Introduction

The extensive utilisation of PET plastic bottles has given rise to resource and environmental concerns in recent years. To mitigate these issues, one approach is to design bottles with optimised material and mechanical property distributions. This strategy aims to minimise unnecessary material usage in production while meeting performance requirements during usage. Another option involves incorporating recycled PET (rPET) in bottle manufacturing. However, the quality of rPET varies significantly due to factors like feedstock sources, sorting methods, collection practices, and recycling techniques. Successfully integrating rPET into bottle production systems necessitates adapting existing systems to accommodate varying grades of rPET.

In this context, the measurement of mechanical properties in produced bottles proves invaluable. It serves the dual purpose of achieving optimal mechanical property distributions in PET bottles and facilitating quality monitoring throughout the production of rPET bottles. This proactive approach contributes to sustainable practices and aligns with the broader goal of minimising the environmental impact associated with PET bottle manufacturing.

Plastic bottles are typically produced using the stretch blow moulding (SBM) process, where a preform, which is a test-tube-shaped specimen, is first heated above its glass transition temperature. It is then stretched in its axial direction by a stretching rod and in the hoop direction by pressurised air to form a bottle in the mould. Due to the complex biaxial deformation behaviour of PET material in the SBM process, biaxial stretching tests of PET sheets are often used in a laboratory environment as a more controllable way to investigate the effects of different recyclable contents and process conditions (such as temperature, rate of stretch, and stretch ratio) on the room temperature mechanical properties of formed bottles [1]. In such tests, efficient methods for measuring mechanical properties can significantly reduce the time required to evaluate new materials and conditions.

However, currently, there are only a limited number of techniques available for measuring the mechanical properties of polymer sheets. Tensile testing, as one of the most conventional and widely accepted methods for measuring mechanical properties, has been used for a long time in polymer characterisation [2, 3]. Apart from Young’s modulus, Nguyen et al. [4] also identified Poisson’s ratio and shear modulus from heterogeneous tensile tests carried out on PET samples cut from plastic bottles, using digital image correlation [5] combined with the virtual field method [6]. For thin polymer films where traditional tensile tests are no longer feasible, the indentation test serves as an alternative method. For example, Du et al. [7] identified the Young’s modulus of polycarbonate (PC) and polystyrene (PS) films with a thickness of \({10} \ \mu \text {m}\) by using atomic force microscopy (AFM) indentation measurements. The bulge test is another frequently used method for characterising polymer materials. For instance, Lin et al. [8] retrieved the Poisson’s ratio, biaxial modulus, and residual stress of polyimide (PI) films from their 3D membrane deformation distribution obtained via fringe projections in bulge tests.

In addition to the quasi-static testing methods described above, non-destructive dynamic methods have also been developed to measure the mechanical properties of polymer materials under specific conditions, such as high temperature and high pressure. Lotfalian et al. [9] examined the correlation between Young’s modulus of a polymer layer and the changes in resonance angle in a surface plasmon resonance structure. They also conducted practical experiments to measure Young’s modulus of silicone rubber. Ma et al. [10] calculated the dependence of natural frequencies of polymer capsules on Young’s modulus and Poisson’s ratio numerically. They then developed a combined method of resonant ultrasound spectroscopy (RUS) and laser-based resonant ultrasound spectroscopy (LRUS) to determine the elastic modulus of the capsules. Yamamoto et al. [11] determined the dynamic Young’s moduli and dam** ratios of polymer composites by analysing the absolute value of Fourier spectra and the natural frequencies obtained from impulse excitation tests.

The ultrasonic measurement method, as a non-destructive technique, has found wide applications in identifying the mechanical properties of various materials. The application of conventional ultrasonic testing for material characterisation relies on the transmission of bulk waves, which exist in infinite homogeneous bodies and propagate indefinitely without being interrupted by boundaries or interfaces. One of the most common techniques is to cut cubes of the material and measure the speed of bulk waves in different directions. Analytical formulas are then used to invert the collected wave speed information to determine the effective elastic constants of the material [12].

