Guided Waves in Plates and Rods

Abstract

In this chapter we consider elastic wave modes which propagate in composites with finite boundaries. There are those waves that exist between the two plane parallel boundaries of a homogeneous anisotropic solid. We consider that well-known problem, as well as waves in an elastic anisotropic rod, specifically an individual graphite fiber. Composite laminates seen in applications are essentially all multilayered structures, and in many cases can be considered periodically layered. So, we also take up the subject of guided waves in layered plates in later chapters. In a plate geometry, as illustrated in Fig. 5.1, we choose the propagation direction to be parallel to the x1 axis and the x3 axis to be normal to the plate surfaces. This geometry is particularly significant for composite materials since, by design, laminates are often locally planar in nature. While the solutions we find are appropriate for flat plates, with some modifications they describe wave motion in gently curved structures as well. Clear and mathematically straightforward descriptions of the characteristics of plate waves exist for isotropic media. The results obtained for isotropic media are not, however, directly applicable to most composites. We begin by considering the behavior of waves in a uniaxial composite laminate. In later chapters we generalize the calculation to layered orthotropic media, concentrating on the results and physical interpretation rather than the algebraic details. To begin a description of waves in plates, let us consider the possible polarizations of particle motion. Let the plate surfaces lie in the (x1, x2) plane of mirror symmetry with the origin dividing the plate thickness in half, as shown in Fig. 5.1. Then, we will at first assume the wave to be uniform in the x2 direction and propagating in the x1 direction, and (x1, x3) is the plane of symmetry. Particle motion can occur along any axis. Note that in this restricted symmetry, shear partial waves polarized along the x2 axis will have no component of particle motion normal to the plate surfaces. Partial waves are a concept introduced by Rayleigh to acknowledge that a superposition of both shear and longitudinal particle motion is generally needed to produce plate waves polarized in the vertical plane.