elastoplasticmodelbase

ElastoplasticModelBase

Bases: ConstitutiveModelBase

Base class for elastoplastic contact laws.

cached_ `instance-attribute`

cached_: bool

dpadsiga `instance-attribute`

dpadsiga: VectorXf

dqadtaua `instance-attribute`

dqadtaua: VectorXf

dtauadsiga `instance-attribute`

dtauadsiga: MatrixXf

plasticMethod `instance-attribute`

plasticMethod: str

The plastic method

state `instance-attribute`

state: str

State of the elastoplastic stress integration, "none", "elastic_trial", "elastic_to_yield", "elastoplastic", or "plastic_correction"

CPPM

CPPM(K: ndarray, deps: ndarray, sv0: StateVariable, sv: StateVariable)

Solve plastic problem (plastic strain) with the closest point projection method (CPPM)

PARAMETER	DESCRIPTION
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

CSL

CSL(sv0: StateVariable, sv: StateVariable) -> float

Calculate the critical state line

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`float`	Critical void ratio

ExplicitCPA

ExplicitCPA(K: ndarray, deps: ndarray, sv0: StateVariable, sv: StateVariable)

Solve plastic problem (plastic strain) with the explicit or cutting plane algorithm (CPA)

PARAMETER	DESCRIPTION
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

RKn

RKn(
    n: int,
    K: ndarray,
    deps: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
    revert: bool = False,
) -> tuple[ndarray, ndarray, ndarray, ndarray, ndarray]

Perform a Runge-Kutta increment

PARAMETER	DESCRIPTION
`n`	Order of the Runge-Kutta method TYPE: `int`
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`
`revert`	Whether to revert the state variables after the increment for estimation TYPE: `bool` DEFAULT: `False`

RETURNS	DESCRIPTION
`Elastoplastic stiffness matrix, stress, stress increment, plastic multiplier, and plastic strain increment`

RungeKutta

RungeKutta(
    K: ndarray,
    deps: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
    order: int = None,
)

Solve plastic problem (plastic strain) with the Runge-Kutta method

PARAMETER	DESCRIPTION
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`
`order`	Order of the Runge-Kutta method, by default None TYPE: `int` DEFAULT: `None`

abstractSetup

abstractSetup()

beginConstantStressStateScope

beginConstantStressStateScope(sv0: StateVariable, sv: StateVariable)

Begin the scope of a constant stress state which can be used to cache intermediate variables for the plastic correction, such as the derivatives of the yield and plastic potential functions, and hardening law.

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

beginConstantStressStateScope1

beginConstantStressStateScope1(sv0: StateVariable, sv: StateVariable)

Begin the scope of a constant stress state which can be used to cache intermediate variables for the plastic correction, such as the derivatives of the yield and plastic potential functions, and hardening law.

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

consistentElastoplasticStiffness

consistentElastoplasticStiffness(
    K: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
    check_elastic_state: bool = True,
    check_positive_definite: bool = None,
    check_non_singularity: bool = None,
) -> ndarray

continuumElastoplasticStiffness

continuumElastoplasticStiffness(
    K: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
    check_elastic_state: bool = True,
    check_positive_definite: bool = None,
    check_non_singularity: bool = None,
) -> ndarray

converged

converged(sv0: StateVariable, sv: StateVariable) -> bool

Whether the current stress state can be accepted as converged, used for elastoplastic substepping.

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

d2gdsig2

d2gdsig2(dfdsig: ndarray, sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate second-order derivative of potential function with respect to the stress:

\[ \frac{\partial^2 G}{\partial \boldsymbol{\sigma}^2} \]

PARAMETER	DESCRIPTION
`dfdsig`	Derivative of yield function with respect to the stress TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`d2gdsig2`	Derivative of potential function with respect to the stress TYPE: `ndarray`

d2gdsig20

d2gdsig20(dfdsig: ndarray, sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate second-order derivative of potential function with respect to the stress:

\[ \frac{\partial^2 G}{\partial \boldsymbol{\sigma}^2} \]

PARAMETER	DESCRIPTION
`dfdsig`	Derivative of yield function with respect to the stress TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`d2gdsig2`	Derivative of potential function with respect to the stress TYPE: `ndarray`

d2gdsig21

d2gdsig21(dfdsig: ndarray, sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate second-order derivative of potential function with respect to the stress:

\[ \frac{\partial^2 G}{\partial \boldsymbol{\sigma}^2} \]

