SalehaniIrani 3D Coupled Traction separation law

3D Coupled (3DC) cohesive law of Salehani and Irani with no damage

Description

This class implements the non-stateful traction separation law proposed by Salehani and Irani (2018). This material model is an extension to 3 dimensions of the 2-dimensional traction-separation law proposed by Xu and Needleman Xu and Needleman (1993). This traction separation law should only be used for monotonic loading conditions as it will not produce realistic results for unloading and reloading. This model can be used for 1D, 2D and 3D problems.

The traction separation relationship is defined by: $var element = document.getElementById("moose-equation-5e7617b8-eaf3-4014-99bc-4acb9d1a4aa7");katex.render("T_i = \\frac{\\phi_i}{\\delta_i}\\frac{\\Delta_i}{\\delta_i} \\exp [ -\\sum_{j=1}^{d}(\\frac{\\Delta_j}{\\delta_j})^{\\alpha}]", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-2ea6e9ba-5dc9-454e-9bda-f97e2d5e5b89");katex.render("i", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-256905e6-6534-46ea-a526-5bacdd1328d5");katex.render("j", element, {displayMode:false,throwOnError:false});$ are indices representing the displacement jump component with the index 1 being associated with the opening direction, $var element = document.getElementById("moose-equation-8f897d7e-5215-44f8-b2eb-905759b231a3");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ is a model parameter with values $var element = document.getElementById("moose-equation-fba3fc35-c73e-4030-81f1-83b78c2dea7c");katex.render("\\alpha = 1", element, {displayMode:false,throwOnError:false});$ if $var element = document.getElementById("moose-equation-fb7f4b29-d881-4e0f-aeb7-5fe64d43ed79");katex.render("j==1", element, {displayMode:false,throwOnError:false});$ or $var element = document.getElementById("moose-equation-80309b3f-89b0-4ff9-8c62-a9b5d6ef1bdd");katex.render("\\alpha = 2", element, {displayMode:false,throwOnError:false});$ if $var element = document.getElementById("moose-equation-8ed79684-25e1-4e02-ac49-bfe4dd8edcb3");katex.render("j == 2,3", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-62634608-2e3b-431a-8a32-f5403f071466");katex.render("d", element, {displayMode:false,throwOnError:false});$ is a parameter representing the number of dimensions of the problem, $var element = document.getElementById("moose-equation-fb601322-e9dc-4067-b961-e195eaeeb8b4");katex.render("\\Delta_i", element, {displayMode:false,throwOnError:false});$ is the current gap value and $var element = document.getElementById("moose-equation-47cb2c39-a3e2-4340-9968-ab6b8431bff2");katex.render("\\delta_i", element, {displayMode:false,throwOnError:false});$ is the characteristic length of separation related to the maximum sustainable traction. The symbol $var element = document.getElementById("moose-equation-649ce6ed-3c78-4719-9737-5d436af7f448");katex.render("\\phi_i", element, {displayMode:false,throwOnError:false});$ represents the work of separation and is defined as $var element = document.getElementById("moose-equation-7a53ce97-df6d-4f7a-ac68-07472546f6ff");katex.render("\\phi_i = T_{i,max} \\lambda \\delta_i", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-621d186d-62c0-424d-8be6-dcc05ea3eca0");katex.render("\\lambda = e", element, {displayMode:false,throwOnError:false});$ if $var element = document.getElementById("moose-equation-a5e42782-0320-4881-a246-04d521fc6bbf");katex.render("i==1", element, {displayMode:false,throwOnError:false});$ or $var element = document.getElementById("moose-equation-95634b53-b68d-411d-9562-0cbaea9e4818");katex.render("\\lambda = \\sqrt{2 e}", element, {displayMode:false,throwOnError:false});$ if $var element = document.getElementById("moose-equation-721778b1-8069-4d88-a15d-a5a47cbf0399");katex.render("i == 2,3", element, {displayMode:false,throwOnError:false});$ . The parameter $var element = document.getElementById("moose-equation-dac32b74-e633-4310-bce9-edc94c0d30bf");katex.render("T_{i,max}", element, {displayMode:false,throwOnError:false});$ represents the maximum allowed traction that the interface can withstand in the $var element = document.getElementById("moose-equation-c371828d-03b9-4d3c-a8af-c67a34e52b9a");katex.render("i-th", element, {displayMode:false,throwOnError:false});$ direction. Note that the values of maximum allowed traction can be different in the normal and tangential directions, however $var element = document.getElementById("moose-equation-e7f1bc83-e61d-4a45-a5c0-e2a4dc98460f");katex.render("T_{2,max}", element, {displayMode:false,throwOnError:false});$ is assumed to be equal to $var element = document.getElementById("moose-equation-e0de9bb5-bdfd-419e-a572-7be6a6bc36b8");katex.render("T_{3,max}", element, {displayMode:false,throwOnError:false});$ . The same restrictions on $var element = document.getElementById("moose-equation-7eba83c2-6b64-43f5-a3bf-32cc15a0f036");katex.render("T_{i,max}", element, {displayMode:false,throwOnError:false});$ apply to $var element = document.getElementById("moose-equation-e1068f26-b255-4fc6-96ba-bd1ffcdb82b1");katex.render("\\delta_i", element, {displayMode:false,throwOnError:false});$

Examples

[./czm_3dc]
  type = SalehaniIrani3DCTraction
  boundary = 'Block0_Block1 Block1_Block2'
  normal_gap_at_maximum_normal_traction = 1
  tangential_gap_at_maximum_shear_traction = 0.5
  maximum_normal_traction = 500
  maximum_shear_traction = 300
  base_name = 'czm_b012'
[../]

References

Mohsen Khajeh Salehani and Nilgoon Irani. A coupled mixed-mode cohesive zone model: an extension to three-dimensional contact problems. arXiv preprint arXiv:1801.03430, 2018.

