2. This question concerns the effect of heating a solid spherical ball by hot gas. We analyse the thermoelastic stresses in a single spherical ball of radius b and initial

temperature To, while the gas has a temperature Tw. The numbers which are relevant to this question are T_{0}=293 K, \quad T_{w}=600 K, \quad b=0.025 m, \quad \alpha=10^{-5} K^{-1}, E=200 G P a a) The thermoelastic stresses in an isolated sphere are analysed. The linear elastic equations for the solid stresses (orr, 0ee) are related to the strain field (err, E9e)through \left[\begin{array}{c} \sigma_{r r} \\ \sigma_{\theta \theta} \end{array}\right]=\frac{E}{(1+v)(1-2 v)}\left[\begin{array}{cc} 1-v & v \\ v & 1-v \end{array}\right]\left[\begin{array}{c} \epsilon_{r r} \\ \epsilon_{\theta \theta} \end{array}\right]-\frac{E \alpha T}{1-2 v}\left[\begin{array}{l} 1 \\ 1 \end{array}\right] Provide a physical justification for the equilibrium conditions \frac{d \sigma_{r r}}{d r}+\frac{2}{r}\left(\sigma_{r r}-\sigma_{\theta \theta}\right)=0 b) The strain is related to the radial displacement u through \epsilon_{r r}=\frac{d u}{d r}, \epsilon_{\theta \theta}=\frac{u}{r} Show that the radial displacement u satisfies \frac{d}{d r}\left(\frac{1}{r^{2}} \frac{d\left(r^{2} u\right)}{d r}\right)=\alpha\left(\frac{1+v}{1-v}\right) \frac{d T}{d r} and the general solution is u=\frac{(1+v) \alpha}{r^{2}(1-v)} \int_{0}^{r} T r^{2} d r+C_{1} r+\frac{C_{2}}{r^{2}} c) Show that the radial stress due to the temperature field is \sigma_{r r}=\frac{2 \alpha E}{1-v}\left(\frac{1}{b^{3}} \int_{0}^{b} \operatorname{Tr}^{2} \mathrm{~d} r-\frac{1}{r^{3}} \int_{0}^{r} \operatorname{Tr}^{2} d \mathrm{r}\right) and calculate the azimuthal stress. d) The effect of temperature on the stress field is analysed using a simplified temperature distribution of where T varies from To to Tw at a distance R from the centre of the sphere: T = To (r R) Using your derived expressions for stress, calculate how the radial and azimuthal stress (orr and o) vary with r with the temperature distribution given. e) As the spherical ball heat up R decreases. Plot numerically the variation of radialRand azimuthal stresses with r for = 0.1,0.2, 0.5 and 0.8 (using the values given atthe start of the question).b f) Discuss briefly the thermal cracking that can occur when the spheres are heated or cooled rapidly, focusing on the mode of failure and the characteristic measures of when this can occur. (250 words maximum).