UPuZrPorosityEigenstrain

Computes the swelling, porosity and eigenstrain from gasous and low-temperature mechanisms in UPuZr.

Description

UPuZrPorosityEigenstrain computes the swelling contributions, total swelling, porosity, and voluetric eigenstrain from swelling arising from gaseous and low-temperature swelling U-Pu-Zr systems. It is designed to be used in conjunction with a gaseous swelling model such as in UPuZrGaseousSwelling and a low-temperature swelling model such as in UPuZrLowTemperatureSwelling, which compute swelling increments from the aforementioned mechanisms. UPuZrPorosityEigenstrain is a joiner class needed to set an overall porosity threshold incorporating multiple swelling mechanisms, above which fission gas venting occurs.

The porosity interconnection threshold is a value above which no further porosity-driven swelling can occur in the U-Pu-Zr fuel. Initial or fabrication porosity is included in this porosity interconnection threshold.

The maximum generated porosity, $var element = document.getElementById("moose-equation-37cc1016-9d8f-4bb1-966f-49def3b128e9");katex.render("P_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ is found as $var element = document.getElementById("moose-equation-0497ab62-3a2e-44e4-a03e-b1687d0f4869");katex.render(" P_{max}^{generated} = P_{terminating} - P^{initial},", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-3cdc95e1-2d96-4c20-9da8-16f186d0516f");katex.render("P_{terminating}", element, {displayMode:false,throwOnError:false});$ is the user-specified maximum porosity for porosity interconnection and $var element = document.getElementById("moose-equation-cd5f5c5a-173a-4288-989e-c03675caebda");katex.render("P^{initial}", element, {displayMode:false,throwOnError:false});$ is the initial or fabrication porosity.

The maximum generated swelling, $var element = document.getElementById("moose-equation-f30ed75e-9cf3-4ba0-92ec-5bdcc87f486f");katex.render("\\theta_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ ,is calculated from $var element = document.getElementById("moose-equation-11a15dcf-18cd-463d-a683-2e6792980325");katex.render("P_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ as $var element = document.getElementById("moose-equation-b99b533b-b72e-48a3-8e17-bc03f4a21b6e");katex.render(" \\theta_{max}^{generated} = \\frac{P_{max}^{generated}}{1 - P_{max}^{generated}}.", element, {displayMode:true,throwOnError:false});$

The swelling increments computed by UPuZrLowTemperatureSwelling and UPuZrGaseousSwelling may cause the calculated swelling to increase over $var element = document.getElementById("moose-equation-2df821f1-0146-4ba8-8824-3a5451e4d972");katex.render("\\theta_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ . This is handled by reducing the swelling increments until $var element = document.getElementById("moose-equation-e0666a3a-264a-43bb-ac5a-d20a7239b4ff");katex.render("\\theta_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ is reached, and preference is given to the gaseous swelling mechanism. First, it is determined if the gaseous swelling increment alone is sufficient to cause the total swelling to exceed $var element = document.getElementById("moose-equation-1364e03f-7ffc-4122-9372-3a8fa4a72868");katex.render("\\theta_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ . If so, the low-temperature swelling increment is reduced to zero and the gaseous swelling increment is reduced to a value just sufficient to achieve $var element = document.getElementById("moose-equation-03869c43-1b2d-4b8f-afb3-d977447320c6");katex.render("\\theta_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ . If the gaseous swelling increment alone is not sufficient to exceed $var element = document.getElementById("moose-equation-c9b6054b-7fc3-4eac-bddb-e619e56bbb0b");katex.render("\\theta_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ , then its value is retained and the low-temperature swelling increment is reduced to a value just sufficient to achieve $var element = document.getElementById("moose-equation-017caed9-0fdf-4f95-a505-29177e8e2267");katex.render("\\theta_{max}^{generated}", element, {displayMode:false,throwOnError:false});$ . From these adjusted swelling increments, the swelling from each contribution is calculated, and the total swelling, $var element = document.getElementById("moose-equation-1a84a109-2836-45ec-a4ea-b98ac023c0cd");katex.render("\\theta", element, {displayMode:false,throwOnError:false});$ , is calculated by summing the swelling from each contribution.

