**3. Numerical analysis**

This section discusses the numerical simulation carried out for detection of subsurface delamination in a four layered GFRP plate with circular delamination at the central layer. The numerical model of the delaminated GFRP plate was carried out using ABAQUS 6.14 software. Each layer of the GFRP plate is considered to be of 0.5 mm thickness with the total thickness of plate as 2 mm. The area of the plate considered for numerical simulation is 180 180 mm<sup>2</sup> with a delamination between second and third layer. Thus, the residual thickness of the damage required for calculation of analytical LDR frequency is 1 mm. The diameter of the circular delamination is taken as 12 mm while the location of the delamination is at x = 120 mm and y = 60 mm, taking left bottom corner of the plate as the origin. **Figure 2** shows the schematic of the delaminated plate with circular delamination.

The material properties of GFRP plate used for carrying out the explicit temperature displacement analysis are given in **Table 1**. The rule of mixtures was implemented for calculating the material properties of GFRP specimen from the individual property of glass fibre and epoxy [29]. The simulation is carried out by providing sliding contact interaction properties at the defect interface. Moreover, the friction properties are included on the delamination area while the rest of the plate area is provided with tie constraints for all layers. The initial

#### **Figure 2.**

*Schematic of GFRP plate with circular delamination at the central layer.*


#### **Table 1.**

*Material properties for GFRP plate [28].*

ambient temperature was considered as 293 K for all nodes of the plate. The number of elements considered for carrying out the present numerical investigation is 42,269 10-noded tetrahedral elements. First, the analytical LDR frequency of the delamination is calculated from the relation presented in previous section, followed by the steady state dynamic analysis to confirm the LDR frequency. The steady state analysis is performed on a frequency limit such that the analytical LDR calculated falls within the range of the analysis. The mode shape obtained from the contour plot of steady state analysis confirms the exact LDR frequency required for performing explicit temperature-displacement analysis. Subsequently, the explicit coupled temperature-displacement analysis is carried out followed by heat transfer analysis to detect the defect using LDR based vibro-thermography.

The explicit coupled temperature-displacement analysis is performed on the GFRP plate using single periodic LDR excitation confirmed from the steady state analysis. The force amplitude for carrying out the analysis is taken as 50 N in the thickness direction. The excitation time of 10 ms is used for vibrating the plate at its LDR frequency with a fixed time increment of 5 <sup>10</sup><sup>8</sup> s. The time increment is calculated according to the CFL condition for explicit analysis [27]. The 10-noded modified thermally coupled second order tetrahedron (C3D10MT) mesh elements are used for the analysis. Subsequently, a node on the top surface of the plate positioned at centre of the damage area is selected to determine the amplitude of vibration during excitation. In the analysis history output, nodes at delamination layer are selected for plotting temperature profiles. The temperature gradient obtained at the end of explicit coupled temperature-displacement analysis is further used as predefined field for a heat transfer analysis. Finally, a 10 ms heat transfer analysis is performed with 10-noded quadratic heat transfer tetrahedron (DC3D10) mesh type for this step. The transient response of the heat transfer step is captured at the delamination layer as well as the top surface of the plate.
