MCElasticityTensor

Calculates the Young's modulus and Poisson's ratio for (U,Pu)C (mixed mono-carbide) fuel as a function of temperature, porosity, and plutonium content.

Description

The MCElasticityTensor material model computes the elasticity tensor for mixed carbide (MC) (monocarbides) using for the Young's modulus and Poission's ratio as reported by Preusser (1982). The Young's modulus for UC and (U $var element = document.getElementById("moose-equation-f9db8d82-2c24-4ad4-a0df-c0adf47b0f2c");katex.render("_{0.8}", element, {displayMode:false,throwOnError:false});$ ,Pu $var element = document.getElementById("moose-equation-4f56e531-d9fe-4a4b-a587-d774bb07b3a2");katex.render("_{0.2}", element, {displayMode:false,throwOnError:false});$ )C, is computed in units of Pa as: $var element = document.getElementById("moose-equation-286bba1c-1d1e-448c-87ad-cd2dd68b8987");katex.render(" E_{UC} = 2.15\\times 10^{11} \\left(1 - 2.3 \\phi\\right)\\left(1-0.92\\times 10^{-4}\\left[T-298\\right]\\right), \\\\ E_{(U_{0.8},Pu_{0.2})C} = 2.02\\times 10^{11} \\left(1 - 1.54 \\phi\\right)\\left(1-0.92\\times 10^{-4}\\left[T-298\\right]\\right),", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-4846d82d-319b-4f4d-bff8-593307351ed2");katex.render("293 K \\leq T \\leq 1523.15", element, {displayMode:false,throwOnError:false});$ K is the temperature in K and $var element = document.getElementById("moose-equation-361ad56e-3862-42f9-bad1-465da8874db6");katex.render("0\\leq \\phi \\leq 0.3", element, {displayMode:false,throwOnError:false});$ is the porosity. For compositions that are not exactly UC or (U $var element = document.getElementById("moose-equation-06bcb117-cd52-4482-b6e6-fa7ff53753e0");katex.render("_{0.8}", element, {displayMode:false,throwOnError:false});$ ,Pu $var element = document.getElementById("moose-equation-3ff432b5-a117-4ff8-82d5-fed62d01b016");katex.render("_{0.2}", element, {displayMode:false,throwOnError:false});$ )C, a linear interpolation is used in the absence of better data: $var element = document.getElementById("moose-equation-efab1d81-d7f1-4230-b37a-170d933ac72e");katex.render(" E_{MC} = E_{UC} (1 - 5 \\alpha) + E_{(U_{0.8},Pu_{0.2})C} (5 \\alpha)", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-9ada3b18-aced-444d-9bb7-1a32cf004913");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is the atomic fraction of plutonium relative to the actinides in the mixed monocarbide, i.e., (U $var element = document.getElementById("moose-equation-e04ccad0-4b82-40ea-9e8f-08d5c7add4b7");katex.render("_{1-\\alpha}", element, {displayMode:false,throwOnError:false});$ ,Pu $var element = document.getElementById("moose-equation-868ba998-07f4-4a1b-a082-cab1729b4e89");katex.render("_{\\alpha}", element, {displayMode:false,throwOnError:false});$ )C. Note that for $var element = document.getElementById("moose-equation-1a247f9d-6876-4ad4-b347-28ccb3f1bb60");katex.render("\\alpha = 0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-8c3760f2-8f21-4156-a5d0-eacca1285ca7");katex.render("E_{MC} = E_{UC}", element, {displayMode:false,throwOnError:false});$ , and for $var element = document.getElementById("moose-equation-4890bbf6-c11a-4d5f-b1ca-90a17cc6891f");katex.render("\\alpha = 0.2", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-6a417015-5785-469f-9754-007b6df0cab9");katex.render("E_{MC} = E_{(U_{0.8},Pu_{0.2})C}", element, {displayMode:false,throwOnError:false});$ . Such a linear approximation to the plutonium content should be used with caution, but is reasonable between 0 and 0.5 relative atom fraction of Pu.

