Compute Small Strain

Compute a small strain.

Description

The material ComputeSmallStrain is designed for linear elasticity problems, which calculates the small, total strain. This material is useful for verifying material models with hand calculations because of the simplified strain calculations.

Linearized small strain theory assumes that the gradient of displacement with respect to position is much smaller than unity, and the squared displacement gradient term is neglected in the small strain definition to give: $var element = document.getElementById("moose-equation-2227733e-564a-4550-bb85-241bb0e568f6");katex.render("\\epsilon = \\frac{1}{2} \\left( u \\nabla + \\nabla u \\right) \\quad when \\quad \\frac{\\partial u}{ \\partial x} << 1", element, {displayMode:true,throwOnError:false,macros:{"\\bs":"\\boldsymbol{#1}","\\normal":"\\bs{n}","\\grad":"\\bs{\\nabla}","\\divergence":"\\grad \\cdot","\\norm":"\\left\\lVert#1\\right\\rVert","\\abs":"\\left\\lvert#1\\right\\rvert","\\tr":"\\text{tr}","\\dev":"\\text{dev}","\\sym":"\\text{sym}","\\diff":"\\text{ d}#1","\\activepart":"{\\left< A \\right>}","\\inactivepart":"{\\left< I \\right>}","\\Gc":"{\\mathcal{G}_c}","\\strain":"\\bs{\\varepsilon}","\\stress":"\\bs{\\sigma}","\\macaulay":"\\left<#1\\right>","\\body":"\\Omega","\\bodyboundary":"{\\partial\\body}","\\ep":"{\\varepsilon^p}","\\ep0":"{\\varepsilon_0^p}","\\epdot":"{\\dot{\\varepsilon}}^p","\\epdot0":"{\\dot{\\varepsilon}}_0^p","\\bfa":"\\boldsymbol{a}","\\bfb":"\\boldsymbol{b}","\\bfc":"\\boldsymbol{c}","\\bfd":"\\boldsymbol{d}","\\bfe":"\\boldsymbol{e}","\\bff":"\\boldsymbol{f}","\\bfg":"\\boldsymbol{g}","\\bfh":"\\boldsymbol{h}","\\bfi":"\\boldsymbol{i}","\\bfj":"\\boldsymbol{j}","\\bfk":"\\boldsymbol{k}","\\bfl":"\\boldsymbol{l}","\\bfm":"\\boldsymbol{m}","\\bfn":"\\boldsymbol{n}","\\bfo":"\\boldsymbol{o}","\\bfp":"\\boldsymbol{p}","\\bfq":"\\boldsymbol{q}","\\bfr":"\\boldsymbol{r}","\\bfs":"\\boldsymbol{s}","\\bft":"\\boldsymbol{t}","\\bfu":"\\boldsymbol{u}","\\bfv":"\\boldsymbol{v}","\\bfw":"\\boldsymbol{w}","\\bfx":"\\boldsymbol{x}","\\bfy":"\\boldsymbol{y}","\\bfz":"\\boldsymbol{z}","\\bfA":"\\boldsymbol{A}","\\bfB":"\\boldsymbol{B}","\\bfC":"\\boldsymbol{C}","\\bfD":"\\boldsymbol{D}","\\bfE":"\\boldsymbol{E}","\\bfF":"\\boldsymbol{F}","\\bfG":"\\boldsymbol{G}","\\bfH":"\\boldsymbol{H}","\\bfI":"\\boldsymbol{I}","\\bfJ":"\\boldsymbol{J}","\\bfK":"\\boldsymbol{K}","\\bfL":"\\boldsymbol{L}","\\bfM":"\\boldsymbol{M}","\\bfN":"\\boldsymbol{N}","\\bfO":"\\boldsymbol{O}","\\bfP":"\\boldsymbol{P}","\\bfQ":"\\boldsymbol{Q}","\\bfR":"\\boldsymbol{R}","\\bfS":"\\boldsymbol{S}","\\bfT":"\\boldsymbol{T}","\\bfU":"\\boldsymbol{U}","\\bfV":"\\boldsymbol{V}","\\bfW":"\\boldsymbol{W}","\\bfX":"\\boldsymbol{X}","\\bfY":"\\boldsymbol{Y}","\\bfZ":"\\boldsymbol{Z}"}});$ For more details on the linearized small strain assumption and derivation, see a Continuum Mechanics text such as Malvern (1969) or Bower (2009), specifically Chapter 2.

