## 4.4. Rate-Independent Plasticity

Plasticity is used to model materials subjected to loading beyond their elastic limit. As shown in the following figure, metals and other materials such as soils often have an initial elastic region in which the deformation is proportional to the load, but beyond the elastic limit a nonrecoverable plastic strain develops:

Unloading recovers the elastic portion of the total strain, and if the load is completely removed, a permanent deformation due to the plastic strain remains in the material. Evolution of the plastic strain depends on the load history such as temperature, stress, and strain rate, as well as internal variables such as yield strength, back stress, and damage.

To simulate elastic-plastic material behavior, several constitutive models for plasticity are provided. The models range from simple to complex. The choice of constitutive model generally depends on the experimental data available to fit the material constants.

The following rate-independent plasticity material model topics are available:

### 4.4.1. Understanding the Plasticity Models

The constitutive models for elastic-plastic behavior start with a decomposition of the total strain into elastic and plastic parts and separate constitutive models are used for each. The essential characteristics of the plastic constitutive models are:

• The yield criterion that defines the material state at the transition from elastic to elastic-plastic behavior.

• The flow rule that determines the increment in plastic strain from the increment in load.

• The hardening rule that gives the evolution in the yield criterion during plastic deformation.

The following topics concerning plasticity theory, behavior and model definition are available:

#### 4.4.1.1. Nomenclature

Following are the common symbols used in the rate-independent plasticity theory documentation:

SymbolDefinitionSymbolDefinition
Identity tensorAnisotropic directional yield strength
StrainYoung's Modulus
Elastic strainElasto-plastic tangent
Plastic strainElasto-plastic tangent in direction i
Plastic strain componentsPlastic tangent
Effective plastic strainPlastic tangent in direction i
Accumulated equivalent plastic strainHill yield surface coefficients
StressHill yield surface directional yield ratio
Stress componentsGeneralize Hill yield surface coefficients
Principal stressesGeneralized Hill constant
Stress minus back stressGeneralized Hill tensile and compressive yield strength
Yield stressPlastic work
Anisotropic yield stress in direction iUniaxial plastic work
Initial yield stressDrucker-Prager yield surface constant
Initial yield stress in direction iDrucker-Prager plastic potential constant
Equivalent plastic stressMohr-Coulomb cohesion
Von Mises effective stressMohr-Coulomb internal friction angle
User input strain-stress data pointMohr-Coulomb flow internal friction angle
Magnitude of plastic strain incrementExtended Drucker-Prager yield surface pressure sensitivity
Effective stress functionExtended Drucker-Prager plastic potential pressure sensitivity
Yield functionExtended Drucker-Prager power law yield exponent
Plastic potentialExtended Drucker-Prager power law plastic potential exponent
Translation of yield surface (back stress)Extended Drucker-Prager hyperbolic yield constant
Set of material internal variablesExtended Drucker-Prager hypobolic plastic potential constant

#### 4.4.1.2. Strain Decomposition

From Figure 4.1: Stress-Strain Curve for an Elastic-Plastic Material, a monotonic loading to gives a total strain . The total strain is additively decomposed into elastic and plastic parts:

The stress is proportional to the elastic strain :

and the evolution of plastic strain is a result of the plasticity model.

For a general model of plasticity that includes arbitrary load paths, the flow theory of plasticity decomposes the incremental strain tensor into elastic and plastic strain increments:

The increment in stress is then proportional to the increment in elastic strain, and the plastic constitutive model gives the incremental plastic strain as a function of the material state and load increment.

#### 4.4.1.3. Yield Criterion

The yield criterion is a scalar function of the stress and internal variables and is given by the general function:

 (4–4)

where represents a set of history dependent scalar and tensor internal variables.

Equation 4–4 is a general function representing the specific form of the yield criterion for each of the plasticity models. The function is a surface in stress space and an example, plotted in principal stress space, as shown in this figure:

Stress states inside the yield surface are given by and result in elastic deformation. The material yields when the stress state reaches the yield surface and further loading causes plastic deformation. Stresses outside the yield surface do not exist and the plastic strain and shape of the yield surface evolve to maintain stresses either inside or on the yield surface.

#### 4.4.1.4. Flow Rule

The evolution of plastic strain is determined by the flow rule:

where is the magnitude of the plastic strain increment and is the plastic potential.

When the plastic potential is the yield surface from Equation 4–4, the plastic strain increment is normal to the yield surface and the model has an associated flow rule, as shown in this figure:

These flow rules are typically used to model metals and give a plastic strain increment that is proportional to the stress increment. If the plastic potential is not proportional to the yield surface, the model has a non-associated flow rule, typically used to model soils and granular materials that plastically deform due to internal frictional sliding. For non-associated flow rules, the plastic strain increment is not in the same direction as the stress increment.

Non-associated flow rules result in an unsymmetric material stiffness tensor. Unsymmetric analysis can be set via the NROPT command. For a plastic potential that is similar to the yield surface, the plastic strain direction is not significantly different from the yield surface normal, and the degree of asymmetry in the material stiffness is small. In this case, a symmetric analysis can be used, and a symmetric material stiffness tensor will be formed without significantly affecting the convergence of the solution.

#### 4.4.1.5. Hardening

The yield criterion for many materials depends on the history of loading and evolution of plastic strain. The change in the yield criterion due to loading is called hardening and is defined by the hardening rule. Hardening behavior results in an increase in yield stress upon further loading from a state on the yield surface so that for a plastically deforming material, an increase in stress is accompanied by an increase in plastic strain.

Two common types of hardening rules are isotropic and kinematic hardening. For isotropic hardening, the yield surface given by Equation 4–4 has the form:

where is a scalar function of stress and is the yield stress.

Plastic loading from to increases the yield stress and results in uniform increase in the size of the yield surface, as shown in this figure:

This type of hardening can model the behavior of materials under monotonic loading and elastic unloading, but often does not give good results for structures that experience plastic deformation after a load reversal from a plastic state.

For kinematic hardening, the yield surface has the form:

where is the back stress tensor.

As shown in the following figure, the back stress tensor is the center (or origin) of the yield surface, and plastic loading from to results in a change in the back stress and therefore a shift in the yield surface:

Kinematic hardening is observed in cyclic loading of metals. It can be used to model behavior such as the Bauschinger effect, where the compressive yield strength reduces in response to tensile yielding. It can also be used to model plastic ratcheting, which is the buildup of plastic strain during cyclic loading.

Many materials exhibit both isotropic and kinematic hardening behavior, and these hardening rules can be used together to give the combined hardening model. Other hardening behaviors include changes in the shape of the yield surface in which the hardening rule affects only a local region of the yield surface, and softening behavior in which the yield stress decreases with plastic loading.

#### 4.4.1.6. Large Deformation

The plasticity constitutive models are applicable in both small-deformation and large-deformation analyses. For small deformation, the formulation uses engineering stress and strain. For large deformation (NLGEOM,ON), the constitutive models are formulated with the Cauchy stress and logarithmic strain.

#### 4.4.1.7. Output

Output quantities specific to the plastic constitutive models are available for use in the POST1 database postprocessor (/POST1) and in the POST26 time-history results postprocessor (/POST26).

The equivalent stress (label SEPL) is the current value of the yield stress evaluated from the hardening model.

The accumulated plastic strain (label EPEQ) is a path-dependent summation of the plastic strain rate over the history of the deformation:

where is the magnitude of the plastic strain rate.

The stress ratio (label SRAT) is the ratio of the elastic trial stress to the current yield stress and is an indicator of plastic deformation during an increment. If the stress ratio is:

> 1

A plastic deformation occurred during the increment.

