Plastic Truss

Computes the stress and strain for a truss element with plastic behavior defined by either linear hardening or a user-defined hardening function.

Description

The PlasticTruss model implements J2 plasticity for 1D truss elements. The axial elongation of the element is obtained from $var element = document.getElementById("moose-equation-2e736133-394f-46fc-b74e-fcc7e73b5832");katex.render("\\varepsilon = \\frac{\\Delta l}{l_0} = \\frac{(l-l_0)}{l_0} \\, ,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-cb9e92a9-b971-481a-8783-ba612f60556a");katex.render("\\delta l", element, {displayMode:false,throwOnError:false});$ is the total stretch computed from the updated length $var element = document.getElementById("moose-equation-2015b5d9-bc37-41bb-9c3b-328ae6106115");katex.render("l", element, {displayMode:false,throwOnError:false});$ and the old length $var element = document.getElementById("moose-equation-fe096242-0c70-47d2-886a-7056e9d12be4");katex.render("l_0", element, {displayMode:false,throwOnError:false});$ of the truss element. In case of linear elastic material, the axial stress $var element = document.getElementById("moose-equation-bf5cf02d-66a8-4ed3-ba96-54d4548f9616");katex.render("\\sigma_a", element, {displayMode:false,throwOnError:false});$ is computed as $var element = document.getElementById("moose-equation-94a83a8e-b8e0-42dd-8bab-24ca880d490a");katex.render("\\sigma_a = E \\frac{\\Delta l}{l_0} \\, .", element, {displayMode:true,throwOnError:false});$ Here, the nonlinear behavior of the truss is implemented using a simple J2 plasticity model that can use either simple linear hardening or a user-defined function to define the hardening behavior. The strain increment is obtained from the total stretch update $var element = document.getElementById("moose-equation-5d1532cc-831f-4cb5-a3f0-8dde55edff46");katex.render("\\Delta \\varepsilon = \\varepsilon - \\varepsilon_{old} \\, .", element, {displayMode:true,throwOnError:false});$ The trial stress is estimated as $var element = document.getElementById("moose-equation-7b910aad-de7a-4c8d-b59c-8084b0d33b2c");katex.render("\\sigma^{tr} = \\sigma_{old} + E \\Delta \\varepsilon \\, .", element, {displayMode:true,throwOnError:false});$

The yield condition is determined as $var element = document.getElementById("moose-equation-16ea2f08-eb39-4427-80e7-06c78f8531b5");katex.render("f = |\\tilde \\sigma^{tr}| - E \\Delta \\varepsilon^p - r -\\sigma_y = 0 \\, ,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-7339e8da-53d7-41ea-a43e-1bd6f393a46b");katex.render("E", element, {displayMode:false,throwOnError:false});$ is the Young's modulus, $var element = document.getElementById("moose-equation-dcb311ab-bf6f-4803-992e-e2ba5cbb2402");katex.render("r", element, {displayMode:false,throwOnError:false});$ is the hardening function and $var element = document.getElementById("moose-equation-bb1c3a39-aeca-40f0-b09d-b43af8b60213");katex.render("\\sigma_y", element, {displayMode:false,throwOnError:false});$ is the yield stress. When the trial stress is outside of the yield envelop the stresses are brought down using the iterative Newton method. In the case of linear hardening, the hardening function is defined as $var element = document.getElementById("moose-equation-e7ee9203-71b3-4fb7-afa5-5d6604891dc9");katex.render("r=h |\\varepsilon^p|", element, {displayMode:false,throwOnError:false});$ with $var element = document.getElementById("moose-equation-04ed2c68-a10a-4ea0-a1d7-b61c19d9198c");katex.render("h", element, {displayMode:false,throwOnError:false});$ being the hardening constant. In this case, the hardening variable at the beginning of the iterative process is obtained as $var element = document.getElementById("moose-equation-3ca1426d-4710-4a8b-8f1f-d9206b6d9690");katex.render("r^{(k)} = r_{old} + h (\\Delta \\varepsilon^p)^{(k)} \\, .", element, {displayMode:true,throwOnError:false});$ The plastic strain increment is computed as $var element = document.getElementById("moose-equation-cb7f7311-f539-4b08-a692-5bdfb01bcb83");katex.render("d\\Delta \\varepsilon^p = \\frac{f}{\\frac{df}{d\\Delta \\varepsilon^p}} = \\frac{(|\\tilde \\sigma^{tr}| - E (\\Delta \\varepsilon^p)^{(k)} - r^k -\\sigma_y)}{E + h} = 0 \\, .", element, {displayMode:true,throwOnError:false});$ The plastic strain for the next iteration is updated $var element = document.getElementById("moose-equation-9b5d316f-79b0-421f-b0cb-b33e996b1808");katex.render("(\\Delta \\varepsilon^p)^{(k+1)} = (\\Delta \\varepsilon^p)^{(k)} + d\\Delta \\varepsilon^p \\, .", element, {displayMode:true,throwOnError:false});$ The iterative process continues until the updated stress lies on the yield curve. Then the elastic strain is updated as $var element = document.getElementById("moose-equation-4bbb844a-da0f-43c4-a032-ee49a717ff39");katex.render("\\Delta \\varepsilon^e = \\Delta \\varepsilon - \\Delta \\varepsilon^p \\, .", element, {displayMode:true,throwOnError:false});$ The updated axial stress is calculated $var element = document.getElementById("moose-equation-3511fa6b-be0a-4b99-8798-2750404ddd0a");katex.render("\\sigma_a = \\sigma_{old} + E \\Delta \\varepsilon^e \\, .", element, {displayMode:true,throwOnError:false});$

