## 6 NDT Applications
### The Acoustic Emission Propagation in Guided Wave  
In the present application, three typical acoustic emission (AE) events are simulated using a lattice discrete element method (LDEM) coupled with the traditional finite element method (FEM). In the problem analyzed, the dispersion curves are used to improve the identification of AE in this specific geometry. The acoustic emission techniqueThis is a promising method for nondestructive quantitative evaluation of damage progression. Acoustical emission is a phenomenon that could be caused by a structural change in the material, provoking the propagation of transient elastic-waves, due to a release of strain energy. These waves travel towards the surface of the structures detected by surface transductors that transform the released strain energy into electrical signals [49]. These signals have frequencies that typically range from kHz to MHz. Commonly in AE testing, parameters such as arrival time, velocity, hits, count, rise time, amplitude, duration, and frequency are recorded to characterize the signals. From the temporal and spatial distribution of the signals represented by using the cited parameters, both damage evolution and localization of AE sources in the specimens can be determined [50]. The lattice discrete element method (LDEM) is the name given in this paper to the version of the discrete element method used, originally proposed by [PERSON] [51]. This methodology is based on the representation of solid material using a cubic arrangement of uniaxial elements that can only carry axial loads. In this paper, a cubic module is illustrated in Figure 18a and how to compose a structure is shown in Figure 18b. Moreover, the mass of the solid is lumped at the nodes in the spatial discretization. In the discrete model, each node have as degrees of freedom three displacements in the coordinate directions (\(x\),\(y\),\(z\)). Considering an elastic isotropic material, the explicit expressions that link the mechanical properties with the elements properties, are presented in Equation (7). \[\eta=\frac{9\
u}{4-8\
u},EA_{n}=EL_{c}{}^{2}\frac{(9+8\eta)}{2(9+12\eta)},EA_{d }=\frac{2\sqrt{3}}{3}A_{n}, \tag{7}\]  
In this Equation, \(E\) represents the Young's modulus, \(\
u\) the Poisson's ratio, \(A_{i}\) the transversal areas of the elements, where \(i=n\) represent the normal, and \(i=d\) the diagonal elements, and finally \(L_{c}\) represents the size of the basic cubic cell. The spatial discretization achieved allowed us to obtain a well-known matrix motion equation presented in Equation (8). \[M\ddot{x}(t)+C\dot{x}(t)+F_{r}(t)-P(t)=0, \tag{8}\]  
Figure 18: The lattice discrete element method (LDEM); (a) basic cubic module, (b) generation of a prismatic body, and (c) bilinear constitutive model where the damage is taken into account. where \(x\) represents a vector of nodal displacements, \(M\) represents the mass matrix that in the present case is diagonal, moreover, \(C\) is the damping matrix (also will be assumed as diagonal), \(F_{r}\) and \(P\) represent the vector of the internal and external forces respectively, acting on the nodal masses. The dots in the Equation (8) represent the time differentiation, that is \((\dot{*})=d(*)/dt\). If \(M\) and \(C\) are diagonal, the Equation (8) is uncoupled and the explicit central finite difference scheme is extremely suitable for making time domain integration. The nodal coordinates being updated every step, large displacements computation naturally and efficiently. The link between axial force and axial strain in the elements represent quasi-fragile material behavior based on the bilinear law originally proposed by [PERSON] [52]. This law is represented in Figure 18.c, by applying this constitutive law, the specific fracture energy (\(Gf\)), is directly proportional to the area below the bilinear constitutive law. The random nature of the model could be introduced considering Gf as a 3D random field with a Weibull probability distribution.