Sensitivity analysis for the eddy current testing of carbon fiber reinforced plastic materials

1. Introduction

Carbon fiber reinforced plastic (CFRP) materials are widely used in automotive and aircraft industries thanks to their unique characteristics. Despite their light weight, they have excellent mechanical properties, heat and corrosion resistance, which make them preferable in various applications [1]. One layer of the CFRP lamina consists of μm thick carbon fibers glued into resin, interwoven with each other. In practice, several layers are stacked onto each other in different orientations to provide a uniform strength. The presence of the occasionally appearing material defects, however, greatly reduces the mechanical strength, hence their detection is of key importance.

In the industry, several non-destructive testing (NDT) methods exist for different types of material flaws, e.g., ultrasonic testing (UT) and radiographic testing (RT, X-ray) are suitable for foreign matter detection and crack observation, or laser excited thermography [2] for delamination detection. In our study, eddy current testing (ECT) is in focus, which was found to be effective for detecting fiber-related defects, such as waviness, misorientations, and fiber breaks [3,4]. This method is based on the directional-dependent, i.e., anisotropic conductivity of the CFRP material, a property that can be exploited to inspect the state of the CFRP—and estimate its remaining lifetime—by evaluating an ECT scan.

In order to efficiently detect the fiber-related defects, the ECT setup—especially the measuring coil [4,5]—should be optimized to gain as much information on the fibers as possible. This, however, requires a great understanding of the fiber response. Previous electromagnetic models [1,6] that used a homogenized bulk conductivity tensor [7] are inherently unable to predict the correct ECT signal, since they neglect the inhomogeneous distribution of the fibers originating from the manufacturing inaccuracies.

In this work, we build a 3D finite element (FE) model (using COMSOL Multiphysics) of a typical ECT measurement configuration based on a recent work proposed in [8]. This model will be subsequently used for parametric studies to draw conclusions regarding the spatially varying fiber distribution and the dynamic range of the ECT signal in order to maximize the fiber information yielding from the ECT scan.

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Fig. 1.

Schematic representation of the magnetic flux density around a coil and the induced eddy currents in the CFRP specimen (left), and the geometric model (right).

Due to the arrangement of the fibers in the CFRP layers, the conductivity of a single layer is direction-dependent, which is usually described by a homogenized anisotropic conductivity tensor [7] where 𝜎 _L  ≈ 39000 S/m denotes the longitudinal, 𝜎 _T  ≈ 7.9 S/m the transversal and 𝜎 _cp  ≈ 0.63 S/m the cross-plane component of the conductivity [6,8].

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Fig. 2.

Comparison of the absolute value of the current distribution in the homogeneous (left) and inhomogeneous 𝜎 case (right).

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Fig. 3.

Inhomogeneous fiber distribution in the direction perpendicular to the fibers. Left: schematic drawing of the fiber arrangement, right: the longitudinal component of the conductivity tensor, 𝜎 _L .

In the general ECT setup, the alternating magnetic field of an excitation coil closely placed to the CFRP lamina induces eddy currents inside the specimen as shown in Fig. 1 (left). The change of the primary magnetic field caused by the magnetic field of the eddy currents can be detected by measuring the change in the impedance or voltage of the ECT coil. The geometric model in Fig. 1 (right) consists of three parts: the coil, the CFRP specimen, and the air as the background. For the coil domain, we applied the built-in homogenized multiturn coil model of COMSOL with the parameters R_out = 1.5 mm and R_in = 1 mm for the outer and inner radius, respectively, and h = 1 mm for its height. The number of turns is N = 8, and a current excitation of 1 A at f = 20 MHz was applied. The elongated shape of the current distribution shown in Fig. 2 (left) is the direct consequence of the fibers being placed in the x direction.

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Fig. 4.

Normalized relative voltages of the probe along the x = 0 scanning line.

