1. **State the problem:** We need to select a material for thermal protection tiles that can withstand a temperature rise of 400°C during re-entry, minimizing thermal stress and maintaining structural integrity. 2. **Understand the criteria:** - **Yield Strength (MPa):** Higher values mean the material can resist higher stress without permanent deformation. - **Elastic Modulus (GPa):** Higher values mean the material is stiffer. - **Thermal Expansion (×10⁻⁶/°C):** Lower values minimize thermal stress caused by expansion. - **Toughness (MJ/m³):** Higher values indicate better ability to absorb energy without fracturing. 3. **Evaluate each material considering thermal stress:** Thermal stress $$\sigma$$ from temperature change $$\Delta T$$ is approximated by $$\sigma = E \cdot \alpha \cdot \Delta T$$ where $$E$$ is Elastic Modulus, $$\alpha$$ is Thermal Expansion coefficient. Calculate thermal stress for each material at $$\Delta T = 400^\circ C$$: - Alumina: $$300\times10^9~Pa \cdot 8\times10^{-6}/^\circ C \cdot 400 = 300~GPa\cdot 0.0032 = 1.216~GPa$$ (Using units conversion this seems off; correct calculation below) Actually, rewrite calculation correctly as: $$\sigma = E \times \alpha \times \Delta T$$ Given $$E$$ in GPa and $$\alpha$$ in $$10^{-6}/^\circ C$$, compute $$\sigma$$ in MPa: $$\sigma = E (GPa) \times 10^3 (MPa/GPa) \times \alpha \times 10^{-6} \times \Delta T$$ Calculate: - Alumina: $$380 \times 10^3 \times 8 \times 10^{-6} \times 400 = 380000 \times 0.000008 \times 400 = 380000 \times 0.0032 = 1216\ MPa$$ - Silicon Carbide: $$410000 \times 0.000004 \times 400 = 410000 \times 0.0016 = 656\ MPa$$ - Zirconia: $$200000 \times 0.00001 \times 400 = 200000 \times 0.004 = 800\ MPa$$ - Silicon Nitride: $$300000 \times 0.0000028 \times 400 = 300000 \times 0.00112 = 336\ MPa$$ 4. **Compare thermal stress to yield strength:** The material's - Alumina: thermal stress (1216 MPa) > yield strength (300 MPa) → fails - Silicon Carbide: thermal stress (656 MPa) > yield strength (400 MPa) → fails - Zirconia: thermal stress (800 MPa) > yield strength (250 MPa) → fails - Silicon Nitride: thermal stress (336 MPa) < yield strength (800 MPa) → withstands 5. **Account for toughness:** Silicon Nitride has lower toughness (8 MJ/m³) than Silicon Carbide (15 MJ/m³) and Alumina (10 MJ/m³) but has the highest yield strength and lowest thermal expansion, critical under these conditions. **Final Answer: Silicon Nitride is the most suitable material as it withstands the thermal stress induced by the 400°C temperature rise and maintains structural integrity.**