Journal of Nondestructive Evaluation, Vol. 24, No. 4, December 2005 ( C© 2005)DOI: 10.1007/s10921-005-8783-9

Dynamic Piezoresistivity Calibration for Eddy CurrentNondestructive Residual Stress Measurements

Feng Yu1 and Peter B. Nagy1,2

Received March 16, 2005; revised August 13, 2005

It has been recently demonstrated [M. P. Blodgett and P. B. Nagy, J. Nondestruct. Eval. 23,107 (2004)] that eddy current conductivity measurements can be exploited for near-surfaceresidual stress assessment in surface-treated nickel-base superalloy components. To quanti-tatively assess the prevailing residual stress from eddy current conductivity measurements,the piezoresistivity coefficients of the material must be first determined using known exter-nal applied stresses. These calibration measurements are usually conducted on a referencespecimen of the same material using cyclic uniaxial loads between 0.1 and 10 Hz, which isfast enough to produce adiabatic conditions. Therefore, the question arises whether dynamiccalibration measurements can be used or not for accurately assessing the sensitivity of theeddy current method for static residual stress. It is demonstrated in this paper that such dy-namic calibration measurements should be corrected for the thermoelastic effect, which isalways positive, i.e., it increases the conductivity in tension, when the material cools down,and reduces it in compression, when the material heats up. For low-conductivity titanium andnickel-base engine alloys the thermoelastic corrections are relatively modest at ≈5–10%, butfor high-conductivity aluminum alloys the difference between the adiabatic and isothermalproperties could be as high as 50%.

KEY WORDS: Eddy current; shot peening; residual stress.

1. INTRODUCTION

Different mechanical surface treatments, such asshot peening, laser-shock peening, and low-plasticityburnishing, are widely used to improve the resistanceto fatigue and foreign-object damage in criticalmetal components, including gas-turbine engines,via introducing beneficial compressive near-surfaceresidual stresses. However, the fatigue life improve-ment gained via surface enhancement is usuallynot accounted for in current engine component lifemanagement processes because of the highly uncer-tain nature of the absolute level and depth profile

1 Department of Aerospace Engineering and EngineeringMechanics, University of Cincinnati, Cincinnati, Ohio 45221-0070.

2 Corresponding author; e-mail: peter.nagy@uc.edu

of the protective residual stress distribution. Theseuncertainties arise partly because of manufacturingprocess variations and partly because of subsequentthermo-mechanical relaxation during extendedservice at elevated temperatures. There would bea significant potential for increasing the predicteddamage tolerance capabilities of fracture-criticalcomponents if residual stress considerations wereactually incorporated into life prediction method-ologies. One of the main reasons why the beneficialeffects of mechanical surface treatments cannotbe accounted for in life prediction models is thelack of available nondestructive evaluation (NDE)methods that are capable of quantitatively assess-ing the actual residual stress levels present in thematerial.

The characteristic dependence of electrical resis-tivity on stress, the so-called piezoresistive effect, has

1430195-9298/05/1200-0143/0 C© 2005 Springer Science+Business Media, Inc.

144 Yu and Nagy

been thought to be very promising for residual stressmeasurements in metals for a long time, though theseexpectations have remained largely unfulfilled as faras surface-treated components are concerned.(1–9)

In most metals the stress-dependence of the elec-trical conductivity is rather weak and the primaryresidual stress effect is rather difficult to separatefrom the secondary cold work effect. It has been re-cently demonstrated that eddy current conductivitymeasurements are particularly well suited for near-surface residual stress assessment in surface-treatednickel-base superalloy components.(10) The feasibil-ity of the eddy current method critically depends onan important common feature exhibited by nickel-base superalloys, namely that the parallel and normalpiezoresistivity coefficients are both negative. In thispaper we will show that this behavior is rather rare,but not unique to this class of materials. For exam-ple, pure copper, which is often used in fundamentalstudies of piezoresistivity, behaves similarly.

