# Mech UPuZr

Calculates mechanical properties and thermal expansion of U-Pu-Zr fast reactor fuel

warning:Deprecated Solid Mechanics Material

The functionality of this solid mechanics material is being replaced in the TensorMechanics system:

## Description

MechUPuZr is used to calculate mechanical properties and thermal expansion, as well to model thermal and irradiation creep behavior of U-Pu-Zr fast reactor fuel. If porosity and porosity_material are both defined an error will result. The user has the choice of not supplying any porosity, supplying a variable for porosity or couple to a material that calculates the porosity.

Due to the redistribution of zirconium in the fuel, and the need to model both U-Zr and U -Pu-Zr fuel, correlations included in MechUPuZr must be applicable to any concentration of plutonium or zirconium. Unfortunately, the available models for the elastic properties of U-Pu-Zr alloys are generally incomplete. Regardless, some correlations can be made given the sparse data, as described in the IFR Property Evaluation Working Group (1988). These correlations are described below.

## Elastic Constants

The correlation for Young's modulus can be derived using data from U-11Pu-6.3Zr cast fuel pins as: (1) where E=1.6 10 MPa is the Young's modulus for pure U at the reference temperature of 588 K, P is the fractional porosity, W is the zirconium weight fraction, W is the plutonium weight fraction, T is the temperature in K, and T = 1405 K is the melting temperature of pure uranium.

The correlation for Poisson's ratio can be derived from the U-Zr correlation as: (2) where is the temperature in K. = 0.24 is the Poisson's ratio of pure U at the reference temperature of 588 K.

## Thermal expansion

Using the available thermal expansion data for U-Zr (Saller et al., 1956) and U-Pu-Zr (IFR Property Evaluation Working Group, 1988), a model can be created that accounts for any variation in or . In general, the thermal expansion of U-Zr and U-Pu-Zr fuels can be described as a high and low temperature temperature region of nearly linear expansion, separated by a jump in the total expansion corresponding to the phase transition. For U-Zr, the high and low temperature regions can be described as, (3) where the terms are constant coefficients, is the temperature in K, and is the mole fraction of zirconium. In U-Pu-Zr fuels, the high and low temperature regions can be determined by, (4) where is the plutonium mole fraction, is calculated using, (5) and, (6)

The coefficients used for and in the high and low temperature regions are listed in Table 1.

Table 1: Coefficients for the U-Pu-Zr thermal expansion model.

CoefficientLow temperatureHigh temperature
-0.29500.9325
0.7236-.003032
0.99800.9738
0.29070.5636
0.16500.01890
-1.152-2.099
0.9091-16.44
0.387523.97
2.5043-8.002
-14.41431.65
35.19-48.12
2.636-3.330

Given the mathematical representations of the high and low temperature regions, simple linear interpolation can be used to link the two regions: (7) where is calculated at the starting and ending transition temperatures as noted.

The start and end temperatures of the transition regions can be estimated from the data as, (8)

The precision of the model can be estimated through the difference between the measured and estimated expansion, as well as the percent difference between the measured and estimated expansion. The model described above results in a standard deviation of the differences of 0.03, (2.4% for percent difference), an average difference of , (2% for percent difference), and an maximum difference of 0.1, (18% for percent difference).

## Example Input Syntax

[./creep]
type = MechUPuZr
block = 1
temp = temp
fission_rate = fission_rate
porosity = 0
youngs_modulus = 7.5E10
poissons_ratio = .3
model_creep = true
calc_youngs = false
calc_poissons = false
formulation = linear
[../]
(test/tests/mechTests/mechUPuZr/3d_test.i)

## Input 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.

• 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.

• 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

• thermal_expansion_functionThermal expansion coefficient as a function of temperature.

C++ Type:FunctionName

Description:Thermal expansion coefficient as a function of temperature.

• 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.

• 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.

• 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)

• X_PuCoupled plutonium atom fraction

C++ Type:std::vector

Description:Coupled plutonium atom fraction

• 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

• 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.

