High-Temperature Mechanical Behaviors of SiO2-Based Ceramic Core for Directional Solidification of Turbine Blades

The high-temperature mechanical behaviors of SiO2-based ceramic cores for the directional solidification of turbine hollow blades were investigated. Isothermal uniaxial compression tests of ceramic core samples were conducted on a Gleeble-1500D mechanical simulator with an innovative auxiliary thermal system. The stress–strain results and macro- and micro- structures of SiO2-based ceramic cores were investigated experimentally. The microstructures were characterized by the scanning electron microscope (SEM). Based on the experimental data, a nonlinear constitutive model for high temperature compressive damage was established. The statistical results of Weibull moduli show that the stability of hot deformation increases with the increase of temperature. The fracture type of the SiO2-based core samples is brittle fracture, but when the temperature exceeds 1400 °C, the mechanical behavior exhibits thermo-viscoelastic and viscoplastic property. Under high-temperature (>1400 °C) and stress conditions, the strength of the ceramic core is weakened owing to the viscous slip of SiO2, which is initially melted at the temperature of 1400 °C. The comparison results between the predictions of nonlinear model and experimental values indicate that the model is applicable.


Introduction
In response to the increasing worldwide need for reliable, low-cost, and environmentally compatible generation of energy, the new generation of H-class gas turbines (GT) is developed [1,2]. Ni-based single-crystal (SX) superalloy turbine blades, which are the key hot-end assemblies of the gas turbine engines, can be produced by using the directional solidification (DS) technology [3]. The complex inner cavity formed by the ceramic core provides the possibility for the development of the hollow blade cooling technology. Nowadays, due to the complex thermal stress-strain interactions during DS, the size of blades appears imprecise, and the ceramic core even appears cracked. As a result, the performance of the SiO 2 -based ceramic core directly affects the dimensional accuracy of the SX hollow turbine blade. Therefore, the high-temperature mechanical properties of the SiO 2 -based ceramic core are crucial for the preparation of SX hollow turbine blade.
There are some studies focused on the high-temperature mechanical behaviors of SiO 2 -based ceramic cores [4]. Xu et al. [5] investigated the flexural strength of silica-based ceramic cores doped with different silica nanopowders at 1540 • C. The results showed that large quantities of cristobalite were crystallized at 1540 • C, which might provide enhanced mechanical property in the casting. Niu et al. [6] found that the ceramic cores with 3 wt% mullite fibers showed excellent properties, such as flexural strength being 22.3 MPa at 1550 • C, owing to fiber reinforcing. At the same time, there are two opposite conclusions about the high-temperature properties of SiO 2 -based ceramic core. For instance, Thermal process of ceramic shell/core during directional solidification is shown in Figure 1a. Since the tendency of core breakage is mainly concentrated at temperatures above 800 • C, the hot compression temperatures of ceramic cores were set at 700 • C (ST700), 1100 • C (ST1100), and 1400 • C (ST1400). The sample ST25 was tested at 25 • C. The average size of the ceramic cores is 14.77 mm in diameter and 15.25 mm in length (ϕ14.77 mm × 15.25 mm). The stain rate of 0.001 s −1 was chosen. After taking into consideration the inhomogeneity of the ceramic core, we tested three parallel samples for each deformation temperature. The whole process can be represented by Figure 1 and Table 2.

High-Temperature Experimental System for Mechanical Behaviors
The Gleeble system has been widely employed in the research of material constitutive model [14,15]. It mainly includes three parts: heating system, mechanical system, and computer control system. The heating system forms a current loop with a loaded metal sample (as a resistor) to heat the metal sample. The heating rate and heating temperature are varied by controlling the current in the sample. Therefore, the Gleeble system is generally unable to measure the high-temperature mechanical properties of non-conductive materials, such as ceramic materials. In order to realize the high-temperature mechanical measurement of non-conductive materials on the Gleeble simulator, we designed and developed an auxiliary thermal device that could expand the measurement material range of the Gleeble simulator [16]. The schematic diagram of the auxiliary thermal system of the Gleeble testing is shown in Figure 2. The measured temperature can reach 1600 • C. The temperature control accuracy is ±4 • C.

