The direct piezoelectric effect consists of an electric polarization in a fixed direction when the piezoelectric crystal is deformed. The polarization is proportional to the deformation and causes an electric potential difference over the crystal.
The inverse piezoelectric effect, on the other hand, constitutes the opposite of the direct effect. This means that an applied potential difference induces a deformation of the crystal.
In the above relations, the naming convention used in piezoelectricity theory is assumed, so that the structural strain is denoted by S, and the stress is denoted by
T. Thus, the naming convention differs in piezoelectricity theory compared to structural mechanics theory.
You find all the necessary material data inputs within the Piezoelectric Material feature under the Solid Mechanics interface, which are added automatically when you add a predefined Piezoelectricity multiphysics interface. Such node can be also added manually under any Solid Mechanics interface similar to all other material model features. The piezoelectric material uses the Voigt notation for the anisotropic material data, as customary in this field. More details about the data ordering can be found in
Orthotropic and Anisotropic Materials section.
where the ρV is the electric charge concentration. The electric field is computed from the electric potential
V as
In both Equation 3-65 and
Equation 3-94, the constitutive relations
Equation 3-62 are used, which makes the resulting system of equations closed. The dependent variables are the structural displacement vector
u and the electric potential
V.
Since the relations in Equation 3-66 are linear, they posses the following time-harmonic wave solutions:
where k =
kn is the wave number vector,
n is the direction vector that defines the wavefront propagation direction. The wavefront is an imaginary line connecting solid particles of the same phase. The velocity of such wavefront in the direction normal to it is given by the phase velocity
c = ω/k.
For example, the pressure wave propagating in the X-axis direction is a particular solution, for which
The shear wave propagation in the X-axis direction and with
XY-plane polarization is a solution such that
where un,j is the wave polarization vector that is the eigenvector corresponding to the eigenvalue solution
of the Christoffel’s equation.
For these cases, it is thus necessary that ηX is positive in order for the material to be thermodynamically stable with the chosen signs.
The sign of the coupling losses requires more considerations, and the chosen sign must be considered as a definition. All values of ηe and
ηd does not necessarily have to be positive. For some simple theoretical cases with isotropic loss factors, it can however be shown that the definition above is reasonable:
It is also possible to define the electrical conductivity of the piezoelectric material, σ. Electrical conductivity appears as an additional term in the variational formulation (weak equation form). The conductivity does not change during transformation between the formulations.
Using the functionality available under the Piezoelectric Material feature and Solid Mechanics interface, one can define initial stress (S0), initial strain (
ε0), and remanent electric displacement (
Dr) for models. In the constitutive relation for piezoelectric material these additions appear in the stress-charge formulation: