Which of the plot(s) shown is/are valid Mohr’s circle representations of a plane stress state in a material? (The center of each circle is indicated by O.)

The correct options are M1 and M3.

To determine which of the given plots represent valid Mohr's circles for a plane stress state in a material, we need to analyze the fundamental property of Mohr's circle.

In Mohr's circle for plane stress:
1. The horizontal axis represents the normal stress ($\sigma$).
2. The vertical axis represents the shear stress ($\tau$).

For any general state of plane stress characterized by normal stresses ${\sigma }_x$ and ${\sigma }_y$ and shear stress ${\tau }_{xy}$, the center of the Mohr's circle, $O$, has coordinates given by:

$\left({\sigma }_{\text{avg}},0\right)=\left(\frac{{\sigma }_x+{\sigma }_y}{2},0\right)$

Because the shear stress coordinate of the center is always zero (${\tau }_{\text{center}}=0$), the center $O$ of Mohr's circle must always lie on the horizontal normal stress ($\sigma$) axis. It can never be positioned above or below this axis.

Now, let's analyze each of the plots shown in the image:

1. Plot M1:
The center $O$ lies exactly at the origin, which is on the horizontal $\sigma$-axis. This represents a state of pure shear (where ${\sigma }_x=-{\sigma }_y$). Since the center is on the $\sigma$-axis, M1 is a valid Mohr's circle.

2. Plot M2:
The center $O$ is located in the first quadrant, above the horizontal axis where $\tau >0$. Since the center does not lie on the $\sigma$-axis, M2 is invalid.

3. Plot M3:
The center $O$ lies on the horizontal $\sigma$-axis (to the right of the origin, with the circle tangent to the vertical $\tau$-axis). Since the center is on the $\sigma$-axis, M3 is a valid Mohr's circle.

4. Plot M4:
The center $O$ lies on the vertical $\tau$-axis, above the horizontal axis ($\tau >0$). Since the center does not lie on the $\sigma$-axis, M4 is invalid.

Thus, the valid Mohr's circle representations are indeed M1 and M3.