Total strain theories are path independent: in MOOSE, path independence means that the total strain, from the beginning of the entire simulation, is used to calculate stress and other material properties. Incremental theories, on the other hand, use the increment of strain at timestep to calculate stress. Because the total strain formulation ComputeSmallStrain is path independent, no old values of strain or stress from the previous timestep are stored in MOOSE. For a comparison of total strain vs incremental strain theories with experimental data, see Shammamy and Sidebottom (1967).

Example Input File Syntax

The small strain calculator can be activated in the input file through the use of the SolidMechanics QuasiStatic Physics, as shown below.

[Physics]
  [SolidMechanics]
    [QuasiStatic]
      [./block1]
        strain = SMALL #Small linearized strain, automatically set to XY coordinates
        add_variables = true #Add the variables from the displacement string in GlobalParams
      [../]
    []
  []
[]

note:Use of the Solid Mechanics QuasiStatic Physics Recommended

The Solid Mechanics Physics is designed to automatically determine and set the strain and stress divergence parameters correctly for the selected strain formulation. We recommend that users employ the Solid Mechanics Physics whenever possible to ensure consistency between the test function gradients and the strain formulation selected.

Although not recommended, it is possible to directly use the ComputeSmallStrain material in an input file.

[./strain]
  type = ComputeSmallStrain
[../]

Input Parameters

displacementsThe displacements appropriate for the simulation geometry and coordinate system
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
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
Unit:(no unit assumed)
Controllable:No
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 blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Unit:(no unit assumed)
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaryThe list of boundaries (ids or names) from the mesh where this object applies
C++ Type:std::vector<BoundaryName>
Unit:(no unit assumed)
Controllable:No
Description:The list of boundaries (ids or names) from the mesh where this object 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
Unit:(no unit assumed)
Controllable:No
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 computeQpProperties() 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
Unit:(no unit assumed)
Options:NONE, ELEMENT, SUBDOMAIN
Controllable:No
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 computeQpProperties() for the 0th quadrature point, and then copy that value to the other qps. Evaluations on element qps will be skipped
declare_suffixAn optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
eigenstrain_namesList of eigenstrains to be applied in this strain calculation
C++ Type:std::vector<MaterialPropertyName>
Unit:(no unit assumed)
Controllable:No
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
Unit:(no unit assumed)
Controllable:No
Description:Optional material property holding a global strain tensor applied to the mesh as a whole
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
Unit:(no unit assumed)
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.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
volumetric_locking_correctionFalseFlag to correct volumetric locking
Default:False
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:Flag to correct volumetric locking

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Unit:(no unit assumed)
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Unit:(no unit assumed)
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
Unit:(no unit assumed)
Controllable:No
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
Unit:(no unit assumed)
Controllable:No
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<std::string>
Unit:(no unit assumed)
Controllable:No
Description:List of material properties, from this material, to output (outputs must also be defined to an output type)
outputsnone Vector of output names where you would like to restrict the output of variables(s) associated with this object
Default:none
C++ Type:std::vector<OutputName>
Unit:(no unit assumed)
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

References

A. F. Bower. Applied Mechanics of Solids. CRC press, 2009.

Lawrence E Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, 1969.

MR Shammamy and OM Sidebottom. Incremental versus total-strain theories for proportionate and nonproportionate loading of torsion-tension members. Experimental Mechanics, 7(12):497–505, 1967.

