ComputeSimoHughesJ2PlasticityStress

The Simo-Hughes style J2 plasticity.

Overview

This class provides a hyperelastic Neo-Hookean stress update with J2 plasticity Simo and Hughes (2006), Borden et al. (2016).

Following a multiplicative decomposition of the deformation gradient $var element = document.getElementById("moose-equation-1c5ff3d2-13cf-4c53-afc9-44b88baed18e");katex.render(" \\begin{aligned} F_{iJ} &= F^e_{iK} F^p_{KJ}, \\\\ b^e_{ij} &= F^e_{iK} F^e_{Kj}, \\end{aligned}", element, {displayMode:true,throwOnError:false});$ and a yield surface with associated flow rule $var element = document.getElementById("moose-equation-ba37602c-77b1-4703-b24d-25bc7a3c79a2");katex.render(" \\begin{aligned} \\phi:=\\tau_{ij}n_{ij} - H(\\varepsilon^p) \\leq 0, \\quad \\dot{\\varepsilon}^p \\geq 0, \\quad \\phi \\dot{\\varepsilon} = 0, \\\\ L_v(b^e_{ij}) = -\\frac{2}{3} \\dot{\\varepsilon}^p b^e_{kk} n_{ij}, \\quad n_{ij} = \\sqrt{\\frac{3}{2}} s_{ij} (s_{kl}s_{kl})^{-1/2}, \\end{aligned}", element, {displayMode:true,throwOnError:false});$ the stress-strain relation is defined as $var element = document.getElementById("moose-equation-0d99337c-22d6-447e-b25d-8fc2f460cd3e");katex.render(" \\begin{aligned} P_{iJ} &= \\tau_{ij}F^{-1}_{Jj}, \\\\ \\tau_{ij} &= \\frac{1}{2}K\\left( J^2-1 \\right) \\delta_{ij} + G \\left( b^e_{ij} - \\frac{1}{3}b^e_{kk}\\delta_{ij} \\right). \\end{aligned}", element, {displayMode:true,throwOnError:false});$

In the above definitions, Einstein notation is used. $var element = document.getElementById("moose-equation-f5aa19a6-235e-440a-a90a-d1084f9c10e0");katex.render("F", element, {displayMode:false,throwOnError:false});$ is the deformation gradient, $var element = document.getElementById("moose-equation-36991645-9fab-4286-8a3b-9c8cac94afa6");katex.render("F^e", element, {displayMode:false,throwOnError:false});$ is the elastic deformation gradient, $var element = document.getElementById("moose-equation-1b0e73dd-02d2-4f30-97c7-96ebe16506de");katex.render("F^p", element, {displayMode:false,throwOnError:false});$ is the plastic deformation gradient, $var element = document.getElementById("moose-equation-b25401d5-94d3-4ce7-add9-ac0f142c6df7");katex.render("b^e", element, {displayMode:false,throwOnError:false});$ is the elastic counterpart of the left Cauchy-Green strain, $var element = document.getElementById("moose-equation-bb289b46-4450-4dd8-8ef1-b746038c7fd0");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ is the yield surface (yield function), $var element = document.getElementById("moose-equation-451bb464-f479-46f1-983c-ee400b984862");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ is the Kirchhoff stress, $var element = document.getElementById("moose-equation-d73955d9-f10d-490f-86db-6516e3c35fc0");katex.render("n", element, {displayMode:false,throwOnError:false});$ is the direction of plastic flow, $var element = document.getElementById("moose-equation-f511630e-bdee-4454-8ae6-50b336f7ba1c");katex.render("H", element, {displayMode:false,throwOnError:false});$ is the hardening function, $var element = document.getElementById("moose-equation-1619b9f1-330f-4960-a2ad-7c04e6cc23da");katex.render("\\varepsilon^p", element, {displayMode:false,throwOnError:false});$ is the effective (equivalent) plastic strain, $var element = document.getElementById("moose-equation-3e2f266e-1f01-4cd1-9e88-04cc061445ee");katex.render("s", element, {displayMode:false,throwOnError:false});$ is the deviatoric part of the Kirchhoff stress, $var element = document.getElementById("moose-equation-2e0c5963-76de-408c-951d-c3c3efbfe1ce");katex.render("P", element, {displayMode:false,throwOnError:false});$ is the first Piola-Kirchhoff stress, $var element = document.getElementById("moose-equation-43aec12a-41fe-4305-8939-09cb3268a5ff");katex.render("K", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-9dc78a31-7247-49d0-bed7-338c47ee663f");katex.render("G", element, {displayMode:false,throwOnError:false});$ and bulk and shear moduli, and $var element = document.getElementById("moose-equation-cdc7e664-f989-415a-b43a-eecfbee84b4a");katex.render("J", element, {displayMode:false,throwOnError:false});$ is the determinant of the deformation gradient.

