ADCoupledFissionGasViscoplasticityStressUpdate

Description

ADCoupledFissionGasViscoplasticityStressUpdate computes an inelastic volumetric strain to account for gaseous swelling in nuclear fuel via a bubble surface force balance model. This model was originally developed for U-Zr and U-Pu-Zr fuel, thus some of the assumptions are directly related to observations from metallic fuel. However, the model may be applicable to other advanced fuels that exhibit large bubble sizes such as U-Mo, carbides, or nitrides. The inelastic strain contribution is calculated via a bubble-surface force-balance by assuming the bubbles remain in equilibrium depending on the coupled temperature and the stress state of the solid surrounding the bubble. However, ADCoupledFissionGasViscoplasticityStressUpdate 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.

ADCoupledFissionGasViscoplasticityStressUpdate inherits from the base class FissionGasViscoplasticityStressUpdateBase, which contains several common models for fission gas swelling using viscoplasticity methods. ADCoupledFissionGasViscoplasticityStressUpdate 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 bubble-surface force-balance technique to calculate the size of bubbles is formulated in Matthews and Unal (2019) which is based on discussions in Olander (1976). The fundamental assumption utilized is that the ratio of gas atoms to vacancies in a given bubble is in equilibrium such that the bubbles are not over- or under-pressurized. This assumption is supported by the extremely high mobility of vacancies in metallic nuclear fuel, as well as the high number of vacancies created during fission fragment damage events in nuclear fuel.

Since the large swelling in metallic fuel is a consequence of the isotropic, high-temperature phase behavior, the following model will focus on the behavior in this region, with an eye towards adapting this model to capture the behavior of the other regions (e.g. non-spherical bubbles) as future work.

Several assumptions are utilized to simplify the model, the details of which will be expanded upon in the Theory section:

• The bubble is assumed to be in equilibrium, e.g. not over- or under-pressurized;

• The bubble exhibits a purely hydrostatic stress on the solid material surrounding the bubble;

• The size of extremely small bubbles converge on the space provided per atom given by the xenon van der Waal's constant, e.g. xenon is not assumed to form a solid;

• Only one, non-constant bubble size is assumed at any given quadrature point;

• Bubbles are assumed to be spherical,

• The concentration of bubbles is assumed to remain constant in time, but may vary in space,

• All fission gas will be assumed to act like xenon,

• Bubble growth is monotonic, e.g. bubbles are not allowed to shrink;

• Knock-out of fission gas back into the solid does not occur;

• Interconnection to the plenum is defined by local conditions, e.g. percolation is ignored,

• The accumulation of fission gas can be treated as a diffusion limited process.

The bulk swelling due to fission gas bubbles can be calculated from the bubble radius and concentration as, (1) where is volume, is the largest bubble size, and is the bubble size density. As a first approximation, all bubbles can be assumed to have the same radius, allowing for the simplification of the integral in Eq. (1), an assumption that may be relaxed later. In addition, the fission gas will be assumed to be comprised only of xenon gas, an assumption typically applied in many fission gas models.

Eq. (1) is the foundational relationship that defines bubble swelling in solids. In general, models that capture fission gas swelling calculate either the size of the bubble (i.e. ) and fix , or fix the bubble size and vary the bubble density. Since the bubbles in nuclear fuel accumulate fission gas and grow during irradiation, the bubble concentration is fixed and an evolving bubble size will be calculated here.

For the model described here, the bubble size will be calculated via a bubble surface force balance constraint, (2) Here, captures the pressure exerted on the bubble surface by the gas atoms contained within the bubble, and corresponds to the pressure exerted via the surface tension of the bubble and the stress in the surrounding material. In other-words, Eq. (2) assumes that the bubbles are not over- or under-pressurized. This assumption is justified based on the rapid mobility of vacancies in U-Pu-Zr and high concentration of defects due to the irradiation likely resulting in uranium vacancies quickly equalizing any over- or under-pressurization of gas bubbles.

