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Rate-independent plasticity is characterized by the irreversible straining that occurs in a material once a certain level of stress is reached. The plastic strains are assumed to develop instantaneously, that is, independent of time. The ANSYS program provides seven options to characterize different types of material behaviors. These options are:

Material Behavior Option

Bilinear Isotropic Hardening

Multilinear Isotropic Hardening

Nonlinear Isotropic Hardening

Classical Bilinear Kinematic Hardening

Multilinear Kinematic Hardening

Nonlinear Kinematic Hardening

Anisotropic

Drucker-Prager

Cast Iron

User Specified Behavior (see User Routines and Non-Standard Uses of the

*Advanced Analysis Techniques Guide*and the*Guide to ANSYS User Programmable Features*)

Except for User Specified Behavior (**TB**,USER), each
of these is explained in greater detail later in this chapter. Figure 4.1: "Stress-Strain Behavior of Each of the Plasticity Options" represents
the stress-strain behavior of each of the options.

Plasticity theory provides a mathematical relationship that characterizes the elastoplastic response of materials. There are three ingredients in the rate-independent plasticity theory: the yield criterion, flow rule and the hardening rule. These will be discussed in detail subsequently. Table 4.1: "Notation" summarizes the notation used in the remainder of this chapter.

The yield criterion determines the stress level at which yielding is
initiated. For multi-component stresses, this is represented as a function
of the individual components, f({σ}), which can be interpreted as an equivalent
stress σ_{e}:

(4–4) |

where:

{σ} = stress vector |

Variable | Definition | ANSYS Output Label |
---|---|---|

{ε^{el}} | elastic strains | EPEL |

{ε^{pl}} | plastic strains | EPPL |

{ε^{tr}} | trial strain | |

equivalent plastic strain | EPEQ[1] | |

{σ} | stresses | S |

σ_{e} | equivalent stress | |

σ_{y} | material yield parameter | |

σ_{m} | mean or hydrostatic stress | HPRES |

equivalent stress parameter | SEPL | |

λ | plastic multiplier | |

{α} | yield surface translation | |

κ | plastic work | |

C | translation multiplier | |

[D] | stress-strain matrix | |

E_{T} | tangent modulus | |

F | yield criterion | |

N | stress ratio | SRAT |

Q | plastic potential | |

{S} | deviatoric stress |

When the equivalent stress is equal to a material yield parameter σ_{y},

(4–5) |

the material will develop plastic strains. If σ_{e} is
less than σ_{y}, the material is elastic and the stresses
will develop according to the elastic stress-strain relations. Note that
the equivalent stress can never exceed the material yield since in this case
plastic strains would develop instantaneously, thereby reducing the stress
to the material yield. (Equation 4–5) can be plotted
in stress space as shown in Figure 4.2: "Various Yield Surfaces" for some of the
plasticity options. The surfaces in Figure 4.2: "Various Yield Surfaces" are known
as the yield surfaces and any stress state inside the surface is elastic,
that is, they do not cause plastic strains.

The flow rule determines the direction of plastic straining and is given as:

(4–6) |

where:

λ = plastic multiplier (which determines the amount of plastic straining) |

Q = function of stress termed the plastic potential (which determines the direction of plastic straining) |

If Q is the yield function (as is normally assumed), the flow rule is termed associative and the plastic strains occur in a direction normal to the yield surface.

The hardening rule describes the changing of the yield surface with progressive yielding, so that the conditions (i.e. stress states) for subsequent yielding can be established. Two hardening rules are available: work (or isotropic) hardening and kinematic hardening. In work hardening, the yield surface remains centered about its initial centerline and expands in size as the plastic strains develop. For materials with isotropic plastic behavior this is termed isotropic hardening and is shown in Figure 4.3: "Types of Hardening Rules" (a). Kinematic hardening assumes that the yield surface remains constant in size and the surface translates in stress space with progressive yielding, as shown in Figure 4.3: "Types of Hardening Rules" (b).

The yield criterion, flow rule and hardening rule for each option are summarized in Table 4.2: "Summary of Plasticity Options" and are discussed in detail later in this chapter.

**Table 4.2 Summary of Plasticity Options**

Name | TB Lab | Yield Criterion | Flow Rule | Hardening Rule | Material Response |
---|---|---|---|---|---|

Bilinear Isotropic Hardening | BISO | von Mises/Hill | associative | work hardening | bilinear |

Multilinear Isotropic Hardening | MISO | von Mises/Hill | associative | work hardening | multilinear |

Nonlinear Isotropic Hardening | NLISO | von Mises/Hill | associative | work hardening | nonlinear |

Classical Bilinear Kinematic Hardening | BKIN | von Mises/Hill | associative (Prandtl- Reuss equations) | kinematic hardening | bilinear |

Multilinear Kinematic Hardening | MKIN/KINH | von Mises/Hill | associative | kinematic hardening | multilinear |

Nonlinear Kinematic Hardening | CHAB | von Mises/Hill | associative | kinematic hardening | nonlinear |

Anisotropic | ANISO | modified von Mises | associative | work hardening | bilinear, each direction and tension and compression different |

Drucker- Prager | DP | von Mises with dependence on hydrostatic stress | associative or non- associative | none | elastic- perfectly plastic |

Extended Drucker-Prager | EDP | von MIses with dependence on hydrostatic stress | associative or non- associative | work hardening | multilinear |

Cast Iron | CAST | von Mises with dependence on hydrostatic stress | non- associative | work hardening | multilinear |

Gurson | GURS | von Mises with dependence pressure and porosity | associative | work hardening | multilinear |

If the equivalent stress computed using elastic properties exceeds the material yield, then plastic straining must occur. Plastic strains reduce the stress state so that it satisfies the yield criterion, (Equation 4–5). Based on the theory presented in the previous section, the plastic strain increment is readily calculated.

