Use rate-dependent plasticity to model the rate-dependent plastic behavior of materials. Similar to rate-independent plasticity, the material remains elastic when the stress is below the yield stress. Above the yield stress, the material behavior is similar to creep and higher plastic deformation rates are proportional to a higher equivalent stress in the material.
The following topics related to rate-dependent plasticity are available:
For further information about rate-dependent plastic (viscoplastic) material options, see Viscoplasticity Model in the Structural Analysis Guide.
The Perzyna model has the following form:
The Peirce model has the following form:
where 
is the number of terms, 
is the effective stress, 
is the equivalent plastic strain rate, 
are the strain rate hardening parameters, 
are the material viscosity parameters, and 
is the static yield stress.
As the material viscosity () approaches 
, the strain rate hardening () approaches zero, or the equivalent plastic strain rate
() approaches zero, the solution approaches the static
(rate-independent) solution.
When the strain rate hardening is very small, the Peirce model has less difficulty converging as compared to the Perzyna model.
These models can be combined with an isotropic hardening model for hardening of the
static yield stress (). Anisotropic viscoplastic behavior is simulated by combining the
Hill yield surface.
For the Perzyna and Peirce constants, the number of terms () is determined from the number of material parameters defined in the
table. Up to 20 terms are allowed and the material constants (defined via
TBDATA) are:
| Constant | Meaning | Property | Unit | Range |
|---|---|---|---|---|
| C1 |
| Strain rate hardening exponent | Dimensionless |
|
| C2 |
| Viscosity | 1 / Time |
|
| . . . | . . . | . . . | . . . | . . . |
| C(2 -1) |
| Strain rate hardening exponent | Dimensionless |
|
| C(2) |
| Viscosity | 1 / Time |
|
The EVH model has the following form:
where is the number of terms, is the effective stress, is the equivalent plastic strain rate, are the strain rate hardening parameters, 
are the material viscosity parameters, and 
is the static yield stress:
Where 
is the initial yield stress, 
is the linear hardening coefficient, 
is the exponential hardening coefficient, and 
is the saturation rate.
For the EVH constants, the number of terms () is determined from the number of material parameters defined in the
table. Up to 20 terms are allowed and the material constants (defined via
TBDATA) are:
| Constant | Meaning | Property | Unit | Range |
|---|---|---|---|---|
| C1 |
| Initial yield stress | Stress |
|
| C2 |
| Linear hardening coefficient | Stress |
|
| C3 |
| Exponential hardening coefficient | Stress | -- |
| C4 |
| Exponential saturation rate | Dimensionless |
|
| C5 |
| Strain rate hardening exponent | Dimensionless |
|
| C6 |
| Viscosity | Stress × Timem1 |
|
| . . . | . . . | . . . | . . . | . . . |
| C(2+3) |
| Strain rate hardening exponent | Dimensionless |
|
| C(2+4) |
| Viscosity | Stress × Timem1 |
|
Specify the EVH model as follows: TB,RATE,,,,EVH
Equivalent plastic strain rate-dependent isotropic hardening can be defined for bilinear isotropic hardening (TB,PLASTIC,,,,BISO), multilinear isotropic hardening (TB,PLASTIC,,,,MISO), and nonlinear isotropic hardening (TB,NLISO,,,,VOCE / POWER).
Specify the desired isotropic hardening model (via the appropriate TB command), then define tabular data at various equivalent plastic strain rates (TBFIELD,PLSR). It is good practice to begin with rate-independent data at zero equivalent plastic strain rate.
The isotropic hardening data can be a function of other supported field variables (such as TEMP) in addition to the equivalent plastic strain rate.
Example 4.22: Rate-Dependent Multilinear Isotropic Hardening
TB,PLAS,1,,,MISO ! Define miso curves for different rates TBFIELD,PLSR,0. ! Define miso curve for zero rate TBPT,DEFI,0.0,15.0 TBPT… TBFIELD,PLSR,rate2 ! Define miso curve for rate2 TBPT,DEFI,0.0,18.624 TBPT… TBFIELD,PLSR,rate3 ! Define miso curve for rate3 TBPT,DEFI,0.0,20.483 TBPT… …
The direct tabular entry rate-dependent model supports valid material model combinations. You can use it as an alternative to rate-dependent isotropic hardening combinations of TB,PLASTIC,,,,BISO, TB,PLASTIC,,,,MISO, and TB,NLISO,,,,VOCE / POWER with TB,RATE.
