Investigation of stress effect on combinational harmonics generation of low-frequency S 0 Lamb wave

Abstract

This study experimentally and numerically explores the effect of initial stresses on the generation of combinational harmonics due to the nonlinear mixing of low-frequency $S_0$ Lamb waves. The experimental study considers different levels of uniaxial tensile stress. The results indicate that the sum combinational harmonic has a much higher sensitivity to the prestress level, compared to the corresponding acoustoelastic effect. In the numerical study, material nonlinearity is modelled through a 5-constant hyperelastic constitutive equation derived from Murnaghan’s strain energy function. The finite element (FE) simulation results for a quasi-one-dimensional FE model agree with analytical results for bulk-wave mixing, thereby validating the two-step computational procedure for simulating wave propagation in the presence of a prestress. The same procedure is applied to a three-dimensional FE model to investigate different biaxial prestress conditions. The results indicate the feasibility of using the nonlinearity parameter derived from measurements of combinational harmonics to identify the principal stress directions and magnitudes in biaxially prestressed structures, provided that measurements are carried out for a sufficient sampling of wave propagation directions.

Introduction

Metallic plate-like structures have been extensively used in different types of high-value engineering assets such as aircraft, warehouses and oil tanks. These high-value assets operate in various environments and are typically subjected to loading cycles with different magnitudes. The service life of structures can be significantly affected by changes in the service environment, such as temperature changes which can be characterized as changes in initial stress. ¹ Therefore, it is crucial to monitor such stress changes during operation.

Conventional stress monitoring methods, such as bulk-wave inspection techniques, ² strain gauges and digital image correlation (DIC) ³ can only monitor a limited area of the structure and often require downtime for instrumentation and access to the critical area of the component. The latter can impose significant costs and reduce the efficiency of the operation.

Over the last decades, guided wave inspection techniques have been widely used in the field of structural health monitoring (SHM) and nondestructive evaluation (NDE). Early studies of guided waves were based on linear features, such as the time of flight (ToF), the amplitudes of reflected and/or transmitted waves from various forms of damage.^4–7 However, the linear features of the guided waves have relatively limited sensitivity to microscale defects or material degradation that characterize early-fatigue damage, and they are not sensitive to initial stress ⁸ or pre-stress.

In recent years, many studies have focused on non-linear guided-wave techniques, which provide an improved sensitivity for detecting and monitoring microscale defects and material degradation characteristic of early stage damage, ⁹ based on the non-linear features associated with material non-linearities ¹⁰ and contact acoustic nonlinearity (CAN). ¹¹ The generation of higher harmonics is one of the most commonly used non-linear features.^12,13 However, one of thottlenecks that limits the practical implementation of higher harmonics is the influence of unavoidable non-damage-induced non-linearities, such as instrumentation non-linearity and background noise. In addition, imperfect contact between the actuating/receiving transducer and the specimen could also contribute to a certain level of non-linearities due to CAN. ¹⁴ Distinguishing unavoidable non-damage-induced non-linearities from damage-induced nonlinearities is necessary in practical applications, but this is very challenging.

To address the limitation of higher harmonics generation, the generation of combinational harmonics due to ultrasonic wave mixing phenomenon has attracted significant research interest, which is immune to non-damage induced non-linearities. Li et al. ¹⁵ investigated the effect of synchronism on the combinational harmonic generation associated with two collinear Lamb waves mixing. Hasanian and Lissenden ¹⁶ studied the condition of internal resonance of the mixing of ultrasonic-guided waves in a plate. Compared with non-collinear and counter propagating guided wave mixing techniques, collinear-guided wave mixing technique provides a relatively large size of mixing zone, which is beneficial for defects detection/monitoring in a large-scale area. On the other hand, combinational harmonics have been used to assess different types of defects, such as low-speed impact damage, ¹⁷ localized material degradation, ¹⁸ fatigue cracks,^19,20 delamination^21,22 and debonding. ²³ Hughes et al. ²⁴ investigated wave mixing of edge-guided waves, and found that the amplitudes of the second harmonics are small when excited as part of a dual-frequency input, relative to separate single-frequency inputs, suggesting that energy is being transferred to the combination harmonic frequency. They also found that the amplitude of the sum frequency combinational harmonic exhibited the clearest trend of a linear increase with propagation distance compared with the second harmonics and difference frequency combinational harmonic, thereby indicating a key advantage of wave mixing for measurement of material nonlinearity.

The present work aims to investigate the effect of uniaxial and biaxial prestress on combinational harmonics arising from the non-linear mixing of Lamb waves due to dual frequency inputs. In contrast, most recent studies on the effect of stress on non-linear responses of guided waves have mainly focused on the second harmonic,^25–27 while the research on combinational harmonics ²⁸ remains limited. The objective of the present work is to assess the performance of the sum combinational harmonic generated by Lamb-wave mixing to monitor the stress state in plate-like structures. This study will include a comparison with the measured results based on the acoustoelastic effect, which was investigated for stress monitoring in the literature.^29,30 The outcomes of the current study can be very helpful for improving the accuracy of current and future SHM systems.

The article is structured as follows: Section ‘Theoretical background’ summarizes some analytical results for bulk-wave mixing in the presence of prestress based on a 5-constant hyperelastic constitutive equation derived from Murnaghan’s strain energy function. These results provide a benchmark for validating the computational procedure employed later, as well as providing a guide to what can be expected for Lamb wave mixing in the presence of initial stress, for which analytical results are not currently available. An experimental study employing a uniaxial tensile stress is presented in section ‘Experimental evaluation’, and a computational study of Lamb wave mixing in the presence of a biaxial prestress is presented in section ‘Numerical case study’. Finally, the findings and conclusions of this study are summarized in section ‘Discussion and conclusions’.

