# Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly
## 3 Numerical Simulations  
It is quite challenging to get the analytic non-linear responses of Lamb waves propagating along the tri-layer adhesive with randomly distributed micro-cracks in the adhesive layer. Therefore, numerical simulations are employed instead in this work. The commercial software Abaqus/EXPLICIT is used, which is employed by the central difference method and conditionally stable. A finite element model in two dimensions is set up for the tri-layer adhesive structure with micro-cracks distributing in a random way within the adhesive layer. As shown in Figure 3, the middle part of the adhesive layer with a length of \(L\) is the area with micro-crack damage. A dynamical displacement disturbance is exerted on the left side of the model in order to generate the Lamb wave in this tri-layer adhesive structure. After the interaction of Lamb wave with the micro-cracks distributing randomly in the damaged area, the wave signals propagating toward the direction of positive \(x\)-axis will be collected at signal detection positions 1 and 2. According to the dispersion curves shown in Figure 2, the fundamental mode S\({}_{1}\) can be excited when the frequency of the dynamical displacement disturbance at the left side is 0.52 MHz. This dynamical disturbance is expressed as:  
\[u(x,t)=\frac{1}{2}B\bigg{[}1-\cos\bigg{(}\frac{2\pi ft}{15}\bigg{)}\bigg{]} \tag{6}\]  
Figure 2: The dispersion curve of the tri-layer adhesive structure. (**a**) Phase velocity versus the frequency and (**b**) Group velocity versus the frequency. The solid lines represent the symmetric modes; the dashed lines represent the anti-symmetric modes. where \(B\) is the amplitude of the dynamical disturbance with \(B=1\times 10^{-5}\) mm in the simulations; and \(f\) is the frequency. The Lamb wave propagates in the positive \(x\)-direction, and interacts with the micro-cracks distributing randomly at the micro-crack damaged area of the adhesive layer. As a result, Lamb waves with the second component (S\({}_{2}\) mode) propagating with the same phase velocity will thus be generated. Most waves will penetrate across a damaged area with a micro-crack because the dimension of the micro-crack is far smaller than the wavelengths of S\({}_{1}\) and S\({}_{2}\) mode waves. The transmitted waves will be collected at positions 1 and 2. In the present simulation, linear elastic constitutive models of aluminum and epoxy are employed in FEM models. Material properties of aluminum are \(E=72.8\)GPa, \(\rho=2665\)kg/m\({}^{3}\), \(v=0.33\); and those of epoxy are \(E=1\)GPa, \(\rho=1000\)kg/m\({}^{3}\) and \(v=0.45\). The dimension of the model is 500 mm \(\times\) 3.3 mm, where the thickness of the aluminum plate is 1.5 mm and that of the adhesive layer is 0.3 mm. The length of the micro-crack damage zone is \(L\), which ranges from 4-16 mm. The micro-cracks are assumed to be distributed in the space with an area \(A\) at the micro-crack region of the adhesive layer. In the simulation, we assume that the length of each micro-crack is \(2a\). The crack density is defined using a dimensionless parameter as:  
\[c=\frac{Na^{2}}{A} \tag{7}\]  
where \(N\) is the number of micro-cracks. The center and direction of the micro-crack, which are variables affecting the random distribution of the micro-cracks, can be represented by the function of the probability density. In this numerical simulation, the variable of the function is uniformly random. Consider the surfaces of the micro-cracks as contacting surfaces which will separate from each other when Lamb waves pass through them while being unable to interpenetrate into each other. A contact model is thus employed to represent the clapping and slipping behavior of the micro-crack. To account for the influence of the surface friction to the Lamb wave propagation, the Coulomb friction law is employed in modeling the relative sliding of the micro-crack surfaces.