Piezoelectric capability (TB,PIEZ) is available with the coupled-field elements. (See Material Model Support for Elements for piezoelectricity.) Material properties required for the piezoelectric effects include the dielectric (relative permittivity) constants, the elastic coefficient matrix, and the piezoelectric matrix.

You can define the piezoelectric stress matrix at constant strain [e] (TBOPT = 0) or the piezoelectric strain matrix [d] (TBOPT = 1). The [e] matrix is typically associated with the input of the anisotropic elasticity in the form of the stiffness matrix [c], and the permittivity at constant strain [ε^S]. The [d] matrix is associated with the input of compliance matrix [s] and permittivity at constant stress [ε^T]. Select the appropriate matrix form for your analysis using the TB,PIEZ command.

The full 6 x 3 piezoelectric matrix relates terms x, y, z, xy, yz, xz to x, y, z via 18 constants as shown:

For 2D problems, a 4 x 2 matrix relates terms ordered x, y, z, xy via 8 constants (e₁₁, e₁₂, e₂₁, e₂₂, e₃₁, e₃₂, e₄₁, e₄₂). The order of the vector is expected as {x, y, z, xy, yz, xz}, whereas for some published materials the order is given as {x, y, z, yz, xz, xy}. This difference requires the piezoelectric matrix terms to be converted to the expected format.

You can define up to 18 constants (C1-C18) with TBDATA commands (6 per command):

In a piezoelectric analysis using PLANE222, PLANE223, SOLID225, SOLID226, or SOLID227, you can define temperature-dependent piezoelectric coefficients (with TBOPT = 0) by using tabular input: TBDATA,STLOC,%tabname%, where tabname is the name of a table array parameter (*DIM) with TEMP as a primary variable. For more information, see Defining Linear Material Properties Using Tabular Input.

See Piezoelectric Analysis in the Coupled-Field Analysis Guide for more information about this material model.

Elements with piezoresistive capabilities use the TB,PZRS command to calculate the change in electric resistivity produced by elastic stress or strain. Material properties required to model piezoresistive materials are electrical resistivity, the elastic coefficient matrix, and the piezoresistive matrix.

You can define the piezoresistive matrix either in the form of piezoresistive stress matrix [π] (TBOPT = 0) or piezoresistive strain matrix [m] (TBOPT = 1).

The piezoresistive stress matrix [π] uses stress to calculate the change in electric resistivity due to piezoresistive effect, while the piezoresistive strain matrix [m] (TBOPT = 1) uses strain to calculate the change in electric resistivity. See Piezoresistivity in the Mechanical APDL Theory Reference for more information.

The full 6x6 piezoresistive matrix relates the x, y, z, xy, yz, xz terms of stress to the x, y, z, xy, yz, xz terms of electric resistivity via 36 constants:

For 2D problems, a 4x4 matrix relates terms ordered x, y, z, xy via 16 constants.

The order of the vector is expected as {x, y, z, xy, yz, xz}, whereas for some published materials the order is given as {x, y, z, yz, xz, xy}. This difference requires the piezoresistive matrix terms to be converted to the expected format.

See Piezoresistive Analysis in the Coupled-Field Analysis Guide for more information on this material model.

Elements with magnetic capability use the TB table to input points characterizing B-H curves.

Initialize the curves with the TB,BH command. Use TBPT commands to define up to 500 points (H, B). The constants (X, Y) entered on TBPT (two per command) are:

Specify the system of units (MKS or user defined) with EMUNIT, which also determines the value of the permeability of free space. This value is used with the relative permeability property values (MP) to establish absolute permeability values. The defaults (also obtained for Lab = MKS) are MKS units and free-space permeability of 4 πE-7 Henries/meter. You can specify Lab = MUZRO to define any system of units, then input free-space permeability.

Inputting multiple B-H curves to define temperature dependency is not supported.

For soft magnetic materials, temperature dependency of the B-H curve can be modeled by setting TBOPT = TCF on the TB,BH command. Use TBTEMP (or TBFIELD,TEMP) and TBDATA commands to define the thermal coefficient vs. temperature. The thermal coefficient is used to modify the B-H curve:

TB,BH,MATID,,,TCF
TBTEMP,TempValue(or TBFIELD,TEMP,TempValue)
TBDATA,1,C1

The C1 value should be within the range of 0 to 1.

This method of defining temperature dependency for B-H curves is not valid for permanent magnets.

