Residual stress determination in aluminium using electromagnetic acoustic transducers

Residual stress determination in aluminium using electromagnetic acoustic transducers AV. CLARK JR and J.C. MOULDER

The residual stresses in a shrink-fit specimen were measured ultrasonically, using shear-horizonal (SH) waves transmitted and received by noncontacting electromagnetic acoustic transducers. The presence of stress induces a small change in the velocity of the SH-waves. The difference in velocities of orthogonally polarized SH-waves (acoustic birefringence) was measured with a simple time-interval averaging system; this velocity difference can be related to the difference of principal stresses. The presence of material anisotropy (texture) in the specimen also causes relative velocity changes comparable to stressinduced changes. A simple method was used to remove the anisotropydependent component of the total relative velocity change, thereby allowing a determination of residual stress. This method consisted of measuring the birefringence in unstressed reference specimens and subtracting it from the birefringence measured at stressed locations. For the specimen used here, good agreement between theoretical and experimental values of stress was obtained. KEYWORDS:

ultrasonics,

acoustic

birefringence,

Introduction Advances in accuracy and precision of relative velocity measurements have now made it possible to consider the use of ultrasonics as a stress measurement tool. The presence of stress causes a small but measurable change in ultrasonic velocity. Several theories have been developed that allow components of stress to be calculated from velocity measurements using longitudinal and/or shear wavesL*2,3. Measurements using longitudinal waves have probably been the most popular, because of the ease of coupling the sound from the transducer into the specimen. Generally, longitudinal-wave measurements are analogous to measurements of the isopachics in photoelasticity; that is, the change in longitudinal wave velocity is proportional to the sum of the principal stresses1,2.3. Ref. 4 summarizes many such measurements on specimens such as centre- and edgecracked panels. and a panel with a central hole, all in far-field tension. Another application is the use of longitudinal waves to measure the axial stress in threaded fasteners5. Measurement of shear-wave velocities corresponds to the use of isochromatics in photoelasticity. That is. assuming the material is sufficiently isotropic, the The

authors

of Standards,

are at the

Fracture

Boulder,

Colorado

and

Deformation

80303,

USA.

Division, Paper

National

received

19

Bureau June

1985.

0041-624X/85/06025347 ULTRASONICS.

NOVEMBER

1985

$03.00

stress

measurement

difference in the velocities of shear waves polarized along the principal stress directions gives the difference of principal stresses:

VI - v2 = a(q - u2) = B

(1)

M(Vt + Vd

Here V, is the velocity of the wave polarized along the or-direction and a is the acoustoelastic constant. The normalized difference in shear wave velocities is called the acoustic birefringence, B. Since the waves travel through the same thickness of material. we also have 2(t, - r2)/(f, + t,) = -B. where t, is the transit time of a wave polarized along the or-direction. Crecraft6 was one of the first to attempt to use shear waves for stress measurement using the acoustic birefringence method to measure residual stress in a deformed bar and ring HSU’ measured the difference of radial and hoop stresses in a disc under diametral compression. In both Refs 6 and 7 it was recognized that any material anisotropy due to such fabrication processes as rolling and extrusion could also cause birefringence, even in the absence of stress. For example. in rolled aluminium alloy plates, it is typically found that pure-mode shear-horizontal waves propagate only when polarized parallel or perpendicular to the rolling direction. The velocities of these orthogonally polarized waves are different: the unstressed birefringence can typically be equivalent to the effect of about 100 MPa of stress. 0 1985

Butterworth

& Co (Publishers)

