2020 Volume 61 Issue 12 Pages 2371-2377
For predicting hot tearing during casting process, viscosity properties and constitutive equations in the semi-solid region are required. In this research, viscosity properties of two alloys, which are Al–5 mass%Mg alloy with Ti–B grain-refiner and Al–2 mass%Cu alloy with Ti–B grain-refiner, are estimated using the image-based modelling method as suggested by Matsushita et al. As to the mechanical properties of the solid phase in semi-solid region, it is found that by using the temperature dependent value rather than the value just below the solidus temperature, the prediction accuracy of the viscosity value in semi-solid region has increased. By comparing the obtained numerical properties with the experimental properties, it is found that the method can predict the viscosity properties of Al–5 mass%Mg alloy and Al–2 mass%Cu alloy in the semi-solid region at the solid fraction where hot tears are likely to occur.
This Paper was Originally Published in Japanese in J. JILM 70 (2020) 187–193.
In recent years, due to the demand for improving mechanical properties such as ductility and yield strength, it has become necessary to choose alloy composition that manifests high hot tearing susceptibility. Thermal stress analyses have been tried for predicting occurrence of hot tearing during casting with FEM (finite element method).1–4)
In the temperature that hot tearing occurs, the alloy shows a viscous body, where solid and liquid coexisting and the stress depend on the strain rate. Therefore, viscoelastic constitutive equation is required to predict hot tearing. Viscosity properties in semi-solid region have been obtained experimentally with several strain rate-controlled tensile tests.5–10)
However, there are two problems in this experimental method. Firstly, many alloys are known to have a brittle temperature range (BTR).11–13) Because these alloys show both viscous and brittle behavior in this range, they fractured with small strain before reaching steady-state stress. As a result, it is difficult to obtain viscosity properties of those kind of alloys. Secondly, as the solidified microstructure changes accordingly with the alloy composition and cooling rate, tensile tests in these conditions are required to obtain the viscosity properties.14,15)
To overcome these problems, several studies have been conducted to estimate stress-strain curves during partial solidification using numerical methods.16–23) Among them, Phillion et al.16,17) experimentally validated the elastic region of the stress-strain curve, while Sheykhjaberi et al.23) compared the experimentally obtained yield stress with the numerical value.
However, except for Matsushita et al.,24) there are no studies that obtain viscosity properties by stress analysis and compare the numerical properties with the experimental properties. Matsushita et al.24) obtained numerical viscosity properties of Al–5 mass%Mg alloy using the image-based FEM.
In this suggested method, solid-liquid models were obtained by binarizing the micrograph of alloy that was water quenched in semi-solid region. The viscosity properties in semi-solid region were obtained by stress analyses of tensile tests for the models using the mechanical property of the solid and liquid phases. Since models that were used in the analyses are from the real microstructures, the microstructural models can reflect each casting condition such as alloy composition and cooling rate.
Furthermore, Matsushita et al.24) experimentally validated the viscosity properties. However, it is not clarified whether this method can be applied to the other alloys. It should be considered inappropriate to use the value of temperature just under the solidus temperature as the mechanical property of the solid phase in the semi-solid region. This is because it is considered that the mechanical properties of the solid phase changes as the solid fraction change and the temperature also changes in the semi-solid region.
Therefore, in this study, the viscosity properties of Al–5 mass%Mg alloy and Al–2 mass%Cu alloy with Ti–B grain-refiner (Table 1) are estimated using the image-based modelling method as suggested by Matsushita et al.24) Numerical values are then compared with the experimental values.25)
The following two points are examined in order to verify this method.