For consistent property calculations, the correlation for Poisson's ratio $var element = document.getElementById("moose-equation-429891d7-f703-4cd7-9ad7-95699fcbe04e");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ should be computed from the same data as $var element = document.getElementById("moose-equation-43f0004d-f660-4897-a8d1-6fa5ecb75576");katex.render("E", element, {displayMode:false,throwOnError:false});$ , so we again use Preusser (1982): $var element = document.getElementById("moose-equation-024aaf1f-e506-4ebf-8f7d-b210fe093837");katex.render(" \\nu = 0.288 - 0.286 \\phi,", element, {displayMode:true,throwOnError:false});$ which is valid for $var element = document.getElementById("moose-equation-8ad0bc76-f420-41ec-b9b0-8f31805f2ab2");katex.render("0.05 \\leq \\phi \\leq 0.27", element, {displayMode:false,throwOnError:false});$ according to Padel et al. (1969), but can reasonably be extended to $var element = document.getElementById("moose-equation-d18dcbb6-1807-4c01-8165-0a794ce2fe91");katex.render("0 \\leq \\phi \\leq 0.27", element, {displayMode:false,throwOnError:false});$ (Nickerson and Kastenberg, 1975).

This material is assumed to be isotropic, and $var element = document.getElementById("moose-equation-cdc9b946-0406-4bbb-855d-3191b7f86705");katex.render("E", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-82ee7f4d-a6e5-42f9-b6a0-e163d964c39c");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ are therefore used to generate an isotropic elasticity tensor. The values for $var element = document.getElementById("moose-equation-02cd5e35-1fe6-4b09-b4bc-4341839cb7be");katex.render("E", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-dc63fadb-2081-4573-a223-9f17cb1f986b");katex.render("\\nu", element, {displayMode:false,throwOnError:false});$ are given over the ranges of porosity and temperature shown above, so ValueRangeInterface is used to check that the porosity or temperatures are within this range.

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [elasticity_tensor]
    type = ADMCElasticityTensor<<<{"description": "Calculates the Young's modulus and Poisson's ratio for (U,Pu)C (mixed mono-carbide) fuel as a function of temperature, porosity, and plutonium content.", "href": "MCElasticityTensor.html"}>>>
    temperature<<<{"description": "Coupled temperature"}>>> = temp
    porosity<<<{"description": "Porosity material property name."}>>> = porosity
    X_Pu_to_An_in_MC<<<{"description": "Relative atom fraction of plutonium to actnidies in mono-carbide phase."}>>> = X_Pu_to_An_in_MC
    output_properties<<<{"description": "List of material properties, from this material, to output (outputs must also be defined to an output type)"}>>> = 'youngs_modulus poissons_ratio'
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

Input Parameters

porosityPorosity material property name.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Porosity material property name.
temperatureCoupled temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature

Required Parameters

X_Pu_to_An_in_MC0Relative atom fraction of plutonium to actnidies in mono-carbide phase.
Default:0
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Relative atom fraction of plutonium to actnidies in mono-carbide phase.
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.
elasticity_tensor_prefactorOptional function to use as a scalar prefactor on the elasticity tensor.
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:Optional function to use as a scalar prefactor on the elasticity tensor.
execute_onLINEARThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:LINEAR
C++ Type:ExecFlagEnum
Options:XFEM_MARK, NONE, INITIAL, LINEAR, NONLINEAR_CONVERGENCE, NONLINEAR, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, FINAL, CUSTOM
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
value_range_behaviorEXCEPTIONWhat to do if input value is outside the range of applicability.
Default:EXCEPTION
C++ Type:MooseEnum
Options:ERROR, WARN, IGNORE, EXCEPTION
Controllable:No
Description:What to do if input value is outside the range of applicability.

