The default Mooney-Rivlin option for hyperelasticity (TB,HYPER,,,,MOONEY) enables you to define 2, 3, 5, or 9 parameters via NPTS on the TB command. For example, to define a five-parameter model, issue the TB,HYPER,1,,5,MOONEY command.

The two-parameter Mooney-Rivlin option has an applicable strain of about 100 percent in tension and 30 percent in compression. Compared to the other options, higher orders of the Mooney-Rivlin option may provide better approximation to a solution at higher strain.

TB,HYPER,1,,3,MOONEY     !Activate 3 parameter Mooney-Rivlin data table
TBDATA,1,0.163498        !Define c10
TBDATA,2,0.125076        !Define c01
TBDATA,3,0.014719        !Define c11
TBDATA,4,6.93063E-5      !Define incompressibility parameter
                         !(as 2/K, K is the bulk modulus)

See Mooney-Rivlin Hyperelastic Material (TB,HYPER) for a description of the material constants required for this option.

The Ogden option (TB,HYPER,,,,OGDEN) enables you to define an unlimited number of parameters via the NPTS argument of the TB command. For example, to define a three-parameter model, use TB,HYPER,1,,3,OGDEN.

Compared to the other options, the Ogden option usually provides the best approximation to a solution at larger strain levels. The applicable strain level can be up to 700 percent. A higher parameter value can provide a better fit to the exact solution. It may however cause numerical difficulties in fitting the material constants, and it requires enough data to cover the whole range of deformation for which you may be interested. For these reasons, a high parameter value is not recommended.

TB,HYPER,1,,2,OGDEN      !Activate 2 parameter Ogden data table
TBDATA,1,0.326996        !Define μ1
TBDATA,2,2               !Define α
1
TBDATA,3,-0.250152       !Define μ2
TBDATA,4,-2              !Define α
2
TBDATA,5,6.93063E-5      !Define incompressibility parameter
                         !(as 2/K, K is the bulk modulus)
                         !(Second incompressibility parameter d2 is zero)

Refer to Ogden Hyperelastic Material Constants for a description of the material constants required for this option.

The Neo-Hookean option (TB,HYPER,,,,NEO) represents the simplest form of strain energy potential, and has an applicable strain range of 20-30%.

TB,HYPER,1,,,NEO         !Activate Neo-Hookean data table
TBDATA,1,0.577148        !Define mu shear modulus
TBDATA,2,7.0e-5          !Define incompressibility parameter
                         !(as 2/K, K is the bulk modulus)

Refer to Neo-Hookean Hyperelastic Material for a description of the material constants required for this option.

The polynomial form option (TB,HYPER,,,,POLY) enables you to define an unlimited number of parameters through the NPTS argument of the TB command. For example, to define a 3 parameter model you would issue TB,HYPER,1,,3,POLY.

Similar to the higher order Mooney-Rivlin options, the polynomial form option may provide a better approximation to a solution at higher strain.

For NPTS = 1 and constant c₀₁ = 0, the polynomial form option is equivalent to the Neo-Hookean option (see Neo-Hookean Hyperelastic Option (TB,HYPER,,,,NEO) for an example input listing). Also, for NPTS = 1, it is equivalent to the 2 parameter Mooney-Rivlin option. For NPTS = 2, it is equivalent to the 5 parameter Mooney-Rivlin option, and for NPTS = 3, it is equivalent to the 9 parameter Mooney-Rivlin option (see Mooney-Rivlin Hyperelastic Option (TB,HYPER,,,,MOONEY) for an example input listing).

Refer to Polynomial Form Hyperelastic Material Constants for a description of the material constants required for this option.

The Arruda-Boyce option (TB,HYPER,,,,BOYCE) has an applicable strain level of up to 300%.

