**1. Introduction**

Imaging potential defects and their exact position and size are among the greatest challenges of recent studies in non-destructive testing (NDT) and structural health monitoring (SHM). One of the most efficient techniques is to use ultrasonic waves followed by their processing. In general, two approaches are possible for damage imaging. In the first, the guided wave field is sensed over an inspected area (usually in a non-contact manner) and it is then subjected to further processing using, for example, root mean squares to obtain a useful defect image [1,2]. The second approach is connected with the use of an array of piezoelectric transducers, acting as both actuators and sensors, attached to selected points on an analysed structure. Wave propagation signals are collected and then processed using appropriate damage detection algorithms such as the time of flight (TOF) based triangulation method [3], supported by ellipse or hyperbolic probability imaging [4,5], the time-reversal technique [6], the migration technique [7], phased-array beamforming [8] and finally ultrasound tomography, widely used for inspection of structures made of concrete [9–11], metal [12–14] and composite materials [15,16].

Tomography using guided ultrasound waves has become a popular technique incorporated into the structural health monitoring of elements of civil engineering infrastructure such as plates, shells and pipes [17–22], and also in the monitoring of aircraft components [23]. The increasing use of non-destructive methods for detecting defects in plate elements has resulted in a growing need to increase the effectiveness and efficiency of inspection. When performing tests based on Lamb waves, some practical problems affecting the results may be encountered. These problems are discussed extensively in the literature. Rao et al. [24] carried out a study on monitoring the corrosion of steel plates and the reconstruction of the thickness of corrosion damage. Wang et al. [25] used guided waves to image a hole with a corroded edge. Zhao et al. [26] investigated the imaging of damage in aluminium plates, comparing different tomography algorithms. Leonard et al. [27] analysed Lamb wave tomography on both aluminium and composite plates, with defects

**Citation:** Zieli ´nska, M.; Rucka, M. Imaging of Increasing Damage in Steel Plates Using Lamb Waves and Ultrasound Computed Tomography. *Materials* **2021**, *14*, 5114. https:// doi.org/10.3390/ma14175114

Academic Editor: Andrea Di Schino

Received: 27 July 2021 Accepted: 2 September 2021 Published: 6 September 2021

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of various sizes and thicknesses. The possibility of imaging defects in the form of round and rectangular holes in aluminium and composite plates was tested by Khare et al. [28]. Balvantin and Baltazar [29] analysed the images of an aluminium plate containing damage in the form of a two-stage circular discontinuity, using the multiplicative algebraic reconstruction technique (MART) and back projection. The structural condition monitoring system for composite panels with openings was presented by Prasad et al. [30]. The size of the defect itself is also of great importance in the imaging of tomographic plates, as evidenced by Menke and Abbott [31], Cerveny [32] and Belanger and Cawley [33].

The research described in this article aims to evaluate the use of ultrasound tomography to locate surface damage of varying sizes. The experimental and numerical analyses were carried out on four steel plates, one intact and three containing defects in the form of circular holes with diameters of 2, 5 and 10 cm. The non-destructive inspection was carried out using Lamb waves and ultrasound tomography (UT). The reconstruction of the Lamb wave propagation velocity was performed using both the non-reference approach and a method based on the differences in the transition times between the reference model and the three damaged models with surface defects. In order to improve the image quality, certain methods of tracing the real course of wave paths were used. The influence of mesh density on the possibility of estimating damage size was also assessed.

#### **2. Ultrasound Tomography—Theoretical Background**

Ultrasound tomography imaging allows recreating the internal structure of an examined object using the properties of elastic waves. A schematic process of performing ultrasound tomography is shown in Figure 1. The first step is to divide the test sample into cells called pixels. Each one of them constitutes a discrete area in the tested element. Such a division is shown with a dashed line in Figure 1a. Then, after applying appropriate algorithms, each pixel takes a value representing the speed of the ultrasonic wave propagated through this area.

**Figure 1.** Schematic diagram of velocity reconstruction using ultrasound tomography: (**a**) plate with indicated receivers (*R*), transmission (*T*) points and simulated wave field; (**b**) guided wave propagation time signals; (**c**) ultrasound tomography map.

Image reconstruction is based on information from ultrasound wave signals propagating from transmitting points (*T*) to receiving points (*R*). This information may be, for example, the time of flight (TOF), measured along with the multiple ray paths, determined after appropriate signal processing [34–36]. Each obstacle in the path of the travelling wave changes the propagation time. Among them, we can distinguish defects that delay the wave reaching the receiver, such as air voids or cracks, and those that decreased total velocity along all rays passing through these inclusions. The latter include inclusions made of materials whose propagation velocity is higher than in the surrounding medium. Based on the time of flight of the wave, with the known geometry of the tested object, it is possible to determine wave propagation speed, correlated with the mass density, modulus of elasticity, and Poisson's ratio characteristic for a given material.

The time of flight between the transmitter (*T*) and the receiver (*R*) can be described by a line integral of the transition time distribution along the propagation way, *w*:

$$dt = \int\_0^{w\_1} dt = \int\_0^{w\_1} s dw = \int\_0^1 \frac{1}{v} dw\_\prime \tag{1}$$

where *v* is the average velocity, *w*<sup>1</sup> denotes the distance between transmitter and receiver and *s* is the inverse of the velocity, *v*, referred to as slowness, *s* = 1/*v*. Measurement of the wave travel time along specific paths enables the determination of local velocities in the tested plate. The wave velocity for each cell, *vj*, along the path of the wave from the transmitter to the receiver can be determined from the following formula:

$$t\_i = \sum\_{j=1}^{n} w\_{ij} s\_j, \text{ i } = 1, 2, 3, \dots, m, \ j = 1, 2, 3, \dots, n,\tag{2}$$

where *m* denotes the number of rays, *n* denotes the number of pixels into which the tested sample is divided, *ti* is the time of flight of the guided wave along the *i*-ray, *wij* is the *i*-ray propagation path through the *j*-pixel and *sj* is the slowness at pixel *j*. It is assumed that velocity, *vj*, in individual cells is constant.

One of the techniques for solving this type of system of equations is the algebraic reconstruction technique (ART) [37]. It was used to perform calculations in the conducted research. First, each cell is assigned the same slowness. Its value is calculated as the inverse of the average velocity of ultrasonic wave propagation in the tested material. Based on the initial image created in this way, the iteration process begins and corrections are made.
