Conductivity inversion of unidirectional CFRP laminate based on ECT using neural network

Abstract

For the electrical anisotropy of carbon fiber reinforced polymer (CFRP), conductivity of unidirectional CFRP laminate in three directions was inverted in this paper. The three-dimensional eddy current electromagnetic model of unidirectional composites was constructed by ANSYS software, and the influence of the electrical conductivity of the material on the detection signal of the probe in the longitudinal, transverse and thickness directions was studied. In order to improve the amplitude of the probe output signal induced by the change of conductivity, the optimal detection angle of the eddy current probe was determined. On this basis, the relationship between the conductivity and the detection signal was studied to estimate the initial values of the conductivity based on the experimental data obtained by the eddy current testing (ECT). According to the forward model, the theoretical probe voltage under the estimated conductivity were calculated. The database consisting of conductivity and corresponding theoretical results was built for the neural network to construct the mapping that can estimate conductivity by experimental results. Using neural network for iteration, the conductivity was inverted quickly and precisely.

Keywords

CFRP ECT conductivity inversion forward model neural network

1. Introduction

Due to the excellent properties such as high strength, light weight, high temperature and high pressure resistance, CFRP materials were widely used in aerospace, military and civil industries [1–4]. As the application range of materials continues to expand, the study of its electromagnetic properties is of great significance. Unlike the isotropic material, CFRPs are a heterogeneous multiphase material, whose electrical conductivity is mainly caused by carbon fibers. Therefore, the conductivity of CFRPs depend on the distribution of carbon fibers and the formation of fiber conduction paths, thus the overall performance is anisotropy [5].

Identification of material properties for CFRP is fundamental in laminate structures design. It is particularly important for various applications to know the electrical conductivity of CFRPs, which provides the basis for damage reconstruction. The technology of damage reconstruction is helpful for assessing the severity level and expansion trend of damage. The electrical conductivity is required in constructing the calculation model for studying the relationship between the detection signal and the damage. However, the conductivity of CFRP is not easy to know in advance, and need to be determined by either contact or non-contact method. The contact measurement is mainly based on the probe method which is using the impedance analyzer to measure the resistance between the two stages, and the conductivity can be calculated [6–8]. Although this method has high accuracy, the whole process is cumbersome. Non-contact methods, based on resonant circuits and on the physics of the eddy current, have been proposed to determine the overall conductivity of CFRP [9]. A simple inverse method for measurement of through-thickness conductivity of carbon fiber composites proposed by Mizukami proves the feasibility of the non-contact method [10].

In this paper, unidirectional CFRP laminate is taken as an example to study its electrical anisotropy using ECT method [11]. Firstly, the forward model was constructed by ANSYS software and the influence of conductivity of unidirectional CFRP laminate on the output signal was studied. Then, the ECT system and experimental method adopted in this paper is briefly introduced. Finally, the initial value of conductivity was estimated and then the iteratively updated inversion was carried out through neural network by minimizing the error between the calculated from the forward model and the measured ECT signal.

2. Principle of conductivity inversion

2.1. Research on the forward problem

Conductivity inversion is a typical inverse problem which requires a fast and accurate solution to the positive problem. For the construction of the eddy current forward model, there are two ways in general: the integral method using the theoretical formula and the analysis method using experimental signals [12]. The former’s derivation process is very cumbersome, while the latter requires a large amount of data to maintain robustness. To this end, this paper directly uses the ANSYS software with built-in integral formula to construct the forward model. The whole model can be regarded as an electromagnetic field solving domain. The solution domain contains the excitation source, the conductor region and the air region. The different regions were connected by boundary conditions, as shown in Fig. 1(a) and the corresponding three-dimensional eddy current field model used in this paper is shown in Fig. 1(b). In addition, by computing the benchmark problems formulated by Japan Society for Applied Electromagnetics and Mechanics (JSAEM) [13], the accuracy of the forward model for numerical calculations was verified.

Fig. 1.

Forward model: (a) solving domain; (b) the corresponding 3D model.

