Question

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

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

Answer 1

Answer: (D)

Shear modulus $=\frac{\text{ Shear stress }}{\text{ Shear stress }}$
$\eta =\frac{\text{ Shear stress }}{\varphi }$
$\eta \varphi =$ Shear stress
Strain energy per unit volume $=\frac{1}{2}\times$ shear stress $x$ shear strain
$\Rightarrow \frac{\text{ Strain energy }}{\text{ Volume }}=\frac{1}{2}\times \eta \dot{\eta }\times \varphi$ (Cross multiply volume)
Strain energy $=\frac{1}{2}\eta {\varphi }^{2}V$