Non-baseline identification of delamination in plates using wavelet-aided fractal analysis of two-dimensional mode shapes

Abstract

Delamination is a typical form of damage in composite plates. Identification of delamination in plates has been a focus of increasing research interest in relatively recent years. This study develops a new approach, termed a scale waveform dimension analysis of two-dimensional mode shapes, to identify delamination in composite plates. The scale waveform dimension analysis comprises two components: decomposition of a mode shape into scale mode shapes and waveform dimension analysis of scale mode shapes. The first component acts as a splitter that broadly splits a noisy mode shape into three sets of trend-, noise-, and damage-scale mode shapes, from which the damage scale mode shapes can be selected for use in damage characterization; the second component functions as a detector to detect abnormalities of damage scale mode shapes, to indicate the presence and location of delamination. The efficacy of the method is numerically studied and experimentally examined using composite plates containing small delamination. The results show that the proposed method can identify delamination with great accuracy in noisy conditions, needing no intact baseline mode shapes nor any prior knowledge of either the material properties or boundary conditions of the plate being inspected.

Keywords

Delamination mode shape waveform dimension composite materials noise robustness damage detection vibration noncontact measurement

Introduction

Laminated composite plates are widely applied in civil, aeronautical, and mechanical structures as well as architecture and light industry products, primarily because of their high ratios of strength and stiffness to weight (Jiang et al., 2008; Żak et al., 2001). Delamination is a representative type of damage in composite laminates, most frequently occurring as interlaminar separation of adjoining plies and imperfect bonding (Pines and Purekar, 2010; Żak et al., 2000), which severely jeopardizes the safety of structures (Larrosa et al., 2014). Identification of delamination in composite laminates is a research topic of significant importance.

Representative studies of delamination detection in laminated composite plates are as follows. Todoroki et al. (2002) and Iwasaki and Todoroki (2005) individually diagnosed delamination in graphite/epoxy laminates based on changes in their electric resistance, with the delamination identified correctly. Akhter et al. (2009) used pulsed electronic speckle pattern interferometry (Pulsed-ESPI) to locate delamination in carbon fiber–reinforced plastic laminate plates. Their results showed that the location of delamination was successfully detected. Qiao et al. (2007) acquired curvature modal shapes of a composite plate using a smart sensing system and processed those shapes to identify delamination. Numerical and experimental results indicated that the method identified delamination with great accuracy. Yam et al. (2004) utilized a combination of changes in modal damping and modal strain energy distribution to locate delamination in multilayer composite plates. Their results showed that the region of delamination was predicted accurately. Yan and Yam (2004) employed the energy spectrum of structural dynamic responses decomposed by a wavelet analysis to detect delamination in composite plates and reported accurate detection of the delamination. Wei et al. (2004) proposed a method for detecting delamination in multilayer composites using combined modal parameter analysis with wavelet packet transform. The capability of the method was clearly verified. Sohn et al. (2004) defined a delamination-sensitive index using a wavelet transform to process the measured dynamic responses of a composite plate. The capacity of the index to identify delamination was demonstrated by experimental studies considering varying temperature and boundary conditions. Sohn et al. (2011a) developed a noncontact guided wave imaging system to inspect composites for delamination. The performance of the system was experimentally validated using a 1.8-mm-thick multilayer composite laminate. Okabe et al. (2010) constructed an ultrasonic propagation system with macrofiber composite actuators and fiber Bragg grating sensors. This system efficiently identified delamination in composite laminates based on mode conversion of Lamb waves. Rosalie et al. (2004) detected delaminations in GLARE aluminum plate-like structures relying on the change in group velocity of Lamb waves with a frequency-thickness product, with small delaminations correctly identified. Su and Ye (2004a) created a Lamb wave propagation-based quantitative identification scheme for delamination in carbon fiber–reinforced polymer composite plate-type structures. Their scheme was validated by identifying actual delaminations in CF/EP (T650/F584) quasi-isotropic composite laminates. Su and Ye (2004b) utilized the fundamental symmetric Lamb mode and the delamination-induced basic shear mode in an ultrasonic frequency range to locate delaminations in composite laminates. Satisfactory prediction of the location of delamination was achieved in numerical and experimental cases. Ibarra-Castanedo et al. (2011) exploited an active thermography technique to detect delamination-type defects in glass-reinforced composite panels, and the characteristics of the technique were elucidated. Sohn et al. (2011b) created a signal and image processing algorithm for automatically detecting delamination and disbonding in composite plates, with the merits and the limitations of the algorithm fully presented. Shang et al. (2010) developed a model-based delamination detection methodology using the continuum damage mechanics model and a subset selection technique. Applications of the proposed methodology to an E-glass/epoxy symmetric composite panel with delamination damage were demonstrated numerically and experimentally. Minak et al. (2010) made use of the resonant frequencies of a composite laminate for the purpose of delamination assessment, showing the capacity of resonant frequencies to characterize delamination. Cao et al. (2013b) proposed a new physical concept of a multiscale shear-strain gradient from a vibrational mode shape of a Kirchhoff plate. Their numerical and experimental results indicated that the multiscale shear-strain gradient was capable of detecting delamination in composite laminates through revealing shear-strain singularities within a multiscale frame. Trendafilova et al. (2014) developed a method for identifying delamination in composites based on nonlinear signal correlation. Experiments showed that the method accurately localized delamination in a carbon fiber composite laminate.

