ADStressDivergenceRZTensors

Calculate stress divergence for an axisymmetric problem in cylindrical coordinates.

Description

The kernel ADStressDivergenceRZTensors solves the stress divergence equation for an Axisymmetric problem in the cylindrical coordinate system on a 2D mesh. Forward mode automatic differentiation is used to compute an exact Jacobian.

warning:Symmetry Assumed About the

var element = document.getElementById("moose-equation-ccd6a435-bded-4975-944e-0a8705f47e78");katex.render("z", element, {displayMode:false,throwOnError:false});

-axis

The axis of symmetry must lie along the $var element = document.getElementById("moose-equation-e4882751-716b-4d9e-b437-04e70ef13a05");katex.render("z", element, {displayMode:false,throwOnError:false});$ -axis in a $var element = document.getElementById("moose-equation-91309876-4585-406d-b899-01f705585507");katex.render("\\left(r, z, \\theta \\right)", element, {displayMode:false,throwOnError:false});$ or cylindrical coordinate system. This symmetry orientation is required for the calculation of the residual and of the jacobian, as defined in Eq. (1).

Residual Calculation

The stress divergence kernel handles the calculation of the residual, $var element = document.getElementById("moose-equation-3c831bc1-d40a-4086-82f5-fe4d6d623f2e");katex.render("\\mathbb{R}", element, {displayMode:false,throwOnError:false});$ , from the governing equation and the calculation of the Jacobian using forward mode automatic differentiation. From the strong form of the governing equation for mechanics, neglecting body forces,

$var element = document.getElementById("moose-equation-ac048e12-1e50-4df2-b975-21dd2db0f503");katex.render("\\nabla \\cdot \\sigma = 0", element, {displayMode:true,throwOnError:false});$

the weak form, using Galerkin's method and the Gauss divergence theorem, becomes

$var element = document.getElementById("moose-equation-23a5cf22-e23c-4e13-87a2-1a0463c753a1");katex.render("\\int_S n \\cdot (\\sigma \\psi) dS - \\int_V \\sigma \\cdot \\nabla \\psi dV = 0", element, {displayMode:true,throwOnError:false});$

in which $var element = document.getElementById("moose-equation-1128a2a4-1ca7-4569-bf38-9c692d4f4627");katex.render("\\psi", element, {displayMode:false,throwOnError:false});$ is the test function. The second term of the weak form equation is the residual contribution calculated by the stress divergence kernel.

The 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.

note:Automatic Differentiation Materials

The computation of a correct Jacobian contribution requires the use of compatible automatic differentiation materials ([ADMaterials]).

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.

$(1)var element = document.getElementById("moose-equation-b50d7e69-051f-4638-99c4-ed2934180a5b");katex.render(" \\begin{aligned} \\nabla \\sigma & = \\left[ \\frac{\\partial \\sigma_{rr}}{\\partial r} + \\frac{u_r}{X_r}\\sigma_{\\theta \\theta} + \\frac{\\partial \\sigma_{rz}}{\\partial z} \\right] \\hat{e}_r \\\\ & + \\left[ \\frac{\\partial \\sigma_{zz}}{\\partial z} + \\frac{\\partial \\sigma_{rz}}{\\partial r} \\right] \\hat{e}_z \\end{aligned}", element, {displayMode:true,throwOnError:false});$

note:Notation Order Change

The axisymmetric system changes the order of the displacement vector from $var element = document.getElementById("moose-equation-0deec333-213a-43a0-bbeb-33f8f4164493");katex.render("(u_r, u_{\\theta}, u_z)", element, {displayMode:false,throwOnError:false});$ , usually seen in textbooks, to $var element = document.getElementById("moose-equation-2ebb8c53-4f10-4707-9991-dd547e273244");katex.render("(u_r, u_z, u_{\\theta})", element, {displayMode:false,throwOnError:false});$ . Take care to follow this convention in your input files and when adding extra stresses.

note:Use RZ Coordinate Type

The coordinate type in the [Problem] block of the input file must be set to coord_type = RZ.

Input Parameters

componentAn integer corresponding to the direction the variable this kernel acts in. (0 refers to the radial and 1 to the axial displacement.)
C++ Type:unsigned int
Controllable:No
Description:An integer corresponding to the direction the variable this kernel acts in. (0 refers to the radial and 1 to the axial displacement.)
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
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.
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_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
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