Question

The stress-strain diagram for a steel alloy having an original diameter of 0.50.5 in. and a gage length of 22 in. is given in the figure. Determine approximately the modulus of resilience and the modulus of toughness for the material.

Solution

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The test element has dimensions d=0.5d = 0.5 in and L=2L = 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:

(ur)approx=12σrεr=12401030.001(ur)approx=20 lb in in3\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:

(ut)approx=45100.04(ut)approx=18 kip in in3\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:

{(ur)approx=20 lb in in3\boxed{(u_{r})_{approx}=20\dfrac{ \text{ lb in}}{\text{ in}^3}}}

{(ut)approx=18 kip in in3\boxed{(u_{t})_{approx}=18\dfrac{ \text{ kip in}}{\text{ in}^3}}}

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