< 1

An elastic deformation occurred during the increment.

1

The stress state is on the yield surface.

Alternatively, the output quantities can have special meanings specific to the given material model. For example, in the generalized Hill model, the equivalent plastic strain is given by:

and the equivalent stress is:

For the extended Drucker-Prager model, the accumulated plastic strain (EPEQ) is the summation of:

where is the magnitude of the deviatoric plastic strain increment and is the magnitude of the volumetric plastic strain increment.

#### 4.4.1.8. Resources

1. Hill, R. The Mathematical Theory of Plasticity. New York: Oxford University Press, 1983.

2. Prager, W. “The Theory of Plasticity: A Survey of Recent Achievements.” Proceedings of the Institution of Mechanical Engineers. 169.1 (1955): 41-57.

3. Besseling, J. F. “A Theory of Elastic, Plastic, and Creep Deformations of an Initially Isotropic Material Showing Anisotropic Strain-Hardening, Creep Recovery, and Secondary Creep.” ASME Journal of Applied Mechanics. 25 (1958): 529-536.

4. Owen, D. R. J., A.Prakash, O. C. Zienkiewicz. “Finite Element Analysis of Non-Linear Composite Materials by Use of Overlay Systems.” Computers and Structures. 4.6 (1974): 1251-1267.

5. Rice, J. R. “Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms.” Constitutive Equations in Plasticity. Ed. A. Argon. Cambridge, MA: MIT Press, 1975. 23-79.

6. Chaboche, J. L. “Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity.” International Journal of Plasticity. 5.3 (1989): 247-302.

7. Chaboche, J. L. “On Some Modifications of Kinematic Hardening to Improve the Description of Ratchetting Effects.” International Journal of Plasticity. 7.7 (1991): 661-678.

8. Shih, C. F., D. Lee. “Further Developments in Anisotropic Plasticity.” Journal of Engineering Materials and Technology. 100.3 (1978): 294-302.

9. Valliappan, S., P. Boonlaulohr, I. K. Lee. “Non-Linear Analysis for Anisotropic Materials.” International Journal for Numerical Methods in Engineering. 10.3 (1976): 597-606.

10. Drucker, D. C., W. Prager. “Soil Mechanics and Plastic Analysis or Limit Design.” Quarterly of Applied Mathematics. 10.2 (1952): 157-165.

11. Sandler, I. S, F. L. DiMaggio, G. Y. Baladi. “A Generalized Cap Model for Geological Materials.” Journal of the Geotechnical Engineering Division. 102.7 (1976): 683-699.

12. Schwer, L. E., Y. D. Murray. “A Three-Invariant Smooth Cap Model with Mixed Hardening.” International Journal for Numerical and Analytical Methods in Geomechanics. 18.10 (1994): 657-688.

13. Foster, C., R. Regueiro, A. Fossum, R. Borja. “Implicit Numerical Integration of a Three-Invariant, Isotropic/Kinematic Hardening Cap Plasticity Model for Geomaterials.” Computer Methods in Applied Mechanics and Engineering. 194.50-52 (2005): 5109-5138.

14. Pelessone, D. “A Modified Formulation of the Cap Model.” Technical Report GA-C19579. San Diego: Gulf Atomics, 1989.

15. Fossum, A.F., J. T. Fredrich. “Cap Plasticity Models and Compactive and Dilatant Pre-Failure Deformation.” Pacific Rocks 2000: Rock Around the Rim. Proceedings of the Fourth North American Rock Mechanics Symposium. Eds. J. Girard, M. Liebman, C. Breeds, T. Doe, A. A. Balkema. Rotterdam, 2000: 1169-1176.

16. Gurson, A. L. “Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I--Yield Criteria and Flow Rules for Porous Ductile Media.” Journal of Engineering Materials and Technology. 99.1 (1977): 2-15.

17. Needleman, A. V. Tvergaard. “An Analysis of Ductile Rupture in Notched Bars.” Journal of the Mechanics and Physics of Solids. 32.6 (1984): 461-490.

18. Arndt, S., B. Svendsen, D. Klingbeil. “Modellierung der Eigenspannungen an der Riβspitze mit einem Schädigungsmodell.” Technische Mechanik. 4.17 (1997): 323-332.

19. Besson, J., C. Guillemer-Neel. “An Extension of the Green and Gurson Models to Kinematic Hardening.” Mechanics of Materials. 35.1-2 (2003): 1-18.

20. Hjelm, H. E. “Yield Surface for Grey Cast Iron Under Biaxial Stress.” Journal of Engineering Materials and Technology. 116.2 (1994): 148-154.

21. Chen, W. F., D. J. Han. Plasticity for Structural Engineers. New York: Springer-Verlag, 1988.

22. Mϋhlich, U. and W. Brocks. "On the Numerical Integration of a Class of Pressure-Dependent Plasticity Models Including Kinematic Hardening." Computational Mechanics. 31.6 (2003): 479-488.

### 4.4.2. Isotropic Hardening

During plastic deformation, isotropic hardening causes a uniform increase in the size of the yield surface and results in an increase in the yield stress. The yield criterion has the form:

where is a scalar function of stress and is the yield stress that evolves as a function of the set of material internal variables . This type of hardening can model the behavior of materials under monotonic loading and elastic unloading, but often does not give good results for structures that experience additional plastic deformation after a load reversal from a plastic state.

Three general classes of isotropic hardening models are available: bilinear, multilinear, and nonlinear. Each of the hardening models assumes a von Mises yield criterion, unless an anisotropic Hill yield criterion is defined, and includes an associated flow rule.

Isotropic hardening can also be combined with kinematic hardening and the Extended Drucker-Prager and Gurson models to provide an evolution of the yield stress. For more information, see Material Model Combinations.

The following topics concerning the isotropic hardening material model are available:

#### 4.4.2.1. Yield Criteria and Plastic Potentials

Hardening models assume a von Mises yield criterion, unless an anisotropic Hill yield criterion is defined.

##### 4.4.2.1.1. Von Mises Yield Criterion

The von Mises yield criterion is commonly used in plasticity models for a wide range of materials. It is a good first approximation for metals, polymers, and saturated geological materials. The criterion is isotropic and independent of hydrostatic pressure, which can limit its applicability to microstructured materials and materials that exhibit plastic dilatation.

The von Mises yield criterion is:

 (4–5)

where is the von Mises effective stress, also known as the von Mises equivalent stress,

and is the yield strength and corresponds to the yield in uniaxial stress loading.

In principal stress space, the yield surface is a cylinder with the axis along the hydrostatic line and gives a yield criterion that is independent of the hydrostatic stress, as shown in the following figure:

For an associated flow rule, the plastic potential is the yield criterion in Equation 4–5 and the plastic strain increment is proportional to the deviatoric stress

##### 4.4.2.1.2. Hill Yield Criterion

The Hill yield criterion [1] is an anisotropic criterion that depends on the orientation of the stress relative to the axis of anisotropy. It can be used to model materials in which the microstructure influences the macroscopic behavior of the material such as forged metals and composites.

In a coordinate system that is aligned with the anisotropy coordinate system, the Hill yield criterion given in stress components is:

 (4–6)

The coefficients in this yield criterion are functions of the ratio of the scalar yield stress parameter and the yield stress in each of the six stress components:

where the directional yield stress ratios are the user input parameters and are related to the isotropic yield stress parameter by:

where is the yield stress in the direction indicated by the value of subscript i. The stress directions are in the anisotropy coordinate system which is aligned with the element coordinate system (ESYS). The isotropic yield stress is entered in the constants for the hardening model.

