WB4ThermalExpansionEigenstrain

Computes an eigenstrain due to thermal epxansion for WB4 using a bilinear interpolation of temperature and hydrostatic stress providing an instantaneous coefficient of thermal expansion.

Description

The instantaneous thermal expansion data for tungsten tetraboride (WB4) was obtained from digitizing Figure 13b of Yang et al. (2022). Extreme lower values were not digitized, so temperatures below 200 K may not provide valid coefficients. The digitized values for instantaneous coefficients of thermal expansion are provided in Table 1.

Table 1: Digitized data from Figure 13b of Yang et al. (2022) providing instantaneous $var element = document.getElementById("moose-equation-a7c5d223-dfe0-4574-9e80-17e902713916");katex.render("\\alpha \\times 10^{-6}", element, {displayMode:false,throwOnError:false});$ (K $var element = document.getElementById("moose-equation-16e76988-0cec-45c9-bbf4-8c6527839e37");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ).

				Hydrostatic Stress (GPa)
T (K)	0	5	10	15	20	25	30	35
200	6.304	5.597	5.032	4.531	4.133	3.760	3.490	3.195
250	8.360	7.486	6.792	6.188	5.700	5.251	4.839	4.505
300	9.799	8.861	8.090	7.435	6.869	6.368	5.944	5.572
350	10.929	9.927	9.105	8.411	7.820	7.281	6.805	6.394
400	11.764	10.724	9.863	9.131	8.488	7.936	7.460	7.024
450	12.471	11.366	10.441	9.670	9.015	8.450	7.974	7.563
500	13.049	11.842	10.865	10.094	9.413	8.861	8.385	7.987
550	13.460	12.253	11.251	10.441	9.773	9.195	8.707	8.296
600	13.769	12.535	11.546	10.749	10.043	9.465	8.964	8.514
650	14.013	12.805	11.803	10.968	10.287	9.696	9.182	8.732
700	14.257	13.011	12.009	11.173	10.467	9.863	9.349	8.899
750	14.475	13.216	12.188	11.328	10.608	9.991	9.465	9.015
800	14.681	13.383	12.330	11.482	10.724	10.107	9.555	9.105

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [cte]
    type = ADWB4ThermalExpansionEigenstrain<<<{"description": "Computes an eigenstrain due to thermal epxansion for WB4 using a bilinear interpolation of temperature and hydrostatic stress providing an instantaneous coefficient of thermal expansion.", "href": "WB4ThermalExpansionEigenstrain.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."}>>> = thermal_eigenstrain
    stress_free_temperature<<<{"description": "Reference temperature at which there is no thermal expansion for thermal eigenstrain calculation"}>>> = 320.0
    temperature<<<{"description": "Coupled temperature"}>>> = temperature
    hydrostatic_stress<<<{"description": "Coupled hydrostatic stress"}>>> = hydrostatic_stress
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
    mean_thermal_expansion_coefficient_name<<<{"description": "Name of the mean coefficient of thermal expansion."}>>> = 'mean_cte'
  []
[]

The eigenstrain name must also be passed to the strain calculator created by the QuasiStatic Action, and an example parameter setting is shown below:

[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"}>>> = thermal_eigenstrain
        generate_output<<<{"description": "Add scalar quantity output for stress and/or strain"}>>> = hydrostatic_stress
        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.
hydrostatic_stressCoupled hydrostatic stress
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:Coupled hydrostatic stress
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
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

cte_scale_factor1Scale factor to be applied to the thermal expansion coefficient. Used for calibration and sensitivity studies.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Scale factor to be applied to the thermal expansion coefficient. Used for calibration and sensitivity studies.

Advanced: Scaling Factors 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/wb4_thermal_expansion/ad_coupled.i)
(test/tests/solid_mechanics/wb4_thermal_expansion/exact.i)

References

Ancang Yang, Yonghua Duan, and Mingjun Peng. Effects of temperature and pressure on the mechanical and thermodynamic properties of high-boride WB4 from first-principles predictions. Materials Today Communications, 30:103187, 2022. URL: https://www.sciencedirect.com/science/article/pii/S2352492822000642, doi:10.1016/j.mtcomm.2022.103187.

@article{YANG2022103187,
    author = "Yang, Ancang and Duan, Yonghua and Peng, Mingjun",
    title = "Effects of temperature and pressure on the mechanical and thermodynamic properties of high-boride {WB4} from first-principles predictions",
    journal = "Materials Today Communications",
    volume = "30",
    pages = "103187",
    year = "2022",
    issn = "2352-4928",
    doi = "10.1016/j.mtcomm.2022.103187",
    url = "https://www.sciencedirect.com/science/article/pii/S2352492822000642"
}

(test/tests/solid_mechanics/wb4_thermal_expansion/ad_coupled.i)

# Couples ADWB4ThermalExapnsionEigenstrain to temperature and pressure using Automatic Differentiation
[GlobalParams]
  displacements = 'disp_x'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 5
  []
[]

[Variables]
  [temperature]
    initial_condition = 320.0
  []
[]

