Compute Axisymmetric 1D Small Strain

Compute a small strain in an Axisymmetric 1D problem

Description

The material ComputeAxisymmetric1DSmallStrain calculates the small total strain for 1D Axisymmetric systems and is intended for use with Generalized Plane Strain simulations. This material assumes symmetry about the $var element = document.getElementById("moose-equation-9f4fd48b-db10-4193-806e-bd141b79d177");katex.render("z", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ -axis. This 'strain calculator' material computes the strain within the cylindrical coordinate system and relies on the specialized Axisymmetric RZ kernel to handle the stress divergence calcuation.

warning:Symmetry Assumed About the

var element = document.getElementById("moose-equation-375addf7-163d-47af-abd2-1ca1e54e4de7");katex.render("z", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});

-axis

The axis of symmetry must lie along the $var element = document.getElementById("moose-equation-5c4db006-c6fd-47d9-b2cd-17ba14d86799");katex.render("z", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ -axis in a $var element = document.getElementById("moose-equation-d3aa6581-bcba-4e3b-b29d-17e5edab38f4");katex.render("\\left(r, z, \\theta \\right)", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ cylindrical coordinate system. This symmetry orientation is required for the calculation of the residual and of the jacobian. See StressDivergenceRZTensors for the residual equation and the germane discussion.

1D Axisymmetric Strain Formulation

The axisymmetric model uses the cylindrical coordinates, $var element = document.getElementById("moose-equation-ac90a3a0-42a0-44f8-87fd-719ffb81be89");katex.render("r", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , $var element = document.getElementById("moose-equation-133c2d36-7a37-4f79-8f3d-195c3bdf70f6");katex.render("z", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , and $var element = document.getElementById("moose-equation-000b6c1f-c3c2-4386-b8f4-7417d1639513");katex.render("\\theta", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , where the linear section formed by the $var element = document.getElementById("moose-equation-f57bde8d-f0da-49ec-80fb-331793258e0d");katex.render("r", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ axis is rotated about the $var element = document.getElementById("moose-equation-9f62001c-1824-4021-8a72-e6b0eb14082c");katex.render("z", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ axis in the $var element = document.getElementById("moose-equation-e4ee05a9-7d95-4c42-a3cf-2f2ad464885b");katex.render("\\theta", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ direction.

The definition of a small total linearized strain is $(1)var element = document.getElementById("moose-equation-ba0b04c9-75f6-4ef2-b73a-f8446392d9b8");katex.render(" \\epsilon_{ij} = \\frac{1}{2} \\left( u_{i,j} + u_{j,i} \\right)", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ In this asixymmetric 1D formulation, the strain tensor is diagonal. Eq. (1) is therefore implemented in a straight-forward manner as $(2)var element = document.getElementById("moose-equation-a765bd7f-395c-4033-a06d-49b7783f7af8");katex.render(" \\epsilon_{ij} = \\begin{bmatrix} \\epsilon_{rr} & 0 & 0 \\\\ 0 & \\epsilon_{zz} & 0 \\\\ 0 & 0 & \\epsilon_{\\theta \\theta} \\end{bmatrix}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where the components of the strain tensor in Eq. (2) are given as $(3)var element = document.getElementById("moose-equation-5e645a9f-c440-4352-8f10-a669d0d64d0e");katex.render(" \\begin{aligned} \\epsilon_{rr} & = u_{r,r} \\\\ \\epsilon_{zz} & = \\epsilon|^{op} \\\\ \\epsilon_{\\theta \\theta} & = \\frac{u_r}{X_r} \\end{aligned}", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-77e2c3f1-7e1e-421e-be71-ec6109fb9dd6");katex.render("\\epsilon|^{op}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is a prescribed out-of-plane strain value: this strain value can be given either as a scalar variable or a nonlinear variable. The Generalized Plane Strain problems use scalar variables. The value of the strain $var element = document.getElementById("moose-equation-01b30115-36f2-4f9a-91b8-b65eb8a1a3ee");katex.render("\\epsilon_{\\theta \\theta}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ depends on the displacement and position in the radial direction.

note:Notation Order Change

The axisymmetric system changes the order of the displacement vector from $var element = document.getElementById("moose-equation-989275ac-89ae-44cf-8548-dc58228589cd");katex.render("(u_r, u_{\\theta}, u_z)", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , usually seen in textbooks, to $var element = document.getElementById("moose-equation-0deba9f0-9a87-4ac1-9537-c1abc3c83ea3");katex.render("(u_r, u_z, u_{\\theta})", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ . Take care to follow this convention in your input files and when adding eigenstrains or extra stresses.

