- Email: [email protected]
Ultrasonic measurement of stresses
~ULTRASON1C.S FOR INDUSTRY 1967 1conference paper
Ultrasonic
measurement
of
1
stresses
D. I. Crecraft*
This paper describes the principle of attempts to measure stresses or strains in solids with ultrasonic waves, and records the progress made to date. The obvious application of the method would be the measurement of residual stress.
Photoelastic studies, strain gauges, and even ultrasonics, can indicate the additional strain or stress induced in a specimen by an applied load, but only ultrasonics is capable of detecting the presence of a residual stress or strain.
PHOTOELASTICITY AND SONOELASTICITY
I Axis of I transmitter
The use of ultrasonic waves in stress analysis is analogous to that of light waves in photoelasticity. Non-linearity exists in the stress-strain relationship of all materials--even in their ‘elastic’ regions; Hooke’s Law is, in fact, a very good linear approximation. This means that the elastic ‘constant’ of a solid varies under an applied stress. Since the velocity of a sound wave depends on the appropriate constant, the velocity changes slightly with stress. In photoelasticity the electrical permittivity is strain-dependent and the principal permittivities that occur are proportional to the principal stresses (or strains) and effective on the principal stress axes. Fig 1 shows how a polarized wave entering a transparent material (for example, a model of a part or a coating on the surface of a part) resolves into two components, each polarized along a principal stress axis. These two waves then travel with different velocities, which depend on the different permittivities and, in turn, on the principal stresses. Whereas the emerging waves as seen by the second polarizer are equal and opposite in space, they may or may not have the same phase depending on the number of wavelengths of ‘relative retardation’ they have suffered. This gives rise to the familiar ‘fringes’, the bands of light which appear across the model to connect points that have the same relative retardation. They are actually isobars. It is the velocity difference which makes this technique so powerful, since the actual velocity changes are quite small (about the same as the strain, in fact) and hence difficult to measure with any useful accuracy. We can use the same technique with our stress-dependent sound waves if we use shear waves, which are analogous to light waves in their transverse motion. Fig 1 applies equally well to shear waves, but there is an important difference in the effective resolution. The velocity difference is again of the same order as the difference between the principal stresses, but because the wavelength is typically one thousand times as long (6 X 10e4m for an ultrasonic wave at 4.7MHz in copper,and about 6 x 10-7m for sodium light), a stress that generates one thousand light fringes will only cause one cycle of phase shift between two shear waves. In practice phase shifts much less than this are encountered within the ‘elastic’ regions of metals.
through material
Components of Wave emerging from material
Fig 1 Section through XY plane of material under stress, showing the space intensities of light waves in photo. elasticity and ultrasonic shear waves in sonoelasticity, with ‘crossed’ polarizers
Gate pulse generator
Variable delay
-
-
Fixed delay
c Transmitter drive-pulse generator
Pulsed Hartley oscillator
-+.
ULTRASONIC MEASUREMENTS We now have a basis for measurement. We transmit a shear wave through a specimen, try to find the two principal stress axes in the plane normal to the direction of propagation, and measure the phase shift between the two emergent waves. Let us now consider some measurement of velocity changes,
* School of Engineering Science, University of Warwick, Coventry, CV4 7AL, England
fl
Quartz crystals\
Fig 2 The ‘sing-around’
system
ULTRASONICS April 1968
117
~ULTRASONICS
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ZIVDUS’XY
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1 conference paper
1
caused by applied stresses, which are necessary if we are to use this technique in stress analysis. Fig 2 shows the block diagram of the ‘sing-around’ velocity meter1 used.
8,)
Af -=_-_ppe f C
L
where c is velocity, f is the repetition frequency, p is Poisson’s ratio, and eL is the longitudinal strain. Thus in Fig 4, where the longitudinal strain is negative, the velocity changes have a slightly positive slope compared with the prf changes. The result is that longitudinal waves and perpendicularly polarized shear waves suffer very small changes in velocity. This confirms the futility of attempting to use longitudinal waves for residual stress analysis: measurements would be completely swamped by variations in composition and even by inaccuracies in measurements of path lengths. The longitudinal strain of the bar causes shear strain in planes parallel to the bar axis, so that the effective shear, or rigidity, modulus in these planes changes with the applied stress. The parallel-polarized shear wave shears in these planes and this explains its comparatively large velocity variation. The perpendicularly polarized wave shears in the plane of the bar cross-section, which suffers no shear strain under the axial loading.