Theoretical background

This section presents analytical results for bulk-wave mixing in a prestressed medium, including the associated change in wave speed and the relative non-linearity parameter ${\beta }^{\prime }$ . Combinational harmonics arise from the interaction of multiple incident ultrasonic waves as illustrated schematically in Figure 1, where the input signal consists of two incident waves at the fundamental central frequencies of $f_A$ and $f_B$ . When these incident waves propagate in a non-linear medium, the second harmonics ( $2f_A$ and $2f_B$ ) of the fundamental frequencies, as well as combinational harmonics at the difference ( $f_B$  −  $f_A$ ) and sum ( $f_B$  +  $f_A$ ) frequencies are generated as a consequence of material non-linearity. This phenomenon can be observed in the frequency spectrum by converting the collected time domain signal using the fast Fourier transform (FFT) as shown in Figure 1.

Figure 1.

Schematic of generation of combinational harmonics due to ultrasonic waves mixing.

The relative non-linearity parameter ${\beta }^{\prime }$ can be calculated from the amplitude of the combinational harmonic at the sum frequency, $C_{\mathrm{sum}}^{\prime }$ , as follows ³¹ :

$$\begin{array}{c}{\beta }^{\prime }=\frac{C_{\mathrm{sum}}^{\prime }}{A_1^{\prime }·B_1^{\prime }}\hfill \end{array}$$

(1)

where $A_1^{\prime }$ and $B_1^{\prime }$ are the amplitudes of the inputs at the primary excitation frequencies $f_A$ and $f_B$ , respectively.

Equation of motion

The deformation of a material is described by the deformation gradient tensor $\mathit{F}$ , which relates each material point in the initial undeformed configuration $\mathit{X}$ to its current deformed configuration $\mathit{x}$ . ³² The stress state of a material may be represented using several equivalent stress tensors. The Cauchy stress tensor $\mathbf{\sigma }$ describes the stress relative to the current deformed configuration, while the first Piola–Kirchhoff stress tensor $\mathit{P}$ transforms the stress tensor from the current configuration back to the initial undeformed configuration. The relationships are defined as follows:

$$\mathit{F}=\frac{\partial \mathit{x}}{\partial \mathit{X}},J=det\mathit{F}={\rho }_0/\rho ,\mathit{P}=J\mathbf{\sigma }{\mathit{F}}^{-T}$$

(2)

where $J$ is the determinant of the Jacobian, which represents the volume change due to the deformation, is equal to the ratio of the density in the initial configuration ${\rho }_0$ to the density in the deformed configuration $\rho$ .

In the presence of an initial stress, the total deformation experienced by the material can be regarded as a combination of individual deformations. ³³ We can consider a material that first undergoes a deformation described by the tensor ${\mathit{F}}_1$ followed by a subsequent deformation described by the tensor ${\mathit{F}}_2$ . The overall determinant of the Jacobian of the combined deformation gradient $J$ is the product of the individual Jacobian determinants due to the individual deformations as shown below

$$\mathit{F}={\mathit{F}}_2{\mathit{F}}_1,J=\det {\mathit{F}}_2\det {\mathit{F}}_1=J_2{J}_1$$

(3)

Using Equation (3), the Piola–Kirchhoff stress referred to the basis of the initially stressed state, denoted ${\mathit{P}}^{\left(2\right)}$ , is

$${\mathit{P}}^{\left(2\right)}=J_1^{-1}{\mathit{PF}}_1^{\mathit{T}}=J_2{\mathit{\sigma }F}_2^{-\mathit{T}}$$

(4)

The equation of motion (without body forces) relative to the initially stressed state is

$$\mathrm{div}{\mathit{P}}^{\left(2\right)}=\frac{{\rho }_0}{J_1}\frac{{\partial }^2\mathit{\chi }}{\partial t^2}$$

(5)

where $\mathbf{\chi }$ represents the position field mapping material points from the initial to current configuration. It corresponds to ${\mathit{F}}_2$ written as a function of the coordinates of the initially stressed state $\mathit{x}$ .

Non-linear wave propagation in initially stressed media

If the initial stress is uniform, the right stretch tensor ${\mathit{U}}_1$ corresponding to the deformation gradient tensor of the initial stress state, ${\mathit{F}}_1$ can be represented as a diagonal matrix with the principal stretches as its diagonal elements as follows:

$${\mathit{U}}_1=\mathrm{diag}({\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2,{\mathrm{\Lambda }}_3),J_1={\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3$$

(6)

This diagonal form indicates that the deformation is aligned with the principal axes, and there is no shear component in the initial stress state. A longitudinal plane wave is subsequently imposed upon this prestressed medium. The associated deformation is a one-dimensional (1D) motion

$$\mathbf{\chi }(x,t)=x+\eta u_1(x,t)+{\eta }^2{u}_2(x,t)+\dots$$

(7)

with all other components zero and where $\eta$ ≪1 represents a small parameter to be used in the subsequent perturbation analysis, representing the magnitude of the first-order displacement. The perturbation solution relies upon the assumption that the displacement remains small in comparison to unity. The first- and second-order displacements $u_1(x,t)$ and $u_2(x,t)$ are time-harmonic waves.

A hyperelastic material characterized by the Murnaghan strain energy function can be written as

$$W\left(\mathit{E}\right)=\frac{1}{2}(\lambda +2\mu )i_1^2-2\mu i_2+\frac{1}{3}(l+m)i_1^2-2m{i}_1{i}_2+n{i}_3$$

(8)

where $\mathbf{E}$ is the Green–Lagrange strain tensor. $\lambda$ and $\mu$ are Lame constants or the second-order elastic constants. $l,m,n$ are Murnaghan constants or the third-order elastic constants. $i_1=\mathrm{tr}(\mathbf{E})$ , $i_2=\frac{1}{2}[i_1^2-\mathrm{tr}{\left(\mathit{E}\right)}^2]$ , $i_3=det(\mathit{E})$ are strain invariants. The corresponding stress components ${\mathit{P}}^{(2)}$ can be calculated via Equation (4), leading to

$$P_{11}^{\left(2\right)}={\alpha }_1\frac{\partial \chi }{\partial x}+{\alpha }_3{\left(\frac{\partial \chi }{\partial x}\right)}^3+{\alpha }_5{\left(\frac{\partial \chi }{\partial x}\right)}^5$$

