Low-frequency air-coupled transducer based damage detection in composite materials

In this paper results of simulations of non-contact elastic wave generation in the composite panel based on acoustic to elastic wave transformation are presented. For this purpose simulations of acoustic wave generation and processing are based on the FEM method in COMSOL. Elastic wave generation and propagation are based on the spectral element method (SEM) in the time domain. The SEM model utilises time-varying acoustic pressure distributions calculated in the FEM. The SEM allows to simulate the interactions of elastic waves with the delamination. Damage localization is based on RMS elastic wave energy maps. In this research a panel made of carbon fibre-reinforced polymer composite is investigated. Research related to low-frequency air-coupled transducer (ACT) is presented. The utilisation of low-frequency waves allows for the reduction of the effects of the wave attenuation in composite material. The proposed combination of FEM and SEM gives an efficient tool for the simulation of non-contact wave generation for non-destructive testing analysis.


Introduction
The non-contact measurement method is very useful in the field of non-destructive testing (NDT).There is no need for transducer bonding and surface modification.Different methods are utilized in NDT but very popular is the method which relies on elastic guided waves.Elastic waves are excited in plates, and shells and propagate between parallel surfaces as guided waves.The damage located in the structure is a source of changes in the wave propagation.Analysis of the wave propagation change allows the detection of structural discontinuities such as damage.
One of the non-contact elastic wave generation methods is based on air-coupled transducers (ACT) [1].The ACT generates acoustic waves in the air that are later transformed into elastic waves at the boundary of the air/specimen.In the case of ACT for elastic waves with frequencies from tens of kHz [2], [3] up to hundreds of kHz [1], [4] are utilized.High attenuation of elastic waves in composite materials is observed.To overcome this problem, low frequencies of waves should be utilised [4].
In the case of ACT-based elastic wave generation very important are numerical simulations.They allow us to better understand the phenomena related to acoustic wave generation in the air by ACT as well as the process of acoustic to elastic wave transformation.In the case of modelling of acoustic wave generation and propagation in the air, the finite element method (FEM) is utilised [5].For example, the authors of [5] developed a two-dimensional theoretical model of ACT in COMSOL Multiphysics software and analysed single as well as multiple ACTs.
Very important is also the process of acoustic to elastic wave transformation.To simulate this process and the behaviour of a fully non-contact air-coupled NDT system authors of [5] also employed FEM (COMSOL software).They analysed the elastic wave generation in thin isotropic plates.The whole model was divided into transmission, elastic wave propagation, and reception phase.The authors investigated the dependency of the generated Lamb waves on different transmitter parameters and incidence angle.
On the other hand, the detailed elastic wave propagation analysis allows us to better understand the wave-structure interactions.For this purpose (FEM) [6], [7], the semi-analytical finite element (SAFE) method [8], and the spectral element method (SEM) [9] are utilised.The SEM method in the frequency domain was utilised to model wave propagation in the stiffened structures with different cross-sections [10].The SEM in the time domain was utilised for the investigation of stiffener influence on wave propagation and mode conversion [6].
In this paper, we propose to utilise the simulation of acoustic wave generation in the air based on FEM and elastic wave propagation based on SEM in the time domain.In the case of FEM the COMSOL software is utilised with the model of ACT.Based on the calculated acoustic wavefield the acoustic pressure distribution is extracted.Next, the pressure distribution is converted to loads in the SEM model of the composite panel with delamination.The elastic wave analysis is performed by in-house SEM software developed in MATLAB.It performs parallel computation with the use the graphic processor units (GPUs).The overall methodology is presented in Figure 1.Based on calculated elastic wave propagation damage detection and localisation process is performed based on RMS energy maps.

Numerical models
In this research, we utilised the FEM model for modelling of acoustic wave generation by ACT and SEM model for analysis of elastic wave generation and propagation in the carbon fibre reinforced polymer (CFRP) panel.The panel had dimension 500 mm x 500 mm x 3.9 mm.The specimen consisted of 40 prepreg of uniaxial TDS-75 g/m 2 layers and the IMP503Z epoxy resin.The panel had a unidirectional fibre reinforcement in all layers: [0]40.
In the case of FEM, the model of ACT was developed.A resonant transducer with a frequency of 40 kHz was modelled.The ACT had a diameter of 16 mm and a length of 12 mm.Model of the ACT is visible in In the case of SEM, the model of the panel with the delamination was investigated.Delamination was simulated by the node separation in the SEM model.The excitation signal was in the form of five cycles of sine modulated by Hann window with a carrier frequency of 40 kHz.

