Guided wave based techniques are among the promising techniques for structural health monitoring due to their ability to detect damage with high precision, ability to scan large area and low power consumption. However, practical implementation of these techniques to structures with relatively complex geometries such as stiffened plates demand further research due to several challenges. Some of such challenges include the presence of stiffeners and edges in close vicinity causing multiple reflections of waves, possibility of multiple damages and lack of baseline data. To address all these issues, a modified hyperbola based approach using mode converted signal and a data based matching scheme has been proposed in this paper. A stiffened plate is taken as the example structure. Corrosion like defects have been considered as the damages. Each panel of the stiffened plate has been scanned separately through phased array actuation. The effectiveness and robustness of the proposed approach have been shown through a case study using 3D finite element simulation data.

## 1.Introduction

Lamb wave based techniques are finding growing application to defect detection in the plate like structures due to associated advantages like less attenuation, low power consumption, no requirement of transducer movement, etc Generally, piezoelectric patches are used for actuation and sensing of Lamb waves in plate like structures. Detailed reviews of the developments in this field can be found in [1, 2]. Two patches fitted at the two opposite surfaces in a chosen location of a plate and actuated with the same or opposite voltage signal can generate symmetric or antisymmetric Lamb waves. In many applications, an array of such actuators is actuated separately with a chosen time delay to ensure predominant wave propagation in a certain direction. This procedure is called phased array beam forming [3].

Delay-And-Sum (DAS), ellipse based method, direct arrival, hyperbola based method, time reversal method and probability based methods, etc are among the various methods widely used by researchers for damage detection in plates from Lamb wave propagation data [4–12]. In the DAS method, differential signals across various sensor-actuator paths are shifted according to their time-of-flight (TOF). The signal thus obtained from each actuator-sensor pair is squared and averaged at each spatial location. Researchers like Khodaei and Aliabadi [6], Michaels [13], showed the use of DAS method with various modifications. DAS algorithms have limitations in case of multiple defects and their efficacy depends on the availability of baseline signals. Xu *et al* [14] used a sparse Bayesian learning strategy to estimate the sparse traveling distance of overlapped dispersive Lamb waves for dispersion compensation. The signal was then used for multiple defect imaging using DAS algorithm. However, the dependency on the baseline signal was still present. The hyperbola based imaging technique uses the difference of arrival time of damage scattered waves at two sensors [15–17] and is not baseline free.

The time reversal method is a baseline free technique for Lamb wave based damage detection [18–21]. It is based on the rational that an original wave can be reconstructed at its source point if its forward wave recorded at another point is time reversed and emitted back to the source point. The received output signal contains two parts—the directly propagating wave and the scattered wave from the defect. The scattered wave signal is used for baseline-free imaging to localize damages. Some of the other baseline free techniques proposed by various researchers for plate damage detection can be found in [22–25]. However, the efficacy of these techniques in the presence of multiple damages and multiple reflections of the wave through edge and stiffeners has not been adequately addressed yet. In many studies, absorbing boundaries have been incorporated artificially to minimize reflections from boundaries. This may not always be feasible in case of in service structures. Ebrahimkhanlou, Dubuc and Salamone [26] proposed a model based imaging algorithm incorporating edge reflected Lamb waves through ray and mode conversion tracking for plate damage detection. Yang, Lee and Cheng [27] detected defects in coated and buried pipes using the torsional mode of guided waves. Recently Kumar and Sunny [28] proposed a data-driven approach using intrinsic mode-based damage feature for the detection of multiple structural damages from Lamb wave propagation data. However, the approach depends strongly on baseline data. Huang *et al* [29] proposed a multi patch baseline free damage imaging method for damage detection in composite plates. Performance evaluation of such a technique in the presence of multiple damages, especially in a one dimensional wave propagation path is still missing. Perfetto *et al* [30] studied sensitivity of guided wave propagation characteristics to damage in the presence of quasi-static load using the finite element method. Due to the presence of multiple damages and multiple reflections from stiffeners, edges, etc in the structure, time domain signal data obtained from sensors consist of overlapped wave packets. It becomes very complex to separate damage-related response signals without pristine data [9, 31, 32].