Finally, the porosity, $var element = document.getElementById("moose-equation-372190c8-017a-42b7-a921-7f1ad61fb08b");katex.render("P", element, {displayMode:false,throwOnError:false});$ , is computed as $var element = document.getElementById("moose-equation-16c4b2f4-3b07-4ee2-be2b-b1e39e129b40");katex.render(" P = P^{initial} + \\frac{\\theta}{1 + \\theta}", element, {displayMode:true,throwOnError:false});$ and the volumetric swelling eigenstrain is computed from $var element = document.getElementById("moose-equation-980a36d5-97b4-4eea-903a-8ad70d3ed30b");katex.render("\\theta", element, {displayMode:false,throwOnError:false});$ .

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [total_porosity]
    type = UPuZrPorosityEigenstrain<<<{"description": "Computes the swelling, porosity and eigenstrain from gasous and low-temperature mechanisms in UPuZr.", "href": "UPuZrPorosityEigenstrain.html"}>>>
    initial_porosity<<<{"description": "Initial or fabrication porosity"}>>> = 0.1
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
    output_properties<<<{"description": "List of material properties, from this material, to output (outputs must also be defined to an output type)"}>>> = 'total_swelling porosity gas_swelling lowT_swelling'
    eigenstrain_name<<<{"description": "Material property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator."}>>> = porosity_eigenstrain
  []
[]

UPuZrPorosityEigenstrain must be used in conjunction with the low-temperature swelling model:

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [lowT_swelling]
    type = UPuZrLowTemperatureSwelling<<<{"description": "Computes swelling increment due to low-temperature swelling in UPuZr.", "href": "../UPuZrLowTemperatureSwelling.html"}>>>
    temperature<<<{"description": "Coupled temperature variable"}>>> = temperature
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

as well as the gaseous swelling model:

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [gas_swelling]
    type = UPuZrGaseousSwelling<<<{"description": "Computes a swelling increment due to gas swelling in UPuZr.", "href": "../UPuZrGaseousSwelling.html"}>>>
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
    temperature<<<{"description": "Coupled temperature variable"}>>> = temperature
    bubble_number_density<<<{"description": "Material property name for the number density of intragranular bubbles, [bubbles/m^3]"}>>> = 1e20
  []
[]

and with the solid mechanics quasi-static action to apply the calculated eigenstrain:

[Physics<<<{"href": "../../../syntax/Physics/index.html"}>>>]
  [SolidMechanics<<<{"href": "../../../syntax/Physics/SolidMechanics/index.html"}>>>]
    [QuasiStatic<<<{"href": "../../../syntax/Physics/SolidMechanics/QuasiStatic/index.html"}>>>]
      [all]
        strain<<<{"description": "Strain formulation"}>>> = FINITE
        add_variables<<<{"description": "Add the displacement variables"}>>> = true
        eigenstrain_names<<<{"description": "List of eigenstrains to be applied in this strain calculation"}>>> = 'porosity_eigenstrain'
        generate_output<<<{"description": "Add scalar quantity output for stress and/or strain"}>>> = 'strain_xx strain_yy'
      []
    []
  []
[]

Input Parameters

eigenstrain_nameMaterial property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.
C++ Type:std::string
Controllable:No
Description:Material property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.

Required Parameters

base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
C++ Type:std::string
Controllable:No
Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
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
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.
initial_porosity0Initial or fabrication porosity
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Initial or fabrication porosity
interconnection_terminating_porosity0.25Porosity at which fission gas release finishes
Default:0.25
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Porosity at which fission gas release finishes

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

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

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.
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
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.

Material Property Retrieval Parameters

Input Files

(test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_porosity_eigenstrain/test.i)

References

No citations exist within this document.

(test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_porosity_eigenstrain/test.i)

# This tests UPuZrPorosityEigenstrain, a swelling model for UPuZr metal fuel
# that incorporates low temperature swelling.
#
# The low temperature swelling model is new. It assumes a temperature-dependent
# swelling rate that is linearly proportional to burnup. The total amount of
# porosity generated in the material is limited to a user-defined value.

# This test calculates the low temperature and gas swelling contributions as well as
# the total eigenstrain at 750 K for nine time steps (to t=9e6 s)  until the porosity
# threshold is reached.  To accelerate the test, initial porosity = 0.1. The computed
# values match the analytical values until t=8e6, at which point 1) the gaseous swelling
# smoothing activates and 2) the total swelling would exceed the swelling threshold
# and the logic to adjust the swelling increments is activated.