A common characteristic of most existing methods for identifying delamination is that they usually indicate delamination by addressing discrepancies between signals from the laminated composite plate under inspection and its pristine counterpart. The signal from the pristine counterpart serves as a baseline for delamination characterization. Unfortunately, a pristine counterpart is commonly unavailable in real-life delamination diagnosis, severely hampering the feasibility of delamination detection. To address this limitation, this study focuses on developing a non-baseline scheme for identifying delamination by fully integrating noncontact high-resolution vibration measurement techniques and advanced signal processing theories including fractal analysis (Cao et al., 2005, 2006; Farhidzadeh et al., 2013; Moustafa et al., 2014; Moustafa and Salamone, 2012) and wavelet transforms (Cao and Qiao, 2008; Masoumi and Ashory, 2014; Quek et al., 2001; Sohn et al., 2004; Wang and Deng, 1999; Wei et al., 2004; Wu and Wang, 2011). First, two new concepts, scale mode shape (SMS) and scale waveform dimension (SWD), are elaborated. Second, use of the joint SMS and SWD to identify delamination in composite plates is numerically demonstrated, with emphasis on the method’s robustness against noise. Finally, the efficiency of the proposed method is experimentally validated in identifying thermally caused delamination in an actual composite plate, the mode shapes of which are acquired through noncontact high-resolution measurement achieved by a scanning laser vibrometer (SLV).

SMSs for isolating damage component

Notate a mode shape for a rectangular composite plate $W : (x_{i, j}, y_{i, j}, w_{i, j})$ , with $w_{i, j}$ the deflection amplitude at the grid point ( $x_{i, j}$ , $y_{i, j}$ ). To gain insight into the mode shape, $W$ is processed by a wavelet transform

W^{s} : (x_{i, j}, y_{i, j}, {w_{i, j}}^{s}) = W : (x_{i, j}, y_{i, j}, w_{i, j}) \otimes ψ_{s} (x, y)

(1)

where $ψ_{s} (x, y)$ is a two-dimensional (2D) wavelet with s being the scale parameter, the symbol ⊗ denotes the convolution, and $w_{i, j}^{s}$ is the deflection of the point ( $x_{i, j}$ , $y_{i, j}$ ) at the scale s. $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ is defined as the SMS, conveying the s-scale component of $W : (x_{i, j}, y_{i, j}, w_{i, j})$ .

Compared with other types of wavelet, the wavelet $g_{s}^{2, 2} (x, y)$ , defined by the second-order mixed derivatives of Gaussian function, has some prominent properties suited to damage detection, including symmetry, smoothness, differentiability, localizability, and explicit mathematical expressions (Xu et al., 2013), as given by

\begin{matrix} g_{s}^{2, 2} (x, y) = \frac{1}{3 \sqrt{π / 2}} \frac{1}{s} (4 {(\frac{x}{s})}^{2} - 2) (4 {(\frac{y}{s})}^{2} - 2) \\ e^{- ({(x / s)}^{2} + {(y / s)}^{2})} \end{matrix}

(2)

Using $g_{s}^{2, 2} (x, y)$ to specify $ψ_{s} (x, y)$ in equation (1), giving

W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s}) = W : (x_{i, j}, y_{i, j}, w_{i, j}) \otimes g_{s}^{2, 2} (x, y)

(3)

Equation (3) indicates that the mode shape $W : (x_{i, j}, y_{i, j}, w_{i, j})$ of a laminated composite plate can be transformed by the wavelet $g_{s}^{2, 2} (x, y)$ into a series of SMSs $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ . In terms of the Heisenberg’s uncertainty property of wavelets (Mallat, 1998), the lower-s SMSs related to wavelets with narrower supports convey higher frequency components of the mode shape, and the higher-s SMSs relevant to wavelets with wider supports embody lower frequency components of the mode shape. Consequently, with the increase in s, a sequence of SMSs occurs that represent mode shape components of progressively increased frequency content.