In a recent paper the authors presented an inver-sion method capable of predicting the sought depthprofile of the frequency-independent intrinsic elec-trical conductivity of the specimen from the mea-sured frequency-dependent apparent eddy currentconductivity.(11) Then, the depth profile of the elec-trical conductivity can be converted into the soughtresidual stress profile if the appropriate stress coef-ficient of the electrical conductivity is known froman independent calibration measurement. These cal-ibration measurements are usually done using thesame eddy current instrumentation on an appropri-ate reference specimen of the same material in anMTS machine under cyclic uniaxial loading. In orderto apply the thereby measured dynamic stress coeffi-cients to the inversion of static residual stresses, wemust better understand the subtle difference in howthe presence of dynamic and static stresses affect theelectrical conductivity of metals.

Essentially all earlier studies on the piezoresis-tivity of different materials in the literature wereconducted by conventional contact bulk resistivitymeasurements using galvanic coupling. In contrast,in most NDE applications the near-surface electri-cal conductivity variation caused by the piezoresis-tive effect is measured by noncontacting eddy currentmethods. In this paper we investigate the relation-ship between the isothermal piezoresistivity coeffi-cients measured under static uniaxial tension or hy-drostatic pressure using galvanic coupling and theadiabatic electroelastic coefficients measured underdynamic uniaxial loading using either nondirectional

circular or directional elliptical eddy current coils.We will introduce a thermoelastic correction to ac-count for the differences between the piezoresistiv-ity coefficients determined during static and dynamicmeasurements. We will investigate the thermoelasticeffect on the measured electroelastic coefficients ofseveral materials, including pure copper (C 11000),two aluminum alloys (Al 2024 and Al 7075), twonickel-base superalloys (Waspaloy and IN 718), anda common titanium alloy (Ti-6Al-4V). We will showthat in high-conductivity metals (copper and alu-minum alloys) dynamic calibration measurementsconducted above 0.01 Hz really must be correctedfor the thermoelastic effect that causes a secondaryconductivity variation due to the strong temperature-dependence of the electrical conductivity. The ther-moelastic effect is always positive, i.e., it increasesthe conductivity in tension, when the material coolsdown, and reduces it in compression, when the mate-rial heats up. On the other hand, in low-conductivityengine materials the correction is less significant be-cause the electrical conductivity is much less sensi-tive to temperature oscillations, therefore the ther-moelastic contribution is rather small relative to theprimary conductivity variation caused directly by thealternating stress.

2. PIEZORESISTIVITY

In the presence of elastic stress (τ) the elec-trical resistance (ρ) and conductivity (σ) ten-sors of an otherwise isotropic conductor becomeslightly anisotropic (σ = ρ−1). In general, the stress-dependence of the electrical resistivity can bedescribed by the fourth-order piezoresistivity (π)tensor.(12–18) For easier comparison to our eddy cur-rent measurements we are going to consider thestress-induced change in the electrical conductivityrather than in the electrical resistivity. In direct anal-ogy to the well-known acoustoelastic coefficients, awidely used NDE terminology for the stress coeffi-cient of the acoustic velocity, we are going to referto the stress coefficient of the electrical conductivityas the electroelastic coefficient. Also, for simplicityof comparison between different materials and load-ing conditions, we are going to normalize the stresscomponents to Young’s modulus E as follows

�σ1/σ0

�σ2/σ0

�σ3/σ0

=

κ11 κ12 κ12

κ12 κ11 κ12

κ12 κ12 κ11

τ1/Ei

τ2/Ei

τ3/Ei

. (1)

Dynamic Piezoresistivity Calibration 145

Here �σi = σi − σ0 (i = 1, 2, 3), σ0 = 1/ρ0, denotesthe electrical conductivity in the absence of stress andκ11 = −Eπ11 and κ12 = −Eπ12 are the unitless par-allel and normal electroelastic coefficients, respec-tively, where π11 and π12 are the corresponding par-allel and normal piezoresistivity coefficients.