(9)

$$\begin{array}{c}\multicolumn{1}{c}{{\alpha }_1=\frac{{\mathrm{\Lambda }}_1}{4{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3}[2\lambda ({\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-3)-4\mu +l{({\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-3)}^2}\\ \multicolumn{1}{c}{+2m({\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-{\mathrm{\Lambda }}_2^2{\mathrm{\Lambda }}_3^2)+n(({\mathrm{\Lambda }}_2^2-1)({\mathrm{\Lambda }}_3^2-1)]}\end{array}$$

(10)

$${\alpha }_3=\frac{{\mathrm{\Lambda }}_1^3}{2{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3}[\lambda +2\mu +l({\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-3)-2m]$$

(11)

$$\begin{array}{c}{\alpha }_5=\frac{{\mathrm{\Lambda }}_1^5}{4{\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_3}(l+2m)\hfill \end{array}$$

(12)

where the coefficients ${\alpha }_1$ , ${\alpha }_3$ and ${\alpha }_5$ are functions of the material properties ( $\lambda ,\mu ,l,m,n)$ and the principal stretches $({\mathrm{\Lambda }}_1,{\mathrm{\Lambda }}_2,{\mathrm{\Lambda }}_3)$ . The equation governing 1D wave motion is given by

$$\frac{\partial P_{11}^{\left(2\right)}}{\partial x}=\frac{{\rho }_0}{J_1}\frac{{\partial }^2\chi }{\partial t^2}$$

(13)

Acoustoelastic effect on wave speed

Considering the first-order displacement consisting of a mixed wave containing the two frequencies ${\omega }_A$ ( $=2\mathrm{\pi }{f}_A)$ and ${\omega }_B$ ( $=2\mathrm{\pi }{f}_B)$ with a relative phase φ and wave speed $c$

$$u_1(x,t)=A_1\cos \left[{\omega }_A(t-\frac{x}{c})\right]+B_1\cos [{\omega }_B(t-\frac{x}{c})+\phi ]$$

(14)

Undertaking a regular perturbation analysis based on the parameter $\eta$ , Equation (13) becomes an equation for the wave speed $c$ of an infinitesimal amplitude wave in a pre-stretched medium.

$$\begin{array}{c}\multicolumn{1}{c}{{\rho }_0{c}^2=\frac{1}{2}(\lambda +2\mu ){\mathrm{\Lambda }}_1^2(3{\mathrm{\Lambda }}_1^2+{\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-3)}\\ \multicolumn{1}{c}{-\mu {\mathrm{\Lambda }}_1^2({\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-2)}\\ \multicolumn{1}{c}{+\frac{1}{4}l{\mathrm{\Lambda }}_1^2({\mathrm{\Lambda }}_1^2+{\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-3)(5{\mathrm{\Lambda }}_1^2+{\mathrm{\Lambda }}_2^2+{\mathrm{\Lambda }}_3^2-3)}\\ \multicolumn{1}{c}{+\frac{1}{2}m{\mathrm{\Lambda }}_1^2(5{\mathrm{\Lambda }}_1^4-6{\mathrm{\Lambda }}_1^2-{\mathrm{\Lambda }}_2^2({\mathrm{\Lambda }}_3^2-1)+{\mathrm{\Lambda }}_3^2)}\\ \multicolumn{1}{c}{+\frac{1}{4}n{\mathrm{\Lambda }}_1^2({\mathrm{\Lambda }}_1^2-1)({\mathrm{\Lambda }}_3^2-1)}\end{array}$$

(15)

If there is no pre-stretch, ${\mathrm{\Lambda }}_1={\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_3=1$ , Equation (15) recovers the familiar linear elastic result for the speed of a longitudinal wave ${\rho }_0{c}^2=\lambda +2\mathrm{\mu }$ . If the pre-stretch consists of a unidirectional stress applied in the direction of wave propagation, the principal stretches can be expressed as

$${\mathrm{\Lambda }}_1=1+\frac{\lambda +\mu }{\mu (3\lambda +2\mu )}{\sigma }_1$$

(16)

$${\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_3=1-\frac{\lambda }{2\mu (3\lambda +2\mu )}{\sigma }_1$$

(17)

Substituting Equation (15) into Equation (14), the first-order estimate for the wave speed c is presented as

$${\rho }_0{c}^2=\lambda +2\mu +(5+\frac{4(\lambda +m)}{3\mu }+\frac{2(3l+2m-4\lambda )}{3(3\lambda +2\mu )}){\sigma }_1$$

(18)

It agrees with the corresponding results in Destrade and Ogden ³⁴ though the results in that work were presented in terms of the density in the initially stressed state $\rho ={\rho }_0/J_1$ , where $J_1=1+{\sigma }_1/(3\lambda +2\mu )$ . It should be noted that the wave speed $c$ , in Equations (15) and (18) is relative to the initially stressed state rather than the stress-free reference state.

Change of material non-linearity induced byprestress

Continuing with the analysis of wave mixing as defined in Equation (14), the second-order terms in Equation (13) allow the determination of the displacement amplitude of the second harmonic component $A_2$ relative to the fundamental amplitude $A_1$ . For this purpose, analogous to the single-frequency case discussed in Norris, ³⁵ the second-order displacement at a propagation distance x is taken in the following form:

$$\begin{array}{c}\multicolumn{1}{c}{u_2(x,t)=x{A}_2\cos [2{\omega }_A(t-\frac{x}{c})+{\phi }_A]}\\ \multicolumn{1}{c}{+x{B}_2\cos [2{\omega }_B(t-\frac{x}{c})+{\phi }_B]}\\ \multicolumn{1}{c}{+x{C}_2\cos [({\omega }_B+{\omega }_A)(t-\frac{x}{c})+{\phi }_C]}\\ \multicolumn{1}{c}{+x{D}_2\cos [({\omega }_B-{\omega }_A)(t-\frac{x}{c})+{\phi }_D]}\end{array}$$

