The Measurement of Applied and Residual Stresses in Metals Using Ultrasonic Waves

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J. Sound vz%. (1967) 5 I), 173-192

THE MEASUREMENT OF APPLIED AND RESIDUAL STRESSESIN METALS USING ULTRASONIC WAVES

D. I. CRECRAFT~School of Engineering Science, The University of Warwick, Coventry, England

Received 25 February I 966)

The results of measurements of the stress-induced velocity variations of both longi-tudinal and shear ultrasonic waves in the megacycle region are quoted for steel, aluminiumand copper, and the corresponding third-order elastic constants calculated.

The methods of measuring the small velocity changes due to both stress and preferentialgrain alignment are discussed, and some attempts at separating the two effects so thatresidual stress can be non-destructively detected are described.

I. INTRODUCTIONResidual stresses are becoming of ever greater imp ortance to the engineer subjectingmetallic materials to loads approaching their yield stresses, since, under these conditions,an unfavourable residual stress can add to the applied stress and cause failure of a criticalcomp onent in service. Whilst the material is, in general, taken through many stress-relieving heat processes before attaining the final shape of the componen t, the final heatprocess w ill seldom involve sufficiently high temperatures for complete stress relief, andthe final machining process w ill almost certainly introduce further residual stresses. Inthis context the critical stages in the manu facture are, therefore, the final stages afterwhich n o further work can be done on the componen t to relieve residual stresses. There isthus a need for a non-destructive means of detecting residual stresses in finished com-ponents. The only established technique depends on the use of X-rays to measure theBragg reflection from, and hence the strain in, the metallic lattice of the first few atomiclayers from the surface. The residual stress is then calculated from the residual strain,using the accepted values of elastic constants.

As a method of measuring residual stresses w ithin m etallic materials, rather than ontheir surfaces, the use of ultrasonic waves has been suggested. It is know n that ultrasonicwave velocities in mater ials are sensitive to stress within the materials, and hence thetechnique wou ld be analogous to photoelasticity, which is used in the analysis of stressdistribution in transparent materials. Because of this analogy the words sonoelastic(Elion I)) and acoustoelastic (Benson and Raelson 2)) have been used to describethe effect, and sonoelasticity is used here to describe the use of ultrasonic shear (trans-verse) stress waves in close analogy with the polarized transverse light waves of photo-elasticity.

2. PHOTO ELASTICITY AND SONOELASTICITYIn photoelasticity plane polarized light is obtained from a polarizer and passed througha parallel faced t ransparent solid at normal incidence. Figure I shows how the wave splitsinto two comp onents, each polarized parallel to the axes of the principal stresses in the

t Dr. Crecraft was at the Department of Electronic Electrical Engineering, University of Birminghamwhile most of this work was done.

173



I74 D. I. CRECRAFTplane normal to the propagation direction. In general, the two comp onents travel withdifferent velocities, since the effect of stress (e xcep t in the special case wh en the principalstresses are identical) is to render th e material electrically anisotro pic: i.e., the perm ittivityvaries with direction, giving rise to principal perm ittivity axes coinciden t with theprincipal stress axes, and the material is said to be optically birefringent. The emerginglight is viewed throu gh a second polarizer, often crossed with the first, and Figure Ishows that this passes compo nents of the two waves which are equal and opposite inspace, but which have a relative time-phase shift due to the difference in their velocitiesin the stressed mater ial. This interference effect causes the light intensity at a given poin t

Axisof Itransmitterpoloriser , Principalstressaxis (Xl

emerging from moteriolFigure I. Section through XY plane of material under stress, showing the space intensities oflight waves in photoelasticity and ultrasonic shear waves in sonoelasticity, w ith crossedpolarizers.

to reach a maxim um whenever the stress is sufficient to cause 180~ phase shift, and toreduce to zero when an integral num ber of cycles of phase shift have been encountered.Points over the surface of the second polarizer at the same intensity appear to be joined bybands or fringes as they are comm only called, and these represent isobars in thestressed material. Since at any point the relative retardation , as it is called, betw eenthe two wave componen ts is the result of all the stresses encountered in passing throu ghthe solid, and since the velocity chang es are very small, the velocity difference indicatesthe average stress over the thickness of the solid at this point. M oreover, the two com-ponents passed by the second polarizer are maxim um when the principal stress axes lieat 45 to the polarizer axes, and zero when either stress axis coincides with either polarizeraxis; thus the directions of the stress axes can be found by rotation of the solid o r of thepolarizers. The relationship betw een velocity difference and stress is linear, and Table I(Heywood (3)) h ows values of two coefficients used in photoelastic studies for threetransparent materials representing the wide range encountered. The stress-opticalcoefficient, S, is defined as the fractional velocity difference per unit stress: i.e.,

where c, and c2 are the two velocities under stressed conditions, cO is the velocity in theunstressed material, and P and Q are the principal stresses in the plane normal to pro-paga tion. The fringe-stress coefficient, F, is defined as the stress required to produce one



STRESSMEASUREMENTBY ULTRASONICS 17.5fringe: i.e., on e cycle of phase difference between the two waves, in unit thickness ofmaterial. Thus

F = P-Q)xN

where z is the thickness of the material, and N is the number of cycles of phase difference.It is readily shown that F=h/S, and mu st therefore be quoted at the wavelength A.The elastic stress in a material also affects th e velocity of an elastic wave . In the case ofa solid, both comp ressional (longitudinal) and shear (transverse) waves will propagate,and the analogy w ith photoelasticity is in the use of shear waves. The velocity is given byc = (C/p), where p is the density and C is the appropriate elastic constant, which forshear waves is the shear mod ulus I*. Whilst the change in density due to strain has asmall effect, the velocity change caused by the effective elastic m odulus changing with

TABLE IPhotoelastic and sonoelastic coeficients

Material S(i~~.~lb-l) (lb in.-2)

