The true stress (σ) - true strain ( ε ) diagram of a strain hardening material is shown in figure. First, there is loading up to point A, i.e., up to stress of 500 MPa and strain of 0.5. Then from point A, there is unloading up to point B, i.e., to stress of 100 MPa. Given that the Young’s modulus E = 200 GPa, the natural strain at point B (ε_B ) is _________ (correct to three decimal places).

The correct answer is 0.498.

Step-by-Step Explanation:

1. Identify key values from the diagram:
Based on the provided true stress-true strain graph:

• At point A (before unloading), the true stress is ${\sigma }_A=500\text{MPa}$ and the true strain is ${\epsilon }_A=0.5$, which are marked by the dashed lines extending to the axes.
• At point B (after partial unloading), the true stress is ${\sigma }_B=100\text{MPa}$ and the true strain is denoted as ${\epsilon }_B$.
• The Young's modulus of the material is given as $E=200\text{GPa}=200\times {10}^3\text{MPa}$.

2. Understand the unloading behavior:
During the unloading phase of a strain-hardening material, the unloading path AB is elastic and is represented by a straight line. The slope of this unloading line is equal to the Young's modulus ($E$) of the material.

Therefore, we can write the relation for the slope of the line AB as:

$E=\frac{{\sigma }_A-{\sigma }_B}{{\epsilon }_A-{\epsilon }_B}$

3. Rearrange to solve for the strain at point B (${\epsilon }_B$):

${\epsilon }_A-{\epsilon }_B=\frac{{\sigma }_A-{\sigma }_B}{E}$

4. Substitute the given numerical values into the equation:

$0.5-{\epsilon }_B=\frac{500-100}{200\times {10}^3}$

$0.5-{\epsilon }_B=\frac{400}{200000}$

$0.5-{\epsilon }_B=0.002$

5. Calculate the final strain at B:

${\epsilon }_B=0.5-0.002=0.498$

Thus, the natural strain at point B (correct to three decimal places) is 0.498.