Piezoelectric Behavior

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 elements, which have both displacement and electrical potential as nodal variables.

This page discusses:

Defining a Piezoelectric Material
Alternative Forms of the Constitutive Equations
Specifying Dielectric Material Properties
Specifying Piezoelectric Material Properties
Energy Balance Considerations
Elements

Defining a Piezoelectric Material

A piezoelectric material responds to an electric potential gradient by straining, while stress causes an electric potential gradient in the material. This coupling between electric potential gradient and strain is the material's piezoelectric property. The material will also have a dielectric property so that an electrical charge exists when the material has a potential gradient. Piezoelectric material behavior is discussed in Piezoelectric analysis.

The mechanical properties of the material must be modeled by linear elasticity (Linear Elastic Behavior). The mechanical behavior can be defined by

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

in terms of the piezoelectric stress coefficient matrix, $e_{m i j}^{φ}$ , or by

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

in terms of the piezoelectric strain coefficient matrix, $d_{m k l}^{φ}$ . The electrical behavior is defined by

q_{i} = e_{i j k}^{φ} ε_{j k} + D_{i j}^{φ (ε)} E_{j},

where

$σ_{i j}$: is the mechanical stress tensor;
$ε_{i j}$: is the strain tensor;
$q_{i}$: is the electric “displacement” vector;
$D_{i j k l}^{E}$: is the material's elastic stiffness matrix defined at zero electrical potential gradient (short circuit condition);
$e_{m i j}^{φ}$: is the material's piezoelectric stress coefficient matrix, defining the stress $σ_{i j}$ caused by the electrical potential gradient $E_{m}$ 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);
$d_{m k l}^{φ}$: 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 (an alternative interpretation is given later in this section);
$φ$: is the electrical potential;
$D_{i j}^{φ (ε)}$: is the material's dielectric property, defining the relation between the electric displacement $q_{i}$ and the electric potential gradient $E_{j}$ for a fully constrained material; and
$E_{i}$: is the electrical potential gradient vector, $- \partial φ / \partial x_{i}$ .

The material's electrical and electro-mechanical coupling behaviors are, thus, defined by its dielectric property, $D_{i j}^{φ (ε)}$ , and its piezoelectric stress coefficient matrix, $e_{m i j}^{φ}$ , or its piezoelectric strain coefficient matrix, $d_{m k l}^{φ}$ . These properties are defined as part of the material definition (Material Data Definition).

Alternative Forms of the Constitutive Equations

Alternative forms of the piezoelectric constitutive equations are presented in this section. These forms of the equations involve material properties that cannot be used directly as input for Abaqus/Standard. However, they are related to the Abaqus/Standard input through simple relations that are presented in Piezoelectric analysis. The intent of this section is to draw connections between the Abaqus/Standard terminology and input to that used commonly in the piezoelectricity community. The mechanical behavior can also be defined by

σ_{i j} = D_{i j k l}^{q} (ε_{k l} - g_{m k l}^{φ} q_{m})

in terms of the piezoelectric coefficient matrix $g_{m i j}^{φ}$ , and the stiffness matrix $D_{i j k l}^{q}$ , which defines the mechanical properties at zero electrical displacement (open circuit condition). Likewise, the electrical behavior can also be defined by

q_{i} = d_{i j k}^{φ} σ_{j k} + D_{i j}^{φ (σ)} E_{j}

in terms of the dielectric matrix $D_{i j}^{φ (σ)}$ for an unconstrained material or by

q_{i} = D_{i m}^{φ (σ)} g_{m j k}^{φ} σ_{j k} + D_{i j}^{φ (σ)} E_{j},

where

$D_{i j k l}^{q}$: is the material's elastic stiffness matrix defined at zero electrical displacement;
$d_{i j k}^{φ}$: is the material's piezoelectric strain coefficient matrix used earlier, and based on the equations, may alternatively be interpreted as the electrical displacement $q_{i}$ caused by the stress $σ_{j k}$ at zero electrical potential gradient;
$g_{m k l}^{φ}$: is the material's piezoelectric coefficient matrix, which can be interpreted as defining either the strain $ε_{k l}$ caused by the electrical displacement $q_{m}$ in an unconstrained material or the electrical potential gradient $E_{m}$ caused by the stress $σ_{k l}$ at zero electrical displacement; and
$D_{i j}^{φ (σ)}$: is the material's dielectric property, defining the relation between the electric displacement $q_{i}$ and the electric potential gradient $E_{j}$ for an unconstrained material.

