1

Two-dimensional analysis of

subharmonic ultrasound generation at

closed cracks by damped double nodes

Kazushi YAMANAKA, Yohei SHINTAKU,

Miyuki OGUMA and Yoshikazu OHARA

Tohoku University (Japan)

2

Content of this talk

• Background

“Why high depth resolution is required for

nonlinear imaging methods”

• New simulation method for nonlinear ultrasound

• Effect of amplitude, incidence angle and residual

stress

• Comparison with Experiments

• Conclusion

3

Background

If the crack is closed by

（residual stress, oxide film）

Important structures such as nuclear power plant is degraded by

cracks including

・Stress Corrosion Crack

（SCC)

・Fatigue Crack

Problem

Open Region

O

A

B Closed Region

Weld Parts

Underestimation of crack

SCC in

stainless steel

SCC in

Ni-based alloy

To solve this, accurate crack depth

measurement is essential.

4

Report on Primary Loop

Recirculation piping

Feb 26, 2003 Japanese

Government organization

６－１ p13.

『UT：1.0mm ⇒

Real depth: 8.5mm』

Onagawa No1 Plant

Miyagi (near Sendai)

Real depth (mm)

Est

imat

ed d

epth

by U

T (

mm

)

Example of crack sizing error in atomic power

plant in Japan (worst case)

5

Nonlinear Ultrasound

Closed Crack

ω

Small Amplitude

Underestimation

Overlook

Transmission

ω

Large Amplitude

ω

Superharmonics ＜ Subharmonics

Closed Cracks ,

Transducer, Couplant Closed Cracks

Possible to detect closed cracks

Generation of Nonlinear Ultrasound

Superharmonics 2ω, 3ω,・・・ [1]

Subharmonics ω/2, ω/3,・・・ [2-4]

[1] O. Buck, et al., Appl. Phys. Lett. (1978).

[2] I. Solodov, et al., Acoust. Phys. (1993).

[3] K. Yamanaka, et al., JJAP. (2004).

[4] Y. Ohara, et al., Ultrasonics. (2006).

Contact Vibration

(Opening and Closing)

Crack Plane

Source

Accurate imaging of

closed cracks with

high selectivity

Selectivity of closed cracks

6

1D oscillator model

・K. Yamanaka, T. Mihara, T. Tsuji, Jpn. J.

Appl. Phys. 43 (2004) 3082.

・Y. Ohara, T. Mihara, K. Yamanaka,

Ultrasonics 44 (2006) 194.

・R. Sasaki, T. Ogata, Y. Ohara, T. Mihara, K.

Yamanaka, Jpn. J. Appl. Phys. 44 (2005) 4389.

0 5 10 15

-10

0

10

Crack position x / �Ð

mF

orce f

(arb

.Un

it.)

COD )sin)(( tatx�̀��

0 5 10 15

-10

0

10

0 5 10 15

-10

0

10

Crack position x / �Ð

mF

orce f

(arb

.Un

it.)

COD )sin)(( tatx�̀���̀��

Configuration of

crack and probe can

not be expressed in

detail.

Short burst of even

a few cycles can

show subharmonics.

m k

Time t, a.u.

Dis

pla

cem

ent

of

mas

s

7

Practical

thick

specimen

(40 mm)

Subharmonic Phased Array for Crack Evaluation (SPACE)

8

Experiments

on a closed

fatigue crack

in Al alloy

(μs) Time

②tip diffraction echo

20

–30

–20

–10

0

15

Load:p

Am

pli

tud

e

①lateral wave ③reflection echo from slit

③

②

①

P

0kgf

150kgf

0kgf

(⊿ｐ＝10kgf)

(⊿ｐ＝-10kgf)

Mag.Mag.

