The hot-cracking susceptibility of Cu-Ni-Si alloys was quantitatively evaluated through an I-beam test for the first time. The test results of the two cooling conditions changed as functions of the mold temperature, and the minimum restraint distance necessary for hot cracking increased as the cooling rate decreased. The pressure drop in the liquid phase at the dendrite gap was calculated from the test results using the Rappaz–Drezet–Gremaud (RDG) model. By comparing the calculated pressure drop to the cavitation depression, the appropriate coalescence solid fraction to reproduce the experimental results was estimated to be 0.918–0.919 and 0.906–0.921 under rapid (21.8–21.9°C/s) and slow (4.1–6.9°C/s) cooling conditions, respectively. No significant difference was observed in the coalescence solid fraction, at which the dendrite branches merges, between the two cooling conditions used in this study.

Appropriate coalescence solid fraction of Cu-Ni-Si alloy estimated by the Rappaz–Drezet–Gremaud model at a (a) rapid cooling rate and (b) slow cooling rate.

In recent years, the demand for copper alloys with high electrical conductivity has surged owing to the miniaturization of electronic components with high electrical current densities. In addition, because high strength is required to support miniaturization, precipitation-type copper alloys with a good balance between electrical conductivity and strength have attracted attention. Among them, Cu-Ni-Si alloys with Ni₂Si precipitation phase are the most important copper alloys because of their high conductivity–strength balance [1–5].

However, one of the issues encountered in the production of metals, including copper alloys, is the cracking of ingots and billets during casting [6, 7]. In particular, the cracking in the semisolid region, where liquid and solid phases coexist, is called hot cracking. Suppressing hot cracking to maintain quality during casting is crucial. Thus, many studies on hot cracking of metals, in particular, experimental evaluations of the behavior of hot cracking, have been conducted [8–11]. Such evaluation methods include semisolid tensile testing, in which a part of the specimen is melted and subjected to tensile loading, and I-beam and ring mold testing, in which a molten metal is poured into I-beam molds or ring molds, and strain is induced by the surrounding solidification shrinkage.

However, the hot cracking of copper alloys has been rarely explored. Oya et al. [12–16] conducted I-beam and ring mold tests on conventional copper alloys, such as brass and Cu-Al alloys. In these studies, the effects of additive elements and dendrite shape on hot cracking were discussed; however, the strain in the semisolid region was not obtained. Yoshida et al. [17] also conducted I-beam tests to evaluate the castability of lead-free bronze and investigated the effects of the added amounts of lead and bismuth; however, the strain in the semisolid region was not obtained, similar to Oya’s group studies. In addition, all these studies were conducted on solid solution-type copper alloys, and to our knowledge there are no reports on the evaluation of hot cracking of precipitation-type copper alloys, such as Cu-Ni-Si alloys, under conditions that simulate casting. Therefore, it is extremely important to investigate the effects of hot cracking in Cu-Ni-Si alloys, which is currently attracting considerable attention. In addition, because the purpose of this study was to evaluate hot-cracking behavior during casting, the experimental conditions must be close to the cooling conditions of the casting process, and the strain must be obtained in the semisolid region.