TB,HYPER,1,,,BOYCE      !Activate Arruda-Boyce data table
TBDATA,1,200.0           !Define initial shear modulus
TBDATA,2,5.0             !Define limiting network stretch
TBDATA,3,0.001           !Define incompressibility parameter
                         !(as 2/K, K is the bulk modulus)

Refer to Arruda-Boyce Hyperelastic Material Constants for a description of the material constants required for this option.

The Gent option (TB,HYPER,,,,GENT) has an applicable strain level of up to 300 percent.

TB,HYPER,1,,,GENT      !Activate Gent data table
TBDATA,1,3.0            !Define initial shear modulus
TBDATA,2,42.0           !Define limiting I1 - 3
TBDATA,3,0.001          !Define incompressibility parameter
                        !(as 2/K, K is the bulk modulus)

Refer to Gent Hyperelastic Material Constants for a description of the material constants required for this option.

The Yeoh option (TB,HYPER,,,,YEOH) is a reduced polynomial form of the hyperelasticity option TB,HYPER,,,,POLY. An example of a 2 term Yeoh model is TB,HYPER,1,,2,YEOH.

Similar to the polynomial form option, the higher order terms may provide a better approximation to a solution at higher strain.

For NPTS = 1, the Yeoh form option is equivalent to the Neo-Hookean option (see Neo-Hookean Hyperelastic Option (TB,HYPER,,,,NEO) for an example input listing).

TB,HYPER,1,,2,YEOH      !Activate 2 term Yeoh data table
TBDATA,1,0.163498            !Define C1
TBDATA,2,0.125076            !Define C2
TBDATA,3,6.93063E-5          !Define first incompressibility parameter

Refer to Yeoh Hyperelastic Material Constants for a description of the material constants required for this option.

The Blatz-Ko option (TB,HYPER,,,,BLATZ) is the simplest option for simulating the compressible foam type of elastomer. This option is analogous to the Neo-Hookean option of incompressible hyperelastic materials.

TB,HYPER,1,,,BLATZ      !Activate Blatz-Ko data table
TBDATA,1,5.0            !Define initial shear modulus

Refer to Blatz-Ko Foam Hyperelastic Material Constants for a description of the material constants required for this option.

The Ogden compressible foam option (TB,HYPER,,,,FOAM) simulates highly compressible foam material. An example of a 3 parameter model is TB,HYPER,1,,3,FOAM. Compared to the Blatz-Ko option, the Ogden foam option usually provides the best approximation to a solution at larger strain levels. The higher the number of parameters, the better the fit to the experimental data. It may however cause numerical difficulties in fitting the material constants, and it requires sufficient data to cover the whole range of deformation for which you may be interested. For these reasons, a high parameter value is not recommended.

TB,HYPER,1,,2,FOAM      !Activate 2 parameter Ogden foam data table
TBDATA,1,1.85              !Define μ1
TBDATA,2,4.5               !Define α
1
TBDATA,3,-9.20             !Define μ2
TBDATA,4,-4.5              !Define α
2
TBDATA,5,0.92              !Define first compressibility parameter
TBDATA,6,0.92              !Define second compressibility parameter

Refer to Ogden Compressible Foam Hyperelastic Material Constants for a description of the material constants required for this option.

The response function hyperelastic option (TB,HYPER,,,,RESPONSE) works with experimental data (TB,EXPE).

The TB,HYPER command's NPTS argument defines the number of terms in the volumetric potential function. The data table includes entries for the deformation limit cutoff for the stiffness matrix as the first entry, and the volumetric potential function incompressibility parameters starting in the third position. (The second position in the data table is unused.)

TB,HYPER,1,,2,RESPONSE      ! Activate Response Function data table
TBDATA,1,1E-4               ! Define deformation limit cutoff
TBDATA,3,0.002              ! Define first incompressibility parameter
TBDATA,4,0.00001            ! Define second incompressibility parameter

For a description of the material constants required for this option, see Response Function Hyperelasticity (TB,HYPER,,,,RESPONSE) in the Material Reference. For detailed information about response functions determined via experimental data, see Experimental Response Functions in the Mechanical APDL Theory Reference.

