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Controlling Factor for Maximum Tensile Stress and Elongation of Aluminum Alloy during Partial Solidification
Ryosuke TakaiRei HiroharaNaoki EndoYoshihiro NagataToshimitsu OkaneMakoto Yoshida
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2019 年 60 巻 11 号 p. 2406-2415

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Abstract

To predict hot tearing of direct chill casting ingot, both the tensile constitutive behavior and elongation of alloy are inevitable during partial solidification. For predicting both the maximum true stress σss and the elongation εelong regardless of alloy systems, their dominant factor was examined in terms of the solidification microstructure. For an Al–Mg and an Al–Cu alloys, (i) temperature T dependences of the maximum true stress and elongation (σss = f(T) and εelong = f(T)) and (ii) dihedral angle θ of liquid phase formed at grain boundary were measured experimentally. Then, fraction of solid cohesion C was determined by the Campbell’s model using the angle.

Firstly, the solid fraction dependence of the tensile properties (σss = f(fs) and εelong = f(fs)) were compared between the two alloys. The two dependences differ with each other. Secondly, the fraction of solid cohesion dependences of the tensile properties (σss = f(C) and εelong = f(C)) were compared and the result shows that the two dependences were consistent with each other. The fraction of solid cohesion enables to explain the difference in solid fraction dependence of the tensile properties for the two alloys. The result demonstrates that the dihedral angle should be essential to predict the two tensile properties of alloy during partial solidification.

 

This Paper was Originally Published in Japanese in J. JILM 69 (2019) 255–262.

Fig. 8 Solid cohesion dependence of tensile properties for two alloys in partially solidified state: (a) normalized stress and (b) elongation.

1. Introduction

Wrought aluminum alloy generally includes various elements to improve material properties such as strength at room temperature. In direct chill casting, which produces an alloy ingot, increased casting speed is desired to improve productivity. However, these operations sometimes entail enhanced risks of hot tearing occurrence in ingots.13) In recent years, finite element method (FEM) thermal stress analysis of casting during solidification has become useful to ascertain the processing parameters for producing sound ingots without hot tearing. For predicting thermal stress, hot tearing tendency and its occurrence,411) mechanical properties such as strength, true stress–strain curve, and the ductility are indispensable for alloy in a semi-solid state. For the former, both the strength and the true stress–true strain curve indicate the mechanical basis for a constitutive model.411) For the latter, the temperature dependence of the ductility reveals the brittle temperature range (BTR), which is a criterion of hot tearing occurrence.46) Therefore, not estimation but experimental measurements of the actual phenomena indicating the mechanical properties above are indispensable for reasonable analysis and hot tearing prediction.

Nevertheless, for existing and newly developed alloys of various chemical compositions, it is unreasonable to expect to obtain their tensile properties in a semi-solid state. Therefore, irrespective of the alloy system, it is strongly desired that mechanical properties of various alloys be described in a semi-solid state.

Earlier works not only experimentally obtained the mechanical properties of several alloys in a semi-solid state;1218) they also presented various constitutive models using parameter reflecting the morphology of the solidification microstructure.9,15,16,19,20) For the former, some experimentally obtained results13,18) indicates that the behaviors of strength development against the solid fraction differ depending on alloy system. Figure 1 is a schematic demonstrating that the solid fraction unable to describe the difference. In curve (1), the increasing rate of strength against the solid fraction is higher immediately above the solid fraction where strength starts to arise, i.e. immediately below the zero strength temperature (ZST). Then, the change in the rate is small toward the solidification end. By the contrast, in curve (2), the rate of increase is lower at the solid fraction where strength starts to arise. Then, the change in the rate is relatively large immediately before the solidification end.

Fig. 1

Classification of strength development with the increase in solid fraction for various wrought aluminum alloys in semi-solid state.

