The von Mises stress at a point in a body subjected to forces is proportional to the square root of the

The correct option is distortional strain energy per unit volume.

To understand why this is correct, let us break down the concept of strain energy and how it relates to the von Mises yield criterion.

When a material is subjected to a general state of stress, the total strain energy per unit volume ($U_{\mathrm{total}}$) stored in the material can be divided into two distinct parts:
1. Dilatational strain energy ($U_v$): The energy associated with a change in volume (hydrostatic or spherical state of stress) without changing the shape.
2. Distortional strain energy ($U_d$): The energy associated with a change in shape (deviatoric state of stress) without changing the volume.

Mathematically, the total strain energy per unit volume is expressed as:
$U_{\mathrm{total}}=U_v+U_d$

According to the Distortion Energy Theory (also known as the von Mises yield criterion), yielding in a ductile material begins when the distortional strain energy per unit volume ($U_d$) under a complex state of stress reaches the distortional strain energy per unit volume at the yield point in a simple tension test.

For a three-dimensional principal stress state (${\sigma }_1$, ${\sigma }_2$, ${\sigma }_3$), the distortional strain energy per unit volume is given by the formula:
$U_d=\frac{1+\nu }{6E}[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2]$
where $E$ is the Young's modulus and $\nu$ is the Poisson's ratio.

The von Mises stress (${\sigma }_v$) is defined as:
${\sigma }_v=\sqrt{\frac{1}{2}[{({\sigma }_1-{\sigma }_2)}^2+{({\sigma }_2-{\sigma }_3)}^2+{({\sigma }_3-{\sigma }_1)}^2]}$

By comparing the equations for distortional strain energy ($U_d$) and von Mises stress (${\sigma }_v$), we can see that:
$U_d=\frac{1+\nu }{3E}{\sigma }_v^2$

Rearranging this equation to solve for the von Mises stress (${\sigma }_v$) gives:
${\sigma }_v=\sqrt{\frac{3E}{1+\nu }{U}_d}$

Since $E$ and $\nu$ are constant material properties, we can write:
${\sigma }_v\propto \sqrt{U_d}$

This directly proves that the von Mises stress is proportional to the square root of the distortional strain energy per unit volume.