A measured mode shape for a composite plate commonly broadly consists of three portions: the noise component, the damage component, and the trend component, identified by the higher, intermediate, and lower frequency contents, respectively (Bai et al., 2014; Mallat and Hwang, 1992). Their respective SMSs yielded by equation (3) occupy smaller-, intermediate-, and greater- scale contents in light of the inverse relation between frequency and scale in a wavelet definition (Mallat and Hwang, 1992; Masoumi and Ashory, 2014). That being the case, the damage component can be managed by successively adjusting the scale parameter s in equation (3). Similarly, proper isolation of the damage component of mode shapes of a beam by applying the same principle was achieved by Cao and Qiao (2008). Therefore, SMSs lay the foundation for distinguishing the damage component from the noise and trend components in measured mode shapes.

SWD for damage characterization

Through the SMS analysis described above, a particular $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ at $s = s$ can properly be selected from the SMSs, conveying damage information by way of largely excluding noise and trend components in a measured mode shape $W : (x_{i, j}, y_{i, j}, w_{i, j})$ . In this section, $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ is further processed by an SWD-based scheme to characterize damage.

SWD

Waveform fractal analysis is introduced to develop a method of inspecting $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ for damage. To facilitate the use of waveform fractal analysis, $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ is first decomposed into SMS lines by the regime

W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s}) \approx W_{x}^{s} : (x_{i, j}, y_{j}, w_{i, j}^{s}) + W_{y}^{s} : (x_{i}, y_{i, j}, w_{i, j}^{s})

(4)

where $W_{x}^{s} : (x_{i, j}, y_{j}, w_{i, j}^{s})$ is a set of SMS lines in thex-direction, indexed by $y_{j}$ , $j = 1, 2, \dots, M$ , and $W_{y}^{s} : (x_{i}, y_{i, j}, w_{i, j}^{s})$ is a set of SMS lines in the y-direction, indexed by $x_{i}$ , $i = 1, 2, \dots, N$ , as schematically illustrated in Figure 1.

Damage detection is performed through detecting the abnormality of $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ , which is achieved by scrutinizing the singularity of each SMS line using Katz’s fractal dimension (KFD) (Cao et al., 2013a; Katz, 1988). The KFD is defined on a one-dimensional (1D) signal $Y_{j}, 1 < j \leq N$ as

Γ = \frac{\log_{10} (n)}{\log_{10} (n) + \log_{10} (d / L)}

(5)

where $n = N - 1$ is the number of increments of the waveform of N sampling points, L is the sum of all increments, and $d = max [dist (Y_{1}, Y_{j})]$ is the maximum distance between the first point $Y_{1}$ and any other point $Y_{j}, 1 < j \leq N$ . The $Γ$ is a measure with values ranging from 1 to 2 that quantifies the topological complexity of the analyzed signal from the perspective of fractal dynamics (Bai et al., 2012).

Let $U^{s} : (χ_{k}, w_{k}^{s})$ be a general instance of SMS lines of $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ . In particular, in the case of $χ = x$ and k = i, $U^{s} : (χ_{k}, w_{k}^{s})$ is an SMS line of $W_{x}^{s} : (x_{i, j}, y_{j}, w_{i, j}^{s})$ in the x-direction; in the case of $χ = y$ and k = j, $U^{s} : (χ_{k}, w_{k}^{s})$ is an SMS line of $W_{y}^{s} : (x_{i}, y_{i, j}, w_{i, j}^{s})$ in the y-direction. Let $V^{s} : {(x_{p}, w_{p}^{s}) |}_{p = 1}^{q}$ be a segment of $U^{s} : (χ_{k}, w_{k}^{s})$ , covering q sampling points with the midpoint $χ_{m}$ . An improved KFD defined on $V^{s}$ as an alternative to $Y$ is given as

Γ_{χ}^{s} (V^{s} : {(x_{p}, w_{p}^{s}) |}_{p = 1}^{q}) = \frac{\log_{10} (q)}{\log_{10} (q) + \log_{10} (d_{χ}^{s} / L_{χ}^{s})}

(6)

where

d_{χ}^{s} = max (\sqrt{{(χ_{k} - χ_{k - 1})}^{2} + {(w_{k}^{s} - w_{k - 1}^{s})}^{2}}); L_{χ}^{s} = \sum (\sqrt{{(χ_{i} - χ_{i - 1})}^{2} + {(w_{i}^{s} - w_{i - 1}^{s})}^{2}})

(7)

Equations (6) and (7) define the SWD, which quantifies the complexity of $V^{s} : {(x_{p}, w_{p}^{s}) |}_{p = 1}^{q}$ centered at $χ_{m}$ . The SWD inherits the sensitivity of the KFD to abnormality of the signal being inspected, while overcoming its weakness of susceptibility to noise.