In the constitutive model of the semi-solid state presented in earlier works, the fraction of the grain boundary area covered by liquid film fLGB20) and/or solid cohesion C15,16) are used as parameters that reflect the material strength. Definition of the parameters are presented below.   

\begin{equation} f_{\text{LGB}} = \frac{A_{\text{wet$\,$GB}}}{A_{\text{total$\,$GB}}} \end{equation} (1)
  
\begin{equation} C = \frac{A_{\text{contact$\,$GB}}}{A_{\text{total$\,$GB}}} = 1 - f_{\text{LGB}} \end{equation} (2)
When the grain shape in the semi-solid microstructure is approximated as a hexagonal array in Fig. 2(a), the schematic images of Awet GB and Acontact GB are presented respectively as thick lines in Fig. 2(c) and (d). Atotal GB is the total area of grain boundary. Haaften et al.20) presented one model with the parameter as shown below.   
\begin{equation} \sigma_{\text{ss}} = \sigma_{\text{s}}(1 - f_{\text{LGB}}) = \sigma_{\text{s}}C \end{equation} (3)
  
\begin{equation} f_{\text{LGB}}\ \text{or}\ C = \mathrm{f}(f_{\text{s}}(T),\theta) \end{equation} (4)
  
\begin{equation} \sigma_{\text{s}} = \left(\frac{\dot{\varepsilon}_{\text{s}}}{A_{0}}\right)^{\frac{1}{n_{\text{se}}}}\exp \left(\frac{Q}{n_{\text{se}}RT_{\text{ab}}}\right) \end{equation} (5)
In the equation above, σss, σs and $\dot{\varepsilon }_{\text{s}}$ respectively present the maximum true stress in the semi-solid state, the flow stress, and the strain rate in the solid state of an alloy. In addition, nse, Q, A0, R, and Tab respectively express the stress exponent, the activation energy, the creep parameter, the gas constant, and the absolute temperature. Moreover, θ is the interfacial contact angle at which liquid phase formed at the grain boundary. In the model, the semi-solid stress can be predicted from the stress in the solid state multiplied by the fraction of the grain boundary area covered by liquid film (1 − fLGB (or the fraction of solid cohesion C)). The stress in the solid state can be ascertain from the general creep law21,22) in eq. (5). For example, σss equals 0 when fLGB = 1 (C = 0). And σss equals σs when fLGB = 0 (C = 1). Ludwig et al.15,16) presented a model incorporating solid cohesion using parameter C. Through the two models15,16,20) above, the semi-solid strengths were discussed in terms of the fraction of solid cohesion C (or the fraction of area covered by liquid film). Earlier works can be classified into the two groups presented in Table 1.

Fig. 2

Idealized microstructure in semi-solid state of alloy: (a) solid grains in the form of regular hexagons with liquid phases, (b) second-phase particle at grain boundary triple junction,25) (c) grain boundary area Awet GB wet by liquid phase (thick line) and (d) grain boundary area Acontact GB where grains contact each other (thick line).

Table 1 Earlier works discussing relation between parameters C (fraction of solid cohesion) and fLGB (fraction of grain boundary area covered by liquid film) and strength for wrought aluminum alloys in semi-solid state.

(1) Earlier works determining parameter fLGB and/or C using experimentally obtained strength measurements

Giraud et al.23) and Bai et al.24) obtained the tensile strength of wrought aluminum alloy in a semi-solid state from measurements taken during experimentation. They derived the solid fraction dependence of the fraction of the grain boundary area covered by liquid film fLGB (= f(fs)) by substituting the strength into eq. (3). Then, using the parameter, they elucidated the relation between the liquid film morphology and the strength. Similarly, Ludwig et al.15,16) and Subroto et al.16) also ascertained the solid fraction dependence of the solid cohesion C (= 1 − (1 − fs)p) using both experimental strength and a constitutive model.15,16) However, to verify the values of the derived parameters fLGB and/or C, no one has compared these parameters fLGB and/or C with those of the actual solidification microstructure.

(2) Earlier works predicting alloy strength in a semi-solid state using the fraction of grain boundary area covered by liquid film fLGB determined by the dihedral angle

Theoretical models25,26) such as that by Campbell can elucidate the fraction of the grain boundary area covered by liquid film fLGB and/or the fraction of solid cohesion C against each solid fraction (eq. (4)) if the dihedral angle of liquid phase formed at grain boundary is known. Using one of the models,26) Haaften et al.20) determined the parameter fLGB against each liquid fraction using both θ = 0° and 180°, which are the two extreme values in the range of theoretically possible angles. Then, by substituting the derived parameters fLGB (eq. (4)) into eq. (3), they respectively predicted the maximum stress σss in a semi-solid state for each dihedral angle. Consequently, they show that the experimentally obtained value is generally in the range of the two predicted values. However, the two dihedral angles they used have no experimental basis. Therefore, no earlier work examined prediction of the maximum true stress in semi-solid state of alloy using the experimentally measured dihedral angle. Still, the ductility of alloy in the semi-solid state (or BTR) is an experimental basis for the cracking criterion. Therefore, the value should also be a factor along with the maximum true stress in hot tearing prediction. Earlier works12,2729) suggest that the ductility, as well as the maximum true stress, is closely related to the solid cohesion. However, no report of the literature describes a study examining the quantitative relation between them.