The Anand option offers a unified plasticity model requiring no combination with other material models.
Details for the Anand option appear in Anand Viscoplasticity Option in the Mechanical APDL Theory Reference.
This option requires nine material constants input via the data-table command (TBDATA) in the order shown:
| Constant | Meaning | Property | Units |
|---|---|---|---|
| C1 | s0 | Initial value of deformation resistance | Stress |
| C2 | Q/R | Q = Activation energy | Energy / Volume |
| R = Universal gas constant | Energy / Volume temperature | ||
| C3 | A | Pre-exponential factor | 1 / Time |
| C4 | xi | Stress multiplier | Dimensionless |
| C5 | m | Strain rate sensitivity of stress | Dimensionless |
| C6 | h0 | Hardening / softening constant | Stress |
| C7 |
| Coefficient for deformation resistance saturation value | Stress |
| C8 | n | Strain rate sensitivity of saturation (deformation resistance) value | Dimensionless |
| C9 | a | Strain rate sensitivity of hardening or softening | Dimensionless |
Specify this option (TBOPT = ANAND)
as follows: TB,RATE,,,9,ANAND
The Anand model supports plane strain, axisymmetric and full three-dimensional element behavior.
The creep strain rate, 
, can be a function of stress, strain, temperature, and neutron flux
level. Libraries of creep strain rate equations are included in Implicit Creep Equations.
Enter the constants shown in these equations using TB,CREEP and
TBDATA as described below. These equations (expressed in
incremental form) are characteristic of materials being used in creep design
applications (see the Mechanical APDL Theory Reference for details).
For a list of the elements that support creep behavior, see Material Model Support for Elements.
Three types of creep equations are available:
Primary creep
Secondary creep
Irradiation induced creep
You can define the combined effects of more than one type of creep using the implicit
equations specified via TBOPT = 11 or 12, or a user-defined
creep equation.
Mechanical APDL analyzes creep via the implicit method. It is robust, fast, accurate, and
recommended for general use, especially with problems involving large creep strain and
large deformation. The method has provisions for including temperature-dependent
constants. (Temperature dependency can also be incorporated via the Arrhenius option
(TBFT,,,,Option4).)
The program can model pure creep, creep with isotropic hardening plasticity, and creep with kinematic hardening plasticity, using both von Mises and Hill potentials. (See Material Model Combinations.) Where the material model combination with creep is supported, any of the implicit creep models are valid.
The following topics about the creep equations are available:
Also see Creep Model in the Structural Analysis Guide.
Ansys, Inc. recommends using implicit creep over explicit creep for better computational accuracy and efficiency.
Enter an implicit creep equation via TBOPT
(TB). Enter the value of TBOPT
corresponding to the equation:
Table 4.2: Implicit Creep Equations
|
Creep Model
( | Name | Equation | Type | |
|---|---|---|---|---|
| 1 | Strain Hardening |
| C1>0 | Primary |
| 2 | Time Hardening |
| C1>0 | Primary |
| 3 | Generalized Exponential |
 , 
| C1>0, C5>0 | Primary |
| 4 | Generalized Graham |
| C1>0 | Primary |
| 5 | Generalized Blackburn |
| C1>0, C3>0, C6>0 | Primary |
| 6 | Modified Time Hardening |
| C1>0, C3>-1 | Primary |
| 7 | Modified Strain Hardening |
| C1>0 | Primary |
| 8 | Generalized Garofalo |
| C1>0 | Secondary |
| 9 | Exponential form |
| C1>0 | Secondary |
| 10 | Norton |
| C1>0 | Secondary |
| 11 | Combined Time Hardening |
| C1>0, C3>-1, C5>0 | Primary + Secondary |
| 12 | Rational polynomial |
| C2>0 | Primary + Secondary |
| 13 | Generalized Time Hardening |
| Primary | |
| 100 | --- | User Creep | --- | --- |
where:
| εcr = equivalent creep strain |
 = change in equivalent creep strain with respect to
time |
| σ = equivalent stress |
| T = temperature (absolute). The offset temperature (from TOFFST), is internally added to all temperatures for convenience. |
| C1 through C12 = constants defined via TBDATA |
| t = time at end of substep |
| e = natural logarithm base |
Specify the user-defined creep option by setting
TBOPT = 100 and using TB,STATE to
specify the number of state variables for the user-defined creep subroutine.