(19)

where $A_2$ , $B_2$ , $C_2$ and $D_2$ are coefficients representing displacement amplitude of the second harmonic, and the combinational harmonics at sum and difference frequency, and each component has a corresponding phase $\phi$ . Substituting Equation (19) into the equation of motion (5) allows the identification of each coefficient and demonstrates that the assumed form of the solution for $u_2$ is correct. Using the expression for the stress Equation (9), the second-order terms derived from the equation of motion Equation (13) leads to

$$\frac{A_2}{A_1^2}={\omega }_A^2\beta ,\frac{B_2}{B_1^2}={\omega }_B^2\beta$$

(20)

$$\frac{C_2}{A_1{B}_1}=\frac{D_2}{A_1{B}_1}=2{\omega }_A{\omega }_B\beta$$

(21)

$${\phi }_A=0,{\phi }_B=2\phi ,$$

(22)

$${\phi }_C=\phi ,{\phi }_D=\pi -\phi$$

(23)

where the non-linearity parameter $\beta$ is given by

$$\begin{array}{c}\multicolumn{1}{c}{\beta =\frac{{\mathrm{\Lambda }}_1^4}{{\rho }_0{c}^4}}\\ \multicolumn{1}{c}{(\frac{3}{8}(\lambda +2\mu )+\frac{1}{8}l(5{\mathrm{\Lambda }}_1^2+3{\mathrm{\Lambda }}_2^2+3{\mathrm{\Lambda }}_3^2-9)+\frac{1}{4}m(5{\mathrm{\Lambda }}_1^2-3))}\end{array}$$

(24)

with the wave speed $c$ given by Equation (18). For the case of a uniaxial prestress applied in the direction of wave propagation, this result can be recast more conveniently in terms of the prestress ${\sigma }_1$ instead of the principal stretches induced by the prestress as

$$\begin{array}{c}\multicolumn{1}{c}{\beta =\frac{1}{{\rho }_0{c}^4}[\frac{3}{8}(\lambda +2\mu )+\frac{1}{4}(l+2m)}\\ \multicolumn{1}{c}{+(\frac{3}{2}+\frac{l+3m+\lambda }{2\mu }+\frac{5l+6m-4\lambda }{4(3\lambda +2\mu )}){\sigma }_1]}\end{array}$$

(25)

If ${\sigma }_1=0$ (there is no pre-stretch), then a result equivalent to the expression for the second-order non-linearity parameter associated with longitudinal waves in unstressed media is recovered. ³⁵

The parameter $\beta$ is related to ${\beta }^{\prime }$ as defined earlier in Equation (1) through $\beta ={\beta }^{\prime }/x$ . The preceding results will be used for validating the numerical simulation results in section ‘Stress effect on longitudinal-bulk waves mixing’, and they also provide a guide for the case of Lamb wave mixing in the presence of pre-stress, for which analytical results are not currently available.

Experimental evaluation

This section describes the procedures and details of the experimental study of stress effects on combinational harmonic generation and discuss the main findings from the experimental study.

Uniaxial tension test

A dogbone-shaped plate-like specimen was subjected to a uniaxial tensile prestress in an electrohydraulic testing machine (INSTRON 1342) as illustrated in Figure 2(a). The specimen was made from a 1.6 mm thick aluminium alloy (7075-T651) plate, with an elastic modulus of 70.76 GPa. The in-plane dimensions of the specimen are $830\times 500$ mm. Two strain gauges (TML FLA-5-23) were attached to the rear surface of the test plate to measure the change of strain at different levels of tensile load. Strain Gauge 1 was positioned in alignment with the designated ultrasonic waves excitation area, while Strain Gauge 2 was positioned within the assigned measurement area on the frontal aspect of the plate as shown in Figure 2(a). The applied load was increased from 0 to 27 kN with intervals of 5.4 kN. The collected data from strain gauges are listed in Table 1. Figure 2(b) presents a linear regression analysis depicting the relationship between the average strain across the area of interest and the applied tensile force. The results show that the change of strain at Strain Gauges 1 and 2 are similar, indicating that the stress is uniformly distributed within the area of interest. Also, the maximum stress level within the area of interest remains far below the tensile yield strength of aluminium 7075-T651 (≈503 MPa).

Figure 2.

(a) Schematic of strain gauge location and (b) applied tensile force versus measured strain in the proposed excitation and measurement area for ultrasonic testing.

Table 1.

Measured strain excitation and measurement areas at different stress levels.

Stage	Tensilestress (MPa)	Uniaxialtension (kN)	Averaged data fromStrain Gauge 1 ( $\mathrm{\mu }\mathrm{\epsilon }$ )	Standarddeviation	Averaged datafrom StrainGauge 2 ( $\mathrm{\mu }\mathrm{\epsilon }$ )	Standarddeviation
1	10	5.4	131.7	1.5	131.3	1.2
2	20	10.8	273.0	2.0	267.7	1.5
3	30	16.2	416.0	2.0	406.3	1.2
4	40	21.6	559.3	1.5	544.3	1.5
5	50	27.0	704.0	1.0	685.0	1.0

Actuating and measuring Lamb waves

Figure 3 schematically shows the experimental setup for actuating and measuring Lamb waves. First, a signal generator (National Instrument PXle-5412) was used to generate the Hann-windowed input tone-bursts. The choice of centre frequencies and number of cycles for these tone bursts are discussed in the next subsection. The tone-burst signal was amplified by a signal amplifier (Ciprian HVA-800-A). Next, the amplified signal was sent to a piezoceramic transducer (PZT) (Ferroperm Pz 27), which was bonded to the surface of the aluminium plate using conductive epoxy.

Figure 3.

Schematic of the experimental setup for actuating and measuring Lamb waves.

The out-of-plane displacements at the measurement points were measured by a scanning laser Doppler vibrometer (Polytec PSV-400-M2-20) in 1D mode. The measurement distance between the laser head and the specimen was fixed at 1 m. Each measured signal was obtained at a sampling frequency of 25.6 MHz. To improve the accuracy of the measurement, the measured signals were averaged by 500 acquisitions. To enhance the signal-to-noise ratio, the measurement area was polished by sandpaper and then coated with a thin layer of aluminium base coat reflective paint. Furthermore, a low-pass filter with a cut-off frequency of 800 kHz was applied to the measured signals. Figure 4 shows the experimental setup of the specimen under uniaxial tension test.

Figure 4.

Experimental setup of the specimen under uniaxial tension test.