Summarizing a review of the earlier works presented above, at first, the experimentally obtained result revealed that the solid fraction dependence of the strength of alloys in a semi-solid state depends on the alloy systems1318) as presented in Fig. 1. Secondly, the constitutive models incorporate the fraction of solid cohesion C and the fraction of grain boundary area covered by liquid film fLGB = f(fs,θ) as a factor to determine the strength.15,16,20) Therefore, to verify experimentally that the fraction of solid cohesion C is a dominant factor of the semi-solid tensile properties, the relation between (a) the fraction of solid cohesion C derived from the experimentally measured dihedral angle and (b) the experimentally obtained semi-solid tensile properties (σss = f(C) and εelong = f(C)) should be compared for two alloys with different alloy systems. Then, if the two curves of one alloy are consistent with those of the other alloy, we can explain the solid fraction dependence of the semi-solid tensile properties irrespective of alloy systems. Therefore, to clarify the effects of the fraction of solid cohesion determined by the measured dihedral angle on the tensile properties, the present study examines the following two points.

  1. (i)    Experimental measurements of both the dihedral angle of liquid phase formed at the grain boundary and the tensile properties (maximum true stress and elongation) in a partially solidified state.
  2. (ii)    The relation between the solid cohesion derived using the above dihedral angle and the experimentally obtained tensile properties (maximum true stress and elongation).

2. Experimental

2.1 Alloys

An Al–2 mass%Cu and an Al–5 mass%Mg alloy were used for alloys with the two alloy systems. An Al–5Ti–1B master alloy (as grain refiner) was added to the alloys after melting for the two reasons below.

  1. (i)    Constitutive behavior of aluminum alloy in a semi-solid state depends on the grain size.12) To clarify the alloy system effects on constitutive behavior, the grain size difference between the two alloys should be minimized to the greatest degree possible.
  2. (ii)    As described in subsection 2.3.5, the shape of the primary phase during solidification is assumed as regular hexagonal. Fine equiaxed dendritic microstructure rather than a coarse equiaxed one should be closer to such a simplified microstructure morphology.

2.2 Device for the tensile test after partial solidification

The tensile testing device3035) we used enables one to obtain the true stress–true strain curve of aluminum alloy in the partially solidified state. Figure 3(a) depicts the specimen dimensions. Because details of the testing procedures are presented in earlier reports,3035) only brief outlines are presented below.

Fig. 3

(a) Dimensions of the test specimen and (b) typical true stress-true strain curve obtained by the testing device.

  1. (i)    The test specimen is cast by pouring molten material (720°C) into the mold. The test specimen is then constrained by both the fixed rod and the pull rod embedded in the specimen.
  2. (ii)    The tensile test is performed at a given tensile speed at a given temperature during solidification of the test specimen.
  3. (iii)    The area (7.5 mm × 15 mm) depicted in Fig. 3(a) is observed using a high-speed video camera during the test. True strain is obtained by measuring the displacement of two dendrite heads selected as markers.
  4. (iv)    A true stress–true strain curve is obtained by time synchronization of the strain data and load data.

In the same manner as described in earlier works,33,35) the true stress and true strain were calculated under the assumption that deformation of the test specimen conforms to the law of constant volume. Calculating the true stress requires the initial cross-sectional area. The specimens solidified without tensile testing were prepared in preliminary tests. Then, the measured the cross-sectional areas were 198 mm2 for the Al–Mg alloy and 186 mm2 for the Al–Cu alloy respectively. The values were used as the initial cross-sectional area.33,35)

Figure 3(b) presents a schematic of a true stress–true strain curve obtained using the testing device. The two semi-solid tensile properties, i.e. the maximum true stress σss and the elongation εelongation are determined from the curve as the following. The maximum true stress is the peak value of the curve. However, after the true stress reaches the peak, the stress decreases gradually with increasing true strain. The curves show no remarkable fracture strain. Therefore, the elongation was defined for this study as the true strain corresponding to the maximum true stress in this study.

