- variableThe name of the variable that this Kernel operates on
C++ Type:NonlinearVariableName

Description:The name of the variable that this Kernel operates on

- componentAn integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z; note in this kernel disp_x refers to the radial displacement and disp_y refers to the axial displacement.)
C++ Type:unsigned int

Description:An integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z; note in this kernel disp_x refers to the radial displacement and disp_y refers to the axial displacement.)

- displacementsThe string of displacements suitable for the problem statement
C++ Type:std::vector

Description:The string of displacements suitable for the problem statement

# Stress Divergence RZ Tensors

Calculate stress divergence for an axisymmetric problem in cylinderical coordinates.

## Description

The kernel `StressDivergenceRZTensors`

solves the stress divergence equation for an Axisymmetric problem in the cylindrical coordinate system on a 2D mesh.

The axis of symmetry must lie along the -axis in a or cylindrical coordinate system. This symmetry orientation is required for the calculation of the residual and of the jacobian, as defined in Eq. (4).

The `StressDivergenceRZTensors`

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/Stress Divergence page.

## Residual Calculation

The stress divergence kernel handles the calculation of the residual, , from the governing equation and the calculation of the Jacobian. From the strong form of the governing equation for mechanics, neglecting body forces, (1) the weak form, using Galerkin's method and the Gauss divergence theorem, becomes (2) in which 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**:

(3) which is nonzero for .

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.

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.

In cylindrical coordinates, the divergence of a rank-2 tensor includes mixed term contributions. In the axisymmetric model we assume symmetric loading conditions, in addition to the zero out-of-plane shear strains, so that the residual computation is simplified.

(4)

The calculation of the Jacobian is similarly complex, requiring up to four terms in the calculation of the diagonal entries.

The axisymmetric system changes the order of the displacement vector from , usually seen in textbooks, to . Take care to follow this convention in your input files and when adding extra stresses.

## Example Input File

The coordinate type in the Problem block of the input file must be set to ** COORD_TYPE = RZ**.

Using the tensor mechanics master action, as shown

```
[Modules/TensorMechanics/Master]
[./all]
strain = FINITE
add_variables = true
block = 1
[../]
[]
```

(modules/tensor_mechanics/test/tests/2D_geometries/2D-RZ_finiteStrain_test.i)the `StressDivergenceRZTensors`

kernel will be automatically built when the coordinate system in the Problem block is specified for the axisymmetric RZ system,

```
[Problem]
coord_type = RZ
[]
```

(modules/tensor_mechanics/test/tests/2D_geometries/2D-RZ_finiteStrain_test.i)and only two displacement variables are provided:

```
[GlobalParams]
displacements = 'disp_r disp_z'
[]
```

(modules/tensor_mechanics/test/tests/2D_geometries/2D-RZ_finiteStrain_test.i)## Input Parameters

- temperatureThe name of the temperature variable used in the ComputeThermalExpansionEigenstrain. (Not required for simulations without temperature coupling.)
C++ Type:std::vector

Description:The name of the temperature variable used in the ComputeThermalExpansionEigenstrain. (Not required for simulations without temperature coupling.)

- base_nameMaterial property base name
C++ Type:std::string

Description:Material property base name

- use_finite_deform_jacobianFalseJacobian for corotational finite strain
Default:False

C++ Type:bool

Description:Jacobian for corotational finite strain

- out_of_plane_directionzThe direction of the out_of_plane_strain variable used in the WeakPlaneStress kernel.
Default:z

C++ Type:MooseEnum

Description:The direction of the out_of_plane_strain variable used in the WeakPlaneStress kernel.

- volumetric_locking_correctionFalseSet to false to turn off volumetric locking correction
Default:False

C++ Type:bool

Description:Set to false to turn off volumetric locking correction

- thermal_eigenstrain_namethermal_eigenstrainThe eigenstrain_name used in the ComputeThermalExpansionEigenstrain.
Default:thermal_eigenstrain

C++ Type:std::string

Description:The eigenstrain_name used in the ComputeThermalExpansionEigenstrain.

- out_of_plane_strainThe name of the out_of_plane_strain variable used in the WeakPlaneStress kernel. Required only if want to provide off-diagonal Jacobian in plane stress analysis using weak formulation.
C++ Type:std::vector

Description:The name of the out_of_plane_strain variable used in the WeakPlaneStress kernel. Required only if want to provide off-diagonal Jacobian in plane stress analysis using weak formulation.

- 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

### Optional Parameters

- enableTrueSet the enabled status of the MooseObject.
Default:True

C++ Type:bool

Description:Set the enabled status of the MooseObject.

- 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

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.)

- use_displaced_meshTrueWhether 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:True

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

- 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

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.)

- 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

### Advanced Parameters

- vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime

C++ Type:MultiMooseEnum

Description:The tag for the vectors this Kernel should fill

- extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector

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

Description:The tag for the matrices this Kernel should fill

- extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector

Description:The extra tags for the matrices this Kernel should fill

### Tagging Parameters

## Input Files

- modules/porous_flow/examples/tutorial/11_2D.i
- modules/tensor_mechanics/test/tests/jacobian/thermal_coupling_rz.i