Role of transducer inertia in generation,sensing,and time-reversal process of Lamb waves in thin plates with surface-bonded piezoelectric transducers

Abstract

An analytical model is presented for the generation, sensing, and time-reversible process of Lamb waves in thin isotropic plates with surface-bonded piezoelectric wafer transducers, incorporating the shear-lag effect of the bonding layer and inertia effects of the system in transducer modeling. A one-dimensional dynamic shear-lag model for the actuator-plate interaction is used to obtain the shear stress distribution at the actuator-plate interface. The Lamb wave solution for the plate under this shear traction excitation is obtained using the two-dimensional (2D) elasticity equations. A consistent sensor-plate interaction model incorporating the shear-lag and inertia effects is developed to determine the induced sensor voltage from the Lamb strain at the plate surface. The model is validated by comparing it with the 2D coupled piezoelasticity-based finite element simulation and experimental data. Detailed parametric studies are conducted to illustrate the effect of inclusion of inertia of actuator, sensor, adhesive, and plate in the transducer modeling on the Lamb wave generation, sensing, time reversibility, and the system’s best reconstruction frequency, and to ascertain how various geometrical and material parameters of the system influence the same. The developed closed-form solution will be immensely useful for the design of Lamb wave based structural health monitoring systems.

Keywords

Lamb wave time reversal process dynamic shear-lag model transducer inertia best reconstruction frequency structural health monitoring

Introduction

Lamb wave based techniques have attracted increasing interest in the last two decades for structural health monitoring (SHM) of thin plate/shell-type structures encountered in aircraft (Dalton et al., 2001), space vehicles (Giurgiutiu et al., 2011; Huang, 2001), pipelines (Huan et al., 2020), pressure vessels (Hu et al., 2020), wind turbine blades (Druffner et al., 2011) etc. The advantages of using ultrasonic Lamb waves for damage detection in such thin-walled structures are its ability to travel long distances at high speed, ease of activation and sensing, ability to detect both surface and internal damages of small size, and low energy consumption. Piezoelectric patches/wafers and piezoelectric fibre reinforced composites (MFCs) have emerged as the most preferred transducers for actuating and sensing Lamb waves because they are very thin, non-intrusive, minimally invasive, and inexpensive, and can be conveniently surface-mounted on existing structures or embedded in laminated structures using adhesives (Collet et al., 2011; Giurgiutiu, 2003). An analytical model which can accurately capture the dynamic interaction between the piezoelectric actuator/sensor and the host structure, and provide an accurate solution for actuation, propagation and sensing of Lamb waves, is in fact essential for the effective and optimal design of such guided wave based SHM systems.

The initial studies on the analytical solutions of Lamb wave actuation, propagation, and sensing assumed the strain transfer from the actuator to the plate to take place at the edges of the actuator. It is known as the pin-force model, which is essentially the result of two premises: (i) the actuator has a negligible thickness and (ii) the bonding between the actuator and the plate is perfect. Both these premises, however, do not represent the real situation, because of which the actual shear transfer is spread over an area near the edge (Kapuria and Kumari, 2013). This interfacial stress distribution, of course, has a large influence on Lamb wave generation and sensing (Shevtsova et al., 2019).

Lin and Yuan (2001) presented an analytical solution for propagation of fundamental antisymmetric mode ( $A_{0}$ ) Lamb wave in an integrated actuator-plate-sensor system using the pin-force model for actuation and the Mindlin plate theory for solving the propagation problem in the plate. The first order laminate theory assumption was used to compute the radial moment at the edge of the circular actuator. The sensor model assumed a perfect bonding, very thin sensor and uniform strain over the sensor area. Rose and Wang (2004) employed the Mindlin plate theory to develop analytical source solutions for $A_{0}$ mode Lamb wave due to point moment and force doublet source excitations and used them to obtain the response for circular and narrow rectangular shaped actuators following the pin-force approximation. For high frequency waves, the inplane displacements may not follow a linear distribution across the thickness of the plate, as assumed in the Mindlin theory. Consequently, later efforts can been seen to use the three dimensional (3D) elasticity based solutions for wave propagation in isotropic plates under the plane strain condition (Giurgiutiu, 2005; Wang et al., 2004) as well as for the general 3D case (Raghavan and Cesnik, 2005). These studies too used the pin-force model for actuation. Raghavan and Cesnik (2005) proposed a sensor model in which the strain in the transducer was assumed to be equal to that on the surface of the plate. This was based on the assumption that the sensor is highly compliant compared to the host plate and there is rigid bonding between the sensor and the plate, which is generally not true in practice. Lanza di Scalea et al. (2007) used the same approximation to obtain a frequency domain solution for the electric potential induced in surface-bonded rectangular piezoelectric sensors due to Rayleigh and Lamb waves under harmonic and arbitrary excitations.

The important role that the adhesive layer between the piezoelectric transducer and the host structure plays has been experimentally established for the electromechanical impedance and resonance frequency of the transducers (Islam and Huang, 2014; Qing et al., 2006), actuation and sensing of Lamb waves (Ha and Chang, 2010; Islam and Huang, 2016; Pohl et al., 2012; Sirohi and Chopra, 2000) as well as its time reversibility (Agrahari and Kapuria, 2016a). The first actuator-to-structure stress transfer model which incorporated the effect of the adhesive layer was presented by Crawley and Luis (1987) for a beam-type host structure with two collocated piezoelectric actuator patches bonded on both surfaces of the beam. The model, known as the shear-lag model, was developed for quasi-static actuation, considering Euler-Bernoulli theory for the beam and membrane-type behaviour of the thin transducer. Giurgiutiu (2005) modified the model for the case where a single actuator is bonded to one side of the beam, which in general generates both symmetric and anti-symmetric modes of guided waves simultaneously. Lanza di Scalea and Salamone (2008) used this quasi-static shear-lag model for stress transfer between the transducer and the host plate to improve the previous pin-force model based solution (Giurgiutiu, 2005) for actuation and sensing of Lamb waves in a thin isotropic plate with surface-bonded piezoelectric transducers under an harmonic excitation. However, the solution for the sensor potential suffers from inconsistencies and does not satisfy the appropriate boundary conditions.

The shear-lag model for the transducer-plate interaction used in the aforementioned studies is based on a plane stress assumption, while the wave propagation solution corresponds to the plane strain condition of the plate. A recent study based on a 2D shear-lag solution for the interfacial shear stresses for a rectangular transducer (Kapuria and Agrahari, 2016) has, however, shown that for a square-shaped piezoelectric transducer, the shear stress distribution at the transducer-plate interface matches with the stress distribution obtained from the 1D shear-lag model developed considering the plane strain assumption for the transducer and the plate. Based on this observation, Kapuria and Agrahari (2018) presented a closed-form time domain analytical solution for actuation and sensing of Lamb waves in plates with surface-bonded actuators and sensors using the plane strain 1D shear-lag model for the transducers. In this formulation, a consistent model for the sensor was developed satisfying the appropriate boundary conditions. In another development, Shan et al. (2017) presented a theoretical framework for the Lamb wave generation, propagation and sensing considering the adhesive nonlinearity. The model, however, did not provide a closed-solution for the sensor output, and it too employed the plane stress 1D shear-lag model for the transducer.

Although in Lamb wave based SHM applications, the piezoelectric transducers are excited dynamically, all the earlier studies employed quasi-static solutions for transducer-structure interaction for actuation and sensing, neglecting the inertia of the transducer and structure. However, the inertia effect of the transducers can not be neglected for high frequency excitations, when the wavelength is comparable to the length of the transducer (Wang and Huang, 2006). Studies have shown that the Lamb wave response obtained using the quasi-static shear-lag solution differs significantly from detailed elasticity based FE solutions at the high frequencies generally used for SHM (Kapuria and Agrahari, 2018). Therefore, it is necessary to develop an analytical solution for Lamb wave generation and sensing, considering the inertia effect in the actuator and sensor models.

An analytical actuator model for the dynamic stress transfer between a piezoelectric actuator and an isotropic host plate subjected to guided waves was developed by Huang et al. (2010) considering the actuator inertia. The model directly coupled the 1D equation of inplane motion of the thin actuator with the 2D elasticity based wave equation for the plate in terms of the interfacial shear stress. An approximate numerical solution of the coupled integral equations was obtained using the method of collocation. The study did not include the important effect of the bonding layer. Han et al. (2008) presented a similar solution in the integral form for dynamic response of a sensor bonded to an elastic half-space, which modelled the bonding layer. Like the actuator model, this sensor model too did not offer a closed-form solution.

