# UOvar element = document.getElementById("moose-equation-89954227-ca13-4d37-b330-87c13e2b250b");katex.render("_2", element, {displayMode:false,throwOnError:false}); Isotropic Dislocation Density AuxKernel

Calculates 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.

## 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).

Table 1: Values for coefficients in the isotropic dislocation density expression ( Eq. 2)

ParameterFitted ValuePhysical Interpretation
$\alpha_1 / \vec{b}$
$R_c / \vec{b}$

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

[./dislocation_density]
type = UO2IsotropicDislocationDensity
variable = dislocation_density
burnup = burnup
execute_on = 'initial linear'
[../]
(test/tests/dislocation_density/isotropic_dislocation_density_burnup_aux.i)

## Input Parameters

• variableThe name of the variable that this object applies to

C++ Type:AuxVariableName

Description:The name of the variable that this object applies to

### Required Parameters

• initial_dislocation_density6e+13Dislocation density of as-recieved fuel (1/m^2)

Default:6e+13

C++ Type:double

Description:Dislocation density of as-recieved fuel (1/m^2)

• effective_dislocation_velocity0.00028Smeared isotropic parameter representing dislocation velocity as a function of burnup

Default:0.00028

C++ Type:double

Description:Smeared isotropic parameter representing dislocation velocity as a function of burnup

• 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

• execute_onLINEARThe list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.

Default:LINEAR

C++ Type:ExecFlagEnum

Description:The list of flag(s) indicating when this object should be executed, the available options include NONE, INITIAL, LINEAR, NONLINEAR, TIMESTEP_END, TIMESTEP_BEGIN, FINAL, CUSTOM.

• 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:BurnupFunctionName

Description:Burnup function

• dislocation_multiplication_factor2.5e+09Smeared isotropic parameter used to determine the multiplication of dislocations to create new dislocations

Default:2.5e+09

C++ Type:double

Description:Smeared isotropic parameter used to determine the multiplication of dislocations to create new dislocations

• burnupCoupled Burnup, in units of fissions/atoms-U

C++ Type:std::vector

Description:Coupled Burnup, in units of fissions/atoms-U

• dislocation_annhiliation_interaction_radius10Number of burgers' vectors in the radius of interaction between two annhiliating dislocations

Default:10

C++ Type:double

Description:Number of burgers' vectors in the radius of interaction between two annhiliating dislocations

### Optional Parameters

• 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.

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• 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.