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Chapter 16: Problem 3

A large-particle composite consisting of tungsten particles within a copper matrix is to be prepared. If the volume fractions of tungsten and copper are 0.70 and \(0.30,\) respectively, estimate the upper limit for the specific stiffness of this composite given the data that follow. $$\begin{array}{lcc}\hline & \begin{array}{c}\text {Specific} \\\\\text {Gravity}\end{array} & \begin{array}{c}\text {Modulus of} \\\\\text {Elasticity (GPa)}\end{array} \\\\\hline \text { Copper } & 8.9 & 110 \\\\\text { Tungsten } & 19.3 &407 \\\\\hline\end{array}$$.

Short Answer

Expert verified
In step 1 we calculated the modulus of elasticity (E) of the composite as: \(E_{composite} = 0.70 \times 407 + 0.30 \times 110\) \(E_{composite} = 284.9 + 33\) \(E_{composite} = 317.9\) In step 2 we calculated the specific gravity (SG) of the composite as: \(SG_{composite} = 0.70 \times 19.3 + 0.30 \times 8.9\) \(SG_{composite} = 13.51 + 2.67\) \(SG_{composite} = 16.18\) In step 3 we calculate the specific stiffness (SS) of the composite: \(SS_{composite} = \frac{E_{composite}}{SG_{composite}}\) \(SS_{composite} = \frac{317.9}{16.18}\) \(SS_{composite} \approx 19.64\) The upper limit for the specific stiffness of the composite is approximately 19.64.

Step by step solution

01

Calculate the stiffness of the composite using the Rule of Mixtures

Using the Rule of Mixtures, the modulus of elasticity (E) of the composite can be calculated as: \(E_{composite} = V_{Tungsten} \times E_{Tungsten} + V_{Copper} \times E_{Copper}\) where \(V_{Tungsten}\) and \(V_{Copper}\) are the volume fractions of tungsten and copper, respectively, and \(E_{Tungsten}\) and \(E_{Copper}\) are their respective modulus of elasticity. Plugging in the values, we get: \(E_{composite} = 0.70 \times 407 + 0.30 \times 110\) Calculate the value of \(E_{composite}\).
02

Calculate the specific gravity of the composite

To find the specific gravity (SG) of the composite, we can use a weighted average of the specific gravities of tungsten and copper, as follows: \(SG_{composite} = V_{Tungsten} \times SG_{Tungsten} + V_{Copper} \times SG_{Copper}\) where \(SG_{Tungsten}\) and \(SG_{Copper}\) are the specific gravities of tungsten (19.3) and copper (8.9), respectively. Plugging in the values, we have: \(SG_{composite} = 0.70 \times 19.3 + 0.30 \times 8.9\) Calculate the value of \(SG_{composite}\).
03

Calculate the specific stiffness of the composite

Now that we have found the modulus of elasticity and specific gravity of the composite, we can calculate its specific stiffness (SS) as follows: \(SS_{composite} = \frac{E_{composite}}{SG_{composite}}\) Using the values obtained in steps 1 and 2, plug in and calculate the value for \(SS_{composite}\), which is the upper limit for the specific stiffness of the composite.

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Most popular questions from this chapter

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

For a continuous and oriented fiber-reinforced composite, the moduli of elasticity in the longitudinal and transverse directions are \(33.1\) and \(3.66\) GPa ( \(4.8 \times 10^{6}\) and \(\left.5.3 \times 10^{5} \mathrm{psi}\right)\), respectively. If the volume fraction of fibers is \(0.30\), determine the moduli of elasticity of fiber and matrix phases.

Briefly describe laminar composites. What is the prime reason for fabricating these materials?

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).

(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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