The hardening rule states that the yield criterion changes with work hardening and/or with kinematic hardening. Incorporating these dependencies into (Equation 4–5), and recasting it into the following form:

(4–7) |

where:

κ = plastic work |

{α} = translation of yield surface |

κ and {α} are termed internal or state variables. Specifically, the plastic work is the sum of the plastic work done over the history of loading:

(4–8) |

where:

and translation (or shift) of the yield surface is also history dependent and is given as:

(4–9) |

where:

C = material parameter |

{α} = back stress (location of the center of the yield surface) |

(Equation 4–7) can be differentiated so that the consistency condition is:

(4–10) |

Noting from (Equation 4–8) that

(4–11) |

and from (Equation 4–9) that

(4–12) |

(Equation 4–10) becomes

(4–13) |

The stress increment can be computed via the elastic stress-strain relations

(4–14) |

where:

[D] = stress-strain matrix |

with

(4–15) |

since the total strain increment can be divided into an elastic and plastic part. Substituting (Equation 4–6) into (Equation 4–13) and (Equation 4–15) and combining (Equation 4–13), (Equation 4–14), and (Equation 4–15) yields

(4–16) |

The size of the plastic strain increment is therefore related to the total increment in strain, the current stress state, and the specific forms of the yield and potential surfaces. The plastic strain increment is then computed using (Equation 4–6):

(4–17) |

An Euler backward scheme is used to enforce the consistency condition (Equation 4–10). This ensures that the updated stress, strains and internal variables are on the yield surface. The algorithm proceeds as follows:

The material parameter σ

_{y}(Equation 4–5) is determined for this time step (e.g., the yield stress at the current temperature).The stresses are computed based on the trial strain {ε

^{tr}}, which is the total strain minus the plastic strain from the previous time point (thermal and other effects are ignored):(4–18) where the superscripts are described with Understanding Theory Reference Notation and subscripts refer to the time point. Where all terms refer to the current time point, the subscript is dropped. The trial stress is then

(4–19) The equivalent stress σ

_{e}is evaluated at this stress level by (Equation 4–4). If σ_{e}is less than σ_{y}the material is elastic and no plastic strain increment is computed.If the stress exceeds the material yield, the plastic multiplier λ is determined by a local Newton-Raphson iteration procedure (Simo and Taylor(155)).

{Δε

^{pl}} is computed via (Equation 4–17).The current plastic strain is updated

(4–20) where:

= current plastic strains (output as EPPL) and the elastic strain computed

(4–21) where:

ε ^{el}= elastic strains (output as EPEL)The stress vector is:

(4–22) where:

{σ} = stresses (output as S) The increments in the plastic work Δκ and the center of the yield surface {Δα} are computed via (Equation 4–11) and (Equation 4–12) and the current values updated

(4–23) and

(4–24) where the subscript n-1 refers to the values at the previous time point.

For output purposes, an equivalent plastic strain (output as EPEQ), equivalent plastic strain increment Δ (output with the label “MAX PLASTIC STRAIN STEP”), equivalent stress parameter (output as SEPL) and stress ratio N (output as SRAT) are computed. The stress ratio is given as

(4–25) where σ

_{e}is evaluated using the trial stress . N is therefore greater than or equal to one when yielding is occurring and less than one when the stress state is elastic. The equivalent plastic strain increment is given as:(4–26) The equivalent plastic strain and equivalent stress parameters are developed for each option in the next sections.

Note that the Euler backward integration scheme in step 4 is the radial return algorithm (Krieg(46)) for the von Mises yield criterion.

The tangent or elastoplastic stress-strain matrix is derived from the local Newton-Raphson iteration scheme used in step 4 above (Simo and Taylor(155)). It is therefore the consistent (or algorithmic) tangent. If the flow rule is nonassociative (F ≠ Q), then the tangent is unsymmetric. To preserve the symmetry of the matrix, for analyses with a nonassociative flow rule (Drucker-Prager only), the matrix is evaluated using F only and again with Q only and the two matrices averaged.

Multilinear Isotropic Hardening and Bilinear Isotropic Hardening

These options use the von Mises yield criterion with the associated
flow rule and isotropic (work) hardening (accessed with **TB**,MISO
and **TB**,BISO).

The equivalent stress (Equation 4–4) is:

(4–27) |

where {s} is the deviatoric stress (Equation 4–37).
When σ_{e} is equal to the current yield stress σ_{k} the
material is assumed to yield. The yield criterion is:

(4–28) |

For work hardening, σ_{k} is a function of the
amount of plastic work done. For the case of isotropic plasticity assumed
here, σ_{k} can be determined directly from the equivalent
plastic strain of (Equation 4–42) (output
as EPEQ) and the uniaxial stress-strain curve as depicted in Figure 4.4: "Uniaxial Behavior".
σ_{k} is output as the equivalent stress parameter
(output as SEPL). For temperature-dependent curves with the MISO option,
σ_{k} is determined by temperature interpolation of
the input curves after they have been converted to stress-plastic strain format.

