I

NASA/TM--2001-210697

Quantifying Residual Stresses by Means

of Thermoelastic Stress Analysis

Andrew L. Gyekenyesi

Ohio Aerospace Institute, Brook Park, Ohio

George Y. Baaklini

Glenn Research Center, Cleveland, Ohio

February 2001
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NASA/TMm2001-210697

Quantifying Residual Stresses by Means

of Thermoelastic Stress Analysis

Andrew L. Gyekenyesi

Ohio Aerospace Institute, Brook Park, Ohio

George Y. Baaklini

Glenn Research Center, Cleveland, Ohio

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Glenn Research Center

February 2001
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Available electronically at h_p://gltrs._c.nasa.gov/GLTRS
Quantifying Residual Stresses by Means of Thermoelastic Stress Analysis

Andrew L. Gyekenyesi

Ohio Aerospace Institute

Brook Park, Ohio 44142

George Y. Baaklini

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

ABSTRACT

This study focused on the application of the Thermoelastic Stress Analysis (TSA) technique as a tool for assessing the

residual stress state of structures. TSA is based on the fact that materials experience small temperature changes when

compressed or expanded. When a structure is cyclically loaded, a surface temperature profile results which correlates to the

surface stresses. The cyclic surface temperature is measured with an infrared camera. Traditionally, the amplitude of a TSA

signal was theoretically defined to be linearly dependent on the cyclic stress amplitude. Recent studies have established that

the temperature response is also dependent on the cyclic mean stress (i.e., the static stress state of the structure). In a previous

study by the authors, it was shown that mean stresses significantly influenced the TSA results for titanium and nickel based

alloys.t This study continued the effort of accurate direct measurements of the mean stress effect by implementing various

experimental modifications. In addition, a more in-depth analysis was conducted which involved analyzing the second

harmonic of the temperature response. By obtaining the amplitudes of the first and second harmonics, the stress amplitude

and the mean stress at a given point on a structure subjected to a cyclic load can be simultaneously obtained. The

experimental results showed good agreement with the theoretical predictions for both the first and second harmonics of the

temperature response. As a result, confidence was achieved concerning the ability to simultaneously obtain values for the

static stress state as well as the cyclic stress amplitude of structures subjected to cyclic loads using the TSA technique. With

continued research, it is now feasible to establish a protocol that would enable the monitoring of residual stresses in structures

utilizing TSA.

1. INTRODUCTION

Thermoelastic stress analysis (TSA) is a full-field, non-contacting technique for surface stress mapping of structures. TSA is

based on the fact that materials experience a temperature change when compressed or expanded (i.e., experience a change in

volume). If the load causing the volumetric change is removed and the material returns to its original temperature and shape,

the process is deemed reversible. This reversibility is achieved when a material is loaded elastically at a high enough rate so

as to eliminate significant conduction of heat. The thermodastic temperature change, in steel for instance, as a result of an

applied cyclic load of 1.0 MPa (145 psi) is on the order of 0.001 °C.2

Lord Kelvin 3 first quantified an analytical relationship between the change in temperature and the change in stress. The

formulation is as follows

(1)

= - K b'/1 (2)

where S T is the cyclic change in temperature; a is the coefficient of linear thermal expansion; To is the absolute temperature

of the specimen; t_It is the change in the sum of the principal stresses; p is the material density; Cp is the specific heat at

constant pressure; and K is the thermoelastic constant. Note that the formulations of Equations (1) and (2) indicate that S T is

NASA/TM--2001-210697 1
independent of the mean stress. Therefore, 6 T is assumed to remain constant for a given stress range, 6 Ij, and absolute

temperature, To, regardless of the appfied mean stress.

In recent years, experimental data as well as theoretical formulations have shown a mean stress dependence concerning the

thermoelastic constant, K. Such stress dependence was illustrated experimentaI16y by Machin et al. 4 and Dunn et al. 5 The

theoretical explanation for this non-linear response was defined by Wong et al. Additional experimental evidence of the

mean stress effect was provided in Wong et al. 7 In Wong et al. 6, the reformulated theory shows that the mean stress

dependence of the thermoelastic parameter (earlier referred to as a constant) is fully accounted for by the temperature

dependence of the elastic moduli of the material.

In this paper, further experimental verification of the mean stress effect is provided. This includes both a direct measure of

the TSA signal as a function of the mean stress as well as an in-depth analysis of the first and second harmonics of the TSA

signal. The theory, as described in the following section, shows that the thermoelastic response of a structure subjected to a

pure sinusoidal mechanical load with a frequency, o% produces a TSA signal with frequency components at the primary

frequency, o%as well as the second harmonic, 2ca The first harmonic of the thermal response is a function of the cyclic stress

amplitude and the mean stress while the second harmonic is a function of the square of the stress amplitude. By obtaining the

TSA amplitudes of the frrst and second harmonics, the stress amplitude and the mean stress at a given point on a structure

subjected to a cyclic load can be simultaneously obtained. Furthermore, comparisons are made between the theoretical

predictions and the experimental data. Once confidence is achieved concerning the measured TSA response, then steps can be

taken for developing empirical relationships between the TSA signal and the cyclic stress state of various materials.

