ABSTRACT
This chapter presents wave propagation in graded material structures and one such example of graded material structure is the Functionally Graded Material (FGM) structures. Since these structures are inhomogeneous, the wave propagation in these structures is also inhomogeneous, means that the waves, as it propagates, will attenuate. First, the concept of FGM structures is introduced followed by the method of modeling them. FGM structures will have continuously varying material properties over its spatial directions. The grading of materials in 1D waveguides can either be in the direction of wave propagation (which we call the lengthwise gradation) or in the direction perpendicular to the direction of the wave propagation (which we call depth-wise gradation). Hence, two different variation of material properties as a function of the axial coordinate is assumed in this chapter, one is exponential variation and the second is the variation based on the power law. The inhomogeneity in these structures will always lead to variable coefficient equations, that require special methods to solve the wave equations. First, the wave propagation in lengthwise graded rod is presented, followed by spectral analysis on the depth-wise graded FGM beam. Here, the equations are derived based on higher-order assumptions that includes both shear deformation and deformation due to lateral contraction. Depth-wise graded case does not yield variable coefficient differential equations and hence the equations could be solved through spectral analysis to obtain both the wavenumbers and their speeds. As before, this model introduces cut-off frequencies in both shear and contraction modes. The spectrum and dispersion relations are plotted for power law assumption of material properties, for different values of exponent. Next, wave propagation in the lengthwise graded beam is presented and the governing equation in this model is a variable coefficient differential equation. Assuming both Young's Modulus and density vary similarly with exponential variation, the governing equation reduces to constant-coefficient type and hence wavenumber and group speed relations are derived and plotted. The analysis shows that choosing the grading parameter approprietly, one can control the type of higher-order modes that can be allowed to propagate in this FGM waveguide. The last part of this chapter deals with wave propagation in 2D FGM waveguide, wherein, again using partial wave technique, wave equations are solved, and spectrum relations are obtained. Here, the main emphasis is to identify the main difference between the wave propagation in homogeneous and inhomogeneous 2D waveguides. The chapter ends with a summary, the details of the MATLAB Codes provided and some exercise problems.