Photoelastic values at h= 5893 A (Heywood) Glass 1.87 x 10-s 1.24 x 103Celluloid 9.45 x 10 8 2.45 x 102Gelatin 1.66 x IO ~ 1.4 x 10 lSonoelastic values atf= 4.7 M C/S (Crecraft) Nickel-steel 594 x 10-s 475 x 105Copper 9.0 x 10-s 2.15 x 10~Aluminium 2.41 x 10-7 I.08 x 105

stress can be mu ch larger, which implies non-linearity in the stress-strain relationship,within the so-called elastic region. This will be discussed later. The complete analogywith photoelasticity occurs when a shear wave is propagated through a stressed solid.The latter becomes elastically anisotropic, with the axes of principal elastic constantscoincident with the principal stress axes, and the wave splits into two componentspolarized parallel to the stress axes, so that Figure I applies to this case also. The materialis then said to be sonically birefringent. Continuing the analogy, the stress-sonic coefficientand fringe-stress coefficient may be defined as in the case of photoelasticity, and somevalues determined by the author are shown in Table I for comparison. It is interesting tonote that the stress-optical coefficient for Celluloid has practically the same value as thestress-sonic coefficient for copper, but that Celluloid is over one thousand times as sensitiveto stress in term s of the fringe-stress coefficient. The difference is attributable to thedifference in wav elengths, since F = X/S. The photoelastic values in Table I are quotedfor a wavelength of 393 A , or about 6 x IO- m, whereas the ultrasonic waves at 4.7 MC /Shave X = 6 x IO-~ m in copper. Thus in photoelasticity the low S values are compe nsatedby small wavelengths, and so large numbers of fringes can be obtained for small stresses.In sonoelasticity, on the other hand, the larger wavelengths give low phase shifts, whichmeans tha t small phase angles must b e measured accurately if stresses are to be detectedat all in metals. The use of higher ultrasonic frequencies with smaller wavelengths leadsto excessive attenuation. Furthermore, while the resolution across the field of view in thephotoelastic case is limited only by the very small wavelengths, in the sonoelastic case



176 D. I. CRECRAFTconsiderations of beam shaping and attenuation at present lead to the use of sources andreceivers several wavelengths across, which in practice means a resolution of severalmillimetres.

3. METHODS OF MEASUREMENT OF STRESS INDUCED VELOCITY CHANGESThe direct measurem ent of the phase shift between the two shear wave comp onents

propagating in a metal under applied stress will yield the values of S and F and these canthen be used in determination of residual stresses. How ever, it is not possible to tell fromthis measurem ent how much the individual velocities have changed. Consequ ently,method s of measuring small changes in the velocities of both shear and longitudinalwaves were considered so that the changes in the elastic constants under stress cou ld beestimated.Earlier work on the non-linear stress-strain relationship was carried o ut by Bridgman(4) and others, using very high hydrostatic pressures. Hughe s and Kelly (5) propagatedboth shear and longitudinal waves through solids stressed both hydrostatically anduniaxially, and showed that a set of three different wave velocity change measurem entswas sufficient to calculate the three third-order elastic constants 1, and n introduced byMu rnaghan (6) in his second-order elastic theory. For velocity chang e measurem ents inPolystyrene, Arm co Iron and Pyrex Glass, Hughes and Kelly used the pulse-echotechnique entailing measurem ent by oscilloscope time-base of the time taken for a pulseof ultrasonic waves to travel through the specimen. This method lacks precision, and arefinement used by Bergman and Shahbender (7) and Elion I) involved simultaneoustransmission of a pulse through the specimen and through an ultrasonic delay line set togive the same delay as the specimen in the unstressed condition. This enabled them tomake comparative measurem ents of transit time differences with higher time-base speeds,and led to resolution of velocity changes of IO-~ in 4 in. thick aluminium.

McSkimin andAn dreatch((8), (9)) used the p u se superposition method , wherein thepulse repetition frequency (p.r.f.) is adjusted so that the period is close to an integralmultiple of the transit time. It is then possible for waves within succeeding receivedpulses to arrive in phase, w ith a consequent increase in the amplitudes observed on theoscilloscope. The transit time is calculated from the p.r.f., the echo number and inter-polation from a num ber of readings. Quo ted accuracies were of the order of one part in104 for measurem ents on fused silica and single crystals of quartz and germanium .Various interferometer techniques have been used wherein the electrical signals from atrain of pulse echoes produced by multiple reflections in the specimen are added to thosefrom ano ther train, w ith velocity differences leading to phase cancellation of particularechoes. Rollins ((IO), I I)) used an arrangement, based on the technique of Espinola andWaterman IZ), in which echoes from a stressed specimen were added to those from anidentical unstressed specimen. This technique is quite sensitive, Espinola and Waterm anclaiming velocity resolution of 5 X I O 6 in their measurem ents of temperature inducedvariations in the elastic co nstan ts of steel, fused silica, and alkali halide crystals. Rollinsestimated a stress resolution of 20 lb ine2 in his measurem ents on 2s aluminium. Rollins(10) also used an arrangement in which two shear wave echo trains occur within thetested specimen, generated by one transducer shearing at 45 to the applied stress axis,so that the two components of Figure I are generated. Smith (13) used separate trans-ducers, one each side of a nickel-steel specimen, to generate the two principal shear waves,with calibrated variable electrical delay between them so that exact echo cancellation waspossible. The transducers were placed opposite one another so that both launched wavesthrough the same volume of the specimen. Their axes of polarization were aligned one



STRESS MEASUREMENT BY ULTRASONICS I77parallel to, and one normal to, the applied stress axis so that each produced a principalwave polarized perpendicular to, and hence unaffected by, the other.