[Kernels]
  [diffusion]
    type = ADHeatConduction
    variable = temperature
  []
  [dtime]
    type = ADHeatConductionTimeDerivative
    variable = temperature
  []
[]

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

[Functions]
  [pressure_change]
    type = PiecewiseLinear
    x = '0.0 200.0 800.0 1200.0 60000'
    y = '-35e9 -30e9 -5e9 5e9 -50e9'
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [Pressure]
    [push]
      boundary = right
      function = pressure_change
      use_automatic_differentiation = true
    []
  []
  [temp_left]
    type = ADDirichletBC
    variable = temperature
    value = 500
    boundary = left
  []
  [right_temp]
    type = ADDirichletBC
    variable = temperature
    value = 500
    boundary = right
  []
[]

[Materials]
  [elasticity_tensor] # Not real values
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 300.0e9
    poissons_ratio = 0.3
  []
  [cte]
    type = ADWB4ThermalExpansionEigenstrain
    eigenstrain_name = thermal_eigenstrain
    stress_free_temperature = 320.0
    temperature = temperature
    hydrostatic_stress = hydrostatic_stress
    outputs = all
    mean_thermal_expansion_coefficient_name = 'mean_cte'
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
  [therm]
    type = ADGenericConstantMaterial # Fake numbers
    prop_names = 'density thermal_conductivity specific_heat'
    prop_values = '3000 120 1600'
  []
[]

[Postprocessors]
  [avg_T]
    type = ElementAverageValue
    variable = temperature
  []
  [disp_x_right]
    type = SideAverageValue
    boundary = right
    variable = disp_x
  []
  [avg_hydrostatic]
    type = ElementAverageValue
    variable = hydrostatic_stress
    execute_on = 'initial timestep_end'
  []
  [avg_cte]
    type = ElementAverageMaterialProperty
    mat_prop = 'mean_cte'
    execute_on = 'timestep_end'
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  num_steps = 10
  dt = 600
  nl_abs_tol = 1e-19
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/wb4_thermal_expansion/ad_coupled.i)

# Couples ADWB4ThermalExapnsionEigenstrain to temperature and pressure using Automatic Differentiation
[GlobalParams]
  displacements = 'disp_x'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 5
  []
[]

[Variables]
  [temperature]
    initial_condition = 320.0
  []
[]

[Kernels]
  [diffusion]
    type = ADHeatConduction
    variable = temperature
  []
  [dtime]
    type = ADHeatConductionTimeDerivative
    variable = temperature
  []
[]

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

[Functions]
  [pressure_change]
    type = PiecewiseLinear
    x = '0.0 200.0 800.0 1200.0 60000'
    y = '-35e9 -30e9 -5e9 5e9 -50e9'
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [Pressure]
    [push]
      boundary = right
      function = pressure_change
      use_automatic_differentiation = true
    []
  []
  [temp_left]
    type = ADDirichletBC
    variable = temperature
    value = 500
    boundary = left
  []
  [right_temp]
    type = ADDirichletBC
    variable = temperature
    value = 500
    boundary = right
  []
[]

[Materials]
  [elasticity_tensor] # Not real values
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 300.0e9
    poissons_ratio = 0.3
  []
  [cte]
    type = ADWB4ThermalExpansionEigenstrain
    eigenstrain_name = thermal_eigenstrain
    stress_free_temperature = 320.0
    temperature = temperature
    hydrostatic_stress = hydrostatic_stress
    outputs = all
    mean_thermal_expansion_coefficient_name = 'mean_cte'
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
  [therm]
    type = ADGenericConstantMaterial # Fake numbers
    prop_names = 'density thermal_conductivity specific_heat'
    prop_values = '3000 120 1600'
  []
[]

[Postprocessors]
  [avg_T]
    type = ElementAverageValue
    variable = temperature
  []
  [disp_x_right]
    type = SideAverageValue
    boundary = right
    variable = disp_x
  []
  [avg_hydrostatic]
    type = ElementAverageValue
    variable = hydrostatic_stress
    execute_on = 'initial timestep_end'
  []
  [avg_cte]
    type = ElementAverageMaterialProperty
    mat_prop = 'mean_cte'
    execute_on = 'timestep_end'
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  num_steps = 10
  dt = 600
  nl_abs_tol = 1e-19
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/wb4_thermal_expansion/ad_coupled.i)

# Couples ADWB4ThermalExapnsionEigenstrain to temperature and pressure using Automatic Differentiation
[GlobalParams]
  displacements = 'disp_x'
[]

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 5
  []
[]

[Variables]
  [temperature]
    initial_condition = 320.0
  []
[]

[Kernels]
  [diffusion]
    type = ADHeatConduction
    variable = temperature
  []
  [dtime]
    type = ADHeatConductionTimeDerivative
    variable = temperature
  []
[]