• fission_rate_materialfission_rate_materialFission rate material property name

Default:fission_rate_material

C++ Type:MaterialPropertyName

Description:Fission rate material property name

• A_Zr0.091224Atomic weight of zirconium [kg/mol]

Default:0.091224

C++ Type:double

Description:Atomic weight of zirconium [kg/mol]

• 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.

• plenum_pressureThe name of the plenum_pressure postprocessor value.

C++ Type:PostprocessorName

Description:The name of the plenum_pressure postprocessor value.

• model_creepTrueFlag for creep model

Default:True

C++ Type:bool

Description:Flag for creep model

• youngs_modulusYoung's modulus of the material.

C++ Type:double

Description:Young's modulus of the material.

• 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.

• calc_youngsTrueFlag to calculate Youngs Modulus or use given value

Default:True

C++ Type:bool

Description:Flag to calculate Youngs Modulus or use given value

• disp_xThe x displacement

C++ Type:std::vector

Description:The x displacement

• use_material_fission_rateFalseFlag to use the material 'fission_rate_material' instead of variable fission rate

Default:False

C++ Type:bool

Description:Flag to use the material 'fission_rate_material' instead of variable fission rate

• gamma_transition923Gamma phase transition temperature.

Default:923

C++ Type:double

Description:Gamma phase transition temperature.

• 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

• 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

• A_Pu0.244Atomic weight of plutonium [kg/mol]

Default:0.244

C++ Type:double

Description:Atomic weight of plutonium [kg/mol]

• youngs_modulus_functionYoung's modulus as a function of temperature.

C++ Type:FunctionName

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

• strain_zzThe zz strain

C++ Type:std::vector

Description:The zz strain

• A_U0.238029Atomic weight of uranium [kg/mol]

Default:0.238029

C++ Type:double

Description:Atomic weight of uranium [kg/mol]

• 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

• hydrostatic_stressCoupled Hydrostatic Stress

C++ Type:std::vector

Description:Coupled Hydrostatic Stress

• bulk_modulusThe bulk modulus for the material.

C++ Type:double

Description:The bulk modulus for the material.

• 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.

• compute_material_timestep_limitFalseWhether to compute the matl_timestep_limit material property

Default:False

C++ Type:bool

Description:Whether to compute the matl_timestep_limit material property

• 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

• 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

• volumetric_modelsVolumetric models to apply

C++ Type:std::vector

Description:Volumetric models to apply

• max_its30Maximum number of Newton iterations

Default:30

C++ Type:unsigned int

Description:Maximum number of Newton iterations

• X_ZrCoupled zirconium atom fraction

C++ Type:std::vector

Description:Coupled zirconium atom fraction

• open_pore_compressibility_factor0The name of the open pore compression alpha factor

Default:0

C++ Type:MaterialPropertyName

Description:The name of the open pore compression alpha factor

• compute_JIntegralFalseWhether to compute the J Integral.

Default:False

C++ Type:bool

Description:Whether to compute the J Integral.

• calc_poissonsTrueFlag to calculate Poissons ratio or use given value

Default:True

C++ Type:bool

Description:Flag to calculate Poissons ratio or use given value

• 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.

• 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-11Absolute convergence tolerance for Newton iteration

Default:1e-11

C++ Type:double

Description:Absolute convergence tolerance for Newton iteration

• porosity0Porosity material property name

Default:0

C++ Type:MaterialPropertyName

Description:Porosity material property name

• relative_tolerance1e-08Relative convergence tolerance for Newton iteration

Default:1e-08

C++ Type:double

Description:Relative convergence tolerance for Newton iteration

• 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

• 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

• 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. Compiled by: IFR Property Evaluation Working Group. Metallic fuels handbook. Argonne National Laboratory, June 1988.[BibTeX]
2. H A Saller, R F Dickerson, and W E Murr. Uranium Alloys for High-Temperature Application. Technical Report BMI-1098, Divison of Technical Information Extension, U.S. Atomic Energy Commission, June 1956.[BibTeX]