# Stress-induced Densification Hot Pressing UOvar element = document.getElementById("moose-equation-86c64866-0801-45f1-b416-21457c5f4d43");katex.render("_2", element, {displayMode:false,throwOnError:false});

Models the stress induced densification of UO2

warning:Deprecated Solid Mechanics Material

The functionality of this solid mechanics material is being replaced in the TensorMechanics system by UO2HotPressingCreepUpdate for creep and UO2HotPressingPlasticityUpdate for instantaneous plasticity.

## Description

The model HotPressingUO2 accounts for creep and/or instantaneous plastic flow surrounding pores in ceramic UO2 fuel pellet, which under pressure, can reduce pore volume and consequently reduce fuel porosity and increase fuel density. This is a densification mechanism of UO fuel under compressive stresses, which contributes to fuel densification in addition to the irradiation induced densification. Such mechanical densification process is more pronounced at high temperatures with high creep rate or plastic deformations, and it is also referred to as hot-pressing.

The mathematical model of hot-pressing of ceramic UO was described by Rashid et al. (1974). By using an analogy of close-packed spherical shells in infinite media under hydrostatic compression, model on the stress-induced densification based on the mechanism of instantaneous plasticity and creep was derived. This section describes the implementation of the hot-pressing model based on the mechanisms of creep and plasticity as follows. The new material class HotPressingUO2 in the Bison code is inherited from CreepUO2 described here.

## Creep

Creep is modeling with a power-law form (1) The tangential creep rate of porous media, with density , at the pore surface is given as in Rashid et al. (1974). (2) And, the volumetric creep rate is (3) Hot-pressing parameter is defined as (4) where the hydrostatic pressure (Pa), is the exponent in the power law creep equation, is the leading coefficient in the power law creep equation, is the fractional density (dimensionless), is the creep rate, and the subscripts and represent tangential and volumetric components respectively.

Figure 1: Hot-pressing parameter versus fractional density

This material parameter is used in the hot-pressing model for the volumetric creep of UO. The creep of UO involves several mechanisms, and in their mathematical descriptions, different exponent could be used for the different mechanisms. The volumetric creep strain implemented in Bison code is assumed to be the combination of all the creep strains together. From Eq. 4, the hot-pressing parameter depends on the fuel density. With the increase of fuel density, the parameter would be reduced; when fractional density approaches 1.0, the parameter approaches zero, and the densification would essentially be terminated, i.e., the volumetric creep strain rate in Eq. 3 becomes zero. A plot of the hot-pressing parameter versus initial density at different is shown in Figure 1.

## Instantaneous Plasticity

In Rashid et al. (1974), yield criterion of UO is described by a modified Mohr-coulomb criterion. Eq. 5 through Eq. 9 summarize the criterion (Rashid et al., 1974): (5) The incremental plastic strain is given as: (6) where (7) The volumetric strain increment is (8) defines a material parameter that relates to the yield stress of 100% UO density. (9) where is the yield stress of UO at initial density , and is the yield stress at 100% UO density. The determination of needs experimental data on the yield stress of UO with different porosities, which is scarce in the literatures. Instead, an approximation is made in the code by using a constant ratio of =0.95. The resultant equation of in Bison is (10)

however is not defined in Rashid et al. (1974). To implement the model in Bison, a new flow rule is used. The yield criterion is formulated as: (11) The effective stress is derived as: (12) The new flow rule used in Bison is provided in following equations. (13) (14)

### Yield Stress Model

The hot-pressing or mechanical densification under instantaneous plastic flow depends on the yield strength of UO, which is currently not available in Bison. A linear hardening material model is used for modeling the yield stress of UO. The yield stress for the linear hardening material is (15) where is the initial yield stress, is the new yield stress (effective stress), is the effective incremental plastic strain, and is the hardening modulus. The incremental effective plastic strain and new yield stress are computed in Bison using a radial return method.

