The use of non-collinear mixing for nonlinear ultrasonic detection of plasticity and fatigue Anthony J.Croxford,a)Paul D.Wilcox,and Bruce W.Drinkwater Department of Mechanical Engineering,University of Bristol,Bristol BS81TR,United Kingdom
ford@bristol.ac.uk,p.wilcox@bristol.ac.uk,
b.drinkwater@bristol.a
c.uk
Peter B.Nagy
Department of Aerospace Engineering,University of Cincinnati,Cincinnati,Ohio45221
peter.nagy@uc.edu
Abstract:This letter reports on the application of the non-collinear mixing
technique to the ultrasonic measurement of material nonlinearity to assess
plasticity and fatigue damage.Non-collinear mixing is potentially more at-
tractive for assessing material state than other nonlinear ultrasonic tech-
niques because system nonlinearities can be both independently measured
and largely eliminated.Here,measurements made on a sample after plastic
deformation and on a sample subjected to low-cycle fatigue show that the
non-collinear technique is indeed capable of measuring changes in both,
and is therefore a viable inspection technique for these types of material
degradation.
©2009Acoustical Society of America
P ACS numbers:43.35.Zc,43.35.Yb,43.25.Zx[JM]
Date Received:July29,2009Date Accepted:August26,2009
1.Introduction
Nonlinear ultrasonic measurements enable the detection of the onset of plastic deformation and fatigue damage at an earlier stage than conventional linear nondestructive testing(NDT)tech-niques,which have insufficient sensitivity to the changes in the microstructure brought on by dislocation movements.Finite-deformation elastic theory introduces three independent con-stants,referred to as third order elastic constants(TOECs),which describe the nonlinear stress-strain behavior in an isotropic material.1,2Different sets of independent TOECs have been pro-posed by various authors,including A,B,and C used by Landau and Lifshitz,1which are a linear combination of the l,m,and n Murnaghan constants.2
Of practical interest is the dependence of TOECs on the level of plastic strain or fa-tigue damage induced dislocation accumulation in a material.Various ultrasonic methods of measuring material nonlinearity have been developed.Thefirst makes use of the so-called acousto-elastic effect.3,4In this case the nonlinear behavior manifests itself through variations in ultrasonic propagation velocity with applied strain.Through the application of different wave types and the measurement of velocity in unstrained and strained states all three TOECs can be measured.One problem with this technique is the difficulty of measuring the small changes in propagation time and distance accurately enough to allow the velocity,and from that the TO-ECs,to be determined.A second problem is the necessity of loading a specimen to measure the changes in velocity.
The second and perhaps most widely reported method for interrogating material non-linearity is the harmonic generation technique.5–8If ultrasonic energy at one frequency is in-jected into a material,harmonics of the input frequency are generated due to nonlinearity as the ultrasound propagates.By measuring the magnitude of the harmonics the degree of material nonlinearity can be quantified.There is a considerable body of experimental evidence that
a͒Author to whom correspondence should be addressed.
shows a strong correlation between the normalized harmonic amplitude and the amount of fa-tigue damage6or plastic deformation7in a material.The major measurement difficulty with the harmonic generation method as a NDT technique lies in isolating the causes of nonlinearity. Specifically,amplifiers,transducers,and coupling methods are all contributors to the measured harmonic,often on a scale greater than the material nonlinearity itself.Thus it is practically very difficult to determine if the measured nonlinearity is due to the material or the equipment.
A third technique,which is the main subject of this paper,for TOEC measurement was first proposed by Jones and Kobett,9and experimentally observed by Rollins.10This approach is based on the fact that material nonlinearities cause interaction between two intersecting ultra-sonic waves.11Under cer
tain circumstances,this can lead to the generation of a third wave with a frequency and wavevector equal to the sum of the incident wave frequencies and wavevectors, respectively.Theoretically,there are several incident wave combinations that can achieve this; however,practical material constraints to the theory lead to the interaction of two shear waves generating a longitudinal wave as the most useful case.
The non-collinear mixing technique has two important advantages over the conven-tional nonlinear ultrasonic harmonic generation technique.First,it is much less sensitive to system nonlinearities due to spatial selectivity(the nonlinear interaction is limited to the region where the incident beams intersect),modal selectivity(the nonlinear mixing signal is a different mode to the incident waves),frequency selectivity(the mixing signal frequency can be sepa-rated from harmonics of the incident waves if the driving frequencies are chosen to be unequal), and directional selectivity(the mixing signal propagates in a different direction form the mixed ones and their higher harmonics).Second,unlike the harmonic generation techniques,the level of the underlying system nonlinearity can be measured directly by summing the responses to each of the incident waves excited separately,that is,without the interaction present.
