Evaluating grain size in polycrystals with rough surfaces by corrected ultrasonic attenuation

Ultrasonics 78 (2017) 23–29

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Evaluating grain size in polycrystals with rough surfaces by corrected ultrasonic attenuation Xiongbing Li a, Xiaoqin Han a, Andrea P. Arguelles b, Yongfeng Song a, Hongwei Hu c,⇑ a

School of Traffic and Transportation Engineering, Central South University, Changsha, Hunan 410075, China X-wave Innovations, Inc., Gaithersburg, MD 20878, USA c College of Automotive and Mechanical Engineering, Changsha University of Science & Technology, 410114, China b

a r t i c l e

i n f o

Article history: Received 16 January 2017 Received in revised form 11 February 2017 Accepted 23 February 2017 Available online 27 February 2017 Keywords: Nondestructive evaluation Grain size Ultrasonic attenuation Multi-Gaussian beam Rough surface

a b s t r a c t Surface roughness of a sample has a great effect on the calculated grain size when measurements are based on ultrasonic attenuation. Combining modified transmission and reflection coefficients at the rough interface with a Multi-Gaussian beam model of the transducer, a comprehensive correction scheme for the attenuation coefficient is developed. An approximate inverse model of the calculated attenuation, based on Weaver’s diffuse scattering theory, is established to evaluate grain size in polycrystals. The experimental results showed that for samples with varying surface roughness and matching microstructures, the fluctuation of evaluated average grain size was ±1.17 lm. For polished samples with different microstructures, the relative errors to optical microscopy were no more than ±3.61%. The presented method provides an effective nondestructive tool for evaluating the grain size in metals with rough surfaces. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Grain size is an important parameter for characterizing the microstructure of metals. Grain refinement is an effective method for improving the tensile and fatigue strength of Inconel 718 [1], whereas grain growth can increase the creep resistance and yield strength of Ni-Co-based alloys [2]. However, processes such as melting, heat treatment, or machining can result in grain sizes that deviate from the design specifications. Hence, effective nondestructive methods to evaluate the grain size of metals are highly desirable. In polycrystalline media, elastic wave scattering is directly dependent on grain size. Various authors [3–5] have focused on analytically establishing the correlation between scattering and grain structure. For example, Weaver [5] presented a forward model that relates grain size to attenuation due to scattering. Therefore, in principle, the grain size of metals can be determined by measuring the materials’ attenuation coefficient. However, accurate experimental estimates of the materials’ attenuation coefficient can be challenging to obtain. The rough surface of a sample can cause diffusion of ultrasonic waves, which decreases the amplitude of reflected echoes [6]. Furthermore, diffraction losses

⇑ Corresponding author. E-mail address: [email protected] (H. Hu). http://dx.doi.org/10.1016/j.ultras.2017.02.018 0041-624X/Ó 2017 Elsevier B.V. All rights reserved.

occur due to beam divergence and spreading depending on the ultrasonic setup. Therefore, it is necessary to formulate corrections for these losses in order to correlate measured ultrasonic attenuation to material microstructure accurately. Ogilvy [7] has reviewed the literatures pertinent to wave scattering from rough surfaces, including early theoretical models and experimental investigations. The perturbation technique and the Kirchhoff approximation models were compared in his work, which showed the importance of Kirchhoff approximation for rough surface scattering. Nagy and Adler [8] modified the ultrasonic transmission and reflection coefficients for a randomly rough interface by using the elastic wave equation and the Kirchhoff approximation. Reed et al. [9] formulated a correction for the attenuation coefficient when evaluating the porosity in materials with rough surface, but his work failed to include diffraction losses induced by the sound field, which resulted in overestimated porosity measurements. Wydra et al. [10] and Zeng et al. [11] used the Lommel diffraction correction method when investigating the effect of grain size on attenuation in pure niobium and copper alloys, respectively. Their results are in good agreement with classical stochastic scattering theory. However, the Lommel diffraction correction method is only suited for planar transducers. Zhang et al. [12] proposed a grain size ultrasonic evaluation method with diffraction correction based on the Gaussian beam theory, which could evaluate grain size in samples with different curvatures. However, this method

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X. Li et al. / Ultrasonics 78 (2017) 23–29

neglected the effects of surface roughness, interface reflections and transmission coefficients, which resulted in decreased accuracy in the grain size estimates. According to the analysis above, a correction method for experimental calculation of attenuation to account for diffraction and interfaces losses is proposed. The diffraction loss correction is derived based on the Multi-Gaussian beam (MGB) theory. The interface losses induced by the rough surface are corrected using modified transmission and reflection coefficients. Finally, the effects of surface roughness and the water path (diffraction losses) on calculated ultrasonic attenuation are evaluated to study their effect on grain size estimates.