The outline of obtaining a stress-strain curve by tensile stress analysis is described below.
| \begin{equation} \sigma_{\text{x}}^{\text{element}} = f_{\text{s}} \sigma_{\text{x}}^{\text{sol}} + f_{\text{l}} \sigma_{\text{x}}^{\text{liq}} \end{equation} | (1) |
| \begin{align} \varepsilon_{\text{x}}^{\text{element}} &= f_{\text{s}} (\varepsilon_{\text{elastic,x}}^{\text{sol}} + \varepsilon_{\text{plastic,x}}^{\text{sol}}) \\ &\quad + f_{\text{l}}(\varepsilon_{\text{elastic,x}}^{\text{liq}} + \varepsilon_{\text{plastic,x}}^{\text{liq}}) \end{align} | (2) |
| \begin{equation} \sigma_{\text{x}}^{\text{ave.}} = \sum\sigma_{\text{x}}^{\text{element}}\big/\sum(f_{\text{s}} + f_{\text{l}}) \end{equation} | (3) |
| \begin{equation} \varepsilon_{\text{x}}^{\text{ave.}} = \sum \varepsilon_{\text{x}}^{\text{element}}\big/\sum(f_{\text{s}} + f_{\text{l}}) \end{equation} | (4) |
Schematic of the boundary conditions (Matsushita, 2019).
Material properties of solid and liquid phase were defined separately. Matsushita et al. used value of temperature just under the solidus temperature as the mechanical properties of the solid phase. Among them, the viscosity properties were obtained by the creep test,26) and the Young’s modulus and Poisson’s ratio were calculated by JMatPro6.0, an alloy physical properties calculation software.
However, it is considered that the mechanical properties of the solid phase change as the solid fraction and the temperature also change in the semi-solid region. Therefore, in this study, as the mechanical properties of the solid phase, the value just under the solidus temperature was compared with temperature-dependent value.
As temperature-dependent value, mechanical properties of the solid phase in semi-solid region is difficult to obtain experimentally. Therefore, they were extrapolated from temperature-dependent viscosity properties of pure aluminum,27) and viscosity properties just under the solidus of Al–5Mg alloy and Al–2Cu alloy. Young’s modulus and Poisson’s ratio of the solid phase in semi-solid region were extrapolated from values under the solidus of each alloy calculated with JMatPro6.0 (Table 2).
The liquid phase was defined as the elastic perfectly plastic properties with yield strength of 10E-3 MPa so that liquid phase bears almost zero strength, as described by Matsushita et al.24) The Poisson’s ratio was defined as 0.45 to achieve convergence although 0.5 is ideal for newton fluid, as stated by Matsushita et al.24) and Phillion et al.16,17)
2.3 Method to obtain viscosity propertiesThe outline of obtaining viscosity properties from stress-strain curve is described as follows.24) For alloy during partial solidification, strain hardening can be ignored and plastic strain can be omitted from total strain (eq. (5)). Also, eq. (8) can be derived by integrating the Hooke’s law of elastic constitutive eq. (6) and the Norton’s law of viscous constitutive eq. (7).
| \begin{equation} \varepsilon_{\text{total}} = \varepsilon_{\text{elastic}} + \varepsilon_{\text{creep}} \end{equation} | (5) |
| \begin{equation} \varepsilon_{\text{elastic}} = \sigma/E \end{equation} | (6) |
| \begin{equation} \dot{\varepsilon}_{\text{creep}} = k \sigma^{n_{\text{eff}}} \end{equation} | (7) |
| \begin{equation} \log(\dot{\varepsilon}_{\text{total}} - \dot{\sigma}/E) = n_{\text{eff}}\log\sigma + \log k \end{equation} | (8) |
Then, neff was calculated from the gradient of the linear approximation, and log k was calculated from the intercept. (Hereinafter, we will refer to log k as k.)
The tensile speed of Al–5Mg alloy was 0.001 mm/s, referring to Matsushita et al.24) Analysis of Al–2Cu alloy was conducted in two conditions. First, the tensile speed was set to 0.001 mm/s so that it will has the same strain rate as Al–5Mg alloy. Second, it was set to 0.300 mm/s, to have similar logarithm of stress value with the former. Table 3 shows the analysis condition for both alloys.
