1. Introduction
Contemporary technologies are employing composite plate materials developed by using various innovative materials (nanostructures, mica structures, etc.). Such materials are being used in the fields of electronics, specifically, in heating, lighting and electricity control devices and mechanics including the aerospace industry wherever exceptional physical properties are required, e.g., in compasses, rockets, oxygen breathing equipment, and hearing aids. In many cases, the manufacturing technologies do not allow employing contact measurement methods for quality control of the manufactured products [
1,
2,
3]. However, ultrasonic methods enable investigation of key properties of materials that cannot be assessed by merely using traditional methods. For these purposes, ultrasonic guided waves propagating within the interior of the objects under investigation may be employed. The guided waves are already used for non-destructive testing and evaluation (NDT) of sheet-type materials. Usually, they are excited when the vibrations of an acoustic transducer are transferred to the investigated item by a direct contact via a narrow gap of liquid. However, contact liquids are impossible to use in many cases as the investigated item may be contaminated, damaged or otherwise negatively affected. This is relevant in the cases of paper, high-density polyethylene (HDPE), membranes, GFRP, GLARE, CFRP and other composite structures [
4,
5].
The guided waves in sheet-type materials may be excited via an air gap in two different ways depending on the ratio of propagation velocities of ultrasonic waves in the material and air [
6,
7]. When the ultrasound velocity in air is lower than the phase speed of the guided waves in the object, the optimal incident angle of the ultrasonic wave propagating from the air onto the plate exists at which the biggest amplitude of the excited guided wave is obtained. Then, the propagating guided wave in the plate material radiates a leaky wave into air which may be used for air-coupled reception [
8]. This mode is commonly applied for detecting defects inside composite materials as the leakage is increasing at the defective zone. The attenuation of acoustic waves in air and in the object under investigation increases with the frequency. The losses of ultrasonic signals may be reduced by generating guided waves while using lower frequencies. We used novel air-coupled ultrasonic techniques based on the excitation of guided waves or by phased arrays at lower frequencies, or by deflected ultrasonic transducers at higher frequencies depending if the phase velocity of the mode is higher or lower than the velocity in air. It should be remarked that both methods of the excitation and reception of guided waves are innovative; neither of them has been explored yet [
9,
10].
Usually, for non-destructive evaluation, zero-order Lamb wave antisymmetric A
0 and symmetric S
0 modes are used [
4,
11,
12]. However, application of higher-order modes could allow better detection and characterization of defects characteristic of planar plastic and composite structures. There are already some publications describing application of higher-order guided waves for detection of wall thickness reduction and crack-like defects, however only in metallic structures [
13,
14,
15,
16]. For example, P. Khalili and P. Cawley analyzed symmetric S
0 mode at ~1.5 MHz-mm and shear horizontal SH
0 and SH
1 modes at ~3 MHz-mm in the 10 mm thick steel plate and determined that they can be suitable for detection of shallow defects [
15]. Authors for that exploited SH
1 waves, but parasitic SH
0 was excited as well, and this can complicate the inspection process and, therefore, limit defect detectability. Authors showed that the SH
1 mode was sensitive to gradual and smaller defects down to 10% of wall thickness loss [
15]. In another paper, P. Khalili and F. Cegla presented research focused on application of the SH
1 wave for inspection of the 10 mm thick aluminum plate with crack-like defects [
16]. By means of analytical predictions and finite element modeling, it was shown that a signal dominated by the SH
1 mode can be generated using a single electromagnetic acoustic transducer with a permanent periodic magnet. It was also shown that, by studying the reflection coefficient of the SH
1 mode, the pure SH
1 mode can be used to detect defects as shallow as a 5% thickness loss from a 500 mm standoff. Other authors in their articles examined the spectra of acoustic oscillations generated by interdigital transducers in a plate made of LiNbO
3 piezo crystals. They found that along with the zeroth and higher-order modes, this spectrum also contains odd harmonics of the same modes. When the velocities are close, the amplitude–frequency response modes and harmonics can be too close and even interfere with each other, distorting the entire spectrum of acoustic vibrations. The authors suggested that plates up to three wavelengths thick and transducers with bandwidths up to 5% and operating frequencies below the first harmonic are most suitable for practical use [
17,
18]. Other authors studied the influence of thin interlayers on the behavior of elastic guided waves in isotropic laminate structures, paying particular attention to the application of the obtained results to determine the mechanical properties of such sublayers and to the evaluation of the integrity of the interlayer contact using a laser Doppler vibrometer [
19,
20,
21,
22]. The authors claimed that the use of effective boundary conditions can explain the observed behavior of high-order elastic guided waves in the considered laminate structures (i.e., emergence of mode pairs, mapping of closed-form boundary frequencies, etc.). Unlike these authors, our aim was to apply ultrasonic air-coupled transducers to excite and obtain higher-order Lamb wave modes.
