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GATE 2024 MT – Question 43

Mechanical Behaviour of Materials · Elasticity stress and strain · 2 marks · Multiple choice

The initially stress-free isotropic metal cuboid shown has thermal expansion coefficient $\alpha$, Young’s modulus $E$, and Poisson ratio $\nu$. Its dimensions along X, Y, Z are $a,b,c$. It is rigidly constrained against X expansion but free in Y and Z. After a temperature increase $\Delta T$, the change in its Y dimension is ______. Assume linear elasticity and infinitesimal thermal/mechanical strains.

Diagram for GATE 2024 MT question 43
  1. $b(1-\nu)\alpha\Delta T$
  2. $b(1+\nu)\alpha\Delta T$
  3. $b\alpha\Delta T$
  4. $b(1+\alpha)\Delta T$

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Show answer and explanation

Correct answer: (B) $b(1+\nu)\alpha\Delta T$

Explanation

With free Y/Z faces, σy=σz=0.

The X constraint gives 0=σx/E+αΔT, so σx=−EαΔT.

Then εy=−νσx/E+αΔT=(1+ν)αΔT.

The Y dimension changes by **b(1+ν)αΔT, B**.