For more information about this material option, see Additional Guidelines for Defining Regional Material Properties and Real Constants in the Low-Frequency Electromagnetic Analysis Guide

Current technology coupled-field elements with piezoelectric and electrostatic-structural capabilities use the TB,DPER table to input the matrix of relative electric permittivity. (See Material Model Support for Elements for Anisotropic Electric Permittivity.) Alternatively, you can use the MP command to input electric permittivity as constants PERX, PERY, and PERZ.

The full 3 x 3 electric permittivity matrix:

relates x, y, z components of electric field intensity {E} to the x, y, z components of electric flux density {D} via 6 constants. For 2D problems, a 2x2 matrix relates terms ordered x, y via three constants (ε₁₁ ε₂₂ ε₁₂). The program will make the [ε] matrix symmetric.

You can define electric permittivity at constant strain [ε^S] (TBOPT = 0) or constant stress [ε^T] (TBOPT = 1)

The program converts matrix [ε^T] to [ε^S] using piezoelectric strain and stress matrices.

Define the permittivity matrix [ε] via the TB family of commands, as follows:

3D Input

2D Input

In a piezoelectric analysis using PLANE222, PLANE223, SOLID225, SOLID226, or SOLID227, you can define temperature-dependent permittivity coefficients (with TBOPT = 0) by using tabular input: TBDATA,STLOC,%tabname%, where tabname is the name of a table array parameter (*DIM) with TEMP as a primary variable. For more information, see Defining Linear Material Properties Using Tabular Input.

Current-technology coupled-field elements with piezoelectric capability use the TB,AVIS table to input the matrix of viscosity coefficients. Anisotropic viscosity is used in a transient or harmonic piezoelectric analysis to account for anisotropic viscous damping. For a list of elements that use this model, see Material Model Support for Elements. For constitutive relations involving anisotropic viscosity, see Piezoelectrics in the Theory Reference. For more information on using TB,AVIS in a piezoelectric analysis, see Piezoelectric Analysis in the Coupled-Field Analysis Guide.

The full 6 x 6 viscosity coefficient matrix:

relates the stress and strain rate vector components.

The order of the vector components is expected to be {x, y, z, xy, yz, xz}. That is:

and ,

whereas for some published materials the order is given as {x, y, z, yz, xz, xy}. This difference requires the [η] matrix to be converted to the expected format. Use the TBDATA command to input up to 21 constants, forming the lower or upper triangle of the symmetric matrix [η]. For 2D problems, a 4 x 4 matrix relates the stress and elastic strain rate vector components ordered {x, y, z, xy} via 10 constants (η₁₁, η₂₁, η₃₁, η₄₁, η₂₂, η₃₂, η₄₂, η₃₃, η₄₃, η₄₄). Using this input, the program makes the viscosity matrix [η] symmetric. The resulting symmetric matrix should be positive semidefinite.

The TB,AVIS input can be either in the form of viscosity matrix [η] (TBOPT = 0) or fluency matrix [ζ] (TBOPT = 1). Both forms use the same data table input. The viscosity matrix [η] is typically input together with the stiffness matrix [c] (TB,ANEL or TB,ELAS,,,,AELS), while the fluency matrix [ζ] is commonly used together with the compliance matrix [s] (TB,ANEL). If the fluency option is selected with the TB,AVIS command, the program converts the fluency matrix [ζ] to viscosity matrix [η] using the stiffness or compliance matrix input on TB,ANEL. A zero-frequency limit is assumed during the fluency to viscosity conversion.

Define the viscosity coefficient matrix [η] via the TB family of commands, as follows:

3D Input

2D Input

! Viscosity coefficients for Lithium Niobate (N/m**2 s)
et11= 0.6547e-3 
et12= 0.2275e-3
et13= 0.2499e-3
et14=-0.0687e-3 
et33= 0.3377e-3    
et44= 0.1765e-3   
et66= (et11-et12)/2

! Viscosity matrix in IEEE format
!
!   [et11  et12 et13  et14  0    0 ]        
!   [et12  et11 et13 -et14  0    0 ]        
!   [et13  et13 et33    0   0    0 ]        
!   [et14 -et14  0    et44  0    0 ]        
!   [ 0    0   0        0 et44 et14]        
!   [ 0    0   0        0 et14 et66]        
!
! Viscosity matrix in Mechanical APDL format
!
!   [et11  et12 et13   0  et14  0 ]        
!   [et12  et11 et13   0 -et14  0 ]        
!   [et13  et13 et33   0   0    0 ]        
!   [ 0     0    0   et66  0  et14]        
!   [et14 -et14  0    0  et44   0 ]    
!   [ 0     0    0   et14  0  et44]     

! Anisotropic viscosity table input
tb,AVIS,1,,,0  
tbda,1,et11,et12,et13,,et14
tbda,7,et11,et13,,-et14
tbda,12,et33
tbda,16,et66,,et14
tbda,19,et44
tbda,21,et44

Reference

Viscosity coefficients for Lithium Niobate (N/m**2 s) are referenced from:

Bajak, I. L., McNab, A., Richter, J., & Wilkinson, C. D. W. (1981). Attenuation of acoustic waves in lithium niobate. The Journal of the Acoustical Society of America. 69(3), 689-95.

