Transverse isotropy

Remarks

Example: with axis $e = e_3$ and Voigt notation, the stiffness matrix of a transversely isotropic material has the form $$\begin{pmatrix} C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\ C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\ C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\ 0 & 0 & 0 & C_{44} & 0 & 0 \\ 0 & 0 & 0 & 0 & C_{44} & 0 \\ 0 & 0 & 0 & 0 & 0 & C_{66} \end{pmatrix}, \qquad C_{66} = \tfrac{1}{2}(C_{11} - C_{12}),$$ with the $5$ independent constants $C_{11}, C_{12}, C_{13}, C_{33}, C_{44}$. Non-example: a cubic material has $C_{11} = C_{33}$, $C_{12} = C_{13}$ and $C_{66} = C_{44}$. If $C_{44} \ne \tfrac{1}{2}(C_{11} - C_{12})$, the rotation by $\pi/4$ about $e_3$ changes $C$. So such a cubic material is not transversely isotropic. An isotropic material is transversely isotropic for every axis; many authors exclude this case. The plane normal to $e$ is the plane of isotropy.