Piezoelectric Analysis

Coupled piezoelectric problems:

are those in which an electric potential gradient causes straining, while stress causes an electric potential gradient in the material;
are solved using an eigenfrequency extraction, modal dynamic, static, dynamic, or steady-state dynamic procedure;
require the use of piezoelectric elements and piezoelectric material properties;
can be performed for continuum problems in one, two, and three dimensions; and
can be used in both linear and nonlinear analysis (however, in nonlinear analysis the piezoelectric part of the constitutive behavior is assumed to be linear).

This page discusses:

Piezoelectric Response
Procedures Available for Piezoelectric Analysis
Initial Conditions
Boundary Conditions
Loads
Predefined Fields
Material Options
Elements
Output
Limitations
Input File Template

Piezoelectric Response

The electrical response of a piezoelectric material is assumed to be made up of piezoelectric and dielectric effects:

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

where

$φ$: is the electrical potential,
$q_{i}$: is the component of the electric flux vector (also known as the electric displacement) in the ith material direction,
$e_{i j k}^{φ}$: is the piezoelectric stress coupling,
$ε_{i j}$: is a small-strain component,
$D_{i j}^{φ (ε)}$: is the material's dielectric matrix for a fully constrained material, and
$E_{i}$: is the negative of the gradient of the electrical potential along the ith material direction, $- \partial φ / \partial x_{i}$ .

Defining piezoelectric and dielectric properties is discussed in Piezoelectric Behavior. The theoretical basis of the piezoelectric analysis capability in Abaqus is defined in Piezoelectric analysis.

Procedures Available for Piezoelectric Analysis

Piezoelectric analysis can be carried out with the following procedures:

Initial Conditions

Initial conditions of piezoelectric quantities cannot be specified. See Initial Conditions for a description of the initial conditions that can be applied in static or dynamic procedures.

Boundary Conditions

The electric potential at a node (degree of freedom 9) can be prescribed using a boundary condition (see Boundary Conditions). Displacement and rotation degrees of freedom can also be prescribed by using boundary conditions as described in the relevant static and dynamic analysis procedure sections.

Boundary conditions can be prescribed as functions of time by referring to amplitude curves (see Amplitude Curves).

In an eigenfrequency extraction step (Natural Frequency Extraction) involving piezoelectric elements, the electric potential degree of freedom must be constrained at least at one node to remove singularities from the dielectric part of the element operator. If there are multiple disconnected piezoelectric regions in the model, the electric potential degree of freedom must be constrained at least at one node in each of these regions.

Loads

Both mechanical and electrical loads can be applied in a piezoelectric analysis.

Applying Mechanical Loads

The following types of mechanical loads can be prescribed in a piezoelectric analysis:

Concentrated nodal forces can be applied to the displacement degrees of freedom (1–6); see Concentrated Loads.
Distributed pressure forces or body forces can be applied; see Distributed Loads.

Applying Electrical Loads

The following types of electrical loads can be prescribed, as described in Electromagnetic Loads:

Concentrated electric charge.
Distributed surface electric charge and body electric charge.

Loading in Mode-Based and Subspace-Based Procedures

Electrical charge loads should be used only in conjunction with residual modes in the eigenvalue extraction step, due to the “massless” mode effect. Since the electrical potential degrees of freedom do not have any associated mass, these degrees of freedom are essentially eliminated (similar to Guyan reduction or mass condensation) during the eigenvalue extraction. The residual modes represent the static response corresponding to the electrical charge loads, which will adequately represent the potential degree of freedom in the eigenspace.

Predefined Fields

The following predefined fields can be specified in a piezoelectric analysis, as described in Predefined Fields:

Although temperature is not a degree of freedom in piezoelectric elements, nodal temperatures can be specified. The specified temperature affects only temperature-dependent material properties, if any.
The values of user-defined field variables can be specified. These values affect only field-variable-dependent material properties, if any.

Material Options

The piezoelectric coupling matrix and the dielectric matrix are specified as part of the material definition for piezoelectric materials, as described in Piezoelectric Behavior. They are relevant only when the material definition is used with coupled piezoelectric elements.

The mechanical behavior of the material can include linear elasticity only (Linear Elastic Behavior).

Each material definition can have a material damping coefficient assigned for procedures where damping can be part of the solution. For piezoelectric materials you can specify piezoelectric damping. You can define stiffness proportional viscous and structural damping by providing damping coefficients for the displacement (mechanical), piezoelectric coupling, and dielectric parts of the damping operator. If you specify piezoelectric damping to define stiffness proportional viscous damping, you cannot specify material damping to define stiffness proportional viscous damping, and vice versa. The same applies for stiffness proportional structural damping.

Elements

Piezoelectric elements must be used in a piezoelectric analysis (see Choosing the Appropriate Element for an Analysis Type). The electric potential, $φ$ , is degree of freedom 9 at each node of these elements. In addition, regular stress/displacement elements can be used in parts of the model where piezoelectric effects do not need to be considered.

Output

The following output variables are applicable to the electrical solution in a piezoelectric analysis:

Element integration point variables:

EENER: Electrostatic energy density.
EPG: Magnitude and components of the negative of the electrical potential gradient vector, $- \partial φ / \partial x$ .
EPGM: Magnitude of the electrical potential gradient vector.
EPGn: Component n of the negative of the electrical potential gradient vector (n=1, 2, 3).
EFLX: Magnitude and components of the electrical flux (displacement) vector, $q$ .
EFLXM: Magnitude of the electrical flux (displacement) vector.
EFLXn: Component n of the electrical flux (displacement) vector (n=1, 2, 3).

Whole element variables:

CHRGS: Values of distributed electrical charges.
ELCTE: Total electrostatic energy in the element, $\int_{v} EENER d v$ .

Nodal variables:

EPOT: Electrical potential degree of freedom at a node.
RCHG: Reactive electrical nodal charge (conjugate to prescribed electrical potential).
CECHG: Concentrated electrical nodal charge.

Limitations

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 is generated due to the constraints. This results in an equivalent increase in the total energy of the model.

Input File Template

HEADING
…
MATERIAL, NAME=matl
ELASTIC
Data lines to define linear elasticity
PIEZOELECTRIC
Data lines to define piezoelectric behavior
PIEZOELECTRIC DAMPING, BETA
Data lines to define piezoelectric damping
DAMPING, ALPHA= $α_{R}$ 
DIELECTRIC
Data lines to define dielectric behavior

…
AMPLITUDE, NAME=name
Data lines to define amplitude curve for defining concentrated electric charge
**
STEP, (optionally NLGEOM)
STATIC
** or DYNAMIC, FREQUENCY, MODAL DYNAMIC, 
** STEADY STATE DYNAMICS (, DIRECT or , SUBSPACE PROJECTION)
BOUNDARY
Data lines to define boundary conditions on electrical potential and
displacement (rotation) degrees of freedom
CECHARGE, AMPLITUDE=name
Data lines to define time-dependent concentrated electric charges
DECHARGE and/or DSECHARGE
Data lines to define distributed electric charges
CLOAD and/or DLOAD and/or DSLOAD
Data lines to define mechanical loading
END STEP