Hill Plasticity Stress Update

This class uses the generalized radial return for anisotropic plasticity model.This class can be used in conjunction with other creep and plasticity materials for more complex simulations.

Description

This class computes Hill plasticity via a generalized radial return mapping algorithm Versino and Bennett (2018). It requires that the elastic behavior of the material is isotropic, whereas any departure from the yield function is anisotropic. The Hill yield function can be defined as: (1) where is the deviatoric stress tensor in Voigt form, is the anisotropy (Hill) tensor, is the temperature and is an internal parameter that can be used, for example, to prescribe strain hardening through a plasticity modulus. Hill's tensor is defined as a six by six matrix using the unitless constants , , , , and as following:

(2)

The model currently uses power law hardening:

(3)

where is the hardening constant and is the hardening exponent. The hardening exponent has a default value of 1.0 and this value can be modified by the user by setting the parameter "hardening_exponent". The "yield_stress" () and "hardening_constant" () are the required parameters to be supplied by the user.

Verification

With being the yield condition, the deviatoric stress and the plastic multiplier, the plastic strain rate is given as:

(4)

Eq. (4) is a statement of normality condition, i.e., the associative flow rule. Using Eq. (1), the associative flow rule in Eq. (4) can be written as:

(5)

which can be written in incremental form as:

(6)

If is the internal state variable associated with isotropic hardening, and is the energy conjugate to the , the is written as:

(7)

The thermodynamic variable is here (see Versino and Bennett (2018) for details). Equation Eq. (7) can be written in integrated incremental form as:

(8)

Using Eq. (8) in Eq. (6), taking the partial derivative of with respect to , and applying chain rule, we obtain:

(9)

(10)

For the case where flow rule involves linear hardening with hardening constant :

(11)

Substituting Eq. (11) in Eq. (10) we obtain:

(12)

The left hand side of Eq. (12) gives the slope of the stress vs plastic strain curve. This equation can be written for the direct components of stress () as:

(13)

For a uniaxial tensile test with monotonically increasing load, the stress always lies on the yield surface once the plastic deformation starts (for loading in x-direction = ) which simplifies Eq. (13) to:

(14)

Similarly, for uniaxial loading in y-direction we obtain:

(15)

These simplified equations (Eq. (14) & Eq. (15)) can be used in the verification tests for this material model.

warningwarning

The combination of elastic isotropy and plastic anisotropy should be solved by the more efficient HillPlasticityStressUpdate class.

The effective plastic strain increment is obtained within the framework of a generalized (Hill plasticity) radial return mapping, see GeneralizedRadialReturnStressUpdate.

Example Input File Syntax

[Materials]
  [trial_plasticity]
    type = ADHillPlasticityStressUpdate
    hardening_constant = 2000.0
    yield_stress = 0.001
    absolute_tolerance = 1e-14
    relative_tolerance = 1e-12
    base_name = trial_plasticity
    internal_solve_full_iteration_history = true
    max_inelastic_increment = 2.0e-6
    internal_solve_output_on = on_error
  []
[]
(moose/modules/solid_mechanics/test/tests/anisotropic_plasticity/ad_aniso_plasticity_y.i)

Input Parameters

  • effective_inelastic_strain_nameeffective_plastic_strainName of the material property that stores the effective inelastic strain

    Default:effective_plastic_strain

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:Name of the material property that stores the effective inelastic strain

  • hardening_constantHardening constant (H) for anisotropic plasticity

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Hardening constant (H) for anisotropic plasticity

  • inelastic_strain_rate_nameplastic_strain_rateName of the material property that stores the inelastic strain rate

    Default:plastic_strain_rate

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:Name of the material property that stores the inelastic strain rate

  • yield_stressYield stress (constant value) for anisotropic plasticity

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Yield stress (constant value) for anisotropic plasticity

Required Parameters

  • absolute_tolerance1e-11Absolute convergence tolerance for Newton iteration

    Default:1e-11

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Absolute convergence tolerance for Newton iteration

  • acceptable_multiplier10Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress

    Default:10

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress

  • base_nameOptional parameter that defines a prefix for all material properties related to this stress update model. This allows for multiple models of the same type to be used without naming conflicts.