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

(assessment/carbide/EBRII/WSA32/analysis/base.i)
(test/tests/solid_mechanics/mc_elasticity_tensor/exact.i)

References

G M Nickerson and W E Kastenberg. Preliminary assessments of carbide fuel pins during mild overpower transients (journal article) | osti.gov. In Trans. Am. Nucl. Soc., v. 22, 417–418. San Francisco, 11 1975. American Nuclear Society 1975 winter meeting. URL: https://www.osti.gov/biblio/4069540-preliminary-assessments-carbide-fuel-pins-during-mild-overpower-transients.

@inproceedings{NICKERSON1975417,
    author = "Nickerson, G M and Kastenberg, W E",
    address = "San Francisco",
    booktitle = "Trans. Am. Nucl. Soc., v. 22",
    month = "11",
    pages = "417-418",
    publisher = "American Nuclear Society 1975 winter meeting",
    title = "Preliminary assessments of carbide fuel pins during mild overpower transients (Journal Article) | OSTI.GOV",
    url = "https://www.osti.gov/biblio/4069540-preliminary-assessments-carbide-fuel-pins-during-mild-overpower-transients",
    year = "1975"
}

A. Padel, Ch. De Novion, A. Padel, and Ch. De Novion. Constantes elastiques des carbures, nitrures et oxydes d'uranium et de plutonium. Journal of Nuclear Materials, 33:40–51, 1969. URL: https://ui.adsabs.harvard.edu/abs/1969JNuM...33...40P/abstract, doi:10.1016/0022-3115(69)90006-3.

@article{PADEL196940,
    author = "Padel, A. and Novion, Ch. De and Padel, A. and Novion, Ch. De",
    doi = "10.1016/0022-3115(69)90006-3",
    issn = "0022-3115",
    issue = "1",
    journal = "Journal of Nuclear Materials",
    pages = "40-51",
    title = "Constantes elastiques des carbures, nitrures et oxydes d'uranium et de plutonium",
    volume = "33",
    url = "https://ui.adsabs.harvard.edu/abs/1969JNuM...33...40P/abstract",
    year = "1969"
}

Timm Preusser. Modeling of carbide fuel rods. Nuclear Technology, 57(3):343–371, 1982. URL: https://doi.org/10.13182/NT82-A26303, arXiv:https://doi.org/10.13182/NT82-A26303, doi:10.13182/NT82-A26303.

@article{preusser:1982,
    author = "Preusser, Timm",
    title = "Modeling of Carbide Fuel Rods",
    journal = "Nuclear Technology",
    volume = "57",
    number = "3",
    pages = "343-371",
    year = "1982",
    publisher = "Taylor \& Francis",
    doi = "10.13182/NT82-A26303",
    URL = "https://doi.org/10.13182/NT82-A26303",
    eprint = "https://doi.org/10.13182/NT82-A26303"
}

(test/tests/solid_mechanics/mc_elasticity_tensor/exact.i)

# This tests the formation of the ADMCElasticityTensor due to changing temperature porosity and plutonium content.
# The exact solution is calculated as a function and solved simultaneously.
# The primary metric is the maxmium difference in postprocessors between the calculated and
# analytical solutions for Young's modulus and Poisson's ratio.

[Problem]
  solve = false
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[AuxVariables]
  [temp]
    initial_condition = 300
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0 100'
    y = '300 900'
  []
  [porosity_ramp]
    type = PiecewiseLinear
    x = '0      1      100'
    y = '0.0515 0.0515 0.20'
  []
  [X_Pu_to_An_in_MC_ramp]
    type = PiecewiseLinear
    x = '0 25  50 75  100'
    y = '0 0.3 0  0.3 0'
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = temp_ramp
  []
[]

[Materials]
  [X_Pu_to_An_in_MC]
    type = ADGenericFunctionMaterial
    prop_names = 'X_Pu_to_An_in_MC'
    prop_values = 'X_Pu_to_An_in_MC_ramp'
    outputs = all
  []
  [elasticity_tensor]
    type = ADMCElasticityTensor
    temperature = temp
    porosity = porosity
    X_Pu_to_An_in_MC = X_Pu_to_An_in_MC
    output_properties = 'youngs_modulus poissons_ratio'
    outputs = all
  []
  [porosity]
    type = ADGenericFunctionMaterial
    prop_names = porosity
    prop_values = porosity_ramp
    outputs = all
  []
[]

