Piezoelectric #

The piezoelectric calculation can give you

1. effective stiffness of the system
2. effective permittivity of the system
3. effective piezoelectric of the system
4. strain distribution
5. stress distribution
6. electric field distribution
7. electric polarization distribution
8. electric displacement

Equation solved #

The coupled equation of

$\nabla \cdot D = 0$, where $D = \epsilon_0 \epsilon_r E + d \sigma$

$\nabla \cdot \sigma = 0$, where $\sigma = C(\epsilon - d^T E)$

• D Electric displacement
• $\epsilon_0$ vaccum permittivity
• $\epsilon_r$ relative permittivity
• E Electric field
• $\sigma$ The stress
• $\epsilon$ The strain
• d Piezoelectric coefficient
• $d^T$ The transpose of piezoelectric coefficient, or converse piezoelectric coefficient

Input options #

The following options are needed for the dielectric calculation.

• input->system The choice of calculation system must be piezoelectric
• input->external->electricField External applied electric field (V/m), if you want to calculate the electric field distribution within the composite
  • x Applied electric field along x
  • y Applied electric field along y
  • z Applied electric field along z
• input->external->elastic/type Either stress or strain, this will determine what external constraints are expected.
• input->external->elastic/stress External applied stress constraint, if you want to calculate the stress/strain distribution within the composite
  • tensor11$\sigma_{11}$
  • tensor12$\sigma_{12}$
  • tensor13$\sigma_{13}$
  • tensor22$\sigma_{22}$
  • tensor23$\sigma_{23}$
  • tensor33$\sigma_{33}$
• input->external->elastic/strain External applied strain constraint, if you want to calculate the stress/strain distribution within the composite
  • tensor11$\epsilon_{11}$
  • tensor12$\epsilon_{12}$
  • tensor13$\epsilon_{13}$
  • tensor22$\epsilon_{22}$
  • tensor23$\epsilon_{23}$
  • tensor33$\epsilon_{33}$
• input->solver->ref A property tensor for the reference phase called permittivity and a tensor of stiffness are needed. Tensor definition is explained [here]

<system>
    <type>piezoelectric</type>
    <external>
        <electricField>
            <x>0</x>
            <y>0</y>
            <z>2e7</z>
        </electricField>
        <elastic>
            <type>strain</type>
            <strain>
                <tensor11>0.0</tensor11>
                <tensor22>0.0</tensor22>
                <tensor33>0.0</tensor33>
                <tensor23>0.0</tensor23>
                <tensor13>0.001</tensor13>
                <tensor12>0.0</tensor12>
            </strain>
        </elastic>
    </external>
    <solver>
        <ref>
            <tensor>
                <name>permittivity</name>
                <rank>2</rank>
                <pointGroup>custom</pointGroup>
                <component>
                    <value>100</value>
                    <index>11</index>
                    <index>22</index>
                    <index>33</index>
                </component>
            </tensor>
            <tensor>
                <pointGroup>Oh</pointGroup>
                <name>stiffness</name>
                <rank>4</rank>
                <component>
                    <value>1000</value>
                    <index>1111</index>
                </component>
                <component>
                    <value>1200</value>
                    <index>1122</index>
                </component>
                <component>
                    <value>1100</value>
                    <index>1212</index>
                </component>
                <component>
                    <value>1100</value>
                    <index>1221</index>
                </component>
            </tensor>
        </ref>
    </solver>
</system>

Output files #

There are a total of 10 possible output files for the dielectric calculation

1. effDielectricPermittivity.dat
2. effElasticStiffness.dat
3. effPiezoelectric.dat
4. avElectricVariables.dat
5. avElasticVariables.dat
6. elePtntl.00000000.dat
7. eleField.00000000.dat
8. eleDspl.00000000.dat
9. stress.00000000.dat
10. strain.00000000.dat