Figure 2 and Fig. 3 show an example of the binarized microstructure model and the stress distribution of tensile direction in the Al–5Mg and Al–2Cu alloys, respectively. In Fig. 2(b) and 3(b), black region indicates the solid phase and white region indicates the liquid phase. Meanwhile in Fig. 2(c) and 3(c), the arrow indicates tensile direction. Furthermore, only the stress distribution of solid phase is shown because the liquid phase does not substantially bear the load. Relatively, high stress is generated within the solid-solid connecting area in both alloys as shown in Fig. 2(c) and 3(c).
Example of numerical results (Al–5Mg fs = 0.91, 0.001 mm/s): (a) microstructure; (b) binarized moedl; (c) x-axis stress.
Example of numerical results (Al–2Cu fs = 0.90, 0.001 mm/s): (a) microstructure; (b) binarized moedl; (c) x-axis stress.
Figure 4 shows an example of the stress-strain curve obtained from the analysis. At the same strain rate (0.001 mm/s) and solid fraction, the stress of Al–2Cu alloy is lower than that of Al–5Mg alloy in the strain range of 0 to 0.003. This will be caused by higher temperature at solidus for Al–2Cu alloy, since the parameters are given as a function of temperature.
Example of stress-strain curves obtained from the analyses Al–5Mg and Al–2Cu.
In the stress-strain curve, when assigning excessive tensile rate on the analysis, the Lagrange model that is surrounding the model will break. Thus, the strain rate in the analysis can not be defined to that of the experiment (0.01 mm/s)25) and the stress-strain curve of the experiment and analysis are not compared here.
From several stress-strain curves obtained by varying the solid fraction, viscosity properties i.e., effective power law coefficient neff and material constant k, are calculated using eq. (8).
3.2 Viscosity propertiesFigure 5 to Fig. 7 show effective power law coefficient neff and material constant k of Al–5Mg alloy and Al–2Cu alloy. Figure 5 compares the temperature-dependent and temperature-constant conditions of Al–5Mg alloy, while Fig. 6 demonstrates that of Al–2Cu alloy (both are at tensile speed of 0.001 mm/s). Figure 7 shows the comparison of experimental and analysis result, which were obtained by using temperature-dependent mechanical properties of the Al–2Cu alloys under different tensile speed conditions. In both alloys, the neff values decrease while the k values increase with respect to the decrease of solid fraction, which are consistent with the experimental values. Details will be described in the next chapter.
Comparison of the Al–5Mg viscosity properties obtained from analyses with temperature dependent, temperature constant, and experiments. (a) Effective power law coef. (b) Material constant, k.
Comparison of the Al–2Cu viscosity properties obtained from analyses 0.001 mm/s with temperature dependent, temperature constant, and experiments. (a) Effective power law coef. (b) Material constant, k.
Comparison of the Al–2Cu viscosity properties obtained from analyses 0.001 mm/s and 0.300 mm/s. (a) Effective power law coef. (b) Material constant, k.
In Fig. 5(a) and Fig. 6(a), analysis result of both temperature-dependent and temperature-constant conditions of Al–5Mg alloy and Al–2Cu alloy show good agreement with the experimental value.
This will be resulted from that the range of neff value of solid phase used in the analysis was 2.4 to 2.8 for the Al–5Mg alloy and 4.6 to 4.8 for the Al–2Cu alloy, in other words neff of solid phase has small temperature dependency.
4.1.2 Material constant kIn Fig. 5(b), the calculated k value of Al–5Mg alloy using temperature-dependent properties shows good agreement with the experimental value, whereas that of temperature-constant properties is lower than experimental value. Also, in Fig. 6(b), calculated k value of Al–2Cu alloy using temperature-dependent and temperature-constant properties, are lower than experimental value. However, the analysis values of temperature-dependent are relatively close to the experimental value.
The definition of the material constant k is based on the creep strain rate at 1 MPa stress as shown in the eq. (7), thus a lower k value indicates that the alloy exhibit, less viscous deformability. On this account, as the temperature of solid phase increases, the easiness of viscous deformation is poorly reflected when using the mechanical properties just under the solidus temperature. This is manifested by the lower analysis value as compared to the experimental value. Furthermore, among the mechanical properties such as Poisson’s ratio, Young’s modulus, neff, and k, it is found that the k value significantly influenced the analysis result.