Regarding the testing conditions, the pouring temperature and the metal mold temperature are, respectively, 720°C and 400°C. Consequently, the average cooling rate at the temperatures between the liquidus and the non-equilibrium eutectic solidus temperature35) are 0.7 K/s. The tensile speed is set as 0.02 mm/s. Tensile tests were conducted at two temperature ranges: (1) above the eutectic solidus (in semi-solid state) at each temperature (2) below the solidus at one temperature (in solid state). For the latter, both the maximum true stress and the corresponding true strain rate obtained at the temperature just below solidus will be used to ascertain creep parameter A0 in eq. (5). The determination presented above enables prediction of the flow stress in the solid state of the alloys, which passed the temperature history of as solidified rather than reheating. Then, eq. (5) is used for predicting the maximum true stress during solidification in section 4.1.

2.3 Microstructural characterization

Microstructural characterizations were performed for the two alloys by measuring the grain size, the secondary dendrite arm spacing (DAS II) and the dihedral angle of the liquid phase formed at the grain boundary. The sample for the characterizations was obtained from the center part of the specimen. Section A–A′ depicted in Fig. 3(a) was mirror-polished and etched with 2 mass% sodium hydroxide solutions; then the microstructure was characterized using the following steps.

2.3.1 Grain size

Earlier works12) have shown that the constitutive behavior of alloy in semi-solid state depends on the grain size. Therefore, the grain size was measured for the two alloys. Their differences are discussed in section 3.2. The grain size was measured through eq. (4). Here, 1500 µm was used as value L for the two alloys.   

\begin{equation} \skew3\bar{d}_{\text{g}} = \sqrt{\frac{4L^{2}}{N_{\text{g}}\cdot \pi}} \end{equation} (6)
Here, $\skew3\bar{d}_{\text{g}}$, L2 and Ng are respective the effective grain size, the representative area and the number of grains in the area.

2.3.2 Secondary dendrite arm spacing (DAS II)

DAS II is used for calculating the solid fraction for each temperature (fs = f(T)) considering diffusion of the major solute element in solid phase. As described in reports of earlier works,33,35,36) DAS II was measured by dividing the distance between the center of secondary dendrite arms by the number of arms. The total number of measured arms was greater than 40.

2.3.3 Dihedral angle of liquid phase formed at the grain boundary

The dihedral angles were measured to calculate the fraction of solid cohesion C against each solid fraction (C = f(fs,θ)), as described in subsection 2.3.5. A few methods exist for measuring the angle.3) For this study, we adopted a method37,38) that measures the dihedral angle of the eutectic product formed at the grain boundary after solidification ends. Figures 4(a) and (b) present schematics of the measuring method. In this method, the eutectic product shape is assumed to represent the shape of the liquid phase immediately above the temperature of non-equilibrium eutectic solidus during solidification. The measuring procedure included the following three steps. (1) As depicted in Fig. 4(a), the coordinates of five points (o, a, b, c and d) were measured along the interface between the primary phase and the eutectic product. Point “o” is the triple junction of a eutectic product and the adjacent two primary phases. (2) The two cycles were determined respectively as passing through points o–a–b and o–c–d. (3) As depicted in Fig. 4(a), the two tangents l1 and l2 of each cycle were determined with passage through the point “o”. Then, the angle was determined arithmetically as formed by the two tangents. Through the procedure described above, 100 dihedral angles were measured for each alloy. Then their cumulative distributions (the relation between cumulative rate and the dihedral angle) were derived. The angle corresponding to the medium value (the cumulative rate of 50%) of the curves was ascertained as the representative value θrep of the dihedral angle. Regarding the Al–5 mass%Mg alloy (the chemical composition is shown in Table 2) used in the tensile testing, the eutectic product was recognized in the specimen but the amount was too small to measure the angle. Therefore, in this study, the dihedral angle was measured though the specimen of Al–10 mass%Mg alloy (the chemical composition is shown in Table 2), which should have a larger amount of the eutectic product compared to that of the Al–5 mass%Mg alloy. Then, the value was used as the angle of Al–5 mass%Mg. An earlier report29) revealed that the dihedral angle of an aluminum alloy is independent of the amount of the major solute element.