SWD analysis

By regarding $V^{s} : {(x_{p}, {w_{p}}^{s}) |}_{p = 1}^{q}$ as a sliding window, we can move it to cover an SMS line $U^{s} : (χ_{k}, w_{k}^{s})$ step by step, evaluating the SWD in terms of equations (6) and (7) at each step. After the window slides across the SMS line, an SWD trajectory can be produced that depicts the SMS line’s complexity profile. Because damage can induce local change in the complexity of SWD, the SWD trajectory is capable of signaling the occurrence of damage by its abnormalities.

Notate $ℑ_{x}$ and $ℑ_{y}$ the operation of applying the sliding window individually to all the SMS lines of $W_{x}^{s} : (x_{i, j}, y_{j}, w_{i, j}^{s})$ and $W_{y}^{s} : (x_{i}, y_{i, j}, w_{i, j}^{s})$ , respectively. When $ℑ_{x}$ is performed, a set of SWD trajectories can be produced, forming an x-SWD surface $Π_{x}^{s}$ at the scale s

Π_{x}^{s} = ℑ_{x} (W_{x}^{s} : (x_{i, j}, y_{j}, w_{i, j}^{s})), j = 1, \dots, M

(8)

In the same way, $ℑ_{y}$ can give rise to a y-SWD surface $Π_{y}^{s}$ at the scale s from $W_{y}^{s} : (x_{i}, y_{i, j}, w_{i, j}^{s})$

Π_{y}^{s} = ℑ_{y} (W_{y}^{s} : (x_{i}, y_{i, j}, w_{i, j}^{s})), i = 1, \dots, N

(9)

$Π_{x}^{s}$ and $Π_{y}^{s}$ display the profiles of complexity of the $W_{x}^{s} : (x_{i, j}, y_{j}, w_{i, j}^{s})$ and $W_{y}^{s} : (x_{i}, y_{i, j}, w_{i, j}^{s})$ , respectively.

Delamination characterization

Delamination, a typical form of damage in a composite plate, can locally change the complexity of $W^{s} : (x_{i, j}, y_{i, j}, w_{i, j}^{s})$ , causing abnormalities in $Π_{x}^{s}$ and/or $Π_{y}^{s}$ . Owing to its directionality, ${Π_{x}}^{s}$ favorably reveals the feature of delamination in the y-direction but not the x-direction, and vice versa for $Π_{y}^{s}$ . To consolidate the capability of characterizing delamination in both the x- and y-directions, fusion of $Π_{x}^{s}$ and $Π_{y}^{s}$ by simple averaging is adopted to form an x-y-SWD surface

Π_{x + y}^{s} = \frac{1}{2} (Π_{x}^{s} + Π_{y}^{s})

(10)

Thus, $Π_{x + y}^{s}$ produces a more sophisticated indication of damage than either $Π_{x}^{s}$ or $Π_{y}^{s}$ .

Numerical demonstration

Numerical model

Functionally graded materials (FGMs) are advanced materials in which the material properties vary with position in the component (Yang and Shen, 2001). FGMs have thermomechanical properties changing with position along the thickness direction (Chakraborty et al., 2003). Therefore, when FGMs are subjected to varying temperature along the thickness, the expansion or contraction of the different layers is not uniform but changes with the mixing ratio. This effect in turn causes shear stresses in the interface between different layers, thereby causing shear delamination of the material (Dag et al., 2004; Dag and Ilhan, 2008; Yıldırım et al., 2008).

Use of the SWD analysis to identify delamination is demonstrated on functionally graded plates made of SUS304 and Si₃N₄. The functional grade is achieved by gradually adjusting the mixture ratio of SUS304 to Si₃N₄, to obtain different material properties of Young’s modulus E, Poisson’s ratio v, and mass density $ρ$ . The variations in E, v, and ρ conform to the following general rules (Yang and Shen, 2001)

E = (E_{suf} - E_{mid}) {(\frac{2 | h |}{H})}^{λ} + E_{mid}

(11)

v = (v_{suf} - v_{mid}) {(\frac{2 | h |}{H})}^{λ} + v_{mid}

(12)

ρ = (ρ_{suf} - ρ_{mid}) {(\frac{2 | h |}{H})}^{λ} + ρ_{mid}

(13)

where the subscript suf labels the upper or lower surfaces and mid marks the midplane of a plate; h is the offset from the midplane and H is the thickness of the plate; $E_{suf}$ , $E_{mid}$ , $v_{suf}$ , $v_{mid}$ , $ρ_{suf}$ , and $ρ_{mid}$ are all functions of temperature T (K) as reported by Love (1952); and λ is the volume fraction index.