Several remarks:

The flow rule is consistent with the principal of maximum dissipation.
The flow rule determines the strain up to a spin. It is normalized by the constraint $var element = document.getElementById("moose-equation-43689e1c-2f03-4419-b414-dfb479febf48");katex.render("n_{ij}n_{ij} = 3/2", element, {displayMode:false,throwOnError:false});$ under uniaxial assumptions.
The plastic flow is purely isochoric, i.e. $var element = document.getElementById("moose-equation-24dfe08b-c867-4b4d-abba-d7d8273c6536");katex.render("\\det(F^p) = 1", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-98b79dbf-a05a-4c2e-ac32-db68e4b624b4");katex.render("J = \\det(F^e)", element, {displayMode:false,throwOnError:false});$ .

Algorithm

First, the incremental deformation gradient and its volume-preserving counterpart are computed as $var element = document.getElementById("moose-equation-bb728b5f-dab8-481f-8494-f45f7a2d66ae");katex.render(" \\begin{aligned} f_{ij} &= F_{iK}\\hat{F}^{-1}_{Kj}, \\\\ \\bar{f}_{ij} &= \\det(f)^{-1/3} f_{ij}. \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Assuming an elastic step, i.e. $var element = document.getElementById("moose-equation-1df865d2-cac9-4d41-9519-e9028b0926e2");katex.render("\\Delta \\varepsilon^p = 0", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-42ad9be4-1258-43b0-862d-3060d31f8bfb");katex.render(" \\begin{aligned} \\bar{b}^{e,tr}_{ij} &= \\bar{f}_{ik} \\hat{\\bar{b}}^e_{kl} \\bar{f}_{jl}, \\\\ \\bar{I}^e &= \\bar{b}^{e,tr}_{ii}/3, \\\\ s^{tr}_{ij} &= G \\left( \\bar{b}^{e,tr}_{ij} - \\dfrac{1}{3}\\bar{b}^{e,tr}_{kk}\\delta_{ij} \\right), \\\\ n_{ij} &= \\sqrt{\\dfrac{3}{2}} s^{tr}_{ij}(s^{tr}_{kl}s^{tr}_{kl})^{-1/2}. \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Then, for each iteration in the return mapping, we do $var element = document.getElementById("moose-equation-c40ee4f7-ab6d-406d-8c78-0e0f21dfca95");katex.render(" \\begin{aligned} R^{rm} &= s^{tr}_{ij}n_{ij} - 3G \\Delta\\varepsilon^p \\bar{I}^e - H(\\hat{\\varepsilon}^p + \\Delta\\varepsilon^p), \\\\ J^{rm} &= - 3G \\bar{I}^e - H'(\\hat{\\varepsilon}^p + \\Delta\\varepsilon^p), \\\\ \\Delta\\varepsilon^p &\\leftarrow \\Delta\\varepsilon^p - R^{rm}/J^{rm}. \\end{aligned}", element, {displayMode:true,throwOnError:false});$ Then we update the current configuration $var element = document.getElementById("moose-equation-7305e28f-44c3-455c-90de-00e94c684820");katex.render(" \\begin{aligned} \\varepsilon^p &= \\hat{\\varepsilon}^p + \\Delta \\varepsilon^p, \\\\ \\bar{b}^e_{ij} &= \\bar{b}^{e,tr}_{ij} - 2\\Delta\\varepsilon^p \\bar{I}^e n_{ij}, \\\\ s_{ij} &= G\\left( \\bar{b}^{e}_{ij} - \\dfrac{1}{3}\\bar{b}^{e}_{kk}\\delta_{ij} \\right), \\\\ \\tau_{ij} &= \\dfrac{1}{2}K(J^2-1)\\delta_{ij} + s_{ij}, \\\\ P_{iJ} &= \\tau_{ij} F^{-1}_{Jj}. \\end{aligned}", element, {displayMode:true,throwOnError:false});$

Example Input File Syntax

The follow example configures a large deformation Neo-Hookean model with J2 plasticity and linear hardening.

[Materials]
  [elastic_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = ${E}
    poissons_ratio = ${nu}
  []
  [compute_strain]
    type = ComputeLagrangianStrain
    displacements = 'disp_x disp_y disp_z'
  []
  [flow_stress]
    type = DerivativeParsedMaterial
    property_name = flow_stress
    expression = '320+688*effective_plastic_strain'
    material_property_names = 'effective_plastic_strain'
    additional_derivative_symbols = 'effective_plastic_strain'
    derivative_order = 2
    compute = false
  []
  [compute_stress]
    type = ComputeSimoHughesJ2PlasticityStress
    flow_stress_material = flow_stress
  []
[]

Input Parameters

flow_stress_materialThe material defining the flow stress
C++ Type:MaterialName
Controllable:No
Description:The material defining the flow stress

