ZryCreepTulkkiHayesHoppeUpdate

Computes the viscoelastic primary creep and secondary thermal Hayes and Kassner creep and the Hoppe 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): $var element = document.getElementById("moose-equation-fdf9d763-01a0-4d83-84db-68f94026464a");katex.render("\\varepsilon_{p}\\left(t\\right)=C\\left(\\sigma_{h}-\\sigma_{0}\\right)\\left[1-\\exp{\\left(-\\frac{t-t_{0}}{\\tau} \\right)}\\right]", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-ea8ef28f-2b5e-4264-a6bd-16d3d8f1e1a9");katex.render("C", element, {displayMode:false,throwOnError:false});$ is a constant, $var element = document.getElementById("moose-equation-b68a89ed-cec9-4d4e-b00b-0d0f2257d2c4");katex.render("\\sigma_{h}", element, {displayMode:false,throwOnError:false});$ is the hoop stress, $var element = document.getElementById("moose-equation-9dc768ba-6602-4166-8cf3-624b0634b3d7");katex.render("\\sigma_{0}", element, {displayMode:false,throwOnError:false});$ is the initial state with zero primary creep, $var element = document.getElementById("moose-equation-265c9d29-d063-4146-8032-c32948bbf0d5");katex.render("\\tau", element, {displayMode:false,throwOnError:false});$ is the characteristic time scale of the primary creep and $var element = document.getElementById("moose-equation-8a660d3f-2381-4a73-975b-25711523ed5a");katex.render("t_{0}", element, {displayMode:false,throwOnError:false});$ 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, $var element = document.getElementById("moose-equation-da2fb89a-d176-41a2-b0cf-2afa26657f9a");katex.render("\\sigma_{int}\\left(t\\right)", element, {displayMode:false,throwOnError:false});$ . The time evolution of $var element = document.getElementById("moose-equation-f9337fc0-cf3b-410a-9198-d0d038afb5ee");katex.render("\\sigma_{int}", element, {displayMode:false,throwOnError:false});$ describes the relaxation of the internal state of the system toward the steady state determined by the externally applied stress and is expressed as: $var element = document.getElementById("moose-equation-4c8c1876-10cd-4ad5-b7f4-f67293e9418b");katex.render("\\sigma_{int}\\left(t\\right)=\\sigma_{h}-\\left(\\sigma_{h}-\\sigma_{0}\\right)\\exp{\\left(-\\frac{t-t_{0}}{\\tau} \\right)}", element, {displayMode:true,throwOnError:false});$ Here, $var element = document.getElementById("moose-equation-bc13c4a4-c6c6-4849-9566-4946391e3b8e");katex.render("\\sigma_{int}(t=t_{0})=\\sigma_{0}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-54eb0b21-7138-4a69-85d5-8b52b58d84a3");katex.render("\\sigma_{int}(t\\rightarrow\\infty)=\\sigma_{h}", element, {displayMode:false,throwOnError:false});$ . The two equations can be written in the form such that there is no dependence on the initial values $var element = document.getElementById("moose-equation-9d71b7e7-bd0c-4eb6-82c7-fa89117558bd");katex.render("\\sigma_{0}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-0d36c3f0-705e-48e9-80e8-ff1ab2808e91");katex.render("t_{0}", element, {displayMode:false,throwOnError:false});$ and can be integrated into a finite difference form: $var element = document.getElementById("moose-equation-6f5b2330-a241-4e18-8631-929211faeca5");katex.render("\\varepsilon_{p}\\left(t+\\triangle t\\right) = \\varepsilon_{p}\\left(t\\right) + C\\left(\\sigma_{h}-\\sigma_{int}\\right)\\left[1-\\exp{\\left(-\\frac{\\triangle t}{\\tau} \\right)}\\right]", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-1b57ca03-34fc-46c0-8861-beb1037b5623");katex.render("\\sigma_{int}\\left(t+\\triangle t \\right) = \\sigma_{h} + \\left[\\sigma_{int}\\left(t\\right] - \\sigma_{h}\\right) \\exp{\\left(-\\frac{\\triangle t}{\\tau}\\right)}", element, {displayMode:true,throwOnError:false});$

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 Pérez-Prado (2000).

Hayes-Kassner Secondary Thermal Creep

The secondary thermal creep strain rate equation implemented in BISON is $(1)var element = document.getElementById("moose-equation-2f1e04cb-87b6-48d7-b611-10cb57d5fd3a");katex.render(" \\dot{\\epsilon}_{ss} = A_{0} {\\left(\\frac{\\sigma_m}{G}\\right)}^n \\exp{\\left({\\frac{-Q}{RT}}\\right)}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-63f24cd8-2a6c-494e-bb73-3125466ea04c");katex.render("\\dot{\\epsilon}_{ss}", element, {displayMode:false,throwOnError:false});$ is the effective thermal creep rate (1/s), $var element = document.getElementById("moose-equation-bf38bd16-dcad-4ad9-bec0-834d13420fa5");katex.render("\\sigma_m", element, {displayMode:false,throwOnError:false});$ is the effective (Mises) stress (Pa), $var element = document.getElementById("moose-equation-29101b82-e996-4a2c-89ab-6f6c8560f8f5");katex.render("Q", element, {displayMode:false,throwOnError:false});$ is the activation energy (J/mol), $var element = document.getElementById("moose-equation-60c49dbf-a4f3-4b73-b072-69520e22a5fc");katex.render("R", element, {displayMode:false,throwOnError:false});$ is the universal gas constant (J/mol-K), $var element = document.getElementById("moose-equation-6a249806-3dd0-4c9f-bf03-45fd4e7d8f04");katex.render("T", element, {displayMode:false,throwOnError:false});$ is the temperature (K), $var element = document.getElementById("moose-equation-ec61898d-974b-4234-bc95-e51d2300165c");katex.render("G", element, {displayMode:false,throwOnError:false});$ is the shear modulus (Pa), and $var element = document.getElementById("moose-equation-913d25da-af01-41a4-9454-87175a49a076");katex.render("A_0", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-b990bc3c-1df5-4474-9fa7-89fd36d4597c");katex.render("n", element, {displayMode:false,throwOnError:false});$ are material constants.