Assuming the fibers direction of unidirectional CFRP laminate parallels to the x-axis of the cartesian coordinate system, then the conductivity of the laminate can be represented by a diagonal matrix [14]: $\begin{eqnarray}\displaystyle {\sigma}=\left[\begin{array}{@{}ccc@{}}{\sigma}_{l} & 0 & 0\\ 0 & {\sigma}_{t} & 0\\ 0 & 0 & {\sigma}_{z}\end{array}\right] & & \displaystyle\end{eqnarray}$ (1) where σ_l and σ_t denote the conductivity in the directions along and transverse to the fibers, σ_z denotes the conductivity in the thickness direction. Thus, conductivity inversion becomes a multi-parameter optimization problem. To improve the accuracy of the inversion, this paper involves two parts: the influence of conductivity in the three directions on output signal and the probe angle θ which is the most sensitive to the change of each conductivity. Since the sample is centrally symmetric, the probe angle θ used in this paper is 0°, 30°, 45°, 60°, and 90°, respectively. At the above several angles, the probe voltages on different frequencies of different conductivities are simulated separately. After then, the obtained results in the case of θ = 0 were selected to analyze the relationship between the probe voltage and the frequency, as shown in Fig. 2, where the initial conductivity σ is: (σ_l, σ_t, σ_z) = (30000, 4.8, 1.2) S/m. Then, conductivity in the three directions were reduced by one-third respectively which means the conductivity after the change were: (20000, 4.8, 1.2) S/m, (30000, 3.2, 1.2) S/m and (30000, 4.8, 0.8) S/m. Obviously, a linear relationship between the voltage signal and the frequency is obtained in Fig. 2, which can be fitted by a relationship: $\begin{eqnarray}\displaystyle V=a\times f+b & & \displaystyle\end{eqnarray}$ (2) where V represents probe voltage and f represents excitation frequency.

Fig. 2.

The relation between voltage and frequency at the probe angle of 0°.

As shown in Fig. 2, there is a certain relationship between the conductivity and frequency in each direction. Also, when conductivity in the three directions were all reduced by one-third, the value of the coefficient varies differently, which indicates that the different conductivities have different influences on the probe voltage. According to the change of the coefficient, it’s clearly that the transverse conductivity σ_t in unidirectional CFRP has the greatest influence on the probe voltage due to its anisotropy.

To find the most sensitive probe angle to conductivity changes in three directions, a parameter 𝛼 is defined to represent the rate of change: $\begin{eqnarray}\displaystyle {\alpha}=\frac{|a_{c}-a_{i}|}{a_{i}} & & \displaystyle\end{eqnarray}$ (3) where a_c represents the coefficient a obtained in a changed conductivity under a certain direction as shown in Eq. (2), and a_i represents the coefficient a obtained in the initial conductivity under the same certain direction as a_c. Eq. (3) can be used to calculate the rate of change 𝛼 at different probe angles, and the corresponding results are shown in Table 1.

Table 1

The rate of chage 𝛼 at different probe angles

Conductivity (S/m)	Probe angle (°)
	0	30	45	60	90
σ_l	0.01813	0.00323	0.04968	0.02174	0.01583
σ_t	0.26499	0.27742	0.23913	0.25217	0.25858
σ_z	0.05858	0.05806	0.06211	0.06522	0.06593

According to Table 1, for conductivity in different directions, the largest value of 𝛼 was obtained at different angle θ. The larger 𝛼 is, the more sensitive the direction is to the change of conductivity. Thus, it’s clearly that the most accurate probe angle for σ₁, σ_t and σ_z are 45°, 30° and 90°, respectively.

2.2. Relationship between probe signal and conductivity

To construct the relationship between conductivity and probe signal, the trend of the coefficient with the conductivity changing at the corresponding probe angle obtained from the above analysis was analyzed. It should be noted that when the coefficient was studied as a function of a certain conductivity, the other two conductivities in different directions remain unchanged. The obtained results are shown in Fig. 5, where the trend of the coefficient a changing with conductivity in the three directions is clearly and the value of conductivity could be estimated according to the experimental data.

2.3. BP neural network

As mentioned before, the inversion for conductivity can be transformed to an optimization problem to minimize the objective function: $\begin{eqnarray}\displaystyle {\varepsilon}=\mathop{\sum }_{i=1}^{3}|a_{\exp }-a_{sim}| & & \displaystyle\end{eqnarray}$ (4) where a_exp and a_sim are the coefficients fitted by experimental and theoretical results, respectively. Considering that there are three directions of conductivity, a sum was used here. To minimize the error of the function, a BP neural network for iteration was used in this paper

As a kind of multi-layer feedforward network, a typical BP neural network consist of input layer hidden layer and output layer, and the structure of network used in the algorithm is shown in Fig. 4.