I
‘8 to axis
(-I,, --(-a_$ -M_*_, ‘8
4.5 MHz shear wave polo ised parallel to a L
f
8’
c/d 419 0
2
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4
6
6
1012
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Compressive
.I
I
1416
stress
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The difference in the zero-stress prf’s of the shear waves has no significance, since triggering was taking place at different parts of the received echoes in each case, although we shall see later that a difference in velocities does exist. Fig 5 shows measurements on a bar of the same nickel-steel as above, and confirms that the same effects occur in tension. A slight difference in slopes is attributable to, the use of different machines for tensile and compressive loading. Figs 6 and 7 show similar results obtained in aluminium and copper. Note that here the sing-around repetition period was measured rather than the frequency, so that all the slopes have opposite signs to those in Fig 4. There is some advantage in using the multiple-period averaging facility of a counter/timer, because much better resolution can be obtained in a much shorter time than that given by a frequency measurement when the prf is in the region of 1OOkHzor less. Most of the above measurements illustrate a linear relationship between applied stress and ultrasonic velocity, and it seems likely that such curvature.as does exist in this case is caused by bending of the specimen in Fig 4, and creep and/or disruption of the transducer bonds in Figs 6 and 7. C
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(b) ‘Singing around’ after 3 trips (a) Not ‘singing around’ Fig 3 ‘Sing-around’ oscillograms (d) As (c) but showing the transmitter pulse (lowertrace) (c) (b) expanded X 5 (e) ‘Singing around’ after 5 trips . (f) ‘Sfnging around’ after ‘7 trips Upper traces output from receiver crystal; lower traces output from echo amplifier (except in (d))
118
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Fig 4 Variation of ‘sing-around’ prf with stress in nickel steel S/NTV (1 lbf/in2 = 6.9 x 103N/m2)
b
a r
lOV/cm 1 +
‘I\* 4.5MHz shear wave polarized perpendicular of stress
/
Measurements were made in metallic specimens, each of which was in the form of a bar. Piezoelectric crystals were mounted on two opposite faces at the centre, and waves passed through the specimen. Uniaxial stress was applied along the axis of the bar. The crystals used were first longitudinalwave types and then shear-wave types, mounted first with their polarization parallel to the axis and then perpendicular to it. In other words, in all cases a wave was propagated normal to the principal stress axes and, in the case of the shear waves,polarization was parallel to one and then the other principal axis, the second principal stress being zero.
AC
wave
‘8
A selected echo from the transmitted pulse is used to retrigger the transmitter so that the system recycles, or ‘sings around’, with a period of repetition which equals the time taken for the selected echo to arrive at the receiving transducer. The echo is selected manually by adjustment of the variable delay in the ‘sing-around’ echo selector. The astable multivibrator is used to start the cycle of operations and is switched off when the system is correctly retriggering. The repetition frequency (prf), or period, is monitored by a digital counter/timer, and it is this which gives high resolution. However, absolute accuracy will be impaired by delays in triggering on the selected cycle of the selected echo and in the electronic circuitry. Fig 3 shows the oscillograms obtained.
Fig 4 shows measurements on nickel steel. Note the small fractional repetition frequency change. To find the acoustic velocity change we must allow for the change in path length caused by the transverse strain of the bar undergoing longitudinal strain:
2.5MHzlongitudinal ‘8.)
ULTRASONICS April 1968
10 pseclcm -
lULTRASONZCS FOR INDUSTRY 1967 I conferenceDaDer
I
THIRD-ORDER ELASTIC CONSTANTS The linear variation of velocity with stress is explicable in terms of an expression for strain energy with third-order coefficients in addition to the first- and second-order coefficients. Upon differentiation with respect to strain, this expression becomes an expression for stress in terms of a first-order coefficient (corresponding to initial hydrostatic strain), products of strains and second-order coefficients (the latter thus being the familiar elastic constants), and products of second-degree strains and third-order coefficients. These third-order coefficients can be shown2 to explain a linear relation between stress and acoustic velocity, providing the strain and velocity changes are small-which of course they are. The values of these constants3 are of no interest in the present discussion. wave
polarized
SHEAR-WAVE PROPAGATION FOR MEASUREMENT OF RESIDUAL EFFECTS We have seen that we must measure the velocity difference between two shear waves, that is, the birefringence, in order to estimate stress. Moreover, these waves must traverse the same path and hence must be launched at normal incidence to the surface of the specimen. This means that a shear wedge or any other mode-conversion technique cannot be used at the surface. On the other hand,an oil film will not support shear waves, and a solid bond is obviously out of the question in a practical situation, even if we did not consider the fact that the transducer must be turned in order to change the polarization angle; and variations in the thickness of bond would swamp the birefringence measurements. The problem was solved by using the pressure-coupled transducer shown in Fig 8. This is pressed onto the surface with an apparent interfacial pressure which is below the elastic limit of either the material (steel) of the coupling piece of the transducer or the specimen; it causes nearperfect transmission of the shear wave generated by the piezoelectric disc mounted within it (Fig 9).
4.7 MHz
8
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wave /’
15.5901 0
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Fig 6 Variation of ‘sing-around’ aluminium (1 lbf = 4.4N)
’
’ 2500
period with stress in
35.030 Shear wave polarized to stress 4.5 MHz
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35.020 E
parahel 3 trips
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.