It should be noted that the past thermoelastic data (references 4 through 7) were produced utilizing TSA systems with

infrared (IR) cameras based on a single detector. This single detector camera incorporates a network of scanning mirrors to

provide an image of an area. In references 4 through 7, the scanning devices were disengaged and only the information from

a single point on the specimens was analyzed. For this study (as well as the data produced by the authors in reference 1), the

TSA system uses an IR camera with a 128 x 128 focal plane array (FPA) of detectors. As a result, the reported IR detector

response is based on the average of an array of detectors representing a larger two-dimensional area on the specimen surface.

To obtain the first and second harmonics of a TSA signal composed of an array of IR detectors, hardware modifications of

the current TSA system were required and are detailed in the Experimental Methods section.

Because this research continued the efforts of reference 1, the same metallic alloys were utilized. These included the titanium

base alloys, TIMETAL 21S and Ti-6A1-4V, as well as a nickel base alloy, Inconel 718. These materials are utilized by the

aviation industry in propulsion components due to their relative lightweight, high strength and stiffness, as well as their

property retention at elevated temperatures.

During manufacturing, certain components, which are subjected to a cyclic fatigue environment, are fabricated so as to

contain compressive residual stresses on the surface. These compressive stresses inhibit the nucleation of cracks. As a result

of overloads and elevated temperature excursions, the induced residual stresses dissipate while the component is still in

service, in turn, lowering its resistance to crack initiation. Once confidence is achieved concerning reliable TSA

measurements of the mean stress effect, research can focus on the application of the method to residual stress assessment.

Such measurements will assist in the characterization of materials in the laboratory as well as in-situ monitoring of the

current residual stress state in actual structural components during fabrication and service.

2. THEORY

The revised thermoelastic equation (derived in Wong et al. 6) relating the rate of temperature change and the rate of change in

the stress state in a homogeneous Hookean material under adiabatic conditions can be written as

poC, E'- aT ' i' L E ar E_gTJi=_ °'''d'" (3)

where T is the thermodynamic temperature (Kelvin); _; are the principal stresses (N/m2); lj is the sum of the principal

stresses, (N/m2); Po is the density (kg/m3); Cc is the specific heat under constant strain (N m/°C kg); a is the coefficient of

thermal expansion (°Ct); E is the Young's modulus (N/m2); and v is the Poisson's ratio. The dotted symbols represent

NASA/TM--2001-210697 2
derivativeswithrespect totime.Noticethatforthe reformulated thermoelastic equation, the elastic moduli are not assumed

constant, but are taken as functions of temperature. Hence, the difference when comparing the new formulation to the

classical thermoelastic theory of Equations (1) and (2). It is also seen that the temperature response is now dependent on both

the stress rate and the stress state.

Following the procedure set forth in Wong et al. 7, the uniaxial case is taken as an example. The uniaxial case is defined as

having

o-11 =I1, 0.22 =o'33 =0 (4)

and

6"11 = ]1, 022 = (#33 -----0 (5)

Utilizing Equations (4) and (5), Equation (3) simpfifies to

poCe--_=- a 0.11 (rll E 2 OT

(6)

In keeping with the approach taken by Wong et al. 7, a sinusoidal stress input of the following form is defined

O"=0",n +0"am p sincot (7)

where o'm and 0"amp are the mean and amplitude of the applied stress, respectively, and co is the loading frequency. Note that

for ease of presentation the subscripts on the stress terms have been omitted with the understanding that or= 0.11. Next,

Equation (7) is substituted into Equation (6) to obtain the following solution for the uniaxial case

PoCt--T--f= ot E2- _-_ 0"m ampsincot 4E 2 0T (0.a'np)2COS2(-Ot

(8)

Note that the thermal response of a structure subjected to a pure sinusoidal mechanical excitation produces a thermal

signature function that has components at both the primary loading frequency, o_ as well as the second harmonic, 2oa The

first harmonic of the thermal response is a function of the cyclic stress amplitude and the mean stress while the second

harmonic is a function of the square of the stress amplitude. It is now apparent that by obtaining the first and second

harmonics, it is possible to obtain both the cyclic stress amplitude and the mean stress.

Lastly, comparing Equations (1) and (2) with Equations (6) and (8), the effective K for a homogeneous Hookean material

subjected to a uniaxial stress is derived as

K= -----0._ p Ce )-2

E 2 OT

(9)

Equation (9) shows that the thermoelastic parameter is a linear function of the mean stress, 0"m-A normalized measure of the

thermoelastic parameter's mean stress dependence can be expressed as

1 OK -1 bE

Ko 00"m a E 20T

(10)

where

c(

K o =-- (1 l)

P Ce

NASA/TM--2001-210697 3
Equation(10)simplygivestheslopeof thethermoelasticconstant(inadimensionless, normalizedform)in respect to the

mean stress, tyro.Note that Equation (11) is the zero mean stress solution of Equation (9).