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

Irradiation Secondary Creep

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: $var element = document.getElementById("moose-equation-d6939f68-23e0-488c-a16c-8a996509f95a");katex.render("\\dot{\\epsilon}_{ir} = C_{0} {\\Phi}^{C_1} {\\sigma_m}^{C_2}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-756081be-2938-49a7-b263-168924f5a01f");katex.render("\\dot{\\epsilon}_{ir}", element, {displayMode:false,throwOnError:false});$ is the effective irradiation creep rate (1/s), $var element = document.getElementById("moose-equation-2f76f108-eb63-463d-b649-caa6aebdc335");katex.render("\\Phi", element, {displayMode:false,throwOnError:false});$ is the fast neutron flux (n/m $var element = document.getElementById("moose-equation-405627fe-62aa-407f-bf91-ebcf5d101a20");katex.render("^2", element, {displayMode:false,throwOnError:false});$ -s), $var element = document.getElementById("moose-equation-1c67e693-873f-4184-bd3b-8d7ecd93ee94");katex.render("\\sigma_m", element, {displayMode:false,throwOnError:false});$ is the effective (Mises) stress (MPa), and $var element = document.getElementById("moose-equation-f41e7386-f6af-4e07-b70b-c278550b1cdb");katex.render("C_0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-35ec5f9f-7c44-4ceb-adaa-ea2512caf88d");katex.render("C_1", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-bd95a6db-ea6a-4cf6-ab84-d9be4e64de97");katex.render("C_2", element, {displayMode:false,throwOnError:false});$ are material constants. The material constant values $var element = document.getElementById("moose-equation-fe0815ff-86f0-45c2-95a3-8dad90247cac");katex.render("C_0", element, {displayMode:false,throwOnError:false});$ = 3.557 $var element = document.getElementById("moose-equation-96041a50-5f0a-43b1-bebd-dbaeaa9ffd4f");katex.render("\\times", element, {displayMode:false,throwOnError:false});$ 10 $var element = document.getElementById("moose-equation-dfa79bfa-a66f-4f5f-8397-464ae59ccd45");katex.render("^{-24}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-0f80563c-0a79-41cb-beeb-4de1e940be07");katex.render("C_1", element, {displayMode:false,throwOnError:false});$ = 0.85, and $var element = document.getElementById("moose-equation-0472ddcb-b713-46c9-b9d4-9dae8138c57d");katex.render("C_2", element, {displayMode:false,throwOnError:false});$ = 1.0 are for stress relief annealed zircaloy.

Example Input Syntax

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [zrycreep]
    type = ZryCreepTulkkiHayesHoppeUpdate<<<{"description": "Computes the viscoelastic primary creep and secondary thermal Hayes and Kassner creep and the Hoppe irradiation creep for Zircaloy cladding.  This material must be run in conjunction with ComputeMultipleInelasticStress.", "href": "ZryCreepTulkkiHayesHoppeUpdate.html"}>>>
    temperature<<<{"description": "The coupled temperature (K)"}>>> = temp
    fast_neutron_flux<<<{"description": "The fast neutron flux"}>>> = fast_neutron_flux
    model_irradiation_creep<<<{"description": "Set true to activate irradiation induced creep"}>>> = true
    model_thermal_creep<<<{"description": "Set true to activate steady state thermal creep"}>>> = true
    zircaloy_material_type<<<{"description": "Type of zircaloy material properties to use in calculating creep. Note: ESCORE_IRRADIATIONGROWTHZR4 is not valid."}>>> = RECRYSTALLIZATION_ANNEALED
  []
[]

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

[Materials<<<{"href": "../../../syntax/Materials/index.html"}>>>]
  [stress]
    type = ComputeMultipleInelasticStress<<<{"description": "Compute state (stress and internal parameters such as plastic strains and internal parameters) using an iterative process.  Combinations of creep models and plastic models may be used.", "href": "../ComputeMultipleInelasticStress.html"}>>>
    tangent_operator<<<{"description": "Type of tangent operator to return.  'elastic': return the elasticity tensor.  'nonlinear': return the full, general consistent tangent operator."}>>> = elastic
    inelastic_models<<<{"description": "The material objects to use to calculate stress and inelastic strains. Note: specify creep models first and plasticity models second."}>>> = 'zrycreep'
  []
[]