From the review it is observed that literature simultaneously dealing with all the complexities due to lack of baseline data, presence of multiple damages and structural features like stiffeners etc are not adequate. To address these issues, we propose a modified version of the hyperbola based technique which is baseline free, capable of detecting multiple damages in the presence of multiple guided wave reflections due to stiffeners and edges. The novelties of the work lie in proposition of (i) the separation of the antisymmetric components from the sensor signals to impart baseline free detection capability to the hyperbola based method which is baseline dependent in its traditional form and (ii) a data based matching scheme to deal with the complexities associated with multiple damages and multiple reflections due to edges and stiffeners. This technique relies on scanning different radial directions from the actuation location separately through the popular phased array technique [21, 33, 34]. A plate with two mutually perpendicular stiffeners has been considered for case study. High fidelity 3D finite element (FE) simulation data have been used. Finally, the robustness of the scheme in dealing with experimental uncertainties has been shown through wave propagation data obtained from a simulation model with artificially added errors in actuator placement. Sections 2 and 3 of the paper describe the example structure considered and the 3D FE model. Input waveform selected for actuation is described in section 4. Although phased array based actuation is described in detail in the literature, a brief description of the present context is given in section 5 to make paper self contained. The proposed detection technique is explained in section 6 comprehensively. Finally, results for different cases are described through 4 subsections of section 7 and the paper is concluded in section 8.

## 2.Structure for damage detection

A stiffened plate of dimension 1200 mm **×** 1200 mm **×** 3 mm made of aluminum alloy AA2024 T3 (density- 2750 kg m^{−3}, Young's modulus- 72.4 GPa, Poisson's ratio- 0.33) was considered as the case study for damage detection. As shown in figure 1, two stiffeners divide the plate into four panels. In this plate, each panel has the same dimension. However, it is not a requirement. Each panel has separate sets of actuators and sensors. Figure 1(b) shows a zoomed view of one representative panels. Each panel has 17 pairs of PZT patches. Each pair refers to two PZTs fitted at two opposite surfaces of the plate at the same location. Among those 17 PZT pairs, nine PZT pairs were used as actuators and the rest eight pairs were used as sensors. The actuators were placed with interspacing of 14 mm. A 450 mm **×** 450 mm area was considered for investigation of damage as shown by a black dashed line in figure 1(b). Circular holes of diameter 10 mm and depth less than the plate thickness were considered as the damages. In figure 1(b), *D*_{i } refers to *i*th damage, *A*_{i }refers to *i*th actuator location and *Si* refers to *i*th sensor location. Each panel was hypothetically divided into four quadrants denoted by Q1, Q2, Q3 and Q4.

## 3.Finite element modeling

A 3D FE model of the stiffened plate described in the previous section was developed in ABAQUS. Eight noded linear brick elements (C3D8R) of length 1.5 mm was used to discretize the plate and stiffeners. The element length was taken to be one-tenth of the wavelength of a transverse wave in the plate [7]. Lead Zirconate Titanate PZT-5H was used for actuation and sensing the lamb wave response and material properties are shown in table 1. The PZT patches were meshed using 8 noded linear piezoelectric brick elements (C3D8E) of length 1 mm. A boundary condition of zero potential was applied at the surface of all PZT patches contacting with the plate. On the free surfaces of the actuator patches electric potential of the waveform shown in figure 3 and magnitude 20 V was applied. This ensures the generation of S0 mode. For sensing, electric potential was extracted from the free surface of sensor PZT patches. A constrained termed as 'EQUATION CONSTRAINT' was applied at the free surface nodes of each sensor patch to ensure all the nodes of the free surface have equal electric potential. A tie constraint was used to connect PZT patches with the plate. Dynamic implicit analysis with free-free boundaries was used for this study. Time increment for analysis was chosen as per CFL (Courant, Friedrichs, and Lewy) criteria and found to be 25 *μ*s [7].