3. EXPERIMENTAL METHODS

Three metallic alloys with various pre-test heat treatments were analyzed for this study. Two were titanium base and one was

a Nickel base. The first titanium alloy, sheet TIMETAL 21S, was heat-treat stabilized at 621 °C (1150 °F) for 8 hours. The

second titanium alloy, Ti-6A1-4V in sheet form, was annealed at 788 °C (1450 °F) for 15 minutes and then air cooled. The

Nickel base Inconel 718 sheet plates were solution treated at 1038 °C (1900 °F) and quick cooled. The material was then

aged at 718 °C (1325 °F) for 8 hours. Next, it was cooled at a rate of 56 °C (100 °F/hour) to 621°C (1150 °F) and held for

8 hours. This was followed by a quick cool.

The TIMETAL 21S was supplied as round, reduced gage section coupons machined from 0.79 cm (0.31 in.) thick, flat plates.

The coupons had a constant diameter gage length of 1.91 cm (0.75 in.) and a nominal gage diameter of 0.635 cm (0.25 in.).

The Ti-6AI-4V and Inconel 718 plates, measuring 17.8 cm x 17.8 cm (7 in. x 7 in.), were cut into straight-sided specimens

using the EDM (Electrical Discharge Machining) process. The overall length of the specimens was 17.8 cm (7 in.) and the

width was 2.54 cm (1 in.). The Ti-6A1-4V had a nominal thickness of 0.203 cm (0.08 in.), while the Inconel 718 had a

nominal thickness of 0.178 cm (0.07 in.). For each material three specimens were tested.

The liquid nitrogen cooled infrared camera of the TSA system employs a 128 x I28 focal plane array of InSb detectors (3 to

5 pm sensitivity). The system operates by recording a periodic temperature change, as viewed by the IR camera, of a

specimen subjected to a cyclic mechanical load (see Figure 1). A reference signal from the load cell is used by the software to

allow it to monitor only the true thermoelastic change in temperature and to disregard any noise and environmental effects

that do not correlate with the reference signal's primary frequency. This is accomplished by using a lock-in detection method

(see Reference 8 for an explanation). The lock-in technique filters the thermal response allowing the analysis of only the TSA

frequency component that corresponds to the primary frequency of the reference signal. To further improve the TSA signal to

noise ratio, the IR signal is monitored, accumulated and averaged over a period of hundreds or thousands of load cycles.

When an image is captured, the computer display depicts a dimensionless, digitized value of the average camera signal range

corresponding to the cyclic load range for each of the 16384 pixels. For typical structural stress measurements, the

dimensionless digital IR values are correlated to a known cyclic stress amplitude (e.g., as measured by a strain gage which

represents a finite, local area) and then utilized to map the surface stress amplitudes of the entire structure. Further details

concerning the TSA system and the specific settings are given in the Appendix.

Two techniques were utilized to compare the experimental results with the theory. First, direct experimental measurements

were conducted for analyzing the mean stress effect. The raw data for these direct measurements were produced in reference

1 and were then modified during this study using newly created temperature correction curves. In addition, the second

harmonic of the TSA signal was collected and analyzed.

For convenience the experimental procedure for the direct measurements of the mean stress effect is repeated here. The

measurements were accomplished by comparing the camera IR value at various discrete mean stresses while maintaining

fixed cyclic stress amplitudes. The test platform employed for inducing the mechanical excitation was a digitally controlled

(+ 50 kN dynamic / + 100 kN static) servo-hydraulic test system. The experiments were conducted in load control with a

10-Hertz sinusoidal waveform. Each of the three materials were subjected to a cyclic stress range of 70 MPa (10 ksi). The

round TIMETAL 21S specimens had mean stresses varying from -276 MPa (-40 ksi) to 276 MPa (40 ksi). For the straight

sided Ti-6AI-4V specimens, the mean stress ranged from 0 to 276 MPa (40 ksi). Finally, the Inconel 718 was subjected to

discretely varying mean stresses from 0 to 345 MPa (50 ksi). At least three repetitions were conducted for each individual

specimen (except for the TIMETAL 21S, where only one repetition was performed). A test repetition involved stepping

through each of the discrete mean stresses and recording an IR signal. The mean stress stepping order was randomized for

each test. The camera distance and angle were held constant throughout the experiments. The camera's line of sight was

perpendicular to the specimen face, while the distance from the camera lens to the specimen face was maintained at 12.38 cm

(4.88 in.). Lastly, it should be noted that all loads were well within the materials' elastic regimes, hence, no plastic

deformation occurred.