Weaver gives a general expression for the attenuation coefficient of longitudinal wave propagation aL as [5,13,14]

aL ¼ aLL þ aLT

ð1Þ

with L

aLT ¼

p

8 6 f 40 q2 c3L c5T

R þ1 1

g~ LL ðhps ÞN1 ðhps Þdðcos hps Þ;

1

g~ LT ðhps Þ½N2 ðhps Þ  N1 ðhps Þdðcos hps Þ;



;

L3

p

2

½

1þk2L L2 þk2T L2 2kL kT L2

2

cos hps 

;

ð3Þ

 6 1 þ 525 cos2 hps þ 525 cos4 hps ;  12 þ 525 cos2 hps : 525 9

525 2 24

p

Z þ

8 >
m q > :

4 8 4 f 0 L3 2 525c3L 2

ð2Þ

where the subscripts LL and LT correspond with the contributions of scattering into longitudinal and transverse waves, respectively. q is the density of the sample, f 0 is the center frequency of the transducer, cL and cT are the longitudinal and transverse velocities, ^ and respectively, and hps is the angle between the incident wave p the scattered wave ^s. The integral in Eq. (2) accounts for the energy ~ LT ðhps Þ are the spa~ LL ðhps Þ and g lost in all scattering directions [13]. g tial Fourier transforms of the two-point spatial correlation function gðjr  r0 jÞ ¼ expðjr  r0 j=LÞ for longitudinal and shear wave scattering, respectively. g represents the probability that two points r and r0 lie within a given grain and L is the spatial correlation length, which relates to the grain size [15]. For equiaxed grains, as depicted ~ LT are given by [13] ~ LL and g in Fig. 1, g

2

ð4Þ

where m ¼ c11  c12  2c44 is anisotropy coefficient, where c11, c12, and c44 are the single crystal elastic constants of the material. Substituting Eqs. (2)–(4) into Eq. (1), the longitudinal attenuation coefficient for equiaxed polycrystals can be written as

aL ðLÞ ¼

R þ1

L3

p2 ½1þ2k2L L2 ð1cos hps Þ

where kL ¼ 2pf 0 =cL , kT ¼ 2pf 0 =cT are the longitudinal and transverse wave numbers, respectively. It is important to note that the grains are assumed entirely equiaxed; however, the grains in real materials are irregular and only approximately equiaxed. The terms N 1 ðhps Þ and N 2 ðhps Þ in Eq. (2) are the covariance of the elastic moduli, which have been defined for crystallites of arbitrary symmetries elsewhere [14]. Using Voigt averaging, these terms can be expressed for crystallites of cubic symmetry as [13,16]

N2 ðhps Þ ¼ m

2.1. Attenuation model

6 4

g~ LT ðhps Þ ¼ g~ LL ðp^ kL  ^skT Þ ¼

N1 ðhps Þ ¼ m2

2. Theory and correction models

aLL ¼ 8qp2 cf80

g~ LL ðhps Þ ¼ g~ LL ðp^ kL  ^skL Þ ¼

þ1

1

þ1

9 þ 6 cos2 hps þ cos4 hps h i2 dðcos hps Þ 2 1 c5L 1 þ 2kL L2 ð1  cos hps Þ 9 = 15 þ 6 cos2 hps  cos4 hps dðcos hps Þ : 2 2 2 ; c5 ½1 þ k L2 þ k L2  2k k L2 cos h  T

