# How do polarizations work?

## Description

The properties of a piezoelectric material are defined with respect to a coordinate system that we will name M1, M2, M3. This coordinate system represents the "natural" directions of the material. The properties you define (s11e, s12e, d33, etc.) are based on this coordinate system.

An ATILA finite element model is based on a global coordinate system X, Y, Z, upon which all the elements of the model are based. Because the piezoelectric materials can take different orientations in space, it is necessary to define this orientation. The "Polarization" is a special boundary condition that is used to define the orientation of piezoelectric materials in space, i.e. to define the orientation of M1, M2, M3 with respect to X, Y, Z.

To facilitate the definition of the polarization orientation, local coordinate systems (local axes) are typically used (x, y, z). Each polarization condition can be defined with respect to one local coordinate system. Therefore, this defines the orientation of M1, M2, M3 with respect to x, y, z.

There are three types of polarization conditions: Cartesian, Cylindrical and Spherical. The first one, Cartesian, is used in materials that are uniformly polarized. The other two, Cylindrical and Spherical, are reserved for materials that have a radial polarization, either with respect to an axis or a point.

## Cartesian Polarization

Cartesian polarizations apply to materials that have a uniform polarization, such as a thickness polarization in a plate. In such cases, the material directions M1, M2, M3 directly match the local axes x, y, z.

## Cylindrical Polarization

This type of condition applies to materials that are polarized radially with respect to an axis. Piezoelectric tubes are a very common example. Here, the direction z of the local coordinate system defines the axis of the cylinder, as well as the direction M2 of the material. The direction M3 of the material is radial with respect to the cylinder. Finally, M2 is in tangential to the cylinder.

## Spherical Polarization

Not tested

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