Determination of Cyclic Tension Fatigue of Al 4Cu 1Mg Alloy Using Ultrasonic Shear Waves

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Determination of Cyclic-Tension Fatigue of Al-4Cu-1Mg Alloy

Using Ultrasonic Shear Waves

H. Yamagishi

1;*

_{and M. Fukuhara}

2 1

Central Research Institute, Toyama Industrial Technology Center, Takaoka 933-0981, Japan 2_{Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan}

Cyclic-tension fatigue of aluminum alloy, Al-4Cu-1Mg, was determined by an analysis of diﬀracted SH and SV waves, passing through the surface and reﬂecting the bottom of the specimen, respectively. The propagation time of SH waves and the internal friction of SV waves begins to decrease and increase from fatigue damage ratio (¼N=Nf) of 0.5, respectively, suggesting noticeable increase of movable dislocation.

From the analysis of SH waves along the direction to the loading axis, the damping ratio decreases and the phase advances with increasing of the degree of fatigue. Assumed from SH wave ﬂux model, the former behavior correlates well with increasing of the residual stress by cyclic tension. The latter may be characterized by viscoelasticity. These results show that ultrasonic transmission method is useful probe for evaluation of fatigue in aluminum alloy. [doi:10.2320/matertrans.48.550]

(Received November 15, 2006; Accepted January 22, 2007; Published February 25, 2007)

Keywords: fatigue, aluminum alloy, cyclic tension, nondestructive inspection, acoustic characteristics

1. Introduction

When engineering metals are subjected to ﬂuctuating loads in service, they are liable to fracture by fatigue, the most

common of all causes of engineering failure.1–3)Thus there

have been extensive eﬀorts to relate observed data using

X-ray diﬀraction,4–6)positron annihilation,7,8)laser diﬀusion,9)

ultrasonic transmission,10–12) acoustic emission,13,14)

hard-ness15) methods to evaluate the actual fatigue level in

materials. However, conventional methods are not necessa-rily satisfactory to predict the damage of fatigued materials for ﬁeld application.

To develop a nondestructive inspection technique for fatigue, we investigated acoustic characteristics of an aluminum alloy, using horizontally polarized shear wave and vertically polarized shear wave (here after referred to as SH and SV wave, respectively). The shear waves are known as sensitive to even atomic or molecule level material

change.16)_{SH wave has polarized plane which is parallel to}

the interface when it propagates with the angle of incident, that is, the transmission wave is easy to propagate under the target surface. And as further advantage, it does not transform its wave mode from shear wave to longitudinal wave, just

remains shear wave mode.17) However, having some great

advantages, this method needs adequate condition of meas-uring such as surface roughness, pressing load, load-holding

time18)_{and so on, because it is contact method. Utilizing the}

advantage of this wave, we must keep these precautions in mind.

In this study, we selected the aluminum alloy as fatigued material, because it is the most commonly used light-metal that fatigue evaluation has not fully developed.

2. Experimental

2.1 Material

The material used in this study was Al-4Cu-1Mg alloy

(JIS-A2024, temper T3) supplied by Corus Aluminum Walzprodukte GmbH. The chemical composition and me-chanical prosperities of material are shown in Table 1 and Table 2, respectively. The specimen conﬁguration, which originally designed for fatigue tests, is shown in Fig. 1. The specimen is a ﬂat-rolled bar with a narrow part for break.

2.2 Experimental methods

Specimen was subjected to a cycle stress amplitude of 233 MPa under load control with 30 Hz frequency in monaxial tensile mode at room temperature, to apply fatigue damage. The fatigue tests were interrupted at various cycles to obtain fatigued specimens having diﬀerent damages. After

speciﬁed number of cycles,N=Nf ¼0, 0.03, 0.16, 0.32, 0.48,

0.64, 0.80, 0.96 (N: number of cycles,Nf: number of cycles to

failure =3:12105_{), the waveform deviation between the}

two contact lines in the center sections of the specimens (Fig. 1) was measured by the use of a bridge with an ultrasonic transmitter/receiver set (Fig. 2). Ultrasonic wave pattern analysis was made by diagnosis and analyzing apparatus (Toshiba Tungaloy).

