| General Matrix Structure |
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| The entries in the constitutive matrices for piezoelectric materials are subscript-labeled as follows: |
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Compliance
Matrix is symmetric. |
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Piezoelectric Coupling | ||||||||||||||||||||||||||||||||||
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Permittivity
Matrix is symmetric, and diagonal for most materials. |
| Subscript Ordering |
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Constitutive data is presented in a 3-axis cartesian coordinate system, denoted by {x, y, z}.
The ordering of the 6 stress (and 6 strain) variables follows the convention used by crystallographers. This ordering determines the layout of the compliance matrix s, and determines the column ordering in the coupling matrix d. The first 3 entries are the direct stresses along the x, y, and z axes, respectively. The final 3 entries are the shear stresses around the x, y, and z axes, respectively. The electric field (and charge displacement) variable ordering is straightforward. The 3 entries correspond to the electric field component along the x, y, and z axes, respectively. This ordering determines the layout of the permittivity matrix e, and determines the row ordering in the coupling matrix d.
For polarized piezoelectric materials (e.g. PZT), the poling direction is assumed to lie along the z axis.
Caution! Some finite element codes use a different ordering scheme for their stress and strain variables. Such a scheme would require the data in the s and d matrices to be re-arranged accordingly. Watch out for this! |