Results -acoustic wave propagation
The first step was related to the analysis of acoustic wave generation in the air by ACT using the COMSOL simulations.Results from the simulation were processed in order to obtain the energy map.The root mean square energy map for acoustic waves (  ) could be calculated using the following formula: In Figure 2 the calculated RMS of the acoustic wavefield generated by ACT in the air was presented.This map presents the directional characteristics of generated acoustic waves by ACT.In Figure 3 frame from the acoustic wave propagation in the air was presented.Next, based on the numerical results, the acoustic pressure on the flat surface is extracted.This surface was at a distance of 25 mm from the front side of ACT.This is utilized for applying the loads to the SEM model.Time-varying pressure is integrated over spectral elements and applied to the nodes as the force excitation for the SEM model.
Elastic wave generation as a result of the conversion of acoustic waves is connected to the Snell law.Important is the angle of acoustic wave generation θ (ACT slope).In this research, we investigated the angles θ=0° and θ=30°.In the case of θ=0° the ACT is perpendicular to the specimen and due to this generates the symmetrical wave pattern.The θ=30° is very close to the optimal angle for A0 wave mode generation in the direction across the reinforcing fibres in our specimen.It was observed that the amplitude of generated elastic waves increased for the optimal ACT slope angle calculated based on phase velocities of wave propagation.More information about the influence of the ACT angle on the investigated specimen can be found in [11].
In Figure 4 the RMS of the pressure on the flat surface was presented.In Figure 4a) the pressure distribution for the ACT at angle θ=0° was presented.The symmetrical distribution could be noticed.In Figure 4b) the pressure distribution for the ACT at angle θ=30° was presented.In this case, the ACT is directed towards the right side of the figure.It could be noticed that there is a directivity of pressure distribution related to the direction of ACT.In Figure 4c) the pressure distribution for ACT also at angle θ=30° was presented.However, in this case, the ACT is directed towards the top side of the figure.It could be noticed that there is also a clear directivity of pressure distribution (acoustic waves focusses in the direction corresponding to the ACT head position).Calculated time-varying acoustic pressure wavefields for the mentioned ACT cases were utilized as an excitation in the SEM model.

Results -elastic wave propagation
In Figure 5 frames from animations of elastic wave propagation in the CFRP specimen for three cases of ACT orientation were presented.It needs to be underlined that the CFRP sample had the unidirectional reinforcing fibre orientation.In the results in Figure 5, the fibre orientation is vertical.Propagation of fundamental A0 wave mode is observed according to the previous results presented in [11].The effect of reinforcing fibre orientation could be noticed as the difference in elastic wave velocity in the direction along the fibres and across.This is noticed by the elliptical wavefront shape in Figure 5.In the case of Figure 5 ACT was oriented perpendicular to the specimen (θ=0°).The wave front is elliptical but a symmetrical distribution of elastic wave amplitudes could be noticed.Moreover, it could be noticed that amplitudes of elastic waves are larger in the vertical and horizontal directions.However, larger amplitudes are observed in the vertical direction that correspond to the direction of reinforcing fibres.
In Figure 6 the ACT was oriented at angle θ=30° in the direction towards the right side of the specimen.It could be noticed that elastic waves are focused in this direction.The amplitudes on the right side are larger than on the left.In Figure 7 the ACT was oriented at angle θ=30° in the direction towards the top side of the specimen.It could be noticed that elastic waves are focused in this direction.The amplitudes on the upper side are larger than on the bottom one.
To analyse the influence of the ACT slope angle θ and its direction on the wave generation, the RMS energy of elastic waves was calculated for the three investigated cases.The RMS energy map for elastic waves (  ) could be calculated using the following formula: where: (, , ) -vibration velocity signals (full wavefield), time [s], T -time duration of signal.
For better visualization of the distribution of elastic wave energy, only the first half of time frames were taken for calculation of   .In Figure 8 the time-limited   wave energy maps were presented.In Figure 8a) the   wave energy map for the ACT at θ=0° was presented.Elliptical and symmetric distribution of energy could be observed.In Figure 8b) the RMS energy map for the ACT at θ=30°, directed towards the right was presented.It could be noticed that elastic wave energy is focused in the direction on the right side of the panel.There is significantly lower energy on the left side of the panel than on the right.In the case of Figure 8c) the   waves energy map for the ACT at θ=30° directed towards the top was presented.The elastic wave energy is focused in the direction towards the top side of the panel.There is significantly lower energy on the bottom side of the panel than on the top.The next step is related to the analysis of the influence of elastic wave energy focusing on the delamination detection performance.

