Compute the longitudinal tensile strength of an aligned glass fiber-epoxy matrix composite in which the average fiber diameter and length are \(0.015 \mathrm{~mm}\left(5.9 \times 10^{-4}\right.\) in. \()\) and \(2.0 \mathrm{~mm}(0.08\) in. \()\), respectively, and the volume fraction of fibers is \(0.25\). Assume that (1) the fiber-matrix bond strength is \(100 \mathrm{MPa}(14,500 \mathrm{psi}),(2)\) the fracture strength of the fibers is \(3500 \mathrm{MPa}\left(5 \times 10^{5} \mathrm{psi}\right)\), and (3) the matrix stress at composite failure is \(5.5 \mathrm{MPa}(800 \mathrm{psi})\).

First, calculate the aspect ratio (\(l/d\)) of the fibers, which is the ratio of the fiber length (\(l\)) to its diameter (\(d\)). \(l/d = \cfrac{2.0 \; \text{mm}}{0.015 \; \text{mm}} = 133.33\)

Next, we need to calculate the load transfer efficiency (\(\eta_L\)) using the following formula: \(\eta_L = 1 - \cfrac{2}{\pi \times (l/d)}\) Substitute the aspect ratio from step 1: \(\eta_L = 1 - \cfrac{2}{\pi \times 133.33} = 0.9925\)

Now, compute the fiber strength efficiency (\(\eta_{\sigma}\)) by dividing the bond strength by the fracture strength of the fiber: \(\eta_{\sigma} = \cfrac{100 \; \text{MPa}}{3500 \; \text{MPa}} = 0.0286\)

Finally, we can determine the tensile strength (\(\sigma_c\)) of the composite using this formula: \(\sigma_c = V_f \times \eta_L \times \eta_{\sigma} \times \sigma_f + (1 - V_f) \times \sigma_m\) where \(V_f\) is the volume fraction of fibers, \(\sigma_f\) is the fracture strength of the fibers, and \(\sigma_m\) is the matrix stress at composite failure. Plugging in the given values for \(V_f\), \(\sigma_f\), and \(\sigma_m\) and the calculated values for \(\eta_L\) and \(\eta_{\sigma}\): \(\sigma_c = 0.25 \times 0.9925 \times 0.0286 \times 3500 \; \text{MPa} + (1 - 0.25) \times 5.5 \; \text{MPa} = 72.875 \; \text{MPa}\) The longitudinal tensile strength of the aligned glass fiber-epoxy matrix composite is approximately \(72.875 \; \text{MPa}\).