NASA/TM--2001-210697 4
Next,acomparisonwasconductedbetweentheoryandexperiment thatconcernedtheanalysisofthesecondharmonicofthe

thermoelastictemperatureresponse. SincetheTSAsystemfiltersall componentsof theTSAsignalexcepttheprimary

frequencyof thereferencesignal,hardwarewasdevelopedwhichallowedtheelectronicstofocusinonthesecondharmonic

oftheIR signal.Thereferencesignalfromtheloadcellwaspassedthroughalinearfrequencydoubler(developedforthis

studyandplacedin-linewiththereferencesignal)priortoitsusebytheTSAsystem, thereby,allowingtheTSAsystemto

focusin onthesecondharmonic. In theearlierstudiesreferredto in theIntroduction, a FastFourierTransform(FFT)

spectrumanalyzerwasemployedfor obtainingtheamplitudeof theTSAsignal'ssecondharmonic.Thiswaspossible

becausetheIR signalwascomposedof onlya singledetector.Here,16,384detectorsweremonitoredby thelock-in

electronics. TomaintainthepracticalityoftheIRarrayTSAsystem,thelock-inelectronicswereutilizedduringthecapture

of thesecondharmonic. AnFFTspectrumanalyzerwasusedthroughout thestudyto monitor the purity of the sinusoidal

loading function of the servo-hydraulic test system. Furthermore, only the TIMETAL 21S coupons were utilized for the

production of the second harmonic TSA data. This is due to the fact that the TIMETAL 21S specimen's geometry allowed

for compressive loads. As a result, larger cyclic amplitudes were achieved which assisted in the analysis of the rather small

second harmonic TSA signal.

Test preparation included painting all the specimens with an ultra-flat black paint to improve surface emissivity. In addition,

K-type thermocouples were spot welded to the back of each specimen (slightly below the gage section) so as to allow

temperature monitoring throughout the tests. Since the IR flux is dependent on the source (i.e., specimen) temperature, an

empirical equation was used to correct the camera signals to values corresponding to a specimen temperature of 23°C

(73.4°F). In reference 1 a single temperature correction curve, acquired from another reference, was utilized for all the TSA

data. Here, three empirical relationships were developed corresponding to each of the three material systems. The TIMETAL

21S specimens were cycled at various fixed amplitudes (103 or 207 MPa; 15 or 30 ksi) while maintaining a zero mean stress.

For the Ti-6A1-4V and Inconel 718 specimens, the amplitudes were maintained at either 69 or 103 MPa (10 or 15 ksi). The

mean stresses for these flat specimens were set equal to the their respective amplitudes so as to avoid any compressive loads.

During any particular fixed amplitude test, the absolute specimen temperature was varied between 21 and 27°C (70 and 80°F)

while the TSA response was monitored. Heating was achieved by employing a standard hand-held heat gun. Finally, each set

of data was normalized by its respective value at 23°C, thereby, allowing the data to reduce to a single line for each material

system.

4. RESULTS

Figure 2 displays a typical thermoelastic image as captured by the software. The dimensionless IR camera signal value used

throughout this study is an average of an image box in the gage area of the specimen. For the TIMETAL 21S, the box

consisted of 300 pixels representing the individual IR detectors (10 wide x 30 long) which were focused on the center portion

of the specimen. The boxed area of interest was 0.318 cm x 0.953 cm (0.125 in. x 0.375 in.). For the larger Ti-6A1-4V

and Inc 718 specimens, the image box consisted of 3500 detectors (50 x 70), representing a 1.56 cm x 2.15 cm (0.616 in. x

0.846 in.) area.

The values in Table 1 are the linear regression constants for the temperature correction data corresponding to each of the

three material systems. Figure 3 displays the data used to produce the linear regression fit for TIMETAL 21S. The

temperature correction curve was defined as

S

523°C = (q + r T) (12)

where Se3_c and S are the corrected and original IR signals, respectively, T is the specimen temperature in degrees Celsius,

and q and r are the linear regression constants from Table 1. All IR data presented in this paper are in the temperature

corrected form. The relationship of Equation (12) is physically a function of the specimen's IR flux, and assumed to be

independent of the subject material. Even so, a cautious approach should be taken on its general use since it is dependent on

various experimental factors (e.g., thermocouple location when calibrating TSA response to the monitored temperature). The

variations in the regression constants are apparent in Table 1.

In the following paragraphs various normalized mathematical expressions are defined and utilized for both the experimental

data as well as for the theoretical formulations. The normalized expressions are required so as to allow a direct comparison

between the experimental data, which is in the form of non-dimensional digital values representing the IR camera's electrical

NASA/TM--2001-210697 5
output, and the theoretical formulations, which give a direct value of the temperature range resulting from the thermoelastic

effect. The goal here is to verify the experimental behavior with the theory, thereby providing confidence in the experimental

measurements. Once confidence is achieved, direct empirical relationships can be developed between the IR camera's digital

values and a structure's stress state.

