fig. 3(bef)), the alloy was in mushy zone, and the amount of solid phase gradually increased. Furthermore, the a-Al dendrites became coarser and more irregular. When solidification was finished (Fig. 3(g)), the microstructure of the alloy consisted mainly of well-developed a-Al dendrites and eutectic structure, with no liquid phase. 3.2. Rheological behaviour during continuous cooling Fig. 2. Rheological experimental procedure.
3. Results 3.1. Microstructures of partially solidified A356 alloy Fig. 3 presents the optical micrographs of the partially solidified alloy quenched at different temperatures. Both liquid and solid phases can be found in the mushy zone of A356 alloy before quenching. After quenching, the liquid phase transforms into extremely fine microstructure as a result of the fast cooling rate. Whereas the solid phases formed at normal cooling rate before quenching remain coarse similar to the morphology of the solid
The effects of temperature and cooling rate on the rheological behaviour of the mushy zone are shown in Fig. 4. At the temperature above 595 C (region I in Fig. 4), the shear stress and the apparent viscosity remains unchanged. In the temperature range of 595e568 C (region II in Fig. 4), both shear stress and apparent viscosity increase with decreasing temperature. Cooling rate has an obvious effect on the rheological behaviour of the mushy zone, where increased shear stress and apparent viscosity can be observed in by increasing cooling rate at the same temperature. 3.3. Thixotropic behaviour 3.3.1. Shear stress and apparent viscosity The evolution of apparent viscosities during the shear time tests
Table 2 Rheological experimental parameters. Test
Temperature ( C)
Shear rate (s1)
Continuous cooling tests Shear time tests
610e560 610, 605, 600, 595, 590, 585, 580
10 0.1, 0.5, 1, 5, 10, 100, 1000
Fig. 3. Optical micrographs of the partially solidified A356 alloy quenched at (a) 610 C, (b) 600 C, (c) 590 C, (d) 580 C, (e) 570 C, (f) 560 C, and (g) 550 C; (h) enlargement of the area in the box of (b).
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rule as that of apparent viscosity according to the definition of the apparent viscosity [29]. Peak shear stress and steady-state shear stress existed during the shear time tests corresponding to the peak viscosity and the steady-state viscosity, respectively.
Fig. 4. Variations of shear stress with temperature at a shear rate of 10 s1.
are plotted in Fig. 5. An “overshoot” phenomenon can be discovered: for each apparent viscosity curve, it firstly increases until reaching a peak value, then decreased and finally comes to a steady-state value. The “overshoot” phenomenon is more obvious when the temperature is lower. Since the shear rate and the temperature were kept constant during the shear time tests, the evolution of shear stress with respect to time should follow the same
3.3.2. Peak and steady-state shear stress/viscosity The variations of steady-state shear stress, steady-state viscosity, peak shear stress and peak viscosity with shear rate and temperature are shown in Figs. 6 and 7, respectively. The shear stress and viscosity values were average values calculated according to the three tests data in each condition. Within the testing temperature range 580e610 C, the steady-state shear stress decreases at first and then increases with increasing shear rate when the shear rate is higher than 100 s1 (Fig. 6(a)). The steady-state viscosity and peak viscosity decrease with increasing shear rate at 580e610 C (Fig. 6(b) and (d)). The peak shear stress, steady-state shear stress, peak viscosity and steady-state viscosity all increase with decreasing temperature (Fig. 7). 4. Discussion 4.1. Microstructural evolution during solidification During solidification, the microstructure of the A356 alloy evolves with decreasing temperature, which determines the
Fig. 5. Effects of shear time and shear rate on viscosity at (a) 610 C, (b) 600 C, (c) 590 C and (d) 580 C.
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Fig. 6. (a) Steady-state shear stress, (b) steady-state viscosity (c) peak shear stress and (d) peak viscosity versus shear rate.
rheological behaviour of the material. Therefore, understanding the microstructural evolution of A356 alloy during solidification is crucial to explain the rheological behaviour of the mushy zone of partially solidified alloy. The enthalpy change during solidification shows that there are three exothermic peaks during solidification (Fig. 8(a)): Peak A (612 - 564 C) corresponds to the formation of a-Al dendrite. Peak B (564 - 549 C) corresponds to the formation of Al-Si binary eutectic and the precipitation of Fe-rich phase. Peak C (549 - 540 C) corresponds to Al-Mg2Si-Si ternary eutectic reaction [30]. Then the components of the solid phase in each stage of solidification can be summarized accordingly. The solid phase existing in the mushy zone is mainly a-Al phase at the initial stage (612 - 564 C) of solidification. In the interim and terminal stage (564 - 540 C), with the precipitation of eutectic phase and Fe-rich phases, the main components of the solid phases change from a-Al to a-Al, eutectic phase and Fe-rich phases. Comparison of solid fraction obtained by metallographic and enthalpy change method is shown in Fig. 8(b), which shows a good consistency of the two methods at 612 - 564 C. While the solid fraction obtained by metallographic method is obviously lower at the late stage (564 - 540 C) of solidification because the metallographic method does not take phases other than a-Al as solid phase. On the contrary, the enthalpy change method takes into account all the solid phases during calculating solid fraction. As a conclusion, the microstructures of partially solidified quenched
alloy can accurately reflect the evolution of the solid phase (a-Al) during solidification A356 alloy at 612 - 564 C. As shown in Fig. 9, the mushy zone during solidification of A356 alloy undergoes three stages according to the content, size and morphology of the a-Al phase in the partial solidification and quenching experiment. 4.2. Rheological behaviour of the mushy zone Apparent viscosity is a parameter that characterizes the fluidity of a fluid. Apparent viscosity is determined by shear rate and the shear stress which can be described by the following equation [29]:
ha ¼
t g_
(1)
where ha is the apparent viscosity, t is the shear stress and g_ is the shear rate. With temperature decreasing from 610 C to 595 C, the mushy zone is at the free-flowing stage (Fig. 9(a)). The rheological behaviour of the mushy zone is essentially of the liquid phase where the influence of dispersed small a-Al islands can be neglected. Therefore, temperature has little effect on the shear stress and apparent viscosity of the mushy zone (Figs. 4 and 7). When the temperature decreases from 595 C to 570 C, the mushy zone is in the dendrite coherency stage (Fig. 9(b)). The rheological
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Fig. 7. (a) Steady-state shear stress, (b) steady-state viscosity, (c) peak shear stress and (d) peak viscosity versus temperature.
Fig. 8. (a) Enthalpy change of A356 alloy during solidification and (b) solid fraction during solidification of A356.
behaviour is greatly affected by the increased content and the coarsening of the solid phase, leading to the increase of shear stress and apparent viscosity (Figs. 4 and 7). When temperature is below 570 C, the mushy zone enters the solid network stage (Fig. 9(c)), causing a dramatic rise of the shear stress and the apparent
viscosity (Fig. 4). Moreover, the cooling rate affects the rheological behaviour of the A356 alloy during solidification by alerting the morphology of a-Al phase. The a-Al phase tends to grow into a dendritic shape rather than spherical or rosette shape at high cooling rates [6,31],
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In order to further investigate the thixotropic behaviour of the mushy zone of partially solidified A356 alloy, the proportional coefficient of peak shear stress (viscosity) and steady-state shear stress (viscosity) is defined as thixotropic factor kt (kh) to characterize the thixotropic strength with the following equations:
Fig. 9. Schematic diagram of the evolution of mushy zone during solidification of A356 alloy. a. Free-flowing stage (610 - 590 C, Fig. 9(a)): Small sized a-Al phase emerges, and the liquid phase can flow freely with a-Al islands dispersed in it. b. Dendrite coherency stage (590 - 570 C, Fig. 9(b)): The content of a-Al phase increases, and the dendrites starts to grow. The small a-Al coalesce, and the liquid phase can flow in the gap of the big a-Al islands. c. Solid network stage (570 - 550 C, Fig. 9(c)): The content of a-Al phase increases rapidly with well developed dendrites. The dendrites intercalated with each other to form a solid network, and the liquid phase separated by the coherent solid network can not flow freely.