The simplest example of the piezoresistivity ef-fect is exhibited by ordinary strain gauges. Underuniaxial stress (τ1 = τua and τ2 = τ3 = τ0), the so-called gauge factor ηua is defined as the ratio of therelative resistance change (�R/R0)ua and the axialstrain εua = τua/E, so that ηua = (1 + 2ν) − κ11, whereν denotes Poisson’s ratio. It is well known that thegauge factor is usually significantly higher than thefirst term of macroscopic geometrical origin, whichindicates that κ11 is negative. It should be mentionedthat κ11 itself includes a 2ν − 1 geometrical compo-nent due to the changing number density of the freeelectrons for a constant total number of electrons,which increases the total geometrical contribution inηua to exactly 2.(19)

Another well-known example of the piezore-sistivity effect in conducting metals is the pressuredependence of the electrical conductivity under hy-drostatic conditions (τ1 = τ2 = τ3 = −p), when thegauge factor is ηhs = − (1 − 2ν) − (κ11 + 2κ12). Thepressure coefficient ηhs is positive in most structuralmetals, although some exceptions do occur.(20,21) Be-sides the well-known cases of uniaxial and hydro-static stress, there are numerous special cases thathave been discussed in the literature. For example,wire and foil strain gauges are often used as pressuresensors between two compressed surfaces.(15–17,22)

One of the main advantages of using noncon-tacting eddy current techniques to measure the elec-trical conductivity change in metals under differentstress conditions is that, assuming that the speci-men is sufficiently large with respect to both theprobe diameter and the frequency-dependent eddycurrent penetration depth, edge and thickness ef-fects are negligible and there are no geometricalcontributions to the observed piezoresistivity ef-fect. When a conventional nondirectional circulareddy current probe is used to measure the aver-age electrical conductivity σ0 under uniaxial stress(τ1 = τua and τ2 = τ3 = 0), the effective electroelas-tic coefficient is κ0 = (κ11 + κ12) /2. In contrast, if adirectional eddy current probe is used to measurethe weighted average of the parallel σ1 and nor-mal σ2 electrical conductivities under uniaxial stress,the effective electroelastic coefficient will also bea weighted average of the parallel κ11 and normal

κ12 electroelastic coefficients. Elliptical and racetrackcoils usually have a modest aspect ratio of 4–5,(23) butmeandering winding magnetometers(24) are shapedlike standard strain gauges and offer a much sharperdirectionality, and therefore allow us to measure κ11

and normal κ12 essentially independent of each other.In the very important case of shot-peened or oth-erwise surface-treated metals, essentially isotropicplane stress (τ1 = τ2 = τip and τ3 = 0) conditions oc-cur. Then, regardless whether nondirectional circularor directional elliptical probes are used, the effectiveelectroelastic coefficient is κ11 + κ12.

Ideally, regardless of which particularexperimental method is used, all the above describedtechniques should yield the same two independentelectroelastic coefficients for an isotropic material.Copper might be sought of as a good basis forcomparison because its piezoresistive propertieshave been studied by numerous researchers us-ing various techniques. Actually, the range of theparallel piezoresistive coefficient κ11 reported forpure copper in the literature covers a rather widerange from −0.7 to −1.1.(21,25–30) We are going toshow that one of the main reasons for this variationis probably that in the case of pure copper thedifference between the adiabatic and isothermalcoefficients, i.e., the thermoelastic correction, is verysubstantial. Since some of the experimenters did notrecognize the importance of the difference betweenslow dynamic (adiabatic) and truly static (isotropic)conditions, the measured numbers tend to fall in awide range which cannot be sufficiently explained byrandom experimental uncertainties only.

3. THERMOELASTIC EFFECT

Because of the thermoelastic effect, under adi-abatic conditions, a harmonic uniaxial stress oscilla-tion τua causes a proportional harmonic temperatureoscillation(31)

�T = − αT0

ρCpτua, (2)

where α is the linear coefficient of thermal expansion,T0 is the average temperature, ρ is the mass densityof the material, and Cp is the specific heat at constantpressure. This simple approximate equation neglectsthe temperature-dependence of the elastic modulus,which is relatively small in comparison to the lin-ear coefficient of thermal expansion.(32) The electri-cal conductivity of the material is an approximatelylinear function of temperature

146 Yu and Nagy

Table I. Material Properties of Pure Copper, Two Aluminum Alloys, and Three High-Temperature Engine Alloys

Material

Property Copper Al 2024 Al 7075 Waspaloy IN 718 Ti-6Al-4V

σ0 (106 s m−1) 58 17.7 18.7 0.87 0.81 0.58β (10−3 K−1) 3.9 1.75 1.92 0.21 0.19 0.29E (109 Pa) 120 73.1 71.7 211 208 114α (10−6 K−1) 17 23.2 21.6 12.2 12.8 8.6ρ (103 kg m−3) 8.9 2.8 2.8 8.1 8.2 4.4Cp (J kg−1 K−1) 390 875 960 520 435 526κth 0.69 0.37 0.33 0.038 0.042 0.037

Note. The thermoelastic coefficient, κth, was calculated from Eq. (4).