[Executioner]
  type = Transient
  num_steps = 10
  dt = 10
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
    execute_on = 'initial timestep_end'
  []
  [porosity]
    type = ElementAverageValue
    variable = porosity
    execute_on = 'initial timestep_end'
  []
  [X_Pu_to_An_in_MC]
    type = ElementAverageValue
    variable = X_Pu_to_An_in_MC
    execute_on = 'initial timestep_end'
  []
  [youngs_avg]
    type = ElementAverageValue
    variable = youngs_modulus
    execute_on = 'initial timestep_end'
  []
  [youngs_exact]
    type = ParsedPostprocessor
    pp_names = 'temp porosity X_Pu_to_An_in_MC'
    expression = 'youngs_uc := 2.15e11 * (1.0 - 2.3 * porosity) * (1.0 - 0.92e-4 * (temp - 298));
                youngs_upuc := 2.02e11 * (1.0 - 1.54 * porosity) * (1.0 - 0.92e-4 * (temp - 298));
                youngs_uc * (1.0 - 5.0 * X_Pu_to_An_in_MC) + youngs_upuc * (5.0 * X_Pu_to_An_in_MC)'
    execute_on = 'initial timestep_end'
  []
  [poissons_avg]
    type = ElementAverageValue
    variable = poissons_ratio
    execute_on = 'initial timestep_end'
  []
  [poissons_exact]
    type = ParsedPostprocessor
    pp_names = 'porosity'
    expression = '0.288 - 0.286 * porosity'
    execute_on = 'initial timestep_end'
  []
  [youngs_diff]
    type = ParsedPostprocessor
    pp_names = 'youngs_avg youngs_exact'
    expression = '(youngs_avg - youngs_exact) / youngs_exact'
    outputs = none
    execute_on = 'initial timestep_end'
  []
  [youngs_max_diff]
    type = TimeExtremeValue
    postprocessor = youngs_diff
    value_type = abs_max
    execute_on = 'initial timestep_end'
  []
  [poissons_diff]
    type = ParsedPostprocessor
    pp_names = 'poissons_avg poissons_exact'
    expression= '(poissons_avg - poissons_exact) / poissons_exact'
    outputs = none
    execute_on = 'initial timestep_end'
  []
  [poissons_diff_max_diff]
    type = TimeExtremeValue
    postprocessor = poissons_diff
    value_type = abs_max
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
[]

(assessment/carbide/EBRII/WSA32/analysis/base.i)

initial_fuel_density = 13630.0

[GlobalParams]
  density = ${initial_fuel_density}
  order = SECOND
  family = LAGRANGE
  energy_per_fission = 3.2e-11 # J/fission
  volumetric_locking_correction = false
  displacements = 'disp_x disp_y'
  value_range_behavior = IGNORE
[]

[Problem]
  type = ReferenceResidualProblem
  reference_vector = 'ref'
  extra_tag_vectors = 'ref'
  group_variables = 'disp_x disp_y'
[]

[Mesh]
  coord_type = RZ
  use_displaced_mesh = false
  [smeared_pellet_mesh]
    type = FuelPinMeshGenerator
    clad_thickness = 0.508e-3
    pellet_outer_radius = 4.301e-3
    pellet_height = 343e-3
    clad_top_gap_height = 43.5e-3
    clad_gap_width = 0.145e-3
    bottom_clad_height = 8e-3
    top_clad_height = 8e-3
    clad_bot_gap_height = 0.2e-3 # unknown
    clad_mesh_density = customize
    pellet_mesh_density = customize
    nx_p = 6
    ny_p = 260
    nx_c = 4
    ny_c = 260
    ny_cu = 3
    ny_cl = 3
    pellet_quantity = 1
    elem_type = QUAD8
  []
  patch_size = 30
  patch_update_strategy = auto
  partitioner = centroid
  centroid_partitioner_direction = y
[]

