Chapter 15.16, Problem 1QP

To determine

To plot:

The graph of elastic modulus versus the volume percent of tungsten (WC) using upper–and lower–bound expressions.

Answer to Problem 1QP

The graph of elastic modulus versus the volume fraction of tungsten $\left(V_{WC}\right)$ using upper–and lower–bound modulus of elasticity was plotted and shown Figure 1.

Explanation of Solution

Given:

The modulus of elasticity of cobalt $\left(\text{Co}\right)$ is, $E_c=200\text{GPa}\left(30\times {10}^6\text{psi}\right)$.

The modulus of elasticity of tungsten $\left(\text{WC}\right)$ is, $E_{\text{WC}}=700\text{GPa}\left(102\times {10}^6\text{psi}\right)$.

Explanation:

Write the expression for upper–bound modulus of elasticity.

$E\left(u\right)=E_{Co}{V}_{Co}+E_{WC}{V}_{WC}$ (I)

Write the expression for lower–bound modulus of elasticity.

$E\left(l\right)=\frac{E_{Co}{E}_{WC}}{V_{Co}{E}_{WC}+V_{WC}{E}_{Co}}$ (II)

Here, $E_{Co}$ is the elastic modulus of cobalt, $V_{Co}$ is the volume fraction of cobalt, $E_{WC}$ is the elastic modulus of tungsten and $V_{WC}$ is the volume fraction of tungsten.

Conclusion:

Calculate the upper–bound modulus of elasticity.

When, the volume fraction of cobalt is, $V_{Co}=10\%$ and the volume fraction of tungsten is, $V_{Co}=90\%$.

Substitute 200 GPa for $E_{Co}$, 10% for $V_{Co}$, 700 GPa for $E_{WC}$ and 90% for $V_{WC}$ in Equation (I).

$\begin{array}{c}E\left(u\right)=\left(200\text{GPa}\right)\left(\frac{10}{100}\right)+\left(700\text{GPa}\right)\left(\frac{90}{100}\right)\\ =20+630\\ =650\text{GPa}\end{array}$

When, the volume fraction of cobalt is, $V_{Co}=90\%$ and the volume fraction of tungsten is, $V_{Co}=10\%$.

Substitute 200 GPa for $E_{Co}$, 90% for $V_{Co}$, 700 GPa for $E_{WC}$ and 10% for $V_{WC}$ in Equation (I).

$\begin{array}{c}E\left(u\right)=\left(200\text{GPa}\right)\left(\frac{90}{100}\right)+\left(700\text{GPa}\right)\left(\frac{10}{100}\right)\\ =180+70\\ =250\text{GPa}\end{array}$

Similarly, the upper–bound modulus of elasticity for different volume fractions are calculated and shown in Table 1.

$V_{Co}$	$V_{WC}$	$E\left(u\right)$
0	100	700
10	90	650
20	80	600
30	70	550
40	60	500
50	50	450
60	40	400
70	30	350
80	20	300
90	10	250
100	0	200

Table 1

Calculate the lower–bound modulus of elasticity.

When, the volume fraction of cobalt is, $V_{Co}=10\%$ and the volume fraction of tungsten is, $V_{Co}=90\%$.

Substitute 200 GPa for $E_{Co}$, 10% for $V_{Co}$, 700 GPa for $E_{WC}$ and 90% for $V_{WC}$ in Equation (II).

$\begin{array}{c}E\left(l\right)=\frac{\left(200\text{GPa}\right)\left(700\text{GPa}\right)}{\left(\frac{10}{100}\right)\left(700\text{GPa}\right)+\left(\frac{90}{100}\right)\left(200\text{GPa}\right)}\\ =\frac{140000}{70+180}\\ =560\text{GPa}\end{array}$

When, the volume fraction of cobalt is, $V_{Co}=90\%$ and the volume fraction of tungsten is, $V_{Co}=10\%$.

Substitute 200 GPa for $E_{Co}$, 90% for $V_{Co}$, 700 GPa for $E_{WC}$ and 10% for $V_{WC}$ in Equation (I).

$\begin{array}{c}E\left(l\right)=\frac{\left(200\text{GPa}\right)\left(700\text{GPa}\right)}{\left(\frac{90}{100}\right)\left(700\text{GPa}\right)+\left(\frac{10}{100}\right)\left(200\text{GPa}\right)}\\ =\frac{140000}{630+20}\\ =215.3846\text{GPa}\end{array}$

Similarly, the lower–bound modulus of elasticity for different volume fractions are calculated and shown in Table 2.

$V_{Co}$	$V_{WC}$	$E\left(l\right)$
0	100	700
10	90	560
20	80	466.67
30	70	400
40	60	350
50	50	311.11
60	40	280
70	30	254.55
80	20	233.33
90	10	215.38
100	0	200

Table 2

Refer Table 1 and Table 2.

Take the volume fraction of tungsten $\left(V_{WC}\right)$ in x–axis and take upper–bound modulus of elasticity $\left[E\left(u\right)\right]$, lower–bound modulus of elasticity $\left[E\left(l\right)\right]$ in y–axis.

Plot the graph of elastic modulus versus the volume fraction of tungsten $\left(V_{WC}\right)$ using upper–and lower–bound modulus of elasticity as shown Figure 1.

Thus, the graph of elastic modulus versus the volume fraction of tungsten $\left(V_{WC}\right)$ using upper–and lower–bound modulus of elasticity was plotted and shown Figure 1.