Question

The strain energy stored in a body of volume $V$ due to shear strain $\varphi$ is (shear modulus is $G$)

• A. $\frac{{\varphi }^{2}V}{2G}$
• B. $\frac{G\varphi V^2}{2}$
• C. $\frac{\varphi V}{2G}$
• D. $\frac{1}{2}G{\varphi }^{2}V$

Answer 1

The correct option is D $\frac{1}{2}G{\varphi }^{2}V$

We know that,
$\text{Shear modulus}=\frac{\text{Shear stress}}{\text{Shear strain}}$
$G=\frac{\text{Shear stress}}{\varphi }$
$\Rightarrow G\varphi =\text{Shear stress}$
Now, Strain energy per unit volume$=\frac{1}{2}\times \text{Shear stress}\times \text{Shear strain}$
$\Rightarrow \frac{\text{Strain energy}}{\text{Volume}\left(V\right)}=\frac{1}{2}\times G\varphi \times \varphi$
$\Rightarrow$Strain energy $=\frac{1}{2}G{\varphi }^{2}V$