The capability of modeling electric force coupling in elastic dielectrics exists in the following elements:
| PLANE223 - 2-D 8-Node Coupled-Field Solid |
| SOLID226 - 3-D 20-Node Coupled-Field Solid |
| SOLID227 - 3-D 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([359])).
 | (10–59) |
where:
| {E} = electric field intensity vector |
| {D} = electric flux density vector |
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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:
 | (10–60) |
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:
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These derivatives are obtained by applying the chain rule to the following expression for the electric force:
|
| where: |
| [B] = strain-displacement matrix (see Equation 2–44) |
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The strong (matrix) coupling between structural and electric equations in the finite element system (Equation 10–60) allows the linear perturbation modal and harmonic analyses to be used following a nonlinear static or transient analysis.
Note that the finite element system (Equation 10–60) is symmetric due to the negative sign assigned to the electric equation.