# Zircaloy Power Law Hardening Plasticity Update

Calculates the plastic strain as a function of strain rate for Zircaloy cladding. Note: This material must be run in conjunction with both ComputeMultipleInelasticStress and ZryElasticityTensor.

## Description

ZryPlasticityUpdate calculates the plastic strain for zircaloy cladding materials as a function of temperature, fluence, strain rate, material cold work factor, and as-recieved oxygen concentration. This material, which must be run in conjunction with ComputeMultipleInelasticStress calculates the plastic strain, the elastic strain, and the resulting stress for zircaloy materials. This material should also be used with the ZryElasticityTensor material.

After yield, the stress-strain relationship follows a power law model (1) where K is the strength coefficient, n is the strain hardening exponent, m is the strain rate exponent and is the strain rate. Note that the total strain ( ) is used in the above expression.

The yield stress ( ) is calculated as the non-zero intersection of the power law hardening equation and Hooke's law and is given by (2) In this analytical model, the Young's modulus, E, is a function of temperature of the cladding, fast neutron fluence, cold work factor and oxygen concentration. The Young's modulus is calculated using the MATPRO material model CELMOD in ZryElasticityTensor.

warning

The Young's modulus calculation is completed in the ZryElasticityTensor class; therefore, it is imperative to use ZryElasticityTensor with the MATPRO options set to true (matpro_youngs_modulus = true and matpro_poissons_ratio = true) when using ZryPlasticityUpdate for the simulation to produce accurate results.

Because this class uses the J2 radial return mapping algorithm, the stress after yield needs to be written in terms of the plastic strain () instead of the total strain (). This formulation can be achieved by in the von Mises stress space by substituting for the elastic strain. Then the stress-plastic strain relation after yield can be written as (3) The hardening modulus can then be obtained as (4)

This material contains two model options to calculate the constants used in the power law plasticity equation: the default PNNL model is based on the report from Geelhood et al. (2008), and a second MATPRO model based on EPRI's Pre-SW Falcon Version 31 code.

### PNNL Model

The correlations for the plasticity power law hardening relations in this model are taken from Geelhood et al. (2008). The strength coefficient, K, strain hardening exponent, n, and strain rate exponent, m, are functions of the cladding temperature, fast neutron fluence, fast neutron flux and cold work factor. To account for the effect of annealing, the MATPRO material model CANEAL is used correct the cold work factor and fast neutron fluence. The reader is referred to Sections 2.2, 2.3, and 2.4 of Geelhood et al. (2008) for the specific equations: these equations are piecewise in both temperature and fluence.

### MATPRO Model

This model option uses MATPRO equations to find the strength coefficient, K, strain hardening exponent, n, and strain rate exponent, m, as a function of the cladding temperature, fast neutron fluence, fast neutron flux and cold work factor, similar to Allison et al. (1993). Note that a fixed strain rate of 1e-3 m/s is used in this model.

## Example Input Syntax


[./zry_plasticity]
type = ZryPlasticityUpdate
block = 1
fast_neutron_flux = fast_neutron_flux
fast_neutron_fluence = fast_neutron_fluence
initial_fast_fluence = 1.0e22
temperature = temp
cold_work_factor = 0.01
plasticity_model_type = MATPRO
[../]

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


[./stress]
type = ComputeMultipleInelasticStress
tangent_operator = elastic
inelastic_models = 'zry_plasticity'
block = 1
[../]

This class uses the MATPRO relations for the elasticity tensor constants to calculate the yield stress, therefore, the input file must also include the elasticity tensor class, ZryElasticityTensor, as shown below:

[./zry_elasticity_tensor]
type = ZryElasticityTensor
matpro_poissons_ratio = true
matpro_youngs_modulus = true
temperature = temp
fast_neutron_fluence = fast_neutron_fluence
cold_work_factor = 0.01
initial_fast_fluence = 1.0e22
block = 1
[../]

## Input Parameters

• fast_neutron_fluenceThe fast neutron fluence

C++ Type:std::vector

Description:The fast neutron fluence

• temperatureTemperature of the cladding (K)

C++ Type:std::vector

Description:Temperature of the cladding (K)

• fast_neutron_fluxThe fast neutron flux

C++ Type:std::vector

Description:The fast neutron flux

### Required Parameters

• cold_work_factor0cold work factor - between 0.0 and 0.75

Default:0

C++ Type:double

Description:cold work factor - between 0.0 and 0.75

• 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

• initial_fast_fluence0Initial fast fluence to be used with the MATPRO model.

Default:0

C++ Type:double

Description:Initial fast fluence to be used with the MATPRO model.

• 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.

• max_its30Maximum number of Newton iterations

Default:30

C++ Type:unsigned int

Description:Maximum number of Newton iterations

• oxygen_concentration0Average oxygen concentration excluding oxide layer - average oxygen concentration of as received cladding (Kg O2/ Kg Zr)

Default:0

C++ Type:double

Description:Average oxygen concentration excluding oxide layer - average oxygen concentration of as received cladding (Kg O2/ Kg Zr)

• 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

• zircaloy_alloy_type4Type of Ziracloy alloy to be used: Zircaloy-2 or -4

Default:4

C++ Type:unsigned int

Description:Type of Ziracloy alloy to be used: Zircaloy-2 or -4

• 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

• strain_rateFixed strain rate value

C++ Type:double

Description:Fixed strain rate value

• plasticity_model_typePNNLThe type of correlation to use to calculate the elastic constants for the ziracloy cladding. Choices are: PNNL MATPRO

Default:PNNL

C++ Type:MooseEnum

Description:The type of correlation to use to calculate the elastic constants for the ziracloy cladding. Choices are: PNNL MATPRO

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

Default:effective_plastic_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

• 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. C. M. Allison, G. A. Berna, R. Chambers, E. W. Coryell, K. L. Davis, D. L. Hagrman, D. T. Hagrman, N. L. Hampton, J. K. Hohorst, R. E. Mason, M. L. McComas, K. A. McNeil, R. L. Miller, C. S. Olsen, G. A. Reymann, and L. J. Siefken. SCDAP/RELAP5/MOD3.1 code manual, volume IV: MATPROâ€“A library of materials properties for light-water-reactor accident analysis. Technical Report NUREG/CR-6150, EGG-2720, Idaho National Engineering Laboratory, 1993.[BibTeX]
2. K.J. Geelhood, C.E. Beyer, and W.G. Luscher. PNNL Stress/Strain correlation for Zircaloy. Technical Report PNNL-17700, Pacific Northwest National Laboratory, 2008.[BibTeX]