Example Input File

note:Use RZ Coordinate Type

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

The common use of the ComputeAxisymmetric1DSmallStrain class is with the Generalized Plane Strain system; this type of simulation uses the scalar strain variables

[./strain]
  type = ComputeAxisymmetric1DSmallStrain
  eigenstrain_names = eigenstrain
  scalar_out_of_plane_strain = scalar_strain_yy
[../]

/opt/civet/build_0/moose/modules/tensor_mechanics/test/tests/1D_axisymmetric/axisymm_gps_small.i

# this test checks the asixymmetric 1D generalized plane strain formulation using incremental small strains

[GlobalParams]
  displacements = disp_x
  scalar_out_of_plane_strain = scalar_strain_yy
[]

[Problem]
  coord_type = RZ
[]

[Mesh]
  file = line.e
[]

[Variables]
  [./disp_x]
  [../]
  [./scalar_strain_yy]
    order = FIRST
    family = SCALAR
  [../]
[]

[AuxVariables]
  [./strain_xx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./strain_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./strain_zz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_xx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_zz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./temp]
    initial_condition = 580.0
  [../]
[]

[Functions]
  [./temp]
    type = PiecewiseLinear
    x = '0   1   2'
    y = '580 580 680'
  [../]
  [./disp_x]
    type = PiecewiseLinear
    x = '0 1'
    y = '0 2e-6'
  [../]
[]

[Kernels]
  [./TensorMechanics]
  [../]
[]

[Modules]
  [./TensorMechanics]
    [./GeneralizedPlaneStrain]
      [./gps]
      [../]
    [../]
  [../]
[]

[AuxKernels]
  [./strain_xx]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_xx
    index_i = 0
    index_j = 0
  [../]
  [./strain_yy]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_yy
    index_i = 1
    index_j = 1
  [../]
  [./strain_zz]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_zz
    index_i = 2
    index_j = 2
  [../]
  [./stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_i = 0
    index_j = 0
    execute_on = timestep_end
  [../]
  [./stress_yy]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_yy
    index_i = 1
    index_j = 1
    execute_on = timestep_end
  [../]
  [./stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_i = 2
    index_j = 2
    execute_on = timestep_end
  [../]
  [./temp]
    type = FunctionAux
    variable = temp
    function = temp
    execute_on = 'timestep_begin'
  [../]
[]

[BCs]
  [./no_x]
    type = DirichletBC
    boundary = 1
    value = 0
    variable = disp_x
  [../]
  [./disp_x]
    type = FunctionDirichletBC
    boundary = 2
    function = disp_x
    variable = disp_x
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 3600
    poissons_ratio = 0.2
  [../]

  [./strain]
    type = ComputeAxisymmetric1DSmallStrain
    eigenstrain_names = eigenstrain
    scalar_out_of_plane_strain = scalar_strain_yy
  [../]

  [./thermal_strain]
    type = ComputeThermalExpansionEigenstrain
    thermal_expansion_coeff = 1e-8
    temperature = temp
    stress_free_temperature = 580
    eigenstrain_name = eigenstrain
  [../]

  [./stress]
    type = ComputeLinearElasticStress
  [../]
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-6
  nl_max_its = 15
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-10
  start_time = 0
  end_time = 2
  num_steps = 2
[]

[Outputs]
  exodus = true
  console = true
[]

which uses a scalar variable for the coupled out-of-plane strain; the arguement for the scalar_out_of_plane_strain parameter is the name of the scalar strain variable:

[./scalar_strain_yy]
  order = FIRST
  family = SCALAR
[../]

/opt/civet/build_0/moose/modules/tensor_mechanics/test/tests/1D_axisymmetric/axisymm_gps_small.i

# this test checks the asixymmetric 1D generalized plane strain formulation using incremental small strains

[GlobalParams]
  displacements = disp_x
  scalar_out_of_plane_strain = scalar_strain_yy
[]