High-Temperature Experimental System for Mechanical Behaviors
The Gleeble system has been widely employed in the research of material constitutive model [14,15]. It mainly includes three parts: heating system, mechanical system, and computer control system. The heating system forms a current loop with a loaded metal sample (as a resistor) to heat the metal sample. The heating rate and heating temperature are varied by controlling the current in the sample. Therefore, the Gleeble system is generally unable to measure the high-temperature mechanical properties of non-conductive materials, such as ceramic materials. In order to realize the high-temperature mechanical measurement of non-conductive materials on the Gleeble simulator, we designed and developed an auxiliary thermal device that could expand the measurement material range of the Gleeble simulator [16]. The schematic diagram of the auxiliary thermal system of the Gleeble testing is shown in Figure 2. The measured temperature can reach 1600 °C. The temperature control accuracy is ±4 °C.

High-Temperature Experimental System for Mechanical Behaviors
The Gleeble system has been widely employed in the research of material constitutive model [14,15]. It mainly includes three parts: heating system, mechanical system, and computer control system. The heating system forms a current loop with a loaded metal sample (as a resistor) to heat the metal sample. The heating rate and heating temperature are varied by controlling the current in the sample. Therefore, the Gleeble system is generally unable to measure the high-temperature mechanical properties of non-conductive materials, such as ceramic materials. In order to realize the high-temperature mechanical measurement of non-conductive materials on the Gleeble simulator, we designed and developed an auxiliary thermal device that could expand the measurement material range of the Gleeble simulator [16]. The schematic diagram of the auxiliary thermal system of the Gleeble testing is shown in Figure 2. The measured temperature can reach 1600 °C. The temperature control accuracy is ±4 °C. Schematic diagram of the auxiliary thermal system (① ceramic sample, ② compression bars, ③ silicon carbide screw tube, ④ corundum tube, ⑤ insulation fiber box, ⑥ S-type thermocouple for temperature control, ⑦ temperature S-type thermocouple for temperature calibration, and ⑧ temperature control cabinet).  3 silicon carbide screw tube, 4 corundum tube, 5 insulation fiber box, 6 S-type thermocouple for temperature control, 7 temperature S-type thermocouple for temperature calibration, and 8 temperature control cabinet). Figure 3 shows the stress-strain curves variation ranges of the samples ST25, ST700, ST1100, and ST1400, from which can be found ST25, ST700, and ST1100 are all brittle fractures, while ST1400 shows thermo-viscoelastic and viscoplastic property. The average compressive strengths of ST25, Materials 2020, 13, 4579 4 of 11 ST700, and ST1100 are 51.83, 55.82, and 80.46 MPa, respectively, as shown in Figure 4. It is worth noting that the compressive strength of ST1100 is higher than that of ST25 and ST700. This is mainly due to the conversion of α-cristobalite to β-cristobalite. The densities of α-cristobalite and β-cristobalite are about 2.32 and 2.22 g/cm 3 , respectively, and the expansion of the β-cristobalite volume makes the sample more compact [17]. More α-cristobalite is transformed into β-cristobalite at 1100 • C, and the strength of β-cristobalite is stronger than that of α-cristobalite [18]. At the same time, Zener pinning would be more pronounced at higher temperatures [9].