**Table 1.**Material properties of PZT-5H.

Electric permittivity | Piezoelectric | Elastic properties | Density | |||
---|---|---|---|---|---|---|

D11 | 1.505e-8 | d1,13 | 7.41e-10 | E1 | 60.61 GPa | 7500 kg m^{−3} |

D22 | 1.505e-8 | d1,23 | 7.41e-10 | E2 | 60.61 GPa | |

D33 | 1.301e-8 | d3,11 | −2.4e-10 | E3 | 48.31 GPa | |

d3,22 | −2.4e-10 | Nu12 | 0.289 | |||

d3,33 | 5.93e-10 | Nu13 | 0.512 | |||

Nu23 | 0.512 | |||||

G12 | 23.5 GPa | |||||

G13 | 23 GPa | |||||

G23 | 23 GPa |

## 4.Selection of input waveform

Figure 2 shows the variation of group velocities *V*_{g } and phase velocities *V*_{p } of basic symmetric (S0) and basic antisymmetric (A0) waves with frequency obtained through GUIGUW software [35] for an aluminum alloy AA2024 T3 plate of thickness 3 mm. Based on this, a tone burst voltage signal V (t) of 5.5 cycles with central frequency of 200 kHz as shown in figure 3 was used for actuation to ensure the presence of only A0 and S0 waveform with reasonably low dispersion of both A0 and S0 signals.

## 5.Phased array beam forming technique

Phased array technique for beam forming uses a series of closely placed actuators in a linear, circular or cruciform pattern [3]. Each actuator is actuated with a time delay so that waves from various actuators enable the propagating wave to intensify in the desired direction by constructive interference in the plate. Figure 4 shows a linear array of 9 actuators and beam propagating at an angle of *θ* with an array axis in both directions of the array.

The sequential time delay is calculated using the following equation [3].

Where represents time delay in the excitation of two nearby actuators, *d* is spacing between actuators, *θ* is angle between the required direction of beam and actuator array axis and is the phase speed of excitation signal at central frequency for the mode of actuation. The beam propagates along two directions which are mirror images about the array axis. As the number of actuators in the array increases, the intensity of the beam increases [3]. The spacing between the actuators has to be chosen properly in order to avoid any unwanted beam propagating in other directions. Usually, the spacing *d* is chosen to be half of the wavelength of the excitation signal (*λ*) as shown below.

Where *f*_{c } = Central frequency of excitation signal

*v*_{c } = Speed of excitation signal at central frequency

The time delay between the actuator signals is shown through figure 5.

## 6.Algorithm for locating the defects

Within each panel damages in each of the quadrants Q1 to Q4 are detected separately. This is achieved by varying the beam propagation direction *θ*. For example, by varying *θ* from 0° to 90°, damages in Q1 are detected. The algorithm for detecting damages within a quadrant is shown by a flow chart in figure 6. Each of the steps are described below.

### 6.1.Extraction of A0 and S0 modes

At each sensor location, the wave propagation data is obtained in the form of voltage signal history extracted from the two PZTs fitted at the two opposite surfaces of the plate. The separated symmetric and antisymmetric components are (f+g)/2 and (f−g)/2 respectively, where f and g are the signals obtained from the PZTs fitted at the top and bottom surface respectively.

### 6.2.Distinction between direct and reflected waves

The travel path of a scattered wave from a defect to sensor is shown in figure 7. In each of these travel paths, S0 wave generated by phased array actuators travels predominantly in a certain direction. If there is any damage in that path, mode conversion of the wave takes place. Hence, the damage scattered wave contains both S0 and A0 waves. Our interest is in the mode converted A0 wave.

Here, four possible travel paths of wave from actuator to sensor have been considered. In the first travel path, mode converted A0 wave from damage directly reaches a sensor from the damage. Maximum possible direct arrival time of mode converted scattered wave (A0) at a sensor from a defect can be estimated using the equation below.

Here *x*_{i-j } represents the distance between the locations *i* and *j*, and represent the co-ordinate locations of actuator, defect and sensor respectively and *θ* is the angle between beam propagation direction and actuator array. and are the group velocity of S0 and A0 modes of Lamb wave for frequency of interest which can be obtained using a test or simulation as the case may be.