Ltd

253

Theories that explicitly account for the effect of material anisotropy on ultrasonic velocity have been developed recentlyz~3~8*9~‘o.These theories allow measurements to be made on specimens for which the principal stress directions are unknown a priori. For example. Okada measured the shear stress in a plate with a central hole in remote tension”. Clark et al. measured the shear stress around the crack tip of a fracture specimen, and obtained the stress intensity factor’*. Other measurements of technological importance have been reported by Frankel et al.r3 and also by Fukuoka et alr4. In Ret 13 the hoop stress in an autofrettaged cylinder was measured ultrasonically, with good agreement with x-ray measurements. In Ref. 14 the principal stress difference in a patch-welded disc was measured with the acoustic birefringence method and gave good agreement with destructive. strain-gauge measurements. All of the acoustic birefringence measurements described above used piezoelectric shear-wave transducers. To transmit ultrasound from the transducer into the specimen a thin layer of couplant (usually viscous) is interposed. Variations in the thickness of this layer are sources of error since they cause changes in the apparent arrival time of the shear wave. Since the acoustoelastic constant, a, in (1) is small (typically of the order of l0-5 MPa-* for aluminium a110ys’5~‘6. small changes in arrival times can lead to large stress errors. Hence, care must be exercized to minimize couplant thickness variation, An attractive alternative is to use non-contacting electromagnetic acoustic transducers ( EMATs) which require no couplant. The advantages of EMATs have been demonstrated by Blessing et al.“. who showed that there was a marked improvement in precision when shear wave measurements were performed with EMATs rather than a contacting piezoelectric transducer. All of the above studies. whether conducted with longitudinal or shear waves, consider only the case of waves propagating normal to the plate surfaces. There are two other techniques which exploit the ability of EMATs to launch shear waves that are polarized parallel to the plate surfaces (SH-waves) and propagate at an angle to the plate normal The method of Thompson et al.‘* uses a low-frequency EMAT to fill a thin plate with SH-waves. The plate acts as a waveguide. and only the lowest-order mode is propagated. A fixture holds transmitting and receiving EMATs at a fixed separation. By orienting the fixture parallel and perpendicular to the direction of an applied uniaxial stress and measuring the difference in arrival times, the effect of the unstressed birefringence can be removed and the stress measured unambiguously. In the method of King and Fortunkor9. transmitting and receiving EMATs in a fixture are also used. However. the shear wave is now propagated into the plate over a range of angles of incidence relative to the plate normal and the arrival times of the resulting echoes (from the back surface of the plate) arc measured. This is done with the fixture axis oriented along both principal stress directions. The difference in arrival times is now a function of angle of incidence. By measuring arrival times at two (or more) angles of incidence at a given location, the unstressed bire-

254

fringence can be removed stress determined.

and the principal

shear

Recently. Blessing et al. proposed a shrink-fit ring specimen as a calibration standard for ultrasonic stress measurement’7~20. The specimen was assembled from ring and plug sections cut from an extruded rod of 2024-T35 1 aluminium. Extrusion induces anisotropy (caused by texture) that is approximately, if not exactly, transversely isotropic. The extrusion direction will be an axis of symmetry: a shear wave propagating in this direction (thickness dimension of the specimen) will have the same unstressed velocity for all polarization directions. The shrink-tit specimen contains a known residual stress state and can be used to correlate results of ultrasonic stress measurements with other stress measurement techniques. To this end Blessing et al. measured the changes in time-of-flight of shearhorizontal waves and qualitatively compared their results with the known stress state. For a transversely isotropic material. the stress-velocity relation is given by8

ve- vr -VO

tr - tfl to

(2)

a(ao - ur) = -

where Ve is the shear-wave velocity for a wave polarized in the 0 (tangential) direction, V, is the unstressed velocity, and t, is the time of flight for a wave polarized in the radial direction. Blessing and coworkers did not obtain the value of a, and so did not transform their time-of-night measurements to stress. More recently, Jackson*’ measured the stresses in this shrink-lit specimen using a dry-contact shear wave system. He obtained a value for the acoustoelastic constant and hence was able to compare the experimentally obtained stresses with the theoretical values. The agreement was good. In this paper, we report measurements of the velocity difference Vo - V,. for a shrink-fit specimen of a somewhat different design and use the independently determined acoustoelastic constant to obtain the principal shear stress u,g - a,. We then compare the results with theoretical predictions. Furthermore, we integrate the stress equilibrium equation (3) to obtain a, from measured values of u,g - a,; we then add the calculated Us to the measured a~ - a, to obtain the hoop stress.

