Non-destructive evaluation of formability in cold rolled steel sheets using the SH0-mode plate wave by electromagnetic acoustic transducer

Ultrasonics 39 (2001) 335±343

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Non-destructive evaluation of formability in cold rolled steel sheets using the SH0-mode plate wave by electromagnetic acoustic transducer Riichi Murayama * Department of Intelligent Mechanical Engineering, Faculty of Engineering, Fukuoka Institute of Technology, Waziro-higashi 3-30-1, higasi-ku, Fukuoka 811-0295, Japan Received 1 March 2000; accepted 23 March 2001

Abstract We evaluated the relation between the Lankford value (r-value) and the velocity of cold rolled steel sheets using the fundamental shear horizontal plate wave (SH0 -mode plate wave) and the fundamental symmetrical Lamb wave (S0 -mode Lamb wave) as a function of the propagation angle. The SH0 -mode plate wave at the traveling angle of 45° showed the opposite tendency compared with that of other traveling angles or other ultrasonic modes. Using the velocity equations derived by accounting for texture and cubic crystal anisotropy, we found that the velocity at 45° was strongly in¯uenced by the f1 0 0g crystallographic plane compared with that of other ultrasonic conditions. Next, the measurement accuracy for the formability of cold rolled steel sheets using the S0 -mode Lamb wave and the SH0 -mode plate wave was estimated to ®nd that the evaluation accuracy using the SH0 -mode plate wave became poor as compared with the case of the S0 -mode Lamb wave. For this, the e€ect of the ultrasonic velocity anisotropy to the propagation direction was the main cause, because the anisotropy of the SH0 -mode plate wave was bigger than that of the S0 -mode Lamb wave. Finally, we con®rmed that the evaluation precision could be improved by using the SH0 -mode EMAT with a narrow-width sensor coil to transmit and receive the sharp ultrasonic beam. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Formability; Lankford value; Lamb wave; Shear horizontal plate wave; Electromagnetic acoustic transducer; Cold rolled steel sheet

1. Introduction The non-destructive evaluation of cold rolled steel sheets using ultrasonic velocity is being established. Available ultrasonic modes are: longitudinal wave, transverse wave, S0 -mode Lamb wave, and SH0 -mode plate wave. Especially, use of the S0 -mode Lamb wave has been intensively researched in Europe [1], USA [2, 3], and Japan [4]. However, the SH0 -mode plate wave is not so used as the S0 -mode Lamb wave, although it has been indicated that the SH0 -mode plate wave enables the evaluation with a small number of sensors. We then tried to simultaneously evaluate the measurement accuracy using these two waves. As a result, we found that the SH0 -mode plate wave was poor in the evaluation precision as compared with the S0 -mode Lamb wave. *

Tel.: +81-92-606-4214. E-mail address: murayama@®t.ac.jp (R. Murayama).

This paper discusses the reason for this and suggests how to improve the measurement precision.

2. Measurement principle of Lankford value (r-value) in cold rolled steel sheet The Lankford value of cold rolled steel sheet is the plastic distortion ratio reduced from ln…w0 =w1 †= ln…t0 =t1 † (after here called the r-value) [5] of the widths …w0 ; w1 † and thicknesses …t0 ; t1 † before and after giving a 15±20% deformation of the tensile specimen. Generally used is the average r-value (ˆ …r0 ‡ 2r45 ‡ r90 †=4), which is computed from the r-value …r0 ; r45 ; r90 † to the directions of the 0°, 45°, and 90°. The average r-value of cold rolled steel sheets is high for sheets with many f1 1 1g and few f1 0 0g crystallographic planes lying parallel to the rolling plane. The ultrasonic velocity shows changes in accordance with the texture. Therefore the average

0041-624X/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 1 - 6 2 4 X ( 0 1 ) 0 0 0 6 8 - 3

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R. Murayama / Ultrasonics 39 (2001) 335±343

r-value can be estimated from the changes in ultrasonic velocity. The texture is described quantitatively using the crystallite orientation distribution function (CODF), which is a probability density function and de®nes how the crystallites are oriented as a whole in the bulk of a polycrystalline. The independent variables are three Euler angles (n, /, w) de®ning the correlation of two Cartesian coordinate systems, one being ®xed to a crystallite and the other to the specimen of a cold rolled steel sheet. As analogous to Fourier expansion, the CODF is expanded in terms of generalized spherical harmonics [6], W …n; /; w† ˆ