35.010
Shear wave polarized perpedicular to stress 4% MHz 3 trips
z 2 34.660 &Z .p -0 g
t
to stress
3 trips
15.600
It was found convenient to use the transducer in the common transmit/receive connection, so that the complication of mounting two transducers was avoided, and the coupling force was applied simply via the steel ball. At this stage the sing-around system could not cope with a common transmit/receive connection nor with the multiple echoes which occurred within the transducer when coupling was not quite perfect. Measurements were therefore made by using the fastest time base speed of a delayed-sweep
perpendicular
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08
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Longitudinal wave 3 MHz 5 trips
A 27.990 Shear wave r’ polarized parallel $ to stress axis ti
34850
I Fig 5 Variation of ‘sing-around’ prf with compressive and tensile stress in nickel steel (1 tonf/in2’ = 15.4 X 10sN/m2)
1
0
’
’
’
’
’
’
’
’
’
’
16 Cort?pressive’stress
Fig 7 Variation of ‘sing-around’
~03tbf/in$
period with stress in copper
oscilloscope. The transducer was pressed onto the surface of the specimen and the echo observed. It was found possible, on a reasonably well-finished surface, to adjust the pressure so that the transducer could be rotated when a good echo was observable. With the small degree of birefringence encountered, the two resolved components of the transmitted wave suffer less than one cycle phase shift, and the received signal simply appears as one echo with a phase shift of maximum and minimum values as the transducer is rotated. ULTRASONICS April 1968
119
~ULT’RASOAUCSFOR NDUSI’RY 1967 1conference
paper ]
It can be shown that, if the actual relative phase is $, then the apparent phase (Y of the received wave varies from zero to 4, as the transducer polarization is rotated from a direction parallel to one principal axis to the other at right-angles to this, following the law o = @sin28
0.250 in steel ball
Cap
(2)
where f3is the angle through which the transducer polarization is turned. So the accuracy of the reading does not depend critically on the angle of polarization: on the other hand, the location of the principal axis is not very precise. This means that the value of the birefringence and the approximate location of the principal axes can be determined fairly quickly. With a time base speed of lOOnsec/cm, a display resolution of lmm corresponding to lOnsec,and a lin thick steel specimen with an echo-return time of about 16psec, the resolution is afractionalvelocitydifference of about 6 X 10-d corresponding to a stress of about 5tonf/in.a
Holes for connecting wires Fig 8 Single-crystal
transducer (lin = 2. 5cm)
PREFERRED ORIENTATION Application of the above technique to the bars on which the sing-around measurements were made revealed immediately degrees of residual birefringence of the same order as, or larger than, that due to applied stresses. These residual effects were constant throughout the length of the bars and clearly indicated elastic anisotropy, caused by preferential alignment of the crystallographic axes of their grains. The anisotropic acids of symmetry was parallel to the bar axis in each case, presumably because of the rolling/extrusion process used in shaping the bar, and in all cases but one (the copper bar), the wave which was polarized parallel to the axis propagated with the greater velocity. It is well known that rolling, extrusion and drawing processes produce elastic anisotropy. The increased strength of wires is due to preferential alignment of the grains. But in many engineered components, where efforts have been made to keep preferred orientation small, anisotropy cannot be measured with any better a tool than ultrasonics. Even x-rays are limited to a resoltuion of a few percent preferred orientation, and that only near the surface. Ultrasonic birefringence can readily detect 0. l%, and even this small amount has the same birefringence as a stress of about 5tonf/in2 (-75 x 10sN/m2) in steel. The observed fractional velocity differences ranged from 0. 2% in nickel-steel to 17% in a piece of Nimonic alloy (NIM 80A). In general, then, attempts to measure residual stress will always be confused by the existence of birefringence, which is caused by preferred grain orientation. Such grain orientation cannot be eliminated, since no other method is available for detecting it with a resolution which even approaches that needed to measure stress. We can always measure applied stresses, of course, because we then simply look for the increase in birefringence when the load is applied. There may be cases when this would be useful, since a measure of the average stress over the path length is obtained, but in general a strain gauge measuring the surface stress is of more value. It is worth pointing out that surface-applied stress can be measured by ultrasonic surface waves, by using an ultrasonic goniometer4 to measure the variation in velocity of the surface waves. PLASTIC DEFORMATION Measurement of the birefringence of a nickel-steel specimen, subject to about 4% permanent longitudinal strain, showed a rise from 0.2% before to 0.4% after strain.and after removal of the tensile stress. Since this level of plastic strain cannot induce appreciable grain alignment, it seems clear that the increased birefringence was due to dislocation movement. Many other workers in this field feel that dislocation effects are largely responsible for elastic anisotropy and the consequent birefringence. The interesting point here is that compressive (uniaxial) plastic strain might also be expected to increase birefringence, whereas birefringence is sensitive to the sign of elastic strain. This led to an attempt to measure residual stresses in a plastically bent bar. This was again nickel steel, and an 120
ULTRASONICS April 1968
10000
20000
Interface pressure [lbf/in2] I
0
I
500 Transducer
1000 load [lbf]
Fig 9 Pressure-coupling characteristics of V4in 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’
initial constant birefringence along the length and across the width of the bar was determined by the echo return time of the parallel-polarized wave; this was 30nsec less than that of the perpendicularly polarized wave. Fig 10 shows the idealized stress distribution during bending on the assumption that all parts of the cross-section suffer plastic strain, and the theoretical residual stress distribution after removal of the load. Under plastic bending, the maximum stress is the yield stress. Upon removal of the load, the inner part of the bar, in an effort to straighten, exerts elastic stress on the outer layers, which are then stressed elastically with a stress opposite to that which caused their deformation; the inner layers are stressed elastically in the same sense as before, with a theoretical maximum equal to the yield stress on the centre, or neutral, axis. The original 30nsec were subtracted from the measurements made after bending (shown in Fig 10 in units of lonsec), and a further 25nsec were then subtracted from all readings except that at the centre, on the basis of the above argument that plastic deformation, regardless of sign, increases birefringence. The resultant set of figures, in units of 1Onsec corresponding to 5tonf/in2 (as shown before) represents an almost balanced stress distribution, with a maximum stress of 50 tonf/in2 in the region where the yield stress of SO tonf/ in2 is the maximum possible. This then appears to confirm
ULTRASONICS
FOR
lNDuSTRY
that birefringence increases, regardless of the sign of plastic strain, but this is of limited help generally in separating texture effects from residual stress effects.