In order to accurately capture , a nucleation model must be considered that includes local state variables. However, through inspection of fuel cross-sectional micrographs, the concentration of fission gas bubbles seems to be roughly constant within a phase across different irradiation conditions, with few large bubbles in the -phase, many small bubbles in the -phase, and many oblong pores in the -phase. Future models would benefit from temperature and phase dependent values of , as well as consideration for the non-spherical porosity in the -phase.

The pressure exerted by the gas within a bubble is a complex function that depends on the surface energy of the material, number of gas atoms, local stress, and temperature. Following the discussion by Olander (1976), the radius of the bubble can be calculated using the van der Waal's equation of state to formulate the pressure inside a gas bubble: (3) where is the number of moles of gas in the bubble, is the Boltzmann constant, is temperature, is the volume of a single bubble, and is the volume occupied by a single Xe gas atom. By taking the limit of Eq. (3) as the radius decreases, Eq. (3) ensures that the bubble radius never exceeds the minimum bubble size , determined by the volume of a single atom , (4) This is a consequence of Eq. (3), where as , .

Balancing the pressure of the bubble is the pressure induced by the surface tension and the stress state of the surrounding solid material, (5) where is the surface tension of the material and is the hydrostatic stress at the bubble surface, with signifying tension. The equilibrium bubble size can be determined by setting Eq. (3) equal to Eq. (5) and multiplying out all denominators to avoid asymptotes, (6)

The highly nonlinear Eq. (6) can be calculated by solving for using an inner Newton-Rapson method. When Eq. (6) is satisfied, the bubble is in equilibrium with the surrounding material. The bubble radius can then be directly utilized in Eq. (1) to determine the volumetric bubble swelling increment. Details for specific values in Eq. (6) are described in the following sections. Note, the models for porosity interconneciton, fission gas production, and fission gas release are included in FissionGasViscoplasticityStressUpdateBase

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [gas_swelling]
    type = ADCoupledFissionGasViscoplasticityStressUpdate<<<{"description": "Computes the change in fuel pellet volume due to gaseous fission product buildup using viscoplasticity methods with a bubble surface force balance model coupled to temperature and the stress state of the surrounding material.", "href": "ADCoupledFissionGasViscoplasticityStressUpdate.html"}>>>
    diffusion_model<<<{"description": "Diffusion model used to compute fission gas accumulation by bubbles. Explicit uses explicit diffusion, implicit integrates the flux of atoms over time, and infinite assumes all atoms are born in bubbles."}>>> = implicit
    gas_diffusivity_name<<<{"description": "Material property for the fission gas diffusivity"}>>> = D_g
    max_inelastic_increment<<<{"description": "The maximum inelastic strain increment allowed in a time step"}>>> = 1e-03
    # below are same as ADSimpleFissionGasViscoplasticityStressUpdate
    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
    outputs<<<{"description": "Vector of output names where you would like to restrict the output of variables(s) associated with this object"}>>> = all
  []
[]

ADCoupledFissionGasViscoplasticityStressUpdate must be used 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

• (test/tests/solid_mechanics/ad_coupled_fission_gas_viscoplasticity/rodlet_full_upuzr_elasticity.i)
• (test/tests/solid_mechanics/ad_coupled_fission_gas_viscoplasticity/rodlet_diff.i)
• (test/tests/solid_mechanics/ad_coupled_fission_gas_viscoplasticity/rodlet_full.i)
• (test/tests/solid_mechanics/ad_coupled_fission_gas_viscoplasticity/rodlet.i)
• (test/tests/solid_mechanics/upuzr_hot_pressing/interconnectivity_decrease_exact.i)
• (test/tests/solid_mechanics/ad_coupled_fission_gas_viscoplasticity/rodlet_vanderwaals.i)
• (test/tests/solid_mechanics/ad_coupled_fission_gas_viscoplasticity/rodlet_hydro.i)
• (test/tests/solid_mechanics/ad_coupled_fission_gas_viscoplasticity/exact.i)

References

References