RATE is necessary to activate implicit creep for specific
elements.
For temperature-dependent constants, define the temperature using
TBTEMP for each set of data. Then, define constants C1 through
Cn using TBDATA (where
n is the number of constants, and depends on the selected
creep model).
The following example shows how you would define the implicit creep model
represented by TBOPT = 1 at two temperature
points.
TB,CREEP,1,,,1 ! Activate creep data table, specify creep model 1 TBTEMP,100 ! Define first temperature TBDATA,1,c11,c12,c13,c14 ! Creep constants c11, c12, c13, c14 at first temp. TBTEMP,200 ! Define second temperature TBDATA,1,c21,c22,c23,c24 ! Creep constants c21, c22, c23, c24 at second temp.
By default, coefficients are linearly interpolated for temperatures that fall between user-defined TBTEMP values. For some creep models, where the change in coefficients spans several orders of magnitude, this linear interpolation might introduce inaccuracies in solution results. Use enough curves to accurately capture the temperature dependency.
In some cases, a linear-logarithmic interpolation of certain parameters may be better suited. The default interpolation method can therefore be modified (TBIN). For example, to use this method for parameter C1 (linear in temperature, logarithmic in C1), the input is as follows:
TB,CREEP,1,,,1 ! Activate creep data table, specify creep model 1 TBIN,SCAL,TEMP,1,LINE,LOG ! Switch to linear-logarithmic interpolation for parameter C1 TBTEMP,100 ! Define first temperature TBDATA,1,c11,c12,c13,c14 ! Creep constants c11, c12, c13, c14 at first temp. TBTEMP,200 ! Define second temperature TBDATA,1,c21,c22,c23,c24 ! Creep constants c21, c22, c23, c24 at second temp.
The other parameters continue to use linear-linear interpolation.
When a temperature is above or below the range of defined temperature values, the coefficients for the nearest temperature are used with no extrapolation.
For a list of elements that can be used with this material option, see Material Model Support for Elements.
See Creep in the Structural Analysis Guide for more information about this material option.
Enter an explicit creep equation by
setting TBOPT = 0 (or by leaving
TBOPT blank) within the TB command,
then specifying the constants associated with the creep equations
(TBDATA).
Specify primary creep with constant C6. Primary Explicit Creep Equation for C6 = 0, through Primary Explicit Creep Equation for C6 = 100, show the available
equations. Select an equation with the appropriate value of
C6 (0 to 15). If C1
0, or if T + Toffset 
0, no primary creep is calculated.
Specify secondary creep with constant C12. Secondary Explicit Creep Equation for C12 = 0 and Secondary Explicit Creep Equation for C12 = 1 show the available
equations. Select an equation with the appropriate value of
C12 (0 or 1). If C7
0, or if T + Toffset 
0, no secondary creep is calculated. Also, primary creep equations
C6 = 9, 10, 11, 13, 14, and 15 bypass any secondary creep
equations, as secondary effects are included in the primary part.
Specify irradiation induced creep with constant C66. Irradiation Induced Explicit Creep Equation for C66 =
5 shows the single equation currently available; select it
with C66 = 5. This equation can be used with equations
C6 = 0 to 11. Enter the constants into the data table as
indicated by their subscripts. If C55 
0 and C61 
0, or if T + Toffset 
0, no irradiation-induced creep is calculated.
A linear stepping function is used to calculate the change in the creep strain
within a time step (Δ εcr = (
)(Δt)). The creep strain rate is evaluated at the condition
corresponding to the beginning of the time interval and is assumed to remain
constant over the time interval. If the time step is less than 1.0e-6, then no creep
strain increment is calculated. Primary equivalent stresses and strains are used to
evaluate the creep strain rate. For highly nonlinear creep strain vs. time curves,
specify a small time step if using the explicit creep algorithm. A creep time-step
optimization procedure is available for increasing the time step automatically
whenever possible. A nonlinear stepping function (based on an exponential decay) is
also available (C11 = 1), used it with caution, as it can
underestimate the total creep strain where primary stresses dominate. This function
is available only for creep equations C6 = 0, 1 and 2.