Determination of Lamb wave mode and input signal

As a basis for selecting a suitable Lamb wave mode for the present investigation, Figure 5(a) and (b) show the phase velocity and group velocity dispersion curves of Lamb waves propagating in a 1.6 mm thick aluminium 7075-T651 plate, respectively. ²⁹ It can be observed that the change of phase and group velocity of fundamental symmetric Lamb wave mode ( $S_0$ ) is minimal in the low-frequency range (lower than 500 kHz), while the variation of phase and group velocity for the fundamental antisymmetric mode Lamb wave ( $A_0$ ) is much larger in this frequency range. In the low-frequency range, the maximum phase velocity difference of $S_0$ is 0.94% satisfying the threshold value of the quasi-phase matching condition, ³⁶ which is necessary to ensure the cumulative growth of second harmonics as well as combination harmonics, by ensuring a non-zero power flux.^37–41 Therefore, the low-frequency $S_0$ mode is selected in the present work as the most attractive wave mode for monitoring the change of material non-linearity due to prestress.

Figure 5.

(a) Phase velocity and (b) group velocity dispersion curve of Lamb waves propagation in 1.6 mm thick aluminium 7075-T651.

To select optimized excitation frequencies for the generation of combinational harmonics, a mode tuning experiment was conducted. Six-cycle Hann-windowed tone bursts, with centre frequencies ranging from 150 to 350 kHz in steps of 10 kHz, were generated by a PZT. The excitation voltage was fixed at ±200 V. Figure 6 shows the resulting amplitude of the excited $S_0$ measured by the laser Doppler vibrometer. It can be observed that the amplitude of $S_0$ increases with the frequency in the range of 150 to 220 kHz and followed by a decrease from 220 to 310 kHz. Based on these results, the primary excitation frequencies $f_A$ and $f_B$ were selected as 180 and 270 kHz in view of the following considerations. First, both selected excitation frequencies have relatively high and similar amplitudes in the tested frequency range, which can reduce the effect of measurement noise. Second, the difference between the two selected excitation frequencies is sufficiently large to avoid an undesirable overlap between the sum harmonic and the individual second harmonics, thereby facilitating accurate quantification of these harmonics.

Figure 6.

Mode tuning curve of S_0 generated by PZT.

The number of cycles of the two input tone bursts ( $N_A$ and $N_B$ ) were selected as six and nine, respectively. This is because this combination of cycles can provide high clarity of generation of combinational harmonic in the frequency domain, which is supported by a high value of wave mixing separation index ( $S$ ) of 30. The S index is defined as ⁴²

$$S=(f_B-f_A)-(\frac{f_A}{N_A}+\frac{f_B}{N_B})$$

(26)

When the $S$ index is a negative value, substantial overlap occurs between the harmonics of fundamental waves, posing a considerable challenge for frequency domain analyses. In contrast, when the $S$ index is a positive value, an increase in the $S$ index value can lead to increased separation between the two fundamental harmonics, which results in less overlapping between the sum combinational harmonic at sum frequency ( $f_A+f_B$ ) and the second harmonics ( $2f_A\mathrm{and}2f_B)$ . The experimental results from Zhu et al. ⁴² showed that an $S$ index over 20 indicates a high clarity in the generation of the sum combinational harmonic. Figure 7 presents the input signal used in this study and its single-frequency components.

Figure 7.

Schematic of the input signal used in this study.

Experimental results of ultrasonic-guided wave testing

Determination of input power and ascertaining the source of non-linear response

In this section, the determination of input power for ultrasonic testing and the evaluation of the source of non-linear responses in the experiment is discussed. To select an optimized input power level, the input signal was tested at voltage levels of 40, 80, 120, 160 and 200 V. The out-of-plane displacement was measured at 200 mm from the edge of the PZT. The amplitude of the combinational harmonic at the sum frequency was found to be linearly proportional to the product of the amplitudes of two fundamental harmonics as illustrated in Figure 8(a). In the subsequent test, the excitation voltage was fixed at 200 V because no obvious errors were observed at this input power level as shown in Figure 8(a) and a high input power is beneficial for enhancing the non-linear response.

Figure 8.

(a) Linear fitting of the amplitude of sum combinational harmonic ( $C_{\mathrm{sum}}^{\prime }$ ) versus the product of amplitudes of two fundamental harmonics ( $A_1^{\prime }\times B_1^{\prime })$ in different excitation voltages (b) normalized ${\beta }^{\prime }$ versus propagation distance for 200 V excitation voltage.

To investigate the relationship between the ${\beta }^{\prime }$ value of the sum combinational harmonic and propagation distance, 11 measurement points with 5 mm intervals were designated between 200 and 250 mm away from the edge of the PZT. Figure 8(b) shows the relationship between ${\beta }^{\prime }$ of measured signals and the wave propagation distance. It shows that the ${\beta }^{\prime }$ value increases linearly with the propagation distance. Overall, the results demonstrate that the material non-linearity of the specimen is the main contribution to the measured non-linear responses.

Assessment of uniaxial tensile stress using linear and non-linear features of ${\mathit{S}}_0$

In this section, the data collected at the measurement point 200 mm away from the edge of the PZT in different uniaxial tensile stresses is discussed. During the assessment, the uniaxial tension provided by INSTRON is based on the results of the uniaxial tension test described in section ‘Uniaxial tension test’. The specimen was loaded from 0 to 50 MPa with 10 MPa intervals. In each stage, four sets of data were collected and then averaged.

Figure 9 presents the measured time-domain out-of-placement displacement signals at different tensile stress levels. It can be observed that both $S_0$ and $A_0$ were generated in the experiment. Only the $S_0$ mode was selected by time gating for the following analysis as indicated in Figure 9. The magnified view in Figure 10(a) highlights the influence of applied uniaxial tensile stresses on the propagation of $S_0$ . It can be observed that the phase delay increases with the applied uniaxial tensile stress. To quantify the acoustoelastic effect due to prestress, the cross-correlation function was used to determine the time lag for both unstressed and prestressed signals, and hence, the ToF of $S_0$ at different stress levels can be determined. The group velocities of $S_0$ at different stress levels can be determined from the propagation distance divided by the corresponding ToF. Figure 10(b) presents the group velocity at various uniaxial tensile pre-stress levels. It can be observed that from 0 to 50 MPa, the change of wave speed of $S_0$ due to the acoustoelastic effect is only around 0.35%. This minor change of wave speed means that it is very challenging to utilize the acoustoelastic effect for stress monitoring in practical applications.