Input Parameters

absolute_tolerance1e-11Absolute convergence tolerance for Newton iteration
Default:1e-11
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Absolute convergence tolerance for Newton iteration
acceptable_multiplier10Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress
Default:10
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Factor applied to relative and absolute tolerance for acceptable convergence if iterations are no longer making progress
adaptive_substeppingFalseUse adaptive substepping, where the number of substeps is successively doubled until the return mapping model successfully converges or the maximum number of substeps is reached.
Default:False
C++ Type:bool
Controllable:No
Description:Use adaptive substepping, where the number of substeps is successively doubled until the return mapping model successfully converges or the maximum number of substeps is reached.
automatic_differentiation_return_mappingFalseWhether to use automatic differentiation to compute the derivative.
Default:False
C++ Type:bool
Controllable:No
Description:Whether to use automatic differentiation to compute the derivative.
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
Controllable:No
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.
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object applies
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 computeQpProperties() 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
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
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 computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
fast_neutron_fluxThe fast neutron flux
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The fast neutron flux
max_creep_increment0.001Maximum creep strain increment allowed by accuracy time step criterion
Default:0.001
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Maximum creep strain increment allowed by accuracy time step criterion
max_inelastic_increment0.0001The maximum inelastic strain increment allowed in a time step
Default:0.0001
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The maximum inelastic strain increment allowed in a time step
maximum_number_substeps25The maximum number of substeps allowed before cutting the time step.
Default:25
C++ Type:unsigned int
Controllable:No
Description:The maximum number of substeps allowed before cutting the time step.
model_irradiation_creepTrueSet true to activate irradiation induced creep
Default:True
C++ Type:bool
Controllable:No
Description:Set true to activate irradiation induced creep
model_thermal_creepTrueSet true to activate steady state thermal creep
Default:True
C++ Type:bool
Controllable:No
Description:Set true to activate steady state thermal creep
outputThe reporting postprocessor to use for the max_iterations value.
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:The reporting postprocessor to use for the max_iterations value.
relative_tolerance1e-08Relative convergence tolerance for Newton iteration
Default:1e-08
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Relative convergence tolerance for Newton iteration
temperatureThe coupled temperature (K)
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The coupled temperature (K)
use_substep_integration_errorFalseIf true, it establishes a substep size that will yield, at most,the creep numerical integration error given by substep_strain_tolerance.
Default:False
C++ Type:bool
Controllable:No
Description:If true, it establishes a substep size that will yield, at most,the creep numerical integration error given by substep_strain_tolerance.
use_substeppingNONEWhether and how to use substepping
Default:NONE
C++ Type:MooseEnum
Options:NONE, ERROR_BASED, INCREMENT_BASED
Controllable:No
Description:Whether and how to use substepping
zircaloy_material_typeSTRESS_RELIEF_ANNEALEDType of zircaloy material properties to use in calculating creep. Note: ESCORE_IRRADIATIONGROWTHZR4 is not valid.
Default:STRESS_RELIEF_ANNEALED
C++ Type:MooseEnum
Options:STRESS_RELIEF_ANNEALED, RECRYSTALLIZATION_ANNEALED, PARTIAL_RECRYSTALLIZATION_ANNEALED, ZIRLO, M5, ESCORE_IRRADIATIONGROWTHZR4
Controllable:No
Description:Type of zircaloy material properties to use in calculating creep. Note: ESCORE_IRRADIATIONGROWTHZR4 is not valid.

Optional Parameters

apply_strainTrueFlag to apply strain. Used for testing.
Default:True
C++ Type:bool
Controllable:No
Description:Flag to apply strain. Used for testing.
control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
effective_inelastic_strain_nameeffective_creep_strainName of the material property that stores the effective inelastic strain
Default:effective_creep_strain
C++ Type:std::string
Controllable:No
Description:Name of the material property that stores the effective inelastic strain
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Controllable:No
Description:The seed for the master random number generator
substep_strain_tolerance0.1Maximum ratio of the initial elastic strain increment at start of the return mapping solve to the maximum inelastic strain allowable in a single substep. Reduce this value to increase the number of substeps
Default:0.1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Maximum ratio of the initial elastic strain increment at start of the return mapping solve to the maximum inelastic strain allowable in a single substep. Reduce this value to increase the number of substeps
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
Controllable:No
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.

Advanced Parameters

creeprate_scale_factor1scaling factor for total creep rate. Used for calibration and sensitivity studies
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:scaling factor for total creep rate. Used for calibration and sensitivity studies

Advanced: Scaling Factors Parameters

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
Controllable:No
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.
internal_solve_output_onon_errorWhen to output internal Newton solve information
Default:on_error
C++ Type:MooseEnum
Options:never, on_error, always
Controllable:No
Description:When to output internal Newton solve information

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<std::string>
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(test/tests/solid_mechanics/zry_creep/tulkki_hayes_hoppe/prim_irrad_therm_Zr4.i)
(test/tests/solid_mechanics/zry_creep/tulkki_hayes_hoppe/prim_irrad_therm_Zr2.i)

References

T. A. Hayes and M. Kassner. Creep of zirconium and zirconium alloys. Metallurgical and Materials Transactions A, 37A:2389–2396, 2006.

@ARTICLE{hayes2006,
    author = "Hayes, T. A. and Kassner, M.",
    title = "Creep of Zirconium and Zirconium Alloys",
    journal = "Metallurgical and Materials Transactions A",
    year = "2006",
    volume = "37A",
    pages = "2389-2396"
}

N. E. Hoppe. Engineering model for zircaloy creep and growth. In Proceedings of the ANS-ENS International Topical Meeting on LWR Fuel Performance, 157–172. Avignon, France, April 21-24, 1991.

@INPROCEEDINGS{hoppe91,
    author = "Hoppe, N. E.",
    title = "Engineering model for zircaloy creep and growth",
    booktitle = "Proceedings of the ANS-ENS International Topical Meeting on {LWR} Fuel Performance",
    year = "1991",
    pages = "157-172",
    address = "Avignon, France",
    month = "April 21-24,"
}

ME Kassner and M-T Pérez-Prado. Five-power-law creep in single phase metals and alloys. Progress in Materials Science, 45(1):1–102, 2000.

