ZrO2ElasticityTensor

Computes the Young's Modulus and Poisson's ratio for zirconium oxide.

Description

The ZrO2ElasticityTensor material model, determines the Young's Modulus and Poisson's ratio of zirconium oxide using the MAPTRO (Allison et al., 1993) correlations.

The Poisson's ratio for ZrO $var element = document.getElementById("moose-equation-bfe5e252-f79a-4e37-8d25-c29a1198268b");katex.render("_2", element, {displayMode:false,throwOnError:false});$ is constant and assumed to be equal to 0.3. The Young's Modulus is a temperature dependent function that depends upon the solidus temperature of the oxide. The solidus temperature is a function of the atomic fraction of oxygen in the oxide. The atomic fraction of oxygen is calculated from the oxygen to metal ratio as follows: $var element = document.getElementById("moose-equation-8312d405-d5ea-4609-823e-7c0a393c7e5f");katex.render("x = \\frac{OM}{1+OM}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-90563659-1768-4b86-bfe3-5719a80c1bd7");katex.render("x", element, {displayMode:false,throwOnError:false});$ is the atomic fraction of oxygen and $var element = document.getElementById("moose-equation-8064ba81-c996-4828-b9c9-59bd25c6f7a4");katex.render("OM", element, {displayMode:false,throwOnError:false});$ is the oxygen-to-metal ratio. Here it is assumed that the oxide that is formed is pure ZrO $var element = document.getElementById("moose-equation-93d7f1cc-df15-488e-8b1d-07a054c553bf");katex.render("_2", element, {displayMode:false,throwOnError:false});$ and therefore $var element = document.getElementById("moose-equation-de1b378f-c3e8-458a-a62a-620a8009288b");katex.render("OM", element, {displayMode:false,throwOnError:false});$ = 2 corresponding to an atomic fraction of oxygen of $var element = document.getElementById("moose-equation-321d79ef-9d71-4d77-bb1a-d7d15484c411");katex.render("x", element, {displayMode:false,throwOnError:false});$ =0.667. Given this value the solidus temperature $var element = document.getElementById("moose-equation-5f18a548-80ed-4de6-ad6c-745c5a8713e5");katex.render("T_{sol}", element, {displayMode:false,throwOnError:false});$ is given by: $var element = document.getElementById("moose-equation-951e491d-24ce-4f2b-8b11-ca819bae76e9");katex.render("T_{sol} = -11572.454 + 21818.181x", element, {displayMode:true,throwOnError:false});$

Thus, using the known atomic fraction of oxygen the solidus temperature used in the Young's Modulus equations below is $var element = document.getElementById("moose-equation-2058498e-0335-4825-a04d-5bb1cc6eb12f");katex.render("T_{sol}", element, {displayMode:false,throwOnError:false});$ = 2980.27 K. The correlation for Young's Modulus is: $(1)var element = document.getElementById("moose-equation-9b8a862c-19ca-4c82-9d99-cc24596f4c3f");katex.render(" E = \\begin{cases} -3.77\\times 10^7 T + 1.637\\times 10^{11} \\text{ Pa} & \\text{for } 300 < T < 1478K \\text{ (monoclinic phase)} \\\\ -8.024\\times 10^7 T + 2.255\\times 10^{11} \\text{ Pa} & \\text{for } 1478 \\leq T < T_{sol} \\text{ (tetragonal and cubic phase)} \\\\ 1.0 & \\text{for } T_{sol} \\leq T \\end{cases}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-086e1c3a-5dd2-4c53-8170-63d349002fdd");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature in K.

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [elasticity_tensor]
    type = ZrO2ElasticityTensor<<<{"description": "Computes the Young's Modulus and Poisson's ratio for zirconium oxide.", "href": "ZrO2ElasticityTensor.html"}>>>
    temperature<<<{"description": "Coupled temperature"}>>> = temperature
  []
[]

Input Parameters

temperatureCoupled temperature
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled temperature

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.
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.

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/zro2_mechanics/elasticity.i)

References

C. M. Allison, G. A. Berna, R. Chambers, E. W. Coryell, K. L. Davis, D. L. Hagrman, D. T. Hagrman, N. L. Hampton, J. K. Hohorst, R. E. Mason, M. L. McComas, K. A. McNeil, R. L. Miller, C. S. Olsen, G. A. Reymann, and L. J. Siefken. SCDAP/RELAP5/MOD3.1 code manual, volume IV: MATPRO-A library of materials properties for light-water-reactor accident analysis. Technical Report NUREG/CR-6150, EGG-2720, Idaho National Engineering Laboratory, 1993.