[Problem]
  coord_type = RZ
[]

[Mesh]
  file = line.e
[]

[Variables]
  [./disp_x]
  [../]
  [./scalar_strain_yy]
    order = FIRST
    family = SCALAR
  [../]
[]

[AuxVariables]
  [./strain_xx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./strain_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./strain_zz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_xx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_zz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./temp]
    initial_condition = 580.0
  [../]
[]

[Functions]
  [./temp]
    type = PiecewiseLinear
    x = '0   1   2'
    y = '580 580 680'
  [../]
  [./disp_x]
    type = PiecewiseLinear
    x = '0 1'
    y = '0 2e-6'
  [../]
[]

[Kernels]
  [./TensorMechanics]
  [../]
[]

[Modules]
  [./TensorMechanics]
    [./GeneralizedPlaneStrain]
      [./gps]
      [../]
    [../]
  [../]
[]

[AuxKernels]
  [./strain_xx]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_xx
    index_i = 0
    index_j = 0
  [../]
  [./strain_yy]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_yy
    index_i = 1
    index_j = 1
  [../]
  [./strain_zz]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_zz
    index_i = 2
    index_j = 2
  [../]
  [./stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_i = 0
    index_j = 0
    execute_on = timestep_end
  [../]
  [./stress_yy]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_yy
    index_i = 1
    index_j = 1
    execute_on = timestep_end
  [../]
  [./stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_i = 2
    index_j = 2
    execute_on = timestep_end
  [../]
  [./temp]
    type = FunctionAux
    variable = temp
    function = temp
    execute_on = 'timestep_begin'
  [../]
[]

[BCs]
  [./no_x]
    type = DirichletBC
    boundary = 1
    value = 0
    variable = disp_x
  [../]
  [./disp_x]
    type = FunctionDirichletBC
    boundary = 2
    function = disp_x
    variable = disp_x
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 3600
    poissons_ratio = 0.2
  [../]

  [./strain]
    type = ComputeAxisymmetric1DSmallStrain
    eigenstrain_names = eigenstrain
    scalar_out_of_plane_strain = scalar_strain_yy
  [../]

  [./thermal_strain]
    type = ComputeThermalExpansionEigenstrain
    thermal_expansion_coeff = 1e-8
    temperature = temp
    stress_free_temperature = 580
    eigenstrain_name = eigenstrain
  [../]

  [./stress]
    type = ComputeLinearElasticStress
  [../]
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-6
  nl_max_its = 15
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-10
  start_time = 0
  end_time = 2
  num_steps = 2
[]

[Outputs]
  exodus = true
  console = true
[]

Input Parameters

displacementsThe displacements appropriate for the simulation geometry and coordinate system
C++ Type:std::vector
Options:
Description:The displacements appropriate for the simulation geometry and coordinate system

Required Parameters

base_nameOptional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
C++ Type:std::string
Options:
Description:Optional parameter that allows the user to define multiple mechanics material systems on the same block, i.e. for multiple phases
blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector
Options:
Description:The list of block ids (SubdomainID) that this object will be applied
boundaryThe list of boundary IDs from the mesh where this boundary condition applies
C++ Type:std::vector
Options:
Description:The list of boundary IDs from the mesh where this boundary condition applies
computeTrueWhen false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
Default:True
C++ Type:bool
Options:
Description:When false, MOOSE will not call compute methods on this material. The user must call computeProperties() after retrieving the MaterialBase via MaterialBasePropertyInterface::getMaterialBase(). Non-computed MaterialBases are not sorted for dependencies.
constant_onNONEWhen ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
Default:NONE
C++ Type:MooseEnum
Options:NONE ELEMENT SUBDOMAIN
Description:When ELEMENT, MOOSE will only call computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps.When SUBDOMAIN, MOOSE will only call computeSubdomainProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
eigenstrain_namesList of eigenstrains to be applied in this strain calculation
C++ Type:std::vector
Options:
Description:List of eigenstrains to be applied in this strain calculation
global_strainOptional material property holding a global strain tensor applied to the mesh as a whole
C++ Type:MaterialPropertyName
Options:
Description:Optional material property holding a global strain tensor applied to the mesh as a whole
out_of_plane_strainNonlinear variable for axisymmetric 1D problem
C++ Type:std::vector
Options:
Description:Nonlinear variable for axisymmetric 1D problem
scalar_out_of_plane_strainScalar variable for axisymmetric 1D problem
C++ Type:std::vector
Options:
Description:Scalar variable for axisymmetric 1D problem
subblock_index_providerSubblockIndexProvider user object name
C++ Type:UserObjectName
Options:
Description:SubblockIndexProvider user object name
volumetric_locking_correctionFalseFlag to correct volumetric locking
Default:False
C++ Type:bool
Options:
Description:Flag to correct volumetric locking