L

T

L T

ps

ð5Þ The relationship between attenuation, grain size and frequency in Eq. (5) can be used to evaluate the grain size from attenuation measurements. 2.2. Experimental attenuation coefficient corrections The accuracy of the measured attenuation coefficient plays a key role in the evaluation of grain size. In this paper, the experimentally obtained attenuation coefficient will be denoted atotal . For computing efficiency, a temporal approach is used to calculate attenuation, as given by

atotal ¼

1 V FW lnðj jÞ; 2H V BW

ð6Þ

where V FW and V BW are the peak values of the front wall (FW) and back wall (BW) echoes, respectively, and H is the thickness of the sample. Generally, the experimentally measured attenuation atotal is a combination of material attenuation due to scattering and interface and diffraction losses. The authors, using a MGB model, presented a correction method for diffraction losses in the attenuation coefficient [12]. The diffraction correction coefficient from FW to BW can be rewritten as

adiff ðf 0 Þ ¼

θ ps

L

pˆ

sˆ

1 t FW ðf 0 Þ lnðj jÞ: 2H tBW ðf 0 Þ

In Eq. (7), tFW ðf 0 Þ and tBW ðf 0 Þ are the acoustoelastic transfer functions, i.e., the integrations of dimensionless particle vibration velocities v FW and v BW over the surface of the transducer, given by

  R v FW ðf 0 Þ k jyj2 exp i f2F ds; S v 0 ðf 0 Þ   R k jyj2 ðf 0 Þ exp i f2F ds; tBW ðf 0 Þ ¼ 2S S vvBW 0 ðf 0 Þ

tFW ðf 0 Þ ¼ 2S

Fig. 1. Schematic of the equiaxed crystal texture within a superalloy.

ð7Þ

ð8Þ

where y are the spatial points of the transducer surface, S is the surface area of the transducer, F is the focal length (F ¼ 1 for the unfocused transducer), and v 0 is the initial particle vibration velocity at the transducer surface. The expressions in Eq. (8) have been given in

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X. Li et al. / Ultrasonics 78 (2017) 23–29

2

3  0  R12 2 2 2 2  2h k þ 2h k ln 0 0 0 FW f BW L 1 T R T 7 asurf ðf 0 ; hFW ; hBW Þ ¼ 6 4  12 21 21  5: 2H þ h2 þ h2 ðk  k Þ2 =2 L f FW BW

detail by Schmerr and Song [17]. Because the transmission and reflection losses will be later included in a separate correction factor, v FW ðf 0 Þ=v 0 ðf 0 Þ and v BW ðf 0 Þ=v 0 ðf 0 Þ are redefined here without the reflection and transmission coefficients of the rough fluidsolid interfaces, as [12,17]

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det½M2 ðWÞj det½M 1 ðWÞj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det½M2 ð0Þj det½M 1 ð0Þj   4pf 0 W  exp i þ ipf 0 yT ½M 2 ðWÞj y ; cf

ð14Þ Eq. (14) is appropriate for hFW and hBW smaller than the ultrasonic wavelength. Finally, the material attenuation coefficient can be written in terms of the measured attenuation and diffraction and surface roughness corrections as

15 v FW ðf 0 Þ X ¼ v 0 ðf 0 Þ j¼1

ð9Þ

aL ¼ atotal  adiff ðf 0 Þ  asurf ðf 0 ; hFW ; hBW Þ:

and

In the next section, the dependence of ultrasonic attenuation

aL on grain size and the effects of the correction factors will be dis-

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 15 det½M 4 ðWÞj det½M 3 ðHÞj det½M 2 ðHÞj v BW ðf Þ X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ v 0 ðf Þ j¼1 det½M4 ð0Þj det½M3 ð0Þj det½M 2 ð0Þj qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   det½M 1 ðWÞj 4pfW 4pfH  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  exp i þi þ ipf yT ½M 4 ðWÞj y ; cf cL det½M 1 ð0Þj

