### 3.1 Roughness-modified Model

A multi-Gaussian beam model can describe the propagation of sound beams and acoustic fields through several interfaces and inside different media [27]. An ultrasonic testing system model can accurately present the echo responses of scatters, such as those from flat bottom holes, side-drilled holes, and cracks, under different detection conditions [28]. Wang et al. established an ultrasonic testing model for micro-defects in metal balls, and conducted a related experimental verification [29].

#### 3.1.1 Ultrasonic Testing Model for Micro-crack

Shear waves are generally used to distinguish the inside scatters of workpieces with higher accuracy and sensitivity in ultrasonic water immersion testing. In this study, a general ultrasonic immersion testing model with a rough surface was established to describe the echo responses of micro-cracks, as shown in Figure 2. In the following, \(\rho_{m}\), \(c_{pm}\), and \(c_{sm}\) (*m* = 1, 2) represent the density, velocity of the longitudinal wave, and velocity of the shear wave in the coupled water/workpiece, respectively. The focal length of a line-focused transducer with a diameter of *a* and nominal frequency *f* is *F*. The depth and width of the micro-crack are *h* and *W,* respectively, and \(Z_{1}\), \(Z_{2}\), \(Z_{3}\), \(Z_{4}\), and \(Z_{5}\) represent the different propagation distances of the incident sound beam inside the liquid and solid, respectively. \(\theta_{1}^{p}\) and \(\theta_{2}^{s}\) denote the incident angle and refraction angle, respectively.

The micro-crack depth is normally less than five percent of the workpiece thickness; this is of the same order of magnitude as the wavelength of the shear wave. It is necessary to consider that in practical inspection, the conditions of both the front and back surfaces might change (e.g., become rough). The echo response of a crack in an isotropic material can be explicitly written as Eq. (5) [30].

$$V(\omega ) = \frac{{2\beta (\omega )}}{{S_{T} }}\int\limits_{{S_{T} }} {v_{{}}^{{back}} } (\omega ){\text{d}}S,$$

(5)

where, \(V(\omega )\) denotes the average vibration velocity on the transducer surface caused by the echo wave; \(\beta (\omega )\) represents the system efficiency factor, and can be computed using the devolution of a reference signal; \(S_{T}\) describes the transducer surface, and \(v_{{}}^{back} (\omega )\) indicates the transient vibration velocity of the transducer surface as caused by a back-surface micro-crack. The roughness-modified time-domain signal of the micro-crack can be further calculated by using an inverse Fourier transform of the frequency-domain response.

#### 3.1.2 Roughness-modified Model and Essential Components

A pressure wave generated from the immersed focused piezoelectric transducer is obliquely launched to the rough front surface of the workpiece, and the oblique incident angle is commonly set to 17°. The emitted wave follows the path shown in Figure 2, where the beam passes through the liquid-solid interface, reflects at the vertical side of the surface crack with a depth of *h* and length of *lc*, reflects once more the back surface of the workpiece, propagates through the solid-liquid interface, and finally reaches the transducer face. However, the beam could also travel the path in a reversed order, where the beam reflects at the back surface first. The total response from the surface crack comprises the contributions of the reflected waves. Subsequently, setting a point on the vertical side of the crack as (*y*, *z*), the total truncated receiving velocity can be defined as Eq. (6) [30].

$$v^{{back}} = \left\{ {\begin{array}{*{20}c} {v_{{side}}^{{back}} + v_{{bottom}}^{{back}} } & {{\text{0}} \le z \le h, - lc/2 \le y \le lc/2,} \\ 0 & {{\text{elesewhere }.}} \\ \end{array} } \right.$$

(6)

The velocity response generated on the focused transducer face from the wave that reflects first from the vertical side of the crack can be characterized by a multi-Gaussian beam as Eq. (7).