10 m

Crack Tip

9

Image of open and closed crack in aluminum alloy （A7075）

5mm

Tip Tip

Back 5mm

Tip Tip

Sample B （Kmax=14, Kmin=2

kgf/mm3/2）

Sample A（Kmax=17, Kmin=2

kgf/mm3/2）

f/2 f

f/2 f

Back

>

Open

f/2 f

<

Closed

f/2 f

10

f/2 f

5mm

A

112 MPa

A

B

84 MPa

A

B

C

29 MPa

Dependence of Crack Images on Crack Closure Stress

29MPa

(2kN)

Load

f/2 f 84MPa

(9kN)

f/2 f 112MPa

(12kN)

・Closure stress of more than 100MPa

・The boundary between the open and closed region at B became ambiguous →The linear scattering source diminished

・ Imaging of the change in crack state with varying closure

stress

・ Subharmonic images always gave an accurate crack depth

・Imaging of various parts of crack as well as crack tip 8.9

8.9

8.9

6.8

3.1

(a) (b)

(c) (d)

(e) (f)

(g)

11

Measurement accuracy of crack depth measured with SPACE

Y. Ohara, T. Mihara, R. Sasaki, T. Ogata, S. Yamamoto, Y. Kishimoto, K. Yamanaka,

Appl. Phys. Lett. 90 (2007) 011902.

0 10 200

10

20

True crack depth [mm]

Mea

su

red

cra

ck d

ep

th [m

m]

Fatigue crack(A7075)

f f/2Fatigue crack (SUS316L)

SCC1 (SUS304)

SCC2 (SUS304)

112 MPa

29 MPa

84 MPa

Fundamental images → Underestimation

Subharmonic images → Accurate crack depths with a measurement accuracy

of approximately 1 mm

Y. Ohara, S. Yamamoto, T. Mihara, K. Yamanaka, Jpn. J. Appl. Phys., 47 (2008) 3908.

12

Damped double node (DDN) model

to reproduce subharmonic generation

in bulk material

Applied to finite difference time

domain (FDTD) simulation

New simulation method for

nonlinear ultrasound

13

43

4

w

w

3T

3T1T

3T

w

3T

w

w

1T

1T

1T

u

u

5T

u

5T

u

5T

z

w

w

3T

3T1T

3T

w

3T

w

w

1T

1T

1T

u

u

5T

u

5T

u

5T

i

w

w

3T

3T1T

3T

w

3T

w

w

1T

1T

1T

u

u

5T

u

5T

u

5T

2

1i

2

1i

2

1j

j

2

1j

x

Finite difference time domain (FDTD) method

,53

x

T

z

T

t

w

z

T

x

T

t

u 51

,13111

z

wc

x

uc

t

T ,3313

3

z

wc

x

uc

t

T

x

w

z

uc

t

T 55

5

x

zAssuming cubic crystal and Uy=0, we

apply Hooke’s law

And Newton’s second law, alternatively

For example, particle velocity w,

k

ji

k

ji

k

ji

k

ji

kk

ji

TTx

t

TTz

tww

ji

,5,15

2

1,

2

13

2

1,

2

131

,2

1,

2

1

The density of particle velocity

node at free boundary is made

proportional to the number of the

stress node connected to it.

(M. Sato, 2007)

unit

cell

14

Closed crack Open crack

kkk

ji jijiwww

,2

1,

2

1

1

,2

1

0w

)(21,

2

1,

2

1

2

1,

2

13,

2

1

1

,2

1

k

ji

k

ji

k

jiPL

k

ji

k

jiwwTVww

Opening

Closing

2

1

2

1,

2

132

1

2

1,

2

132

1

32

1 k

ji

k

ji

k

M TTT

damping

thM TT3

5T

3T

w

u

5T

3Tu

5T

j

1T

1T

1j

w

w

1j

2

1j

2

1j

5T

3T

w

u

5T

3Tu

5T

1T

1T

w

w

5T

3T

w

u

5T

3Tu

5T

1T

1T

w

w

1i i 1i2

1i

2

1i 1i

5T 5T

3T

w w

u u

5T

3Tu

5T

j

1T

1T

1j

5T

u

u

5T

w

w

w

w w

i 1i

1j

2

1j

2

1j

2

1i

2

1i

ww

u

1T

3T1T

3T

1T

3T

ww

Damped double node (DDN) model

Criteria of crack

opening/closing

・If T3M ≦ Tth, the crack continues to be

closed state.

・If T3M > Tth, the crack becomes open.

・If ⊿w>0, the crack continues to be open.

・If ⊿w≦0, the crack becomes closed.

The grid is separated to dual nodes, and the

viscous damping is proportional to the particle

velocity difference between w- and w+.

Crack plane has normal grid which

is the same as no defect area.

