MCThermalExpansionEigenstrain

Computes an eigenstrain due to thermal expansion for (U,Pu)C (mixed monocarbides) using a function that describes the mean thermal expansion as a function of temperature.

Description

This material model computes an eigenstrain tensor that accounts for thermal expansion in mixed carbide (MC). This model uses the correlation from Preusser (1982) to compute the mean coefficient of linear thermal expansion $var element = document.getElementById("moose-equation-ce5df5ec-2592-41c2-bf2c-7577eb4fca8a");katex.render("\\bar{\\alpha}", element, {displayMode:false,throwOnError:false});$ : $var element = document.getElementById("moose-equation-b961d631-62da-4fdb-9368-0908a76f7135");katex.render(" \\bar{\\alpha} = 1.007\\times 10^{-5} + 1.17\\times 10^{-9} (T - 273.15)", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-32e61985-9360-4880-b1de-fcc0b0f90e0f");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in K and $var element = document.getElementById("moose-equation-8c70c28e-5af3-4e27-a45e-8c61a09e31cf");katex.render("\\bar{\\alpha}", element, {displayMode:false,throwOnError:false});$ is given in 1/K. The equation is independent of porosity because porosity does not have a significant influence on thermal expansion (Blank, 2006). Likewise, the U/Pu ratio does not cause significant changes in $var element = document.getElementById("moose-equation-b4511b5e-94fc-4cd3-a3fd-972dd66f810e");katex.render("\\bar{\\alpha}", element, {displayMode:false,throwOnError:false});$ (Matzke, 1986).

This model makes use of methods similar to ComputeMeanThermalExpansionFunctionEigenstrain to compute the thermal strain tensor from the linear thermal expansion function.

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [thexp]
    type = ADMCThermalExpansionEigenstrain<<<{"description": "Computes an eigenstrain due to thermal expansion for (U,Pu)C (mixed monocarbides) using a function that describes the mean thermal expansion as a function of temperature.", "href": "MCThermalExpansionEigenstrain.html"}>>>
    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."}>>> = thexp
    stress_free_temperature<<<{"description": "Reference temperature at which there is no thermal expansion for thermal eigenstrain calculation"}>>> = 540
    temperature<<<{"description": "Coupled temperature"}>>> = temp
  []
[]

The eigenstrain name must also be passed to the strain calculator:

[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"}>>> = SMALL
        decomposition_method<<<{"description": "Methods to calculate the finite strain and rotation increments"}>>> = EigenSolution
        add_variables<<<{"description": "Add the displacement variables"}>>> = true
        eigenstrain_names<<<{"description": "List of eigenstrains to be applied in this strain calculation"}>>> = thexp
        use_automatic_differentiation<<<{"description": "Flag to use automatic differentiation (AD) objects when possible"}>>> = true
      []
    []
  []
[]

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.
stress_free_temperatureReference temperature at which there is no thermal expansion for thermal eigenstrain calculation
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Reference temperature at which there is no thermal expansion for thermal eigenstrain calculation

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.
mean_thermal_expansion_coefficient_nameName of the mean coefficient of thermal expansion.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:Name of the mean coefficient of thermal expansion.
temperatureCoupled temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature
use_old_temperatureFalseFlag to optionally use the temperature value from the previous timestep.
Default:False
C++ Type:bool
Controllable:No
Description:Flag to optionally use the temperature value from the previous timestep.

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
thermal_expansion_scale_factor1Scaling factor on the thermal expansion strain. This input parameter can be used to perform sensitivity analysis on thermal expansion.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scaling factor on the thermal expansion strain. This input parameter can be used to perform sensitivity analysis on thermal expansion.
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/mc_thermal_expansion/ad.i)
(test/tests/solid_mechanics/mc_thermal_expansion/nonad.i)
(assessment/carbide/EBRII/WSA32/analysis/base.i)
(test/tests/solid_mechanics/mc_thermal_expansion/exact.i)

References

Hubert Blank. Nonoxide Ceramic Nuclear Fuels, chapter, pages. John Wiley & Sons, Ltd, 2006. URL: https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527603978.mst0108, doi:10.1002/9783527603978.mst0108.