thus increase shear stress and apparent viscosity of the mushy zone (Fig. 4). As a typical shear thinning non-Newtonian fluid, the shear rate has a significant effect on the rheological behaviour of the mushy zone of partially solidified A356 alloy. The dendrites collapse adequately by high shear rate, leading to smaller sizes and a rounder shape of the solid particles at higher shear rate. As a result, apparent viscosity decreases with increasing shear rate (Fig. 6(b, d)). In this respect, the shear thinning phenomenon (apparent viscosity decreases with increasing shear rate) of the mushy zone can be observed, which demonstrates that the mushy zone of partially solidified A356 alloy is a pseudoplastic fluid. The mushy zone of partially solidified A356 alloy shows thixotropic behaviour due to the presence of solid particles exist within it. This means the shear stress (apparent viscosity) is timedependent during shearing progress (Fig. 5). At the beginning of shearing, the shear stress (apparent viscosity) increases rapidly due to the presence of dendritic a-Al or solid network. With continued shearing, the dendritic a-Al breaks into parts and the solid network gradually collapse, resulting in the decrease of shear stress (apparent viscosity). Finally, the shear stress (apparent viscosity) tends to be stable when the dendritic a-Al is completely broken down and the solid network comes to a complete collapse and a new equilibrium of the microstructure is reached.
kt ¼
tp ts ts
(2)
kh ¼
hp hs hs
(3)
where tp (hp) is the peak shear stress (viscosity) and ts (hs) is the steady-state shear stress (viscosity). The thixotropic factors are different at different temperatures and shear rates. The thixotropic factors increase from 2.7 to 87.3 with shear rate increasing from 0.1 s1 to 100 s1 at the temperature of 580 C (Fig. 10(a)). That is because the higher shear rates break the dendrites more thoroughly, which leading lower shear stress (viscosity) at the steadystate stage. Yet the peak stress (viscosity) is mainly determined by the strength of solid network but not shear rate. Therefore, the thixotropic factors increase with increasing shear rate. Nevertheless, it seems that because of the inaccurate peak shear stress at high shear rates (the peak shear stress is missed due to low data sampling frequency), there is a dramatically drop of thixotropic factors beyond 100 s1 at 580 C (Fig. 10(a)). The thixotropic factors decrease with increasing temperature, and the phenomenon is more obvious at higher shear rate (Fig. 10(b)). The thixotropic factor decreases from 87.3 to 0.1 with temperature increasing from 580 C to 610 C at 100 s1. The reason is that the solid network is stronger when the temperature is lower, and therefore, the corresponding peak stress (viscosity) needed to break the solid network becomes higher. But the steady-state shear stress (viscosity) is mainly related to the shear rate and solid fraction in the mushy zone. The correlation between steady-state shear stress (viscosity) and temperature is weaker compared with the correlation between peak shear stress (viscosity) and temperature. Therefore, the thixotropic factors decrease with increasing temperature. The thixotropic factors are very small (0.3e2.5) and basically not related to the shear rate at 610 C and 600 C (Fig. 10(a)). At this temperature, the mushy zone is at the free-flowing stage with no solid network. There is no collapse of solid network in the shearing process, where the peak stress and the steady-state stress do not differ greatly, resulting in small thixotropic factors. Moreover, due
Fig. 10. Relationships between thixotropic factor and (a) shear rate, (b) temperature.
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to the weak pseudoplasticity of the mushy zone at this temperature, the effects of shear rate on the peak stress (viscosity) and steady-state stress (viscosity) are also insignificant, resulting in constant thixotropic factors with shear rate changing. Comparison of viscosities measured in this work with those in previous works on Al-Si alloys is presented in Fig. 11. Owing to the rotational rheometry, the rheological behaviour of the alloy at low solid fractions (fs < 0.41) is studied in the present work rather than that at high solid fractions (fs > 0.45) in the previews works. The steady-state viscosity values in this study are significantly lower than those from the previous works. This is thought to be associated with the microstructures of the alloys during tests. There is more liquid entrapment within the semi-solid slurries prepared by et al. [33] and partial melting (Hu et al. [16], Liu et al. [32], Loue Yurko et al. [18]) than that prepared by partial solidification in this work, resulting in higher viscosities in the references than that in the present work [16]. What's more, the tests conducted with isothermal compression, back extrusion and drop forge viscometer are for transient behaviour rather than steady-state behaviour. As a consequence, the current steady-state viscosity values are lower than the values in the references. On the other hand, the measurements for transient behaviour in the references lead to constantly changing of the shear rate, in which the viscosity will never reach to peak or steady-state viscosity but to a value between them. As a result, the peak viscosity values (fs ¼ 0.41 and 0.33) in this study is higher than those (fs > 0.45) in the previous works even at lower solid fractions. In conclusion, rotational rheometry used in the present study is more reliable to measure the rheological behaviour of the mushy zone, especially that at low solid fraction. 5. Constitutive modelling of the mushy zone of partially solidified A356 alloy The thixotropic and shear thinning rheological behaviour of the mushy zone of partially solidified A356 alloy at the temperature of 580e610 C and the shear rate of 0.1e100 s1 can be modelled according to the phenomenological method. The proposed model consists of a combination of a viscoelastic Maxwell model and a
pulse model to describe its constitutive relationship [34]. 5.1. Modelling of steady-state viscosity and peak viscosity To obtain the constitutive model, steady-state and peak shear stresses should be modelled first. According to Fig. 6(b), when the shear rate is 0.1e100 s1, the relationship between steady-state viscosity/peak viscosity and the shear rate follows the power law model, namely ha ¼ K,g_ n1 [35], where K is the consistency factor and n-1 is the flow exponent. Therefore, the experimental data is fitted to Eq. (6) and the results are shown in Fig. 12 and Table 3.
log10 ha ¼ ðn 1Þlog10 g_ þ log10 K
(6)
The parameters n-1 and log10K have linear and three-stage relationship with temperature, respectively. The evolutions of the above two parameters with respect to temperature are fitted using the linear fitting and the Boltzmann fitting, respectively. The fitting results are shown in Fig. 13. Eqs. (7) and (8) show the expressions of n-1 as a function of temperature for steady-state and peak viscosities, respectively.
n 1 ¼ 0:00706T 5:381
(7)
n 1 ¼ 0:00866T þ 4:225
(8)
Where T is temperature. The Boltzmann relationships between log10K and temperature for steady-state viscosity and peak viscosity are shown as Eq. (9) and Eq. (10), respectively.
log10 K ¼
2:350 þ 3:480 1 þ exp½ðT 591:954Þ=2:664
(9)
log10 K ¼
3:434 þ 3:594 1 þ exp½ðT 595:066Þ=2:899
(10)
Substituting Eq. (7) and Eq. (9), Eq. (8) and Eq. (10) into Eq. (6), the models of steady-state viscosity and peak viscosity with shear rate and temperature can be obtained as Eqs. (11) and (12),
Fig. 11. Comparison of viscosities versus shear rate obtained by various experimental methods and conditions.
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Fig. 12. Experimental results and fitting curves of (a) steady-state viscosity, (b) peak viscosity.
