Thermal Shock Cracking Behavior of a Cylinder Specimen with an Internal Penny-Shaped Crack Based on Non-Fourier Heat Conduction

Abstract

The thermal shock cracking of solids is analyzed for a long cylinder subjected to a sudden change of temperature on its outer surface, based on a generalized heat conduction model in which the concepts of phase lags of temperature gradient and heat flux are introduced. The temperature field and associated thermal stress for an un-cracked cylinder are obtained in closed form. Then the thermal stress with an opposite sign is loaded on the crack surface to formulate the crack problem. The thermal stress intensity factor is deduced and given by a Fredholm integral equation. The cracking behavior is discussed and thermal shock resistance of the cylinder is evaluated according to the stress criterion and the fracture mechanics criterion, separately. The effects of phase lags of temperature gradient and heat flux and the crack size on the thermal shock resistance of the cylinder are also discussed.

Appendices

Appendix 1

The constants $C_{1m} , C_{2m}$, and $C_{3m} (m=1, 2)$ are given as

$$\begin{aligned} C_{1m}= & {} c_{11} d_{1m} +c_{13} d_{2m} \lambda _m ,\end{aligned}$$

(35)

$$\begin{aligned} C_{2m}= & {} c_{13} d_{1m} +c_{33} d_{2m} \lambda _m,\end{aligned}$$

(36)

$$\begin{aligned} C_{3m}= & {} c_{44} \left( {d_{1m} \lambda _m -d_{2m} } \right) . \end{aligned}$$

(37)

The constants $f_m (m=1, 2)$ are given as

$$\begin{aligned} \left\{ {{\begin{array}{l} {f_1 } \\ {f_2 } \\ \end{array} }} \right\} =\left[ {{\begin{array}{ll} {d_{21} }&{}\quad {d_{22} } \\ {C_{31} }&{}\quad {C_{32} } \\ \end{array} }} \right] ^{-1}\left\{ {{\begin{array}{l} 1 \\ 0 \\ \end{array} }} \right\} . \end{aligned}$$

(38)

The function $\beta _m (s,x)$ is given as

$$\begin{aligned} \left\{ {\beta _m (s,x)} \right\}= & {} \left[ {\Delta (s)} \right] ^{-1}\left\{ {{\begin{array}{l} {\alpha _1 (s,x)} \\ {\alpha _2 (s,x)} \\ \end{array} }} \right\} ,\nonumber \\ \Delta (s)= & {} \left[ {{\begin{array}{l} {\frac{C_{1m} }{\lambda _m }sI_0 (bs/\lambda _m )-\frac{\left( {c_{11} -c_{12} } \right) }{b}d_{1m} I_1 (bs/\lambda _m )} \\ {\frac{C_{3m} }{\lambda _m }sI_1 (bs/\lambda _m )} \\ \end{array} }} \right] , \end{aligned}$$

(39)

where

$$\begin{aligned} \alpha _1 (s,x)= & {} \sum _{m=1}^2 \left[ {\frac{C_{1m} f_m s}{\lambda _m^2 }L_0 \left( {-\frac{s}{\lambda _m }b} \right) -\frac{\left( {c_{11} -c_{12} } \right) }{b}\frac{d_{1m} f_m }{\lambda _m }L_1 \left( {-\frac{s}{\lambda _m }b} \right) } \right] \nonumber \\&\times \sinh \left( {-\frac{s}{\lambda _m }x} \right) ,\end{aligned}$$

(40)

$$\begin{aligned} \alpha _2 (s,x)= & {} \sum _{m=1}^2 {\frac{C_{3m} f_m s}{\lambda _m^2 }L_1 \left( {-\frac{s}{\lambda _m }b} \right) \sinh \left( {-\frac{s}{\lambda _m }x} \right) } , \end{aligned}$$

(41)

and where $L_{i} (i=0, 1)$ are the ith modified Bessel function of the second kind.

Appendix 2

The material contents are given by beryllium oxide [25]: elastic modulus, $E_r =397\,\hbox {GPa}, E_z =450\,\hbox {GPa}, G_{rz} =153\,\hbox {GPa}$, Poisson’s ratios, $v_{r\theta } =0.32, v_{r\theta } =0.16;$ temperature-strain coefficients, $\alpha _{r} = 2.3\times 10^{-6}\,\mathrm{K}^{-1}, \alpha _{z} = 4.4\times 10^{-6}\,\mathrm{K}^{-1}$.

Accordingly, the stiffness coefficients $c_{ij}$ and temperature-stress coefficients $\chi _{ii}$ are as follows:

$$\begin{aligned} c_{11}= & {} 46.3\times 10^{10}\,\mathrm{N}{\cdot }{\mathrm{m}^{-2}}, \quad c_{12} = 16.2\times 10^{10}\,\mathrm{N}{\cdot }{\mathrm{m}^{-2}}, \\ c_{13}= & {} 10.0\times 10^{10}\,\mathrm{N}{\cdot }{\mathrm{m}^{-2}}, \quad c_{33} = 48.2\times 10^{10}\,\mathrm{N}{\cdot }{\mathrm{m}^{-2}}, \\ c_{44}= & {} 15.3\times 10^{10}\,\mathrm{N}{\cdot }{\mathrm{m}^{-2}}, \quad \chi _{11} = 0.107\times 10^{6}\,\hbox {N}{\cdot }\hbox {m}^{-2}{\cdot }{\mathrm{K}^{-1}}, \\ \chi _{33}= & {} 0.212\times 10^{6}\,\hbox {N}{\cdot }\hbox {m}^{-2}{\cdot }{\mathrm{K}^{-1}} \end{aligned}$$

Following the material properties $c_{ij}$ and $\chi _{ii}$, the contents $\mu$ can be calculated by computer: $\mu =-2.14\times 10^{^{11}}\,\hbox {N}{\cdot }\hbox {m}^{-3}{\cdot }{\mathrm{K}^{-1}}$.