Another widely used ultrasonic characterisation method is based on the application of guided Lamb waves. Unlike bulk waves, guided Lamb waves propagate through thin plates and are the result of the reflection, refraction, and mode conversion of bulk waves from the surfaces of the plate. During propagation, guided Lamb waves can cover the entire thickness of a plate or shell structure over a considerable distance, in contrast to the limited area covered by ultrasonic bulk waves just below the transducer [13]. Consequently, in bulk wave inspection, the transducer needs to be moved along the surface to collect data, whereas, in the application of guided Lamb waves, a single probe position is sufficient to inspect the structure. Guided Lamb waves are extensively employed for identifying property parameters of various materials, especially isotropic metals such as steel and aluminium. These metals have well-documented material properties that are readily available [14,15,16]. In recent years, researchers have also advocated for the use of this technique in the study of other materials. For example, Dahmen et al. [17] measured phase velocity dispersion curves of Lamb waves propagating in olive wood using air-coupled transducers and obtained seven out of nine orthotropic elastic constants of the material. Kažys et al. [18] identified both Young’s modulus and thickness of thin plastic polyvinylchloride (PVC) films by analysing the phase velocities of both A0 and S0 Lamb wave modes. Cui et al. [19] characterised the elastic properties of unidirectional, quasi-isotropic, and anisotropic fibre-reinforced composite laminates using dispersion curve matching of relevant guided wave modes and simulated annealing optimisation.

From the review of the literature above, it is evident that the majority of current methods for measuring the mechanical properties of polymer materials are destructive and necessitate specific sample geometries, thereby limiting their efficiency. On the contrary, as a non-destructive technique, the ultrasonic measurement method based on guided Lamb waves has found wide applications in in-situ structural health monitoring and material characterisation. However, there are few studies currently devoted to its application on polymer materials. Therefore, this research aims to develop an approach based on the application of guided Lamb waves to achieve rapid and accurate identification of the elastic modulus of PET sheets. This paper is organised as follows: "Semi-analytical Finite Element Method" describes a numerical method used to calculate the dispersion curves of PET sheets. In "Methodology", the methodology of the proposed approach is described. In "Experimental Investigation", the experimental setup and modulus measurement results using the proposed approach are presented. Conclusions are given in "Conclusions".

Semi-analytical Finite Element Method

Dispersion curves are a set of curves that represent the propagation of ultrasonic wave modes in a specific material. These curves are determined by the geometric information and material property parameters. Therefore, they can be considered as a distinctive fingerprint that contains information about the material. Hence, these curves can be interpreted using specific methods to obtain the underlying property parameters of the material. To calculate the dispersion curves in different waveguides, various modelling methods have been proposed from decades ago, including stiffness matrix method [20], transfer matrix method [21] and spectral collocation method [22], etc. Among them, the semi-analytical finite element method (SAFEM) has been widely utilised due to its advantages in solving arbitrary cross-section waveguide problems [23]. For example, Kalkowski et al. [24] modelled elastic waves propagating in fluid-filled embedded/submerged pipes using SAFEM with high-order spectral elements. In the work of Hayashi et al. [25], the SAFEM was used to calculate the phase velocity, group velocity, and displacement of guided waves in a square bar and rail as well. Bartoli et al. [23] extended this technique to account for material dam** and applied the SAFEM to calculate dispersion curves for anisotropic viscoelastic layered plates, composite-to-composite adhesive joints, and railroad tracks.

In this paper, the SAFEM is used to calculate the numerical group velocity dispersion curves for PET sheets. For the problem of plane wave propagation in a plate, a one-dimensional discretisation across the plate thickness is sufficient. The plate thickness, denoted as h, is discretised into layers that are modelled using quasi 3-node 1D elements while an analytical solution is adopted in the wave propagation direction which reduces the computational cost. The quadratic shape functions for the three-node line element are as follows:

$$\begin{aligned} \begin{aligned} N_1&=\frac{l^2-l}{2} \\ N_2&=1-l^2 \\ N_3&=\frac{l^2+l}{2} \end{aligned} \end{aligned}$$
(1)

where l is the variable in the local coordinate system for the element. And the local coordinates of the 3 nodes are \(l=-1\), \(l=0\), and \(l=1\), respectively.

The derivation of SAFEM formulation is based on the principle of virtual work for deformable elastic bodies without external traction:

$$\begin{aligned} \int _{V} \delta \textbf{u}^{\top } \rho \ddot{\textbf{u}} \textrm{d} V+\int _{V} \delta \varvec{\epsilon }^{\top } \varvec{\sigma } \textrm{d} V=0 \end{aligned}$$
(2)

where V is the volume of the medium, \(\textbf{u}\) is the displacement vector, \(\rho\) is the mass density, \(\ddot{\bullet }\) serves as the second derivative with respect to time, \(\varvec{\epsilon }\) and \(\varvec{\sigma }\) are the strain and stress vector, respectively.

