MATERIALS TRANSACTIONS
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ISSN-L : 1345-9678
Correcting the Ultrasonic Scattering Attenuation Coefficient of a Metal Using an Equivalent Medium Layer
Xiaoqin HanXiongbing LiYongfeng SongPeijun NiYiwei Shi
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2016 Volume 57 Issue 10 Pages 1729-1734

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Abstract

The surface roughness and diffraction attenuation of a metal have an effect on the measurement accuracy of the ultrasonic scattering attenuation coefficient. In order to correct the scattering attenuation coefficient, the rough surface of the sample and its neighbouring couplant are assumed to be an equivalent medium layer, namely the sample is a multi-layered medium composed of a layer of the substrate medium and two layers of equivalent mediums. Based on the Lommel diffraction correction coefficient and the parameters of the equivalent medium layer, the expressions are developed for a circular planar piston transducer's sound field in a multi-layered structure with equivalent medium layers. As a result, a correction model of the scattering attenuation coefficient is established by using the surface roughness and diffraction attenuation. AISI 304 stainless steel samples with different surface roughnesses are used to conduct the ultrasonic experiment. The results show that the attenuation coefficient without correction increases in proportion to the roughness; and the average relative error is up to 182.8% compared to the theoretical attenuation coefficient, while the average relative error is only 1.28% after correction. This indicates that the model can limit the negative effects of the roughness and the sound field diffraction on the extraction of the scattering attenuation coefficient. Consequently, the corrected attenuation coefficient can improve the accuracy and reliability in the nondestructive evaluation of the microstructure.

1. Introduction

The scattering of ultrasonic waves from the grains or pores in metals can be measured by the attenuation of the waves. Therefore, the microstructures of the metals, such as grain size and porosity, can be evaluated by the scattering attenuation coefficient13). However, the coupling and sound field diffraction can affect the measurement accuracy of the scattering attenuation coefficient in practical applications. For example, a material's rough surface can lead to the diffuse reflection of ultrasonic waves, which can worsen the coupling effect and reduce the reflection echo4), thus resulting in a large error in measurement of the scattering attenuation. And the experiment showed that the scattering attenuation is proportional to the material's roughness5). Therefore, it is necessary to effectively eliminate the error caused by the surface roughness of the material for the ultrasonic attenuation evaluation of the microstructure.

Nagy et al. studied the effect of surface roughness on the ultrasonic attenuation evaluation of the porosity, but only took into account the scattering signal which singly-transmitted from the back-wall6). Chen et al. presented an analytical model of double transmission signal attenuation using Fresnel approximation and the phase-screen approximation. The effects of the rough interface on the signal attenuation were also investigated, but this model did not give a corresponding method for correcting the scattering attenuation7). Reed correlated the ultrasonic scattering attenuation of a sample with rough surfaces when evaluating the porosity, but did not take into consideration the scattering attenuation induced by the sound field5). In addition, Rogers et al. used the Lommel diffraction coefficient method to represent the ultrasonic sound pressure produced by the circular piston planar transducer8); Wydra deduced the scattering attenuation coefficient from the Lommel diffraction coefficient on this basis, but this model does not take the effect of the roughness into consideration and has low accuracy and insufficient sensitivity for practical application9).

In this paper, a roughness correction model for the ultrasonic scattering attenuation coefficient is deduced based on the equivalent multi-layered medium, which can correct the error caused by the coupling and diffraction attenuation.

2. The Correction Model Based on Equivalent Medium Layer

The sound pressure received by the circular planar piston transducer propagating in a single medium can be expressed with the Lommel diffraction coefficient as:   

\[P(x,f_0) = P_0 D(s) \exp (-\alpha_s x)\](1)
Where   
\[D(s) = 1 - \exp [-(2\pi/s)i] \cdot [J_0(2\pi/s) + iJ_1(2\pi/s)]\](2)
Where $P$ denotes the average sound pressure that the transducer receives at the sound path $x$, $P_0$ is the initial sound pressure on the surface of the transducer and $\alpha_s$ represents the scattering attenuation coefficient when propagating in the medium. $D(s)$ is the Lommel diffraction coefficient, $s = 2\pi x/ka^2$ is the effective area of the transducer, the radius of the transducer is $a$, $k = 2\pi f_0/c_f$ is the wave number at the center frequency $f_0$ of the transducer, $c_f$ is the P-wave velocity of the fluid, and $J_0$ and $J_1$ are Bessel functions of order zero and one respectively.

