ComputeMeanThermalExpansionFunctionEigenstrain

Computes eigenstrain due to thermal expansion using a function that describes the mean thermal expansion as a function of temperature

Description

This model computes the eigenstrain tensor resulting from isotropic thermal expansion where the temperature-dependent thermal expansion is defined by a user-supplied function that describes the mean thermal expansion coefficient $var element = document.getElementById("moose-equation-88709794-443c-4f0a-8b31-e5c5333e55e5");katex.render("\\bar{\\alpha}", element, {displayMode:false,throwOnError:false});$ as a function of temperature, $var element = document.getElementById("moose-equation-536855c1-4abc-418e-aebc-0e496f5a1a75");katex.render("T", element, {displayMode:false,throwOnError:false});$ . This function is defined relative to a reference temperature, $var element = document.getElementById("moose-equation-c5146f81-8560-471f-8225-ddfb54ac3711");katex.render("T_{ref}", element, {displayMode:false,throwOnError:false});$ , such that the total expansion at a given temperature relative to the refererence temperature is $var element = document.getElementById("moose-equation-a10fbcda-e87f-4285-a33d-063dd49f95aa");katex.render("\\bar{\\alpha}(T-T_{ref})", element, {displayMode:false,throwOnError:false});$ . Following the notation of Niffenegger and Reichlin (2012), $var element = document.getElementById("moose-equation-50bc11b9-53b5-4e11-86a6-3703d3b7beb8");katex.render("\\bar{\\alpha}_{(T_{ref},T)}", element, {displayMode:false,throwOnError:false});$ is defined as:

$var element = document.getElementById("moose-equation-dbbc4593-664c-4af1-9a14-45e7b9d82628");katex.render("\\bar{\\alpha}_{(T_{ref},T)} = \\frac{L_{(T)} - L_{(T_{ref})}}{L_{(T_{ref})}(T-T_{ref})}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-251581af-02f9-4dc1-bc43-31f0ff18e625");katex.render("L_{T}", element, {displayMode:false,throwOnError:false});$ is the length of a body at the current temperature, and $var element = document.getElementById("moose-equation-173c3259-7769-428a-9dbf-b6ff38a655ef");katex.render("L_{T_{ref}}", element, {displayMode:false,throwOnError:false});$ is the length of that body at the reference temperature.

It is important to emphasize that this reference temperature is tied to the definition of the thermal expansion function, and differs in general from the stress-free temperature for a specific simulation. For the general case where the stress-free temperature, $var element = document.getElementById("moose-equation-9697d1b7-dd7a-4ea5-afab-432bef1ffd47");katex.render("T_{sf}", element, {displayMode:false,throwOnError:false});$ , differs from the reference temperature, the total thermal expansion eigenstrain is computed as:

$var element = document.getElementById("moose-equation-837cd2cc-db8b-4ba4-b36f-e2e6f555d0ff");katex.render("\\boldsymbol{\\epsilon}^{th} = \\frac{\\bar{\\alpha}_{(T_{ref},T)}(T-T_{ref}) - \\bar{\\alpha}_{(T_{ref},T_{sf})}(T_{sf}-T_{ref})} {1 + \\bar{\\alpha}_{(T_{ref},T_{sf})}(T_{sf}-T_{ref})} \\cdot \\boldsymbol{I}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-24e0150f-2143-443d-80db-84d9fdc6719e");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the current temperature and $var element = document.getElementById("moose-equation-56ea55d9-99e5-4176-b404-da16d40c2698");katex.render("\\boldsymbol{I}", element, {displayMode:false,throwOnError:false});$ is the identity matrix. Note that the denominator in this equation is a correction to account for the ratio of $var element = document.getElementById("moose-equation-1d4c42c1-01c8-4093-b1b7-1e88bbaa8ca1");katex.render("L_{(T_{sf})}", element, {displayMode:false,throwOnError:false});$ to $var element = document.getElementById("moose-equation-0af8e5c0-32cf-4b57-8015-838e3365a4e2");katex.render("L_{(T_{ref})}", element, {displayMode:false,throwOnError:false});$ . As discussed in Niffenegger and Reichlin (2012), that ratio is very close to 1, so it is not strictly necessary to include that correction, but it is done here for completeness.