Figure 4 shows the thermoelastic response for the TIMETAL 21S specimens as a function of the mean stress. It is seen that a

change in the mean stress from -276 MPa (-40 ksi) to 276 MPa (40 ksi) induced, on average, a 21% increase in the IR signal.

Subsequently, the data of the three specimens were pooled and fitted with a finear, first order regression. The constants of the

regression fit are presented in Figure 4, The regression constants were used to develop the experimentally determined

measure of the mean stress dependence of the thermoelastic parameter (see F_.quation (10)). This was accomplished by

dividing the slope of the regression line by the intercept of the same regression fine. Table 2 compares the experimentally

measured normalized mean stress effect with the theoretical value obtained using Equation (10) and material properties from

reference 9. The comparison indicated a 2.4% difference between the theoretical and experimental values. This result shows a

slight improvement over the values obtained in reference 1 where the error between the theory (using the new material

properties from reference 9) and experiment (using the old temperature correction curve) was 4.1%. Also, a comparison was

conducted concerning the means of the pooled data at the various discrete cyclic mean stresses. Using the student t-test

(unpaired) with a 95% confidence level, the smallest statistically significant observable difference in mean stresses for the

pooled TIMETAL 21S specimens was calculated as 207 MPa (30 ksi). This result is the same as that obtained in reference 1.

Figures 5 and 6 show the thermoelastic responses for the Ti-6AI-4V and Inconel 718 specimens as functions of the mean

stresses. Because no statistically significant differences existed between the specimens at the various discrete mean stresses,

the data were pooled for each material type. The regression constants are indicated in the figures. These constants were again

used to calculate the normalized mean stress effect as given in Table 2. As a result of using the new temperature correction

curves, the errors between the theoretical and experimental values were reduced. The error for the Ti-6Al-4V data is now

35% which compares with a 40% error in reference 1. For the Ti-6A1-4V specimens, an increase of 276 MPa (40 ksi) in the

mean stress altered the IR signal by 8%. The smallest statistically significant difference in mean stresses for the pooled

Ti-6Al-4V specimens was calculated as 69 MPa (10 ksi). These last two values are equivalent to those given in reference 1.

For the Inconel 718 specimens the error was significantly reduced concerning the mean stress dependence. It should be noted

from Figure 6, that there exist two high value outliers at the 0 mean stress, which do not appear to fit the pattern of the data.

The increased signals for these points may have been caused by a slight buckling of the relatively thin specimens during the

negative load portion of the fully reversed cycle. If the camera were focused on the compressive side of the bend, the

IR signal would have an increased value as a consequence of the additive compressive bending stress. As was done in

reference 1, the two outlier points were removed from the analysis. Therefore the regression constants shown in Figure 6

were obtained by fitting a linear regression through the reduced data. Using the new regression constants it is seen that the

difference between the theoretical and experimental mean stress dependence (Table 2) is 5.5% as compared to 20% in

reference 1. An increase of 345 MPa (50 ksi) mean stress resulted in a 4% change in the IR signal. This is equivalent to the

reference 1 result. Lastly, a slight improvement was seen when comparing the means at each of the various mean stresses.

Here, a 207 MPa (30 ksi) change in mean stress was recognized as being statistically significant as compared to 241 MPa

(35 ksi) in reference 1.

Next, the experimental results for the TIMETAL 21S specimens were compared with the theoretical prediction of Equation

(8). Presented in Figures 7 and 8 are the first harmonic and second harmonic response curves, respectively. It is seen that the 7

first harmonic is linearly related to the stress amplitude. In keeping with the approach of Wong et al., an equation of the

form, S_ = a (Ao') was used to describe the data. A curve fit of the data in Figure (7) yielded a slope of a = 103.0 MPa "l.

Now focusing on the second harmonic response of Figure (8), it can be seen that the IR response is linearly proportional to

the square of the cyclic amplitude and can best be described by an equation of the form, $2o_= b (Ao" 2). The calculated slope

was b = 6.53 x 10-3 MPa "2. Next, the ratio of the slopes of the experimentally determined first and second harmonics was

calculated as b/a = 0.634 x 10.4 MPa 1. This was done so as to allow a comparison with Equation (8). The theoretical ratio,

using Equation (8) with a zero mean stress, is as follows:

(13)

NASA/TM--2001-210697 6
UtilizingtheTIMETAL21SmaterialpropertiesinTable2(obtainedfromCastellietal.9),Equation(13)givesapredicted

valueof b/a = 0.846 x 10-4 MPa -1. The difference between the theoretical and experimental values is 25%. The following

section discusses the experimental results.

5. DISCUSSION AND CONCLUSION

The mean stress dependence has been shown for the current material systems, TIMETAL2 IS, Ti-6AI-4V, and Inconel 718,

both experimentally in Figures 4 through 6 as well as theoretically by Equation (10). The errors between the theory and

experiment are reasonable and present an improvement over the errors reported in the previous study by the authors

(reference 1). The improvements are a result of using temperature correction curves defined during this study. Note that for

TIMETAL 21S more accurate material properties were utilized for the theoretical solution. In addition, there was an

improvement in the standard deviations of the data in the mean stress dependence curves of Figures 4 through 6.

