# Thermal and Irradiation Creep for UO2

Models the creep behavior of UO2

warning:Deprecated Solid Mechanics Material

The functionality of this solid mechanics material is being replaced in the TensorMechanics system by UO2CreepUpdate.

## Description

A model for combined secondary thermal creep and irradiation creep of UO fuel is available, with the creep rate modeled as a function of time, temperature, effective stress, density, grain size, fission rate, and oxygen to metal ratio (O/M). The constitutive relation is taken from the MATPRO FCREEP material model (Allison et al. (1993)) and given as (1) where is the creep rate (1/s), is the effective (Mises) stress (Pa), is the temperature (K), is the fuel density (percent of theoretical), is the grain size (, is the volumetric fission rate (fissions/-s), are the activation energies (J/mol), and is the universal gas constant (8.3143 J/mol-K).

The first term in Eq. 1 represents diffusional thermal creep and is applicable to low stress and low temperature conditions. The second term in Eq. 1 represents thermal dislocation or power-law creep and is applicable to high stress and high temperature conditions. Note that irradiation creep is included in both the first and third terms, as both are functions of the fission rate (). Irradiation enhances normal diffusional thermal creep (at elevated temperatures) as indicated in the first term. Irradiation creep also occurs at lower temperatures where thermal creep is not active, and is accounted for by the third term in Eq. 1.

Material constants for the thermal creep terms, as given in the MATPRO documentation (Allison et al. (1993)), are shown in the table below.

Table 1: Parameters used in the UO Creep Model

ParameterValue

The activation energies for the thermal creep terms (Q and Q) are strongly dependent upon the fuel oxygen to metal ratio and, in MATPRO, are defined using the Arrhenius type relations (2) (3)

where the energies are given in J/mole and

(4)

This function is plotted in Figure 1.

Figure 1: The function defining the dependence of the activation energies for thermal creep on the UO oxygen to metal ratio.

In MATPRO, a transition stress is defined to govern the transition between the first (low stress) and second (high stress) regions. When the applied stress is larger than the transition stress, the applied stress is used in the power-law relation and the transition stress is used in the linear creep relation. When the applied stress is lower than the transition stress, the applied stress is used in the linear relation and the power-law contribution is zero. Mai et al. (2010) investigated the MATPRO transition approach in comparison to experimental data and concluded that a better fit to the data could be achieved by simply ignoring the transition stress and applying both the low and high stress terms in all cases. This approach has been adopted in Bison.

The low temperature irradiation creep (third term in Eq. 1) requires further discussion. As the equation indicates, the original MATPRO formulation included exponential temperature dependence in this term. After a careful review of the MATPRO documentation and supporting literature, it is clear that this temperature dependency was based on a single experimental study (Brucklacher et al. (1973)) involving only a very limited number of experiments. Additionally, this early study specified a relatively narrow applicability range for the correlation (523 < T(K) < 773), which was not enforced in the MATPRO implementation. Several other studies, both early (Brucklacher and Dienst (1972), Perrin (1972), Solomon (1973), and Dienst (1977)) and more recent (Sakai et al. (2011), Sakai (2013), and Szoke and Tverberg (2014)) did not observe temperature dependency for low temperature irradiation creep, reporting creep rate variation as a function of only stress and fission rate. The most extensive data for irradiation creep of UO has been generated at the Halden Research Reactor, resulting in the following correlation (Sakai (2013)): (5) where A = .

The default model in Bison, which is referred to as the MATPRO-Halden model, replaces the third term in Eq. 1 with Eq. 5, thus removing temperature dependency. Note that the temperature dependent formulation can still be specified simply by providing the original MATPRO material constants (given in the table below) as input parameters.

Table 2: Irradiation creep material parameters for the original MATPRO model which included temperature dependence

ParameterValue
1/K

The procedure outlined previously for time-independent plasticity was used to implement time-dependent plasticity (creep). Young's modulus, Poisson's ratio, and the coefficient of thermal expansion can each be specified in two ways. The values can be given directly, or computed using MATPRO correlations.

## Example Input Syntax


[./creep]
type = CreepUO2
block = 1
disp_x = disp_x
disp_y = disp_y
disp_z = disp_z
temp = temp
fission_rate = fission_rate
youngs_modulus = 2.e11
poissons_ratio = .3
oxy_to_metal_ratio = 2.0
[../]

## Input Parameters

• densityInitial fuel density

C++ Type:double

Description:Initial fuel density

### Required Parameters

• constitutive_modelConstitutiveModel to use (optional)

C++ Type:std::string

Description:ConstitutiveModel to use (optional)

• cracking_stress0The stress threshold beyond which cracking occurs. Must be positive.

Default:0

C++ Type:double

Description:The stress threshold beyond which cracking occurs. Must be positive.

• isotropic_crackingFalseFlag for activating isotropic cracking model

Default:False

C++ Type:bool

Description:Flag for activating isotropic cracking model

• matpro_poissons_ratioFalseFlag for using MATPRO to compute Poisson's ratio

Default:False

C++ Type:bool

Description:Flag for using MATPRO to compute Poisson's ratio

• cracking_releaseabruptThe cracking release type. Choices are abrupt (default) and exponential.