High-Temperature Mechanical Properties
due to the conversion of α-cristobalite to β-cristobalite. The densities of α-cristobalite and βcristobalite are about 2.32 and 2.22 g/cm 3 , respectively, and the expansion of the β-cristobalite volume makes the sample more compact [17]. More α-cristobalite is transformed into β-cristobalite at 1100 °C, and the strength of β-cristobalite is stronger than that of α-cristobalite [18]. At the same time, Zener pinning would be more pronounced at higher temperatures [9].
In the elastic stage, the stress-strain curve of sample ST25 has a good linear-elastic regime. With the increase of temperature, the stress-strain curves of ST700 and ST1100 have a large amplitude and demonstrate a certain degree of dispersion. The elastic moduli of ST25, ST700, ST1100, and ST1400 are 2726, 2259, 2316, and 1442 MPa, respectively. The elastic modulus of the SiO2-based ceramic core shows a decreasing trend when the temperature is increased. There is a small change in the range of 25~1100 °C, while the elastic modulus decreases rapidly at the range of 1100~1400 °C. The stressstrain curve of ST1400 at the viscoplastic stage is narrow, indicating that the high-temperature experimental result achieves high repeatability and reproducibility. However, the overall hightemperature mechanical property of the sample ST1400 decreases significantly. On the other hand, it can be found from Figure 4 that the dispersion properties of compressive strengths vary greatly with the increase of temperature. The samples ST25 and ST700 demonstrate large dispersion, while samples ST1100 and ST1400, especially sample ST1400, show less dispersion. To quantify this result, a universal empirical model called Weibull approach is introduced. The threeparameter Weibull distribution function can be simplified to a two-parameter Weibull distribution function without affecting its accuracy [10,19]: where is the failure probability, is shape factor, and is Weibull modulus. Generally speaking, the larger m indicates the material is more uniform and less dispersion. From Figure 5, it can be found the Weibull moduli of ST25, ST700, ST1100, and ST1400 are 12.24, 10.87, 18.65, and 38.39, respectively. Obviously, ST1400 demonstrates the largest modulus, indicating that the hot deformation of sample at 1400 °C tends to exhibit certain stable and repeatable characteristic. At the same time, as the moduli of ST700 and ST25 are relatively small, it can be concluded that the deformation at 700 and 25 °C will be quite unstable. In fact, these results furtherly confirm the difference of stress-strain curves in Figure 3 from the angle of data. In the elastic stage, the stress-strain curve of sample ST25 has a good linear-elastic regime. With the increase of temperature, the stress-strain curves of ST700 and ST1100 have a large amplitude and demonstrate a certain degree of dispersion. The elastic moduli of ST25, ST700, ST1100, and ST1400 are 2726, 2259, 2316, and 1442 MPa, respectively. The elastic modulus of the SiO 2 -based ceramic core shows a decreasing trend when the temperature is increased. There is a small change in the range of 25~1100 • C, while the elastic modulus decreases rapidly at the range of 1100~1400 • C. The stress-strain curve of ST1400 at the viscoplastic stage is narrow, indicating that the high-temperature experimental result achieves high repeatability and reproducibility. However, the overall high-temperature mechanical property of the sample ST1400 decreases significantly.
On the other hand, it can be found from Figure 4 that the dispersion properties of compressive strengths vary greatly with the increase of temperature. The samples ST25 and ST700 demonstrate large dispersion, while samples ST1100 and ST1400, especially sample ST1400, show less dispersion. To quantify this result, a universal empirical model called Weibull approach is introduced. The three-parameter Weibull distribution function can be simplified to a two-parameter Weibull distribution function without affecting its accuracy [10,19]: where P(x) is the failure probability, σ 0 is shape factor, and m is Weibull modulus. Generally speaking, the larger m indicates the material is more uniform and less dispersion. From Figure 5, it can be found the Weibull moduli of ST25, ST700, ST1100, and ST1400 are 12.24, 10.87, 18.65, and 38.39, respectively. Obviously, ST1400 demonstrates the largest modulus, indicating that the hot deformation of sample at 1400 • C tends to exhibit certain stable and repeatable characteristic. At the same time, as the moduli of ST700 and ST25 are relatively small, it can be concluded that the deformation at 700 and 25 • C will be quite unstable. In fact, these results furtherly confirm the difference of stress-strain curves in Figure 3 from the angle of data. where is the failure probability, is shape factor, and is Weibull modulus. Generally speaking, the larger m indicates the material is more uniform and less dispersion. From Figure 5, it can be found the Weibull moduli of ST25, ST700, ST1100, and ST1400 are 12.24, 10.87, 18.65, and 38.39, respectively. Obviously, ST1400 demonstrates the largest modulus, indicating that the hot deformation of sample at 1400 °C tends to exhibit certain stable and repeatable characteristic. At the same time, as the moduli of ST700 and ST25 are relatively small, it can be concluded that the deformation at 700 and 25 °C will be quite unstable. In fact, these results furtherly confirm the difference of stress-strain curves in Figure 3 from the angle of data.