Dugnani (2016) attempted a modification to the classical quasi-static shear-lag model of Crawley and Luis (1987), introducing the transducer inertia. However, to maintain a similar solution like the original study, the interfacial shear stress was expressed as a single term solution (considering symmetry), irrespective of the fact that the governing deferential equation was of the fourth order. Furthermore, he neglected the inertia term of the host structure, which too has a notable effect on the sensor response. Recently, Kapuria et al. (2019) proposed a consistent extension of the quasi-static shear-lag model by considering the inertia of the transducer, the host plate as well as the bond layer, which yielded a closed-form solution for the dynamic stress transfer between the harmonically excited piezoelectric actuator and the isotropic host plate. The model has been developed considering the plane strain condition for both the actuator and the host plate, following the inferences made in Kapuria and Agrahari (2016). A comparison with the detailed 2D FE solutions showed that the dynamic shear-lag model predicts dynamic interfacial shear stress distribution accurately at high excitation frequencies. Such a dynamic shear-lag model is at present not available for sensors. Likewise, to the best of authors’ knowledge, no closed-form analytical solution incorporating the dynamic effects of the constituent elements in the transducer models has been reported so far for generation and sensing of Lamb waves in thin isotropic plates with surface-bonded piezoelectric transducers.

The time reversal process (TRP) of Lamb waves (Fink, 1992) has gained immense popularity as a baseline-free damage detection technique for thin-walled structures (Kannusamy et al., 2020; Park et al., 2007; Poddar et al., 2011). In the TRP, the input response can be reconstructed at the actuator location after emitting back the time reversed version of the forward response obtained at the sensor location, in the absence of damage. The damage can thus be identified by comparing the reconstructed signal with the original input signal. An accurate analytical solution for the TRP of Lamb wave is very important for the design of SHM systems based on the TRP and also to arrive at theoretical estimation of the frequency of best reconstruction at which the time reversal method would be most effective (Agrahari and Kapuria, 2016a, 2016b). No such model including inertia effect in the transducer models is available at present.

In this paper, we present a closed-form analytical solution for generation and sensing of Lamb waves in an isotropic plate with an actuator and sensor, adhesively bonded to its surface, incorporating shear-lag and inertia effects in transducer modeling. We study the problem of Lamb wave propagation in the plate in two dimensions under the plane strain condition, considering plane waves. However, a two dimensional (2D) solution for the transducer-bond layer-plate system for actuation and sensing of Lamb waves would be very complex. A simple but effective way of overcoming this complexity is to model the transducers and the plate using the 1D panel theories so as to ascertain the shear stress induced on the plate by the actuator and the sensory potential induced in the sensor by the strain developed in the plate due to Lamb waves. Such 1D models for the actuator and sensor were employed earlier without considering the inertia of the transducer, plate and the adhesive (Kapuria and Agrahari, 2018), which plays an important role on the accurate prediction of the actuated shear stress and sensor potential. Besides, the previous sensor models did not satisfy the stress-free condition at both edges of the sensor. In the present formulation, the shear stress induced at the actuator-plate interface is obtained using the dynamic shear-lag model developed previously by the authors (Kapuria et al., 2019) considering the inertia effect. The solution for the Lamb wave displacement field is obtained for this shear excitation in the frequency domain. Next, a consistent sensor-plate interaction model is developed for the first time considering shear-lag and system inertia effects, and satisfying the appropriate boundary conditions at the sensor.

The sensor model transforms the displacement field on the plate surface at the sensor location into the induced sensor potential. The reconstructed signal after the TRP is computed in the frequency domain by making use of the transfer function obtained from the forward sensor response. Finally, the time domain responses for both forward propagation and the TRP are obtained using the inverse Fourier transform. The model is validated by comparing the results with piezoelasticity based FE solutions. A detailed parametric study is performed to understand the effects of inertia of the actuator, sensor, adhesive, and plate in the transducer modeling on the Lamb wave actuation, sensing, time reversibility, and the system’s best reconstruction frequency.

Actuator-plate interaction model considering inertia

Consider an isotropic plate of thickness $h$ with a pair of square thin piezoelectric patch transducers of length $a$ and thickness $h_{p}$ , bonded to its upper surface using adhesive layers of thickness $h_{a}$ . Figure 1 illustrates a schematic view of the transducer-adhesive-host plate system. The transducers are bonded at a center-to-center distance of $d$ in a pitch-catch configuration to excite and sense Lamb waves in the plate. The results from the static shear-lag solution for the interfacial shear stress for a square patch actuator bonded to an isotropic plate (Kapuria and Agrahari, 2016), have shown that the 1D solution obtained considering both the transducer and the plate under the plane strain condition is, in fact, in close agreement with the 2D solution. Accordingly, we adopt the same condition in both actuator and sensor models.

Figure 1.

Transducer-adhesive-plate system in pitch-catch mode.

The piezoelectric material is considered to be of class mm2 symmetry with the poling direction along the $z$ -axis. $s_{ij}$ and $d_{3 i}$ denote, respectively, the short-circuit elastic compliances and the piezoelectric free-strain coefficients of the material. The elastic constants $s_{ij}$ are obtained from the engineering constants $Y_{i}$ (Young’s moduli) and $ν_{ij}$ (Poisson’s ratios) as

s_{11} = 1 / Y_{1}, s_{12} = - ν_{21} / Y_{1} = - ν_{12} / Y_{2}, s_{22} = 1 / Y_{2}

(1)

The Young’s modulus and Poisson’s ratio of the isotropic plate are $Y_{s}$ and $ν_{s}$ , respectively and the shear modulus of the adhesive layer is $G_{a}$ . The mass densities of the plate, transducers, and adhesive are $ρ$ , $ρ_{p}$ , and $ρ_{a}$ respectively.

Consider that PZT-A (Figure 1) is actuated with an electric potential signal $ϕ_{a} (t)$ applied at the top surface while its bottom surface is grounded. The excitation induces inplane strain $ε_{x}$ in the actuator due to the converse piezoelectric effect, which gets transmitted to the plate via the bonding layer in terms of the interfacial shear stress $τ$ as shown in Figure 2. The plane strain conditions ( $ε_{y} = 0$ , $γ_{yz} = 0$ , $γ_{yz} = 0$ ) and the assumption that the transverse normal stress is negligible for thin plates ( $σ_{z} \approx 0$ ), reduces the three dimensional constitutive relation for $ε_{x}$ for the piezoelectric transducer to

ε_{x} = {\bar{s}}_{11} σ_{x} + {\bar{d}}_{31} E_{z}

(2)

where ${\bar{s}}_{11} = s_{11} - s_{12}^{2} / s_{22}$ and ${\bar{d}}_{31} = d_{31} - d_{32} s_{12} / s_{22}$ , and $E_{z} = - ϕ_{s} / h_{p}$ is the electric field along the $z$ -direction. Equation (2) is also valid for an elastic medium with $d_{ij} = 0$ .

Figure 2.

Stress transfer between transducer and host plate, and variations of inplane displacement across different elements: (a) free body diagrams and (b) variations of inplane displacement u.

In the 1D model, the displacement field in the plate is assumed to follow the classical Kirchhoff theory approximations, while the piezoelectric transducer is assumed to undergo a membrane deformation with a uniform inplane displacement across the thickness. Let $u_{1}$ , $ε_{x}^{(1)}$ , and $σ_{x}^{(1)}$ denote the inplane displacement, inplane strain and stress, respectively, at the top of the host plate and $u_{2}$ , $ε_{x}^{(2)}$ , and $σ_{x}^{(2)}$ denote their values at the bottom of the transducer (which are of course uniform across its thickness). Figure 2(b) illustrates the variations of the inplane displacement across the plate, adhesive, and transducer layers. With the Kirchhoff plate assumptions, the equation of motion for the plate (Figure 2) at its top surface can be obtained as

h σ_{x, x}^{(1)} + α τ = ρ h {\overset{\cdot\cdot}{u}}_{1}, - a / 2 \leq x \leq a / 2

(3)

where a subscript comma followed by $x$ , that is, ( ) $, x$ , denotes partial differentiation with respect to $x$ , and an over dot (·) denotes partial differentiation with respect to time $t$ . $α$ is called the modal participation factor whose value is 4 in the present case when both inplane (symmetric) and flexural (anti-symmetric) waves are actuated. However, if two collocated piezoelectric transducers are installed on both sides of the plate, $α = 1$ when only symmetric mode is excited, whereas $α = 3$ when only antisymmetric mode is excited.

As the inplane stiffness of the adhesive layer is neglected, its inertia in the longitudinal direction is assumed to be resisted by the piezoelectric transducer and hence added to the inertia term in the equation of motion of the transducer. Assuming a linear variation of the inplane displacement $u$ across the thickness of the adhesive layer (Figure 2(b)), its inertia per unit area in the longitudinal direction is $I_{a} = ρ_{a} h_{a} ({\overset{\cdot\cdot}{u}}_{1} + {\overset{\cdot\cdot}{u}}_{2}) / 2$ . Thus, on neglecting bending, the equation of motion for the piezoelectric transducer (Figure 2) inclusive of the adhesive inertia is obtained as

h_{p} σ_{x, x}^{(2)} - τ = (ρ_{p} h_{p} + ρ_{a} h_{a} / 2) {\overset{\cdot\cdot}{u}}_{2} + ρ_{a} h_{a} {\overset{\cdot\cdot}{u}}_{1} / 2

(4)

The shear stress-strain and strain-displacement relations for the bonding layer under pure shear yields

τ = (u_{2} - u_{1}) / β

(5)

where $β = h_{a} / G_{a}$ is the shear compliance parameter of the adhesive layer.