@inbook{blank:2006,
    author = "Blank, Hubert",
    publisher = "John Wiley \\& Sons, Ltd",
    isbn = "9783527603978",
    title = "Nonoxide Ceramic Nuclear Fuels",
    booktitle = "Materials Science and Technology",
    chapter = "",
    pages = "",
    doi = "10.1002/9783527603978.mst0108",
    url = "https://onlinelibrary.wiley.com/doi/abs/10.1002/9783527603978.mst0108",
    year = "2006"
}

Hj Matzke. Science of advanced LMFBR fuels. Elsevier Science Pub. Co. Inc, 1986.

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_thermal_expansion/exact.i)

# This tests the formation of the ADMCThermalExpansionEigenstrain due to changing temperature.
# Exact solution is calculated as a function in the input file.

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[AuxVariables]
  [temp]
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0 100'
    y = '300 2500'
  []
[]

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

[Physics]
  [SolidMechanics/QuasiStatic/all]
    strain = SMALL
    decomposition_method = EigenSolution
    add_variables = true
    eigenstrain_names = thexp
    use_automatic_differentiation = true
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = ADDirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = ADDirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
[]

[Materials]
  [elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [thexp]
    type = ADMCThermalExpansionEigenstrain
    eigenstrain_name = thexp
    stress_free_temperature = 540
    temperature = temp
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
[]

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

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [disp_x_right]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  []
  [thexp_exact]
    type = ParsedPostprocessor
    pp_names = 'temp'
    expression = 'SFT := 540; T0 := 298; alpha_T := 1.007e-5 + 1.17e-9 * (temp - 273.15);
                alpha_SFT := 1.007e-5 + 1.17e-9 * (SFT - 273.15);
                (alpha_T * (temp - T0) - alpha_SFT * (SFT - T0)) / (1 + alpha_SFT * (SFT - T0))'
  []
  [thexp_diff]
    type = ParsedPostprocessor
    pp_names = 'disp_x_right thexp_exact'
    expression = '(disp_x_right - thexp_exact) / thexp_exact'
    outputs = none
  []
  [thexp_max_diff]
    type = TimeExtremeValue
    postprocessor = thexp_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

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

# This tests the formation of the ADMCThermalExpansionEigenstrain due to changing temperature.
# Exact solution is calculated as a function in the input file.

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[AuxVariables]
  [temp]
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0 100'
    y = '300 2500'
  []
[]

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

[Physics]
  [SolidMechanics/QuasiStatic/all]
    strain = SMALL
    decomposition_method = EigenSolution
    add_variables = true
    eigenstrain_names = thexp
    use_automatic_differentiation = true
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = ADDirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = ADDirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
[]

[Materials]
  [elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [thexp]
    type = ADMCThermalExpansionEigenstrain
    eigenstrain_name = thexp
    stress_free_temperature = 540
    temperature = temp
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
[]

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

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [disp_x_right]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  []
  [thexp_exact]
    type = ParsedPostprocessor
    pp_names = 'temp'
    expression = 'SFT := 540; T0 := 298; alpha_T := 1.007e-5 + 1.17e-9 * (temp - 273.15);
                alpha_SFT := 1.007e-5 + 1.17e-9 * (SFT - 273.15);
                (alpha_T * (temp - T0) - alpha_SFT * (SFT - T0)) / (1 + alpha_SFT * (SFT - T0))'
  []
  [thexp_diff]
    type = ParsedPostprocessor
    pp_names = 'disp_x_right thexp_exact'
    expression = '(disp_x_right - thexp_exact) / thexp_exact'
    outputs = none
  []
  [thexp_max_diff]
    type = TimeExtremeValue
    postprocessor = thexp_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

(test/tests/solid_mechanics/mc_thermal_expansion/ad.i)

# This tests the formation of the ADMCThermalExpansionEigenstrain due to changing temperature for three dimensions.

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[Variables]
  [temp]
    initial_condition = 298
  []
[]