Experimental data for the model is entered via the TB,EXPE command, where the TBOPT argument specifies the type of data to be input:

The response function hyperelastic model must include experimental data for at least one of the listed deformations.

Any combination of uniaxial tension, equibiaxial tension, or planar shear is also valid. For incompressible and nearly incompressible materials, uniaxial compression can be used in place of equibiaxial tension.

Combined uniaxial tension and compression data can be used to model material behavior that is different in tension than in compression. When combined tension and compression data is defined, it cannot be combined with the uniaxial tension, equibiaxial tension or planar shear.

Volumetric behavior is specified with either experimental data or a polynomial volumetric potential function. Incompressible behavior results if no volumetric model or data is given.

Volumetric experimental data is input as two values per data point with volume ratio as the independent variable and pressure as the dependent variable. For uniaxial tension, uniaxial compression, equibiaxial tension, planar shear, and combined uniaxial tension and compression deformations, the experimental data is entered in either of these formats:

The input format must be consistent within the table for an individual experimental deformation, but can change between tables for different experimental deformations.

TB,EXPERIMENTAL,1,,,UNITENSION  ! Activate uniaxial data table
TBFIELD,TEMP,21                 ! Temperature for following data
TBPT,, 0.0,   0.0               ! 
TBPT,, 0.2,  1.83               ! 1st data point
TBPT,, 1.0,  5.56               ! 2nd data point
TBPT,, 4.0,  17.6               ! 3rd data point

TB,EXPERIMENTAL,1,,,SHEAR       ! Activate planar shear data table
TBFIELD,TEMP,21                 ! Temperature for following data
TBPT,,  0.0,   0.0              ! 
TBPT,, 0.24,  2.69              ! 1st data point
TBPT,, 0.96,  6.32              ! 2nd data point
TBPT,,  4.2,  19.7              ! 3rd data point
TBPT,,  5.1,  27.4              ! 4th data point

TB,EXPERIMENTAL,1,,,UNIAXIAL            ! Activate uniaxial data table
TBFIELD,TEMP,21                 ! Temperature for following data
TBPT,,   0.0,   0.0             ! 
TBPT,,  0.01,  0.0915           ! 1st tension data point
TBPT,,   0.2,  1.83             ! 2nd tension data point
TBPT,,   1.0,  5.56             ! 3rd tension data point
TBPT,, -0.01,  -0.0915          ! 1st compression data point
TBPT,,  -0.2,  -3.66            ! 2nd compression data point
TBPT,,  -1.0,  -22.24           ! 3rd compression data point

For all input data, the zero stress-strain point should be entered as a data point; otherwise, interpolation or extrapolation of the data to zero strain should yield a value of zero stress. For combined tension and compression data, the initial slope of the data should be the same in tension and compression.

Data outside the experimental strain values are assumed to be constant; therefore, all experimental data should cover the simulated deformation range as measured by the first deformation invariant (expressed by Equation 4–141 in the Mechanical APDL Theory Reference).

The following table shows various I₁ values and the corresponding engineering strains in each experimental deformation for an incompressible material:

For example, a simulation that includes deformation up to I₁ = 10.0 requires experimental data in uniaxial tension up to about 206 percent engineering strain, biaxial tension to 123 percent, uniaxial compression to -80 percent, and planar shear to 198 percent. The values in the table were obtained by solving Equation 4–197 for uniaxial tension, Equation 4–206 for biaxial tension, Equation 4–213 for planar shear (all described in the Mechanical APDL Theory Reference), and converting the biaxial tension strain to equivalent uniaxial compression strain.

Experimental data that does not include the lateral strain are assumed to be for incompressible material behavior; however, this data can be combined with a volumetric potential function to simulate the behavior of nearly incompressible materials. Combining incompressible experimental data with a volumetric model that includes significant compressibility is not restricted, but should be considered carefully before use in a simulation.