4. THE SING AROUND SYSTEMSince the accurate m easurement of ultrasonic velocity in a convenient-sized specimenrequires accurate measurem ent of a small time of a few microseconds, a natural step is to

convert this time to frequency, which can be measured with the required level of accuracy.The technique, described by Cedrone and Curran (14) is known as the sing-aroundmethod and basically involves transmitting a pulse through the specimen from one side

Figure 2. The sing around system.and using the electrical signal from the receiving transducer on the other side to retriggerthe transmitter, so that a pulse of sound energy sings-around the system at a p.r.f.determined by the ultrasonic transit tim e. A refined version devised by Myers, Mack innonand Hoare (I 5), allows the ultrasonic path length to be effectively increased by triggeringfrom an echo among the multiple reflections. This has the additional advantage that themethod will work in materials with low attenuation, where triggering on the first receivedp&e would be disrupted by succeeding echoes. A very sophisticated version of this byForgacs (16) is quoted as measu ring velocity with a precision of one part in IO'. Themethod is not intended primarily for the accurate m easurement of absolute velocity, sincethe measured transit time includes delays in the electronic triggering system. If the delaysare sufficiently constant, however, very small changes in transit time may be detected.The form of the sing-around system used in the present work is show n in Figure 2.Operation is started by first closing the supply switch to the astable multivibrator so thatthe transmitter is triggered at a comparatively low p.r.f. Figure 3(a) show s the first threereceived pulses, representing one, three and five traverses of the specimen, with, on thelower trace, the echo pulses triggered by those in the echo amplifier and fed to the gate,The position of the gate pulse, which is set by the manual control on the monostablemultivibrator providing the variable delay, is show n in Figure 3 by marker pips . Thedelay control is then turned to bring the gate pulse into coincidence with the requiredecho pulse, whereupon the latter is fed to the transmitter drive-pulse generator, causingretriggering of the transmitter. Providing sufficient loop gain then exists in the complete

12



178 D. I. CRJZCRAFTfeed-back loop, and this can be achieved by adjustm ent of various controls, then theselected echo pulse will sing-around the loop. Figure 3(b) show s retriggering takingplace from the second received pulse [the centre pulse in both Figure 3(a) and (b)]:i.e., after the transmitted pulse has made three trips through the specimen. The extrapulse which appears between the first and second received pulses [the third from the leftin Figure 3(b)] is the third received pulse from the previous transmission. Finally, theastable multivibrator is turned off, and the sing-around p.r.f. is read on the digital

0.5 V/cm t O~tiV/Crn T

IO V/cm t IOOVkm t

(a)

O,SV/cm t

IOVhll T

(b)

(cl

Gat:-&e 5 pseckm - Cdl I.0 flsec/cm -----+

05Vkm t

IOV/cm t

5 flseckm W (e)

O.fjV/cm tIO V/cm

t

l.Opsec/cm __) (1) IO psedcm ----+

Figure 3. Sing-around oscillograms: (a) not singing-around ; (b) singing-around after3 trips; (c) (b) expanded x 5 ; (d) as (c) but showing the transmitter pulse (lower trace) ; (e)singing-around after 5 trips; (f) singing-around after 7 trips. U pper traces, output fromreceiver crystal; lower traces, output from ech o amp lifier (excep t in (d)).frequency meter of the counter-timer type. W ith a counter gating time of I set, and atypical total ultrasonic transit time of zo psec for three traverses of the specimen, thep.r.f. will be 50 kc/s and the resolution 2 x IO -5. Greater precision can be obtained byextending the gating time to, say, IO set, but it is often m ore convenient to set the counterto measure the time required for, say, 10~ sing-around cycles. In this example, thistime will be 0.2 set, but a I MC/S counter-timer will measure this in units of I psec, givinga resolution of 5 x I0 -j. The limit to the resolution is determined by the stability of theequipment and is about IO-~ under laboratory conditions.4.1. FACTORS AFFECTING THE SING-AROUND P.R.F. IN A STRESSED BAR

If a long metal bar is stressed longitudinally by tensile or compressive loading at itsends, the distribution of stress will be practically uniform throughout the path of anultrasonic wave propagated across the bar through its centre. As stress is applied to thebar, the p.r.f. u) of the sing-around system w ill change inversely as the transit time(t), the latter change being caused both by velocity (c) change and path length change



STRESS MEA SU REMEN T BY U LTRA SO N ICS 79due to strain. The velocity change will, in turn, be caused by changes both in the elastic constants (C) and in the density (p) due to strain.The conventional definition of strain is

AL dL ALeL = -_j --as--0L L L

where L is the length of the bar a nd AL is a small increase in length due to applied stress.In analogy with th is, we may define the fractional change in any quantity x asAX dx Axe = --em-.-.- as--oX X X

so that, for instance, ef = -e,.The total path length nT = ct where T is the thickness of the bar an d n is a factor dependingon the echo used for retriggering. The total derivative of this gives the lateral strain as

giving the fractional change in velocity ase, = eT-et,

= -peL+efwhere p is Poissons ratio. Since

(1)c = (qpy,

e, = *ec - ep,and the fractional change in density, from linear elastic theory, isep = -eL(I--2p),

giving the fractional change in elastic constant asec = 2ef eL. (2)

Equations (I) and (2) show that if both the p.r.f. and the length of the bar appear (to afirst approximation) to vary linearly with stress, then both the velocity and the elastic constant mus t vary linearly with stress. Since the linear elastic theory is valid only to afirst approximation, the analysis is strictly true only to a first approximation, but theinaccuracy will only be of the same order as the departure from linearity in the stress-strainrelationship, which in the elastic strain region is very sm all.

5. MEASUREM ENTS OF STRESS-INDUCED VELOCITY CHANGES5. I . EXPERIMENTAL DETAILS

The sing-around system was used to make measurements of the changes in velocity,due to applied uniaxial stress, of longitudinal and shear waves propagated across the centreof bars of nickel-steel (S/NTV), high conductivity copper (999% ) and comm ercialpurity aluminium (99.5%). Th e sh ear waves were polarized, first parallel to, and thenperpendicular to, the applied stress axis which was parallel with the axis of the bar ineach case.The applied stress was confined to the elastic region in each case. Com pressive stresseswere applied to a 12 in. long, I in. square section nickel-steel bar, a 6 in. long, I in. squarecopper bar, an d a 2.7 in. long, 4 in. x Q in. section aluminium bar, by a hydraulic press,



180 D. I. CRECRAFTthe load being measured by the hydraulic pressure gauge. Tensile stress was applied to a12 in. long, I in. x Q in. section tensile test piece, machined from the same stock as theabove nickel-steel bar. This test piece was also subjected to compressive loading.

