# Assessment of the Length and Depth of Delamination-Type Defects Using Ultrasonic Guided Waves
## 3 Estimation of Reference Dependencies
### Numerical and Experimental Verification of Reference Dependencies
#### 3.2.1 Numerical Verification  
At first, the reference dependency estimation technique is verified numerically. A total number of 50 2D linear structural mechanics FE models were employed for an anisotropic GFRP plate with the dimensions of \(x=2000\) mm, \(y=4\) mm. In this case, the plain strain model was implemented by assuming that the object is infinite in the \(z\) direction. In each of the 50 models, delamination-type defects  
Figure 6: Magnitude variation of transmitted A\({}_{0}\) mode versus frequency at different defect depths in case the length of defect is equal to \(x_{1}=50\) mm (**a**) and \(x_{2}=70\) mm (**b**).  
Figure 7: Magnitude variation of transmitted A\({}_{0}\) mode versus frequency at different defect depths in case the length of defect is equal to \(x_{3}=90\) mm (**a**) and \(x_{4}=110\) mm (**b**).  
were introduced 1.5 mm below the surface (\(h_{1}\) = 1.5 mm, \(h_{2}\) = 2.5 mm) by separating the appropriate nodes of the mesh. The length of delamination l varied from 10 to 500 mm with an increment of 10 mm, thus yielding a total number of 50 numerical models. The delamination was centered horizontally along the \(x\) axis of the investigated sample for all the defect lengths. Throughout the simulations, 2D structural solid plane-42 finite elements were used. The spatial size of the element was equal to 0.5 mm, which corresponds to 26 nodes per wavelength if the slowest A\({}_{0}\) mode at 80 kHz central frequency is considered. The excitation signal in all cases was a Gaussian envelope tone burst of three cycles and a central frequency of 80 kHz. Such an excitation regime was deliberately selected to converge with the experimental verification, which will be described further in this section. A\({}_{0}\) mode was introduced by applying out-of-plane displacements to the first node column of the model located on the sample edge (see Figure 8). Such a force excites both fundamental A\({}_{0}\) and S\({}_{0}\) modes at a given frequency band and acts similarly to a simple thickness mode transducer. The integration step in the time domain was 0.625 \(\upmu\)s, which is 1/20 of the period at 80 kHz central frequency. Throughout the simulation, a generalized stiffness matrix was used to describe a symmetric GFRP laminate (biaxial: 0\({}^{\circ}\) and 90\({}^{\circ}\)bias: \(\pm\)45\({}^{\circ}\)/biaxial: 0\({}^{\circ}\) and 90\({}^{\circ}\)s), while employing the research conducted by [PERSON] et al. [33] Hence, the stiffness matrix was obtained by summing up the contribution of each ply in terms of their respective thicknesses. The following material properties of GFRP were used: [PERSON]'s modulus (\(E_{\text{x}}\) = 10 GPa, \(E_{\text{z}}\) = 35.7 GPa); [PERSON]'s ratio (\(v_{\text{x}\text{z}}\) = 0.325, \(v_{\text{z}\text{x}}\) = 0.091, \(v_{\text{y}\text{x}}\) = 0.35); shear modulus \(G_{\text{x}\text{z}}\) = 2.8 GPa; density \(\rho\) = 1800 kg/m\({}^{3}\). The sketch of the numerical model used in this study is presented in Figure 8.  
The variable monitored in this study was the vertical component of particle velocity (\(y\)) at the nodal locations before (\(x_{1}\) = 500 mm, \(y_{1}\) = 4 mm) and beyond (\(x_{2}\) = 1500 mm, \(y_{2}\) = 4 mm) the defect (see Figure 8). The obtained peak to peak amplitude variations versus the length of delamination are presented in Figure 9. In the same figure, the predicted curve obtained by using Equation (11) with identical parameters is presented as the solid line.  
Figure 8: Sketch of the numerical model used for the validation of predictions (the sketch of the model is not to scale).  
Figure 9: Magnitude variation of A\({}_{0}\) mode due to the size of the defect at a fixed depth: comparison between analytical calculations and finite element modeling.