Figure 9.

Experimentally measured time domain signals at different tensile stress levels.

Figure 10.

(a) Measured time domain signals of $S_0$ and (b) calculated group velocity of the measured $S_0$ at different tensile stress levels.

Figure 11(a) shows the corresponding frequency domain data. In contrast to the acoustoelastic effect, the effect of applied tensile stress on the amplitude of the sum combinational harmonic can be clearly observed in the frequency domain. Figure 11(b) presents the relationship between ${\beta }^{\prime }$ (at a propagation distance of 200 mm) and the applied tensile stress. It is evident that the value of ${\beta }^{\prime }$ is highly correlated to the applied tensile stress, which is supported by a high R-square value of 0.9055. From 0 to 50 MPa uniaxial tensile prestress, the value of ${\beta }^{\prime }$ increased by around 35%. The result indicates that the sum combinational harmonic has a higher sensitivity to the presence of prestress, compared with the acoustoelastic effect, thereby highlighting its suitability for in-situ stress monitoring.

Figure 11.

(a) Measured frequency domain signals at different tensile stress levels and (b) relative nonlinearity parameter at a propagation distance of 200 mm versus applied uniaxial tensile stresses.

Numerical case study

In this section, a two-step procedure is employed for modelling the effect of an initial prestress, coupled with material non-linearity, on (i) wave speed, and (ii) the amplitude of the mixed-frequency non-linear response due to multiple frequency inputs. Material nonlinearity is modelled by a 5-constant hyperelastic constitutive equation with the parameter values listed in Table 2, and it is implemented through a VUMAT subroutine in ABAQUS/Explicit.^27,32,43 This two-step FE approach is first investigated for longitudinal bulk-wave motion in section ‘Stress effect on longitudinal-bulk waves mixing’, and the results are compared with the analytical predictions in sections ‘Acoustoelastic effect on wave speed’ and ‘Change of material nonlinearity induced by prestress’ for validation purposes. The same approach is then applied to investigate the effect of a biaxial prestress on $S_0$ Lamb wave in the following subsections.

Table 2.

Material properties of aluminium 7075-T651.^44,45

Density		Lame constants		Third-order elastic constants
${\rho }_0$ (kg/m3)	$\lambda$ (GPa)	$\mu$ (GPa)	$l$ (GPa)	$m$ (GPa)	$n$ (GPa)
2704.00	51.60	26.60	−252.20	−325.00	−315.20

Stress effect on longitudinal-bulk waves mixing

A quasi-1D FE model was first developed for simulating longitudinal bulk wave by applying symmetric boundary conditions (YSYMM, $U_2=U{R}_1=U{R}_3=0$ ) to the top and bottom edges of a strip-like domain as indicated in Figure 12. The quasi-1D FE model was meshed by four-node plane strain elements with reduced integration (CPE4R). The mesh size is $0.5\times 0.8$  mm². The material properties of the model are listed in Table 2.

Figure 12.

Schematic of FE model for simulating longitudinal bulk wave.

To model pre-stress in the explicit dynamic simulation, the simulation was done in two steps. In Step 1, a quasi-static analysis was employed to model a gradually increasing applied load that reaches a specified plateau value as indicated in Figure 13. This approach minimizes the influence of the transient effect induced by the applied pre-stress. ²⁷ Once the desired pre-stress level was achieved, a longitudinal bulk wave was generated in Step 2 by applying 5 μm displacement ( $u_x)$ to all nodes at the left end of the specimen as indicated in Figure 12. The time step of each increment was automatically determined by the ABAQUS/Explicit solver, which the maximum time step was lesser than $1/20f_{\max }$ . Six measurement points with equal intervals were set from 50 to 300 mm away from the excitation location as indicated by the red dots in Figure 12.

Figure 13.

Quasi-static loading curve in Step 1 (prestress) of the simulation.

The results of FE models with linear and non-linear material properties are compared. The linear FE model accounts only for linear material properties and does not incorporate material nonlinearity characterized by the third-order elastic constants. In contrast, the non-linear FE model includes both linear material properties and the third-order elastic constants, allowing for a more comprehensive analysis. The results of linear and non-linear FE simulations are first compared under the stress-free condition. Data from the first measurement point is used for comparison. Figure 14(a) and (b) compare the linear and non-linear FE results in the time and frequency domain, respectively. In the time domain, no significant difference between linear (red solid line) and nonlinear FE (blue dot-dash line) results can be observed. In the frequency domain, apart from primary excitation frequencies ( $f_A$ and $f_B$ ), no other non-linear guided wave components can be observed from the linear FE results, while harmonics appear at double frequencies ( $2f_A$ and $2f_B$ ), difference frequency ( $f_B$  −  $f_A$ ) and sum frequency ( $f_A$ +  $f_B$ ) in the non-linear FE results.

Figure 14.

Comparison of linear material and non-linear material properties FE results in (a) time and (b) frequency domain.

Figure 15(a) and (b) present the stress effect on linear and non-linear FE simulated results. The red solid line represents the collected signal in the absence of prestress, while the blue dashed line represents the response in the presence of a 200 MPa uniaxial tensile stress, which is parallel to the wave propagation direction. The phase shift due to the stress effect cannot be observed from the linear FE results as shown in Figure 15(a), confirming that there is no change of wave speed due to a prestress within the framework of linear elasticity. In contrast, compared with the signal collected from the stress-free condition, the phase shift due to the applied prestress can be seen in the longitudinal bulk-wave signal simulated by the nonlinear FE model, which can be observed more clearly in the zoom-in view of Figure 15(b).

Figure 15.

Comparison of time domain signals with and without stress effect simulated by FE model with (a) linear and (b) non-linear material properties.

