Research on Time-Reversal Focusing Imaging Method to Evaluate a Multi-Layer Armor Composite Structure

Abstract

Armor composite structures have attracted interest in structural health monitoring (SHM) for their applications in damage localization. The signal propagation and the frequency dispersion features of the Lamb wave signal on thick armor composite structures are more complicated than their counterparts on other composite plates. In this research, a time-reversal localization and imaging method for impact localization of armor composite structures is proposed. First, composite sandwich structures were designed that are typically composed of ballistic-resistant ceramic materials as the face panel and a composite material as the core layer, sandwiched between metal materials serving as the backplate. The results show that the proposed method can validate the position of impact efficiently, and radial error is within 4.12 mm and 5.39 mm in single-damage and multi-damage imaging localization, respectively.

1. Introduction

Composite structures are increasingly utilized in various armor applications due to their superior characteristics. However, the performance and behavior of these structures may be compromised by damage or impacts incurred during manufacturing, service, or maintenance [1]. Cracking, impact damage, and interlayer delamination commonly manifest during the long-term utilization of armor composites [2,3,4,5]. Failure to promptly detect and repair such damage can lead to structural degradation, reducing the residual strength and fatigue performance of the materials, thus posing significant safety risks. Consequently, real-time impact monitoring for armor composite structures is imperative. Nowadays, the methods of SHM for composite structures include terahertz radiation [6], X-rays [7], piezoelectric sensors, etc.

Lightweight, highly functional, and intelligent armor protection technology utilizing single pure metals has been under development since the 1960s [8]. Presently, in order to withstand projectile attacks and corrosion, armor composite sandwich structural plating is typically composed of bulletproof ceramic material as the panel, metal material as the back plate, and an interlayer laminated with composite materials, offering high hardness, superior bulletproof performance, low density, and robust elastic-plastic characteristics. Additionally, new materials have been incorporated into armor composites [9]. During shooting tests, alumina ceramics have demonstrated the ability to enhance the bulletproof capabilities of armor composites, attributed to their high elasticity performance and stability in oxidation and reduction atmospheres at temperatures up to 1950 °C. Another material, carbon fiber, can dissipate the kinetic energy of projectiles through tensile failure, thus decelerating the velocity of warheads owing to its low fracture elongation [10]. Furthermore, armor composite is segmented into a backplane and panel. The back plate must absorb residual energy through deformation, necessitating high stiffness and bending strength. Consequently, metal materials such as armor steel, aluminum alloy, and titanium alloy are employed as armor backplating materials, owing to their low density, high specific strength, strong corrosion resistance, structural bearing capacity, and bullet resistance.

The discontinuity of transverse stress between multi-layer materials results in directional changes in wave velocity, and beam mixture contributes to the phenomenon of wave velocity propagation bending and angular changes in waveform distortion. Greater thicknesses can induce more complex signal multi-modality and dispersion phenomena, thereby amplifying the attenuation of monitoring signals and presenting challenges in damage monitoring. Indeed, the thickness of multi-layer armor composite materials can exceed 5 mm [2,3,4,5]. In the case of sandwich armor composite materials, the presence of multiple layers and substantial thicknesses, along with the discontinuity of transverse stress between the layers, can induce directional changes in wave velocity. Greater thickness results in more pronounced multi-modal and dispersion phenomena of Lamb wave signals, exacerbating the attenuation of monitoring signals and complicating the accurate determination of Lamb wave propagation speed.

As a promising technology for quantitatively identifying damage in composite structures, guided Lamb wave monitoring is widely employed in structural health monitoring (SHM) using piezoelectric sensors. Toyama [11] detected damage to a T800H/3631 carbon fiber composite structure with a thickness of 1 mm based on the variation in arrival time in the S₀ mode along the 0° direction and monitored the amplitude change in the earliest arriving wave packet to determine the size of the damaged area. Sharif Khodaei [12] impacted a 4.65 mm thick carbon fiber composite material consisting of 12 layers at different positions and trained a neural network to monitor the positions of impacts with varying energy levels. P.T. Coverley [13] conducted a localization impact study on a 3 mm thick graphite fiber epoxy resin composite structure by incorporating angle variation into triangulation and genetic algorithms. P. Ochoa [14] monitored damage in carbon fiber composite materials with a thickness of 2.24 mm under energy impacts of 3 J, 5 J, and 10 J.

As a representative algorithm for armor SHM, the time-reversal focusing method is instrumental in overcoming challenges by enabling self-focusing of signals with minimal prior knowledge of the monitored structures. K. Luo [15] proposed a new adaptive time-reversal method with a damage-imaging algorithm for calculating damage index values through local adaptive analysis of Lamb wave time-reversal signals. LP. Huang [16] investigated an improved time-reversal method to mitigate the effects of the time-reversal operator by modulating the response signal in the forward path and the reconstructed signal in the time-reversed path. R.L. Lucena [17] proposed a method combining the time-reversal method with a spectral element method for structural damage detection. To reduce economic burden and simplify installation, Z.Q. Yang [18] developed a small number of transducers equipped with a concrete-adapted time-reversal imaging algorithm using implantable technology. J.Z. Wang [19] introduced a nonlinear Lamb wave time-reversing method and highlighted the differences between the physical nonlinear time-reversed method and the conventional virtual time-reversed algorithm for detecting and quantifying fatigue cracks.

In summary, all the aforementioned composite structures have small thicknesses and consist of a single layer. Although the above research using materials with small thicknesses can cause complex signal multi-modality and dispersion phenomena, the time-reversal method can overcome the attenuation of Lamb wave signals and locate damage exactly. However, there is currently limited research on the time-reversal method for the health monitoring of multiple multi-layer and large-thickness composite structures. Therefore, it is worth exploring the proposed method on armor composite structure.

In this paper, to address the issue of inaccurate localization resulting from the multi-modality and dispersion of Lamb wave signals in large-thickness composite structures, this research proposes and verifies a time-reversal algorithm for damage detection in armor composite structures. The arrival time of Lamb waves in a sandwich armor composite, comprising alumina ceramic, T700 Carbon Fiber, and TC4 titanium alloy, is calculated using wavelet transform. Time reversal is employed to mitigate differences caused by propagation paths and the influence of armor composite anisotropy, thereby enhancing imaging resolution. This approach facilitates damage monitoring and visualization under virtual loading. The structure of this study is organized as follows: Section 2 elucidates the principles of time-reversal algorithms, and Section 3 introduces the imaging method. Section 4 details the selection of materials for each layer. Finally, single-damage and multi-damage impact localization and imaging validation experiments are conducted on an armor composite plate.