Using equation (5), the constitutive equation (2) and the strain-displacement relations, equations (3) and (4) can be expressed as

\frac{h}{{\bar{s}}_{11}^{s}} u_{1, xx} + α (u_{2} - u_{1}) / β = ρ h {\overset{\cdot\cdot}{u}}_{1}

(6)

\begin{matrix} \frac{h_{p}}{{\bar{s}}_{11}^{p}} u_{2, xx}^{(2)} - (u_{2} - u_{1}) / β = (ρ_{p} h_{p} + ρ_{a} h_{a} / 2) {\overset{\cdot\cdot}{u}}_{2} \\ + ρ_{a} h_{a} {\overset{\cdot\cdot}{u}}_{1} / 2 \end{matrix}

(7)

The subscript/superscript $s$ and $p$ are used to represent the variables related to the structure and the piezoelectric transducer, respectively.

For a harmonic excitation $ϕ_{a} (t) = {\tilde{ϕ}}_{a} e^{- i ω t}$ , the solution shall also be harmonic:

u_{1} (x, t) = u_{1} (x) e^{- i ω t}, u_{2} (x, t) = u_{2} (x) e^{- i ω t}

(8)

where $ω$ is the angular frequency. Substituting $u_{2}$ from equations (6) into (7) and using equation (8) yields a fourth order ordinary differential equation for $u_{1}$ for the transducer region ( $- a / 2 \leq x \leq a / 2$ )

u_{1, xxxx} - Γ_{0} u_{1, xx} - Γ_{1} u_{1} = 0

(9)

where

\begin{matrix} Γ_{0} = (\frac{α {\bar{s}}_{11}^{s}}{β h} + \frac{{\bar{s}}_{11}^{p}}{β h_{p}}) - (ρ {\bar{s}}_{11}^{s} + ρ_{p} {\bar{s}}_{11}^{p} + \frac{ρ_{a} h_{a} {\bar{s}}_{11}^{p}}{2 h_{p}}) ω^{2} \\ = Γ^{2} - γ_{d_{1}} \\ Γ_{1} = \frac{{\bar{s}}_{11}^{p} {\bar{s}}_{11}^{s}}{h h_{p} β} (ρ h + α ρ_{p} h_{p} + α ρ_{a} h_{a}) ω^{2} \\ - (ρ_{p} + \frac{ρ_{a} h_{a}}{2 h_{p}}) ρ {\bar{s}}_{11}^{s} {\bar{s}}_{11}^{p} ω^{4} \\ = Γ_{d} - γ_{d_{2}} \end{matrix}

(10)

$Γ$ is the conventional static shear-lag parameter (Crawley and Luis, 1987; Kapuria and Agrahari, 2018). Considering that the geometry and loading are symmetric and therefore retaining only the antisymmetric terms, the solution of equation (9) is obtained as

u_{1} = C_{2} \sin (λ_{1} x) + C_{4} \sinh (λ_{2} x), - a / 2 \leq x \leq a / 2

(11)

where

\begin{array}{l} λ_{1} = {(\frac{{(Γ_{0}^{2} + 4 Γ_{1})}^{1 / 2} - Γ_{0}}{2})}^{1 / 2} \\ λ_{2} = {(\frac{{(Γ_{0}^{2} + 4 Γ_{1})}^{1 / 2} + Γ_{0}}{2})}^{1 / 2} \end{array}

(12)

and $C_{i}$ are arbitrary constants. Substituting equations (11) and (8) into equation (6), the displacement in the transducer is obtained as

u_{2} = γ_{1} C_{2} \sin (λ_{1} x) + γ_{2} C_{4} \sinh (λ_{2} x)

(13)

where

\begin{matrix} γ_{1} = 1 - \frac{β h ρ ω^{2}}{α} + \frac{β h λ_{1}^{2}}{α {\bar{s}}_{11}^{s}}, γ_{2} = 1 - \frac{β h ρ ω^{2}}{α} - \frac{β h λ_{2}^{2}}{α {\bar{s}}_{11}^{s}} \end{matrix}

(14)

In the region beyond the actuator patch ( $| x | \geq a / 2$ ), $τ = 0$ in equation (3). For this region, the solution for $u_{1}$ takes the form

u_{1} = C_{1}^{'} e^{i λ_{0} x_{1}}, 0 < x_{1} < \infty

(15)

where $λ_{0} = ω \sqrt{ρ {\bar{s}}_{11}^{s}}$ , $C_{1}^{'}$ is an arbitrary constant and $x_{1} = x - a / 2$ .

The stress-free boundary condition at the free edges of the actuator yields

u_{2, x} = {\bar{d}}_{31} E_{z}, at x = \mp a / 2

(16)

The inplane displacement and stress in the host plate need to be continuous at the interface between the transducer region and free region of the plate at $x = a / 2$ , that is, $x_{1} = 0$ . These continuity conditions are

u_{1} |_{x = a / 2} = u_{1} |_{x_{1} = 0}, σ_{x}^{(1)} |_{x = a / 2} = σ_{x}^{(1)} |_{x_{1} = 0}

(17)

The three unknown constants $C_{2}$ , $C_{4}$ , and $C_{1}^{'}$ appearing in the solution given by equations (11), (13) and (15) are obtained from the three conditions given by equations (16) and (17) .

Finally, substituting the resulting expressions for $u_{1}$ and $u_{2}$ in equation (5), the solution for interfacial shear stress $τ$ for the harmonic excitation is obtained as

\begin{matrix} τ = τ_{1} {\tilde{ϕ}}_{a} \\ [α_{2} (ρ ω^{2} {\bar{s}}_{11}^{s} - λ_{1}^{2}) \sin (λ_{1} x) - α_{1} (ρ ω^{2} {\bar{s}}_{11}^{s} + λ_{2}^{2}) \sinh (λ_{2} x)] \end{matrix}

(18)

where

\begin{matrix} τ_{1} = (\frac{h {\bar{d}}_{31}}{α h_{p} {\bar{s}}_{11}^{s}}) \\ (\frac{1}{α_{2} γ_{1} λ_{1} \cos (λ_{1} a / 2) - α_{1} γ_{2} λ_{2} \cosh (λ_{2} a / 2)}) \end{matrix}

(19)

and

\begin{matrix} α_{1} = λ_{1} \cos (λ_{1} a / 2) - i λ_{0} \sin (λ_{1} a / 2) \\ α_{2} = λ_{2} \cosh (λ_{2} a / 2) - i λ_{0} \sinh (λ_{2} a / 2) \end{matrix}

(20)

Elasticity solution for Lamb wave propagation

The wave equations for an isotropic plate (Figure 1) under plane strain condition can be expressed in terms of displacement potentials as

\begin{matrix} ϕ_{, xx} + ϕ_{, zz} = \frac{1}{{c_{L}}^{2}} \overset{\cdot\cdot}{ϕ}, ψ_{, xx} + ψ_{, zz} = \frac{1}{{c_{T}}^{2}} \overset{\cdot\cdot}{ψ} \end{matrix}

(21)

where $ϕ (x, z, t)$ and $ψ (x, z, t)$ are the potential functions, and $c_{L} = \sqrt{\frac{λ + 2 μ}{ρ}}$ and $c_{T} = \sqrt{\frac{μ}{ρ}}$ are longitudinal (pressure) and transverse (shear) wave speeds, respectively. $λ$ and $μ$ are the Lame constants which are expressed in terms of the elastic modulus and Poisson’s ratio as $λ = \frac{Y_{s} ν_{s}}{(1 + ν_{s}) (1 - 2 ν_{s})}$ , $μ = \frac{Y_{s}}{2 (1 + ν_{s})}$ . The displacements $u$ and $w$ and stresses $σ_{z}$ and $τ_{zx}$ are related to the displacement potentials by

\begin{matrix} u = ϕ_{, x} + ψ_{, z}, σ_{z} = λ ϕ_{, xx} + (λ + 2 μ) ϕ_{, zz} - 2 μ ψ_{, xz} \\ w = ϕ_{, z} - ψ_{, x}, τ_{zx} = μ (2 ϕ_{, xz} - ψ_{, xx} + ψ_{, zz}) \end{matrix}

(22)