[Variables]
  [disp_x]
  []
  [disp_y]
  []
  [temp]
    initial_condition = 350
  []
[]

[AuxVariables]
  [contact_pressure]
  []
  [layered_side_contact_pressure]
    block = pellet
    order = CONSTANT
    family = MONOMIAL
  []
  [pellet_wall_temp]
    order = CONSTANT
    family = MONOMIAL
  []
  [radial_heat_flux]
    order = CONSTANT
    family = MONOMIAL
  []
  [clad_inner_wall_temp]
    order = CONSTANT
    family = MONOMIAL
  []
  [volumetric_strain]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [power_history]
    type = PiecewiseLinear
    x = '0 1e5   70e6 70100000' # unknown, final burnup was 12.2 %
    y = '0 1     1    0'
  []
  [axial_peaking_factors]
    type = PiecewiseLinear
    axis = y
    x = '0    46e-3 111e-3 200e-3 248e-3 313e-3 352e-3'
    y = '86e3 86e3  97e3   100e3  96e3   87e3   87e3'
  []
  [fission_func]
    type = ParsedFunction
    symbol_names = 'axial_peaking_factors power_history'
    symbol_values = 'axial_peaking_factors power_history'
    expression = 'linear_heating_rate := axial_peaking_factors * power_history;
             pellet_cross_sectional_area := 3.14159 * pow( 4.301e-3, 2 );
             volumetric_power := linear_heating_rate / pellet_cross_sectional_area;
             energy_per_fission := 3.2e-11;
             volumetric_power / energy_per_fission'
  []
  [burnup_func] # Burnup func is needed because BurnupAux doesn't currentl export an ADMaterialProperty.
    type = PiecewiseLinear
    x = '0 1e5   70e6  70100000'
    y = '0 0     0.122 0.122'
  []

  [coolant_press_ramp]
    type = PiecewiseLinear
    x = '0       70100000'
    y = '0.151e6 0.151e6' # unknown, Na coolant
  []
  [coolant_inlet_temp]
    type = PiecewiseLinear
    x = '0   1e5 70e6 70100000'
    y = '350 644 644  350'
  []
  [coolant_outlet_temp]
    type = PiecewiseLinear
    x = '0   1e5 70e6 70100000'
    y = '350 746 746  350'
  []
  [coolant_wall_temp]
    type = ParsedFunction
    symbol_values = 'coolant_inlet_temp coolant_outlet_temp'
    symbol_names = 'coolant_inlet_temp coolant_outlet_temp'
    expression = 'rod_length := 343e-3 + 43.5e-3 + 2 * 8e-3;
             delta_temp := coolant_outlet_temp - coolant_inlet_temp;
             interpolated_temp := coolant_inlet_temp + y * delta_temp / rod_length;
             max( min( interpolated_temp, coolant_outlet_temp ), coolant_inlet_temp )'
  []
  [radial_heat_flux]
    type = ParsedFunction
    symbol_values = 'coolant_wall_temp power_history axial_peaking_factors'
    symbol_names = 'coolant_wall_temp power_history axial_peaking_factors'
    expression = 'linear_heating_rate := axial_peaking_factors * power_history;
             pellet_radius := 4.301e-3;
             pellet_circumference := 3.14159 * pellet_radius * 2;
             linear_heating_rate / pellet_circumference'
  []
  [clad_inner_wall_temp]
    type = ParsedFunction
    symbol_values = 'radial_heat_flux coolant_wall_temp'
    symbol_names = 'radial_heat_flux coolant_wall_temp'
    expression = 'conductivity_316ss := 22;
             clad_thickness := 0.51e-3;
             coolant_wall_temp + radial_heat_flux / conductivity_316ss * clad_thickness'
  []
  [pellet_wall_temp]
    type = ParsedFunction
    symbol_values = 'radial_heat_flux coolant_wall_temp clad_inner_wall_temp'
    symbol_names = 'radial_heat_flux coolant_wall_temp clad_inner_wall_temp'
    expression = 'conductivity_gap := 1;
             gap_thickness := 0.145e-3;
             clad_inner_wall_temp + radial_heat_flux / conductivity_gap * gap_thickness'
  []