Tensile stress

Crack opening displacement

100 MPa

15

Effect of amplitude, incidence

angle and residual stress

16

Normal incidence

# of element 32 cntered at (0, 150) Frequency 8MHz; time step 2 ns Wavelength λ=0.69 mm; node step 0.02mm Crack depth d= 3mm ; d/λ=4.4 (deep crack) Incident angle 0° Particle displacement 20, 40 and 80 nm Focus position (150, 1200) Compression stress 100 MPa

z O

crack length : 200

x

32 element array

20μm/grid

Comparison of amplitude

Amplitude 20nm

Amplitude 80nm

17

Wgain=-1.86=40nm Γ=0.7

0 1 2

0

100

200

0 1 2

0

100

0 1 2–200

–100

0

Time t (us)

CO

D Δ

W (

nm

)W

+ (

nm

)W

– (

nm

)

0 10 20 30 400

50

100

0 10 20 30 400

50

100

0 10 20 30 40

50

100

Frequency f (MHz)

CO

D (

dB

)W

+ (

dB

)W

–(d

B)

Low amplitude input wave 40nm

No COD

Spectral analysis of small amplitude incident wave

18

0 1 2

0

100

200

0 1 2

0

100

0 1 2–200

–100

0

Time t (us)

CO

D Δ

W (

nm

)W

+ (

nm

)W

– (

nm

)

0 10 20 30 400

50

100

0 10 20 30 400

50

100

0 10 20 30 40

50

100

Frequency f (MHz)

CO

D (

dB

)W

+ (

dB

)W

–(d

B)

Wgain=-3.68=80nm Γ=0.7

Clear subharmonics in

transmission side

High amplitude input wave 80nm

Finite COD

Spectral analysis of large amplitude incident wave

19

Generation of subharmonic waves for w+.

We succeeded in reproducing the subharmonic waves by the 2D

model of closed cracks with compression residual stress.

Wavelet analysis of large amplitude incident wave

0 25 50 75 100 125 150 175

0

20

40

60

80

100

120

A B C D E

0 1 2

0

100

200

0 1 2

0

100

0 1 2–200

–100

0

Time t (μ s)

CO

D Δ

W (

nm

)W

+ (

nm

)W

– (

nm

)

80nm

Time t ( s)

CO

D Δ

w

(nm

) w

+ (

nm

) w

- (

nm

)

Incident side

COD

Transmit side

f /2

Δw

w +

w -

20

0 1 2

0

100

200

0 1 2

0

100

0 1 2–200

–100

0

Time t (us)

CO

D Δ

W (

nm

)W

+ (

nm

)W

– (

nm

)

Wgain=40nm Γ=0.7 No compression stress Tzz=0

COD when 40nm input

Clear superharmonics

with unclear subharmonics

0 10 20 30 400

50

100

0 10 20 30 400

50

100

0 10 20 30 40

50

100

Frequency f (MHz)

CO

D (

dB

)W

+ (

dB

)W

–(d

B)

Envelop detection,

mechanical diode (DC effect)

Effect of compression stress

40nm incident wave

21

Oblique incidence

21

z O

crack length : 200

x

20μm/grid

Amplitude dependence

Amp 20nm

Amp 80nm

Amp 160nm

Closed crack

at the center

# of element 32 cntered at (0, 150) Frequency 8MHz; time step 2 ns Wavelength λ=0.69 mm; node step 0.02mm Crack depth d= 3mm ; d/λ=4.4 Incident angle 56.3° Particle displacement 20nm, 80nm, 160nm Focus position (150, 1200) Compression stress 100 MPa

22

0 1 2 3

0

100

200

0 1 2 3

0

100

0 1 2 3–200

–100

0

Time t (us)

CO

D Δ

W (

nm

)W

+ (

nm

)W

– (

nm

)

Time t

(μs)

CO

D Δ

w

(nm

)

w +

(nm

)

w -

(nm

)

0 25 50 75 100 125 150 175

0

20

40

60

80

100

120

Time t (µs) 0 3.2

f /2

Δw

w +

w -

Simulation of oblique incidence

Subharmonic components were observed more clearly in

the oblique incidence than in the normal incidence.