The boundary conditions at the top and the bottom surfaces of the plate are

σ_{zz} |_{z = \pm h / 2} = 0, τ_{zx} |_{z = h / 2} = τ (x, t), τ_{zx} |_{z = - h / 2} = 0

(23)

For the harmonic electric potential and consequently harmonic interfacial shear excitation $τ (x, t) = τ e^{- i ω t}$ , the wave solution will be harmonic:

ϕ (x, z, t) = ϕ (x, z) e^{- i ω t}, ψ (z, z, t) = ψ (x, z) e^{- i ω t}

(24)

We employ the Fourier transform in $x$ direction to obtain the solution of the wave equations, which is defined by

\tilde{f} {k} = F {f (x)} = \int_{- \infty}^{\infty} f (x) e^{- ikx} dx

(25)

and the inverse transform is defined by

f (x) = F^{- 1} {\tilde{f} (k)} = \frac{1}{2 π} \int_{- \infty}^{\infty} \tilde{f} (k) e^{ikx} dk

(26)

where $k$ denotes the wavenumber. Using equation (25), the wave equation (21) and boundary conditions in equation (23) are transformed into the wavenumber domain as

\begin{matrix} {\tilde{ϕ}}_{, zz} + p^{2} \tilde{ϕ} = 0, {\tilde{ψ}}_{, zz} + q^{2} \tilde{ψ} = 0 \end{matrix}

(27)

{\tilde{σ}}_{zz} |_{z = \pm h / 2} = 0, {\tilde{τ}}_{zx} |_{z = + h / 2} = \tilde{τ}, {\tilde{τ}}_{zx} |_{z = - h / 2} = 0

(28)

where $p^{2} = \frac{ω^{2}}{{c_{L}}^{2}} - k^{2}$ , and $q^{2} = \frac{ω^{2}}{{c_{T}}^{2}} - k^{2}$ . Applying equations (25) into (18), the Fourier transform $\tilde{τ}$ of $τ$ is obtained as

\begin{matrix} \tilde{τ} = τ_{1} {\tilde{ϕ}}_{a} [α_{2} (ρ ω^{2} {\bar{s}}_{11}^{s} - λ_{1}^{2}) \int_{- \infty}^{\infty} \sin (λ_{1} x) e^{- ikx} dx \\ - α_{1} (ρ ω^{2} {\bar{s}}_{11}^{s} + λ_{2}^{2}) \int_{- \infty}^{\infty} \sinh (λ_{2} x) e^{- ikx} dx] \\ = 2 τ_{1} {\tilde{ϕ}}_{a} i [\frac{α_{2} (ρ ω^{2} {\bar{s}}_{11}^{s} - λ_{1}^{2})}{k^{2} - λ_{1}^{2}} \\ {k \sin (λ_{1} a / 2) \cos (ka / 2) - λ_{1} \cos (λ_{1} a / 2) \sin (ka / 2)} \\ - \frac{α_{1} (ρ ω^{2} {\bar{s}}_{11}^{s} + λ_{2}^{2})}{k^{2} + λ_{2}^{2}} \\ {k \sinh (λ_{2} a / 2) \cos (ka / 2) - λ_{2} \cosh (λ_{2} a / 2) \sin (ka / 2)}] \end{matrix}

(29)

Equation (27) has trigonometric functions as its solution whose arbitrary constants are determined from the boundary conditions given by equation (28). Applying inverse Fourier transform to the resulting solution in the wavenumber domain and using the residue theorem, the inplane displacement $u$ and strain $ε_{x}$ at the top surface of the plate is obtained in the physical domain as

\begin{matrix} u_{1} (x) = u (x, h / 2) = u_{1}^{S} (x) + u_{1}^{A} (x) \\ ε_{x}^{(1)} (x) = ε_{x} (x, h / 2) = ε_{x}^{S} (x) + ε_{x}^{A} (x) \end{matrix}

(30)

with

\begin{matrix} u_{1}^{S} (x) = \frac{- i}{2 μ} \sum_{k^{S}} \frac{\tilde{τ} (k^{S}) N_{s} (k^{S})}{k^{S} D_{S}^{'} (k^{S})} e^{i k^{S} x} \\ u_{1}^{A} (x) = \frac{- i}{2 μ} \sum_{k^{A}} \frac{\tilde{τ} (k^{A}) N_{A} (k^{A})}{k^{A} D_{A}^{'} (k^{A})} e^{i k^{A} x} \\ ε_{x}^{S} (x) = \frac{1}{2 μ} \sum_{k^{S}} \frac{\tilde{τ} (k^{S}) N_{s} (k^{S})}{D_{S}^{'} (k^{S})} e^{i k^{S} x} \\ ε_{x}^{A} (x) = \frac{1}{2 μ} \sum_{k^{A}} \frac{\tilde{τ} (k^{A}) N_{A} (k^{A})}{D_{A}^{'} (k^{A})} e^{i k^{A} x} \end{matrix}

(31)

and

\begin{matrix} N_{S} (k) = kq (k^{2} + q^{2}) \cos (ph / 2) \cos (qh / 2) \\ D_{S} (k) = (k^{2} - q^{2})^{2} \cos (ph / 2) \sin (qh / 2) \\ + 4 k^{2} pq \sin (ph / 2) \cos (qh / 2) \\ N_{A} (k) = - kq (k^{2} + q^{2}) \sin (ph / 2) \sin (qh / 2) \\ D_{A} (k) = (k^{2} - q^{2})^{2} \sin (ph / 2) \cos (qh / 2) \\ + 4 k^{2} pq \cos (ph / 2) \sin (qh / 2) \end{matrix}

(32)

where $k^{S}$ and $k^{A}$ denote the roots of the equations $D_{S} (k) = 0$ and $D_{A} (k) = 0$ , respectively for a given value of $ω$ . $S$ and $A$ denote the symmetric and antisymmetric Lamb wave modes, respectively. Substitution of the expression of $\tilde{τ}$ from equations (29) in (31) yields

\begin{matrix} u_{1}^{S} (x) = \frac{τ_{1} {\tilde{ϕ}}_{a}}{μ} \sum_{k^{S}} \frac{τ_{2} (k^{S}) N_{s} (k^{S})}{k^{S} D_{S}^{'} (k^{S})} e^{i k^{S} x} \\ u_{1}^{A} (x) = \frac{τ_{1} {\tilde{ϕ}}_{a}}{μ} \sum_{k^{A}} \frac{τ_{2} (k^{A}) N_{A} (k^{A})}{k^{A} D_{A}^{'} (k^{A})} e^{i k^{A} x} \\ ε_{x}^{S} (x) = \frac{i τ_{1} {\tilde{ϕ}}_{a}}{μ} \sum_{k^{S}} \frac{τ_{2} (k^{S}) N_{s} (k^{S})}{D_{S}^{'} (k^{S})} e^{i k^{S} x} \\ ε_{x}^{A} (x) = \frac{i τ_{1} {\tilde{ϕ}}_{a}}{μ} \sum_{k^{A}} \frac{τ_{2} (k^{A}) N_{A} (k^{A})}{D_{A}^{'} (k^{A})} e^{i k^{A} x} \end{matrix}

(33)

where

\begin{matrix} τ_{2} (k) = \frac{α_{2} (ρ ω^{2} {\bar{s}}_{11}^{s} - λ_{1}^{2})}{k^{2} - λ_{1}^{2}} \\ {k \sin (λ_{1} a / 2) \cos (ka / 2) - λ_{1} \cos (λ_{1} a / 2) \sin (ka / 2)} \\ - \frac{α_{1} (ρ ω^{2} {\bar{s}}_{11}^{s} + λ_{2}^{2})}{k^{2} + λ_{2}^{2}} \\ {k \sinh (λ_{2} a / 2) \cos (ka / 2) - λ_{2} \cosh (λ_{2} a / 2) \sin (ka / 2)} \end{matrix}

(34)

Sensor-plate interaction model considering inertia

Having obtained the solution for the Lamb wave at the top surface of the plate, we need to determine the electric potential induced in the sensor. The governing equation of motion developed in equation (7) for the actuator holds good for the sensor (PZT-B of Figure 1) as well:

\frac{h_{p}}{{\bar{s}}_{11}^{p}} u_{2, xx}^{(2)} - (u_{2} - u_{1}) / β = (ρ_{p} h_{p} + ρ_{a} h_{a} / 2) {\overset{\cdot\cdot}{u}}_{2} + ρ_{a} h_{a} {\overset{\cdot\cdot}{u}}_{1} / 2

(35)

Substitution of equations (8) into (35) yields

u_{2, xx} - δ_{1} u_{2} = - δ_{2} u_{1}

(36)

where

\begin{matrix} δ_{1} = \frac{{\bar{s}}_{11}^{p}}{h_{p}} {\frac{1}{β} - (ρ_{p} h_{p} + \frac{ρ_{a} h_{a}}{2}) ω^{2}} \\ δ_{2} = \frac{{\bar{s}}_{11}^{p}}{h_{p}} (\frac{1}{β} + \frac{ρ_{a} h_{a}}{2} ω^{2}) \end{matrix}

