Out-of-plane and Circumferential Modes

If 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 represents a static in-plane prestress solution:

This can be obtained by a standard static analysis using 2D geometry for the cross section with the corresponding boundary conditions.

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

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

There are two alternative approaches. The wavelength, 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 and possible prestress 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 and speed for the wave that can propagate out-of-plane for a given frequency under given in-plane prestress condition. Such interpretation of the perturbation solution is sometimes called a signaling problem.

Circumferential Modes

A standard 2D axially symmetric representation of the structure geometry assumes zero twist displacement component together with independence of the circumferential position

. The following 3D solution form represent an extension of these assumptions:

where m is a circumferential mode number (or azimuthal mode number) that can only have integer values to obey the axially symmetric nature of the corresponding 3D problem. Thus,

The circumferential wave number can be introduced as

The static prestress solution

has zero twist component

and it can be obtained using a standard static analysis in 2D axially symmetric geometry.

The displacement vector

can have nonzero values in all three components, which are functions of the radial and axial positions. For a given circumferential mode number m, it can be found using an eigenfrequency analysis in a 2D axially symmetric geometry. Hence,

and the perturbation solution becomes

This represents eigenmodes in the corresponding 3D structure, which can be computed assuming certain constraints on the axis and possible static prestress without twist and independent off the position along the axis.