Failure model for HT-9 cladding. Contains multiple models for steady state (burnup calculations) and transient operations.

## Description

FailureCladHT9 is the model for HT9 cladding failure during both long and short transients. Long transient failure is traditionally predicted with the Cumulative Damage Fraction (CDF) method; however, short-time CDF correlation data is also provided if desired for shorter transients. Short transients track cavity growth along grain boundaries with a Constrained Cavity Growth (CCG) with Diffusion and Creep with Sliding (D&CS) mechanism.

A few creep-fracture failure models are available in Bison similar to the approach used in Karahan and Buongiorno (2010). Slow transients, such as burnup, are handled by the "steady state" Cumulative Damage Fraction (CDF) model using the long-time Dorn parameter correlation (Nam et al., 1998). Fast transients are handled with the Constrained Cavity Growth (CCG) with Diffusion and Creep with Sliding (D&CS) model (Tvergaard, 1985) or the short-time Dorn parameter correlation (Nam et al., 1998).

### Cumulative Damage Fraction

The CDF model compares the time-to-rupture value with experimentally obtained results as a function of stress and absolute temperature. When this value (CDF) equals one, the material has failed. (1) The time of rupture function in hours is found in (2) where is 154 kcal/mole for long-time data and 70.17 kcal/mole for short-time data, is the Boltzmann constant, and is the absolute temperature in K. is the Dorn parameter coefficient ranging from 3.915 10 at 650 C to 1 at 600 C for long-time data and 2.778 10 for all short-time data. The Dorn parameter comes from the curve fits (3) (4) for long-time and short-time correlations, respectively, where is the hoop stress in Pa.

For the long-time correlation with temperatures outside of the range of the Dorn parameter coefficient , the value is set to the appropriate end value to avoid extrapolation. Interpolation of the value is conducted within the temperature range. If both correlations are chosen, then the longer rupture time result is taken as the actual time of rupture (DiMelfi et al., 1993). When choosing both correlations, erroneous values are possible. Both correlations are recommended when simulation values fall within the range where the long and short correlations overlap.

### Constrained Cavity Growth

The CCG with D&CS model calculates the crack radius of periodic cavities along grain boundaries. The cavity centers are spaced equally at a distance of . Failure occurs when . The correlation is very sensitive to the user supplied value of . The short CDF model may also be used for short time to failure simulations where the value of is unknown.

The crack radius growth rate is related to the cavity volume growth rate by (5) with being defined as (6) where being the ratio of the grain boundary free energy to twice the grain surface free energy.

The volume growth rate is the sum of the rigid grain growth rate and power law creeping material growth rate for . (7) (8)

The average normal stress is . The sintering stress can be calculated as . The grain boundary diffusion parameter is where is the boundary diffusivity, is the atomic volume, is the Boltzmann constant, and is the absolute temperature in K. The area fraction of the grain boundary is determined with (9) where . The Von Mises stress is . The hydrostatic (mean) stress is . The effective creep strain- rate is . With the assumption that the material follows a power law creep, is the value of the power, , and .

The crack length is assumed to begin at a minimum value. The crack length is never allowed to fall below this value. The crack may shrink after it has grown. However, after failure has occurred, the crack is assumed to be permanent and can no longer shorten. The crack length is found by taking the calculated growth rate and multiplying by the current time increment.

## Example Input Syntax


[./longHT9_failure]
method = cdf_long
boundary = '1 2 3'
temperature = temp
hoop_stress = stress_zz # Since 2D-RZ
[../]
(examples/metal_fuel/uzr/x447.i)

## Input Parameters

C++ Type:std::vector

C++ Type:std::vector

### Required Parameters

Default:6.02214e+23

C++ Type:double

• atomic_volume1.18e-29Atomic Volume (m^3)

Default:1.18e-29

C++ Type:double

Description:Atomic Volume (m^3)

• boundary_free_energy0.85Grain boundary free energy (J/m^2)

Default:0.85

C++ Type:double

Description:Grain boundary free energy (J/m^2)

• b3.5e-06Distance from cavity center to midplane between cavities (m)

Default:3.5e-06

C++ Type:double

Description:Distance from cavity center to midplane between cavities (m)

• computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

Default:True

C++ Type:bool

Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the Material via MaterialPropertyInterface::getMaterial(). Non-computed Materials are not sorted for dependencies.