2. Principle of Time-Reverse Algorithms

Damages occurring in a structure can act as excitation signals generated by wave sources within the structure, conveying information about the damages. When an excitation signal is generated at a specific point as a wave source within the structure, it propagates in various directions through vibrations. Observation points are positioned at other locations within the structure to capture response signals. By conducting time reversal in the time domain on the received response signals and employing them as new excitation signals at the observation points, a reconstructed signal is obtained at the original wave source. The reconstructed signal demonstrates focusing characteristics. Reverse focusing of the signal is accomplished based on the principles of acoustic reciprocity and the time reversal invariance of the linear elastodynamic wave equation.

Utilizing the piezoelectric effect of piezoelectric components and the propagation characteristics of Lamb waves in plate structures, analytical models were developed for the $S_0$ and $A_0$ modes of Lamb waves propagated in isotropic plate structures. The expression for the Lamb wave unit impulse response in a plate structure is given as follows:

$$\epsilon (x,t){|}_{y=d}=-j\frac{a{\tau }_0}{G}\sin \left(k_{A_0}a\right)\frac{N_A\left(k_{A_0}\right)}{D_A^{\prime }\left(k_{A_0}\right)}\times e^{-j\left(k_{A_0}x-{\omega }_{0}t\right)}-j\frac{a{\tau }_0}{G}\sin \left(k_{s_0}a\right)\frac{N_S\left(k_{S_0}\right)}{D_S^{\prime }\left(k_{S_0}\right)}\times e^{-j\left(k_{s_0}x-{\omega }_{0}t\right)}$$

(1)

Among the terms, $\epsilon (x)$ represents the strain solution for Lamb waves, where $x$ denotes the distance of Lamb wave propagation. $a$ represents the radius of the piezoelectric sensor, and ${\tau }_0$ represents the distributed shear force at the edge of the piezoelectric sensor. $k_{S_0}$ and $k_{A_0}$ represent the wave numbers for the $S_0$ and $A_0$ mode, respectively, as obtained from Equations (2) and (3). The coefficients $N_s$, $N_A$, $D_S$, and $D_A$ are determined from Equations (4)–(7).

$$k_{S_0}=\frac{{\omega }_0}{V_{p{\omega }_0{S}_0}}$$

(2)

$$k_{A_0}=\frac{{\omega }_0}{V_{p{\omega }_0{A}_0}}$$

(3)

$$N_S=kq\left(k^2+q^2\right)\cos (pd)\cos (qd)$$

(4)

$$N_A=kq\left(k^2+q^2\right)\sin (pd)\sin (qd)$$

(5)

$$D_S={\left(k^2-q^2\right)}^2\cos (pd)\sin (qd)+4k^{2}qp\sin (pd)\cos (qd)$$

(6)

$$D_A={\left(k^2-q^2\right)}^2\sin (pd)\cos (qd)+4k^{2}qp\cos (pd)\sin (qd)$$

(7)

where $V_{p{\omega }_0{S}_0}$ and $V_{p{\omega }_0{A}_0}$ represent the phase velocities of their respective modes.

Equation (1) can be interpreted as the sum of the amplitude terms associated with the $S_0$ mode and the $A_0$ mode, along with the phase term modulation, which can be simplified as follows:

$$\epsilon (x,t)=A\left({\omega }_0\right)\times e^{-j\left(k_{A_0}x-{\omega }_{0}t\right)}+S\left({\omega }_0\right)\times e^{-j\left(k_{s_0}x-{\omega }_{0}t\right)}$$

(8)

where

$$A\left({\omega }_0\right)=-j\frac{a{\tau }_0}{G}\frac{\sin \left(k_{A_0}a\right)}{k_{A_0}}\frac{N_A\left(k_{A_0}\right)}{D_A^{\prime }\left(k_{A_0}\right)}$$

(9)

$$S\left({\omega }_0\right)=-j\frac{a{\tau }_0}{G}\frac{\sin \left(k_{S_0}a\right)}{k_{S_0}}\frac{N_S\left(k_{S_0}\right)}{D_S^{\prime }\left(k_{S_0}\right)}$$

(10)

Performing a Fourier transform on Equation (8) yields the following:

$$H(x,\omega )={{\displaystyle \int }}_{-\infty }^{+\infty }A\left({\omega }_0\right)\times e^{-j\left(k_{A_0}x-{\omega }_{0}t\right)}{e}^{-j\omega t}dt+{{\displaystyle \int }}_{-\infty }^{+\infty }S\left({\omega }_0\right)\times e^{-j\left(k_{s_0}x-{\omega }_{0}t\right)}{e}^{-j\omega t}dt$$

(11)

The frequency response function of Equation (11) is the following:

$$H(x,\omega )=\left[\pi A\left({\omega }_0\right)\times e^{-j{k}_{A_0}x}+\pi S\left({\omega }_0\right)\times e^{-j{k}_{s_0}x}\right]\times \left[\delta \left(\omega -{\omega }_0\right)+\delta \left(\omega +{\omega }_0\right)\right]$$

(12)

where $\delta$ is the unit impulse function. For single-frequency or narrowband Lamb waves, Equation (12) can be expressed as follows:

$$H(x,\omega )=\pi A(\omega )\times e^{-j{k}_{A_0}x}+\pi S(\omega )\times e^{-j{k}_{s_0}x}$$

(13)

At lower frequencies of Lamb waves, when the $A_0$ mode predominates in propagation, Equation (13) can be approximately simplified as follows:

$$H_{A_0}(x,\omega )\approx \pi A(\omega )\times e^{-j{k}_{A_0}x}$$

(14)

