# Zircaloy Tulkki Primary, Hayes Thermal, and Hoppe Irradiation Creep

Calculates the viscoelastic primary creep and secondary thermal Hayes and Kassner creep and the Hoope irradiation creep for Zircaloy cladding. This material must be run in conjunction with ComputeMultipleInelasticStress.

## Description

This model, ZryCreepTulkkiHayesHoppeUpdate, is for cases where cladding experiences multiple load reversals and load drops. It has been established through experiments (McGrath, 1996; Murty, 1999) that primary creep is re-initiated upon sudden load drops and reversals. In case of a viscoelastic material, the primary creep is the recoverable creep. The secondary creep (thermal and irradiation creep) is calculated using the Hayes and Kassner (2006) model for thermal creep and Hoppe (1991) model for irradiation creep.

The model is purely empirical and the calculation of primary viscoelastic creep strain does not result in any change to the total creep strain. In other words it extracts the primary creep strain upon re-initialization due to load drops or reversals.

## Primary Creep

The primary creep strain is approximately described by a function of the form (Tulkki and Ikonen, 2014): (1) where is a constant, is the hoop stress, is the initial state with zero primary creep, is the characteristic time scale of the primary creep and is the time when the change of stress from its initial to final value occurs.

In order to keep a record of the stress history of several sequential stress changes, the internal state of the system is characterized by a single time-dependent 'stress-like' variable, . The time evolution of describes the relaxation of the internal state of the system toward the steady state determined by the externally applied stress and is expressed as: (2) Here, and . The two equations can be written in the form such that there is no dependence on the initial values and and can be integrated into a finite difference form: (3)

(4)

## Thermal Creep in Standard Operating Conditions

The standard operating temperature thermal creep model used to calculate the secondary thermal creep with a power-law model is based on the work of Hayes and Kassner (2006). Hayes and Kassner found that zircaloy creep rate can be described with a conventional five-power creep law, as described in the review by Kassner and PÃ©rez-Prado (2000).

### Hayes-Kassner Secondary Thermal Creep

The secondary thermal creep strain rate equation implemented in Bison is (5) where is the effective thermal creep rate (1/s), is the effective (Mises) stress (Pa), is the activation energy (J/mol), is the universal gas constant (J/mol-K), is the temperature (K), is the shear modulus (Pa), and and are material constants.

The shear modulus value is taken directly from the elasticity tensor within Bison.

Irradiation-induced creep of cladding materials is based on an empirical model developed by Hoppe (1991) that relates the creep rate to the current fast neutron flux and stress. The specific relation implemented is: (6) where is the effective irradiation creep rate (1/s), is the fast neutron flux (n/m-s), is the effective (Mises) stress (MPa), and , , and are material constants. The material constant values = 3.557 10, = 0.85, and = 1.0 are for stress relief annealed zircaloy.

## Example Input Syntax


[./zrycreep]
type = ZryCreepTulkkiHayesHoppeUpdate
temperature = temp
fast_neutron_flux = fast_neutron_flux
model_thermal_creep = true
zircaloy_material_type = recrystalization_annealed
[../]

ZryCreepTulkkiHayesHoppeUpdate must be run in conjunction with the inelastic strain return mapping stress calculator as shown below:


[./stress]
type = ComputeMultipleInelasticStress
tangent_operator = elastic
inelastic_models = 'zrycreep'
[../]

## Input Parameters

• relative_tolerance1e-08Relative convergence tolerance for Newton iteration

Default:1e-08

C++ Type:double

Description:Relative convergence tolerance for Newton iteration

• max_inelastic_increment0.0001The maximum inelastic strain increment allowed in a time step

Default:0.0001

C++ Type:double

Description:The maximum inelastic strain increment allowed in a time step

• temperatureThe coupled temperature (K)

C++ Type:std::vector

Description:The coupled temperature (K)

• base_nameOptional parameter that defines a prefix for all material properties related to this stress update model. This allows for multiple models of the same type to be used without naming conflicts.

C++ Type:std::string

Description:Optional parameter that defines a prefix for all material properties related to this stress update model. This allows for multiple models of the same type to be used without naming conflicts.

• outputThe reporting postprocessor to use for the max_iterations value.

C++ Type:PostprocessorName

Description:The reporting postprocessor to use for the max_iterations value.

• model_thermal_creepTrueSet true to activate steady state thermal creep

Default:True

C++ Type:bool

Description:Set true to activate steady state thermal creep

• max_its30Maximum number of Newton iterations

Default:30

C++ Type:unsigned int

Description:Maximum number of Newton iterations

• zircaloy_material_typestress_relief_annealedType of zircaloy material properties to use in calculating creep. Choices are: stress_relief_annealed recrystalization_annealed partial_recrystallization_annealed zirlo

Default:stress_relief_annealed

C++ Type:MooseEnum

Description:Type of zircaloy material properties to use in calculating creep. Choices are: stress_relief_annealed recrystalization_annealed partial_recrystallization_annealed zirlo

• acceptable_multiplier10Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress

Default:10

C++ Type:double

Description:Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress

Default:True

C++ Type:bool

Description:Set true to activate irradiation induced creep

• fast_neutron_fluxThe fast neutron flux

C++ Type:std::vector

Description:The fast neutron flux

• absolute_tolerance1e-11Absolute convergence tolerance for Newton iteration

Default:1e-11

C++ Type:double

Description:Absolute convergence tolerance for Newton iteration

• 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

• max_creep_increment0.001Maximum creep strain increment allowed by accuracy time step criterion

Default:0.001

C++ Type:double

Description:Maximum creep strain increment allowed by accuracy time step criterion

• 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

### Optional Parameters

• effective_inelastic_strain_nameeffective_creep_strainName of the material property that stores the effective inelastic strain

Default:effective_creep_strain

C++ Type:std::string

Description:Name of the material property that stores the effective inelastic strain

• 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

• creeprate_scale_factor1scaling factor for total creep rate. Used for calibration and sensitivity studies

Default:1

C++ Type:double

Description:scaling factor for total creep rate. Used for calibration and sensitivity studies

• internal_solve_output_onon_errorWhen to output internal Newton solve information

Default:on_error

C++ Type:MooseEnum

Description:When to output internal Newton solve information

• internal_solve_full_iteration_historyFalseSet true to output full internal Newton iteration history at times determined by internal_solve_output_on. If false, only a summary is output.

Default:False

C++ Type:bool

Description:Set true to output full internal Newton iteration history at times determined by internal_solve_output_on. If false, only a summary is output.

### Debug Parameters

• 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. T. A. Hayes and M. Kassner. Creep of zirconium and zirconium alloys. Metallurgical and Materials Transactions A, 37A:2389â€“2396, 2006.[BibTeX]
2. N. E. Hoppe. Engineering model for zircaloy creep and growth. In Proceedings of the ANS-ENS International Topical Meeting on LWR Fuel Performance, 157â€“172. Avignon, France, April 21-24, 1991.[BibTeX]
3. ME Kassner and M-T PÃ©rez-Prado. Five-power-law creep in single phase metals and alloys. Progress in Materials Science, 45(1):1â€“102, 2000.[BibTeX]
4. M. A. McGrath. In-reactor creep behavior of zircaloy-2 under variable loading conditions in ifa-585. Technical Report HWR-471, Halden, March 1996.[BibTeX]
5. K. L. Murty. Creep studies for zircaloy life prediction in water reactors. JOM, 51(10):32–39, 1999.[BibTeX]
6. V. Tulkki and T. Ikonen. Modeling of zircaloy cladding primary creep during load drop and reversal. Journal of Nuclear Materials, 445(1):98–103, 2014.[BibTeX]