The index of susceptibility to hot cracking is called hot cracking susceptibility. In addition to the experimental evaluation described in the previous section, many models have been proposed to predict hot-cracking susceptibility [6, 7, 18–27]. In previous studies, hot cracking was often discussed based on the assumption that it occurs at specific solid fractions. For example, Clyne et al. [22] assumed that hot cracking occurs when the solid fraction is in the range of 0.9–0.99 in the late solidification stage. The authors proposed that the hot-cracking susceptibility is higher when t_V/t_R is large, where t_R denotes the transit time between solid fractions in the range of 0.4–0.9 in which stage healing of the liquid phase occurs, and t_V denotes the transit time between solid fractions in the range of 0.9–0.99 in which stage hot cracking occurs. Rappaz et al. [23] assumed that cavitation is generated in the liquid phase at the dendrite gap for unidirectionally solidified materials owing to the pressure drop, and that these cavitations are the initiations of hot cracking. The authors proposed a model to predict the hot-cracking susceptibility using an externally applied strain rate based on these assumptions. Morishita et al. [25] assumed that hot cracking does not occur below the solid fraction, where sufficient healing of the liquid phase occurs, and that hot cracking occurs between the solid fractions in the region (region II), where healing is insufficient and the fracture strain is small. The authors proposed that the hot-cracking susceptibility is higher when ΔT₂ and ΔR₂/ΔT₂ are large, where ΔT₂ is defined by the temperature range in region II, and ΔR₂ is defined by the difference in a temperature drop per unit solid fraction in the region II. Easton et al. [26] assumed that for rare-earth-doped Mg alloys, hot cracking occurs between the coherent and coalescence solid fractions. The coherent solid fraction refers to the minimum solid fraction at which the dendrites grow and liquid healing cannot occur geometrically, whereas the coalescence solid fraction refers to the minimum solid fraction at which the dendrites bond to each other and have sufficient strength. The authors showed that the effects on the hot-cracking susceptibility were small for the coherent solid fraction and extremely large for the coalescence solid fraction. On the other hand, Kou et al. [27] assumed that hot cracking occurs at the end of solidification and formulated the hot cracking susceptibility at the grain boundaries between columnar crystals using solid fraction and temperature.

Among the aforementioned models, those of Rappaz et al. [23] and Kou et al. [27] have been widely accepted. Using the model of Kou et al., which is similar to the models of Morishita et al. [25] and Easton et al. [26], hot-cracking susceptibility can be easily evaluated using data on the temperature dependence of the solid fraction. However, this index is not a quantitative value and is unsuitable for determining whether hot cracking will occur. However, the model proposed by Rappaz et al. [23] is called the Rappaz–Drezet–Gremaud (RDG) model, which requires many variables and physical properties, such as the microstructure, strain rate data, and critical pressure of the liquid phase at the dendrite gap for the generation of cavitation. Although the model is not suitable for a simple analysis, it is capable of quantitative evaluation to determine whether hot cracking will occur. In fact, in a study by Suyitno et al. [28], the RDG model was evaluated to reproduce each experimental result. In this study, we decided to use the RDG model, which allows for a quantitative consideration of the experimental results.

The details of the RDG model are described below [23]. The RDG model allows the calculation of the pressure drop in the hot-cracking prediction model in the solid fraction range from zero to one, that is, from the solidus to the liquidus temperature. However, based on previous studies, it was assumed that hot cracking occurred in the high-solid-fraction region, and the pressure drop in other solid-fraction regions is negligible. In this study, it was assumed that hot cracking occurred between the coherent and coalescence solid fractions.

In the RDG model, p_m is the pressure of the liquid phase in the region where there is a sufficient liquid phase at the dendrite tip. The pressure of the liquid phase at the dendrite gap decreases as it approaches the dendrite root. The amount of pressure drop that decreases up to the dendrite root is defined as Δp_max, which can be calculated from the solidification shrinkage term and the external strain term caused by external restraints using the following equation. Because the hot-cracking region was assumed to be between the coherent and coalescence solid fractions, the range of integration for each term was also defined by the corresponding temperature, from the coalescence temperature = T_c to the coherent temperature = T_l.

where λ₂ is the secondary dendrite arm spacing (S-DAS), G is the temperature gradient, β is the solidification shrinkage factor calculated by ρ_s/ρ_l − 1, μ is the viscosity, Δp_max is the pressure drop of the liquid phase at the dendrite gap, v is the solidification rate, f_s(T) is the solid fraction at each temperature, and $\dot{\varepsilon }_{p}(T)$ is the strain rate applied to the liquid phase at the dendrite gap perpendicular to the dendrite growth direction.

Assuming that the strain rate is constant within the temperature range of the equation, it can be expressed as follows,

The condition under which hot cracking occurs is expressed according to eq. (4) using Δp_c, the difference between p_m and p_c, indicating the pressure at which cavitation occurs.