PARAMETER	DESCRIPTION
`dfdsig`	Derivative of yield function with respect to the stress TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`d2gdsig2`	Derivative of potential function with respect to the stress TYPE: `ndarray`

dfdeps

dfdeps(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the strain:

\[ \frac{\partial F}{\partial \boldsymbol{\varepsilon}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdeps`	Derivative of yield function with respect to the strain TYPE: `ndarray`

dfdeps0

dfdeps0(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the strain:

\[ \frac{\partial F}{\partial \boldsymbol{\varepsilon}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdeps`	Derivative of yield function with respect to the strain TYPE: `ndarray`

dfdeps1

dfdeps1(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the strain:

\[ \frac{\partial F}{\partial \boldsymbol{\varepsilon}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdeps`	Derivative of yield function with respect to the strain TYPE: `ndarray`

dfdepsp

dfdepsp(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the plastic strain:

\[ \frac{\partial F}{\partial \boldsymbol{\varepsilon}^p} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdepsp`	Derivative of yield function with respect to the plastic strain TYPE: `ndarray`

dfdepsp0

dfdepsp0(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the plastic strain:

\[ \frac{\partial F}{\partial \boldsymbol{\varepsilon}^p} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdepsp`	Derivative of yield function with respect to the plastic strain TYPE: `ndarray`

dfdepsp1

dfdepsp1(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the plastic strain:

\[ \frac{\partial F}{\partial \boldsymbol{\varepsilon}^p} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdepsp`	Derivative of yield function with respect to the plastic strain TYPE: `ndarray`

dfdsig

dfdsig(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the stress:

\[ \frac{\partial F}{\partial \boldsymbol{\sigma}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdsig`	Derivative of yield function with respect to the stress TYPE: `ndarray`

dfdsig0

dfdsig0(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the stress:

\[ \frac{\partial F}{\partial \boldsymbol{\sigma}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdsig`	Derivative of yield function with respect to the stress TYPE: `ndarray`

dfdsig1

dfdsig1(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of yield function with respect to the stress:

\[ \frac{\partial F}{\partial \boldsymbol{\sigma}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dfdsig`	Derivative of yield function with respect to the stress TYPE: `ndarray`

dgdsig

dgdsig(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of potential function with respect to the stress:

\[ \frac{\partial G}{\partial \boldsymbol{\sigma}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dgdsig`	Derivative of potential function with respect to the stress TYPE: `ndarray`

dgdsig0

dgdsig0(sv0: StateVariable, sv: StateVariable) -> ndarray

Calculate derivative of potential function with respect to the stress:

\[ \frac{\partial G}{\partial \boldsymbol{\sigma}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`dgdsig`	Derivative of potential function with respect to the stress TYPE: `ndarray`

dgdsig1

dgdsig1(
    dfdsig: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
    associated: bool = None,
) -> ndarray

Calculate derivative of potential function with respect to the stress:

\[ \frac{\partial G}{\partial \boldsymbol{\sigma}} \]

PARAMETER	DESCRIPTION
`dfdsig`	Derivative of yield function with respect to the stress TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`
`associated`	Whether to override the option to use associated flow rule TYPE: `bool` DEFAULT: `None`

RETURNS	DESCRIPTION
`dgdsig`	Derivative of potential function with respect to the stress TYPE: `ndarray`

dsigadsig

dsigadsig(sv0: StateVariable, sv: StateVariable) -> ndarray

Derivative of the modified stress with respect to the stress:

\[ \frac{\partial \boldsymbol{\sigma}^\prime}{\partial \boldsymbol{\sigma}} \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

elasticTrial

elasticTrial(
    K: ndarray, deps: ndarray, sv0: StateVariable, sv: StateVariable
) -> float

Perform the elastic trial.

PARAMETER	DESCRIPTION
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`float`	Yield function value after the elastic trial

endConstantStressStateScope

endConstantStressStateScope(sv0: StateVariable, sv: StateVariable)

End the scope of a constant stress state.

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

endConstantStressStateScope1

endConstantStressStateScope1(sv0: StateVariable, sv: StateVariable)

End the scope of a constant stress state.

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

integrateToYieldSurface

integrateToYieldSurface(
    K: ndarray, deps: ndarray, sv0: StateVariable, sv: StateVariable
) -> tuple[bool, float]

Integrate stress state to the yield surface elastically, solve f(sig0 + alpha * K * deps) = 0

PARAMETER	DESCRIPTION
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`Whether the stress state is plastic, and the alpha parameter`

interpolateFabricTensor

interpolateFabricTensor(
    sv0: StateVariable,
    sv: StateVariable,
    combined_stress_tensor_type: str | None = None,
)

Interpolate fabric tensor towards critical state

isCPA

isCPA() -> bool

Whether tha plastic method is CPA.

isCPPM

isCPPM() -> bool

Whether tha plastic method is CPPM.

isExplicit

isExplicit() -> bool

Whether tha plastic method is explicit.

isRungeKutta

isRungeKutta() -> bool

Whether tha plastic method is RungeKutta.

nonCoaxialDirection

nonCoaxialDirection(F: ndarray, n: ndarray, m: ndarray) -> ndarray

Non-coaxial direction

orthogonalDirection

orthogonalDirection(F: ndarray, n: ndarray) -> ndarray

Orthogonal direction

plasticModulus

plasticModulus(dgdsig: ndarray, sv0: StateVariable, sv: StateVariable) -> float

Plastic modulus in the equation of plastic multiplier including dgdsig, coefficient of plastic multiplier:

\[ \sum\frac{\partial F}{\partial\kappa}\frac{\partial\kappa}{\partial\boldsymbol{\varepsilon}^p} \frac{\partial G}{\partial\sigma} \]