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

[Functions]
  [pressure_change]
    type = PiecewiseLinear
    x = '0.0 200.0 800.0 1200.0 60000'
    y = '-35e9 -30e9 -5e9 5e9 -50e9'
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = disp_x
    value = 0
    boundary = left
  []
  [Pressure]
    [push]
      boundary = right
      function = pressure_change
      use_automatic_differentiation = true
    []
  []
  [temp_left]
    type = ADDirichletBC
    variable = temperature
    value = 500
    boundary = left
  []
  [right_temp]
    type = ADDirichletBC
    variable = temperature
    value = 500
    boundary = right
  []
[]

[Materials]
  [elasticity_tensor] # Not real values
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 300.0e9
    poissons_ratio = 0.3
  []
  [cte]
    type = ADWB4ThermalExpansionEigenstrain
    eigenstrain_name = thermal_eigenstrain
    stress_free_temperature = 320.0
    temperature = temperature
    hydrostatic_stress = hydrostatic_stress
    outputs = all
    mean_thermal_expansion_coefficient_name = 'mean_cte'
  []
  [stress]
    type = ADComputeLinearElasticStress
  []
  [therm]
    type = ADGenericConstantMaterial # Fake numbers
    prop_names = 'density thermal_conductivity specific_heat'
    prop_values = '3000 120 1600'
  []
[]

[Postprocessors]
  [avg_T]
    type = ElementAverageValue
    variable = temperature
  []
  [disp_x_right]
    type = SideAverageValue
    boundary = right
    variable = disp_x
  []
  [avg_hydrostatic]
    type = ElementAverageValue
    variable = hydrostatic_stress
    execute_on = 'initial timestep_end'
  []
  [avg_cte]
    type = ElementAverageMaterialProperty
    mat_prop = 'mean_cte'
    execute_on = 'timestep_end'
  []
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  num_steps = 10
  dt = 600
  nl_abs_tol = 1e-19
[]

[Outputs]
  exodus = true
[]

(test/tests/solid_mechanics/wb4_thermal_expansion/exact.i)

# Compares B4CThermalExapnsionEigenstrain to exact calculations of thermal expansion coefficient.

# Hand calculations integrating the instantaneous CTE over the change in temperature match BISON's
# average mean CTE over the temperature change. This is most easily shown when temperature change is
# close to the stress_free_temperature.
# For stress_free_temperature = 300.0 K:
# | Hydrostatic Stress (GPa) | Mean CTE x 10^-6|
# |         0                |   9.8           |
# |        -5                |   8.9           |
# |       -10                |   8.1           |
# |       -15                |   7.4           |
# |       -20                |   6.9           |
# |       -25                |   6.4           |
# |       -30                |   5.9           |
# |       -35                |   5.6           |
# BISON will get close to these number when changing the temperature from 299.5 to 300.5 K at constant
# hydrostatic stress.
# This test can be modified to show that, but currently is testing the whole range with varying temperatures
# and pressures.

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

[Mesh]
  use_displaced_mesh = false
  [mesh]
    type = ExamplePatchMeshGenerator
    dim = 3
    x_length = 1
    y_length = 1
    z_length = 1
  []
[]

[AuxVariables]
  [temperature]
    initial_condition = 300.0
  []
  [hydro]
    initial_condition = -35e9
  []
[]

[Functions]
  [temp_ramp]
    type = PiecewiseLinear
    x = '0.0 20.0 80.0 100.0'
    y = '300.0 350 750 900'
  []
  [pressing]
    type = PiecewiseLinear
    x = '0.0 20.0 80.0 100.0'
    y = '-35e9 -30e9 -5e9 5e9'
  []
[]

[AuxKernels]
  [temp_aux]
    type = FunctionAux
    variable = temperature
    function = temp_ramp
  []
  [hydro_aux]
    type = FunctionAux
    variable = hydro
    function = pressing
  []
[]

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

[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
  []
[]

[Materials]
  [elasticity_tensor] # Not real values
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 300.0e9
    poissons_ratio = 0.3
  []
  [cte]
    type = WB4ThermalExpansionEigenstrain
    eigenstrain_name = thermal_eigenstrain
    stress_free_temperature = 299.5
    temperature = temperature
    hydrostatic_stress = hydro
    outputs = all
    mean_thermal_expansion_coefficient_name = 'mean_cte'
    cte_scale_factor = 2.54
  []
  [stress]
    type = ComputeLinearElasticStress
  []
[]

[Postprocessors]
  [avg_T]
    type = ElementAverageValue
    variable = temperature
    execute_on = 'initial timestep_end'
  []
  [avg_hydrostatic]
    type = ElementAverageValue
    variable = hydro
    execute_on = 'initial timestep_end'
  []
  [disp_x_right]
    type = SideAverageValue
    boundary = right
    variable = disp_x
    execute_on = 'initial timestep_end'
  []
  [avg_cte]
    type = ElementAverageMaterialProperty
    mat_prop = 'mean_cte'
    execute_on = 'timestep_end'
  []
[]

[Executioner]
  type = Transient
  num_steps = 10
  dt = 10
  nl_abs_tol = 1e-19
[]

[Outputs]
  csv = true
[]

Description
Example Input Syntax
Input Parameters
Input Files
References