Stress Divergence Tensors

Stress divergence kernel for the Cartesian coordinate system

Description

The StressDivergenceTensors kernel calculates the residual of the stress divergence for 1D, 2D, and 3D problems in the Cartesian coordinate system. This kernel can be automatically created with the TensorMechanics Master Action. Use of the tensor mechanics master action is recommended to ensure the consistent setting of the _use_displaced_mesh_ parameter for the strain formulation selected. For a detailed explanation of the settings for _use_displaced_mesh_ in mechanics problems and the TensorMechanics Master Action usage, see the Introduction/StressDivergence page.

Residual Calculation

The stress divergence kernel handles the calculation of the residual, $var element = document.getElementById("moose-equation-f01b2adb-8de6-4c17-bbe2-c8d7839b01bb");katex.render("\\mathbb{R}", element, {displayMode:false,throwOnError:false});$ , from the governing equation and the calculation of the Jacobian. From the strong form of the governing equation for mechanics, neglecting body forces, $var element = document.getElementById("moose-equation-8e9be3e2-18e3-4c4e-8006-302eb21d5648");katex.render("\\nabla \\cdot \\sigma = 0", element, {displayMode:true,throwOnError:false});$ the weak form, using Galerkin's method and the Gauss divergence theorem, becomes $var element = document.getElementById("moose-equation-0b4c9cdb-1ab6-433b-8412-d3160e71e911");katex.render("\\int_S n \\cdot (\\sigma \\psi) dS - \\int_V \\sigma \\cdot \\nabla \\psi dV = 0", element, {displayMode:true,throwOnError:false});$ in which $var element = document.getElementById("moose-equation-3bea289f-369d-4a55-9210-c6aacf758d50");katex.render("\\psi", element, {displayMode:false,throwOnError:false});$ is the test function. The second term of the weak form equation is the residual contribution calculated by the stress divergence kernel.

The calculation of the Jacobian can be approximated with the elasticity tensor if the simulation solve type is JFNK:

$var element = document.getElementById("moose-equation-9fc083cf-1b21-45f0-8404-6579313237d6");katex.render("\\frac{d(\\sigma_{ij} \\cdot d(\\psi) / dx_j)}{du_k} = \\frac{d(C_{ijmn} \\cdot du_m / dx_n \\cdot d\\psi /dx_j)}{du_k}", element, {displayMode:true,throwOnError:false});$ which is nonzero for $var element = document.getElementById("moose-equation-c5998855-5e91-476a-9b10-a7a0eecb073a");katex.render("m == k", element, {displayMode:false,throwOnError:false});$ .

If the solve type for the simulation is set to NEWTON the finite deformation Jacobian will need to be calculated. Set the parameter use_finite_deform_jacobian = true in this case.

note:Use of the Tensor Mechanics Master Action Recommended

The use_displaced_mesh parameter must be set correcting to ensure consistency in the equilibrium equation: if the stress is calculated with respect to the deformed mesh, the test function gradients must also be calculated with respect to the deformed mesh. The Tensor Mechanics MasterAction is designed to automatically determine and set the parameter correctly for the selected strain formulation. We recommend that users employ the Tensor Mechanics MasterAction whenever possible to ensure consistency between the test function gradients and the strain formulation selected.

Use with Planar Models

When used with 2D planar models (plane stres, plane strain, or generalized plane strain), it is used to compute the residuals for the in-plane response. In all of these cases, it assumed that the out-of-plane thickness is 1, and the computation of the in-plane residuals is identical to that for the 3D case.

The only exception to this is the plane stress case with finite deformation, because the out-of-plane thickness change can be significant, and in general is not spatially uniform, so local thickness changes must be accounted for. In this case, the standard residual is multiplied by the modified thickness, $var element = document.getElementById("moose-equation-488ee8a4-ec61-4f55-85a9-049b1aad97f5");katex.render("t", element, {displayMode:false,throwOnError:false});$ , which is computed from the logarithmic out of plane strain $var element = document.getElementById("moose-equation-175b43ec-c252-44ff-91e8-dadc43e5d233");katex.render("\\varepsilon_{oop}", element, {displayMode:false,throwOnError:false});$ as: $var element = document.getElementById("moose-equation-fe3918cf-e75d-46a6-8df1-2f7d7778c49c");katex.render("t = e^{\\varepsilon_{oop}}", element, {displayMode:true,throwOnError:false});$ This correction is made for 2D planar models when the deformed mesh is used by setting use_displaced_mesh = true and out_of_plane_strain is specified.