Compared to the root mean square roughness value used in Ref. 8), the thickness $h_r$ of the equivalent medium layer is assumed as $Rz$, where $Rz$ denotes the maximum height of the profile, namely, the maximum peak-to-valley height within the sampling length $L$10), as shown in Fig. 1. Suppose that $h_r$ is much less than the sample thickness $h$, the equivalent attenuation coefficient $\alpha_r$ and equivalent P-wave velocity $c_r$ of the medium layer are constant when $f_0$ is constant. At this point, the substrate medium can be regarded as another layer of the medium with a perfectly smooth surface. $T_{fr}$, $T_{rs}$, $T_{sr}$ and $T_{rf}$ are, respectively, the transmission coefficients from the fluid to the equivalent medium layer, from the equivalent medium layer to the substrate, from the substrate to the equivalent medium layer, from the equivalent medium layer to the fluid. $T_{fs}$ and $T_{sf}$ are the transmission coefficients from the fluid to the substrate, and from the substrate to the fluid. $R_{fr}$, $R_{rs}$, $R_{sr}$, $R_{rf}$, $R_{fs}$ and $R_{sf}$ are the corresponding reflection coefficients. Assume that the relationships of all the transmission coefficients and the reflection coefficients are $T_{fr} = T_{fs}$, $T_{rf} = T_{sf}$, $R_{fr} = R_{fs}$, $R_{rf} = R_{sf}$, $T_{rs} = T_{sr} = 1$ and $R_{rs} = R_{sr} = 0$.

Fig. 1

Schematic diagram of equivalent medium layer.

The first back-wall echo and the second back-wall echo obtained by the immersion pulse reflection setup are shown in Fig. 2. The upper and lower surface of the sample are manufactured with the same process and can be considered to have the same thickness of the equivalent medium layer. Then the sound pressure received from the first back-wall echo is denoted as:   

\begin{align*} &P_1(x_1,f_0) \\ &= P_0 T_{fr} T_{rs} T_{sr} R_{rf} T_{rs} T_{sr} T_{rf} \exp (-2\alpha_s h - 2\alpha_f x_f - 4\alpha_r h_r) D(s_1) \\ &= P_0 T_{fs} R_{sf} T_{sf} \exp (-2\alpha_s h - 2\alpha_f x_f - 4\alpha_r h_r) D(s_1) \end{align*}(3)
Fig. 2

Schematic diagram of ultrasonic pulse reflection process with the equivalent medium layer.

The sound pressure received from the second back-wall echo is:   

\begin{align*} P_2(x_2,f_0) &= P_0 T_{fr} T_{rs} T_{sr} R_{rf} T_{rs} T_{sr} R_{rf} T_{rs} T_{sr} R_{rf} T_{rs} T_{sr} T_{rf} \\ &\hphantom{={}} \cdot \exp (-4\alpha_s h - 2\alpha_f x_f - 8\alpha_r h_r) D(s_2) \\ &= P_0 T_{fs} R_{sf}^3 T_{sf} \exp (-4\alpha_s h - 2\alpha_f x_f - 8\alpha_r h_r) D(s_2) \end{align*}(4)
Where   
\[\left\{ \begin{array}{@{}l@{}} x_1 = 2x_f + 4(c_r/c_f) h_r + 2(c_s/c_f) h, \\ x_2 = 2x_f + 8(c_r/c_f) h_r + 4(c_s/c_f) h, \\ s_{1,2} = 2\pi x_{1,2}/ka^2. \end{array}\right.\](5)
Where, $P_1$ and $P_2$ are sound pressures received from the first and second back-wall echo with the rough layer. $x_f$ is the water path. $x_1$ and $x_2$ are equivalent sound paths in the fluid with the medium layer. $\alpha_f$ is the attenuation coefficient in the fluid and $\alpha_s$ denotes the actual scattering attenuation coefficient in the substrate. $c_s$ is the P-wave velocity of the solid.

