^{1}

^{2}

^{2}

^{3}

This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (

Elastic waves, especially guided waves, generated by a piezoelectric actuator/sensor network, have shown great potential for on-line health monitoring of advanced aerospace, nuclear, and automotive structures in recent decades. Piezoelectric materials can function as both actuators and sensors in these applications due to wide bandwidth, quick response and low costs. One of the most fundamental issues surrounding the effective use of piezoelectric actuators is the quantitative evaluation of the resulting elastic wave propagation by considering the coupled piezo-elastodynamic behavior between the actuator and the host medium. Accurate characterization of the local interfacial stress distribution between the actuator and the host medium is the key issue for the problem. This paper presents a review of the development of analytical, numerical and hybrid approaches for modeling of the coupled piezo-elastodynamic behavior. The resulting elastic wave propagation for structural health monitoring is also summarized.

Elastic waves, particularly guided waves (GWs), have shown great promise to identify damage in aerospace, aircraft and marine structures. In these applications, piezoelectric materials can be employed as actuators to generate high-frequency elastic waves that carry the structural information, based on the converse piezoelectric effect [

To avoid the difficulties associated with the complicated interfaces between the actuators and the host medium, some simplified actuator models have been used to simulate the actuation process of embedded and bonded thin sheet actuators. The uniform strain model was first developed for a cantilever beam with a layer of PVDF bonded on one side only [

Plate and shell models have also been extensively developed in modeling the piezoelectric structures. Lee and Moon [

The beam and plate models operate with the lowest fundamental (bending and longitudinal) guided wave modes, and thus they provide a simple description of the elastodynamics wave processes in the host structure. Their application is, however, restricted, since they are valid only in a low frequency range where the characteristic wavelength is much greater than the plate or shell thickness. Therefore, models based on the Rayleigh-Lamb equations for an elastic layer and/or a half-space have been attracting much attention to capture the high-order Lamb wave modes or Rayleigh surface waves. Giurgiutiu [

Various numerical simulation tools, such as commercially available finite element (FE) codes, have allowed users to conduct coupled multi-physical field solutions in a relatively convenient way. Nieuwenhuis

Compared with the numerical simulation, the analytical approach to consider the coupled dynamic behavior can give quantitative study of the interfacial stress and its resulting wave propagation. Liu

This paper presents a comprehensive review on the state of the art of modeling techniques for piezoelectric wafer actuators bonded to the elastic medium, particularly some representative analytical, numerical and hybrid approaches to model the coupled piezo-elastodynamic behavior, and some resulting ultrasonic wave phenomenon and applications relevant to SHM are also summarized.

In this section, the approaches that aim to achieve the coupled electromechanical behavior of the piezo-actuators bonded to the host structure are reviewed and summarized. These methods include both analytical, numerical and hybrid schemes.

The Euler-Bernoulli model is one of the earliest models developed for beams actuated by the piezoelectric wafers. The widely used analytical model [_{a}_{b}

However, the shear-lag theory has its intrinsic limitations: (i) the theory assumes linear strain distribution across the beam thickness, and this approximation only applies for low values of the frequency-thickness product of the lowest symmetric (S_{0}) and anti-symmetric (A_{0}) modes, and (ii) the theory cannot capture more than two lowest S_{0} and A_{0} modes with the increase of frequency.

To overcome these critical limitations, Giurgiutiu and Bottai-Santoni [_{0} and A_{0} modes as:
_{s}_{A}_{0} and A_{0} modes, respectively, and _{s}_{A}

When the bonding layer becomes thinner and stiffer, Crawley and de Luis’s [

For a perfectly bonded actuator with the limiting case of an infinitely stiff bonding layer, the shear lag parameter _{0}

The two-dimensional pin-force model for the bonded piezo-actuator was proposed in [

The simplified pin-force model can be readily applied in the host structure as traction boundary conditions to obtain the dynamic response of the system. Under plain strain assumption, Giurgiutiu [

Major limitations of the simplified pin-force model are summarized as follows: (i) The model is a good approximation only if the Young’s modulus and thickness of the actuator are small compared to those of the host structure or the bonding layer is very thin and stiff, (ii) the model can only provide qualitative estimation about the actuation mechanism for low-frequency cases, which needs to be calibrated by either numerical simulation or experimental testing, and (iii) piezoelectric resonance effects cannot be captured in the model [

Recently, Dunn _{Act}_{Act}_{Act}_{Act}

The relationship between the edge force, edge velocity, voltage, and current for the piezo-actuator in the spectral domain can be expressed as [_{Act} is assumed to apply on the edge of the actuator, which leads to a velocity at the actuator edge _{Act}_{Str}_{Str}

The application of the continuity of the displacement at the edge leads to:

However, this model is still based on the pin-force model, which cannot capture the electromechanical interfacial stress distribution between the actuator and the host structure, especially for high-frequency cases.