These are useful relationships that are often seen in the piezoelectric literature. In Piezoelectric analysis the properties $g_{m k l}^{φ}$ , $D_{i j}^{φ (σ)}$ , and $D_{i j k l}^{q}$ are expressed in terms of the properties $d_{m k l}^{φ}$ , $D_{i j}^{φ (ε)}$ , and $D_{i j k l}^{E}$ , that are used as input for Abaqus/Standard.

Specifying Dielectric Material Properties

The dielectric matrix can be isotropic, orthotropic, or fully anisotropic. For non-isotropic dielectric materials a local orientation for the material directions must be specified (Orientations). The entries of the dielectric matrix (what are referred to as “dielectric constants” in Abaqus) refer to what is more commonly known in the literature as the permittivity of the material.

Isotropic Dielectric Properties

The dielectric matrix $D_{i j}^{φ (ε)}$ can be fully isotropic, so that

D_{i j}^{φ (ε)} = D^{φ (ε)} δ_{i j} .

You specify the single value $D^{φ (ε)}$ for the dielectric constant. $D^{φ (ε)}$ must be determined for a constrained material. Isotropic behavior is the default.

Orthotropic Dielectric Properties

For orthotropic behavior you must specify three values in the dielectric matrix ( $D_{11}^{φ (ε)}$ , $D_{22}^{φ (ε)}$ , and $D_{33}^{φ (ε)}$ ).

Anisotropic Dielectric Properties

For fully anisotropic behavior you must specify six values in the dielectric matrix ( $D_{11}^{φ (ε)}$ , $D_{12}^{φ (ε)}$ , $D_{22}^{φ (ε)}$ , $D_{13}^{φ (ε)}$ , $D_{23}^{φ (ε)}$ , and $D_{33}^{φ (ε)}$ ).

Specifying Piezoelectric Material Properties

The piezoelectric material properties can be defined by giving the stress coefficients, $e_{m i j}^{φ}$ (this is the default), or by giving the strain coefficients, $d_{m k l}^{φ}$ . In either case, 18 components must be given in the following order (substitute d for e for strain coefficients):

\begin{matrix} e_{1 11}^{φ}, & e_{1 22}^{φ}, & e_{1 33}^{φ}, & e_{1 12}^{φ}, & e_{1 13}^{φ}, & e_{1 23}^{φ}, \\ e_{2 11}^{φ}, & e_{2 22}^{φ}, & e_{2 33}^{φ}, & e_{2 12}^{φ}, & e_{2 13}^{φ}, & e_{2 23}^{φ}, \\ e_{3 11}^{φ}, & e_{3 22}^{φ}, & e_{3 33}^{φ}, & e_{3 12}^{φ}, & e_{3 13}^{φ}, & e_{3 23}^{φ} \end{matrix}

The first index on these coefficients refers to the component of electric displacement (sometimes called the electric flux), while the last pair of indices refers to the component of mechanical stress or strain.

Thus, the piezoelectric components causing electrical displacement in the 1-direction are all given first, then those causing electrical displacement in the 2-direction, and then those causing electrical displacement in the 3-direction. (Some references list these coupling terms in a different order.)

Converting Double Index Notation to Triple Index Notation

Industry-supplied piezoelectric data often use a double index notation. A double index notation can be converted easily to the required triple index notation in Abaqus/Standard by noting the convention followed in Abaqus for the correspondence between (second-order) tensor and vector notations: the 11, 22, 33, 12, 13, and 23 components of the tensor correspond to the 1, 2, 3, 4, 5, and 6 components, respectively, of the corresponding vector.

Energy Balance Considerations

Abaqus does not account for piezoelectric effects in the total energy balance equation, which can lead to an apparent imbalance of the total energy of the model in some situations. For example, if a piezoelectric truss is fixed at one end point and subjected to a potential difference between its two end points, it deforms due to the piezoelectric effect. Subsequently if the truss is held fixed in this deformed configuration and the potential difference removed, strain energy will be generated due to the constraints. This results in an equivalent increase in the total energy of the model.

Elements

Piezoelectric coupling is active only in piezoelectric elements (those with displacement degrees of freedom and electrical potential degree of freedom 9). See Choosing the Appropriate Element for an Analysis Type.