�σth = −σ0β�T, (3)

where β is the thermal coefficient of the electrical re-sistivity. Combining Eqs. (2) and (3) yields an elec-troelastic coefficient of thermoelastic origin

κth = Eτua

�σth

σ0= αβET0

ρCp. (4)

Because of the thermoelastic effect, under adiabaticconditions, the measured electroelastic coefficients

κ∗11 = κ11 + κth and κ∗

12 = κ12 + κth (5)

are offset by the thermoelastic coefficient κth rela-tive to their respective isothermal values of κ11 andκ12. Table I lists the relevant mechanical, thermal,and electrical properties of the materials tested inour study along with the corresponding values of κth.The principal reason why metals exhibit resistanceagainst the flow of electrical current is the lack of per-fect periodicity in their lattice. In high-conductivitymaterials the main source of the lack of latticeperiodicity is the thermal vibration of the atomsaround their equilibrium positions, therefore theelectrical conductivity is much more temperature-dependent than in low-conductivity precipitationhardened nickel-base superalloys and titanium al-loys. Consequently, the piezoresistivity coefficient ofthermoelastic origin κth is also much smaller in theseengine alloys, which is well illustrated in Table I.

It should be mentioned that, in order to rigor-ously distinguish between adiabatic and isotropic ma-terial properties, in Eq. (1) the adiabatic Young’smodulus Ea should be used instead of the isothermalparameter Ei

1Ea

= 1Ei

− α2T0

ρCp. (6)

However, the difference between the adiabatic andisothermal Young’s muduli is at most ≈0.5%, there-fore it was neglected in our analysis.

4. EXPERIMENTAL METHOD

Figure 1 shows a schematic diagram of the ex-perimental arrangement used to measure the elec-troelastic coefficients of different metals in uniax-ial compression and tension.(10) Both nondirectionalcircular and directional racetrack coil probes wereused for the load frame testing. The racetrack coilswere oriented either parallel or normal to the load-ing direction to observe the directional dependenceof stress on the apparent eddy current conductiv-ity. We found that the electroelastic coefficientsmeasured by the nondirectional circular probe wereequal to the average of those measured by the di-rectional racetrack probe at parallel and normalorientations.(10) For brevity, only the results of thedirectional measurements will be reported in thispaper.

Fig. 1. A schematic diagram of the experimental arrangementused to measure electroelastic coefficients in uniaxial compressionand tension.

Dynamic Piezoresistivity Calibration 147

Fig. 2. An example of (a) the axial stress and (b) the eddy currentconductivity at parallel orientation as functions of time in IN 718.

Figure 2 shows examples of the axial stressand the corresponding eddy current conductivity atparallel orientation in IN 718 as functions of time. Al-ternating axial load was applied to the specimens at acyclic frequency of 0.5 Hz. Although we are mainlyinterested in the effect of compressive stresses onthe electrical conductivity of the specimens, the max-imum tensile load was chosen to be 2–4 times ashigh as the maximum compressive load in order tominimize the possibility of buckling in the slenderrectangular bars used as specimens (w = 12.5 mm,t = 6.35 mm, L ≈ 150 mm). The eddy current inspec-tion frequency (f = 300 kHz in the case of low-conductivity IN 718) was always chosen so that thestandard penetration depth (δ = 1 mm) was muchsmaller than the thickness of the specimens, there-fore the spurious thickness modulation caused by thePoisson effect could be neglected. All measurementswere conducted over a sustained period of approx-imately 2 min so that the adverse effects of randomnoise and thermal drift could be sufficiently reducedvia averaging. It is clear even from the somewhatnoisy raw data shown in Fig. 2 that the parallel elec-troelastic coefficient of IN 718 is negative, that is theconductivity increases in compression.

