# Zircaloy-4 Cladding Failure due to Burst

Models the failure of Zircaloy-4 cladding due to burst under LOCA conditions

## Description

For modeling failure due to burst of Zircaloy-4 claddings during Loss-of-Coolant Accident simulations, the Bison model ZryCladdingFailure offers four different options:

### Overstress Criterion

An overstress criterion, which assumes that the time of burst is reached when the local hoop stress equals a limiting burst stress, (Erbacher et al., 1982): (1) where (MPa) is the hoop stress and (MPa) is the burst stress.

### Plasticity Instability Criterion

A plastic instability criterion, which considers cladding burst at the attainment of a limiting value for the effective plastic strain rate: (2) where is the effective plastic (creep + plasticity) strain rate and is the limiting value. Following Marcello et al. (2014), we set h s.

### Combined Failure Criterion

A combined failure criterion, which establishes that cladding burst occurs when either condition Eq. 1 or Eq. 2 is fulfilled.

note:Combined Failure Criterion Recommended for Low-Stress Simulations

Because the overstress criterion may lead to unsafe predictions in low-stress situations, see Marcello et al. (2014), the combined criterion is recommended.

The calculation of the burst stress follows the work of Erbacher et al. (1982). Based on experimental evidence, the burst stress is considered to depend on the temperature and oxygen concentration in the cladding and is represented by: (3) where (MPa) and (K) are constants determined experimentally, and (dimensionless) is the oxygen weight fraction in the cladding. An oxygen weight fraction at fabrication, , is used following Erbacher et al. (1982). The current oxygen weight fraction is computed based on the oxygen mass gain from the oxidation model ZryOxidation as (4) where (m) is the cladding outer radius, = 6550 kgm the density of the cladding metal, (m) the cladding inner radius, g (kgm) the oxygen mass, see ZryOxidation, and with S (m) as the oxide layer thickness, see ZryOxidation, and R=1.56 is the Pilling-Bedworth ratio for Zircaloy.

The values for the parameters a and b are given in the table for material parameters depending on phase. In the mixed phase () region, linear interpolations of ln(a) and b are made between the values for pure and middle of ( ) phase, and between and pure phase, (Erbacher et al., 1982). The volume fractions of each phase are calculated by the phase transformation model described in ZrPhase.

Table 1: Material parameters used to calculate the burst stress of Zircaloy-4 (Erbacher et al., 1982)

Zr Phasea (MPa)b (K)
830
3000
2300

### Overstrain Criterion

An overstrain criterion, which considers cladding burst once the permanent engineering hoop strain (i.e., creep) exceeds 40%. In Bison true strain is used and therefore the permanent engineering hoop strain limit is given by:

(5) where burst occurs if (6)

## Example Input Syntax


[./failure_criterion]
boundary = right #3
failure_criterion = combined_overstress_and_plastic_instability
hoop_stress = hoop_stress
effective_strain_rate_creep = creep_rate
temperature = temp
fraction_beta_phase = fract_beta_phase
[../]

## Input Parameters

• 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

• 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.

C++ Type:std::vector

• hoop_creep_strainThe hoop creep strain in the cladding (1/s)

C++ Type:std::vector

Description:The hoop creep strain in the cladding (1/s)

• 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

• fraction_beta_phaseVolume fraction of Zr beta phase

C++ Type:std::vector

Description:Volume fraction of Zr beta phase

• fraction_oxygen_gainWeight fraction of gained oxygen in the cladding

C++ Type:std::vector

Description:Weight fraction of gained oxygen in the cladding

• hoop_stressHoop stress in cladding, in Pa

C++ Type:std::vector

Description:Hoop stress in cladding, in Pa

• 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

• effective_strain_rate_creepEffective creep strain rate in the cladding (1/s)

C++ Type:std::vector

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

• effective_strain_rate_plasticityEffective plastic strain rate in the cladding (1/s)

C++ Type:std::vector

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

• 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

• failure_criterioncombined_overstress_and_plastic_instabilityOptions to determine when the cladding material is considered to have burst

Default:combined_overstress_and_plastic_instability

C++ Type:MooseEnum

Description:Options to determine when the cladding material is considered to have burst

### 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. F. J. Erbacher, H. J. Neitzel, H. Rosinger, H. Schmidt, and K. Wiehr. Burst criterion of Zircaloy fuel claddings in a loss-of-coolant accident. In Zirconium in the Nuclear Industry, Fifth Conference, ASTM STP 754, D.G. Franklin Ed., 271â€“283. American Society for Testing and Materials, 1982.[BibTeX]
2. V. Di Marcello, A. Schubert, J. van de Laar, and P. Van Uffelen. The TRANSURANUS mechanical model for large strain analysis. Nuclear Engineering and Design, 276:19â€“29, 2014.[BibTeX]