Constitutive equations

Piezoelectric effect depends on directions.

The reference axis, called axis 3, is taken parallel to the direction of poling. Axes 1 and 2 are defined arbitrarily in order to form a direct coordinate system with axis 3. 4, 5 and 6 represent shear movements around axes 1, 2 and 3 respectively. 

Piezoelectric coefficients

Based on this coordinate system, the piezoelectric effect can be described in a simplified way by matrix coefficients. For actuators, the coefficients “d” (coupling coefficients, 3x6 matrix) and “s^E” (stiffness coefficients, 6x6 matrix) are commonly used.

Main coefficients for our standard materials

Basic piezoelectric equations
 

These coefficients are used to relate the strain “S” (6-components vector) to the stress “T” (6-components vector) and electrical field “E” (3-components vector).

S = s^E.T + d^t.E

In this equation, the “s^E.T” term describes the mechanical compliance of the component, similarly to any mechanical component. The “d^t.E” term describes the piezoelectric effect, i.e. strain generated by electrical field.

In practice in most of the cases only one equation is used.

If you want to know more about piezoelectric equations (other relationships of interest, examples…), our piezo technology course will provide you with more information.

Limitations to the constitutive equations

The above equations are useful for designing a piezo application. However, it must be kept in mind that they represent an approximation. Piezoelectric behaviour is affected by non-linearities (hysteresis, creep, field dependency).

Further information on the non-linearities of our standard materials