- stress_free_temperatureReference temperature at which there is no thermal expansion for thermal eigenstrain calculation
C++ Type:std::vector

Description:Reference temperature at which there is no thermal expansion for thermal eigenstrain calculation

- thermal_expansion_function_reference_temperatureReference temperature for thermal_exansion_function (IMPORTANT: this is different in general from the stress_free_temperature)
C++ Type:double

Description:Reference temperature for thermal_exansion_function (IMPORTANT: this is different in general from the stress_free_temperature)

- thermal_expansion_functionFunction describing the mean thermal expansion as a function of temperature
C++ Type:FunctionName

Description:Function describing the mean thermal expansion as a function of temperature

- 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

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.

# Compute Mean Thermal Expansion Function Eigenstrain

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 as a function of temperature, . This function is defined relative to a reference temperature, , such that the total expansion at a given temperature relative to the refererence temperature is . Following the notation of Niffenegger and Reichlin (2012), is defined as:

(1) where is the length of a body at the current temperature, and 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, , differs from the reference temperature, the total thermal expansion eigenstrain is computed as:

(2) where is the current temperature and is the identity matrix. Note that the denominator in this equation is a correction to account for the ratio of to . 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
[../]
```

(modules/tensor_mechanics/test/tests/thermal_expansion_function/thermal_expansion_function_finite_const_alpha_test.i)The `eigenstrain_name`

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

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

(modules/tensor_mechanics/test/tests/thermal_expansion_function/thermal_expansion_function_finite_const_alpha_test.i)## Input Parameters

- computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.
Default:True

C++ Type:bool

Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

- temperatureCoupled temperature
C++ Type:std::vector

Description:Coupled temperature

- 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

Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases

- boundaryThe list of boundary IDs from the mesh where this boundary condition applies
C++ Type:std::vector

Description:The list of boundary IDs from the mesh where this boundary condition applies

- blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector

Description:The list of block ids (SubdomainID) that this object will be applied

### Optional Parameters

- output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector

Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

- outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object
Default:none

C++ Type:std::vector

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

### Outputs Parameters

- enableTrueSet the enabled status of the MooseObject.
Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

- 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

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.

- control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector

Description:Adds user-defined labels for accessing object parameters via control logic.

- seed0The seed for the master random number generator
Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

- implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

- 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 computeSubdomainProperties() 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

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 computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

### Advanced Parameters

## Input Files

- modules/tensor_mechanics/test/tests/thermal_expansion_function/thermal_expansion_function_small_const_alpha_test.i
- modules/tensor_mechanics/test/tests/thermal_expansion_function/thermal_expansion_function_finite_const_alpha_test.i
- modules/tensor_mechanics/test/tests/domain_integral_thermal/j_integral_2d_mean_ctefunc.i
- modules/tensor_mechanics/test/tests/thermal_expansion_function/thermal_expansion_function_small_linear_alpha_test.i
- modules/tensor_mechanics/test/tests/thermal_expansion_function/thermal_expansion_function_finite_linear_alpha_test.i

## 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.[BibTeX]