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GATE 2024 AE – Question 13

Structures · Strength of materials: Strain energy, Castigliano's principles, three-dimensional Hooke's law, plane stress and strain, Euler buckling · 1 mark · Multiple choice

The three-dimensional stress-strain relationship for an isotropic material is given as

$\begin{Bmatrix}\sigma_{xx}\\\sigma_{yy}\\\sigma_{zz}\\\tau_{yz}\\\tau_{xz}\\\tau_{xy}\end{Bmatrix} = \begin{bmatrix}P&Q&Q&0&0&0\\Q&P&Q&0&0&0\\Q&Q&P&0&0&0\\0&0&0&R&0&0\\0&0&0&0&R&0\\0&0&0&0&0&R\end{bmatrix}\begin{Bmatrix}\varepsilon_{xx}\\\varepsilon_{yy}\\\varepsilon_{zz}\\\gamma_{yz}\\\gamma_{xz}\\\gamma_{xy}\end{Bmatrix}$

where, P, Q and R are the three elastic constants, $\sigma$ and $\tau$ represent normal and shear stresses and $\varepsilon$ and $\gamma$ represent normal and engineering shear strains. Which one of the following options is correct?

  1. $R = \frac{P - Q}{2}$
  2. $R = \frac{Q - P}{2}$
  3. $Q = \frac{P - R}{2}$
  4. $Q = \frac{R - P}{2}$

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Correct answer: (A) $R = \frac{P - Q}{2}$

Explanation

For an isotropic material $P = \lambda + 2G$, $Q = \lambda$ and the shear term is $R = G$ (with engineering shear strain). So $P - Q = 2G = 2R$, which is $R = \frac{P - Q}{2}$.