Specify temperatures used in the creep equations based on an absolute scale
(TOFFST).
Enter temperature and fluence values via BF or BFE. The input fluence (Φt) includes the integrated effect of time. Time input explicitly is not used in fluence calculation. For the usual case of a constant flux (Φ), linearly ramp fluence change.
Temperature-dependent creep constants are not valid for explicit creep. Incorporate other creep options by setting C6 = 100.
Example 4.24: Using the Explicit Creep Equation Defined by C6 =
1
TB,CREEP,1 ! Activate creep data table TBDATA,1,c1,c2,c3,c4,,1 !Creep constants c1, c2, c3, c4 for equation C6=1
The explicit creep constants (entered via TBDATA):
| Constant | Meaning |
|---|---|
| C1-CN | Constants C1, C2, C3, etc. (as defined in Primary Explicit Creep Equation for C6 = 0 to Irradiation Induced Explicit Creep Equation for C66 = 5) These are obtained by curve-fitting test results for your material to the equation you choose. Exceptions are defined below. |
Primary Explicit Creep Equation for C6 = 0
where:
 = change in equivalent strain with respect to time |
 = equivalent strain (sum of elastic and creep strain) |
 = equivalent stress |
 = temperature (absolute). The offset temperature (from
TOFFST) is internally added to all temperatures for
convenience. |
 = time at end of substep |
 = natural logarithm base |
Primary Explicit Creep Equation for C6 = 1
Primary Explicit Creep Equation for C6 = 2
where:
Primary Explicit Creep Equation for C6 = 9
Annealed 304 Stainless Steel:
Double Exponential Creep Equation (C4 = 0)
where:
εx = 0 for σ C2
εx = G + H σ for C2 < σ C3
C2 = 6000 psi (default), C3 = 25000 psi (default) s, r, , G, and H = functions of temperature and stress as described in the reference.
This double exponential equation is valid for Annealed 304 Stainless Steel over a temperature range from 800 to 1100°F. The equation, known as the Blackburn creep equation when C1 = 1, is described completely in the High Alloy Steels. The first two terms describe the primary creep strain and the last term describes the secondary creep strain.
To use this equation, input a nonzero value for C1, C6 = 9.0, and C7 = 0.0. Temperatures should be in °R (or °F with Toffset = 460.0). Conversion to °K for the built-in property tables is done internally. If the temperature is below the valid range, no creep is calculated. Time should be in hours and stress in psi. The valid stress range is 6,000 - 25,000 psi.
Rational Polynomial Creep Equation with Metric Units (C4 = 1)
To use the following standard Rational Polynomial creep equation (with metric units) to calculate εc, enter C4 = 1.0:
where:
c = limiting value of primary creep strain p = primary creep time factor = secondary (minimum) creep strain rate
This standard rational polynomial creep equation is valid for Annealed 304 SS over a temperature range from 427°C to 704°C. The equation is described completely in the High Alloy Steels. The first term describes the primary creep strain. The last term describes the secondary creep strain. The average "lot constant" is used to calculate
.
To use this equation, input C1 = 1.0, C4 = 1.0, C6 = 9.0, and C7 = 0.0. Temperature must be in °C and Toffset must be 273 (because of the built-in property tables). If the temperature is below the valid range, no creep is calculated. Also, time must be in hours and stress in Megapascals (MPa).
Various hardening rules governing the rate of change of creep strain during load reversal may be selected with the C5 value: 0.0 - time hardening, 1.0 - total creep strain hardening, 2.0 - primary creep strain hardening. These options are available only with the standard rational polynomial creep equation.
Rational Polynomial Creep Equation with English Units (C4 = 2)
To use the above standard Rational Polynomial creep equation (with English units), enter C4 = 2.0.
This standard rational polynomial equation is the same as described above except that temperature must be in °F, Toffset must be 460, and stress must be in psi. The equivalent valid temperature range is 800 - 1300°F.
Primary Explicit Creep Equation for C6 = 10
Annealed 316 Stainless Steel:
Double Exponential Creep Equation (C4 = 0)
To use the same form of the Double Exponential creep equation as described for Annealed 304 SS (C6 = 9.0, C4 = 0.0) in Primary Explicit Creep Equation for C6 = 9 to calculate εc, enter C4 = 0.0.
