Question

Consider a cube having elastic modulus E and restrained in all the direction, subjected to increase in temperature $T^{\circ }C$, stress developed at sides of cube will be. $(\alpha$ is the coefficient of thermal expansion and $\mu$ is Poisson's ratio).

• A. $\frac{\alpha ET}{\mu }$
• B. $\frac{\alpha ET}{1-\mu }$
• C. $\frac{\alpha ET}{1-2\mu }$
• D. $\frac{E(1-2\mu )}{\alpha T}$

Answer 1

The correct option is C $\frac{\alpha ET}{1-2\mu }$

Volumetric strain
$\frac{\Delta V}{V}=\frac{p}{K}$, where K is the bulk modulus.
$\Delta V=V^{\prime }-V$
$=(a+a\alpha \Delta T{)}^3-a^3$
$\frac{(a+a\alpha \Delta T{)}^3-a^3}{a^3}=\frac{p}{K}$
$(1+\alpha T{)}^3-1=\frac{p}{K}$
$1+3\alpha T-1=\frac{p}{K}$ (Ignoring higher power of expansion)
$\frac{p}{K}=3\alpha T$
$p=3\alpha T.K$
$=\frac{3\alpha T.E}{3(1-2\mu )}[\because K=\frac{E}{3(1-2\mu )}]$
$=\frac{\alpha TE}{1-2\mu }$