## 7 Influence of the Damage Depth and Size on the Scattering Directivity Pattern  
This section deals with the numerical study to observe the effect of the geometrical size of conical damage on the scattered wave fields in the case of an incident A\({}_{0}\) mode. For this purpose, two different studies have been performed. The first study considers the effect of the through-thickness variation (depth of the cone) on the scattered fields of the incident A\({}_{0}\) mode while fixing the diameter at 30 mm in order to conform with the impact test presented earlier. Four different models with different depths of damaged zones equal to 1/4, 1/2, 3/4 and 1 of the total thickness, i.e., 1.55, 3.1, 4.65 and 6.2 mm, are considered in the 3D-FE models, respectively. The schematic of the different depths is shown in Figure 13a. The four different models are solved for the same incident A\({}_{0}\) mode (100 kHz) using the same properties of the FE model as explained earlier in Section 5. The normalized scattered field corresponding to the A\({}_{0}\) mode is shown in Figure 13b in the case of the abovementioned depths. In all four cases of through-thickness depths, it is observed that there is a negligible amplitude scattered between 120\({}^{\circ}\) and 330\({}^{\circ}\) angles towards the anticlockwise direction. For the first three cases of through-thickness depth equal to 1/4, 1/2 and 3/4, the maximum scattered amplitude is towards the 45\({}^{\circ}\) direction and the extent of the scattered field gets narrow for increasing through-thickness depths of the conical damage. It is also noted that with the increase in through-thickness depth, side lobes start to appear in the scattered displacement field amplitudes and increases with the increase in depth. In the case of all through-thickness depth i.e., for depth equal to 6.2 mm, the side lobes have more scattered amplitude towards 80\({}^{\circ}\) and 10\({}^{\circ}\) directions. In this case, the maximum scattered amplitude is not towards the 45\({}^{\circ}\) direction but is oriented toward the side lobe directions. In all four cases, the directivity diagrams are found to be almost symmetrical along the 225\({}^{\circ}\)- 45\({}^{\circ}\) direction due to the isotropic nature of the plate in the \(X\)Z plane (and symmetrical geometry of the damage). The variation of amplitude of the scattered A\({}_{0}\) mode with through-thickness depths of the conical damaged zone in the direction of 45\({}^{\circ}\) is shown in Figure 14. The amplitude increases with the increase of depth until half the total through-thickness and then starts decreasing and decreases to the minimum value at all through-thickness depths. With this scattered field information for a fixed diameter of the damaged zone, one can gain an idea about the depth of the conical damage zone inside the composite plate. According to the literature, the micro-cracks due to impact damage (ex. matrix cracks) mainly contribute to the scattering of the wavefield in the damage region. The delamination reshapes the wavefield in the damaged and undamaged regions. In particular, it was found that the contribution of the delamination in the scattered wavefield is \(\sim\)50% higher than the one corresponding to the matrix cracks [25]. Figure 13: Effect of the conical depths: (**a**) Schematics of four different through-thickness depths (1.65 mm: red, 3.1 mm: black, 4.65 mm: green and 6.2 mm: blue), (**b**) normalized scattered amplitudes A\({}_{0}\) mode for different depths. In the second case, the effect of the aspect ratio (2 \(r\)/\(\lambda\)) on the scattered wave fields is studied, where \(r\) is the radius of the cone and \(\lambda\) is the wavelength of the A\({}_{0}\) mode. The study is performed by taking a through-thickness depth equal to 5.2 mm. Four different 3D-FE models with aspect ratios equal to 1/2, 1, 3/2 and 2, respectively, have been solved. The schematic for different aspect ratios is shown in Figure 15a.