Piezo-electrics

The piezoelectric crystals exhibit an physical property, that they generate electricity under pressure. It means that their elastic, and electric properties are being coupled.

The coupling assumed be linear, it can be described with a 3rd rank tensor ($d$). The elements of this tensor mean represent, that for unit strain, how large additional polarization being generated, or reversed for given polarization how is going to the material being stressed.

\[ d = \begin{bmatrix} d_{11} & d_{12} & d_{13}& d_{14} & d_{15} & d_{16} \\ d_{21} & d_{22} & d_{23}& d_{24} & d_{25} & d_{26} \\ d_{31} & d_{32} & d_{33}& d_{34} & d_{35} & d_{36} \end{bmatrix} \]

The coupling equation is \[ P_{pz} = d \epsilon \] Due the symmetry of the crystal structure the piezoelectric tensor simplifies, for example for cubic crystals:

\[ d = \begin{bmatrix} 0 & 0 & 0& d_{14} &0 & 0 \\ 0 & 0 & 0& 0& d_{14} & 0 \\ 0 & 0 & 0& 0& 0 & d_{14} \end{bmatrix} \]

Examples

2D Simulations

Crystal under periodic voltage