Measurements were made with I mm thick X-cut quartz crystals driven at 2.5 MC/Sfor the longitudinal waves coupled by oil films, and 0.5 mm thick Y-cut quartz crystalsdriven at 45 to 47 M C/S for the shear waves. The latter were coup led by phenyl salicilate(salol ) in early measurem ents, and later by Aroclor I 254.5 .2 . MEASUREMENTS

In Figure 4, the sing-around p.r.f. in the I in. square section n ickel-steel bar isplotted versus compressive stress for the three w ave types (17), retriggering of the trans-mitter occurring on the second received pulse in each case. The spread of one count inthe readings is due to the ambiguity in the counter-timer technique, both possible readingsbeing obtained at each point.

k* 1 2.5 me/s longitudinal: wave% I\ \ kv 41440- N:-*2*_S 3W)_- 4.5 mcksheor wavez 4, 430 polorised perpendidular ~*~:-~,

- toaxis of stressP5E 41,420- r/i 1

IS 41,410-s 1

41,400 - /,



STRESSMEASUREMENTBY ULTRASON ICS 181part of the received pulse in each case, although it will be seen later that their zero-stressvelocities do differ, due to preferred grain orientation .Figure 5 confirms expectations that the velocity change would be of opposite sign fortensile stress. The sligh t difference in slope between the compressive and tensile regions

t 34,090-p.r.f.k/s) 34,880- r

8:8

34,8?0- //2 ,O , 16 12 8 4 / 8/ 4 , q , I? , 192p

IO 864 /Tension,(ton in-) */ -- Campression,(i031bin-2)

Sheor wovepolarised parollel *Ito stress axis ,/1/

-- 34,050

--34,840

/ --34,830

Figure 5. Variation of sing-around p.r.f. w ith compressive and tensile stress in nickel-steel.

19.720 -\ \ \ Shear w ove polorised parallel19,710 -H 19,700-3Hk 1?690- 10

Longitudinal wove

1500 2000 2500Applied compressive load (lb)

Figure 6. Variation of sing-around period with stress in aluminium.(about 4%) is probably attributable to calibration differences between the hydraulicpress and the tensile testing m achine. The com pressive results show curvature at higherloads, which is probably due to the slight curvature of the test piece observed at these loads.Since the I in. square section bar of the same ma terial showed no such effect, it is mostprobable that the curvature of the specimen caused an increase in the effective crystal bond



182 D. I. CRECRAFTthicknesses, giving rise to a longer p ath length. The linear pa rt of the compressive curvehas the same slope as that for the I in. square bar, confirming the reproducibility ofmeasurem ents made by the sing-around technique.

35.030Shear wove polorised parallel

Shear wove polorised perpendicular,to stress 4.5 me/s 3 trips

0 2 4 6 6 IOCompressive stress (IOlb in?

Figure 7. Variation of sing-around period with stress in copper.

The absolute velocities in the three bars were measured by the pulse-echo method ,and the Lame constants, X and CL,were calculated from

p = p& and h = p(&- 2&J,where cl0 and czo are thelongitudinal and shear wave velocities in the unstressed material.Poissons ratio was calculated from

A* = z(X+u)and equations (I) and (2) were then u sed to find the fractional changes in velocity and inelastic constants , shown in Table 2.The values of the stress-sonic coefficient quoted in Table I were computed from theseresults, with the convention that S is the fractional velocity difference between the shearwave polarized parallel to the stress axis and that polarized perpendicular to the stress axis,per unit compressional stress. Using this convention, the values of S calculated from otherworkers and the authors work on metallic materials are all positive. In the present caseit can be seen from Table 2 that this is because the wave polarized parallel to the stressaxis suffers by far the larger velocity change, and always increases in velocity undercompressive stress. Confirmation of this is not possible from the measurem ents made byothers of birefringence, due to the difference between the two wave velocities.On the macroscopic scale, the much larger velocity change of the parallel-polarizedwave is attributable to the fact that the applied u niaxial stress causes shear strain in theplane in which the wave motion shear occurs, so that the velocity is determined by a shear



STRESS MEASUREMENTBY ULTRASONICS 183modulus which has changed due to applied shear. On the other han d, the perpendicular-polarized wave shears in the plane of the applied uniaxial stress where, in an isotropic body,there is no resultant shear deformation. The fact that in practice the velocity does changein the latter case show s that this macroscopic viewpoint does not provide a completedescription of the situation.

TABLE 2 Sing around measurements of stress induced changes

Specimen

Fractional change in S.A.period (t)Fractional change invelocityS = e,-e,Fractional change in elasticconstantsTolerances

Steel Aluminium Copper

t1, I7tzz 12t, -47Clu -0.6CZZ -0.2C2Y 57

59Cl, 01C2Z 10C Y 12.9

20.1

95 8.200 44-24.1 -46

-6.1 -5.334 - 1.5275 7524. I 9.0

-9.2 -7.99.8 -0.3579 17.6+ 02 rto.2

Figures are parts in 10s per IOOO b in-z. Subscripts: all waves propagatedin x direction, and ~y=longitudinal wave propagated transverse to stressaxis, y; zy =shear wave polarized parallel to stress axis; zz = shear wavepolarized normal to stress axis.

