Generalized Plane Strain

Description

Generalized plane strain extends plane strain by allowing a nonzero constant strain in the out-of-plane direction (Adams and Doner, 1967). Other formulations add two rotational degrees of freedom (ABAQUS, 2014) or include the anticlastic problem associated with out-of-plane shear (Adams and Crane, 1984; Li, 1999).

The solid mechanics module implements the form with one extra degree of freedom representing the direct out-of-plane strain (Li and Lim, 2005).

Formulation

For the usual - model plane, the axis is the out-of-plane direction. Under generalized plane strain conditions, the in-plane stresses are not functions of , and the direct out-of-plane strain is represented by a scalar variable. The model is solved on a 2D domain.

The formulation also works when the model plane is - or -. In those cases the out-of-plane direction is the axis or axis, respectively. In RZ axisymmetry, the out-of-plane generalized plane strain direction is the axial direction.

- plane generalized plane strain problem

In-plane equilibrium equations

The kinematical equations of the generalized plane strain problem are identical to those for the plane stress or strain problems, given as (1)

The equilibrium equations for the generalized plane strain problem in the - plane are given as in where are the body forces.

The constitutive equations in terms of stress-strain relationship are given as where are material's stiffness coefficients using Voigt notation, , and are the material's thermal expansion coefficients and is the change in temperature.

Out-of-plane equilibrium equation

A further condition is required associated with the out-of-plane direction. If a constant strain is prescribed as the deformation compatibility condition in the -direction Alternatively, a force as the stress resultant in the -direction can be prescribed. The condition is the equilibrium condition in -direction, given as (2) The stress resultant conjugates with constant strain .

The formulation above shares the in-plane mechanics of a conventional plane strain problem and adds the scalar degree of freedom for the out-of-plane direction.

When an out-of-plane pressure is prescribed, the scalar residual is written as where is the selected out-of-plane direction. Positive out-of-plane pressure is applied toward the body. In Cartesian coordinates, , , and use , , and , respectively. In RZ axisymmetry, , and the scalar residual is integrated with the RZ coordinate weighting.

Implementation

The out-of-plane strain is a scalar variable in the standard system of equations for a mechanics problem, where and represent the displacement vectors in the and directions, and and represent the corresponding reaction forces. For a two-dimensional model in the - plane, the partitioned linearized system can be written as follows:

The off-diagonal entries are nonzero, but not shown here.

- and - plane generalized plane strain problem

The generalized plane strain formulation can also be used if the two-dimensional model is represented in the - or - planes, rather than the - plane, as is typically the case. If the model lies in those other planes, the calculation of the strain tensor is modified to take into account the fact that the model is in a different plane. Also, the scalar variable used to represent the out-of-plane strain in the generalized plane strain formulation is in a different direction. All other aspects of the formulation are identical to the - plane case.

For the case when the model lies in the - plane, the small-strain kinematic equations for the strain calculation that are equivalent to Eq. (1) for the - plane are expressed as: The generalized plane strain equilibrium equation equivalent to Eq. (2) is expressed as: The same pattern is followed for the - plane case.

MOOSE Objects

Objects available for generalized plane strain:

Objects specific for generalized plane strain:

Objects specific to the AD generalized plane strain path:

Table 1: Legacy generalized plane strain objects created by the action

ObjectRole
Generalized Plane Strain Off DiagonalOff-diagonal coupling between in-plane displacement variables and the scalar out-of-plane strain.
Generalized Plane Strain Scalar KernelScalar out-of-plane equilibrium residual.
Generalized Plane Strain User ObjectResidual and diagonal Jacobian data for the scalar kernel.

Table 2: AD generalized plane strain objects created by the action

ObjectRole
ADGeneralizedPlaneStrainScalar out-of-plane equilibrium residual and AD coupling terms.

The AD path requires AD stress materials. ADGeneralizedPlaneStrain owns the scalar residual assembly through ADKernelScalarBase, including displaced-mesh quadrature weights when requested. The out-of-plane pressure material used by ADGeneralizedPlaneStrain is a regular Real material property, so it is treated as derivative-free in the scalar equation.

