The capability of modeling electric force coupling in elastic dielectrics exists in the following elements:
| PLANE222 - 2D 4-Node Coupled-Field Solid |
| PLANE223 - 2D 8-Node Coupled-Field Solid |
| SOLID225 - 3D 8-Node Coupled-Field Solid |
| SOLID226 - 3D 20-Node Coupled-Field Solid |
| SOLID227 - 3D 10-Node Coupled-Field Solid |
Elastic dielectrics exhibit a deformation when subject to an electrostatic field. The electric body force that causes the deformation can be derived from the Maxwell stress tensor [σM] (Landau and Lifshitz([355])).
 | (10–101) |
where:
| {E} = electric field intensity vector |
| {D} = electric flux density vector |
|
Applying the variational principle to the stress equation of motion and to the charge equation of electrostatics coupled by electric force produces the following finite element equation:
 | (10–102) |
where:
| [K] = element structural stiffness matrix (see [Ke] in Equation 2–58) |
| [M] = element mass matrix (see [Me] in Equation 2–58) |
| [Kd] = element dielectric permittivity coefficient matrix (see [Kvs] in Equation 5–117) |
| [C] = element structural damping matrix (discussed in Damping Matrices) |
| [Cvh] = element dielectric damping matrix (defined by Equation 5–116) |
| {F} = vector of nodal and surface forces (defined by Equation 2–56 and Equation 2–58) |
| {L} = vector of nodal, surface, and body charges (see {Le} in Equation 5–117) |
The electrostatic softening matrix [Keu] and the coupling matrix [KeV] are calculated as derivatives of the nodal electric force {Fe} with respect to displacement and voltage:
|
|
These derivatives are obtained by applying the chain rule to the following expression for the electric force (output as FMAG):
|
| where: |
| [B] = strain-displacement matrix (see Equation 2–44) |
|
| vol = element volume |
The strong (matrix) coupling between structural and electric equations in the finite element system (Equation 10–102) allows the linear perturbation modal and harmonic analyses to be used following a nonlinear static or transient analysis.
Note that for KEYOPT(4) = 0, 1, 2, and 4, the finite element system (Equation 10–102) is symmetric due to the negative sign assigned to the electric equation.
The electroelastic analysis results are similar to those computed
in a piezoelectric analysis ( Piezoelectrics) with the coupling matrix
derived from the Maxwell stress vector as summarized in the following
table.
Table 10.5: Electroelastic Analysis Results
| Equation | |||
|---|---|---|---|
| Element result | Output as | Nonlinear static and transient analyses | Linear perturbation harmonic and modal analyses |
| Stress {T} | S | Equation 2–60 | Equation 10–51 and Equation 10–52 with |
| Electric flux density {D} | D | Equation 5–75 | |
| Electric current density {J} | JS | Equation 10–74 with  | |
| Energy | |||
| Elastic UE | NMISC,1 (UE) | Equation 10–75 | Equation 10–78 |
| Dielectric UD | NMISC,2 (UD) | Equation 10–76 | Equation 10–79 |
| Mutual UM | NMISC,3 (UM) | Equation 10–77 with | Equation 10–80 with |
| Total stored | SENE | UE + UD | |
| Kinetic | KENE | Equation 14–342 | Equation 14–344 |
| Damping | DENE | Equation 14–354 | Equation 14–355 |
| Heat generation rate[a] | JHEAT | Equation 10–86 | Equation 10–87 and Equation 10–88 |
[a] Supported losses (see Table 10.4: Losses in Piezoelectric Analysis) are:
Structural isotropic viscous damping (input as MP,BETD or TB,SDAMP,,,,BETD)
Structural isotropic hysteretic damping (input as MP,DMPR, MP,DMPS, or TB,SDAMP,,,,STRU)
Electric isotropic loss tangent (input as MP,LSST), anisotropic loss tangent (input as TB,DLST) and electrical resistivity (input as MP,RSVX/Y/Z).