Both the Voce(253) hardening
law, and the nonlinear power hardening law can be used to model nonlinear
isotropic hardening. The Voce hardening law for nonlinear isotropic hardening
behavior (accessed with **TB**,NLISO,,,,VOCE) is specified
by the following equation:

(4–29) |

where:

k = elastic limit |

R_{o}, ,
b = material parameters characterizing the isotropic hardening behavior of
materials |

= equivalent plastic strain |

The constitutive equations are based on linear isotropic elasticity, the von Mises yield function and the associated flow rule. The yield function is:

(4–30) |

The plastic strain increment is:

(4–31) |

where:

λ = plastic multiplier |

The equivalent plastic strain increment is then:

(4–32) |

The accumulated equivalent plastic strain is:

(4–33) |

The power hardening law for nonlinear isotropic
hardening behavior (accessed with **TB**,NLISO,,,,POWER) which
is used primarily for ductile plasticity and damage is developed in the Gurson's Model:

(4–34) |

where:

σ_{Y} = current yield strength |

σ_{0} = initial yield strength |

G = shear modulus |

is the microscopic equivalent plastic strain and is defined by:

(4–35) |

where:

ε^{p} = macroscopic plastic strain tensor |

= rate change of variables |

σ = Cauchy stress tensor |

: = inner product operator of two
second order tensors |

f = porosity |

This option uses the von Mises yield criterion with the associated flow
rule and kinematic hardening (accessed with **TB**,BKIN).

The equivalent stress (Equation 4–4) is therefore

(4–36) |

where: {s} = deviatoric stress vector |

(4–37) |

where:

{α} = yield surface translation vector (Equation 4–9) |

Note that since (Equation 4–36) is dependent on the
deviatoric stress, yielding is independent of the hydrostatic stress state.
When σ_{e} is equal to the uniaxial yield stress,
σ_{y}, the material is assumed to yield. The yield
criterion (Equation 4–7) is therefore,

(4–38) |

The associated flow rule yields

(4–39) |

so that the increment in plastic strain is normal to the yield surface. The associated flow rule with the von Mises yield criterion is known as the Prandtl-Reuss flow equation.

The yield surface translation is defined as:

(4–40) |

where:

G = shear modulus = E/(2 (1+ν)) |

E = Young's modulus (input as EX on MP command) |

ν = Poisson's ratio (input as PRXY or NUXY on MP command) |

The shift strain is computed analogously to (Equation 4–24):

(4–41) |

where:

(4–42) |

where:

E = Young's modulus (input as EX on MP command) |

E_{T} = tangent modulus from the bilinear uniaxial
stress-strain curve |

The yield surface translation {ε^{sh}}
is initially zero and changes with subsequent plastic straining.

The equivalent plastic strain is dependent on the loading history and is defined to be:

(4–43) |

where:

= equivalent plastic strain for this time point (output as EPEQ) |

= equivalent plastic strain from the previous time point |

The equivalent stress parameter is defined to be:

(4–44) |

where:

= equivalent stress parameter (output as SEPL) |

Note that if there is no plastic straining ( =
0), then is equal to the yield stress. only has meaning during the initial, monotonically
increasing portion of the load history. If the load were to be reversed after
plastic loading, the stresses and therefore σ_{e} would
fall below yield but would register
above yield (since is nonzero).

This option (accessed with **TB**,MKIN and **TB**,KINH)
uses the Besseling(53) model also called the sublayer
or overlay model (Zienkiewicz(54)) to characterize
the material behavior. The material behavior is assumed to be composed of
various portions (or subvolumes), all subjected to the same total strain,
but each subvolume having a different yield strength. (For a plane stress
analysis, the material can be thought to be made up of a number of different
layers, each with a different thickness and yield stress.) Each subvolume
has a simple stress-strain response but when combined the model can represent
complex behavior. This allows a multilinear stress-strain curve that exhibits
the Bauschinger (kinematic hardening) effect (Figure 4.1: "Stress-Strain Behavior of Each of the Plasticity Options" (b)).

The following steps are performed in the plasticity calculations:

The portion of total volume for each subvolume and its corresponding yield strength are determined.

The increment in plastic strain is determined for each subvolume assuming each subvolume is subjected to the same total strain.

The individual increments in plastic strain are summed using the weighting factors determined in step 1 to compute the overall or apparent increment in plastic strain.

The plastic strain is updated and the elastic strain is computed.

The portion of total volume (the weighting factor) and yield stress for each subvolume is determined by matching the material response to the uniaxial stress-strain curve. A perfectly plastic von Mises material is assumed and this yields for the weighting factor for subvolume k

(4–45) |

where:

w_{k} = the weighting factor (portion of total
volume) for subvolume k and is evaluated sequentially from 1 to the number
of subvolumes |

E_{Tk} = the slope of the kth segment of the stress-strain
curve (see Figure 4.5: "Uniaxial Behavior for Multilinear Kinematic Hardening") |

Σ_{wi} = the sum of the weighting factors
for the previously evaluated subvolumes |

The yield stress for each subvolume is given by

(4–46) |

where (ε_{k}, σ_{k})
is the breakpoint in the stress-strain curve. The number of subvolumes corresponds
to the number of breakpoints specified.