Required Parameters

absolute_tolerance1e-11Absolute convergence tolerance for Newton iteration
Default:1e-11
C++ Type:double
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
Controllable:No
Description:Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress
automatic_differentiation_return_mappingFalseWhether to use automatic differentiation to compute the derivative.
Default:False
C++ Type:bool
Controllable:No
Description:Whether to use automatic differentiation to compute the derivative.
base_nameMaterial property base name
C++ Type:std::string
Controllable:No
Description:Material property base name
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
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>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Controllable:No
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
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
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
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
elasticity_tensorelasticity_tensorThe name of the elasticity tensor.
Default:elasticity_tensor
C++ Type:MaterialPropertyName
Controllable:No
Description:The name of the elasticity tensor.
large_kinematicsFalseUse a large displacement stress update.
Default:False
C++ Type:bool
Controllable:No
Description:Use a large displacement stress update.
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
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
Controllable:No
Description:Relative convergence tolerance for Newton iteration

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
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
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
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
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
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
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
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>
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>
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

Michael J Borden, Thomas JR Hughes, Chad M Landis, Amin Anvari, and Isaac J Lee. A phase-field formulation for fracture in ductile materials: finite deformation balance law derivation, plastic degradation, and stress triaxiality effects. Computer Methods in Applied Mechanics and Engineering, 312:130–166, 2016.

@article{borden2016phase,
    author = "Borden, Michael J and Hughes, Thomas JR and Landis, Chad M and Anvari, Amin and Lee, Isaac J",
    title = "A phase-field formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects",
    journal = "Computer Methods in Applied Mechanics and Engineering",
    volume = "312",
    pages = "130--166",
    year = "2016",
    publisher = "Elsevier"
}

Juan C Simo and Thomas JR Hughes. Computational inelasticity. Volume 7. Springer Science & Business Media, 2006.

@book{simo2006computational,
    author = "Simo, Juan C and Hughes, Thomas JR",
    title = "Computational inelasticity",
    volume = "7",
    year = "2006",
    publisher = "Springer Science \\& Business Media"
}

(moose/modules/tensor_mechanics/test/tests/lagrangian/materials/correctness/hyperelastic_J2_plastic.i)

E = 6.88e4
nu = 0.25

[GlobalParams]
  large_kinematics = true
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [disp_z]
  []
[]

[Mesh]
  [msh]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 1
    ny = 1
    nz = 1
  []
[]

[Kernels]
  [sdx]
    type = TotalLagrangianStressDivergence
    variable = disp_x
    component = 0
    displacements = 'disp_x disp_y disp_z'
  []
  [sdy]
    type = TotalLagrangianStressDivergence
    variable = disp_y
    component = 1
    displacements = 'disp_x disp_y disp_z'
  []
  [sdz]
    type = TotalLagrangianStressDivergence
    variable = disp_z
    component = 2
    displacements = 'disp_x disp_y disp_z'
  []
[]

[BCs]
  [fix_x]
    type = DirichletBC
    variable = disp_x
    boundary = 'left'
    value = 0.0
  []
  [fix_y]
    type = DirichletBC
    variable = disp_y
    boundary = 'bottom'
    value = 0.0
  []
  [fix_z]
    type = DirichletBC
    variable = disp_z
    boundary = 'back'
    value = 0.0
  []
  [pull_x]
    type = FunctionDirichletBC
    variable = disp_x
    boundary = 'right'
    function = 't'
    preset = false
  []
[]

[Materials]
  [elastic_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = ${E}
    poissons_ratio = ${nu}
  []
  [compute_strain]
    type = ComputeLagrangianStrain
    displacements = 'disp_x disp_y disp_z'
  []
  [flow_stress]
    type = DerivativeParsedMaterial
    property_name = flow_stress
    expression = '320+688*effective_plastic_strain'
    material_property_names = 'effective_plastic_strain'
    additional_derivative_symbols = 'effective_plastic_strain'
    derivative_order = 2
    compute = false
  []
  [compute_stress]
    type = ComputeSimoHughesJ2PlasticityStress
    flow_stress_material = flow_stress
  []
[]

[Postprocessors]
  [sxx]
    type = ElementAverageValue
    variable = sxx
    execute_on = 'INITIAL TIMESTEP_END'
  []
  [exx]
    type = ElementAverageValue
    variable = exx
    execute_on = 'INITIAL TIMESTEP_END'
  []
[]

[AuxVariables]
  [sxx]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = RankTwoAux
      rank_two_tensor = cauchy_stress
      index_i = 0
      index_j = 0
    []
  []
  [exx]
    family = MONOMIAL
    order = CONSTANT
    [AuxKernel]
      type = RankTwoAux
      rank_two_tensor = total_strain
      index_i = 0
      index_j = 0
    []
  []
[]

[Executioner]
  type = Transient

  solve_type = NEWTON
  line_search = none

  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'

  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-10

  start_time = 0.0
  dt = 5e-4
  end_time = 1e-1
[]

[Outputs]
  csv = true
[]