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Chapter 16: Problem 15

\(16.15\) Compute the longitudinal strength of an aligned carbon fiber-epoxy matrix composite having a \(0.25\) volume fraction of fibers, assuming the following: (1) an average fiber diameter of \(10 \times 10^{-3} \mathrm{~mm}\left(3.94 \times 10^{-4}\right.\) in. \()\), (2) an average fiber length of \(5 \mathrm{~mm}(0.20\) in.), (3) a fiber fracture strength of \(2.5 \mathrm{GPa}\) \(\left(3.625 \times 10^{5} \mathrm{psi}\right),(4)\) a fiber- matrix bond strength of \(80 \mathrm{MPa}(11,600 \mathrm{psi}),(5)\) a matrix stress at fiber failure of \(10.0 \mathrm{MPa}(1450\) psi), and (6) a matrix tensile strength of \(75 \mathrm{MPa}(11,000 \mathrm{psi})\)

Short Answer

Expert verified
In summary, the longitudinal strength of the aligned carbon fiber-epoxy matrix composite can be calculated using the rule of mixtures. Given the fiber volume fraction of 0.25, fiber fracture strength of 2.5 GPa, and matrix stress at fiber failure of 10.0 MPa, the composite strength is determined to be 632.5 MPa.

Step by step solution

01

Determine the volume fraction of the matrix

To determine the volume fraction of the matrix, we need to calculate: $$V_m = 1 - V_f$$ since the fractions of the fibers and matrix sum up to 1. Thus, $$V_m = 1 - 0.25 = 0.75$$
02

Determine the strength of the fibers and matrix

To compute the composite strength, we need the strength of the fibers (\(\sigma_f\)) and that of the matrix (\(\sigma_m\)). We are given the following information: - Fiber fracture strength: \(2.5 \mathrm{GPa}\) (\(3.625 \times 10^{5} \mathrm{psi}\)) - Matrix stress at fiber failure: \(10.0 \mathrm{MPa}\) (\(1450\) psi) We will use these values as the strengths of the fibers and the matrix, respectively.
03

Apply the rule of mixtures

Using the values determined in the previous steps, we can now apply the rule of mixtures to compute the longitudinal strength of the composite: $$\sigma_c = \sigma_mV_m + \sigma_fV_f$$ $$\sigma_c = (10.0 \mathrm{MPa})(0.75) + (2.5 \mathrm{GPa})(0.25)$$ Calculating the result: $$\sigma_c = (7.5 \mathrm{MPa}) + (625 \mathrm{MPa})$$ $$\sigma_c = 632.5 \mathrm{MPa}$$ Therefore, the longitudinal strength of the aligned carbon fiber-epoxy matrix composite is \(632.5 \mathrm{MPa}\).

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