Waves in Periodically Layered Composites

Composite materials, unless they are quite thin, often include periodic layering, where laminated plates composed of alternating uniaxial plies in two or more directions result in more evenly distributed in-plane stiffness. The oriented plies can generally be reduced to a unit cell geometry which repeats throughout the laminate and is composed of sublayers each having highly directional in-plane stiffness, but identical out-of-plane properties. As the transverse isotropy of a uniaxial fibrous ply derives from the geometry of the two-phase material, composite laminates of these plies will have microscopic elastic stiffness tensors which change only in the plane of the laminate, as we saw in Chapter 1. The elastic properties normal to the laminate surface remain unchanged from ply to ply. In this chapter we take up the subject of waves in periodically layered plates. Unusual guided wave dispersion effects have been observed experimentally in periodically layered plates. Shull et al. found, for guided waves polarized in the vertical plane in plates of alternating aluminum and aramid–epoxy composites, that dispersion never scales with the frequency–thickness product, as it would in homogeneous isotropic, or layered transversely isotropic, plates. Instead, grouping of the mode curves has been observed. In an attempt to understand this behavior in terms of periodic layering, Auld et al. have analyzed the simpler case of SH wave propagation in periodically layered plates and have demonstrated that these observed phenomena can be attributed to the pass band and stop band structure caused by the periodic layering. In this section, we will show that Floquet modes play a critical role in the behavior of guided waves in plates that are periodically layered. To analyze the problem, we apply an extension of the stiffness matrix method of the previous chapter. Floquet modes, which are the characteristic modes for the infinite periodically layered medium, can be thought of as the analogy—in a periodically layered medium—to the quasilongitudinal and quasishear modes for the infinite homogeneous medium. On the topic of infinite periodic media, many calculations, both approximate and exact, have been performed to model elastic wave propagation in this important class of structures.