Detection of delamination via numerical simulation is presented using a delaminated plate of length 1 m, width 1 m, and thickness 0.012 m in the x-, y-, and z-directions. This plate is made of mixed SUS304 and Si₃N₄. The delamination area is 0.05 m × 0.05 m, 1/400 of the area of the plate, and centered at the point $x = 0.325 m$ , $y = 0.575 m$ , and $z = 0.0024 m$ . The specific material parameters at $T = 300 K$ , $E_{suf} = 2.0778 \times 10^{11} Pa$ , $E_{mid} = 3.2227 \times 10^{11} Pa$ , $v_{suf} = 0.3178$ , $v_{mid} = 0.2400$ , $ρ_{suf} = 8166 kg / m^{3}$ , and $ρ_{mid} = 2370 kg / m^{3}$ , and λ = 3 (Yang and Shen, 2001) are adopted to frame the general rules expressed in equations (11)–(13), from which E, v, and ρ for each layer are determined as marked by the points in Figure 2. The numerical model of the plate consists of 100 × 100 × 10 three-dimensional (3D) eight-node solid structural SOLID185 elements and is built using the commercial software ANSYS^®. The delamination is created using a null-thickness interface, on which coincident nodes of identical coordinate values but separate structural elements are distributed. Figure 3(a) shows the finite element mesh of the plate with the delamination area, together with the zoomed-in detail of the delamination (Figure 3(b)).

Modal analysis of the delaminated plate is performed to generate mode shapes, to which white Gaussian noise is added to produce noisy mode shapes with the noise level specified by the signal-to-noise ratio (SNR): $SNR = 20 \log_{10} (A_{s} / A_{n})$ , where A_s and A_n represent the root-mean-square magnitudes of the numerical mode shape and the additional noise, respectively. By way of illustration, Figure 4(a) to (c) shows three mode shapes at the typical noise level of SNR 70 dB at natural frequencies 1237.1, 1394.6, and 1601.8 Hz, respectively.

Delamination identification

The mode shape of SNR 70 dB at frequency 1394.6 Hz in Figure 4(b) is arbitrarily selected to illustrate the method developed above. Identification of the delamination is performed by flexibly exploiting the joint SMS and SWD to first isolate the delamination component and then characterize the delamination.

Isolation of delamination component

When the scale parameter s increases from a small beginning, s = 0.5, into a large scale, s = 30, a particular s-interval [5, 25] can be recognized that gives rise to a process of gradual variation in delamination information from weakness to strength and back to weakness, with the noise located at smaller scales and trend information at greater scales within that interval. The occurrence is manifested by the planform of $Π_{x + y}^{s}$ for the SMS at each scale, as typified in Figure 5(a) to (i) for s = 5, 7.5, 10, 12.5, 15, 17.5, 20, 22.5, and 25, respectively. In each $Π_{x + y}^{s}$ , the cluster of higher intensity patches signals the delamination, and the accuracy of identification can be evaluated by the concentration of higher intensity patches. It can be observed by slightly adjusting s that the accuracy of recognition increases from s = 5 to 13 and decreases from s = 13 to 25, testifying that s = 13 specifies a particular SMS $W^{13}$ that conveys the richest delamination information, so that it is used to characterize the delamination. The underlying mechanism of this process is attributed to the multiscale property of the Gaussian wavelet transform that controls the scale content of the SMS.

Delamination characterization

The SWD analysis of $W^{13}$ leads to the x-SWD surface $Π_{x}^{13}$ , y-SWD surface $Π_{y}^{13}$ , and x-y-SWD surface $Π_{x + y}^{13}$ (Figure 6(a) to (c), respectively), in each of which the sharply rising singular peak clearly signifies the occurrence of delamination. Moreover, in the planform of the $Π_{x + y}^{13}$ (Figure 6(d)), the cluster of higher intensity patches arising from the singular peak in Figure 6(c), standing out from the surrounding area, distinctly indicates the distribution of the delamination.

Comparison with existing methods

Comparison with natural frequency–based method

As a typical dynamic feature, natural frequency is widely used to portray damage in composite structures. In particular, several researchers (Minak et al., 2010; Zhang et al., 2014) have verified the capability of natural frequencies to characterize delamination in beams, with the change in natural frequencies indicating the delamination state clearly. Unfortunately, preliminary analysis indicates that the delamination (Figure 3) in the functionally graded plate is so small that it causes negligible change in natural frequencies. As listed in Table 1, the delamination-caused highest relative change ratio of the first 10 natural frequencies is 0.28%, insufficient for delamination identification applications.

