# Triple correlation for detection of damage-related nonlinearities in composite structures

## Abstract

Nonlinear effects in vibration responses are investigated for the undamaged composite plate and the composite plate with a delamination. The analysis is focused on higher harmonic generation in vibration responses for various excitation amplitude levels. This effect is investigated using the triple correlation technique. The dynamics of composite plate was modelled using two-dimensional finite elements and the classical lamination theory. The doubled-node approach was used to model delamination area. Mode shapes and natural frequencies were estimated based on numerical models. Next, the delamination divergence analysis was used to obtain relative displacements for delaminated plies. Experimental modal analysis test was carried out to verify the numerical models. The two strongest vibration modes as well as two vibration modes with the smallest and largest motion level of delaminated plies were selected for nonlinear vibration test. The Fisher criterion was employed to verify the effectiveness and confidence level of the proposed technique. The results show that the method can be used not only to reveal nonlinearities, but also to reliably detect impact damage in composites. These results are confirmed using the statistical analysis.

### Keywords

Composites Damage-related nonlinearities Impact damage detection Nonlinear vibration Higher harmonics generation Triple correlation## 1 Introduction

Composite materials are widely used in many engineering applications due to their interesting properties such as low weight, fatigue strength, resistance to corrosion and design flexibility. It is well known that performance of composite structures can be significantly reduced by manufacturing defects and service-induced damage. The latter is often caused by low-velocity impacts that can lead to sub-surface, hidden damage due to layered structure of composites. This type of damage can take various forms such as matrix cracks (occurs parallel to the fibres due to compression, tension or shear), fibre breakages (due to tension), bucking (due to compression) or delamination (produced by inter-laminar stress). Low-velocity impact damage modifies structural parameters and has a knockdown effect on the residual strength of composite structures. This is one of the major reasons why monitoring for possible structural damage is important in composites, as discussed in [1]. Various methods have been developed for damage detection in composite structures. Altogether, these methods can be classified as passive and active [2]. The former includes methods based on acoustic emission [3, 4], operational load monitoring [1, 5] and impact detection techniques [1, 6, 7, 8, 9, 10]. The latter covers various methods based on vibration analysis [11, 12, 13, 14] and non-destructive techniques, such X-ray [15, 16], shearography [1, 17], vibro-thermography [18, 19], ultrasonic and acousto-ultrasonic testing [20, 21], and guided ultrasonic waves [22, 23, 24, 25, 26]. Recent years have brought research interest in various damage-related nonlinear phenomena that can be observed in ultrasonic responses and vibration characteristics. It is usually anticipated that the sensitivity of nonlinear approaches is much better than the equivalent linear techniques.

Nonlinear acoustics methods utilise higher, super and sub-harmonic generation, frequency shifting, signal modulations, modulation transfer or hysteretic behaviour. Recent examples in the field—related to composite materials—include studies based on higher harmonics generation [27, 28, 29], nonlinear Lamb waves [30], nonlinear vibro-acoustic wave modulations [31, 32, 33, 34, 35, 36], non-classical, nonlinear modulation transfer [37], analysis of fast/slow dynamics [38] and bispectral analysis [39, 40].

It is well known that in vibration analysis, changes to natural frequencies, mode shapes, curvatures, flexibility coefficients and input-output characteristics (e.g. frequency response function, transfer function) can be used to detect structural damage, as reviewed in [11, 12, 13, 14]. An excellent overview of vibration-based methods—that also cover nonlinear approaches—can be found in [41, 42, 43]. It is important to note that delamination detection in composites—based on vibration analysis—is a challenging task. This is mainly due to the fact that delamination in composites has no parallel to damage mechanisms in other materials [11, 44]. In addition, detection of small delaminations in composite specimens is questionable, as discussed in [1]. Recent examples related to composite materials include studies of nonlinear response characteristics [45], reciprocity analysis [46, 47], curvatures and mode shapes [48], modal filtering [49], nonlinear time series [50], nonlinear interactions [51] and higher-order spectra [40, 52]. The latter is particularly attractive for the detection of small quadratic nonlinearities.

The paper presents the application of the novel triple correlation technique—based on the short-time Fourier transform (STFT)—for damage detection in composite structures. Monitored damaged structures are excited modally using single-input harmonic excitation. Various vibration modes and different amplitude levels of excitations are applied. The proposed higher-order statistic is employed to correlate the fundamental harmonic with higher harmonics that result from structural damage-related nonlinearities. The Fisher criterion is used to establish the optimal excitation amplitude for damage detection.

The paper starts with the theoretical background. Section 2 illustrates simple models to explain nonlinear behaviour of composite delaminated plates. The proposed triple correlation coefficient is briefly described in Sect. 3. The composite specimen used is described in Sect. 4. Numerical simulations performed to select vibration modes for the experimental nonlinear analysis are presented in Sect. 5. This work involves modal analysis and divergence analysis of delaminated plies. The latter is used to establish vibration modes that lead to dominant out-of-plane and in-plane motion of delaminated plies. Section 6 describes the entire experimental work undertaken. Intact and delaminated composite plates are used to illustrate the performance of the method. Damage detection results are presented in Sect. 7. Finally, the paper is concluded in Sect. 8.