Microstructures Evolution
As is well-known, the mechanical properties are always associated with microstructures evolution. Figure 6 exhibits the macrostructural investigations of the samples (ST25, ST700, ST1100, and ST1400) fracture surfaces. The difference in the fracture patterns of ST25, ST700, ST1100, and ST1400 can be clearly distinguished (Figure 6a-d). ST25, ST700, and ST1100 are crushed brittle

Microstructures Evolution
As is well-known, the mechanical properties are always associated with microstructures evolution. Figure 6 exhibits the macrostructural investigations of the samples (ST25, ST700, ST1100, and ST1400) fracture surfaces. The difference in the fracture patterns of ST25, ST700, ST1100, and ST1400 can be clearly distinguished (Figure 6a-d). ST25, ST700, and ST1100 are crushed brittle fractures. As the temperature increases, the average size of residual pieces increases. When the strain of ST1400 reaches 0.04, the sample can maintain a substantially complete shape, and only minor fragments occur on the cylindrical surface. At the same time, a sliding plane of approximately 45 • with the bottom surface of the cylindrical sample can be clearly seen. It is this kind of viscous slip that causes the stress plateau of ST1400 after the strain is greater than 0.02 at 1400 • C. The main reason of this sticky slip can be explained in the microstructure morphology.
Materials 2020, 13, x FOR PEER REVIEW 6 of 11 fractures. As the temperature increases, the average size of residual pieces increases. When the strain of ST1400 reaches 0.04, the sample can maintain a substantially complete shape, and only minor fragments occur on the cylindrical surface. At the same time, a sliding plane of approximately 45° with the bottom surface of the cylindrical sample can be clearly seen. It is this kind of viscous slip that causes the stress plateau of ST1400 after the strain is greater than 0.02 at 1400 °C. The main reason of this sticky slip can be explained in the microstructure morphology.  Figure 7 shows the XRD patterns of various samples after hot compression at different temperatures. It can be found that all samples are composed of zircon and α-cristobalite. With the increase of deformation temperature, the peak intensities of two phases have a little change. In order to clarify the phase distribution in the microstructure, EDS point elemental analysis was employed and the results are shown in Figure 8. As shown in Figure 8, the point 1 with gray-black color is SiO2 and the point 2 with bright white color is ZrSiO4 in the image of backscattered electron (BSE).   fractures. As the temperature increases, the average size of residual pieces increases. When the strain of ST1400 reaches 0.04, the sample can maintain a substantially complete shape, and only minor fragments occur on the cylindrical surface. At the same time, a sliding plane of approximately 45° with the bottom surface of the cylindrical sample can be clearly seen. It is this kind of viscous slip that causes the stress plateau of ST1400 after the strain is greater than 0.02 at 1400 °C. The main reason of this sticky slip can be explained in the microstructure morphology.  Figure 7 shows the XRD patterns of various samples after hot compression at different temperatures. It can be found that all samples are composed of zircon and α-cristobalite. With the increase of deformation temperature, the peak intensities of two phases have a little change. In order to clarify the phase distribution in the microstructure, EDS point elemental analysis was employed and the results are shown in Figure 8. As shown in Figure 8, the point 1 with gray-black color is SiO2 and the point 2 with bright white color is ZrSiO4 in the image of backscattered electron (BSE).    