5. ELECTROELASTIC MEASUREMENTSIN HIGH-CONDUCTIVITY ALLOYS

To minimize the spurious edge effects, we used a19-mm-diameter racetrack probe to measure κ∗

11 andκ∗

12 on a copper specimen (w = 25.4 mm, t = 6.3 mm,L = 230 mm) at f = 20 kHz, where the standard pen-etration depth δ ≈ 0.5 mm was much smaller than thethickness of the specimen. It should be noted that al-though the eddy current frequency could affect theaccuracy of electroelastic measurements, the paralleland normal electroelastic coefficients are essentiallyfrequency independent.(10) The cycling frequency ofthe alternating load applied to the specimen was1 Hz. On the basis of the decay time constant of theheated and then air-cooled specimen, we found thatadiabatic conditions prevail above ≈0.01 Hz cyclingfrequency. For the pure copper specimens used inour electroelastic measurements, at T0 = 300 K and apeak-to-peak uniaxial stress level of τua = 167 MPa,the peak-to-peak temperature change |�T| on thesurface of the specimen predicted by Eq. (2) is about0.24 K.

The actual temperature variation was measuredwith a calibrated infrared sensor. Figure 3 shows (a)the axial stress and (b) the resulting temperature os-cillation in copper. The experimental temperature

Fig. 3. An example of (a) the axial stress and (b) temperature os-cillation in pure copper (theoretical �T = 0.24 K, measured �T =0.23 K).

148 Yu and Nagy

Fig. 4. Measured temperature dependence of electrical resistivityin pure copper (relative to T0 = 300 K, β = 3.8 × 10−3 K−1).

oscillation was |�T| = 0.23 K, which is in good agree-ment with the theoretically predicted value. It shouldbe mentioned that the yield strength of pure “elec-tronic” copper (C 11000) changes from 70 to 365 MPadepending on heat treatment. Our own measure-ments in the copper bars used in our tests showeda yield strength of 280 MPa, therefore the 167 MPapeak stress level used in our experiments was roughly60% of the yield strength.

We also measured the temperature coefficientof copper using a 4-point resistance meter. Figure 4shows the measured temperature dependence ofelectrical resistivity in pure copper from T0 = 300 Kto Tmax = 350 K. From the essentially linear slope ofthe electrical resistivity versus temperature curve, wefound the temperature coefficient to be β ≈ 3.8 ×10−3 K−1, also in good agreement with our expecta-tions. Using the parameters listed in Table I, the ther-moelastic component of the adiabatic electroelasticcoefficient is κth ≈ 0.69, a very substantial value dueto the very strong temperature-dependence of theelectrical conductivity in pure copper.

Figure 5 shows conductivity versus stress resultsfor pure copper at both normal and parallel orienta-tions. The symbols represent the experimental data,whereas the solid lines are best-fitting linear regres-sions. The adiabatic electroelastic coefficients ob-tained from the linear fits are κ∗

11 = − 0.39 and κ∗12 =

−0.11, therefore the corrected isothermal parame-ters are κ11 = −1.08 and κ12 = −0.8. These valuesare in good agreement with the previously reviewednumbers from the literature. In high-conductivitypure copper the absolute value of κth is dominantin the measured adiabatic electroelastic coefficients.From Table I we can conclude that the difference

Fig. 5. Electroelastic measurements in copper using a directionaleddy current probe parallel and normal to the applied uniaxialload.

of κth between different materials mostly depends onthe variation of the thermal coefficient of the electri-cal conductivity β, which in turn is mainly determinedby the absolute electric conductivity σ0 of the mate-rial. Intuitively, it seems that the thermoelastic cor-rection factor of the isothermal electroelastic coeffi-cients is roughly proportional to electric conductivityof the material.