Table 3 Fitting parameters of steady-state viscosity and peak viscosity. Fitting parameters Steady-state viscosity
Peak viscosity
n-1 log10K R2 n-1 log10K R2
580 C
585 C
590 C
595 C
600 C
605 C
610 C
1.306 5.779 0.999 0.936 7.00 0.968
1.294 5.647 0.999 0.764 7.067 0.914
1.179 5.076 0.998 0.850 6.284 0.976
1.093 3.994 0.999 0.827 5.530 0.876
1.128 3.687 0.998 0.944 3.917 0.976
1.167 3.510 0.997 1.038 3.781 0.976
1.078 3.406 0.999 1.126 3.643 0.998
Fig. 13. Evolutions of fitting parameters n-1, log10K for (a) steady-state viscosity, (b) peak viscosity with respect to temperature.
respectively.
hs ðg_ ; TÞ ¼ 10
2:350 ð0:00706T5:381Þlog10 g_ þ1þexp½ðT591:954Þ=2:664 þ3:480
hp ðg_ ; TÞ ¼ 10
g
3:434 ð0:00866Tþ4:225Þlog10 _ þ1þexp½ðT595:066Þ=2:899 þ3:594
(11) (12)
Contrast of experimental results with models from Eq. (11) and Eq. (12) is shown in Fig. 14. It can be seen that the models could accurately reflect the variation of steady-state viscosity and peak viscosity with temperature and shear rate changing. According to Eqs. (1), (11) and (12), the expressions of steadystate shear stress and peak shear stress as a function of shear rate and temperature can be obtained:
ts ðg_ ; TÞ ¼ g_ ,hs ðg_ ; TÞ
(13)
tp ðg_ ; TÞ ¼ g_ ,hp ðg_ ; TÞ
(14)
The above results show that not only the relationship between steady-state viscosity and shear rate, but also the relationship between peak viscosity and shear rate obeys the power law model at the temperature of 580e610 C and the shear rate of 0.1e100 s1. That is, the rheological behaviour of the mushy zone obeys the power law even if the dendritic a-Al is unbroken at the free-flowing and dendrite coherency stage i.e. before solid network forming. Moreover, steady-state and peak viscosity both can be modelled based on the power law model coupling with temperature.
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Fig. 14. Comparisons of experimental results with model from (a) (b) Eq. (11) and (c) (d) Eq. (12).
5.2. Modelling of creep shear stress and overshoot shear stress The Maxwell model consisting of a Hooke spring and a Newtonian dashpot is used to describe the properties of a viscoelastic fluid lying between the viscous fluid and the elastic fluid. Here the Maxwell model is used to characterize the rheological behaviour without the “overshoot” phenomenon, namely creep shear stress [36]:
t t_ þ ¼ g_ h G
simulated by JMatPro V7.0 software. The simulate temperature is from 700 C to 10 C with an interval of 1 C. The Scheil-Gulliver solidification model is used. The simulation result is shown in Fig. 15.
(15)
where t is the shear stress, h is the viscosity, t_ is the shear stress rate, G is the shear modulus. Relaxation time lf which is a measure of the time taken for the stress to relax is introduced; the shorter the relaxation time is, the more rapid the stress relaxes [36,37].
lf ¼
hs
(16)
G
Then Eq. (15) can be written as:
t_ ¼ Gg_
t lf
(17)
In order to obtain the shear modulus of the mushy zone of partially solidified A356 alloy, the solidification of A356 alloy was
Fig. 15. Evolution of shear modulus of A356 alloy with respect to temperature simulated using JMatPro.
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With a correction factor a ¼ 102 introduced to conform the result of simulation with the experimental results, the relationship between shear modulus (Pa) and temperature in the temperature range of 580e610 C based on the simulation is shown as:
GðTÞ ¼ a 4:291 1023 exp
T 67141 14:095
(18)
According to Eq. (16) and Eq. (18), relaxation time lf can be calculated:
lf ðg_ ; TÞ ¼
hs ðg_ ; TÞ
(19)
GðTÞ
Solving Eq. (17), the creep shear stress tcreep based on Maxwell model is obtained:
"
tcreep ¼ Glf g_ 1 exp
t
!#
lf
" ¼ ts 1 exp
!#
t
(20)
lf
where t is the shear time. Substituting Eq. (13) into Eq. (20), the viscoelastic model of mushy zone A356 alloy can be established:
"
tcreep ðg_ ; T; tÞ ¼ ts ðg_ ; TÞ 1 exp
t
(21)
lf ðg_ ; TÞ
¼ 1 exp
t
1þexp
g_ t 100:38 log10 g_ þ0:77
P
g_ t t1
P
g_ t exp t2
(22)
where tovershoot is the overshoot shear stress, t1 is a constant related to the shear rate, t2 is a constant and P is an attenuation index of the pulse peak. t1 is determined by Eq. (19).
log10 ð t1 Þ ¼ 0:38 log10 g_ þ 0:77
(23)
Substituting Eq. (14) into Eq. (22), the overshoot shear stress based on the pulse model can be obtained:
tovershoot ðg_ ; T; tÞ ¼ tp ðg_ ; TÞ 1 exp
g_ t t1
P
g_ t exp t2
(24)
5.3. Constitutive model Combining the Maxwell model and the pulse model (Eq. (21) with Eq. (24)), a constitutive model of the mushy zone of partially solidified A356 alloy can be established:
2:350
ð0:00706T5:381Þlog10 g_ þ
2 þ 1 exp
tovershoot ¼ tp , 1 exp
3 þ3:480 7 7 T591:954 7 2:664 g_ 7 7 7 5
2
!#6 6 6 610 lf 6 6 4
Besides, a pulse model is employed to characterize the “overshoot” phenomenon, namely overshoot shear stress [38]. The pulse model is as follows:
!#
tðg_ ; T; tÞ ¼ tcreep ðg_ ; T; tÞ þ tovershoot ðg_ ; T; tÞ "
1151
3
(25)
3:434 þ3:594 7 6 ð0:00866T4:225Þlog10 g_ þ 7 6 T595:066 1þexp 7 2:899 g_ t 6 610 7 _ exp g 7 t2 6 6 7 4 5
Referring to the relationship between shear rate/shear strain and time which follows g ¼ g_ ,t, the constitutive model can also be written as:
tðg_ ; T; gÞ ¼ tcreep ðg_ ; T; gÞ þ tovershoot ðg_ ; T; gÞ 2
"
g ¼ 1 exp _ f gl
!#6 6 6 610 6 6 4
ð0:00706T5:381Þlog10 g_ þ
1þexp
2 þ 1 exp
g 100:38 log10 g_ þ0:77
P
3 þ3:480 7 7 T591:954 7 2:664 g_ 7 7 7 5
2:350
3
3:434 þ3:594 7 6 ð0:00866T4:225Þlog10 g_ þ 7 6 T595:066 1þexp 7 2:899 g 6 610 7 _ exp g 7 t2 6 6 7 4 5
(26)
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The parameters in the constitutive model obtained by fitting are: t2 ¼ 5 and P ¼ 0.377. The comparison between the constitutive model and the experimental data is shown in Fig. 16. It can be seen that the constitutive model is in good agreement with the constitutive relation of the mushy zone of partially solidified A356 alloy at different temperatures and shear rates. The “overshoot” thixotropic behaviour of the mushy zone is reflected well by the constitutive model.
5.4. Application of the constitutive model to filling process To validate the constitutive model, it is employed to describe the mushy zone during the filling process of A356 alloy. Fig. 17 shows the three-dimensional model of the filling process. The simulation consists of a mould, A356 melt and a plunger. The cavity of the mould is step-shaped, and the plunger moves up vertically to push the melt filling the mould. ABAQUS/CAE 6.14 is used to simulate the filling process. The process parameters in the simulation are shown in Table 4. Fig. 18 shows the simulation results of mould filling process using the constitutive model. The flow front is not flat but shows an “arched” geometry during the filling process of the alloy in mushy zone (Fig. 18(b and c)). The “arch” is thought to attribute to the viscoelasticity of the mushy zone but not surface tension. Different from liquid state, surface tension in the mushy zone can be ignored owing to the solid particles and network. Moreover, the curvature of the flow front caused by surface tension is usually small in a large cavity (80e200 mm in the model), which is not consist with the large curvature of the “arch” in Fig. 18(c). During the filling process of viscoelastic fluid, the normal stress difference caused by shearing of the fluid drives the mushy zone to squeeze to the middle [29] (Fig. 18(b)), leading to the formation of the “arch” in the simulation results. The phenomenon has been also observed in the mould filling process of semi-solid aluminium other castings [16,39]. Besides, extrusion swelling of semi-solid aluminium alloy caused by the same above mentioned reason has also been observed by Neag et al. [39]. In conclusion, the simulation results indicate that the constitutive model is able to describe the rheological behaviour of A356 alloy in mushy zone. Moreover, the constitutive model has potential to be used to describe the forming process of other aluminium alloys in mushy zone with corresponding rheological parameters.