The user-defined option (TB,HYPER,,,,USER) enables you to use the subroutine UserHyper to define the derivatives of the strain-energy potential with respect to the strain invariants. For more information, see Subroutine UserHyper (Writing Your Own Isotropic Hyperelasticity Laws) in the Programmer's Reference.

The Hencky option (TB,HYPER,,,,HENCKY) represents the extension of small strain isotropic linear elasticity (Hooke’s law) to finite strains. Although the Hencky model can be used as a standalone hyperelastic material model, it is typically used with finite-strain plasticity.

TB,HYPER,1,,,HENCKY  ! Activate Hencky data table
TBDATA,1,80.19       ! Define mu shear modulus
TBDATA,2,164.21      ! Define K bulk modulus

For more information, including a description of the material model constants required for this option, see Hencky Hyperelasticity (TB,HYPER,,,,HENCKY) in the Material Reference.

To use implicit creep, create the creep material data table (TB,CREEP) and select a creep equation (TBOPT):

TB,CREEP,1,1,4,2
TBTEMP,100
TBDATA,1,C1,C2,C3,C4

You can input other creep expressions by using the UserCreep subroutine and setting TBOPT = 100.

You can define the number of state variables (TB,STATE):

TB,STATE,1,,5

You can simultaneously model creep (TB,CREEP) and isotropic, bilinear kinematic, and Hill anisotropy options to simulate more complex material behaviors. See Material Model Combinations for the combination possibilities. Also, see Material Model Combination Examples in the Material Reference.

An implicit creep analysis requires the RATE solution command with Option = ON.

Example 8.19: Time-Hardening Creep Analysis

/SOLU                 ! First load step, apply mechanical loading
RATE,OFF              ! Creep analysis disabled
TIME,1.0E-8           ! Time period set to a very small value
...
SOLV                  ! Solve this load step
                      ! Second load step, no further mechanical load
RATE,ON               ! Creep analysis enabled
TIME,100              ! Time period set to the desired value
...
SOLV                  ! Solve this load step

Mechanical loading is applied in the first load step, and RATE is set to OFF to bypass the creep strain effect. Because the time period in this load step affects the total time thereafter, the time period for this load step should be small. This example uses a value of 1.0E-8 seconds. The second load step is a creep analysis. RATE is set to ON. Here, the mechanical loading is kept constant, and the material creeps as time increases.

For most materials, the creep strain rate changes significantly at an early stage. A general recommendation, therefore, is to use a small initial incremental time step, then specify a large maximum incremental time step (DELTIM or NSUBST).

For implicit creep, examine the effect of the time increment on the results carefully, as the program does not enforce any creep ratio control by default. You can enforce a creep-limit ratio (CUTCONTROL,CRPLIMIT or CRPLIM). A recommended value for a creep-limit ratio ranges from 1 to 10. The ratio may vary according to material, so select a value based on your own experimentation to gain the required performance and accuracy. For larger analyses, it is a good idea to first perform a time-increment convergence analysis on a simple small size test.

To help you determine the coefficients for all implicit creep options, use TBFT to compare your experimental data with existing material data curves and fit your curve for use when defining the creep model (TB,CREEP). Full TBFT command capability is available except for plotting.

For more information, see Implicit Creep Equations in the Material Reference.

To use explicit creep, create the creep material data table (TB,CREEP with TBOPT = 0) and select a creep equation by adding the appropriate constant as an argument (TBDATA):

TB,CREEP,1 , , ,0
TBDATA,1,C1,C2,C3,C4, ,C6

You can input other creep expressions by using the UserCr subroutine and defining constant C6 = 100 (TBDATA).

You can define the number of state variables (TB,STATE):

TB,STATE,1,,5

For highly nonlinear creep strain vs. time curves, explicit creep requires a small time step. Creep strains are not calculated if the time step is less than 1.0e-6. A creep time-step optimization procedure is available (AUTOTS and CUTCONTROL,CRPLIMIT ) for adjusting the time step automatically as appropriate.

For more information, see Explicit Creep Equations.