The capability of modeling piezoelectric response exists in the following elements:

Constitutive Equations of Piezoelectricity

In linear piezoelectricity the equations of elasticity are coupled to the charge equation of electrostatics by means of piezoelectric constants (IEEE Standard on Piezoelectricity([89])):

(10–41)

(10–42)

or equivalently

(10–43)

where:

Equation 10–41 and Equation 10–42 are the usual constitutive equations for structural and electrical fields, respectively, except for the coupling terms involving the piezoelectric matrix [e].

The elasticity matrix [c] is the usual [D] matrix described in Structural Fundamentals (input using the MP commands). It can also be input directly in uninverted form [c] or in inverted form [c]^-1 as a general anisotropic symmetric matrix (input using TB,ANEL):

(10–44)

The piezoelectric stress matrix [e] (input using TB,PIEZ with TBOPT = 0) relates the electric field vector {E} in the order X, Y, Z to the stress vector {T} in the order X, Y, Z, XY, YZ, XZ and is of the form:

(10–45)

The piezoelectric matrix can also be input as a piezoelectric strain matrix [d] (input using TB,PIEZ with TBOPT = 1). ANSYS will automatically convert the piezoelectric strain matrix [d] to a piezoelectric stress matrix [e] using the elasticity matrix [c] at the first defined temperature:

(10–46)

The orthotropic dielectric matrix [ε^S] uses the electrical permittivities (input as PERX, PERY and PERZ on the MP commands) and is of the form:

(10–47)

The anisotropic dielectric matrix at constant strain [ε^S] (input using TB,DPER,,,,0 command) is used by PLANE223, SOLID226, and SOLID227 and is of the form:

(10–48)

The dielectric matrix can also be input as a dielectric permittivity matrix at constant stress [ε^T] (input using TB,DPER,,,,1). The program will automatically convert the dielectric matrix at constant stress to a dielectric matrix at constant strain:

(10–49)

where:

Derivation of Piezoelectric Matrices

After the application of the variational principle and finite element discretization (Allik([81])), the coupled finite element matrix equation derived for a one element model is:

(10–50)

where:

[K] = element stiffness matrix (defined by Equation 2–58)

[M] = element mass matrix (defined by Equation 2–58)

[C] = element structural damping matrix (discussed in Damping Matrices)

{F} = vector of nodal and surface forces (defined by Equation 2–56 and Equation 2–58)

[Kd] = element dielectric permittivity coefficient matrix ([Kvs] in Equation 5–117 or [Kvh] in Equation 5–116)

{L} = vector of nodal, surface, and body charges (defined by Equation 5–117)

[B] = strain-displacement matrix (see Equation 2–44)

[Cvh] = element dielectric damping matrix (defined by Equation 5–116)

{εth} = thermal strain vector (as defined by equation Equation 2–3)

{N} = element shape functions

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Note:  In a strongly coupled thermo-piezoelectric analysis (see Equation 10–17), the electric potential and temperature degrees of freedom are coupled by:

where:

{α} = vector of coefficient of thermal expansion.

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Energy Calculation

In static and transient piezoelectric analyses, the PLANE223, SOLID226, and SOLID227 element instantaneous energies are calculated as:

(10–51)

(10–52)

(10–53)

where:

= stored elastic strain energy (output as an NMISC element item UE).

= electromechanical or mutual energy (output as an NMISC element item UM)

= dielectric energy (output as an NMISC element item UD)

In a harmonic piezoelectric analysis, the time-averaged element energies are calculated as:

(10–54)

(10–55)

(10–56)

where:

= complex conjugate of the elastic strain

= complex conjugate of the electric field intensity

The real parts of equations (Equation 10–54) and (Equation 10–56) represent the average stored elastic and dielectric energies, respectively. The imaginary parts represent the average elastic and electric losses. Therefore, the quality factor can be calculated from the total stored energy as:

(10–57)

where:

= number of piezoelectric elements

The total stored energy + is output as SENE. The factor can therefore be derived from the real and imaginary records of SENE summed over the piezoelectric elements.

The mutual energy can be used to calculate the electromechanical coupling coefficient as:

(10–58)