## 2 Nonlinear coupling in a composite delaminated plates: theoretical background

Nonlinearity is a common feature in vibration characteristic of nearly all engineering structures. Vibration/modal analysis often exhibits nonlinear symptoms due to material behaviour, boundary conditions, measurement chain or structural damage. The latter is of interest in the current investigations. The physical mechanism behind various types of nonlinearities is manifold and includes global effects—such as imperfection of atomic lattices (e.g. intrinsic or material nonlinearity)—and/or local effects—such as contact interaction between structural elements or structural damage. Material nonlinearity is a subject of many investigations since the early 1960s. Various methods have been developed to detect material imperfections such as for example micro-cracks in materials. Local nonlinearity can produce nonlinear effects in the form of higher, sub- and super-harmonics, frequency mixing and shifting, modulation transfer, slow dynamics effects or reverberation, as discussed in [53].

Simple models presented in this section are sufficient to explain the process of higher harmonics generation. It is important to note that other models—such are asymmetrical dynamic fracture [57], bifurcation [58], discontinuities [59] or contact interaction [60] models—can be also used to explain the physics of investigated nonlinear phenomena. In practice, dynamic behaviour of damaged delaminated plate is much more complicated. A good example of multiple delamination dynamics—based on analytic modelling—is given in [61].

## 3 Triple correlation coefficient based on the short-time Fourier transform

The triple correlation (TC) was firstly investigated in the early 1960s to examine non-Gaussian random processes. The early applications of the technique includes statistical examining of laser spectroscopy and ocean waves [62, 63], fatigue and condition [64, 65] monitoring. The triple correlation is somehow less popular than the standard (i.e. doubled) correlation and is mainly used when multiple observations—embedded in additive noise and corrupted by trends—are present in analysed signals.

*va*r of the complex-value quantity in Eq. (15) is defined as

## 4 Delaminated composite plate

## 5 Numerical simulations

This section describes numerical simulations undertaken to analyse the divergence of delamination and the dynamics of the delaminated composite plate. Firstly, the FE model of the delaminated composite plate is briefly described. Then the divergence and modal analyses performed are explained. Numerical simulations were used to select vibration modes for the nonlinear vibration test described in Sect. 6.

### 5.1 Finite element model of the delaminated plate

*MSC.Patran*FE preprocessor. The plate was discretised using 800000 3-D linear hexahedral elements (Fig. 5). The desired ply stacking sequence— [\(0_{3}/90_{3}\)]s—was reproduced using four elements across the thickness of the plate, so that each finite element represented three plies with the same orientation, as shown in Fig. 5. Two local coordinate systems were created: \((x_{0}, y_{0}, z_{0})\) and \((x_{90}, y_{90}, z_{90})\) to account for different ply orientations. The orthotropic material properties for the \(0^{\circ }\) and \(90^{\circ }\) plies were defined in these local coordinate systems.

### 5.2 Modal analysis

*{u}*is the displacements vector, \(\left\{ { \ddot{u} } \right\} \) is the accelerations vector.

The *MSC.Nastran* FE solver was used to perform computations. The normal modes solution (SOL103) was applied to find vibration mode shapes of the delaminated composite plate in the frequency range from 0 to 1000 Hz.

### 5.3 Delamination divergence analysis

## 6 Experimental vibration tests

This section describes the experimental work undertaken for damage detection. Firstly, experimental modal analysis was performed. The results from modal analysis and the delamination divergence analysis—described in Sect. 5.3—were used to select excitation frequencies for the nonlinear vibration test. This test was performed to reveal damage-related nonlinearities.

### 6.1 Experimental modal analysis

The composite plate was suspended using elastic cords to minimise undesired effects from boundaries. White noise excitation was used in the experimental modal analysis. The excitation signal was generated using a built-in signal generator of the in *Polytec**PSV-400* laser vibrometer and amplified by an *Electronics PAHV 2000* high-voltage amplifier. The excitation was introduced to the plate through a surface-bonded *NOLIAC CMAP04* stack actuator. A *Polytec PSV-400* laser vibrometer was used for non-contact measurement of vibration responses.

*Polytec PSV 8.8*software. The amplitude of the FRF—presented in Fig. 9—displays a series of components corresponding to frequencies of various vibration modes illustrated in Fig. 10. The modal assurance criteria (MAC) coefficient was used to correlate these modes with the results obtained from numerical simulations. The results show that the fourth and fifth vibration modes are the strongest vibration modes investigated.

1st vibration mode (79 Hz)—produces very little or virtually no movement of delaminated plies;

4th vibration mode (334 Hz)—the strongest vibration mode with the dominant out-of-plane motion of delaminated plies;

5th vibration mode (365 Hz)—the second strongest vibration with the dominant in-plane motion of delaminated plies;

8th vibration mode (736 Hz)—produces the largest level of motion of delaminated plies.