# Time | lowT_swelling | gas_swelling | porosity | eigenstrain_xx = eigenstrain_yy
# 1e6     2.3098e-2      5.5566e-4      1.2311e-1      7.7927e-3
# 2e6     4.6196e-2      1.5716e-3      1.4559e-1      1.5554e-2
# 3e6     6.9294e-2      2.8873e-3      1.6732e-1      2.3232e-2
# 4e6     9.2392e-2      4.4452e-3      1.8829e-1      3.0810e-2
# 5e6     1.1549e-1      6.2124e-3      2.0850e-1      3.8282e-2
# 6e6     1.3859e-1      8.1664e-3      2.2797e-1      4.5645e-2
# 7e6     1.6169e-1      1.0291e-2      2.4674e-1      5.2897e-2
# 8e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2
# 9e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[AuxVariables]
  [temperature]
    order = CONSTANT
    family = MONOMIAL
    [InitialCondition]
      type = ConstantIC
      value = 750
    []
  []
  [porosity_eigenstrain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [porosity_eigenstrain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[AuxKernels]
  [temp]
    type = ConstantAux
    variable = temperature
    value = 750
  []
  [eigenstrain_xx_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_xx
    index_i = 0
    index_j = 0
  []
  [eigenstrain_yy_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_yy
    index_i = 1
    index_j = 1
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = 'porosity_eigenstrain'
    generate_output = 'strain_xx strain_yy'
  []
[]

[Functions]
  [fission_rate_func]
    type = ParsedFunction
    expression = '2.59e19'
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
[]

[Materials]
  [fission_rate]
    type = GenericFunctionMaterial
    prop_names = 'fission_rate'
    prop_values = 'fission_rate_func'
    outputs = all
  []
  [burnup]
    type = UPuZrBurnup
    initial_X_Zr = 0.225
    initial_X_Pu = 0.163
    density = 15800
    outputs = all
  []

  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
  []
  [fuel_elasticity_tensor]
    type = UPuZrElasticityTensor
    X_Zr = 0.225
    X_Pu = 0.163
    temperature = temperature
    porosity = porosity
  []
  [lowT_swelling]
    type = UPuZrLowTemperatureSwelling
    temperature = temperature
    outputs = all
  []
  [gas_swelling]
    type = UPuZrGaseousSwelling
    outputs = all
    temperature = temperature
    bubble_number_density = 1e20
  []
  [total_porosity]
    type = UPuZrPorosityEigenstrain
    initial_porosity = 0.1
    outputs = all
    output_properties = 'total_swelling porosity gas_swelling lowT_swelling'
    eigenstrain_name = porosity_eigenstrain
  []
[]

[Executioner]
  type = Transient
  solve_type = Newton

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 60
  l_tol = 1e-5

  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-7
  num_steps = 9
  dt = 1e6
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_porosity_eigenstrain/test.i)

# This tests UPuZrPorosityEigenstrain, a swelling model for UPuZr metal fuel
# that incorporates low temperature swelling.
#
# The low temperature swelling model is new. It assumes a temperature-dependent
# swelling rate that is linearly proportional to burnup. The total amount of
# porosity generated in the material is limited to a user-defined value.

# This test calculates the low temperature and gas swelling contributions as well as
# the total eigenstrain at 750 K for nine time steps (to t=9e6 s)  until the porosity
# threshold is reached.  To accelerate the test, initial porosity = 0.1. The computed
# values match the analytical values until t=8e6, at which point 1) the gaseous swelling
# smoothing activates and 2) the total swelling would exceed the swelling threshold
# and the logic to adjust the swelling increments is activated.

# Time | lowT_swelling | gas_swelling | porosity | eigenstrain_xx = eigenstrain_yy
# 1e6     2.3098e-2      5.5566e-4      1.2311e-1      7.7927e-3
# 2e6     4.6196e-2      1.5716e-3      1.4559e-1      1.5554e-2
# 3e6     6.9294e-2      2.8873e-3      1.6732e-1      2.3232e-2
# 4e6     9.2392e-2      4.4452e-3      1.8829e-1      3.0810e-2
# 5e6     1.1549e-1      6.2124e-3      2.0850e-1      3.8282e-2
# 6e6     1.3859e-1      8.1664e-3      2.2797e-1      4.5645e-2
# 7e6     1.6169e-1      1.0291e-2      2.4674e-1      5.2897e-2
# 8e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2
# 9e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[AuxVariables]
  [temperature]
    order = CONSTANT
    family = MONOMIAL
    [InitialCondition]
      type = ConstantIC
      value = 750
    []
  []
  [porosity_eigenstrain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [porosity_eigenstrain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[AuxKernels]
  [temp]
    type = ConstantAux
    variable = temperature
    value = 750
  []
  [eigenstrain_xx_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_xx
    index_i = 0
    index_j = 0
  []
  [eigenstrain_yy_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_yy
    index_i = 1
    index_j = 1
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = 'porosity_eigenstrain'
    generate_output = 'strain_xx strain_yy'
  []
[]

