Generalized Plane Strain

warningwarning:Deprecated

The generalized plane strain capability described here belongs to the legacy mechanics kernels, which are being phased out. For new models, use the homogenization system provided by the Lagrangian kernels instead.

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 out-of-plane strain (Li and Lim, 2005).

Formulation

With generalized plane strain, the model is solved for a 2D domain. For the usual - model plane, the axis is the out-of-plane direction. The out-of-plane strain is represented by a scalar variable.

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

When generalized plane strain is used for axisymmetric models, the solution domain is 1D, and the out-of-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 included as an additional unknown 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:

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.

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]