UPuZrGaseousSwelling

Description

UPuZrGaseousSwelling computes a swelling increment to account for swelling arising from the presence of gaseous fission products in U-Pu-Zr systems. It is designed to be used in conjunction with a low-temperature swelling model such as in UPuZrLowTemperatureSwelling as well as with a joiner class, UPuZrPorosityEigenstrain, that computes the swelling, the porosity, and volumetric eigenstrain from the swelling increments. The joiner class is needed to set an overall porosity threshold incorporating multiple swelling mechanisms, above which fission gas venting occurs. The derivation for the gaseous swelling model for U-Pu-Zr used here was originally presented in Olander (1976). The model is able to simplify the gaseous swelling into an analytical solution by making the following assumptions:

• Fission gas diffusivity is infinite so that all fission gas is born in the bubbles;

• The number density of bubbles is constant and their size perfectly spherical;

• The gas within the bubbles can be treated as an ideal gas;

• The gas/vacancy ratio in the bubbles is in equilibrium, i.e. the bubbles are not over- or under-pressurized;

• Re-solution of fission gas atoms from the bubbles is neglected;

• The solid is assumed to be un-stressed.

Following these assumptions, the number of gas atoms in each bubble at some time can be calculated directly from the total number of fissions, (1) where is the yield of gas atoms per fission, is the number density of bubbles (bub/m), is a fixed constant representing the amount of fission gas retained in the solid, and is the average number of fissions between and , where is the fission rate density.

Also included in Eq. (1) is the impact of interconnectivity of the bubbles on the total number of gas atoms per bubble. In the absence of a mechanistically informed interconnection model, the total bubble interconnection can be estimated from the porosity by, (2) where and is the interconnection initiating and terminating porosity respectively, and is the bulk porosity at time . and can be set via the input parameters interconnection_initiating_porosity and interconnection_terminating_porosity. Eq. (2) ensures a linear transition of interconnection from 0 to 1.0, capturing the smooth transition from non-interconnected to fully interconnected porosity.

The bubble radius can be simply calculated as a function of the total number of fissions, where is the Boltzmann constant, is the temperature (K), is the surface tension of the fuel (J/m) (Karahan, 2009), is the gaseous fission product yield, is the fission rate density, is time, and is the number density of bubbles (bubbles/m).

Given the radius, the total swelling can be calculated as, (3)

The swelling increment, , is calculated as where is the gaseous swelling from the previous time step.

Several checks are performed on the value calculated by Eq. (3), ensuring that swelling due to gas bubbles only increases.

Example Input Syntax

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [gas_swelling]
    type = UPuZrGaseousSwelling<<<{"description": "Computes a swelling increment due to gas swelling in UPuZr.", "href": "UPuZrGaseousSwelling.html"}>>>
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
    temperature<<<{"description": "Coupled temperature variable"}>>> = temperature
    bubble_number_density<<<{"description": "Material property name for the number density of intragranular bubbles, [bubbles/m^3]"}>>> = 1e20
  []
[]

UPuZrGaseousSwelling must be used in conjunction with the low-temperature swelling class:

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [lowT_swelling]
    type = UPuZrLowTemperatureSwelling<<<{"description": "Computes swelling increment due to low-temperature swelling in UPuZr.", "href": "UPuZrLowTemperatureSwelling.html"}>>>
    temperature<<<{"description": "Coupled temperature variable"}>>> = temperature
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

and in conjunction with the joiner class to calculate swelling, porosity, and eigenstrain:

[Materials<<<{"href": "../../syntax/Materials/index.html"}>>>]
  [total_porosity]
    type = UPuZrPorosityEigenstrain<<<{"description": "Computes the swelling, porosity and eigenstrain from gasous and low-temperature mechanisms in UPuZr.", "href": "solid_mechanics/UPuZrPorosityEigenstrain.html"}>>>
    initial_porosity<<<{"description": "Initial or fabrication porosity"}>>> = 0.1
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
    output_properties<<<{"description": "List of material properties, from this material, to output (outputs must also be defined to an output type)"}>>> = 'total_swelling porosity gas_swelling lowT_swelling'
    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."}>>> = porosity_eigenstrain
  []
[]

and with the solid mechanics quasi-static action to apply the calculated eigenstrain:

[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"}>>> = FINITE
        add_variables<<<{"description": "Add the displacement variables"}>>> = true
        eigenstrain_names<<<{"description": "List of eigenstrains to be applied in this strain calculation"}>>> = 'porosity_eigenstrain'
        generate_output<<<{"description": "Add scalar quantity output for stress and/or strain"}>>> = 'strain_xx strain_yy'
      []
    []
  []
[]

Input Parameters

Input Files

• (test/tests/solid_mechanics/upuzr_eigenstrains/upuzr_porosity_eigenstrain/test.i)

References