@article{shammamy1967incremental,
    author = "Shammamy, MR and Sidebottom, OM",
    title = "Incremental versus total-strain theories for proportionate and nonproportionate loading of torsion-tension members",
    journal = "Experimental Mechanics",
    volume = "7",
    number = "12",
    pages = "497--505",
    year = "1967",
    publisher = "Springer"
}

(moose/modules/solid_mechanics/tutorials/basics/part_1.1.i)

#Tensor Mechanics tutorial: the basics
#Step 1, part 1
#2D simulation of uniaxial tension with linear elasticity

[GlobalParams]
  displacements = 'disp_x disp_y'
[]

[Mesh]
  file = necking_quad4.e
[]

[Physics/SolidMechanics/QuasiStatic]
  [./block1]
    strain = SMALL #Small linearized strain, automatically set to XY coordinates
    add_variables = true #Add the variables from the displacement string in GlobalParams
  [../]
[]

[Materials]
  [./elasticity_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 2.1e5
    poissons_ratio = 0.3
  [../]
  [./stress]
    type = ComputeLinearElasticStress
  [../]
[]

[BCs]
  [./left]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./bottom]
    type = DirichletBC
    variable = disp_y
    boundary = bottom
    value = 0.0
  [../]
  [./top]
    type = DirichletBC
    variable = disp_y
    boundary = top
    value = 0.05
  [../]
[]

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

[Executioner]
  type = Steady

  solve_type = 'NEWTON'

  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type -sub_pc_type -pc_asm_overlap -ksp_gmres_restart'
  petsc_options_value = 'asm lu 1 101'
[]

[Outputs]
  exodus = true
  perf_graph = true
[]

(moose/modules/solid_mechanics/test/tests/elastic_patch/elastic_patch_quadratic.i)

# Patch Test for second order hex elements (HEX20)
#
# From Abaqus, Verification Manual, 1.5.2
#
# This test is designed to compute constant xx, yy, zz, xy, yz, and zx
#  stress on a set of irregular hexes.  The mesh is composed of one
#  block with seven elements.  The elements form a unit cube with one
#  internal element.  There is a nodeset for each exterior node.

# The cube is displaced on all exterior nodes using the functions,
#
#    ux = 1e-4 * (2x + y + z) / 2
#    uy = 1e-4 * (x + 2y + z) / 2
#    ux = 1e-4 * (x + y + 2z) / 2
#
#  giving uniform strains of
#
#    exx = eyy = ezz = 2*exy = 2*eyz = 2*exz = 1e-4
#
#
# Hooke's Law provides an analytical solution for the uniform stress state.
#  For example,
#
#    stress xx = lambda(exx + eyy + ezz) + 2 * G * exx
#    stress xy = 2 * G * exy
#
#   where:
#
#    lambda = (2 * G * nu) / (1 - 2 * nu)
#    G = 0.5 * E / (1 + nu)
#
# For the test below, E = 1e6 and nu = 0.25, giving lambda = G = 4e5
#
# Thus
#
#    stress xx = 4e5 * (3e-4) + 2 * 4e5 * 1e-4 = 200
#    stress xy = 2 * 4e5 * 1e-4 / 2 = 40
#
[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]
[Mesh]
  file = elastic_patch_quadratic.e
[] # Mesh

[Functions]
  [./xDispFunc]
    type = ParsedFunction
    expression = 5e-5*(2*x+y+z)
  [../]
  [./yDispFunc]
    type = ParsedFunction
    expression = 5e-5*(x+2*y+z)
  [../]
  [./zDispFunc]
    type = ParsedFunction
    expression = 5e-5*(x+y+2*z)
  [../]
[] # Functions

[Variables]
  [./disp_x]
    order = SECOND
    family = LAGRANGE
  [../]
  [./disp_y]
    order = SECOND
    family = LAGRANGE
  [../]
  [./disp_z]
    order = SECOND
    family = LAGRANGE
  [../]
[] # Variables

