Discussion
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, σ and τ represent normal and shear stresses and ε and γ represent normal and engineering shear strains. Which one of the following options is correct?
Answer: A
The correct relation is \(R = \frac{P - Q}{2}\).
For an isotropic material, Hooke's law in Lamé form, with similar equations for the other components, is
\[ \begin{aligned} \sigma_{xx} &= (\lambda + 2\mu)\varepsilon_{xx} \\ &\quad + \lambda(\varepsilon_{yy} + \varepsilon_{zz}) \\ \tau_{yz} &= \mu\gamma_{yz} \end{aligned} \]
Comparing with the matrix, \(P = \lambda + 2\mu\), \(Q = \lambda\) and \(R = \mu\),
so \(P - Q = 2\mu = 2R\), giving \(R = (P - Q)/2\). This is the isotropy condition: of the three constants only two are independent.
\((Q - P)/2\) would make the shear modulus negative, since \(P \gt Q\) for any real material. The two options for \(Q\) rearrange the relation incorrectly; the correct rearrangement is \(Q = P - 2R\).
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