Fig. 4

Measurement of dihedral angle of eutectic products: (a) schematic of eutectic phase,38) (b) schematic of measurement, (c) eutectic products of Al2Cu+α primary phase in Al–2 mass%Cu alloy, (d) eutectic products of Al8Mg5+α primary phase in Al–10 mass%Mg alloy and (e) cumulative curve of observed dihedral angles.

Table 2 Chemical composition (mass percent) of alloys.

2.3.4 Temperature dependence of solid fraction (fs = f(T))

Several theoretical models exist to ascertain the temperature dependence of the solid fraction. First, the Scheil–Gulliver model39,40) was used to ascertain the solid fraction of the Al–5 mass%Mg at various temperatures. Results show that the fraction of the eutectic product composed by both α solid solution and Al–Mg intermetallic compound was around 4%.32) However, we confirmed experimentally from microstructure observations that the fraction of the eutectic product was clearly lower than 4%. One reason for the difference is exclusion of the back diffusion of the major solute element from enriched liquid phase by microsegregation into solid phase in the Scheil–Gulliver model. Therefore, the Clyne–Kurz model41) was used to ascertain the solid fraction with consideration of the back diffusion, as in an earlier study.35)   

\begin{equation} f_{\text{s}}(T) = \left(\frac{1}{1 - 2\Omega k_{0}} \right) \left\{ 1 - \left(\frac{T_{\text{m}} - T}{T_{\text{m}} - T_{\text{l}}}\right)^{(1 - 2\Omega k_{0})/(k_{0} - 1)}\right\} \end{equation} (7)
  
\begin{equation} \Omega = \alpha \left[1 - \exp \left(-\frac{1}{\alpha}\right)\right] - \frac{1}{2}\exp \left(-\frac{1}{2\alpha}\right) \end{equation} (8)
  
\begin{equation} \alpha = \frac{4D_{\text{s}}t_{\text{f}}}{\lambda_{2}{}^{2}} \end{equation} (9)
Therein, Tm, T1, and k0 are respective the melting points of pure aluminum, the liquidus temperature, and the equilibrium partition coefficient. Also, Ds and tf respectively denote the diffusion coefficient of the solute element in pure aluminum, and the solidification time. Then, λ2 and α are the secondary dendrite arm spacing and the solidification parameter. For Al–2 mass%Cu alloy, the temperature dependence of solid fraction was determined using the Scheil–Gulliver model (Ω = 0 in eq. (7)(9)). Consequently, the fraction of eutectic product (composed by both α solid solution and θ phase of Al–Cu intermetallic) was around 3%.3) The predicted amount was almost equivalent to the measured value in the solidification microstructure of the specimen. This rough equivalence is expected to result from the lower value of the equilibrium partition coefficient k0 in the Al–Cu (k0 = 0.15) compared to the Al–Mg (k0 = 0.35) alloy, which caused higher segregation of the major solutal element in the liquid phase on the solid-liquid interface. Therefore, in this study, the solid fraction determined using the Scheil–Gulliver model was used for the Al–2 mass%Cu alloy.

Table 3 presents the parameters used to calculate the solid fraction. Both DAS-II and the solidification time tf were measured through experimentation. The temperatures of both melting Tmand the liquidus Tl and the equilibrium partition coefficients k0 for the two alloys were derived, respectively, from the equilibrium diagram of Al–Mg and Al–Cu alloy systems42,43) respectively.

Table 3 Material parameters of alloys to predict flow stress in solid state, solid fraction and solid cohesion.

