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

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

- component0An integer corresponding to the direction the variable this kernel acts in. (0 for x, 1 for y, 2 for z)
Default:0

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)

- 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 RSpherical Tensors

Calculate stress divergence for a spherically symmetric 1D problem in polar coordinates.

## Description

The kernel `StressDivergenceRSphericalTensors`

solves the stress divergence equation for a spherically symmetric system on a 1D mesh.

The `COORD_TYPE`

in the Problem block of the input file must be set to RSPHERICAL.

The `StressDivergenceRSphericalTensors`

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, , 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.

The stress divergence in spherical coordinates includes contributions from the normal polar and azimuthal stresses even in the 1D case. After simplifying for the 1D case, the spherical stress divergence reduces to (4)

In deriving the weak form of this equation, the second term in Eq. (4) goes to zero and the residual contribution in the `StressDivergenceRSphericalTensors`

kernel becomes (5)

## Example Input File syntax

Using the tensor mechanics master action, as shown

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

(modules/tensor_mechanics/test/tests/1D_spherical/finiteStrain_1DSphere_hollow.i)the `StressDivergenceRSphericalTensors`

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

```
[Problem]
coord_type = RSPHERICAL
[]
```

(modules/tensor_mechanics/test/tests/1D_spherical/finiteStrain_1DSphere_hollow.i)and only a single displacement variable is provided:

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

(modules/tensor_mechanics/test/tests/1D_spherical/finiteStrain_1DSphere_hollow.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

## 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`

## Input Files

- modules/tensor_mechanics/test/tests/ad_elastic/rspherical_incremental_small_elastic-noad.i
- modules/tensor_mechanics/test/tests/ad_elastic/rspherical_finite_elastic-noad.i
- modules/tensor_mechanics/test/tests/ad_elastic/rspherical_small_elastic-noad.i