# 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: $\frac{\partial}{\partial t} = 0$

<analysis>
<static/>
</analysis>


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

<analysis>
<transient>
<numSteps>100</numSteps>
<deltaT>3e-06</deltaT>
</transient>
</analysis>


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

<analysis>
<harmonic>
<frequencyList>
<freq value="100"/>
<freq value="200"/>
<freq value="300"/>
<\frequencyList>
<\harmonic>
<\analysis>


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

<analysis>
<eigenFrequency>
<numModes>10</numModes>
<freqShift>0</freqShift>
<writeModes normalization="max">yes</writeModes>
<!-- scale them to a maximum displacement of 1 -->
</eigenFrequency>
</analysis>


## 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

...
</pdeList>
<couplingList>
<direct>
<piezoDirect>
<regionList>
<region name="V"/>
</regionList>
</piezoDirect>
</direct>
</couplingList>
...


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.

## Material¶

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

 <piezo>
<piezoCoupling>
<linear>
<tensor dim1="3" dim2="6">
<real>
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
</real>
</tensor>
</linear>
</piezoCoupling>
</piezo>


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


## 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)

## Tutorials¶

Piezoelectric Unit-Cube

Coming soon...