### 6.2 Nonlinear vibration test

*Polytec*

*PSV-400*laser vibrometer amplified by an

*Electronics PAHV 2000*high-voltage amplifier. The excitation was introduced to the plate through a surface-bonded

*NOLIAC CMAP04*stack actuator. Vibration responses were acquired using a

*Polytec PSV-400*laser vibrometer and data acquisition system. Measurements were taken in 25 symmetrically and equally spaced locations on the plates. Figure 11a shows the experimental set-up used for damage detection. The measuring grid is illustrated in Fig. 11b.

## 7 Impact damage detection results

The mean value of triple correlation coefficient was calculated for vibration response data from all measuring locations. The 20 s time record of acquired signal was divided into 1 s segments. The Hamming and rectangular windows were employed as internal and external windows, respectively. To avoid possible leakage of information, the 50 % overlapping between segments was used.

The same excitation used for the delaminated plate—that exhibits very little or virtually no relative movement between delamination plies—leads to monotonically increasing values of the triple correlation for the amplitude excitation levels above 40–50 V. However, the triple correlation values are always smaller than 0.2, and the corresponding Fisher criterion is relatively small (always smaller than 12).

The excitation of the strongest fourth vibration mode (Fig. 13c, d) separates the triple correlation curves for the undamaged and damaged composite plate quite well. However, the relevant Fisher criterion coefficient is always smaller than 7 for all amplitude excitation levels, giving very little confidence in the results. Also, the values of triple correlation for the undamaged and damaged composite plate are always smaller than 0.1 and 0.21, respectively, for all analysed excitation amplitudes.

The excitation corresponding to the frequency of the fifth vibration mode (Fig. 13e, f) produces the best damage detection results. The values of triple correlation for the undamaged plate are always smaller than 0.07 for all amplitude levels investigated. The triple correlation increases from the initial value of 0.12 to the maximum value of 0.63 for the excitation amplitude of 10 and 70 V, respectively. Also, the triple correlation curves for the undamaged and damaged composite plate are always well separated, and the relevant values of the Fisher criterion remain relatively large (i.e. larger than 20) when amplitude excitation levels are larger than 35 V. This gives good confidence level in the damage detection results. The largest value 45 of the Fisher criterion coefficient is reached for the excitation amplitude equal to 70 V. It is important to note that similarly to the fourth mode, the fifth mode is one of the two strongest vibration modes investigated. However, in contrast to the fourth mode, the fifth mode is dominated by the in-plane—rather than out-of-plane—motion of delaminated plies.

When the plates are excited using the frequency of the weakest eighth vibration mode (Fig. 13g, h), damage detection results are also very good. The triple correlation curves are well separated for all amplitude excitation levels investigated. The best results are obtained when the plates are excited with amplitudes larger than 60 V. These excitation levels give triple correlation values larger than 0.6 for the damaged plate. The relevant values of the Fisher criterion are always larger than 20 giving the maximum of 58 for the excitation amplitude equal to 70 V. It is important to recall that this excitation produces the strongest in-plane motion of delaminated planes. Interestingly, when the plate is not damaged, the eighth vibration mode produces the largest (monotonically increasing with excitation amplitude) levels of triple correlation.

## 8 Conclusions

Nonlinear effects in vibration responses were studied for the undamaged composite plate and the composite plate with a delamination. These effects were investigated using the triple correlation. The analysis was focused on higher harmonic generation in vibration responses for three different scenarios associated with the movement of delaminated plies, i.e. no motion, out-of-plane and in-plane motions. Various excitation amplitude levels were used in these investigations.

The selection of excitation frequency is important for the generation of higher harmonics that are associated with delamination. The results show that the strongest vibration mode does not need to lead to the strongest nonlinear effect.

Movement of delaminated plies enhances nonlinear effects. This behaviour is particularly observed when weaker vibration modes and smaller excitation amplitudes are used.

The most significant nonlinear effect has been observed for the in-plane motion of the delaminated plies. When delaminated plies produce the out-of-plane motion (or very little motion at all), damage-related nonlinearities are much weaker. This suggests that the nonlinear mechanism of higher harmonics generation due to damage is associated with dissipation (friction and/or hysteresis) rather than with elasticity.

When the plate is delaminated, higher amplitude levels of excitation produce stronger nonlinear effects, as expected. However, when the in-plane motion of delaminated plies is involved, even relatively small amplitude excitation levels lead to relatively strong nonlinearities. Interestingly, large excitation amplitudes lead to nonlinear effects in the undamaged composite plate as expected.

The strongest nonlinear effect for the undamaged plate has been observed when the plate was excited with the large amplitudes of the eighth vibration mode. This effect needs to be explained and requires further investigations.

In summary, the work presented shows that triple correlation can be used effectively not only to reveal nonlinear coupling between the fundamental and higher harmonics, but also to reliably detect relatively small delaminations in impacted composite plates. The work also demonstrates that when the triple correlation is combined with the Fisher criterion analysis, the method can be used to establish the best (or optimal) parameters, i.e. frequencies and amplitudes levels of excitation, leading to more confident damage detection results. Finally, it is also clear that further research modelling and experimental work are required to confirm all the above findings.

## Notes

### Acknowledgments

The work presented in this paper was supported by funding from the research Project No. N501158640, sponsored by the Polish National Science Centre.

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