5.3. THIRD-ORDER ELASTIC CONSTANTSThe linear, or first-order, elastic theory derives from an expression for the strain energyin terms of strain products of the first and second degree, w ith corresponding first-orderand second-order coefficients. This expression is differentiated with respect to strain inorder to produce an expression for stress in terms of the first-order coefficient (corre-

spond ing to initial hydrostatic strain) a nd the products of the strains and the second-ordercoefficients. The latter are thus the familiar elastic constants of linear theory. If the expres-sion for the strain energy is now considered with additional third-order strain products,then the first differentiation yields, in addition to the above terms, second-degree strainproducts with third-order coefficients. A further differentiation with respe ct to straingives the elastic constants in terms of the second-order coefficients and the products ofthe third-orde r coefficients and the strains.

Murnaghan (6) introduced three third-order constants, I, m and n for the case of anormally isotropic body, taking the Lam e constants, X and CL,as the second-order co-efficients. Hug hes and Kelly (5) derived expression s for elastic wave velocities in termsof these five constants in seven cases, two of hydrostatic pressure and five of simpleuniaxial stress with the stress axis, wav e propag ation direction, and polarization (in theshear wave cases) coinciding with one or other of the axes of an orthogonal system.

Three of these expressions, of interest in the present case, arePO&J A 2p-go[ 2- (m+X++) 1,



184 D. I. CRJZCRAFTPO& = CC-T3Ko [m ;; h zpI ,

PO& = p- i-[m-(%)n-zh].In these, p. is the density and Ko = A+ 2~/3 is the bulk modulus, under zero stressconditions. T is the uniaxial compressive stress, wh ich is applied in the direction givenby the second subscripts on the velocities. Of the first subscripts, I refers to longitudinal

waves and 2 refers to shear waves, the latter being polarized in they direction. Propagationis in the x direction in all cases.In general, c=co(I -e,), and since e, is very small (the largest observed value beingabout IO-~) to the required level of accuracyc2 = c (1 - ze,),

so that the velocities given by equations (I), (2) and (3) should vary linearly with stress,as is confirmed by the present measurem ents.The Lam 6 and Mu rnagha n constants derived from the three wave velocities m easured

during the present work are show n in Table 3 together with those calculated by HughesTABLE 3

Lam C and Murnaghan constants bars x 10~)

Material Lam6 constantsx II Mumaghan constants1 m n Source

Polystyrene 0.2889 0.1381 - 1.89* 000 I + 0001 f. 0.32Armco-iron 11 8.20 - 34.8

4 ko.10 +6.5Pyrex I353 275 I4_c0.003 * 0.03 +4-oNickel-steel S/NTV 10.90 8.17 -5.6_co*1 f 002 f 20Copper (999%) 10.40 4.60 542+0.1 + 001 +3oAluminium (99 %) 6.10 2.49 -4.7ko.1 * 001 f 2.5

- I33 - 1.00 Hughes and Kelly+ 0.29 +0*14- 103.0 110L z 0 +I10

9.2 42f5.0 f35-67.1 - 78.5 Crecraftf 0.6 + 0.7- 372 -40.1+ 0.5 +0.5- 342 - 24.8+ 10 rt 10

and Kelly (5) from their measurem ents on three other elastic materials. It should be notedthat n is given by equations (2) and (3) as

= -(@+ I)&where S is the stress-sonic coefficient, and thus n can be determined from measuremen tsof birefringence alone. The values for nickel-steel differ somew hat from those publishedpreviously by the author (17), since they include corrections for slight errors in theabsolute values of velocities upon wh ich previous calculations were based.



STRESS MEASUREMENT BY ULTRASONICS 16. RESIDUAL ANISOTROP Y

6. I THE PROBLEM OF LAUNCHING SHEAR WAVESIt has been seen that shear wave velocities are affected by stress to an extent determined

by their direction of polarization with respect to the principal stress axes in the planenormal to their direction of propagation. This is extremely important in the determinationof residual stresses, since, in analog y with pho toelasticity, the difference in velocitybetween the two principal shear waves indicates the magnitude of the stress, and theirdirections of polarization indicate the principal stress axes. Moreover, an accuratemeasurem ent of ultrasonic path length is unnecessary. The use of longitudinal waveswou ld involve the measu rement of path length an d transit time to within one part in tenthousand even to detect large stresses, would not indicate the stress axes, and would bevery sensitive to slight changes in material properties from sample to sample.For these reasons, the use of shear waves is imperative, and these w aves must enter thespecimen at normal incidence so that, when the transducer is turned to change the angleof polarization, the path through the specimen is unaltered. Thus it is not possible to usethe familiar shear-wedge, used in ultrasonic flaw detection, wherein shear waves aregenerated by mode conversion of a longitudinal wave incident at an angle to the boundarybetween the specimen and an oil film interposed between the wedge and the specimen.An attemp t to launch shear waves directly from an appropriate transducer will result intransverse slip if a liquid couplant is used, and is only completely successful if the trans-ducer is cemented to the surface. This latter arrangemen t is impracticable since changingof the polarization direction involves different bo nd thicknesses, apart from the timeinvolved. There are some very viscous liquids which have sufficient dynamic shearstiffness to couple shear waves through a very thin film. Aroclor 1254 (a chlorinateddiphenyl; Mon santo Chem icals Ltd.) is one of these and at 3 MC /S its use involves about6 dB loss per transmission compa red with a solid cement, phenyl salicylate ( salol ).

In an attempt to launch shear waves at normal incidence without cementing to thespecimen and to improve on the loss involved w ith liquid couplants, a transducer assemblywas devised (IS), relying o n interface pressure for coupling. A Y-cut, or rotated Y -cut,qua rtz crystal, or lead-zircon ate-titanate disc electrically polarized for thickness shear, iscemented by phenyl salicylate to a metal coupling piece . This is pressed hard againstthe surface of the specimen by the application of force to the cover or cap as shown inFigure 8. In this way it is possible to apply an interface pressure high enoug h for almostcomplete transmission of the incident energy without the application of injurious stressesto the crystal. The loading force is applied via the steel ball, which ensures symm etricalpressure distribution at the interface. Figure 9 shows how the shear energy transmittedthrough the interface increases with apparent interface pressure for a steel coupling pieceon a ground nickel-steel surface. It can be seen that an apparent pressure of about 20,000lb inW2 is sufficient to cause com plete suppression of reflections from the interface andalmost lossless transmission. This requires a load of IOOO lb, since the front face has4 in. diameter.