@TECHREPORT{hagrman93,
    author = "Allison, C. M. and Berna, G. A. and Chambers, R. and Coryell, E. W. and Davis, K. L. and Hagrman, D. L. and Hagrman, D. T. and Hampton, N. L. and Hohorst, J. K. and Mason, R. E. and McComas, M. L. and McNeil, K. A. and Miller, R. L. and Olsen, C. S. and Reymann, G. A. and Siefken, L. J.",
    editor = "Hagrman, D. T.",
    title = "{SCDAP/RELAP5/MOD3.1} Code Manual, Volume {IV}: {MATPRO}-{A} Library of Materials Properties for Light-Water-Reactor Accident Analysis",
    institution = "Idaho National Engineering Laboratory",
    year = "1993",
    number = "NUREG/CR-6150, EGG-2720"
}

(test/tests/solid_mechanics/zro2_mechanics/elasticity.i)

# This test case is prepared to test the elastic moduli of zirconium oxide
# using the MATPRO correlation for temperatures ranging from 300 to 2500 K.
#
# The geometry represents a ring oxide with inner radius of 0.005 m, outer radius
# of 0.0055 m, and a height of 0.01 m.
#
#  An axial load is applied to the top of the ring with a value of -100 MPa.
#  The strain in the radial direction can be calculated using the Young's modulus
#  and the Poisson's ratio from the applied 100.0 MPa pressure.
#
#  epsilon_{rr} = nu * sigma_{axial} / E
#
#  The Young's Modulus (E) for zirconium oxide is calculated via:
#
#  For 300 < T < 1478 K:
#
#  E = -3.77e7 * T + 1.637e11
#
#  For 1478 <= T < 2980.27 K:
#
#  E = -8.024e7 * T + 2.255e11
#
#  For T > 2980.27 K:
#
#  E = 1.0
#
#  The Poisson's ratio of zirconium oxide is taken as 0.3.
#
#  The results of BISON compared to the analytical solution for an increase in
#  temperature under constant pressure are summarized:
#
#                          BISON             Analytical
#    Temperature (K)    strain_rr (m/m)    strain_rr (m/m)
#        520              2.0819e-4          2.0819e-4
#        740              2.2091e-4          2.2091e-4
#        960              2.3528e-4          2.3528e-4
#       1180              2.5165e-4          2.5165e-4
#       1400              2.7047e-4          2.7047e-4
#       1620              3.1410e-4          3.1410e-4
#       1840              3.8531e-4          3.8531e-4
#       2060              4.9829e-4          4.9829e-4
#       2280              7.0501e-4          7.0501e-4
#       2500              1.2048e-3          1.2048e-3

[Mesh]
  coord_type = RZ
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0.005
    xmax = 0.0055
    ymin = 0
    ymax = 0.01
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Variables]
  [temperature]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
[]

[Physics]
  [SolidMechanics]
    [QuasiStatic]
      [all]
        strain = FINITE
        block = 0
        add_variables = true
        generate_output = 'elastic_strain_xx'
      []
    []
  []
[]

[Functions]
  [temperature_function]
    type = PiecewiseLinear
    x = '0       10'
    y = '300   2500'
  []
  [pressure_function]
    type = PiecewiseLinear
    x = '0  10'
    y = '1   1'
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  [top_pressure] #Simplified BC to apply pressure in only disp_y
    type = Pressure
    boundary = top
    variable = disp_y
    function = pressure_function
    factor = 100e6
  []

  [u_bottom_fix]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  []

  [temp_bc]
    type = FunctionDirichletBC
    variable = temperature
    boundary = 'bottom top left right'
    function = temperature_function
  []
[]

[Materials]
  [elasticity_tensor]
    type = ZrO2ElasticityTensor
    temperature = temperature
  []
  [stress]
    type = ComputeFiniteStrainElasticStress
  []
  [clad_density]
    type = StrainAdjustedDensity
    strain_free_density = 5800.0
  []
  [thermal]
    type = ZrO2Thermal
    temperature = temperature
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  petsc_options       = '-snes_ksp_ew'
  petsc_options_iname = '-ksp_gmres_restart'
  petsc_options_value = '101'

  line_search = 'none'

  l_max_its  = 100
  nl_max_its = 100
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-8
  l_tol      = 1e-8
  start_time = 0.0
  end_time   = 10
  dt         = 1
[]

