_{2}-SiC Ceramics after High Temperature Oxidation

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The effects of oxidation on heat transfer and mechanical behavior of ZrB_{2}-SiC ceramics at high temperature are modeled using a micromechanics based finite element model. The model recognizes that when exposed to high temperature in air ZrB_{2}-SiC oxidizes into ZrO_{2}, SiO_{2}, and SiC-depleted ZrB_{2} layer. A steady-state heat transfer analysis was conducted at first and that is followed by a thermal stress analysis. A “global-local modeling” technique is used combining finite element with infinite element for thermal stress analysis. A theoretical formulation is developed for calculating the thermal conductivity of liquid phase SiO_{2}. All other temperature dependent thermal and mechanical properties were obtained from published literature. Thermal stress concentrations occur near the pore due to the geometric discontinuity and material properties mismatch between the ceramic matrix and the new products. The predicted results indicate the development of thermal stresses in the SiO_{2} and ZrO_{2} layers and high residual stresses in the SiC-depleted ZrB_{2} layer.

Ultrahigh temperature ceramics (UHTCs) such as zirconium diboride and hafnium diboride (ZrB_{2} and HfB_{2}) have been proposed for thermal protection of hypersonic aerospace vehicles, which may be exposed to temperatures above 1500°C in oxidizing environments. These materials are chemically and physically stable above 1600°C and have melting points above 3000°C [_{2} because of its lower theoretical density is attractive for aerospace applications [_{2} (s)) to air at elevated temperatures results in its oxidation to solid zirconia (ZrO_{2} (s)) and liquid boria (B_{2}O_{3} (l)). The oxidation resistance of ZrB_{2} (s) can be improved by adding SiC (s) to promote the formation of a silica-rich scale. At high temperature, above 1100°C, SiC (s) oxidizes by reaction to form SiO_{2} (l) which has a lower volatility and a higher melting point and viscosity compared with B_{2}O_{3} (l) [_{2}-SiC at 1500°C in air. The oxide scales that form on ZrB_{2}-SiC consist of an outer layer of SiO_{2}, a middle layer of porous ZrO_{2}, sometimes filled with SiO_{2}, and a layer of SiC-depleted ZrB_{2} adjoining the unoxidized ZrB_{2}-SiC at around 1500°C [_{2}-SiC ceramics to predict the thicknesses of the above three new productions. For temperatures below ~1600°C, an external glassy SiO_{2} layer forms and completely fills in pores of the porous ZrO_{2} scale whereas at higher temperatures, the glassy scale recedes due to evaporation of SiO_{2} (l) so that it only partially fills the pores in the ZrO_{2} layer.

The region of particular interest, from a mechanical perspective, is the interface between the pores and the corner of the pores in the ZrO_{2} scale. The pore itself may or may not be filled with liquid SiO_{2} (l). The interface therefore consists of three materials (ZrO_{2} scale, solid/liquid SiO_{2}, and SiC-depleted ZrB_{2} layer) of significantly different thermal and mechanical properties. This thermomechanical mismatch and geometric discontinuity would lead to residual stresses, and additional stress concentrations during the cool-down process from the processing temperature, thereby leading to potential cracking. Some researchers [_{2}-SiC, have indeed shown cracking in the ZrO_{2} scale.

The purpose of this study is to develop a thermal and mechanical simulation model for ZrB_{2}-SiC ceramics after oxidation. A steady-state heat transfer analysis was conducted using finite element analysis (FEA) modeling. An adpative remeshing technique is employed in both heat transfer and thermal stress analysis. A “global-local modeling” technique is used to combine finite element with infinite element for the thermal stress and the stress concentration analysis near a pore. Temperature, thermal, and residual stress distributions will be presented.

To simplify the problem, the ZrO_{2} scale was assumed to be of uniform thickness with regularly distributed pores. The pores were assumed to be straight, columnar in structure without tortuosity. A cylindrical representative volume unit (CRVU) was constructed and further treated as a two-dimensional (2D, pseudo-3D) axisymmetric problem subjected to local heating as shown in Figure

A schematic of a model with cylindrical representative volume unit (CRVU) subjected to local heating.

The oxide scale that forms on ZrB_{2}-SiC consists of an outer layer of SiO_{2}, a middle layer of porous ZrO_{2}, and a layer of SiC-depleted ZrB_{2} next to the unoxidized ZrB_{2}-SiC [_{2}-SiC ceramic after oxidation at high temperature were created as shown in Figure _{2}, ZrO_{2}, ZrB_{2} (SiC-depleted) and part of the ZrB_{2}-SiC base near the pore. The temperature dependent dimensions of the ZrO_{2} scale (crystalline oxide), glassy SiO_{2}, and SiC-depleted ZrB_{2} layer in ZrB_{2}-20 vol% SiC were obtained from the chemical oxidation model [_{2}-SiC ceramic is treated as a macroscale continuous solid with properties of a predetermined ratio of 4 : 1 of ZrB_{2} to SiC (ZrB_{2}-20 vol% SiC).

