# Weakening Detection of Composite Structure Adhesive Layer Based on Nonlinear Guided Waves
## 2 Theoretical Background  
Figure 1 illustrates the geometric model of a three-layer composite structure of an upper aluminum plate, a middle acrylic resin adhesive layer, and a lower polystyrene layer. In this model, \(d_{1}\), \(d_{2}\), and \(d_{3}\) represent the thicknesses of the upper, middle, and lower media, respectively, while \(d\) denotes the total thickness. The \(x\)-axis extends along the direction of the plate, and the \(z\)-axis is perpendicular to the plate direction. The derivation of the dispersion equation for the three-layer composite structure presented uses the mathematical transfer matrices model to analyze the interfacial characteristics of guided waves within the composite. By combining potential functions with interfacial properties, the dispersion equation is solved. The potential functions defining the propagation of ultrasonic waves in solids are used to construct the potential functions \(\Phi\) and \(\psi\) for longitudinal and shear waves. The amplitudes of longitudinal and shear waves in the positive and negative \(z\)-directions can serve as the basis for further expressing \(\Phi\) and \(\psi\) as \(\Phi^{+}\), \(\Phi^{-}\), \(\psi^{+}\), and \(\psi^{-}\). These can be used with boundary conditions to generate the displacement components \(u_{x}\) and \(u_{z}\) and the stress components \(\sigma_{x}\) and \(\sigma_{z}\). The equations are thus made simpler and can be expressed in matrix form by replacing the displacement components \(u_{x}\) and \(u_{z}\) and the stress components \(\sigma_{x}\) and \(\sigma_{z}\) with the potential functions \(\Phi\) and \(\psi\) shown as follows [23]:  
\[ \begin{bmatrix}u_{x}\\ u_{z}\\ \sigma_{x}\\ \sigma_{z}\end{bmatrix} = \begin{bmatrix}\psi^{+}\\ \phi^{-}\\ \psi^{+}\\ \psi^{-}\end{bmatrix}  \tag{1}\]  
Figure 1: Geometric model of a three-layer composite structure. \[\mathbf{D}= \begin{bmatrix}\exp(\mathrm{i}zk\sqrt{\alpha^{2}-1})&0&0&0\\ 0&\exp(-\mathrm{i}zk\sqrt{\alpha^{2}-1})&0&0\\ 0&0&\exp(\mathrm{i}zk\sqrt{\beta^{2}-1})&0\\ 0&0&0&\exp(-\mathrm{i}zk\sqrt{\beta^{2}-1})\end{bmatrix}  \tag{3}\]  
\[ \begin{array}{ll}\alpha=C_{P}/\sqrt{(\lambda+2\mu)/\rho}\\ \beta=C_{P}/\sqrt{\mu/\rho}\\ k=2\pi f/C_{P}\end{array}  \tag{4}\]  
The \(\mathbf{D}\) matrix represents the material stiffness matrix, which is related to the transmission characteristics of interlayer interface stress and displacement; the \(\mathbf{E}\) matrix is the diagonal matrix associated with the frequency and phase velocity of the material; \(C_{P}\) is the guided wave's phase velocity; \(\lambda\) and \(\mu\) are the Lame constants; \(i\) is the imaginary unit; \(\rho\) is the upper layer density; \(f\) is the ultrasonic frequency; and \(z\) is the coordinate along the local coordinate system axis in Equations (2)-(4). The following are the boundary conditions for the stress \(\sigma\) and displacement \(u\) on each layer's top and lower surfaces:  
\[ \begin{array}{ll}u_{x(1)}^{-}=u_{x(2)}^{+},u_{x(2)}^{-}=u_{x(3)}^{+},u_{z(1) }^{-}=u_{z(2)}^{+},u_{z(2)}^{-}=u_{z(3)}^{+}\\ \sigma_{x(1)}^{-}=\sigma_{x(2)}^{+},\sigma_{x(2)}^{-}=\sigma_{x(3)}^{+},\sigma _{z(1)}^{-}=\sigma_{z(2)}^{+},\sigma_{z(2)}^{-}=\sigma_{z(3)}^{+}\end{array}  \tag{5}\]  
where the subscripted digits \"1\", \"2\", and \"3\" stand for the first, second, and third layers, respectively, while the superscripts \"+\" and \"-\" indicate the upper and bottom surfaces, respectively. The transfer matrix corresponding to the first layer can be calculated as:  
\[\mathbf{T}_{1}=\mathbf{D}_{1,U}\times\mathbf{D}_{1,L}^{-1} \tag{6}\]  
The transfer matrix associated with the second layer can be computed as follows:  
\[\mathbf{T}_{2}=\mathbf{D}_{2,U}\times\mathbf{D}_{2,L}^{-1} \tag{7}\]  
The computation of the transfer matrix for the third layer is carried out as:  
\[\mathbf{T}_{3}=\mathbf{D}_{3,U}\times\mathbf{D}_{3,L}^{-1} \tag{8}\]  
The first layer transfer matrix is denoted by \(\mathbf{T_{1}}\), the second layer by \(\mathbf{T_{2}}\), and the third layer by \(\mathbf{T_{3}}\).