$$\begin{gathered} v_{{side}}^{{back}} \text{(}\omega \text{,}\user2{y}_{s} \text{) = }\sum\limits_{{r = 1}}^{N} \user2{d} \frac{{A_{r} }}{{1 + \text{(}\frac{{iB_{r} z_{1} }}{{D_{R} }}\text{)}}}T_{{12}}^{{s:p}} R_{{23}}^{{s:s}} R_{{23}}^{{s:s}} T_{{34}}^{{p:s}} \cdot \hfill \\ \frac{{\sqrt {\text{det}\user2{M}_{2}^{s} \text{(0)}} }}{{\sqrt {\text{det}\user2{M}_{2}^{s} \text{(}z_{2} \text{)}} }}\frac{{\sqrt {\text{det}\user2{M}_{3}^{s} \text{(0)}} }}{{\sqrt {\text{det}\user2{M}_{3}^{s} \text{(}z_{3} \text{)}} }}\frac{{\sqrt {\text{det}\user2{M}_{4}^{s} \text{(0)}} }}{{\sqrt {\text{det}\user2{M}_{4}^{s} \text{(}z_{4} \text{)}} }}\frac{{\sqrt {\text{det}\user2{M}_{5}^{p} \text{(0)}} }}{{\sqrt {\text{det}\user2{M}_{5}^{p} \text{(}z_{5} \text{)}} }} \cdot \hfill \\ \exp [ik_{{p1}} (z_{1} + z_{5} ) + ik_{{s2}} (z_{2} + z_{3} + z_{4} ) + \frac{{ik_{{s2}} }}{2}\user2{y}_{s} ^{\text{T}} [\user2{M}_{5}^{p} (z_{5} )]_{{\text{r}}} \user2{y}_{s} ]. \hfill \\ \end{gathered}$$

(7)

In the case where the wave reflects first from the back surface, the velocity response of the transducer can be given as Eq. (8).

$$\begin{gathered} v_{bottom}^{back} \text{(}\omega \text{,}{\varvec{y}}_{b} \text{) = }\sum\limits_{r = 1}^{N} {\varvec{d}} \frac{{A_{r} }}{{1 + \text{(}\frac{{iB_{r} z_{1} }}{{D_{R} }}\text{)}}}T_{12}^{s:p} R_{23}^{s:s} R_{23}^{s:s} T_{34}^{p:s} \cdot \hfill \\ \frac{{\sqrt {\text{det}{\varvec{M}}_{2}^{s} \text{(0)}} }}{{\sqrt {\text{det}{\varvec{M}}_{2}^{s} \text{(}z_{4} \text{)}} }}\frac{{\sqrt {\text{det}{\varvec{M}}_{3}^{s} \text{(0)}} }}{{\sqrt {\text{det}{\varvec{M}}_{3}^{s} \text{(}z_{3} \text{)}} }}\frac{{\sqrt {\text{det}{\varvec{M}}_{4}^{s} \text{(0)}} }}{{\sqrt {\text{det}{\varvec{M}}_{4}^{s} \text{(}z_{2} \text{)}} }}\frac{{\sqrt {\text{det}{\varvec{M}}_{5}^{p} \text{(0)}} }}{{\sqrt {\text{det}{\varvec{M}}_{5}^{p} \text{(}z_{1} \text{)}} }} \cdot \hfill \\ \exp [ik_{p1} (z_{1} + z_{5} ){ + }ik_{s2} (z_{2} + z_{3} + z_{4} ){ + }\frac{{ik_{s2} }}{2}{\varvec{y}}_{b}^{\text{T}} [{\varvec{M}}_{5}^{p} (z_{1} )]_{r} {\varvec{y}}_{b} ], \hfill \\ \end{gathered}$$

(8)

where, \({\varvec{y}}_{s}\) and \({\varvec{y}}_{b}\) denote the points on the transducer surface corresponding to the point on the vertical side of the micro-crack; \({\varvec{d}}\) represents the polarization vector of the longitudinal wave or shear wave, and \(k_{pm}\), \(k_{sm}\)(*m* = 1, 2) are the wave numbers for two types of waves in different media; \(A_{r}\) and \(B_{r}\) are the Wen and Breazeale coefficients, respectively. The terms \(T_{12}^{s:p} ,R_{23}^{s:s} ,R_{23}^{s:s} ,T_{34}^{p:s}\) are the transmission and reflection coefficients within the propagation process, and represent the main impact factors containing the surface roughness for the echo responses. The ABCD transfer matrices \({\varvec{M}}_{m}^{p} (z_{n} )\), \({\varvec{M}}_{m}^{s} (z_{n} )\) proposed by Huang et al. [31] can express the propagation, transmission, and reflection of the Gaussian beam.