Figure 16(a) compares the theoretical solution for the wave speed (Equation (14)) with non-linear FE simulation results for various tensile stress levels, whereas Figure 16(b) compares the theoretical non-linearity parameter given by Equation (24) with the non-linear FE simulation results obtained at the first measurement point for various tensile stress levels. It can be observed that the theoretical solution results agree with the non-linear FE predictions. This demonstrates that the effect of prestress on the linear and non-linear features of ultrasonic wave can be successfully simulated by VUMAT with the implementation of Murnaghan’s hyperelastic material model.

Figure 16.

Comparison of theoretical and non-linear material properties FE results of the change of (a) longitudinal bulk wave speed, and (b) normalized ${\beta }^{\prime }$ due to applied uniaxial tensile stress.

3D FE simulation of ${\mathit{S}}_0$ in a prestressed plate

This section describes the 3D FE model for investigating the stress effect on the combinational harmonics of $S_0$ . Figure 17 schematically shows the 3D FE model. By taking advantage of symmetry, only a quarter of the aluminium 7075-T651 plate was modelled, and symmetric conditions were applied to both the x–z and y–z planes of the model. The dimension of the model is $400\times 400\times 1.6$  mm³. Eleven measurement points with intervals of 5 mm were evenly allocated from 200 to 250 mm away from the edge of the excitation area along 0°, 22.5°, 45°, 67.5° and 90° directions, respectively. To ensure computational accuracy for simulating the Lamb waves, it is necessary to have at least eight elements per wavelength and a minimum of 4 to 5 nodes in the thickness direction. ⁴⁶ In this study, the highest frequency of interest was 450 kHz, for which the $S_0$ mode has a phase velocity ( $C_p$ ) of 5378 m/s, and a wavelength of 12 mm. Thus, the in-plane dimensions of the elements in FE were selected as $0.5\times 0.5$  mm² with a thickness of 0.2 mm. This size ensures that there are at least 20 elements per wavelength and eight layers of elements in the thickness direction. Eight-node 3D brick elements with reduced integration (C3D8R) were employed to mesh the model. The excitation area was a quarter of a circle, and the radius was the same as the size of the PZT used in the experiment. The excitation area was located on the top and bottom surfaces of the plate. To generate $S_0$ , 5 μm out-of-plane displacement was applied to the excitation area shown in Figure 17.

Figure 17.

Schematic of 3D FE model for simulating propagation of $S_0$ Lamb wave in plate.

Figure 18(a) and (b) compares the numerically calculated and experimentally measured time and frequency domain signals from the first measurement point in the initial condition, respectively. The red solid line represents the experimentally measured data, while numerical calculated results of the nonlinear FE is labelled as the blue dot-dash line. To directly compare nonlinear FE and experimental data, both numerical and experimental data were normalized by their absolute maximum value, respectively. Figure 18(a) indicates that the FE calculated and experimentally measured data have a similar pattern and arrival time of the $S_0$ Lamb wave. Furthermore, Figure 18(b) shows a good agreement between experimentally measured and numerical simulated results in the frequency domain. This demonstrated that ABAQUS/Explicit with a VUMAT subroutine implementing hyperelastic material properties can provide accurate predictions of non-linear Lamb wave responses in the frequency bandwidth of interest ( $f_A$ , $f_B$ and $f_A$ +  $f_B$ ).

Figure 18.

Comparison of experimental and non-linear FE results in (a) time and (b) frequency domain.

Parametric study using 3D FE model

Effect of biaxial prestressed conditions

In this section, the effect of different biaxial prestress combinations and various stress magnitudes on the wave speed and ${\beta }^{\prime }$ of sum combinational harmonic are investigated using the non-linear 3D FE model. The principal stresses ${\sigma }_x$ and ${\sigma }_y$ are applied in the 0° and 90° directions as shown in Figure 19, and they are related via ${\sigma }_x=\lambda {\sigma }_y$ , where $\lambda$ is the biaxial stress ratio. Six different biaxial pre-stressed cases are investigated and summarized in Table 3. In Cases 1 and 2, the pre-stress in the 90° direction ( ${\sigma }_y$ ) is fixed at 50 MPa tension, while the pre-stress in the 0° direction ( ${\sigma }_x$ ) is 50 MPa and 100 MPa in tension, respectively. In Cases 3 and 4, ${\sigma }_y$ is fixed at 50 MPa in compression, while ${\sigma }_x$ is 50 and 100 MPa in compression, respectively. Finally, in Cases 5 and 6, ${\sigma }_y$ is fixed at 50 MPa in tension with ${\sigma }_x$ is 50 and 100 MPa in compression, respectively.

Figure 19.

Schematic of 3D FE model for simulating propagation of $S_0$ Lamb wave in a pre-stressed plate.

Table 3.

Summary of biaxial pre-stress combinations

Type	Case	λ	${\sigma }_x$ (MPa)	${\sigma }_y$ (MPa)
Biaxial tension	Case 1	1	+50	+50
	Case 2	2	+100	+50
Biaxial compression	Case 3	1	−50	−50
	Case 4	2	−100	−50
${\sigma }_x$ is compression, ${\sigma }_y$ is tension	Case 5	−1	−50	+50
	Case 6	−2	−100	+50

The effect of biaxial stresses on both the wave speed and ${\beta }^{\prime }$ of the sum combinational harmonic were investigated. To directly compare the change of the wave speed and ${\beta }^{\prime }$ values in different propagation angles due to various biaxial pre-stresses, the group velocity ( $C_g$ ) and ${\beta }^{\prime }$ values are normalized by their corresponding values in the case of stress-free condition, respectively.

Figure 20(a) to (c) show the normalized $C_g$ in different propagation directions under the cases of biaxial tension, biaxial compression and the combination of tension and compression, respectively. The black dot-dashed line with the star marker corresponds to the stress-free condition, which provides a benchmark showing the same value in different directions due to the assumed material isotropy. In biaxial tension cases (Cases 1 and 2), regardless of the value of λ, values of ${\beta }^{\prime }$ from all directions are always smaller than the corresponding value in the stress-free condition, as shown in Figure 20(a). In contrast, all values (Cases 3 and 4) are larger than the benchmark in all directions can also be observed in biaxial compression cases as shown in Figure 20(b). In Figure 20(c), the cases of 0° in compression and 90° in tension (Cases 5 and 6), the values of normalized wave speed decreases with the propagation direction, changing from 0° to 90°, which also presents a sinusoidal dependence between propagation angle and the change in wave speed. This phenomenon is consistent with the results of biaxial prestress experiment that reported by Shi et al. ⁴⁷

Figure 20.