By utilizing a linear frequency doubler in line with the reference signal, the second harmonic of the IR signal was

successfully measured using the TSA system's lock-in electronics. As a result, the practicality of obtaining simultaneous

solutions for a 128 x 128 array of IR detectors was maintained. The linear relationship for TIMETAL 21S observed in Figure

7 agrees with the theory where the slope, a, is directly related to the thermoelastic parameter, K. Changing the mean stress

would result in a changing slope according to Equation (10). Therefore, it is possible at this point to utilize plots of the first

harmonic such as the one in Figure 7 and to use the slopes of such curves to empirically solve for the mean stress. However, a

value for the stress amplitude is required. This is only practical when characterizing coupons in a laboratory environment or

when testing complex structures which have been fitted with strain gages.

It is seen in Figure 8 that the second harmonic data has a larger spread than the first harmonic data of Figure 7. This is due to

the fact that the second harmonic signal is only about 2% of the primary signal. As a result, the second harmonic IR signal is

closer to the noise band of the system especially when utilizing small cyclic amplitudes. Also, it is seen that some aberrations

in the data did occur. Note that the second harmonic IR values at the 103 MPa (15 ksi) level are, on average, smaller than the

IR values at 34 MPa (5 ksi) (see Figure 8). No explanation for this behavior is available, although, it could be due to the low

value of the IR signal and its proximity to the noise band of the TSA system. Although the error between the theoretical value

of b/a and the experimental b/a is reasonable, improvements can be achieved. The current error is attributed to the sensitivity

of the theoretical value to the material properties, and the fact that the material properties utilized here were obtained from

reference 9 and not directly measured. Therefore, it appears that confidence was achieved concerning the accuracy of the

experimentally obtained data. It is now reasonable to develop empirical equations that directly correlate the TSA results

(i.e., the non-dimensional digital values of the IR camera) to the cyclic stress amplitudes and mean stress states of

components.

Difficulties do arise with the introduction of multiaxial stress_,. The multiaxial problem is very complex, although, some

attempts have been made to address the issue of solving for the individual stress components. Past researchers have combined

the bulk stress TSA data with numerical methods to obtain stress separation in limited cases. Also, the possibility does exist

for combining the results of TSA, which provides information concerning the sum of the principal stresses, and

photoelasticity, which provides information on the differences and directions of principal stresses. With on going research in

these areas, the TSA method does show promise as a residual stress assessment tool.

APPENDIX

The thermoelastic stress analysis was conducted using the DeltaTherm 1000 system manufactured by Stress Photonics

Incorporation. The following system software settings were utilized during 1R data acquisition concerning the first and

second harmonics of Figures 7 and 8: Accumulation Time = 4.7 seconds; Gain = 1; and AC Channel Integration Time

constant = 120 seconds. Collection time for any given test condition was 240 seconds. This consisted of 120 seconds to

collect the first harmonic and then activating the frequency doubler and obtaining a 120 seconds reading for the second

harmonic. Note the timer was reset after each 120 second capture. Lastly, the data which was obtained from reference 1

had an Accumulation Time = 4.7; Gain=2; and an AC Channel Integration Time constant = 300 seconds (see Figures 4

through 6).

NASA/TM_2001-210697 7
REFERENCES

1. A.L. Gyekenyesi and G.Y. Baaklini, "Thermoelastic Stress Analysis: The Mean Stress Effect in Metallic Alloys,"

Proceedings of SPIE Conference on the NDE of Aging Materials and Composites, Newport Beach, CA, March I-4,

1999.

2. J.R. Lesniak, and B.R. Boyce, "A High-Speed Differential Thermographic Camera," SEM Spring Conference

Proceedings, Baltimore, Maryland, June 6-8, 1994, pp. 491-497.

3. W. Thomson (Lord Kelvin), "On the Dynamical Theory of Heat," Trans. Roy. Soc. Edinburgh, Volume 20, 1853,

pp. 261-283.

4. A.S. Machin, J.G. Sparrow, and M.G. Stimson, "Mean Stress Dependence of the Thermoelastic Constant," Strain,

Volume 23, 1987, pp. 27-30.

5. S.A. Dunn, D. Lombardo, and J.G. Sparrow, "The Mean Stress Effect in Metallic Alloys and Composites," SPIE Stress

and Vibration." Recent Developments in Industrial Measurement and Analysis, Volume 1084, 1989, pp. 129-142.

6. A.K. Wong, R. Jones, and J.G. Sparrow, "Thermoelastic Constant or Thermoelastic Parameter?" J. Phys. Chem. Solids,

Volume 48, 1987, pp. 749-753.

7. A.K. Wong, J.G. Sparrow, and S.A. Dunn, "On the Revised Theory of the Thermoelastic Effect," J. Phys. Chem. Solids,

Volume 49, 1988, pp. 395-400.