During manufacturing, the carbon fibers do not remain parallel to each other and contact points will be present between them [3]. The latter changes the uniform fiber distribution, hence fiber-rich and fiber-rare regions are formed as shown in Fig. 3. This variation can be considered periodic and can be approximated with the first harmonic as proposed in [8], thus, the conductivity tensor is sinusoidally modulated in the longitudinal direction as a function of y (the transversal direction), i.e., where k = 2 π λ is the wavenumber and 𝜆 is the wavelength of the spatial variation of the conductivity. With parameter d—which will later denote the coil’s position—we can easily shift the conductivity tensor in the y direction. By taking into account the inhomogeneity of the conductivity tensor, the distribution of the current density changes as shown in Fig. 2 (right). During the ECT scan, the measured voltage of the probe—assuming a constant 1 A current excitation—does not remain a constant value, it becomes a spatially varying quantity due to the spatial variation of the fiber distribution. Hence, the former contains information about the latter which is especially important for the proper evaluation of the CFRP specimen. In the following, several parametric studies are performed to examine the dynamic range of the coil voltage and its connection to the fiber distribution in order to maximize the extracted information regarding the fine variations of the fiber distribution. The examined parameters are the spatial variation of the 𝜎 tensor (𝜆), the diameter of the coil (D_coil = 2R_coil), the number of turns (N) and the relative position of the coil (d) in y direction.

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Fig. 5.

Dynamic range of the probe signal with varying 𝜆. The highest dynamic range is observed at the 𝜆 = 10 mm case.

During the numerical simulations, a virtual scanning was applied [8], i.e., the conductivity tensor was shifted instead of the coil, hence only the generation of a single 3D mesh was required. A single simulation required cca. 8 × 10⁵ mesh elements (yielding ∼5 × 10⁶ degrees of freedom) and took 4 min on a desktop computer with an 8-core i7–9700 K processor and 64 GB RAM.

For the first set of simulations, the size of the coil was kept fixed at D_coil = 3 mm, while the spatial variation of fiber distribution varied from 𝜆 = 10 mm to 𝜆 = 2 mm. The simulated coil voltages are normalized as V norm = V coil − V ref Im ( V ref ) , where V_ref is the coil voltage in air. As the results of the 1D scanning, the obtained coil voltage is indeed periodic as shown in Fig. 4. In all cases, the maximal eddy currents—that result in a high real value and low imaginary value—belong to those coil positions, where the middle of it is placed directly above a high conductivity region.

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Fig. 6.

Dynamic range of the ECT signal as the function of the coil’s radius for different 𝜆 values.

The minimal eddy currents, however, do not belong exactly to the minimal, i.e., 𝜎 _L  = 0 positions. Furthermore, for lower 𝜆 values, the extreme values of the voltage curves are in entirely different positions, thanks to the unique interaction of the “natural” eddy current paths (that the coil would induce in a homogeneous specimen) and the varying conductivity.

A single period of the results (i.e., d ∈ { 0 … λ } ) are summarized in Fig. 5. It can be seen that the highest dynamic range belongs to the lowest D_coil∕𝜆 value, which corresponds to our a priori expectation: the smaller the coil compared to 𝜆, the more easily it can detect the small variations of the conductivity, i.e., it increases the dynamic range of V_coil and hence the detectability of the fiber distribution.

One could easily conclude from this study, that the measuring coil in an ECT setup should therefore be as small as possible, however, this neglects the limitation in the number of turns. Smaller coils have fewer turns, which limits their magnetizing properties and ultimately, it limits the dynamic range of the ECT signal.

To reveal this phenomenon, a last parametric study was performed with a CFRP specimen having a λ ∈ { 2 mm , 2.5 mm , 3 mm } spatial variation in the conductivity. The coil’s size varied in the { 0.625 mm : 0.125 mm : 3.25 mm } range with a wire diameter of d_wire = 0.125 mm, hence 8 more turns were added at each iteration.

The results in Fig. 6 revealed that an optimal coil size indeed exists and the maximal dynamic range was found at 𝜆∕D_coil ≈ 0.45 at each case. Even though adding more turns increases the induced voltage, when R_coil becomes comparable to 𝜆, an additional increase in the coil’s size makes it difficult to detect the fine variation of the conductivity. Or equivalently, using a smaller coil with fewer turns for a given CFRP specimen will decrease the dynamic range of the measured signal if the coil becomes too small. We note, that in a real measurement, V_coil depends on further parameters, such as signal-to-noise ratio, coil geometry, and the distribution of the windings. Nevertheless, these initial studies revealed that the optimal coil design is not straightforward and we might need to even consider the CFRP specimen itself (and the expected spatial variation of the fibers) in order to maximize the available fiber information obtained by the ECT scanning.