Piezoelectrics

Certain materials are atomically structured such that they can deform and polarize as they do so. This is a useful property that allows us to convert energy between mechanical and electrical domains. Examples of such materials include $SiO_{2}$ (Quartz) and $BaTiO_{3}$ (Ceramic).

A Compression force on Quartz Generating a a Charge Dipole

When dealing with piezoelectric materials we are usually concerned about the relative amount, direction, and type of force that corresponds to a particular polarization vector. The relation between tense forces ( $T_1, T_2, T_3$ ) and shear forces ( $T_4, T_5, T_6$ ) are represented in a piezoelectric coefficient matrix ( $D$ ) with a polarization output vector shown below:

$\begin{bmatrix} P_1\\P_2\\P_3 \end{bmatrix} = \begin{bmatrix} d_{11}&d_{12}&d_{13}&d_{14}&d_{15}&d_{16} \\ d_{21}&d_{22}&d_{23}&d_{24}&d_{25}&d_{26} \\ d_{31}&d_{32}&d_{33}&d_{34}&d_{35}&d_{36} \\ \end{bmatrix} \begin{bmatrix} T_1\\T_2\\T_3\\T_4\\T_5\\T_6 \end{bmatrix}$

This equation is, of course reversible as in $T = D^T*P$.