In order to measure the difference in the velocities of the two principal waves over agiven path through a specimen, the transducer is pressed onto the surface and used as acomm on transmitter-receiver to observe the transit time of a pulse reflected back from aparallel o pposite face. The load applied to the transducer affects both velocities equallyand has no effect on the birefringence due to the principal stresses in the plane normal topropagation, which are those resolved by this method . Since the velocity difference isusually very small, the relative phase of the received echoes being less than one cycle,birefringence as such is not apparent and as the transducer is rotated the velocity appears



186 D. I. CRECRAFTsimply to have maxim um and minimum values 90 apart. Under these conditions it can beshown that, if the relative phase difference is 4, then the apparent phase E of the received

n. sieel boll\

Holes for connecting wires

Figure 8. Single-crystal transducer.

10.000 20,000lnterfoce pressure (lb/in*)

0 500 1000Transducer load (lb)

1500

Figure 9. Pressure-coupling characteristics of 4 in. steel transducer on polished steel surface.Curve A, polished transducer with liquid; curve B, polished transducer on dry surface; curve C,transducer face as turned.

wave will vary from zero to +, as the transducer polarization is rotated from a directionparallel to one anisotropy axis to the other at right-angles to this, following the law

where 0 is the kgle turned through. The accuracy of the stress reading does not, therefore,depend critically on the angle of polarization, since da/d@ is small when 0 is near zeroand 90. On the other hand, the location of the principal stress axes is not very precise, and



STRESS MEASUREMENT BY ULTRASONICS 87consequently there is no need for more than three or four readings at 45 intervals underthese circumstances.In use, the transducer is loaded by either a small hand press or an hydraulic press. Thelatter has the advantage that the load can be monitored by measurem ent of the hydraulicpressure, ensuring the same interface conditions and strain in the specimen at every angularposition of the transducer.The velocity measurem ent process is simplified further by the use of a four-crystaltransducer, which generates two shear waves with orthogonally disposed polarizations,obviating the need for rotation in many cases. All four crystals are polarized along radii ofthe transducer and in use one of a pair of diametrically opposite crystals is used to generatea shear w ave and the other to receive the reflection from the back-wall of the specimen.The transmitter and receiver are then switched to the other pair of crystals, which transm itand receive a wave polarized in a direction at 90 to the first.

Since the sing-around system is not readily app licable to a comm on transmit-receivetransducer, velocity differences have been observed on a delayed time-base oscilloscopewith a time-base calibration of IOO nsec cm- l. Taking the display resolution as I m mgives IO nsec time resolution which, in the case of the I n. thick nickel-steel specim enwith a round-trip time of 1542 psec, is ractional velocity difference of 6.32 x IO-~corresponding to a stress resolution of 485 ton inF2, or about 5 ton inm2.6.2. PREFERRED ORIENTATION

Early measurem ents with the pressure-coupled transducer revealed elastic anisotropyin all the specimens used previously for measurem ents of stress-induced changes. Theanisotropy was very small and constant througho ut each specimen, with a principal axisparallel to the bar axis, and evidently due to preferential grain alignment induced byrolling or extrusion. The fact that the anisotropy was constant througho ut each specimenmeant that it could not be attributed to residual stress, since the forces in a closed systemmust sum to zero.

The fractional velocity differences observed were o-27 n both nickel-steel specimens,0.55~h in the copper bar, and 3.3% through the thickness (0.5 in.) and 2.6% through thewidth (0.75 in.) of the aluminium bar. In addition, a Duralum in bar gave a difference of0*3% whilst a Nimon ic alloy piece exhibited a difference of 1.4 and another (NIM 8oA)of 17*oO/~.n all cases except that of the coppe r b ar, the wave polarized parallel to the baraxis propagated with the greater velocity.Preferred orientation in the direction of working is well know n, and is readily measur-able in extreme cases, such as drawn wires, by static measurem ent of elastic constantsparallel to, and perpendicular to, the extrusion or rolling axis. Surface effects are measur-able by X-ray techniques, but the proportion of grains lined up in a given direction mustbe at least a few per cent to be noticeable. Ultrasonic birefringence is certainly the mostsensitive metho d for bulk measurem ents, since velocity differences of the order of 0.1 y.are readily observable. Sullivan and Papadakis (19) have also observed small preferredorientation in rolled steel by ultrasonic birefringence with crystals cemented to the surfacewith polarization at 45 to the rolling direction. Imagining a mode l with some grainsaligned parallel to the rolling direction, embedd ed in a sea of random ly-orientated grains,they derived the expression

ts-ty = L&rc,)(c,c,)-1,where t, and tY are the transit times of the two waves polarized parallel to and transverseto the rolling direction, is the path length, K is the fraction of grains aligned preferentially,



D I. CRECRAFTand c, and cY are the shear w ave velocities in a single crystal, or grain, given by

with polarizations along a [IOO] and a [I O] direction respectively. Rearranging theexpression in terms o f fractional velocity change it becomes

k _ -C. 'xcYco cy-cxIf the values of the elastic constants of iron are used, this gives k = 0.4% as the proportionof grains in the nickel-steel bar with their [I IO] direction parallel t o the bar axis. Sullivanand Papadakis observed o* I o/oalignment.Even for surface m easurements an ultrasonic metho d is the most sensitive, the ultra-

sonic goniometer developed by Bradfield (20) measuring accurately the surface wavevelocity, and hence the surface elasticity, in various directions.6.3. PLASTIC DEFORMATION