Theory Ultrasonic

The shrink-fit specimen we used differs from that of Refs 17, 20 and 21 in several respects. Our specimen consists of a rolled plate of 25 mm thick 6061-T651 aluminium. The plate is a square, with 686 mm sides. and with a 100 mm hole bored in the centre. A stainless steel plug 102 mm in diameter was cooled with liquid nitrogen and inserted into the hole and then allowed to warm to room temperature. A rolled aluminium plate is usually slightly and typically displays orthotropic symmetry

ULTRASONICS.

anisotropic (nine

NOVEMBER

1985

independent second-order elastic constants). For the rolled 6061-T651 aluminium used in our experiments. a shear-horizontal wave polarized parallel to the rolling direction travels faster than a wave polarized transversely, for waves propagating through the plate thickness in the unstressed condition. This is also true for aluminium alloys such as 2024-T351 and 7075-T6: the rolling direction is the ‘fast axis’ and the transverse direction is the ‘slow axis’. Because of the slightly orthotropic symmetry. the stress-velocity relation is now more complicated than (1) and (2). However. when the material symmetry directions coincide with the principal stress axes..we have

aluminium. The simpler technique employed here requires an accuracy of4 x IO+ to achieve the same stress resolution in this aluminium alloy. To make the comparison between the off-axis technique and our conventional technique unambiguous, we have used the same transducers that were used by King and Fortunkorg. In this way. differences in stresses measured with the two techniques will not be due to differences in transducer design or differences in the ultrasonic gauge length. In fact we also used the same shrink-fit specimen that was used by King and Fortunko. Our time measurement system was, however, different from theirs. as will be explained later. Elasticity

where V, is the velocity in the unstressed state of a wave polarized in the r (radial) direction. The shear waves propagate in the thickness direction at normal incidence. Note the presence of the unstressed birefringence. B, = ( VS, - V,)/V,. This term arises because the shear modulus along the rolling direction differs (slightly) from the shear modulus transverse to the rolling direction. Consequently, to obtain a~ - a,. the unstressed hire fringence must be known or measured One method of dealing with this problem was developed by King and Fortunkorg. They discovered that if the SH-waves are made to propagate through the plate at an angle @ to the normal then (4) can be generalized to

ve- vr -

v,-

veo- vrO i vo

1

co2 @+ a(@)(oe - 0,)

(5)

Note that the birefringence and acoustoelastic constant both become functions of the angle of incidence. @. By measuring velocity differences at two (or more) values of @. the value of 00 - a, can be determined. Results of such experiments are reported in Ref. 19, with good agreement with theory. Another method of removing the unstressed birefringence is simply to measure it on unstressed reference specimens. making enough measurements average out any effects of specimen inhomogeneity.

NOVEMBER

1985

00 =

pi ,(R,A)2

_

P + WY1

1l

(6) pi % =

[(R/A)2 - l]

]I - (R/r)*

1

where 2A and 2R are the inner and outer diameters of the ring respectively. and r is the distance from the centre of the plug The difference in principal stresses is 2P.A * (R/r)* = $--

Ue - 0, =

(7)

where we let R ‘k 340 mm and A Q R In assembling the shrink-fit specimen. strain gauges were placed on the plate before the (cooled) plug was inserted to measure the actual residual stress. The coefficient P,A’ could then be evaluated from strain gauge data and used to calculate as - a, from (7).

to

We have chosen to test the latter approach because it has several attractive features when compared to the offaxis technique. First_ the off-axis technique increases the effective ultrasonic gauge length, since two transducers are required (the gauge length is the distance between transducers). The gauge length for the normal-incidence technique is just the transducer aperture. Consequently. the normal-incidence technique will give better spatial resolution of the stress field Second. the off-axis technique requires special transducers (usually EMATs for SH-waves). whereas the normal-incidence technique can use either conventional piezoelectric transducers or EMATs. Third the off-axis technique requires increased resolution in time-of-flight measurements compared with the more conventional technique we explore here (see Appendix). For example, the method used in the King and Fortunko experimentsI requires an accuracy of 2 X lop5 in relative time measurement to give + 10 MPa uncertainty in stress for 6061-T651

ULTRASONICS.