Normalized wave speed of (a) biaxial tension, (b) biaxial compression, (c) combination of tension (in 90° direction) and compression (at 0° direction), and normalized ${\beta }^{\prime }$ of (d) biaxial tension, (e) biaxial compression, (f) combination of tension (in 90° direction) and compression (in 0° direction).

Figure 20(d) to (f) presents the normalized ${\beta }^{\prime }$ values of the measurement point at 200 mm from excitation area in different propagation directions under biaxial tension, biaxial compression and combination of tension and compression, respectively. It can be observed the patterns of ${\beta }^{\prime }$ values change in different biaxial pre-stress conditions, which are opposite to the trend of the changes of wave speed. The observed trends are similar to the results that reported in the past studies of acoustoelastic effect on bulk wave ⁴⁸ and guided waves. ²⁶ This phenomenon suggests that both the acoustoelastic effect and the combinational harmonics can be utilized to identify the type of stress and major stress direction in biaxially pre-stressed structures. However, it can be observed that the combinational harmonic has higher sensitivity to the pre-stress level. Furthermore, it should be noted that the wave speed or ${\beta }^{\prime }$ value in anisotropic biaxially pre-stressed cases (Cases 2, 4, 5 and 6) can be same as other loading cases in particular propagation angles. The propagation angles having the same value of wave speed or ${\beta }^{\prime }$ with other loading cases are highly correlated with the biaxial prestress ratio $\lambda$ . This phenomenon indicates that it is necessary to measure data from different propagation angles to avoid incorrect identification of stress conditions in structures.

Discussion and conclusions

This study has investigated the effect of initial stresses on the generation of combinational harmonics due to non-linear $S_0$ Lamb waves mixing. An experimental evaluation has been carried out on a 1.6 mm thick aluminium alloy (7075-T651) plate under uniaxial tensile stress with various magnitudes. The experimental results showed that when the applied uniaxial tensile prestress increased from 0 to 50 MPa ( $\approx 700\mathrm{\mu }\mathrm{\epsilon }$ ), the change in wave speed of $S_0$ due to acoustoelastic effect is approximately 0.35%. In contrast, the ${\beta }^{\prime }$ value increased by more than 35% at the corresponding pre-stress level, which is significantly greater than the change of the wave speed caused by the acoustoelastic effect. This phenomenon suggests that the combinational harmonics is more sensitive than the wave speed for in-situ stress monitoring.

The change in wave speed due to initial stress aligns with the studies on the acoustoelastic effect in the literature. Gandhi et al. ³⁰ analytically and experimentally investigated the acoustoelastic effect on the propagation of $S_0$ , $A_1$ and $S_1$ wave in a 6.35 mm thick aluminium plate. From 0 to 50 MPa uniaxial tensile stress, the wave speed of $S_0$ experienced a change of 0.4%. Their results for the acoustoelastic effect on the wave speed of $S_0$ are consistent with the corresponding results in this study. In addition, the results of present study are also consistent with the prediction of SAFEDC. ²⁹ On the other hand, the change in ${\beta }^{\prime }$ due to initial stress in the present work is relatively gradual compared to the results of Hughes et al. ²⁵ They investigated the effect of initial stress on the second harmonic of a Rayleigh wave propagating in a 47.5 mm thick aluminium block. They observed a change of 116% in ${\beta }^{\prime }$ value when the applied compressive stress is increased from 22 to 32 MPa. This discrepancy may be attributed to the fact that non-linear features of different wave modes exhibit varying sensitivities to initial stress levels. Further comparative studies are needed to fully understand the sensitivity of non-linear features across different wave modes in response to initial stresses.

In the numerical study, material non-linearity was modelled through a 5-constant hyperelastic constitutive equation, derived from Murnaghan’s strain energy function. The results from a quasi-1D FE model have showed good agreement with the corresponding theoretical predictions, thereby providing a validation for the two-step computational procedure. The same procedure was then applied to a 3D FE model for simulating the propagation of $S_0$ Lamb wave in a biaxially prestressed aluminium plate. In contrast to studies in the literature on the effects of biaxial stress conditions on non-linear responses of guided waves, which only focused on the parallel and orthogonal directions relative to the applied stress direction,^26,28 the current work has also investigated different oblique directions relative to the applied stresses directions. The 3D FE simulation results have demonstrated the feasibility of using combinational harmonics to identify the principal stress direction and types of stress state in a biaxially prestressed structure. Meanwhile, the results have revealed the necessity of measuring data from multiple propagation angles and evaluating the trend to ensure accurate stress identification and avoid misinterpretation of the measured results, especially when the applied stresses directions are unknown.

Furthermore, it is worthwhile to note that an inverse method proposed by Shi et al. ⁴⁷ could be utilized to estimate the magnitudes of arbitrary biaxial stress by analysing the change of phase velocity as a function of propagation angles. It is anticipated that this method could be useful for reconstructing biaxial stress state based on the change of combinational harmonics since the effect of biaxial prestress on wave speed is inversely proportional to the corresponding changes in non-linearity parameter as shown in Figure 20.

Despite the encouraging results, it is important to note that several factors shall be also taken into account when developing a robust stress monitoring system for future in-situ applications. In the experimental setup, the use of a laser vibrometer requires precise optical conditions and accurate alignment, which limits its suitability for in-situ monitoring. It needs further research to address these challenges and improve the practicality of the proposed method. Additionally, the current work assumes that the aluminium specimen is isotropic. However, anisotropy could also be introduced due to the material texture, which develops during the rolling process in manufacturing. ⁴⁹ These factors should be taken into account in future analyses to ensure accuracy of the measured results under real-world conditions.

In summary, the outcomes of this study have demonstrated the feasibility of using combinational harmonics of low-frequency $S_0$ for in-situ stress monitoring. The findings of this study could be utilized for the development and improvement of the robustness of future SHM systems.