  [contact_pressure_func]
    type = ConstantFunction
    value = 12e6
  []
  [plenum_pressure]
    type = ConstantFunction
    value = 12.4e6 # initial fill
  []

  [porosity_ramp]
    type = PiecewiseLinear
    x = '0    1e5  70e6  70100000'
    y = '0.14 0.14 0.145 0.145' # Pellets start at 88.1% TD, but no data on final porosity.
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [fuel]
    block = pellet
    strain = SMALL
    incremental = true
    extra_vector_tags = 'ref'
    eigenstrain_names = 'fuel_thermal_expansion fuel_swelling'
    use_automatic_differentiation = true
  []
  [clad]
    block = clad
    strain = SMALL
    incremental = true
    extra_vector_tags = 'ref'
    use_automatic_differentiation = true
  []
[]

[Kernels]
  [gravity]
    type = ADGravity
    variable = disp_y
    value = -9.81
    extra_vector_tags = 'ref'
  []
  [heat]
    type = ADHeatConduction
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_ie]
    type = ADHeatConductionTimeDerivative
    variable = temp
    extra_vector_tags = 'ref'
  []
  [heat_source]
    type = ADFissionRateHeatSource
    variable = temp
    block = pellet
    fission_rate = fission_rate
    extra_vector_tags = 'ref'
  []
[]

[AuxKernels]
  [contact_pressure_aux]
    type = FunctionAux
    variable = contact_pressure
    function = contact_pressure_func
  []
  [layered_side_contact_pressure]
    type = SpatialUserObjectAux
    variable = layered_side_contact_pressure
    user_object = layered_side_contact_pressure
    block = pellet
  []

  [pellet_wall_temp]
    type = FunctionAux
    variable = pellet_wall_temp
    function = pellet_wall_temp
  []
  [radial_heat_flux]
    type = FunctionAux
    variable = radial_heat_flux
    function = radial_heat_flux
  []
  [clad_inner_wall_temp]
    type = FunctionAux
    variable = clad_inner_wall_temp
    function = clad_inner_wall_temp
  []
  [volumetric_strain]
    type = ADRankTwoScalarAux
    rank_two_tensor = total_strain
    variable = volumetric_strain
    scalar_type = VolumetricStrain
  []
[]

[BCs]
  [no_x_all]
    type = DirichletBC
    variable = disp_x
    boundary = 12
    value = 0.0
  []
  [no_y_fuel]
    type = DirichletBC
    variable = disp_y
    boundary = 20
    value = 0.0
  []
  [no_y_clad]
    type = DirichletBC
    variable = disp_y
    boundary = 1
    value = 0.0
  []
  [Pressure]
    [coolantPressure]
      boundary = '1 2 3'
      function = coolant_press_ramp
    []
    [plenumPressure]
      boundary = 9
      function = plenum_pressure
    []
  []
  [clad_outer_temp]
    type = FunctionDirichletBC
    boundary = '1 2 3'
    function = coolant_wall_temp
    variable = temp
  []
  [pellet_to_clad_cooling]
    type = FunctionDirichletBC
    boundary = 10
    function = pellet_wall_temp
    variable = temp
  []
[]