The Hill yield criterion defines a surface in six-dimensional stress space and the flow direction is normal to the surface. The plastic strain increments in the anisotropy coordinate system are:

The Hill surface, used with a hardening model, replaces the default von Mises yield surface.

After defining the material data table (TB,HILL), input the required constants (TBDATA):

ConstantMeaningProperty
C1R11Yield stress ratio in X direction
C2R22Yield stress ratio in Y direction
C3R33Yield stress ratio in Z direction
C4R12Yield stress ratio in XY direction
C5R23Yield stress ratio in YZ direction
C6R13Yield stress ratio in XZ direction

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

Example 4.2:  Hill Surface
```/prep7
MP,EX,1,20.0E5                   ! ELASTIC CONSTANTS
MP,NUXY,1,0.3

TB,HILL,1                        ! HILL TABLE
TBDATA,1,1.0,1.1,0.9,0.85,0.9,0.80
```

###### 4.4.2.1.2.1. Separated Hill Potentials for Plasticity and Creep

This capability is reserved for use with the material combination of Chaboche nonlinear kinematic hardening with implicit creep.

To define separated Hill potentials for the plastic yielding and the creep flow (TB,HILL,,,,PC), issue two TBDATA commands, one to set constants C1-C6 defining the Hill yield surface for plasticity, the other to set constants C7-C12 defining the Hill potential for the creep direction.

Example 4.3:  Defining Hill Surfaces for Plasticity and Creep
```/prep7
MP,EX,1,20.0E5                     ! ELASTIC CONSTANTS
MP,NUXY,1,0.3

TB,HILL,1,1,,PC                    ! HILL data table for plasticity and creep
TBDATA,1,1.0,1.1,0.9,0.85,0.9,0.80 ! Plasticity Hill parameters
TBDATA,7,1.0,1.0,1.0,1.0 ,1.0,1.0  ! Creep Hill parameters
```

#### 4.4.2.2. General Isotropic Hardening Classes

Support is available for these general classes of isotropic hardening:

##### 4.4.2.2.1. Bilinear Isotropic Hardening

Bilinear isotropic hardening is described by a bilinear effective stress versus effective strain curve. The initial slope of the curve is the elastic modulus of the material. Beyond the user-specified initial yield stress , plastic strain develops and stress-vs.-total-strain continues along a line with slope defined by the user-specified tangent modulus . The tangent modulus cannot be less than zero or greater than the elastic modulus.

###### 4.4.2.2.1.1. Defining the Bilinear Isotropic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,BISO), input the following constants (TBDATA):

ConstantMeaning Property
C1Yield stress
C2Tangent modulus

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

Example 4.4:  Bilinear Isotropic Hardening
```/prep7
MPTEMP,1,0,500            ! Define temperatures for Young's modulus
MPDATA,EX,1,,14E6,12e6
MPDATA,PRXY,1,,0.3,0.3
TB,BISO,1,2               ! Activate a data table
TBTEMP,0.0                ! Temperature = 0.0
TBDATA,1,44E3,1.2E6       ! Yield = 44,000; Tangent modulus = 1.2E6
TBTEMP,500                ! Temperature = 500
TBDATA,1,29.33E3,0.8E6    ! Yield = 29,330; Tangent modulus = 0.8E6
```

##### 4.4.2.2.2. Multilinear Isotropic Hardening

The behavior of multilinear isotropic hardening is similar to bilinear isotropic hardening except that a multilinear stress versus total or plastic strain curve is used instead of a bilinear curve.

The multilinear hardening behavior is described by a piece-wise linear stress-total strain curve, starting at the origin and defined by sets of positive stress and strain values, as shown in this figure:

The first stress-strain point corresponds to the yield stress. Subsequent points define the elastic plastic response of the material.

###### 4.4.2.2.2.1. Defining the Multilinear Isotropic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,PLASTIC,,,,MISO), input the following constants (TBDATA):

ConstantMeaningProperty
XStrain value
YStress value

The stress-plastic strain data points are entered into the table via the TBPT command.

Temperature-dependent data can be defined (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP). Interpolation between temperatures occurs via stress-vs.-plastic-strain.

Example 4.5:  Multilinear Hardening with Plastic Strain
```/prep7
MPTEMP,1,0,500            ! Define temperature-dependent EX,
MPDATA,EX,1,,14.665E6,12.423e6
MPDATA,PRXY,1,,0.3

TB,PLASTIC,1,2,5,MISO     ! Activate TB,PLASTIC data table
TBTEMP,0.0                ! Temperature = 0.0
TBPT,DEFI,0,29.33E3       ! Plastic strain, stress at temperature = 0
TBPT,DEFI,1.59E-3,50E3
TBPT,DEFI,3.25E-3,55E3
TBPT,DEFI,5.91E-3,60E3
TBPT,DEFI,1.06E-2,65E3
TBTEMP,500                ! Temperature = 500
TBPT,DEFI,0,27.33E3       ! Plastic strain, stress at temperature = 500
TBPT,DEFI,2.02E-3,37E3
TBPT,DEFI,3.76E-3,40.3E3
TBPT,DEFI,6.48E-3,43.7E3
TBPT,DEFI,1.12E-2,47E3
```

##### 4.4.2.2.3. Nonlinear Isotropic Hardening

Power law and Voce equations are available to model nonlinear isotropic hardening.

###### 4.4.2.2.3.1. Power Law Nonlinear Isotropic Hardening

The power law equation has a user-defined initial yield stress and exponent N. The current yield stress is given by solving the following equation:

where G is the shear modulus determined from the user defined elastic constants and is the accumulated equivalent plastic strain.

Defining the Power Law Nonlinear Isotropic Hardening Model

For the power law hardening model, define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,NLISO,,,,POWER), input the following constants (TBDATA):

ConstantMeaningProperty
C1Initial yield stress
C2NExponent

The exponent N must be positive and less than 1.

Temperature-dependent data can be defined (`NTEMP` on the TB command), with temperatures specified for the subsequent set of constants (TBTEMP).

Example 4.6:  Power Law Nonlinear Isotropic Hardening
```/prep7
TB,NLISO,1,2,,POWER
TBTEMP,100               ! Define first temperature
TBDATA,1,275,0.1
TBTEMP,200               ! Define second temperature
TBDATA,1,275,0.1
```

###### 4.4.2.2.3.2. Voce Law Nonlinear Isotropic Hardening

The Voce hardening model is similar to bilinear isotropic hardening, with an exponential saturation hardening term added to the linear term, as shown in this figure:

The evolution of the yield stress for this model is specified by the following equation:

where the user-defined parameters include , the difference between the saturation stress and the initial yield stress, , the slope of the saturation stress and, , the hardening parameter that governs the rate of saturation of the exponential term.

Defining the Voce Law Nonlinear Isotropic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,NLISO,,,,VOCE), input the following constants (TBDATA):

ConstantMeaningProperty
C1Initial yield stress
C2Linear coefficient
C3Exponential coefficient
C4Exponential saturation parameter

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the subsequent set of constants (TBTEMP).

Example 4.7:  Voce Nonlinear Isotropic Hardening
```/PREP7
TB,NLISO,1,2,,VOCE         ! Activate NLISO data table
TBTEMP,40                  ! Define first temperature
TBDATA,1,280,7e3,155,7e2   ! Constants at first temperature
TBTEMP,60                  ! Define second temperature
TBDATA,1,250,5e3,120,3e2   ! Constants at second temperature
```

##### 4.4.2.2.4. Isotropic Hardening Static Recovery

Static recovery, also known as thermal recovery, of the isotropic yield stress is dependent on time, temperature and the current yield stress. The rate of isotropic hardening can be separated into work hardening and static recovery, as follows:

where the first term on the right side of the equation represents the rate of work hardening, and the second defines the rate of static recovery where , and are material parameters, is the temperature, and is the current yield stress. A lower limit yield stress can be defined for static recovery such that:

where is the user defined static recovery threshold.

