Compute the longitudinal strength of an aligned carbon fiber-epoxy matrix composite having a 0.20 volume fraction of fibers, assuming the following: (1) an average fiber diameter of \(6 \times 10^{-3} \mathrm{mm}\left(2.4 \times 10^{-4} \mathrm{in.}\right)\) (2) an average fiber length of \(8.0 \mathrm{mm}(0.31 \text { in. })\) (3) a fiber fracture strength of \(4.5 \mathrm{GPa}\) \(\left(6.5 \times 10^{5} \mathrm{psi}\right),(4)\) a fiber-matrix bond strength of \(75 \mathrm{MPa}(10,900 \mathrm{psi}),(5)\) a matrix stressat composite failure of \(6.0 \mathrm{MPa}(870 \mathrm{psi})\) and (6) a matrix tensile strength of \(60 \mathrm{MP}\) \((8,700 \mathrm{psi})\)

We must first determine the critical fiber length (\(l_c\)) for the composite. This can be calculated using the formula: \(l_c = \frac{\sigma_{f} D}{2 \tau_{c}}\) where \(\sigma_{f}\) is the fiber fracture strength, \(D\) is the average fiber diameter, and \(\tau_{c}\) is the fiber-matrix bond strength. Using the given values: \(\sigma_{f} = 4.5 \mathrm{GPa} = 4.5 \times 10^{9} \mathrm{Pa}\) \(D = 6 \times 10^{-3} \mathrm{mm} = 6 \times 10^{-6} \mathrm{m}\) \(\tau_{c} = 75 \mathrm{MPa} = 75 \times 10^{6} \mathrm{Pa}\) Calculating \(l_c\): \(l_c = \frac{(4.5 \times 10^9 Pa)(6 \times 10^{-6} m)}{2(75 \times 10^6 Pa)}\)

The efficiency factor, \(\eta\), can be calculated using the formula: \(\eta = \frac{l}{l + l_c}\) where \(l\) is the average fiber length. The given value for \(l\) is: \(l = 8.0 \mathrm{mm} = 8.0 \times 10^{-3} \mathrm{m}\) Using the critical fiber length \(l_c\) from Step 1, we can calculate the efficiency factor: \(\eta = \frac{l}{l + l_c}\)

To compute the composite longitudinal strength, we'll use the Rule of Mixtures equation: \(\sigma_{c} = \eta V_f \sigma_{f} + (1 - V_f) \sigma_{m}\) where \(\sigma_{c}\) is the composite longitudinal strength, \(\eta\) is the efficiency factor calculated in Step 2, \(V_f\) is the volume fraction of fibers, \(\sigma_{f}\) is the fiber fracture strength, and \(\sigma_{m}\) is the matrix stress at composite failure (given as \(6.0 \mathrm{MPa}\)). Using the given values and the calculated efficiency factor: \(V_f = 0.20\) \(\sigma_{m} = 6.0 \mathrm{MPa} = 6.0 \times 10^6 \mathrm{Pa}\) Calculating \(\sigma_{c}\): \(\sigma_{c} = \eta V_f \sigma_{f} + (1 - V_f) \sigma_{m}\) After calculating the composite longitudinal strength, we will have the solution to the given problem.