Calculating the solid fraction using the Clyne–Kurz model41) requires diffusion coefficient Ds of the major solute element (magnesium in this study) in the matrix (aluminum in this study) at a representative temperature Trep. In this study, the temperature is defined as explained hereinafter. Regarding the diffusion of the major solute element segregated in the liquid phase into the solid phase, the diffusion is expected to be much higher at the final stage of solidification rather than at the initial stage because of the higher gradient of the concentration in the major solute element. Therefore, the representative temperature Trep was determined using eq. (10).   

\begin{equation} T_{\text{rep}} = \mathrm{f}(T_{\text{l}},T_{\text{e}}) = (T_{\text{l}} - T_{\text{e}})0.3 + T_{\text{e}} \end{equation} (10)
Here, Te is the non-equilibrium eutectic solidus temperature. The diffusion coefficient of magnesium in aluminum at the temperature was obtained from a report of an earlier study.44)

2.3.5 Solid fraction dependence of fraction of solid cohesion (C = f(fs))

Campbell’s model25) was used to calculate the fraction of solid cohesion at various solid fractions. The model is expressed as shown below.   

\begin{equation} C = 1 - f_{\text{LGB}} = 1 - 2.64\left(\frac{1 - f_{\text{s}}}{k}\right)^{0.5} \end{equation} (11)
  
\begin{align} k &= \sqrt{3} + \frac{3}{\tan(30 - \theta_{\text{rep}}/2)} \\ &\quad - \left(\frac{30 - \theta_{\text{rep}}/2}{60}\right)\frac{\pi}{\sin^{2}(30 - \theta_{\text{rep}}/2)} \end{align} (12)
Through the model, the geometry of grains in the solidification microstructure was assumed as the two-dimensional regular hexagon depicted in Figs. 2(a) and 2(b). Then, the geometry of the liquid film formed at grain boundary was determined theoretically using both the solid fraction and the dihedral angle. For the dihedral angle, the experimental values were used as described in subsection 2.3.3. Table 3 presents the parameters used to calculate the fraction of solid cohesion.

3. Experimental Results

3.1 Temperature dependences of the maximum true stress and elongation

Figures 5(a) and 5(b) show the maximum true stress and elongation at various temperatures. The ZST and the zero ductility temperature (ZDT), which can be determined from the graph, are both critical parameters3,45) for evaluating hot tearing susceptibility of alloy. Figure 5(a) shows that the ZSTs of Al–2 mass%Cu and Al–5 mass%Mg alloy are approximately 630°C and 610°C, respectively. By contrast, Fig. 5(b) presents that the ductility increases at the higher temperature (lower solid fraction) side. Therefore, ZST cannot be ascertained for the two alloys. The mechanism of the increase in the ductility is discussed in subsection 4.2.2. Temperatures in these graphs will be replaced with the fraction of solid cohesion for the discussion in Chapter 4.

Fig. 5

Temperature dependence of tensile properties for two alloys in partially solidified state: (a) maximum true stress (plot) with predicted stress in solid phase (dotted line) and (b) elongation.

3.2 Solidification microstructure

The grain sizes were, respectively, 147 ± 6 µm and 126 ± 8 µm for the Al–2 mass%Cu alloy and the Al–5 mass%Mg alloy. The difference between the two alloys are regarded to have no remarkable effect on the tensile properties during their partial solidification. The DAS II of Al–5 mass%Mg alloy was 44 ± 6 µm.

Regarding the dihedral angle, Figs. 4(c) and 4(d) show examples of the eutectic product observed through an optical microscope. In the figures, the dihedral angles were measured using the method described in subsection 2.3.3. Figure 4(e) shows the cumulative distribution curves derived from the measured angles. The representative dihedral angle θrep of the Al–Mg alloy (the median value in the cumulative distribution function) is larger than that of the Al–Cu alloy. These measured angles in this study are well consistent with those measured in an earlier work38) for commercial Al–Mg and Al–Cu alloys.

Next, the solid fraction against various temperatures (fs = f(T)) and the fraction of solid cohesion against the solid fraction (C = f(fs,θ)) were determined. Figure 6(a) shows the solid fraction calculated using eq. (7). Then, Fig. 6(b) presents the fraction of the solid cohesion calculated using eq. (11). Table 3 presents values of the parameters for their calculation. Figure 6(b) shows that the cohesion starts at a certain solid fraction (C = 0). Then the structure reaches to a perfect cohesion (C = 1) toward the end of solidification, which suggests that the solid fraction corresponding to the ZST is C = 0 and that the solid fraction is lower when the dihedral angle is larger. The temperatures corresponding to C = 0 are 632°C and 605°C, respectively, for Al–2 mass%Cu and Al–5 mass%Mg alloy. These values show good agreement with the ZST of the two alloys described in section 3.1. Both the solid fraction and the fraction of cohesion determined in this section are used in the next chapter to discuss the control of factors of the two tensile properties during partial solidification.