An approximate solution for the static recovery of the yield stress over a time step is to integrate the static recovery rate and incorporate the work hardening rate into the initial condition. The yield stress is then given by the solution to:

with the initial condition:

where is the yield stress at the end of the previous increment and is the increment in yield stress due to work hardening over the current time step.

###### 4.4.2.2.4.1. Defining the Isotropic Static Recovery

Static recovery of the isotropic yield stress can be used with the combined creep and Chaboche nonlinear hardening material. The material parameters are defined via a TB material table with `Lab` = PLASTIC and `TBOPT` = ISR.

ConstantMeaning Property
C1Coefficient
C2Temperature Coefficient
C3Exponential
C4Recovery threshold

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

Example 4.8:  Isotropic Hardening Static Recovery
```/prep7

YOUNGS = 30e3   ! Young’s Modulus
NU     = 0.3    ! Poisson’s ratio
SIGMA0 = 18.0E0 ! Initial yield stress

mp,ex,1,YOUNGS
mp,prxy,1,NU

TB,CHAB,1,1,1    ! Chaboche kinematic hardening
TBDATA,1,sigma0,
TBDATA,2,1e+3,0

TB,CREEP,1,1, ,1,1   ! creep model
tbdata,1,            ! No creep strain

TB,PLASTIC,1,,,MISO  ! Multilinear isotropic hardening
TBPT,DEFI,0.0,SIGMA0
TBPT,DEFI,0.001,35.0
TBPT,DEFI,0.002,40.0
TBPT,DEFI,0.010,60.0

TB,PLASTIC,1,,4,ISR  ! Isotropic hardening static recovery
TBDATA,1,1D-5,1,2,SIGMA0
```

### 4.4.3. Kinematic Hardening

During plastic deformation, kinematic hardening causes a shift in the yield surface in stress space. In uniaxial tension, plastic deformation causes the tensile yield stress to increase and the magnitude of the compressive yield stress to decrease. This type of hardening can model the behavior of materials under either monotonic or cyclic loading and can be used to model phenomena such as the Bauschinger effect and plastic ratcheting.

The yield criterion has the form:

where is a scalar function of the relative stress and is the yield stress. The relative stress is:

 (4–7)

where the back stress is the shift in the position of the yield surface in stress space and evolves during plastic deformation.

The general classes of kinematic hardening models are bilinear, multilinear, and nonlinear. The hardening models assume a von Mises yield criterion, unless an anisotropic Hill yield criterion is defined, and includes an associated flow rule.

Kinematic hardening can also be combined with isotropic hardening and the Gurson model to provide an evolution of the yield stress. For more information, see Material Model Combinations.

The following topics concerning the kinematic hardening material model are available:

#### 4.4.3.1. Yield Criteria and Plastic Potentials

Kinematic hardening uses the von Mises yield criterion with an associated flow rule unless a Hill yield surface is defined. If a Hill yield surface is defined, the kinematic hardening model uses it for both the yield criterion and as the plastic potential to give an associated flow rule.

#### 4.4.3.2. General Kinematic Hardening Classes

Support is available for these general classes of kinematic hardening:

##### 4.4.3.2.1. Bilinear Kinematic Hardening

The back stress tensor for bilinear kinematic hardening evolves so that the effective stress versus effective strain curve is bilinear. The initial slope of the curve is the elastic modulus of the material and beyond the user specified initial yield stress , plastic strain develops and the back stress evolves so that stress versus total strain continues along a line with slope defined by the user specified tangent modulus . This tangent modulus cannot be less than zero or greater than the elastic modulus.

For uniaxial tension followed by uniaxial compression, the magnitude of the compressive yield stress decreases as the tensile yield stress increases so that the magnitude of the elastic range is always , as shown in this figure:

The back stress is proportional to the shift strain :

where G is the elastic shear modulus and the shift strain is numerically integrated from the incremental shift strain which is proportional to the incremental plastic strain:

where

and is Young's Modulus and is the user-defined tangent modulus [2]. The incremental plastic strain is defined by the associated flow rule for the von Mises or Hill potential given in Yield Criteria and Plastic Potentials with the stress given by the relative stress .

###### 4.4.3.2.1.1. Defining the Bilinear Kinematic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,BKIN), input the following constants (TBDATA):

ConstantMeaning Property
C1Initial yield stress
C2Tangent modulus

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

This model can be used with the TB command’s `TBOPT` option:

• For `TBOPT` ≠ 1, no stress relaxation occurs with an increase in temperature. This option is not recommended for non-isothermal problems.

• For `TBOPT` = 1, Rice's hardening rule [5] is applied, which takes stress relaxation with temperature increase into account.

Example 4.9:  Bilinear Kinematic Hardening
```/prep7
MPTEMP,1,0,500            ! Define temperatures for Young's modulus
MPDATA,EX,1,,14E6,12e6
MPDATA,PRXY,1,,0.3,0.3
TB,BKIN,1,2,2,1           ! Activate a data table with TBOPT=1
! stress relaxation with temperature
TBTEMP,0.0                ! Temperature = 0.0
TBDATA,1,44E3,1.2E6       ! Yield = 44,000; Tangent modulus = 1.2E6
TBTEMP,500                ! Temperature = 500
TBDATA,1,29.33E3,0.8E6    ! Yield = 29,330; Tangent modulus = 0.8E6
```

##### 4.4.3.2.2. Multilinear Kinematic Hardening

The back stress tensor for multilinear kinematic hardening evolves so that the effective stress versus effective strain curve is multilinear with each of the linear segments defined by a set of user input stress-strain points, as shown in this figure:

The model formulation is the sublayer or overlay model of Besselling and Owen, Prakash and Zienkiewicz in which the material is assumed to be composed of a number of sublayers or subvolumes, all subjected to the same total strain. The number of subvolumes is the same as the number of input stress-strain points, and the overall behavior is weighted for each subvolume where the weight is given by:

where is the tangent modulus for segment of the stress-strain curve.

The behavior of each subvolume is elastic-perfectly plastic, with the uniaxial yield stress for each subvolume given by:

where is the input stress-strain point for subvolume k.

For plane stress, is used to calculate the weights and subvolume yield stresses. The resulting stress-strain behavior is exact for uniaxial stress states but can deviate from the defined stress-strain values for general deformations.

The default yield surface is the von Mises surface, and each subvolume yields at an equivalent stress equal to the subvolume uniaxial yield stress. A Hill yield criterion can be used in which each subvolume yields according to the Hill criterion with the subvolume uniaxial yield as the isotropic yield stress and the subvolume anisotropic yield condition determined by the Hill surface.

The subvolumes undergo kinematic hardening with an associated flow rule and the plastic strain increment for each subvolume is the same as that for bilinear kinemtatic hardening (TB,BKIN). The total plastic strain is given by:

For more information, see Specialization for Multilinear Kinematic Hardening in the Mechanical APDL Theory Reference.

###### 4.4.3.2.2.1. Defining the Multilinear Kinematic Hardening Model

Define the isotropic or anisotropic elastic behavior via MP commands. To specify the hardening behavior, define the material data table (TB,PLAS,,,,KINH) and input the constants (TBPT) as stress vs. total strain points or as stress vs. plastic strain points.

