Measuring stress intensity factors during fatigue crack growthusing thermoelasticity

F. A. D IAZ, E. A. PATTERSON, R. A. TOMLINSON and J. R. YATES

Department of Mechanical Engineering, The University of Sheffield, Mappin Street, Sheffield S10 3JD, UK

Received in final form 26 February 2004

A B S T R A C T Thermoelastic stress analysis has been developed in recent years as a direct method ofinvestigating the crack tip stresses in a structure under cyclic loading. This is a conse-quence of the fact that stress intensity factors obtained from thermoelastic experimentsare determined from the cyclic stress field ahead of a fatigue crack, rather than inferredfrom measurement of the crack length and load range. In the present paper the results offatigue crack growth tests performed on welded ferritic steel plates are reported. Fromthe results it can be observed that the technique is sensitive to the effects of crack closureand the presence of tensile and compressive residual stresses due to welding.

Keywords DeltaTherm; differential thermography; fatigue cracks; stress intensityfactor (SIF); thermoelastic stress analysis (TSA).

N O M E N C L A T U R E a = crack lengthA = thermoelastic constant

Cp = specific heat at constants pressureE = elastic modulus

Kmax = stress intensity factor at maximum loadKmin = stress intensity factor at minimum load

Kopen = stress intensity factor at the opening loadK res = stress intensity factor due to residual stresses

m(x, a) = weight functionm1, m2 = weight function coefficients

R = ratio of the minimum to the maximum load appliedS = magnitude of the thermoelastic signal

S0 = residual stress distribution along the crack lineT = absolute temperature of the specimen

W = specimen widthr, θ = notation for polar coordinates

x = distance along the crack liney = vertical distance from the crack lineα = coefficient of thermal expansion

�K = stress intensity factor range�K eff = effective stress intensity factor rangeεx, εy = strains in the Cartesian directions

ρ = densityν = Poisson’s ratio

σ 1, σ 2 = principal stressesσ open = opening stress

σ x, σ y = stresses in the Cartesian directions

Correspondence: J. R. Yates. E-mail: j.yates@sheffield.ac.uk

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572 F. A . D IAZ et al.

I N T R O D U C T I O N

Thermoelastic stress analysis (TSA) is a non-contact ex-perimental technique that provides full-field stress mapsfrom the surface of a loaded component. This techniquehas been developed in recent years as a direct method ofinvestigating the crack tip stresses in a structure undercyclic loading. The principle of this technique is based onthe thermoelastic effect.1 This phenomenon arises fromthe fact that when a structural component is cyclicallyloaded, the structure experiences small, reversible tem-perature changes. If adiabatic and reversible conditionsare achieved, these temperature changes are proportionalto the first stress invariant.

These temperature variations, which are of the order oftens of milliKelvins in metallic materials, can be measuredat the surface of a component using infrared detectors.Thus, by measuring the variations in temperature due tothe applied cyclic load, the sum of the principal stressescan be determined from the detector output:

�(σ1 + σ2) = AS, (1)

where A is the thermoelastic constant and S is the magni-tude of the thermoelastic signal.

At the same time, an expression for the sum of principalstresses in the vicinity of the crack tip location can be de-rived from mathematical models, and from that the stressintensity factor can be estimated.

Stress intensity factors obtained from thermoelastic ex-periments are those that represent the crack tip stress fieldduring cyclic loading rather than those inferred from themeasurements of the crack length and maximum and min-imum loads, which is the case for the majority of othertechniques. The thermoelastic data therefore contain in-formation about any crack face contact, or closure, oc-curring during the loading cycle. With TSA, the thermalresponse of the specimen due to the application of an ex-ternal load cycle is experimentally obtained, and conse-quently, the actual crack driving forces can be experimen-tally determined and the effective stress intensity factorevaluated. This fact makes it potentially possible to eval-uate the effects of crack closure from thermoelastic im-ages of cracked components. Therefore direct full-fieldmeasurement of the crack tip stresses by thermoelasticityoffers the opportunity of determining the magnitude ofthe closure stresses in a growing fatigue crack.