FEA models for ZrB_{2}-SiC ceramic after oxidation. (a) Lower temperature (external layer of SiO_{2}), and (b) higher temperature (partial evaporation of the glassy SiO_{2}).

The heat conduction equation for an axisymmetric problem can be expressed as

The heat flux condition is given by

The temperature dependent thermal and mechanical properties of the solid phases, needed for the heat transfer and mechanical analyses, can be found in the literature or databases [_{2} (l) and the elastic constants cannot be found. As such, the temperature dependence of the thermal conductivity of liquid SiO_{2} (l) and the elastic constants have to be predicted based on thermodynamics and some available test data. The predictive methods used for calculating the above properties are outlined in the next section. The cylindrical representative volume unit with equivalent pore diameter was treated as a 2D axisymmetric model (pseudo-3D). The modeling involves a steady-state heat transfer analysis representing local heat-up to calculate the temperature distribution and then a transient heat transfer analysis for 30 minutes representing a cool-down event to calculate the residual temperature distribution. The resulting temperature distributions were then applied to a thermomechanical finite element model to calculate the thermal stress distribution in the material. Adaptive remeshing technique was employed for the heat transfer analysis to improve accuracy. A “global-local modeling”, along with the adaptive remeshing technique, is used to combine finite element with infinite element for thermal stress analysis. The procedure is summarized in Figure

Flow diagram showing the procedure and steps for thermal and mechanical analyses.

As mentioned earlier, the thermal conductivity and elastic constants of liquid phases of SiO_{2} are not readily available. In an earlier work [_{2} at a given temperature. The following thermal conductivity equation for a liquid by Hirschfelder et al. [

In the above equation, _{2} was reported in [_{2} were given in [_{2} was found in [_{2} was calculated using (

Using the temperature dependent values for density and speed of sound in liquid of SiO_{2}, the bulk and shear moduli _{2} were calculated using the Newton-Laplace equation [_{2} instead of the viscous properties because the stress state in liquid phase of SiO_{2} was not of interest in the present study.

A 2D (pseudo-3D) 4-node linear axisymmetric heat transfer quadrilateral element was used in the thermal analysis. Heat flux was used as an error indicator variable to control the adaptive remeshing rule [

Two steps were used in the heat transfer analyses. The first step in the heat transfer analysis was a steady-state analysis representing local heating at the top surface to calculate the temperature distribution. The second step was a transient heat transfer analysis for 30 minutes representing a cooling event to room temperature to predict residual temperature distribution. The surface heating temperature was set as ^{2}·K) during the heating representing a high speed fluid flow and 100 W/(m^{2}·K) during cooling assuming a cooler fluid flow next to a solid boundary in air. The surface film coefficient was set as 100 W/(m^{2}·K) at all other boundaries during both heating and cooling.

The heating, cooling, and sink temperature conditions used in the analysis.

The calculated temperature distributions in the body after surface heating temperatures of 1780 K and 2240 K are shown in Figures _{2} layer is 1168 K (1492 K) which is less than the applied heating temperature of 1780 K (2240 K). This is due to the effect of the surface film coefficient on heat transfer between a fluid and a solid and the thermal conduction at the boundaries. The temperatures at the interface between the outer SiO_{2} layer and the ZrO_{2}, and at the interface between the oxide scale and ZrB_{2}, are 1160 K (1432 K) and 1148 K (1404 K), respectively. The temperature at the bottom surface is 1124 K (1370 K) which is much less than the heating temperature applied at the top surface. The temperatures at locations shown in Figures _{2} layer. This deviation could be due to the increase in ZrO_{2} layer thickness accompanied by a decrease in SiO_{2} layer thickness. Figure _{2}-SiC after steady-state analysis for heating to 2240 K. It is seen that a heat flux concentration occurs at the pore corner due to the geometric discontinuity and thermal conductivity mismatch.

Predicted temperature distribution of ZrB_{2}-SiC after steady-state analysis at heating to 1780 K.

Predicted temperature distribution of ZrB_{2}-SiC after steady-state analysis for heating to 2240 K.

Predicted heat flux distribution of ZrB_{2}-SiC after steady-state analysis for heating to 2240 K.