(37)

Considering that $u_{1}$ is known from the Lamb wave solution given by equation (30), equation (36) reduces to a second order ordinary differential equation, whose complementary solution is given by

u_{2}^{c} (x') = {\bar{C}}_{1} \sinh (\bar{λ} x') + {\bar{C}}_{2} \cosh (\bar{λ} x')

(38)

where $\bar{λ} = \sqrt{δ_{1}}$ and $x' = x - d$ denotes the $x$ -coordinate with respect to the center of the sensor patch. ${\bar{C}}_{1}$ and ${\bar{C}}_{2}$ are arbitrary constants to be determined from the boundary conditions. The particular solution to equation (36) is obtained as

u_{2}^{p} (x') = {\bar{γ}}_{1} u_{x}^{S} (d + x') + {\bar{γ}}_{2} u_{x}^{A} (d + x')

(39)

where

{\bar{γ}}_{1} = \frac{δ_{2}}{{k^{S}}^{2} + {\bar{λ}}^{2}}, {\bar{γ}}_{2} = \frac{δ_{2}}{{k^{A}}^{2} + {\bar{λ}}^{2}}

(40)

Thus, the complete solutions of the inplane displacement and strain of the sensor become

\begin{matrix} u_{2} = {\bar{C}}_{1} \sinh (\bar{λ} x') + {\bar{C}}_{2} \cosh (\bar{λ} x') \\ + {\bar{γ}}_{1} u_{x}^{S} (d + x') + {\bar{γ}}_{2} u_{x}^{A} (d + x') \end{matrix}

(41)

\begin{matrix} ε_{x}^{(2)} = \bar{λ} {\bar{C}}_{1} \cosh (\bar{λ} x') + \bar{λ} {\bar{C}}_{2} \sinh (\bar{λ} x') \\ + {\bar{γ}}_{1} ε_{x}^{S} (d + x') + {\bar{γ}}_{2} ε_{x}^{A} (d + x') \end{matrix}

(42)

where $ε_{x}^{S}$ and $ε_{x}^{A}$ are as defined in equation (33).

The stress-free boundary conditions at the free edge of the sensor yields

ε_{x}^{(2)} = 0 at x' = - a / 2, a / 2

(43)

Substitution of equations (42) into (43) yields

\begin{matrix} \bar{λ} {\bar{C}}_{1} \cosh (\bar{λ} a / 2) - \bar{λ} {\bar{C}}_{2} \sinh (\bar{λ} a / 2) \\ + {\bar{γ}}_{1} ε_{x}^{S} (d - a / 2) + {\bar{γ}}_{2} ε_{x}^{A} (d - a / 2) = 0 \\ \bar{λ} {\bar{C}}_{1} \cosh (\bar{λ} a / 2) + \bar{λ} {\bar{C}}_{2} \sinh (\bar{λ} a / 2) \\ + {\bar{γ}}_{1} ε_{x}^{S} (d + a / 2) + {\bar{γ}}_{2} ε_{x}^{A} (d + a / 2) = 0 \end{matrix}

(44)

which are solved for constants ${\bar{C}}_{1}$ and ${\bar{C}}_{2}$ to obtain

{\bar{C}}_{1} = - \frac{{\bar{γ}}_{1} ε_{0}^{S} + {\bar{γ}}_{2} ε_{0}^{A}}{2 \bar{λ} \cosh (\bar{λ} a / 2)}, {\bar{C}}_{2} = - \frac{{\bar{γ}}_{1} ε_{d}^{S} + {\bar{γ}}_{2} ε_{d}^{A}}{2 \bar{λ} \sinh (λ a / 2)}

(45)

where

\begin{matrix} ε_{0}^{S} = ε_{x}^{S} (d + a / 2) + ε_{x}^{S} (d - a / 2) \\ ε_{0}^{A} = ε_{x}^{A} (d + a / 2) + ε_{x}^{A} (d - a / 2) \\ ε_{d}^{S} = ε_{x}^{S} (d + a / 2) - ε_{x}^{S} (d - a / 2) \\ ε_{d}^{A} = ε_{x}^{A} (d + a / 2) - ε_{x}^{A} (d - a / 2) \end{matrix}

(46)

The transverse electric displacement $D_{z}$ at the sensor surface for the plane strain case is given by

D_{z} = {\bar{d}}_{31} σ_{x} + {\bar{ϵ}}_{33} E_{z}

(47)

where ${\bar{ϵ}}_{33} = ϵ_{33} - d_{32}^{2} / s_{22}^{p}$ , $ϵ_{33}$ being the dielectric constant of the sensor along $z$ -direction. Using equation (2), equation (47) can be expressed as

D_{z} = {\bar{e}}_{31} ε_{x} + {\bar{η}}_{33} E_{z}

(48)

where ${\bar{e}}_{31} = {\bar{d}}_{31} / {\bar{s}}_{11}^{p}$ and ${\bar{η}}_{33} = {\bar{ϵ}}_{33} - {\bar{d}}_{31}^{2} / {\bar{s}}_{11}^{p}$ . Considering that the top surface of the sensor is under open circuit condition and is also equipotential as it is electroded, we have

\int_{- a / 2}^{a / 2} D_{z} dx' = 0

(49)

On substitution of equations (48) into (49) yields the potential ${\bar{ϕ}}_{s}$ induced at the top surface of the sensor as

{\tilde{ϕ}}_{s} = S_{p} \int_{- a / 2}^{a / 2} ε_{x}^{(2)} dx'

(50)

\Rightarrow {\tilde{ϕ}}_{s} = S_{p} {u_{2} (a / 2) - u_{2} (- a / 2)}

(51)

where $S_{p} = \frac{{\bar{e}}_{31} h_{p}}{{\bar{η}}_{33} a} = \frac{h_{p}}{a} \frac{d_{31} Y_{p}}{ϵ_{33} (1 - v_{12}^{p}) - 2 d_{31}^{2} Y_{p}}$ (for class 6 mm symmetry). Using equation (41) in conjunction with equations (45) and (51) yields

\begin{matrix} {\tilde{ϕ}}_{s} = S_{p} [- \frac{\tanh (\bar{λ} a / 2)}{\bar{λ}} ({\bar{γ}}_{1} ε_{0}^{S} + {\bar{γ}}_{2} ε_{0}^{A}) + {\bar{γ}}_{1} {u_{x}^{S} (d + a / 2) \\ - u_{x}^{S} (d - a / 2)} + {\bar{γ}}_{2} {u_{x}^{A} (d + a / 2) - u_{x}^{A} (d - a / 2)}] \end{matrix}

(52)

Note that when the inertia is neglected in the sensor model, $δ_{1} = δ_{2} = \frac{{\bar{s}}_{11}^{p}}{β h_{p}}$ , and ${\bar{γ}}_{1} = \frac{{\bar{λ}}^{2}}{{k^{S}}^{2} + {\bar{λ}}^{2}}$ , ${\bar{γ}}_{2} = \frac{{\bar{λ}}^{2}}{{k^{A}}^{2} + {\bar{λ}}^{2}}$ with $\bar{λ} = \sqrt{\frac{{\bar{s}}_{11}^{p}}{β h_{p}}}$ in the above expression for ${\tilde{ϕ}}_{s}$ .

Equation (52) can be expressed in the form

{\tilde{ϕ}}_{s} = H (ω) {\tilde{ϕ}}_{a}

(53)

where $H (ω)$ is the transfer function between the input actuation and the sensor output. For an arbitrary input potential $ϕ_{a} (t)$ , ${\tilde{ϕ}}_{a} (ω)$ is obtained by performing Fourier transform as

{\tilde{ϕ}}_{a} (ω) = \int_{- \infty}^{\infty} ϕ_{a} (t) e^{i ω t} dt

(54)

Having obtained ${\tilde{ϕ}}_{s} (ω)$ using equation (53) in the frequency domain, its inverse Fourier transform yields the sensor response in the time domain as

ϕ_{s} (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{ϕ}}_{s} (ω) e^{- i ω t} d ω

(55)

Time-reversal response of Lamb wave

In the time-reversal process, the forward response is reversed in time and then reemitted at the sensor (PZT-B). The time reversal operation in the time domain is equivalent to taking complex conjugate of the signal in the frequency domain. Thus, using equation (53), the time reversed version of the forward response in frequency domain is obtained as

{\tilde{ϕ}}_{tr} (ω) = {\tilde{ϕ}}_{s}^{*} (ω) = H^{*} (ω) {\tilde{ϕ}}_{a}^{*} (ω)

(56)

where a superscript * denotes complex conjugate of a variable. The reconstructed response at the actuation point is obtained again by using equation (53) as

{\tilde{ϕ}}_{rc} (ω) = H (ω) {\tilde{ϕ}}_{tr} (ω)

(57)

which, on substitution of equation (56), yields

{\tilde{ϕ}}_{rc} (ω) = H (ω) H^{*} (ω) {\tilde{ϕ}}_{a}^{*} (ω) = | H (ω) |^{2} {\tilde{ϕ}}_{a}^{*} (ω)

(58)

Finally, the reconstructed signal in the time domain is obtained by applying the inverse Fourier transform to ${\tilde{ϕ}}_{rc}$ as

ϕ_{rc} (t) = \frac{1}{2 π} \int_{- \infty}^{\infty} {\tilde{ϕ}}_{rc} (ω) e^{- i ω t} d ω .