Table 1.

Effect of delamination on first 10 natural frequencies.

Modal order	1	2	3	4	5	6	7	8	9	10
Intact (Hz)	19.202	43.484	118.81	145.30	162.76	279.47	337.29	351.53	380.44	485.07
Delaminated (Hz)	19.149	43.447	118.54	145.08	162.55	279.17	336.5	350.79	379.76	484.37
Relative change ratio (%)	0.28	0.09	0.23	0.15	0.13	0.11	0.23	0.21	0.18	0.14

Comparison with existing fractal-based method

The proposed method with joint SMS and SWD is compared with the existing method that directly applies the KFD to mode shape lines to produce an x-y surface $Π_{x + y}$ exclusive of the scale concept. The comparison is individually performed on mode shapes of SNR 70 dB at natural frequencies 1237.1 and 1601.8 Hz. Delamination identification results represented by the planforms of $Π_{x + y}^{13}$ and $Π_{x + y}$ , obtained using the proposed and existing methods to process the mode shape of SNR 70 dB at frequency 1237.1 Hz, are presented in Figure 7(a) and (b), respectively. In the planform of $Π_{x + y}^{13}$ (Figure 7(a)), the cluster of higher intensity patches, contrasting with the surrounding area, distinctly indicates the distribution of the delamination, whereas in the planform of $Π_{x + y}$ (Figure 7(b)), a group of discontinuous horizontal strips resulting from the ridges of the mode shape (Figure 4(a)) dominates the planform, from which no features can be discerned to reflect delamination. A similar observation can be found from the mode shapes at the natural frequency 1601.8 Hz, as shown in Figure 8. Thus, the proposed method outperforms the existing method in identifying delamination in noisy conditions.

Experimental validation

Experimental setup

A glass fiber–reinforced polymer (GFRP) square plate (Figure 9(a)) of length 440 mm, width 440 mm, and depth 1.5 mm, containing a small area of thermal delamination, is used as a test specimen. This specimen consists of two-ply unidirectional glass fiber mats oriented in the x-direction, with local separation of two plies forming the delamination, as enclosed by the ellipse in Figure 9. The local separation is fabricated by locally heating the plate using a hot air blower with the temperature of output air set to 500°C. With 6-s heating, the two plies instantly separate from each other, engendering the delamination as depicted in Figure 9(b).

Vibration of the specimen is created using a circular 10-mm-diameter piezoelectric lead zirconate titanate actuator, located at the geometrical center of the plate (Figure 9(a)), to excite the plate harmonically and perpendicularly. When the specimen vibrates stably, an SLV (Polytec PSV-400) (Figure 10) is employed to scan its non-heated surface, to register the isochronous transverse velocities at 385 × 383 sampling points evenly distributed across the surface. The array of isochronous transverse velocities constitutes mode shapes for the plate, as illustrated in Figure 10(b) for the mode shape at the natural frequency of 494 Hz.

Results

By adjustment of the scale parameter s, the s-interval with delamination components is recognized as [5, 15] for the mode shape at natural frequencies 494 Hz. From this interval, $s = 10$ specifies the proper SMS, $W^{10}$ , probably carrying the richest delamination information. Consequently, $W^{10}$ is adopted for use in characterization of the delamination, giving the x-y surface $Π_{x + y}^{10}$ with the planform shown in Figure 11(a). A cluster of much higher intensity patches can be clearly observed toward the right of the planform, clearly delineating the distribution of the delamination. Interestingly, a secondary cluster of small higher intensity patches can be discerned in the upper middle of the planform. Tracing this cluster of patches, some concealed original stripe-like defects (Figure 11(b)) are dramatically unveiled by pressing the surface point by point using a nail. These observations validate that the proposed method can localize delaminations in actual composite plates with great accuracy.

Conclusion

New physical quantities, the SMS and SWD, are proposed with the aim of developing technologies for the identification of delamination; a new method for identifying delaminations in composite plates is established based on these two quantities. The performance of the method is numerically demonstrated using mode shapes of a delaminated functionally graded plate, with particular emphasis on its robustness against noise. Moreover, the applicability of the method to the identification of delamination is experimentally validated using a GFRP plate containing a thermal delamination, with mode shapes measured by SLV-based noncontact measurement. From the study, the major observations obtained are as follows:

SMSs describe mode shapes of a composite plate at the scale level, entailing damage information concentrated at several particular SMSs that are largely free of interference from the noise and trend of the mode shape.

SWD has the distinctive capability of characterizing delamination by revealing the singularities of an SMS from the perspective of waveform fractal dynamics.