Fig. 17. Three-dimensional model of the filling process.
Table 4 Process parameters in the mould filling simulation. Parameters
Value
Mould inner diameters Plunger diameter Melt height Melt initial temperature Mould initial temperature Heat transfer coefficient Plunger velocity
80e200 mm 200 mm 90 mm 610 C 400 C 1500 W/m2$K 10 mm/s
6. Conclusions The current study focuses on the influence of microstructure, temperature and rheological parameters (shear rate and shear strain/time) on the rheological behaviour of the mushy zone of partially solidified A356 alloy. Particularly, a constitutive model of the mushy zone has been established based on the rheological experiments. The main conclusions obtained are as follows: (1) The solidification of A356 alloy is divided into three stages according to the content, size and morphology of a-Al phases: free-flowing stage (610 - 590 C), dendrite coherency
Fig. 16. Comparison of constitutive modelling and experimental data at (a) 0.1 s1, (b) 590 C.
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Fig. 18. Simulation result of mould filling process using the constitutive model at (a) 0.39 s, (b) 2.63 s, (c) 5.25 s and (d) 7.50 s after start of filling.
(2)
(3)
(4)
(5)
stage (590 - 570 C) and solid network stage (570 - 550 C). The apparent viscosity changed as a result of the evolution of a-Al phase. The thixotropic strength increases with increasing shear rate from 0.1 s1 to 100 s1 and with decreasing temperature from 610 C to 580 C. The maximum thixotropic factor of 87.3 was observed at 100 s1 and 580 C. The thixotropic strength is weak (thixotropic factors from 0.3 to 2.5) and basically not related to the shear rate at 610 C and 600 C. The peak viscosity, as well as the steady-state viscosity, obeys the power law model at temperature form 580 Ce610 C, shear rate from 0.1 s1 to 100 s1. Peak viscosity and steadystate viscosity were modelled using a power law model with the coupling of temperature and shear rate. A constitutive model of the mushy zone of partially solidified A356 alloy taking into consideration temperature, shear rate, shear tress and shear strain/time as parameters at temperature form 580 Ce610 C and shear rate from 0.1 s1 to 100 s1 is established. The constitutive model is successfully applied to simulate the filling process of the alloy in mushy zone, and the viscoelastic filling behaviour of the alloy was observed as well as that in other experiment. The constitutive model can provide a solution for characterizing the forming behaviour of other aluminium alloys in mushy zone with corresponding rheological parameters.
Acknowledgment This work was supported by the National Key research project [No. 2016YFB0300901]. References [1] M.A. Bayoumi, M.I. Negm, A.M. El-Gohry, Microstructure and mechanical properties of extruded AleSi alloy (A356) in the semi-solid state, Mater. Des.
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Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Rheological behaviour of partially solidified A356 alloy: Experimental study and constitutive modelling Zhen Ma a, Huarui Zhang a, c, *, Xiaoli Zhang b, Xiaoyan Wu a, Hanwei Fu a, Lina Jia a, Hu Zhang a, c, ** a b c
School of Materials Science and Engineering, Beihang University, Beijing, 100191, China Army Military Transportation University, Tianjin, 300000, China Ningbo Institute of Technology, Beihang University, Ningbo, 315832, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 April 2019 Received in revised form 26 June 2019 Accepted 27 June 2019 Available online 28 June 2019
Rheological behaviour of the mushy zone is dependent on the microstructure and related to parameters such as temperature, shear rate and shear strain/time. However, constitutive modelling of the rheological behaviour is difficult due to the diversity of the parameters. This study focuses on the influence of the microstructure and the above parameters on the rheological behaviour of partially solidified A356 alloy and establishing a constitutive model accordingly. During solidification, the evolution of a-Al phase leads to the changes of the apparent viscosity of mushy zone. Steady-state viscosity and peak viscosity measured by rotational rheometry reveal the thixotropy of the mushy zone. The thixotropic strength is found to increase with increasing shear rate and decreasing temperature. The peak viscosity, as well as the steady-state viscosity, is found to obey a power law model and is modelled using the power law model with the coupling of temperature and shear rate. Furthermore, a constitutive model with temperature, shear rate, shear tress and shear strain/time as parameters which describes the rheological behaviour of the mushy zone of partially solidified A356 alloy accurately and completely is established. Comparing with previously reported experimental results, the FEM simulation employing the constitutive model works well in enabling visualization of the viscoelastic filling process of A356 alloy in mushy zone. The constitutive model provides a solution for characterizing the forming behaviour of other mushy zone/semi-solid aluminium alloys with corresponding rheological parameters. © 2019 Elsevier B.V. All rights reserved.
Keywords: Mushy zone Partially solidified A356 aluminum alloy Microstructural evolution Rheological behaviour Constitutive model
1. Introduction Cast Al-Si alloys are widely used in aerospace, automotive and other fields owing to their excellent mechanical properties, outstanding corrosion resistance and good castability [1e4]. A356 alloy is a type of commercial cast hypoeutectic Al-Si alloy with extensive applications [5]. Cast defects such as segregation, porosity, and hot cracking are easily formed due to the flow of liquid and solid during the filling and feeding processes of casting [6]. This flow is determined by the rheological behaviour of mushy
* Corresponding author. School of Materials Science and Engineering, Beihang University, Beijing, NO.37 Xueyuan Road, 100191, China. ** Corresponding author. School of Materials Science and Engineering, Beihang University, Beijing, 100191, China. E-mail addresses: [email protected] (H. Zhang), [email protected] (H. Zhang). https://doi.org/10.1016/j.jallcom.2019.06.345 0925-8388/© 2019 Elsevier B.V. All rights reserved.