In the thermal stress analysis, the layout of infinite elements and finite elements, as well as the displacement constraints for the stress analysis shown in Figure _{2} layer. The temperature at this location is about 1166 K (Figure _{2} is sensitive to tensile stress with an average tensile strength of _{2} layer. The maximum value of the maximum principal stresses of 568 MPa occurs at the upper corner of the pore in the ZrO_{2} layer and is less than the flexural strength (900 MPa) of ZrO_{2} [_{2} is 451 MPa and occurs near the lower corner of the pore. This is higher than the measured bend strength of ZrB_{2} [_{2}-SiC is 191 MPa, located near the lower corner of the pore. The flexural strength of ZrB_{2}-SiC is 1000 MPa [

The infinite element, the finite element, and the boundary conditions for the thermal stress analysis.

Maximum principal stresses distribution in the enlarged area near pore after steady-state thermal analysis at 1780 K.

Heating-up

Cool-down

The distribution in the maximum principal stresses near the pore for steady-state heating to 2240 K and cool-down from 2240 K to 293 K are shown in Figures _{2} (2702 MPa) and ZrO_{2} (2224 MPa) near the pore. This may initiate tensile cracking. These results are consistent with the experimental observations by Levine et al. [_{2} + 20 vol.% SiC ceramic tests. Their results show that both pores and cracks appeared in the ZrO_{2} when oxidized in air at 1927°C for ten 10-min cycles. The highest maximum principal stress near the lower corner of the pore in the ZrB_{2} layer is 657 MPa shown in Figure _{2} as indicated above. For the cool-down case, the maximum principal stresses shown in Figure _{2} layer.

Maximum principal stresses distribution in the enlarged area near pore after steady-state thermal analysis at 2240 K.

Heating-up

Cool-down

The results were also obtained for additional temperatures. The variation of the maximum principal stress at indicated locations in Figures _{2}-SiC matrix is relatively small and does not vary much with heating temperature _{2}-SiC decreases with the increasing heating temperature _{2}-30 vol% SiC composites using neutron diffraction. Their results indicated that stresses begin to accumulate at about 1673 K during cool-down from the processing temperature of 2172 K. The stress increased to an average compressive stress of 880 MPa in the SiC phase and to an average tensile stress of 450 MPa in the ZrB_{2} phase. By using the rule of mixtures for 34 vol% SiC, the stress in the SiC (880 MPa) converts to an equivalent stress of 453 MPa which is very close to the measured stress of 450 MPa [

Thermal and residual maximum principal stress at indicated locations for different heating temperatures.

Thermal stress

Residual stress

A “global-local modeling” technique is used combining finite element with infinite element for thermal stress analysis for the oxidation effects on heat transfer and mechanical behavior of ZrB_{2}-SiC ceramics at high temperature. Thermal conductivity was calculated for the liquid phase of SiO_{2} based on a theoretical formulation. The predicted temperature at the top surface of the outer SiO_{2} layer is less than the applied heating temperature due to the surface film coefficient effect on the heat transfer between a fluid and a solid and the thermal conduction at the boundaries. An increase in ZrO_{2} layer thickness, accompanied by a decrease in SiO_{2} layer thickness, during oxidation will affect heat transfer in the body. Heat flux concentration occurs at the pore corner due to the geometric discontinuity and the material property mismatch. Thermal and residual stress concentrations occur near the pore due to geometric discontinuity and the material properties mismatch between the ceramic matrix and the new products. Thermal stresses in the surface oxide layers consisting of SiO_{2} and ZrO_{2}, are higher than their respective materials strengths. Thermal and residual stresses in the layer of a new oxidation product of SiC-depleted ZrB_{2} layer for both heating and cooling cases are higher than the material strength. Therefore, it is expected that damage may initiate in the layers of new oxidation products.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This project was funded under subcontract 10-S568-0094-01-C1 through the Universal Technology Corporation under prime contract number FA8650-05-D-5807. The authors are grateful to the technical support on the program by the Air Force Research Laboratory and specifically to Dr. Mike Cinibulk at AFRL for both his collaboration and guidance.

_{2}-based ultra-high temperature monolithic and fibrous monolithic ceramics

_{2}-SiC oxidation: formation of a SiC-depleted region

_{2}volatility diagram

_{2}-SiC during the oxidation in air

_{2}-SiC composites

_{2}and ZrB

_{2}-SiC Ceramics at high temperature

_{2}up to 2300 K from Brillouin scattering measurements

_{2}and ZrB

_{2}

_{2}-SiC

_{2}-SiC composites