[AuxVariables]
  [./stress_xx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_yy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_zz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_xy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_yz]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./stress_zx]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./elastic_energy]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./vonmises]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./hydrostatic]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./firstinv]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./secondinv]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./thirdinv]
    order = CONSTANT
    family = MONOMIAL
  [../]
[] # AuxVariables

[Kernels]
  [SolidMechanics]
    use_displaced_mesh = true
  [../]
[]

[AuxKernels]
  [./stress_xx]
    type = RankTwoAux
    rank_two_tensor = stress
    index_i = 0
    index_j = 0
    variable = stress_xx
  [../]
  [./stress_yy]
    type = RankTwoAux
    rank_two_tensor = stress
    index_i = 1
    index_j = 1
    variable = stress_yy
  [../]
  [./stress_zz]
    type = RankTwoAux
    rank_two_tensor = stress
    index_i = 2
    index_j = 2
    variable = stress_zz
  [../]
  [./stress_xy]
    type = RankTwoAux
    rank_two_tensor = stress
    index_i = 0
    index_j = 1
    variable = stress_xy
  [../]
  [./stress_yz]
    type = RankTwoAux
    rank_two_tensor = stress
    index_i = 1
    index_j = 2
    variable = stress_yz
  [../]
  [./stress_zx]
    type = RankTwoAux
    rank_two_tensor = stress
    index_i = 2
    index_j = 0
    variable = stress_zx
  [../]
  [./elastic_energy]
    type = ElasticEnergyAux
    variable = elastic_energy
  [../]
  [./vonmises]
    type = RankTwoScalarAux
    rank_two_tensor = stress
    scalar_type = VonMisesStress
    variable = vonmises
  [../]
  [./hydrostatic]
    type = RankTwoScalarAux
    rank_two_tensor = stress
    scalar_type = Hydrostatic
    variable = hydrostatic
  [../]
  [./fi]
    type = RankTwoScalarAux
    rank_two_tensor = stress
    scalar_type = FirstInvariant
    variable = firstinv
  [../]
  [./si]
    type = RankTwoScalarAux
    rank_two_tensor = stress
    scalar_type = SecondInvariant
    variable = secondinv
  [../]
  [./ti]
    type = RankTwoScalarAux
    rank_two_tensor = stress
    scalar_type = ThirdInvariant
    variable = thirdinv
  [../]
[] # AuxKernels

[BCs]
  [./all_nodes_x]
    type = FunctionDirichletBC
    variable = disp_x
    boundary = '1 2 3 4 6 7 8 9 10 12 15 17 18 19 20 21 23 24 25 26'
    function = xDispFunc
  [../]
  [./all_nodes_y]
    type = FunctionDirichletBC
    variable = disp_y
    boundary = '1 2 3 4 6 7 8 9 10 12 15 17 18 19 20 21 23 24 25 26'
    function = yDispFunc
  [../]
  [./all_nodes_z]
    type = FunctionDirichletBC
    variable = disp_z
    boundary = '1 2 3 4 6 7 8 9 10 12 15 17 18 19 20 21 23 24 25 26'
    function = zDispFunc
  [../]
[] # BCs

[Materials]
  [./elast_tensor]
    type = ComputeIsotropicElasticityTensor
    youngs_modulus = 1e6
    poissons_ratio = 0.25
  [../]
  [./strain]
    type = ComputeSmallStrain
  [../]
  [./stress]
    type = ComputeLinearElasticStress
  [../]
[] # Materials

[Executioner]
  type = Transient

  #Preconditioned JFNK (default)
  solve_type = 'PJFNK'

  nl_rel_tol = 1e-6

  l_max_its = 20

  start_time = 0.0
  dt = 1.0
  num_steps = 1
  end_time = 1.0
[] # Executioner

[Outputs]
  [./out]
    type = Exodus
    elemental_as_nodal = true
  [../]
[] # Outputs