The presented examples deal with non-destructive testing of metallic and ceramics samples, elastic properties of which are very different from other plastic and composite materials. Excitation and reception of higher-order modes in such materials also should be different. The excitation process depends on the spatial distribution of the surface displacements of the sample, which should be excited by an ultrasonic transducer.
Symmetric modes possess two displacement components: vertical and longitudinal. The amplitude of the longitudinal component at low frequencies is many times greater than the amplitude of the vertical component on the surface of the sample. It means that excitation of symmetric modes is also more complicated than antisymmetric modes in composite materials. Asymmetric modes are more versatile because the amplitudes of the vertical and longitudinal displacement components are of a similar order. It allows exciting such modes using both methods: the contact and the air-coupled.
In the case of the air-coupled excitation, it is necessary to find the frequency range and the angle of incidence of the ultrasonic wave in air, at which the required guided wave mode in the plastic sample is optimally excited. Those parameters can be determined from dispersion curves of guided waves propagating in the object under test. In most cases, the dispersion curves for particular materials are not known in advance, and therefore they should be calculated or measured beforehand. For that, elastic constant of the materials is necessary to know, which usually is found from the measured phase velocities in investigated plastic or composite materials [
23,
24,
25,
26,
27,
28]. It is necessary to point out that in the case of materials with high attenuation, phase velocities are frequency-dependent even in bulk specimens, and therefore methods based on reconstruction of the phase velocity dispersion curves should be used [
17,
18]. In our case, attenuation is not very high, and therefore the elastic constants of the materials were determined using methods described in [
25,
26,
27,
28].
However, until now, majority of publications have been devoted to the analysis of application of zero-order antisymmetric modes for detection of various defects in composite, metal and plastic materials. Application of the higher-order Lamb wave modes through an air gap is very limited because their selective excitation and reception are much more complicated. For this purpose, air-coupled ultrasonic transducers can be applied; however, at the moment, they are mainly exploited for the excitation and reception of the lowest Lamb wave modes. Thus, there is a lack of research demonstrating excitation of high-order asymmetric and symmetric modes for defect detection in various composite materials.
The objective of the paper is investigation of the excitation and reception of higher-order guided Lamb wave modes in plastic and composite plates.
The paper is organized as follows.
Section 2 reviews the properties of higher-order guided wave modes and presents the theoretical background. The experimental investigations performed by contacts and air-coupled methods are reported in
Section 3. Conclusions and discussion can be found in
Section 4.
2. Properties of Higher-Order Guided Wave Modes
To analyze the physics of higher-order guided Lamb wave modes, we chose high-density polyethylene (HDPE). It is prepared from ethylene by a catalytic process [
10,
29]. It possesses the absence of branching resulting in a more closely packed structure with a higher density and somewhat higher chemical resistance than LDPE. Compared to engineering plastics, it has lower thermal and mechanical properties, such as tensile stress and flexural and compressive strength. Compared to high-molecular-weight PE, it is more rigid, and its resistance to continuous shock is therefore lower [
30,
31].
PE-300 is also somewhat harder and opaquer, and it can withstand rather higher temperatures (120° Celsius for short periods, 110° Celsius continuously). High-density polyethylene lends itself particularly well to blow molding. HDPE Polyethylene 300 has the following properties: rigidity, resilience, durability, chemical and moisture tolerance, gas permeability, ease of processing and ease of formation. PE-300 plastic is used for wastewater treatment plants, for production of various tanks, water slides, petroleum and chemical receptacles, shelves, and partitions and for the bottling industry, including production of snow arena guard rails, playgrounds, supporting/guard bars, security systems for motorcycles, quadricycles, motor vehicles, etc. It is easy to weld or form by heat treatment. The low friction coefficient and its non-hygroscopicity make it suitable for use as a bearing or mechanical parts with low loads, even when operating in water [
32,
33,
34].