To verify this trend in other high-conductivitymetals, we conducted additional electroelastic mea-surements on two common aluminum alloys (Al 2024and Al 7075), whose conductivity is roughly one-third of that of copper. Because of the lower electri-cal conductivity of these aluminum alloys, the stan-dard penetration depth was about δ ≈ 0.8 mm atthe same inspection frequency f = 20 kHz, which isstill much smaller than the thickness of the speci-men (t = 6.3 mm). All other experimental conditionswere identical to those used in our measurementson pure copper. Figures 6 and 7 show eddy cur-rent conductivity versus stress results for Al 2024 andAl 7075, respectively, at both parallel and normal ori-entations. Again, the symbols represent experimen-tal data while the solid and dashed lines are best-fitting linear regressions. It is apparent from theseresults that the parallel adiabatic electroelastic coef-ficients of the two aluminum alloys are positive andthe normal coefficients are negative. This behavior isquite different from copper, in which both paralleland normal electroelastic coefficients are negative.To verify whether the sign difference of parallel andnormal electroelastic coefficients is common or notin other aluminum alloys, future research is neededon a larger number of different alloys. On the ba-sis of the physical properties listed in Table I, it is

Dynamic Piezoresistivity Calibration 149

Fig. 6. Electroelastic measurements in Al 2024 using a directionaleddy current probe parallel and normal to the applied uniaxialload.

obvious that the required thermoelastic correctionsare much smaller in aluminum alloys than in purecopper, which supports our conclusion that the ab-solute electrical conductivity is the most influentialfactor determining the difference between the adia-batic and isothermal electroelastic coefficients.

6. ELECTROELASTIC MEASUREMENTSIN ENGINE ALLOYS

Our main interest in this research is to de-termine the isothermal electroelastic coefficients ofhigh-temperature engine alloys for the ultimate ob-jective of quantitatively assessing the prevailingresidual stress in surface-treated components fromeddy current conductivity measurements. We se-lected two nickel-base superalloys (Waspaloy and

Fig. 7. Electroelastic measurements in Al 7075 using a directionaleddy current probe parallel and normal to the applied uniaxialload.

Fig. 8. Electroelastic measurements in Waspaloy using a direc-tional eddy current probe parallel and normal to the applied uni-axial load.

IN 718) and a popular titanium alloy (Ti-6Al-4V)as the model materials of engine alloys to conductelectroelastic measurements. These materials exhibitroughly two orders of magnitude lower electricalconductivity than pure copper, i.e., one order of mag-nitude larger eddy current penetration depth at agiven inspection frequency. Therefore, these mea-surements were conducted at a higher inspection fre-quency of f = 300 kHz so that the standard pene-tration depth (δ ≈ 1 mm) is again much smaller thanthe thickness of the specimens (t = 6.35 mm). Alter-nating axial load was applied to the specimens at acyclic frequency of 0.5 Hz. Like before, these mea-surements were conducted over a sustained periodof approximately 2 min so that the adverse effects ofrandom noise and thermal drift could be sufficientlyreduced via averaging.

Figures 8–10 show eddy current conductiv-ity versus stress results for Waspaloy, IN 718, and

Fig. 9. Electroelastic measurements in IN 718 using a directionaleddy current probe parallel and normal to the applied uniaxialload.

150 Yu and Nagy

Fig. 10. Electroelastic measurements in T1-6A1-4V using a direc-tional eddy current probe parallel and normal to the applied uni-axial load.

Ti-6Al-4V, respectively, at both parallel and nor-mal orientations. The symbols represent experi-mental data while the solid and dashed lines arebest-fitting linear regressions. The electroelastic be-havior of Ti-6Al-4V is quite different from thoseof the two nickel-base superalloys.(10) In the caseof Ti-6Al-4V, under compressive loading the elec-trical conductivity in the load direction (paral-lel orientation) progressively decreases, while forIN 718 and Waspaloy it increases. In comparison,in the normal direction the electrical conductiv-ity increases in all three engine materials undercompression.

As expected, the thermoelastic correction fac-tor of the electroelastic coefficients is fairly low inthe case of low-conductivity engine alloys, and itis almost trivial in comparison with the large ther-moelastic factors needed to correct the differencebetween adiabatic and isothermal electroelastic coef-ficients in pure copper and aluminum alloys of muchhigher electrical conductivity. Considering the exper-imental uncertainty of our measurements, the smalldifference between adiabatic and isothermal elec-troelastic coefficients in nickel-base superalloys andTi-6Al-4V is rather insignificant in the nondestruc-tive evaluation of near-surface residual stress distri-butions in surface-treated components. In summary,Table II lists the measured adiabatic electroelasticcoefficients (κ∗