Outside the plug region our shrink-fit specimen can be idealized as a ring under uniform internal pressure Pi (except close to the plate edges). Therefore. in the elastic region we have22

Experiment Measurement

system

We used a simple time-interval averaging system similar to that described by Lacy and Daniels23. An amplifier sends a gated cw pulse of centre frequency 710 kHz to a transmitting EMAT. When the voltage of the gating pulse reaches a selected level a START signal is sent to a timer. which has a 100 MHz clock rate. The ultrasonic signal is received by a second EMAT, gated (to look at a particular echo of interest) and amplified. When this amplified signal reaches a selected voltage. a STOP signal is sent to the timer. which averages over a large number of time-of-flight measurements (typically IO’-104). The system has a precision of about +l ns for lo4 measurements, with the transducers fixed in place. This represents the precision of arrival time measurement by the electronics. Slight changes in the position of the transducers cause additional imprecision in arrival-time measurements as will be discussed later.

255

The characteristics of the EMATs have in detail elsewhere’9%24. Suffice it to say purposes we had adequate signal-to-noise determine arrival times for the first and We note that for the transmitter-receiver the first echo propagates at an angle Q second at an angle of 18.5”.

been described that for our ratios to second echoes. pair we used. = 33.8”, the

Plan of the experiments

We measured the value of’ao - a, along an axis parallel to the rolling direction (called the 0” axis) and orthogonal to the rolling direction (90” axis). King and Fortunko have obtained data along this latter axis. using their off-axis technique to remove the dependence on the unstressed birefringence19. We compared the accuracy of our more conventional method with theirs for evaluating the difference in stresses. ag - a, The values of the acoustoelastic constants were obtained for our specimen by King and Fortunko19. Therefore. all that was necessary was to measure B = (Vo - VJV, at the stressed location of interest and subtract an average value of &, co?@ measured at several reference-specimen locations. Sources

of error

To measure the birefringencc S = -(lo - t,)/t,. WC first oriented the transducers so that the shear-wave polarization direction was in the &direction. and measured an arrival time, tg. The transducers were then rotated through 90” and r,. measured. We repeated this process several times (typically four to six times) and took the average value of(te - tr)/t,. This procedure was repeated for locations both on stressed and unstressed specimens. Typically there was a smaller standard deviation in (to - t,)/t,, on the stressed specimen than on the unstressed specimens. In the unstressed specimens, the standard deviations were typically -tX ns in a 25 ps travel time. or a precision of about 3 X 1OY. equivalent to a stress uncertainty of about 9 MPa (using the acoustoelastic constant for the first echo).

until reaching a location with a clean baseline; the arrival time measurements were then made at this location. The other major source of error in determining a@ - a, is material inhomogeneity. The method used requires measuring (tg - t,)/t, on the reference (unstressed) specimen to obtain B, co?@. However, it is well known that this quantity. though small. is inhomogeneous. Using a value of B, cos*@ measured at a single (unstressed) location can lead to significant errors in determining stresses. We attempt to compensate for inhomogeneity by measuring arrival times at various locations on the reference specimens and in some cases by using several reference specimens. The success of this method will be demonstrated for axes parallel and perpendicular to the rolling direction of the shrink-fit specimen. Results 90” Axis We used two slightly different techniques to measure B - & co?@ along this axis. The first technique was to measure II0 co?@ at four different locations on the reference specimen and calculate the average value. which was (43 -t 4) X IOF. The birefringence was then measured at locations on the stressed plate. and the average value of B, cos’@ subtracted to obtain as - a,. These values are shown as open circles in Fig. I. The second technique was to measure B at the stressed location of interest and then measure B, co?@ at an arbitrary location on the reference specimen. This was done each time we made a measurement of ~0 - a, Consequently. each value of as - a, is calculated using values of B, which differ slightly from one measurement to the next. The values of us - a, calculated using this technique are shown as crosses in Fig I. It is interesting to note that the average of the nine values of B, cos*@ measured with this technique was (41 -t 4) X 10p4. 150