Example Input File syntax

The Cartesian StressDivergenceTensors is the default case for the tensor mechanics master action

[Modules]
  [./TensorMechanics]
    [./Master]
      [./all]
        strain = FINITE
        add_variables = true
      [../]
    [../]
  [../]
[]

Either 1, 2, or 3 displacement variables can be used in the stress divergence calculator for the Cartesian system.

Input Parameters

componentAn integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z)
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z)
displacementsThe string of displacements suitable for the problem statement
C++ Type:std::vector<VariableName>
Controllable:No
Description:The string of displacements suitable for the problem statement
variableThe name of the variable that this residual object operates on
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the variable that this residual object operates on

Required 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
coupled_variablesVector of nonlinear variable arguments this object depends on
C++ Type:std::vector<VariableName>
Controllable:No
Description:Vector of nonlinear variable arguments this object depends on
eigenstrain_namesList of eigenstrains used in the strain calculation. Used for computing their derivatives for off-diagonal Jacobian terms.
C++ Type:std::vector<MaterialPropertyName>
Controllable:No
Description:List of eigenstrains used in the strain calculation. Used for computing their derivatives for off-diagonal Jacobian terms.
out_of_plane_directionzThe direction of the out_of_plane_strain variable used in the WeakPlaneStress kernel.
Default:z
C++ Type:MooseEnum
Options:x, y, z
Controllable:No
Description:The direction of the out_of_plane_strain variable used in the WeakPlaneStress kernel.
out_of_plane_strainThe name of the out_of_plane_strain variable used in the WeakPlaneStress kernel.
C++ Type:std::vector<VariableName>
Controllable:No
Description:The name of the out_of_plane_strain variable used in the WeakPlaneStress kernel.
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
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.
temperatureThe name of the temperature variable used in the ComputeThermalExpansionEigenstrain. (Not required for simulations without temperature coupling.)
C++ Type:std::vector<VariableName>
Controllable:No
Description:The name of the temperature variable used in the ComputeThermalExpansionEigenstrain. (Not required for simulations without temperature coupling.)
use_finite_deform_jacobianFalseJacobian for corotational finite strain
Default:False
C++ Type:bool
Controllable:No
Description:Jacobian for corotational finite strain
volumetric_locking_correctionFalseSet to false to turn off volumetric locking correction
Default:False
C++ Type:bool
Controllable:No
Description:Set to false to turn off volumetric locking correction

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_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime, system
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
Options:nontime, time
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging 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>
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>
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.)
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

The stress divergence family of automatic differentiation kernels

ADStressDivergenceTensors - Stress divergence kernel for Cartesian coordinates Problem/coord_type=XYZ (default)
ADStressDivergenceRZTensors - Stress divergence kernel for 2D cylindrical coordinates Problem/coord_type=RZ
ADStressDivergenceRSphericalTensors - Stress divergence kernel for 1D spherical coordinates Problem/coord_type=RSPHERICAL

(moose/modules/tensor_mechanics/test/tests/finite_strain_elastic/finite_strain_elastic_new_test.i)

[Mesh]
  type = GeneratedMesh
  dim = 3
  elem_type = HEX8
[]

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Functions]
  [./tdisp]
    type = ParsedFunction
    expression = '0.01 * t'
  [../]
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./all]
        strain = FINITE
        add_variables = true
      [../]
    [../]
  [../]
[]

[BCs]
  [./symmy]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0
  [../]
  [./symmx]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0
  [../]
  [./symmz]
    type = DirichletBC
    variable = disp_z
    boundary = back
    value = 0
  [../]
  [./tdisp]
    type = FunctionDirichletBC
    variable = disp_z
    boundary = front
    function = tdisp
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ComputeElasticityTensor
    C_ijkl = '1.684e5 0.176e5 0.176e5 1.684e5 0.176e5 1.684e5 0.754e5 0.754e5 0.754e5'
    fill_method = symmetric9
  [../]
  [./stress]
    type = ComputeFiniteStrainElasticStress
  [../]
[]

[Preconditioning]
  [./smp]
    type = SMP
    full = true
  [../]
[]

[Executioner]
  type = Transient
  dt = 0.05

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'

  petsc_options_iname = -pc_hypre_type
  petsc_options_value = boomeramg
  nl_abs_tol = 1e-10
  nl_rel_step_tol = 1e-10
  dtmax = 10.0
  nl_rel_tol = 1e-10
  end_time = 1
  dtmin = 0.05
  num_steps = 10
  nl_abs_step_tol = 1e-10
[]

[Outputs]
  exodus = true
[]