@article{kassner2000,
    author = "Kassner, ME and P{\'e}rez-Prado, M-T",
    title = "Five-power-law creep in single phase metals and alloys",
    journal = "Progress in Materials Science",
    volume = "45",
    number = "1",
    pages = "1-102",
    year = "2000",
    publisher = "Pergamon"
}

M. A. McGrath. In-reactor creep behavior of zircaloy-2 under variable loading conditions in ifa-585. Technical Report HWR-471, Halden, March 1996.

@TECHREPORT{HWR-471,
    author = "McGrath, M. A.",
    title = "In-reactor Creep behavior of Zircaloy-2 Under Variable Loading Conditions in IFA-585",
    institution = "Halden",
    year = "1996",
    number = "HWR-471",
    month = "March"
}

K. L. Murty. Creep studies for zircaloy life prediction in water reactors. JOM, 51(10):32–39, 1999.

@article{murty1999creep,
    author = "Murty, K. L.",
    title = "Creep studies for zircaloy life prediction in water reactors",
    journal = "{JOM}",
    volume = "51",
    number = "10",
    pages = "32-39",
    year = "1999",
    publisher = "Springer"
}

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.

@article{tulkki2014,
    author = "Tulkki, V. and Ikonen, T.",
    title = "Modeling of Zircaloy cladding primary creep during load drop and reversal",
    journal = "Journal of Nuclear Materials",
    volume = "445",
    number = "1",
    pages = "98-103",
    year = "2014",
    publisher = "Elsevier"
}

(test/tests/solid_mechanics/zry_creep/tulkki_hayes_hoppe/prim_irrad_therm_Zr2.i)

#'--------------------------------------------------------------------------------------------------------
# REGRESSION TEST FOR THE MODEL: ZryCreepTulkkiHayesHoppeUpdate [RECRYSTALLIZATION_ANNEALED / Zr-2]
# ================================================================================================
#
# PARAMETERS TESTED/CHECKED:
# ==========================
# A) Primary Creep using Tulkki Correlation (Tulkki, V., and T. Ikonen. "Modeling of Zircaloy cladding primary creep # during load drop and #                                              reversal." Journal of Nuclear Materials 445.1 # (2014): 98-103.)
#
#
# B) Secondary Creep: Irradiation + Thermal Creep
#        i) Irradiation Creep (Hoppe - BISON Theory Manual 8.12.1)
#       ii) Thermal Creep (Hayes & Kassner - BISON Theory Manual 8.12.2.1)
#
#
# GEOMETRY:
#      Ri = 0.005 m
#      Ro = 0.0055 m
#      H  = 0.01 m
#
# NO. OF ELEMENTS: Single element
#
# LOADING/INTERNAL PRESSURE (Pa):
#
#               |
#               |
#          8.0E6|___________________ "A"
#               |                   |
#               |                   |
#               |                   |
#            Pi |                   |
#               |                   |
#               |                   |
#               |                   |
#               -----------------------------
#               0        time(s)    7.3e5
# -------------------------------------------------------------------------------------------------------
#
# CALCULATIONS AT POINT "A":
# ==========================
# A) Primary Creep
#   C = 1.92E-06
#   Tau = 360000
#   sig_1 = 8.00E+07
#   sig_0 = 0
#   t = 7.30E+05
#   t_0 = 0
#   e_prim = 0.000133382    FROM BISON, e_prim = 0.00013283
#            -----------                         ----------
#
# B) Secondary Creep
#    Temperature = 630K
#    Von-Mises = 82.00 MPa or 8.2e7 Pa
#    Time (s) = 7.30E+05
#    i) Irradiation Creep
#       C0 = 1.65E-24
#       C1 = 0.85
#       C2 = 1
#       FNF = 3.00E+17
#       Irrad Strain rate (1/s) = 2.7E-11
#       e_irr = 0.0000197
#               ---------
#
#   ii) Thermal Creep
#       A0 =   3.14E+24
#       G =    3.08E+10
#       n =    4.8
#       Q =    2.70E+05
#       R =    8.3143
#       Thermal Strain rate (1/s) = 5.67546E-11
#       e_th = 0.0000414
#              ---------
#
#    e_s = e_irr + e_th = 0.0000612     FROM BISON, e_s = 0.0000603
#                         ---------                       ---------
#--------------------------------------------------------------------------------------------------------'

[Mesh]
  coord_type = RZ
  [mesh]
    type = FileMeshGenerator
    file = clad_rz.e
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Variables]
  [disp_x]
    order = FIRST
    family = LAGRANGE
  []
  [disp_y]
    order = FIRST
    family = LAGRANGE
  []
  [temp]
    order = FIRST
    family = LAGRANGE
    initial_condition = 630.0
  []
[]

[AuxVariables]
  [fast_neutron_flux]
    order = FIRST
    family = LAGRANGE
  []
  [stress_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [vonmises]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [prim_tulkki]
    order = CONSTANT
    family = MONOMIAL
  []
  [pr_sc_creep]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [pressure_ramp]
    type = PiecewiseConstant
    data_file = int_pressure.csv
    format = columns
    scale_factor = 1.0
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = finite
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
[]