2967
Compressive -
1 conference paper 1
Tensile )
FREQUENCY-SENSITIVE EFFECTS Recent work by P. Mahadevans has shown that stress-induced birefringence is insensitive to the ultrasonic frequency used over the range 0. 5-6. OMHz, but that texture-induced, or residual, birefringence changes markedly with frequency. It can be shown that a visco-elastically damped vibrating string model of a vibrating dislocation, which is pinned at each end, affects the rigidity, or shear, modulus so that at low frequencies, where the damping is ineffective, the modulus is constant. With rising frequency, the damping makes the dislocation look even ‘stiffer’, with a consequent rise in modulus proportional to frequency. At even higher frequencies, unattainable in practice because of excessive attenuation, the modulus levels out again, because the dislocation cannot follow the alternating stress field.
a
-50 +7.5 + 10 - 10
An extension of this hypothesis, to a material that has suffered preferential dislocation alignment by uniaxial strain, shows that the change in rigidity modulus with frequency will be direction-dependent. The difference in the rigidity moduli, and hence the ultrasonic birefringence, will be frequencydependent.
+5.0 +5.0
Fig 11 is representative of Mahadevan’s measurements, a full account of which is yet to be published. This shows that normal ultrasonic testing frequencies are in the mid-frequency range, and that the lower frequency range occurs below about 500kHz. It is hoped that measurement of birefringence on a particular specimen at a few frequencies, covering the low and mid-frequency ranges, will be sufficient to estimate residual stress. The slope in the mid-frequency range is proportional to the difference in dislocation densities parallel to the principal anisotropy axes, and this should predict a low frequency birefringence figure. A departure from this figure at low frequencies should indicate the presence of additional birefringence, caused by stress.
b
C
CONCLUSION The problem of detecting residual stresses, in the general case, is still not solved, but it seems that we may have a solution for some cases. This remains to be proved.
positions
d
Fig 10 The bent bar
ACKNOWLEDGEMENTS The author is indebted to Mr P. Mahadevan for permission to discuss his work prior to publication; to Professor D. G. Tucker of the University of Birmingham where most of the author’s work was carried out; to Professor L. Kay, now of the University of Canterbury, NZ; and to Rolls-Royce Ltd, on whose behalf, and with whose financial support, all the work described was performed.
(a)
Stress distribution during plastic strain
(b)
Stress distribution after removal of load
(c)
Sonoelastic readings. (Compression 5 tonf/i$ = 77 x 10sN/m2)
1.0 =
(d) Readings after subtration of 3.0 owing to original preferred orientation
REFERENCES Myers, A., Mackinnon, L., Hoare, F. E. ‘Modifications to standard pulse techniques for ultrasonic velocity measurements’, Journal of the Acoustical Society of America, Vol 31, No 2 (1959) Hughes, D. S., Kelly, J. L. ‘Second-order elastic deformation of solids’, Physical Review, Vol 92, No 5 (1953)
/
g
//
Crecraft, D. I. ‘The measurement of applied and residual stresses in metals using ultrasonic waves’, Journal of Sound and Vibration, Vol 5, No 1 (1967) Bradfield, G. ‘Use in industry of elasticity measurements with the help of mechanical vibrations’, National Physical Laboratory Notes on Applied Science, No 30, (Her Majesty’s Stationery Office, London 1964) Mahadevan, P. Nature, Vol 211, No 5049 (August 1966) p 621
I
0.5
I
1.0
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2.0
3.0 Frequency
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4.0
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50
I
60
[MHz]
Fig 11 Typical variation of birefringence
with frequency
ULTRASONICS April 1968
121
Ultrasonic
measurement
of
1
stresses
D. I. Crecraft*
This paper describes the principle of attempts to measure stresses or strains in solids with ultrasonic waves, and records the progress made to date. The obvious application of the method would be the measurement of residual stress.