A constant time value *t*_{sp } = 0.000015 s is added considering the spread of the pulse due to dispersion.

Three indirect travel paths of wave from an actuator to a sensor are possible. The first possible indirect path is denoted as t_{ref1}. In this travel path, mode converted A0 wave reaches a boundary and from the boundary reaches a sensor. The minimum time of arrival of a mode converted A0 wave through such a path considering all the various damage positions is

The distance and can be determined with the geometry of the plate and co-ordinates of actuator, sensors and defects in the beam propagation path.

The second indirect path is denoted as t_{ref2}. In this path, an actuator generated wave reaches damage via boundary. After that, mode converted wave reaches a sensor. The minimum time of arrival of a mode converted A0 wave through such a path considering all the various damage positions is given by

Third possibility is mode conversion of actuator generated wave by stiffener and arrival of the mode converted wave at the sensor. The minimum time of arrival of such reflection is estimated by

For the damage location ( ) to be along the beam propagation direction following equation needs to be satisfied.

Hence, in equations 4 to 7, and should be related through equation 8. For the configuration under consideration, table in annexure 1 shows the values of and at sensors of quadrant-1 for possible defects along various beam propagation directions in quadrant-1. This table is applicable to all quadrants by analogy. It can be observed that of any sensor has the least value for any propagation direction of the beam. It means reflections arrive after the arrival of direct scattered waves from defects. Hence, all the peaks within can be considered as direct scattered waves from defects and the number of such peaks indicates the number of defects in that direction. It can be noted that there are several other possible indirect travel paths. For example, S0 wave may reach damage via an edge and from the damage mode converted wave may reach the sensor via the same or a different edge. But it is obvious that the minimum arrival time of such wave is higher than

Therefore, a knowledge base consisting of for different sensors for various beam propagation angle *θ* in each quadrant is created. Presence of peaks within the time interval [0, ] in the A0 components of sensor signals in the quadrant under investigation indicates the presence of damages in the beam propagation direction *θ*.

### 6.3.Defect localization equations

Through phased array excitation it is ensured that beam intensity in the direction of *θ* is significantly more compared to other directions. Hence, any defect in this direction scatters more intense mode converted waves as compared to defects in other directions. By changing the value of *θ* i.e., the angle of beam propagation, whole plate can be scanned and whichever direction gives strong scattered waves to nearby sensors, confirms the angular location of defect.

The difference between times of arrival (ToA) of damage scattered A0 at two different sensor locations is equal to difference of distance between defect and sensors divided by speed of A0 wave (figure 8). For example, considering known sensor locations and with unknown defect location this difference between arrival time can be written as:

represents the difference between times of arrival (ToA) of defect scattered A0 signals at sensors *Si* and *Sj*. This is an equation of a hyperbola with variables Similar equations can be obtained for all pairs of sensor locations. If the time delays between various pairs of sensor locations correspond to the same defect the curves intersect at a common point. The point represents the defect location if it lies on the line corresponding to angle of propagation, i.e., if it satisfies equation 8.

### 6.4.Date based matching scheme for identification of peaks from same defect at each sensor location

Once, the number of peaks within is found, ToAs of these peaks are recorded. When the plate contains multiple defects, the mode converted signal at a sensor location contains multiple peaks, one corresponding to each defect. But the signal doesn't indicate which pulse corresponds to which defect. For example, in case of two defects say D1 and D2, time delays between peaks of sensor signal S1 and S2 can have four values Δt_{S1D1-S2D1}, Δt_{S1D2-S2D1,} Δt_{S1D1-S2D2,} Δt_{S1D2-S2D2,} where Δt_{SiDj-SkDl} represents time delay between peak at sensor Si due to damage Dj and peak at sensor Sk due to damage Dl. Out of these four values only Δt_{S1D1- S2D1} and Δt_{S1D2- S2D2} are to be used for the localization of D1 and D2. However, it is not known *a priori* which of these values correspond to Δt_{S1D1-S2D1} and Δt_{S1D2-S2D2.} If all the time delay values are used for drawing the hyperbolas, the extra curves when used for image creation may give artifacts especially when the number of defects are more in number. Hence, it is required to identify peaks from same defect at each sensor location before damage localization using the hyperbola equations. This is done as follows.