However, the values of the coherent and coalescence solid fractions required for the above calculations differ from those in previous studies.

Morishita et al. [25] determined the solid fraction range within which hot cracking occurs (0.75–0.95). They commented that the lower limit is the solid phase fraction at which healing of the liquid phase disappears (coherent solid fraction), and then stated that it is equal to the solid fraction corresponding to ZDT. Based on previous studies by Nagaumi et al. [29, 30] and Mizukami et al. [31], the authors suggested that the solid fraction of ZDT is in the range of 0.70–0.80, and used the middle value of 0.75. They also stated that the upper limit was the point at which the fracture strain increased significantly; based on the work of Mizukami et al. [11], the upper limit of the solid fraction was determined to be 0.95. Using dendrite photographs acquired by Kruz et al., Rappaz et al. [23, 32] stated that hot cracking occurs in structures where a liquid-phase film remains in the dendrite interstices, as observed at a solid fraction of 0.82, and coalescence occurs in the structure observed at a solidus ratio of 0.94. Grasso et al. [33] also noted from dendrite observations of SCN-acetone alloys that the value of the solid fraction differed depending on whether coalescence occurred intragranularly or intergranularly, with 0.95 for the former and 0.99 for the latter. Easton et al. [26] stated that it is reasonable to use a value of 0.70 for the coherent solid fraction and 0.98 for the coalescence solid fraction. Kou et al. [27] stated that it is reasonable to use a solid fraction in the range of 0.87–0.94 for hot cracking to occur.

As shown by Easton et al. [26], the influence of the coherent solid fraction was not significant; therefore, a value in the range of 0.7–0.8 is acceptable for predicting the hot-cracking susceptibility using the RDG model. However, the exact value of the coalescence solid fraction has not yet been determined, although it has a significant influence on the calculation results, which is an issue intended to be addressed in this study.

Based on previous studies, this study conducted an I-beam test to evaluate the hot-cracking susceptibility of Cu-Ni-Si alloys, which (to our knowledge) has not been performed in the past. In addition, the RDG model was used to propose a method for predicting the hot-cracking susceptibility of Cu-Ni-Si alloys. Because the RDG model requires strain rates in the semisolid region, in-situ observations were conducted using a high-speed camera during the tests. Another problem with the RDG model is that the coalescence solid fraction is not uniquely determined. Therefore, we calculated the pressure drop Δp_max for several coalescence solid fractions and compared it to the cavitation depression Δp_c, to estimate the appropriate coalescence solid fraction that can reproduce the I-beam test results. Furthermore, because the coalescence solid fraction is expected to be affected by the shape of the dendrites, it is also expected to change with cooling conditions. Therefore, I-beam tests were conducted under two cooling conditions, and the dependence of the coalescence solid fraction on the cooling conditions was also discussed. The purpose of this study is to experimentally evaluate the hot-cracking susceptibility of Cu-Ni-Si alloys during casting and propose a prediction method using the RDG model.