Photoelastic studies, strain gauges, and even ultrasonics, can indicate the additional strain or stress induced in a specimen by an applied load, but only ultrasonics is capable of detecting the presence of a residual stress or strain.
PHOTOELASTICITY AND SONOELASTICITY
I Axis of I transmitter
The use of ultrasonic waves in stress analysis is analogous to that of light waves in photoelasticity. Non-linearity exists in the stress-strain relationship of all materials--even in their ‘elastic’ regions; Hooke’s Law is, in fact, a very good linear approximation. This means that the elastic ‘constant’ of a solid varies under an applied stress. Since the velocity of a sound wave depends on the appropriate constant, the velocity changes slightly with stress. In photoelasticity the electrical permittivity is strain-dependent and the principal permittivities that occur are proportional to the principal stresses (or strains) and effective on the principal stress axes. Fig 1 shows how a polarized wave entering a transparent material (for example, a model of a part or a coating on the surface of a part) resolves into two components, each polarized along a principal stress axis. These two waves then travel with different velocities, which depend on the different permittivities and, in turn, on the principal stresses. Whereas the emerging waves as seen by the second polarizer are equal and opposite in space, they may or may not have the same phase depending on the number of wavelengths of ‘relative retardation’ they have suffered. This gives rise to the familiar ‘fringes’, the bands of light which appear across the model to connect points that have the same relative retardation. They are actually isobars. It is the velocity difference which makes this technique so powerful, since the actual velocity changes are quite small (about the same as the strain, in fact) and hence difficult to measure with any useful accuracy. We can use the same technique with our stress-dependent sound waves if we use shear waves, which are analogous to light waves in their transverse motion. Fig 1 applies equally well to shear waves, but there is an important difference in the effective resolution. The velocity difference is again of the same order as the difference between the principal stresses, but because the wavelength is typically one thousand times as long (6 X 10e4m for an ultrasonic wave at 4.7MHz in copper,and about 6 x 10-7m for sodium light), a stress that generates one thousand light fringes will only cause one cycle of phase shift between two shear waves. In practice phase shifts much less than this are encountered within the ‘elastic’ regions of metals.
through material
Components of Wave emerging from material
Fig 1 Section through XY plane of material under stress, showing the space intensities of light waves in photo. elasticity and ultrasonic shear waves in sonoelasticity, with ‘crossed’ polarizers
Gate pulse generator
Variable delay
-
-
Fixed delay
c Transmitter drive-pulse generator
Pulsed Hartley oscillator
-+.
ULTRASONIC MEASUREMENTS We now have a basis for measurement. We transmit a shear wave through a specimen, try to find the two principal stress axes in the plane normal to the direction of propagation, and measure the phase shift between the two emergent waves. Let us now consider some measurement of velocity changes,
* School of Engineering Science, University of Warwick, Coventry, CV4 7AL, England
fl
Quartz crystals\
Fig 2 The ‘sing-around’
system
ULTRASONICS April 1968
117
~ULTRASONICS
FOR
ZIVDUS’XY
1967
1 conference paper
1
caused by applied stresses, which are necessary if we are to use this technique in stress analysis. Fig 2 shows the block diagram of the ‘sing-around’ velocity meter1 used.
8,)
Af -=_-_ppe f C
L
where c is velocity, f is the repetition frequency, p is Poisson’s ratio, and eL is the longitudinal strain. Thus in Fig 4, where the longitudinal strain is negative, the velocity changes have a slightly positive slope compared with the prf changes. The result is that longitudinal waves and perpendicularly polarized shear waves suffer very small changes in velocity. This confirms the futility of attempting to use longitudinal waves for residual stress analysis: measurements would be completely swamped by variations in composition and even by inaccuracies in measurements of path lengths. The longitudinal strain of the bar causes shear strain in planes parallel to the bar axis, so that the effective shear, or rigidity, modulus in these planes changes with the applied stress. The parallel-polarized shear wave shears in these planes and this explains its comparatively large velocity variation. The perpendicularly polarized wave shears in the plane of the bar cross-section, which suffers no shear strain under the axial loading.