###### 4.4.3.2.2.2. Specifying the Constants
ConstantMeaning Property
P1Strain value
P2Stress value

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

When entering temperature-dependent stress-strain points, the set of data at each temperature must have the same number of points. Thermal softening for the multilinear kinematic hardening model is the same as that for bilinear kinematic hardening (TB,BKIN) with Rice's hardening rule.

Entering Stress vs. Plastic Strain Points

After defining the material data table (TB,PLASTIC,,,,KINH), enter the stress-strain points (TBPT).

No segment slope can be larger than the slope of the previous segment.

Example 4.10:  Multilinear Kinematic Hardening with Stress vs. Plastic Strain
```/prep7
TB,PLASTIC,1,2,3,KINH        ! Define the material data table
TBTEMP,20.0                  ! Temperature = 20.0
TBPT,,0.0,1.0                ! Plastic Strain = 0.0000, Stress = 1.0
TBPT,,0.1,1.2                ! Plastic Strain = 0.1000, Stress = 1.2
TBPT,,0.2,1.3                ! Plastic Strain = 0.2000, Stress = 1.3
TBTEMP,40.0                  ! Temperature = 40.0
TBPT,,0.0,0.9                ! Plastic Strain = 0.0000, Stress = 0.9
TBPT,,0.0900,1.0             ! Plastic Strain = 0.0900, Stress = 1.0
TBPT,,0.129,1.05             ! Plastic Strain = 0.1290, Stress = 1.05
```

##### 4.4.3.2.3. Nonlinear Kinematic Hardening

The nonlinear kinematic hardening model is a rate-independent version of the kinematic hardening model proposed by Chaboche [6][7]. The model allows the superposition of several independent back stress tensors and can be combined with any of the available isotropic hardening models. It can be useful in modeling cyclic plastic behavior such as cyclic hardening or softening and ratcheting or shakedown.

The model uses an associated flow rule with either the default von Mises yield criterion or the Hill yield criterion if it is defined. The relative stress given by Equation 4–4 is used to evaluate the yield function, and the back stress tensor is given by the superposition of a number of evolving kinematic back stress tensors:

where n is the number of kinematic models to be superposed.

The evolution of each back stress model in the superposition is given by the kinematic hardening rule:

where and are user-input material parameters, is the plastic strain rate, and is the magnitude of the plastic strain rate.

During a solution, if there is a change in temperature over an increment (non-isothermal loading), the back stress terms are scaled in a manner similar to that of bilinear kinematic hardening with Rice's hardening rule [5].

###### 4.4.3.2.3.1. Defining the Nonlinear Kinematic Hardening Model

Define the material data table (TB,CHABOCHE) and input the following constants (TBDATA):

ConstantMeaning Property
C1Initial yield stress
C2Material constant for first kinematic model
C3Material constant for first kinematic model
C4Material constant for second kinematic model
C5Material constant for second kinematic model
.........
C(2n)Material constant for last kinematic model
C(1+2n)Material constant for last kinematic model

Temperature-dependent data can be defined (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

Set the TB command’s `NPTS` value equal to n, the number of superimposed kinematic hardening models.

Example 4.11:  Nonlinear Kinematic Hardening
```/prep7
TB,CHABOCHE,1,1,3          ! Activate Chaboche data table with
! 3 models to be superposed
! Define Chaboche material data
TBDATA,1,18.8              ! C1 - Initial yield stress
TBDATA,2,5174000,4607500   ! C2,C3	- Chaboche constants for 1st model
TBDATA,4,17155,1040        ! C4,C5	- Chaboche constants for 2nd model
TBDATA,6,895.18,9          ! C6,C7	- Chaboche constants for 3rd model
```

#### 4.4.3.3. Kinematic Hardening Static Recovery

Static recovery (also known as thermal recovery) for kinematic hardening is included in the Chaboche nonlinear kinematic hardening model by modifying the evolution of the back stress tensor components:

where the last term on the right side of the equation is the rate of static recovery of the kinematic back stress component with and the material parameters for kinematic static recovery, and is the von Mises effective back stress component.

##### 4.4.3.3.1. Defining the Kinematic Static Recovery

Static recovery of the kinematic back stress can be used with the combined creep and Chaboche nonlinear hardening material. The material parameters are defined via a TB material table with `Lab` = PLASTIC and `TBOPT` = KSR. The number of material parameter sets should correspond to the number of superimposed Chaboche back stress terms. Undefined values default to 0.0.

ConstantMeaning Property
C1Coefficient
C2Temperature Coefficient

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

Example 4.12:  Kinematic Hardening Static Recovery
```/prep7

YOUNGS = 30e3   ! Young’s Modulus
NU     = 0.3    ! Poisson’s ratio
SIGMA0 = 18.0E0 ! Initial yield stress

mp,ex,1,YOUNGS
mp,prxy,1,NU

TB,CHAB,1,1,1    ! Chaboche kinematic hardening
TBDATA,1,sigma0,
TBDATA,2,1e+4,0

TB,CREEP,1,1, ,1,1   ! creep model
tbdata,1,            ! No creep strain

TB,PLASTIC,1,,1,KSR  ! Kinematic hardening static recovery
TBDATA,1,1e-3,2.0,
```

### 4.4.4. Generalized Hill

The generalized Hill plasticity model is an anisotropic model that accounts for different yield strength in tension than in compression. The anisotropic and asymmetric tension-compression behavior can be useful in modeling textured metals with low crystallographic symmetries. Examples of such materials include titanium and zirconium alloys, and composite materials with oriented microstructures such as natural and processed wood products and fiber-matrix composites.

The model includes a yield surface that is a specialization of the Hill yield surface [8], an anisotropic work-hardening rule [9], and an associated flow rule. In a coordinate system that is aligned with the anisotropy coordinate system, the generalized Hill yield criterion given in stress components is:

 (4–8)

where the coefficients are functions of the parameter and the tensile and compressive yield stress:

 (4–9)

where and are the user-defined magnitudes of the tension and compression yield strength, respectively. The subscripts on the tension and compression yield stresses correspond to the Voigt notation coordinate directions .

From the assumption of incompressible plastic deformation, the mixed subscript coefficients are given by:

 (4–10)

The strength differential coefficients are:

 (4–11)

which, for incompressibility, must satisfy:

Setting the coefficient , then:

and the coefficients in the yield criterion from Equation 4–8 are determined from Equation 4–9 through Equation 4–11, and the user-input tension and compression yield stresses.

Due to the incompressibility assumption,

 (4–12)

and for a closed yield surface,

 (4–13)

Equation 4–12 and Equation 4–13 must be satisfied throughout the evolution of yield stresses that result from plastic deformation. The program checks these conditions through 20 percent equivalent plastic strain, but you must ensure that conditions are satisfied if the deformation exceeds that range.

A bilinear anisotropic work hardening rule is used to evolve the individual components of tension and compression yield stresses. For a general state of deformation with a bilinear hardening law, the plastic work is:

where is the effective stress at initial yield, and is the current effective yield stress. For uniaxial stress , the plastic strain is:

and the plastic work is:

where the plastic slope is the slope of the stress versus plastic strain. The uniaxial plastic work is equivalent to the effective plastic work if:

 (4–14)

where the plastic tangent is related to the user input tangent modulus by:

Equation 4–14 is then the isotropic hardening evolution equation for the tensile and compressive yield stress components.

#### 4.4.4.1. Defining the Generalized Hill Model

Define the isotropic or anisotropic elastic behavior via MP commands. Specify the generalized Hill material parameters by defining the material data table (TB,ANISO) and entering the following input (TBDATA):

ConstantMeaningProperty
C1-C3Tensile yield stresses in the material x, y, and z directions
C4-C6Tangent moduli of tension in material x, y, and z directions
C7-C9Compressive yield stresses in the material x, y, and z directions
C10-C12Tangent moduli of compression in material x, y, and z directions
C13-C15Shear yield stresses in the material xy, yz, and xz directions
C16-C18Tangent moduli in material xy, yz, and xz directions

Temperature-dependent parameters are not allowed.