It is important to note that the evidence of correlation between material degradation (e.g.,fatigue or pl
asticity)and nonlinear ultrasonic phenomena that has been reported is based mainly on evidence from the harmonic generation technique.In this configuration,only longi-tudinal waves can be used,and the harmonic amplitude is a function of all three TOECs(A,B, and C)or alternatively two of Murnaghan’s three TOECs(l and m).However,the non-collinear technique based on the interaction of two shear waves to produce a longitudinal wave was shown by Jones and Kobett9and Taylor and Rollins11to lead to a longitudinal wave amplitude that depends only on TOECs A and B(or the m and n Murnaghan TOECs).
What has not been studied to date is whether the particular combination of the two TOECs probed by the non-collinear technique is sensitive to fatigue and plasticity,and there-fore whether the non-collinear technique can be used for NDT of fatigue damage.The purpose of this letter is to demonstrate that the non-collinear mixing technique can indeed detect changes due to plasticity and fatigue damage,and therefore has the potential to be used as a NDT technique.
2.Experimental arrangement
Experimental measurements were performed on an Al2014-T4aluminum alloy specimen.Fig-ure1shows the basic experimental arrangement.Two intersecting shear waves are generated using o
blique incidence shear transducers made of longitudinal transducers of5MHz nominal center frequency mounted on60°Perspex wedges.Within the volume of intersection a third longitudinal wave is generated due to nonlinear interaction.Once generated,this wave propa-gates through the material in a conventional manner and is detected by the receiver.The receiver was a normal-incidence longitudinal transducer of10MHz nominal center frequency.The ex-citation signals were generated using a digital oscilloscope/signal generator and the detected interaction wave(after amplification)was recorded using the same instrument.The excitation signals were amplified using a power amplifier,resulting in signals with amplitude of approxi-mately60V p-p.
The excitation signals to both input transducers were20-cycle,Hanning-windowed tone bursts with center frequencies of5.5MHz.Using the same driving frequency for both incident waves removes one of the advantages of the non-collinear technique(frequency sepa-
ration)although the following results show that ample suppression of system nonlinearities is
still achieved.The recorded data were digitally filtered using an 11MHz center frequency,2
MHz bandwidth bandpass filter.The use of incident waves of the same frequency significantly
simplifies the experimental apparatus as only one common driving signal needs to be generated.
It also simplifies the experimental geometry since with both incident waves excited at equal and
opposite angles,the resulting interaction wave is generated perpendicular to the specimen sur-
face.The latter point means that the receiver can be placed on either the top or bottom surface of
the specimen.For the purpose of this investigation the single sided arrangement was considered
more suitable as it reflects the limited access likely to be encountered in practical applications.
Note that the vertical position of the interaction zone can be readily moved by altering the
separation of the input transducers.
Throughout all stages of experimentation each test comprised three measurements:
one with each input transducer excited individually and one with both excited simultaneously.
The signals recorded when the input transducers were excited individually were summed and
the amplitude of this signal at the expected arrival time of the interaction wave used to estimate
the level of remnant system nonlinearity.Figure 2(a)shows an example of time-domain re-
sponse obtained from an as-manufactured sample when both input transducers are excited si-
multaneously.The pulse in the window labeled first reflection is the first received interaction
wave after it has been reflected off the bottom surface of the sample.The subsequent pulses in
the windows labeled second and third reflections correspond to reverberations of the interaction
wave between the sample surfaces.In the following,the peak amplitude in the window corre-
generated
sponding to the first reflection is taken as the measure of material nonlinearity.Figure 2(b)
shows the equivalent time-domain signal obtained by summing the responses from each of the
two input transducers excited separately.The amplitude of the signal in the window labeled first
reflection in Fig.2(b)is due to the combined effect of all nonlinearities in the measurement
system (e.g.,reflections due to sidelobes of harmonics in the transmitted shear waves).It can be
seen that the system nonlinearity is an order of magnitude smaller than the material nonlinear-
ity,yielding a signal-to-noise ratio of 30,even in the as-manufactured sample,which is ex-
pected to contain the lowest nonlinearity anyway.
It is worth noting the presence of the signal at 2.5ϫ10−5S in Fig.2(a)resulting from
second harmonic generation on a longitudinal wave propagating through the specimen and off
of the back wall.If a conventional harmonic generation technique were employed this signal
would be impossible to differentiate from any potential equipment nonlinearity,whereas using
the non-collinear technique the interaction wave is spatially
separated.