zone i.e. semi-solid state alloy during solidification of the alloy and by the casting parameters [6e9]. In this respect, understanding the rheological behaviour of mushy zone is crucial for obtaining appropriate casting process to eliminate defects. So far, several works have been published on the rheological behaviour of mushy zone or semi-solid state of aluminium alloys. Flemings [10] studied the rheological behaviour of semi-solid Al6.5Si alloys and found that the apparent viscosity decreased with increasing temperature and shear rate. The effect of compositional variation on rheological behaviour of semi-solid Al-Si-Mg alloy at a solid fraction (fs) of 0.45 was reported by Paes et al. [11], showing that increasing silicon and decreasing Mg content lead to lower apparent viscosity. Previous studies indicate that the mushy zone of aluminium alloys is a non-Newtonian fluid whose apparent viscosity varies with shear rate, temperature i.e. solid fraction and composition [10e12]. However, the previous rheological investigations of aluminium alloys are mostly based on the
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experiments of parallel-plate compression [2,13,14], where the material is assumed Newtonian fluid when calculating the rheological parameters [15,16], and the experiments are carried out based on partial melting of the alloys. Therefore, these experiments are unable to accurately reflect the rheological behaviour of the alloys during solidification. Rotational rheometry based on partial solidification, which takes materials as non-Newtonian fluid [17,18], is considered to have high accuracy and good applicability for investigating the rheological behaviour of the alloys during solidification. Using this method, the thixotropic behaviour of aluminium alloys in mushy zone can be measured [19,20]. For instance, Brabazon et al. [19] studied the thixotropic behaviour of Al-4Si alloy in mushy zone at a solid fraction of 0.36 and found that the thixotropic behaviour could be evaluated by peak and steady-state viscosities. The peak and steady-state viscosities are thought to result from the collapse, agglomeration and disagglomeration of solid particles during continuous shearing [16,20e22]. Quaak's study [23] on A356 alloy confirmed that the relationship between steady-state viscosity and shear rate obey a power law model, which is widely accepted and confirmed by other studies [19,20,24]. However, the mathematic relationship between the peak viscosity and shear rate is still unclear. Moreover, the thixotropic behaviour of mushy zone aluminium alloys indicates that the rheological behaviour of mushy zone is not only dependent on shear rate, temperature and composition, but also on shear strain/time. Therefore, to accurately describe the rheological behaviour, constitutive modelling of mushy zone taking into consideration temperature, shear rate and shear strain/time as parameters is of great importance. Some models have been established to describe the rheological behaviour of mushy zone, which are fundamental for the numerical simulation of casting processes [16,25e28]. Hu et al. [16] established a time-dependent power law model for 319s alloy considering the influence of geometry based on compression tests. Ludwig et al. [27] obtained a compressible constitutive model of AA5182 alloy considering the state of cohesion of mushy zone based on shear and tensile tests. However, previous rheological models are mostly incomplete due to the neglect of thixotropic behaviour. Given the diversity of parameters, a constitutive model that is capable of describing the rheological behaviour of mushy zone accurately and completely needs to be developed. In the present study, a rheological investigation of the mushy zone of partially solidified A356 is performed in a wide temperature range (580e610 C) and shear rate range (0.1e1000 s1) using rotational rheometry. The relationship between rheological behaviour and temperature, shear rate, shear strain/time and microstructure of the mushy zone of partially solidified A356 alloy was investigated. A power law model for the peak and steady-state viscosities are obtained with the coupling of temperature and shear rate. Furthermore, a constitutive model of the mushy zone of partially solidified A356 alloy is established and applied to simulate the filling process of the alloy in mushy zone.
chemical composition listed in Table 1. Partial solidification and quenching experiment was conducted along with differential scanning calorimetry (DSC) analysis using STA-449F3 to study the microstructural evolution of A356 alloy during solidification. During the partial solidification and quenching experiment, samples with a volume of 14 10 5 mm were first cut from A356 alloy ingots and placed in crucibles after polishing and ultrasonic cleaning; the samples were then heated to 700 C at a heating rate of 5 C/min in a tube furnace and isothermally treated for 5 min. The samples were finally cooled to 610 - 550 C at a cooling rate of 5 C/min followed by rapid quenching in water. Rheological experiments were performed using a hightemperature rotational rheometer MCR 502 with a concentriccylinder device (Fig. 1). During the experiment, the external cup remained static, and the internal bob rotated at a speed corresponding to the set shear rate in order to shear the samples inside the cup. The concentric-cylinder device was located in a heating chamber equipped with a hot air circulation device to provide a high-temperature environment with an accuracy of ±1 C. Inert atmosphere was maintained within the heating chamber using nitrogen gas to prevent sample oxidation. The rheological tests were based on the partial solidification of A356 alloy. Samples cut from A356 alloy ingots were polished and ultrasonically cleaned. Fig. 2 shows the procedure of the rheological tests. Two different types of rheological tests, continuous cooling tests and shear time tests, were performed to characterize the rheological behaviour of partially solidified A356 alloy. The continuous cooling tests were adopted to study the effect of temperature on rheological behaviour. The shear time tests were performed to reveal the thixotropic behaviour of the alloy. Three specimens were prepared for each condition in the shear time tests to ensure repeatability and accuracy of the results. During the continuous cooling tests, three different cooling rates were used with the shear rate being kept constant, whilst the shear time tests were performed with 6 combinations of temperature and shear rate. The shear time was 200 s for each shear time test. Table 2 summarizes the parameters of the rheological tests.
2. Experimental procedure The material studied in this research is A356 alloy with the
Fig. 1. Concentric-cylinder device of high-temperature rotational rheometer.
Table 1 A356 alloy composition. Element
Si
Mg
Fe
Cu
Ti
Sr
Zn
Ni
Al
Wt. %
7.128
0.346
0.111
0.089
0.145
0.0144
0.006
0.005
Balance
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phases before quenching. As shown in Fig. 3(h), the larger a-Al dendrites are considered to be the resultant solid phase transformed from the mushy zone, while other microstructures (such as extremely fine a-Al and eutectic microstructures) are considered to be liquid phase of the mushy zone. At the beginning of solidification (Fig. 3(a)), the alloy was in liquid state. With decreasing quenching temperature (Fig. 3(bef)), the alloy was in mushy zone, and the amount of solid phase gradually increased. Furthermore, the a-Al dendrites became coarser and more irregular. When solidification was finished (Fig. 3(g)), the microstructure of the alloy consisted mainly of well-developed a-Al dendrites and eutectic structure, with no liquid phase. 3.2. Rheological behaviour during continuous cooling Fig. 2. Rheological experimental procedure.
3. Results 3.1. Microstructures of partially solidified A356 alloy Fig. 3 presents the optical micrographs of the partially solidified alloy quenched at different temperatures. Both liquid and solid phases can be found in the mushy zone of A356 alloy before quenching. After quenching, the liquid phase transforms into extremely fine microstructure as a result of the fast cooling rate. Whereas the solid phases formed at normal cooling rate before quenching remain coarse similar to the morphology of the solid
The effects of temperature and cooling rate on the rheological behaviour of the mushy zone are shown in Fig. 4. At the temperature above 595 C (region I in Fig. 4), the shear stress and the apparent viscosity remains unchanged. In the temperature range of 595e568 C (region II in Fig. 4), both shear stress and apparent viscosity increase with decreasing temperature. Cooling rate has an obvious effect on the rheological behaviour of the mushy zone, where increased shear stress and apparent viscosity can be observed in by increasing cooling rate at the same temperature. 3.3. Thixotropic behaviour 3.3.1. Shear stress and apparent viscosity The evolution of apparent viscosities during the shear time tests
Table 2 Rheological experimental parameters. Test
Temperature ( C)
Shear rate (s1)
Continuous cooling tests Shear time tests
610e560 610, 605, 600, 595, 590, 585, 580
10 0.1, 0.5, 1, 5, 10, 100, 1000
Fig. 3. Optical micrographs of the partially solidified A356 alloy quenched at (a) 610 C, (b) 600 C, (c) 590 C, (d) 580 C, (e) 570 C, (f) 560 C, and (g) 550 C; (h) enlargement of the area in the box of (b).
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rule as that of apparent viscosity according to the definition of the apparent viscosity [29]. Peak shear stress and steady-state shear stress existed during the shear time tests corresponding to the peak viscosity and the steady-state viscosity, respectively.
Fig. 4. Variations of shear stress with temperature at a shear rate of 10 s1.
are plotted in Fig. 5. An “overshoot” phenomenon can be discovered: for each apparent viscosity curve, it firstly increases until reaching a peak value, then decreased and finally comes to a steady-state value. The “overshoot” phenomenon is more obvious when the temperature is lower. Since the shear rate and the temperature were kept constant during the shear time tests, the evolution of shear stress with respect to time should follow the same
3.3.2. Peak and steady-state shear stress/viscosity The variations of steady-state shear stress, steady-state viscosity, peak shear stress and peak viscosity with shear rate and temperature are shown in Figs. 6 and 7, respectively. The shear stress and viscosity values were average values calculated according to the three tests data in each condition. Within the testing temperature range 580e610 C, the steady-state shear stress decreases at first and then increases with increasing shear rate when the shear rate is higher than 100 s1 (Fig. 6(a)). The steady-state viscosity and peak viscosity decrease with increasing shear rate at 580e610 C (Fig. 6(b) and (d)). The peak shear stress, steady-state shear stress, peak viscosity and steady-state viscosity all increase with decreasing temperature (Fig. 7). 4. Discussion 4.1. Microstructural evolution during solidification During solidification, the microstructure of the A356 alloy evolves with decreasing temperature, which determines the
Fig. 5. Effects of shear time and shear rate on viscosity at (a) 610 C, (b) 600 C, (c) 590 C and (d) 580 C.
