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.)

Correct Options: M1 and M3

Mohr's circle is a graphical representation used to determine the normal and shear stress components acting on any inclined plane within a stressed material. In the $\sigma -\tau$ coordinate system shown in the plots:
1. The horizontal axis represents the normal stress ($\sigma$).
2. The vertical axis represents the shear stress ($\tau$).

For a state of plane stress, the coordinates of the center of Mohr's circle, $O$, are defined as:
$$\left({\sigma }_{\text{c}},{\tau }_{\text{c}}\right)=\left(\frac{{\sigma }_x+{\sigma }_y}{2},0\right)$$
where ${\sigma }_x$ and ${\sigma }_y$ represent the normal stresses on two mutually perpendicular planes. Since the shear stress coordinate of the center is always zero (${\tau }_{\text{c}}=0$), the center of any valid Mohr's circle must lie on the horizontal normal stress axis ($\sigma$-axis).

Let us analyze the four plots shown in the image by observing the location of the center $O$ relative to the axes:
- Plot M1: The center $O$ is located exactly at the origin of the coordinate system (where $\sigma =0$ and $\tau =0$). Since the origin lies on the $\sigma$-axis, this is a valid representation (e.g., pure shear state where ${\sigma }_x=-{\sigma }_y$). Hence, M1 is valid.
- Plot M2: The center $O$ lies in the first quadrant, above the $\sigma$-axis, meaning it has a non-zero shear stress value (${\tau }_{\text{c}}>0$). This is physically impossible for a Mohr's circle. Hence, M2 is invalid.
- Plot M3: The center $O$ lies directly on the horizontal $\sigma$-axis to the right of the origin. This represents a valid state of plane stress where the average normal stress is positive. Hence, M3 is valid.
- Plot M4: The center $O$ lies on the vertical $\tau$-axis above the origin, meaning ${\sigma }_{\text{c}}=0$ and ${\tau }_{\text{c}}>0$. Since the center does not lie on the $\sigma$-axis, this representation is physically impossible. Hence, M4 is invalid.

Consequently, only M1 and M3 are valid Mohr’s circle representations of a plane stress state.