I
‘8 to axis
(-I,, --(-a_$ -M_*_, ‘8
4.5 MHz shear wave polo ised parallel to a L
f
8’
c/d 419 0
2
I
I
I
I
4
6
6
1012
I
Compressive
.I
I
1416
stress
I
[lG3tbf/in2
J
J
The difference in the zero-stress prf’s of the shear waves has no significance, since triggering was taking place at different parts of the received echoes in each case, although we shall see later that a difference in velocities does exist. Fig 5 shows measurements on a bar of the same nickel-steel as above, and confirms that the same effects occur in tension. A slight difference in slopes is attributable to, the use of different machines for tensile and compressive loading. Figs 6 and 7 show similar results obtained in aluminium and copper. Note that here the sing-around repetition period was measured rather than the frequency, so that all the slopes have opposite signs to those in Fig 4. There is some advantage in using the multiple-period averaging facility of a counter/timer, because much better resolution can be obtained in a much shorter time than that given by a frequency measurement when the prf is in the region of 1OOkHzor less. Most of the above measurements illustrate a linear relationship between applied stress and ultrasonic velocity, and it seems likely that such curvature.as does exist in this case is caused by bending of the specimen in Fig 4, and creep and/or disruption of the transducer bonds in Figs 6 and 7. C
I
I
1
I
0.5Vlc m
O.SV/cm JLXYiI
1OVlcm i
0.5Vlcm t
--we
0.5Vlcm t
100Vlcm t
pb--
1OVlcm t l.Opsec/cm -
T
m
c-trc--t---+--4c w
R
0.5Vlcm
1
‘-t--+-t_tt3--r-
lOV/cm
1
m
5 pscc km -
(b) ‘Singing around’ after 3 trips (a) Not ‘singing around’ Fig 3 ‘Sing-around’ oscillograms (d) As (c) but showing the transmitter pulse (lowertrace) (c) (b) expanded X 5 (e) ‘Singing around’ after 5 trips . (f) ‘Sfnging around’ after ‘7 trips Upper traces output from receiver crystal; lower traces output from echo amplifier (except in (d))
118
I
162022
Fig 4 Variation of ‘sing-around’ prf with stress in nickel steel S/NTV (1 lbf/in2 = 6.9 x 103N/m2)
b
a r
lOV/cm 1 +
‘I\* 4.5MHz shear wave polarized perpendicular of stress
/
Measurements were made in metallic specimens, each of which was in the form of a bar. Piezoelectric crystals were mounted on two opposite faces at the centre, and waves passed through the specimen. Uniaxial stress was applied along the axis of the bar. The crystals used were first longitudinalwave types and then shear-wave types, mounted first with their polarization parallel to the axis and then perpendicular to it. In other words, in all cases a wave was propagated normal to the principal stress axes and, in the case of the shear waves,polarization was parallel to one and then the other principal axis, the second principal stress being zero.
AC
wave
‘8
A selected echo from the transmitted pulse is used to retrigger the transmitter so that the system recycles, or ‘sings around’, with a period of repetition which equals the time taken for the selected echo to arrive at the receiving transducer. The echo is selected manually by adjustment of the variable delay in the ‘sing-around’ echo selector. The astable multivibrator is used to start the cycle of operations and is switched off when the system is correctly retriggering. The repetition frequency (prf), or period, is monitored by a digital counter/timer, and it is this which gives high resolution. However, absolute accuracy will be impaired by delays in triggering on the selected cycle of the selected echo and in the electronic circuitry. Fig 3 shows the oscillograms obtained.
Fig 4 shows measurements on nickel steel. Note the small fractional repetition frequency change. To find the acoustic velocity change we must allow for the change in path length caused by the transverse strain of the bar undergoing longitudinal strain:
2.5MHzlongitudinal ‘8.)
ULTRASONICS April 1968
10 pseclcm -
lULTRASONZCS FOR INDUSTRY 1967 I conferenceDaDer
I
THIRD-ORDER ELASTIC CONSTANTS The linear variation of velocity with stress is explicable in terms of an expression for strain energy with third-order coefficients in addition to the first- and second-order coefficients. Upon differentiation with respect to strain, this expression becomes an expression for stress in terms of a first-order coefficient (corresponding to initial hydrostatic strain), products of strains and second-order coefficients (the latter thus being the familiar elastic constants), and products of second-degree strains and third-order coefficients. These third-order coefficients can be shown2 to explain a linear relation between stress and acoustic velocity, providing the strain and velocity changes are small-which of course they are. The values of these constants3 are of no interest in the present discussion. wave
polarized
SHEAR-WAVE PROPAGATION FOR MEASUREMENT OF RESIDUAL EFFECTS We have seen that we must measure the velocity difference between two shear waves, that is, the birefringence, in order to estimate stress. Moreover, these waves must traverse the same path and hence must be launched at normal incidence to the surface of the specimen. This means that a shear wedge or any other mode-conversion technique cannot be used at the surface. On the other hand,an oil film will not support shear waves, and a solid bond is obviously out of the question in a practical situation, even if we did not consider the fact that the transducer must be turned in order to change the polarization angle; and variations in the thickness of bond would swamp the birefringence measurements. The problem was solved by using the pressure-coupled transducer shown in Fig 8. This is pressed onto the surface with an apparent interfacial pressure which is below the elastic limit of either the material (steel) of the coupling piece of the transducer or the specimen; it causes nearperfect transmission of the shear wave generated by the piezoelectric disc mounted within it (Fig 9).
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wave /’
15.5901 0
’
’
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’
’
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’ 2000 [Ibf]
co$%sivFPdbd
Fig 6 Variation of ‘sing-around’ aluminium (1 lbf = 4.4N)
’
’ 2500
period with stress in
35.030 Shear wave polarized to stress 4.5 MHz
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35.020 E
parahel 3 trips
18\ $‘8
.