23

0 1 2

–0.0001

0

0.0001

0.0002

0 1 2

–0.0001

0

0.0001

0.0002

0 1 2

–0.0001

0

0.0001

0.0002

Time t (μ s)

T3M

T31+

T31–

0 1 2

–0.0001

0

0.0001

0.0002

0 1 2

–0.0001

0

0.0001

0.0002

0 1 2

–0.0001

0

0.0001

0.0002

Time t (μ s)

T3M

T31+

T31–

(b) γ=0.7 (a) γ=0

Effect of damping on the stress waveform High frequency noise disturbs clear

observation of superharmonics Damping enhances subharmonics

24 (b) γ=0.7

0 25 50 75 100 125 150 175

0

20

40

60

80

100

120

0 25 50 75 100 125 150 175

0

20

40

60

80

100

120

(a) γ=0

f /2 is

enhanced

by the

damping

Freq

ue

ncy

f

(MH

z)

4

Time t (µs) Time t (µs) 0 0 3.2 3.2

8 f f/2 4

8

8 f f/2 4

8

8 f f/2 4

8

Δw

w +

w -

Δw

w +

w -

Effect of damping

25

Comparison with experiments

26

33

192 [mm]

10mm

Side A

Side B Weld

Weld

144

200

Weld

SCC

SCC was formed in a sensitized type 304 stainless steel pipe in

high temperature pressurized water (280℃) in an autoclave.

After that, we cut the pipe, and then, machined it for ultrasonic

testing.

SCC (type 304 stainless steel, high temperature pressurized water)

SCC

27

10mm

1 32

Focal point

(-12.6, 15.0)

FA SA

B

10mm

1 32 1 32

The SCC was not imaged because of strong linear scattering at various coarse grains.

Measurement condition Imaging results PZT array transducer

(5 MHz, element pitch 0.5 mm,

32 elements)

Input wave：7MHz, 3cylcle, 150V

The SCC was clearly imaged.

(Focal point: -12.6、15.0)

A

To confirm the subharmonic generation, we performed the frequency analysis of the sum of phase-shifted waves at response A and B.

7.0 ± 1.2

MHz

3.5 ±1.2

MHz

Imaging of SCC by single array SPACE

Single Array SPACE is useful in imaging closed crack with high selectivity.

Ohara, Horinouchi, Yamanaka et al,

Session 17 (Wed), 27 (Thu), QNDE2011

28

0 1 2 3 4 5 6 7 8 9 100

1

7 8 9 10–0.1

0

0.1Sum of phase-shifted waves at response A

Volt

age

(V)

Time ( s) Frequency（MHz）

ampli

tude

FA

7 10

7.0

(f)

3.5

(f/2)

Time ( s)

Fre

quen

cy (

MH

z)

(-15.5, 19.5)

wavelet

0

15

Phase-shifted waves：Waveform distortion was

observed around 8.7 s.

FFT：Peak at subharmonic frequency (3.5MHz) was

clearly observed.

Wavelet transformation：Generation of subharmonic

component separated from fundamental component

was confirmed.

SA

Frequency analysis of sum of phase-shifted waves at response A

29

Comparison between experiment and simulation

0 25 50 75 100 125 150 175

0

20

40

60

80

100

120

0 25 50 75 100 125 150 175

0

20

40

60

80

100

120

8.5 11.7 10 7

f (7MHz)

f/2

(3.5 MHz) freq

uen

cy (

MH

z)

0

15

time ( s) time ( s)

f

(8 MHz)

f/2

(4 MHz) freq

uen

cy (

MH

z)

0

15

0 3.2 3.2 0 time ( s) time ( s)

W+ W+

Response A Response B

Normal

incidence

Oblique

incidence

Experiment

Simulation

・In experiment, the subharmonic generation was not local.

・At this point, the result in oblique incidence was closer to experimental results than that in

normal incidence.

・We succeeded in reproducing the subharmonic generation similar to experimental observation.

30

Conclusion • We introduced our works on a nonlinear imaging method,

subharmonic phased array for crack evaluation (SPACE) to significantly improve the depth measurement accuracy of closed cracks.

• We developed a two-dimensional model to reproduce subharmonic generation at closed cracks using damped double nodes (DDNs).

• Numerical simulation using finite-difference time-domain (FDTD) method was performed and successfully compared with experimental waveforms and time–frequency wavelet analysis.

• The DDN model together with SPACE will enhance the testing condition and probes to significantly improve the depth measurement accuracy of dangerous cracks.

• It will contribute to enhance safety of atomic power plants and other important structures.

31

Thank you

For your attention