Thermally induced delamination and buckling of a ceramic coating with temperature-dependent material properties from porous substrate at high temperatures

Abstract

Ceramic coatings are ideal materials for thermal protection systems such as the thermal shield of the space shuttle. In a high-temperature environment, material properties of ceramics strongly depend on the temperature. The severe mismatch of material properties between the ceramic coating and the substrate can result in progressive mechanical failure of thermal protection system. This paper investigates delamination and buckling behaviors between a temperature-dependent ceramic coating and a porous substrate. The shear stress intensity factor at the tips of the delamination crack and buckling region are derived. Based on the stress intensity factor, the critical temperature of the coating buckling from the substrate is obtained. A fitting formula of the critical buckling temperature with respect to the length-to-thickness ratio of the coating, and the buckling region is obtained. It is found that the effect of the temperature dependence of material properties on delamination and buckling is more significant for higher temperatures than for lower temperatures. The critical temperatures of delamination and buckling are overestimated if the temperature dependence of material properties is neglected. The critical temperatures of delamination and buckling increase with the porosity of the substrate.

Appendix A

With the substitution of Eq. (7) into Eq. (6), one can obtain the following equation:

$$\begin{aligned} \frac{2}{\pi }\sum \limits _{j=0}^\infty {A_{j} } \int _{-1}^1 {\frac{V_{j} (\zeta )}{(\bar{{x}}-\zeta )\sqrt{1-\zeta ^{2}} }} \hbox {d}\zeta +\frac{E_{\mathrm{s}} (L-a)}{2E_{\mathrm{c}} H}\sum \limits _{j=0}^\infty {A_{j} } \int _{\bar{{x}}}^1 {\frac{V_{j} (\zeta )}{\sqrt{1-\zeta ^{2}} }\hbox {d}\zeta } =\frac{E_{\mathrm{s}} }{1-\upsilon }\int _{T_{0} }^{T_{1} } {\alpha _{\mathrm{c}} (T)\hbox {d}T}. \end{aligned}$$

(A1)

By introducing an intermediate variable $\xi$ which satisfies $\zeta =\hbox {cos}\xi$, $0\le \xi \le \pi$, the Chebyshev polynomials of the first kind can be expressed as $V_{j}(\xi )=\hbox {cos}(j\xi )$. Thus, the second integral at the left hand side of Eq. (A1) reduces to $\int _{\arccos \bar{{x}}}^0 {\frac{\cos (j\xi )}{\sqrt{1-\cos ^{2}(\xi )} }} \hbox {d}\cos (\xi )=\frac{1}{j}\sin \left( {j\arccos (\bar{{x}})} \right)$. In addition, the Chebyshev polynomials of the first kind and the second kind hold $\pi U_{j-1} (\bar{{x}})=\int _{-1}^1 {V_{j} (\zeta )/(\bar{{x}}-\zeta )/\sqrt{1-\zeta ^{2}} } \hbox {d}\zeta ,j\ge 1$. Thus, Eq. (A1) can be rewritten as

$$\begin{aligned} 2\sum \limits _{j=1}^\infty {A_{j} } U_{j-1} (\bar{{x}})+\frac{E_{\mathrm{s}} (L-a)}{2E_{\mathrm{c}} H}\sum \limits _{j=1}^\infty {\frac{1}{j}A_{j} } \sin \left( {j\arccos (\bar{{x}})} \right) =\frac{E_{\mathrm{s}} }{1-\upsilon }\int _{T_{0} }^{T_{1} } {\alpha _{\mathrm{c}} (T)\hbox {d}T}. \end{aligned}$$

(A2)

The Chebyshev polynomials of the second kind can be expressed in terms of the trigonometric functions as $U_{j-1} (\bar{{x}})=\sin \left( {j\arccos (\bar{{x}})} \right) /\sin \left( {\arccos (\bar{{x}})} \right)$. Therefore, Eq. (A2) reduces to

$$\begin{aligned} 2\sum \limits _{j=1}^\infty {A_{j} } \frac{\sin \left( {j\arccos (\bar{{x}})} \right) }{\sin \left( {\arccos (\bar{{x}})} \right) }+\frac{E_{\mathrm{s}} (L-a)}{2E_{\mathrm{c}} H}\sum \limits _{j=1}^\infty {\frac{A_{j} }{j}} \sin \left( {j\arccos (\bar{{x}})} \right) =\frac{E_{\mathrm{s}} }{1-\upsilon }\int _{T_{0} }^{T_{1} } {\alpha _{\mathrm{c}} (T)\hbox {d}T}. \end{aligned}$$

(A3)