The increment in plastic strain for each subvolume is computed using a von Mises yield criterion with the associated flow rule. The section on specialization for bilinear kinematic hardening is followed but since each subvolume is elastic-perfectly plastic, C and therefore {α} is zero.

The plastic strain increment for the entire volume is the sum of the subvolume increments:

(4–47) |

where:

N_{sv} = number of subvolumes |

The current plastic strain and elastic strain can then be computed for the entire volume via (Equation 4–20) and (Equation 4–21).

The equivalent plastic strain (output as EPEQ) is defined by (Equation 4–43) and equivalent stress parameter (output as SEPL) is computed by evaluating the input stress-strain curve at (after adjusting the curve for the elastic strain component).

The material model considered is a rate-independent version of the nonlinear
kinematic hardening model proposed by Chaboche(244, 245) (accessed with **TB**,CHAB). The constitutive
equations are based on linear isotropic elasticity, a von Mises yield function
and the associated flow rule. Like the bilinear and multilinear kinematic
hardening options, the model can be used to simulate the monotonic hardening
and the Bauschinger effect. The model is also applicable to simulate the
ratcheting effect of materials. In addition, the model allows the superposition
of several kinematic models as well as isotropic hardening models. It is
thus able to model the complicated cyclic plastic behavior of materials, such
as cyclic hardening or softening and ratcheting or shakedown.

The model uses the von Mises yield criterion with the associated flow rule, the yield function is:

(4–48) |

where:

R = isotropic hardening variable |

According to the normality rule, the flow rule is written:

(4–49) |

where:

λ = plastic multiplier |

The back stress {α} is superposition of several kinematic models as:

(4–50) |

where:

n = number of kinematic models to be superposed. |

The evolution of the back stress (the kinematic hardening rule) for each component is defined as:

(4–51) |

where:

C_{i}, γ_{i}, i = 1, 2,
... n = material constants for kinematic hardening |

The associated flow rule yields:

(4–52) |

The plastic strain increment, (Equation 4–49) is rewritten as:

(4–53) |

The equivalent plastic strain increment is then:

(4–54) |

The accumulated equivalent plastic strain is:

(4–55) |

The isotropic hardening variable, R, can be defined by:

(4–56) |

where:

k = elastic limit |

R_{o}, ,
b = material constants characterizing the material isotropic hardening behavior. |

The material hardening behavior, R, in (Equation 4–48) can also be defined through bilinear or multilinear isotropic hardening options, which have been discussed early in Specialization for Hardening.

The return mapping approach with consistent elastoplastic tangent moduli that was proposed by Simo and Hughes(252) is used for numerical integration of the constitutive equation described above.

There are two anisotropic plasticity options in ANSYS. The first option
uses Hill's(50) potential theory (accessed by **TB**,HILL
command). The second option uses a generalized Hill potential theory (Shih
and Lee(51)) (accessed by **TB**,
ANISO command).

The anisotropic Hill potential theory (accessed by **TB**,HILL)
uses Hill's(50) criterion. Hill's criterion is
an extension to the von Mises yield criterion to account for the anisotropic
yield of the material. When this criterion is used with the isotropic hardening
option, the yield function is given by:

(4–57) |

where:

σ_{0} = reference yield stress |

= equivalent plastic strain |

and when it is used with the kinematic hardening option, the yield function takes the form:

(4–58) |

The material is assumed to have three orthogonal planes of symmetry. Assuming the material coordinate system is perpendicular to these planes of symmetry, the plastic compliance matrix [M] can be written as:

(4–59) |

F, G, H, L, M and N are material constants that can be determined experimentally. They are defined as:

(4–60) |

(4–61) |

(4–62) |

(4–63) |

(4–64) |

(4–65) |

The yield stress ratios R_{xx}, R_{yy},
R_{zz}, R_{xy}, R_{yz} and
R_{xz} are specified by the user and can be calculated
as:

(4–66) |

(4–67) |

(4–68) |

(4–69) |

(4–70) |

(4–71) |

where:

= yield stress values |

Two notes:

The inelastic compliance matrix should be positive definite in order for the yield function to exist.

The plastic slope (see also (Equation 4–42)) is calculated as:

(4–72) |

where:

E_{x} = elastic modulus in x-direction |

E_{t} = tangent modulus defined by the hardening
input |

The generalized anisotropic Hill potential theory (accessed by **TB**,ANISO)
uses Hill's(50) yield criterion, which accounts
for differences in yield strengths in orthogonal directions, as modified by
Shih and Lee(51) accounting for differences in yield
strength in tension and compression. An associated flow rule is assumed and
work hardening as presented by Valliappan et al.(52)
is used to update the yield criterion. The yield surface is therefore a distorted
circular cylinder that is initially shifted in stress space which expands
in size with plastic straining as shown in Figure 4.2: "Various Yield Surfaces" (b).