Equation (14) represents the simplified expression of the frequency response function of the transfer function for the single-frequency or low-frequency narrowband Lamb waves propagating in a plate structure in the $A_0$ mode. As depicted in Figure 1, with $n+1$ piezoelectric sensors affixed to the plate structure, assuming the frequency response of the transfer function from the source location 0 to the $i$th piezoelectric sensor for propagating Lamb waves is denoted as $H_{0i}$, and the excitation signal emitted by the source is represented as $E_0$, the response signal received by each piezoelectric sensor is expressed as follows:

$$E_i=H_{0i}{E}_0$$

(15)

By time-reversing the response signals received at each receiving end (by taking the complex conjugate in the frequency domain) and subsequently using the corresponding receiving end as the excitation point, each piezoelectric sensor re-excites the time-reversed response signal of the Lamb wave it originally received back into the structure. Consequently, the signal received at the original source’s piezoelectric sensor is as follows:

$$E_0^{\prime }={\displaystyle \sum _{i=1}^n{H}_{i0}}{E}_i^{*}$$

(16)

where $E_i^{*}$ represents the conjugate form of $E_0$, indicating the time-reversed signal of the source signal, and $H_{i0}$ represents the conjugate form of $H_{0i}$. According to the reciprocity principle of acoustic wave propagation, exchanging the positions of the excitation end and the receiving end results in the transfer function between the sensors and the source being independent of the propagation direction and remaining unchanged, denoted as $H_{i0}=H_{0i}$. Therefore, Equation (16) can be expressed as follows:

$$E_0^{\prime }={\displaystyle \sum _{i=1}^n{H}_{i0}{H}_{i0}^{*}{E}_0^{*}}$$

(17)

${\displaystyle \sum _{i=1}^n{H}_{i0}{H}_{i0}^{*}}$ in Equation (17) can be transformed into matrix form:

$$E_0^{\prime }=(H\ast H^{*})E_0^{*}=|H{|}^2{E}_0^{*}$$

(18)

Among these terms, $H$ represents the transmission function matrix of signal communication, $|H{|}^2$ denotes the proportion of the propagation function of the sound wave in the medium, which is the accumulation of the common function, and its value must be a positive real function. According to Equation (18), if the sound source incentive signal is a low-frequency narrowband Lamb wave signal, the term $|H{|}^2$ can be expressed by ${\pi }^2|A(\omega ){|}^2$. Equation (18) can be expressed as follows:

$$E_0^{\prime }=n{\pi }^2|A(\omega ){|}^2{E}_0^{*}$$

(19)

Taking the Fourier transform of Equation (19) yields the following:

$$e_0^{\prime }(t)=\frac{1}{2\pi }{{\displaystyle \int }}_{-\infty }^{+\infty }|\mathit{H}{|}^2{E}_0^{*}{e}^{j\omega t}d\omega =\left[n\frac{\pi |A(\omega ){|}^2}{2}\right]e_0(-t)$$

(20)

It can be observed that the Lamb wave signal excited from the source is re-excited to the corresponding piezoelectric sensors after time reversal. Moreover, the focused signal $e_0^{\prime }(t)$ at the original source is amplified by a certain factor compared to the source excitation signal $e_0(t)$. This process represents signal focusing and amplification. Equations (19) and (20) are derived under the assumption of low-frequency narrowband Lamb wave excitation, where the $A_0$ mode predominates as the dominant propagating mode in the structure. They are not applicable to structures where the dominant mode of propagation is the $S_0$ mode or for wideband Lamb waves. When the Lamb wave excited in the structure is not primarily the $A_0$ mode, the $|H{|}^2$ term can be expressed using Equation (21).

$$|\mathit{H}{|}^2={\pi }^2\left[\begin{array}{l}|A(\omega ){|}^2+|S(\omega ){|}^2+S(\omega )A^{*}(\omega )\times e^{-j\left(k_{s_0}-k_{s_0}\right)x}\\ +A(\omega )S^{*}(\omega )\times e^{-j\left(k_{s_0}-k_{s_0}\right)x}\end{array}\right]$$

(21)

The focused signal $E_0^{\prime }$, obtained by substituting Equation (21) into Equation (18), demonstrates wavefront focusing. However, interference signals in the form of side lobes appear on both sides of the wavefront, presenting a symmetric pattern as depicted in Figure 2. Analyzing the focused signal presents challenges.

The synthesized signal of the excitation signals from each piezoelectric sensor propagating at the structural health location $h$, i.e., the non-source location, can be represented as follows:

$$E_h^{\prime }={\displaystyle \sum _{t=1}^n{H}_{ih}{E}_i^{*}}={\displaystyle \sum _{t=1}^n{H}_{ih}{H}_{0i}^{*}{E}_0^{*}}$$

(22)

The synthesized signal at the structural health location $h$, obtained by substituting Equation (14) into Equation (22), is given as follows:

$$E_h^{\prime }=\left[{{\displaystyle \sum }}_{i=1}^n{\pi }^2|A(\omega ){|}^2\times e^{-j\left(k_{A_0}{x}_{ih}-k_{A_0}{x}_{i0}\right)}\right]E_0^{*}$$

(23)

Based on Equations (19), (20), and (23), it can be observed that the synthesized signal $E_h^{\prime }$ at the structural health location $h$ is smaller than the focused signal $E_0^{\prime }$ at the original source location. The time-reversed signals, when propagated through the structure, reach their maximum value at the original source location, where they are superimposed and focused, resulting in an increased signal magnitude. However, at non-source locations in the healthy region, no significant focusing occurs, leading to smaller signal magnitudes. This facilitates the identification of the source location.

