SimpleFissionGasViscoplasticityStressUpdate

Description

SimpleFissionGasViscoplasticityStressUpdate computes a volumetric strain to account for gaseous swelling in U-Pu-Zr metal fuel systems. This material is similar to UPuZrGaseousEigenstrain for which a bubble size is computed assuming infinite diffusivity of fission gas, an un-stressed bulk, and an ideal gas equation of state. However, SimpleFissionGasViscoplasticityStressUpdate utilizes the viscoplasticity methods inherited by FissionGasViscoplasticityStressUpdateBase. This is different than the historical approach for fission gas swelling that utilized the eigenstrain system. By using the viscoplasticity methods, this model can be coupled to other inelastic strains such as creep, which are accumulated by ADComputeMultipleInelasticStress. The porosity rate of change is then computed from the trace of the inelastic strain rate by ADPorosityFromStrain: In this way, the effects of multiple sources of inelastic strain can be accumulated into a single porosity value.

SimpleFissionGasViscoplasticityStressUpdate inherits from the base class FissionGasViscoplasticityStressUpdateBase, which contains several common models for fission gas swelling using viscoplasticity methods. SimpleFissionGasViscoplasticityStressUpdate only provides the model to calculate the inelastic strain and the absorption of gas in the bubbles. Consequently, the Theory section only focuses on the inelastic strain calculation and gas absorption model, while FissionGasViscoplasticityStressUpdateBase contains the remaining underlying fission gas inventory and interconnectivity models.

Theory

The derivation for the fuel swelling model for U-Pu-Zr used here is based on formulations presented in Olander (1976). The model is able to simplify the gaseous swelling porosity into an analytical solution by making the following assumptions:

• A constant fraction of gas remains dissolved in the fuel at all times;

• Of the remaining produced fission gas that does not remain dissolved, gas atom diffusivity is infinite such that all non-dissolved fission gas is born in the bubbles;

• The concentration of bubbles is constant and their shape is 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 force balance on the surface of the bubble does not account for the stress state of the surrounding solid.

Using methods in FissionGasViscoplasticityStressUpdateBase,the produced fission gas density is calculated at a rate proportional to the fission rate density. Given that a infinite diffusivity is assumed here, the newly produced fission gas is instantly partitioned into the dissolved fission gas density , (1) where is the constant fraction of fission gas that is assumed to be present in the solid. is calculated using interconnection independent and dependent terms, where is the fraction of bubbles interconnected. See FissionGasViscoplasticityStressUpdateBase for more details on the interconnectivity model. The remainder of the produced fission gas is then assumed to be instantly deposited in the bubbles, leading to a simple expression for the number of moles of fission gas in each bubble , (2) where is the concentration of bubbles. Given the amount of gas in each bubble, the bubble radius can be computed by assuming an ideal gas equation of state, (3) where is the ideal gas constant, is the temperature, and is the surface tension of the fuel.

The effective volumetric strain increment is essentially the change in the volume of the bubble multiplied by the bubble concentration, (4) where is a user defined scalar that can be used to artificially increase the strain increment. Here, the is included to account for bubble interconnection during which their gas contents are vented to the plenum, and their growth is terminated.

Given the effective strain increment, the applied inelastic strain is calculated as, (5) where is the anisotropic strain tensor, given as, (6) where is the anisotropic_factor input parameter.

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [gas_swelling]
    type = ADSimpleFissionGasViscoplasticityStressUpdate<<<{"description": "Computes the change in fuel pellet volume due to gaseous fission product buildup using viscoplasticity methods by assuming all gas is born into uniformly sized bubbles.", "href": "SimpleFissionGasViscoplasticityStressUpdate.html"}>>>
    fission_rate<<<{"description": "Name of material property for gas source rate"}>>> = fission_rate
    temperature<<<{"description": "Coupled temperature variable"}>>> = temp
    bubble_concentration<<<{"description": "Name of material property for bubble number density"}>>> = 3e14
    initial_bubble_concentration<<<{"description": "Function describing the initial bubble concentration"}>>> = 3e14
    interconnection_initiating_porosity<<<{"description": "Porosity value for which interconnectivity begins"}>>> = 0.25
    interconnection_terminating_porosity<<<{"description": "Porosity value for which interconnectivity finishes"}>>> = 0.3
    compute_interconnectivity<<<{"description": "Flag to calculate interconnectivity interconnectivity"}>>> = true
    scalar<<<{"description": "Scale factor to be applied to the gaseous swelling strain. Used for calibration and sensitivity studies"}>>> = 1
    initial_fgm_dissolved<<<{"description": "Initial concentration of dissolved fission gas in mols"}>>> = 0
    surface_energy<<<{"description": "Surface energy of material (J/m^2)"}>>> = 1.2
    fission_gas_yield<<<{"description": "Yield of fission gas atoms per fission"}>>> = 0.3
    retained_gas_fraction<<<{"description": "Retained gas fraction that does not contribute to gaseous swelling"}>>> = 0.2
    interconnection_dependent_retained_gas_fraction<<<{"description": "Additional retained gas fraction before interconnection."}>>> = 0.2
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

SimpleFissionGasViscoplasticityStressUpdate must be used in with ADComputeMultipleInelasticStress

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [fuel_stress]
    type = ADComputeMultipleInelasticStress<<<{"description": "Compute state (stress and internal parameters such as plastic strains and internal parameters) using an iterative process.  Combinations of creep models and plastic models may be used.", "href": "../ADComputeMultipleInelasticStress.html"}>>>
    inelastic_models<<<{"description": "The material objects to use to calculate stress and inelastic strains. Note: specify creep models first and plasticity models second."}>>> = 'gas_swelling'
  []
[]

Input Parameters

Input Files

• (assessment/metallic_fuel/FBTA/analysis/steady_state/common_inputs/test_base.i)
• (test/tests/solid_mechanics/simple_fission_gas_viscoplasticity/exact.i)
• (assessment/metallic_fuel/EBRII/X447/analysis/enhancement/x447_enh_base.i)
• (test/tests/solid_mechanics/simple_fission_gas_viscoplasticity/old_method_comparison.i)
• (test/tests/solid_mechanics/simple_fission_gas_viscoplasticity/rodlet.i)
• (assessment/metallic_fuel/EBRII/X423/analysis/x423_vp_base.i)
• (test/tests/solid_mechanics/upuzr_hot_pressing/ad.i)
• (assessment/metallic_fuel/WPF/analysis/X425_T418/X425_base.i)
• (test/tests/solid_mechanics/simple_fission_gas_viscoplasticity/rodlet_typical.i)
• (assessment/metallic_fuel/FBTA/analysis/steady_state/AK181/181193/IFR1_181193.i)
• (assessment/metallic_fuel/WPF/analysis/FM-1/FM_base.i)

References