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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 \(0.60\) and \(0.40\), respectively, estimate the upper limit for the specific stiffness of this composite given the data that follow. \begin{tabular}{lcc} \hline & Specific Gravity & Modulus of Elasticity \((\boldsymbol{G P a})\) \\ \hline Copper & \(8.9\) & 110 \\ Tungsten & \(19.3\) & 407 \end{tabular}

Short Answer

Expert verified
Answer: The upper limit for the specific stiffness of the tungsten-copper composite is estimated to be 17.598 (GPa/g⋅cm⁻³).

Step by step solution

01

Identify relevant information

The given information is the volume fractions of tungsten and copper, the specific gravity of each, and the modulus of elasticity of each. We want to find the upper limit for the specific stiffness of the composite.
02

Calculate the specific stiffness of each material

Specific stiffness is the ratio of the modulus of elasticity to the specific gravity. To find the specific stiffness of copper and tungsten individually, we will calculate this ratio for each material. For Copper: Specific Stiffness (Copper) = Modulus of Elasticity (Copper) / Specific Gravity (Copper) Specific Stiffness (Copper) = \(110 \, GPa / 8.9 = 12.360 \, (GPa/g \cdot cm^{-3})\) For Tungsten: Specific Stiffness (Tungsten) = Modulus of Elasticity (Tungsten) / Specific Gravity (Tungsten) Specific Stiffness (Tungsten) = \(407 \, GPa / 19.3 = 21.090 \, (GPa/g \cdot cm^{-3})\)
03

Use Rule of Mixtures formula

Now, we will use the Rule of mixtures formula to estimate the upper limit of specific stiffness for the composite. Specific Stiffness (Composite) = (Volume Fraction of Tungsten \(\times\) Specific Stiffness (Tungsten)) + (Volume Fraction of Copper \(\times\) Specific Stiffness (Copper)) For given problem Volume fraction of Tungsten = \(0.60\) and volume fraction of Copper = \(0.40.\) Specific Stiffness (Composite) = \((0.60 \times 21.090) + (0.40 \times 12.360) = 12.654 + 4.944 = 17.598 \, (GPa/g \cdot cm^{-3})\)
04

Interpret the result

The upper limit for the specific stiffness of the tungsten-copper composite is estimated to be \(17.598 \, (GPa/g \cdot cm^{-3})\). This value represents the maximum specific stiffness that the composite material can theoretically achieve, assuming a linear relationship between the properties of the composite and its constituent materials.

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