[Materials]
  [fission]
    type = ADGenericFunctionMaterial
    prop_names = fission_rate
    prop_values = fission_func
    outputs = all
  []
  [burnup]
    type = ADGenericFunctionMaterial
    block = pellet
    prop_names = burnup
    prop_values = burnup_func
    outputs = all
  []
  [fuel_thermal]
    block = pellet
    type = ADMCThermal
    temperature = temp
    porosity = porosity
    X_Pu = 0.0
    outputs = all
  []
  [fuel_porosity]
    block = pellet
    type = ADGenericFunctionMaterial
    prop_names = porosity
    prop_values = porosity_ramp
    outputs = all
  []
  [fuel_elasticity_tensor]
    block = pellet
    type = ADMCElasticityTensor
    temperature = temp
    porosity = porosity
    output_properties = 'youngs_modulus poissons_ratio'
    outputs = all
  []
  [fuel_thermal_expansion]
    block = pellet
    type = ADMCThermalExpansionEigenstrain
    eigenstrain_name = fuel_thermal_expansion
    stress_free_temperature = 350
    temperature = temp
  []
  [fuel_creep]
    block = pellet
    type = ADMCCreepUpdate
    temperature = temp
    fission_rate = fission_rate
    outputs = all
  []
  [fuel_swelling]
    block = pellet
    type = ADMCVolumetricSwellingEigenstrain
    eigenstrain_name = fuel_swelling
    temperature = temp
    porosity = porosity
    contact_pressure = layered_side_contact_pressure
    burnup = burnup
    outputs = all
  []
  [fuel_radial_return_stress]
    block = pellet
    type = ADComputeMultipleInelasticStress
    inelastic_models = 'fuel_creep'
  []
  [fuel_density]
    block = pellet
    type = ADStrainAdjustedDensity
    strain_free_density = ${initial_fuel_density}
  []

  [clad_elasticity_tensor]
    block = clad
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 190e9
    poissons_ratio = 0.265
  []
  [clad_stress]
    block = clad
    type = ADComputeLinearElasticStress
  []
  [clad_thermal]
    block = clad
    type = ADSS316Thermal
    temperature = temp
  []
  [clad_density]
    block = clad
    type = ADStrainAdjustedDensity
    strain_free_density = 8000
  []
[]

[UserObjects]
  [layered_side_contact_pressure]
    # This reads the contact pressure at the pellet OD and propagates it inward so that it can
    # be used in ADMCVolumetricSwellingEigenstrain for inner elements as well.
    type = LayeredSideAverage
    direction = y
    num_layers = 100
    variable = contact_pressure
    execute_on = linear
    boundary = 10
  []
[]

[Preconditioning]
  [SMP]
    type = SMP
    full = true
  []
[]

[Dampers]
  [limitT]
    type = MaxIncrement
    max_increment = 100.0
    variable = temp
  []
  [limitX]
    type = MaxIncrement
    max_increment = 1e-5
    variable = disp_x
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  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 = 50
  l_tol = 8e-3
  nl_max_its = 15
  nl_rel_tol = 1e-4
  nl_abs_tol = 1e-10

  start_time = -200
  n_startup_steps = 1
  end_time = 70100000
  dtmin = 1
  dtmax = 2e6

  [Quadrature]
    order = fifth
    side_order = seventh
  []

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 2e2
    time_t = ' 0   1e5 70e6 70100000'
    time_dt = '1e2 1e2 1e2  1e2'
    optimal_iterations = 10
    growth_factor = 2
    cutback_factor = 0.5
    iteration_window = 1
    linear_iteration_ratio = 100
  []
[]

[Postprocessors]
  [_dt]
    type = TimestepSize
  []
  [num_lin_it]
    type = NumLinearIterations
  []
  [num_nonlin_it]
    type = NumNonlinearIterations
  []
  [burnup]
    block = pellet
    type = ADElementAverageMaterialProperty
    mat_prop = burnup
  []
  [FCT]
    type = NodalExtremeValue
    boundary = 12
    variable = temp
    execute_on = 'initial timestep_end'
  []
  [volumetric_strain]
    type = ElementAverageValue
    block = pellet
    variable = volumetric_strain
  []
  [pellet_outer_disp_x]
    type = NodalExtremeValue
    boundary = 10
    variable = disp_x
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  color = true
  exodus = true
  perf_graph = true
  csv = true
  [console]
    type = Console
    max_rows = 25
    time_step_interval =  1
    output_linear = true
  []
  [chkfile]
    type = CSV
    hide = 'num_nonlin_it num_lin_it _dt'
  []
[]

(test/tests/solid_mechanics/mc_elasticity_tensor/exact.i)

# This tests the formation of the ADMCElasticityTensor due to changing temperature porosity and plutonium content.
# The exact solution is calculated as a function and solved simultaneously.
# The primary metric is the maxmium difference in postprocessors between the calculated and
# analytical solutions for Young's modulus and Poisson's ratio.