Direct time of arrival of scattered A0 signal for all possible defect locations of a quadrant for a propagation angle is estimated with the knowledge of group velocities of S0 and A0 modes at all three sensor locations of the quadrant using the following equation.

Estimated radial distance of damage from the center of the plate is given as:

The table in annexure 2 shows time of arrival and radial distance for a propagation angle of 120°. Similar Tables were constructed for various propagation angles.

The peaks corresponding to same defect in two different sensor signals can be identified using the table. If the distance corresponding to time of arrival of a peak in a sensor signal matches or is close to the distance corresponding to time of arrival of some peak in the other sensor, then this indicates that both of these peaks are due to scattering from the same defect. Hence, the subtraction of ToAs of these peaks can correctly identify the locus of the defect through the hyperbolic algorithm mentioned in the previous section. This eliminates the extra hyperbolic curves and hence avoids artifacts.

Therefore, a knowledge base like the one shown in the table in annexure 2 needs to be created *a priori*. And peaks due to the same defect at all the sensors need to be correctly identified using the above described data based matching scheme with the help of the knowledgebase before defect localization.

### 6.5.Generation of the defect image

For each location on the plate equation 8 can be modified and written in the form of a function as shown below.

Each location on plate gives one value of 'f'. When the co-ordinates matches with a point on the curve represented by equation 8, the value of the function 'f' becomes zero or minimum and inverse (1/f) of the function becomes infinity or maximum. Since each location on the plate gives one value of f, we obtain one matrix f. A function 'g' is defined below for each value of f. Matrix of g is normalized with respect to maximum value of 'g' to obtain a matrix P containing the pixel values. That is

The matrix P is used to create an image using the Matlab function 'imshow'.

Similarly, pixel values for each hyperbolic curve corresponding to each sensor-location pair are calculated using the equations below

Finally the total pixel value of each point is calculated as Matlab software uses the above pixel values to plot images of the curves and straight line corresponding to angular location of the beam with the help of function 'imshow'. The point of intersection where most of the curves pass through has the highest pixel value and the darkest spot represents the defect location in the image.

## 7.Results and analysis

### 7.1.Defects in Quadrant-4

Quadrant-4 contains three defects D2, D5 and D6 as shown in figure 1(b). A screenshot of animation after FE analysis with propagation angle of 120° is shown in figure 9 where A0 signal generated from three defects can be seen. Scatters from defects which are far from propagation direction of the beam have negligible amplitudes. In the second screenshot (figure 10), the beam can be observed to encounter the stiffener and be reflected by boundaries. The reflected beam can be seen traveling in a direction other than propagation direction as per Snell's law. The A0 scattered waves from stiffener is also seen. All the mode converted scatters are clearer in figure 11.

Mode converted A0 signal at S5, S6 and S7 sensor locations for angles of propagation 110°, 120°, 135° and 145° is shown in figure 12. Signals of 110° propagation angle show one peak within of S4 and S6 each and 2 peaks within of S5 as seen in figure 13. Hyperbolic curves are plotted using ToAs of these peaks and result obtained is shown in figure 13. The hyperbolic curve *f*_{ij } corresponds to the curve drawn from data of sensors *Si* and *Sj*.

In figure 13(b) it can be observed that this point of intersection of hyperbolic curves does not lie on the dark farm line (propagation direction), i.e., equation 8 corresponding to *θ* = 110^{o} indicating that this is not the true location of the defect.

Signals of 120° propagation angle show three peaks within of S4 and S5 each and 2 peaks within of S6 as seen in figure 14(a). Time of arrival of peaks is taken from these plots (figure 14(a)). Corresponding estimated distances for propagation direction of 120° is recorded from table in annexure 2 and reported in table 2 below. Distances 179 mm from S4, 173.2 from S5 and 179 from S6 are close indicating that corresponding peaks are from same defect. The other group of distances 63.5 mm, 80.8 mm and 80.8 mm are from another defect.