To evaluate the hot-cracking susceptibility, an I-beam test was performed. Schematics are shown in Fig. 1. The shape of the specimen was an I-beam according to Oya et al. [10]. An insulation sheet with a width of 10 mm and a thickness of 1 mm (BSSR paper, Isolite Insulating Products) was applied to the surface of the mold at the longitudinal center of the parallel section to serve as the final solidification region. This was performed to load the tensile strain at the center of the length via solidification shrinkage from the surroundings. Mold-1 shown in the figures was sloped to widen the opening to facilitate pouring. The mold temperature was varied for testing under two different cooling conditions. Under the fast-cooling rate condition, the mold temperature at the start of pouring was equal to the room temperature (RT). Conversely, under the slow cooling rate condition, the mold temperature at the onset of pouring was equal to 700°C; it was set by inserting a heater (mold-2: GLC2202, surface plate: GLC4901, Hakko) into the mold and surface plate. The mold temperature was controlled by inserting a K-thermocouple (Kawaso Electric Industrial Co., Ltd.) and a controller (DGC2330, Hakko). SUS430 was used for the surface plate and molds-2 with heaters at 700°C, and FCD450 for the other molds (mold-1, mold-2 at RT). The molds with heaters inserted were covered with insulation wool during the test to facilitate the temperature increase. Additionally, insulation (No. 5625-A, TOMBO) was placed on the left, right, and bottom parts of the surface plate to suppress heat dissipation. At the center of the longitudinal section, a R thermocouple (Kawaso Electric Industrial Co., Ltd.) was inserted approximately 2 mm from the insulation sheet to measure the molten metal temperature. A test was conducted to measure the temperature gradients under each cooling condition. The R-thermocouple was set to approximately 4 mm from the insulation sheet, and the temperature gradient was determined by measuring the temperature difference between the two thermocouples and dividing it by the distance difference of 2 mm. The signal from the R thermocouple was acquired every 0.1 s by a data logger (NR-500, Keyence). In addition, a high-speed camera (FASTCAM Nova R2, Photron) with a single focal length lens (EF180 mm F3.5L, Canon) was attached to observe the sample’s surface and measure the strain at the central part of the specimen along the longitudinal direction. The resolution of the high-speed camera was 1280 × 960, the distance per pixel was approximately 15 µm, and the shooting speed was set to 50 frames per second.

Fig. 1

Pattern diagram of I-beam test conducted at 700°C. K thermocouple, heater and some of heat resistant materials were not used for RT conditions. (online color)

To evaluate the hot-cracking susceptibility, it is necessary to determine the critical conditions that serve as the boundary for hot cracking to occur. In the I-beam test, it was possible to vary the amount of shrinkage from the surroundings with the different strains applied by changing the value of the longitudinal length of mold-2 (L). Tests were conducted with L = 20, and 30 mm at a mold temperature set to the RT and L = 30, and 45 mm at 700°C. Herein, 20 mmL denotes the condition of L = 20 mm.

The composition was a typical Cu-Ni-Si alloy, Cu-2.3Ni-0.55Si (mass%). Electrolytic Cu, electrolytic Ni, and pure Si were melted in a carbon crucible in an atmospheric melting furnace (FIH-503VM; Fuji Electronic Industrial Co., Ltd.). The molten metal (at a temperature of 1300 ± 50°C) was poured into the mold. To ensure workability, a heat-resistant brick was placed in mold-1 during the pouring process. Strain analysis was performed using image analysis software (Dipp Motion V, DITECT) based on the video data captured by the high-speed camera. In the acquired video, the longitudinal displacement was measured using the difference in brightness at specific points within the area where the insulation sheet was applied. Displacements were acquired every 0.02 s at a mold temperature set to RT, and every 0.06 s at 700°C. A combination of two points with a distance of 3.5 ± 1 mm was selected from each point, and the time variation in the distance in the longitudinal direction between the two points was obtained. The time variation in the longitudinal strain was calculated by dividing the variation in distance from the onset time of the strain calculation by the initial distance.

The tested sample surfaces were observed under an optical microscope (VHX-8000, Keyence) to determine the occurrence of hot cracking. The fracture surfaces of the hot-cracked specimens were analyzed by optical microscopy and scanning electron microscopy (SEM) (JCM-7000, JEOL). Under each condition, the specimen was cut vertically near the center of its length, and the macro- and microstructures of the sections were observed under an optical microscope.

The thermodynamic calculation software Thermo-Calc (version 2022b, TCCU5), which is Cu-based Alloy Database, and MOBCU5, which is Cu-based Alloy Mobility Database, were used to calculate the relationship between the solid fraction and temperature using the Scheil–Gulliver model.

The relationship between temperature and solid fraction was calculated using the Thermo-Calc for Cu-2.3Ni-0.55Si (Fig. 2). The values in the high-solid fraction region are expressed as solid fraction (Fs) values of 0.75 at 1075°C and 0.98 at 1002°C.

Fig. 2

Calculated relationship between solid fraction and temperature using Thermo-Calc.

