# Asymptotic Analytical Solution on Lamb Waves in Functionally Graded Nano Copper Layered Wafer
## 2 Statement of the Problem  
### Governing Equations  
In this study, the Lamb waves in two types of FG nano copper layered wafer are considered (Figure 1). The wave propagation direction is the positive direction of the \(x_{1}\) axis, and the thickness is along the \(x_{3}\) axis. If the grain size of the FG nano copper wafer varies with thickness and is a function of \(x_{3}\), then the corresponding material parameters, including the elastic parameters and mass density, are not constants, but are functions of \(x_{3}\). Figure 1 presents a simple FG nano copper wafer with thickness \(h\) and a monotonously changing grain size, whereas Figure 1 displays a multilayered FG nano copper wafer composed of several simple wafers with interfaces that have continuous grain sizes. The thickness of each simple wafer is \(h\), and the total thickness is \(H\). Therefore, the grain size is alternately increasing and decreasing.  
Denoting the displacement component along the \(x_{j}\) direction as \(u_{j}\), the relationship between the strain and the displacement is expressed as:  
\[\varepsilon_{ij}=\frac{1}{2}\big{(}u_{i,j}+u_{j,i}\big{)} \tag{1}\]  
where \(\varepsilon_{ij}\) is the strain tensor, and the comma followed by subscript \(i\) indicates space differentiation with respect to the corresponding coordinate \(x_{i}\).  
The constitutive equation for elastic materials is written as:  
\[\sigma_{ij}=c_{ijkl}\varepsilon_{kl} \tag{2}\]  
where \(\sigma_{ij}\) is the stress tensor, \(c_{ijkl}\) is the elastic parameter, and the repeated index in the subscript implies summation with respect to that index. The two independent elastic parameters for isotropic elastic materials can be expressed by the Lame parameters \(\lambda\) and \(\mu\). The elastic parameters of the FG nano copper wafer that depend on the grain size are not constants, but are functions of position.  
The motion equation is expressed as:  
\[\sigma_{ij,j}=\rho\ddot{u_{i}} \tag{3}\]  
where \(\rho\) is the mass density, which depends on the grain size and is a function of \(x_{3}\).  
The displacement components of the Lamb waves propagating in the FG nano copper wafer are expressed as:  
\[u_{1}=u_{1}(x_{1},x_{3},t),u_{2}=0,u_{3}=u_{3}(x_{1},x_{3},t) \tag{4}\]  
and the governing equations for the mechanical displacements are defined as:  
\[\begin{split}&(\lambda+2\mu)\frac{\partial^{2}u_{1}}{\partial x_{1} ^{2}}+\lambda\frac{\partial^{2}u_{3}}{\partial x_{1}\partial x_{3}}+\mu\Big{(} \frac{\partial^{2}u_{3}}{\partial x_{1}\partial x_{3}}+\frac{\partial^{2}u_{1 }}{\partial x_{2}\partial x_{2}^{2}}\Big{)}+\frac{d\mu}{\partial x_{3}}\Big{(} \frac{\partial u_{1}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{3}} \Big{)}=\rho\frac{\partial^{2}u_{1}}{\partial t^{2}}\\ &(\lambda+2\mu)\frac{\partial^{2}u_{3}}{\partial x_{3}^{2}}+ \lambda\frac{\partial^{2}u_{1}}{\partial x_{1}\partial x_{3}}+\mu\Big{(} \frac{\partial^{2}u_{1}}{\partial x_{1}\partial x_{3}}+\frac{\partial^{2}u_{ 3}}{\partial x_{1}^{2}}\Big{)}+\frac{d\lambda}{\partial x_{3}}\frac{\partial u _{1}}{\partial x_{1}}+\frac{d(\lambda+2\mu)}{\partial x_{3}}\frac{\partial u _{3}}{\partial x_{3}}=\rho\frac{\partial^{2}u_{3}}{\partial t^{2}}\end{split} \tag{5}\]