How to Use Generalized Plane Strain

The GeneralizedPlaneStrainAction can be used to set up a generalized plane strain model. The QuasiStaticSolidMechanicsPhysics which considers the GeneralizedPlaneStrainAction as Meta-Action can also be used.

The following test-spec block shows a modern QuasiStatic small-strain input run in AD mode.

  [generalized_plane_strain_small_ad]
    type = 'Exodiff'
    input = 'generalized_plane_strain_small.i'
    exodiff = 'generalized_plane_strain_small_out.e'
    custom_cmp = 'generalized.exodiff'
    prereq = generalized_plane_strain_small
    cli_args = 'Physics/SolidMechanics/QuasiStatic/all/use_automatic_differentiation=true '
               'Postprocessors/react_z/type=ADMaterialTensorIntegral '
               'Materials/elastic_tensor/type=ADComputeIsotropicElasticityTensor '
               'Materials/thermal_strain/type=ADComputeThermalExpansionEigenstrain '
               'Materials/stress/type=ADComputeLinearElasticStress'
    requirement = 'The system shall support an automatic-differentiation generalized plane strain mechanics solution'
    design = 'ADComputePlaneSmallStrain.md'
  []
(modules/solid_mechanics/test/tests/generalized_plane_strain/tests)

The following standalone action input uses AD mode and lets the action create the missing scalar out-of-plane strain variable.

[Physics<<<{"href": "../../syntax/Physics/index.html"}>>>]
  [SolidMechanics<<<{"href": "../../syntax/Physics/SolidMechanics/index.html"}>>>]
    [GeneralizedPlaneStrain<<<{"href": "../../syntax/Physics/SolidMechanics/GeneralizedPlaneStrain/index.html"}>>>]
      [gps]
        use_automatic_differentiation<<<{"description": "Use automatic differentiation to assemble the generalized plane strain equation and its coupling terms"}>>> = true
        use_displaced_mesh<<<{"description": "Whether to use displaced mesh"}>>> = true
        displacements<<<{"description": "The displacement variables"}>>> = 'disp_x disp_y'
        scalar_out_of_plane_strain<<<{"description": "Scalar variable for the out-of-plane strain (in y direction for 1D Axisymmetric or in z direction for 2D Cartesian problems)"}>>> = scalar_strain_zz
        out_of_plane_pressure_function<<<{"description": "Function used to prescribe pressure (applied toward the body) in the out-of-plane direction (y for 1D Axisymmetric or z for 2D Cartesian problems)"}>>> = traction_function
        pressure_factor<<<{"description": "Scale factor applied to prescribed out-of-plane pressure (both material and function)"}>>> = 1e5
      []
    []
  []
[]
(modules/solid_mechanics/test/tests/generalized_plane_strain/generalized_plane_strain_auto_scalar.i)

References

  1. ABAQUS. ABAQUS/CAE User's Manual, 27.1.2 Choosing the element's dimensionality, Version 6.14. 2014.[Export]
  2. Donald F. Adams and David A. Crane. Finite element micromechanical analysis of a unidirectional composite including longitudinal shear loading. Computers & Structures, 18(6):1153–1165, 1984. URL: http://www.sciencedirect.com/science/article/pii/0045794984901603, doi:https://doi.org/10.1016/0045-7949(84)90160-3.[Export]
  3. Donald F. Adams and Douglas R. Doner. Transverse normal loading of a unidirectional composite. Journal of Composite Materials, 1(2):152–164, 1967. URL: https://doi.org/10.1177/002199836700100205, doi:10.1177/002199836700100205.[Export]
  4. Shuguang Li. On the unit cell for micromechanical analysis of fibre-reinforced composites. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 455(1983):815–838, 1999. URL: http://rspa.royalsocietypublishing.org/content/455/1983/815, doi:10.1098/rspa.1999.0336.[Export]
  5. Shuguang Li and Szu-Hui Lim. Variational principles for generalized plane strain problems and their applications. Composites Part A: Applied Science and Manufacturing, 36(3):353–365, 2005. URL: http://www.sciencedirect.com/science/article/pii/S1359835X04001885, doi:https://doi.org/10.1016/j.compositesa.2004.06.036.[Export]