@article{salehani2018coupled,
    author = "Salehani, Mohsen Khajeh and Irani, Nilgoon",
    title = "A coupled mixed-mode cohesive zone model: An extension to three-dimensional contact problems",
    journal = "arXiv preprint arXiv:1801.03430",
    year = "2018"
}

X-P Xu and A Needleman. Void nucleation by inclusion debonding in a crystal matrix. Modelling and Simulation in Materials Science and Engineering, 1(2):111, 1993.

@article{xu1993void,
    author = "Xu, X-P and Needleman, A",
    title = "Void nucleation by inclusion debonding in a crystal matrix",
    journal = "Modelling and Simulation in Materials Science and Engineering",
    volume = "1",
    number = "2",
    pages = "111",
    year = "1993",
    publisher = "IOP Publishing"
}

(moose/modules/tensor_mechanics/test/tests/cohesive_zone_model/czm_multiple_action_and_materials.i)

[Mesh]
  [./msh]
    type = GeneratedMeshGenerator
    dim = 3
    nx = 1
    ny = 1
    nz = 4
    zmin = 0
    zmax = 4
  [../]
  [./subdomain_id]
    type = ElementSubdomainIDGenerator
    input = msh
    subdomain_ids = '0 1 2 3'
  []
  [./split]
    type = BreakMeshByBlockGenerator
    input = subdomain_id
    split_interface = true
  []
  [add_side_sets]
    input = split
    type = SideSetsFromNormalsGenerator
    normals = '0 -1  0
               0  1  0
               -1 0  0
               1  0  0
               0  0 -1
               0  0  1'
    fixed_normal = true
    new_boundary = 'y0 y1 x0 x1 z0 z1'
  []
[]

[GlobalParams]
  displacements = 'disp_x disp_y disp_z'
[]

[Functions]
  [./stretch]
    type = PiecewiseLinear
    x = '0 1'
    y = '0 100'
  [../]
[]

[Constraints]
  [x1]
    type = EqualValueBoundaryConstraint
    variable = disp_x
    secondary = 'x1'    # boundary
    penalty = 1e6
  []
  [y1]
    type = EqualValueBoundaryConstraint
    variable = disp_y
    secondary = 'y1'    # boundary
    penalty = 1e6
  []
[]

[BCs]
  [./fix_x]
    type = DirichletBC
    preset = true
    value = 0.0
    boundary = x0
    variable = disp_x
  [../]
  [./fix_y]
    type = DirichletBC
    preset = true
    value = 0.0
    boundary = y0
    variable = disp_y
  [../]
  [./fix_z]
    type = DirichletBC
    preset = true
    value = 0.0
    boundary = z0
    variable = disp_z
  [../]
  [./back_z]
    type = FunctionNeumannBC
    boundary = z1
    variable = disp_z
    use_displaced_mesh = false
    function = stretch
  [../]
[]


[Modules/TensorMechanics/CohesiveZoneMaster]
  [./czm_ik_012]
    boundary = 'Block0_Block1 Block1_Block2'
    base_name = 'czm_b012'
  [../]
  [./czm_ik_23]
    boundary = 'Block2_Block3'
    base_name = 'czm_b23'
  [../]
[]

[Materials]
  # cohesive materials
  [./czm_3dc]
    type = SalehaniIrani3DCTraction
    boundary = 'Block0_Block1 Block1_Block2'
    normal_gap_at_maximum_normal_traction = 1
    tangential_gap_at_maximum_shear_traction = 0.5
    maximum_normal_traction = 500
    maximum_shear_traction = 300
    base_name = 'czm_b012'
  [../]
  [./czm_elastic_incremental]
    type = PureElasticTractionSeparationIncremental
    boundary = 'Block2_Block3'
    normal_stiffness = 500
    tangent_stiffness = 300
    base_name = 'czm_b23'
  [../]
  # bulk materials
  [./stress]
    type = ADComputeFiniteStrainElasticStress
  [../]
  [./elasticity_tensor]
    type = ADComputeIsotropicElasticityTensor
    youngs_modulus = 200e4
    poissons_ratio = 0.3
  [../]
[]

[Modules]
  [./TensorMechanics]
    [./Master]
      [./all]
        strain = FINITE
        add_variables = true
        use_finite_deform_jacobian = true
        use_automatic_differentiation = true
        generate_output = 'stress_xx stress_yy stress_zz stress_xy stress_yz stress_xz'
      [../]
    [../]
  [../]
[]

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

[Executioner]
  # Executioner
  type = Transient

  solve_type = 'NEWTON'
  line_search = none
  petsc_options_iname = '-pc_type '
  petsc_options_value = 'lu'
  nl_rel_tol = 1e-10
  nl_abs_tol = 1e-6
  l_max_its = 20
  start_time = 0.0
  dt = 0.25
  dtmin = 0.25
  num_steps =1
[]

[Outputs]
  exodus = true
[]