35.010
Shear wave polarized perpedicular to stress 4% MHz 3 trips
z 2 34.660 &Z .p -0 g
t
to stress
3 trips
15.600
It was found convenient to use the transducer in the common transmit/receive connection, so that the complication of mounting two transducers was avoided, and the coupling force was applied simply via the steel ball. At this stage the sing-around system could not cope with a common transmit/receive connection nor with the multiple echoes which occurred within the transducer when coupling was not quite perfect. Measurements were therefore made by using the fastest time base speed of a delayed-sweep
perpendicular
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1
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8 8/
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08
08 A
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A 27.990 Shear wave r’ polarized parallel $ to stress axis ti
34850
I Fig 5 Variation of ‘sing-around’ prf with compressive and tensile stress in nickel steel (1 tonf/in2’ = 15.4 X 10sN/m2)
1
0
’
’
’
’
’
’
’
’
’
’
16 Cort?pressive’stress
Fig 7 Variation of ‘sing-around’
~03tbf/in$
period with stress in copper
oscilloscope. The transducer was pressed onto the surface of the specimen and the echo observed. It was found possible, on a reasonably well-finished surface, to adjust the pressure so that the transducer could be rotated when a good echo was observable. With the small degree of birefringence encountered, the two resolved components of the transmitted wave suffer less than one cycle phase shift, and the received signal simply appears as one echo with a phase shift of maximum and minimum values as the transducer is rotated. ULTRASONICS April 1968
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~ULT’RASOAUCSFOR NDUSI’RY 1967 1conference
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It can be shown that, if the actual relative phase is $, then the apparent phase (Y of the received wave varies from zero to 4, as the transducer polarization is rotated from a direction parallel to one principal axis to the other at right-angles to this, following the law o = @sin28
0.250 in steel ball
Cap
(2)
where f3is the angle through which the transducer polarization is turned. So the accuracy of the reading does not depend critically on the angle of polarization: on the other hand, the location of the principal axis is not very precise. This means that the value of the birefringence and the approximate location of the principal axes can be determined fairly quickly. With a time base speed of lOOnsec/cm, a display resolution of lmm corresponding to lOnsec,and a lin thick steel specimen with an echo-return time of about 16psec, the resolution is afractionalvelocitydifference of about 6 X 10-d corresponding to a stress of about 5tonf/in.a
Holes for connecting wires Fig 8 Single-crystal
transducer (lin = 2. 5cm)
PREFERRED ORIENTATION Application of the above technique to the bars on which the sing-around measurements were made revealed immediately degrees of residual birefringence of the same order as, or larger than, that due to applied stresses. These residual effects were constant throughout the length of the bars and clearly indicated elastic anisotropy, caused by preferential alignment of the crystallographic axes of their grains. The anisotropic acids of symmetry was parallel to the bar axis in each case, presumably because of the rolling/extrusion process used in shaping the bar, and in all cases but one (the copper bar), the wave which was polarized parallel to the axis propagated with the greater velocity. It is well known that rolling, extrusion and drawing processes produce elastic anisotropy. The increased strength of wires is due to preferential alignment of the grains. But in many engineered components, where efforts have been made to keep preferred orientation small, anisotropy cannot be measured with any better a tool than ultrasonics. Even x-rays are limited to a resoltuion of a few percent preferred orientation, and that only near the surface. Ultrasonic birefringence can readily detect 0. l%, and even this small amount has the same birefringence as a stress of about 5tonf/in2 (-75 x 10sN/m2) in steel. The observed fractional velocity differences ranged from 0. 2% in nickel-steel to 17% in a piece of Nimonic alloy (NIM 80A). In general, then, attempts to measure residual stress will always be confused by the existence of birefringence, which is caused by preferred grain orientation. Such grain orientation cannot be eliminated, since no other method is available for detecting it with a resolution which even approaches that needed to measure stress. We can always measure applied stresses, of course, because we then simply look for the increase in birefringence when the load is applied. There may be cases when this would be useful, since a measure of the average stress over the path length is obtained, but in general a strain gauge measuring the surface stress is of more value. It is worth pointing out that surface-applied stress can be measured by ultrasonic surface waves, by using an ultrasonic goniometer4 to measure the variation in velocity of the surface waves. PLASTIC DEFORMATION Measurement of the birefringence of a nickel-steel specimen, subject to about 4% permanent longitudinal strain, showed a rise from 0.2% before to 0.4% after strain.and after removal of the tensile stress. Since this level of plastic strain cannot induce appreciable grain alignment, it seems clear that the increased birefringence was due to dislocation movement. Many other workers in this field feel that dislocation effects are largely responsible for elastic anisotropy and the consequent birefringence. The interesting point here is that compressive (uniaxial) plastic strain might also be expected to increase birefringence, whereas birefringence is sensitive to the sign of elastic strain. This led to an attempt to measure residual stresses in a plastically bent bar. This was again nickel steel, and an 120
ULTRASONICS April 1968
10000
20000
Interface pressure [lbf/in2] I
0
I
500 Transducer
1000 load [lbf]
Fig 9 Pressure-coupling characteristics of V4in 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’
initial constant birefringence along the length and across the width of the bar was determined by the echo return time of the parallel-polarized wave; this was 30nsec less than that of the perpendicularly polarized wave. Fig 10 shows the idealized stress distribution during bending on the assumption that all parts of the cross-section suffer plastic strain, and the theoretical residual stress distribution after removal of the load. Under plastic bending, the maximum stress is the yield stress. Upon removal of the load, the inner part of the bar, in an effort to straighten, exerts elastic stress on the outer layers, which are then stressed elastically with a stress opposite to that which caused their deformation; the inner layers are stressed elastically in the same sense as before, with a theoretical maximum equal to the yield stress on the centre, or neutral, axis. The original 30nsec were subtracted from the measurements made after bending (shown in Fig 10 in units of lonsec), and a further 25nsec were then subtracted from all readings except that at the centre, on the basis of the above argument that plastic deformation, regardless of sign, increases birefringence. The resultant set of figures, in units of 1Onsec corresponding to 5tonf/in2 (as shown before) represents an almost balanced stress distribution, with a maximum stress of 50 tonf/in2 in the region where the yield stress of SO tonf/ in2 is the maximum possible. This then appears to confirm
ULTRASONICS
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that birefringence increases, regardless of the sign of plastic strain, but this is of limited help generally in separating texture effects from residual stress effects.
