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Constitutive equations

Piezoelectric coefficients and directions

The polar, or 3 axis, is defined parallel to the direction of poling within the ceramic. This direction is established during manufacturing process by a high DC voltage that is applied between a pair of electroded faces to activate the material. In many cases these electrodes are also used in operation, so the field is always applied in direction 3. Directions 1 and 2 are physically equivalent so they can be defined arbitrarily, perpendicular to direction 3 and to each other. The directions termed 4, 5 and 6 correspond to tilting (shear) motions around axes 1, 2 and 3 respectively.

In shear elements, the poling electrodes are later removed and replaced by electrodes deposited on a second pair of faces. In this event, the 3 axis is not altered, but is then parallel to the electroded faces found on the finished element. Operating field is therefore applied in direction 1 (or 2). In such devices, the wanted mechanical stress or strain is shear around axis 5.

Piezoelectric materials are characterized by several coefficients. Piezoelectric coefficients with double subscripts link electrical and mechanical quantities. The first subscript gives the direction of the electric field associated with the voltage applied, or the dielectric charge produced. The second subscript gives the direction of the mechanical stress or strain.

The piezoelectric constants relating the mechanical strain produced by an applied electric field are termed the piezoelectric deformation constants, or the “d” coefficients. They are expressed in meters per volt [m/V]. Conversely, these coefficients which are also called piezoelectric charge constants may be viewed as relating the charge collected on the electrodes to the applied mechanical stress. The units can therefore also be expressed in Coulombs per Newton [C/N].

In addition, several piezoelectric material constants may be written with a “superscript” which specifies either a mechanical or an electrical boundary condition. The superscripts are T, E, D, and S, signifying: 

  • T=constant stress=mechanically free
  • E=constant field=short circuit
  • D=constant electrical displacement=open circuit
  • S=constant strain=mechanically clamped

Here are three examples of parameters used in the piezoelectric equations together with an explanation of their notation:

Basic piezoelectric equations

There are different ways of writing the fundamental equations of the piezoelectric materials, depending on which variables are of interest. The two most common forms are (the superscript t stands for matrix-transpose):

These matrix relationships are widely used for finite element modelling. For analytical approaches, in general only some of the relationships are useful so the problem can be further simplified. For example this relationship, extracted from line 3 of the first matrix equation, describes strain in direction 3 as a function of stress and field.

Just like any other elastic material, strain is proportional to the applied stress. But in addition, a piezoelectric term is present, so an "electric strain" can be superimposed to the "elastic strain".

Limitations of the linear constitutive equations

There are a number of limitations of the linear constitutive equations. The piezoelectric effect is actually non-linear in nature due to hysteresis and creep.

Furthermore, the dynamics of the material are not described by the linear constitutive equations. Piezoelectric coefficients are temperature dependant. Piezoelectric coefficients show a strong electric field dependency. 

The linear constitutive equations above are applicable for low electric field only!

These non-linearities are described in more details in Properties of piezoceramic material at high field.

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