When there are two sources of excitation within the structure, as shown in Figure 3, $n+1$ piezoelectric sensors are still attached to the plate structure. Assuming that the frequency response of the transfer function from the source positions 0 and 1 to the $i_{th}$ piezoelectric sensor, denoted as $H_{0\mathrm{i}}$ and $H_{1i}$ for Lamb wave propagation, respectively, and the excitation signals emitted from the source positions 0 and 1 are $E_0$ and $E_1$, respectively, the response signal received by each piezoelectric sensor can be expressed as follows:

$$E_i=H_{0i}{E}_0+H_{1i}{E}_1$$

(24)

Similarly, by time-reversing the response signals from each receiver and utilizing the corresponding receiver as the excitation source, the signal received on the piezoelectric sensor at the original source can be defined as follows:

$$E_0^{\prime }={\displaystyle \sum _{i=2}^n{H}_{i0}{E}_i^{*}}$$

(25)

By substituting Equation (24) into Equation (25), where $H_{0i}=H_{i0}$, we can obtain the following:

$$E_{C_1}^{\prime }={\displaystyle \sum _{i=2}^n{H}_{i0}{H}_{0i}^{*}{E}_0^{*}}+{\displaystyle \sum _{i=2}^n{H}_{i0}{H}_{1i}^{*}{E}_1^{*}}$$

(26)

$$E_{C_2}^{\prime }={\displaystyle \sum _{i=2}^n{H}_{i1}{H}_{1i}^{*}{E}_1^{*}}+{\displaystyle \sum _{i=2}^n{H}_{i1}{H}_{0i}^{*}{E}_0^{*}}$$

(27)

Based on Equation (14), Equations (26) and (27) can be expressed as follows [15,16,17]:

$$E_0^{\prime }=n{\pi }^2|A\left({\omega }_c\right){|}^2{E}_0^{*}+{\pi }^2|A\left({\omega }_c\right){|}^2{\displaystyle \sum _{i=2}^n{e}^{-j{k}_{A_0}\left(x_{i0}-x_{i1}\right)}{E}_2^{*}}$$

(28)

$$E_1^{\prime }=n{\pi }^2|A\left({\omega }_c\right){|}^2{E}_1^{*}+{\pi }^2|A\left({\omega }_c\right){|}^2{\displaystyle \sum _{i=2}^n{e}^{-j{k}_{A_0}\left(x_{i1}-x_{i0}\right)}{E}_0^{*}}$$

(29)

From Equations (28) and (29), it can be inferred that the signal demonstrates focalization at both source locations 0 and 1, while also introducing additional sidelobes. As the number of sources increases, the sidelobes gradually amplify and begin to influence the focalized signals, ultimately causing the source locations to become indistinguishable and rendering them unidentifiable.

3. Principles of Image Method

Based on Equation (14), it is known that for low-frequency narrowband Lamb wave signal propagation, the dominant mode of propagation is the $A_0$ mode. The transfer function can be divided into two components: the magnitude component $\pi A({\omega }_c)$ and the phase component $e^{-j{k}_{A0}x}$. In the principle of time reversal, these two components exert distinct influences on signal focalization. When the time-reversed signal propagates back to the original source location, Equations (19) and (20) indicate that the signals undergo constructive interference. During signal focalization, the phase component of the transfer function is compensated. The magnitude component of the transfer function represents the magnitude of the reconstructed signal waveform after focalization, and the phase component plays a crucial role in determining whether the signal can achieve focalization after time reversal. Therefore, phase modulation can be applied to Lamb wave response signals to achieve a similar effect of time-reversed focalization, thus enabling localization. The key to phase modulation lies in acquiring the propagation velocity of Lamb waves in the structure, which is utilized to modulate the phase of the signal. This eliminates the need to acquire the transfer function along the propagation path, thereby reducing the demand for experimental or modeled transfer function determination and facilitating the implementation of time reversal in software.

As depicted in Figure 4, $n$ piezoelectric sensors are arranged on the structure. At position $s(x,y)$, there exists an unknown sound source. The response signals that can be received by each piezoelectric sensor are extracted using the Shannon continuous complex wavelet transform to obtain narrowband signals with the desired center frequency. Let $E_i(\omega )$ denote the frequency response of the narrowband signal, with a center frequency of ${\omega }_c$. The distances from position $s(x,y)$ to each piezoelectric sensor are denoted as $r_1$, $r_2$, …, $r_n$. The phase of the narrowband signal $E_i(\omega )$ is delayed by $e^{j{k}_{A0}(-r_i)}$ and then synthesized. The synthesized signal is expressed as follows:

$$V_S={\displaystyle \sum }_{i=1}^n{E}_i{(\omega )}^{*}{e}^{-j{k}_{A_0}(\omega )r_i}$$

(30)

By performing the inverse Fourier transform on the synthesized signal $V_S$ and taking its modulus, the following can be obtained:

$$\left|v_s(t,x,y)\right|=\left|\frac{1}{2\pi }{\displaystyle \sum }_{i=1}^n{{\displaystyle \int }}_{-\infty }^{+\infty }{E}_i{(\omega )}^{*}{e}^{-j{k}_{{\varphi }_0}(\omega )r_i}{e}^{j\omega t}d\omega \right|$$

(31)

In this context, $v_s(t,x,y)$ signifies the phase-synthesized signal, $e^{-j{k}_{{\varphi }_0}(\omega )r_i}=e^{-j{r}_i\omega /c_{pA0}}$, and $c_{pA0}$ denotes the phase velocity of Lamb wave propagation.

According to the dispersion characteristics of narrowband Lamb waves, for the narrowband signal $E_i(\omega )$, Equation (31) can be expressed as follows [18]:

$$\begin{array}{l}|v_s(t,x,y)|=|\frac{1}{2\pi }{\displaystyle \sum _{i=1}^n{{\displaystyle \int }}_{-\infty }^{+\infty }\left({\displaystyle \sum _{{\omega }_0\in \left[{\omega }_l,{\omega }_h\right]}{E}_i{\left({\omega }_0\right)}^{*}{e}^{-j{r}_i{\omega }_0/c_{p{\omega }_0{A}_0}}}\right)e^{j\omega t}d\omega }|\\ =\frac{1}{2\pi }|{\displaystyle \sum _{i=1}^n[{\displaystyle \sum _{{\omega }_0\in \left[{\omega }_l,{\omega }_h\right]}{e}_i^{\prime }\left(\tau -t-r_i/c_{p{\omega }_0{A}_0}\right)}}]|\end{array}$$

(32)

In this context, $E_i({\omega }_0)$ represents the frequency component of $E_i(\omega )$ at frequency ${\omega }_0$, and $e_i^{\prime }$ denotes the temporal representation of $E_i({\omega }_0)$. $c_{p{\omega }_0{A}_0}$ is the phase velocity corresponding to the frequency signal with a frequency of ${\omega }_0$. The center frequency of the narrowband signal $E_i(\omega )$ is ${\omega }_c$, with a frequency range of $[{\omega }_l,{\omega }_h]$. According to Equation (32), the magnitude of the synthesized signal $v_s(t,x,y)$ can be obtained, and this magnitude is used as the corresponding pixel value at position $s(x,y)$. By following a specific step size and the previously mentioned calculation process, the magnitude at each position within the region is obtained. Then, the corresponding pixel values are presented in the form of an image, representing the imaging of the sound source. The areas in the image with larger and more concentrated pixel values indicate the location of the sound source.