From eqs. (3) and (4), the scattering attenuation coefficient with correction can be obtained as follows   

\[\alpha_s = \frac{1}{2h} \ln \left[ \frac{P_1(x_1,f_0)}{P_2(x_2,f_0)} \frac{R_{sf}^2D(s_2)}{D(s_1)} \right] - 2\alpha_r \frac{h_r}{h}\](6)

Equation (6) indicates there are two parts in the effect of the surface roughness on the scattering attenuation coefficient. The first part is contained in the Lommel diffraction coefficient, resulting in an increase in the equivalent sound path of ultrasonic waves. It can be found from eq. (5) that this attenuation is not caused by scattering, and therefore it shall be eliminated from the measurement of the scattering attenuation coefficient. The second part is directly related to the equivalent attenuation coefficient $\alpha_r$ and equivalent medium layer thickness, the higher the surface roughness value, the larger the attenuation in the equivalent medium layer. This attenuation is not induced by the scattering attenuation. Equation (6) shows that the ultrasonic scattering attenuation coefficient can be corrected by determining the surface roughness when parameters of the equivalent medium layer are known.

The calibration method for the equivalent sound velocity $c_r$ and equivalent attenuation coefficient $\alpha_r$ will be introduced below. As shown in Fig. 3, the propagation process of the ultrasonic wave in the equivalent medium layer is divided into two parts, i.e. from the fluid to the medium layer and from the medium layer to the substrate, thus, the following equation could be obtained by matching the propagation time as:   

\[\frac{h_r}{c_r} = \frac{h_r/2}{c_f} + \frac{h_r/2}{c_s}\](7)
Equation (7) is simplified as   
\[c_r = \frac{2}{1/c_f + 1/c_s}\](8)
It can be seen from eq. (8) that $c_r$ is between the sound velocity of fluid and that of a solid.
Fig. 3

Schematic diagram of the ultrasonic wave propagating in the equivalent medium layer.

The attenuation coefficient of the equivalent medium layer of the reference sample is denoted by $\alpha_r^{Ref}$, on the basis of the constant condition of the equivalent attenuation coefficient we could write:   

\[\left\{ \begin{array}{@{}l@{}} \displaystyle \frac{1}{2h} \ln \left[ \frac{P_1}{P_2} \frac{R_{sf}^2 D(s_2)}{D(s_1)} \right] - 2\alpha_r \frac{h_r}{h} \\ \displaystyle\quad = \frac{1}{2h^{Ref}} \ln \left[ \frac{P_1^{Ref}}{P_2^{Ref}} \frac{R_{sf}^2 D^{Ref}(s_2)}{D^{Ref}(s_1)} \right] - 2\alpha_r^{Ref} \frac{h_r^{Ref}}{h^{Ref}} \\ \displaystyle \alpha_r = \alpha_r^{Ref} \end{array}\right.\](9)
The following equation can be obtained from eq. (9)   
\[\alpha_r = \frac{\displaystyle h\ln \left[ \frac{P_1^{Ref}}{P_2^{Ref}} \frac{R_{sf}^2 D^{Ref}(s_2)}{D^{Ref}(s_1)} \right] - h^{Ref} \ln \left[ \frac{P_1}{P_2} \frac{R_{sf}^2 D(s_2)}{D(s_1)} \right]}{4(h_r^{Ref}h - h_r h^{Ref})}\](10)

Therefore, the equivalent attenuation coefficient $\alpha_r$ can be calibrated by eq. (10).

In addition, in order to emphasize the relationship between the attenuation coefficient $\alpha$ without correction and the medium layer $h_r$, the attenuation coefficient is defined from eq. (6)   

\[\alpha = \frac{1}{2h} \ln \left[ \frac{P_1(x_1,f_0)}{P_2(x_2,f_0)} \right] = \alpha_s + 2\alpha_r \frac{h_r}{h} - \frac{1}{2h} \ln \left[ \frac{R_{sf}^2 D(s_2)}{D(s_1)} \right]\](11)

In order to analyze the difference between the attenuation coefficients with and without correction under different surface roughnesses more intuitively, the scattering attenuation coefficient $\alpha_s = 14{.}9864\,{\rm m/s}$, equivalent attenuation coefficient $\alpha_r = 1542{.}6\,{\rm m/s}$, equivalent sound velocity $c_r = 2366{.}396\,{\rm m/s}$, $h = 1{.}9971\,{\rm cm}$, $c_s = 5950\,{\rm m/s}$, $c_f = 1480\,{\rm m/s}$, $R_{sf} = 0{.}9381$ and $f_0 = 10\,{\rm MHz}$ are substituted into eq. (11). The theoretical result is shown in Fig. 4.