Default:abrupt

C++ Type:std::string

Description:The cracking release type. Choices are abrupt (default) and exponential.

• bulk_modulusThe bulk modulus for the material.

C++ Type:double

Description:The bulk modulus for the material.

• formulationElement formulation. Choices are: Nonlinear3D NonlinearRZ AxisymmetricRZ SphericalR Linear PlaneStrain NonlinearPlaneStrain

C++ Type:MooseEnum

Description:Element formulation. Choices are: Nonlinear3D NonlinearRZ AxisymmetricRZ SphericalR Linear PlaneStrain NonlinearPlaneStrain

• cracking_beta1The coefficient used in the exponetional model.

Default:1

C++ Type:double

Description:The coefficient used in the exponetional model.

• burnup_functionBurnup function

C++ Type:BurnupFunctionName

Description:Burnup function

• compute_InteractionIntegralFalseWhether to compute the Interaction Integral.

Default:False

C++ Type:bool

Description:Whether to compute the Interaction Integral.

• thermal_expansion_reference_temperatureReference temperature for mean thermal expansion function.

C++ Type:double

Description:Reference temperature for mean thermal expansion function.

• stress_free_temperatureThe stress-free temperature. If not specified, the initial temperature is used.

C++ Type:double

Description:The stress-free temperature. If not specified, the initial temperature is used.

• q30Activation energy for irradiation creep, divided by gas constant (1/K)

Default:0

C++ Type:double

Description:Activation energy for irradiation creep, divided by gas constant (1/K)

• youngs_modulusYoung's modulus of the material.

C++ Type:double

Description:Young's modulus of the material.

• oxy_to_metal_ratio2Oxygen to metal ratio

Default:2

C++ Type:double

Description:Oxygen to metal ratio

• initial_stressThe initial stress tensor (xx, yy, zz, xy, yz, zx)

C++ Type:std::vector

Description:The initial stress tensor (xx, yy, zz, xy, yz, zx)

• poissons_ratioPoisson's ratio for the material.

C++ Type:double

Description:Poisson's ratio for the material.

• compute_methodThe method used in the stress calculation.

C++ Type:MooseEnum

Description:The method used in the stress calculation.

• increment_calculationRashidApproxThe algorithm to use when computing the incremental strain and rotation (RashidApprox or Eigen). For use with Nonlinear3D/RZ formulation.

Default:RashidApprox

C++ Type:std::string

Description:The algorithm to use when computing the incremental strain and rotation (RashidApprox or Eigen). For use with Nonlinear3D/RZ formulation.

• strain_zzThe zz strain

C++ Type:std::vector

Description:The zz strain

• rod_ave_lin_powAverage linear power function

C++ Type:FunctionName

Description:Average linear power function

• volumetric_locking_correctionTrueSet to false to turn off volumetric locking correction

Default:True

C++ Type:bool

Description:Set to false to turn off volumetric locking correction

• output_iteration_infoFalseSet true to output sub-newton iteration information

Default:False

C++ Type:bool

Description:Set true to output sub-newton iteration information

• matpro_thermal_expansionFalseFlag for using MATPRO to compute the thermal expansion coefficient

Default:False

C++ Type:bool

Description:Flag for using MATPRO to compute the thermal expansion coefficient

• 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

• youngs_modulus_functionYoung's modulus as a function of temperature.

C++ Type:FunctionName

Description:Young's modulus as a function of temperature.

• active_crack_planesPlanes on which cracks are allowed (0,1,2 -> x,z,theta in RZ)

C++ Type:std::vector

Description:Planes on which cracks are allowed (0,1,2 -> x,z,theta in RZ)

• disp_zThe z displacement

C++ Type:std::vector

Description:The z displacement

• disp_yThe y displacement

C++ Type:std::vector

Description:The y displacement

• cracking_residual_stress0The fraction of the cracking stress allowed to be maintained following a crack.

Default:0

C++ Type:double

Description:The fraction of the cracking stress allowed to be maintained following a crack.

• shear_modulusThe shear modulus of the material.

C++ Type:double

Description:The shear modulus of the material.

• scalar_strain_zzThe zz strain (scalar variable)

C++ Type:std::vector

Description:The zz strain (scalar variable)

• thermal_expansion_function_typeType of thermal expansion function. Choices are: instantaneous mean

C++ Type:MooseEnum

Description:Type of thermal expansion function. Choices are: instantaneous mean

• disp_rThe r displacement

C++ Type:std::vector

Description:The r displacement

• appended_property_nameName appended to material properties to make them unique

C++ Type:std::string

Description:Name appended to material properties to make them unique

• a77.78e-37Coefficient on irradiation creep term

Default:7.78e-37

C++ Type:double

Description:Coefficient on irradiation creep term

• outputThe reporting postprocessor to use for the max_iterations value.