Therefore, it is found that using a temperature-dependent value for the material constant k of the solid phase in semi-solid region have to be more accurate than using the value just under the solidus temperature, as proposed by Matsushita et al.25)
On the other hand, it is considered that simply using a temperature-dependent value is not sufficient for the k value of in the case Al–2Cu alloy. A detailed discussion on improving the analysis for Al–2Cu alloys is described in the next section.
4.2 Influence of tensile speed on the viscosity propertiesIn Fig. 7(a), the calculated neff value shows good agreement with the experimental value for both 0.001 mm/s and 0.300 mm/s tensile speed conditions, but in Fig. 7(b), the k value is different between 0.001 mm/s and 0.300 mm/s tensile speed. In other words, the k value shows good agreement with the experimental value for 0.300 mm/s tensile speed, but that of 0.001 mm/s tensile speed has the difference of about 0.80 from the experimental value when solid fraction range is less than 0.95.
In order to explain this deviation, Fig. 8 shows the relationship of eq. (8), in which the left side of that is plotted on y-axis, while log x from the right side is plotted on x-axis.
Comparison of $\log (\dot{\varepsilon }_{\textit{total}} - \dot{\sigma }/E)$ vs. log σ obtained from the analyses 0.001 mm/s and 0.300 mm/s of Al–2Cu.
Since the viscosity properties in eq. (8) is not a function of strain rate, even if the tensile speed is varied and the strain rate changes, the viscosity properties value do not change in the formula.
However, in Fig. 8, the data range of the Al–2Cu alloy (tensile speed 0.001 mm/s), which stress is smaller than that of Al–5Mg, exists at a position away from the y-intercept because the x-axis is the logarithm of stress value. The k value is calculated from the y-intercept of the linear approximation, and the interpolation distance by extrapolation of the linear approximation becomes longer as the data range away from the intercept, thus may resulted in the increase of extrapolation error. On the other hand, assigning excessive tensile speed will result in the breaking of Lagrange frame, and eventually diverging the analysis.
Therefore, in order to obtain the k value that is close to the experimental value, it is desirable to apply high tensile speed as long as the Lagrange frame is not destroyed.
4.3 Comparison of analysis and experimental result of maximum stressWhen predicting hot tearing by thermal stress analysis, in order to analyze the stress distribution inside the ingot of semi-continuous casting, the max. stress generated during viscous deformation in the semi-solid region must match the experimental value.
Therefore, the max. stress computed using the calculated neff and k, was compared with experimental value obtained by Takai et al.28) The max. stress is expressed by eq. (9) by rearranging eq. (7).
| \begin{equation} \sigma_{\text{max}} = (\dot{\varepsilon}_{\text{max}}/k)^{1/n_{\text{eff}}} \end{equation} | (9) |
Comparison of the maximum stress obtained from analyses and experiments. (a) Al–5Mg, (b) Al–2Cu.
As a result, when the solid fraction is more than 0.93, the max. stress coincides with the experimental value, but when the solid fraction is lower than that, especially in the Al–2Cu alloy, the analysis value tends to have higher max. stress than the experimental value.
As presented by Takai et al., elongation value was at minimum in the solid fraction of 0.93 to 0.95 for the Al–5Mg alloy, and 0.95 to 0.97 for the Al–2Cu alloy, where hot tearing is likely to occur in these solid fraction range (BTR). As stated in the introduction, it is generally difficult to experimentally obtain the viscosity properties in BTR.
Therefore, it is imposed that the proposed method in this study can be used to predict the viscosity properties of Al–5Mg alloy and Al–2Cu alloy in semi-solid region.
In this study, viscosity properties of Al–5Mg and Al–2Cu alloy with Ti–B grain-refiner are predicted using finite element analysis based on quenched solidified structure, and experimentally validated.
The results are described as below.
