Piezoelectricity

A piezoelectric material:

is one in which an electrical field causes the material to strain, while stress causes an electric potential gradient;
provides linear relations between mechanical and electrical fields; and
is used in piezoelectric analyses, in which both displacement and electrical potential are solution variables.

Piezoelectric

A piezoelectric material responds to an electric potential gradient by straining, while stress causes an electric potential gradient in the material. The coupling between an electric potential gradient and mechanical strain is simulated using a piezoelectric material option.

Note: The material will also have a dielectric property so that an electrical charge exists when the material has a potential gradient (see Dielectric).

The coupled electrical-mechanical behavior can be defined in terms of the elastic stiffness matrix and the piezoelectric stress coefficient matrix, $e_{m i j}^{φ}$ , as

σ_{i j} = D_{i j k l}^{E} ε_{k l} - e_{m i j}^{φ} E_{m}

or in terms of the elastic stiffness matrix and the piezoelectric strain coefficient matrix,

d_{m k l}^{φ}

, as

σ_{i j} = D_{i j k l}^{E} {(ε}_{k l} - d_{m k l}^{φ} E_{m})

where

σ_{i j}

is the mechanical stress tensor,

ε_{k l}

is the strain tensor,

D_{i j k l}^{E}

is the material's elastic stiffness matrix defined at zero electrical potential gradient,

e_{m i j}^{φ}

is the material's piezoelectric stress coefficient matrix, defining the stress

σ_{i j}

caused by the electrical potential gradient

E_{m} = - \frac{\partial φ}{\partial x_{m}}

in a fully constrained material (it can also be interpreted as the electrical displacement

q_{m}

caused by the applied strain

ε_{i j}

at a zero electrical potential gradient),

φ

is the electrical potential,

d_{m k l}^{φ}

is the material's piezoelectric strain coefficient matrix, defining the strain

ε_{k l}

caused by the electrical potential gradient

E_{m}

in an unconstrained material.

The elastic mechanical properties of a piezoelectric material must use linear elasticity (see Introduction to Linear Elasticity).

The components of the mechanical stress tensor $σ_{i j}$ due to the electrical potential gradient $E_{m}$ , labeled here as $σ_{i j}^{φ}$ , can be computed by multiplying the piezoelectric stress coefficient matrix, $e_{m i j}^{φ}$ with $E_{m}$

{\begin{matrix} \begin{matrix} σ_{11}^{φ} \\ σ_{22}^{φ} \end{matrix} \\ σ_{33}^{φ} \\ σ_{12}^{φ} \\ σ_{13}^{φ} \\ σ_{23}^{φ} \end{matrix}} = {\begin{matrix} \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{111}^{φ} & \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{211}^{φ} & {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{311}^{φ} \end{matrix} \end{matrix} \\ \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{122}^{φ} & \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{222}^{φ} & {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{322}^{φ} \end{matrix} \end{matrix} \\ \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{133}^{φ} & \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{233}^{φ} & {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{333}^{φ} \end{matrix} \end{matrix} \\ \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{112}^{φ} & \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{212}^{φ} & {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{312}^{φ} \end{matrix} \end{matrix} \\ \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{113}^{φ} & \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{213}^{φ} & {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{313}^{φ} \end{matrix} \end{matrix} \\ \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{123}^{φ} & \begin{matrix} {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{223}^{φ} & {\begin{matrix} \begin{matrix} ε \end{matrix} \end{matrix}}_{323}^{φ} \end{matrix} \end{matrix} \end{matrix}} {\begin{matrix} E_{1} \\ E_{2} \\ E_{3} \end{matrix}} .

You can define the piezoelectric behvaior by specifying either the its piezoelectric stress coefficient matrix, $e_{m i j}^{φ}$ , or its piezoelectric strain coefficient matrix, $d_{m k l}^{φ}$ .

Table 1. Type=Stress
Input Data	Description
E111, E122,…,E323	Piezoelectric stress coefficient matrix components, $e_{m i j}^{φ}$ .
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.

Table 2. Type=Strain
Input Data	Description
D111, D122,…,D323	Piezoelectric strain coefficient matrix components, $d_{m k l}^{φ}$ .
Use temperature-dependent data	Specify material parameters that depend on temperature. A Temperature field appears in the data table.
Number of field variables	Specifies material parameters that depend on one or more independent field variables. A Field column appears in the data table. For more information, see Specifying Material Data as a Function of Temperature and Independent Field Variables.