(59)

Time reversibility

The time reversibility of the system at a particular excitation frequency is obtained by comparing the main wave mode of the reconstructed response with the original input signal after normalizing with respect to corresponding peak amplitude. The time reversibility is quantified by percent similarity between the reconstructed and the input signal based on $L_{2}$ norm error given by:

\begin{matrix} % Similarity = \\ (1 - \sqrt{\int_{t_{i}}^{t_{f}} {[V (t) - I (t)]}^{2} dt / \int_{t_{i}}^{t_{f}} I^{2} (t) dt}) \times 100 \end{matrix}

(60)

where $V$ and $I$ denote reconstructed and input response respectively and $t_{i}$ and $t_{f}$ denotes the starting and end time instances of the signal under comparison. The initial and final time instance of the main mode waveform of the reconstructed signal is given by

t_{i} = t_{m} - t_{sl} / 2, t_{f} = t_{m} + t_{sl} / 2

where $t_{m}$ and $t_{sl}$ denotes the time instances of the peak of the main mode of the reconstructed signal and the width of the input signal respectively.

Results and discussion

Validation

To validate the proposed analytical model, a detailed FE simulation is performed for Lamb wave actuation, propagation, and sensing using the commercially available FE software ABAQUS 6.14 standard. Figure 3 shows the transducer-adhesive-plate system considered for this purpose, with $h = 3$ mm, $h_{a} = 50$ $μ$ m, $h_{p} = 0.1$ mm, $a = 8$ mm, and $d = 300$ mm. The material properties of the aluminum plate, araldite adhesive, and the SP-5H piezoelectric transducers are listed in Table 1. A 2D plane strain FE model is developed for the analysis. The plate and the adhesive layer are modelled using the eight-node biquadratic plane strain quadrilateral element (CPE8R) with reduced integration. Similarly, the piezoelectric transducers are discretized using the eight-node biquadratic plane strain piezoelectric quadrilateral element (CPE8RE), with reduced integration. A mesh size of 0.5 mm × 0.5 mm is considered for the aluminum plate (Figure 4). A finer mesh with element length of 0.1 mm is considered for the piezoelectric transducers and the adhesive layers, to accurately capture the transducer-plate interaction. The same fine mesh of 0.1 mm is also adopted for the plate in and around the transducer patch region. The transducers and the adhesive layers are discretized by two and three elements in the thickness direction, respectively. The chosen mesh satisfies the requirement of minimum of 20 elements per shortest wavelength for accurate solution of wave propagation problems. The bottom surfaces of the piezoelectric transducers are electrically grounded, and the potential response is obtained at the top surface of the sensor after applying an equipotential constraint. The numerical analysis is conducted using the dynamic implicit scheme, with an integration time step of 100 ns for the excitations with central frequency up to 300 kHz and 50 ns for excitations beyond that. The chosen time step satisfies the requirement of at least 20 time steps per one wave-cycle of the excitation for good temporal resolution.

Figure 3.

Transducer-adhesive-plate system considered for analysis.

Table 1.

Material properties.

Material	$Y_{1}$ (GPa)	$Y_{2}$ (GPa)	$Y_{3}$ (GPa)	$G_{23}$ (GPa)	$G_{13}$ (GPa)	$G_{12}$ (GPa)	$ν_{23}$	$ν_{13}$	$ν_{12}$	ρ (Kg $m^{- 3}$ )
Aluminum^a	70.30	70.30	70.30	26.42	26.42	26.42	0.33	0.33	0.33	2700
Adhesive^b	4.70	4.70	4.70	1.67	1.67	1.67	0.4	0.4	0.4	1700
SP-5H^c	66.67	66.67	47.62	23.00	23.00	23.50	0.51	0.51	0.29	7500
	$d_{31}$ (×10⁻¹² mV⁻¹)	$d_{32}$ (×10⁻¹² mV⁻¹)	$d_{33}$ (×10⁻¹² mV⁻¹)	$d_{15}$ (×10⁻¹² mV⁻¹)	$d_{24}$ (×10⁻¹² mV⁻¹)		$ϵ_{11} / ϵ_{0}$ ^d	$ϵ_{22} / ϵ_{0}$	$ϵ_{33} / ϵ_{0}$
SP-5H^c	−265.0	−265.0	550.0	741	741		3100	3100	3400

Agrahari and Kapuria (2016b).

Epoxy (2008).

Sparkler (2018).

Electrical permittivity of free space $ϵ_{0} = 8.854 \times 10^{- 12}$ F/m (Hooker, 1998).

Figure 4.

Mesh configuration shown for a portion of actuator patch in the 2D FE model.

For the forward response, PZT-A (Figure 3) is actuated with a Hann window modulated five cycle tone burst signal of 20 V amplitude. The input signal is given by $V (t) = \frac{V_{0}}{2} [1 - \cos (2 π t / T_{H})] \sin (2 π ft)$ , where $f$ denotes the central frequency of the excitation and $T_{H}$ represents the Hann window length obtained by using number of cycles in the tone burst $N_{B}$ as $T_{H} = N_{B} / f$ . For the reconstructed response, PZT-B is actuated with time reversed scaled version of the forward signal at the same location and the electric potential output is recorded at PZT-A.

First, to validate the proposed model in respect of Lamb wave generation and propagation, the inplane strain $ε_{x}$ generated at the plate’s top surface below the center of the sensor patch is compared with the 2D FE simulation results. Figure 5 depicts this comparison for 100, 200, and 300 kHz excitations, which also includes the analytical response obtained neglecting the inertia effects in the transducer modeling. The present results considering inertia effects show excellent agreement with the 2D FE solution for excitation frequencies upto 200 kHz, thereby validating the present actuator-plate interaction model. For the higher excitation frequency, the present solution shows small differences with the 2D FE solution in the amplitudes of $S_{0}$ and $A_{0}$ modes, which could be due to the neglect of bending and/or the peel stress at the actuator-plate interface in the actuator model. Clearly, the neglect of the inertia terms of the actuator, adhesive layer and the plate in the actuator model has a significant effect on the results, for excitation frequencies of 200 kHz and above. It reduces the amplitude of the strain waveforms for both $S_{0}$ and $A_{0}$ modes, but in different proportions depending on the excitation frequency.

Figure 5.

Longitudinal strain $ε_{x} (d, h / 2)$ at the top surface of the host plate at sensor location under five-cycle Hann window modulated tone burst with central frequency of: (a) 100 kHz, (b) 200 kHz, and (c) 300 kHz.

The sensor potential $ϕ_{s}$ obtained from the present solution is compared next in Figure 6 with the 2D FE solution. The voltage response obtained considering the inertia effect shows very good agreement with the 2D FE solution for excitations of upto 300 kHz frequency considered here. This validates the accuracy of the proposed sensor-plate dynamic interaction model. The neglect of the inertia terms in the transducer models has a similarly significant effect on the sensor response as observed in the case of Lamb strain.

Figure 6.

Sensor potential $ϕ_{s}$ under five-cycle Hann window modulated tone burst with central frequency of: (a) 100 kHz, (b) 200 kHz, and (c) 300 kHz.

For further validation, the present solution is also compared with the experimental results presented by Sharma et al. (2020) for a similar system as in Figure 3 with the same plate and adhesive layer thickness but with transducers of 0.25 mm thickness. The transducer size used in the experiment is 10 mm, but a reduced size of 9 mm is considered for the 2D FE simulation and the analytical solution to compensate for the possible weak bonding around the edges of the transducer in the experimental set-up (Sharma et al., 2020). A recent study (Agrahari and Kapuria, 2016a) has shown that although a full-field 3D FE solution differs from the simplified plane strain 2D FE solution in terms of actual magnitude of the voltage response, the two solutions match well when they are normalized with respect to their respective peaks. Therefore, we compare the present analytical and 2D FE solutions with the experimental data after normalization. Figure 7 depicts this comparison for an excitation frequency of 200 kHz, which shows good agreement between the three results.

Figure 7.

Comparison of present solution with 2D FE solution and experimental data for a 200 kHz modulated five-cycle tone burst excitation in a 3 mm thick plate.

