Simulation of Thermal and Mechanical Response of (Zr,W)B2 Ceramic after Oxidation

Jun Wei1,Maryam K. Dehdashti2,Lokeswarappa R. Dharani3, K. Chandrashekhara4, Gregory E. Hilmas5, and William G. Fahrenholtz6

Missouri University of Science and Technology, Rolla, MO 65409, USA

1junwei@mst.edu, 2mk7y8@mst.edu, 3dharani@mst.edu, 4chandra@mst.edu, 5ghilmas@mst.edu, 6billf@mst.edu

Keywords: Ceramic,High-temperature, Oxidation, Tungsten, FEA, Stress.

Abstract. This study relates to a micromechanics based finite element model of the effect of

oxidation on heat transfer and mechanical behavior of a (Zr,W)B2 ceramic at high temperature. An

adaptive remeshing technique is employed in both heat transfer and thermal stress analysis models.

A “global-local modeling” technique is used to combine finite elements with infinite elements for

thermal stress analysis. Temperature and thermal stress distributions in the ceramic and the oxides

are presented.

Introduction

Thermal protection materials on hypersonic aerospace vehicles will be exposed to temperatures

of 1500°C and above. Ultra-High-Temperature Ceramics (UHTCs) such as zirconium diboride

(ZrB2) have been proposed as candidates for such applications. Oxidation of solid ZrB2in air will

result in its oxidation to solid porous zirconia (ZrO2) and liquid boria (B2O3) [1-6]. The typical

approach to improving the oxidation resistance of ZrB2 (s) is to add SiC (s) to promote the

formation of a silica-rich scale [7, 8]. More recently, WC additions have been shown to improve the

oxidation resistance of ZrB2 ceramics. The results showed that WC additions changed the

morphology of the zirconia scale formed during oxidation, improving the oxidation resistance of

ZrB2 [9, 10].

The material and geometric changes that take place as a result of high temperature oxidation of

the original ceramic affect the heat transfer and mechanical response. How the oxidation affects the

heat transfer and mechanical behavior of ZrB2 based ceramics after oxidation at high temperature

needs to be understood before this material can be effectively deployed in hypersonic space vehicle

structures. The present study models a ceramic consisting of ZrB2 + 8 mol% W after oxidation at

high temperature. A micromechanics based finite element (FE) model accounting for the effect of

oxidation on heat transfer and mechanical behavior is presented. A “global-local modeling”

approach, along with an adaptive remeshing technique, is used to combine finite elements with

infinite elements for thermal stress analysis.

Methods

A micromechanics based FE models the oxidation effects on the heat transfer and mechanical

behavior of a (Zr,W)B2 ceramic. A cylindrical representative volume unit with an equivalent

ellipsoid is treated as a 2D axisymmetric model (pseudo-3D). A steady state heat transfer analysis

was conducted at the heat-up step followed by a transient heat transfer analysis for 30 minutes at the

cool down step. The resulting temperature distributions were applied to the subsequent thermal

stress analysis model to calculate the thermal stress distribution.

Applied Mechanics and Materials Vols. 446-447 (2014) pp 40-44Online available since 2013/Nov/08 at www.scientific.net© (2014) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.446-447.40

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Model

Experimental observations show that there are three stages of oxidation for ZrB2. In stage I, for

temperatures less than 1300°C, (Zr,W)B2 forms a two-layered oxide scale. The outer layer consists

of glassy B2O3 and the inner layer contains porous ZrO2+ WO3 while the pores are filled with B2O3.

In stage II, at temperatures between 1300°C and 1500 °C, considerable evaporation of B2O3 takes

place resulting in a higher weight gain and thinner glassy layer. In stage III, at temperatures greater

than 1500°C, rapid evaporation of B2O3 takes place leading to higher weight gains and a thicker

porous scale composed of ZrO2 and WO3. Based on the experimental observations, a two

dimensional (pseudo-3D) axisymmetric model of (Zr,W)B2 after oxidation was created. Figure 1(a)

shows the model for heating at 1173 K and 1573 K while Fig. 1(b) corresponds to heating at 1873 K

with an equivalent ellipse. For the material properties, it was assumed that the unoxidized matrix

consists of onlyZrB2as the amount of the W is small. The glassy layer consists of B2O3 and the

inner layer of ZrO2 plus WO3 contains ellipticalpores that are filled with B2O3. The WO3content is

11.5vol% in the porous layer. The ellipticalpore radius equals the semi-minor axis of the ellipsoid.