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Fig. 6. (a) Steady-state shear stress, (b) steady-state viscosity (c) peak shear stress and (d) peak viscosity versus shear rate.
rheological behaviour of the material. Therefore, understanding the microstructural evolution of A356 alloy during solidification is crucial to explain the rheological behaviour of the mushy zone of partially solidified alloy. The enthalpy change during solidification shows that there are three exothermic peaks during solidification (Fig. 8(a)): Peak A (612 - 564 C) corresponds to the formation of a-Al dendrite. Peak B (564 - 549 C) corresponds to the formation of Al-Si binary eutectic and the precipitation of Fe-rich phase. Peak C (549 - 540 C) corresponds to Al-Mg2Si-Si ternary eutectic reaction [30]. Then the components of the solid phase in each stage of solidification can be summarized accordingly. The solid phase existing in the mushy zone is mainly a-Al phase at the initial stage (612 - 564 C) of solidification. In the interim and terminal stage (564 - 540 C), with the precipitation of eutectic phase and Fe-rich phases, the main components of the solid phases change from a-Al to a-Al, eutectic phase and Fe-rich phases. Comparison of solid fraction obtained by metallographic and enthalpy change method is shown in Fig. 8(b), which shows a good consistency of the two methods at 612 - 564 C. While the solid fraction obtained by metallographic method is obviously lower at the late stage (564 - 540 C) of solidification because the metallographic method does not take phases other than a-Al as solid phase. On the contrary, the enthalpy change method takes into account all the solid phases during calculating solid fraction. As a conclusion, the microstructures of partially solidified quenched
alloy can accurately reflect the evolution of the solid phase (a-Al) during solidification A356 alloy at 612 - 564 C. As shown in Fig. 9, the mushy zone during solidification of A356 alloy undergoes three stages according to the content, size and morphology of the a-Al phase in the partial solidification and quenching experiment. 4.2. Rheological behaviour of the mushy zone Apparent viscosity is a parameter that characterizes the fluidity of a fluid. Apparent viscosity is determined by shear rate and the shear stress which can be described by the following equation [29]:
ha ¼
t g_
(1)
where ha is the apparent viscosity, t is the shear stress and g_ is the shear rate. With temperature decreasing from 610 C to 595 C, the mushy zone is at the free-flowing stage (Fig. 9(a)). The rheological behaviour of the mushy zone is essentially of the liquid phase where the influence of dispersed small a-Al islands can be neglected. Therefore, temperature has little effect on the shear stress and apparent viscosity of the mushy zone (Figs. 4 and 7). When the temperature decreases from 595 C to 570 C, the mushy zone is in the dendrite coherency stage (Fig. 9(b)). The rheological
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Fig. 7. (a) Steady-state shear stress, (b) steady-state viscosity, (c) peak shear stress and (d) peak viscosity versus temperature.
Fig. 8. (a) Enthalpy change of A356 alloy during solidification and (b) solid fraction during solidification of A356.
behaviour is greatly affected by the increased content and the coarsening of the solid phase, leading to the increase of shear stress and apparent viscosity (Figs. 4 and 7). When temperature is below 570 C, the mushy zone enters the solid network stage (Fig. 9(c)), causing a dramatic rise of the shear stress and the apparent
viscosity (Fig. 4). Moreover, the cooling rate affects the rheological behaviour of the A356 alloy during solidification by alerting the morphology of a-Al phase. The a-Al phase tends to grow into a dendritic shape rather than spherical or rosette shape at high cooling rates [6,31],
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In order to further investigate the thixotropic behaviour of the mushy zone of partially solidified A356 alloy, the proportional coefficient of peak shear stress (viscosity) and steady-state shear stress (viscosity) is defined as thixotropic factor kt (kh) to characterize the thixotropic strength with the following equations:
Fig. 9. Schematic diagram of the evolution of mushy zone during solidification of A356 alloy. a. Free-flowing stage (610 - 590 C, Fig. 9(a)): Small sized a-Al phase emerges, and the liquid phase can flow freely with a-Al islands dispersed in it. b. Dendrite coherency stage (590 - 570 C, Fig. 9(b)): The content of a-Al phase increases, and the dendrites starts to grow. The small a-Al coalesce, and the liquid phase can flow in the gap of the big a-Al islands. c. Solid network stage (570 - 550 C, Fig. 9(c)): The content of a-Al phase increases rapidly with well developed dendrites. The dendrites intercalated with each other to form a solid network, and the liquid phase separated by the coherent solid network can not flow freely.
thus increase shear stress and apparent viscosity of the mushy zone (Fig. 4). As a typical shear thinning non-Newtonian fluid, the shear rate has a significant effect on the rheological behaviour of the mushy zone of partially solidified A356 alloy. The dendrites collapse adequately by high shear rate, leading to smaller sizes and a rounder shape of the solid particles at higher shear rate. As a result, apparent viscosity decreases with increasing shear rate (Fig. 6(b, d)). In this respect, the shear thinning phenomenon (apparent viscosity decreases with increasing shear rate) of the mushy zone can be observed, which demonstrates that the mushy zone of partially solidified A356 alloy is a pseudoplastic fluid. The mushy zone of partially solidified A356 alloy shows thixotropic behaviour due to the presence of solid particles exist within it. This means the shear stress (apparent viscosity) is timedependent during shearing progress (Fig. 5). At the beginning of shearing, the shear stress (apparent viscosity) increases rapidly due to the presence of dendritic a-Al or solid network. With continued shearing, the dendritic a-Al breaks into parts and the solid network gradually collapse, resulting in the decrease of shear stress (apparent viscosity). Finally, the shear stress (apparent viscosity) tends to be stable when the dendritic a-Al is completely broken down and the solid network comes to a complete collapse and a new equilibrium of the microstructure is reached.
kt ¼
tp ts ts
(2)
kh ¼
hp hs hs
(3)
where tp (hp) is the peak shear stress (viscosity) and ts (hs) is the steady-state shear stress (viscosity). The thixotropic factors are different at different temperatures and shear rates. The thixotropic factors increase from 2.7 to 87.3 with shear rate increasing from 0.1 s1 to 100 s1 at the temperature of 580 C (Fig. 10(a)). That is because the higher shear rates break the dendrites more thoroughly, which leading lower shear stress (viscosity) at the steadystate stage. Yet the peak stress (viscosity) is mainly determined by the strength of solid network but not shear rate. Therefore, the thixotropic factors increase with increasing shear rate. Nevertheless, it seems that because of the inaccurate peak shear stress at high shear rates (the peak shear stress is missed due to low data sampling frequency), there is a dramatically drop of thixotropic factors beyond 100 s1 at 580 C (Fig. 10(a)). The thixotropic factors decrease with increasing temperature, and the phenomenon is more obvious at higher shear rate (Fig. 10(b)). The thixotropic factor decreases from 87.3 to 0.1 with temperature increasing from 580 C to 610 C at 100 s1. The reason is that the solid network is stronger when the temperature is lower, and therefore, the corresponding peak stress (viscosity) needed to break the solid network becomes higher. But the steady-state shear stress (viscosity) is mainly related to the shear rate and solid fraction in the mushy zone. The correlation between steady-state shear stress (viscosity) and temperature is weaker compared with the correlation between peak shear stress (viscosity) and temperature. Therefore, the thixotropic factors decrease with increasing temperature. The thixotropic factors are very small (0.3e2.5) and basically not related to the shear rate at 610 C and 600 C (Fig. 10(a)). At this temperature, the mushy zone is at the free-flowing stage with no solid network. There is no collapse of solid network in the shearing process, where the peak stress and the steady-state stress do not differ greatly, resulting in small thixotropic factors. Moreover, due
Fig. 10. Relationships between thixotropic factor and (a) shear rate, (b) temperature.