[AuxKernels]
  [fast_neutron_flux]
    type = FastNeutronFluxAux
    variable = fast_neutron_flux
    factor = 3e17
  []
  [stress_yy]
   type = RankTwoAux
   rank_two_tensor = stress
   variable = stress_yy
   index_j = 1
   index_i = 1
   execute_on = timestep_end
  []
  [stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [vonmises]
   type = RankTwoScalarAux
   rank_two_tensor = stress
   variable = vonmises
   scalar_type = VonMisesStress
   execute_on = timestep_end
  []
  [elastic_strain_yy]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [elastic_strain_xx]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [elastic_strain_zz]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [creep_strain_yy]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [creep_strain_xx]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [creep_strain_zz]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [prim_tulkki]
    type = MaterialRealAux
    property = primary_viscoelastic_creep
    variable = prim_tulkki
    execute_on = timestep_end
  []
  [pr_sc_creep]
    type = MaterialRealAux
    property = creep_strain_viscoelastic
    variable = pr_sc_creep
    execute_on = timestep_end
  []
[]

[BCs]
  [Pressure]
    [internal_pressure]
      boundary = 1
      factor = 1.0
      function = pressure_ramp
    []
  []
  [u_bottom_fix]
    type = DirichletBC
    variable = disp_y
    boundary =4
    value = 0.0
  []
  [temp_bc]
    type = DirichletBC
    variable = temp
    boundary = '1 2 3 4'
    value = 630.0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 8.0e10
    poissons_ratio = 0.3
  []
  [stress]
    type = ComputeMultipleInelasticStress
    tangent_operator = elastic
    inelastic_models = 'zrycreep'
  []
  [zrycreep]
    type = ZryCreepTulkkiHayesHoppeUpdate
    temperature = temp
    fast_neutron_flux = fast_neutron_flux
    model_irradiation_creep = true
    model_thermal_creep = true
    zircaloy_material_type = RECRYSTALLIZATION_ANNEALED
  []
  [clad_thermal]
    type = HeatConductionMaterial
    block = 1
    thermal_conductivity = 100.0
    specific_heat = 1.0
  []
  [density]
    type = StrainAdjustedDensity
    block = 1
    strain_free_density = 6500
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew'

  petsc_options_iname = '-ksp_gmres_restart'
  petsc_options_value = '101'

  line_search = 'none'

  l_max_its = 100
  nl_max_its = 100
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-6
  l_tol = 1e-5
  start_time = 0.0
  end_time  = 1080001
  dt = 10000
[]

[Outputs]
  exodus = true
  csv = true
[]

(test/tests/solid_mechanics/zry_creep/tulkki_hayes_hoppe/prim_irrad_therm_Zr2.i)

#'--------------------------------------------------------------------------------------------------------
# REGRESSION TEST FOR THE MODEL: ZryCreepTulkkiHayesHoppeUpdate [RECRYSTALLIZATION_ANNEALED / Zr-2]
# ================================================================================================
#
# PARAMETERS TESTED/CHECKED:
# ==========================
# A) Primary Creep using Tulkki Correlation (Tulkki, V., and T. Ikonen. "Modeling of Zircaloy cladding primary creep # during load drop and #                                              reversal." Journal of Nuclear Materials 445.1 # (2014): 98-103.)
#
#
# B) Secondary Creep: Irradiation + Thermal Creep
#        i) Irradiation Creep (Hoppe - BISON Theory Manual 8.12.1)
#       ii) Thermal Creep (Hayes & Kassner - BISON Theory Manual 8.12.2.1)
#
#
# GEOMETRY:
#      Ri = 0.005 m
#      Ro = 0.0055 m
#      H  = 0.01 m
#
# NO. OF ELEMENTS: Single element
#
# LOADING/INTERNAL PRESSURE (Pa):
#
#               |
#               |
#          8.0E6|___________________ "A"
#               |                   |
#               |                   |
#               |                   |
#            Pi |                   |
#               |                   |
#               |                   |
#               |                   |
#               -----------------------------
#               0        time(s)    7.3e5
# -------------------------------------------------------------------------------------------------------
#
# CALCULATIONS AT POINT "A":
# ==========================
# A) Primary Creep
#   C = 1.92E-06
#   Tau = 360000
#   sig_1 = 8.00E+07
#   sig_0 = 0
#   t = 7.30E+05
#   t_0 = 0
#   e_prim = 0.000133382    FROM BISON, e_prim = 0.00013283
#            -----------                         ----------
#
# B) Secondary Creep
#    Temperature = 630K
#    Von-Mises = 82.00 MPa or 8.2e7 Pa
#    Time (s) = 7.30E+05
#    i) Irradiation Creep
#       C0 = 1.65E-24
#       C1 = 0.85
#       C2 = 1
#       FNF = 3.00E+17
#       Irrad Strain rate (1/s) = 2.7E-11
#       e_irr = 0.0000197
#               ---------
#
#   ii) Thermal Creep
#       A0 =   3.14E+24
#       G =    3.08E+10
#       n =    4.8
#       Q =    2.70E+05
#       R =    8.3143
#       Thermal Strain rate (1/s) = 5.67546E-11
#       e_th = 0.0000414
#              ---------
#
#    e_s = e_irr + e_th = 0.0000612     FROM BISON, e_s = 0.0000603
#                         ---------                       ---------
#--------------------------------------------------------------------------------------------------------'

[Mesh]
  coord_type = RZ
  [mesh]
    type = FileMeshGenerator
    file = clad_rz.e
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Variables]
  [disp_x]
    order = FIRST
    family = LAGRANGE
  []
  [disp_y]
    order = FIRST
    family = LAGRANGE
  []
  [temp]
    order = FIRST
    family = LAGRANGE
    initial_condition = 630.0
  []
[]

