Piezoelectric Material
The Piezoelectric Effect
The piezoelectric effect manifests itself as a transfer of electric to mechanical energy and vice versa. It is present in many crystalline materials, while some materials such as quartz, Rochelle salt, and lead titanate zirconate ceramics display the phenomenon strongly enough for it to be of practical use.
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.
Piezoelectric Constitutive Relations
It is possible to express the relation between the stress, strain, electric field, and electric displacement field in either a stress-charge form or strain-charge form:
Stress-Charge
Strain-Charge
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.
The Piezoelectric Material uses the structural mechanics nomenclature. The strain is named ε (instead of S) and the stresses are denoted by either σ or S (instead of T). This makes the names consistent with those used in the other structural mechanics interfaces.
The constitutive relation using COMSOL Multiphysics symbols for the different constitutive forms are thus:
Stress-Charge
(3-50)
The Stress-Charge form is always used in the variational formulation (weak equation form) which COMSOL Multiphysics uses for discretization and computation.
Strain-Charge
(3-51)
Most material data appears in the strain-charge form, and it can be easily transformed into the stress-charge form. In COMSOL Multiphysics both constitutive forms can be used; simply select one, and the software makes any necessary transformations. The following equations transform strain-charge material data to stress-charge data:
(3-52)
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 Piezoelectric Devices 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.
Governing Equations
The equations of Piezoelectricity combine the momentum equation Equation 3-78 with the charge conservation equation of Electrostatics,
(3-53)
where the ρV is the electric charge concentration. The electric field is computed from the electric potential V as
In both Equation 3-53 and Equation 3-78, the constitutive relations Equation 3-50 are used, which makes the resulting system of equations closed. The dependent variables are the structural displacement vector u and the electric potential V.
Wave Propagation
In case of geometric linearity, the governing equations are linear and have the following time-harmonic wave solutions:
where k is the wave number vector that determines the direction of the wave propagation, and c is the phase velocity (or wave speed).
The expressions for the wave speed can be computed analytically for waves of different types, polarizations and directions of propagations. For example, the pressure wave propagating in the X-axis direction is a particular solution, for which
The corresponding pressure wave speed is given by
The shear wave propagation in the X-axis direction and with XY-plane polarization is a solution such that
and the corresponding wave speed is computed as
COMSOL Multiphysics provides predefined variables for the waves speeds for waves of different types and polarizations propagating in the X, Y, and Z directions.
Piezoelectric Dissipation
In order to define dissipation in the piezoelectric material for a time-harmonic analysis, all material properties in the constitutive relations can be complex-valued matrices where the imaginary part defines the dissipative function of the material.
Complex-valued data can be defined directly in the fields for the material properties, or a real-valued material X and a set of loss factors ηX can be defined, which together form the complex-valued material data
The sign in the complex damping terms are defined as
In the four first expressions, the choice of sign is a result of the single physics observations that
•
Strain lags behind stress.
•
Electric displacement (or polarization) lags behind the electric field.
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:
•
Consider a material which in strain-charge form only has mechanical damping. Thus, d is real, and the second transformation law in Equation 3-52shows that e must have a positive loss factor.
•
Consider a material with only coupling loss. Then the second transformation law in Equation 3-52 shows that d and e must have the same loss factor multiplier. From the third transformation law it can be inferred that ηe and ηd in this case must be positive in order for the material to have an appropriate phase lag between D and E.
In a real material, all types of losses occur simultaneously. Then, the criteria for allowable values of the loss factors become complicated, particularly if they are not isotropic.
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.
The energy dissipation modeling is also available in time domain. The options are: dielectric dispersion for the electrical part, and Rayleigh damping for the mechanical and coupling parts of the problem. The total dissipated energy can be computed as a function of time.
Piezoelectric Losses
Initial Stress, Strain, and Electric Displacement
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:
When solving the model, the program does not interpret these fields as a constant initial state, but they operate as additional fields that are continuously evaluated. Thus use these initial field to add, for example, thermal expansion or pyroelectric effects to models.