Fig. 6

Prediction of solid fraction and solid cohesion using theoretical models for two alloys: (a) temperature dependences of solid fraction and (b) solid fraction dependences of solid cohesion.

4. Discussion

4.1 Solid fraction dependences of tensile properties (the maximum true stress and the elongation)

Figures 7(a) and 7(b) show the two tensile properties (normalized maximum true stress and the elongation) with various solid fractions. Figure 7 shows that the tensile properties of the two alloys are mutually different. The result demonstrates that the solid fraction alone can never explain the difference of the two tensile properties during solidification at each solid fraction, which agrees with the earlier works presented in Fig. 1. The reason for normalizing the maximum true stress and the procedure are described below.

Fig. 7

Solid fraction dependence of tensile properties for two alloys in partially solidified state: (a) normalized stress and (b) elongation.

Regarding the temperature for the two alloys at the same solid fraction, Fig. 6(a) shows that the temperature of the Al–Cu alloy is higher than that of the Al–Mg alloy. Therefore, at the same solid fraction, the strength of the solid phase for the Al–Cu alloy is expected to be lower than that for the Al–Mg alloy. To exclude differences of strength in the solid phase, the maximum true stress in the semi-solid alloys was normalized using the following equation.   

\begin{equation} \sigma_{\text{normalized}} (T) \equiv \frac{\sigma_{\text{ss}} (T)}{\sigma_{\text{s}}(T)} \end{equation} (13)
Dotted lines in Fig. 5(a) show the flow stress of solid phase predicted using eq. (5). Table 3 presents the material parameters used for the prediction. Both the stress exponent nse and the apparent activation energy Q during creep deformation were derived from reports of earlier works.15,20) As to the creep parameter A0, an earlier work23) suggests that the value depends strongly on the temperature history before tensile testing, i.e. the value obtained at immediately after solidification (with a test piece not cooled to room temperature) is around one-ten thousandth as large as the value obtained after reheating (with a test piece cooled to room temperature after solidification and reheating). The result indicates that the high temperature strength of the former is higher than the latter, even at the same temperature and strain rate. Therefore, in this study, the as-solidified value A0as-solidified was determined using eq. (14). The mechanism of the temperature history dependence of the parameter A0 has not been clarified yet.23) However, the difference of the amount of porosity and the intensity of the major solute element are possible causes.   
\begin{equation} A_{0}{}^{\text{as-solidified}} = \frac{\dot{\varepsilon}_{\text{s}}{}^{\text{as-solidified}}(T)}{\sigma_{\text{s}}{}^{\text{as-solidified}}(T)^{n_{\text{se}}}}\exp \left(\frac{Q}{RT_{\text{ab}}}\right) \end{equation} (14)
In that equation, σsas-solidified(T) and $\dot{\varepsilon }_{\text{s}}{}^{\text{as-solidified}}(T)$ are the maximum true stress and the corresponding true strain rate obtained at the temperature immediately below solidus using the tensile testing device adopted for this study. The normalized stress shown in Fig. 7(a) was determined using eqs. (4), (13), and (14) above.

4.2 Fraction of solid cohesion dependence of tensile properties

Figure 8(a) and 8(b) show the tensile properties of various fractions of solid cohesion. The fraction of solid cohesion dependences (σss = f(C) and εelong = f(C)) for each alloy show more mutual consistency rather than the solid fraction dependence in Fig. 7ss = f(fs) and εelong = f(fs)). The result demonstrates for the first time that the cohesion parameter derived from the measured dihedral angle enables description of the development of tensile properties (the maximum true stress and the elongation) during the solidification of the two alloys with different alloy systems. Campbell’s model used for this study for calculating the solid cohesion was developed under the condition that the shape of the solid phases are hexagonal. Such a microstructure is expected to be well consistent with the fine equiaxed dendritic structure portrayed in Figs. 4(c) and 4(d). However, for a coarse dendritic microstructure, it is not known whether the tensile properties show similar results to those in Fig. 8, or not. That is left as a task to be investigated in the future studies. In the following sections, details if the maximum true stress and the elongation will be discussed.

Fig. 8

Solid cohesion dependence of tensile properties for two alloys in partially solidified state: (a) normalized stress and (b) elongation.