A system efficiency factor is used to describe the electrical and electromechanical elements in the testing system. It can be calculated by a deconvolution between the frequency-domain response \(V_{0} (\omega )\) and acoustic/elastic transfer function \(t_{A} (\omega )\).

$$\beta (\omega ) = \frac{{V_{0} (\omega )[t_{A} (\omega )]^{*} }}{{\left| {t_{A} (\omega )} \right|^{2} + \varepsilon^{2} \max \left\{ {\left| {t_{A} (\omega )} \right|^{2} } \right\}}}.$$

(9)

The Wiener Filter is used to reduce the sensitivity of the deconvolution noise, and *ε* is a constant taken as 0.03.

The RMS heights of the front and back surfaces in Figure 2 are set to \(\sigma_{1}\) and \(\sigma_{2}\). The reflection of the incident wave mainly occurs at the solid-liquid interface of the back surface, and the reflection coefficient \(R_{23}^{s:s}\) can be explicitly presented based on the phase-screen approximation as follows:

$$R_{23}^{s:s} = R_{0}^{s} (\omega ,\theta_{1}^{p} )\exp \left\{ { - 9(Rq_{2} k_{s2} \cos \theta_{2}^{s} )^{2} } \right\}.$$

(10)

The transmission coefficients \(T_{12}^{s:p}\) and \(T_{34}^{p:s}\) on the front surface can be given as Eqs. (11) –(12).

$$T_{12}^{s:p} = T_{0}^{s:p} (\omega ,\theta_{1}^{p} ) {\text{exp}}\left\{ { - \frac{{5}}{{2}}Rq_{1}^{2} {[}k_{s2} \text{cos}(\theta_{2}^{s}) - k_{p1} \text{cos}(\theta_{1}^{p} )]^{2} } \right\},$$

(11)

$$T_{34}^{p:s} \text{ = }T_{1}^{p:s} {(}\omega ,\theta_{2}^{s} \text{)exp}\left\{ { - \frac{5}{2}Rq_{1}^{2} {[}k_{p1} {cos(}\theta_{1}^{p} {) - }k_{{s\text{2}}} \text{cos(}\theta_{2}^{s} {)]}^{2} } \right\},$$

(12)

where, \(R_{0}^{s} (\omega ,\theta_{1}^{p} )\), \(T_{0}^{s:p} {(}\omega ,\theta_{1}^{p} {)}\), and \(T_{1}^{p:s} {(}\omega ,\theta_{2}^{s} {)}\) represent the reflection and transmission coefficients under the condition of smooth surfaces, respectively. By substituting Eqs. (10)–(12) into Eqs. (7)–(8), the total velocity response of the transducer surface can be acquired. Subsequently, the frequency-domain response of the transducer for micro-cracks under the effects of a rough surface can be modeled from Eq. (6), in combination with the system efficiency factor from Eq. (9).

### 3.2 Validation and Comparison between Analytical Model and Simulation Model

#### 3.2.1 Simulation Model and Validation of Analytical Model

The finite element method can efficiently analyze elastic wave scattering, including that involving a multiple-physics coupled field [32]. Numerical models have been established to detect wall thinning and defects from corroded surfaces using ultrasonic guided waves [33], and to investigate the wave scattering on rough interfaces [34].