[Functions]
  [fission_rate_func]
    type = ParsedFunction
    expression = '2.59e19'
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
[]

[Materials]
  [fission_rate]
    type = GenericFunctionMaterial
    prop_names = 'fission_rate'
    prop_values = 'fission_rate_func'
    outputs = all
  []
  [burnup]
    type = UPuZrBurnup
    initial_X_Zr = 0.225
    initial_X_Pu = 0.163
    density = 15800
    outputs = all
  []

  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
  []
  [fuel_elasticity_tensor]
    type = UPuZrElasticityTensor
    X_Zr = 0.225
    X_Pu = 0.163
    temperature = temperature
    porosity = porosity
  []
  [lowT_swelling]
    type = UPuZrLowTemperatureSwelling
    temperature = temperature
    outputs = all
  []
  [gas_swelling]
    type = UPuZrGaseousSwelling
    outputs = all
    temperature = temperature
    bubble_number_density = 1e20
  []
  [total_porosity]
    type = UPuZrPorosityEigenstrain
    initial_porosity = 0.1
    outputs = all
    output_properties = 'total_swelling porosity gas_swelling lowT_swelling'
    eigenstrain_name = porosity_eigenstrain
  []
[]

[Executioner]
  type = Transient
  solve_type = Newton

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 60
  l_tol = 1e-5

  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-7
  num_steps = 9
  dt = 1e6
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_porosity_eigenstrain/test.i)

# This tests UPuZrPorosityEigenstrain, a swelling model for UPuZr metal fuel
# that incorporates low temperature swelling.
#
# The low temperature swelling model is new. It assumes a temperature-dependent
# swelling rate that is linearly proportional to burnup. The total amount of
# porosity generated in the material is limited to a user-defined value.

# This test calculates the low temperature and gas swelling contributions as well as
# the total eigenstrain at 750 K for nine time steps (to t=9e6 s)  until the porosity
# threshold is reached.  To accelerate the test, initial porosity = 0.1. The computed
# values match the analytical values until t=8e6, at which point 1) the gaseous swelling
# smoothing activates and 2) the total swelling would exceed the swelling threshold
# and the logic to adjust the swelling increments is activated.

# Time | lowT_swelling | gas_swelling | porosity | eigenstrain_xx = eigenstrain_yy
# 1e6     2.3098e-2      5.5566e-4      1.2311e-1      7.7927e-3
# 2e6     4.6196e-2      1.5716e-3      1.4559e-1      1.5554e-2
# 3e6     6.9294e-2      2.8873e-3      1.6732e-1      2.3232e-2
# 4e6     9.2392e-2      4.4452e-3      1.8829e-1      3.0810e-2
# 5e6     1.1549e-1      6.2124e-3      2.0850e-1      3.8282e-2
# 6e6     1.3859e-1      8.1664e-3      2.2797e-1      4.5645e-2
# 7e6     1.6169e-1      1.0291e-2      2.4674e-1      5.2897e-2
# 8e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2
# 9e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[AuxVariables]
  [temperature]
    order = CONSTANT
    family = MONOMIAL
    [InitialCondition]
      type = ConstantIC
      value = 750
    []
  []
  [porosity_eigenstrain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [porosity_eigenstrain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[AuxKernels]
  [temp]
    type = ConstantAux
    variable = temperature
    value = 750
  []
  [eigenstrain_xx_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_xx
    index_i = 0
    index_j = 0
  []
  [eigenstrain_yy_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_yy
    index_i = 1
    index_j = 1
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = 'porosity_eigenstrain'
    generate_output = 'strain_xx strain_yy'
  []
[]

[Functions]
  [fission_rate_func]
    type = ParsedFunction
    expression = '2.59e19'
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
[]