SWD analysis underpinned by the SMSs can identify delamination with great accuracy in noisy conditions, requiring no intact mode shapes as baseline nor any prior knowledge of either material properties or boundary conditions of the plate being inspected.

With all these features, the proposed method holds promise for the development of practical technologies for damage diagnosis in composite structures.

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: The authors gratefully acknowledge the financial support provided by the Natural Science Foundations of China (No. 11172091), the Fundamental Research Funds for the Central Universities (Grant Nos. 2014B03914 and 2012B05814) and the China Postdoctoral Science Foundation (No. 2014 M560386).

References

Akhter

Jung

Chang

. (2009) Location of delamination in laminated composite plates by pulsed laser holography. Optics and Lasers in Engineering 47: 584–588.

Bai

Cao

. (2012) Fractal dimension analysis of higher-order mode shapes for damage identification of beam structures. Mathematical Problems in Engineering 2012: 454568.

Bai

Ostachowicz

Cao

. (2014) Crack detection in beams in noisy conditions using scale fractal dimension analysis of mode shapes. Smart Materials and Structures 23(6): 065014.

Cao

Qiao

(2008) Integrated wavelet transform and its application to vibration mode shapes for the damage detection of beam-type structures. Smart Material Structures 17: 055014.

Cao

Ostachowicz

Bai

. (2013a) Fractal mechanism for characterizing singularity of mode shape for damage detection. Applied Physics Letters 103(22): 221906.

Cao

Ostachowicz

Radzieński

. (2013b) Multiscale shear-strain gradient for detecting delamination in composite laminates. Applied Physics Letters 103(10): 101910.

Cao

Ren

Qiao

(2006) Nondestructive assessment of reinforced concrete structures based on fractal damage characteristic factors. Journal of Engineering Mechanics: ASCE 132(9): 924–931.

Cao

Ren

Wang

(2005) A method of detecting seismic singularities using combined wavelet with fractal. Chinese Journal of Geophysics: Chinese Edition 48(3): 672–679.

Chakraborty

Gopalakrishnan

Reddy

(2003) A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Sciences 45(3): 519–539.

10.

Dag

Ilhan

(2008) Mixed-mode fracture analysis of orthotropic functionally graded material coatings using analytical and computational methods. Journal of Applied Mechanics: Transactions of the ASME 75(5): 051104.

11.

Dag

Yildirim

Erdogan

(2004) Interface crack problems in graded orthotropic media: analytical and computational approaches. International Journal of Fracture 130: 471–496.

12.

Farhidzadeh

Dehghan Niri

Moustafa

. (2013) Damage assessment of reinforced concrete structures using fractal analysis of residual crack patterns. Experimental Mechanics 53(9): 1607–1619.

13.

Ibarra-Castanedo

Avdelidis

Grinzato

. (2011) Delamination detection and impact damage assessment of GLARE by active thermography. International Journal of Materials and Product Technology 41: 5–16.

14.

Iwasaki

Todoroki

(2005) Statistical evaluation of modified electrical resistance change method for delamination monitoring of CFRP plate. Structural Health Monitoring 4(2): 119–136.

15.

Jiang

Leung

CKY

(2008) 3×3 coupler-based Mach-Zehnder interferometer and its application in the delamination detection in FRP composite. Journal of Intelligent Material System and Structures 19(4): 497–507.

16.

Katz

(1988) Fractals and the analysis of waveforms. Computers in Biology and Medicine 18(3): 145–156.

17.

Larrosa

Lonkar

Chang

(2014) In situ damage classification for composite laminates using Gaussian discriminant analysis. Structural Health Monitoring 13(2): 190–204.

18.

Love

AEH

(1952) A Treatise on the Mathematical Theory of Elasticity. 4th ed. Cambridge: Cambridge University Press.

19.

Mallat

(1998) A Wavelet Tour of Signal Processing. London: Academic Press.

20.

Mallat

Hwang

(1992) Singularity detection and processing with wavelets. IEEE Transactions on Information Theory 38: 617–643.

21.

Masoumi

Ashory

(2014) Damage identification from uniform load surface using continuous and stationary wavelet transforms. Latin American Journal of Solids and Structures 11: 738–754.

22.

Minak

Palazzetti

Trendafilova

. (2010) Localization of a delamination and estimation of its length in a composite laminate beam by the VSHM and pattern recognition methods. Mechanics of Composite Materials 46(4): 387–394.

23.

Moustafa

Salamone

(2012) Fractal dimension based Lamb wave tomography algorithm for damage detection in plate-like structures. Journal of Intelligent Material Systems and Structures 23(11): 1269–1276.

24.