1 l l X X X 0

Wlmn Zlmn …n† exp… imw†

mˆ l nˆ l

 exp… in/† Z Wlmn ˆ

0

2p

Z

2p

Z

0

‡1 1

…1†

W …n; w; /†Zlmn …n† exp…imw†

 exp…in/† dn dw d/

…2†

where Zlmn are the generalized Legendre functions. The expansion coecients, Wlmn , are called orientation distribution coecients (ODCs). For cubic materials with orthorhombic texture, some ODCs vanish and many others are mutually p dependent. The independent ODCs are W000 ˆ 1=…4 2p2 †, W4m0 …m ˆ 0; 2; 4†, etc. The CODF is used to weight the orientation dependence of singlecrystal quantities and integrate them over the total range of Euler angles in order to obtain the macroscopic counterparts for a textured metal. The basic ODCs, W000 , determine the isotropic property. All other ODCs contribute to the anisotropy. The second order elastic constants, Cij , of polycrystalline materials are obtained by integrating the products of elastic constants in the monocrystal and CODFs in any direction over the total range of Euler angles. The ultrasonic velocity dependent to the propagation direction is calculated using elastic equations with these elastic constants [7]. First, fundamental symmetrical Lamb …S0 † wave velocity is given by: 1=2

VS0 …h† ˆ VS0 …1 D† ‡ …c=qVS0 †…s0 W400 ‡ s2 W420 cos 2h ‡ s4 W440 cos 4h† …3† The fundamental shear horizontal (SH0 ) plate wave velocity is given by: VSH0 …h† ˆ VSH0 ‡ …c=qVSH0 †…q0 W400 ‡ q4 W440 cos 4h†

…4†

where c…ˆ C11 C12 2C44 † denotes the elastic anisotropy inherent in a cubic single-crystal, VS0 is the S0 -mode Lamb wave ultrasonic velocity …ˆ f4l…k ‡ l†=q…k ‡ 1=2 2l†g † in the case without texture, VSH0 is the SH0 -mode 1=2 plate wave …ˆ …l=q† † in the case without texture, q is the density of steel, the angle h measures the propagation

Table 1 Elastic constants and coecients of S0 -mode Lamb wave and SH0 mode plate wave velocity equations Monocrystal elastic constants (GPa) C11 C12 C44

232.2 135.6 117.0

Theory Hill Polycrystal elastic constants (GPa) k ‡ 2l l c c=l Coecients of velocity equations S0 S2 S4 q0 q4

277.0 81.7 136.3 1.668 6.08 9.18 6.67 1.60 13.34

p …ˆ f2 2p2 = direction from the rolling direction, s0 p 35gf3 ‡ 16k…k ‡ l†=…k ‡ 2l†2 g†, s2 …ˆ f 8 5p2 =35g p  f…3k ‡ 2l†=…k ‡ 2l†g†, s4p …ˆ4p2 =…35†2 †, q0 …ˆ 4p2 2= 35† and q4 …ˆ …8=35†p2 = 70† are functions of Pois2 2 son's ratio and D…ˆ fk=…k ‡ 2l† …kd† =3g is the correction term for the velocity decrease due to the ®nite thickness/wavelength ratio. The values of C11 , C12 , C44 , k, l, c, c=l, s0 , s2 , s4 , q0 , q4 are shown in Table 1. Next, we could transfer Eqs. (3) and (4) to Eqs. (5) and (6), in the case when we ®xed the distance between the transmitter and the receiver. The transit time by the S0 -mode Lamb wave velocity is given by: TS0 …h† ˆ L=VS0 …h†  ˆ …L=V0 † 1 ‡ D=2

…c=qV02 †…s0 W400

‡ s2 W420 cos 2h ‡ s4 W440 cos 4h†



…5†

The transit time by the SH0 -mode plate wave velocity is given by: TSH0 …h† ˆ L=VSH0 …h† ˆ …L=VSH0 †f1