To know the distributions of higher-order modes in those materials, one must first know their exact viscoelastic properties, which were measured at the Physical Acoustics department (APY), Bordeaux, France. Dimensions of the sample selected for measurements were 200 × 200 mm. The experimental equipment and set-up are presented in
Figure 1 and
Figure 2. The experimental set-up consists of a LeCroy 9450A Dual 300 kHz oscilloscope (LeCroy, Geneva, Switzerland), Olympus 5077PR square wave pulse/receiver (Bucharest, Romania), Newport motion controller model MM4006 (Champaign, IL, USA), V 305 2.25 MHz/0.75 111,280 unfocussed immersion transmitter ultrasonic transducer (Sofranel, Sartrouville, France) and V 305 2.25 MHz/0.75″ 11,282 receiver ultrasonic transducer (Sofranel, Sartrouville, France). The diameter of the ultrasonic transducers is 0.75″, receiver attenuation is −40 dB, and excitation voltage is
Uex = 100 V. Two different materials were experimentally investigated: high-density polyethylene (HDPE) and glass fiber (GFRP) composite material. Before measuring the viscoelastic properties, dimensions of the samples were measured, and the volumes of the samples were calculated. The samples were weighed on a scale: the weight of the polyethylene (HDPE) sample is 237.30 g, and the weight of the glass fiber (GFRP) sample is 407.31 g. The sample densities were calculated using the measurement data. The measured viscoelastic properties and elastic parameters of the HDPE sample are presented in
Table 1 and
Table 2. The calculated densities are given in the corresponding tables (
Table 2 and Table 4). The viscoelastic properties of HDPE and GFRP material were measured in the water tank, the water temperature was 23 °C, and scanning was performed from 0° to 72° by step 2°. The GFRP material was measured using 1 MHz ultrasonic transducers. The features and appearance of the ultrasonic transducers are the same as the 2.25 MHz ultrasonic transducers. The used measurement method is widely known and described in the following articles [
25,
26,
27,
28,
33,
35].
The dispersion curves were calculated using two different methods: the semi-analytical finite element (SAFE) method (Institute of Ultrasound, Kaunas, Lithuania) and the PROPAG software (APY Department, Bordeaux, France). The dispersion curves were verified by experimental investigations. They were carried out on a PE-300 specimen with clearly expressed properties, which were measured in the APY department. In this sample, the antisymmetric modes A
0 and A
1 and symmetric S
0 and S
1 modes were excited and signals recorded; the measurement investigations of the PE-300 sample were performed by the ULTRALAB system (Institute of Ultrasound, Kaunas) using the experimental set-up presented in the previous section. The experimental set-up was the same as used for investigation of the PE-300 sample and measurements for glass fiber (GFRP) composite material. Pultruded glass fiber reinforced polymer (GFRP) has light weight, good strength and excellent resistance to corrosion [
36,
37]. The GFRP material was measured using 1 MHz ultrasonic transducers. The measured viscoelastic properties and elastic parameters of the HDPE sample are presented in
Table 1 and
Table 2. The measured viscoelastic properties and elastic parameters of the GFRP composite material are presented in
Table 3 and
Table 4. Young’s modulus and Poisson’s ratio are given with the accuracy of 10
−4 to compare the two different calculation methods of the dispersion curves. Note that this accuracy is only used for calculation to get the best possible match of results.
The measurements showed that the PE-300 sample is an isometric material, and then the other viscoelastic properties are C
22 = C
11, C
33 = C
11, C
44 = C
66, C
55 = C
66, C
12 = C
11 − 2 × C
66, C
13 = C
12, C
23 = C
12. The elastic parameters of the PE-300 sample are presented in
Table 2. The elastic parameters were calculated of the measured viscoelastic properties in the HDPE sample. From the measured results follows that HDPE material is isotropic material and GFRP material is anisotropic material. Knowing the exact elastic properties of the material, we can calculate the phase and group velocities of the modes and the mode distributions.
