First and Second Order Equations for Piezoelectric Crystals Expressed in Tensor Form

E O L O T R O P I C substances have been used for a wide variety of elastic piezoelectric, dielectric, pyroelectric, temperature expansive, piezooptic and electro-optic effects. While most of these effects m a y be found treated in various publications 1 there does not appear to be any integrated treatment of them by the tensor method which greatly simplifies the method of writing and manipulating the relations between fundamental quantities. Other short hand methods such as the matrix method 2 can also be used for all the linear effects, but for second order effects involving tensors higher t h a n rank four, tensor methods are essential. Accordingly, it is the purpose of this paper to present such a derivation. The notation used is that agreed upon by a committee of piezoelectric experts under the auspices of the Institute of Radio Engineers. In the first part the definition of stress and strain are given and their interrelation, the generalized Hookes law is discussed. The modifications caused by adiabatic conditions are considered. When electric fields, stresses, and temperature changes are applied, there are nine first order effects each of which requires a tensor to express the resulting constants. The effects are the elastic effect, the direct and inverse piezoelectric effects, the temperature expansion effect, the dielectric effect, the pyroelectric effect, the heat of deformation, the electrocaloric effect, and the specific heat. There are three relations between these nine effects.