The cooling curves for each test condition are shown in Fig. 3. When the mold temperature increased from RT to 700°C, the cooling rate decreased. However, when the mold temperatures were equal, the cooling curves were similar regardless of the restraint distance. Figure 4 shows the temperature gradient of the cooling curve. The value of the temperature gradient increased when the slope of the cooling curve increased. The average values of the cooling rate and temperature gradient were obtained in the range of Fs of 0.75–0.98, which is the high-solid fraction region. The cooling rate was calculated by dividing the temperature difference by the time difference for Fs values in the range of 0.75–0.98. The temperature gradient was calculated using the averages of six values at Fs = 0.75, 0.80, 0.85, 0.90, 0.95, and 0.98 because there were large differences in the values at each temperature setting. When the dendrites grew stably, the solidification rate was obtained by dividing the cooling rate by the temperature gradient. The temperature measurement points were approximately 2 mm away from the insulation sheet attached to the mold, and the dendrites were assumed to have grown stably; therefore, the aforementioned method was applied. The average cooling rate, temperature gradient, and solidification rate are summarized in Table 1. It was shown that the cooling rate and temperature gradient were approximately four times and three times higher, respectively, when the mold temperature was set to the RT compared with the findings at 700°C.

Fig. 3

Cooling curves at all the test conditions. (online color)

Fig. 4

Cooling curves and temperature gradient at (a) RT-30 mmL and (b) 700°C-30 mmL. (online color)

Table 1 Cooling rate, temperature gradient, and solidification rate values at all tested conditions.

The surface of the sample was directly observed during the solidification process using a high-speed camera. Figure 5 shows an image at Fs = 0.75 for a test with the mold temperature set to RT at a restraint distance of 30 mm. In this image, the vertical direction is the longitudinal direction. Solidification proceeded from the surface of the heat-resistant sheet in the horizontal direction of the image. In contrast, the tensile strain was applied in the vertical direction owing to the surrounding solidification shrinkage.

Fig. 5

Sample appearance of RT-30 mmL during solidification. Markers shown as P-1 and P-2 in this figure represent the example of measurement position of the strain. (online color)

As previously mentioned, in hot cracking, the solid fraction region that should be focused on the applied strain is the high-solid fraction region. Therefore, the strain was calculated by extracting a video of the time corresponding to Fs = 0.75–0.98. Figure 6 shows the time variation in the strain calculated between the two points shown in the example. The time at Fs = 0.75 was set to 0 s and the positive value of strain was in the tensile direction. It was confirmed that the strain was loaded in the tensile direction as the solidification progressed.

Fig. 6

Strain-rate curve of RT-30 mmL.

The strain curve at Fs = 0.75–0.98 was linearly approximated, and the strain rate was calculated from the slope of the curve. The strain rate loaded under each test condition was calculated by averaging three or more strain rates. The RT strain rates of 20 and 30 mmL were 1.06 × 10⁻³ and 2.46 × 10⁻³ s⁻¹, and those of 700°C of 30 and 45 mmL were 0.12 × 10⁻³ and 0.61 × 10⁻³ s⁻¹, respectively. The strain rate increased as a function of the restraining distance, indicating that the expected test could be performed. In addition, as the mold temperature increased, the applied strain rate decreased because the solidification time increased.

To determine whether hot cracks occurred, the samples were observed from the top under each test condition. Figure 7 shows a magnified view of the central part of the specimen along the longitudinal direction. At a mold temperature equal to RT, no hot cracking occurred at a restraint distance of 20 mm, but cracking occurred at 30 mm. When the mold temperature was raised to 700°C, no hot cracking occurred at a restraint distance = 30 mm, but cracking occurred at 45 mm. For each mold temperature condition, hot cracking occurred when the restraint distance increased. Based on these findings, it was found that the shortest restraint distance at which hot cracking occurred was 30 mm and 45 mm at the mold temperatures equal to RT and 700°C, respectively.

