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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

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) Write an expression for the modulus of elasticity for a hybrid composite in which all fibers of both types are oriented in the same direction. (b) Using this expression, compute the longitudinal modulus of elasticity of a hybrid composite consisting of aramid and glass fibers in volume fractions of \(0.25\) and \(0.35\), respectively, within a polyester resin matrix \(\left[E_{m}=4.0\right.\) GPa \(\left.\left(6 \times 10^{5} \mathrm{psi}\right)\right]\)

(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.

Cite one similarity and two differences between precipitation hardening and dispersion strengthening.

For a polymer-matrix fiber-reinforced composite: (a) List three functions of the matrix phase. (b) Compare the desired mechanical characterisics of matrix and fiber phases. (c) Cite two reasons why there must be a strong bond between fiber and matrix at their interface.
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