This equation, also described in High Alloy Steels, differs from the Annealed 304 SS equation in that the built-in property tables are for Annealed 316 SS, the valid stress range is 4000 - 30,000 psi, C2 defaults to 4000 psi, C3 defaults to 30,000 psi, and the equation is called with C6 = 10.0 instead of C6 = 9.0. Temperatures should be in °R (or °F with Toffset = 460.0). Conversion to °K for the built-in property tables is done internally. If the temperature is below the valid range (800 to 1100 °F), no creep is calculated.
Rational Polynomial Creep Equation with Metric Units (C4 = 1)
To use the same form of the standard Rational Polynomial creep equation with metric units as described for Annealed 304 SS (C6 = 9.0, C4 = 1.0) in Primary Explicit Creep Equation for C6 = 9, enter C4 = 1.0.
This standard rational polynomial equation, also described in High Alloy Steels, differs from the Annealed 304 SS equation in that the built-in property tables are for Annealed 316 SS, the valid temperature range is 482 - 704°C, and the equation is called with C6 = 10.0 instead of C6 = 9.0. The hardening rules for load reversal described for the C6 = 9.0 standard Rational Polynomial creep equation are also available. The average "lot constant" from High Alloy Steels is used in the calculation of
.
Rational Polynomial Creep Equation with English Units (C4 = 2)
To use the previous standard Rational Polynomial creep equation with English units, enter C4 = 2.0.
This standard rational polynomial equation is the same as described above except that the temperatures must be in °F, Toffset must be 460, and the stress must be in psi (with a valid range from 0.0 to 24220 psi). The equivalent valid temperature range is 900 - 1300°F.
Primary Explicit Creep Equation for C6 = 11
Annealed 2 1/4 Cr - 1 Mo Low Alloy Steel:
Modified Rational Polynomial Creep Equation (C4 = 0)
To use the following Modified Rational Polynomial creep equation to calculate εc, enter C4 = 0.0:
A, B, and
are functions of temperature and stress as described in the reference.
This modified rational polynomial equation is valid for Annealed 2 1/4 Cr -1 Mo Low Alloy steel over a temperature range of 700 - 1100°F. The equation is described completely in the Low Alloy Steels. The first term describes the primary creep strain and the last term describes the secondary creep strain. No modification is made for plastic strains.
To use this equation, input C1 = 1.0, C6 = 11.0, and C7 = 0.0. Temperatures must be in °R (or °F with Toffset = 460.0). Conversion to °K for the built-in property tables is done internally. If the temperature is below the valid range, no creep is calculated. Time should be in hours and stress in psi. Valid stress range is 1000 - 65,000 psi.
Rational Polynomial Creep Equation with Metric Units (C4 = 1)
To use the following standard Rational Polynomial creep equation (with metric units) to calculate εc, enter C4 = 1.0:
where:
c = limiting value of primary creep strain p = primary creep time factor = secondary (minimum) creep strain rate
This standard rational polynomial creep equation is valid for Annealed 2 1/4 Cr - 1 Mo Low Alloy Steel over a temperature range from 371°C to 593°C. The equation is described completely in the Low Alloy Steels. The first term describes the primary creep strain and the last term describes the secondary creep strain. No tertiary creep strain is calculated. Only Type I (and not Type II) creep is supported. No modification is made for plastic strains.
To use this equation, input C1 = 1.0, C4 = 1.0, C6 = 11.0, and C7 = 0.0. Temperatures must be in °C and Toffset must be 273 (because of the built-in property tables). If the temperature is below the valid range, no creep is calculated. Also, time units are specified in hours and stress units in Megapascals (MPa). The hardening rules for load reversal described for the C6 = 9.0 standard Rational Polynomial creep equation are also available.
Rational Polynomial Creep Equation with English Units (C4 = 2)
To use the above standard Rational Polynomial creep equation with English units, enter C4 = 2.0.
This standard rational polynomial equation is the same as described above except that temperatures must be in °F, Toffset must be 460, and stress must be in psi. The equivalent valid temperature range is 700 - 1100°F.