Fig.1.͑Color online ͒Experimental arrangement.
The responses of the three transducers used were calibrated to absolute values using a
heterodyne laser interferometer.The purpose of this calibration was to enable a theoretical
value of interaction wave amplitude for the as-manufactured material to be estimated for com-
parison with that measured experimentally.Values of −25.22ϫ1010,−32.5ϫ1010,and
−35.12ϫ1010N/m 2for the TOECs for single crystal aluminum were taken from literature.12
The approximate amplitude of the interaction wave was then estimated using the expression
(Table I,case I)provided by Taylor and Rollins,11yielding a value of 3.2ϫ10−12m.The mea-
sured amplitude of the interaction wave for the intact sample was 2ϫ10−12m.This is suffi-
ciently close to the estimated value to give confidence to the basic soundness of the non-
collinear measurement technique.The difference is believed to be primarily due to the
theoretical values being calculated for a single crystal and therefore not taking into account the
polycrystalline nature of the real sample.This calibration procedure illustrates the means by
which the non-collinear technique could be used for making absolute measurements.
Having confirmed that the measured nonlinear interaction wave was of similar ampli-
tude to that predicted theoretically,experiments were carried out to investigate changes in the
magnitude of the interaction wave as the material was subjected to both quasi-static plastic
strain and low-cycle fatigue damage.The first sample was used to investigate the effect of plas-
tic deformation.Strain gauges were bonded to opposing faces of the sample directly above the
region of interaction and the sample was loaded in a tensile test machine.After removing the
load,the residual plastic strain was measured using the strain gauges and the specimen was
removed from the test machine for the non-collinear measurements to be performed.Each set of
non-collinear measurements corresponded to eight measurement points along the length of
the
Fig.2.͑Color online ͒Time-domain signals obtained from as-manufactured sample corresponding to ͑a ͒the total
response when the shear transducers are excited simultaneously and ͑b ͒the sum of the responses when the shear
transducers are excited separately.The colored lines show the calculated arrival times of the first,second,and third
reflections.
sample.The process was repeated using the same specimen subjected to progressively higher
loads to obtain non-collinear mixing data at successively higher levels of plastic strain.
In order to measure the material nonlinearity independent of the excitation level the
amplitude of the interaction wave A 3was normalized to the product of the two input amplitudes
A 1and A 2measured in volts.
␹=A 3A 1A 2.͑1͒
Figure 3shows the measured values of ␹as a function of residual strain normalized to the intact
value in order to make the change clearer.The key point to observe in this graph is the increase
in ␹by around 30%with residual strain,indicating that the non-collinear approach is sensitive
to plasticity.Each point on the graph represents the mean of the eight individual measurements
made for that particular plastic strain level.Between each measurement the transducer fixture
was completely removed and the specimen cleaned.The error bars represent one standard de-
viation of the measurements.
A second specimen was tested under low-cycle fatigue conditions.These correspond
to cyclically straining the material to beyond its yield point but significantly below its failure
stress.Typical fatigue life under these conditions is less than 100cycles.In the example pre-
sented here the sample was stressed between 0%and 110%of yield (420MPa)in blocks of 10
cycles.Initially,this stress level led to a residual strain of 2%,significantly below the failure
strain,but still high enough to result in a low-cycle fatigue failure.The results of this test are
shown in Fig.4.
It can be seen that ␹initially increases rapidly with the number of cycles.Beyond 20
cycles the rate of increase drops significantly due to work-hardening in the material,and this is
in line with published data in literature.8In this experiment the error bars get larger with in-
creasing number of cycles indicating a higher degree of variability in the measurements.This
can be attributed to taking measurements along the whole of the test section rather than at a
single location.End effects near the points of attachment to the tensile test machine may well
result in more localized fatigue damage,hence increasing the variability of measurements.
3.Conclusions
These results demonstrate that the non-collinear technique is sensitive to both plasticity and
fatigue damage in a similar way to collinear harmonic generation.This is despite the non-
collinear technique being sensitive to a combination of different TOECs to the harmonic gen-
eration technique.Because of its intrinsically better rejection of spurious system nonlinearities,
Residual Strain (%)
N o r m a l i z e d N o n l i n e a r i t y ,χ/χi n t e r a c t Fig.3.͑Color online ͒Change in non-collinear measurement parameter ␹with increasing plastic strain.

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