The apparatus and a block diagram for the measurement are presented schematically in Fig. 2. Facing narrow-band

width SH wave transducers, with probe area of 6mm

6mm, were used as the transmitter and the receiver. The two

facing transducers in the bridge were arranged by a ditch of 10 mm width. The incident and receiving angles of the SH

waves were ﬁxed at 21 _{which gives maximum receiving}

amplitude derived from following formula:

Sin1 V1

¼Sin2

V2

ð1Þ

whereSin1 andSin2are incident and refracting angles for

boundary normal, respectively. V1 and V2 are shear wave

velocities of contacting material of piezoelectric element and

pure aluminum, respectively.2is designed for surfacewave,

i.e., critical angle 90. In order to prevent propagation loss

due to high frequency and signal broading due to low frequency, a frequency of 5 MHz was selected as center

*_{Corresponding author, E-mail: [email protected]}

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frequency of transducer. The transducer was contacted at one edge of specimen under a pressure of 3.1 MPa, by water-free

naphthenic hydrocarbon oil (Tungsonic Oil H19)).

[image:2.595.50.549.85.124.2]

Figure 3 shows propagation behavior of SH wave in thin plate using this probe. This result shows that the effective depth of SH wave is up to around 7.5 mm and the influence of reflected wave at the bottom surface of specimen, A2024T3 thickness 6.0 mm, is confirmed around 7–8th wavelet.

Table 2 Mechanical prosperities of material.

Temper 0.2% Proof stress [MPa]

Tensile strength [MPa]

Elongation [%]

T3 307 446 17.2

Direction A

Direction B Loading axis

(=Rolling direction)

t6

Fig. 1 Conﬁguration of the tension test specimen (dimension in mm).

Specimen (A2024T3 t6)

AD Converter

Wave Memory

Filter

Peak Detector

Spectrum–Analyzer

Receiver

Transmitter

SH sensor

(Specimen)

(SH sensor)

(Pressing Jig)

Pressing

Jig

Weight

Pulser / Receiver

10

22

6 (mm)

Fig. 2 Schematic diagram for measurement.

Scale 0.500 us/div Start Time 11.000 us Scale 0.500 us/div Start Time 11.000 us Scale 0.500 us/div Start Time 11.000 us

A2017

A2024T3

diffracted wave

reflected wave

reflected wave _{reflected wave}

diffracted wave

reflected wave

multiple

reflected wave multiple reflected wave

thickness: 7.5 mm

thickness: 6.0 mm

thickness: 5.0 mm

thickness: 4.0 mm

thickness: 3.0 mm

thickness: 6.0 mm

thickness: 3.0 mm specimen

for test

Fig. 3 Propagation behavior of SH wave in thin plate. Table 1 Chemical composition of material [mass%].

Material Si Fe Cu Mn Mg Cr Zn Ti Ti+Zr Others

each

[image:2.595.51.284.130.239.2] [image:2.595.305.548.172.650.2] [image:2.595.57.283.294.769.2]

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Therefore we decided to analyze the waveform until 3rd wavelet to avoid the influence of reflected wave. In this case it can be estimated the effective depth of SH wave is up to around 4 mm.

After application of a given cyclic load, we analyzed the wave patterns of SH waves propagated through the fatigued specimens. Figure 4 and Figure 5 show transitions of receiving waveform with fatigue damage ratio in direction A and its superposition corrected to be equivalent maximum amplitude, respectively. Apparently, it is hard to ﬁnd

informative shift except for the time advance of N=Nf ¼

0:80. The transition of direction B is much the same to

direction A. Since the receiving waveform varies delicately with the degree of fatigue, we extracted the speciﬁc parameters, propagation time, damping ratio and phase from receiving waveform for more advanced analysis.

