UO2IsotropicDislocationDensity

Description

This auxkernel is used to calculate the dislocation density in UO fuel as a function of burnup. The calculated dislocation density is intended to be used in conjunction with a simplified cluster dynamics model that describes the evolution of intra-grannular bubbles in the fuel. The dislocation density measure calculated in this auxkernel does not contribute to the strain.

This auxkernel model was developed by simplifying the Continuum Dislocation Dynamics (CDD) model developed by Li et.al. to calculate a smeared-out effective total dislocation density (Li et al., 2014). The effective total dislocation density is treated as the sum of the dislocation densities on each of the slip planes in the crystal. The prominent simplifying assumption is that the density of immobile dislocation is comparable to the density of mobile dislocations. Applying this assumption, the evolution of dislocation density equations reduces to the form of the classical Kocks' dislocation density model while retaining the physical basis of the parameters (Kocks, 1976). (1)

The analytical solution of Eq. (1) is reformulated as a function of burnup, , by taking the dislocation driving force, , to be proportional to burnup. The dislocation density expression is defined as (2) where the coefficients relate to physical lattice parameters from the CDD model, as shown in the table below, and the initial dislocation density is taken as = 6.0e13 from Noirot (2006). The coefficient includes a conversion factor from time (sec) to burnup in (GWd/tU) which is on the order of 10. The values of the coefficients in Eq. (2) were determined by fitting the model to experimental data for dislocation density under increasing burnup (Nogita and Une, 1994).

Figure 1: Comparison between the experimental results and the present model predictions of dislocation density as a function of burnup.

In Table 1 represents the initial shear slip rate, is the dislocation multiplication coefficient, is the burger's vector of the lattice, and is the capture radius for dislocation mutual annihilation.

The parameters in Table 1 were selected by emphasizing the fit for the first five experimental data points, taken below 45 GWd/tU. The Ham experimental approach applied in Nogita and Une (1994) is known to underestimate dislocation densities in extremely tangled dislocation networks; therefore, we allowed the fit of the parameters to deviate above the experimentally measured 83 GWd/tU dislocation densities where tangled networks were observed. The present model does, however, capture the saturation of the dislocation density at higher burnup, as shown in Figure 1.

The alignment of the current model parameters with the expected physically based values demonstrates the connection of the present model to the fuel microstructure.

Example Input Syntax

[AuxKernels<<<{"href": "../../syntax/AuxKernels/index.html"}>>>]
  [dislocation_density]
    type = UO2IsotropicDislocationDensity<<<{"description": "Computes an effective isotropic dislocation density for UO2 fuel by smearing the sum of the dislocations across all of the slip planes; uses a Kocks-type dislocation density model.", "href": "UO2IsotropicDislocationDensity.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = dislocation_density
    burnup<<<{"description": "Coupled Burnup, in units of fissions/atoms-U."}>>> = burnup
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial linear'
  []
[]

Input Parameters

Input Files

• (test/tests/dislocation_density/isotropic_dislocation_density_burnup_aux.i)

References