The calculations were performed by two different methods: using the semi-analytical finite element (SAFE) method (Ultrasound Institute, Kaunas) and compared with the simulated dispersion curves in the APY department (Bordeaux, France). The simulated dispersion curves of the phase and group velocities, for the 4 mm HDPE plate, are shown in
Figure 3. In
Figure 3a are presented phase velocities and in
Figure 3b group velocities of various modes. The black solid lines show velocities simulated by the SAFE method in the Ultrasound Institute, Kaunas, and the various color dots indicate velocities obtained in the APY department, Bordeaux, using the PROPAG software [
27,
38,
39,
40]. The presented results demonstrate a very good correspondence of the results obtained by two different methods.
The dispersion curves of the Lamb waves were calculated using the semi-analytical finite element (SAFE) method [
41]. The SAFE method can be used not only for isotropic materials but also for layered anisotropic materials. The dispersion curves of the phase and group velocities, for the 4 mm GFRP plate, are shown in
Figure 4, respectively. The direction used in calculation by the SAFE method for anisotropic GFRP material was XY.
Knowing the phase velocities of the sample from the dispersion curves, it is possible to determine at what angle of incidence the higher-order modes should be excited, for example, the antisymmetric mode A
1 and symmetric mode S
1. Let us take a closer look at the HDPE sample. Usually, for air-coupled excitation and reception of the antisymmetric Lamb wave modes, ultrasonic transducers are oriented to the plate structure under test at the optimum incidence angle
θi. There is the frequency-dependent phase velocity of the guided Lamb wave modes in the structure under a sample. The value of the optimum incidence angle
θi according to Snell’s law (Equation (1)) depends on the ratio of the phase velocities
Vair/Vph(f). For the case when the ultrasound phase velocity in the structure is greater than or equal to the sound velocity in air
Vph(f) ≥
Vair, the incidence angle
θi can be calculated as shown in Equation (1) [
36]:
where
Vph is the phase velocity of the selected mode in the HDPE plate found from the dispersion curves, and
Vair is the velocity of sound in air. The ultrasound velocity in air at 21 °C was 343.21 m/s. From the dispersion curves, the following values were obtained for the 4 mm HDPE plate at 300 kHz excitation frequency: the phase velocity of
A1 mode is 1508 m/s, and the group velocity is 756 m/s. Using the phase velocity, the angle of incidence of
θi = 22.96° was calculated using Snell’s law. The wavelength in air at 300 kHz was 1.4 mm, the wavelength of the
A0 mode in the HDPE plate at 300 kHz was 2.93 mm, the wavelength of the
A1 mode was 5.02 mm, and the wavelength of the
S0 mode was 3.12 mm. The distributions of particle displacements across the 4 mm thickness of the sample at the 300 kHz of the antisymmetric modes
A0 and
A1 and symmetric
S0 mode are presented in
Figure 5,
Figure 6 and
Figure 7. The amplitude values in
Figure 7e are normalized with respect to the maximum displacement values.
The antisymmetric modes A
0 and A
1 and symmetric S
0 mode distributions of particle displacements across the 4 mm thickness of the sample by 300 kHz are presented in
Figure 5,
Figure 6 and
Figure 7. From the presented results follows that in those modes, there are two components: vertical and longitudinal (tangential). In this paper, we present both components. The normal component of the antisymmetric modes
A0 and
A1 has the same sign on both sides, with a large amplitude on the surface. The normal component of the symmetrical
S0 mode has opposite signs on both sides and a large amplitude on the surface. Due to this fact, it can be assumed that at 300 kHz frequency, the
A0,
A1 and
S0 modes can be excited through the air gap with ultrasonic air-coupled transducers. In the next
Section 3, we present the experimental investigation performed by two different methods: contact and air-coupled methods.