Figure 6 shows a representative SH waveform pattern. We measured the position of the 1st wavelet as propagation time

T1. The damping ratios,P1=P2andS1=S2, were calculated

from amplitudes and its integrated area, respectively. And we use conveniently the waveform from 1st to 3rd wavelets for fast fourier transform (FFT) to determine the main frequency

f and phase’ at f. The phase’ was calculated using the

following formula:

¼arctan Imð!Þ Reð!Þ

ð2Þ

where!¼2f.Reð!ÞandImð!Þare the real and imaginary

parts of wave passing through the sample.

The internal friction at intersection of parallel and Scale 0.500 us/div Start Time 11.000 us

Scale 0.500 us/div Start Time 11.000 us

N/N

f

: 0

N/N

_f

: 0.32

N/N

f

: 0.64

N/N

_f

: 0.96

Fig. 4 Transition of receiving waveform with fatigue damage ratio.

12.1 12.2 12.3 12.4 12.5 12.6 12.7

−0.6

−0.4

−0.2 0.0 0.2 0.4 0.6

Amplitude

,

a

/

V

Propagation time, t/µs

12 13 14

−0.6

−0.4

−0.2 0.0 0.2 0.4 0.6

Amplitude

,

a

/

V

Propagation time, t/ _µs

Direction A N/Nf: 0.00

N/N_f: 0.03 N/Nf: 0.16

N/Nf: 0.32

N/Nf: 0.48

N/N_f: 0.64 N/Nf: 0.80

N/Nf: 0.96

Direction A N/Nf: 0.00

N/Nf: 0.03

N/Nf: 0.16

N/Nf: 0.32

N/Nf: 0.48

N/Nf: 0.64

N/Nf: 0.80

N/Nf: 0.96

[image:3.595.98.242.65.696.2] [image:3.595.316.542.75.434.2]

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perpendicular directions to the loading axis,i.e., direction A and B, respectively shown in Fig. 1, was measured by a SV wave with frequency of 5 MHz at room temperature. The

logarithmic dampingwas determined by the form:

¼ln A1

A2

ð3Þ

whereA1andA2are 1st and 2nd amplitudes of the receiving

waves, respectively.

The internal friction Qs1 is deﬁned by the following

formula:

Qs1 ¼

Vsln

_A

1 A2

2l f ð4Þ

whereVsis the sound velocity of SV wave.lis the thickness

of the specimen and f is the center frequency.

3. Results and Discussion

The relationships of propagation time and internal friction

for fatigue damage ratio N=Nf are shown in Fig. 7. The

propagation times for parallel and perpendicular directions to the loading axis increase smoothly, respectively, and then,

from aroundN=Nf ¼0:5, decreases absurdly up toN=Nf ¼

0:8 with increasing of N=Nf. For both directions, a

sub-sequent beading makes the propagation time lengthen. On the other hand, the internal friction does not change distinctly up

to around N=Nf ¼0:5, but noticeably increases for beading

overN=Nf ¼0:5. Therefore, the sharp change in propagation

time correlates with the rapid increase in internal friction. Here we consider a relation between fatigued degree and internal friction in terms of energy consumption by

disloca-tion modisloca-tion. Koehler,20)Granato and Lu¨cke21)produced KGL

model expressing dislocation motion by assuming elastic

string model under the condition of viscoelasticity. Accord-ing to KGL model, the internal friction is leaded by the

following equations:22)

A@ 2_u @t2 þB

@u

@t C @2_u

@x2 ¼b ð5Þ

whereu,A,B,Candbare displacement, eﬀective mass per

unit length, viscous resistance, line tension and Burgers vector, respectively. The solution of normal vibration is expressed in the following form:

u¼4b0

A sin

Lx

_exp½_i_ð_!t

0Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð!2

0!2Þ

2_{þ ð}_!d_Þ2

q ;