[Postprocessors]
  [elastic_strain_rr]
    type = ElementAverageValue
    variable = elastic_strain_xx
  []
  [temp]
    type = AverageNodalVariableValue
    variable = temperature
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/zro2_mechanics/elasticity.i)

# This test case is prepared to test the elastic moduli of zirconium oxide
# using the MATPRO correlation for temperatures ranging from 300 to 2500 K.
#
# The geometry represents a ring oxide with inner radius of 0.005 m, outer radius
# of 0.0055 m, and a height of 0.01 m.
#
#  An axial load is applied to the top of the ring with a value of -100 MPa.
#  The strain in the radial direction can be calculated using the Young's modulus
#  and the Poisson's ratio from the applied 100.0 MPa pressure.
#
#  epsilon_{rr} = nu * sigma_{axial} / E
#
#  The Young's Modulus (E) for zirconium oxide is calculated via:
#
#  For 300 < T < 1478 K:
#
#  E = -3.77e7 * T + 1.637e11
#
#  For 1478 <= T < 2980.27 K:
#
#  E = -8.024e7 * T + 2.255e11
#
#  For T > 2980.27 K:
#
#  E = 1.0
#
#  The Poisson's ratio of zirconium oxide is taken as 0.3.
#
#  The results of BISON compared to the analytical solution for an increase in
#  temperature under constant pressure are summarized:
#
#                          BISON             Analytical
#    Temperature (K)    strain_rr (m/m)    strain_rr (m/m)
#        520              2.0819e-4          2.0819e-4
#        740              2.2091e-4          2.2091e-4
#        960              2.3528e-4          2.3528e-4
#       1180              2.5165e-4          2.5165e-4
#       1400              2.7047e-4          2.7047e-4
#       1620              3.1410e-4          3.1410e-4
#       1840              3.8531e-4          3.8531e-4
#       2060              4.9829e-4          4.9829e-4
#       2280              7.0501e-4          7.0501e-4
#       2500              1.2048e-3          1.2048e-3

[Mesh]
  coord_type = RZ
  [mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0.005
    xmax = 0.0055
    ymin = 0
    ymax = 0.01
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Variables]
  [temperature]
    order = FIRST
    family = LAGRANGE
    initial_condition = 300.0
  []
[]

[Physics]
  [SolidMechanics]
    [QuasiStatic]
      [all]
        strain = FINITE
        block = 0
        add_variables = true
        generate_output = 'elastic_strain_xx'
      []
    []
  []
[]

[Functions]
  [temperature_function]
    type = PiecewiseLinear
    x = '0       10'
    y = '300   2500'
  []
  [pressure_function]
    type = PiecewiseLinear
    x = '0  10'
    y = '1   1'
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  [top_pressure] #Simplified BC to apply pressure in only disp_y
    type = Pressure
    boundary = top
    variable = disp_y
    function = pressure_function
    factor = 100e6
  []

  [u_bottom_fix]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  []

  [temp_bc]
    type = FunctionDirichletBC
    variable = temperature
    boundary = 'bottom top left right'
    function = temperature_function
  []
[]

[Materials]
  [elasticity_tensor]
    type = ZrO2ElasticityTensor
    temperature = temperature
  []
  [stress]
    type = ComputeFiniteStrainElasticStress
  []
  [clad_density]
    type = StrainAdjustedDensity
    strain_free_density = 5800.0
  []
  [thermal]
    type = ZrO2Thermal
    temperature = temperature
  []
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  petsc_options       = '-snes_ksp_ew'
  petsc_options_iname = '-ksp_gmres_restart'
  petsc_options_value = '101'

  line_search = 'none'

  l_max_its  = 100
  nl_max_its = 100
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-8
  l_tol      = 1e-8
  start_time = 0.0
  end_time   = 10
  dt         = 1
[]

[Postprocessors]
  [elastic_strain_rr]
    type = ElementAverageValue
    variable = elastic_strain_xx
  []
  [temp]
    type = AverageNodalVariableValue
    variable = temperature
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  exodus = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References