8. J.R. Lesniak and B.R. Boyce, "Differential Thermography Applied to Structural Integrity Assessment," Proceedings of

the International Conference on Thermal Sensing and Imaging Diagnostic Applications, Orlando, Florida, April 19-21,

1995, Vol. 2473, pp. 179-189.

9. M.G. Castelli, S.M. Arnold, and A.F. Saleeb, "Specialized Deformation Tests for the Characterization of a Viscoplastic

Model: Application to a Titanium Alloy," NASA TM-106268, in print.

10. M.G. Castelli, B.A. Lerch, and D.J. Keller, "A Comparison of Deformation Behaviors of HIPed Foil and Sheet Titanium

Alloys," HITEMP Review 1999: Advanced High Temperature Engine Materials Technology Project, NASA CP208915, Vol. 2, 1999.

11. "Metals Handbook: Desk Edition," 9 th edl, 1-EEl Boyer and T.L. Gall, edL, Amelffcan Society for Metals, 1997.

12. "High Temperature, High Strength Nickel Base Alloys," Inco Limited, New York, NY, 1977.

TABLE 1. LINEAR REGRESSION CONSTANTS

FOR TEMPERATURE CORRECTION CURVES.

Material Intercept,

Norm TSA Signal*

(q: Eq. 12)

TIMETAL 21S 0.337

Ti-6AI-4V 0.0236

Inconel 718 -0.0479

Slope,

(Norm TSA)/(Temp. °C)

(r: % 12)

0.0288

0.0425

0.0455

*TSA signals normalized by respective TSA value at 23°C (see Figure 3).

TABLE 2. COMPARISON OF THEORETICAL AND

EXPERIMENTAL MEAN STRESS DEPENDENCE OF K.

Material* a, Ez3°c, a E/3 T

10"6/°K GPa MPa/°K

TIMETAL 21S 8.24

Ti-6AI-4V 9.06

Inconel 718 12.2

116

120

205

Theory

Eq. (10)

MPa -_

-57.7

-37.6 3.39 x 10 -4

-61.8 4.'7"5 x-10-4

I

1.i2 x 10-4

Experiments

MPa -1

3.47 x 10-4

3.07 x 10-4

1.18 x 10_

"Matenal properties obtained from references 9 through 12.

7

NASA/TM--2001-210697 8
Load Actuator

Applies load (force) to test objectN

._o..---+

_.+....-'"

,,, I--................s.o+oaI I+e_

Camera _.1 ..... (Scan "Field") .l_..Object

Detector Signal (Camera .-,.=-SPU)

k._ "_ 1=

Camera-Control & Feedback I -

(Camera+-_SPU) _

=l

+u Signal

Processing

Unit

J Processed Detector-Data

(SPU --_Computer)

Camera-Control 8 Feedback

(SPU .e-->Computer)

J

Load (Reference) Signal

) Video 1

Signal

II

. .on,o, IJ

Figure 1 Schematic diagram of TSA system (DeltaTherm 1000 User's Manual).

Figure 2 Typical IR test image for a TIMETAL 21S specimen. The rectangular box within the specimen indicates the area where the

average signal was obtained. The scale displays the dimensionless digital values of the IR camera signal.

NASA/TM--2001-210697 9
1.14

1.12

1.10

•_ 1.08

"t 1.06

re 1.04

i 1.02 1.00

Z 0.98

0.96

0._

0._ .......

20 21 22 23 24 25 26 27

TIMETAL 21S

ml,

Y = 0.337 + 0.0288 * X

r = 0.99 O.._ •

o

Temperature, °C

28

Figure 3 Temperature dependence of IR signal for TIMETAL 21S specimens. Data consists of eight tests utilizing three specimens. All

tests normalized by corresponding IR signals at 23°C. Equation represents first order linear regression fit of all data.

m

t_

e-

_m

t_

gg

w

7800

7600 -

7400-

7200 -

7000 -

6800 -

6600 -

6400-

6200 -

58oo i

-4OO

TIMETAL 21S

Y=6854+2.38*X ..•

r = 0.93 _11

• "

• Specimen 1

Specimen 2

• Specimen 3

I I I t t I I

-300 -200 -100 0 100 200 300 4O0

Mean stress, MPa

Figure 4 Mean stress dependence of IR signal for TIMETAL 21S utilizing three specimens. (G_mp= 34 MPa (5 ksi)). Equation represents

first order linear regression fit of all data.

NAS A/TM--2001-210697 10
g

m

ea"

m

9200

9000-

8800 -

8600 -

8400 -

8200 -

8000 -

7800 -

7600-

Ti-6AI-4V

Y = 8136 + 2.50" X

r = 0.85

A Q

e

[]

o ®

D

Specimen 1

O Specimen 2

Specimen 3

@ A •

OA•

•ADO

1 I t I I I I I

-50 0 50 100 150 200 250 300

Mean stress, MPa

Figure 5 Mean stress dependence of IR signal for Ti-6A]-4V utilizing three specimens and three repetitions each (a_,,p = 34 MPa (5 ksi)).

