# Uvar element = document.getElementById("moose-equation-1581047d-47f4-458b-9c3e-b5f1b41f09c7");katex.render("_3", element, {displayMode:false,throwOnError:false});Sivar element = document.getElementById("moose-equation-efd9163f-6058-4536-8832-7b7f10a85941");katex.render("_2", element, {displayMode:false,throwOnError:false}); Volumetric Swelling Eigenstrain

Calculates and sums the change in fuel pellet volume due to densification and fission product release. This class applies a volumetric strain correction before adding the strain from this class to the diagonal entries of the eigenstrain tensor.

## Description

Swelling due to solid fission products, gaseous fission products, and densification all contribute to the change in volume of a fuel pellet. The contributions from all three of these components are modeled in U3Si2VolumetricSwellingEigenstrain.

## Densification of the Fuel

USi is expected to experience densification similar to UO. Thus, the fuel densification is computed using the ESCORE empirical model (Rashid et al., 2004) given by: (1) where is the densification strain, is the total densification that can occur (given as a fraction of theoretical density), is the burnup, and is the burnup at which densification is complete. The parameter is dependent on temperature: (2) Note that in Eq. 2 the temperature variable, , is given in Celcius.

## Fission Product Swelling

### Empirical Finlay Model

Since the data for USi is limited, an empirical expression for the swelling of USi was determined using data from Figure 3 of M. R. Finlay (2004). The swelling of fuel particles was calculated by Finlay using the results of miniplate irradiation tests. To convert Finlay's data (fission density) to FIMA, a value of 10.735 g/cm was used as the heavy metal density, equivalent to 95% theoretical heavy metal density. Based on Finlay's data the volumetric strain can be written as a function of burnup: (3) where is the volumetric strain at a given burnup Bu. The burnup is in units of FIMA. The quadratic equation for the total volumetric strain is then decoupled into its solid and gaseous components. The solid swelling is a linear function of burnup based upon the data of Hofman and Ryu (1989) using the same conversion procedure from fission density to burnup given above: (4) which results in a gaseous swelling contribution given by the following quadratic function of burnup: (5)

### Uvar element = document.getElementById("moose-equation-9f5f1009-dc9c-4794-ae2b-efde1f7479e6");katex.render("_3", element, {displayMode:false,throwOnError:false});Sivar element = document.getElementById("moose-equation-d05a6e3c-af7a-431a-83dc-4813c2fac355");katex.render("_2", element, {displayMode:false,throwOnError:false}); Coupled Fission Gas Release and Swelling

In addition to the empirical correlation for gaseous swelling described above the gaseous swelling component for USi can be calculated by coupling to U3Si2FissionGas. The theoretical bases of the volumetric strain due to gaseous fission product for this model is found on the U3Si2FissionGas page.

### Argonne Model

A third model for gaseous swelling utilizes U3Si2TricubicInterpolationUserObject to calculate the total gaseous swelling.

## Example Input Syntax


[./fuel_swelling]
type = U3Si2VolumetricSwellingEigenstrain
temperature = T
burnup = burnup
complete_burnup = 10
eigenstrain_name = volumetric_swelling
gaseous_swelling_type = U3SI2FG
save_densification = true
save_solid_swelling = true
[../]
(test/tests/tensor_mechanics/u3si2_eigenstrains/u3si2_vswelling/VSwellingU3Si2_mech_test_tm.i)

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


[Modules]
[./TensorMechanics]
[./Master]
[./all]
strain = FINITE
eigenstrain_names = 'fuelthermal_strain volumetric_swelling'
[../]
[../]
[../]
[]
(test/tests/tensor_mechanics/u3si2_eigenstrains/u3si2_vswelling/VSwellingU3Si2_mech_test_tm.i)

## 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

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.

• densityInitial fuel density

C++ Type:double

Description:Initial fuel density

### Required Parameters

• gaseous_swelling_objectName of the UserObject that is used to calculate the gaseous swelling. Must only be supplied when using the Argonne gaseous swelling model.