Combining the time-harmonic assumption and the finite element discretisation, the particle displacements of any point in an element can be written as:

$$\begin{aligned} \textbf{u}^{(e)}(x, y, z, t)=\textbf{N}(y, z) \textbf{q}^{(e)} \cdot e ^{i(\xi x-\omega t)} \end{aligned}$$
(3)

where \(\textbf{N}(y, z)\) is the shape function matrix:

$$\begin{aligned} \textbf{N}(y, z)=\left[ \begin{array}{ccccccccc} N_1 &{} 0 &{} 0 &{} N_2 &{} 0 &{} 0 &{} N_3 &{} 0 &{} 0 \\ 0 &{} N_1 &{} 0 &{} 0 &{} N_2 &{} 0 &{} 0 &{} N_3 &{} 0 \\ 0 &{} 0 &{} N_1 &{} 0 &{} 0 &{} N_2 &{} 0 &{} 0 &{} N_3 \end{array}\right] \end{aligned}$$
(4)

\(\textbf{q}^{(e)}\) is the element nodal displacement vector:

$$\begin{aligned} \textbf{q}^{(e)}=\left[ \begin{array}{lllllllll} U_{x 1}&U_{y 1}&U_{z 1}&U_{x 2}&U_{y 2}&U_{z 2}&U_{x 3}&U_{y 3}&U_{z 3} \end{array}\right] ^{\textrm{T}} \end{aligned}$$
(5)

\(\xi\) and \(\omega\) are the wavenumber and angular frequency, respectively.

The corresponding strain and stress vectors can be obtained from the following equations:

$$\begin{aligned} \varvec{\varepsilon }^{(e)}=\left[ \textbf{L}_x \frac{\partial }{\partial x}+\textbf{L}_y \frac{\partial }{\partial y}+\textbf{L}_z \frac{\partial }{\partial z}\right] \textbf{u}^{(e)} \end{aligned}$$
(6)
$$\begin{aligned} \varvec{\sigma }^{(e)}=\textbf{C}^{(e)} \varvec{\varepsilon }^{(e)} \end{aligned}$$
(7)

where

$$\begin{aligned} \textbf{L}_x=\left[ \begin{array}{lll} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \end{array}\right] , \quad \textbf{L}_y=\left[ \begin{array}{lll} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 0 \\ 1 &{} 0 &{} 0 \end{array}\right] , \quad \textbf{L}_z=\left[ \begin{array}{lll} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 \\ 0 &{} 1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \end{array}\right] \end{aligned}$$
(8)

and \(\textbf{C}^{(e)}\) is the elementary stiffness matrix.

In typical biaxial stretching tests, PET sheets are usually deformed in two modes: equibiaxial (EB) deformation and constant width (CW) deformation [26]. In EB deformation, the square specimen is simultaneously and uniformly deformed in both directions, resulting in an isotropic material property. In contrast, in CW deformation, the specimen is only deformed in the X direction (machine direction) and not in the Y direction (transverse direction), resulting in an orthotropic material property. Without loss of generality, the calculation of dispersion curves in the following identification steps assumes an orthotropic material property for stretched PET sheets, without considering dam**.

Substituting equations (3), (6) and (7) into equation (2), the following eigenvalue problem can be obtained after proper matrix manipulations (see [23] for more details):

$$\begin{aligned} \left[ \textbf{K}_{1}+\xi \hat{\textbf{K}}_{2}+\xi ^{2} \textbf{K}_{3}-\omega ^{2} \textbf{M}\right] _{M} \hat{\textbf{U}}=0 \end{aligned}$$
(9)

where \(\textbf{K}_{1}\), \(\textbf{K}_{2}\) and \(\textbf{K}_{3}\) are stiffness matrices, \(\textbf{M}\) is the mass matrix and \(\hat{\textbf{U}}\) is the global nodal displacement vector, the subscript M is the number of total degrees of freedom (dof) of the system. By assigning real values to \(\xi\), equation (9) can be solved as a standard eigenvalue problem in \(\omega (\xi )\). Group velocity dispersion curves can then be calculated from its definition:

$$\begin{aligned} c_{g}=\frac{d \omega }{d \xi } \end{aligned}$$
(10)

Figure 1 shows the group velocity dispersion curves calculated by using the above SAFEM and typical material property parameters and thickness of an isotropic PET sheet obtained from an equibiaxial stretch of a square specimen. The parameters used in this calculation are shown in Table 1. Here, the product of frequency and thickness is used as the unit of the horizontal axis to show the influence of both geometry and frequency on the group velocity of Lamb waves. In this way, the results can be scaled linearly. For example, these curves can represent wave modes in a 0.4 mm thick plate in the frequency range up to 5 MHz, or equally, for a 0.2 mm thick plate in the frequency range up to 10 MHz [27].

As can be seen from Fig. 1, there are two types of Lamb wave modes: symmetric modes presented in red and anti-symmetric modes presented in blue. The labels adjacent to the curves indicate the names of the wave modes. The first letter in the name is either A or S, depending on whether the particle displacements are symmetric or anti-symmetric about the centre line of the plate. The second part of the identifier is the mode number, which distinguishes between modes within each type. The symmetric and anti-symmetric zero-order modes (S0 and A0 modes) deserve special attention because they exist at all frequencies and in most practical situations they carry more energy than the higher-order modes.

Fig. 1
figure 1

The group velocity dispersion curves of PET material

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Table 1 Parameters used for group velocity calculation
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Sensitivity Analysis

In order to select the proper wave mode to be used in the Young’s modulus identification procedure, a sensitivity analysis was performed first. Only the two fundamental zero-order (A0 and S0) modes were involved in this analysis considering their relatively simple wave structure which makes the excitation and detection easier. The influence of nine orthotropic elastic engineering constants on group velocity dispersion curves of PET sheets was studied separately. The initial values of these constants are given in Table 2. It was assumed that the variation range of each parameter relative to the initial value is ±30%. The material density of PET was set as 1300 \(\mathrm {kg/m^3}\) and the thickness of the material was 0.3 mm. The dispersion curves for guided waves propagating in the principal direction 1 and 2 (the axes of material symmetry) were then calculated using SAFEM described above, and the results are shown in Figs. 2 and 3 in which blue curves show the changes of dispersion curves caused by the increase of the corresponding elastic constants while the red curves present the changes of dispersion curves caused by the decrease of them.

Table 2 Initial values of nine elastic constants used in the sensitivity analysis; Units of modulus [GPa]
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As can be seen from Figs. 2 and 3, the group velocity of the S0 and A0 mode in the principal direction 1 is only influenced by \(E_{11}\), \(E_{22}\), \(v_{12}\) and \(G_{13}\) while that in the principal direction 2 is only influenced by \(E_{11}\), \(E_{22}\), \(v_{12}\) and \(G_{23}\). Another feature is, compared with the A0 mode, the S0 mode is more sensitive to the changes of elastic constants except for the two shear moduli \(G_{13}\) and \(G_{23}\). In the meantime, the S0 mode is nearly non-dispersive in the studied frequency range (0-500 kHz) which is a conducive feature for its application. On the one hand, it will facilitate signal processing and improve the accuracy of group velocity measurement in practical experiments as the waveform will not be distorted after propagating a specific distance. On the other hand, only one data point needs to be measured from experiments to represent the entire S0 group velocity dispersion curve, which provides the basis for fast and efficient modulus measurement. Based on the above analysis, the S0 mode was selected as the wave mode to be used in the following identification procedures. Furthermore, it is evident that the S0 mode is most sensitive to the changes of \(E_{11}\) in the principal direction 1 and \(E_{22}\) in the principal direction 2. Therefore, Young’s moduli in these two principal directions were selected as the potential material parameters to be identified in later steps.

Fig. 2
figure 2

The influence of the variation of nine elastic constants on the dispersion curves for the principal direction 1

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Fig. 3
figure 3

The influence of the variation of nine elastic constants on the dispersion curves for the principal direction 2

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Methodology

Based on SAFEM for the rapid calculation of the numerical group velocity of the S0 mode Lamb waves, an optimisation method can be proposed to identify Young’s modulus of the material. A flowchart of the proposed method with the main steps is present in Fig. 4. The input material parameters of the SAFEM model are updated in the iterations until the difference between numerical group velocity and experimental group velocity is minimised, which gives the sought values of the material parameters.

Fig. 4
figure 4

The flowchart of the proposed method for the property identification

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Extraction Method of Group Velocity

As can be seen from Fig. 4, an important step in the proposed method is the measurement of group velocity from experiments. This is also an indispensable step for any ultrasonic measurement technique based on the application of dispersion curves. Over recent years, different methods have been developed to extract dispersion curves from experimental measurements. The most widely used technique to measure dispersion curves is the 2D fast Fourier transform (2D-FFT) [28], which can convert the time and position information in measured data to frequency and wavenumber information. However, this technique requires the acquisition of signals at many measuring points along the wave propagation path to get a good resolution on the frequency and wavenumber scale. Apart from this method, different time-frequency analysis (TFA) methods are also used for the determination of dispersion curves, such as the short-time Fourier transform (STFT), the wavelet transform (WT) and the Wigner-Ville distribution (WVD) [29]. However, these TFA methods often suffer from some drawbacks which limit their application to specific cases [30].

