- componentAn integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the direction in order parameter space this kernel acts in (e.g. for unrotated functionals 0 for q_x, 1 for q_y, 2 for q_z).
- polar_xThe x component of the polarization
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The x component of the polarization
- polar_yThe y component of the polarization
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The y component of the polarization
- variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Unit:(no unit assumed)
Controllable:No
Description:The name of the variable that this residual object operates on
CubicParentFlexoelectricFullPDerivative
Calculates the residual of the full elastic energy, electrostrictive AND flexoelectric, w.r.t. the polarization, for a cubic parent phase. Replaces CubicParentElasticPDerivative when flexoelectricity is on.
Overview
The elastic energy density of a cubic parent phase with electrostrictive and flexoelectric eigenstrains is
with the electrostrictive and flexoelectric eigenstrains
where is the total strain, the elastic compliance and the flexocoupling tensor. This form is the exact Legendre transform of the Gibbs functional of Gu et al. (2014), including the term.
This kernel computes the residual of the polarization equation due to this energy,
which contains both the electrostrictive and the flexoelectric contributions. It is used instead of CubicParentElasticPDerivative, never together with it; with the two are identical.
The residual, the diagonal and off-diagonal Jacobian, the block (when displacements is given) and the blocks with a SCALAR mean strain (when global_strain is given) are all generated from the single energy density above, so they are exact derivatives of each other and of the energy computed by the corresponding postprocessor.
commentnote:Conventions
mu11, mu12 and mu44 are the tensor components , and of the flexocoupling tensor; the compliances s11, s12, s44 are in Voigt form (). With the sign above, the flexocoupling computed from the published of Gu et al. (2014) is entered directly and gives their tail-to-tail Néel component. The flexoelectric eigenstrain is added to the electrostrictive one (no CompositeEigenstrain weight of ). The deprecated NV/RI flexoelectric objects of earlier development versions carried the shear slot as , i.e. four times too large, and expected the opposite sign.
warningwarning:Literature flexocoupling values
Published flexocoupling tensors come in several index conventions. In particular the displacement-gradient form of Maranganti and Sharma (2009) mixes the 12 and 44 components relative to the strain-gradient form used here, and Voigt shear factors of 1, 2 or 4 appear in the literature. Convert to the tensor components of the strain-gradient form before entering them.
Example Input File Syntax
The kernel is applied once per polarization component, in place of CubicParentElasticPDerivative:
[Kernels]
[flexo_x]
type = CubicParentFlexoelectricFullPDerivative
variable = polar_x
component = 0
polar_x = polar_x
polar_y = polar_y
polar_z = polar_z
displacements = 'u_x u_y u_z' # optional: P <- u Jacobian block
global_strain = global_strain # optional: P <-> eps_bar Jacobian blocks
[]
# ... flexo_y (component = 1), flexo_z (component = 2)
[]
It needs the material properties C11 C12 C44, Q11 Q12 Q44, s11 s12 s44 and mu11 mu12 mu44. The rotated counterpart is used in
[./flexo_polar_coupled_x]
type = RotatedCubicParentFlexoelectricFullPDerivative
variable = polar_x
component = 0
block = '0'
[../]Input Parameters
- base_nameMaterial property base name
C++ Type:std::string
Controllable:No
Description:Material property base name
- 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
- displacementsThe displacement variables. Optional; when given, the P<-u Jacobian block is assembled.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The displacement variables. Optional; when given, the P<-u Jacobian block is assembled.
- global_strainThe SCALAR mean strain, if it is being solved for. Optional; when given, the P<->eps_bar blocks are assembled.
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The SCALAR mean strain, if it is being solved for. Optional; when given, the P<->eps_bar blocks are assembled.
- polar_zThe z component of the polarization
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:The z component of the polarization
Optional Parameters
- absolute_value_vector_tagsThe tags for the vectors this residual object should fill with the absolute value of the residual contribution
C++ Type:std::vector<TagName>
Controllable:No
Description:The tags for the vectors this residual object should fill with the absolute value of the residual contribution
- extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the matrices this Kernel should fill
- extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
Description:The extra tags for the vectors this Kernel should fill
- matrix_onlyFalseWhether this object is only doing assembly to matrices (no vectors)
Default:False
C++ Type:bool
Controllable:No
Description:Whether this object is only doing assembly to matrices (no vectors)
- matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Controllable:No
Description:The tag for the matrices this Kernel should fill
- vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Controllable:No
Description:The tag for the vectors this Kernel should fill
Contribution To Tagged Field Data Parameters
- 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.
- diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
- 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
- save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector<AuxVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
- search_methodnearest_node_connected_sidesChoice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).
Default:nearest_node_connected_sides
C++ Type:MooseEnum
Controllable:No
Description:Choice of search algorithm. All options begin by finding the nearest node in the primary boundary to a query point in the secondary boundary. In the default nearest_node_connected_sides algorithm, primary boundary elements are searched iff that nearest node is one of their nodes. This is fast to determine via a pregenerated node-to-elem map and is robust on conforming meshes. In the optional all_proximate_sides algorithm, primary boundary elements are searched iff they touch that nearest node, even if they are not topologically connected to it. This is more CPU-intensive but is necessary for robustness on any boundary surfaces which has disconnections (such as Flex IGA meshes) or non-conformity (such as hanging nodes in adaptively h-refined meshes).
- 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
- 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
- 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
References
- Yijia Gu, Menglei Li, Anna N. Morozovska, Yi Wang, Eugene A. Eliseev, Venkatraman Gopalan, and Long-Qing Chen.
Flexoelectricity and ferroelectric domain wall structures: Phase-field modeling and DFT calculations.
Phys. Rev. B, 89:174111, 2014.
doi:10.1103/PhysRevB.89.174111.[Export]
- R. Maranganti and P. Sharma.
Atomistic determination of flexoelectric properties of crystalline dielectrics.
Phys. Rev. B, 80:054109, 2009.
doi:10.1103/PhysRevB.80.054109.[Export]