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:Optional parameter that defines a prefix for all material properties related to this stress update model. This allows for multiple models of the same type to be used without naming conflicts.

  • blockThe list of blocks (ids or names) that this object will be applied

    C++ Type:std::vector<SubdomainName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The list of blocks (ids or names) that this object will be applied

  • boundaryThe list of boundaries (ids or names) from the mesh where this object applies

    C++ Type:std::vector<BoundaryName>

    Unit:(no unit assumed)

    Controllable:No

    Description:The list of boundaries (ids or names) from the mesh where this object applies

  • constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

    Default:NONE

    C++ Type:MooseEnum

    Unit:(no unit assumed)

    Options:NONE, ELEMENT, SUBDOMAIN

    Controllable:No

    Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

  • declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.

  • hardening_exponent1Hardening exponent (n) for anisotropic plasticity

    Default:1

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Hardening exponent (n) for anisotropic plasticity

  • max_inelastic_increment0.0001The maximum inelastic strain increment allowed in a time step

    Default:0.0001

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:The maximum inelastic strain increment allowed in a time step

  • max_integration_error0.0005The maximum inelastic strain increment integration error allowed

    Default:0.0005

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:The maximum inelastic strain increment integration error allowed

  • prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

    C++ Type:MaterialPropertyName

    Unit:(no unit assumed)

    Controllable:No

    Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.

  • relative_tolerance1e-08Relative convergence tolerance for Newton iteration

    Default:1e-08

    C++ Type:double

    Unit:(no unit assumed)

    Controllable:No

    Description:Relative convergence tolerance for Newton iteration

  • use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

  • use_transformationTrueWhether to employ updated Hill's tensor due to rigid body or large deformation kinematic rotations. If an initial rigid body rotation is provided by the user in increments of 90 degrees (e.g. 90, 180, 270), this option can be set to false, in which case the Hill's coefficients are extracted from the transformed Hill's tensor.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Whether to employ updated Hill's tensor due to rigid body or large deformation kinematic rotations. If an initial rigid body rotation is provided by the user in increments of 90 degrees (e.g. 90, 180, 270), this option can be set to false, in which case the Hill's coefficients are extracted from the transformed Hill's tensor.

Optional Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

    Description:Adds user-defined labels for accessing object parameters via control logic.

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:Yes

    Description:Set the enabled status of the MooseObject.

  • implicitTrueDetermines whether this object is calculated using an implicit or explicit form

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Determines whether this object is calculated using an implicit or explicit form

  • seed0The seed for the master random number generator

    Default:0

    C++ Type:unsigned int

    Unit:(no unit assumed)

    Controllable:No

    Description:The seed for the master random number generator

  • use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

  • internal_solve_full_iteration_historyFalseSet true to output full internal Newton iteration history at times determined by `internal_solve_output_on`. If false, only a summary is output.

    Default:False

    C++ Type:bool

    Unit:(no unit assumed)

    Controllable:No

    Description:Set true to output full internal Newton iteration history at times determined by `internal_solve_output_on`. If false, only a summary is output.

  • internal_solve_output_onon_errorWhen to output internal Newton solve information

    Default:on_error

    C++ Type:MooseEnum

    Unit:(no unit assumed)

    Options:never, on_error, always

    Controllable:No

    Description:When to output internal Newton solve information

Debug Parameters

  • output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

    Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

  • outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object

    Default:none

    C++ Type:std::vector<OutputName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

References

  1. Daniele Versino and Kane C Bennett. Generalized radial-return mapping algorithm for anisotropic von mises plasticity framed in material eigenspace. International Journal for Numerical Methods in Engineering, 116(3):202–222, 2018.[BibTeX]