Piezoceramic Material Data: Subscript Ordering
engineering fundamentals Piezo: Subscript Ordering
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General Matrix Structure
The entries in the constitutive matrices for piezoelectric materials are subscript-labeled as follows:
s11 s12 s13 s14 s15 s16
s22 s23 s24 s25 s26
s33 s34 s35 s36
s44 s45 s46
s55 s56
s66
Compliance
Matrix is symmetric.
d11 d12 d13 d14 d15 d16
d21 d22 d23 d24 d25 d26
d31 d32 d33 d34 d35 d36
Piezoelectric Coupling
e11 0 0
e22 0
e33
Permittivity
Matrix is symmetric, and diagonal for most materials.
Subscript Ordering
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!

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