The larger error at the unstressed locations occurs because the transducers were usually closer to a reflecting boundary. The off-axis SH-waves are guided. travelling along inside the specimen until they hit an edge. and then reflect Consequently. the ‘ultrasonic reverberations’ from previous pulses are sometimes present when the direct echo arrives at the receiving transducer. The coherent interference between these reverberations and the echo causes a change in the apparent arrival time of the echo.

120

90

60

Note that the presence of reverberations causes the precision of the measurements of 00 - a, to be worse (by about an order of magnitude) than the precision of the electronics in measuring arrival time. We attempted to minimize the effect of reverberations by several means. We tried changing the pulse repetition rate to obtain the cleanest baseline signal between the first three echoes. This was taken as an indication that the reverberations were minimized (perhaps by destructive interference) during the time measurements were taken. Another method, useful only for obtaining (to - t,.)/t, on the unstressed specimen. was to shift the transducer location in small increments

256

30

0 :

6

9

12

15

Position Fig. 1 values. of

Plot of oB -

90”

axe.

SolId

theoretical

values

obtained

by using

B,,. Crosses are expenmental

values

obtalned

by measuring

locatIons

on reference

24

km)

lone represents

are expenmental

arbitrary

Circles

0, along

21

19

an average

Bo

value

at

spec!men

ULTRASONICS.

NOVEMBER

1985

Most of the values of a~ - a, in Fig 1 obtained with the first technique are lower than the theoretical predictions, and values obtained with the second technique tended to be high. Recall that the average value of & cos2@ used in the first technique was 43 x 10p4; for the second technique, & cos2@ was 41 x 10e4. Consequently, we have a different ‘dc’ level for the two techniques, which may cause them to give different values of us - a,. at the same location. Errors caused by the ultrasonic reverberation mentioned previously are also present We calculated the average error obtained from all of the measurements plotted in Fig. 1. That is. we took the average ae - a, at each stressed location ri (as measured by our two techniques) and calculated the absolute value of the error at ri. We then averaged these errors over all locations: 1

N

c N

i=

I I(00 - UT)~- (Ue -

‘6

Ur)tIi

I

I

I

I

I

9

12

15

18

21

Position

1

Here superscripts e and t refer to experimental and theoretical values, respectively, and N = 9 is the number of measurement locations. The average error for all measurements shown in Fig. 1 is 6 MPa. In contrast, we found that the average error (calculated as above) for the King and Fortunko dataI was about 14 MPa. Based on the results shown in Fig. 1, we conclude that our method of measuring B - B, cos*@ gives reasonably accurate values of ue - u, (within about 6 MPa of true values. on average). 0” Axis Along this axis, we used one of the same techniques employed along the 90” axis. We measured B at a stressed location. then measured B,, cosz@ at an arbitrary location on an unstressed reference specimen and then subtracted the two values to get (us - u&z(@). We used four unstressed specimens: two pieces of aluminum cut from the same stock as the stressed plate and two corners of the stressed plate itself (the stresses are approximately zero there, since the plate edges are stress free).

(

4

(cm1

Fig. 2 Plot of ct - rr along 0’ axis. Solid line represents theoretical values. Circles are average values of measurements made at four reference locations

boundary. we terminated our measurements and assumed u, a0 there (from (6). u, -5

at r = 23 cm MPa there).