[Problem]
  solve = false
[]

[Mesh]
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
  []
[]

[AuxVariables]
  [temp]
    initial_condition = 300
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0 100'
    y = '300 900'
  []
  [porosity_ramp]
    type = PiecewiseLinear
    x = '0      1      100'
    y = '0.0515 0.0515 0.20'
  []
  [X_Pu_to_An_in_MC_ramp]
    type = PiecewiseLinear
    x = '0 25  50 75  100'
    y = '0 0.3 0  0.3 0'
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temp
    function = temp_ramp
  []
[]

[Materials]
  [X_Pu_to_An_in_MC]
    type = ADGenericFunctionMaterial
    prop_names = 'X_Pu_to_An_in_MC'
    prop_values = 'X_Pu_to_An_in_MC_ramp'
    outputs = all
  []
  [elasticity_tensor]
    type = ADMCElasticityTensor
    temperature = temp
    porosity = porosity
    X_Pu_to_An_in_MC = X_Pu_to_An_in_MC
    output_properties = 'youngs_modulus poissons_ratio'
    outputs = all
  []
  [porosity]
    type = ADGenericFunctionMaterial
    prop_names = porosity
    prop_values = porosity_ramp
    outputs = all
  []
[]

[Executioner]
  type = Transient
  num_steps = 10
  dt = 10
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
    execute_on = 'initial timestep_end'
  []
  [porosity]
    type = ElementAverageValue
    variable = porosity
    execute_on = 'initial timestep_end'
  []
  [X_Pu_to_An_in_MC]
    type = ElementAverageValue
    variable = X_Pu_to_An_in_MC
    execute_on = 'initial timestep_end'
  []
  [youngs_avg]
    type = ElementAverageValue
    variable = youngs_modulus
    execute_on = 'initial timestep_end'
  []
  [youngs_exact]
    type = ParsedPostprocessor
    pp_names = 'temp porosity X_Pu_to_An_in_MC'
    expression = 'youngs_uc := 2.15e11 * (1.0 - 2.3 * porosity) * (1.0 - 0.92e-4 * (temp - 298));
                youngs_upuc := 2.02e11 * (1.0 - 1.54 * porosity) * (1.0 - 0.92e-4 * (temp - 298));
                youngs_uc * (1.0 - 5.0 * X_Pu_to_An_in_MC) + youngs_upuc * (5.0 * X_Pu_to_An_in_MC)'
    execute_on = 'initial timestep_end'
  []
  [poissons_avg]
    type = ElementAverageValue
    variable = poissons_ratio
    execute_on = 'initial timestep_end'
  []
  [poissons_exact]
    type = ParsedPostprocessor
    pp_names = 'porosity'
    expression = '0.288 - 0.286 * porosity'
    execute_on = 'initial timestep_end'
  []
  [youngs_diff]
    type = ParsedPostprocessor
    pp_names = 'youngs_avg youngs_exact'
    expression = '(youngs_avg - youngs_exact) / youngs_exact'
    outputs = none
    execute_on = 'initial timestep_end'
  []
  [youngs_max_diff]
    type = TimeExtremeValue
    postprocessor = youngs_diff
    value_type = abs_max
    execute_on = 'initial timestep_end'
  []
  [poissons_diff]
    type = ParsedPostprocessor
    pp_names = 'poissons_avg poissons_exact'
    expression= '(poissons_avg - poissons_exact) / poissons_exact'
    outputs = none
    execute_on = 'initial timestep_end'
  []
  [poissons_diff_max_diff]
    type = TimeExtremeValue
    postprocessor = poissons_diff
    value_type = abs_max
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References