[AuxVariables]
  [fast_neutron_flux]
    order = FIRST
    family = LAGRANGE
  []
  [stress_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [vonmises]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [prim_tulkki]
    order = CONSTANT
    family = MONOMIAL
  []
  [pr_sc_creep]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [pressure_ramp]
    type = PiecewiseConstant
    data_file = int_pressure.csv
    format = columns
    scale_factor = 1.0
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = finite
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
[]


[AuxKernels]
  [fast_neutron_flux]
    type = FastNeutronFluxAux
    variable = fast_neutron_flux
    factor = 3e17
  []
  [stress_yy]
   type = RankTwoAux
   rank_two_tensor = stress
   variable = stress_yy
   index_j = 1
   index_i = 1
   execute_on = timestep_end
  []
  [stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [vonmises]
   type = RankTwoScalarAux
   rank_two_tensor = stress
   variable = vonmises
   scalar_type = VonMisesStress
   execute_on = timestep_end
  []
  [elastic_strain_yy]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [elastic_strain_xx]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [elastic_strain_zz]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [creep_strain_yy]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [creep_strain_xx]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [creep_strain_zz]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [prim_tulkki]
    type = MaterialRealAux
    property = primary_viscoelastic_creep
    variable = prim_tulkki
    execute_on = timestep_end
  []
  [pr_sc_creep]
    type = MaterialRealAux
    property = creep_strain_viscoelastic
    variable = pr_sc_creep
    execute_on = timestep_end
  []
[]

[BCs]
  [Pressure]
    [internal_pressure]
      boundary = 1
      factor = 1.0
      function = pressure_ramp
    []
  []
  [u_bottom_fix]
    type = DirichletBC
    variable = disp_y
    boundary =4
    value = 0.0
  []
  [temp_bc]
    type = DirichletBC
    variable = temp
    boundary = '1 2 3 4'
    value = 630.0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 8.0e10
    poissons_ratio = 0.3
  []
  [stress]
    type = ComputeMultipleInelasticStress
    tangent_operator = elastic
    inelastic_models = 'zrycreep'
  []
  [zrycreep]
    type = ZryCreepTulkkiHayesHoppeUpdate
    temperature = temp
    fast_neutron_flux = fast_neutron_flux
    model_irradiation_creep = true
    model_thermal_creep = true
    zircaloy_material_type = RECRYSTALLIZATION_ANNEALED
  []
  [clad_thermal]
    type = HeatConductionMaterial
    block = 1
    thermal_conductivity = 100.0
    specific_heat = 1.0
  []
  [density]
    type = StrainAdjustedDensity
    block = 1
    strain_free_density = 6500
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew'

  petsc_options_iname = '-ksp_gmres_restart'
  petsc_options_value = '101'

  line_search = 'none'

  l_max_its = 100
  nl_max_its = 100
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-6
  l_tol = 1e-5
  start_time = 0.0
  end_time  = 1080001
  dt = 10000
[]

[Outputs]
  exodus = true
  csv = true
[]

(test/tests/solid_mechanics/zry_creep/tulkki_hayes_hoppe/prim_irrad_therm_Zr4.i)

#'--------------------------------------------------------------------------------------------------------
# REGRESSION TEST FOR THE MODEL: ZryCreepTulkkiHayesHoppeUpdate [STRESS_RELIEF_ANNEALED / Zr-4]
# =============================================================================================
#
# PARAMETERS TESTED/CHECKED:
# ==========================
# A) Primary Creep using Tulkki Correlation (Tulkki, V., and T. Ikonen. "Modeling of Zircaloy cladding primary creep # during load drop and #                                              reversal." Journal of Nuclear Materials 445.1 # (2014): 98-103.)
#
#
# B) Secondary Creep: Irradiation + Thermal Creep
#        i) Irradiation Creep (Hoppe - BISON Theory Manual 8.12.1)
#       ii) Thermal Creep (Hayes & Kassner - BISON Theory Manual 8.12.2.1)
#
#
# GEOMETRY:
#      Ri = 0.005 m
#      Ro = 0.0055 m
#      H  = 0.01 m
#
# NO. OF ELEMENTS: Single element
#
# LOADING/INTERNAL PRESSURE (Pa):
#
#               |
#               |
#          8.0E6|___________________ "A"
#               |                   |
#               |                   |
#               |                   |
#            Pi |                   |
#               |                   |
#               |                   |
#               |                   |
#               -----------------------------
#               0        time(s)    7.3e5
# -------------------------------------------------------------------------------------------------------
#
# CALCULATIONS AT POINT "A":
# ==========================
# A) Primary Creep
#   C = 5.0E-06
#   Tau = 144000
#   sig_1 = 8.00E+07
#   sig_0 = 0
#   t = 7.30E+05
#   t_0 = 0
#   e_prim = 0.000397486    FROM BISON, e_prim = 0.000397344
#            -----------                         -----------
#
# B) Secondary Creep
#    Temperature = 630K
#    Von-Mises = 82.00 MPa or 8.2e7 Pa
#    Time (s) = 7.30E+05
#    i) Irradiation Creep
#       C0 = 3.557E-24
#       C1 = 0.85
#       C2 = 1
#       FNF = 3.00E+17
#       Irrad Strain rate (1/s) = 5.81E-11
#       e_irr = 0.0000424
#               ---------
#
#   ii) Thermal Creep
#       A0 =   3.14E+24
#       G =    3.08E+10
#       n =    5.0
#       Q =    2.70E+05
#       R =    8.3143
#       Thermal Strain rate (1/s) = 1.73437E-11
#       e_th = 0.0000127
#              ---------
#
#    e_s = e_irr + e_th = 0.0000551     FROM BISON, e_s = 0.0000543
#                         ---------                       ---------
#--------------------------------------------------------------------------------------------------------'