Example Input File Syntax

[./thermal_expansion_strain1]
  type = ComputeMeanThermalExpansionFunctionEigenstrain
  block = 1
  thermal_expansion_function = cte_func_mean
  thermal_expansion_function_reference_temperature = 0.5
  stress_free_temperature = 0.0
  temperature = temp
  eigenstrain_name = eigenstrain
[../]

The eigenstrain_name parameter value must also be set for the strain calculator, and an example parameter setting is shown below:

[Modules/TensorMechanics/Master]
  [Modules/TensorMechanics/Master]
    [Modules/TensorMechanics/Master]
      [./all]
        strain = FINITE
        add_variables = true
        eigenstrain_names = eigenstrain
        generate_output = 'strain_xx strain_yy strain_zz'
      [../]
    []
  []
[]

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>
Controllable:No
Description:Reference temperature at which there is no thermal expansion for thermal eigenstrain calculation
thermal_expansion_functionFunction describing the mean thermal expansion as a function of temperature
C++ Type:FunctionName
Controllable:No
Description:Function describing the mean thermal expansion as a function of temperature
thermal_expansion_function_reference_temperatureReference temperature for thermal_exansion_function (IMPORTANT: this is different in general from the stress_free_temperature)
C++ Type:double
Controllable:No
Description:Reference temperature for thermal_exansion_function (IMPORTANT: this is different in general from the stress_free_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
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
Controllable:No
Description:Name of the mean coefficient of thermal expansion.
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
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.
temperatureCoupled temperature
C++ Type:std::vector<VariableName>
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
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

References

Markus Niffenegger and Klaus Reichlin. The proper use of thermal expansion coefficients in finite element calculations. Nuclear Engineering and Design, 243:356–359, February 2012. doi:10.1016/j.nucengdes.2011.12.006.

@article{niffenegger2012proper,
    author = "Niffenegger, Markus and Reichlin, Klaus",
    doi = "10.1016/j.nucengdes.2011.12.006",
    journal = "Nuclear Engineering and Design",
    month = "February",
    pages = "356--359",
    title = "The proper use of thermal expansion coefficients in finite element calculations",
    volume = "243",
    year = "2012"
}

(moose/modules/tensor_mechanics/test/tests/thermal_expansion_function/finite_const.i)

# This tests the thermal expansion coefficient function using both
# options to specify that function: mean and instantaneous.  There
# two blocks, each containing a single element, and these use the
# two variants of the function.

# In this test, the instantaneous CTE function has a constant value,
# while the mean CTE function is an analytic function designed to
# give the same response.  If \bar{alpha}(T) is the mean CTE function,
# and \alpha(T) is the instantaneous CTE function,

# \bar{\alpha}(T) = 1/(T-Tref) \intA^{T}_{Tsf} \alpha(T) dT

# where Tref is the reference temperature used to define the mean CTE
# function, and Tsf is the stress-free temperature.

# This version of the test uses finite deformation theory.
# The two models produce very similar results.  There are slight
# differences due to the large deformation treatment.

[Mesh]
  file = 'blocks.e'
[]

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

[AuxVariables]
  [./temp]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Modules/TensorMechanics/Master]
  [./all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = eigenstrain
    generate_output = 'strain_xx strain_yy strain_zz'
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = disp_x
    boundary = 3
    value = 0.0
  [../]

  [./bottom]
    type = DirichletBC
    variable = disp_y
    boundary = 2
    value = 0.0
  [../]

  [./back]
    type = DirichletBC
    variable = disp_z
    boundary = 1
    value = 0.0
  [../]
[]

[AuxKernels]
  [./temp]
    type = FunctionAux
    variable = temp
    block = '1 2'
    function = temp_func
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1e6
    poissons_ratio = 0.3
  [../]
  [./small_stress]
    type = ComputeFiniteStrainElasticStress
  [../]
  [./thermal_expansion_strain1]
    type = ComputeMeanThermalExpansionFunctionEigenstrain
    block = 1
    thermal_expansion_function = cte_func_mean
    thermal_expansion_function_reference_temperature = 0.5
    stress_free_temperature = 0.0
    temperature = temp
    eigenstrain_name = eigenstrain
  [../]
  [./thermal_expansion_strain2]
    type = ComputeInstantaneousThermalExpansionFunctionEigenstrain
    block = 2
    thermal_expansion_function = cte_func_inst
    stress_free_temperature = 0.0
    temperature = temp
    eigenstrain_name = eigenstrain
  [../]
[]

