Investigation of layer interface model of multi-layer structure using semi-analytical and FEM analysis for eddy current pulsed thermography

The use of a multi-layer structure is widely recognized in aerospace engineering due to the fact that structural and functional properties can be implemented by designing geometric structure. Eddy current pulsed thermography (ECPT) is one of the crucial NDT techniques to inspect and evaluate the defects in composite multi-layer structure due to the volumetric heating nature. Thus, it is important to investigate the scattering electromagnetic wave into the interface of multi-layer structure to improve the detectability and evaluation capbility of an ECPT system. In this work, the conductivity tensor form is used to describe the scattering EM wave in the 3D FEM model to investigate the fiber orientation influence on each layer interface. The semi-analytical model is used to prove the concept and both results are validated by experimental studies with dedicated samples. The findings can be applied for simulating the scattering electromagnetic behavior in the interface of the multi-layer structure.


I INTRODUCTION
The use of CFRP materials has become increasingly popular among conventional engineered materials due to their extraordinary mechanical and thermal properties, such as high strength-to-weight ratio, corrosion resistance, improved fatigue performance and low coefficient of thermal expansion. Multilayer composites are a broad and important group of structural and functional materials whose properties may vary over a very wide range. The possibility of combining in one monolithic material layers of a different nature that exhibit markedly different physical properties makes it possible to construct materials for very different functional purposes, including impact and high temperature, heat and corrosionresistance, heat-conducting, and heat-protective.
Eddy Current Pulsed Thermography (ECPT) has been recently proposed for multi-layer composite structure evaluation. When ECPT is applied to CFRP, the stimulation can be considered volumetric since the electrical conductivity is relatively low. Depending on the excitation frequency of the Eddy Current (EC) and the sample thickness, the typically achieved skin depths are greater than the sample thickness itself, or at least comparable to it [1]. In addition, due to the multi-physics nature of the ECT, the electrical and thermal properties can be evaluated simultaneously in one experiment. ECT is also less influenced by the surface conditions of the SUT such as emissivity, roughness, etc. [2]. Moreover, ECT can be exploited to evaluate the barely visible impact damage on composites [3] and the presence of delamination [4,5].
In order to understand the electromagnetic and thermal behaviour of the induction thermography. Simulation and modelling are needed. The current challenges for modelling inductive thermography are the anisotropic conductivity of the material especially the through-thickness electrical conductivity [6]. The electrical conductivity of CFRPs depends strongly on the orientation of carbon fibers: the longitudinal conductivity (parallel to the fiber direction, ) is the highest; while the transverse conductivity (perpendicular to the fibers, ) is relatively lower and on the same order of magnitude as the electrical conductivity along the thickness of the specimen ( ) [6]. Additionally, this anisotropic electrical conductivity is further compounded by a strong dependence on the presence of interfaces between adjacent plies. These interfaces vary in size, physical composition, and chemical composition and therefore result in uncertain value [7]. In addition, (according to the thickness of CFRP laminates) can be greatly affected by the presence of interfaces between layers and the lamination of the individual plies. For instance, the stacking sequence tends to increase the dispersion of measurements [8]. Thus, using to interpret the interface condition of the laminated composites can be the solution to model the anisotropic behaviour of the composites.
Due to the important number of carbon fibers impregnated in each layer, it is very difficult to take into account the real geometry in the simulation. The composite layer is then replaced by a homogenized one [9]. Moreover, as the composite sheets have a small thickness compared with their other dimensions, shell elements can be used to reduce the number of unknowns. The case of three-dimensional (3-D) induction heating simulation of composite plate with equivalent anisotropic conductivities has been presented [10]. In the meantime, CEA LIST has recently developed a new semi-analytical method for the computation of the 3D primary fields induced by an eddy current probe in a homogeneous conductor presenting a local perturbation of the geometry [11]. This approach is an extension of the curvilinear coordinate method (CCM), which is efficient for the computation of the fields scattered by 2D diffraction gratings enlightened by a plane wave or perfectly conductive random surfaces [12,13].
This full text paper was peer-reviewed at the direction of IEEE Instrumentation and Measurement Society prior to the acceptance and publication.
Investigation of layer interface model of multi-layer structure using semi-analytical and FEM analysis for eddy current pulsed thermography In this work, 3-D induction heating simulation model of a multilayer anisotropic composite materials is proposed. The real geometry of multilayer composite materials with an equivalent anisotropic individual layer is considered. A global equivalent model is then introduced to consider the different fibbers' orientations. In addition, to model the scattering electromagnetic field penetrating which contains the internal reflections occurring inside every layer composing the structure, the conductivity tensor in perpendicular direction in FEM model is obtained by approximating the output of semi-analytical method.
The rest of the paper is organized as follows: Section II introduces the methodology of FEM model and semianalytical method to simulate the electromagnetic behaviour of CFRP. Section III presents the experimental setup. Section IV gives the results and analysis and conclusion is introduced in Section V.