**Table 2.**Time delay calculation based on matching estimate distances.

Sensor location | Peak No | ToA (μs) | Corresponding estimated distance (mm) | Time delays (Based on matching estimated distance) | Average distance of defects from center of plate along propagation direction of 120° (mm) | Actual distance of defects (mm) |
---|---|---|---|---|---|---|

S4 | 1 | 67.2 | 63.5, 103.9 | S4(1)-S5(2) | (63.5 + 80.8 + 80.8)/3 = 75.03 | 76 |

2 | 87.3 | 179 | S4(2)-S5(1) | |||

3 | 102 | 213.6 | S4(3)-S5(3) | |||

S5 | 1 | 67.8 | 173.2 | S4(1)-S6(1) | (179 + 173.2 + 179)/3 = 177.06 | 188 |

2 | 84.9 | 80.8, 230.9 | S4(2)-S6(2) | |||

3 | 95.4 | 17, 254 | S4(3)-S6(2) | |||

S6 | 1 | 82.5 | 80.8 | S5(1)-S6(2) | ||

2 | 117.9 | 179.0 | S5(2)-S6(1) | |||

S5(3)-S6(2) |

Thus, 3 hyperbolic equations are obtained from time delays between peaks of S4 and S5. The set of 3 equations obtained is called f45 here to represent that these equations are a result of time delays between peaks of sensors S4 and S5. The curves corresponding to these equations are hyperbolas shown in figure 14(b). Similarly, three more equations are obtained by time delays between peaks of S5 and S6 and called f56 here. And three more equations represented as f46 are obtained from time delay between peaks of S4 and S6. These curves show two points where at least one curve from each of f45, f56 and f46 intersect. And these intersection points lie in equation 8 for propagation angle 120° indicating that these are true locations of defects. Defect image is shown in figure 15. The defects are located by two darkest spots lying on straight line corresponding to 120°.

Signals of 135° propagation angle shows one peak within of S4, S5 and S6 each. When curves are plotted, the point of intersection was found to lie on angular location (figure 16). This confirms the correctness of the location. The accuracy of localization of defects D2, D3, D4 is shown in table 3 by comparing the actual and detected damage locations.

**Table 3.**Co-ordinates of defect locations (Actual and estimated) in millimeters (mm).

Defect | Actual | Estimated | Error |
---|---|---|---|

D1 | (70, 150) | (70.1, 148) | (0.1, 2) |

D2 | (160, −100) | (160, −96.6) | (0, 3.4) |

D3 | (−130, 65) | (−130.1, 64) | (0.1, 1) |

D4 | (−40, −160) | (−43, −172) | (3, 12) |

D5 | (65, −40) | (70.1, −44) | (5.1, 4) |

D6 | (100, −100) | (102.1, −96) | (2.1, 4) |

### 7.2.Defects in Quadrant-1, Quadrant-2 and Quadrant-3

In Q1, one peak was observed within of S2 and S3 and two peaks in S4 were observed within as shown in figures 17(a) and 18(a). Similarly in Q2 and Q3, one peak was seen within of each of the sensors in the respective quadrants as shown in figures 19(a) and 20(a). Using these peaks, defects are localized by modified hyperbolic equations as shown in figure17(b) to figure 20(b). Also obtained damaged locations were compared with the actual ones as provided in table 3. Errors in localization of different damages are also shown in table 3.

### 7.3.Asymmetry in actuation

In real life, there can be asymmetry in actuation due to several unavoidable reasons such as different thickness of adhesives used to bond the PZT patches, error in fixing the PZT patches at the same location at the top and bottom surfaces etc To check the robustness of this scheme in the presence of such errors, FE simulations were performed for pure symmetry in actuation and partial asymmetry in actuation. In purely symmetric actuation, input voltages of the same magnitude and same profile (figure 3) were applied at the top and bottom PZT patches to ensure the generation of pure S0 wave. Partial asymmetric actuation involved the actuation of the top and bottom PZT patches by voltages of same profile (figure 3) but with a magnitude difference to ensure the existence of A0 wave along with S0 wave. Figure 21 shows the comparison of envelopes of A0 signals of sensor S1 and S3 corresponding to propagation angle 30° for purely symmetric actuation and partial symmetric actuation with magnitude differences of 5%, 10% and 20%. It can be observed that the number of peaks and location of peaks within are not affected by actuation asymmetry level of upto 10%. . Subsequently, with an increase in magnitude difference to 20%, change in the number of peaks can be observed. Therefore, it can be concluded from this investigation that asymmetry level upto 10% in actuation is acceptable.