## Example Input Syntax


[./hotpressing]
type = HotPressingUO2
block = 1
disp_r = disp_x
disp_z = disp_y
temp = temp
fission_rate = fission_rate
input_yield_stress = 100
input_hardening_modulus = 1000
youngs_modulus = 2.e5
poissons_ratio = .3
oxy_to_metal_ratio = 2.0
model_creep = false
debug_output = false
[../]
(test/tests/hotpressing_uo2/hotpressing_test_plasticity.i)

## 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.

• model_creepTrueSet true to turn on creep model

Default:True

C++ Type:bool

Description:Set true to turn on creep model

• 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.

• input_hotpressingFalseFlag for using input hot pressing parameter

Default:False

C++ Type:bool

Description:Flag for using input hot pressing parameter

• stress_rel_error0.001Relative error in the stress iteration loop

Default:0.001

C++ Type:double

Description:Relative error in the stress iteration loop

• youngs_modulus_functionYoung's modulus as a function of temperature.

C++ Type:FunctionName

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

• 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.

• debug_outputFalseFlag for turn on debug output

Default:False

C++ Type:bool

Description:Flag for turn on debug output

• 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.

• input_hardening_modulus0Input UO2 hardening modulus

Default:0

C++ Type:double

Description:Input UO2 hardening modulus

• 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)

• rel_error0.001Relative error in the iteration loop to compute trial pressure

Default:0.001

C++ Type:double

Description:Relative error in the iteration loop to compute trial pressure

• 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

• use_flow_ruleTrueUse modifed flow rule for modified Mohr-Columnb yield function

Default:True

C++ Type:bool

Description:Use modifed flow rule for modified Mohr-Columnb yield function

• tempCoupled Temperature

C++ Type:std::vector

Description:Coupled Temperature

• 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.

• disp_xThe x displacement

C++ Type:std::vector

Description:The x displacement

• rod_ave_lin_powAverage linear power function

C++ Type:FunctionName

Description:Average linear power function

• stress_abs_error0.01Absolute error in the stress iteration loop

Default:0.01

C++ Type:double

Description:Absolute error in the stress iteration loop

• output_iteration_infoFalseSet true to output sub-newton iteration information

Default:False

C++ Type:bool

Description:Set true to output sub-newton iteration information

• hotpressing_alpha0Input hot pressing parameter alpha

Default:0

C++ Type:double

Description:Input hot pressing parameter alpha

• 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

• input_yield_stress0Input UO2 yield stress

Default:0

C++ Type:double

Description:Input UO2 yield stress

• 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.

• model_hotpressingTrueFlag to turn on hot pressing model; equivalent to CreepUO2 when set as false

Default:True

C++ Type:bool

Description:Flag to turn on hot pressing model; equivalent to CreepUO2 when set as false

• 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

• strain_zzThe zz strain

C++ Type:std::vector

Description:The zz strain

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

Default:7.78e-37

C++ Type:double

• outputThe reporting postprocessor to use for the max_iterations value.

C++ Type:PostprocessorName

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

• has_hotpressing_plasticityFalseFlag to turn on instantaneous plasticity model

Default:False

C++ Type:bool

Description:Flag to turn on instantaneous plasticity model

• dep_matl_propsNames of material properties this material depends on.

C++ Type:std::vector

Description:Names of material properties this material depends on.

• abs_error1e-06Absolute error in the iteration loop to compute trial pressure

Default:1e-06

C++ Type:double

Description:Absolute error in the iteration loop to compute trial pressure

• 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

• max_iteration1000Maximum iteration number

Default:1000

C++ Type:int

Description:Maximum iteration number

• 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.

• stress_max_its1000Maximum iteration number for stress iteration loop

Default:1000

C++ Type:unsigned int

Description:Maximum iteration number for stress iteration loop

• 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.

• hotpressing_nu0Input hot pressing parameter

Default:0

C++ Type:double

Description:Input hot pressing parameter

• 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

• lambdaLame's first parameter for the material.

C++ Type:double

Description:Lame's first parameter for the material.

• 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

• compute_pressureTrueCompute trial pressure

Default:True

C++ Type:bool

Description:Compute trial pressure

### 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. Y. R. Rashid, H. T. Tang, and E. B. Johansson. Mathematical treatment of hot pressing of reactor fuel. Nuclear Engineering Design, 29:1–6, 1974.[BibTeX]