In this study, a two-dimensional simulation model for the ultrasonic inspection of micro-cracks containing rough surfaces was established to acquire the echo response, as shown in Figure 3. The model consisted of a virtual focused transducer, water, and a steel plate with a thickness of 10 mm. The piezoelectric focused transducer was formed by a wafer and arc lens, and was 6 mm in diameter and 25 mm in length. The coupled water thickness was 20 mm, and the incident angle of the pressure wave was 17°. The velocity of the longitudinal wave in water was 1480 m/s, and that of the shear wave in steel was 3230 m/s. A broadband modulated pulse as an excitation signal was applied to an external circuit. The signal can be expressed as Eq. (13).

$$f(t) = Q \cdot \exp ( - \frac{{(t/2 - \mu )^{2} }}{{\tau^{2} }}) \cdot \sin (2\uppi f_{0} t),$$

(13)

where, *Q* denotes the reference amplitude; the center frequency \(f_{0}\) was set to 5 MHz. The values of *μ* and *τ* corresponding to the translation and standard deviation were \(1/f_{0}\) and \(1/2f_{0}\), respectively.

The solid-liquid interfaces, including the front surface, back surface, and arc lens surface in the simulation model, were set as acoustic-structure coupled boundaries. The other boundaries were set as radiation boundaries in the water region and low-reflection boundaries in the solid region. Several grid control regions were intensively meshed near the rough surfaces and micro-cracks to improve the calculation accuracy. The grid density was eight elements per wavelength in the control region and six elements in other regions. The random rough surfaces generated as described in Section 2.1, which tend to have a Gaussian distribution, were imported into the simulation model.

A cylindrical reference reflector was used to replace the steel plate in Figure 3 to establish an ultrasonic immersion testing model, and to acquire the system efficiency factor. The parameters of the focused transducer, coupled water, and workpiece were consistent with the simulation model for the micro-crack. A smooth surface was used as the front surface of the reference reflector, and the echo signal is presented in Figure 4a. Figure 4b further shows the system efficiency factor as obtained from the deconvolution reference signal according to Eq. (9). Taking a back-surface micro-crack with a depth of 0.5 mm as an example, the back and front surfaces were set to be smooth. The echo responses acquired from the simulation and analytical models are shown in Figure 4c. It can be seen that the waveform and amplitude of the predicted signal of the micro-crack are similar to those of the simulated response. The simulated signal is slightly larger than the predicted response, and the amplitude difference is 0.8 dB. Therefore, the analytical response of the ultrasonic testing model for the micro-cracks was considered as consistent with the simulation results.

#### 3.2.2 Comparison under Different Rough Surfaces

The above analysis validated the consistency between the analytical model and numerical simulation. The microtopography fluctuations of the rough surfaces in accordance with the Gaussian distribution showed randomness; hence, it was necessary to verify the reasonability of the ultrasonic testing model for micro-cracks under the effects of rough surfaces. A back-surface micro-crack with depth of 0.5 mm was consistently used to obtain the echo responses under different surface roughness values. The back surface was initially set to a smooth surface, and the RMS heights of random rough front surfaces were set to 0 μm, 5 μm, 15 μm, 25 μm, 35 μm, and 50 μm, respectively. In another case, the front surface was alternatively smooth, whereas the range for the RMS height values of the rough back surfaces was the same as that of the first condition. Ten simulated signals corresponding to each RMS height were computed for comparison with the predicted response for the micro-crack from the ultrasonic testing model.

Figure 5a shows the amplitude variation of the simulation responses and predicted curve of the analytical model for the back-surface micro-crack under different front surface roughness values. The amplitudes of the micro-crack acquired from the analytical model and numerical simulations, as normalized by the echo signal from the smooth front surface, decrease nonlinearly with an increase in the RMS height. The mean values of the simulated amplitudes are in good agreement with the predicted results from the ultrasonic testing model for the rough front surfaces, and the error between the two responses is within 0.9 dB. The simulated results are also consistent with the predicted curve for the rough back surface, and the maximum error between the mean value of the simulated amplitudes and predicted responses is 2.3 dB, as shown in Figure 5b. Therefore, it can be observed that the ultrasonic immersion testing model can accurately and rapidly calculate the echo responses of micro-cracks under rough surface conditions.