## 8.Conclusions

A baseline free technique has been developed for the detection of multiple structural damages. A plate with mutually perpendicular stiffeners has been considered as the case study. Each panel has been scanned separately through phased array beam steering. Response obtained at each sensor location has been decomposed into symmetric and antisymmetric parts. Next, using hyperbola based scheme multiple damages have been detected. Proper positioning of sensors and a data based matching scheme have been proposed to eliminate the effects of multiple reflections from edges and stiffeners. The efficiency of the scheme in detecting multiple damages in the presence of multiple geometric features like stiffeners, edges and absence of baseline data at all the panels has been demonstrated through high fidelity 3D simulation data. The robustness of this technique was shown through adding partial asymmetry in actuation. It was shown that presence of partial A0 wave component due to partial asymmetry does not alter the number of peaks in the A0 signal envelope. If a defect doesn't create any asymmetry along the plate thickness, wave will propagate through the defect without any mode conversion. In such case, the proposed technique will not be able to detect the defect. Example of such a defect is corrosion of same thickness, area and shape at both side of the plate. This is a limitation of this technique. However, the possibility of occurrence of such symmetric defect is very low. Future extension of this work includes experimental implementation, application to anisotropic structures like fiber reinforced plastic structures.

**Annexure 1.**Time of arrival of direct signals and reflections for 600 **×** 600 mm plate (in * μ *

**s**).

S2 | S3 | S4 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Prop. Angle | ||||||||||||

5 | 73.0 | 140.0 | 96.9 | 99.7 | 114.0 | 152.0 | 137.0 | 121.0 | 147.0 | 207.0 | 172.0 | 169.0 |

10 | 79.2 | 140.0 | 101.0 | 103.0 | 110.0 | 149.0 | 130.0 | 115.0 | 146.0 | 191.0 | 166.0 | 166.0 |

15 | 84.4 | 142.0 | 107.0 | 108.0 | 104.0 | 147.0 | 124.0 | 110.0 | 143.0 | 184.0 | 163.0 | 164.0 |

20 | 91.4 | 145.0 | 113.0 | 116.0 | 98.4 | 146.0 | 118.0 | 107.0 | 141.0 | 180.0 | 160.0 | 163.0 |

25 | 101.0 | 149.0 | NA | 125.0 | 97.4 | 145.0 | NA | 105.0 | 141.0 | 178.0 | NA | 164.0 |

30 | 111.0 | 155.0 | NA | 137.0 | 98.0 | 146.0 | NA | 107.0 | 141.0 | 177.0 | NA | 166.0 |

35 | 123.0 | 161.0 | NA | 151.0 | 98.4 | 147.0 | NA | 113.0 | 143.0 | 176.0 | NA | 170.0 |

40 | 136.0 | 167.0 | NA | 168.0 | 98.8 | 151.0 | NA | 123.0 | 146.0 | 174.0 | NA | 178.0 |

45 | 153.0 | 171.0 | NA | 189.0 | 105.0 | 157.0 | NA | 139.0 | 153.0 | 171.0 | NA | 189.0 |

50 | 146.0 | 174.0 | NA | 170.0 | 99.5 | 151.0 | NA | 113.0 | 136.0 | 167.0 | NA | 152.0 |

55 | 143.0 | 176.0 | NA | 167.0 | 99.8 | 147.0 | NA | 108.0 | 123.0 | 161.0 | NA | 141.0 |

60 | 141.0 | 177.0 | NA | 165.0 | 100.0 | 146.0 | NA | 106.0 | 111.0 | 155.0 | NA | 131.0 |

65 | 141.0 | 177.0 | NA | 163.0 | 100.0 | 145.0 | NA | 105.0 | 101.0 | 149.0 | NA | 123.0 |

70 | 142.0 | 178.0 | 160.0 | 163.0 | 101.0 | 146.0 | 118.0 | 107.0 | 92.6 | 145.0 | 113.0 | 115.0 |

