In an aligned and continuous carbon fiberreinforced nylon 6,6 composite, the fibers are to carry \(97 \%\) of a load applied in the longitudinal direction. (a) Using the data provided, determine the volume fraction of fibers required. (b) What is the tensile strength of this composite? Assume that the matrix stress at fiber failure is \(50 \mathrm{MPa}\) (7250 psi).

Short Answer

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A: The relationship between the volume fraction of fibers and the stress in the fibers and matrix can be described using the rule of mixtures as follows: $$\sigma_c = \sigma_f \times V_f + \sigma_m \times (1 - V_f)$$ In this case, since the fibers are carrying 97% of the load, the relationship can be represented as: $$0.97 \sigma_c = \sigma_f \times V_f$$

Step by step solution

01

Determine the Volume Fraction of Fibers

The rule of mixtures states that for a composite material, the applied load is shared by the different components in proportion to their volume fractions. In this case, 97% of the load is to be carried by the carbon fibers, which means the volume fraction of fibers (Vf) is needed. The rule of mixtures equation is: $$\sigma_c = \sigma_f \times V_f + \sigma_m \times (1 - V_f)$$ Where \(\sigma_c\) is the stress in the composite, \(\sigma_f\) is the stress in the fibers, \(V_f\) is the volume fraction of fibers, and \(\sigma_m\) is the stress in the matrix. In this case, we want the fibers to carry 97% of the load, so the equation becomes: $$0.97 \sigma_c = \sigma_f \times V_f$$ We don't know the applied stress (\(\sigma_c\)) yet, so the equation cannot be solved directly. However, we do know that the matrix stress at fiber failure is 50 MPa, so we can use this value to determine the tensile stress at failure in the composite: $$\sigma_c = \frac{\sigma_f \times V_f}{0.97}$$ Now plug in the matrix stress at failure (50 MPa) value: $$50 \mathrm{MPa} = \frac{\sigma_f \times V_f}{0.97}$$ Rearrange for the volume fraction of fibers (\(V_f\)): $$V_f = \frac{50 \mathrm{MPa} \times 0.97}{\sigma_f}$$
02

Calculate the Tensile Strength of the Composite

To determine the tensile strength of the composite, we need to know the tensile strengths of the fibers and matrix. However, these values are not provided in the exercise. Without the provided tensile strengths of the fibers and matrix, we cannot calculate the tensile strength of the composite. We need this information to complete the problem.

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Most popular questions from this chapter

A continuous and aligned fiber-reinforced composite is to be produced consisting of 45 vol\% aramid fibers and 55 vol \(\%\) polycarbonate matrix; the mechanical characteristics of these two materials are as follows: The stress on the polycarbonate matrix when the aramid fibers fail is \(35 \mathrm{MPa}\) (5075 psi). For this composite, compute the following: (a) The longitudinal tensile strength (b) The longitudinal modulus of elasticity

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 \(5.5 \mathrm{MPa}(800 \mathrm{psi})\).

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(a) Verify that Equation 16.11, the expression for the ratio of fiber load to matrix load \(\left(F_{f} / F_{m}\right)\), is valid. (b) What is the \(F_{f} / F_{c}\) ratio in terms of \(E_{f}, E_{m}\), and \(V_{f}\) ?

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