There is no glassy B2O3 layer at 1873 K. There are two pore dimensions, a = 1.0 and 0.5 µm, c =

2.8 µm for 1173K and 8.8 µm for at 1573K, respectively. There are two pore dimensions of a = 1.0

and 0.5 µm, and c = 26.3 µm at 1873K.

(a) (b)

Fig. 1. Two dimensional (pseudo-3D) axisymmetric model of (Zr,W)B2 after oxidation: (a) heating

at 1173 K and 1573 K; (b) heating at 1873 K.

Results

The temperature dependent properties ofZrB2, ZrO2 and B2O3 are available in [11, 12]. The

properties of WO3 are obtained using different methods. Temperature dependent specific heat was

obtained from [13]. Temperature dependent coefficient of thermal expansion and density were

calculated from [14]. Thermal conductivity was taken as 1.63 W/m•K at room temperature from

[15]. Room temperature Young’s modulus and shear modulus were taken to be 311 GPa and 123

GPa, respectively [16]. The properties of ZrO2+WO3 were calculated using the composite spheres

model [17].

Two separate FE models are developed. In the heat transfer analysis a steady state procedure was

used to reach a given heating temperature, followed by a 30 minute cooling transient analysis [12]

to room temperature of 293 K. In the present study, three typical heating temperatures were applied,

1173, 1573 and 1873 K. Finite element and infinite element models were created based on the

model shown in Fig. 1.The heating temperature was applied at the local surfaces as shown in Fig.

1,colored with yellow or red. For the thermal stress analysis the axisymmetric displacement

boundary conditions were applied on the related sides in Fig. 1. Temperature contours for heating to

1173, 1573and 1873 K are shown in Fig. 2, respectively. The heat flux contours for heating at 1173

Kare shown in Fig. 3indicating the heat flux concentration. Figure 4shows the maximum principal

stress contours for heating at 1173 K indicating the stress concentration. To simplify, only the

maximum principal stresses near the pore at the sampled locations are presented. The sampled

locations are shown in Fig. 1(a) indicated with numbers1 to 4.

Applied Mechanics and Materials Vols. 446-447 41

Figure 5 shows the maximum principal stress against heating temperature while reaching steady-

state at three locations for pore radii equal to 0.5 µm and 1 µm, respectively.Location-4 is in the

liquid phase. It is seen that the stress at locations 1 and 3 changed with the pore radius at a heating

temperature of 1573 K. The stresses vary between 231 MPa (tension) and -119 MPa (compression).

Figure 6 shows the residual maximum principal stress against heating temperature after 30 minutes

of cooling to 293K at four locations for pore radiiof0.5 µm and 1 µm, respectively. It is seen that

the residual stress does not change much for this small change in pore radius. Location-4 is not

shown for 1873K because the B2O3 had totally evaporated at that temperature.

(a) (b) (c)

Fig. 2. Temperature contours for heating to (a) 1173 K, (b) 1573K and (c) 1873 K.

Fig. 3. Heat flux contours for heating at 1173

K show the heat flux concentration.

Fig. 4. Maximum principal stress contours for

the heating at 1173 K show the stress

concentration.

42 Advanced Research in Material Science and Mechanical Engineering

Fig. 5. Maximum principal stress vs. heating temperature while reaching steady-state at three

locations (Location-1 to Location-3 corresponding to numbers 1 to 3 in Fig. 1(a)) for the pore radii

equal to (a) 1 µm and (b) 0.5 µm.

Fig. 6. Residual maximum principal stress vs heating temperature after 30 minutes of cooling to

293K at four locations (Location-1 to Location-4 corresponding to numbers 1 to 4 in Fig. 1(a)) for

the pore radii equal (a) 1 µm and (b) 0.5 µm.

Summary

A micromechanics based finite element model of oxidation effects on heat transfer and mechanical

behavior of a (Zr,W)B2 ceramic at high temperature was conducted. Temperature, heat flux, thermal

and residual stress distributions are presented and heat flux and thermal stress concentrations occur

at the pore corner due to material properties mismatch between the ceramic base and the new

products.

Acknowledgments

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 for 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.

Applied Mechanics and Materials Vols. 446-447 43

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