Fig. 7

Appearances at the center of specimens after solidification at all tested conditions: (a) RT-20 mmL, (b) RT-30 mmL, (c) 700°C-30 mmL, and (d) 700°C-45 mmL. (online color)

To estimate the temperature range in which hot cracking occurred, the fractured surface was observed. The entire fracture surface was observed using an optical microscope, and an area approximately 2 mm from the insulation sheet was observed using SEM. The results are presented in Fig. 8. A dendrite structure was observed on the fracture surface, regardless of the mold temperature. In addition, torn structures, such as voids, were not observed on the dendrite surface. These results indicate that hot cracking occurred in the temperature range where the liquid phase remained in the dendrite gaps. It was confirmed that the cracking observed in this test was hot cracking caused by the tensile strain applied in the semisolid region, which is consistent with the targeted cracking morphology.

Fig. 8

Fracture surface. (a)–(c) RT-30 mmL, and (d)–(f) 700°C-45 mmL. (online color)

The prediction of hot-cracking susceptibility involves various assumptions depending on the used model. The RDG model used in this study was based on the strain applied perpendicular to the direction of growth for a dendrite structure that grew in one direction. In this test, the strain from solidification shrinkage was induced in the longitudinal direction of the specimen; thus, the dendrite growth must be perpendicular to this direction. Under all the tested conditions, the macrostructure grew in the vertical and horizontal directions, and no equiaxed or columnar crystals elongated in the vertical direction were observed. When the strain was loaded in the vertical direction, it was confirmed that the directions of dendrite growth and strain were perpendicular, as expected.

The etched microstructure was observed using an optical microscope to evaluate the dendrite microstructure. The results are presented in Fig. 9. The observation point was located approximately 2 mm from the insulation sheet. Dendrites were observed in grains with macrostructures extending in the horizontal direction, and the horizontal direction in the figure represents the direction of dendrite growth. From these photographs, the primary dendrite arm spacing (P-DAS) and S-DAS were determined at RT and 700°C. The P-DASs at RT and 700°C were 80.7 and 107.9 µm, and the S-DASs at RT and 700°C were 39.4 and 53.6 µm, respectively. Both P-DAS and S-DAS exhibited higher values under high-mold temperature conditions. P-DAS and S-DAS are reportedly negatively correlated with the cooling rate and temperature gradient [34–36]. Consequently, the higher values of P-DAS and S-DAS under high-mold-temperature conditions, with a low cooling rate and temperature gradient as shown in Table 1, are reasonable.

Fig. 9

Microstructures at all the tested conditions: (a), (b) RT-20 mmL, and (c) 700°C-45 mmL. (online color)

Hot-cracking susceptibility was evaluated using the RDG model based on the experimental results obtained in the previous section. Table 2 presents the physical properties used in the RDG model. The symbols ρ_s and ρ_l denote the density of solid and liquid phase, for which the experimental values for pure copper at 926.85°C and 1126.85°C were used, respectively [37, 38]. μ denotes the viscosity, for which the experimental value at the melting point of pure copper was used [39]. The cavitation pressure, p_c, is an unknown parameter in the RDG model with a value which varies depending on the previous study. Rappaz et al. [23] validated their model using Δp_c value of 2.0 kPa for Al-Cu alloys. They stated that this Δp_c value is an appropriate value with reference to the calculations of Ampuero et al. [40]. Drezet et al. [41] also performed calculations for the case of Al billet casting and showed that Δp_c = 5.6 kPa. Grasso et al. [42] calculated Δp_c for SCN-acetone alloys using the RDG model and found that Δp_c = 4.0 kPa when an average strain rate of 4 × 10⁻⁵/s⁻¹ was used, but Δp_c increased to 30 kPa when the strain rate of 4 × 10⁻⁴/s⁻¹ was used, which was increased owing to local deformation at grain boundaries and other factors. Bellet et al. [43] demonstrated p_c = 90 kPa for steel arc welding. Few studies have discussed p_c in the RDG model for copper alloys. In particular, there are no previous studies on Cu-Ni-Si alloys, and p_c = 84.3 kPa was reported for welded Cu-Cr-Zr alloy [44]. Our group considered that the alloy type should be matched and decided to use the p_c value of 84.3 kPa for future discussions, which was reported for the copper alloy. The p_m refers to the metallostatic pressure of molten metal at the dendrite tip. Because hot cracking at the top of the sample was observed in this study, it was not necessary to consider the weight of the molten metal, and a standard atmospheric pressure of 101.3 kPa was used for p_m. The presence of hot cracking was determined by comparing Δp_c to the calculated Δp_max from the RDG model equation.

