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Piezoelectric PDE

The mechanical and electrical partial differential equation are directly coupled (volume-coupling) via the material law.

Coupling via linear constitutive law

The consitutive relations can be written in serveral, related forms (both formulations are given here in Voigt-notation). In the d-form, \begin{equation} \mathbf{s}=\mathbf{S}\mathbf{\sigma} + \mathbf{d} \mathbf{E} \mathrm{,} \end{equation} \begin{equation} \mathbf{D}=\mathbf{d} \mathbf{\sigma} + \mathbf{\epsilon^{\sigma}} \mathbf{E} \mathrm{,} \end{equation} and in the e-from, \begin{equation} \label{MLM} \mathbf{\sigma}=\mathbf{C} \mathbf{s} - \mathbf{e} \mathbf{E} \mathrm{,} \end{equation} \begin{equation} \label{MLE} \mathbf{D}=\mathbf{e} \mathbf{s} + \mathbf{\epsilon^{s}} \mathbf{E} \mathrm{.} \end{equation} The strain is denoted as \mathbf{s}, while \mathbf{\sigma} is the stress tensor (both in Voigt-notation). The electric field is \mathbf{E} and the electric flux density is \mathbf{D}.

In the finite-element-methode the \mathbf{e}-formulation is used. The piezoelectric coupling tensor \mathbf{e} looks usually as follows,

\begin{equation} \mathbf{e}= \left( \begin{array}{rrrrrr} 0 & 0 & 0 & 0 & e_{\mathrm{15}} & 0 \\ 0 & 0 & 0 & e_{\mathrm{15}} & 0 & 0 \\ e_{\mathrm{31}} & e_{\mathrm{31}} & e_{\mathrm{33}} &0 &0 &0\\ \end{array} \right) \mathrm{.} \end{equation}

Both formulations are related, the stiffness tensor \mathbf{C} (for piezoceramics usually transversally isotropic) is the inverse of the compliance tensor \mathbf{S}, \begin{equation} \mathbf{C} = \mathbf{S}^{-1} \mathbf{,} \end{equation} and the relation between the \mathbf{d}-tensor and \mathbf{e}-tensor is given by, \begin{equation} \mathbf{d}=\mathbf{e} \mathbf{C}^{-1} \mathrm{.} \end{equation} The permittivity tensor at constant strain \mathbf{\epsilon}^\sigma can be calculated out of the permittivity tensor at constant strain \mathbf{\epsilon}^s, \begin{equation} \mathbf{\epsilon}^\sigma= \mathbf{\epsilon}^s + \mathbf{e} \mathbf{C}^{-1} \mathbf{e}^{\mathrm{T}} \mathrm{.} \end{equation}

Governing equations

If we insert eq. \ref{MLM} into the mechanical PDE and eq. \ref{MLE} into the electrical PDE, while introducing the differential operator \mathcal{B},

\begin{equation} \mathcal{B}= \left( \begin{array}{rrrrrr} \frac{\partial}{\partial x} & 0 & 0 & 0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial y} \\ 0 & \frac{\partial}{\partial y} & 0 & \frac{\partial}{\partial z} & 0 & \frac{\partial}{\partial x} \\ 0 & 0 & \frac{\partial}{\partial z} & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 \\ \end{array} \right) \mathrm{,} \end{equation}

using the linearised strain-displacement relation, \begin{equation} \mathbf{S}=\mathcal{B} \mathbf{u} \mathrm{,} \end{equation} and the definition of the electric potential \begin{equation} \mathbf{E} = - \nabla \phi \mathrm{,} \end{equation} we arrive at the coupled mechanical (\ref{CPDEM}) and electrical (\ref{CPDEE}) PDE in their strong formulation,

\begin{equation} \label{CPDEM} \rho \mathbf{\ddot{u}} - \mathcal{B}^T \left( \mathbf{C} \mathcal{B} \mathbf{u} - \mathbf{e}^T \nabla \phi \right) = \mathbf{\displaystyle{f}}_{\mathrm{V}} \mathrm{,} \end{equation}
\begin{equation} \label{CPDEE} \nabla \cdot \left( \mathbf{e} \mathcal{B} \mathbf{u} - \mathbf{\epsilon} \nabla \phi \right)= q_{\mathrm{e}} \mathrm{.} \end{equation}

Analysis Types

There are different analysis types availible for piezoelectric simulations:

Static analysis: \frac{\partial}{\partial t} = 0


Transient analysis: \frac{\partial}{\partial t} \neq 0


**Harmonic analysis: ** \frac{\partial}{\partial t}(\cdot) = j\omega (\cdot)

                <freq value="100"/>
                <freq value="200"/>
                <freq value="300"/>

Eigenfrequenzy analysis: \frac{\partial}{\partial t}(\cdot) = j\omega (\cdot) and RHS =0

            <writeModes normalization="max">yes</writeModes>
            <!-- scale them to a maximum displacement of 1 -->

PDE types

Within the command we define the induvidual PDEs with their boundary conditions, etc. ... ...

Defining the coupling

The mechanical and electrostatic PDE are coupled directly over the volume. This is done (in the XML-file) after the pdeList

                <region name="V"/>

Boundary conditions and Loads

For the piezoelctric coupled system the boundary conditions from the mechanics and the electrostatics PDE can be applied. The boundary conditions and loads can be defined for each PDE.


Additionally to the mechanical and electrical properties, the couple tensor in e-form must be given.

      <tensor dim1="3" dim2="6">
      0.0000e-003  0.0000e-003  0.0000e-003  0.0000e-003 12.7174e+000 0.0000e-003
      0.0000e-003  0.0000e-003  0.0000e-003 12.7174e+000  0.0000e-003 0.0000e-003
     -5.1714e+000 -5.1714e+000 15.1005e+000  0.0000e-003  0.0000e-003 0.0000e-003

An example material file is availible under tutorials for Piezoelectric Unit-Cube.

The polarization direction inside the material can be rotated in respect to the global coordinate system (default is \alpha=\beta=\gamma=0).

<domain geometryType="3d">
        <region name="V" material="PZT-4">
            <matRotation alpha="0.0" beta="0" gamma="0"/>

Postprocessing results

The typical postprocessing results for the mechanical and electrical PDE can be defined in each PDE.

Modelling assumptions

There a certain modelling assumptions made:

  • Small strains
  • Linear piezoelectricity
  • Perfect, homogeneous polarization
  • Zero resistance assumed in electrodes
  • Electrodes are often assumed mechanically not significant (infinitely thin)


Piezoelectric Unit-Cube


Coming soon...