Moustafa

Dehghan Niri

Farhidzadeh

. (2014) Corrosion monitoring of post-tensioned concrete structures using fractal analysis of guided ultrasonic waves. Structural Control and Health Monitoring 21(3): 438–448.

25.

Okabe

Fujibayashi

Shimazaki

. (2010) Delamination detection in composite laminates using dispersion change based on mode conversion of Lamb waves. Smart Materials and Structures 19(11): 115013.

26.

Pines

Purekar

(2010) Damage detection in thin composite laminates using piezoelectric phased sensor arrays and guided lamb wave interrogation. Journal of Intelligent Material Systems and Structures 21(10): 995–1010.

27.

Qiao

Lestari

. (2007) Curvature mode shape-based damage detection in composite laminated plates. Composite Structures 80(3): 409–428.

28.

Quek

Wang

Zhang

. (2001) Sensitivity analysis of crack detection in beams by wavelet technique. International Journal of Mechanical Sciences 43(12): 2899–2910.

29.

Rosalie

Vaughan

Bremner

. (2004) Variation in the group velocity of Lamb waves as a tool for the detection of delamination in GLARE aluminium plate-like structures. Composite Structures 66(1–4): 77–86.

30.

Shang

Yun

Qiao

(2010) Delamination identification of laminated composite plates using a continuum damage mechanics model and subset selection technique. Smart Materials and Structures 19(5): 055024.

31.

Sohn

Dutta

Yang

. (2011a) Delamination detection in composites through guided wave field image processing. Composites Science and Technology 71(9): 1250–1256.

32.

Sohn

Dutta

Yang

. (2011b) Automated detection of delamination and disbond from wavefield images obtained using a scanning laser vibrometer. Smart Materials and Structures 20: 045017.

33.

Sohn

Park

Wait

. (2004) Wavelet-based active sensing for delamination detection in composite structures. Smart Materials and Structures 13(1): 153–160.

34.

(2004a) Lamb wave-based quantitative identification of delamination in CF/EP composite structures using artificial neural algorithm. Composite Structures 66(1–4): 627–637.

35.

(2004b) Fundamental Lamb mode-based delamination detection for CF/EP composite laminates using distributed piezoelectrics. Structural Health Monitoring 3: 43–68.

36.

Todoroki

Tanaka

Shimamura

(2002) Delamination monitoring of graphite/epoxy laminated composite plate of electric resistance change method. Composites Science and Technology 62: 1151–1160.

37.

Trendafilova

Palazzetti

Zucchelli

(2014) Delamination assessment in structures made of composites based on general signal correlation. International Journal of Structural Stability and Dynamics. DOI: 10.1142/S0219455414400227.

38.

Wang

Deng

(1999) Damage detection with spatial wavelets. International Journal of Solids and Structures 36(23): 3443–3468.

39.

Wei

Yam

Cheng

(2004) Detection of internal delamination in multi-layer composites using wavelet packets combined with modal parameter analysis. Composite Structures64(3–4): 377–387.

40.

Wang

(2011) Experimental studies on damage detection of beam structures with wavelet transform. International Journal of Engineering Science 49(3): 253–261.

41.

Radzieński

Ostachowicz

. (2013) Damage detection in plates using two-dimensional directional Gaussian wavelets and laser scanned operating deflection shapes. Structural Health Monitoring 12(5–6): 457–468.

42.

Yam

Wei

Cheng

(2004) Nondestructive detection of internal delamination by vibration based method for composite plates. Journal of Composite Materials 38: 2183–2198.

43.

Yan

Yam

(2004) Detection of delamination damage in composite plates using energy spectrum of structural dynamic responses decomposed by wavelet analysis. Computers and Structures 82(4–5): 347–358.

44.

Yang

Shen

(2001) Dynamic response of initially stressed functionally graded rectangular thin plates. Composite Structures 54: 497–508.

45.

Yıldırım

Yılmaz

Kadıoğlu

(2008) Delamination of compressively stressed orthotropic functionally graded material coatings under thermal loading. Journal of Applied Mechanics: Transactions of the ASME 75(5): 051106.

46.

Żak

Krawczuk

Ostachowicz

(2000) Numerical and experimental investigation of free vibration of multilayer delaminated composite beams and plates. Computational Mechanics 26(3): 309–315.

47.

Żak

Krawczuk

Ostachowicz

(2001) Vibration of a laminated composite plate with closing delamination. Journal of Intelligent Material Systems and Structures 12(8): 545–551.

48.

Zhang

Shankar

Morozov

. (2014) Vibration-based delamination detection in composite beams through frequency changes. Journal of Vibration and Control. 1077546314533584.