Here, after considering the feasibility and simplicity in practical experiments, a method (termed as cross-correlation method) proposed by Crespo et al. [31] was used to extract group velocity information from collected signals, which is based on Hilbert transform and cross-correlation method. This method only requires two signals measured at two different positions for the calculation of group velocity which provides a basis for fast property measurements. The signal envelopes are first obtained using Hilbert transform and then the cross-correlation algorithm is used to measure the similarity of these two envelopes as a function of the time delay between them. The time of flight (TOF) between two measurement points is subsequently identified as the time delay that maximises the cross-correlation value. Although Draudviliene et al. [32] mentioned in their work that this method is sensitive to mode-shape distortions, this drawback was overcome here since only the nearly non-dispersive S0 mode Lamb waves were generated in practical experiments, which will be shown in the following "Experimental Investigation". At the same time, high-mode waves were not generated in the frequency range of interest (0-500 kHz) for the typical material property of PET material.

Particle Swarm Optimisation

After measuring the group velocity of guided waves propagating in PET sheets, an inverse analysis was carried out to match the experimental group velocities (\(V^{exp}\)) to the numerical group velocities (\(V^{pred}\)) calculated by the SAFEM by iterating the engineering constants, as shown in Fig. 4. Different algorithms have been used to tackle optimisation problems in guided wave applications, such as the genetic algorithm [33], the simulated annealing algorithm [34] and the Nelder-Mead simplex algorithm [35]. Here, a particle swarm optimisation (PSO) algorithm was employed for this inversion analysis due to its capacity to handle non-linear multivariate optimisations [36] and fast convergence rate.

In the optimisation, a vector composed of Young’s modulus \(E_{11}\), \(E_{22}\) and Poisson’s ratio \(v_{12}\) was treated as the position vector (\(\textbf{x}_{i}\)) of particles since it can be seen from Figs. 2 and 3 only these three parameters affect the group velocity of S0 mode Lamb waves. Therefore, in the optimisation, they were updated in the iterations while the other material parameters were assumed as constants. The objective function to be minimised was defined as the mean square error (MSE) between the experimental group velocity (\(V^{exp}\)) and the SAFEM calculated group velocity (\(V^{pred}\)), which can be expressed as:

$$\begin{aligned} F(\textbf{x}_{i})=\frac{1}{n_{f}} \sum _{i=1}^{n_{f}} (V^{exp}_{f_{i}}-V^{pred}_{f_{i}}(\textbf{x}_{i}))^{2} \end{aligned}$$
(11)

where \(f_{i}\) is the frequency point at which the group velocity is evaluated, and \(n_{f}\) is the number of total frequency points.

Independence on Thickness Measurement

As described above, material density and thickness were assumed to be known values in the optimisation process which means their values should be measured in the practical application of the proposed method. However, as demonstrated in Fig. 1, the group velocity dispersion curve of S0 mode Lamb waves is nearly non-dispersive when the product of thickness and frequency is in the range of 0\(\sim\)0.2 MHz\(\cdot\)mm. In general, the thickness of stretched PET sheets is in the range of 0\(\sim\)0.3 mm. On the other hand, the frequency of interest in this study is within 500 kHz. The above analysis indicates that the application of the proposed method is within the approximately non-dispersive region of the dispersion curve of S0 mode in which the group velocity has little dependence on the material thickness. To validate this, the group velocity dispersion curves of S0 mode Lamb waves in an isotropic PET sheet with Young’s modulus of 5000 MPa were calculated using SAFEM with different thickness values. As shown in Fig. 5, in the frequency range of 0\(\sim\)500 kHz, the variation of group velocity is less than 10 m/s when the thickness varies from 0.1 mm to 0.3 mm, and this variation decreases with lower frequency. To get an understanding of the influence of a group velocity variation of 10 m/s on Young’s modulus identification result, group velocity values 2055 m/s and 2045 m/s were input into the optimisation algorithm respectively while the other parameters were kept the same. The output Young’s moduli were 5013 MPa and 4965 MPa respectively, which shows a difference of 48 MPa and a percentage error of 1%.

Fig. 5
figure 5

The influence of thickness on group velocity

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The insensitivity of the group velocity to the material thickness indicates that thickness information is not required with rigorous accuracy in the inverse analysis for Young’s modulus estimation. This would also further improve the efficiency of the proposed method, as the material thickness can be assumed to be a constant value in the inverse analysis while having little effect on the accuracy of Young’s modulus estimation results.