2 …c=qVSH †…q0 W400 0

‡ q4 W440 cos 4h†g

…6†

On the other hand, Wlmn computed for some ideal orientations shown in Table 2 and the following conclusions could be derived [7]. If a f1 1 1g crystallographic plane exists, W400 has a negative value, and a f1 0 0g crystallographic plane leads to a positive value in W400 . This means that a thin sheet with a high r-value leads to a negative large value of W400 , because as the r-value increases, the f1 1 1g crystallographic plane of the cold rolled steel sheet increases and the f1 0 0g crystallographic plane of that decreases. In short, we can evaluate the average r-value by measuring W400 . Eq. (7), calculated from Eq. (5), shows that the average r-value

R. Murayama / Ultrasonics 39 (2001) 335±343 Table 2 ODCs of some ideal orientations (h k l) ‰u v wŠ W400 (1 1 1)[1 1 0] (1 1 1)[1 1 2] (1 1 0)[0 0 1] (1 0 0)[0 1 1] (1 0 0)[0 0 1]

W420

0.0209 0.0209 0.0078 0.0313 0.0313

0 0 0.0246 0 0

W440 0 0 0.014 0.0187 0.0187

(W400 ) is estimated from the S0 -mode Lamb wave velocity at h ˆ 0°, 45°, 90° relative to the rolling direction and Eq. (8), calculated from Eq. (6), shows that the average r-value (W400 ) is estimated from the SH0 -mode plate wave velocities at h ˆ 0° and 45°. We evaluate the average r-value by using the measuring system shown in Fig. 1.  W400 / TS0 …0† ‡ 2TS0 …45† ‡ TS0 …90† =4 …7†  W400 / TSH0 …0† ‡ TSH0 …45† =2

…8†

3. Measurement technique The basic composition of the electromagnetic acoustic transducer (EMAT) system for the S0 -mode Lamb wave and the SH0 -mode plate wave is shown in Fig. 2. The EMAT mainly used the magnetostrictive e€ect [8± 10]. As shown in Fig. 2(a), the magnetostrictive type EMAT for the S0 -mode Lamb wave consists of a meandering coil and an electromagnet which applies the magnetic ®eld in the propagation direction. When the RF current drives the coil, a dynamic ®eld occurs in the direction parallel to the static magnetic ®eld. The spacing of the coil wire coincides with the half wavelength. It also shows the distribution of the e€ective magnetic ®eld, which is the sum of the static and dynamic ®elds and that of the resulting magnetostriction in

337

the ferromagnetic materials. The magnetostriction introduces a longitudinal displacement in the measuring metal sheet. For receiving, when the stress by the S0 -mode Lamb wave changes the permeability of the steel, it results in a ¯ux density change under the static ®eld. An eddy current is then introduced in the sheet, and the meandering coil detects the magnetic ®eld induced by the eddy current. As shown in Fig. 2(b), the SH0 -mode type EMATs consist of a meandering coil and an electromagnet, which applies an orthogonal magnetic ®eld to the propagation direction. The excitation and reception mechanics the same as above, but the magnetostriction generates a shear deformation on the surface to excite the SH wave. The electromagnet was a U-shaped one. The intervals of wiring the sensor coil, which correspond to the wavelength, were 10 mm for the S0 -mode Lamb and 6 mm for the SH0 -mode plate wave. The frequency was determined from the fact that each velocity was approximately 5000 and 3200 m/s, respectively. But the optimal driving frequency was 250 kHz for the S0 -mode Lamb wave and 320 kHz for the SH0 -mode plate wave. The width of the both sensor coils was 30 mm, but changed later. The distance between the transmitter and the receiver was ®xed to 250 mm. The measuring system is shown in Fig. 2(c). The pulser was the frequency-variable type, and fed tone bursts. Signals were acquired by digitizing at 100 MHzsampling rate and an 8-bit resolution. The band path ®lter had an e€ective range from 70 kHz to 1.3 MHz. Typical received signals with 1 mm thick steel sheet are shown in Fig. 3. Fig. 3(a) presents the case of the S0 -mode Lamb wave and Fig. 3(b) is that for the SH0 -mode plate wave. It was demonstrated that both received signals have a sucient signal to noise ratio for measuring the transit time.