11 and κ∗12), the thermoelastic correc-

tion factor (κth) calculated from Eq. (4), and thecorrected isothermal electroelastic coefficients (κ11

and κ12) for the six materials tested in our exper-imental study. It should be mentioned that in thecase of pure copper the thermoelastic correction fac-tor can be also calculated using Eq. (3) from the

Table II. The Measured (Adiabatic) Parallel and Normal Elec-troelastic Coefficients, the Calculated Thermoelastic Coefficientκth and the Corresponding Corrected (Isothermal) Electroelastic

Coefficients

Electroelastic coefficient

Material κ∗11 κ∗

12 κth κ11 κ12

Copper −0.39 −0.11 0.69 −1.08 −0.8Al 2024 1.2 −0.26 0.37 0.83 −0.63Al 7075 1.04 −0.35 0.33 0.71 −0.68Waspaloy −0.48 −0.22 0.038 −0.52 −0.26IN 718 −0.84 −0.65 0.042 −0.88 −0.69Ti-6Al-4V 0.65 −0.55 0.037 0.61 −0.59

measured values of β ≈ 3.8 × 10−3 K−1 and |�T| =0.23 K as κth ≈ 0.63 which is close to, but slightlylower, than the κth ≈ 0.69 value listed in Table II.In relative terms, for pure copper the correctionsfor the parallel and normal electroelastic coefficientsare ≈180 and ≈630%, respectively. For aluminum al-loys, the corrections range from ≈30 to ≈140%. Incomparison, for high-temperature engine alloys thecorresponding corrections are significantly smaller at≈4–8% except for the slightly higher ≈18% value forthe normal electroelastic coefficient in Waspaloy.

7. CONCLUSION

In this paper we investigated the relationshipbetween the conventional isothermal piezoresistivitycoefficients and the corresponding adiabatic coeffi-cients often determined during dynamic calibrationmeasurements for the inversion of nondestructiveeddy current data. The parallel and normal electroe-lastic coefficients of different materials can be sepa-rately measured under uniaxial stress conditions bydirectional eddy current coils. Unlike conventionalcontact bulk resistance measurements using galvaniccoupling, the noncontacting eddy current techniquemeasures the conductivity change in metals free ofspurious geometrical effects, assuming that edge andthickness effects are negligible. However, special at-tention must be paid to the substantial difference be-tween adiabatic and isothermal properties when dy-namic calibration measurements are conducted onreference specimens using cyclic uniaxial loads above0.01 Hz, which is fast enough to produce adiabaticconditions.

Our experimental results in copper showed thatin high-conductivity materials inherent thermoelas-tic temperature oscillations significantly affect thevalue of the measured electroelastic coefficients

Dynamic Piezoresistivity Calibration 151

during dynamic loading, therefore appropriate cor-rections must be introduced. It appears that someof the numbers reported in the literature for copperwere obtained in dynamic measurements, though theavailable data is insufficient to determine whetherthe existing conditions were adiabatic, isothermal, orsomewhere in between. We found that the piezore-sistivity coefficients reported in the literature are allbetween the adiabatic and isothermal values, withthe majority being very close to the isothermal value.Eddy current measurements usually yield a valuevery close to the adiabatic limit because of the re-quirement of using a specimen with sufficiently largecross-sectional area. Similar measurements on alu-minum alloys and high-temperature engine alloysverified that the electrical conductivity of the ma-terial is the dominant factor in the thermoelasticcorrection via the temperature coefficient of theconductivity. Most importantly, we found that forhigh-temperature engine alloys of low electrical con-ductivity, such as nickel-base superalloys and tita-nium alloys, the difference between the isothermaland adiabatic parameters is fairly low, roughly oneorder of magnitude smaller than the measured paral-lel and normal electroelastic coefficients.

ACKNOWLEDGMENTS

This work was performed at the University ofCincinnati in cooperation with the Center for NDEat Iowa State University with funding from the AirForce Research Laboratory through S&K Technolo-gies, Inc. on delivery order number 5007-IOWA-001of the prime contract F09650-00-D-0018. The au-thors express their appreciation to Mark Blodgett ofAFRL/MLLP for valuable discussions.

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