4. Conclusions and Discussion
Various modes of guided waves are already used for non-destructive testing and evaluation of composite structures. It is possible to expect that higher-order symmetric and antisymmetric modes may improve detection and characterization of various defects in plastic and composite materials. For that, first of all, efficient techniques for the excitation and reception of such modes are necessary. The aim of this work was to study the excitation and reception of higher-order guided Lamb wave modes in plastic and composite plates. For excitation of such modes, it is necessary to select the proper frequencies, which depend on the mode type, thickness of the sample and elastic properties of the material. If all those parameters are known, then the necessary frequencies can be found from the dispersion curves. However, in many cases, elastic properties of the material are not known with the necessary accuracy; therefore, we proposed at the very beginning to measure elastic constants and to use them for the calculation of accurate dispersion curves. Viscoelastic properties were measured at the Department of Physical Acoustics (APY), Bordeaux, France. The dimensions of the plastic and composite samples selected for measurements were 200 × 200 mm. Dispersion curves were calculated using two different methods: the semi-analytical finite element (SAFE) method (Ultrasound Institute, Kaunas) and the PROPAG software (APY department, Bordeaux). The theoretical calculations were performed using the semi-analytical finite element method, which made it possible to find out the frequency ranges in which higher-order modes can be excited. Theoretical calculations showed that it is possible to excite higher-order modes at 300 kHz. The dispersion curves also allow selected frequencies to determine phase and group velocities of particular modes.
To excite and receive efficiently higher-order modes through an air gap, the ultrasonic wave in air must be directed to the sample at the angle of incident which depends on the phase velocity of the mode to be excited. It is necessary to keep in mind that phase velocities depend not only on the mode type but also on the frequency used to excite the required mode. Those velocities are found from the calculated (or measured) dispersion curves and are used to determine the deflection angle of the air-coupled ultrasonic transducer to get the necessary angle of the incidence.
It is necessary to point out that in the frequency range where higher-order modes may exist, usually not a single but a few different modes may be excited. It may complicate identification of what modes actually were excited. For identification, we proposed the following identification criteria: the phase velocity cph and the angle of incidence θi at which the maximal amplitude of the required mode is obtained.
During experimental investigations, we checked whether the calculated dispersion curves correspond to the experimental ones and found a good agreement. Those dispersion curves were exploited for determination of the frequencies and incident angles necessary for the excitation and reception of the required higher-order Lamb wave modes. The experimental investigations were performed by two different methods: a contact method using point-type ultrasonic transducers with 0.1 mm aperture and an air-coupled method using air-coupled ultrasonic transducers. Higher-order guided Lamb wave modes were excited and received by the contact ultrasonic transducers operating in the frequency range from 50 kHz to 350 kHz. The amplitudes of higher-order guided Lamb wave modes were measured by the ULTRALAB system at different distances from the excitation region, and in such a way, a B-scan was obtained. The phase velocities were determined from the B-scan applying a 2D spatial and temporal spectrum analysis. The experimental results obtained by the contact method demonstrate that it is feasible to excite higher-order A1 and S1 modes in plastic and composite materials.
For practical applications, more attractive is the air-coupled method; therefore, other experiments directed to the excitation and reception of higher-order modes were carried out using this method. Experiments were performed by one pair of air-coupled ultrasonic transducers manufactured at the Ultrasound Institute of Kaunas University of Technology (UI, KTU). The pair consisted of single-element unfocused air-coupled transducers with an operation frequency of 300 kHz. The obtained results show that two higher-order modes, such as the antisymmetric mode A1 and the symmetric mode S1, were excited. Higher-order modes were identified by the angle of incidence, which was calculated using Snell’s law and the phase and group velocities of the excited modes. In the investigated plastic HDPE plate with 4 mm thickness at the excitation frequency of 300 kHz, the following higher-order A1 and S1 modes were observed. The antisymmetric A1 mode was excited by the air-coupled ultrasonic wave incident at the angle θi = 22.96°. The phase velocity of the excited A1 guided wave was 1508 m/s. Correspondingly, the symmetric S1 mode was excited at the angle of incidence θi = 16.85°. The phase velocity of the guided S1 wave was 2046 m/s. The measured velocities correspond well to the velocities of the corresponding modes in the calculated dispersion curves.
The obtained results clearly demonstrate the possibility to excite and receive efficiently different higher-order guided Lamb wave modes with very different phase velocities. The presented comparison of the experimental results with the simulation results showed a good agreement. The combination of air-coupled excitation and non-contact reception enables a non-destructive evaluation and characterization of moving plastic objects and composite structures. In the future, it is planned to perform measurements in samples of complex geometry to identify and investigate various defects in such specimens using air-coupled phased arrays with novel piezoelectric materials.