Products: Abaqus/Standard Abaqus/CAE
“Defining a concentrated charge,” Section 16.9.27 of the Abaqus/CAE User's Manual
“Defining a surface charge,” Section 16.9.28 of the Abaqus/CAE User's Manual
“Defining a body charge,” Section 16.9.29 of the Abaqus/CAE User's Manual
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).
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, 
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Piezoelectric analysis can be carried out with the following procedures:
Initial conditions of piezoelectric quantities cannot be specified. See “Initial conditions in Abaqus/Standard and Abaqus/Explicit,” Section 32.2.1, for a description of the initial conditions that can be applied in static or dynamic procedures.
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 32.3.1). 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 32.3.1.
Boundary conditions can be prescribed as functions of time by referring to amplitude curves (“Amplitude curves,” Section 32.1.2).
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.
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,” Section 32.4.2.
Distributed pressure forces or body forces can be applied; see “Distributed loads,” Section 32.4.3.
The following types of electrical loads can be prescribed, as described in “Electromagnetic loads,” Section 32.4.5:
Concentrated electric charge.
Distributed surface electric charge and body electric charge.
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.
The following predefined fields can be specified in a piezoelectric analysis, as described in “Predefined fields,” Section 32.6.1:
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.
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 25.5.2. 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 21.2.1).
Piezoelectric elements must be used in a piezoelectric analysis (see “Choosing the appropriate element for an analysis type,” Section 26.1.3). 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:
EENER | Electrostatic energy density. |
EPG | Magnitude and components of the electrical potential gradient vector, |
EPGM | Magnitude of the electrical potential gradient vector. |
EPGn | Component n 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). |
CHRGS | Values of distributed electrical charges. |
ELCTE | Total electrostatic energy in the element, |
EPOT | Electrical potential degree of freedom at a node. |
RCHG | Reactive electrical nodal charge (conjugate to prescribed electrical potential). |
CECHG | Concentrated electrical nodal charge. |
*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
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