Question

A cube having each side of length ‘a’ is constrained in all directions and is heated uniformly so that the temperature is raised to T°C. If α is the thermal coefficient of expansion of the cube material and E is the modulus of elasticity, the stress developed in the cube is
1. \(\frac{{\alpha TE}}{v}\)
2. \(\frac{{\alpha TE}}{{\left( {1 - 2v} \right)}}\)
3. \(\frac{{\alpha TE}}{{2v}}\)
4. \(\frac{{\alpha TE}}{{\left( {1 + 2v} \right)}}\)

Answer 1

Correct Answer - Option 2 : \(\frac{{\alpha TE}}{{\left( {1 - 2v} \right)}}\)

For hydrostatic state stress,

σ_x = σ_y = σ_z = σ

Strain restrained, \(\epsilon_x = \alpha T\).............(1)

Also, \(\epsilon_x = \frac{{\sigma_x - \mu\sigma_y - \mu\sigma_z}}E\) .............(2)

Combining above two equations, we get 

\(\frac{ {\sigma({1 - 2\mu}})}E=\alpha T\)

\(\sigma = \frac{E\alpha T}{(1-2\mu)}\)