A Numerical Solution of the Dispersion Equation of Guided Wave Propagation in N-Layered Media
The study of wave propagation through elastic solid media can be used to carry out non-destructive tests (NDT) of structures. These tests can be used to detect and identify, in many cases, both the actual elastic properties and possible geometric imperfections included in the material damage of a structure [SaGa04]. One important application of high-frequency waves is the characterization of composite materials.
A composite material consists of several thin layers or laminae, and the resultant solid acts as a full plate. The layers can be of different materials, but normally the same material is used across the plate. Within the framework of the science of materials, one important issue is to design and estimate the mechanical properties of a composite material. There is an extensive literature on this topic (see [Jo75], [TsPa68], [Wh87], and [ViSi89]). The subject of composite materials optimal design is also of great interest and has been treated in [GuHa99].
In this chapter, the general theory of high-frequency wave propagation in layered media will be summarized and the dispersion equation will be obtained. The dispersion equation is presented as a result of a generalized nonlinear eigenvalue–eigenvector problem.
The chapter is organized as follows. In Section 5.2, the simple case N = 1, i.e., the well-known Lamb wave propagation is summarized. The dispersion curves obtained there are used to validate some results of the multi-layered general theory described in Section 5.3. Finally, in Sections 5.4 and 5.5, some computational procedures and examples are presented.
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- [FoMa77]Forsythe, G.E., Malcolm, M.A., Moler, C.B.: Computer Methods for Mathematical Computations, Prentice-Hall, Englewood Cliffs, NJ (1977).Google Scholar
- [GuHa99]Gurdal, Z., Haftka, R.T., Hajela, P.: Design and Optimization of Laminated Composite Materials, Wiley, Princeton University Press, Princeton, NJ (1999).Google Scholar
- [Jo75]Jones, R.M.: Mechanics of Composite Materials, McGraw-Hill, New York (1975).Google Scholar
- [LaLi59]Landau, L.D., Lifshitz, E.M.: Theory of Elasticity, Vol. 7, Pergamon Press, New York (1959).Google Scholar
- [MaFi98]Mathews, K.D., Fink, K.D.: Numerical Methods Using Matlab, Prentice-Hall, Englewood Cliffs, NJ (1998).Google Scholar
- [Ro04]Rose, J.L.: Ultrasonic Waves in Solid Media, Cambridge University Press, London (2004).Google Scholar
- [RoDi00]Royer, D., Dieulesaint, E.: Elastic Waves in Solids, Vols. I and II, Springer, Berlin-Heidelberg (2000).Google Scholar
- [SaGa04]Samartín, A., García-Palacios, J., Tabuenca, P.: Structural damage identification using dynamic numerical methods, in Proceedings of the 2004 IASS Symposium on Shell and Spatial Structures from Models to Realization, Montpellier, France (2004), 20-24.Google Scholar
- [StBu80]Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, Springer, New York (1980).Google Scholar
- [TsPa68]Tsai, S.W., Pagano, N.J.: Invariant properties of composite materials, in Composite Materials Workshop, Technomic, Westport, CT (1968), 233-253.Google Scholar
- [ViSi89]Vinson, J.R., Sierakowski, R.L.: The Behavior of Structures Composed of Composite Materials, Martinus Nijhoff, Leiden, The Netherlands (1989).Google Scholar
- [Wh87]Whitney, J.M.: Structural Analysis of Laminate Anisotropic Plates, Technomic, Westport, CT (1987).Google Scholar