In the computation for sound source localization mentioned above, obtaining the phase velocity $c_{p{A}_0}$ corresponding to each frequency component of Lamb wave propagation is necessary. This necessitates performing Equations (30) and (31), which involve Fourier and inverse Fourier transformations. When dealing with wide-ranging monitoring, a large number of piezoelectric sensors is required, which imposes a significant challenge. According to the relationship of the absolute value inequality, Equation (34) satisfies the following:

$$\left|v_s(t,x,y)\right|\approx \left|{\displaystyle \sum }_{i=1}^n{e}_i\left(\tau -t+\frac{r_i}{c_{g{A}_0}}\right)\right|\le {\displaystyle \sum }_{i=1}^n\left|e_i\left(\tau -t+\frac{r_i}{c_{g{A}_0}}\right)\right|$$

(33)

where the frequency component with a center frequency of ${\omega }_c$ from the narrowband signal $E_i(\omega )$ is extracted, with a phase velocity of $c_{p{\omega }_c{A}_0}$. This simplification reduces a certain amount of computation, but it still requires Fourier transformation and inverse Fourier transformation calculations during the solving process. This simplification decreases resolution, particularly when the Lamb wave frequency is low and the frequency band is narrow, leading to reduced resolution.

Based on the dispersion characteristics of narrowband Lamb waves, neglecting the influence of the amplitude coefficient $1/2\pi$, Equation (32) can be simplified as follows:

$$\left|v_s(t,x,y)\right|\approx \left|{\displaystyle \sum }_{i=1}^n{e}_i\left(\tau -t+\frac{r_i}{c_{g{A}_0}}\right)\right|$$

(34)

where $e_i$ represents the narrowband signal extracted through the Shannon continuous complex wavelet transform, and $c_{g{A}_0}$ represents the group velocity of this narrowband signal. This simplification method eliminates the need for Fourier transformation and inverse Fourier transformation calculations, significantly reducing computational requirements. By performing time-domain superposition focusing on the narrowband signals extracted from Lamb response signals using the Shannon continuous complex wavelet transform, it greatly contributes to enhancing resolution.

According to the relationship of the absolute value inequality, Equation (34) satisfies the following:

$$\left|v_s(t,x,y)\right|\approx \left|{\displaystyle \sum }_{i=1}^n{e}_i\left(\tau -t+\frac{r_i}{c_{g{A}_0}}\right)\right|\le {\displaystyle \sum }_{i=1}^n\left|e_i\left(\tau -t+\frac{r_i}{c_{g{A}_0}}\right)\right|$$

(35)

Therefore, the expression for signal phase synthesis is as follows:

$$\left|v_s^{\prime }(t,x,y)\right|={\displaystyle \sum }_{i=1}^n\left|e_i\left(\tau -t+\frac{r_i}{c_{g{A}_0}}\right)\right|$$

(36)

Equation (34) represents the need to first compute the composite signal by summing the focused signals and then calculate the modulus of the composite signal. Equation (36) represents the approach of first computing the modulus of each signal and then performing a focused summation of these moduli. During the signal-focusing process, the resolution of the resulting image is affected by factors such as the propagation distance and velocity of the signal within the structure, as well as the size of the imaging search step. These factors pose challenges to accurately localizing the source of the signal. After obtaining the pixel values for all target points, normalization is performed. When the target cell is not at the location of the damage, the summed amplitude will be smaller compared to the amplitude at the damage location. By applying thresholding, the pixel value at the maximum will indicate the location of the damage.

4. Selection of Materials for Each Layer

In the development of armored composite materials, composite sandwich structures typically consist of ballistic-resistant ceramic materials as the face panel and composite materials as the core layer, sandwiched between metal materials serving as the backplate. This design combines low density and high hardness characteristics, along with excellent properties such as high ballistic resistance, strong plasticity, and corrosion resistance. Although individual ceramics exhibit high hardness, they are brittle and unable to withstand secondary damage. In lightweight armored composites, the ceramic material in the face panel, with its high hardness and compressive strength, forces the projectile to deform or even fracture, thereby significantly reducing its speed. The fiber composite material in the core layer converts the kinetic energy of the projectile into the internal energy of the laminated plate during penetration. The kinetic energy of the projectile causes the fibers to undergo tensile damage, and stress waves propagate in the axial and longitudinal directions of the fibers. Energy is absorbed at the crossing of the fibers, resulting in delamination and fragmentation of the ceramic material. Together with the fragmented projectile, they collectively bear the load on the backplate. The backplate undergoes tensile and shear deformation, as well as interlayer delamination, to absorb the remaining kinetic energy.

4.1. Selection of Panel Materials

Ceramic armor, as a face panel material, possesses characteristics of high hardness, high strength, high toughness, and low density.

Table 1. Comparison of the main performance data for commonly used ceramic materials in armor.

Materials	Alumina	Silicon Carbide	Titanium Boride	Boron Carbide
Density (g/cm3)	3.6–3.9	3.12–3.28	4.5	2.5
Elastic modulus (GPa)	340	408–451	570	400
Vickers hardness (HK)	1800	2500	2600	2900

presents the main performance data of commonly used ceramic materials for armor.