The equivalent stress for this option is redefined to be:

(4–73) |

where [M] is a matrix which describes the variation of the yield stress
with orientation and {L} accounts for the difference between tension and compression
yield strengths. {L} can be related to the yield surface translation {α}
of (Equation 4–36) (Shih and Lee(51))
and hence the equivalent stress function can be interpreted as having an initial
translation or shift. When σ_{e} is equal to a material
parameter K, the material is assumed to yield. The yield criterion (Equation 4–7) is then

(4–74) |

The material is assumed to have three orthogonal planes of symmetry. The plastic behavior can then be characterized by the stress-strain behavior in the three element coordinate directions and the corresponding shear stress-shear strain behavior. Therefore [M] has the form:

(4–75) |

By evaluating the yield criterion (Equation 4–74) for all the possible uniaxial stress conditions the individual terms of [M] can be identified:

(4–76) |

where:

σ_{+j} and σ_{-j} =
tensile and compressive yield strengths in direction j (j = x, y, z, xy, yz,
xz) |

The compressive yield stress is handled as a positive number here.
For the shear yields, σ_{+j} = σ_{-j}.
Letting M_{11} = 1 defines K to be

(4–77) |

The strength differential vector {L} has the form

(4–78) |

and from the uniaxial conditions {L} is defined as

(4–79) |

Assuming plastic incompressibility (i.e. no increase in material volume due to plastic straining) yields the following relationships

(4–80) |

and

(4–81) |

The off-diagonals of [M] are therefore

(4–82) |

Note that (Equation 4–81) (by means of (Equation 4–76) and (Equation 4–79)) yields the consistency equation

(4–83) |

that must be satisfied due to the requirement of plastic incompressibility. Therefore the uniaxial yield strengths are not completely independent.

The yield strengths must also define a closed yield surface, that is, elliptical in cross section. An elliptical yield surface is defined if the following criterion is met:

(4–84) |

Otherwise, the following message is output: “THE DATA TABLE DOES NOT REPRESENT A CLOSED YIELD SURFACE. THE YIELD STRESSES OR SLOPES MUST BE MADE MORE EQUAL”. This further restricts the independence of the uniaxial yield strengths. Since the yield strengths change with plastic straining (a consequence of work hardening), this condition must be satisfied throughout the history of loading. The program checks this condition through an equivalent plastic strain level of 20% (.20).

For an isotropic material,

(4–85) |

and

(4–86) |

and the yield criterion ((Equation 4–74) reduces down to the von Mises yield criterion

(Equation 4–38) with {α} = 0).

Work hardening is used for the hardening rule so that the subsequent yield strengths increase with increasing total plastic work done on the material. The total plastic work is defined by (Equation 4–23) where the increment in plastic work is

(4–87) |

where:

= average stress over the increment |

For the uniaxial case the total plastic work is simply

(4–88) |

where the terms are defined as shown in Figure 4.6: "Plastic Work for a Uniaxial Case".

For bilinear stress-strain behavior,

(4–89) |

where:

= plastic slope (see also (Equation 4–42)) |

E = elastic modulus |

E_{T} = tangent moulus |

(4–90) |

Combining (Equation 4–89) with (Equation 4–88) and solving for the updated yield stress σ:

(4–91) |

Extending this result to the anisotropic case gives,

(4–92) |

where j refers to each of the input stress-strain curves. (Equation 4–92) determines the updated yield stresses by equating the amount of plastic work done on the material to an equivalent amount of plastic work in each of the directions.

The parameters [M] and {L} can then be updated from their definitions (Equation 4–76) and (Equation 4–79) and the new values of the yield stresses. For isotropic materials, this hardening rule reduces to the case of isotropic hardening.

The equivalent plastic strain (output as EPEQ) is computed using the tensile x direction as the reference axis by substituting (Equation 4–89) into (Equation 4–88):

(4–93) |

where the yield stress in the tensile x direction σ_{+x} refers
to the initial (not updated) yield stress. The equivalent stress parameter (output as SEPL) is defined as

(4–94) |

where again σ_{+x} is the initial yield stress.

This option uses the Drucker-Prager yield criterion with either an associated
or nonassociated flow rule (accessed with **TB**,DP). The
yield surface does not change with progressive yielding, hence there is no
hardening rule and the material is elastic- perfectly plastic (Figure 4.1: "Stress-Strain Behavior of Each of the Plasticity Options" (f)
Drucker-Prager). The equivalent stress for Drucker-Prager is

(4–95) |

where:

{s} = deviatoric stress (Equation 4–37) |

β = material constant |

[M] = as defined with (Equation 4–36) |

This is a modification of the von Mises yield criterion ((Equation 4–36) with {α} = {0}) that accounts for the influence of the hydrostatic stress component: the higher the hydrostatic stress (confinement pressure) the higher the yield strength. β is a material constant which is given as

(4–96) |

where:

φ = input angle of internal friction |

The material yield parameter is defined as

(4–97) |

where:

c = input cohesion value |

The yield criterion (Equation 4–7) is then

(4–98) |

This yield surface is a circular cone (Figure 4.2: "Various Yield Surfaces"-c) with the material parameters (Equation 4–96) and (Equation 4–97) chosen such that it corresponds to the outer aspices of the hexagonal Mohr-Coulomb yield surface, Figure 4.7: "Drucker-Prager and Mohr-Coulomb Yield Surfaces".

is readily computed as

(4–99) |

is similar, however β is
evaluated using φ_{f} (the input “dilatancy”
constant). When φ_{f} = φ, the flow rule is
associated and plastic straining occurs normal to the yield surface and there
will be a volumetric expansion of the material with plastic strains. If φ_{f} is
less than φ there will be less volumetric expansion and if φ_{f} is
zero, there will be no volumetric expansion.