!0¼ L

ffiffiffiffi

C A

r

; tan0¼ !d

ð!2 0!2Þ

; d¼B

A

! ð6Þ

whereL is transposition length. On the assumption that all

transposition lengths are the same, the distortion"dcaused by

dislocation displacement is calculated by the following form:

"d ¼

b L

ZL

0

udx ð7Þ

where is dislocation density. Substituting eq. (5) into

eq. (6), the internal friction is leaded by following form:

Q10L

2

w2₀wd

½ðw2₀w2_Þ2_{þ ð}_wd_Þ2; 0¼

8b2 3_CÞ ð8Þ

Scale 0.500 us/div Start Time 11.000 us

T1

P1

P2

Range of FFT

S2

S1

Fig. 6 Example of a receiving waveform using SH sensor.

0.0 0.2 0.4 0.6 0.8 1.0

0.006 0.007 0.008 0.009 0.010 0.011

Fatigue damage ratio, N/N_f

Inter

nal fr

iction,

∆

Qs

−

1

12.185 12.190 12.195 12.200 12.205 12.210

Propagation time

,

t

/

Propagation time

,

t

/

s

µ

Direction A

Direction B

12.365 12.370 12.375 12.380 12.385 12.390

s

µ

[image:4.595.54.284.69.329.2] [image:4.595.314.541.71.398.2]

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where is shear modulus. As from N=Nf ¼0:5 of Fig. 7

according to the internal friction form of eq. (8), the rapid increasing point in internal friction suggests the drastic change of material in terms of fatigue. Thus it seems that measurement of propagation time and/or internal friction become a probe for evaluation of fatigue.

The relationships between fatigue damage ratioN=Nf and

damping ratios,P1=P2andS1=S2, are shown in Fig. 8. The

damping ratios of parallel direction to the loading axis,

direction A, decrease inversely with the increase ofN=Nf. In

particular, the integrated area ratio of amplitude S1=S2

correlates well with the fatigue damage ratio N=Nf in

comparison with damping ratioP1=P2.

Here we consider a relation between fatigued degree and damping ratio in terms of acoustoelasticity. If a

two-dimen-sional x-y plate to which a stress is applied in the xandy

directions of a cartesian coordinate system (x;y;z) (Fig. 9),

where a residual stress exists in the plate, the acoustic energy

ﬂow P of the SH waves propagating in the x-y plane at an

arbitrary anglefrom the norminal (i.e.,z) can be expressed

in the following form;23)

P¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

P2

xþP2y

q

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

2u xz

2

þ 1

2u yz

2

s

ð9Þ

¼tan1Px

Py

¼tan1 yz

xz

ð10Þ

xz¼

@u

@x; yz¼ @u

@y ð11Þ

u¼Aexp½iðxsinctþzÞ ð12Þ

wherePxandPyare energy ﬂux of direction x and direction

y, respectively.uis the displacement of SH wave to direction

z. xzand yzare shearing stress of directionzatxplane and y

plane, respectively. , A, and are shear modulus,

displacement amplitude, number of waves, element of wave number to direction z, respectively. As can be seen from decrease in damping of Fig. 8 (Direction A) and the energy ﬂux angle of eq. (10) and considering the simulation result that the energy ﬂow of SH wave bends to more interior as the

deeper it propagates,24)it is clear that the incident waves shift

to the specimen surface with increasing residual stress as the

degree of fatigue,i.e., the crystal lattice distortion by tensile

stress. So it is summarized that the damping ratio S1=S2

decreases with the increase of cyclic-tension fatigue. As for

the increase of S1=S2 (Direction A) at N=Nf ¼0:64, it is

assumed to be due to noticeable increase of movable dislocation indicated by Fig. 7. Since a movable dislocation consumes energy of SH wave, the interior wave propagating for a longer path is subject to attenuate. So it is summarized

that the damping ratio S1=S2 increases with the increase of

movable dislocation. However, compared to the inﬂuence of residual stress, tensile stress makes the incident waves shift to specimen surface, energy decay by movable dislocation does not seem to carry much weight on damping ratio.