In the final sections work is described on specimens containing residual stresses inducedby inhomog eneous plastic deform ation. In order to investigate the effect of plastic strainin the absence of macroscopic residual stresses, the nickel-steel test-piece describedpreviously was subjected to a few per cent uniaxial tensile plastic strain. Unfortunately,the strain measurements were upset by necking of the specimen in the region of theextensometer. The last reading obtained was 3.25% under load and from extrapolationof the stress-strain curve it seem s likely that the residual strain in the parallel re gion afte rremoval of the load was about 4%. Before strain, the preferred orientation caused afractional velocity difference of betw een 0.20 and 0.23%, the spread representing+ I nsec tolerance on the velocities across the 3 in. section, with the parallel polarized wavepropagating faster. After plastic deformation, and removal of the stress, this differencewas increased to between 0*39O/~ nd 0.42O/~.Since strain o f the order of 50% is necessaryto cause appreciable change in preferred orientation (Doan (zI)), t seems likely that thisincrease in velocity difference was due to dislocation movem ent alone. The wave polarizedparallel to the stress axis propagates as shear in a plane parallel to the deformation direction,and mu st attempt to shear slip planes where some dislocations are pinned following theplastic strain. Since the prep onde rance of slip planes lie at 45 to the principal stress axis,the wave polarized in the plane normal to the deformation direction mu st have a similaraction, but since the plastic strain in this plane is half the principal strain, the increase inmacroscopic shear modu lus due to pinned dislocations is less for this wave than for thatpolarized parallel to the specimens axis.The above argument applies equally w ell to tensile and compressive plastic strain. Thisis of practical imp ortanc e in stress an alysis, since it mea ns tha t the velocity of the parallel-polarized wave always increases with respect to that of the other wave regardless of thesign of the plastic strain, but unde r elastic cond itions is sensitive to the sign of the strain.This is demo nstrated in the next section.Rollins (II) found no further birefringence induced by plastic strain of 0.36% in aCIOI~ steel specim en. Evidently this order of strain causes a fractional velocity differenceof less than o-01 , the apparent resolution of Rollins measurem ent.6.4. RESIDUAL STRESSES IN .4 ENT BAR

In an effort to learn something more of the combined effects of residual stresses andgrain orientation, the I in. square nickel-steel bar was subjected to simple beam loadingsufficient to cause perman ent bending. Under these conditions the upper and lower halves



STRESS MEASUREMENT BY ULTRASONICS 189of the beam su ffer plastic deformation near the centre o f the beam, where the bendingmom ent is greatest (see Figure IO(a)). The stress parallel to the beam axis caused by thebending mom ent falls off from the yield stress in the region of plastic flow to zero at thecentre of the section (the neutral axis) and reverses sign on the other side of the centre(Orowan (22)). Wh en the load is removed, the inner part of the bar, in an effort to straighten,exerts elastic stress on the outer layers, w ith the result that the latter becom e stressedelastically with the opposite stress to that which caused their d eformation, whilst theadjacent layers wh ich suffered only elastic strain are now elastically strained by stressesin the same sense as before, as shown b y Figure IO(b).

Compressive 1 ensileI *(0) Stress distributionduring plasticstrain

; (a).5 -5.0 (b) Stressdistribution= i-7.5 after removal

2 +10

@

of loadg 0z0 -10+5 0k t5.0

9 (b)

Sonoelosticreodmgs(compressionI-0*5ton in- )

Readings aftersubtraction of3 Odue tooriginalpreferred orientationCc) (d)

Figure IO. The bent barMeasu rements made across the centre of the bar are recorded in Figure IO(C), these

representing the difference in round-trip times of the two waves, that polarized parallel tothe bar axis being the faster. The original anisotropy gave rise to a difference of 30 nsec,and this figure subtracted from the readings gives those show n in Figure IO(d). This lastset is not balanced and evidently does not represent the residual stress system . Subtractionof a further 25 nsec from all but the centre reading, to allow for the plastic deformation,yields a more nearly balanced set, which is shown in Figure IO(b) after conversion to stress.(IO nsec represent 5 ton in2, see section 6. I .)

This, then, confirm s the fact that plastic strain cau ses a greater increase in the shearmod ulus in the direction of the greatest strain, regardless of the sign of the strain.6.5. RESIDUAL STRESSES IN A DISTORTED RING

Some residual stresses may exist in circular parts in the form of hoop stresses. Figure I Ishows a specially prepared nickel-steel specimen of approximately 2 in. square sectionand originally forged as a circular ring from the cast billet in such a way that the grainflow was expected to be radially symm etrical, with the flow across the section as shown .The ring was then plastically deformed, hot, between the jaws of a drop forge. T he systemof residual stresses and plastic strains is not expected to be simple, therefore.



190 D. I. CRECRAFTWave round-trip time differences observed across the section at eight positions round

the ring are shown in Figure I I, in units of zo nsec, representing approxima tely 5 ton in.-2in the 2 in. section. The wave polarized in a direction tangential to the ring was the fasterin every case. Positions A, C, E and G are those of greatest deformation and mus t containmost residual anisotropy. The positions B, D, F and H lie midw ay between these and,since each lies between positions of opposite bending mom ent, are assumed to be stress-free, or neutral positions. It is also assumed that no plastic deformation took place here,and so the grain orientation should be as before distortion of the ring.

7

1416

)Figure I . The distorted ring: (a) sonoe lastic readings ; (b) expected grain flow in section.

Although the readings vary considerably across the section at the neutral p ositions, allfour positions are similar, confirming the fact that grain flow during manu facture wasradially sym metrical. The readings have been averaged for each radius and the curve inFigure 12 shows the way the average anisotropy varies across the section, indicatinggreater preferential orientation near the outside edge, as shown by the section in Figure I I.Positions A and E are similar since, in both, the outside edge of the ring has beenplastically stretched and the inside compressed. The two similar sets of readings at thesepositions have been averaged, and the averages of the neutral p ositions at correspondingradii have been subtracted, giving the curve shown in Figure 12. This procedure wasrepeated for the readings at positions C and G, which are also similar, giving the final curvein Figure 12.The two curves for the residual stress regions cannot be true representations of theresidual stresses across the sections, since the bending mom ents do not sum to zero, andin order to achieve a balanced distribution it is necessary to subtract further figures in bothcases. Whilst this again confirms the fact that induced birefringence is independent of thesign of plastic deform ation, it is difficult in this case to find a simple proce dure to yieldbalanced distributions. If the bending operation were of a simple nature, then no plasticdeformation wou ld occur at the centre of the section. The curve for positions C and Gshows this to be so for these positions, but this is certainly not true for positions A and E.