Figure 9b,d,f,h shows the images of the BSE. In the BSE images, the major component of the white-gray phase is ZrSiO4, and the major component of the black-gray phase is SiO2. From the micro-topography of ST25, ST700, and ST1100, it can be seen that the SiO2 particles mainly undergo cleavage fracture, the ZrSiO4 particles mainly undergo dimple fracture. Most penetrated cracks are distributed in larger SiO2 particles. However, there are almost no cleavage fractures in SiO2 particles of the ST1400, and there are little dimple-like ZrSiO4 sections. The surfaces of the small SiO2 particles have a smooth curved shape. The occurrence of material fracture usually has great uncertainly, which is also the reason for the divergence of the stress-strain curves. As mentioned before, the Weibull modulus of ST1400 is much larger than that of ST25, ST700, and ST1100, meaning that the deformation of ST1400 is more stable. Therefore, the microstructure observation results of samples above are relatively consistent with the stress-strain curves.
In the cross-section of ST25, ST700, and ST1100, the surfaces of the large SiO2 particles are clean, and almost no adhesion of fine SiO2 particles is observed. However, in addition to penetrating cracks on the surface of the large SiO2 particles in ST1400, a large number of smooth and fine SiO2 particles are attached to the surface (Figure 9h). It is generally believed that the melting temperature of βcristobalite is 1720 ± 10 °C [17,20]. However, some studies have shown that, when the temperature reaches 1400~1450 °C, it will slowly melt on the surface of the SiO2, and the presence of other elements or impurities may reduce the β-cristobalite transformation temperature [21,22]. Therefore, in the high-temperature environment of 1400 °C, the main reason for the viscous slip of the SiO2-based ceramic core samples is that the surfaces of the fine SiO2 particles are initially melted, which plays a role in lubrication between large particles. The SiO2, which is initially melted at the temperature of 1400 °C, adheres to the surface of the large SiO2 particle. When the temperature drops further to the room temperature, it combines with the large particles, to form a unitary body.   Figure 9b,d,f,h shows the images of the BSE. In the BSE images, the major component of the white-gray phase is ZrSiO 4 , and the major component of the black-gray phase is SiO 2 . From the micro-topography of ST25, ST700, and ST1100, it can be seen that the SiO 2 particles mainly undergo cleavage fracture, the ZrSiO 4 particles mainly undergo dimple fracture. Most penetrated cracks are distributed in larger SiO 2 particles. However, there are almost no cleavage fractures in SiO 2 particles of the ST1400, and there are little dimple-like ZrSiO 4 sections. The surfaces of the small SiO 2 particles have a smooth curved shape. The occurrence of material fracture usually has great uncertainly, which is also the reason for the divergence of the stress-strain curves. As mentioned before, the Weibull modulus of ST1400 is much larger than that of ST25, ST700, and ST1100, meaning that the deformation of ST1400 is more stable. Therefore, the microstructure observation results of samples above are relatively consistent with the stress-strain curves.
In the cross-section of ST25, ST700, and ST1100, the surfaces of the large SiO 2 particles are clean, and almost no adhesion of fine SiO 2 particles is observed. However, in addition to penetrating cracks on the surface of the large SiO 2 particles in ST1400, a large number of smooth and fine SiO 2 particles are attached to the surface (Figure 9h). It is generally believed that the melting temperature of β-cristobalite is 1720 ± 10 • C [17,20]. However, some studies have shown that, when the temperature reaches 1400~1450 • C, it will slowly melt on the surface of the SiO 2 , and the presence of other elements or impurities may reduce the β-cristobalite transformation temperature [21,22]. Therefore, in the high-temperature environment of 1400 • C, the main reason for the viscous slip of the SiO 2 -based ceramic core samples is that the surfaces of the fine SiO 2 particles are initially melted, which plays a role in lubrication between large particles. The SiO 2 , which is initially melted at the temperature of 1400 • C, adheres to the surface of the large SiO 2 particle. When the temperature drops further to the room temperature, it combines with the large particles, to form a unitary body.