C++ Type:PostprocessorName

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

• poissons_ratio_functionPoisson's ratio as a function of temperature.

C++ Type:FunctionName

Description:Poisson's ratio as a function of temperature.

• dep_matl_propsNames of material properties this material depends on.

C++ Type:std::vector

Description:Names of material properties this material depends on.

• disp_xThe x displacement

C++ Type:std::vector

Description:The x displacement

• large_strainFalseWhether to include large strain terms in AxisymmetricRZ, SphericalR, and PlaneStrain formulations.

Default:False

C++ Type:bool

Description:Whether to include large strain terms in AxisymmetricRZ, SphericalR, and PlaneStrain formulations.

• store_stress_olderFalseParameter which indicates whether the older stress state, required for HHT time integration, needs to be stored

Default:False

C++ Type:bool

Description:Parameter which indicates whether the older stress state, required for HHT time integration, needs to be stored

Default:1e-05

C++ Type:double

Description:Fuel grain radius (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.

• tempCoupled Temperature

C++ Type:std::vector

Description:Coupled Temperature

• fission_rateCoupled fission rate

C++ Type:std::vector

Description:Coupled fission rate

• po2_fraction0Weight fraction of PO2

Default:0

C++ Type:double

Description:Weight fraction of PO2

• matpro_youngs_modulusFalseFlag for using MATPRO to compute Young's modulus

Default:False

C++ Type:bool

Description:Flag for using MATPRO to compute Young's modulus

• max_its10Maximum number of sub-newton iterations

Default:10

C++ Type:unsigned int

Description:Maximum number of sub-newton iterations

• cracking_neg_fractionThe fraction of the cracking strain at which a transitition begins during decreasing strain to the original stiffness.

C++ Type:double

Description:The fraction of the cracking strain at which a transitition begins during decreasing strain to the original stiffness.

• thermal_expansion_functionThermal expansion coefficient as a function of temperature.

C++ Type:FunctionName

Description:Thermal expansion coefficient as a function of temperature.

• compute_JIntegralFalseWhether to compute the J Integral.

Default:False

C++ Type:bool

Description:Whether to compute the J Integral.

• max_cracks3The maximum number of cracks allowed at a material point.

Default:3

C++ Type:unsigned int

Description:The maximum number of cracks allowed at a material point.

• 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

• thermal_expansionThe thermal expansion coefficient.

C++ Type:double

Description:The thermal expansion coefficient.

• cracking_stress_functionThe cracking stress as a function of time and location

C++ Type:FunctionName

Description:The cracking stress as a function of time and location

• absolute_tolerance1e-20Absolute convergence tolerance for sub-newtion iteration

Default:1e-20

C++ Type:double

Description:Absolute convergence tolerance for sub-newtion iteration

• relative_tolerance0.0001Relative convergence tolerance for sub-newtion iteration

Default:0.0001

C++ Type:double

Description:Relative convergence tolerance for sub-newtion iteration

• burnupCoupled burnup

C++ Type:std::vector

Description:Coupled burnup

• lambdaLame's first parameter for the material.

C++ Type:double

Description:Lame's first parameter for the material.

### 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. 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. D. Brucklacher and W. Dienst. Creep behavior of ceramic nuclear fuels under neutron irradiation. Journal of Nuclear Materials, 42:285â€“296, 1972.[BibTeX]
3. D. Brucklacher, W. Dienst, and F. Thummler. Creep behavior of oxide fuels under neutron irradiation, Paper 251. In Proceedings REAKTORTAGUNG 1973 des Deutschen Atomforums/KTG held in Karlsruhe, 413â€“416. Apr 10â€“13, 1973.[BibTeX]
4. W. Dienst. Irradiation induced creep of ceramic materials. Journal of Nuclear Materials, 65:1â€“8, 1977.[BibTeX]
5. A. T. Mai, W. F. Lyon, R. O. Montgomery, and R. S. Dunham. An evaluation of the MATPRO fuel creep model using the FALCON fuel analysis code. Trans. Am. Nucl. Soc., 102:888–889, 2010.[BibTeX]
6. J. S. Perrin. Effect of irradiation on creep of UO_2-PuO_2. Journal of Nuclear Materials, 42:101â€“104, 1972.[BibTeX]
7. K. Sakai. The fuel creep test IFA-701: results after four irradiation cycles. Technical Report HWR-1039, OECD Halden Reactor Project, 2013.[BibTeX]
8. K. Sakai, H. Hanakawa, and T. Tverberg. Investigation of fission induced creep of UO2 and Cr-doped fuel in IFA-701. Technical Report HWR-1006, OECD Halden Reactor Project, 2011.[BibTeX]
9. A. A. Solomon. Radiation induced creep of UO_2. Journal of the American Ceramic Society, 56:164â€“171, 1973.[BibTeX]
10. R. Szoke and T. Tverberg. Update on in-pile results from the fuel creep test IFA-701. Technical Report HWR-1092, OECD Halden Reactor Project, 2014.[BibTeX]