STRESS MEASUREMENT BY ULTRASONICS 191It appears that overall plastic strain has taken place in the latter case, and any furtherattempts at analysis would be pure conjecture unless some independent means of separatingresidual stress from the other effects can be found.

(al

OutsidsGcdge,,I 1 I

+ in ihtervals Inside edbe(cl

Figure 12. Anistropy in distorted ring: (a) neutral positions, B, D, F and H; (b) positions Cand G ; (c) positions A and E.7. CONCLUSIONS

Although the problem of residual stress analysis is by no means solved, it has been sho wnthat elastic anisotropy of small orders can be reliably measured, and that in cases whereanisotropy due to preferred grain orientation is of a regular or predictable nature so thatits variation from point to point can be allowed for, then residual stresses can be estimatedfrom the remaining anisotropy not accounted for by grain alignment.

The ap plication of more precise m easurement techniques, such as the sing-aroundsystem, to the pressure-coupled transducer will lead to simple measurem ents of greaterresolution, so that if a way is found for reliable separation of the effects of residual stressand preferred orientation, ultrasonic residual stress analysis will become a useful technique.

ACKNOWLEDGMENTSThe majority of this work was performed in the Electronic and Electrical Engineering

Departm ent of the University of Birmingham, and grateful thanks are due for the helpand advice of the Head of Departm ent, Professor D. G. Tucker and of Dr. L. Kay (nowProfessor at the University of Canterbury, N.Z.).

Thanks are also due to Rolls-Royce Ltd., Derby, for their generous financial supp ortof this project, and in particular to Mr. J. D. Hislop of this compan y for his interest andadvice and his help in supplying specimens for testing.The plastic strain measurem ents on the tensile test piece were carried out in LanchesterCollege, and Mr. G . A. Johnson of the Mechanical Engineering Departm ent is to bethanked for his kind assistance in operating the tensile test machine.



192 D. I. CRECRAFT

I.

2.3.4.

2:7.8.

9.

IO.II.12.13.14.IS.16.17.18.19.20.21.22.

REFERENCESH. A. ELION 1960 Non-destruct. Test. 18, 180. Recent advances in new areas of non-destructivetesting.R. W. BENSONand V. J. RAELSON1959 Product. Engng. 30. Acoustoelasticity.R. B. HEYWOOD 1952 Designing by Photoelasticity. Chapman and Hall.P. W. BRIDGMAN 948 Proc. Am. Acad. Arts Sci. 76, 55. The compression of 39 substances toI oo kgjcm2.D. S. HUGHESand J. L. KELLY 1953 Phys. Rev. 92,s. Second-order elastic deformation of solids.F. D. MURNAGHAN 95 I Finite Deformation of an Elastic Solid. London : J. Wiley.R. H. BERGMANand R. A. SHAHBENDER 958r. Appl. Phys. 29, 1736. Effect of staticallyapplied stresses on the velocity of propagation of ultrasonic waves.J. MCSKIMIN 1961 J. acoust. Sot. Am. 33, I. Pulse superposition method for measuring ultra-sonic wave velocities in solids.J. MCSKIMIN and P. ANDREATCH 962 J. acoust. Sot. Am. 34, 5. Analysis of the pulse super-position method for measuring ultrasonic wave velocities as a function of temperature andpressure.F. R. ROLLINS Dec. 1959 W.A.D.C. tech. Rep. 59-561. Study of methods for non-destructivemeasurement of residual stress.F. R. ROLLINS May 1961 W.A.D.C. tech. Rep. 61-42 Part I. Ultrasonic methods for non-destructive measurement of residual stress.R. P. ESPINOLAand P. C. WATERMAN19583. appl. Phys. 29,718. An unltrasonic interferometerfor measurement of velocity changes in solids.R. T. SMITH 1963 Ultrasonics I, 3. Stress-induced anisotropy in solids-the acoustoelasticeffect.N. P. CEDRONE nd D. R. CURRAN1954J. acoust. Sot. Am. 26,963. Electronic pulse method formeasuring the velocity of sound in liquids and solids.A. MYERS, L. MACKINNONand F. E. HOARE 1959J. acoust. Sot. Am. 31, 2. Modifications tostandard pulse techniques for ultrasonic velocity measurements.R. L. FORGACS196oJ. acoust. Sot. Am. 32, 12. Improvements in the sing-around technique forultrasonic velocity measurements.D. I. CRECRAFT1962 Nature, Lond. 195, No. 4847. Ultrasonic wave velocities in stressed nickel-steel.D. I. C CRAFT 1964J. Sound. Vib. I 4. Launching ultrasonic shear waves into solids at normalincidence by pressure coupling.P. F. SULLIVANand E. P. PAPADAKIS 962J. acoust. Sot. Am. 34. Ultrasonic double refraction inworked metals.G. BRADFIELD 1964 Use in Industry of Elasticity Measurements with the Help of MechanicalVibrations. N.P.L. Notes on Applied Science No. 30. London: H.M.S.O.G. E. DOAN 1953 The Principles of Physical Metallurgy, 3rd Ed. London: McGraw-Hill.E. OROWAN 1959 Internal Stresses and Fatigue in Metals, Gen. Motors Res. Labs. Elsevier.Pub. Co. Causes and effects of internal stresses.

The Measurement of Applied and Residual Stresses in Metals Using Ultrasonic Waves

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