Results -delamination localisation
In this section results of delamination detection and localization are analysed.In Figure 9 the investigated specimen with the dimensions and location of delamination is presented.Delamiantion has dimensions 20 mm x 20 mm.Three investigated ACT orientations are considered.The   wave energy calculated for the whole length of signals (all time frames) was investigated.The results in the form of   wave energy maps are presented in Figure 10.In the case of Figure 10a) the   map for the case of ACT at θ=0° was presented.The location of delamination could be noticed as the square region with elastic wave energy concentration.Moreover, there is difference in energy distribution behind the delamination (on the right side).The energy concentration could be better visible in the magnified plot of   energy in Figure 11a).
In the case of Figure 10b) the   wave energy map for the case of ACT at θ=30° directed towards the right was presented.The location of delamination could be also noticed as the square region with energy concentration.There is also a difference in energy distribution behind the delamination (on the right).A similar energy concentration could be observed in the magnified plot of RMS energy in Figure 11b).However, a slightly larger amplitude was achieved than in the previous case.
In the case of Figure 10c) the   wave energy map for the case of ACT at θ=30° directed towards the top was presented.The location of delamination could be also easily noticed.Moreover, there is also a difference in energy distribution behind the delamination (on the right).The energy concentration in the delamination region (Figure 11c) is similar to previous cases.However, a slightly lower amplitude was achieved than in the previous two cases.It could be noticed that elastic wave energy concentration in delaminated regions is similar for the three investigated cases (Figure 11).The case of ACT at θ=30° directed towards the right (Figure 11c) is optimal from the point of view of the effectiveness of A0 mode generation due to the selected ACT slope angle in a direction across the reinforcing fibres of the specimen [11].Moreover, in this case, ACT is directed towards the delamination.This causes the fact that waves are focused in the direction of delamination.However, there was lowest   amplitude for the case of optimal ACT angle (Figure 11c) comparing to two previous cases (Figure 11a, b).The largest   amplitude were achieved in the case of ACT at θ=30° directed towards the top.This could be caused due to the wave generation in the direction of reinforcing fibres.In the case of the ACT at θ=30° directed towards the right, waves were generated in the direction across the fibres.Due to the symmetric distribution of wave energy in the case of ACT at θ=0° there could be the best choice for the elastic wave generation in the investigated case of low-frequency waves (40 kHz).In the case of higher frequencies, the influence of ACT slope angle on the wave generation and damage localization results needs to be further investigated.

Figure 4 .
Extracted   acoustic pressure distribution for the ACT at an angle: a) 0°perpendicular, b) 30°towards the right, c) 30°towards the top; numerical results (COMSOL).

Figure 5 .
Figure 5. Elastic wave propagation in the panel for the ACT angle: 0° perpendicular; numerical results (SEM).

Figure 6 .
Figure 6.Elastic wave propagation in the panel for the ACT angle: 30° towards the right (across fibres); numerical results (SEM).

Figure 7 .
Figure 7. Elastic wave propagation in the panel for the ACT angle: 30° towards the top (along fibres); numerical results (SEM).

Figure 8 .
Figure 8.The time-limited   energy maps for the ACT at an angle: a) 0°, b) 30°towards the right (across fibres), c) 30°towards the top (along fibres); numerical results (SEM).

7th
International Conference of Engineering Against Failure Journal of Physics: Conference Series 2692 (2024) 012026