Fig. 4

Diagram of relationship between the roughness $h_r$ and attenuation coefficient $\alpha$ without correction.

It can be seen from Fig. 4 that the scattering attenuation coefficient $\alpha_r$ increases in proportion to the roughness $h_r$, but the corrected scattering attenuation coefficient of the theoretical smooth sample is not affected by changes in the roughness and the water path (i.e. the diffraction attenuation).

3. Experiment and Analysis

3.1 Experimental result

AISI 304 stainless steel with the face-centered cubic (FCC) structure was chosen as the experimental material, whose chemical composition is shown in Table 1. First, 5 blanks were acquired via wire cutting with the overall dimension of $\phi 25 \times 20\,{\rm mm}$, they can be considered to be of the same internal microstructure without any other treatment. Then, the upper and lower surface of each sample was processed by process of the electrical discharge machining to obtain a group of samples with surface roughness distribution gradients. The sample with the smallest roughness is used as a reference sample, it was sanded with 2000 grit sandpaper, and polished to approximate smoothness with nylon cloth and $1\,\mu{\rm m}$ alumina solution. The surface roughness of each sample was measured using TR210 TIME roughness tester, the profile arithmetical mean deviation $Ra$, profile maximum height $Rz$, and the thickness $D$ of each sample are measured, as Table 2 shows.

Table 1 The chemical composition of the AISI 304 stainless steel (mass%).
C Mn P S Cr Si Ni N
0.055 1.67 0.028 0.008 18.64 0.38 9.45 0.087
Table 2 Surface roughness and thickness of all samples.
Sample No. Ra/μm Rz/μm $\overline{D}$/cm
#0 0.0613 0.415 1.9897
#1 4.5630 25.305 1.9980
#2 9.2185 51.489 2.0049
#3 10.0190 57.194 2.0122
#4 12.5910 63.248 1.9971

The ultrasonic signal acquisition system adopted in the experiment is displayed in Fig. 5. An Olympus 5072PR ultrasonic pulse generator/receiver was employed; it was connected to a V317-SU-F2.3 planar transducer whose center frequency $f_0$ is $10\,{\rm MHz}$. A high-speed data acquisition card ADLINK PCIe-9852 whose sampling frequency is $200\,{\rm MHz}$ was adopted to obtain the waveforms. DMC2610 PCI bus 6-axis motion control card and 6-DOF motion platform were used to guarantee that the transducer is positioned perpendicular to the sample surface. The ultrasonic data were acquired on the sample #0~#4 with the above system respectively, and 20 sets of ultrasonic signals were acquired for each sample.

Fig. 5

Schematic diagram of the ultrasonic signal acquisition system.

The first and second back-wall echo of the ultrasonic signal of the sample are extracted, then waveforms were processed by Fast Fourier Transform (FFT) to obtain $P_1(x_1,f_0)$ and $P_2(x_2,f_0)$ respectively. Ultrasonic signals extracted from different samples are shown in Fig. 6. It can be seen in Fig. 6 that the amplitude spectrums of the signals are inversely proportional to the surface roughness of the samples.

Fig. 6

Ultrasonic signal amplitude spectrum of samples with different roughness (a) the first back-wall echo (b) the second back-wall echo.

Assuming the sound velocity of water and stainless steel as $1480\,{\rm m/s}$ and $5950\,{\rm m/s}$ respectively, the equivalent sound velocity cr is obtained as $2366{.}396\,{\rm m/s}$ according to eq. (8). Samples #0 and #2 were used as the reference and test sample of the experiment respectively, the equivalent attenuation coefficient $\alpha_r = 1542{.}65\,{\rm NP} {\cdot} {\rm m}^{-1}$ was obtained with eq. (10) by using amplitudes of two waveforms, sample thickness and thickness of the equivalent medium layer at the center frequency of $10\,{\rm MHz}$.

The scattering attenuation coefficient of different samples can be corrected by using eq. (6) and two equivalent parameters. Figure 7 shows the total attenuation coefficient and scattering attenuation coefficient with correction of each sample. As seen in Fig. 7, the total attenuation coefficient without correction increases in proportion to the surface roughness of the sample, which means that the attenuation caused by the surface roughness accounts for an increasing proportion in the total attenuation, but the scattering attenuation coefficient with correction eliminated the attenuation caused by the roughness and remains roughly the same.