cussed based on the developed model (Eqs. (5), (7) and (14)). 3. Numerical results In order to understand the need for the diffraction losses correction factor, the coefficient in Eq. (7) is calculated for various water paths. A transducer with 76.2 mm (3 in.) focal length and 6.35 mm diameter is employed for the simulation. The water paths are chosen as 88.9, 76.2, 63.5, and 50.8 mm (3.5, 3, 2.5, and 2 in.). The thickness of the sample is set as 20 mm. The other parameters are shown in Table 1. The effect of frequency and water path on the diffraction correction coefficient is shown in Fig. 2. The frequency dependence is highly nonlinear. The effect with water path changes with frequency but has a generally increasing trend with increasing water path. The magnitude of the correction factor for the chosen parameters is on the same order of magnitude as the material attenuation coefficient. Hence, the influence of diffraction losses on measured ultrasonic attenuation cannot be neglected. Next, to examine the role of surface roughness on the correction coefficient in Eq. (14), the RMS surface roughness values are varied to 0, 8, 16, and 24 lm. The thickness of the sample is kept as 20 mm. Note that because the wavelength in water should be no

ð10Þ where j is the group number of Gaussian beams [18,19], W is the distance between the surfaces of the transducer and the sample, namely the water path, and cf is the wave velocity in the fluid. M1 , M2 , M3 , and M 4 are all 2  2 complex symmetric matrices, whose explicit parameter definitions can be located in Chapter 9 of Ref. [17]. The second step to obtaining a corrected attenuation coefficient is to introduce the interface losses. Following the concept in Refs. [8,20], attenuation is corrected for surface roughness effects. A surface loss correction coefficient of FW to BW is defined here as

asurf ¼



R12 1 ; ln 2H T 12 R21 T 21

ð11Þ

where R12 and R21 are the reflection coefficients of the fluid-solid and solid-fluid interfaces, respectively. Similarly, T 12 and T 21 are the transmission coefficients of the fluid-solid and solid-fluid interfaces. Considering the surface roughness, the reflection and transmission coefficients are given by [8,20]

h i 8 2 2 > R12 ðhFW Þ ¼ R012 exp 2hFW kf ; > > > > h i > > > < R21 ðhBW Þ ¼ R021 exp 2h2BW k2L ; h i 2 2 0 > > > T 12 ðhFW Þ ¼ T 12 exp hFW ðkf  kL Þ =2 ; > > > h i > > 2 2 : T 21 ðhBW Þ ¼ T 021 exp hBW ðkL  kf Þ =2 :

ð15Þ

ð12Þ

where hFW and hBW are the Root-Mean-Square (RMS) roughness of the two sides of the sample. R012 , R021 , T 012 , and T 021 are the ideal reflection and transmission coefficients of the smooth surface,

R012 ¼ ðZ 2  Z 1 Þ=ðZ 1 þ Z 2 Þ; R021 ¼ ðZ 1  Z 2 Þ=ðZ 1 þ Z 2 Þ; T 012 ¼ 2Z 2 =ðZ 1 þ Z 2 Þ; T 021 ¼ 2Z 1 =ðZ 1 þ Z 2 Þ;

ð13Þ

where Z 1 ¼ qf cf and Z 2 ¼ qcL are the acoustic impedances of the fluid and the sample, and qf is the density of the fluid. Thus, Substituting Eq. (12) into Eq. (11), the surface correction coefficient can be rewritten as

Fig. 2. The diffraction correction spectrum.

Table 1 Parameters used in the model. Density (kg/m3)

Velocity (m/s)

qf

q

cf

cL

cT

1000

7880

1486

5750

3100

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X. Li et al. / Ultrasonics 78 (2017) 23–29

of surface roughness and diffraction. Then, the inverse model of Eq. (5) is used to evaluate the grain size. 4. Experiments and results 4.1. Sample preparation and metallography

Fig. 3. The relationship between surface correction, center frequency of transducer and surface roughness.

less than 24 lm, the highest possible inspection frequency is limited to 10 MHz. Fig. 3 shows that asurf decreases with the increasing center frequency and surface roughness. As a result, when ignoring surface roughness, the surface correction coefficient would be overestimated, and the scattering attenuation coefficient underestimated. In the next section, stainless steel samples are prepared and studied by metallography and ultrasound. Eq. (15) is used to correct the measured attenuation coefficient, eliminating the effects