Example 4.13:  Generalized Hill
```! Define generalized Hill model
/prep7

! Define elastic material properties
mp,ex,1,210
mp,nuxy,1,0.3

! Define anisotropic material properties
tb,aniso,1
tbdata,1,0.33,0.33,0.495  	      ! Tensile yield stress (x,y & z)
tbdata,4,0.21,0.21,0.315         ! Tangent moduli (tensile)
tbdata,7,0.33,0.33,0.495         ! Compressive yield stress (x,y & z)
tbdata,10,0.21,0.21,0.315        ! Tangent moduli (compressive)
tbdata,13,0.1905,0.1905,0.1905   ! Shear yield stress (xy,yz,xz)
tbdata,16,0.105,0.07,0.07        ! Tangent moduli (shear)
```

### 4.4.5. Drucker-Prager

The following topics concerning to Drucker-Prager plasticity are available:

#### 4.4.5.1. Classic Drucker-Prager

The classic Drucker-Prager model [10] is applicable to granular (frictional) material such as soils, rock, and concrete and uses the outer cone approximation to the Mohr-Coulomb law. The input consists of only three constants:

• Cohesion value (> 0)

• Angle of internal friction

• Dilatancy angle

The amount of dilatancy (the increase in material volume due to yielding) can be controlled via the dilatancy angle. If the dilatancy angle is equal to the friction angle, the flow rule is associative. If the dilatancy angle is zero (or less than the friction angle), there is no (or less of an) increase in material volume when yielding and the flow rule is non-associated.

##### 4.4.5.1.1. Defining the Classic Drucker-Prager Model

Define the isotropic or anisotropic elastic behavior via MP commands. Define the material data table (TB,DP) and define up to three constants (TBDATA), as follows:

ConstantMeaningProperty
C1Force/AreaCohesion value
C2Angle (in degrees)Internal friction
C3Angle (in degrees)Dilatancy

Temperature-dependent parameters are not allowed.

Example 4.14:  Classic Drucker-Prager
```MP,EX,1,5000
MP,NUXY,1,0.27
TB,DP,1
TBDATA,1,2.9,32,0    ! Cohesion = 2.9 (use consistent units),
! Angle of internal friction = 32 degrees,
! Dilatancy angle = 0 degrees```

#### 4.4.5.2. Extended Drucker-Prager (EDP)

The extended Drucker-Prager (EDP) material model includes three yield criteria and corresponding flow potentials similar to those of the classic Drucker-Prager model commonly used for geomaterials with internal cohesion and friction. The yield functions can also be combined with an isotropic or kinematic hardening rule to evolve the yield stress during plastic deformation.

The model is defined via one of the three yield criteria combined with any of the three flow potentials and an optional hardening model.

The following topics for defining the EDP material model are available:

##### 4.4.5.2.1. EDP Yield Criteria Forms

The EDP yield criteria include the following forms:

###### 4.4.5.2.1.1. Linear Form

The EDP linear yield criterion form is:

where the user-defined parameters are the pressure sensitivity and the uniaxial yield stress .

Defining the EDP Linear Yield Criterion

After initializing the extended Drucker-Prager linear yield criterion (TB,EDP,,,,LYFUN), enter the following constants (TBDATA):

ConstantMeaningProperty
C1Pressure sensitivity
C2Uniaxial yield stress

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

###### 4.4.5.2.1.2. Power Law Form

The EDP power law yield criteria form is:

where the exponent , pressure sensitivity , and uniaxial yield stress are the user-defined parameters.

Defining the EDP Power Law Yield Criterion

After initializing the extended Drucker-Prager power law yield criterion (TB,EDP,,,,PYFUN), enter the following constants (TBDATA):

ConstantMeaningProperty
C1Pressure sensitivity
C2Exponent
C3Uniaxial yield stress

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

###### 4.4.5.2.1.3. Hyperbolic Form

The EDP hyperbolic yield criteria form is:

where the constant , pressure sensitivity , and uniaxial yield stress are the user-defined parameters.

In the following figure, the hyperbolic yield criterion is plotted and compared to the linear yield criterion shown in the dashed line:

Defining the EDP Hyperbolic Yield Criterion

After initializing the extended Drucker-Prager hyperbolic yield criterion (TB,EDP,,,,HYFUN), enter the following constants (TBDATA):

ConstantMeaningProperty
C1Pressure sensitivity
C2Material parameter
C3Uniaxial yield stress

The constants can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

##### 4.4.5.2.2. EDP Plastic Flow Potentials

Three EDP flow potentials correspond in form to each of the yield criteria. However, the user-defined parameters for the flow potentials are independent of those for the yield criteria, and any potential can be combined with any yield criterion.

The EDP plastic flow potentials include the following forms:

###### 4.4.5.2.2.1. Linear Form

The linear form of the plastic flow potential is:

where is the flow potential pressure sensitivity.

Defining the Linear Plastic Flow Potential

After initializing the material data table (TB,EDP,,,,LFPOT), enter the following constant (TBDATA):

ConstantMeaningProperty
C1Pressure sensitivity

The material behavior can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

###### 4.4.5.2.2.2. Power Law Form

The power law form of the plastic flow potential is:

where the exponent and the pressure sensitivity are user-defined parameters.

Defining the Linear Plastic Flow Potential

After initializing the material data table (TB,EDP,,,,PFPOT), enter the following constants (TBDATA):

ConstantMeaningProperty
C1Pressure sensitivity
C2Exponent

The material behavior can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

###### 4.4.5.2.2.3. Hyperbolic Form

The hyperbolic form of the plastic flow potential is:

where the pressure sensitivity the constant are user-defined parameters.

Defining the Linear Plastic Flow Potential

After initializing the material data table (TB,EDP,,,,HFPOT), enter the following constants (TBDATA):

ConstantMeaningProperty
C1Pressure sensitivity
C2Material parameter

The material behavior can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

##### 4.4.5.2.3. Plastic Strain Increments for Flow Potentials

The plastic strain increment corresponding to each of the plastic flow potentials is:

where is the deviatoric stress:

The dilatation for each of the flow potentials is:

Associated flow is obtained if the plastic potential form and parameters are set equal to the yield criterion.

For the extended Drucker-Prager model, the accumulated plastic strain (EPEQ) is the summation of:

where is the magnitude of the deviatoric plastic strain increment and is the magnitude of the volumetric plastic strain increment.

##### 4.4.5.2.4. Example EDP Material Model Definitions

The following examples show how to define the EDP material model using various yield criteria and flow potentials:

Example 4.15:  EDP -- Linear Yield Criterion and Flow Potential
```/prep7
!!! Define linear elasticity constants
mp,ex,1,2.1e4
mp,nuxy,1,0.45

! Extended Drucker-Prager Material Model Definition

! Linear Yield Function
tb,edp,1,1,2,LYFUN
tbdata,1,2.2526,7.894657

! Linear Plastic Flow Potential
tb,edp,1,1,2,LFPOT
tbdata,1,0.566206```

Example 4.16:  EDP -- Power Law Yield Criterion and Flow Potential
```/prep7
!!! Define linear elasticity constants
mp,ex,1,2.1e4
mp,nuxy,1,0.45

! Extended Drucker-Prager Material Model Definition

! Power Law Yield Function
tb,edp,1,1,3,PYFUN
tbdata,1,8.33,1.5

! Power Law Plastic Flow Potential
tb,edp,1,1,2,PFPOT
tbdata,1,8.33,1.5
```

Example 4.17:  EDP -- Hyperbolic Yield Criterion and Flow Potential
```/prep7
!!! Define linear elasticity constants
mp,ex,1,2.1e4
mp,nuxy,1,0.45

! Extended Drucker-Prager Material Model Definition

! Hyperbolic Yield Function
tb,edp,1,1,3,HYFUN
tbdata,1,1.0,1.75,7.89

! Hyperbolic Plastic Flow Potential
tb,edp,1,1,2,HFPOT
tbdata,1,1.0,1.75
```

#### 4.4.5.3. Extended Drucker-Prager Cap

The EDP Cap material model has a yield criterion similar to the other extended Drucker-Prager yield criteria with the addition of two cap surfaces that truncate the yield surface in tension and compression regions [11]. The model formulation follows that of Schwer and Murray [12] and Foster et al [13] and the numerical formulation is modified from the work of Pelessone [14].