The equivalent plastic strain (output as EPEQ) is defined by (Equation 4–43) and the equivalent stress parameter (output as SEPL) is defined as

(4–100) |

The equivalent stress parameter is interpreted as the von Mises equivalent stress at yield at the current hydrostatic stress level. Therefore for any integration point undergoing yielding (stress ratio (output as SRAT) >1), should be close to the actual von Mises equivalent stress (output as SIGE) at the converged solution.

This option is an extension of the linear Drucker-Prager yield criterion
(input with **TB**,EDP). Both yield surface and the flow potential,
(input with * TBOPT* on

The yield function with linear form (input with * TBOPT* =
LYFUN) is:

(4–101) |

where:

α = material parameter referred to pressure sensitive parameter
(input as C1 on TBDATA command using TB,EDP) |

The yield function with hyperbolic form (input with * TBOPT* =
HYFUN) is:

(4–102) |

where:

a = material parameter characterizing the shape of yield surface (input
as C2 on TBDATA command using TB,EDP) |

The yield function with power law form (input with * TBOPT* =
PYFUN) is:

(4–103) |

where:

b = material parameter characterizing the shape of yield surface (input
as C2 on TBDATA command using TB,EDP): |

Similarly, the flow potential Q for linear form (input with * TBOPT* =
LFPOT) is:

(4–104) |

The flow potential Q for hyperbolic form (input with * TBOPT* =
HFPOT) is:

(4–105) |

The flow potential Q for power law form (input with * TBOPT* =
PFPOT) is:

(4–106) |

The plastic strain is defined as:

(4–107) |

where:

Note that when the flow potential is the same as the yield function,
the plastic flow rule is associated, which in turn results in a symmetric
stiffness matrix. When the flow potential is different from the yield function,
the plastic flow rule is nonassociated, and this results in an unsymmetric
material stiffness matrix. By default, the unsymmetric stiffness matrix (accessed
by **NROPT**,UNSYM) will be symmetricized.

The cap model focuses on geomaterial plasticity resulting from compaction at low mean stresses followed by significant dilation before shear failure. A three-invariant cap plasticity model with three smooth yielding surfaces including a compaction cap, an expansion cap, and a shear envelope is described here.

Geomaterials typically have much higher tri-axial strength in compression than in tension. The cap model accounts for this by incorporating the third-invariant of stress tensor (J3) into the yielding functions.

Functions that will be utilized in the cap model are first introduced. These functions include shear failure envelope function, compaction cap function, expansion cap function, the Lode angle function, and hardening functions. Then, a unified yielding function for the cap model that is able to describe all the behaviors of shear, compaction, and expansion yielding surfaces is derived using the shear failure envelope and cap functions.

A typical geomaterial shear envelope function is based on the exponential format given below:

(4–108) |

where:

I_{1} = first invariant of Cauchy stress tensor |

subscript "s" = shear envelope function |

superscript "y" = yielding related material constants |

σ_{0} = current cohesion-related material
constant (input using TB,EDP with =
CYFUN)TBOPT |

A, β^{y}, α^{y} =
material constants (input using TB,EDP with =
CYFUN)TBOPT |

(Equation 4–108) reduces to the
Drucker-Prager yielding function if parameter "A" is set to zero.
It should be noted that all material constants in (Equation 4–108) are
defined based on I_{1} and J_{2} ,
which are different from those in the previous sections. The effect of hydrostatic
pressure on material yielding may be exaggerated at high pressure range by
only using the linear term (Drucker-Prager) in (Equation 4–108).
Such an exaggeration is reduced by using both the exponential term and linear
term in the shear function. Figure 4.8: "Shear Failure Envelope Functions" shows the configuration
of the shear function. In Figure 4.8: "Shear Failure Envelope Functions" the dots are the
testing data points, the finer dashed line is the fitting curve based on the
Drucker-Prager linear yielding function, the solid curved line is the fitting
curve based on (Equation 4–108), and the coarser dashed line
is the limited state of (Equation 4–108) at very high pressures.
In the figure is the current modified
cohesion obtained through setting I_{1} in (Equation 4–108) to
zero.

The compaction cap function is formulated using the shear envelope function defined in (Equation 4–108).

(4–109) |

where:

H = Heaviside (or unit step) function |

subscript "c" = compaction cap-related function or constant |

R = ratio of elliptical x-axis to y-axis (I_{1} to
J_{2}) |

K_{0} = key flag indicating the current transition
point at which the compaction cap surface and shear portion intersect. |

In (Equation 4–109), Y_{c} is an elliptical
function combined with the Heaviside function. Y_{c} is
plotted in Figure 4.9: "Compaction Cap Function".

This function implies:

When I

_{1}, the first invariant of stress, is greater than K_{0}, the compaction cap takes no effect on yielding. The yielding may happen in either shear or expansion cap portion.When I

_{1}is less than K_{0}, the yielding may only happen in the compaction cap portion, which is shaped by both the shear function and the elliptical function.

Similarly, Y_{t} is an elliptical function combined
with the Heaviside function designed for the expansion cap. Y_{t} is
shown in Figure 4.10: "Expansion Cap Function".

(4–110) |

where:

subscript "t" = expansion cap-related function or constant |

This function implies that:

When I

_{1}is negative, the yielding may happen in either shear or compaction cap portion, while the tension cap has no effect on yielding.When I

_{1}is positive, the yielding may only happen in the tension cap portion. The tension cap is shaped by both the shear function and by another elliptical function.

(Equation 4–110) assumes that Y_{t} is
only a function of σ_{0} and not a function of K_{0} as
I_{1} is set to zero in function Y_{s}.

