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3. Relationship between elastic compliance and elastic stiffness constants for cubic crystals (10 points) Given in class that, for cubic crystals, the elastic stiffness matrix is given by $\begin{pmatrix} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44} \end{pmatrix}$ Show that the corresponding elements of the elastic compliance matrix, $\begin{pmatrix} S_{11} & S_{12} & S_{12} & 0 & 0 & 0 \\ S_{12} & S_{11} & S_{12} & 0 & 0 & 0 \\ S_{12} & S_{12} & S_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{44} \end{pmatrix}$ are given by $S_{11} = \frac{C_{11} + C_{12}}{(C_{11} - C_{12})(C_{11} + 2C_{12})}$ $S_{12} = \frac{-C_{12}}{(C_{11} - C_{12})(C_{11} + 2C_{12})}$ $S_{44} = \frac{1}{C_{44}}$

          3.
Relationship between elastic compliance and elastic stiffness constants for
cubic crystals (10 points)
Given in class that, for cubic crystals, the elastic stiffness matrix is given by
$\begin{pmatrix} C_{11} & C_{12} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{12} & 0 & 0 & 0 \\ C_{12} & C_{12} & C_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{44} \end{pmatrix}$
Show that the corresponding elements of the elastic compliance matrix,
$\begin{pmatrix} S_{11} & S_{12} & S_{12} & 0 & 0 & 0 \\ S_{12} & S_{11} & S_{12} & 0 & 0 & 0 \\ S_{12} & S_{12} & S_{11} & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{44} \end{pmatrix}$
are given by
$S_{11} = \frac{C_{11} + C_{12}}{(C_{11} - C_{12})(C_{11} + 2C_{12})}$
$S_{12} = \frac{-C_{12}}{(C_{11} - C_{12})(C_{11} + 2C_{12})}$
$S_{44} = \frac{1}{C_{44}}$

3.
Relationship between elastic compliance and elastic stiffness constants for
cubic crystals (10 points)
Given in class that, for cubic crystals, the elastic stiffness matrix is given by
< p m a t r i x >
Show that the corresponding elements of the elastic compliance matrix,
< p m a t r i x >
are given by
S11 = (C11 + C12)/((C11 - C12)(C11 + 2C12))
S12 = (-C12)/((C11 - C12)(C11 + 2C12))
S44 = (1)/(C44)

Added by Matthew A.

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