# Asymptotic Analytical Solution on Lamb Waves in Functionally Graded Nano Copper Layered Wafer
## 1 Introduction  
Since the introduction of functionally graded (FG) nano copper in 2011 [1], this novel material with a graded grain size distribution has attracted increasing scientific interest due to its high strength and high ductility [2; 3; 4]. The interest in materials with graded grain size distributions is not limited to nano copper and also includes titanium [5; 6], zirconium [7], metallic glass [8], Fe-Mn austenitic steel [9], and low-carbon steel [10]. The grain refinement mechanism, microstructure characteristics, and material properties of samples of a gradient nano/microstructured surface layer on pure copper are investigated through several experiments [11]. Copper rod samples with gradient grain structures are also developed, measured in terms of microhardness, analyzed through electron backscatter diffraction, and simulated numerically [12]. With the in-depth study on FG nano copper materials [13; 14] and their superiority in terms of strength and ductility, these materials will be widely utilized in engineering fields. The nondestructive evaluation of structures made of these materials is an important research topic in laboratory and engineering applications. The guided wave technique, which is one of the most popular nondestructive evaluation methods is used to analyze Lamb, shear horizontal, Love, and Rayleigh waves. Related previous studies focused on the waves in homogenous structures [15; 16]. Scientists began to investigate waves in structures made of FG materials in the 1990s. Researchers proposed various analytical methods, including directly analytical method, special function solution, the Wentzel-Kramers-Brillouin (WKB) method, the Legendre series method, and the power series method. The directly analytical method is used to solve different wave propagation problems when the material parameters vary with the same exponentialfunction [17; 18]. In this case, the governing differential equation with variable coefficients can be transformed into a differential equation with constant coefficients to obtain an analytical solution. If the governing equations can be decoupled and the material parameter variations follow specific laws, the special function method can be applied to solve wave propagation problems [19; 20; 21]. The WKB method is used to study the horizontal shear waves in different FG layered structures with one displacement component [22; 23]. However, this approach is only applicable to wave propagation problems with large wave numbers [20]. As asymptotic methods with series form, the Legendre [24; 25; 26] and power series methods [27] are utilized to analyze waves in various FG structures. Especially, the Legendre series method has been applied, not only for solving wave propagation behavior of macroscopic structure [28], but also that of microstructure using the combined modified couple stress theory [29; 30]. Numerical analyses of wave propagation in inhomogeneous media are also conducted. The main idea of the numerical solution is to divide the FG medium into a multilayer model. The material parameters in each layer of the model are assumed to be homogenous [31; 32; 33]. The transfer matrix method, which is based on the continuity of the stress and displacement in the interface, is used to solve the wave propagation problem in multilayered structures [15; 34]. This numerical technique can also be applied to solve wave propagation problems in various FG layered structures [35; 36]. Considering that Lamb waves have been widely used in nondestructive evaluation in engineering application, [PERSON] analyzed and compared the similarity and discrepancy of dispersion properties of Lamb wave propagation in both functionally graded and homogeneous plates [37; 38]. Considering the nondestructive evaluation of FG nano metal layered structure, we focused on different guided waves in a single layer FG nano metal wafer or a multilayered FG nano metal wafer. An early report has been published on horizontal shear (SH) waves in these structures [39].