From Table 1, it is evident that boron carbide exhibits the lowest density and the highest Vickers hardness, whereas boron-titanium carbide demonstrates the highest density and elastic modulus. Due to its relatively high elastic modulus, silicon carbide ceramic causes substantial plastic deformation in the projectile, consequently diminishing its kinetic energy during penetration. However, its high cost, ranging from USD 110 to 220 per kilogram, and incapacity to withstand multiple hits limit its extensive application in armored vehicles. Fractures in this ceramic material during protection can readily propagate to adjacent ceramic components, thereby reducing the overall structural protection capability. Titanium boride demonstrates the highest density and elastic modulus, along with a relatively high Vickers hardness. This renders it suitable for use in intermediate armor. Companies such as Ceradyne and Cercom in the United States have employed titanium boride to defend against large-caliber projectiles in the M2 Infantry Fighting Vehicle (IFV). It exhibits strong resistance to violence and minimal armor fragmentation. Boron carbide showcases the best performance parameters, being one of the hardest materials globally and among the most stable compounds. Additionally, it boasts a high melting point, excellent wear resistance, and corrosion resistance, rendering it an ideal material for ceramic armor. It effectively contributes to reducing tank weight and energy consumption. However, its unstable processing and high production costs contribute to its expensive price. Due to the mature sintering process, abundant raw materials, and low production costs, alumina ceramics find wide application in various types of armored vehicles, seats, and belly panels due to their low price and stable material performance. Alumina ceramics contain varying proportions of ${\mathrm{Al}}_2{\mathrm{O}}_3$, resulting in different compressive strength and fracture toughness characteristics.

Table 2. Comparison of the performance parameters of ceramics with different Al₂O₃ content levels.

Performance Parameters	Content Ratio of Al2O3 (%)
	85	90	96	99.5
Density (g/cm3)	3.68	3.76	3.80	3.82
Elastic modulus (GPa)	200	275	344	370
Compressive strength (GPa)	1930	2150	2260	2600
Fracture toughness (MPa·m1/2)	3.1	3.3	3.7	4.6

provides the ceramic performance parameters for different levels of ${\mathrm{Al}}_2{\mathrm{O}}_3$ content. Ceramics with a 99.5% ${\mathrm{Al}}_2{\mathrm{O}}_3$ content exhibit higher compressive strength and fracture toughness. Impact tests conducted by Liang Yanyuan revealed that higher ${\mathrm{Al}}_2{\mathrm{O}}_3$ content correlates with better resistance to projectiles and higher fracture toughness. Thus, this study selected ceramics with a 99.5% ${\mathrm{Al}}_2{\mathrm{O}}_3$ content as the panel material.

4.2. Selection of Interlayer Materials

Commonly used fibers in laminated composite materials include glass fiber, ultra-high-molecular-weight polyethylene fiber (UHMWPE), aramid fiber, and carbon fiber.

Table 3. Major performance indicators of several fiber composite materials.

Materials	Density (g/cm3)	Tensile Strength (MPa)	Elongation (%)	Specific Energy Absorption Rate (J·m2/kg)
Glass fiber	2.5	2400	2.0–4.0	7.0
UHMWPE	0.98	3800	4.0–8.7	34.9
Aramid fiber	1.4	2200	1.4–3.2	9.7
Carbon fiber T700	1.6	4900	2.4	13.1

presents the key performance indicators of several commonly used fiber composite materials.

The table clearly indicates that glass fiber possesses the highest density and the lowest specific energy absorption rate. On the other hand, UHMWPE demonstrates outstanding energy absorption capabilities owing to its well-connected main chain, high degree of molecular alignment, and high crystallinity. This enables it to absorb a significant amount of energy during plastic deformation, resulting in the highest specific energy absorption rate, which characterizes the energy absorption characteristics of materials due to deformation in armor material or structure. UHMWPE exhibits exceptional resistance to high-speed impacts. Aramid fiber has a relatively lower tensile strength, and carbon fiber T700 exhibits the highest tensile strength, enabling it to withstand higher tensile stresses. Furthermore, carbon fiber T700 has a specific energy absorption rate that is only surpassed by UHMWPE. It retains its strength at temperatures above 2000 °C in an oxygen-free environment and exhibits excellent high-temperature resistance. With a low coefficient of thermal expansion (0~$1.1\times {10}^{-6} {\mathrm{K}}^{-1}$), carbon fiber T700 exhibits fatigue and corrosion resistance properties. Therefore, it was chosen as the interlayer material.

4.3. Selection of Backing Materials

In lightweight armor, aluminum alloy and titanium alloy are widely used, with armored steel being the most extensively employed. Aluminum alloy follows closely as the second-largest category of armor materials. It exhibits a lower density and excellent resistance to ballistic impact, but comparatively lower hardness and strength. On the other hand, titanium alloy shows a small difference in strength compared to armored steel, yet it possesses better toughness than aluminum alloy.

Table 4. Comparative analysis of the key parameter indicators for armored steel, aluminum alloy, and titanium alloy.

Materials	Armor Steel	Aluminum Alloy	Titanium Alloy
Density (${\mathrm{g}/\mathrm{cm}}^3$)	7.86	2.7	4.5
Tensile strength (MPa)	1170	350	970
Mass shielding factor ($E_m$)	1	1.0–1.2	1.5

presents the key parameter indicators for armored steel, aluminum alloy, and titanium alloy.

The table illustrates that titanium alloy has a density 60% lower than armored steel and boasts the highest protection coefficient. Additionally, it possesses excellent heat resistance and resistance to low temperatures, enabling it to maintain normal operation for extended periods in environments with temperatures above 550 °C and below −250 °C. Furthermore, it exhibits outstanding corrosion resistance, rendering it an excellent choice for armor metal materials. Hence, titanium alloy is selected as the backing material.

4.4. Selection of Adhesive Materials

The ballistic resistance of armored composite materials correlates closely with the strength and thickness of the interlayer adhesive. Adhesive selection for bonding between different layers primarily considers the bonding between metals and non-metals, as well as ceramics and non-metals. The adhesive effect between adjacent layers can mitigate destructive forms such as delamination, layer cracking, and interlayer separation upon impact, thus reducing the likelihood of interlayer collapse phenomena. In this study, epoxy resin AB adhesive was selected as a transparent adhesive consisting of an epoxy resin-based two-component fast-curing epoxy resin adhesive. It demonstrates excellent bonding strength after curing when used for bonding connections between metal materials, ceramic materials, wood, glass products, and hard plastics. However, its adhesive performance for soft and elastic materials, rubber, and leather items is unsatisfactory. The cured adhesive demonstrates good resistance to acid and alkaline substances as well as moisture, and it has waterproof properties.

Table 5. Parameter indicators after the curing of epoxy resin AB adhesive.

Parameters	Value
Hardness (Shore D)	≥70
Compressive strength (MPa)	≥60
Shear strength (MPa)	≥13
Yield strength (MPa)	≥22

presents the relevant parameter indicators for epoxy resin AB adhesive after curing.