Fig. 1. A way of ®nding the average r-value using S0 -mode Lamb wave velocity at h ˆ 0°, 45°, 90° relative to the rolling direction and the SH0 -mode plate wave velocity at h ˆ 0° and 45°.

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R. Murayama / Ultrasonics 39 (2001) 335±343

Fig. 2. Block diagram of the EMAT system for the S0 -mode Lamb wave and the SH0 -mode plate wave including arrangement of the sensor coil and electromagnet.

Fig. 3. Display of the typical received signal with 1 mm thick steel sheet: (a) The case of the S0 -mode Lamb wave. (b) The case of the SH0 -mode plate wave.

4. Measurement results of the transit time 10 sample sheets of thickness ranging from 0.6 to 1.6 mm and with average r-value ranging from 1.168 to 1.970 (Table 3) were prepared. We denoted each term in Table 3 as the following, r-value as the average r-value measured by using tensile test pieces, fh k lg as the average value of the NDkhh k li axis density at the three position measured by the X-ray refraction method and Dfh k lg as the deviation of the NDkhh k li axis density of three measuring points. Then we measured the transit time at the ®xed distance of 250 mm. Fig. 4(a) shows the

correlation between the average r-value and the transit time …TS0 …0†, TS0 …45†, and TS0 …90†† measured by EMAT for the S0 -mode Lamb wave. Fig. 4(b) gives the results for the SH0 -mode plate wave. We see that the three transit times have negative correlation with the average r-value in the case of the S0 -mode Lamb wave. But, in the case of SH0 -plate wave, TSH0 …0† has a negative correlation and TSH0 …45† has a positive correlation with the average r-value. To explain these results, we studied the relation between the average r-value and NDkhh k li axis density using Table 3. Then it was found, when the average

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Table 3 Speci®cations of sample sheets (plate thickness, the averaged r-value, crystal axis density, deviation of crystal axis density) No.

Thickness (mm)

r-value

f1 1 1g

Df1 1 1g

f1 1 0g

Df1 1 0g

f1 0 0g

Df1 0 0g

1 2 3 4 5 6 7 8 9 10

1.20 1.00 0.60 0.60 1.60 0.60 0.70 0.80 0.70 1.00

1.168 1.170 1.335 1.373 1.620 1.670 1.720 1.905 1.960 1.970

2.324 3.452 5.318 5.165 4.629 6.201 8.248 7.516 7.824 6.506

0.111 0.223 0.224 0.378 0.252 0.287 0.206 0.243 0.529 0.534

0.896 0.741 0.528 0.515 0.554 0.438 0.221 0.209 0.247 0.421

0.005 0.011 0.016 0.030 0.014 0.003 0.005 0.008 0.008 0.026

1.086 1.036 0.976 0.994 1.992 1.091 1.228 1.382 1.264 1.010

0.021 0.035 0.030 0.076 0.044 0.016 0.039 0.068 0.076 0.062

Fig. 4. Correlation of the average r-value and the transit time measured: (a) The correlation between the average r-value and the transit time (TS0 (0), TS0 (45), and TS0 (90)) measured by EMAT for the S0 -mode Lamb wave. (b) The results for the SH0 -mode plate wave.

r-value increases from the measurement results in Table 3, f1 1 1g crystal axis density also, but little change occurs in f1 0 0g crystal axis density when f1 1 0g crystal axis density decreases. Then, in the case of the S0 -mode Lamb wave, we evaluate each term of Eq. (5) using each coecients of Table 1. Term 1, which is the transit time due to the basic velocity in the case of the isotropic steel sheet, is positive. Term 2, D=2, is positive but negligible under the present experimental conditions. The common constant, c/(qVS20 ) of term 3, is positive. S0 W400 of term 3 is negative and always decreases as the average r-value increases, S2 W420 is positive, and S4 W440 is negative. We could assume that S2 W420 is almost unchanged because of W420 's dependence only on the {1 1 0} axis density crystal axis density (see Table 2) and little change in Table 3. Value of S4 W440 , which depends on the {1 1 0} axis density and {1 0 0} axis densities, decreases as the angle shifts from 0°, and then increases approaching 45°. However, the range of S0 W400 variation is much bigger