PARAMETER	DESCRIPTION
`dgdsig`	Derivative of potential function with respect to the stress TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`Kp`	Plastic modulus in the equation of plastic multiplier TYPE: `float`

plasticModulus1

plasticModulus1(
    dgdsig: ndarray, sv0: StateVariable, sv: StateVariable
) -> float

Plastic modulus in the equation of plastic multiplier including dgdsig, coefficient of plastic multiplier:

\[ \sum\frac{\partial F}{\partial\kappa}\frac{\partial\kappa}{\partial\boldsymbol{\varepsilon}^p} \frac{\partial G}{\partial\sigma} \]

PARAMETER	DESCRIPTION
`dgdsig`	Derivative of potential function with respect to the stress TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`Kp`	Plastic modulus in the equation of plastic multiplier TYPE: `float`

potentialFunction

potentialFunction(sv0: StateVariable, sv: StateVariable) -> float

Potential function g. By default, the potential function is the same as the yield function.

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`float`	Value of the potential function

returnMapping

returnMapping(
    K: ndarray,
    dsig: ndarray,
    deps: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
) -> None

Return mapping algorithm to solve plastic problem (plastic strain and stress).

revertElasticTrial

revertElasticTrial(
    K: ndarray, deps: ndarray, sv0: StateVariable, sv: StateVariable
) -> None

Revert the elastic trial.

PARAMETER	DESCRIPTION
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	Current state variables TYPE: `StateVariable`

setPlasticMethod

setPlasticMethod(method: str)

Set the plastic method

PARAMETER	DESCRIPTION
`method`	The plastic method TYPE: `str`

stressStrainIntegrate

stressStrainIntegrate(
    K: ndarray,
    dsig: ndarray,
    deps: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
) -> None

symmetricContinuumElastoplasticStiffness

symmetricContinuumElastoplasticStiffness(
    K: ndarray,
    deps: ndarray,
    sv0: StateVariable,
    sv: StateVariable,
    check_elastic_state: bool = True,
    check_positive_definite: bool = None,
    check_non_singularity: bool = None,
) -> ndarray

updateHardeningVariables

updateHardeningVariables(
    ddlambda: float, ddepsp: ndarray, sv0: StateVariable, sv: StateVariable
)

Update hardening variables:

\[ \mathrm{d}\kappa = \frac{\partial\kappa}{\partial\boldsymbol{\varepsilon}^p}\mathrm{d}\boldsymbol{\varepsilon}^p \]

PARAMETER	DESCRIPTION
`ddlambda`	Plastic multiplier TYPE: `float`
`ddepsp`	Plastic strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

yieldFunction

yieldFunction(sv0: StateVariable, sv: StateVariable) -> float

Calculate yield function. You are required to define the yield function of the contact force and the state variables:

\[ F = F(\boldsymbol{\sigma}, \kappa) \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`float`	Value of the yield function

yieldFunction1

yieldFunction1(sv0: StateVariable, sv: StateVariable) -> float

Calculate yield function. You are required to define the yield function of the contact force and the state variables:

\[ F = F(\boldsymbol{\sigma}, \kappa) \]

PARAMETER	DESCRIPTION
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

RETURNS	DESCRIPTION
`float`	Value of the yield function

yieldFunctionCorrection

yieldFunctionCorrection(
    K: ndarray, deps: ndarray, sv0: StateVariable, sv: StateVariable
)

Update state variables after every plastic increment to keep the stress state on the yield surface

PARAMETER	DESCRIPTION
`K`	The stiffness matrix. TYPE: `ndarray`
`deps`	Strain increment TYPE: `ndarray`
`sv0`	Initial state variables TYPE: `StateVariable`
`sv`	State variables TYPE: `StateVariable`

butcherTableau

butcherTableau(
    n: int, alpha: float, beta: float, fourth_order_type: str = "classic"
) -> tuple[ndarray, ndarray, ndarray]

Construct Butcher tableau for the Runge-Kutta method, a[i, j] = 0 for j >= i, c[0] = 0

PARAMETER	DESCRIPTION
`n`	Order of the Runge-Kutta method, only 1, 2, 3, and 4 are supported. TYPE: `int`
`alpha`	Parameter for 2nd- and 3rd-order Runge-Kutta methods TYPE: `float`
`beta`	Parameter for 2nd- and 3rd-order Runge-Kutta methods TYPE: `float`
`fourth_order_type`	Type of fourth-order Runge-Kutta method, only 'classic', '3/8-rule', and 'ralston' are supported. TYPE: `str` DEFAULT: `'classic'`

RETURNS	DESCRIPTION
`Butcher tableau (a, b, c)`

elastoplasticmodelbase