Experiments are conducted using AISI 304 stainless steel, with the chemical composition 06Cr19Ni10 and face-centered cubic (FCC) structure. First, seven blanks were acquired via wire cutting with dimensions /25  20 mm. Two sets of samples were designed: (1) #A0-A4 were processed to obtain varying surface roughness, and (2) #B1-B2 were heat treated to obtain varying microstructures. The upper and lower surface of samples #A0-A4 were processed by electrical discharge machining to obtain a group of samples with different surface roughness and nearly identical microstructures. #B1 and #B2 were heat treated in a KSL1700X high temperature furnace to alter the microstructure; temperature was held at 1080 °C for two and four hours, respectively, before water quenching. Samples #A0, #B1, and #B2 were sanded with 2000 grit sandpaper to near mirror finish. The surface roughness of each sample was then measured using a TR210 TIME roughness tester, the profile arithmetical mean deviation Ra, profile maximum height Rz, the calculated RMS and the thickness H of each sample were given in Table 2. After the ultrasonic testing was completed, the samples were ground, polished, and etched for 20 min using an etchant of 20% HF + 10% HNO3 + 70% H2O [21]. The metallographic images of the samples were obtained using a Leica DM4000M microscope system, and are shown in Fig. 4. It is evident that the grain sizes of

Table 2 The thickness and roughness parameters of each sample. Sample No.

Ra/lm

Rz/lm

RMS/lm

H/mm

D/lm

A0 A1 A2 A3 A4 B1 B2

0.0613 4.5630 9.2185 10.0190 12.5910 0.0622 0.0609

0.415 25.305 51.489 57.194 63.248 0.405 0.398

0.0426 11.7087 17.4172 18.7278 19.6088 0.0515 0.0442

19.8970 19.9800 20.0490 20.1220 19.9710 19.9983 20.0001

19.4 19.6 18.9 20.8 18.7 71.3 81.9

(a)#A0

(d)#A3

(b)#A1

(e)#A4

(c)#A2

(f)#B1

(g)#B2

Fig. 4. Metallographic images of the samples (a) #A0, (b) #A1, (c) #A2, (d) #A3, (e) #A4, (f) #B1 and (g) #B2.

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X. Li et al. / Ultrasonics 78 (2017) 23–29

the #A0-A4 samples are similar, and the ones in #B1-B2 are significantly larger. The average grain size D of the samples in Table 2 were measured according to the standard of ASTM E112. 4.2. Ultrasonic setup Fig. 5 depicts the ultrasonic setup. An Olympus 5072PR ultrasonic pulse generator/receiver was employed to drive a GEIAP10.6.3 focused transducer with center frequency f 0 = 9.0 MHz and focal length F = 75.0 mm. A manipulator was used to fix the transducer, which was controlled in micro-motion to ensure perpendicularity to the tested surface. The transducer height was adjusted to optimize the water path, which was set equal to the focal length F. The scanning speed in the ultrasonic imaging experiment was 4 mm/s, and the resolution of scan was 0.4 mm.

Motion control card

Control cabinet

4.3. Results and discussion The attenuation coefficient was calculated at each transducer position to create the images in Fig. 6. The experimental attenuation atotal as calculated by Eq. (6) is given in Fig. 6(a), and the corrected attenuation coefficient aL based on Eq. (15) is given in Fig. 6(b) for samples #A0, #A2 and #A4. The red circle depicts edge effects inherent in ultrasonic imaging techniques. The analysis that follows focuses on the center portion where edge effects can be neglected. Because these samples have near-identical microstructures, the experimental results illustrate the effect that surface roughness has on the calculated attenuation. When surface roughness and diffraction are ignored, the images depict a change in attenuation (Fig. 6(a)). When these factors are included, the attenuation coefficients remain roughly the same (Fig. 6(b)).

ADLink PCIe9852 DAQ card

Industrial computer

Motion platform

Olympus 5072PR pulse generator/receiver GAIN 30 dB

Manipulator

Transducer

Fig. 5. Ultrasonic signal acquisition system.

0

0

0

80

20

20

20

60

40

40

40

60

60

60

80

80

80

(a)

0

20

40

60

80

0

20

40

60

80

40 20

0

20

40

60

80

0

0

0

0

20

20

20

20

10

40

40

40

60

60

60

80

80

80

(b)

0

20

40

60

80

0

20

40

60

80

0 -10

0

20

40

60

Fig. 6. The attenuation coefficient images (a) before correction and (b) after correction.