• function_criteriaFunction name providing criteria value.

C++ Type:FunctionName

Description:Function name providing criteria value.

• hydrostatic_stressHydrostatic (mean) stress in cladding

C++ Type:std::vector

• creep_n_power5Power law creeping material constant

Default:5

C++ Type:double

Description:Power law creeping material constant

• boundary_diffusivity1.1e-12Grain boundary diffusivity (m^3/s)

Default:1.1e-12

C++ Type:double

Description:Grain boundary diffusivity (m^3/s)

• surface_free_energy2.1Grain surface free energy (J/m^2)

Default:2.1

C++ Type:double

Description:Grain surface free energy (J/m^2)

Default:7e-08

C++ Type:double

• boltzmann1.38065e-23Boltzmann constant (J/K)

Default:1.38065e-23

C++ Type:double

Description:Boltzmann constant (J/K)

• comparedless_equalOptions for variable _compared_ to criteria: greater_than greater_equal less_equal less_than

Default:less_equal

C++ Type:MooseEnum

Description:Options for variable _compared_ to criteria: greater_than greater_equal less_equal less_than

• von_mises_stressVon Mises stress in the cladding

C++ Type:std::vector

Description:Von Mises stress in the cladding

• constant_criteria0Numerical value providing criteria value.

Default:0

C++ Type:double

Description:Numerical value providing criteria value.

• 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

• variable_checkVariable name which is compared to criteria. Example: Var < 0, true=failed

C++ Type:std::vector

Description:Variable name which is compared to criteria. Example: Var < 0, true=failed

• methodcdf_longFailure method choice. Options: ccg_dcs cdf_long cdf_short cdf_both

Default:cdf_long

C++ Type:MooseEnum

Description:Failure method choice. Options: ccg_dcs cdf_long cdf_short cdf_both

• 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

• eff_strain_rate_creepEffective creep strain rate in cladding (1/s)

C++ Type:std::vector

Description:Effective creep strain rate in cladding (1/s)

### Optional Parameters

• enableTrueSet the enabled status of the MooseObject.

Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

• 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.

• 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.

• seed0The seed for the master random number generator

Default:0

C++ Type:unsigned int

Description:The seed for the master random number generator

• implicitTrueDetermines whether this object is calculated using an implicit or explicit form

Default:True

C++ Type:bool

Description:Determines whether this object is calculated using an implicit or explicit form

• constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

Default:NONE

C++ Type:MooseEnum

Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped

• output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)

C++ Type:std::vector

Description:List of material properties, from this material, to output (outputs must also be defined to an output type)

• outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object

Default:none

C++ Type:std::vector

Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

## References

1. R. J. DiMelfi, E. E. Gruber, J. M. Kramer, and T. H. Hughes. Strength and rupture-life transitions caused by secondary carbide precipitation in HT-9 during high-temperature low-rate mechanical testing. Technical Report ANL/RE/CP--75828, Argonne National Laboratory, December 1993.[BibTeX]
2. A. Karahan and J. Buongiorno. A new code for predicting the thermo-mechanical and irradiation behavior of metallic fuels in sodium fast reactors. Journal of Nuclear Materials, 396:283–293, 2010.[BibTeX]
3. C. Nam, W. Hwang, and D. Sohn. Statistical failure analysis of metallic U-10Zr/HT9 fast reactor fuel pin by considering the Weibull distribution and cumulative damage fraction. Annals of Nuclear Energy, 25(17):1441–1453, 1998.[BibTeX]
4. V. Tvergaard. Effect of grain boundary sliding on creep constrained diffusive cavitation. Journal of the Mechanics and Physics of Solids, 33(5):447–469, 1985.[BibTeX]