than in S4 W440 . Therefore, it is possible to say that the transit time decreases when the average r-value increases in whichever direction it propagates. 5. Evaluation results of formability in cold rolled steel sheets Fig. 5 shows the evaluation results of formability in steel sheets using the S0 -mode Lamb wave or the SH0 mode plate wave. Fig. 5(a) indicates the correlation between the average r-value and the average transit time calculated from the average transit time for the Lamb wave. Fig. 5(b) gives the results for the SH0 -mode plate wave. The correlation coecients were 0.99 for the Lamb wave and 0.91 for SH0 -mode plate wave. Similarly, the evaluation accuracy (standard deviation from the average) was 0.04 for the S0 -mode Lamb wave and 0.13 for the SH0 -mode plate wave. It is obvious that the evaluation error of formability in cold rolled steel sheets

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R. Murayama / Ultrasonics 39 (2001) 335±343

Fig. 5. Correlation of the average r-value and the average transit time: (a) The correlation between the average r-value and the average transit time calculated from the transit time for the S0 -mode Lamb wave. (b) The results for the SH0 -mode plate wave.

using the SH0 -mode plate wave is bigger than that using the S0 -mode Lamb wave. 6. Discussions of error factors We assumed that there are three factors, behind the evaluation error when we evaluate the average r-value using ultrasonic measurements. The ®rst is the error due to the jitter of the electric circuit. We evaluated this factor by repeatedly measuring the transit time under the same measurement conditions, that is, the measurement with holding the EMAT on the sample sheet. The second is the error due to texture distribution in the sheet. The third is the error due to the uncertainty in the traveling direction of the ultrasonic wave. We assumed that the change in ultrasonic velocity for the traveling direction in the SH0 -mode plate wave was di€erent from that of the S0 -mode Lamb wave. At ®rst, we evaluated how much the average r-value change causes the change in the measured transit time. As shown in Fig. 5, the change from 1 to 2 of the average r-value corresponds to 650 ns for the SH0 -mode Lamb wave and 1000 ns for the S0 -mode plate wave. The e€ect of jittering is as follows: 1. It was 4 ns for the S0 -mode Lamb wave, which means a relative error of 0.004 for the average r-value. 2. It was 6 ns for the SH0 -mode plate wave, which means a relative error of 0.01 for the average r-value. Both evaluated values were suciently small compared with that caused by texture. Then, the measurement error due to the electric circuit could not be the main cause of the di€erence in the evaluation precision between both of the ultrasonic modes.

Next, we discussed the in¯uence due to the inhomogeneity of the texture in the steel sheets. We computed the maximum velocity change due to the inhomogeneity of the texture using Eqs. (3) and (4) and Table 3. As a result, the maximum velocity changes ratio of both modes were about 0.1% and almost same. Then we judged that the measurement error due to the inhomogeneity of the texture could be negligible. Finally, we study the spread angle of the ultrasonic beam because of the ®nite dimensions of the transmitter and receiver. The concept is shown in Fig. 6(a), where W is the width of the sensor and L is the distance between the transmitter and receiver. Then the spread angle of the ultrasonic beam was …h ˆ tan 1 …W =2L††. Of course, we have to consider the in¯uence by the traveling distance enlargement. But, it could be evaluated that the in¯uence was the same at any specialized propagation direction as shown in Fig. 6(b), if the velocity was independent of the traveling angle. Contrary, it could be considered that the in¯uence was di€erent at any specialized propagation direction as shown in Fig. 6(c), if the velocity was dependent of the traveling angle. Then it was indicated that the in¯uence was di€erent for the S0 -mode Lamb wave and the SH0 -mode plate wave. Actually we computed the velocity dependence for the traveling angle about the no. 3 test piece in Table 3 using Eqs. (5), (6), (9) and (10). After that, we evaluated the measurement error due to the angle of the ultrasonic beam spread. The results are shown in Fig. 7(a). Obviously, it was indicated that the in¯uence become bigger as the sensor width increased, and that of the SH0 -mode plate wave was bigger than that of the S0 -mode Lamb wave. We transferred the results to the average r-value as shown in Fig. 7(b). Then we could recognize that the measurement error corresponded to an evaluation error of almost 0.1 at the average r-value when the sensor coil