Effect of inertia terms on Lamb wave response

A detailed study is conducted in this section using the developed analytical model to illustrate the effect of the inertia terms in transducer modeling on the Lamb wave strain and sensor response as well as the influence of various geometrical and material parameters on the same. For this purpose, we consider a system with $h = 3$ mm, $h_{a} = 50$ $μ$ m, $h_{p} = 0.25$ mm, $a = 8$ mm, and $d = 300$ mm, and material properties as given in Table 1, unless stated otherwise.

Effect of inertia of individual components

To understand the effect of inertia of the individual components of the system, that is, the plate, adhesive layer, and the piezoelectric transducers on the Lamb wave and the sensor response, the solution is obtained for four different cases: (i) considering inertia of all constituents (All inertia), (ii) neglecting the transducer inertia ( $ρ_{p} = 0$ ), (iii) considering only plate inertia ( $ρ_{p} = ρ_{a} = 0$ ), and (iv) without considering inertia of any constituent (No inertia). The peak amplitudes of Lamb strain $ε_{x} (d, h / 2)$ for these four cases are plotted in Figure 8 as a function of the excitation frequency for both $S_{0}$ and $A_{0}$ modes.

Figure 8.

Effect of inertia on the peak amplitude of the strain $ε_{x}$ at top surface of the host plate at sensor location ( $x = d$ ) for: (a) $S_{0}$ mode and (b) $A_{0}$ mode.

The peak amplitudes of the waveforms in the response signal are obtained from its Hilbert envelope (Sharma et al., 2020). For both $S_{0}$ and $A_{0}$ modes, the inclusion of the inertia term of each constituent element increases the peak amplitude in the frequency range considered. The inertia effect is very small at low excitation frequencies (≤50 kHz), becomes maximum at certain frequency and then reduces. The actuator inertia has the maximum effect on the strain amplitude, whereas the inertia of the adhesive layer has the minimum and negligible effect. The adhesive layer density and thickness are very small and consequently its inertia effect is negligible. But, although $ρ_{p} h_{p}$ ( $= 1.875$ kg/ $m^{2}$ ) for the actuator is less than $ρ h$ ( $= 8.1$ kg/ $m^{2}$ ) of the plate, the transducer inertia has a much larger influence on the strain response, which is due to the lateral spring effect of the adhesive layer. The relative effect of inertia on the strain amplitude is generally higher for the $A_{0}$ mode than the $S_{0}$ mode.

When the inertia effect is neglected in the actuator-plate interaction model, the peak of $S_{0}$ mode strain occurs at an excitation frequency of 320 kHz. However, the inclusion of the plate inertia alone shifts the frequency corresponding to the peak $S_{0}$ mode strain to 340 kHz. Remarkably, further addition of the actuator inertia shifts the frequency of the peak $S_{0}$ mode strain backward to 290 kHz. In contrast, the excitation frequency corresponding to the peak $A_{0}$ mode strain shows a monotonous increase with the cumulative inclusion of the inertia effects of the plate, adhesive, and the actuator.

Figure 9 depicts similar variations of the peak amplitude of the sensor potential response with the excitation frequency for $S_{0}$ and $A_{0}$ modes for the four inertia cases discussed above. Like the Lamb strain case, the transducer inertia has the most significant effect on the peak amplitude of the sensor potential followed by the plate inertia with the adhesive inertia term having a very small effect, for both the Lamb wave modes. Similarly, the inclusion of the plate inertia shifts the excitation frequency corresponding to the peak $S_{0}$ mode potential to right (from 260 to 290 kHz). However, unlike the strain case, further addition of the transducer inertia does not alter the frequency for the peak amplitude, which may be the result of the combined dynamic effect of the actuator and the sensor. For $A_{0}$ mode, an increasing trend of the excitation frequency corresponding to the peak amplitude of the sensor potential is observed with the sequential addition of inertia effect of the constituents, but it is not as prominent as in the case of Lamb strain in the plate.

Figure 9.

Effect of inertia on the peak amplitude of sensor potential for: (a) $S_{0}$ mode and (b) $A_{0}$ mode.

Influence of adhesive layer thickness on inertia effect

To understand how the thickness of the bonding layer influences the inertia effect, the peak amplitude of the Lamb strain for three values of adhesive thickness is plotted against the excitation frequency in Figure 10, for both $S_{0}$ and $A_{0}$ modes, considering and neglecting the inertia effects of the system. Similar plots for the sensor potential are presented in Figure 11. The results show that within the considered range of adhesive thickness from 0.01 to 100 $μ$ m (which is usually adopted in SHM applications), an increase in the adhesive thickness significantly enhances the inertia effect on the Lamb strain and the sensor potential for both modes. The enhancing effect is higher for the sensor potential than the strain in the plate, possibly because of the combined dynamic effect of the actuator and the sensor on the sensor voltage. The increase in the adhesive thickness causes a rightward shift of the excitation frequency at which the peak amplitude is maximum for both strain and sensor potential responses for $S_{0}$ and $A_{0}$ modes.

Figure 10.

Effect of adhesive thickness on inertia effect on the strain at top surface of the host plate at sensor location ( $x = d$ ):(a) $S_{0}$ mode and (b) $A_{0}$ mode.

Figure 11.

Effect of adhesive thickness on inertia effect on the sensor potential: (a) $S_{0}$ mode and (b) $A_{0}$ mode.

Effect of transducer thickness on inertia effect

The variations of the peak amplitude of the $S_{0}$ and $A_{0}$ modes of the Lamb strain and the sensor potential response as a function of the excitation frequency are plotted in Figures 12 and 13, respectively, for three values of the transducer thickness $h_{p} = 0.1$ , 0.25, and 0.5 mm, both by considering and neglecting the system inertia. Similar to the case of adhesive layer, the increase in the transducer thickness significantly amplifies the inertia effect on the peak $S_{0}$ and $A_{0}$ mode amplitudes of both plate strain and sensor potential. Further, the enhancement is higher for the sensor potential than the plate strain due to the combined effect of actuator inertia and sensor inertia in the former case. When the inertia effect is not considered, an increase in the actuator thickness causes a decrease in the peak amplitude of both $S_{0}$ and $A_{0}$ modes of the Lamb strain. This is expected due to lower electric field in thicker actuator for the same input potential. However, when the system inertia is considered, it results in a higher peak amplitude of the Lamb strain for a thicker actuator albeit for a certain range of excitation frequency. Such change in the trend due to the inclusion of the system inertia is, however, not observed for the sensor potential in the probed frequency range (Figure 13). The inertia effect also aggravates the transducer thickness’s influence on the excitation frequency at which maximum peak strain occurs for both $S_{0}$ and $A_{0}$ modes.

Figure 12.

Effect of transducer thickness on inertia effect on the strain at top surface of the host plate at sensor location ( $x = d$ ): (a) $S_{0}$ mode and (b) $A_{0}$ mode.

Figure 13.

Effect of transducer thickness on inertia effect on sensor potential: (a) $S_{0}$ mode and (b) $A_{0}$ mode.

Effect of plate thickness on the inertia effect

The peak amplitude of Lamb strain versus excitation frequency curve for the system with plate thickness of 1.5 and 3 mm are compared in Figure 14, for $S_{0}$ and $A_{0}$ modes, by considering and neglecting the inertia effect. A similar comparison of the sensor potential is presented in Figure 15. The plots show that contrary to the effect of transducer thickness, an increase in the plate thickness causes a reduction in the relative effect of inertia on the strain and sensor potential response for the $S_{0}$ mode. This mode shows a higher peak amplitude for both strain and sensor potential in case of the thinner plate at all frequencies in the probed range, whether or not the inertia is included. However, in case of the $A_{0}$ mode, a thinner plate generates a lower peak amplitude at certain frequencies, for both strain and sensor potential.

Figure 14.

Effect of plate thickness on inertia effect on the strain at top surface of the host plate at sensor location ( $x = d$ ): (a) $S_{0}$ mode and (b) $A_{0}$ mode.

Figure 15.

Effect of plate thickness on inertia effect on sensor potential: (a) $S_{0}$ mode and (b) $A_{0}$ mode.

Effect of inertia on time reversibility

To study the effect of inertia on time reversibility, the percent similarity of the reconstructed signal after TRP obtained using the present model with the original input signal is plotted in Figure 16 as a function of the central excitation frequency, considering (i) the system inertia, (ii) only plate inertia, and (iii) no inertia. The inclusion of the plate inertia improves the time reversibility at all frequencies within the probed frequency range. But, when the transducer inertia is also added, the improvement occurs only over a certain range of frequency, beyond which the percent similarity drops. The frequency at which the percent similarity is maximum (called best reconstruction frequency) shifts leftward from 115 kHz when the system inertia is included to 85 kHz when only the plate inertia is included and to 74 kHz when inertia is neglected. In recent studies (Agrahari and Kapuria, 2016a, 2016b), the best reconstruction frequency $f_{rc}$ has been proposed to be used as the probing frequency for enhanced sensitivity of the TRP based SHM. Contrary to expectation, the inclusion of the transducer inertia raises the maximum percent similarity at the best reconstruction frequency, from 93.6% to 99.3%, although it worsens the time reversibility in most part of the frequency range (below 92 kHz and above 218 kHz).