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to the weak pseudoplasticity of the mushy zone at this temperature, the effects of shear rate on the peak stress (viscosity) and steady-state stress (viscosity) are also insignificant, resulting in constant thixotropic factors with shear rate changing. Comparison of viscosities measured in this work with those in previous works on Al-Si alloys is presented in Fig. 11. Owing to the rotational rheometry, the rheological behaviour of the alloy at low solid fractions (fs < 0.41) is studied in the present work rather than that at high solid fractions (fs > 0.45) in the previews works. The steady-state viscosity values in this study are significantly lower than those from the previous works. This is thought to be associated with the microstructures of the alloys during tests. There is more liquid entrapment within the semi-solid slurries prepared by et al. [33] and partial melting (Hu et al. [16], Liu et al. [32], Loue Yurko et al. [18]) than that prepared by partial solidification in this work, resulting in higher viscosities in the references than that in the present work [16]. What's more, the tests conducted with isothermal compression, back extrusion and drop forge viscometer are for transient behaviour rather than steady-state behaviour. As a consequence, the current steady-state viscosity values are lower than the values in the references. On the other hand, the measurements for transient behaviour in the references lead to constantly changing of the shear rate, in which the viscosity will never reach to peak or steady-state viscosity but to a value between them. As a result, the peak viscosity values (fs ¼ 0.41 and 0.33) in this study is higher than those (fs > 0.45) in the previous works even at lower solid fractions. In conclusion, rotational rheometry used in the present study is more reliable to measure the rheological behaviour of the mushy zone, especially that at low solid fraction. 5. Constitutive modelling of the mushy zone of partially solidified A356 alloy The thixotropic and shear thinning rheological behaviour of the mushy zone of partially solidified A356 alloy at the temperature of 580e610 C and the shear rate of 0.1e100 s1 can be modelled according to the phenomenological method. The proposed model consists of a combination of a viscoelastic Maxwell model and a
pulse model to describe its constitutive relationship [34]. 5.1. Modelling of steady-state viscosity and peak viscosity To obtain the constitutive model, steady-state and peak shear stresses should be modelled first. According to Fig. 6(b), when the shear rate is 0.1e100 s1, the relationship between steady-state viscosity/peak viscosity and the shear rate follows the power law model, namely ha ¼ K,g_ n1 [35], where K is the consistency factor and n-1 is the flow exponent. Therefore, the experimental data is fitted to Eq. (6) and the results are shown in Fig. 12 and Table 3.
log10 ha ¼ ðn 1Þlog10 g_ þ log10 K
(6)
The parameters n-1 and log10K have linear and three-stage relationship with temperature, respectively. The evolutions of the above two parameters with respect to temperature are fitted using the linear fitting and the Boltzmann fitting, respectively. The fitting results are shown in Fig. 13. Eqs. (7) and (8) show the expressions of n-1 as a function of temperature for steady-state and peak viscosities, respectively.
n 1 ¼ 0:00706T 5:381
(7)
n 1 ¼ 0:00866T þ 4:225
(8)
Where T is temperature. The Boltzmann relationships between log10K and temperature for steady-state viscosity and peak viscosity are shown as Eq. (9) and Eq. (10), respectively.
log10 K ¼
2:350 þ 3:480 1 þ exp½ðT 591:954Þ=2:664
(9)
log10 K ¼
3:434 þ 3:594 1 þ exp½ðT 595:066Þ=2:899
(10)
Substituting Eq. (7) and Eq. (9), Eq. (8) and Eq. (10) into Eq. (6), the models of steady-state viscosity and peak viscosity with shear rate and temperature can be obtained as Eqs. (11) and (12),
Fig. 11. Comparison of viscosities versus shear rate obtained by various experimental methods and conditions.
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Fig. 12. Experimental results and fitting curves of (a) steady-state viscosity, (b) peak viscosity.
Table 3 Fitting parameters of steady-state viscosity and peak viscosity. Fitting parameters Steady-state viscosity
Peak viscosity
n-1 log10K R2 n-1 log10K R2
580 C
585 C
590 C
595 C
600 C
605 C
610 C
1.306 5.779 0.999 0.936 7.00 0.968
1.294 5.647 0.999 0.764 7.067 0.914
1.179 5.076 0.998 0.850 6.284 0.976
1.093 3.994 0.999 0.827 5.530 0.876
1.128 3.687 0.998 0.944 3.917 0.976
1.167 3.510 0.997 1.038 3.781 0.976
1.078 3.406 0.999 1.126 3.643 0.998
Fig. 13. Evolutions of fitting parameters n-1, log10K for (a) steady-state viscosity, (b) peak viscosity with respect to temperature.
respectively.
hs ðg_ ; TÞ ¼ 10
2:350 ð0:00706T5:381Þlog10 g_ þ1þexp½ðT591:954Þ=2:664 þ3:480
hp ðg_ ; TÞ ¼ 10
g
3:434 ð0:00866Tþ4:225Þlog10 _ þ1þexp½ðT595:066Þ=2:899 þ3:594
(11) (12)
Contrast of experimental results with models from Eq. (11) and Eq. (12) is shown in Fig. 14. It can be seen that the models could accurately reflect the variation of steady-state viscosity and peak viscosity with temperature and shear rate changing. According to Eqs. (1), (11) and (12), the expressions of steadystate shear stress and peak shear stress as a function of shear rate and temperature can be obtained:
ts ðg_ ; TÞ ¼ g_ ,hs ðg_ ; TÞ
(13)
tp ðg_ ; TÞ ¼ g_ ,hp ðg_ ; TÞ
(14)
The above results show that not only the relationship between steady-state viscosity and shear rate, but also the relationship between peak viscosity and shear rate obeys the power law model at the temperature of 580e610 C and the shear rate of 0.1e100 s1. That is, the rheological behaviour of the mushy zone obeys the power law even if the dendritic a-Al is unbroken at the free-flowing and dendrite coherency stage i.e. before solid network forming. Moreover, steady-state and peak viscosity both can be modelled based on the power law model coupling with temperature.
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Fig. 14. Comparisons of experimental results with model from (a) (b) Eq. (11) and (c) (d) Eq. (12).
5.2. Modelling of creep shear stress and overshoot shear stress The Maxwell model consisting of a Hooke spring and a Newtonian dashpot is used to describe the properties of a viscoelastic fluid lying between the viscous fluid and the elastic fluid. Here the Maxwell model is used to characterize the rheological behaviour without the “overshoot” phenomenon, namely creep shear stress [36]:
t t_ þ ¼ g_ h G
simulated by JMatPro V7.0 software. The simulate temperature is from 700 C to 10 C with an interval of 1 C. The Scheil-Gulliver solidification model is used. The simulation result is shown in Fig. 15.
(15)
where t is the shear stress, h is the viscosity, t_ is the shear stress rate, G is the shear modulus. Relaxation time lf which is a measure of the time taken for the stress to relax is introduced; the shorter the relaxation time is, the more rapid the stress relaxes [36,37].
lf ¼
hs
(16)
G
Then Eq. (15) can be written as:
t_ ¼ Gg_
t lf
(17)
In order to obtain the shear modulus of the mushy zone of partially solidified A356 alloy, the solidification of A356 alloy was
Fig. 15. Evolution of shear modulus of A356 alloy with respect to temperature simulated using JMatPro.
