6.6.3 Piezoelectric analysis

Products: ABAQUS/Standard  ABAQUS/CAE  



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).

Piezoelectric response

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


is the electrical potential,

is the component of the electric flux vector (also known as the electric displacement) in the ith material direction,

is the piezoelectric stress coupling,

is a small-strain component,

is the material's dielectric matrix for a fully constrained material, and

is the gradient of the electrical potential along the ith material direction, .

Defining piezoelectric and dielectric properties is discussed in Piezoelectric behavior, Section 20.6.2. The theoretical basis of the piezoelectric analysis capability in ABAQUS is defined in Piezoelectric analysis, Section 2.10.1 of the ABAQUS Theory Manual.

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, Section 27.2.1, 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, Section 27.3.1). 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. See Boundary conditions, Section 27.3.1.

Boundary conditions can be prescribed as functions of time by referring to amplitude curves (Amplitude curves, Section 27.1.2).

In an eigenfrequency extraction step (Natural frequency extraction, Section 6.3.5 ) 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.


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:

Applying electrical loads

Electrical charge is the conjugate to electrical potential at a node. Concentrated electric charge can be prescribed at nodes (or node sets). Distributed electric charge can be defined on element faces or surfaces.

Specifying concentrated electric charge

Concentrated electric charge is applied to degree of freedom 9.

Input File Usage:           
node number or node set name, , electric charge magnitude


Load module: Create Load: choose Electrical for the Category and Concentrated charge for the Types for Selected Step

Specifying element-based distributed electric charge

You can specify distributed surface charges (on element faces) or body charges (charge per unit volume). For element-based surface charges you must identify the face of the element upon which the charge in prescribed in the charge label. The distributed charge types available depend on the element type. Part VI, Elements,” lists the distributed charges that are available for particular elements.

Input File Usage:           
element number or element set name, charge label, charge magnitude


Load module: Create Load: choose Electrical for the Category and Body charge for the Types for Selected Step

Distributed surface charges in ABAQUS/CAE are always specified as surface-based loads (see below).

Specifying surface-based distributed electric charge

When you specify distributed electric charge on a surface, the element-based surface (see Defining element-based surfaces, Section 2.3.2) contains the element and face information. You must specify the surface name, the electric charge label, and the electric charge magnitude.

Input File Usage:           
surface name, ES, charge magnitude


Load module: Create Load: choose Electrical for the Category and Surface charge for the Types for Selected Step

Modifying or removing electrical loads

Electrical loads can be added, modified, or removed as described in Applying loads: overview, Section 27.4.1.

Specifying a time-dependent electrical load

The magnitude of a concentrated or a distributed electric charge can be controlled by referring to an amplitude curve (see Amplitude curves, Section 27.1.2). If different magnitude variations are needed for different charges, the charge definitions can be repeated, with each referring to its own amplitude curve.

Specifying electric charges in direct-solution steady-state dynamics analysis

In the direct-solution steady-state dynamics procedure, electric charges are given in terms of their real and imaginary components.

Input File Usage:           Use the following options to define electric charges in direct-integration steady-state dynamics analysis:
*CECHARGE, LOAD CASE=1 (real component) or 2 (imaginary component)


Load module: Create Load: choose Electrical for the Category and Concentrated charge, Surface charge, or Body charge for the Types for Selected Step: Magnitude: real component + imaginary componenti

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, Section 27.6.1:

  • 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, Section 20.6.2. 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, Section 17.2.1).


Piezoelectric elements must be used in a piezoelectric analysis (see Choosing the appropriate element for an analysis type, Section 21.1.3). The electric potential, , 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.


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

Element integration point variables:

Electrostatic energy density.


Magnitude and components of the electrical potential gradient vector, .


Magnitude of the electrical potential gradient vector.


Component n of the electrical potential gradient vector (n=1, 2, 3).


Magnitude and components of the electrical flux (displacement) vector, .


Magnitude of the electrical flux (displacement) vector.


Component n of the electrical flux (displacement) vector (n=1, 2, 3).

Whole element variables:

Values of distributed electrical charges.


Total electrostatic energy in the element, .

Nodal variables:

Electrical potential degree of freedom at a node.


Reactive electrical nodal charge (conjugate to prescribed electrical potential).


Concentrated electrical nodal charge.

Input file template

Data lines to define linear elasticity
Data lines to define piezoelectric behavior
Data lines to define dielectric behavior*AMPLITUDE, NAME=name
Data lines to define amplitude curve for defining concentrated electric charge
*STEP, (optionally NLGEOM)
Data lines to define boundary conditions on electrical potential and
displacement (rotation) degrees of freedom
Data lines to define time-dependent concentrated electric charges
Data lines to define distributed electric charges
*CLOAD and/or *DLOAD and/or *DSLOAD
Data lines to define mechanical loading