Figure 16.

Effect of inertia on time reversibility of Lamb wave induced electric potential.

To appreciate the time reversibility of Lamb wave at different frequencies, the main mode waveform of the normalized reconstructed signal is compared with the normalized input signal in Figure 17 for the best reconstruction frequency (115 kHz) and two other values of frequency, 180 and 350 kHz. The results are presented for both full inertia and no inertia cases. The deviation of the reconstructed waveform from the original input signal, caused by the amplitude dispersion, is clearly the least at the best reconstruction frequency and in tune with the percent similarity (Figure 16).

Figure 17.

Comparison of reconstructed signal with original input signal for: (a) 115 kHz (best reconstruction frequency), (b) 180 kHz, and (c) 350 kHz excitation frequency.

Effect of adhesive layer thickness on inertia effect on $f_{rc}$

Figure 18 shows the variation of the best reconstruction frequency ( $f_{rc}$ ) with adhesive layer thickness obtained by considering and neglecting the system inertia in transducer modeling. It is seen that when the inertia effect is not considered, the best reconstruction frequency remains nearly constant at 74 kHz for the adhesive layer thickness up to 60 $μ$ m beyond which it increases to 85 kHz at 70 $μ$ m thickness and remains constant up to a thickness of 100 $μ$ m. However, $f_{rc}$ shows much higher sensitivity to the adhesive layer thickness, when the inertia effect is not neglected. It shows a rise with an increase in the adhesive layer thickness up to 60 $μ$ m and thereafter shows a decrease. The inclusion of inertia effect raises the best reconstruction frequency for all values of adhesive layer thickness.

Figure 18.

Variation of best reconstruction frequency with adhesive layer thickness.

Effect of transducer thickness on inertia effect on $f_{rc}$

The variations of the best reconstruction frequency $f_{rc}$ with the transducer thickness obtained considering and neglecting the system inertia are compared in Figure 19. It is seen that when the system inertia is not considered, similar to the adhesive thickness, transducer thickness has a marginal influence on the best reconstruction frequency, maintaining a constant value for $f_{rc}$ for different ranges of the transducer thickness. However, when the system inertia is included, $f_{rc}$ shows a very strong dependence on the transducer thickness. It increases from 85 kHz for transducer thickness ( $h_{p}$ ) of 0.1 mm to 129 kHz for $h_{p} = 0.3$ mm and reduces on further increase of $h_{p}$ to even below the no-inertia case for $h_{p} = 0.58$ mm.

Figure 19.

Variation of best reconstruction frequency with PZT thickness.

Effect of transducer length on inertia effect on $f_{rc}$

The system inertia greatly influences the nature of dependency of the best reconstruction frequency on transducer length as well (Figure 20). The best reconstruction frequency shows a monotonic decrease with the increase in the transducer length ( $a$ ), when the inertia effect is neglected. However, in the presence of the inertia, $f_{rc}$ attains a peak for $a =$ 6 mm and then reduces with further increase in transducer length. It is seen that in the presence of the inertia effect, the best reconstruction occurs at higher excitation frequencies for a transducer length of higher than 5.5 mm.

Figure 20.

Variation of best reconstruction frequency with PZT length.

Effect of plate thickness on inertia effect on $f_{rc}$

When the inertia of the system is not taken into account, the best reconstruction frequency shows a monotonous increase with the increase in the plate thickness (Figure 21). However, when the inertia effect is considered, the best reconstruction frequency reaches the maximum at a plate thickness of 4 mm and decreases thereafter.

Figure 21.

Variation of best reconstruction frequency with plate thickness.

Conclusions

An analytical model has been developed for actuation and sensing of Lamb waves in an isotropic plate with surface-bonded transducers under plane strain assumption, considering the shear-lag effect due to the bonding layer as well as the inertia effects of all constituents in transducer modeling. For this purpose, a consistent sensor-plate interaction model considering the aforementioned effects and satisfying the boundary conditions of the sensor has been developed for the first time. A closed-form solution is obtained in the frequency domain for the forward response and the TRP. The time domain response is obtained by employing the inverse Fourier transform. The new model has been validated in comparison with the 2D FE analysis and the experimental data. A detailed parametric study is conducted to ascertain the effect of inertia on the forward and reconstructed response after the TRP. The numerical study has revealed the following:

The present analytical solution considering inertia effects shows very good agreement with the detailed 2D FE analysis results for the Lamb strain in the plate and the sensor potential for both $S_{0}$ and $A_{0}$ modes, thereby establishing the accuracy of the actuator and sensor interaction models. The normalized potential response obtained from the proposed model matches well also with the available experimental data, further validating the new model.

The omission of inertia terms, on the other hand, resulted in significant errors in the peak amplitudes of Lamb strain and the sensor potential for both symmetric and antisymmetric modes for excitation frequencies greater than 100 kHz.

The inclusion of system inertia in the transducer models has a large influence on both $S_{0}$ and $A_{0}$ mode responses for the longitudinal strain in the plate as well as the sensor potential, for frequencies greater than 50 kHz. The relative effect of the system inertia on the peak amplitudes of the $S_{0}$ and $A_{0}$ modes does not always increase with increasing frequency of excitation, but attains a peak at a certain frequency (which is different for different modes and response entities) and reduces thereafter.

The transducer inertia has the most significant effect on Lamb wave response for both the symmetric and antisymmetric modes, followed by the plate inertia. As expected, the adhesive layer inertia has a minimal effect even at high excitation frequencies. The system inertia affects also the excitation frequency corresponding to maximum peak amplitude of both the fundamental modes of the Lamb strain and the sensor potential response.

The increase in both adhesive layer and transducer thickness significantly enhances the inertia effect on the peak $S_{0}$ and $A_{0}$ mode amplitudes of both Lamb strain and sensor voltage. The enhancement is higher for the sensor potential than the strain in plate, because the former sees the combined dynamic effect of both actuator and sensor.

Contrary to the effect of adhesive layer thickness and transducer thickness, an increase in the plate thickness causes a reduction in the relative effect of inertia on the Lamb strain and sensor potential response for the $S_{0}$ mode.

The time reversibility of Lamb waves improves on the inclusion of plate inertia over the probed frequency range, whereas the similarity drops for certain frequencies when the transducer inertia is added.

At the best reconstruction frequency, however, the similarity of the reconstructed signal seems to always increase with the inclusion of transducer inertia.

In the absence of the inertia effect, the adhesive layer thickness and the transducer thickness show only marginal influence on the best reconstruction frequency. However, when the system inertia is included, the best reconstruction frequency shows a strong dependency on the adhesive layer thickness as well as the transducer thickness, which reaches a peak at a certain value of both the thickness parameters.

The best reconstruction frequency shows strong dependency on the transducer length and the plate thickness, irrespective of inclusion and omission of the system inertia. However, the nature of the variation changes from a monotonic one in absence of inertia effect to having a peak in its presence.

Finally, the developed closed-form solution will be of immense use to the design of Lamb wave based SHM systems with and without using baseline signals.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: S. Kapuria acknowledges the financial support provided for this study by the Science & Engineering Research Board, Department of Science and Technology, Government of India through J. C. Bose National Fellowship (Grant No. JCB/2018/000025).

ORCID iDs

Santosh Kapuria

A Arockiarajan

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Role of transducer inertia in generation,sensing,and time-reversal process of Lamb waves in thin plates with surface-bonded piezoelectric transducers

Abstract

Keywords

Introduction

Actuator-plate interaction model considering inertia

Elasticity solution for Lamb wave propagation

Sensor-plate interaction model considering inertia

Time-reversal response of Lamb wave

Time reversibility

Results and discussion

Validation

Effect of inertia terms on Lamb wave response

Effect of inertia of individual components

Influence of adhesive layer thickness on inertia effect

Effect of transducer thickness on inertia effect

Effect of plate thickness on the inertia effect

Effect of inertia on time reversibility

Effect of adhesive layer thickness on inertia effect on f rc

Effect of transducer thickness on inertia effect on f rc

Effect of transducer length on inertia effect on f rc

Effect of plate thickness on inertia effect on f rc

Conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Effect of adhesive layer thickness on inertia effect on $f_{rc}$

Effect of transducer thickness on inertia effect on $f_{rc}$

Effect of transducer length on inertia effect on $f_{rc}$

Effect of plate thickness on inertia effect on $f_{rc}$