In this section, we employed the COSMO model to identify and locate the cracks to steel specimens by numerical simulation and experimental tests.

#### 3.1. Numerical Simulation

A two-dimensional model of a steel board embedded with a single crack was simulated by using a commercial software (Comsol Multiphysics V4.3a. COMSOL, Inc., Palo Alto, CA, USA).

Figure 1 shows the schematic illustration of an ultrasonic measurement system in the simulation. Two longitudinal transducers with 60° wedge as the transmitter were typically used to carry out the ultrasonic inspection on the top surface of the steel board with the length of 220 mm and the height of 30 mm. A fixed distance between the two transducers was kept at a constant interval to make sure the first back wall echo was fully collected by the receiver. The single crack with a depth of

d was located at the middle of the steel board along the x direction, i.e., x = 110 mm.

A 0.5 MHz continuous sinusoidal signal with signal-noise ratio (SNR) of 15 dB was applied as the exciting signal. Both the transmitter and the receiver were moved simultaneously to scan the steel board along the x direction. The receiving waveforms, at eight different positions, were spaced by 20 mm on the top of the simulated steel board, recorded, and the corresponding spectrums were then analyzed.

Figure 2a depicts spectrums at eight positions of the simulated steel board with crack length

d = 2 mm.

According to the COSMO algorithm model, the corresponding spectrums, at eight different observation positions, were saved then a group of

z-scores were calculated by Equation (2) after every scanning process, finally 30 groups of

z-score were obtained by scanning repeatedly 30 times. It is clearly shown in

Figure 2b that

z-score of observation points are distributed almost evenly between 0.3 and 1 except

x = 110 mm, while most of

z-score at

x = 110 mm are mainly distributed below 0.4, just right at the crack’s position. It is shown that the distribution of

z-scores could be used to locate and identify cracks or defects in materials, i.e.,

z-score of damaged regions might be below 0.4. However, the conclusions need to be subjected to hypothesis tests to reach statistical significance, which determines whether a null hypothesis can be rejected or retained.

Figure 3a shows the calculated level of significance testing for crack depth of 2, 4, and 5 mm by Equation (4). It could be seen from

Figure 3a that the

p-value has a much smaller value than 0.1 around the crack region from 90 to 150 mm, which suggests that the imperfect structure of this region is significant. To make the comparisons and analysis clearly, an indicator called deviation level is defined as,

i.e., logarithm transformed

p-value, obviously the small T-value indicates little significant probability of crack. The T-value curve of significance testing is shown in

Figure 3b. Obviously, the maximum T-value occurs around the position of crack (

x = 110 mm) for crack depth of

d = 2, 4 and 5 mm, respectively. Furthermore, the maximum T-value increases simultaneously as crack growth, e.g., the maximum T-value is close to 4 when crack depth is 2 mm, and the peak of T-value up to 10 when crack depth equal to 5. This result indicates that the higher level T-value is strongly correlated with the crack depth, which could become an index to exhibit the evolution of crack growth inside materials.

When the crack depth changed from 0 mm to 5 mm, the peaks of T-value around the cracks were obtained and thus the relationship curve between the maximum T-value and crack depth d was plotted, as shown

Figure 4. It can be seen that as the crack depth increased from 0 to 1, the slope of the curve sharply increased. When crack depth was less than 1 mm, the T-value was not larger than 5, which is basically considered the formation stage of crack, due to the relative small change of crack depth, thus this phase is called stage I. As the crack depth gradually expanded form 1.5 mm to 3.7 mm, the maximum T-value increased slowly from 5 to 7, at stage II. When crack depth was larger than 4 mm, at stage III, the value of the curve increased rapidly up to 10, and as high as 2 times than that of stage I, which means that the small cracks had already expanded to macro-cracks. Therefore, the peaks of T-value might track the progression of damage and evaluate the evolution of crack growth.

#### 3.2. Experimental Measurement

A specimen made of Q235 (See

Table 1) with dimensions 800 mm × 250 mm × 20 mm was used in the experimental measurement, as shown in

Figure 5. Four sections embedded with cracks with average depth of 6 mm, 2 mm, 1 mm, and 0.5 mm were manufactured in the specimen, mainly located at 150 mm, 300 mm, 450 mm, and 600 mm, respectively, denoted by B, C, D and E. Additionally, Section A and F represented as undamaged regions located in the two ends of the specimens. A portable TOFD ultrasonic detector (PXUT-920, Nantong Union Digital Tech., China) was used to excite a narrow-pulse acoustic signal with 200 ns in width and stored the echo signals from inspected cracked region. Scanning was manually carried out by a scanner unit with one pair of 5 MHz normal transducer, i.e., the transmitter and the receiver, with 60° wedges for longitudinal waves. Two transducers, spaced 62 mm apart, were located at equidistant over the crack region center, and scanning was done by moving the scanner in the length direction of steel plate parallel to the crack region. The echo signal sampled by the detector, containing 1496 points, was acquired every 0.5 mm along the length direction. After a scanning, a total of 1600 echo signals (A-scan) were obtained and stored in the ultrasonic detector.

COSMO method was applied to analyze this dataset of echo-signal recorded by TOFD ultrasonic detector. Firstly, the Hellinger distance matrix D was constructed using Equation (1), and the row with minimum sum was chosen in the metric D so that the

z-score could be determined using Equation (2).

Figure 6a shows the

z-score distribution for 30 scanning. It is shown that the

z-score in undamaged sections is much larger than those in the region with cracks. For example, the

z-score for positions A and F are about 0.4~1, while those for positions B, C, D and E are below 0.4. The results suggest that the

z-score is closely related to cracks of specimen, just as the simulated results.

By using Equations (1)~(4) and (5), the deviation level T-Value is calculated to make a significant analysis. It can be observed from

Figure 6b that the T-value at B, C, D and E are of high level compared to those in uncracks region. For instance, the T-value at x = 150 up to 13, and T- value at x = 600 sharply increasing from 0 to 7, but T-value of sections without cracks almost equal to 0, far less than T-value at crack region. In addition, T-value increase almost linearly with the depth of cracks. The relationship curve between the peaks of T-value and crack depth is plotted in

Figure 7. It is shown that the peak of T value increases with crack growth from 0 to1.5 mm quickly up to 5, which is exactly in stage I. When the crack depth is larger than 2, the slope of curve go slow but still faster than the simulated results. It is worth noting that the change of slope is not distinct enough to easily recognize stage II or III when depth of crack exceeding 1.5 mm, different from the simulated curve, which might attributed to the result of multi-physical mechanisms.