Skewing of the acoustic wave energy vector in stacked 1-3 anisotropic material systems

Anisotropic material systems occur naturally (such as wood, tissues, etc.), or are engineered as in the case of "brous composites (such as "ber-glass, graphite-epoxy, ceramics matrix, etc.), or can be due to the manufacturing process; like for instance metals with preferred grain orientations (rolled metal, columnar cast stainless steel, etc.). Anisotropy in solid materials has a tendency to change the direction of the acoustic wave energy (group) vector along preferred orientations. Thus, the wave energy vector (also sometimes referred to as power density vector, Poynting vector, power #ow vector) is not necessarily along the wave vector direction [1}3]. This e!ect is often called skewing. In anisotropic materials, the skewing angle, which is the angle between the energy vector and the wave vector, equals zero only when the wave vector is along one of the directions of material symmetry. Since engineered material systems can be designed and fabricated to speci"cations, further discussions in this paper will be limited to strati"ed "ber-reinforced composite materials and more speci"cally to graphite "ber}epoxy resin systems. It has been previously demonstrated that the "ber direction in "ber reinforced composites plays a critical role in the direction of propagation of the acoustic energy [4, 5]. The basic anisotropy of a unidirectional "ber-reinforced laminate can be assumed to be transversely isotropic, and hence be characterized using "ve independent elastic material constants. In many practical instances transversely isotropic layers oriented at di!erent angles with respect to each other can be combined to form multilayered material systems. In the past, such multilayered structures were considered to be orthotropic, mono-clinic, and tri-clinic using e!ective constants methods [6]. Acoustic plane waves when obliquely incident upon a plane interface between any two dissimilar materials (isotropic or anisotropic) will cause some of the incident energy to be re#ected back to the incident medium while the remaining portion will be transmitted at a refracted angle based on Snell's law. Assuming a lossless medium, the sum of the re#ected and the transmitted energy should be equal to the incident energy. But, due to mode conversion, other wave modes are generated which alters this equation, particularly when the mode-converted wave is a guided wave which travels along the structure [7]. The guided waves may also leak energy to the surrounding media. During the generation of guided modes, especially plate wave modes, the specularly re#ected energy approaches a minimum value thus acting as re#ection factor "lter (changing the equation between transmission and re#ection based on the Kramer coincidence principle [1}3]). Hence, by