Compute Crystal Plasticity Thermal Eigenstrain

As is described in ComputeMultipleCrystalPlasticityStress, a thermal deformation gradient is introduced to account for the thermal induced deformations in a finite strain thermo-mechanical problem with crystal plasticity. Accordingly, the total deformation gradient is multiplicatively decomposed into three components as (1) such that , , and (see Li et al. (2019), Ozturk et al. (2016), and Meissonnier et al. (2001)).

To account for more than one thermal eigenstrain, one can optionally decompose the total thermal deformation gradient into multiple components as (2) where a total of different thermal eigenstrains are considered.

The thermal Lagrangian strain that is associated with is computed as (3)

In this class, the evolution of a typical thermal deformation gradient component, (see Eq. (2)) is expressed with respect to its lattice symmetry axis as (4) where is the temperature rate, and is a diagonal tensor for anisotropic thermal expansion coefficients. One can either use the above evolution equation, or create a different crystal plasticity thermal eigenstrain class with a customized constitutive equation by inheriting from the ComputeCrystalPlasticityEigenstrainBase class.

commentnote:Base Class Requirement

Any constitutive eigenstrain model developed for use within the ComputeMultipleCrystalPlasticityStress class must inherit from the ComputeCrystalPlasticityEigenstrainBase class.

We summarize the numerical implementation details that are specialized in crystal plasticity thermal eigenstrain computation in the following.

Numerical Implementation

The calculations of the thermal deformation gradient (or its component ), the Lagrangian strain (or ), and the associated derivative (or ) are included in this ComputeCrystalPlasticityThermalEigenstrain class. Meanwhile, changes are required in the residual and Jacobian calculations in ComputeMultipleCrystalPlasticityStress in order to account for the additional configuration due to thermal expansion. The implementation details are described below.

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1. Time integration for

The evolution of the thermal deformation gradient component is defined by Eq. (4). Similar to the time integration that has been implemented for , we start with a backward difference expression as an approximation for , where the subscripts and identifies the time steps. By substituting into Eq. (4) and rearranging, we have where .

2. Crystal plasticity Jacobian

The stress is updated iteratively by driving the stress residual to zero. The stress residual is computed by where is the second Piola-Kirchhoff stress, the is the elasticity tensor, and is the elastic part of the Cauchy-Green deformation tensor. The crystal plasticity Jacobian is computed by taking the derivative of with respect to . By using the chain rule, we have (5) The computation of each term in the curly bracket will be described in the follows using indicial notations. The is computed as

From Eq. (1), . Therefore, .

we evaluate the last term in Eq. (5) via chain rule: where the evaluation of is based on by taking chain rule. The evaluation of varies based on the form of the flow rule that differs for different material types. The last term is evaluated as where is the schmid tensor for slip system .

3. Elasto-plastic tangent moduli

The elasto-plastic tangent moduli is computed via where the , and remains the same with the inclusion of the thermal deformation gradient. The last term is evaluated as

commentnote:Off-diagonal Jacobian

The off-diagonal Jacobian contributions due to thermal expansion is currently not accounted for in the stress divergence calculation. Improving the Jacobian entries associated with thermal expansion remains an ongoing work.

Example Input File Syntax

[Materials]
  [stress]
    type = ComputeMultipleCrystalPlasticityStress
    crystal_plasticity_models = 'trial_xtalpl'
    eigenstrain_names = "thermal_eigenstrain_1 thermal_eigenstrain_2"
    tan_mod_type = exact
    maximum_substep_iteration = 5
  []

  [thermal_eigenstrain_1]
    type = ComputeCrystalPlasticityThermalEigenstrain
    eigenstrain_name = thermal_eigenstrain_1
    deformation_gradient_name = thermal_deformation_gradient_1
    temperature = temperature
    thermal_expansion_coefficients = '1e-05 2e-05 3e-05' # thermal expansion coefficients along three directions
  []

  [thermal_eigenstrain_2]
    type = ComputeCrystalPlasticityThermalEigenstrain
    eigenstrain_name = thermal_eigenstrain_2
    deformation_gradient_name = thermal_deformation_gradient_2
    temperature = temperature
    thermal_expansion_coefficients = '2e-05 3e-05 4e-05' # thermal expansion coefficients along three directions
  []
[]
(moose/modules/solid_mechanics/test/tests/crystal_plasticity/cp_eigenstrains/multiple_eigenstrains_test.i)

Input Parameters

  • deformation_gradient_nameMaterial property name for the deformation gradient tensor computed by this model.

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:Material property name for the deformation gradient tensor computed by this model.

  • eigenstrain_nameMaterial property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:Material property name for the eigenstrain tensor computed by this model. IMPORTANT: The name of this property must also be provided to the strain calculator.

  • thermal_expansion_coefficientsVector of values defining the constant second order thermal expansion coefficients, depending on the degree of anisotropy, this should be of size 1, 3, 6 or 9

    C++ Type:std::vector<double>

    Unit:(no unit assumed)

    Controllable:No

    Description:Vector of values defining the constant second order thermal expansion coefficients, depending on the degree of anisotropy, this should be of size 1, 3, 6 or 9

Required Parameters

  • base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases

    C++ Type:std::string

    Unit:(no unit assumed)

    Controllable:No

    Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases

  • blockThe list of blocks (ids or names) that this object will be applied

    C++ Type:std::vector<SubdomainName>

    Unit:(no unit assumed)

    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>

    Unit:(no unit assumed)

    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

    Unit:(no unit assumed)

    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.

  • 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.

  • temperatureCoupled temperature variable

    C++ Type:std::vector<VariableName>

    Unit:(no unit assumed)

    Controllable:No

    Description:Coupled temperature variable

  • 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

    Unit:(no unit assumed)

    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.

Optional Parameters

  • control_tagsAdds user-defined labels for accessing object parameters via control logic.

    C++ Type:std::vector<std::string>

    Unit:(no unit assumed)

    Controllable:No

    Description:Adds user-defined labels for accessing object parameters via control logic.

  • enableTrueSet the enabled status of the MooseObject.

    Default:True

    C++ Type:bool

    Unit:(no unit assumed)

    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

    Unit:(no unit assumed)

    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

    Unit:(no unit assumed)

    Controllable:No

    Description:The seed for the master random number generator

  • 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

    Unit:(no unit assumed)

    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

  • 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>

    Unit:(no unit assumed)

    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>

    Unit:(no unit assumed)

    Controllable:No

    Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

References

  1. Jifeng Li, Ignacio Romero, and Javier Segurado. Development of a thermo-mechanically coupled crystal plasticity modeling framework: application to polycrystalline homogenization. International Journal of Plasticity, 119:313–330, 2019.[BibTeX]
  2. FT Meissonnier, EP Busso, and NP O'Dowd. Finite element implementation of a generalised non-local rate-dependent crystallographic formulation for finite strains. International Journal of Plasticity, 17(4):601–640, 2001.[BibTeX]
  3. Deniz Ozturk, Ahmad Shahba, and Somnath Ghosh. Crystal plasticity FE study of the effect of thermo-mechanical loading on fatigue crack nucleation in titanium alloys. Fatigue & Fracture of Engineering Materials & Structures, 39(6):752–769, 2016.[BibTeX]