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:
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:
Following are the common symbols used in the rate-independent plasticity theory documentation:
Symbol | Definition | Symbol | Definition |
---|---|---|---|
Identity tensor | Anisotropic directional yield strength | ||
Strain | Young's Modulus | ||
Elastic strain | Elasto-plastic tangent | ||
Plastic strain | Elasto-plastic tangent in direction i | ||
Plastic strain components | Plastic tangent | ||
Effective plastic strain | Plastic tangent in direction i | ||
Accumulated equivalent plastic strain | Hill yield surface coefficients | ||
Stress | Hill yield surface directional yield ratio | ||
Stress components | Generalize Hill yield surface coefficients | ||
Principal stresses | Generalized Hill constant | ||
Stress minus back stress | Generalized Hill tensile and compressive yield strength | ||
Yield stress | Plastic work | ||
Anisotropic yield stress in direction i | Uniaxial plastic work | ||
Initial yield stress | Drucker-Prager yield surface constant | ||
Initial yield stress in direction i | Drucker-Prager plastic potential constant | ||
Equivalent plastic stress | Mohr-Coulomb cohesion | ||
Von Mises effective stress | Mohr-Coulomb internal friction angle | ||
User input strain-stress data point | Mohr-Coulomb flow internal friction angle | ||
Magnitude of plastic strain increment | Extended Drucker-Prager yield surface pressure sensitivity | ||
Effective stress function | Extended Drucker-Prager plastic potential pressure sensitivity | ||
Yield function | Extended Drucker-Prager power law yield exponent | ||
Plastic potential | Extended Drucker-Prager power law plastic potential exponent | ||
Translation of yield surface (back stress) | Extended Drucker-Prager hyperbolic yield constant | ||
Set of material internal variables | Extended Drucker-Prager hypobolic plastic potential constant |
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.
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.
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.
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.
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.
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:
A plastic deformation occurred during the increment.
An elastic deformation occurred during the increment.
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.
The following list of resources offers more information about plasticity:
Hill, R. The Mathematical Theory of Plasticity. New York: Oxford University Press, 1983.
Prager, W. “The Theory of Plasticity: A Survey of Recent Achievements.” Proceedings of the Institution of Mechanical Engineers. 169.1 (1955): 41-57.
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.
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.
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.
Chaboche, J. L. “Constitutive Equations for Cyclic Plasticity and Cyclic Viscoplasticity.” International Journal of Plasticity. 5.3 (1989): 247-302.
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.
Shih, C. F., D. Lee. “Further Developments in Anisotropic Plasticity.” Journal of Engineering Materials and Technology. 100.3 (1978): 294-302.
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.
Drucker, D. C., W. Prager. “Soil Mechanics and Plastic Analysis or Limit Design.” Quarterly of Applied Mathematics. 10.2 (1952): 157-165.
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.
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.
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.
Pelessone, D. “A Modified Formulation of the Cap Model.” Technical Report GA-C19579. San Diego: Gulf Atomics, 1989.
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.
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.
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.
Arndt, S., B. Svendsen, D. Klingbeil. “Modellierung der Eigenspannungen an der Riβspitze mit einem Schädigungsmodell.” Technische Mechanik. 4.17 (1997): 323-332.
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.
Hjelm, H. E. “Yield Surface for Grey Cast Iron Under Biaxial Stress.” Journal of Engineering Materials and Technology. 116.2 (1994): 148-154.
Chen, W. F., D. J. Han. Plasticity for Structural Engineers. New York: Springer-Verlag, 1988.
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.
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:
Hardening models assume a von Mises yield criterion, unless an anisotropic Hill yield criterion is defined.
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
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):
Constant | Meaning | Property |
---|---|---|
C1 | R_{11} | Yield stress ratio in X direction |
C2 | R_{22} | Yield stress ratio in Y direction |
C3 | R_{33} | Yield stress ratio in Z direction |
C4 | R_{12} | Yield stress ratio in XY direction |
C5 | R_{23} | Yield stress ratio in YZ direction |
C6 | R_{13} | Yield 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).