Furthermore, the roughness-modified ultrasonic testing model was applied to acquire the echo signals of micro-cracks at different depths under rough surfaces. The depths of the back-surface cracks were set to 0.10 mm, 0.25 mm, 0.50 mm, 1.00 mm, 1.50 mm, and 2.00 mm, respectively. The RMS heights of the rough surfaces changed from 0 μm to 100 μm. Figure 6a presents the predicted responses of the different cracks under the rough front surface and smooth back surface. The results indicate that the normalized amplitudes are evidently reduced with increasing roughness, and the change trend remains consistent with the wave scattering condition on a rough surface. Similarly, the predicted signals for these cracks under the rough back surface decrease nonlinearly as the roughness increases, as shown in Figure 6b. It can be observed that the influence of both rough surfaces (with a roughness of less than 15 μm) on the echo signal amplitude is insignificant. However, the corresponding RMS height of the front surface is 37 μm when the amplitudes are attenuated by 10 dB, whereas the RMS height of the rough back surface is 48 μm. It can be seen that the effect of the rough front surface on the micro-crack detection is more significant than that of the rough back surface; this is owing to the greater attenuation of the multiple transmissions of pressure waves than in the reflection process.

### 3.3 Evaluation Method for Detection Accuracy for Micro-crack

The distortion of the time-domain signal of a micro-crack as caused by the roughness-induced attenuations of the coherent transmission and reflection waves can be clearly observed. The signal-to-noise ratio of a defect response is normally used to evaluate the detectability of ultrasonic testing instruments or equipment, and the critical value should be larger than 10 dB to clearly differentiate between defect signals and noise. For convenience of description of the signal-to-noise ratio, a scalar parameter *SAR* is directly defined by the ratio of the defect signal amplitude \(\overline{{A_{s} }}\) to the noise amplitude \(\overline{{A_{n} }}\) from the simulation or experiment, which can be expressed as Eq. (14) [26]. The value of *SAR* equals 3.16 when the signal-to-noise ratio reaches 10 dB.

$$SAR = \frac{{\overline{{A_{s} }} }}{{\overline{{A_{n} }} }}.$$

(14)

A previous study on internal delamination detection under the influence of roughness showed that the noise amplitude is independent of the defect size [26]. Hence, a micro-crack with a 0.5 mm depth continues to be utilized to calculate the mean amplitudes of noise from multiple simulations. Figure 7a reveals that the noise amplitudes only slightly increase with an increase in roughness. Meanwhile, the change trends of the noise amplitudes under the rough back surface remain similar to those of the rough front surface, as shown in Figure 7b. Furthermore, the *SAR* values corresponding to the pulse-echo responses from the simulation and analytical models are obtained for different roughness values of the front and back surfaces, as shown in Figure 8a and b. It can be observed that the predicted responses are consistent with the numerical simulation results. Therefore, the signal-to-noise ratios of the different micro-cracks can be acquired from the roughness-modified ultrasonic testing model.

Consequently, the mean noise amplitude can be acquired by interpolation for each surface roughness, owing to the slight variation in the mean amplitude of the noise signal as calculated from the ultrasonic simulation. Then, a method can be proposed for predicting and evaluating the detection accuracy of micro-cracks under rough surfaces by combining the numerical simulations and analytical model. Several cracks with depths of 0.1 mm, 0.25 mm, 0.50 mm, 1.00 mm, 1.50 mm and 2.00 mm were taken as examples to analyze the evaluation method. Figure 9a shows the amplitude ratio factor *SAR* values of these cracks under the rough front surface as computed by the ultrasonic immersion testing model. Notably, the *SAR* values decrease with increasing roughness, and the corresponding critical RMS heights are 16, 29, 40, 51, 55, and 58 μm, respectively, when the *SAR* value reaches 3.16.

In addition, the changes in the amplitude ratio factor for the different cracks under the effect of rough back surfaces were investigated, and are presented in Figure 9b. The critical RMS heights for these cracks are 15, 38, 54, 65, 71, and 75 μm. Evidently, the change trends between the two rough surfaces are comparable. Moreover, the acoustic attenuation induced by the rough front surface is more intense than that induced by the rough back surface, indicating that transmission gives rise to more wave scattering than reflection. In general, the ultrasonic detection accuracy for the micro-cracks can be accurately and rapidly predicted by the proposed evaluation method. The reasonable detection sensitivity of the ultrasonic immersion testing can be developed according to the micro-crack depth limitation as predicted by the evaluation method under practical surface conditions.