# Propagation of Non-Linear Lamb Waves in Adhesive Joint with Micro-Cracks Distributing Randomly
## 3 Numerical Simulations
The tensile stress exerting on the surfaces of the micro-crack is expressed as [45]:  
\[\mathbf{q}=\sigma_{0}\mathbf{n}+[\tau_{0}+\mu\sigma_{0}\mathrm{H}(-\sigma_{0} )]\mathrm{H}[\tau_{0}+\mu\sigma_{0}\mathrm{H}(-\sigma_{0})]\mathbf{s} \tag{8}\]  
where \(\sigma_{0}\) is the applied normal stress; \(\tau_{0}\) is the shear stress on the surfaces of the micro-cracks; \(\mu\) is the friction coefficient of the contacting surface; \(\mathrm{H}(\bullet)\) is the Heaviside step function; \(\mathbf{n}\) is a unit vector normal to the crack surface; and \(\mathbf{s}\) is in the plane of the crack. In the numerical simulations, the built-in contact models of the Abaqus software are adopted, which are the hard contact law of the normal behavior and the Coulomb friction law of the tangential behavior. To guarantee enough accuracy and stability during the numerical simulation, we will use more than 20 elements within the smallest wavelength of all the Lamb wave modes. The largest element  
Figure 3: Schematic of the Lamb waves propagating through a region with micro-cracks for the tri-layer adhesive structure. sizes, therefore, for the aluminum layer and the adhesive layer are set to be 0.2 mm and 0.08 mm, respectively. Six elements are employed when modeling the surface of the micro-crack. The four-node plane strain (CPE4R) element is employed to construct the FEM model. As to the time step for the numerical simulation, it should satisfy the condition that the Lamb wave does not travel farther than the size of the smallest element within a time step. The stable time step is chosen to be \(\Delta t=5\times 10^{-10}\) s when the numerical accuracy and efficiency are also taken into consideration. Shown in Figure 4 is the finite element modeling and mesh of the region with micro-cracks distributing in a random way. The non-linearity of acoustics caused by the contacting surface of the crack is a clapping effect of the surface and non-linear stress-strain behavior inherent in the contact, which is different from the theory of classical non-linearity. Some theoretical models have been reported for explaining the contacting acoustic non-linearity; for example, the bi-linear stiffness models and hysteretic models [46; 47]. [PERSON] et al. [48] further investigated the bi-linear stiffness models when different effective tensile and compressive moduli of the local micro-crack region are taken into consideration. For non-linear ultrasonic wave propagating within the material with micro-cracks, they suggested that the Acoustic Non-linearity Parameter (ANP) should be defined as:  
\[\beta^{\prime}=\frac{M_{2}}{M_{1}} \tag{9}\]  
where \(\beta^{\prime}\) is the ANP that has a close relationship to the density and friction coefficient of the micro-crack; \(M_{1}\) and \(M_{2}\) are the magnitudes of fundamental and second harmonics of non-linear Lamb wave, respectively. In the following numerical simulations, Equation (9) will be employed in investigating the relationship between the ANP and the micro-crack. Moreover, since the micro-cracks in the model are randomly distributed, numerical simulations were conducted repeatedly to eliminate the random effect of distribution. The averaged ANP based on the results of numerous finite element models versus the number of simulations is shown in Figure 5. Note that the average ANPs are based on the amplitudes of the fundamental and second harmonic waves collected in the \(x\)- and \(z\)-direction. It is clearly seen that when the number of simulations reaches 10, the average ANP tends to be stable. Thus, the ANPs in the following simulations are all the average values of 25 models with micro-cracks distributing randomly. Figure 4: **The FEM model of the micro-crack damage region: The distribution of micro-cracks and the finite element mesh. **