Equation represents first order linear regression fit of all data.

m

m

rn84OO

8200 -

8000 -

7800 -

7600 -

7400 -

7200

Y = 7684 + 0.908 * X

r = 0.64

Inconel 718

e

Outliers _ •

• z_

e o t5

A

I I I I I I

-50 0 50 100 150 200

Specimen 1 O AII

Specimen 20Z_I-I

Specimen 3 OAB

I t I i

250 300 350 400

Mean stress, MPa

Figure 6 Mean stress dependence of IR signal for Inconel 718 utilizing three specimens with three repetitions each (o'a,,p = 34 MPa (5

ksi)). Equation represents first order linear regression fit of data excluding outliers.

NASA/TM--2001-210697 11
35000

TIMETAL 21S

30000 Y = 103.0 X

r = 0.99

25000

20000

"_ 15000

_¢

10000

Specimen 1 •••

5000 S Specimen 2 0/k []

r_ Specimen 3 O _ []

0 I i 1 I J I

0 50 100 150 200 250 300 350

Cyclic stress amplitude, MPa

Figure 7 First harmonic IR signal output at various stress amplitudes ((r,, = 0) for TIMETAL 21S specimens. The plot contains data from

three specimens; each with three repetitions. Equation represents frrst order linear regression fit of all data.

700

6OO

5OO

8

300

tw

2oo

too

TIMETAL 21S

Y = 6.53x10 a * X

r = 0.8t

!

©

A

l _ //_ Specimen1 •,&I

t_ l Specimen 32 ©/\_ _ _, []

I t i i I

0 2e+4 4e+4 60+4 8e+4 I e+5

Square of cyclic stress amplitude, MPa 2

Figure 8 Second harmonic IR signal output at various stress amplitudes (G,. = 0) for TIMETAL 21S specimens. The plot contains data

from three specimens; each with three repetitions. Equation represents first order linear regression fit of all data.

NASA/TM--2001-210697 12

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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

February 2001 Technical Memorandum

4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

Quantifying Residual Stresses by Means of Thermoelastic Stress Analysis

S. AUTHOR(S)

Andrew L. Gyekenyesi and George Y. Baaklini

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135- 3191

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU-728-30-20-00

8. PERFORMING ORGANIZATION

REPORT NUMBER

E-12628

1"0. SPONSORING/MONITORING'

AGENCY REPORTNUMBER

NASA TM--2001-210697

11. SUPPLEMENTARY NOTES

Prepared for the Nondestructive Evaluation of Aging Materials and Composites IV sponsored by the Society of PhotoOptical Instrumentation Engineers, Newport Beach, California, March 8-9, 2000. Andrew L. Gyekenyesi, Ohio Aerospace Institute, 2001 Aerospace Parkway, Brook Park, Ohio 44142 (work funded by NASA Grant NAG3-755); and

George Y. Baaklini, NASA Glenn Research Center. Responsible person, George Y. Baaklini, organization code 5920,

216--433-6016.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -Unlimited

Subject Category: 39 Distribution: Nonstandard

Available electronically at htto:/Idtrs.zrc.nasa._ovlGLTRS

This publication is available from the NASA Center for AeroSpace Information, 301--621-0390.

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

This study focused on the application of the Thermoelastic Stress Analysis (TSA) technique as a tool for assessing the residual stress

state of structures. TSA is based on the fact that materials experience small temperature changes when compressed or expanded.

When a structure is cyclically loaded, a surface temperature profile results which correlates to the surface stresses. The cyclic surface

temperature is measured with an infrared camera. Traditionally, the amplitude of a TSA signal was theoretically defined to be linearly

dependent on the cyclic stress amplitude. Recent studies have established that the temperature response is also dependent on the

cyclic mean stress (i.e., the static stress state of the structure). In a previous study by the authors, it was shown that mean stresses

significandy influenced the TSA results for titanium- and nickel-based alloys. This study continued the effort of accurate direct

measurements of the mean stress effect by implementing various experimental modifications. In addition, a more in-depth analysis

was conducted which involved analyzing the second harmonic of the temperature response. By obtaining the amplitudes of the first

and second harmonics, the stress amplitude and the mean stress at a given point on a structure subjected to a cyclic load can be

simultaneously obtained. The experimental results showed good agreement with the theoretical predictions for both the first and

second harmonics of the temperature response. As a result, confidence was achieved concerning the ability to simultaneously obtain

values for the static stress state as well as the cyclic stress amplitude of structures subjected to cyclic loads using the TSA technique.

With continued research, it is now feasible to establish a protocol that would enable the monitoring of residual stresses in structures

utilizing TSA.

14. SUBJECT TERMS

Thermoelastic stress analysis; Residual stress; Mean stress; SPATE;

Thermoelastic constant; TSA

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