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
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
Options:
Description:Determines whether this object is calculated using an implicit or explicit form
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Options:
Description:The seed for the master random number generator

Advanced Parameters

output_propertiesList of material properties, from this material, to output (outputs must also be defined to an output type)
C++ Type:std::vector
Options:
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names were you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector
Options:
Description:Vector of output names were you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

Input Files

modules/tensor_mechanics/test/tests/1D_axisymmetric/axisymm_gps_small.i

modules/tensor_mechanics/test/tests/1D_axisymmetric/axisymm_gps_small.i

# this test checks the asixymmetric 1D generalized plane strain formulation using incremental small strains

[GlobalParams]
  displacements = disp_x
  scalar_out_of_plane_strain = scalar_strain_yy
[]

[Problem]
  coord_type = RZ
[]

[Mesh]
  file = line.e
[]

[Variables]
  [./disp_x]
  [../]
  [./scalar_strain_yy]
    order = FIRST
    family = SCALAR
  [../]
[]

[AuxVariables]
  [./strain_xx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./strain_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./strain_zz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_xx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_zz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./temp]
    initial_condition = 580.0
  [../]
[]

[Functions]
  [./temp]
    type = PiecewiseLinear
    x = '0   1   2'
    y = '580 580 680'
  [../]
  [./disp_x]
    type = PiecewiseLinear
    x = '0 1'
    y = '0 2e-6'
  [../]
[]

[Kernels]
  [./TensorMechanics]
  [../]
[]

[Modules]
  [./TensorMechanics]
    [./GeneralizedPlaneStrain]
      [./gps]
      [../]
    [../]
  [../]
[]

[AuxKernels]
  [./strain_xx]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_xx
    index_i = 0
    index_j = 0
  [../]
  [./strain_yy]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_yy
    index_i = 1
    index_j = 1
  [../]
  [./strain_zz]
    type = RankTwoAux
    rank_two_tensor = total_strain
    variable = strain_zz
    index_i = 2
    index_j = 2
  [../]
  [./stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_xx
    index_i = 0
    index_j = 0
    execute_on = timestep_end
  [../]
  [./stress_yy]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_yy
    index_i = 1
    index_j = 1
    execute_on = timestep_end
  [../]
  [./stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    variable = stress_zz
    index_i = 2
    index_j = 2
    execute_on = timestep_end
  [../]
  [./temp]
    type = FunctionAux
    variable = temp
    function = temp
    execute_on = 'timestep_begin'
  [../]
[]

[BCs]
  [./no_x]
    type = DirichletBC
    boundary = 1
    value = 0
    variable = disp_x
  [../]
  [./disp_x]
    type = FunctionDirichletBC
    boundary = 2
    function = disp_x
    variable = disp_x
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 3600
    poissons_ratio = 0.2
  [../]

  [./strain]
    type = ComputeAxisymmetric1DSmallStrain
    eigenstrain_names = eigenstrain
    scalar_out_of_plane_strain = scalar_strain_yy
  [../]

  [./thermal_strain]
    type = ComputeThermalExpansionEigenstrain
    thermal_expansion_coeff = 1e-8
    temperature = temp
    stress_free_temperature = 580
    eigenstrain_name = eigenstrain
  [../]

  [./stress]
    type = ComputeLinearElasticStress
  [../]
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-6
  nl_max_its = 15
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-10
  start_time = 0
  end_time = 2
  num_steps = 2
[]

[Outputs]
  exodus = true
  console = true
[]

Description
1 D Axisymmetric Strain Formulation
Example Input File
Input Parameters
Input Files

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