Experimental Investigation

Experimental Setup

An ultrasonic measurement system was established to validate the proposed Young’s modulus identification method, as shown in Fig. 6. In this measurement system, the USB-UT350T (US Ultratek, Inc.) ultrasonic tone burst pulser/receiver was used to generate and collect voltage signals. Three PZT discs (STEINER & MARTINS, Inc.) were used in each test. Among them, one was used for ultrasonic signal excitation and the other two were used to receive signals at two different positions in the propagation path. The switch between the two receiving PZT discs was implemented by a multiplexer (hasseb Inc.). A LabVIEW program was designed for device control and signal processing purposes.

Fig. 6
figure 6

The ultrasonic measurement system

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Accuracy of the Group Velocity Measurement

Tests were first carried out to verify the accuracy of the group velocity measurement by the proposed method. In these tests, a PET sheet was used as the test sample, which was biaxially stretched under a temperature of 100 \(^{\circ }\)C, a stretch ratio of 3 and a strain rate of 16 \(\mathrm {s^{-1}}\). Since the EB deformation mode was used in the biaxial stretching, this PET sheet can be considered to have isotropic material properties.

Before the tests, the attaching positions of the transmitting PZT and two receiving PZTs were marked on the sample first, as shown in Fig. 7(a). During the test, the transmitting PZT and the receiving PZT 1 were fixed while the receiving PZT 2 was moved away from PZT 1 so as to vary the wave propagation distance and study the effect of wave propagation distance on the calculated group velocity. Three wave propagation distances (20 mm, 30 mm and 40 mm) were used in the tests, as shown in Fig. 7(b). Apart from this, measurements were conducted on three different positions as shown by the three red lines in Fig. 7(a), so as to verify the consistency of the measurements.

Fig. 7
figure 7

PZT disc arrangement in the accuracy test

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PZTs were attached to the sheet with the help of a coupling agent (SWC-2, Olympus Corp). The excitation voltage was set to 40 V, and the sampling rate was 50 MHz. Figure 8 shows the signals received from two receiving PZTs when the distance between them is 20 mm. As seen in the figure, the amplitude of the signal received by PZT 2 is much lower than that of PZT 1 due to the effect of material dam**. However, the waveform and its envelope are not distorted after propagating the distance between the two PZTs. This result proves that only S0 mode Lamb waves were generated during the test, as A0 mode Lamb waves are dispersive in the studied frequency range and would distort the collected waveform. This feature also contributes to the accuracy of TOF calculation by the cross-correlation method since this method seeks the time shift that maximises the similarity of two signals, and it is obviously easier to compare signals without shape distortions. Combining the known wave propagation distance and calculated TOF, the group velocity can then be determined.

Fig. 8
figure 8

The signals received at two adjacent measurement points on a PET sheet

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The group velocity measurement was carried out with two different types of PZTs: one with a central frequency of 300 kHz (\(\Phi 7 \times 0.2\) mm) and the other with a central frequency of 450 kHz (\(\Phi 5 \times 0.4\) mm). The same test procedures were used for both, and the group velocities at these two frequencies were calculated for comparison purposes. Table 3 presents the group velocity measurement results from this accuracy test. As seen in the table, the discrepancies between group velocities measured at different wave propagation distances and positions on the sheet are considerably small. This high consistency in group velocity measurement indicates the material property homogeneity of the sample and the repeatability of the proposed method. Another noteworthy feature is that the group velocities measured at 300 kHz and 450 kHz frequencies are essentially the same, consistent with the earlier conclusion that the S0 mode is nearly non-dispersive in the studied frequency range.

Table 3 Measured group velocities from the accuracy test
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Inverse analyses were then conducted using the measured group velocities. Three inversion analyses were performed: the first two with single group velocity data measured at 300 kHz and 450 kHz respectively as the input and the third one with both as inputs. This was done to investigate the effect of the number of frequency data points on the accuracy of Young’s modulus measurement. The values of Young’s modulus identified from the three cases were 5689 MPa, 5727 MPa, and 5717 MPa, respectively. The small differences (less than 1%) between them were caused by the nearly non-dispersive feature of the S0 mode Lamb waves in the studied frequency range. It can be concluded that the benefit of adopting multiple frequency points in the inverse analysis is quite limited compared to the efficiency improvement achieved by using only one frequency point. Thus, only the frequency point of 450 kHz was used in the subsequent measurements.