Nonlinear Constitutive Models for High-Temperature Compressive Damage
It can be seen from Figures 4 and 5 that the compressive strength and modulus are basically negatively correlated with temperature. The macro-effect of temperature on the properties of ceramic core includes two aspects. On the one hand, the intermolecular forces decrease with increasing temperature. On the other hand, the change of structure caused by the variation of temperature will greatly affect the properties of the material, such as thermal mismatch. Therefore, the thermal damage D(T) is employed to describe the temperature effect on the property [23]: where E 0 is elastic modulus at room temperature, and E T is the elastic modulus at T. The elastic modulus denoted by thermal damage is expressed as follows:

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According the analysis of the experimental results at different temperatures, the thermal damage value, D(T), at each temperature point is calculated, as shown in Figure 10. Through data fitting, the expression of thermal damage with temperature variation can be written as follows: Materials 2020, 13, x FOR PEER REVIEW 8 of 11

Nonlinear Constitutive Models for High-Temperature Compressive Damage
It can be seen from Figures 4 and 5 that the compressive strength and modulus are basically negatively correlated with temperature. The macro-effect of temperature on the properties of ceramic core includes two aspects. On the one hand, the intermolecular forces decrease with increasing temperature. On the other hand, the change of structure caused by the variation of temperature will According the analysis of the experimental results at different temperatures, the thermal damage value, D(T), at each temperature point is calculated, as shown in Figure 10. Through data fitting, the expression of thermal damage with temperature variation can be written as follows: According to the author's previous research [24], the continuous damage constitutive model based on Weibull distribution method, at room temperature, is summarized as follows: where is the axial strain, and is a constant. In order to obtain the nonlinear constitutive model for high-temperature compressive damage, the E in Equation (5) can be substituted by , and the formula can be rewritten as follows: The experiment results of typical compression stress-strain of SiO2-based ceramic core and the simulation results based on thermo-visco damage model are presented in Figure 11.
From Figure 11, it can be found that the nonlinear constitutive model has a good generalization property. In other words, this model could reflect the uniaxial compression behaviors of ceramic cores deformed at various temperatures. According to the author's previous research [24], the continuous damage constitutive model based on Weibull distribution method, at room temperature, is summarized as follows: where ε 1 is the axial strain, and ε 0 is a constant. In order to obtain the nonlinear constitutive model for high-temperature compressive damage, the E in Equation (5) can be substituted by E T , and the formula can be rewritten as follows: The experiment results of typical compression stress-strain of SiO 2 -based ceramic core and the simulation results based on thermo-visco damage model are presented in Figure 11.

Conclusions
(1) In the temperature range from 25 to 1400 °C, the elastic moduli of the SiO2-based ceramic cores range from 1442 to 2726 MPa at the elastic stages. The statistical results of Weibull moduli show that the stability of deformation increases with the increase of temperature. (2) The SiO2-based ceramic core samples are all brittle fractures, while, when the temperature exceeds 1400 °C, the mechanical behaviors of the samples are characterized by thermoviscoelastic and viscoplastic properties, which mainly can be ascribed to the initial surface melting of SiO2 fine particles.  From Figure 11, it can be found that the nonlinear constitutive model has a good generalization property. In other words, this model could reflect the uniaxial compression behaviors of ceramic cores deformed at various temperatures.

Conclusions
(1) In the temperature range from 25 to 1400 • C, the elastic moduli of the SiO 2 -based ceramic cores range from 1442 to 2726 MPa at the elastic stages. The statistical results of Weibull moduli show that the stability of deformation increases with the increase of temperature. (2) The SiO 2 -based ceramic core samples are all brittle fractures, while, when the temperature exceeds 1400 • C, the mechanical behaviors of the samples are characterized by thermo-viscoelastic and viscoplastic properties, which mainly can be ascribed to the initial surface melting of SiO 2 fine particles. (3) Nonlinear constitutive model for high-temperature compressive damage is established to predict the hot deformation of ceramic core. The comparison results between the nonlinear model predictions and experimental values indicate that the model is applicable.