[Mesh]
  coord_type = RZ
  [mesh]
    type = FileMeshGenerator
    file = clad_rz.e
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Variables]
  [disp_x]
    order = FIRST
    family = LAGRANGE
  []
  [disp_y]
    order = FIRST
    family = LAGRANGE
  []
  [temp]
    order = FIRST
    family = LAGRANGE
    initial_condition = 630.0
  []
[]

[AuxVariables]
  [fast_neutron_flux]
    order = FIRST
    family = LAGRANGE
  []
  [stress_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [vonmises]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [prim_tulkki]
    order = CONSTANT
    family = MONOMIAL
  []
  [pr_sc_creep] # primary plus secondary creep
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [pressure_ramp]
    type = PiecewiseConstant
    data_file = int_pressure.csv
    format = columns
    scale_factor = 1.0
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = finite
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
[]


[AuxKernels]
  [fast_neutron_flux]
    type = FastNeutronFluxAux
    variable = fast_neutron_flux
    factor = 3e17
  []
  [stress_yy]
   type = RankTwoAux
   rank_two_tensor = stress
   variable = stress_yy
   index_j = 1
   index_i = 1
   execute_on = timestep_end
  []
  [stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [vonmises]
   type = RankTwoScalarAux
   rank_two_tensor = stress
   variable = vonmises
   scalar_type = VonMisesStress
   execute_on = timestep_end
  []
  [elastic_strain_yy]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [elastic_strain_xx]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [elastic_strain_zz]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [creep_strain_yy]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [creep_strain_xx]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [creep_strain_zz]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [prim_tulkki]
    type = MaterialRealAux
    property = primary_viscoelastic_creep
    variable = prim_tulkki
    execute_on = timestep_end
  []
  [pr_sc_creep]
    type = MaterialRealAux
    property = creep_strain_viscoelastic
    variable = pr_sc_creep
    execute_on = timestep_end
  []
[]

[BCs]
  [Pressure]
    [internal_pressure]
      boundary = 1
      factor = 1.0
      function = pressure_ramp
    []
  []
  [u_bottom_fix]
    type = DirichletBC
    variable = disp_y
    boundary =4
    value = 0.0
  []
  [temp_bc]
    type = DirichletBC
    variable = temp
    boundary = '1 2 3 4'
    value = 630.0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 8.0e10
    poissons_ratio = 0.3
  []
  [stress]
    type = ComputeMultipleInelasticStress
    tangent_operator = elastic
    inelastic_models = 'zrycreep'
  []
  [zrycreep]
    type = ZryCreepTulkkiHayesHoppeUpdate
    temperature = temp
    fast_neutron_flux = fast_neutron_flux
    model_irradiation_creep = true
    model_thermal_creep = true
    zircaloy_material_type = stress_relief_annealed
  []
  [clad_thermal]
    type = HeatConductionMaterial
    block = 1
    thermal_conductivity = 100.0
    specific_heat = 1.0
  []
  [density]
    type = StrainAdjustedDensity
    block = 1
    strain_free_density = 6500
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew'

  petsc_options_iname = '-ksp_gmres_restart'
  petsc_options_value = '101'

  line_search = 'none'


  l_max_its = 100
  nl_max_its = 100
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-6
  l_tol = 1e-5
  start_time = 0.0
  end_time  = 1080001
  dt = 10000
[]

[Outputs]
  exodus = true
  csv = true
[]

(test/tests/solid_mechanics/zry_creep/tulkki_hayes_hoppe/prim_irrad_therm_Zr2.i)

#'--------------------------------------------------------------------------------------------------------
# REGRESSION TEST FOR THE MODEL: ZryCreepTulkkiHayesHoppeUpdate [RECRYSTALLIZATION_ANNEALED / Zr-2]
# ================================================================================================
#
# PARAMETERS TESTED/CHECKED:
# ==========================
# A) Primary Creep using Tulkki Correlation (Tulkki, V., and T. Ikonen. "Modeling of Zircaloy cladding primary creep # during load drop and #                                              reversal." Journal of Nuclear Materials 445.1 # (2014): 98-103.)
#
#
# B) Secondary Creep: Irradiation + Thermal Creep
#        i) Irradiation Creep (Hoppe - BISON Theory Manual 8.12.1)
#       ii) Thermal Creep (Hayes & Kassner - BISON Theory Manual 8.12.2.1)
#
#
# GEOMETRY:
#      Ri = 0.005 m
#      Ro = 0.0055 m
#      H  = 0.01 m
#
# NO. OF ELEMENTS: Single element
#
# LOADING/INTERNAL PRESSURE (Pa):
#
#               |
#               |
#          8.0E6|___________________ "A"
#               |                   |
#               |                   |
#               |                   |
#            Pi |                   |
#               |                   |
#               |                   |
#               |                   |
#               -----------------------------
#               0        time(s)    7.3e5
# -------------------------------------------------------------------------------------------------------
#
# CALCULATIONS AT POINT "A":
# ==========================
# A) Primary Creep
#   C = 1.92E-06
#   Tau = 360000
#   sig_1 = 8.00E+07
#   sig_0 = 0
#   t = 7.30E+05
#   t_0 = 0
#   e_prim = 0.000133382    FROM BISON, e_prim = 0.00013283
#            -----------                         ----------
#
# B) Secondary Creep
#    Temperature = 630K
#    Von-Mises = 82.00 MPa or 8.2e7 Pa
#    Time (s) = 7.30E+05
#    i) Irradiation Creep
#       C0 = 1.65E-24
#       C1 = 0.85
#       C2 = 1
#       FNF = 3.00E+17
#       Irrad Strain rate (1/s) = 2.7E-11
#       e_irr = 0.0000197
#               ---------
#
#   ii) Thermal Creep
#       A0 =   3.14E+24
#       G =    3.08E+10
#       n =    4.8
#       Q =    2.70E+05
#       R =    8.3143
#       Thermal Strain rate (1/s) = 5.67546E-11
#       e_th = 0.0000414
#              ---------
#
#    e_s = e_irr + e_th = 0.0000612     FROM BISON, e_s = 0.0000603
#                         ---------                       ---------
#--------------------------------------------------------------------------------------------------------'