We took the values of ae - u, shown in Fig 2 and substituted them into the integral in (8). The integral was evaluated by fitting a cubic spline to all values and then numerically integrating. The result is shown by the open circles in Fig 3. Since we started with an error of 5 MPa at r = 23 cm our values of a, lie above the theoretical values. We note that the integration has smoothed out the effects of errors in measurement of 00 - a, which are randomly distributed about the true values (see Fig 2). For comparison we have also numerically integrated the values of u,g - a, reported by King and Fortunko’g. starting from r = 17 cm. We used the correct value of ur at r = 17 cm as the ‘boundary condition’ in (8). The result is shown by the stars in Fig 3. The agreement

We found that one corner of the plate tended to give values of ue - a, that were low. whereas the other corner gave high values. The same was true of the two specimens cut from the same stock as the plate. that is one specimen gave high values, the other low. Rather than show all the data points generated by using four reference specimens, we show only the average values of ue - u, in Fig, 2. The standard deviations (error bars in Fig 2) are about 15 MPa. Means and standard deviations were calculated from all the values of ue - a, obtained using the various measurements at stressed and unstressed locations. Calculation

-60

of radial and hoop stresses

We return to the stress equilibrium integrate it to obtain

equation

(3) and -751 6

(8) At r = R we have a, = 0 (traction-free boundary). Rather than measure 00 - a, all the way to the

ULTRASONICS.

NOVEMBER 1985

I

I

I

I

I

I

I

9

12

15

IO

21

24

Position

[cm)

Fig. 3 Plot of radial stress or Circles are values obtamed by integrating stress-equilibrwm equation using expenmental values of ‘~0- or along 0” axis. Crosses are values obtained by using data from Ref. 19 In integrating the stress-equilibrium equation. Solid line is theoretlcal value

257

another. Rather, we wish to show that the simple method used here can work in certain situations. Clearly, the method will not work if the microstructure in the reference specimen is different from the microstructure in the stressed specimen. As an example. consider the problem of ultrasonic measurement of residual stresses due to welding. In the heataffected zone (HAZ). the thermal history will cause the microstructure to be different from the base metal. Obviously. errors can result if reference specimens from the base plate are used to measure the unstressed birefrinpence in the HAL

6

I

I

I

I

I

9

12

15

16

21

Position Fig. 4 Circles ~0 -

Plot of hoop are from

IJ~ Crosses

of crB -

oB SolId

expenmental

c, are Integrated

or to gwe

values

stress

gr given

to gwe

data 0;

obtained

values

are values In Ref.

line

24

(cm1

represents along

of cr thus

of vs obtained

theoretical

values.

0” axis. Values

of

obtalned

are added

I” hke fashion,

to 0~

us!ng

19

with theory is good both for our method King and Fortunko.

and that of

We note that the off-axis technique of Ref. 19 will also lead to errors in this situation. The ultrasonic gauge length (distance between transmitting and receiving transducers) may be larger than the width of the HAZ. with resulting loss in spatial resolution of& and (JI - u,. We observe that the reference specimens should be from the same stock as the stressed specimen. If not_ the values of B0 measured on the reference specimen can be different from the values of B,, for the stressed specimen. This effect has been demonstrated in Rcf: Is. where it was found that the unstressed birefringence could differ by about 50% for different thicknesses of mild steel and aluminium alloy plate. Wc expect that some of the differences

To obtain the hoop stress 00. we now acid the calculated value of a, to the measured value of as - a, Since errors in the measured us - a,. are smoothed out in calculating a, we expect that the calculated value ot us will contain more errors than a, This is clearly shown in Fig 4, where we have plotted the results ol our method (circles) anti the method of Ref. 1Y (stars). For 1 I
between theory and experiment are causect by the ‘ultrasonic reverberation’ referred to previously. A simple way to check this hypothesis is to repeat the experiments using normal incidence EMATs. Conclusions We have measured the principal shear stress ug - a, in a shrink-lit specimen by measuring the acoustic birefringence. Since the final birefringence is due to both stress and material anisotropy. it is necessary to measure the unstressed birefringence to account for the latter effect. WC have chosen to measure the unstressed birefringence at a number of reference locations (in contrast to the King and Fortunko technique19). The 75

60 We use this form of a, in the relation (ug - a,) + ur to obtain ug:

us =

I

(10) Proceeding in this way. we find that a = 5.1 MPa. n7 = 2.58 for our data along the YO” axis (Fig I ). in contrast with the theoretical values of (Y= Y.2 MPa. 111= 2. The values of ug obtained are plotted in Fig 5 as a dashed line over the region IO < r B I8 cm. We now see a better fit of theory and experiment. since we have smoothed all the errors in ug and u,. before calculating both a, and ~0. This approach is similar to that of Jacksod’.

