# Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly
## 2 Dispersion Curves of Lamb Waves  
The schematic of a tri-layer adhesive structure composed of two aluminum layers and an adhesive layer is described in Figure 1. Two aluminum plates are joined by an adhesive layer of epoxy resin. The thicknesses of the aluminum layer and adhesive layer are \(h_{1}\) and \(h_{2}\), respectively. It is assumed that the material of each layer is isotropic, elastic and homogenous. The Lamb waves propagate within the tri-layer adhesive structure along the positive \(x\)-axis and the wave polarization is along the \(z\)-axis. The displacements of particles, therefore, occur only in the \(xz\) plane for the Lamb waves. Figure 1: The schematic of the tri-layer adhesive structure. For the tri-layer adhesive structure, the state vector \(V_{U}^{(j)}\) of the upper surface of the \(j\)-th layer can be expressed as:  
\[V_{U}^{(j)}=\left( \begin{array}{c}u_{U}^{(j)}\\ w_{U}^{(j)}\\ \sigma_{zU}^{(j)}\\ \sigma_{zU}^{(j)}\end{array} \right)=P_{U}^{(j)}e^{\text{k}(x-\text{ct})}\left( \begin{array}{c}A^{(j)}\\ B^{(j)}\\ C^{(j)}\\ D^{(j)}\end{array} \right) \tag{1}\]  
where \(u_{U}^{(j)}\), \(w_{U}^{(j)}\), \(\sigma_{zU}^{(j)}\) and \(\sigma_{zxU}^{(j)}\) are the displacements and stresses in \(x\)- and \(z\)-directions, respectively. The subscript \(U\) refers to the upper surface; \(P_{U}^{(j)}\) is the amplitude of the state vector; \(A^{(j)}\), \(B^{(j)}\), \(C^{(j)}\) and \(D^{(j)}\) are constants of the \(j\)-th layer. For this tri-layer adhesive structure, the relation between the state vectors of the upper and lower surfaces for each layer can be expressed as:  
\[V_{L}^{(j)}=T^{(j)}V_{U}^{(j)} \tag{2}\]  
where \(V_{L}^{(j)}\) is the state vector of the lower surface of the \(j\)-th layer; \(T^{(j)}\) is the transfer matrix of the \(j\)-th layer. For the whole tri-layer structure, therefore, state vectors of the upper and lower surfaces can be related as:  
\[\left( \begin{array}{c}u_{L}\\ w_{L}\\ \sigma_{zL}\\ \sigma_{zxL}\end{array} \right)=\text{T}\left( \begin{array}{c}u_{U}\\ w_{U}\\ \sigma_{zU}\\ \sigma_{zxU}\end{array} \right) \tag{3}\]  
where \(T=T^{(1)}T^{(2)}T^{(3)}\). Considering the free surface condition of the upper and lower surface for the tri-layer structure, the following equation can be obtained, according to Equations (1)-(3):  
\[\left[ \begin{array}{cc}T(3,1)&T(3,2)\\ T(4,1)&T(4,2)\end{array} \right]\left( \begin{array}{c}u_{U}\\ w_{U}\end{array} \right)=\left( \begin{array}{c}0\\ 0\end{array} \right) \tag{4}\]  
To avoid the trivial solution of Equation (4), the following expression must be satisfied:  
\[\text{det}\left[ \begin{array}{cc}T(3,1)&T(3,2)\\ T(4,1)&T(4,2)\end{array} \right]=0 \tag{5}\]  
Equation (5) is the dispersion equation of the tri-layer adhesive structure. By solving this equation numerically, the dispersion curves can be plotted, as shown in Figure 2. Mode conversions for the tri-layer adhesive structure can be clearly found, differing from the dispersion curves of a single plate. For the dispersion curves of the tri-layer adhesively-jointed structure, the phase velocities converge to the transversal wave velocity of the adhesive layer. To generate the accumulative second harmonics of Lamb waves, the Lamb wave mode should satisfy the conditions of nonzero power flux from the fundamental to the second harmonic wave and phase velocity matching [43]. Considering that the lower order modes carry more energy than the higher-order modes for the Lamb waves propagating within the tri-layer adhesive structure, the first symmetric mode pair of S\({}_{1}\) and S\({}_{2}\) are chosen as the fundamental and second harmonic waves, respectively. As shown in Figure 2a, when the frequency of the fundamental Lamb waves (Mode S\({}_{1}\)) is selected as 0.52 MHz, accumulative second harmonic waves (Mode S\({}_{2}\)) with the frequency of 1.04 MHz can be generated with the same phase velocity.