CubicParentFEPhaseFieldAction

Overview

Sets up a cubic-parent-phase ferroelectric problem: the three polarization components and the electrostatic potential, the Landau bulk and gradient kernels, the optional electrostatic and electrostrictive couplings, the constant material properties and the free energy postprocessors.

The mechanics are deliberately left to the SolidMechanics QuasiStatic physics, which owns the displacement variables, the strain calculator and the stress divergence kernels. The two actions are coupled through the eigenstrain: this action adds a ComputeCubicParentElectrostrictiveStrain named by eigenstrain_name, and that same name must appear in the SolidMechanics action's eigenstrain_names.

A prescribed mean strain (an epitaxial misfit, say) is supplied as a GenericConstantRankTwoTensor and reaches the strain calculator through [GlobalParams] global_strain. It contributes to total_strain, which is the quantity the cubic-parent polarization kernels read.

electrostatics and elastic are independent, so the action covers the purely polar-elastic problem as well as the full polar-elastic-electric one.

Elastic energy

Two routes are provided, and Ftotal uses the second when it is available:

• Felastic, from CubicParentElasticEnergy, which evaluates the cubic-parent form directly and subtracts only its own eigenstrain.

• Felastic_true, an ElementIntegralVariablePostprocessor over the f_el aux variable that ElasticEnergyAux fills with 1/2 sigma:elastic_strain. Because the strain calculator has already removed every eigenstrain listed in the SolidMechanics action's eigenstrain_names, this is the true elastic energy no matter how many contribute one, with no double counting and no cross term left over.

With a single eigenstrain the two agree to roundoff. They diverge as soon as anything else contributes an eigenstrain, and only Felastic_true stays correct; set add_elastic_energy_aux = false to drop it and fall back to Felastic.

Example Input File Syntax

[Ferret<<<{"href": "../../syntax/Ferret/index.html"}>>>]
  [CubicParentFEPhaseField<<<{"href": "../../syntax/Ferret/CubicParentFEPhaseField/index.html"}>>>]
    electrostatics<<<{"description": "Solve Poisson's equation for the depolarization field and couple it to the polarization."}>>> = true
    elastic<<<{"description": "Couple the polarization to the strain through the cubic-parent electrostrictive energy."}>>> = true
    alpha_ijkl<<<{"description": "Names of the Landau expansion coefficients."}>>> = 'alpha1 alpha11 alpha12 alpha111 alpha112 alpha123 alpha1111 alpha1112 alpha1122 alpha1123'
    alpha_ijkl_val<<<{"description": "Values of the Landau expansion coefficients."}>>> = '${alpha1} -0.073 0.75 0.26 0.61 -3.7 0.0 0.0 0.0 0.0'
    G_ij<<<{"description": "Names of the gradient coefficients."}>>> = 'G110 G11_G110 G12_G110 G44_G110 G44P_G110'
    G_ij_val<<<{"description": "Values of the gradient coefficients."}>>> = '${G110} 0.6 0.0 0.3 0.3'
    Q_ij<<<{"description": "Names of the electrostrictive Q tensor coefficients."}>>> = 'Q11 Q12 Q44'
    Q_ij_val<<<{"description": "Values of the electrostrictive Q tensor coefficients."}>>> = '0.089 -0.026 0.03375'
    C_ij<<<{"description": "Names of the elastic stiffness coefficients. Must be 'C11 C12 C44' (cubic parent)."}>>> = 'C11 C12 C44'
    C_ij_val<<<{"description": "Values of the elastic stiffness coefficients."}>>> = '174.269 79.029 47.62'
    permittivity_val<<<{"description": "Value of the permittivity."}>>> = '${fparse eps_r*eps0}'
  []
[]

Alongside the SolidMechanics physics:

[Physics<<<{"href": "../../syntax/Physics/index.html"}>>>]
  [SolidMechanics<<<{"href": "../../syntax/Physics/SolidMechanics/index.html"}>>>]
    [QuasiStatic<<<{"href": "../../syntax/Physics/SolidMechanics/QuasiStatic/index.html"}>>>]
      [all]
        strain<<<{"description": "Strain formulation"}>>> = SMALL
        add_variables<<<{"description": "Add the displacement variables"}>>> = true
        eigenstrain_names<<<{"description": "List of eigenstrains to be applied in this strain calculation"}>>> = 'ferro'
        generate_output<<<{"description": "Add scalar quantity output for stress and/or strain"}>>> = 'strain_xx strain_yy'
      []
    []
  []
[]

Input Parameters