I

I

I

9

12

15

I 16

Position Discussion Our purpose in performing these measurements to espouse one technique (of removing B,,) over

258

Fig. 5

is not

Dashed

Plot of hoop line

stress

IS generated

Polynomial

IS fltted

Integrated

to obtain

LT,, SolId

usmg

to measured v,. Hoop

values

stress

data

of oH -

rf) IS then

ULTRASONICS.

; 4

(cm1

llne represents

expenmental

I

21

theoretical obtalned

up whxh calculated

values.

along

90”

axIs.

are then from

(10)

NOVEMBER

1985

values of ae - a, obtained with our method are in good agreement with theory: the agreement is at least as good as published data from Ref: 19. The method used here can be implemented with a simple system for making relative velocity measurements. The accuracy required for these measurements is about an order of magnitude less than the accuracy required when using the off-axis technique for the aluminium alloy used in our experiments. Furthermore. we have integrated the stress-equilibrium equation to obtain the radial stress u,. with good agreement with theory. Using these calculated values of a, plus the measured values of ae - a, allowed us to obtain the hoop stress Co. The agreement with theory was good but displayed more error than ~7~We attribute this to the fact that a,. is obtained by ‘ integration of (O,g - c,.)/Y. which smooths any errors in this quantity. Therefore. any errors in 00 - a, tend to be suppressed in a, and accentuated in 00. When we fitted the measured values of 00 - a, with a polynomial before integrating a smoothing occurred which gave experimental values of hoop stress which were in better agreement with theory. By measuring the values of stress. we have demonstrated the utility of the shrink-fit specimen as a possible standard for ultrasonic measurement of residual stress.

2

3

1

s 6

7

8

Y I0

I2

13

In the King and Fortunko approach, measurements of the birefringence B(Q) are made at two angles of incidence @, and C& and the resulting set of equations solved for 00 - a, The solution is given byI

I4

15

HG, 1 co3 @J1- B(@1> co? $2 &,>

(AI)

cos2 @1 - a(@1) cos2 42

I6

where no errors are assumed. If there are errors 6&Q,) and 6&Q,) in the birefringence measurement. then there will be an error S(Oe - Ur) =

lm(e, ) 1~0~2G1 t Im(4, a($2

> m2

41

-

a(@,

>i ~0s~G2

1 cos2

02

642)

Using values of @, = 33.8”. & = 12.6” (first and third echoes for the plate) and values of a(@,) = 3.7 x IV5/MPa and a(@,) = 4.4 x 10-5/MPa19 WC find that the denominator above is equal to 2.9 x IV/MPa. If we assume &3, = 6ti, then (AZ) requires that 6B < 1.8 x IV5 for a stress error of 10 MPa or less. Although this method requires an uncertainty in determining B of less than 2 x lV5. we note that the measurement system used by King and Fortunko19 was capable of providing this level of accuracy in the absence of reverberations. Consequently. if care is taken to avoid the ultrasonic reverberation problem mentioned previously. it may be possible to obtain stresses to within IO MPa using the King and Fortunko method. If the signs of the errors 6B,. and 6rJ, are known to be the same, then the estimate of 6(ue - UJ given by (A2) is conservative and the required accuracy in B may be correspondingly reduced.

ULTRASONICS.

I

II

Appendix

00 - a, =

References

NOVEMBER

1985

17

IX

lY

20

21

27 ‘3

24

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