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Fig. 6. Concept ®gure of evaluation error due to ultrasonic velocity anisotropy to the propagation direction: (a) Con®guration of the EMAT system and the concept of the ultrasonic beam spread (h ˆ tan 1 (W =2L)); (b) The in¯uence at any specialized propagation when the velocity was independent of the traveling angle; (c) The in¯uence at any specialized propagation when the velocity was dependent of the traveling angle.

Fig. 7. The correlation between the computerized measurement error and the sensor coil width: (a) The transit time and the sensor coil width; (b) The compensated r-value and the sensor coil width.

of 30 mm width was used. This result supported the experimental data as shown in Fig. 5(b). Furthermore, it become obvious that a sensor width of less than 5 mm was proper because the measurement error of the transit time was almost same as that of the electric circuit and the texture deviation.

7. E€ect of narrow ultrasonic beam To con®rm experimentally the result of the preceeding discussion, the transit time was again measured using the EMAT with the narrowed width of the sensor coil about the EMAT for the SH0 -mode plate wave. The

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Fig. 8. Example of the averaged r-value evaluation for the narrowed sensor: (a) The relationship between the experimental evaluation accuracy of the average r-value and the sensor width for the SH0 -mode plate wave. (b) The relationship between the average transit time and the average r-value using the EMAT with a 5 mm sensor coil width.

experimental evaluation accuracy of the average r-value with the sensor width (W) of 5, 10, 20 and 30 mm were similarly shown in Fig. 8(a). It was indicated that the measurement accuracy would be improved, as the width of the sensor coil decreased. As a result, Fig. 8(b) shows the relationship between the transit time and the average r-value using the EMAT with a 5 mm sensor coil width for the SH0 -mode plate wave. Comparing the data of Fig. 8(b) with that of Fig. 5(b), the evaluation precision of the average r-value was demonstrated to be dramatically improved. 8. Conclusions The evaluation precision of the average r-value in cold rolled steel sheets was examined by comparing the EMAT for the SH0 -mode plate wave with the EMAT for the S0 -mode Lamb wave. The conclusions were the following, (1) The relation between the average r-value and the transit times (TS0 (0), TS0 (45), and TS0 (90)) measured by the EMAT for the S0 -mode Lamb wave show negative correlation. But the relation between the average r-value and the transit time (TSH0 (0) and TSH0 (45)) measured by the EMAT for the SH0 -mode plate shows a negative correlation for TSH0 (0) and a positive correlation for TSH0 (45). Taking into consideration that the velocity dependence equations using ODCs represents the texture, only the ultrasonic velocity of the SH0 -mode plate wave traveling at 45° direction to the rolling direction strongly depends on the W400 term, which has good correlation with the f1 0 0g crystallographic plane. The results indicate that the velocity of the SH0 -mode plate

wave is sensitive to the propagation direction and f1 0 0g crystallographic plane. (2) Poor evaluation precision of the average r-value was obtained for the SH0 -mode plate wave compared with the case of the S0 -mode Lamb wave using the meander sensor coil of 30 mm width, which means approximately 5° of the ultrasonic spread angle. (3) We conclude that the SH0 -mode plate wave velocity anisotropy of the propagation direction to the rolling direction was larger than that of the S0 -mode Lamb wave. To say in other words, evaluation precision could be improved by using the narrowed sensor coil to make the ultrasonic beam with the narrow ultrasonic spread angle. (4) To con®rm the value of a narrowed beam for theSH0 mode plate wave, we tested it using a few EMATs, which have di€erent sensor coil widths. It was powerful technique for improvement of the evaluation precision of the material more than the EMAT for the S0 -mode Lamb wave. We could demonstrate that the application to material property evaluations using the SH0 -mode plate wave could be improved by considering the proper speci®cations of the EMAT for the SH0 -mode plate wave.

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