Fig. 3.

The relation between the coefficient and conductivity at the probe angel: (a) 45; (b) 30° and (c) 90°.

Fig. 4.

The structure of the neural network.

Fig. 5.

Inversion algorithm using neural network.

In Fig. 4, coefficients of different angles and conductivity in three directions are used for input and output, respectively, to construct a mapping relationship. Based on the BP neural network, the whole process of inversion algorithm was carried out as pictured in Fig. 5. Firstly, conductivity of the unidirectional CFRP laminate in the three directions σ can be estimated according to the coefficent a_exp and the trend obatained in Fig. 3. Also, the corresponding theoretical coefficient a_sim can be calculated by forward model. Then, a database including coefficient and conductivity is built for the initial training of neural network. With the trained neural network, σ can be quickly assessed by a_exp, and then, the corresponding a_sim can be simulated by forward model. When the residual ϵ between a_exp and a_sim is less than the goal, the iterative process is terminated. If not, updating a_sim and corresponding σ to the database, adding one more node for the hidden layer and repeating the above iteration process.

3. Experimental setup and results discussion

3.1. ECT setup

The ECT system for CFRP was presented in Fig. 6(a). It consists of a signal generator (Agilent 33220A) a stepping motor in X-Y plane, a T-R probe, a PC (for control), a DAQ card (for data collecting) and a lock-in amplifier (Signal Recovery SR844), which allows extracting a weak signal buried in background noise The tested unidirectional CFRP laminate is shown in Fig. 6(b). The plate is made up of 8 layers of unidirectional prepreg, and the thickness of each prepreg is about 0.125 mm.

Fig. 6.

Experimental schematic: (a) ECT system; (b) unidirectional CFRP laminate.

3.2. Results of inversion

In the process of experiment, the output signal of the pickup coil was obtained by single point acquisition instead of continuous scanning. To correspond with the simulation results, the probe voltages at different frequencies were measured during experiment process, and the coefficients were obtained by linear relationship fitting. In order to improve the sensitivity of conductivity inversion in the three directions, the T-R probe was used and the signals of T-R probe were measured at 30°, 45° and 90° between the probe and the fiber respectively. The results of the experiment and the corresponding inversion results are shown in Table 2.

Table 2
The result of experiment and inversion

Probe angle (°) Coefficient a Conductivity (S/m)

Experiment Simulation

30 2.33 2.3262 σ_t= 3.77

45 2.26 2.2693 σ_l= 15106

90 2.762 2.7667 σ_z= 0.67

Probe angle (°)	Coefficient a	Conductivity (S/m)
30	2.33	2.3262	σ_t= 3.77
45	2.26	2.2693	σ_l= 15106
90	2.762	2.7667	σ_z= 0.67

According to Table 2, conductivity of unidirectional CFRP laminate obtained in the iterative process is (15034, 3.78, 0.66) S/m. The result shows good agreement with the result (14860, 3.8, 0.6) S/m, which was measured by the two-probe method. In this way, the accuracy of the proposed inversion method has been verified.

4. Conclusion

In this paper, the eddy current method is used to study the conductivity inversion of unidirectional CFRP laminate. Firstly, the three-dimensional eddy current electromagnetic model of unidirectional CFRP was constructed by using ANSYS software, and the influence of the electrical conductivity of this material in the longitudinal, transverse and thickness directions on the output signal of the probe was studied. Secondly, the relationship between the conductivity and the output signal was constructed, and the initial value of the conductivity in three directions were estimated based on the experimental data obtained by the eddy current testing (ECT). Then, the database consisting of conductivity and corresponding theoretical results was built for the neural network to construct the mapping that can estimate conductivity by experimental results. Using neural network for iteration, the conductivity is inverted precisely. Compared with the previous inversion method, the proposed method in this paper does not require a large amount of experimental data, and it has high operability and fast speed.

Footnotes

Acknowledgements

This work was supported by the Natural Science Funding (51875277), Foundation strengthening plan technology funding (2019-JCJQ-JJ-337), State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and astronautics (Grant Nos. MCMS-I-0518K01 & MCMS-I-0519G02), and PAPD.

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