Example Input Syntax

[./truss]
  type = PlasticTruss
  youngs_modulus = 2.0e11
  yield_stress = 500e5
  outputs = 'exodus'
  output_properties = 'elastic_stretch hardening_variable plastic_stretch total_stretch'
[../]

Input Parameters

displacementsThe displacements appropriate for the simulation geometry and coordinate system
C++ Type:std::vector<VariableName>
Controllable:No
Description:The displacements appropriate for the simulation geometry and coordinate system
yield_stressYield stress after which plastic strain starts accumulating
C++ Type:double
Controllable:No
Description:Yield stress after which plastic strain starts accumulating

Required Parameters

absolute_tolerance1e-10Absolute convergence tolerance for Newton iteration
Default:1e-10
C++ Type:double
Controllable:No
Description:Absolute convergence tolerance for Newton iteration
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
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>
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>
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
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
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
Controllable:No
Description:An optional suffix parameter that can be appended to any declared properties. The suffix will be prepended with a '_' character.
hardening_constant0Hardening slope
Default:0
C++ Type:double
Controllable:No
Description:Hardening slope
hardening_functionEngineering stress as a function of plastic strain
C++ Type:FunctionName
Controllable:No
Description:Engineering stress as a function of plastic strain
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.
relative_tolerance1e-08Relative convergence tolerance for Newton iteration
Default:1e-08
C++ Type:double
Controllable:No
Description:Relative convergence tolerance for Newton iteration
temperature273.0Temperature in Kelvin
Default:273.0
C++ Type:std::vector<VariableName>
Controllable:No
Description:Temperature in Kelvin
temperature_ref273Reference temperature for thermal expansion in K
Default:273
C++ Type:double
Controllable:No
Description:Reference temperature for thermal expansion in K
thermal_expansion_coeff0Thermal expansion coefficient in 1/K
Default:0
C++ Type:double
Controllable:No
Description:Thermal expansion coefficient in 1/K
youngs_modulusVariable containing Young's modulus
C++ Type:std::vector<VariableName>
Controllable:No
Description:Variable containing Young's modulus

Optional 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.
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
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_meshFalseWhether 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:False
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

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>
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>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

Outputs Parameters

(moose/modules/tensor_mechanics/test/tests/truss/truss_plastic.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  elem_type = EDGE
  nx = 1
[]

[GlobalParams]
  displacements = 'disp_x'
[]

[Variables]
  [./disp_x]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[AuxVariables]
  [./axial_stress]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./e_over_l]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./area]
    order = CONSTANT
    family = MONOMIAL
  [../]
  [./react_x]
    order = FIRST
    family = LAGRANGE
  [../]
[]

[Functions]
  [./hf]
    type = PiecewiseLinear
    x = '0    0.0001  0.0003  0.0023'
    y = '50e6 52e6    54e6    56e6'
  [../]
[]

[BCs]
  [./fixx1]
    type = DirichletBC
    variable = disp_x
    boundary = left
    value = 0.0
  [../]
  [./load]
    type = FunctionDirichletBC
    variable = disp_x
    boundary = right
    function = 't'
  [../]
[]

[AuxKernels]
  [./axial_stress]
    type = MaterialRealAux
    property = axial_stress
    variable = axial_stress
    execute_on = 'initial TIMESTEP_END'
  [../]
  [./e_over_l]
    type = MaterialRealAux
    property = e_over_l
    variable = e_over_l
    execute_on = 'initial TIMESTEP_END'
  [../]
  [./area]
    type = ConstantAux
    variable = area
    value = 1.0
    execute_on = 'initial timestep_begin'
  [../]
[]

[Postprocessors]
  [./s_xx]
    type = ElementIntegralMaterialProperty
    mat_prop = axial_stress
  [../]
  [./e_xx]
    type = ElementIntegralMaterialProperty
    mat_prop = total_stretch
  [../]
  [./ee_xx]
    type = ElementIntegralMaterialProperty
    mat_prop = elastic_stretch
  [../]
  [./ep_xx]
    type = ElementIntegralMaterialProperty
    mat_prop = plastic_stretch
  [../]
[]

[Executioner]
  type = Transient
  solve_type = 'PJFNK'
  petsc_options = '-snes_ksp_ew'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
  nl_abs_tol = 1e-11
  l_max_its = 20
  dt = 5e-5
  num_steps = 10
[]

[Kernels]
  [./solid]
    type = StressDivergenceTensorsTruss
    component = 0
    variable = disp_x
    area = area
    save_in = react_x
  [../]
[]

[Materials]
  [./truss]
    type = PlasticTruss
    youngs_modulus = 2.0e11
    yield_stress = 500e5
    outputs = 'exodus'
    output_properties = 'elastic_stretch hardening_variable plastic_stretch total_stretch'
  [../]
[]

[Outputs]
  exodus = true
[]