[Mesh]
  coord_type = RZ
  [mesh]
    type = FileMeshGenerator
    file = clad_rz.e
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Variables]
  [disp_x]
    order = FIRST
    family = LAGRANGE
  []
  [disp_y]
    order = FIRST
    family = LAGRANGE
  []
  [temp]
    order = FIRST
    family = LAGRANGE
    initial_condition = 630.0
  []
[]

[AuxVariables]
  [fast_neutron_flux]
    order = FIRST
    family = LAGRANGE
  []
  [stress_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [stress_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [vonmises]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [elastic_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_yy]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_xx]
    order = CONSTANT
    family = MONOMIAL
  []
  [creep_strain_zz]
    order = CONSTANT
    family = MONOMIAL
  []
  [prim_tulkki]
    order = CONSTANT
    family = MONOMIAL
  []
  [pr_sc_creep]
    order = CONSTANT
    family = MONOMIAL
  []
[]

[Functions]
  [pressure_ramp]
    type = PiecewiseConstant
    data_file = int_pressure.csv
    format = columns
    scale_factor = 1.0
  []
[]

[Physics/SolidMechanics/QuasiStatic]
  [all]
    strain = finite
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temp
  []
  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temp
  []
[]


[AuxKernels]
  [fast_neutron_flux]
    type = FastNeutronFluxAux
    variable = fast_neutron_flux
    factor = 3e17
  []
  [stress_yy]
   type = RankTwoAux
   rank_two_tensor = stress
   variable = stress_yy
   index_j = 1
   index_i = 1
   execute_on = timestep_end
  []
  [stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [vonmises]
   type = RankTwoScalarAux
   rank_two_tensor = stress
   variable = vonmises
   scalar_type = VonMisesStress
   execute_on = timestep_end
  []
  [elastic_strain_yy]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [elastic_strain_xx]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [elastic_strain_zz]
    type = RankTwoAux
    rank_two_tensor = elastic_strain
    variable = elastic_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [creep_strain_yy]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_yy
    index_j = 1
    index_i = 1
    execute_on = timestep_end
  []
  [creep_strain_xx]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_xx
    index_j = 0
    index_i = 0
    execute_on = timestep_end
  []
  [creep_strain_zz]
    type = RankTwoAux
    rank_two_tensor = creep_strain
    variable = creep_strain_zz
    index_j = 2
    index_i = 2
    execute_on = timestep_end
  []
  [prim_tulkki]
    type = MaterialRealAux
    property = primary_viscoelastic_creep
    variable = prim_tulkki
    execute_on = timestep_end
  []
  [pr_sc_creep]
    type = MaterialRealAux
    property = creep_strain_viscoelastic
    variable = pr_sc_creep
    execute_on = timestep_end
  []
[]

[BCs]
  [Pressure]
    [internal_pressure]
      boundary = 1
      factor = 1.0
      function = pressure_ramp
    []
  []
  [u_bottom_fix]
    type = DirichletBC
    variable = disp_y
    boundary =4
    value = 0.0
  []
  [temp_bc]
    type = DirichletBC
    variable = temp
    boundary = '1 2 3 4'
    value = 630.0
  []
[]

[Materials]
  [elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 8.0e10
    poissons_ratio = 0.3
  []
  [stress]
    type = ComputeMultipleInelasticStress
    tangent_operator = elastic
    inelastic_models = 'zrycreep'
  []
  [zrycreep]
    type = ZryCreepTulkkiHayesHoppeUpdate
    temperature = temp
    fast_neutron_flux = fast_neutron_flux
    model_irradiation_creep = true
    model_thermal_creep = true
    zircaloy_material_type = RECRYSTALLIZATION_ANNEALED
  []
  [clad_thermal]
    type = HeatConductionMaterial
    block = 1
    thermal_conductivity = 100.0
    specific_heat = 1.0
  []
  [density]
    type = StrainAdjustedDensity
    block = 1
    strain_free_density = 6500
  []
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'

  petsc_options = '-snes_ksp_ew'

  petsc_options_iname = '-ksp_gmres_restart'
  petsc_options_value = '101'

  line_search = 'none'

  l_max_its = 100
  nl_max_its = 100
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-6
  l_tol = 1e-5
  start_time = 0.0
  end_time  = 1080001
  dt = 10000
[]

[Outputs]
  exodus = true
  csv = true
[]

Description
Primary Creep
Thermal Creep in Standard Operating Conditions
Irradiation Secondary Creep
Example Input Syntax
Input Parameters
Input Files
References