75 | 144.0 | 178.0 | 163.0 | 164.0 | 104.0 | 147.0 | 124.0 | 110.0 | 85.6 | 142.0 | 107.0 | 108.0 |

80 | 146.0 | 179.0 | 166.0 | 166.0 | 110.0 | 149.0 | 130.0 | 115.0 | 80.0 | 140.0 | 101.0 | 103.0 |

85 | 150.0 | 179.0 | 169.0 | 169.0 | 116.0 | 152.0 | 136.0 | 121.0 | 76.3 | 139.0 | 95.6 | 99.7 |

Maximum time of arrival of direct scattered A0 wave from defect. Minimum time for arrival of boundary reflection of mode converted A0 from defect. Minimum time for arrival of for wave reflected from boundary and scattered by defect. Minimum time for arrival of mode converted wave after reflection from stiffener.

**Annexure 2.**ToA of defect scattered signal for 120° propagation angle (in *μ*s).

Radial distance (mm) | t_S6_dir (s) | t_S5_dir (s) | t_S4_dir (s) |
---|---|---|---|

17.3 | 74.2 | 95.1 | 71.5 |

23.1 | 74.5 | 94.1 | 70.8 |

28.9 | 74.9 | 93.1 | 70.2 |

34.6 | 75.3 | 92.2 | 69.5 |

40.4 | 75.9 | 91.2 | 69 |

46.2 | 76.5 | 90.2 | 68.4 |

52.0 | 77.2 | 89.2 | 67.9 |

57.7 | 78.1 | 88.3 | 67.5 |

63.5 | 79 | 87.3 | 67.1 |

69.3 | 80 | 86.4 | 66.8 |

75.1 | 81.1 | 85.4 | 66.6 |

80.8 | 82.3 | 84.5 | 66.5 |

86.6 | 83.6 | 83.6 | 66.5 |

92.4 | 85.1 | 82.8 | 66.6 |

98.1 | 86.6 | 81.9 | 66.8 |

103.9 | 88.2 | 81.1 | 67.1 |

109.7 | 89.9 | 80.3 | 67.6 |

115.5 | 91.7 | 79.5 | 68.3 |

121.2 | 93.6 | 78.7 | 69.1 |

127.0 | 95.5 | 78 | 70.1 |

132.8 | 97.6 | 77.4 | 71.3 |

138.6 | 99.7 | 76.8 | 72.6 |

144.3 | 102 | 76.2 | 74.1 |

150.1 | 104 | 75.7 | 75.8 |

155.9 | 107 | 75.4 | 77.6 |

161.7 | 109 | 75.1 | 79.6 |

167.4 | 111 | 74.9 | 81.7 |

173.2 | 114 | 74.9 | 84 |

179.0 | 116 | 75 | 86.3 |

184.8 | 119 | 75.3 | 88.8 |

190.5 | 122 | 75.7 | 91.3 |

196.3 | 124 | 76.5 | 93.9 |

202.1 | 127 | 77.4 | 96.6 |

207.8 | 130 | 78.6 | 99.4 |

213.6 | 133 | 80 | 102 |

219.4 | 135 | 81.6 | 105 |

225.2 | 138 | 83.4 | 108 |

230.9 | 141 | 85.5 | 111 |

236.7 | 144 | 87.7 | 114 |

242.5 | 147 | 90.1 | 117 |

248.3 | 150 | 92.6 | 120 |

254.0 | 153 | 95.2 | 123 |

259.8 | 156 | 97.9 | 126 |

## Acknowledgments

The authors are grateful to the Department of Science and Technology (DST), Science and Engineering Research Board (SERB), India for providing the support under the Early Career Research Award scheme (ECR/2016/001177).

## Data availability statement

The data cannot be made publicly available upon publication because they are not available in a format that is sufficiently accessible or reusable by other researchers. The data that support the findings of this study are available upon reasonable request from the authors.