II LAYER INTERFACE MODELLING
3D FEM method and semi-analytical models are introduced and discussed.

A 3D FEM method for multi-layer CFRP
To model the anisotropic behaviour of composites material, the electrical and thermal conductivities have tensor form as follows [10]: This work considers the conductivity tensor to represent electromagnetic properties of each laminate. The relationship between conductivity tensor and the orientation of the layer is shown in Eq. (3) and (4) respectively. (3) For layers of laminated composites, each layer 's magnetic field can be calculated: The boundary conditions of the magnetic field are as follows: (6) where represents the thickness of layer .
According to Faraday's law combined with the local form of Ohm's law, the anisotropic electrical field of each layer can be obtained as follows: (7) Where is the normal vector and , are scalar values dependable on electrical conductivity tensor is shown in Eq (1) and can be calculated as [14].
After obtaining current density of layer . The heat source of in layer can be obtained as: After the calculation of of each layer. The heat transfer inside the anisotropic material can be solved as: (10) with the boundary condition: (11) Where is the tensor form shown in Eq. (2), is the material density, is the specific heat, is the convective coefficient, is the temperature of layer and is the room temperature.
For all cases, the 3-D electromagnetic and thermal behaviour are solved in COMSOL multi-physics.

B Semi-analytical method
The configuration for the semi-analytical method in this paper is shown in Fig.2. It consists of a multilayer stack with N layers, labelled by p=1,2,...,N of the wave number and thickness . each layer is bounded by the and interfaces in which the eigenmodes are denoted by . At the interface, we have the outgoing waves corresponding to the coefficients and the incoming waves corresponding to the coefficients .The superscripts -and + refers to transmission and reflection respectively. Moreover, the superscripts (up) and (dn) refer to the upper and downer coefficients regarding the interface. Assuming that the and the Nth layers are in air, results from the incident field due to the coil and Thus, the 2D discrete FT along Ox and Oy was adopted: (12) Where (13) Therefore, the partial derivatives become and . We take also the assumption of variable separation: so . The modal representation of the field depends on the truncation orders and of the FT. Numerically, each component is a matrix of dimension L, with Depending on the nature of the considered anisotropic layer, the modal decomposition is presented in [12].  fig. 3(c) shows, the ECPT contains four units, an excitation module with a rectangle coil, an infrared camera, a signal generator, and a computer. In this study, only the operational RMS current and frequency of the excitation module were changed to 300 A and 300 kHz, respectively.The heating induced by eddy current is 500 ms. A 10 mm lift-off between the coil's bottom edge and the top faces of the specimen was kept in all the ROIs' testing.
In addition, as shown in fig 3(a) and fig 3(b), the 0-45 sequence sample is used for the experiment. The parameters for the sample are shown in Table I.  IV RESULTS AND ANALYSIS In order to validate the proposed FEM model, the temperature evolution over time is compared for experimental and simulated data. It is observed that in Fig 4, the experimental data are scattering around the simulated data, which confirms the validation of FEM data and experimental data.   Based on the above analysis, the conclusions are as follows: 1.The scattering electromagnetic field at the interface in multi-layer structure is modelled by the semi-analytical model and the results are used to model the interface in FEM methods.
2.Experimental results for ECPT match with the simulated study. However, it is observed that temperature experimental data at heating stage is slightly lower than simulated output, which can be due to the lack of description of the scattering electromagnetic field in FEM.
3.The scattering EM field in the interface can be modelled by varying the through-thickness conductivity, which is proved in this work.
Future work will continue to investigate the relationship between and different stacking sequences of the composite material