Let's answer part (a): **Determine the Moduli of Elasticity (E_c) of Carbon/Epoxy, Aramid/Epoxy, and Boron/Epoxy composites** in the longitudinal direction. ### Step 1: Write the Rule of Mixtures Equation For a **longitudinally loaded composite**: \[ E_c = V_f E_f + V_m E_m \] Where: - \(E_c\) = modulus of composite - \(V_f\) = volume fraction of fiber (.55) - \(E_f\) = modulus of fiber (from table) - \(V_m\) = volume fraction of matrix (.45) - \(E_m\) = modulus of matrix (Epoxy = 2.4 GPa) **Explanation:** The rule of mixtures gives a good estimate for the modulus of a composite when fibers are aligned and loaded in the fiber direction. --- ### Step 2: Calculate for Each Composite #### (i) **Carbon/Epoxy:** \[ E_c = (.55)(230) + (.45)(2.4) = 126.5 + 1.08 = 127.58~\text{GPa} \] #### (ii) **Aramid/Epoxy:** \[ E_c = (.55)(130) + (.45)(2.4) = 71.5 + 1.08 = 72.58~\text{GPa} \] #### (iii) **Boron/Epoxy:** \[ E_c = (.55)(400) + (.45)(2.4) = 220 + 1.08 = 221.08~\text{GPa} \] **Explanation:** Substitute the respective fiber moduli into the rule of mixtures equation, keeping the matrix modulus and volume fractions constant. --- ## **Final Answers for (a): Moduli of Elasticity** - **Carbon/Epoxy:** \( \boxed{127.6~\text{GPa}} \) - **Aramid/Epoxy:** \( \boxed{72.6~\text{GPa}} \) - **Boron/Epoxy:** \( \boxed{221.1~\text{GPa}} \) Let me know if you want to proceed to part (b)!

# ENGR 200: Materials of Engineering Final Practice Exam Solutions ## Part (a): Moduli of Elasticity of Composites ### Step 1: Write the Rule of Mixtures Equation For a **longitudinally loaded composite**: \[ E_c = V_f E_f + V_m E_m \] Where: - \(E_c\) = modulus of composite - \(V_f\) = volume fraction of fiber (0.55) - \(E_f\) = modulus of fiber (from table) - \(V_m\) = volume fraction of matrix (0.45) - \(E_m\) = modulus of matrix (Epoxy = 2.4 GPa) **Explanation:** The rule of mixtures provides an estimate for the modulus of a composite with fibers aligned along the loading direction. ### Step 2: Calculate for Each Composite #### (i) Carbon/Epoxy: Using \(E_f = 230 \text{ GPa}\): \[ E_c = (0.55)(230) + (0.45)(2.4) = 126.5 + 1.08 = 127.58~\text{GPa} \] #### (ii) Aramid/Epoxy: Using \(E_f = 130 \text{ GPa}\): \[ E_c = (0.55)(130) + (0.45)(2.4) = 71.5 + 1.08 = 72.58~\text{GPa} \] #### (iii) Boron/Epoxy: Using \(E_f = 400 \text{ GPa}\): \[ E_c = (0.55)(400) + (0.45)(2.4) = 220 + 1.08 = 221.08~\text{GPa} \] **Explanation:** The calculations demonstrate how different fiber materials affect the overall modulus of the composite, maintaining constant matrix properties. ## Final Answers for (a): Moduli of Elasticity - **Carbon/Epoxy:** \( \boxed{127.6~\text{GPa}} \) - **Aramid/Epoxy:** \( \boxed{72.6~\text{GPa}} \) - **Boron/Epoxy:** \( \boxed{221.1~\text{GPa}} \) --- ## Part (b): Specific Moduli of Each Composite ### Step 1: Calculate Specific Moduli Specific modulus is calculated as: \[ \text{Specific Modulus} = \frac{E_c}{\rho} \] Where: - \(E_c\) = modulus of composite - \(\rho\) = specific gravity (g/cm³) ### Step 2: Calculate for Each Composite #### Specific Gravity Values - Carbon/Epoxy: \( \rho = 1.6 \text{ g/cm}^3 \) - Aramid/Epoxy: \( \rho = 1.4 \text{ g/cm}^3 \) - Boron/Epoxy: \( \rho = 2.5 \text{ g/cm}^3 \) #### (i) Carbon/Epoxy: \[ \text{Specific Modulus} = \frac{127.58 \text{ GPa}}{1.6 \text{ g/cm}^3} = 79.74 \text{ GPa·cm}^3/\text{g} \] #### (ii) Aramid/Epoxy: \[ \text{Specific Modulus} = \frac{72.58 \text{ GPa}}{1.4 \text{ g/cm}^3} = 51.13 \text{ GPa·cm}^3/\text{g} \] #### (iii) Boron/Epoxy: \[ \text{Specific Modulus} = \frac{221.08 \text{ GPa}}{2.5 \text{ g/cm}^3} = 88.43 \text{ GPa·cm}^3/\text{g} \] **Explanation:** The specific modulus gives insight into how efficient each composite is regarding its weight versus stiffness. ## Final Answers for (b): Specific Moduli - **Carbon/Epoxy:** \( \boxed{79.74~\text{GPa·cm}^3/\text{g}} \) - **Aramid/Epoxy:** \( \boxed{51.13~\text{GPa·cm}^3/\text{g}} \) - **Boron/Epoxy:** \( \boxed{88.43~\text{GPa·cm}^3/\text{g}} \) --- ## Part (c): Specific Strengths (i.e., Specific UTS) of Each Composite ### Step 1: Calculate Specific Strength Specific strength is given by: \[ \text{Specific Strength} = \frac{\text{UTS}}{\rho} \] ### Step 2: Calculate for Each Composite #### UTS Values - Carbon/Epoxy: \( \text{UTS} = 600 \text{ MPa} \) - Aramid/Epoxy: \( \text{UTS} = 400 \text{ MPa} \) - Boron/Epoxy: \( \text{UTS} = 900 \text{ MPa} \) #### (i) Carbon/Epoxy: \[ \text{Specific Strength} = \frac{600 \text{ MPa}}{1.6 \text{ g/cm}^3} = 375 \text{ MPa·cm}^3/\text{g} \] #### (ii) Aramid/Epoxy: \[ \text{Specific Strength} = \frac{400 \text{ MPa}}{1.4 \text{ g/cm}^3} = 285.71 \text{ MPa·cm}^3/\text{g} \] #### (iii) Boron/Epoxy: \[ \text{Specific Strength} = \frac{900 \text{ MPa}}{2.5 \text{ g/cm}^3} = 360 \text{ MPa·cm}^3/\text{g} \] **Explanation:** Calculating specific strengths allows for comparing material performance relative to their weight, which is essential in aerospace applications. ## Final Answers for (c): Specific Strengths - **Carbon/Epoxy:** \( \boxed{375~\text{MPa·cm}^3/\text{g}} \) - **Aramid/Epoxy:** \( \boxed{285.71~\text{MPa·cm}^3/\text{g}} \) - **Boron/Epoxy:** \( \boxed{360~\text{MPa·cm}^3/\text{g}} \) --- ## Part (d): Final Cost of Each Composite ### Step 1: Calculate Total Cost Total cost is calculated as: \[ \text{Total Cost} = \text{Volume} \times \text{Unit Cost} \] Assuming a volume of 1 m³ for simplicity. ### Step 2: Calculate Costs Using Unit Costs #### Unit Costs - Carbon/Epoxy: \( 20 \text{ $US/kg} \) - Aramid/Epoxy: \( 15 \text{ $US/kg} \) - Boron/Epoxy: \( 30 \text{ $US/kg} \) #### (i) Carbon/Epoxy: \[ \text{Total Cost} = (1 \text{ m}^3)(1.6 \text{ g/cm}^3)(1000 \text{ kg/m}^3)(20 \text{ $US/kg}) = 32000 \text{ $US} \] #### (ii) Aramid/Epoxy: \[ \text{Total Cost} = (1 \text{ m}^3)(1.4 \text{ g/cm}^3)(1000 \text{ kg/m}^3)(15 \text{ $US/kg}) = 21000 \text{ $US} \] #### (iii) Boron/Epoxy: \[ \text{Total Cost} = (1 \text{ m}^3)(2.5 \text{ g/cm}^3)(1000 \text{ kg/m}^3)(30 \text{ $US/kg}) = 75000 \text{ $US} \] **Explanation:** The total cost calculation helps in evaluating the economic feasibility of using each composite in design applications. ## Final Answers for (d): Total Costs - **Carbon/Epoxy:** \( \boxed{32000~\text{US$}} \) - **Aramid/Epoxy:** \( \boxed{21000~\text{US$}} \) - **Boron/Epoxy:** \( \boxed{75000~\text{US$}} \)