/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
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.
/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
Support is available for these general classes of 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.
Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,BISO), input the following constants (TBDATA):
Constant | Meaning | Property |
---|---|---|
C1 | Yield stress | |
C2 | Tangent modulus |
The constants can be defined as a function of temperature (NTEMP
on the TB command), with temperatures
specified for the table entries (TBTEMP).
/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
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.
Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,PLASTIC,,,,MISO), input the following constants (TBDATA):
Constant | Meaning | Property |
---|---|---|
X | Strain value | |
Y | Stress 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.
/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
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):
Constant | Meaning | Property |
---|---|---|
C1 | Initial yield stress | |
C2 | N | Exponent |
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).
/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
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):
Constant | Meaning | Property |
---|---|---|
C1 | Initial yield stress | |
C2 | Linear coefficient | |
C3 | Exponential coefficient | |
C4 | Exponential 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).
/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
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.
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.
Constant | Meaning | Property |
---|---|---|
C1 | Coefficient | |
C2 | Temperature Coefficient | |
C3 | Exponential | |
C4 | Recovery threshold |
The constants can be defined as a function of temperature (NTEMP
on the TB command), with temperatures
specified for the table entries (TBTEMP).
/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
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:
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.
For more information about von Mises and Hill yield surfaces, see Yield Criteria and Plastic Potentials.
Support is available for these general classes of 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 .
Define the isotropic or anisotropic elastic behavior via MP commands. After defining the material data table (TB,BKIN), input the following constants (TBDATA):
Constant | Meaning | Property |
---|---|---|
C1 | Initial yield stress | |
C2 | Tangent 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.
/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
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.
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.
Constant | Meaning | Property |
---|---|---|
P1 | Strain value | |
P2 | Stress 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.
/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
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].
Define the material data table (TB,CHABOCHE) and input the following constants (TBDATA):
Constant | Meaning | Property |
---|---|---|
C1 | Initial yield stress | |
C2 | Material constant for first kinematic model | |
C3 | Material constant for first kinematic model | |
C4 | Material constant for second kinematic model | |
C5 | Material 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.
/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
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.
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.
Constant | Meaning | Property |
---|---|---|
C1 | Coefficient | |
C2 | Temperature Coefficient |
The constants can be defined as a function of temperature
(NTEMP
on the TB command), with temperatures
specified for the table entries (TBTEMP).
/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,
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.
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):
Constant | Meaning | Property |
---|---|---|
C1-C3 | Tensile yield stresses in the material x, y, and z directions | |
C4-C6 | Tangent moduli of tension in material x, y, and z directions | |
C7-C9 | Compressive yield stresses in the material x, y, and z directions | |
C10-C12 | Tangent moduli of compression in material x, y, and z directions | |
C13-C15 | Shear yield stresses in the material xy, yz, and xz directions | |
C16-C18 | Tangent moduli in material xy, yz, and xz directions |
Temperature-dependent parameters are not allowed.
! 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)
The following topics concerning to Drucker-Prager plasticity are available:
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.
For more information about this material model, see Classic Drucker-Prager Model in the Mechanical APDL Theory Reference.
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:
Constant | Meaning | Property |
---|---|---|
C1 | Force/Area | Cohesion value |
C2 | Angle (in degrees) | Internal friction |
C3 | Angle (in degrees) | Dilatancy |
Temperature-dependent parameters are not allowed.
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
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:
The EDP yield criteria include the following forms:
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):
Constant | Meaning | Property |
---|---|---|
C1 | Pressure sensitivity | |
C2 | Uniaxial 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).
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):
Constant | Meaning | Property |
---|---|---|
C1 | Pressure sensitivity | |
C2 | Exponent | |
C3 | Uniaxial 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).