2967
Compressive -
1 conference paper 1
Tensile )
FREQUENCY-SENSITIVE EFFECTS Recent work by P. Mahadevans has shown that stress-induced birefringence is insensitive to the ultrasonic frequency used over the range 0. 5-6. OMHz, but that texture-induced, or residual, birefringence changes markedly with frequency. It can be shown that a visco-elastically damped vibrating string model of a vibrating dislocation, which is pinned at each end, affects the rigidity, or shear, modulus so that at low frequencies, where the damping is ineffective, the modulus is constant. With rising frequency, the damping makes the dislocation look even ‘stiffer’, with a consequent rise in modulus proportional to frequency. At even higher frequencies, unattainable in practice because of excessive attenuation, the modulus levels out again, because the dislocation cannot follow the alternating stress field.
a
-50 +7.5 + 10 - 10
An extension of this hypothesis, to a material that has suffered preferential dislocation alignment by uniaxial strain, shows that the change in rigidity modulus with frequency will be direction-dependent. The difference in the rigidity moduli, and hence the ultrasonic birefringence, will be frequencydependent.
+5.0 +5.0
Fig 11 is representative of Mahadevan’s measurements, a full account of which is yet to be published. This shows that normal ultrasonic testing frequencies are in the mid-frequency range, and that the lower frequency range occurs below about 500kHz. It is hoped that measurement of birefringence on a particular specimen at a few frequencies, covering the low and mid-frequency ranges, will be sufficient to estimate residual stress. The slope in the mid-frequency range is proportional to the difference in dislocation densities parallel to the principal anisotropy axes, and this should predict a low frequency birefringence figure. A departure from this figure at low frequencies should indicate the presence of additional birefringence, caused by stress.
b
C
CONCLUSION The problem of detecting residual stresses, in the general case, is still not solved, but it seems that we may have a solution for some cases. This remains to be proved.
positions
d
Fig 10 The bent bar
ACKNOWLEDGEMENTS The author is indebted to Mr P. Mahadevan for permission to discuss his work prior to publication; to Professor D. G. Tucker of the University of Birmingham where most of the author’s work was carried out; to Professor L. Kay, now of the University of Canterbury, NZ; and to Rolls-Royce Ltd, on whose behalf, and with whose financial support, all the work described was performed.
(a)
Stress distribution during plastic strain
(b)
Stress distribution after removal of load
(c)
Sonoelastic readings. (Compression 5 tonf/i$ = 77 x 10sN/m2)
1.0 =
(d) Readings after subtration of 3.0 owing to original preferred orientation
REFERENCES Myers, A., Mackinnon, L., Hoare, F. E. ‘Modifications to standard pulse techniques for ultrasonic velocity measurements’, Journal of the Acoustical Society of America, Vol 31, No 2 (1959) Hughes, D. S., Kelly, J. L. ‘Second-order elastic deformation of solids’, Physical Review, Vol 92, No 5 (1953)
/
g
//
Crecraft, D. I. ‘The measurement of applied and residual stresses in metals using ultrasonic waves’, Journal of Sound and Vibration, Vol 5, No 1 (1967) Bradfield, G. ‘Use in industry of elasticity measurements with the help of mechanical vibrations’, National Physical Laboratory Notes on Applied Science, No 30, (Her Majesty’s Stationery Office, London 1964) Mahadevan, P. Nature, Vol 211, No 5049 (August 1966) p 621
I
0.5
I
1.0
I
I
2.0
3.0 Frequency
I
4.0
I
50
I
60
[MHz]
Fig 11 Typical variation of birefringence
with frequency
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