[Kernels]
  [diffusion]
    type = Diffusion
    variable = temp
  []
  [dt]
    type = TimeDerivative
    variable = temp
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    decomposition_method = EigenSolution
    add_variables = true
    eigenstrain_names = thexp
    temperature = temp
    use_automatic_differentiation = true
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = ADDirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = ADDirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
  [temp_left]
    type = ADFunctionDirichletBC
    variable = temp
    function = '300 + 100*t'
    boundary = left
    preset = false
  []
[]

[Materials]
  [elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [thexp]
    type = ADMCThermalExpansionEigenstrain
    eigenstrain_name = thexp
    stress_free_temperature = 298
    temperature = temp
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
[]

[Executioner]
  type = Transient

  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'

  num_steps = 10
  dt = 1
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [disp_x_right]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/mc_thermal_expansion/nonad.i)

# This tests the formation of the MCThermalExpansionEigenstrain due to changing temperature for three dimensions.

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[Variables]
  [temp]
    initial_condition = 298
  []
[]

[Kernels]
  [diffusion]
    type = Diffusion
    variable = temp
  []
  [dt]
    type = TimeDerivative
    variable = temp
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = SMALL
    decomposition_method = EigenSolution
    add_variables = true
    eigenstrain_names = thexp
    temperature = temp
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = DirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = DirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
  [temp_left]
    type = FunctionDirichletBC
    variable = temp
    function = '300 + 100*t'
    boundary = left
    preset = false
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [thexp]
    type = MCThermalExpansionEigenstrain
    eigenstrain_name = thexp
    stress_free_temperature = 298
    temperature = temp
  []
  [stress]
    type = ComputeLinearElasticStress
  []
[]

[Executioner]
  type = Transient

  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'

  nl_rel_tol = 1e-12
  num_steps = 10
  dt = 1
[]

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [disp_x_right]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  []
[]

[Outputs]
  exodus = 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_thermal_expansion/exact.i)

# This tests the formation of the ADMCThermalExpansionEigenstrain due to changing temperature.
# Exact solution is calculated as a function in the input file.

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = ExamplePatchMeshGenerator
    dim = 3
  []
[]

[AuxVariables]
  [temp]
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0 100'
    y = '300 2500'
  []
[]

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

[Physics]
  [SolidMechanics/QuasiStatic/all]
    strain = SMALL
    decomposition_method = EigenSolution
    add_variables = true
    eigenstrain_names = thexp
    use_automatic_differentiation = true
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [bottom]
    type = ADDirichletBC
    variable = disp_y
    value = 0
    boundary = bottom
  []
  [back]
    type = ADDirichletBC
    variable = disp_z
    value = 0
    boundary = back
  []
[]

[Materials]
  [elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 1
    poissons_ratio = 0.3
  []
  [thexp]
    type = ADMCThermalExpansionEigenstrain
    eigenstrain_name = thexp
    stress_free_temperature = 540
    temperature = temp
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
[]

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

[Postprocessors]
  [temp]
    type = ElementAverageValue
    variable = temp
  []
  [disp_x_right]
    type = SideAverageValue
    variable = disp_x
    boundary = right
  []
  [thexp_exact]
    type = ParsedPostprocessor
    pp_names = 'temp'
    expression = 'SFT := 540; T0 := 298; alpha_T := 1.007e-5 + 1.17e-9 * (temp - 273.15);
                alpha_SFT := 1.007e-5 + 1.17e-9 * (SFT - 273.15);
                (alpha_T * (temp - T0) - alpha_SFT * (SFT - T0)) / (1 + alpha_SFT * (SFT - T0))'
  []
  [thexp_diff]
    type = ParsedPostprocessor
    pp_names = 'disp_x_right thexp_exact'
    expression = '(disp_x_right - thexp_exact) / thexp_exact'
    outputs = none
  []
  [thexp_max_diff]
    type = TimeExtremeValue
    postprocessor = thexp_diff
    value_type = abs_max
  []
[]

[Outputs]
  csv = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References