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,

where a, b, and c are constants. Thus, at the cross section z = 0, one has

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).

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 a, b, and c, one obtains:

Thus, in the leading order approximation, the strains become independent of the out-of-plane coordinate Z.

When a 2D plane strain model represents a cross-section of the structure that has a significant uniform extension in the out-of-plane Z direction, the following 3D solution can be sought in form of the amplitude expansion:

The first term, u0, represents a static in-plane prestress deformation:

The second part of the solution, u1, presents a time-harmonic linear perturbation with an amplitude that can be a function of the in-plane coordinates X and Y. Such a perturbation can be seen as an out-of-plane wave, with a small amplitude that propagates in the Z direction, a wavelength L, and phase velocity c:

Note that in contrast to the prestress solution u0, the perturbation amplitude u1 can have nonzero values in all the displacement components:

There are two alternative approaches. The wavelength L, and thus the wave number kZ, can be considered as a parameter. Then, ω can be computed by an eigenfrequency analysis for the 2D cross section with all three displacement components taken as dependent variables. As a result, one obtains

Thus, the wave speed for the out-of-plane wave is computed as a function of the wavelength L and possible prestress u0 in the material. The dependence of the wave speed on the wavelength is often called dispersion.

Alternatively, the frequency f (and thus ω) can be taken as a parameter. Then, the solution can be computed via eigenvalue analysis with respect to the wave number kZ using the 2D cross section geometry. Hence,

which determine the wavelength L and phase velocity c for the wave that propagates out-of-plane for a given frequency f under given in-plane prestress deformation u0. Such interpretation of the perturbation solution is sometimes called a signaling problem.