[Materials]
  [fission_rate]
    type = GenericFunctionMaterial
    prop_names = 'fission_rate'
    prop_values = 'fission_rate_func'
    outputs = all
  []
  [burnup]
    type = UPuZrBurnup
    initial_X_Zr = 0.225
    initial_X_Pu = 0.163
    density = 15800
    outputs = all
  []

  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
  []
  [fuel_elasticity_tensor]
    type = UPuZrElasticityTensor
    X_Zr = 0.225
    X_Pu = 0.163
    temperature = temperature
    porosity = porosity
  []
  [lowT_swelling]
    type = UPuZrLowTemperatureSwelling
    temperature = temperature
    outputs = all
  []
  [gas_swelling]
    type = UPuZrGaseousSwelling
    outputs = all
    temperature = temperature
    bubble_number_density = 1e20
  []
  [total_porosity]
    type = UPuZrPorosityEigenstrain
    initial_porosity = 0.1
    outputs = all
    output_properties = 'total_swelling porosity gas_swelling lowT_swelling'
    eigenstrain_name = porosity_eigenstrain
  []
[]

[Executioner]
  type = Transient
  solve_type = Newton

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 60
  l_tol = 1e-5

  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-7
  num_steps = 9
  dt = 1e6
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_porosity_eigenstrain/test.i)

# This tests UPuZrPorosityEigenstrain, a swelling model for UPuZr metal fuel
# that incorporates low temperature swelling.
#
# The low temperature swelling model is new. It assumes a temperature-dependent
# swelling rate that is linearly proportional to burnup. The total amount of
# porosity generated in the material is limited to a user-defined value.

# This test calculates the low temperature and gas swelling contributions as well as
# the total eigenstrain at 750 K for nine time steps (to t=9e6 s)  until the porosity
# threshold is reached.  To accelerate the test, initial porosity = 0.1. The computed
# values match the analytical values until t=8e6, at which point 1) the gaseous swelling
# smoothing activates and 2) the total swelling would exceed the swelling threshold
# and the logic to adjust the swelling increments is activated.

# Time | lowT_swelling | gas_swelling | porosity | eigenstrain_xx = eigenstrain_yy
# 1e6     2.3098e-2      5.5566e-4      1.2311e-1      7.7927e-3
# 2e6     4.6196e-2      1.5716e-3      1.4559e-1      1.5554e-2
# 3e6     6.9294e-2      2.8873e-3      1.6732e-1      2.3232e-2
# 4e6     9.2392e-2      4.4452e-3      1.8829e-1      3.0810e-2
# 5e6     1.1549e-1      6.2124e-3      2.0850e-1      3.8282e-2
# 6e6     1.3859e-1      8.1664e-3      2.2797e-1      4.5645e-2
# 7e6     1.6169e-1      1.0291e-2      2.4674e-1      5.2897e-2
# 8e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2
# 9e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[AuxVariables]
  [temperature]
    order = CONSTANT
    family = MONOMIAL
    [InitialCondition]
      type = ConstantIC
      value = 750
    []
  []
  [porosity_eigenstrain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [porosity_eigenstrain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[AuxKernels]
  [temp]
    type = ConstantAux
    variable = temperature
    value = 750
  []
  [eigenstrain_xx_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_xx
    index_i = 0
    index_j = 0
  []
  [eigenstrain_yy_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_yy
    index_i = 1
    index_j = 1
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = 'porosity_eigenstrain'
    generate_output = 'strain_xx strain_yy'
  []
[]

[Functions]
  [fission_rate_func]
    type = ParsedFunction
    expression = '2.59e19'
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
[]

[Materials]
  [fission_rate]
    type = GenericFunctionMaterial
    prop_names = 'fission_rate'
    prop_values = 'fission_rate_func'
    outputs = all
  []
  [burnup]
    type = UPuZrBurnup
    initial_X_Zr = 0.225
    initial_X_Pu = 0.163
    density = 15800
    outputs = all
  []

  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
  []
  [fuel_elasticity_tensor]
    type = UPuZrElasticityTensor
    X_Zr = 0.225
    X_Pu = 0.163
    temperature = temperature
    porosity = porosity
  []
  [lowT_swelling]
    type = UPuZrLowTemperatureSwelling
    temperature = temperature
    outputs = all
  []
  [gas_swelling]
    type = UPuZrGaseousSwelling
    outputs = all
    temperature = temperature
    bubble_number_density = 1e20
  []
  [total_porosity]
    type = UPuZrPorosityEigenstrain
    initial_porosity = 0.1
    outputs = all
    output_properties = 'total_swelling porosity gas_swelling lowT_swelling'
    eigenstrain_name = porosity_eigenstrain
  []
[]