Unlike metals, the yielding and failure behaviors of geomaterials are
affected by their relatively weak (compared to compression) tensile strength.
The ability of a geomaterial to resist yielding is lessened by non-uniform
stress states in the principle directions. The effect of reduced yielding
capacity for such geomaterials is described by the Lode angle β and the
ratio ψ of tri-axial extension strength to compression strength. The Lode
angle β can be written in a function of stress invariants J_{2} and
J_{3}:

(4–111) |

where:

J_{2} and J_{3} = second and
third invariants of the deviatoric tensor of the Cauchy stress tensor. |

The Lode angle function Γ is defined by:

(4–112) |

where:

ψ = ratio of triaxial extension strength to compression strength |

The three-invariant plasticity model is formulated by multiplying J_{2} in
the yielding function by the Lode angle function described by (Equation 4–112).
The profile of the yielding surface in a three-invariant plasticity model
is presented in Figure 4.11: "Yielding Surface in π-Plane".

The cap hardening law is defined by describing the evolution of the
parameter X_{0}, the intersection point of the compaction
cap and the I_{1}axis. The evolution of X_{0} is
related only to the plastic volume strain .
A typical cap hardening law has the exponential form proposed in Fossum and
Fredrich(92):

(4–113) |

where:

X_{i} = initial value of X_{0} at
which the cap takes effect in the plasticity model. |

= maximum possible plastic volumetric strain for geomaterials. |

Parameters and have units of 1/Mpa and 1 Mpa/Mpa, respectively. All constants in (Equation 4–113) are non-negative.

Besides cap hardening, another hardening law defined for the evolution
of the cohesion parameter used in the shear portion described in (Equation 4–108) is
considered. The evolution of the modified cohesion is
assumed to be purely shear-related and is the function of the effective deviatoric
plastic strain γ_{p}:

(4–114) |

The effective deviatoric plastic strain γ_{p} is
defined by its rate change as follows:

(4–115) |

where:

ε_{p} = plastic strain tensor |

"" = rate change of variables |

I = second order identity tensor |

The unified and compacted yielding function for the cap model with three smooth surfaces is formulated using above functions as follows:

(4–116) |

where:

K_{0} = function of both X_{0} and σ_{0} |

Again, the parameter X_{0} is the intersection point
of the compaction cap and the I_{1} axis. The parameter
K_{0} is the state variable and can be implicitly described
using X_{0} and σ_{0} given
below:

(4–117) |

The yielding model described in (Equation 4–116) is used
and is drawn in the J_{2} and I_{1} plane
in Figure 4.12: "Cap Model".

The cap model also allows non-associated models for all compaction cap, shear envelope, and expansion cap portions. The non-associated models are defined through using the yielding functions in (Equation 4–116) as its flow potential functions, while providing different values for some material constants. It is written below:

(4–118) |

where:

(4–119) |

where:

superscript "f" = flow-related material constant |

The flow functions in (Equation 4–118) and (Equation 4–119) are
obtained by replacing β^{y}, α^{y}, , and in (Equation 4–116) and (Equation 4–117) with β^{f}, α^{f}, , and . The nonassociated
cap model is input by using **TB**,EDP with * TBOPT* =
CFPOT.

You can take into account on shear hardening through
providing by using **TB**,MISO, **TB**,BISO, **TB**,NLISO,
or **TB**,PLAS. The initial value of must
be consistent to σ_{i} - A. This input regulates
the relationship between the modified cohesion and the effective deviatoric
plastic strain.

Calibrating the CAP constants σ_{i}, β^{Y},
A, α^{Y}, β^{Y}, α^{F} and
the hardening input for differs significantly
from the other EDP options. The CAP parameters are all defined in relation
to I_{1} and I_{2}, while the other
EDP coefficients are defined according to p and q.

The Gurson Model is used to represent plasticity and damage in ductile porous metals. The model theory is based on Gurson(366) and Tvergaard and Needleman(367). When plasticity and damage occur, ductile metal goes through a process of void growth, nucleation, and coalescence. Gurson’s method models the process by incorporating these microscopic material behaviors into macroscopic plasticity behaviors based on changes in the void volume fraction (porosity) and pressure. A porosity index increase corresponds to an increase in material damage, which implies a diminished material load-carrying capacity.

The microscopic porous metal representation in Figure 4.13: "Growth, Nucleation, and Coalescence of Voids in Microscopic Scale"(a), shows how the existing voids dilate (a phenomenon, called void growth) when the solid matrix is in a hydrostatic-tension state. The solid matrix portion is assumed to be incompressible when it yields, therefore any material volume growth (solid matrix plus voids) is due solely to the void volume expansion.

The second phenomenon is void nucleation which means that new voids are created during plastic deformation. Figure 4.13: "Growth, Nucleation, and Coalescence of Voids in Microscopic Scale"(b), shows the nucleation of voids resulting from the debonding of the inclusion-matrix or particle-matrix interface, or from the fracture of the inclusions or particles themselves.

The third phenomenon is the coalescence of existing voids. In this process, shown in Figure 4.13: "Growth, Nucleation, and Coalescence of Voids in Microscopic Scale"(c), the isolated voids establish connections. Although coalescence may not discernibly affect the void volume, the load carrying capacity of this material begins to decay more rapidly at this stage.