The criterion is a function of the three stress invariants , , and , given by:

where is the deviatoric stress.

Three functions define the surfaces that make up the yield criterion. The shear envelope function is given by:

where is the cohesion related yield parameter and is a user defined material parameter along with , , and . This function reduces to the Drucker-Prager criterion for . For positive values of , the shear failure envelope is evaluated at = 0, which gives the constant value.

The compaction function is itself a function of the shear envelope function and is given by:

where is the Heaviside step function, is a user-input material parameter, and defines the intersection of the compaction surface with the shear envelope, given by:

where is the user-defined value of at the intersection of the compaction cap with , as shown in the following figure:

The compaction function defines the material yield surface when .

The expansion function is a function of the shear envelope function and is given by:

where is a user-input material parameter. The expansion function defines the material yield surface when . The expansion cap function reaches peak value at .

These functions define the yield criterion, given by:

 (4–15)

where is the Lode angle function. The Lode angle is given by:

and the Lode angle function is:

where is a user-defined material parameter, a ratio of the extension strength to compression strength in triaxial loading.

Two methods of isotropic hardening can be used to evolve the yield criterion due to plastic deformation. Hardening of the compaction cap is due to evolution of , which is the intersection of the cap surface with shown in Figure 4.14: Yield Surface for the Cap Criterion. This value evolves due to plastic volume strain , and the relationship is given by [15]:

where is the initial value of , and the user-defined parameters are , , and . The restriction is enforced so that the material does not soften.

Evolution of the yield surface at the intersection of the shear envelope with the expansion cap occurs by combining the cap model with an isotropic hardening model to evolve the value of . The bilinear, multilinear, or nonlinear isotropic hardening function can be used, and the yield stress from the isotropic hardening model must be consistent with the value of calculated from the cap material parameters given by .

The following topics for defining the EDP Cap material model are available:

##### 4.4.5.3.1. Defining the EDP Cap Yield Criterion and Hardening

After initializing the material data table (TB,EDP,,,,CYFUN), enter the following constants (TBDATA):

ConstantMaterialProperty
C1Compaction cap parameter
C2Expansion cap parameter
C3Compaction cap yield pressure
C4Cohesion yield parameter
C5Shear envelope exponent
C6Shear envelope exponential coefficient
C7Shear envelope linear coefficient
C8Ratio of extension to compression strength
C9Limiting value of volumetric plastic strain
C10Hardening parameter
C11Hardening parameter

The yield criterion and hardening behavior can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

##### 4.4.5.3.2. Defining the EDP Cap Plastic Potential

After initializing the material data table (TB,EDP,,,,CFPOT), enter the following constants (TBDATA):

ConstantMaterialProperty
C1Compaction cap parameter
C2Expansion cap parameter
C3Shear envelope exponent
C4Shear envelope linear coefficient

The plastic flow potential can be defined as a function of temperature (`NTEMP` on the TB command), with temperatures specified for the table entries (TBTEMP).

If the plastic flow potential is not defined, the yield surface is used as the flow potential, resulting in an associated flow model.

##### 4.4.5.3.3. Example EDP Cap Material Model Definition

The following example input shows how to define an EDP Cap model by defining the yield criterion, hardening, and plastic flow potential:

Example 4.18:  EDP Cap Model Material Constant Input
```/prep7
! Define linear elasticity constants
mp,ex  ,1,14e3
mp,nuxy,1,0.0

! Cap yield function
tb,edp ,1,1,,cyfun
tbdata,1,2	          ! Rc
tbdata,2,1.5        ! Rt
tbdata,3,-80        ! Xi
tbdata,4,10         ! SIGMA
tbdata,5,0.001      ! B
tbdata,6,2	          ! A
tbdata,7,0.05       ! ALPHA
tbdata,8,0.9        ! PSI

! Define hardening for cap-compaction portion
tbdata,9,0.6        ! W1c
tbdata,10,3.0/1000  ! D1c
tbdata,11,0.0       ! D2c

! Cap plastic flow potential function
tb,edp ,1,1,,cfpot
tbdata,1,2	          ! RC
tbdata,2,1.5        ! RT
tbdata,3,0.001      ! B
tbdata,4,0.05       ! ALPHA
```

### 4.4.6. Gurson

Use the Gurson model to represent plasticity and damage in ductile porous metals [16][17]. When plasticity and damage occur, ductile metal undergoes a process of void growth, nucleation, and coalescence. The model incorporates these microscopic material behaviors into macroscopic plasticity behavior based on changes in the void volume fraction, also known as porosity, and pressure. A porosity increase corresponds to an increase in material damage, resulting in a diminished load-carrying capacity.

The yield criterion and flow potential for the Gurson model is:

where is the von Mises equivalent stress, is the yield stress, , , and are user-input Tvergaard-Needleman constants, and is the modified void volume fraction. The hydrostatic pressure is defined as:

The following additional Gurson model topics are available:

For more information, see Gurson's Model in the Mechanical APDL Theory Reference.

#### 4.4.6.1. Void Volume Fraction

The following figure shows the phenomena of voids at the microscopic scale that are incorporated into the Gurson model:

The void volume fraction is the ratio of void volume to the total volume. A volume fraction of 0 indicates no voids and the yield criterion reduces to the von Mises criterion. A volume fraction of 1 indicates all the material is void. The initial void volume fraction, , is a user-defined parameter, and the rate of change of void volume fraction is a combination of the rate of growth and the rate of nucleation:

From the assumption of isochoric plasticity and conservation of mass, the rate of change of void volume fraction due to growth is proportional to the rate of volumetric plastic strain:

Void nucleation is controlled by either the plastic strain or the stress, and is assumed to follow a normal distribution of statistics.

In the case of strain-controlled nucleation, the distribution is described by the mean strain, , and deviation, . The void nucleation rate due to strain control is given by:

where is the maximum void fraction for nucleated voids, is the effective plastic strain, and the rate of effective plastic strain, , is determined by equating the microscopic plastic work to the macroscopic plastic work:

In the case of stress-controlled nucleation, void nucleation is determined by the distribution of maximum normal stress on the interfaces between inclusions and the matrix, equal to . Stress-controlled nucleation takes into account the effect of triaxial loading on the rate of void nucleation. The void-nucleation rate for stress control is given by:

where distribution of stress is described by the mean stress, and deviation, .

The modified void volume fraction, , is used to model the loss of material load carrying capacity associated with void coalescence. When the current void volume fraction reaches a critical value , the material load carrying capacity decreases rapidly due to coalescence. When the void volume fraction reaches , the load-carrying capacity of the material is lost completely. The modified void volume fraction is given by:

#### 4.4.6.2. Hardening

The Gurson model can be combined with one of the isotropic hardening models to incorporate isotropic hardening of the yield stress in the Gurson yield criterion.