[Executioner]
  type = Transient
  solve_type = Newton

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 60
  l_tol = 1e-5

  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-7
  num_steps = 9
  dt = 1e6
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_porosity_eigenstrain/test.i)

# This tests UPuZrPorosityEigenstrain, a swelling model for UPuZr metal fuel
# that incorporates low temperature swelling.
#
# The low temperature swelling model is new. It assumes a temperature-dependent
# swelling rate that is linearly proportional to burnup. The total amount of
# porosity generated in the material is limited to a user-defined value.

# This test calculates the low temperature and gas swelling contributions as well as
# the total eigenstrain at 750 K for nine time steps (to t=9e6 s)  until the porosity
# threshold is reached.  To accelerate the test, initial porosity = 0.1. The computed
# values match the analytical values until t=8e6, at which point 1) the gaseous swelling
# smoothing activates and 2) the total swelling would exceed the swelling threshold
# and the logic to adjust the swelling increments is activated.

# Time | lowT_swelling | gas_swelling | porosity | eigenstrain_xx = eigenstrain_yy
# 1e6     2.3098e-2      5.5566e-4      1.2311e-1      7.7927e-3
# 2e6     4.6196e-2      1.5716e-3      1.4559e-1      1.5554e-2
# 3e6     6.9294e-2      2.8873e-3      1.6732e-1      2.3232e-2
# 4e6     9.2392e-2      4.4452e-3      1.8829e-1      3.0810e-2
# 5e6     1.1549e-1      6.2124e-3      2.0850e-1      3.8282e-2
# 6e6     1.3859e-1      8.1664e-3      2.2797e-1      4.5645e-2
# 7e6     1.6169e-1      1.0291e-2      2.4674e-1      5.2897e-2
# 8e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2
# 9e6     1.6582e-1      1.0652e-2      2.5000e-1      5.4173e-2

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
  []
[]

[AuxVariables]
  [temperature]
    order = CONSTANT
    family = MONOMIAL
    [InitialCondition]
      type = ConstantIC
      value = 750
    []
  []
  [porosity_eigenstrain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [porosity_eigenstrain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[AuxKernels]
  [temp]
    type = ConstantAux
    variable = temperature
    value = 750
  []
  [eigenstrain_xx_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_xx
    index_i = 0
    index_j = 0
  []
  [eigenstrain_yy_aux]
    type = RankTwoAux
    rank_two_tensor = porosity_eigenstrain
    variable = porosity_eigenstrain_yy
    index_i = 1
    index_j = 1
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = 'porosity_eigenstrain'
    generate_output = 'strain_xx strain_yy'
  []
[]

[Functions]
  [fission_rate_func]
    type = ParsedFunction
    expression = '2.59e19'
  []
[]

[BCs]
  [no_x]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  []
  [no_y]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  []
[]

[Materials]
  [fission_rate]
    type = GenericFunctionMaterial
    prop_names = 'fission_rate'
    prop_values = 'fission_rate_func'
    outputs = all
  []
  [burnup]
    type = UPuZrBurnup
    initial_X_Zr = 0.225
    initial_X_Pu = 0.163
    density = 15800
    outputs = all
  []

  [fuel_stress]
    type = ComputeFiniteStrainElasticStress
  []
  [fuel_elasticity_tensor]
    type = UPuZrElasticityTensor
    X_Zr = 0.225
    X_Pu = 0.163
    temperature = temperature
    porosity = porosity
  []
  [lowT_swelling]
    type = UPuZrLowTemperatureSwelling
    temperature = temperature
    outputs = all
  []
  [gas_swelling]
    type = UPuZrGaseousSwelling
    outputs = all
    temperature = temperature
    bubble_number_density = 1e20
  []
  [total_porosity]
    type = UPuZrPorosityEigenstrain
    initial_porosity = 0.1
    outputs = all
    output_properties = 'total_swelling porosity gas_swelling lowT_swelling'
    eigenstrain_name = porosity_eigenstrain
  []
[]

[Executioner]
  type = Transient
  solve_type = Newton

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 60
  l_tol = 1e-5

  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-7
  num_steps = 9
  dt = 1e6
[]

[Outputs]
  exodus = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References