80

-20

28

X. Li et al. / Ultrasonics 78 (2017) 23–29 Table 3 The grain size with correction and the error. Sample No.

D/lm

s

e/%

A0 A1 A2 A3 A4 B1 B2

20.10 18.85 19.90 19.00 19.63 70.04 80.12

0.7845 0.9201 0.1499 1.6189 1.9820 0.6852 0.7264

3.61 3.83 5.29 5.63 4.97 1.77 2.17

Once the attenuation coefficient images are obtained, the grain size of each position can be calculated by inverting Eq. (5). However, obtaining the inverse function is complicated due to the integral operation. A forward search algorithm could be used but proves too time-consuming for real-time imaging. Therefore, assuming a fixed center frequency and single crystal elastic constants, an approximate inverse function using the nonlinear least squares fitting on experimental data is obtained as follows: Fig. 7. Approximate inverse function based on the nonlinear least squares fitting.

D ¼ 0:02065a1:485 þ 13:38a0:3472 ; L L

ð16Þ

where D is the evaluated grain size (assuming D = 2L). Fig. 7 shows the experimental data and fit. The maximum error is less than

0

50

20

100

0

100

20

80

20

80

40

40

30

60

20

80

10

(a)

0

60

0

20

40

60

80

0

60

40

40

60

20

80

(b)

0

20

40

60

80

0

60

40

40

60

20

80

(c)

0

Fig. 8. The spatial variation of grain size (a) #A1, (b) #B1 and (c) #B2.

Fig. 9. Four kinds of attenuation coefficient.

20

40

60

80

0

X. Li et al. / Ultrasonics 78 (2017) 23–29

0.2 lm, which is acceptable for grain size evaluation. The average computation time of this numerical inverse function is 0.00124 s, while the search algorithm is 0.762 s for one transducer position. In order to test the validity of Eq. (16), the spatial variation of grain sizes are imaged for the polished samples #A0, #B1 and #B2, using the corrected attenuation coefficients (Fig. 8). Visually the grain sizes correspond well with those calculated through metallography. After the corrected grain sizes of all samples are statistically analyzed, they are compared with the grain sizes determined by the metallographic method to further study the effect of surface roughness. Table 3 shows that the relative error, e, is less than ±6% for all samples considered. Furthermore, samples #A0-A4 show good agreement in the grain size estimates even though they have varying surface roughness. The differences between evaluated average grain sizes and the optical microscopy results are 0.70, 0.75, 1.00, 1.17, and 0.93 lm, respectively. Thus, the fluctuation of evaluated average grain size is ±1.17 lm. The fluctuation range of the grain sizes (i.e. the standard deviation, s), as depicted in Fig. 8, increases with increasing surface roughness, which provides an indication of the inherent error in the proposed technique. In other words, although surface roughness still has influence on grain size evaluation locally, its effect on the average grain size evaluation can be neglected when a representative area is studied. Finally, Fig. 9 shows the contributions of grain scattering as well as interface and diffraction losses on the measured attenuation coefficient for samples #A0-A4, based on Eqs. (6), (10) and (15). The measured attenuation coefficients and the interface correction coefficients decrease with the increase of the roughness, which is consistent with the theoretical results in Fig. 3. However, the scattering component of attenuation and the diffraction correction remain constant, as expected. 5. Conclusions Weaver’s model [5] was employed to relate the grain size of polycrystals to the ultrasonic attenuation coefficient. A comprehensive correction method was developed to eliminate the effect of diffraction and surface losses from the experimental measurement of attenuation. The MGB model was used to formulate the diffraction correction coefficient and modified transmission and reflection coefficients were used to correct for surface roughness and interface losses. A fitting inverse evaluation model to Weaver’s model was established by the nonlinear least squares method. Ultrasonic experiments were conducted on 304 stainless steel samples. For the samples with different surface roughness and the same grain size, the corrected attenuation images have good agreement, which indicates that the effect of rough surfaces is mostly eliminated by the developed technique. Good accuracy and reliability of presented method for grain sizing were shown by statistical comparison of the tested samples. The maximum rel-

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