Based on the comparative analysis of the armor materials mentioned above, the panel material selected for this study’s experiment is a ceramic with a 99.5% ${\mathrm{Al}}_2{\mathrm{O}}_3$ content. The interlayer material chosen is carbon fiber T700, and the backing material selected is TC4 titanium alloy. The adhesive selected is epoxy resin AB adhesive.

4.5. Manufacture of Armor Materials

The materials and facilities required for preparing sandwich armored composite materials include $200\times 200\times 1$ mm ceramic plates, T700 carbon fiber sheets, TC4 titanium alloy plates, molds, epoxy resin AB adhesive, paraffin wax, and a high-temperature oven. Figure 5 depicts the final product, with dimensions of $200\times 200\times 5$ mm. The specific preparation process used is as follows:

(1)

Apply a layer of paraffin wax to the inner wall of the mold and allow it to dry completely to ensure surface dryness;

(2)

Place a 1 mm thick TC4 titanium alloy plate at the bottom of the mold. Apply an even layer of epoxy resin AB adhesive to its upper surface. Then, lay a 1 mm thick T700 carbon fiber sheet on top of the adhesive, followed by another layer of epoxy resin AB adhesive on the surface of the T700 carbon fiber sheet. Finally, place a 1 mm thick aluminum oxide plate on top;

(3)

Place weights on the structure to compact the entire surface, and roll to expel air bubbles, ensuring a flat overall structure and enhancing the interlayer bonding strength of the materials;

(4)

Transfer the entire structure to a high-temperature oven and maintain a temperature of 80 °C for 2 h;

(5)

Remove the structure from the mold and trim the surrounding edges to achieve a smooth finish.

5. Experimental Research on the Positioning of Armored Composite Materials

This section focuses on researching the damage modes, such as delamination, debonding, and interlayer cracking, caused by low-speed external impacts in the structure of armored composite materials. To simulate the damage of armored composite materials, absorptive adhesive and mass blocks were applied to the outer surface of the structure. Both single and multiple damages occurring in the structure are monitored, and damage localization imaging was performed using time-reversal algorithms and Shannon continuous complex wavelets. The location of the impact signal source was determined using a sensor array, and a multiple signal classification algorithm was employed to locate the impacts occurring in the near-field region of the array.

5.1. Experimental Research on Single Damage Imaging and Localization

In order to conduct experimental research on damage imaging and localization in armored composite material structures, as shown in Figure 6, a sandwich armored composite material was used as the experimental object. Four piezoelectric sensors are arranged on the structure, with the origin set at sensor 1. The diameter of the PZT sensor is 8 mm. The resonant frequency of PZT is 2 kHz. The coordinates of sensor 2 are (160 mm, 0 mm); sensor 3 is located at (160 mm, 160 mm); and sensor 4 is located at (0 mm, 160 mm). All sensors are connected into the monitoring PC interface, which has one excitation PCI interface card, one eight-channel piezoelectric signal condition PCI interface card, and one processing card. The excitation PCI interface card is in charge of the excitation signal to omit the excitation sensor, and the piezoelectric signal condition PCI card is in charge of receiving the response signal from responding sensors.

The excitation signal was configured as a narrowband modulated five-cycle sinusoidal signal with a center frequency ${\omega }_c$ of 50 kHz. The sampling system was configured with a sampling frequency of 5 $\mathrm{MHz}$ and a total sampling length of 10,000 points, which included a pre-sampling length of 2000 points. The triggering voltage was set to 300 $\mathrm{mV}$. Damage simulation was conducted using weights and absorbing rubber. The damage locations were designated at the central position of the armored composite material structure and the four corner regions, with the specific coordinates provided in

Table 6. Coordinates of simulated damage locations.

Serial Number	Coordinates/mm
Simulated damage 1	(70, 120)
Simulated damage 2	(120, 80)
Simulated damage 3	(130, 130)
Simulated damage 4	(50, 40)
Simulated damage 5	(80, 80)

.

The approximate operational procedure for the damage imaging and localization experiment is illustrated in Figure 7. Initially, the structure is in a healthy status. We set three operation procedures for the experiment. First, the first sensor is set as excitation, and we can obtain response signals from the second, third, and fourth receptors. Then, the second sensor is set as excitation, and we can obtain a response signal from the third and fourth receptors. Finally, the third sensor is set as excitation, and we can obtain a response signal from the fourth receptor. In each setting, the excitation sensor emits the excitation signal, and receptor sensors receive the response signals, which comprises a total of six baseline signals. In contrast, as the structure is in a damaged status, the same channel parameters were maintained, and the signals were repeatedly collected as damage signals in damage status. This process was repeated to acquire a total of 20 sets of signals. In a healthy or damaged status, the response signals from each excitation sensor channel are processed using the Shannon continuous complex wavelet transform to extract the narrowband signals with a center frequency. The damage-scattered signals for each excitation sensor channel were determined. The previously obtained propagation velocity (group velocity $c_g$) of the narrowband signal with ${\omega }_c=50$ $\mathrm{kHz}$ inside the structure was employed, as calculated in the previous experiment. The distances from the hypothetical damage location $(x,y)$ within the structure to the piezoelectric sensors in each excitation sensor channel were computed. Subsequently, the damage-scattered signals were phase-combined, and the amplitude at that location was utilized as the pixel value for the image. The imaging process was conducted for all unit points within the entire region, resulting in the generation of the damage imaging.

Figure 8 depicts the time-domain plots of the narrowband signals with a center frequency of ${\omega }_c=50$ $\mathrm{kHz}$, obtained through the Shannon continuous complex wavelet transform, for the response signals in the 1–2, 1–3, and 1–4 excitation sensor channels under both healthy and damaged conditions of the structure. From the figure, it can be observed that the amplitude of the damaged signal is slightly smaller than that of the healthy signal during propagation, and there is a phase shift, indicating propagation attenuation. Furthermore, in the 1–3 channel, the direct wave arrival time is noticeably delayed compared to the other channels due to the longer propagation distance. Figure 9 presents the damage-scattered signals obtained by subtracting the narrowband response signals from the aforementioned channels.