Fig. 7

Scattering attenuation coefficient with and without correction.

In addition, sample #6 was used to conduct the experiment, and the changing relationship between the scattering attenuation coefficient with and without correction and the water path was obtained with a constant roughness, as shown in Fig. 8. As anticipated, the figure shows that the scattering attenuation coefficient without correction changes with the water path while the attenuation coefficient with correction is relatively stable.

Fig. 8

The changing relationship between the scattering attenuation coefficient with and without correction and the water path.

3.2 Result analysis

First, the metallographic images of the five samples were obtained using the Leica DM4000M microscope system, as shown in Fig. 9. Grain sizes of stainless steel were measured with the linear intercept method according to the American standard11). The results are 17.52 ± 1.2 $\mu {\rm m}$, 18.23 ± 1.0 $\mu{\rm m}$, 17.05 ± 0.8 $\mu{\rm m}$, 18.04 ± 1.1 $\mu{\rm m}$ and 17.60 ± 1.1 $\mu{\rm m}$ respectively. This validates the consistency in the internal microstructure of samples #0~#4. The theoretical scattering attenuation coefficient is 13.6584 ${\rm NP} {\cdot} {\rm m}^{-1}$ according to the density, elastic constants and grain size of the stainless steel sample12). But the attenuation coefficient of all samples measured in the experiment increase due to the effect of the surface roughness as shown in Fig. 7. Table 3 shows the scattering attenuation coefficient of all samples with correction based on eq. (6).

Fig. 9

Metallographical images of the samples. (a) #0, (b) #1, (c) #2, (d) #3, (e) #4.

Table 3 The scattering attenuation coefficient of each sample with correction.
No. $\alpha_s$/NP·m−1 E/%
#1 12.9 ± 1.0 −5.27
#2 13.9 ± 1.2 1.67
#3 14.5 ± 1.0 6.45
#4 13.4 ± 1.1 −1.57

In this paper, the scattering attenuation coefficients with correction stabilize at 13 ${\rm NP} {\cdot} {\rm m}^{-1}$ and the relative error between the observed values and theoretical values do not exceed $\pm7$%. This shows that the scattering attenuation coefficient can effectively be corrected.

Further, in order to verify the rationality of the assumption that the equivalent parameters are constant with different thicknesses of equivalent medium layers within $h_r \ll h$, sample combinations with different roughnesses were selected as the reference and test sample to calibrate the equivalent attenuation coefficient. For example, the equivalent attenuation coefficient of #0 and #3 is $1638{.}62\,{\rm NP} {\cdot} {\rm m}^{-1}$, #0 and #4 is $1561{.}47\,{\rm NP} {\cdot} {\rm m}^{-1}$, which means that the calibrated equivalent attenuation coefficients are basically the same. As the equivalent sound velocity is related to the sound velocities of water and stainless steel, this assumption is reasonable. So according to the constant equivalent parameters within the range of a certain roughness, the method in this paper can be used to correct scattering attenuation coefficients for samples with different roughnesses.

4. Conclusions

An idea of the equivalent multilayer medium is presented in this paper, with which a method for correcting the scattering attenuation coefficient is developed by considering the effect of surface roughness and sound field diffraction. It calibrates the parameters in the equivalent medium layer by using the experimental data of the reference sample, and corrects scattering attenuation coefficients of metals with different surface roughnesses.

The assumption that the equivalent parameter is constant at different thicknesses of equivalent medium layers is verified through the experiment. The experimental results show that corrected scattering attenuation coefficients of the metals with different surface roughnesses and same mean grain size agree well with each other.

Surface roughnesses used in this paper need be determined with a surface roughometer, and future work will be done to develop a method for ultrasonic evaluation of the surface roughness which will be integrated with the correction model in this paper so as to improve its practicability.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61271356 and 51575541), and Zhejiang Key Discipiline of Instrument Science and technology (Grant No. JL150509), and Natual Science Foundation of Zhejiang Province (LY15E050012). The authors would like to express their gratitude to T. D. Ashworth for English writing.

REFERENCES
 
© 2016 The Japan Institute of Metals and Materials
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