Z. Ma et al. / Journal of Alloys and Compounds 803 (2019) 1141e1154
With a correction factor a ¼ 102 introduced to conform the result of simulation with the experimental results, the relationship between shear modulus (Pa) and temperature in the temperature range of 580e610 C based on the simulation is shown as:
GðTÞ ¼ a 4:291 1023 exp
T 67141 14:095
(18)
According to Eq. (16) and Eq. (18), relaxation time lf can be calculated:
lf ðg_ ; TÞ ¼
hs ðg_ ; TÞ
(19)
GðTÞ
Solving Eq. (17), the creep shear stress tcreep based on Maxwell model is obtained:
"
tcreep ¼ Glf g_ 1 exp
t
!#
lf
" ¼ ts 1 exp
!#
t
(20)
lf
where t is the shear time. Substituting Eq. (13) into Eq. (20), the viscoelastic model of mushy zone A356 alloy can be established:
"
tcreep ðg_ ; T; tÞ ¼ ts ðg_ ; TÞ 1 exp
t
(21)
lf ðg_ ; TÞ
¼ 1 exp
t
1þexp
g_ t 100:38 log10 g_ þ0:77
P
g_ t t1
P
g_ t exp t2
(22)
where tovershoot is the overshoot shear stress, t1 is a constant related to the shear rate, t2 is a constant and P is an attenuation index of the pulse peak. t1 is determined by Eq. (19).
log10 ð t1 Þ ¼ 0:38 log10 g_ þ 0:77
(23)
Substituting Eq. (14) into Eq. (22), the overshoot shear stress based on the pulse model can be obtained:
tovershoot ðg_ ; T; tÞ ¼ tp ðg_ ; TÞ 1 exp
g_ t t1
P
g_ t exp t2
(24)
5.3. Constitutive model Combining the Maxwell model and the pulse model (Eq. (21) with Eq. (24)), a constitutive model of the mushy zone of partially solidified A356 alloy can be established:
2:350
ð0:00706T5:381Þlog10 g_ þ
2 þ 1 exp
tovershoot ¼ tp , 1 exp
3 þ3:480 7 7 T591:954 7 2:664 g_ 7 7 7 5
2
!#6 6 6 610 lf 6 6 4
Besides, a pulse model is employed to characterize the “overshoot” phenomenon, namely overshoot shear stress [38]. The pulse model is as follows:
!#
tðg_ ; T; tÞ ¼ tcreep ðg_ ; T; tÞ þ tovershoot ðg_ ; T; tÞ "
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3
(25)
3:434 þ3:594 7 6 ð0:00866T4:225Þlog10 g_ þ 7 6 T595:066 1þexp 7 2:899 g_ t 6 610 7 _ exp g 7 t2 6 6 7 4 5
Referring to the relationship between shear rate/shear strain and time which follows g ¼ g_ ,t, the constitutive model can also be written as:
tðg_ ; T; gÞ ¼ tcreep ðg_ ; T; gÞ þ tovershoot ðg_ ; T; gÞ 2
"
g ¼ 1 exp _ f gl
!#6 6 6 610 6 6 4
ð0:00706T5:381Þlog10 g_ þ
1þexp
2 þ 1 exp
g 100:38 log10 g_ þ0:77
P
3 þ3:480 7 7 T591:954 7 2:664 g_ 7 7 7 5
2:350
3
3:434 þ3:594 7 6 ð0:00866T4:225Þlog10 g_ þ 7 6 T595:066 1þexp 7 2:899 g 6 610 7 _ exp g 7 t2 6 6 7 4 5
(26)
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The parameters in the constitutive model obtained by fitting are: t2 ¼ 5 and P ¼ 0.377. The comparison between the constitutive model and the experimental data is shown in Fig. 16. It can be seen that the constitutive model is in good agreement with the constitutive relation of the mushy zone of partially solidified A356 alloy at different temperatures and shear rates. The “overshoot” thixotropic behaviour of the mushy zone is reflected well by the constitutive model.
5.4. Application of the constitutive model to filling process To validate the constitutive model, it is employed to describe the mushy zone during the filling process of A356 alloy. Fig. 17 shows the three-dimensional model of the filling process. The simulation consists of a mould, A356 melt and a plunger. The cavity of the mould is step-shaped, and the plunger moves up vertically to push the melt filling the mould. ABAQUS/CAE 6.14 is used to simulate the filling process. The process parameters in the simulation are shown in Table 4. Fig. 18 shows the simulation results of mould filling process using the constitutive model. The flow front is not flat but shows an “arched” geometry during the filling process of the alloy in mushy zone (Fig. 18(b and c)). The “arch” is thought to attribute to the viscoelasticity of the mushy zone but not surface tension. Different from liquid state, surface tension in the mushy zone can be ignored owing to the solid particles and network. Moreover, the curvature of the flow front caused by surface tension is usually small in a large cavity (80e200 mm in the model), which is not consist with the large curvature of the “arch” in Fig. 18(c). During the filling process of viscoelastic fluid, the normal stress difference caused by shearing of the fluid drives the mushy zone to squeeze to the middle [29] (Fig. 18(b)), leading to the formation of the “arch” in the simulation results. The phenomenon has been also observed in the mould filling process of semi-solid aluminium other castings [16,39]. Besides, extrusion swelling of semi-solid aluminium alloy caused by the same above mentioned reason has also been observed by Neag et al. [39]. In conclusion, the simulation results indicate that the constitutive model is able to describe the rheological behaviour of A356 alloy in mushy zone. Moreover, the constitutive model has potential to be used to describe the forming process of other aluminium alloys in mushy zone with corresponding rheological parameters.
Fig. 17. Three-dimensional model of the filling process.
Table 4 Process parameters in the mould filling simulation. Parameters
Value
Mould inner diameters Plunger diameter Melt height Melt initial temperature Mould initial temperature Heat transfer coefficient Plunger velocity
80e200 mm 200 mm 90 mm 610 C 400 C 1500 W/m2$K 10 mm/s
6. Conclusions The current study focuses on the influence of microstructure, temperature and rheological parameters (shear rate and shear strain/time) on the rheological behaviour of the mushy zone of partially solidified A356 alloy. Particularly, a constitutive model of the mushy zone has been established based on the rheological experiments. The main conclusions obtained are as follows: (1) The solidification of A356 alloy is divided into three stages according to the content, size and morphology of a-Al phases: free-flowing stage (610 - 590 C), dendrite coherency
Fig. 16. Comparison of constitutive modelling and experimental data at (a) 0.1 s1, (b) 590 C.
Z. Ma et al. / Journal of Alloys and Compounds 803 (2019) 1141e1154
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Fig. 18. Simulation result of mould filling process using the constitutive model at (a) 0.39 s, (b) 2.63 s, (c) 5.25 s and (d) 7.50 s after start of filling.
(2)
(3)
(4)
(5)
stage (590 - 570 C) and solid network stage (570 - 550 C). The apparent viscosity changed as a result of the evolution of a-Al phase. The thixotropic strength increases with increasing shear rate from 0.1 s1 to 100 s1 and with decreasing temperature from 610 C to 580 C. The maximum thixotropic factor of 87.3 was observed at 100 s1 and 580 C. The thixotropic strength is weak (thixotropic factors from 0.3 to 2.5) and basically not related to the shear rate at 610 C and 600 C. The peak viscosity, as well as the steady-state viscosity, obeys the power law model at temperature form 580 Ce610 C, shear rate from 0.1 s1 to 100 s1. Peak viscosity and steadystate viscosity were modelled using a power law model with the coupling of temperature and shear rate. A constitutive model of the mushy zone of partially solidified A356 alloy taking into consideration temperature, shear rate, shear tress and shear strain/time as parameters at temperature form 580 Ce610 C and shear rate from 0.1 s1 to 100 s1 is established. The constitutive model is successfully applied to simulate the filling process of the alloy in mushy zone, and the viscoelastic filling behaviour of the alloy was observed as well as that in other experiment. The constitutive model can provide a solution for characterizing the forming behaviour of other aluminium alloys in mushy zone with corresponding rheological parameters.
Acknowledgment This work was supported by the National Key research project [No. 2016YFB0300901]. References [1] M.A. Bayoumi, M.I. Negm, A.M. El-Gohry, Microstructure and mechanical properties of extruded AleSi alloy (A356) in the semi-solid state, Mater. Des.
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