6.7.2 Piezoelectric analysis

Products: Abaqus/Standard  Abaqus/CAE  

Overview

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:

where

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

Defining piezoelectric and dielectric properties is discussed in Piezoelectric behavior, Section 26.5.2. The theoretical basis of the piezoelectric analysis capability in Abaqus is defined in Piezoelectric analysis, Section 2.10.1 of the Abaqus Theory Guide.

Initial conditions

Initial conditions of piezoelectric quantities cannot be specified. See Initial conditions in Abaqus/Standard and Abaqus/Explicit, Section 34.2.1, 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 in Abaqus/Standard and Abaqus/Explicit, Section 34.3.1). 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 in Abaqus/Standard and Abaqus/Explicit, Section 34.3.1.

Boundary conditions can be prescribed as functions of time by referring to amplitude curves (Amplitude curves, Section 34.1.2).

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.

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:

Applying electrical loads

The following types of electrical loads can be prescribed, as described in Electromagnetic loads, Section 34.4.5:

  • 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, Section 34.6.1:

  • 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 26.5.2. 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 22.2.1).

Elements

Piezoelectric elements must be used in a piezoelectric analysis (see Choosing the appropriate element for an analysis type, Section 27.1.3). 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.

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, .

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, .

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, .



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 will be 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
*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
Your query was poorly formed. Please make corrections.


6.7.2 Piezoelectric analysis

Products: Abaqus/Standard  Abaqus/CAE  

Your query was poorly formed. Please make corrections.

Overview

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).

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Piezoelectric response

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

where

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

Defining piezoelectric and dielectric properties is discussed in Piezoelectric behavior, Section 26.5.2. The theoretical basis of the piezoelectric analysis capability in Abaqus is defined in Piezoelectric analysis, Section 2.10.1 of the Abaqus Theory Guide.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Initial conditions

Initial conditions of piezoelectric quantities cannot be specified. See Initial conditions in Abaqus/Standard and Abaqus/Explicit, Section 34.2.1, for a description of the initial conditions that can be applied in static or dynamic procedures.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Boundary conditions

The electric potential at a node (degree of freedom 9) can be prescribed using a boundary condition (see Boundary conditions in Abaqus/Standard and Abaqus/Explicit, Section 34.3.1). 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 in Abaqus/Standard and Abaqus/Explicit, Section 34.3.1.

Boundary conditions can be prescribed as functions of time by referring to amplitude curves (Amplitude curves, Section 34.1.2).

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.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Loads

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

Your query was poorly formed. Please make corrections.

Applying mechanical loads

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

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Applying electrical loads

The following types of electrical loads can be prescribed, as described in Electromagnetic loads, Section 34.4.5:

  • Concentrated electric charge.

  • Distributed surface electric charge and body electric charge.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

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.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Predefined fields

The following predefined fields can be specified in a piezoelectric analysis, as described in Predefined fields, Section 34.6.1:

  • 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.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

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 26.5.2. 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 22.2.1).

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Elements

Piezoelectric elements must be used in a piezoelectric analysis (see Choosing the appropriate element for an analysis type, Section 27.1.3). 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.

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

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, .

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, .

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, .



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.


Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

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

Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.

Input file template

*HEADING*MATERIAL, NAME=matl
*ELASTIC
Data lines to define linear elasticity
*PIEZOELECTRIC
Data lines to define piezoelectric behavior
*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
Your query was poorly formed. Please make corrections.
Your query was poorly formed. Please make corrections.