[Functions]
  [./cte_func_mean]
    type = ParsedFunction
    symbol_names = 'tsf tref scale' #stress free temp, reference temp, scale factor
    symbol_values = '0.0 0.5  1e-4'
    expression = 'scale * (t - tsf) / (t - tref)'
  [../]
  [./cte_func_inst]
    type = PiecewiseLinear
    xy_data = '0 1.0
               2 1.0'
    scale_factor = 1e-4
  [../]

  [./temp_func]
    type = PiecewiseLinear
    xy_data = '0 1
               1 2'
  [../]
[]

[Postprocessors]
  [./disp_1]
    type = NodalExtremeValue
    variable = disp_x
    boundary = 101
  [../]

  [./disp_2]
    type = NodalExtremeValue
    variable = disp_x
    boundary = 102
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  l_max_its = 100
  l_tol = 1e-4
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-12

  start_time = 0.0
  end_time = 1.0
  dt = 0.1
[]

[Outputs]
  csv = true
[]

(moose/modules/tensor_mechanics/test/tests/thermal_expansion_function/finite_const.i)

# This tests the thermal expansion coefficient function using both
# options to specify that function: mean and instantaneous.  There
# two blocks, each containing a single element, and these use the
# two variants of the function.

# In this test, the instantaneous CTE function has a constant value,
# while the mean CTE function is an analytic function designed to
# give the same response.  If \bar{alpha}(T) is the mean CTE function,
# and \alpha(T) is the instantaneous CTE function,

# \bar{\alpha}(T) = 1/(T-Tref) \intA^{T}_{Tsf} \alpha(T) dT

# where Tref is the reference temperature used to define the mean CTE
# function, and Tsf is the stress-free temperature.

# This version of the test uses finite deformation theory.
# The two models produce very similar results.  There are slight
# differences due to the large deformation treatment.

[Mesh]
  file = 'blocks.e'
[]

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

[AuxVariables]
  [./temp]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Modules/TensorMechanics/Master]
  [./all]
    strain = FINITE
    add_variables = true
    eigenstrain_names = eigenstrain
    generate_output = 'strain_xx strain_yy strain_zz'
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = disp_x
    boundary = 3
    value = 0.0
  [../]

  [./bottom]
    type = DirichletBC
    variable = disp_y
    boundary = 2
    value = 0.0
  [../]

  [./back]
    type = DirichletBC
    variable = disp_z
    boundary = 1
    value = 0.0
  [../]
[]

[AuxKernels]
  [./temp]
    type = FunctionAux
    variable = temp
    block = '1 2'
    function = temp_func
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1e6
    poissons_ratio = 0.3
  [../]
  [./small_stress]
    type = ComputeFiniteStrainElasticStress
  [../]
  [./thermal_expansion_strain1]
    type = ComputeMeanThermalExpansionFunctionEigenstrain
    block = 1
    thermal_expansion_function = cte_func_mean
    thermal_expansion_function_reference_temperature = 0.5
    stress_free_temperature = 0.0
    temperature = temp
    eigenstrain_name = eigenstrain
  [../]
  [./thermal_expansion_strain2]
    type = ComputeInstantaneousThermalExpansionFunctionEigenstrain
    block = 2
    thermal_expansion_function = cte_func_inst
    stress_free_temperature = 0.0
    temperature = temp
    eigenstrain_name = eigenstrain
  [../]
[]

[Functions]
  [./cte_func_mean]
    type = ParsedFunction
    symbol_names = 'tsf tref scale' #stress free temp, reference temp, scale factor
    symbol_values = '0.0 0.5  1e-4'
    expression = 'scale * (t - tsf) / (t - tref)'
  [../]
  [./cte_func_inst]
    type = PiecewiseLinear
    xy_data = '0 1.0
               2 1.0'
    scale_factor = 1e-4
  [../]

  [./temp_func]
    type = PiecewiseLinear
    xy_data = '0 1
               1 2'
  [../]
[]

[Postprocessors]
  [./disp_1]
    type = NodalExtremeValue
    variable = disp_x
    boundary = 101
  [../]

  [./disp_2]
    type = NodalExtremeValue
    variable = disp_x
    boundary = 102
  [../]
[]

[Executioner]
  type = Transient

  solve_type = PJFNK
  l_max_its = 100
  l_tol = 1e-4
  nl_abs_tol = 1e-8
  nl_rel_tol = 1e-12

  start_time = 0.0
  end_time = 1.0
  dt = 0.1
[]

[Outputs]
  csv = true
[]