Primary Explicit Creep Equation for C6 = 12
where:
| C1 = Scaling constant |
| M, N, K = Function of temperature (determined by linear interpolation within table) as follows: |
| Constant | Meaning |
|---|---|
| C5 | Number of temperature values to describe M, N, or K function (2 minimum, 6 maximum) |
| C49 | First absolute temperature value |
| C50 | Second absolute temperature value |
| ... | |
| C48 + C5 | C5th absolute temperature value |
| C48 + C5 + 1 | First M value |
| ... | |
| C48 + 2C5 | C5th M value |
| C48 + 2C5 | C5th M value |
| ... | |
| C48 + 2C5 | C5th M value |
| C48 + 2C5 + 1 | First N value |
| ... | |
| C48 + 3C5 | C5th N value |
| C48 + 3C5 + 1 | First K value |
| ... | |
This power function creep law having temperature dependent coefficients is similar to Equation C6 = 1.0 except with C1 = f1(T), C2 = f2(T), C3 = f3(T), and C4 = 0. Temperatures must not be input in decreasing order.
Primary Explicit Creep Equation for C6 Equals 13
Sterling Power Function:
where:
| εacc = creep strain accumulated to this time (calculated by the program). Internally set to 1 x 10-5 at the first substep with nonzero time to prevent division by zero. |
| A = C1/T |
| B = C2/T + C3 |
| C = C4/T + C5 |
This equation is often referred to as the Sterling Power Function creep equation. Constant C7 should be 0.0. Constant C1 should not be 0.0, unless no creep is to be calculated.
Primary Explicit Creep Equation for C6 = 14
where:
εc = cpt/(1+pt) +
 |
| ln c = -1.350 - 5620/T - 50.6 x 10-6 σ + 1.918 ln (σ/1000) |
| ln p = 31.0 - 67310/T + 330.6 x 10-6 σ - 1885.0 x 10-12 σ2 |
ln  = 43.69 - 106400/T + 294.0 x
10-6 σ + 2.596 ln (σ/1000) |
This creep law is valid for Annealed 316 SS over a temperature range from 800°F to 1300°F. The equation is similar to that given for C6 = 10.0 and is also described in High Alloy Steels.
To use equation, input C1 = 1.0 and C6 = 14.0. Temperatures should be in °R (or °F with Toffset = 460). Time should be in hours. Constants are only valid for English units (pounds and inches). Valid temperature range: 800° - 1300°F. Maximum stress allowed for ec calculation: 45,000 psi; minimum stress: 0.0 psi. If T + Toffset < 1160, no creep is calculated.
Primary Explicit Creep Equation for C6 = 15
General Material Rational Polynomial:
where:
|
 |
 |
 |
This rational polynomial creep equation is a generalized form of the standard rational polynomial equations given as C6 = 9.0, 10.0, and 11.0 (C4 = 1.0 and 2.0). This equation reduces to the standard equations for isothermal cases. The hardening rules for load reversal described for the C6 = 9.0 standard Rational Polynomial creep equation are also available.
Primary Explicit Creep Equation for C6 = 100
A user-defined creep equation is used. Issue TB,STATE to specify the number of state variables for the UserCr subroutine.
Secondary Explicit Creep Equation for C12 = 0
where:
| σ = equivalent stress |
| T = temperature (absolute). The offset temperature (from TOFFST), is internally added to all temperatures for convenience. |
| t = time |
| e = natural logarithm base |
Secondary Explicit Creep Equation for C12 = 1
Irradiation Induced Explicit Creep Equation for C66 = 5
where:
| B = FG + C63 |
 |
 |
| σ = equivalent stress |
| T = temperature (absolute). The offset temperature (from TOFFST) is internally added to all temperatures for convenience. |
| Φ t0.5 = neutron fluence (input via BF or BFE) |
| e = natural logarithm base |
| t = time |
This irradiation induced creep equation is valid for 20% Cold Worked 316 SS over a temperature range from 700° to 1300°F. Constants 56, 57, 58 and 62 must be positive if the B term is included.
Geomaterials under long-term loading can exhibit both plasticity and creep. You can combine the Extended Drucker-Prager and Extended Drucker-Prager Cap plasticity models with the creep models to simulate this behavior.
For more information, see Extended Drucker-Prager (EDP) Creep Model and Cap Creep Model in the Theory Reference.