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Chapter 16: Problem 8

16.8 A continuous and aligned fiber-reinforced composite is to be produced consisting of 30 vol \(\%\) aramid fibers and 70 vol\% of a polycarbonate matrix; mechanical characteristics of these two materials are as follows: \begin{tabular}{lcc} \hline & Modulus of Elasticity [GPa (psi)] & Tensile Strength \([\) MPa (psi)] \\\ \hline Aramid fiber & \(131\left(19 \times 10^{6}\right)\) & \(3600(520,000)\) \\ Polycarbonate & \(2.4\left(3.5 \times 10^{5}\right)\) & \(65(9425)\) \\ \hline \end{tabular} Also, the stress on the polycarbonate matrix when the aramid fibers fail is \(45 \mathrm{MPa}\) (6500 psi). For this composite, compute (a) the longitudinal tensile strength, and

Short Answer

Expert verified
Answer: To find the longitudinal tensile strength of the composite, we need to determine the critical volume fraction (V_c) and compare it with the actual volume fraction (V_f) of aramid fibers. In this case, V_f is 30%. Using the given values, we can calculate V_c as follows: $V_c = \frac{45 - 65}{3600 - 45}$ After calculating V_c, we can determine the tensile strength of the composite using either the Rule of Mixtures formula or the formula for when the fibers fail, depending on whether V_f is less than or equal to V_c.

Step by step solution

01

Identify given values

From the exercise, we have the following values: - Volume fraction of aramid fibers(\(V_f\)): 30 \(\%\) - Volume fraction of polycarbonate matrix(\(V_m\)): 70 \(\%\) - Tensile strength of aramid fibers(\(\sigma_f\)): \(3600\) MPa - Tensile strength of polycarbonate matrix (\(\sigma_m\)): \(65\) MPa - Stress on polycarbonate matrix when the aramid fibers fail (\(\sigma'_m\)): \(45\) MPa
02

Determine the critical volume fraction

In this step, we will find the critical volume fraction (V_c) at which the aramid fibers fail. This can be calculated using the following formula: $$V_c = \frac{\sigma'_m - \sigma_m}{\sigma_f - \sigma'_m}$$ Where \(\sigma'_m\) is the stress at which the aramid fibers fail, and other variables are defined as above. Using the given values: $$V_c = \frac{45 - 65}{3600 - 45}$$
03

Calculate the tensile strength of the composite

Now, we compare the actual volume fraction of aramid fibers (\(V_f\)) with the critical volume fraction (\(V_c\)) to determine the tensile strength of the composite. If \(V_f < V_c\), then the fibers do not fail, and the tensile strength is given by the Rule of Mixtures formula: $$\sigma_c = V_f\sigma_f + V_m\sigma_m$$ If \(V_f \geq V_c\), then the fibers fail, and the tensile strength of the composite is given by: $$\sigma_c = V_f\sigma_f + (1 - V_f)\sigma'_m$$ Based on the calculated critical volume fraction and the given volume fraction of aramid fibers, we can now determine the tensile strength of the composite using one of these formulas.

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Most popular questions from this chapter

16.12 In an aligned and continuous glass fiberreinforced nylon 6,6 composite, the fibers are to carry \(94 \%\) of a load applied in the longitudinal direction. (a) Using the data provided, determine the volume fraction of fibers that will be required. (b) What will be the tensile strength of this composite? Assume that the matrix stress at fiber failure is 30 MPa (4350 psi). \begin{tabular}{lcc} \hline & Modulus of Elasticity ofensile [GPa (psi)] & Strength [MPa(psi)] \\ \hline Glass fiber & \(72.5\left(10.5 \times 10^{6}\right)\) & \(3400(490,000)\) \\\ Nylon 6,6 & \(3.0\left(4.35 \times 10^{5}\right)\) & \(76(11,000)\) \\ \hline \end{tabular}

\(16.15\) 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 \(75 \mathrm{MPa}(11,000 \mathrm{psi})\)

(a) What is the distinction between cement, and concrete? (b) Cite three important limitations that restrict the use of concrete as a structural material. (c) Briefly explain three techniques that, are used to strengthen concrete by reinforcement.

\(16.18\) (a) From the moduli of elasticity data in Table \(16.2\) for glass fiber-reinforced polycarbonate composites, determine the value of the fiber efficiency parameter for each of 20,30 , and 40 vol\% fibers. (b) Estimate the modulus of elasticity for 50 vol \% glass fibers.

16.28 Briefly describe pultrusion, filament winding, and prepreg production fabrication processes; cite the advantages and disadvantages of each.
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