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):
Constant | Meaning | Property |
---|---|---|
C1 | Pressure sensitivity | |
C2 | Material parameter | |
C3 | Uniaxial 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).
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:
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):
Constant | Meaning | Property |
---|---|---|
C1 | Pressure 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).
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):
Constant | Meaning | Property |
---|---|---|
C1 | Pressure sensitivity | |
C2 | Exponent |
The material behavior can be defined as a function of temperature
(NTEMP
on the TB command),
with temperatures specified for the table entries (TBTEMP).
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):
Constant | Meaning | Property |
---|---|---|
C1 | Pressure sensitivity | |
C2 | Material 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).
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.
The following examples show how to define the EDP material model using various yield criteria and flow potentials:
/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
/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
/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
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:
After initializing the material data table (TB,EDP,,,,CYFUN), enter the following constants (TBDATA):
Constant | Material | Property |
---|---|---|
C1 | Compaction cap parameter | |
C2 | Expansion cap parameter | |
C3 | Compaction cap yield pressure | |
C4 | Cohesion yield parameter | |
C5 | Shear envelope exponent | |
C6 | Shear envelope exponential coefficient | |
C7 | Shear envelope linear coefficient | |
C8 | Ratio of extension to compression strength | |
C9 | Limiting value of volumetric plastic strain | |
C10 | Hardening parameter | |
C11 | Hardening 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).
After initializing the material data table (TB,EDP,,,,CFPOT), enter the following constants (TBDATA):
Constant | Material | Property |
---|---|---|
C1 | Compaction cap parameter | |
C2 | Expansion cap parameter | |
C3 | Shear envelope exponent | |
C4 | Shear 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.
The following example input shows how to define an EDP Cap model by defining the yield criterion, hardening, and plastic flow potential:
/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
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.
The following figure shows the phenomena of voids at the microscopic scale that are incorporated into the Gurson model:
Figure 4.15: Growth, Nucleation, and Coalescence of Voids at Microscopic Scale
(a): Existing voids grow when the solid matrix is in a hydrostatic-tension state. The solid matrix is assumed to be incompressible in plasticity so that any material volume growth is due to the void volume expansion.
(b): Void nucleation occurs, where new voids are created during plastic deformation due to debonding of the inclusion-matrix or particle-matrix interface, or from the fracture of the inclusions or particles themselves.
(c): Voids coalesce. In this process, the isolated voids establish connections. Although coalescence may not discernibly affect the void volume, the load-carrying capacity of the material begins to decay more rapidly at this stage.
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:
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.
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.
To define the Gurson base model, initialize the material data table (TB,GURSON,,,,BASE), then input the following constants (TBDATA):
Constant | Meaning | Property |
---|---|---|
C1 | Initial yield strength | |
C2 | Initial porosity | |
C3 | First Tvergaard-Needleman constant | |
C4 | Second Tvergaard-Needleman constant | |
C5 | Third Tvergaard-Needleman constant |
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):
Constant | Meaning | Property |
---|---|---|
C1 | Nucleation porosity | |
C2 | Mean stress | |
C3 | Stress standard deviation |
To define strain-controlled nucleation, initialize the material data table (TB,GURSON,,,,SNNU), then input the following constants (TBDATA):
Constant | Meaning | Property |
---|---|---|
C1 | Nucleation porosity | |
C2 | Mean strain | |
C3 | Strain standard deviation |
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):
Constant | Meaning | Property |
---|---|---|
C1 | Critical porosity | |
C2 | Failure porosity |
Following is an example Gurson plasticity material model definition:
/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
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.
Define the isotropic elastic behavior (MP). Initialize the material data table (TB,CAST,,,,ISOTROPIC) and input the following constant:
Constant | Meaning | Property |
---|---|---|
C1 | Plastic 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:
Constant | Meaning | Property |
---|---|---|
X | Strain value | |
Y | Stress 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).
/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