Current-technology coupled-field elements with piezoelectric capability use the TB,ELST table to input the matrix of elastic loss tangents (factors). Anisotropic elastic loss tangent is used in a harmonic piezoelectric analysis to account for anisotropic hysteretic damping. For elements that use this model, see Material Model Support for Elements. For constitutive relations involving anisotropic elastic loss factors, see Piezoelectrics in the Theory Reference. For more information on using TB,ELST in a piezoelectric analysis, see Piezoelectric Analysis in the Coupled-Field Analysis Guide. Isotropic elastic loss tangent (structural damping coefficient) can be input using the MP,DMPS and TB,SDAMP,,,,STRU commands.

The full 6 x 6 elastic loss tangent matrix :

is used together with the 6 x 6 elastic stiffness matrix [c] to form the imaginary part of the complex stiffness matrix (, where denotes the Hadamard product) in a harmonic piezoelectric analysis. The elastic stiffness matrix [c] can be defined by either specifying the stiffness constants (EX, EY, etc.) with MP commands or by specifying the terms of the elasticity matrix via the TB,ANEL or TB,ELAS data table. The same elastic loss tangent matrix [ψ] applies to the elastic stiffness [c] and elastic flexibility [s] forms.

The order of rows and columns in matrix [ψ] corresponds to the stress or strain vector components ordered {x, y, z, xy, yz, xz}, whereas for some published materials the order is given as {x, y, z, yz, xz, xy}. This difference requires the [ψ] matrix terms to be converted to the expected format. Use the TBDATA command to input up to 21 constants forming the lower or upper triangle of symmetric matrix [ψ]. For 2D problems, a 4 x 4 matrix is input using 10 constants (ψ₁₁, ψ₂₁, ψ₃₁, ψ₄₁, ψ₂₂, ψ₃₂, ψ₄₂, ψ₃₃, ψ₄₃, ψ₄₄) ordered {x, y, z, xy}. Using this input, the program makes the elastic loss tangent matrix [ψ] symmetric. The input of [ψ] and [c] matrices should be such that the resulting imaginary part of [c*] is positive semidefinite.

Define the elastic loss tangent matrix via the TB family of commands, as follows:

3D Input

2D Input

Current-technology coupled-field elements with piezoelectric capability use the TB,DLST table to input the matrix of dielectric loss tangents (factors). Anisotropic dielectric loss tangent is used in a harmonic piezoelectric analysis to account for anisotropic dielectric losses. For a list of elements that use this model, see Material Model Support for Elements. For constitutive relations involving anisotropic electric loss tangent, see Piezoelectrics in the Theory Reference. For more information on using TB,DLST in a piezoelectric analysis, see Piezoelectric Analysis in the Coupled-Field Analysis Guide. Isotropic electric loss tangent can be input using MP,LSST.

The full 3 x 3 dielectric loss tangent matrix:

is used together with the 3 x 3 dielectric permittivity matrix [ε] to form the imaginary part of the complex dielectric permittivity matrix (, where denotes the Hadamard product) in a harmonic piezoelectric analysis. The dielectric permittivity matrix [ε] can be defined by either specifying the permittivity constants (PERX, PERY, and PERZ) with MP commands or by specifying the terms of the permittivity matrix with the TB,DPER data table. The same dielectric loss tangent matrix [φ] applies to the permittivity at constant strain [εS] (input as MP,PERX/Y/Z or TB,DPER with TBOPT = 0) and to the permittivity at constant stress [εT] (input as TB,DPER with TBOPT = 1).

The order of rows and columns in matrix [φ] corresponds to the electric flux or electric field vector components ordered {x, y, z}. Input up to 6 constants (φ11, φ22, φ33, φ12, φ23, φ13) using the TBDATA command as shown below. For 2D problems, a 2 x 2 matrix input needs up to 3 constants (φ11, φ22, φ12). Using this input, the program makes the dielectric loss tangent matrix [ψ] symmetric.

Define the dielectric loss tangent matrix via the TB family of commands, as follows:

3D Input

2D Input