Since interest results of viscoelasticity of aluminum alloy are evaluated from the phase modulation in complex waves,

the relationship between fatigue damage ratio N=Nf and

phaseis shown in Fig. 10. Although the center frequency of

receiving waves does not shift throughout the test, i.e.,

maintains 5.4 MHz for both directions, the phase advances. This means an increase in viscoelasticity during progressive fatigue such as thermal degradation of resin reported by Ref. 25 using SH wave. Although we should investigate why the phase advances with the increase of fatigue, this result is a note of high interest for the possibility of determination of metal’s viscoelasiticity using SH wave.

On the other hand, both damping ratio and phase modulation for perpendicular direction to the loading axis,

directionB, did not show simple characteristics in variation

of N=Nf. The path length for perpendicular direction to the

loading axis is considerably long against the specimen width, 22 mm nearly the specimen width 25 mm, so we must consider the eﬀects of stress gradient and its transition with

increasing of N=Nf. Inﬂecting of ultrasonic wave by stress

0.0 0.2 0.4 0.6 0.8 1.0

1.10 1.11 1.12 1.13 1.14 1.15

Direction A P1/P2 S1/S2

Damping r

atio

, P1/P2

1.36 1.38 1.40 1.42

Damping r

atio

, S1/S2

0.0 0.2 0.4 0.6 0.8 1.0

1.12 1.13 1.14 1.15 1.16 1.17 1.18

Direction B P1/P2 S1/S2

Fatigue damage ratio, N/Nf

Damping r

atio

, P1/P2

1.38 1.40 1.42 1.44 1.46 1.48

Damping r

atio

, S1/S2

Fig. 8 Relationship between fatigue damage ratio and damping ratios.

<

Px

>

<

Py

>

<

P

>

α

Z

y

x

[image:5.595.336.510.71.221.2] [image:5.595.55.284.74.386.2]

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gradient has reported by Ref. 23, 26 and eq. (10). Therefore, it may be assumed that the cyclic transitions of damping ratio and phase modulation for perpendicular direction to the loading axis are due primarily to cyclic changes of the traveling path by inﬂecting due to positioning error of every specimen mounting, process of fatigue itself and so on. In the following paper, we must investigate minutely about this subject using X-ray diﬀraction (XRD) and transmission electron micrography (TEM).

4. Conclusion

In this study, we analyzed the wave patterns of SH and SV waves propagating through cyclic-tension fatigued alloy, Al-4Cu-1Mg, in view of viscoelasticity and acoustoelasticity. The propagation time both parallel and perpendicular direction to the applied stress and the internal friction at the center of specimen decreases and increases as the number of

cycles increases over N=Nf ¼0:5, respectively. This

sug-gests increasing in movable dislocations, i.e., the drastic

change of material in terms of fatigue. Furthermore the damping ratio and the phase decreases and advances with increasing fatigue, respectively. Assumed from SH wave ﬂux model, the residual stress ﬁeld applied by cyclic tension makes the damping ratio decrease. The advance of the phase indicates the possibility of fatigue evaluation in terms of viscoelasticity. On the other hand, both damping ratio and phase modulation along the perpendicular direction to the applied stress, the path length is nearly the specimen width,

did not show simple characteristics in variation ofN=Nf. This

would be estimated due to inﬂection of ultrasonic wave attributed to stress gradient and its transition caused by positioning error of every specimen mounting or process of fatigue itself and so on. As far as we detect acoustic characteristics along the direction to the loading axis, the ultrasonic SH wave method is suitable probe for fatigue evaluation in aluminum alloy.

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0.0 0.2 0.4 0.6 0.8 1.0 6

8 10 12

Phase

,

φ

/r

ad

Direction B

Phase

,