It is desired to produce an aligned carbon fiber-epoxy matrix composite having a longitudinal tensile strength of \(500 \mathrm{MPa}(72,500 \mathrm{psi})\). Calculate the volume fraction of fibers necessary if (1) the average fiber diameter and length are \(0.01 \mathrm{~mm}\left(3.9 \times 10^{-4}\right.\) in.) and \(0.5 \mathrm{~mm}\left(2 \times 10^{-2}\right.\) in.), respectively; (2) the fiber fracture strength is \(4.0\) GPa \(\left(5.8 \times 10^{5} \mathrm{psi}\right)\); (3) the fiber-matrix bond strength is \(25 \mathrm{MPa}\) (3625 psi); and (4) the matrix stress at composite failure is \(7.0 \mathrm{MPa}\) (1000 psi).

The Rule of Mixtures is a weighted mean formula used to estimate the properties of composite materials, given the properties of their individual components. In this case, we're trying to find the contribution of fibers to the overall composite's longitudinal tensile strength. The simplified Rule of Mixtures formula for tensile strength is: \(\sigma_{c} = V_{f} \sigma_{f} + V_{m} \sigma_{m}\) where \(\sigma_{c}\) is the composite tensile strength (500 MPa), \(\sigma_{f}\) is the fiber fracture strength (4.0 GPa), \(\sigma_{m}\) is the matrix stress at composite failure (7.0 MPa), \(V_{f}\) is the volume fraction of fibers, and \(V_{m}\) is the volume fraction of the matrix. Since we have a two-phase material (fibers and matrix), we know that their volume fractions must add up to 1: \(V_{f} + V_{m} = 1\). We can rearrange this equation to find \(V_{m}\) in terms of \(V_{f}\): \(V_{m} = 1 - V_{f}\).

Now we can substitute the given values for tensile strengths and the relationship between volume fractions into the Rule of Mixtures formula: \(500 \mathrm{MPa} = V_{f} \times 4.0 \mathrm{GPa} + (1 - V_{f}) \times 7.0 \mathrm{MPa}\)

To find the value of \(V_f\), we will first convert the units of all the terms into the same units (MPa). \(500 = V_{f} \times 4000 + (1 - V_{f}) \times 7\) Now, we can solve for \(V_{f}\): \(500 = 4000V_{f} + 7 - 7V_{f}\) Combine the terms with \(V_{f}\): \(493 = 3993V_{f}\) To find the value of \(V_{f}\), divide both sides of the equation by 3993: \(V_{f} = \frac{493}{3993} \approx 0.1235\) So, the volume fraction of fibers necessary to achieve the desired longitudinal tensile strength of 500 MPa is approximately 0.1235 or 12.35%.