The stress-strain diagram for a steel alloy having an original diameter of $0.5$ in. and a gage length of $2$ in. is given in the figure. Determine approximately the modulus of resilience and the modulus of toughness for the material.

Solution

VerifiedThe test element has dimensions $d = 0.5$ in and $L = 2$ in. It is necessary to determine modulus of resilience and the modulus of toughness.

From the diagram we read the necessary values for solving the task:

Determination modulus of resilience:

$\begin{align*} (u_{r})_{approx}&=\dfrac{1}{2}\cdot \sigma_{r}\cdot \varepsilon_{r}\\&=\dfrac{1}{2}\cdot40\cdot10^3\cdot0.001\\(u_{r})_{approx}&=20\dfrac{ \text{ lb in}}{\text{ in}^3} \end{align*}$

Determination modulus of toughness us:

$\begin{align*} (u_{t})_{approx}&=45\cdot10\cdot0.04\\(u_{t})_{approx}&=18\dfrac{ \text{ kip in}}{\text{ in}^3} \end{align*}$

The solution is:

{$\boxed{(u_{r})_{approx}=20\dfrac{ \text{ lb in}}{\text{ in}^3}}$}

{$\boxed{(u_{t})_{approx}=18\dfrac{ \text{ kip in}}{\text{ in}^3}}$}