C++ Type:UserObjectName

Description:Name of the UserObject that is used to calculate the gaseous swelling. Must only be supplied when using the Argonne gaseous swelling model.

• originOrigin of cylinder axis of rotation for 2D and 3D Cartesian models that use the Argonne gaseous swelling model.

C++ Type:libMesh::VectorValue

Description:Origin of cylinder axis of rotation for 2D and 3D Cartesian models that use the Argonne gaseous swelling model.

• 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 in Kelvin

C++ Type:std::vector

Description:Coupled temperature in Kelvin

• 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

• include_solid_swellingTrueShould the calculation of volumetric swelling include swelling due to solid fission products

Default:True

C++ Type:bool

Description:Should the calculation of volumetric swelling include swelling due to solid fission products

• save_solid_swellingFalseShould the solid swelling be saved in a material property

Default:False

C++ Type:bool

Description:Should the solid swelling be saved in a material property

• include_gaseous_swellingTrueShould the calculation of volumetric swelling include swelling due to gaseous fission products

Default:True

C++ Type:bool

Description:Should the calculation of volumetric swelling include swelling due to gaseous fission products

• complete_burnup5The burnup at which densification is complete input in units of MWd/kgU

Default:5

C++ Type:double

Description:The burnup at which densification is complete input in units of MWd/kgU

• axis_vectorVector defining direction of cylindrical axis (3D Cartesian models) that use the Argonne gaseous swelling model.

C++ Type:libMesh::VectorValue

Description:Vector defining direction of cylindrical axis (3D Cartesian models) that use the Argonne gaseous swelling model.

• initial_porosity0.05Initial fuel porosity (/)

Default:0.05

C++ Type:double

Description:Initial fuel porosity (/)

• total_densification0.01The densification that will occur given as a fraction of theoretical density

Default:0.01

C++ Type:double

Description:The densification that will occur given as a fraction of theoretical density

• burnupCoupled Burnup

C++ Type:std::vector

Description:Coupled Burnup

• gaseous_swelling_typeFINLAYUse gaseous swelling from FINLAY, ARGONNE, or U3SI2FG

Default:FINLAY

C++ Type:MooseEnum

Description:Use gaseous swelling from FINLAY, ARGONNE, or U3SI2FG

• include_densificationTrueShould the calculation of volumetric swelling include volumetric changes due to densification

Default:True

C++ Type:bool

Description:Should the calculation of volumetric swelling include volumetric changes due to densification

• save_densificationFalseShould the densification be saved in a material property

Default:False

C++ Type:bool

Description:Should the densification be saved in a material property

• 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

• burnup_functionBurnup function

C++ Type:FunctionName

Description:Burnup function

• 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

• 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

• solid_swelling_scaling_factor1Scaling factor to be applied to the solid swelling strain. Used for sensitivity and calibration studies

Default:1

C++ Type:double

Description:Scaling factor to be applied to the solid swelling strain. Used for sensitivity and calibration studies

• gaseous_swelling_scaling_factor1Scaling factor to be applied to the gaseous swelling strain. Used for sensitivity and calibration studies

Default:1

C++ Type:double

Description:Scaling factor to be applied to the gaseous swelling strain. Used for sensitivity and calibration studies

• 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

## References

1. G. L. Hofman and W. S. Ryu. Detailed Analysis of Uranium Silicide Dispersion Fuel Swelling. Technical Report CONF-8909141â€“10, Argonne National Laboratory, 1989.[BibTeX]
2. J.L Snelgrove M. R. Finlay, G. L. Hofman. Irradiation behaviour of uranium silicide compounds. Journal of Nuclear Materials, 325:118–128, 2004.[BibTeX]
3. Y Rashid, R Dunham, and R Montgomery. Fuel Analysis and Licensing Code: FALCON MOD01. Technical Report, Electric Power Research Institute, December 2004.[BibTeX]