Wave dispersion in 2D Unit cell subjected to a periodic boundary

[M_Abubakr]

TL;DR

I have a 2D unit cell which is a Repeated Unit Cell (RUC) of a composite with a fibre and a matrix. Longitudinal excitation is applied of different frequencies. Using Floquet-Blochs Theorem I have to get the dispersion curves for this composite unit cell.

How do I get the wave dispersion for a 2D continuum unit cell subjected to a periodic boundary which is excited longitudinally? I'll be applying forces in ABAQUS with varying frequencies. I have come across Blochs theorem but I can't find any application of it in continuous systems. Every application of it deals with atomic wave functions which I have no idea of how electron states are analogous to mechanical systems.

Can anyone tell me how to start the analysis? I already have the natural frequencies of the unit cell.

 

[FEAnalyst]

Here are some articles that can be helpful for you:
- "A general purpose element-based approach to compute dispersion relations in periodic materials with existing nite element codes" C. Valencia et al.
- "Bloch wave approach for the analysis of sequential bifurcations in bilayer structures" J. Liu et al.
- "Dispersion of Waves in Composite Laminates With Transverse Matrix Cracks, Finite Element and Plate Theory Computations" M. Aberg et al.
- "Dispertion relations for composite structures. Part II. Methods for determining dispersion curves" M. Barski et al.