Table 2 Physical properties of the RDG model.

Equations (2) and (3) were used to calculate Δp_max. The coherent solid fraction was set to 0.75 with reference to Morishita et al. [25]. Because there was no specific value for the coalescence solid fraction, as mentioned above, the calculations were performed under five conditions, whereby the coalescence solid fraction was set to 0.80, 0.85, 0.90, 0.95, and 0.98. The results are shown in Fig. 10 and Fig. 11. Figure 10 provides an overall view of the calculated solid fraction from 0.80 to 0.98, and Fig. 11 shows a magnified view near the intersection with Δp_c. The experimental results indicate that the coalescence solid fraction must be higher than 0.918, which is the intersection point of Δp_c and Δp_max at a mold temperature equal to the RT and a restraint distance equal to 30 mm because hot cracking occurred at the test condition. Conversely, the coalescence solid fraction must be lower than 0.919, which is the intersection point of Δp_c and Δp_max at a mold temperature equal to the RT and a restraint distance equal to 20 mm, as no hot cracking occurred at that test condition. Therefore, it can be determined that the appropriate coalescence solid fraction at a mold temperature equal to the RT is in the range of 0.918–0.919. From the same evaluation, the appropriate coalescence solid fraction at 700°C was determined to be in the range of 0.906–0.921.

Fig. 10

Calculated maximum pressure drop by the RDG model: (a) RT and (b) 700°C. (online color)

Fig. 11

Enlarged figure of maximum pressure drop by the RDG model at (a) RT and (b) 700°C. (online color)

These results indicate that the coalescence solid fraction did not change significantly under these test conditions, even though the cooling conditions were changed by changing the mold temperature. Figure 12 compares the cooling conditions in this test with the temperature gradient and solidification rate during casting and welding [45]. These cooling conditions were in the range where the dendrites grew, and the results were consistent with the microstructural observations. The results also indicated that these cooling conditions were similar to the surface cooling conditions of the casting. This suggests that the test results can be used to evaluate the hot cracking in the semisolid region that occurs on the ingot surface. Compared with the differences in the cooling conditions for welding and casting, the differences in the cooling conditions for this experiment were not significant. Therefore, there was no significant difference in the coalescence solid fraction among the cooling conditions in the test. The coalescence solid fraction estimated in this study was comparable to the coalescence solid fraction results of previous studies presented in the Introduction and was considered to be reasonable. However, as mentioned in this section, the coalescence solid fraction was estimated using the unknown parameter p_c, and more accurate verification will be required in the future.

Fig. 12

Comparison of cooling conditions between this study, welding, and casting [45]. (online color)

Hot cracking susceptibility was quantitatively evaluated for Cu-Ni-Si alloys using the I-beam test for the first time. The results of the tests for the two cooling conditions changed by the mold temperature showed that the minimum restraint distance necessary for hot cracking increased as the cooling rate decreased. The RDG model was used to calculate the pressure drop in the liquid phase of the dendrite gap. By comparison with the cavitation depression, it was estimated that the appropriate coalescence solid fraction required to reproduce the experimental results under the rapid and slow cooling conditions were in the ranges of 0.918–0.919 and 0.906–0.921, respectively. The coalescence solid fraction, which is the solid fraction of dendrite branches merging with each other, was not significantly different between the two cooling conditions in this study. This is presumably because the differences in the two cooling conditions used in this study were not as large as the difference in the cooling conditions between the welding and the casting processes.

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