The signals within the range from 0 to 0.3 ms from the damage scattering signals were selected as direct wave signals. These selected signals underwent processing to calculate the distance from each unit point $(x,y)$ within the monitoring area to the sensor. Subsequently, the time delay of the response signal was calculated. The delayed damage scattering signals, obtained by reversing the order of time, were utilized to construct an image matrix, with the pixel values representing the corresponding amplitude at each moment. The image was scanned with a step size of $0.1\times 0.1$ mm, capturing pixel values for all unit points. After normalizing the pixel values, a damage imaging map was obtained. Figure 10 presents the imaging results for each damage location. The regions with higher pixel values indicate a higher probability of damage presence. The right side of the figure shows the damage imaging map after setting a threshold of 90%. Most pixel values are concentrated in a small ellipse within the damaged area, indicating the relative size range of the damage.

Table 7. Coordinates and error statistics for the imaging localization of each damage location.

Serial Number	Actual Coordinates/mm	Image Coordinates/mm	Radial Error/mm
Damage 1	(70, 120)	(70, 122)	2
Damage 2	(120, 80)	(124, 79)	4.12
Damage 3	(130, 130)	(131, 129)	1.41
Damage 4	(50, 40)	(47, 39)	3.16
Damage 5	(80, 80)	(82, 83)	3.61

provides the coordinates and error statistics for the localization of each damage location. For example, the imaging localization of damage 2 resulted in a center position of (124, 79), and the actual center position of the damage was (120, 80). The horizontal error was 4 mm, the vertical error was 1 mm, and the radial error was 4.12 mm.

5.2. Experimental Study on Multi-Damage Imaging Localization

When multiple locations in armored composite materials are subjected to external impacts, damage incidents are not isolated. Often, multiple damages occur within the structure, making the monitoring of multi-damage scenarios vital. Multiple damages can lead to overlapping Lamb signals, and the Shannon continuous complex wavelet transform offers high time-frequency resolution capabilities. By applying the wavelet transform to Lamb wave signals and subsequently performing time-reversal focusing imaging, effective monitoring can be achieved. Building upon the testing conducted in the previous section, the acquisition parameters remained unchanged, and two damage locations were identified. The locations of the damages are detailed in

Table 8. Coordinates of two simulated damage locations.

Group	Simulated Damage 1 Coordinates/mm	Simulated Damage 2 Coordinates/mm
Group 1	(69, 86)	(34, 53)
Group 2	(148, 135)	(99, 96)
Group 3	(119, 82)	(134, 29)
Group 4	(66, 137)	(54, 52)
Group 5	(90, 90)	(90, 50)

.

The operational procedure for the damage imaging localization experiment remained consistent with the previous section. Figure 11 depicts the time-domain plots of the response signals extracted by the Shannon continuous complex wavelet transform with a center frequency of ${\omega }_c=50$ $\mathrm{kHz}$ from the excitation-sensing channels 1–2, 1–3, and 1–4 under both healthy and damaged states of the structure. From the figure, it is evident that the amplitude of the damaged signal remains slightly smaller than that of the healthy signal during propagation. However, there is a significant phase shift, leading to an increased amplitude of the damage scattering signal obtained by subtracting the narrowband response signals of each sensing channel, as illustrated in Figure 12.

The signal within the range of 0 to 0.3 $\mathrm{ms}$ from the damage scattering signal was chosen as the direct wave signal. The signal in this portion was processed by determining the distance from the sensor to each unit point $(x,y)$ within the monitoring area. The time delay of the response signal was calculated, and the delayed damage scattering signal, obtained by reversing the time, was utilized. The corresponding amplitude value at each time was used as the pixel value of the image matrix. The image was then scanned with a step size of $0.1\times 0.1$ mm to obtain the pixel values of all unit points. After normalizing the pixel values, the damage imaging map was obtained. Figure 13 displays the imaging results of the damage positions in each group. The areas with higher pixel values in the image indicate a higher likelihood of damage. On the right side of the figure, the damage imaging maps are shown after setting a threshold of 90%, with most pixels concentrated in a small ellipse at the location of the damage, indicating the relative size range of the damage.

Table 9. The coordinates and error statistics of the imaging localization for each damage position.

Group	Actual Coordinates/mm	Image Coordinates/mm	Radial Error/mm
Group 1	(70, 90)	(69, 86)	4.12
	(30, 50)	(34, 53)	5
Group 2	(150, 140)	(148, 135)	5.39
	(100, 100)	(99, 96)	4.12
Group 3	(120, 80)	(119, 82)	2.24
	(130, 30)	(134, 29)	4.12
Group 4	(70, 140)	(66, 137)	5
	(50, 50)	(54, 52)	4.48
Group 5	(90, 90)	(89, 88)	2.24
	(90, 50)	(90, 53)	3

provides the coordinates and error statistics for the localization of each damage position. In the second group, the imaging localization of damage 1 yielded a center position of (148, 135), and the actual center position of the damage was (150, 140), with a horizontal error of 2 mm, vertical error of 5 mm, and radial error of 5.39 mm. For damage 2, the imaging localization yielded a center position of (99, 96), and the actual center position of the damage was (100, 100), with a horizontal error of 1 mm, vertical error of 4 mm, and radial error of 4.12 mm.

The imaging and localization of damage in armored composite structures utilize the time-reversal focusing method. The response signal undergoes Shannon continuous complex wavelet transform to extract narrowband signals, and the direct wave portion of the damage scattering signal is chosen for time reversal. The time-reversed signal is delayed by the distance from the unit point in the monitoring area to the piezoelectric sensor, resulting in a damage imaging map of the monitoring area. The positioning errors for single and multiple damages in this method are within 4.12 mm and 5.39 mm, respectively.

6. Conclusions

Lamb wave signals in multi-layer and thick composite structures have multi-mode and dispersion characteristics, which contributes to inaccurate localization. This paper investigates a damage detection method based on a time-reversal focusing algorithm on multi-layer and thick composite structures using a PZT sensor array. First, a composite sandwich structure is designed using ballistic-resistant ceramic materials as the face panel and a composite material as the core layer, sandwiched between metal materials serving as the backplate. Then, we have implemented single-damage imaging and localization as well as multi-damage imaging and localization on thick armor composite structures using the proposed method. The experiment results show that the positioning errors for single and multiple damages in this method are within 4.12 mm and 5.39 mm, respectively.