Tests on PET Sheets

To validate the proposed method for measuring Young’s modulus of PET sheets, biaxial stretching tests were conducted to produce 8 PET sheets under various combinations of stretching process conditions, including temperature, strain rate, and stretch ratio. This allowed us to generate stretched PET sheets with diverse mechanical properties, as the use of different stretching process conditions alters the final microstructure of the produced PET sheets, leading to variations in mechanical properties [3]. Subsequently, the proposed method was employed to measure Young’s modulus of these samples and assess its capability to detect differences in Young’s modulus. The specific combinations of process conditions utilised are detailed in Table 4.

Table 4 Process conditions of PET sheets
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Same as the test procedures used in "Accuracy of the Group Velocity Measurement", the group velocity was measured from three positions on each sample, as shown in Fig. 7(a). However, measurements with only one wave propagation distance of 40 mm were performed here. The average value of the group velocities measured from three positions was used as the final group velocity for Young’s modulus identification. Figure 9 shows the group velocity measurement results, the identified Young’s modulus and the tensile modulus of these samples for validation purposes. Tensile test specimens were cut from the central part of the stretched PET sheets. And the tensile modulus was measured according to International Standard ISO 527-3 which specifies the conditions for determining the tensile properties of plastic films or sheets less than 1 mm thick. The test speed was kept constant at 1 mm/min. Young’s modulus was determined from the slope of the stress–strain curve in the strain interval between 0.05%\(-\)0.25%.

Fig. 9
figure 9

Comparison between tensile modulus and identified modulus

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As shown in Fig. 9, three curves all present the same changing trend which indicates that the influence of process conditions on the resulting Young’s modulus of PET sheets are captured by the tensile modulus and identified modulus. Take Sample 4 as an example, it was produced under a low temperature, a high strain rate and a high stretch ratio. With these process conditions, theoretically, Sample 4 should have the highest modulus among all these samples since these conditions all contribute to a greater degree of orientation of the polymer chains which will result in a higher stiffness [37], and this maximum modulus was captured by its tensile modulus and the identified modulus by the proposed method, as shown in Fig. 9.

However, another obvious feature that can be seen from Fig. 9 is that there is a gap between the black and red curve, which means the modulus of the samples is overestimated by the proposed method, with a minimum percentage error of 7.80% for Sample 8 and a maximum percentage error of 35.95% for Sample 5. The overestimation of elastic modulus measured by dynamic methods is widely reported in studies on other materials as well. For instance, in the work of Fjær et al. [38], potential reasons for the higher dynamic modulus of rocks compared to their static modulus were reviewed, including factors such as strain rate and amplitude of the measurement method, as well as the heterogeneities and anisotropy of rocks. Similarly, in the work of Nasir et al. [39] and Fathi et al. [40], this overestimation in dynamic modulus was also observed for woods, attributed to the viscoelastic behaviour and attenuation of wood. Regarding polymer materials, Ong et al. [41] reported similar results where Young’s moduli of an epoxy adhesive measured by three different ultrasonic methods were all 60% higher than those measured from quasi-static testing. In addition, in the work of Lagakos et al. [42], the elastic moduli of a number of commercially available polymers were proved to be frequency dependent. This frequency dependence was thought to stem from the viscoelastic behaviour and molecular relaxation process of polymeric materials. In quasi-static tests, the molecular chains of the material have enough time to relax, which contributes to its viscous behaviour and leads to a lower modulus. On the contrary, when the loading frequency is high, the molecular chains do not have enough time to relax, and then the viscous response of the material is lost. This in turn results in the material behaving more like an elastic solid with a higher modulus.

Dynamic Mechanical Analysis on PET Sheets

Although the discrepancy between dynamic and static modulus is widely reported for different materials, few studies investigated the reasons for this discrepancy. Here, to further validate the identified Young’s modulus by the proposed method and the frequency dependence of the modulus of PET material, dynamic mechanical analysis (DMA) was carried out on PET sheets to study their viscoelastic behaviour. It should be noted that the load frequency of DMA is much lower compared to that of ultrasonic testing, usually with an upper limit of 100 Hz. Therefore, in order to extend the frequency range of the frequency-dependent analysis to cover the frequencies of the ultrasonic testing, the time-temperature superposition (TTS) principle [43] was used to convert the material characteristics at low temperatures and low frequency to those at room temperature and high frequency.