One major disadvantage of using plate/beam theory is that it can only approximately model the lowest A_{0} Lamb wave modes when the excitation frequency-plate thickness produce is sufficiently low. Therefore, the models employing the Rayleigh-Lamb equations for the elastic host medium attracts more attention to consider high-frequency Lamb waves and Rayleigh surface waves [

The solution of the host structure is based on the elasticity theory. In the model, the actuator thickness is assumed to be very small in comparison with its length, the applied electric filed primarily results in an axial deformation, and the following assumptions can be made: (i) _{y}_{y}_{z}_{vz}_{a}_{a}_{a}_{a}_{a}_{a}_{a}

_{0} and A_{0} modes predicted by the integral model [

Similar approaches to consider the coupled electromechanical behavior can be also found in the literatures [

Numerical simulation techniques have been widely utilized to analyze the elastic wave behavior induced by the piezo-actuators [

Hybrid approaches provide potential solutions to compensate for the disadvantages of pure FE simulation. In the hybrid schemes, the FE solution using piezoelectric elements is only conducted in limited areas (e.g., the piezo-actuation area) to obtain the prescribed excitation, and then combined with analytical guided wave excitation model in the host structure, as shown in

The most fundamental issue surrounding the effective use of piezo-actuators in SHM is the evaluation of the generated wave propagation. Based on the solution of the interfacial stress, the local dynamic response of the host medium generated by the piezo-actuator can be solved by using Fourier transform technique and solving the resulting integral equations in terms of the interfacial stress.

For its simplicity, the pin-force model [_{0} Lamb wave mode at 100 kHz in an aluminum plate by rectangular and circular actuators. It is found that the wave field excited by a rectangular actuator tends to a circular crested wave filed with angularly dependent amplitude at large distances from the actuator, while the circular actuator wave filed spatially attenuates with equally spaced peaks and troughs in the far field. The transient responses to a time-limited signal can be obtained by conducting the inverse Fourier transform of the integral of the product of the harmonic response [

From the point of view of guided wave based SHM, the tuning of a particular mode is quite important, since it allows researchers to address the detection of specific damages with specific wave modes. To find the Lamb strain induced in the plate, Giurgiutiu [_{0} mode is excited very strongly at low frequencies, while the S_{0} mode is barely observed. A preferential excitation spot of the S_{0} mode can be identified around 300 kHz for the current actuator/structural configuration. More detailed description was given in refs. [

Elastic waves have been successfully used in the nondestructive evaluation (NDE) of materials and structures. Since elastic guided waves are sensitive to the material parameters of the host medium and can propagate over long distance, they can be used to detect surface/embedded damages in structures [

To establish the quantitative relation between the surface signals and the embedded damages in materials, efforts have been made both theoretically and experimentally by “propagating” elastic waves back to the damages from the surface. The idea is based on that the wave field is reversible [

Similarly, a reverse wave technique using high-frequency piezo-induced elastic bulky wave propagation was presented to interpret the received elastic wave signals and detect embedded cracks [

Among the various schemes being considered for SHM, elastic waves generated by piezoelectric actuators have particularly shown great promise. In these applications, piezoelectric materials are usually employed as actuators to generate the high-frequency diagnostic elastic waves. To effectively use bonded piezo-actuators in these integrated SHM system, the quantitative evaluation of the induced elastic wave propagation is strongly needed. Accurate characterization of the coupled piezo-elastodynamic behavior between the actuator and the host medium is the key issue for the problem. This paper reviews the state of the art and recent advance of different modeling approaches for piezoelectric wafer actuators bonded to the elastic medium, including analytical, numerical, and hybrid approaches to model the coupled piezo-elastodynamic behavior. Some resulting ultrasonic wave phenomenon and applications relevant to SHM are also summarized. It is demonstrated that the integral model approach [

This review was partly supported by the National Science Foundation Grant No. EPS-0701890 and NASA EPSCoR RID grant.

Illustration of a piezoelectric wafer actuator bonded to the host structure. Taken from Giurgiutiu [

Variation of interfacial shear stress with respect to bond thickness. Taken from [

Free body diagram of the actuator/host structural system. Taken from Dunn

The actuator model with coupled piezo-elastodynamics. Taken from Huang and Sun [

The normalized interfacial shear stress. Taken from Wang and Huang [

The loading frequency effects on the normalized interfacial shear stress. Taken from Wang and Huang [

The comparison of the resulting Lamb waves predicted by the integral model [

FE modeling and meshing of a circular piezo-actuator bonded to a host plate.

Illustration of the scheme of the hybrid approach.

Downward view of normalized harmonic radiation field for out-of-plane surface displacement in an aluminum plate at 100 kHz, the A_{0} mode by a pair of (a) (left) square actuators and (b) (right) circular actuators. Taken from Raghavan and Cesnik [

Comparison between the normalized sensor signals obtained analytically and experimentally for the circular actuator: (a) the S_{0} mode at central frequency of 200 kHz and (b) the A_{0} mode at central frequency of 50 kHz. Taken from Raghavan and Cesnik [

Lamb wave mode tuning with varying excitation frequencies (a) the simplified pin-force model and (b) the experimental testing. Taken from Giurgiutiu [

Image of embedded cracks with various shapes. (a) The linear crack (b) the wedge crack (c) the inclined crack and (d) two collinear cracks. Taken from Wang and Huang [