The evolution equation of porosity is given by

(4–120) |

where:

f = porosity |

= rate change of variables |

The evolution of the microscopic equivalent plastic work is:

(4–121) |

where:

= microscopic equivalent plastic strain |

σ = Cauchy stress |

: = inner product operator of two
second order tensors |

ε^{p} = macroscopic plastic strain |

σ_{Y} = current yielding strength |

The evolution of porosity related to void growth and nucleation can be stated in terms of the microscopic equivalent plastic strain, as follows:

(4–122) |

where:

I = second order identity tensor |

The void nucleation is controlled by either the plastic strain or stress, and is assumed to follow a normal distribution of statistics. In the case of strain-controlled nucleation, the distribution is described in terms of the mean strain and its corresponding deviation. In the case of stress-controlled nucleation, the distribution is described in terms of the mean stress and its corresponding deviation. The porosity rate change due to nucleation is then given as follows:

(4–123) |

where:

f_{N} = volume fraction of the segregated inclusions
or particles |

ε_{N} = mean strain |

S_{N} = strain deviation |

σ_{N} = mean stress |

It should be noted that "stress controlled nucleation" means
that the void nucleation is determined by the maximum normal stress on the
interfaces between inclusions and the matrix. This maximum normal stress is
measured by σ_{Y} + p. Thus, more precisely, the "stress" in
the mean stress σ_{N} refers to σ_{Y} +
p. This relationship better accounts for the effect of tri-axial loading conditions
on nucleation.

Given (Equation 4–120) through (Equation 4–123), the material yielding rule of the Gurson model is defined as follows:

(4–124) |

where:

q_{1}, q_{2}, and q_{3} =
Tvergaard-Needleman constants |

σ_{Y} = yield strength of material |

f*, the Tvergaard-Needleman function is:

(4–125) |

where:

f_{c} = critical porosity |

f_{F} = failure porosity |

The Tvergaard-Needleman function is used to model the loss of material
load carrying capacity, which is associated with void coalescence. When the
current porosity f reaches a critical value f_{c}, the
material load carrying capacity decreases more rapidly due to the coalescence.
When the porosity f reaches a higher value f_{F}, the
material load carrying capacity is lost completely. The associative plasticity
model for the Gurson model has been implemented.

The cast iron plasticity model is designed to model gray cast iron. The microstructure of gray cast iron can be looked at as a two-phase material, graphite flakes inserted into a steel matrix (Hjelm(334)). This microstructure leads to a substantial difference in behavior in tension and compression. In tension, the material is more brittle with low strength and cracks form due to the graphite flakes. In compression, no cracks form, the graphite flakes behave as incompressible media that transmit stress and the steel matrix only governs the overall behavior.

The model assumes isotropic elastic behavior, and the elastic behavior is assumed to be the same in tension and compression. The plastic yielding and hardening in tension may be different from that in compression (see Figure 4.14: "Idealized Response of Gray Cast Iron in Tension and Compression"). The plastic behavior is assumed to harden isotropically and that restricts the model to monotonic loading only.

*Yield Criteria*

A composite yield surface is used to describe the different behavior in tension and compression. The tension behavior is pressure dependent and the Rankine maximum stress criterion is used. The compression behavior is pressure independent and the von Mises yield criterion is used. The yield surface is a cylinder with a tension cutoff (cap). Figure 4.15: "Cross-Section of Yield Surface" shows a cross section of the yield surface on principal deviatoric-stress space and Figure 4.16: "Meridian Section of Yield Surface" shows a meridional sections of the yield surface for two different stress states, compression (θ = 60) and tension (θ = 0).

The yield surface for tension and compression "regimes" are described by (Equation 4–126) and (Equation 4–127) (Chen and Han(332)).

The yield function for the tension cap is:

(4–126) |

and the yield function for the compression regime is:

(4–127) |

where:

p = I_{1} / 3 = σ_{ii} /
3 = hydrostatic pressure |

S = deviatoric stress tensor |

σ_{t} = tension yield stress |

σ_{c} = compression yield stress |

*Flow Rule*

The plastic strain increments are defined as:

(4–128) |

where Q is the so-called plastic flow potential, which consists of the von Mises cylinder in compression and modified to account for the plastic Poisson's ratio in tension, and takes the form:

(4–129) |

(4–130) |

and

where:

ν^{pl} = plastic Poisson's ratio (input
using TB,CAST) |

(Equation 4–130) is for less than 0.5. When ν^{pl} =
0.5, the equation reduces to the von Mises cylinder. This is shown below:

As the flow potential is different from the yield function, nonassociated flow rule, the resulting material Jacobian is unsymmetric.

*Hardening*

The yield stress in uniaxial tension, σ_{t},
depends on the equivalent uniaxial plastic strain in tension, , and the temperature T. Also the yield stress in uniaxial
compression, σ_{c}, depends on the equivalent uniaxial
plastic strain in compression, , and
the temperature T.

To calculate the change in the equivalent plastic strain in tension, the plastic work expression in the uniaxial tension case is equated to general plastic work expression as:

(4–131) |

where:

{Δε^{pl}} = plastic strain vector
increment |

(Equation 4–128) leads to:

(4–132) |

In contrast, the change in the equivalent plastic strain in compression is defined as:

(4–133) |

where:

The yield and hardening in tension and compression are provided using
the **TB**,UNIAXIAL command which has two options, tension
and compression.