The strain energy stored in a body of volume V due to shear S and shear modulus η is

The correct option is ηS²V/2.

Step-by-step derivation:

1. Understand the Terms:
Let $S$ represent the shear strain in the body, $\eta$ be the shear modulus (modulus of rigidity), and $V$ be the volume of the body.

2. Hooke's Law for Shear:
According to Hooke's Law, the shear modulus ($\eta$) is defined as the ratio of shear stress to shear strain:
$\eta =\frac{\text{Shear Stress}}{\text{Shear Strain}}$
Rearranging this equation, we get the expression for shear stress in terms of shear modulus and shear strain:
$\text{Shear Stress}=\eta \times S$

3. Strain Energy Density:
Strain energy stored per unit volume (strain energy density, $u$) is given by the formula:
$u=\frac{1}{2}\times \text{Shear Stress}\times \text{Shear Strain}$

4. Substitute Stress into Energy Density:
Substituting the expression for shear stress ($\eta \times S$) into the density formula:
$u=\frac{1}{2}\times (\eta \times S)\times S$
$u=\frac{\eta S^2}{2}$

5. Total Strain Energy:
The total strain energy ($U$) stored in the entire volume $V$ of the body is the product of the strain energy density and the total volume:
$U=u\times V$
$U=\frac{\eta S^{2}V}{2}$

Thus, the strain energy stored in the body is $\frac{\eta S^{2}V}{2}$, which corresponds to the option ηS²V/2.