4.2.1 Maximum true stress

As presented in Fig. 8(a), irrespective of the alloy type, the stresses in the partially solidified state start to arise at the starting point of cohesion (C = 0); then they increase toward the complete cohesion (C = 0 and fs = 1). The solid line in Fig. 8(a) shows the value predicted by the Haaften’s model.20) As eq. (3) shows, in the model, the normalized stress is equivalent to the fraction of solid cohesion (Normalized σss = C). Therefore, Fig. 8(a) indicates that the Haaften’s model as overestimating the maximum tensile stress compared to the experimental value. The reason for the gap separating the predicted (Haaften model) value and the experimentally obtained value is expected to be the decrease in the fraction of solid cohesion47) by dynamical wetting29,46) during deformation. For this study, the dihedral angles used to obtain the fraction of solid cohesion in Fig. 6(b) were measured from the solidification microstructure under a static condition, not during deformation. Earlier works29,46) have demonstrated that the deformation promotes the wetting of liquid, i.e. decreases the fraction of solid cohesion. Therefore, for more accurate prediction of the maximum stress compared to the Haaften model, the decrease in the fraction of solid cohesion associated with deformation47) should be incorporated into account in the constitutive model.

4.2.2 Elongation

As presented in Fig. 8(b), irrespective of alloy type, the fraction of solid cohesion where the elongations show the minimum is in the range of 0.3–0.6. The result revealed the following two points (i) Hot tearing of the two alloys should occur in the range. (ii) The fraction of solid cohesion C should be a factor related to the brittle temperature range (BTR).

To discuss the mechanism by which the elongation shows a minimum, the fraction of solid cohesion in Fig. 8(b) was divided into three regimes based on earlier works.27,48,49) In stage I (C > 0.6), the semi-solid body has a capacity to sustain the load by applied deformation. Then the creep deformation in the solid phase increases the elongation. In stage III (C < 0.3), the liquid film exists continuously on the grain boundary. Therefore, the deformed zone should be healed by liquid phase so that the elongation increases. In stage II (0.6 ≥ C ≥ 0.3), the two phenomena above are restricted so that the elongation shows the minimum.

Summarizing the discussion above, it was demonstrated that the fraction of solid cohesion estimated using the measured dihedral angle enables a unified description the two tensile properties (maximum true stress and the elongation) in a partially solidified state of the two alloys with respectively differing alloy systems. The fitted curves of the normalized stress and the elongation against the solid cohesion in Fig. 8 make it possible to predict the tensile properties in the semi-solid state of other alloys with a known dihedral angle.

5. Conclusions

For predicting both semi-solid tensile properties (both the maximum true stress σss and the elongation εelong) irrespective of the alloy systems, tensile properties were obtained experimentally for the two alloys with mutually different alloy system. Then, their dominant factors were examined in terms of the morphology of the solidification microstructure. The results are summarized as explained below.

  1. (i)    For both an Al–Mg and Al–Cu alloys, the maximum true stress and elongation were obtained at each temperature during partial solidification. Then the relation between the properties and the solid fraction (σss = f(fs) and (εelong = f(fs)) were compared between the two alloys. Results show that the relation mutually differ, which demonstrates that the solid fraction alone is insufficient to explain the difference.
  2. (ii)    The dihedral angle θ of the liquid phase formed at grain boundary was measured experimentally; then the parameter of solid cohesion C was derived at each solid fraction using Campbell’s model (C = f(fs, θ)). Comparison of the relation between the tensile properties and the parameter Css = f(C) and (εelong = f(C)) between the two alloys demonstrates that the relations are mutually consistent. The results verified experimentally that the dihedral angle can explain the difference of the semi-solid tensile properties of the two alloys.
  3. (iii)    Based on the fitting curve for the relation between parameter C and the experimental tensile properties of the two alloys in semi-solid state, predicting the tensile properties at each temperature (including ZST and BTR) is expected to be possible for other alloys, for which the dihedral angle has been found.

Acknowledgements

Some aluminum alloys used for this study were supplied by both UACJ Corporation and DAIKI Aluminum Industry Co., Ltd. The authors express their sincere gratitude for their support.

REFERENCES
 
© 2019 The Japan Institute of Light Metals
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