Chapter 17, Problem 27P

(a)

To determine

Write the values of modulus of resilience and modulus of toughness for titanium.

The range of modulus of resilience for titanium is $\underset{\_}{2.888\text{MPa}\text{to}4.167\text{MPa}}$.

The range of modulus of toughness for titanium is $\underset{\_}{190.9\text{MPa}\text{to}253\text{MPa}}$.

Explanation of Solution

Refer Table 17.4 “The strength of Selected Materials” in the textbook.

The range for Yield strength of titanium alloys $\left({\sigma }_Y\right)$ is $760\text{MPa}-1000\text{MPa}$.

The range for Ultimate strength of titanium alloy $\left({\sigma }_U\right)$ is $900\text{MPa}-1200\text{MPa}$.

Refer Table 17.2 “Modulus of Elasticity and Shear Modulus of Selected Materials” in the textbook.

The modulus of elasticity of titanium alloy is $E=100\text{GPa}-120\text{GPa}$.

The elongation of titanium alloy is 23%.

Find the range of modulus of resilience $\left(U_r\right)$ using the relation.

$U_r=\frac{{\sigma }_Y^2}{2E}$ (1)

Substitute the minimum and maximum values;

$\begin{array}{c}{U}_r=\frac{{760}^2}{2\times 100\text{GPa}\times \frac{{\text{10}}^{\text{3}}\text{MPa}}{\text{1}\text{GPa}}}\text{to}\frac{{1000}^2}{2\times 120\text{GPa}\times \frac{{\text{10}}^{\text{3}}\text{MPa}}{\text{1}\text{GPa}}}\\ =2.888\text{MPa}\text{to}4.167\text{MPa}\end{array}$

Therefore, the range of modulus of resilience is $\underset{\_}{2.888\text{MPa}\text{to}4.167\text{MPa}}$.

Find the range of modulus of toughness $\left(U_t\right)$ using the relation.

$U_t=\left(\frac{{\sigma }_Y+{\sigma }_U}{2}\right)\times {\epsilon }_f$ (2)

Substitute minimum and maximum values;

$\begin{array}{c}{U}_t=\left(\frac{760+900}{2}\right)\times 0.23\text{to}\left(\frac{1000+1200}{2}\right)\times 0.23\\ =190.9\text{MPa}\text{to}253\text{MPa}\end{array}$

Therefore, the range of modulus of toughness is $\underset{\_}{190.9\text{MPa}\text{to}253\text{MPa}}$.

(b)

To determine

Write the values of modulus of resilience and modulus of toughness for steel.

The range of modulus of resilience for steel is $\underset{\_}{0.304\text{MPa}\text{to}2.381\text{MPa}}$.

The range of modulus of toughness for steel is $\underset{\_}{66.75\text{MPa}\text{to}165\text{MPa}}$.

Explanation of Solution

Refer Table 17.4 “The strength of Selected Materials” in the textbook.

The range for Yield strength of steel $\left({\sigma }_Y\right)$ is $340\text{MPa}-1000\text{MPa}$.

The range for Ultimate strength of steel $\left({\sigma }_U\right)$ is $550\text{MPa}-1200\text{MPa}$.

Refer Table 17.2 “Modulus of Elasticity and Shear Modulus of Selected Materials” in the textbook.

The range for modulus of elasticity of steel (E) is $190\text{GPa}-210\text{GPa}$

The elongation of steel is 15%.

Modulus of resilience:

Substitute the minimum and maximum values in Equation (1).

$\begin{array}{c}{U}_r=\frac{{340}^2}{2\times 190\text{GPa}\times \frac{{\text{10}}^{\text{3}}\text{MPa}}{\text{1}\text{GPa}}}\text{to}\frac{{1000}^2}{2\times 210\text{GPa}\times \frac{{\text{10}}^{\text{3}}\text{MPa}}{\text{1}\text{GPa}}}\\ =0.304\text{MPa}\text{to}2.381\text{MPa}\end{array}$

Therefore, the range of modulus of resilience is $\underset{\_}{0.304\text{MPa}\text{to}2.381\text{MPa}}$.

Modulus of toughness;

Substitute minimum and maximum values in Equation (2);

$\begin{array}{c}{U}_t=\left(\frac{340+550}{2}\right)\times 0.15\text{to}\left(\frac{1000+1200}{2}\right)\times 0.15\\ =66.75\text{MPa}\text{to}165\text{MPa}\end{array}$

Therefore, the range of modulus of toughness is $\underset{\_}{66.75\text{MPa}\text{to}165\text{MPa}}$.