To combine the Gurson model with Chaboche kinematic hardening [22], the yield criterion is modified to:

where is the von Mises equivalent modified relative stress, and is the effective hydrostatic pressure defined as:

where is the modified relative stress, which itself is a function of the modified back stress :

where is the kinematic hardening back stress.

The rate of effective plastic strain equation is also modified to:

For more information, see Gurson Plasticity with Isotropic/Chaboche Kinematic Hardening in the Mechanical APDL Theory Reference.

#### 4.4.6.3. Defining the Gurson Material Model

The Gurson material model requires material parameters for the base model combined with parameters for either strain-controlled or stress-controlled nucleation. Additional input is required to define the void coalescence behavior.

##### 4.4.6.3.1. Defining the Gurson Base Model

To define the Gurson base model, initialize the material data table (TB,GURSON,,,,BASE), then input the following constants (TBDATA):

ConstantMeaningProperty
C1Initial yield strength
C2Initial porosity
C3First Tvergaard-Needleman constant
C4Second Tvergaard-Needleman constant
C5Third Tvergaard-Needleman constant
##### 4.4.6.3.2. Defining Stress- or Strain-Controlled Nucleation

The Gurson base model is combined with either stress- or strain-controlled nucleation.

To define stress-controlled nucleation, initialize the material data table (TB,GURSON,,,,SSNU), then input the following constants (TBDATA):

ConstantMeaningProperty
C1Nucleation porosity
C2Mean stress
C3Stress standard deviation

To define strain-controlled nucleation, initialize the material data table (TB,GURSON,,,,SNNU), then input the following constants (TBDATA):

ConstantMeaningProperty
C1Nucleation porosity
C2Mean strain
C3Strain standard deviation
##### 4.4.6.3.3. Defining the Void Coalescence Behavior

Define the void coalescence behavior after defining the Gurson base model and either the stress- or strain-controlled nucleation behavior.

Initialize the material data table (TB,GURSON,,,,COAL), then input the following constants (TBDATA):

ConstantMeaningProperty
C1Critical porosity
C2Failure porosity
##### 4.4.6.3.4. Example Gurson Model Definition

Following is an example Gurson plasticity material model definition:

Example 4.19:  Gurson Model with Isotropic and Kinematic Hardening
```/prep7
!!! Define linear elasticity constants
mp,ex,1,207.4E3    ! Young modulus (MPa)
mp,nuxy,1,0.3      ! Poisson's ratio

!!! Define parameters related to Gurson model with
!!! the option of strain-controlled nucleation with
!!! coalescence

f_0=1E-8           ! initial porosity
q1=1.5             ! first Tvergaard constant
q2=1.0             ! second Tvergaard constant
q3=2.25            ! third Tvergaard constant = q1^2
f_c=0.15           ! critical porosity
f_F=0.25           ! failure porosity
f_N=0.04           ! nucleation porosity
s_N=0.1            ! standard deviation of mean strain
strain_N=0.3       ! mean strain
sigma_Y=755        ! initial yielding strength (MPa)
power_N=0.1        ! power value for nonlinear isotropic
! hardening power law
!base model
tb,gurson,1,,5,base
tbdata,1,sigma_Y,f_0,q1,q2,q3

! Strain-controlled nucleation
tb,gurson,1,,3,snnu
tbdata,1,f_N,strain_N,s_N

! Coalescence
tb,gurson,1,,2,coal
tbdata,1,f_c,f_F

! Power law isotropic hardening
tb,nliso,1,,2,POWER
tbdata,1,sigma_Y,power_N
```

### 4.4.7. Cast Iron

The cast iron plasticity model is used to model gray cast iron. The microstructure of gray cast iron is a two-phase material with graphite flakes embedded in a steel matrix [20]. The microstructure leads to different behavior in tension and compression. In tension, cracks form due to the graphite flakes and the material is brittle with low strength. In compression, the graphite flakes behave as incompressible media that transmit stress and the steel matrix governs the overall behavior.

The model is isotropic elastic with the same elastic behavior in tension and compression. The yield strength and isotropic hardening behavior may be different in tension and in compression. Different yield criteria and plastic flow potentials are used for tension and compression.

A composite yield surface is used to model different yield behavior in tension and compression. The tension behavior is pressure-dependent and the Rankine maximum stress criterion is used:

where is the uniaxial tension yield stress. The hydrostatic pressure , the von Mises equivalent stress , and the Lode angle are defined as:

where and are the stress invariants:

where is the deviatoric stress.

In compression, the pressure-independent von Mises yield criterion is used:

The following figure shows the composite yield surface:

The yield surfaces are plotted in the meridional plane in which the ordinate and abscissa are von Mises equivalent stress and pressure, respectively.

The evolution of the yield stress in tension and compression follows the piecewise linear stress-strain curves for compression and tension input by the user. The tension yield stress evolves as a function of the equivalent uniaxial plastic strain, . The evolution of the equivalent uniaxial plastic strain is defined by equating the uniaxial plastic work increment to the total plastic work increment:

The compression yield stress evolves as a function of the equivalent plastic strain, , which is calculated from the increment in plastic strain determined by consistency with the yield criterion and the flow potential.

The plastic flow potential is defined by the von Mises yield criterion in compression and results in an associated flow rule. The flow potential in compression is:

In tension, the Rankine cap yield surface is replaced by an ellipsoidal surface defined by:

where is a constant function of the user-defined plastic Poisson's ratio, :

The plastic Poisson's ratio is the absolute value of the ratio of the transverse to the longitudinal plastic strain under uniaxial tension. It determines the amount of volumetric expansion during tensile plastic deformation. The tensile flow potential gives a nonassociated flow model and results in an unsymmetric material stiffness tensor.

#### 4.4.7.1. Defining the Cast Iron Material Model

Define the isotropic elastic behavior (MP). Initialize the material data table (TB,CAST,,,,ISOTROPIC) and input the following constant:

ConstantMeaningProperty
C1Plastic Poisson's ratio

Enter the tensile multilinear hardening stress-strain points into a data table (TB,UNIAXIAL,,,,TENSION). Do the same for the compressive multilinear hardening stress-strain points (TB,UNIAXIAL,,,,COMPRESSION). Enter the tension and compression stress-strain points into their respective tables via the TBPT command, with the compression points being entered as positive values:

ConstantMeaningProperty
XStrain value
YStress value

Enter tension and compression stress-strain points into their respective tables (TBPT), with the compression points entered as positive values.

The plastic Poisson's ratio and stress-strain points can be defined as a function of temperature (`NTEMP` value on the TB command), with individual temperatures specified for the table entries (TBTEMP).

Example 4.20:  Cast Iron
```/prep7
mp,  ex, 1,14.773E6
mp,nuxy, 1,0.2273

! Define cast iron model
TB,CAST,1,,,ISOTROPIC
TBDATA,1,0.04

TB,UNIAXIAL,1,1,5,TENSION
TBTEMP,10
TBPT,,0.550E-03,0.813E+04
TBPT,,0.100E-02,0.131E+05
TBPT,,0.250E-02,0.241E+05
TBPT,,0.350E-02,0.288E+05
TBPT,,0.450E-02,0.322E+05

TB,UNIAXIAL,1,1,5,COMPRESSION
TBTEMP,10
TBPT,,0.203E-02,0.300E+05
TBPT,,0.500E-02,0.500E+05
TBPT,,0.800E-02,0.581E+05
TBPT,,0.110E-01,0.656E+05
TBPT,,0.140E-01,0.700E+05
```