Generalized Plane Strain
One possible extension of the plane strain formulation is to assume that the strains are independent of the out-of -plane coordinate z; that is,
Under the small strain assumption, the above equations have the following 3D solution:
where a, b, and c are constants. Thus, at the cross section z = 0, one has
and
The above conditions differ from the plane strain state only by the fact that the normal out-of -plane strain component can vary linearly throughout the cross section. This approximation is good when the structure is free to expand in the out-of-plane direction, and the possible bending curvature is small with respect to the extents of the structure in the xy-plane.
At the cross section z = 0, the deformation is in-plane and fully characterized by the in-plane displacement components u(x,y) and v(x,y).
In COMSOL Multiphysics, the coefficients a, b, and c in the expression for the εz strain are modeled as extra degrees of freedom that are constant throughout the model (global variables).
Geometric Nonlinearity
In case of geometric nonlinearity, the strains are represented by the Green-Lagrange strain tensor:
Consider the following displacement field expressed in terms of the material coordinates:
Coefficients a, b, and c are assumed to be small. Then, using the above displacement field in the strain tensor expression and dropping quadratic and higher order terms in the coefficients, one obtains:
Thus, in the leading order approximation, the strains become independent of the out-of-plane coordinate Z.