DMA was conducted in tension mode on a rectangular PET film cut from Sample 7 under load frequencies ranging from 0.1 Hz to 10 Hz. Before each test, the test chamber temperature was cooled down to -30 \(\mathrm {^{\circ }C}\) and then gradually increased to 40 \(\mathrm {^{\circ }C}\) during the test. The storage moduli of the sample under increasing temperature were recorded. The test result of Sample 7 is shown in Fig. 10. From the figure, it can be seen that the storage modulus of the material shows a significant decreasing trend with increasing temperature. Another feature is that the storage modulus increases when the loading frequency increases and this feature is more pronounced at low temperatures. This result is consistent with the previous description of the frequency dependence of the modulus.

Fig. 10
figure 10

DMA results of Sample 7

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In order to obtain the modulus of the material at higher loading frequencies using the TTS principle, the modulus curves in Fig. 10 were interpolated every 2 \(^{\circ }\)C in the interval from -26 \(^{\circ }\)C to 38 \(^{\circ }\)C and the storage modulus was plotted with the logarithmic frequency. The final data obtained is shown in Fig. 11. The storage modulus - logarithmic frequency curve at 20 \(^{\circ }\)C was then chosen as the reference curve, the curves above this temperature were shifted to its left and the curves below this temperature were shifted to its right to construct a smooth master curve. The finally obtained master curve is shown in Fig. 12. From this master curve, the storage modulus of Sample 7 can be obtained at any given frequency. The comparison of the modulus of Sample 7 measured by three different methods is shown in Table 5. The percentage error between the tensile modulus and the value measured by the proposed method is 15.5% while the percentage error between the low frequency (0.001 Hz) storage modulus and high frequency (450 kHz) storage modulus is 12.8%. This small percentage error difference can be explained by the slight heterogeneity in the material and different load amplitudes and deformation rates of these methods. Nevertheless, it is clear from the master curve that the storage modulus of the material increases with the increasing loading frequency, which explains the modulus overestimation caused by using the ultrasonic method proposed in this paper.

Fig. 11
figure 11

Storage modulus versus logarithmic frequency at different temperatures

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Fig. 12
figure 12

The master curve constructed by the TTS principle

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Table 5 Comparison of the modulus of Sample 7 measured by three different methods (Unit: MPa)
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Conversion from Dynamic Modulus to Static Modulus

Compared to the dynamic modulus, the tensile modulus is more commonly used as an index to represent the stiffness of a material. Therefore, in order to make the modulus measurements more readily available for further performance analysis, it is necessary to convert the measured dynamic modulus to static modulus. As an empirical method, the linear regression method is the most widely used method to establish the relationship between static modulus measured by traditional methods and dynamic modulus measured from novel methods [44,45,46]. In this study, a total of 39 samples of biaxially stretched PET sheets formed under different process conditions were used, 30 of which were used to build a linear regression model and the other 9 samples were used to test if the model is able to convert dynamic modulus to a value which is comparable with tensile modulus.

The relationship between stretched PET sheets’ dynamic and tensile modulus is shown in Fig. 13. It is evident that there is a strong linear relationship between them and the coefficient of determination is as high as 0.91. The comparison of the identified modulus, tensile modulus and corrected identified modulus based on this relationship of the 9 samples is demonstrated in Fig. 14. It can be seen that the error between the corrected identified modulus and tensile modulus is much smaller than that between the original identified modulus and tensile modulus. Before the correction, the maximum and minimum absolute errors were 1045 MPa and 650 MPa respectively, and the maximum and minimum percentage errors were 31.86% and 14.53%. After the correction, the maximum and minimum absolute errors became 335 MPa and 11 MPa and the maximum and minimum percentage errors became 10.21% and 0.24%, which is within an acceptable level for the measurement of Young’s modulus for PET sheets.

Fig. 13
figure 13

The linear regression model

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Fig. 14
figure 14

Measurement result correction based on the linear regression model

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This result clearly indicates that the method proposed in this paper can be used to measure Young’s modulus of PET sheets combined with the linear regression method. This provides an effective tool to evaluate the change in material modulus caused by the variance of raw material or process conditions in a non-destructive way.

Conclusions

In this paper, an approach was developed to identify Young’s modulus for stretched PET sheets using the group velocity of propagating Lamb waves. The sensitivity analysis confirmed that the group velocity of the zero-order symmetric Lamb waves is most dependent on Young’s modulus. The cross-correlation method was used to extract the group velocity information, and the PSO algorithm successfully retrieved the material’s Young’s modulus. The method was validated through actual experiments, demonstrating high accuracy in measuring the group velocity of Lamb waves. The identified modulus differed from the tensile modulus due to frequency dependence but could be corrected using a linear regression model. The proposed method is non-destructive, efficient, and has the potential for inline property measurement and quality control in bottle manufacturing. Future work will focus on applying this method to PET bottles directly