In order to study the potential of TSA for crack clo-sure evaluation, work has been conducted on welded steelplates (Fig. 1), where the influence of residual stresses dueto welding strongly affects the life of the component. Thepaper starts with an introduction of TSA and the equip-ment used and a short review about how this techniquehas been used by different investigators in the study of

fatigue cracks and crack closure. Subsequently, a novelmethodology for the experimental evaluation of the SIFand crack tip location based on the analysis of thermoe-lastic images is presented and discussed. After that, thismethodology is applied to the experimental calculation ofthe SIF from thermoelastic images captured during fa-tigue tests performed at different R-ratios (R = 0.1, 0.3and 0.6) on single edge notched ferritic steel specimenswith a weld at the notch position. Finally, experimentalresults for the SIF are compared with those derived fromthe use of compliance methods and with those predictedfrom numerical calculations.

T H E T H E R M O E L A S T I C E F F E C T

Thermoelastic stress analysis is based on the thermoe-lastic effect; first reported by Lord Kelvin in 1853. Thethermoelastic effect states that any substance in natureexperiences changes in its temperature when its volumeis changed due to the application of a force: compressiveloads cause an increase in temperature while tensile loadproduces a decrease in temperature. Consequently, if acyclic load is applied to a component there will be a cyclicchange in temperature. Under elastic conditions, thesetemperature variations are normally quite small (tens ofmK) and they are normally ignored in the classical the-ory of elasticity. However, with the use of high preci-sion infrared detectors these temperature changes can bequantified. The thermoelastic effect is a reversible con-version between the mechanical and thermal forms of en-ergy, because the temperature variation will reverse whenthe load is withdrawn. However, this energy conversion isreversible only if the elastic range of the material is not ex-ceeded and there is no significant transport of heat duringloading and unloading of the structure.

Moreover, the thermoelastic theory states that under adi-abatic and reversible conditions, the temperatures vari-ations experienced by the cyclically loaded material areproportional to the sum of principal stresses. The relationbetween the change in temperature due to the applicationof a cyclic loading and the stress range of a linear elasticand homogeneous material can be written as:

�T = −αTρCp

�(σ1 + σ2), (2)

where α is the coefficient of thermal expansion, T is theabsolute temperature of the material, ρ is the density, Cp

is the specific heat at constant pressure and σ 1 and σ 2 arethe principal stresses.

To ensure that the experimentally recorded variationis linear, the load cycle must be fast enough to pre-vent heat transport and thus achieve adiabatic conditions.Truly, adiabatic conditions may be achieved only if the

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MEASURING STRESS INTENSITY FACTORS 573

Fig. 1 (a) Dimensions of the specimen used for a fatigue testing. (b) Illustration showing the location of the strain gauges used for thecalibration of the thermoelastic measurements. (c) Detail of the weld in the specimen.

thermal conductivity of the material is zero or no stressgradients are present in the specimen. However, if the loadfrequency is high enough the thermal diffusion length isreduced and the presence of non-adiabatic effects is min-imized.

D E S C R I P T I O N O F T H E T E C H N I Q U EA N D T H E E Q U I P M E N T

In modern TSA, an infrared camera based on a staring ar-ray of photon detectors is used to measure the temperaturechanges at the surface of the component as a consequenceof an applied cyclic load. For the experiments describedin the current paper, the latest version of DeltaTherm(DeltaTherm 1500, developed by Stress Photonics Inc.)was used. This system is built with an indium-antimonidefocal plane array detector that consists in a matrix of 81 920single detectors that allows thermal data to be capturedsimultaneously from a window of 320 by 256 pixels. Thesensor is refrigerated by a closed circuit cooling system.The device has a spatial resolution of 25 µm when at max-imum optical magnification and a maximum thermal res-olution of 2 mK with a 30 s acquisition time.

The thermoelastic signal is often quite noisy, as the in-frared detector is not only collecting the infrared radi-ation coming from the component due to the thermoe-

lastic effect but also the radiation coming from the back-ground. To reduce such noise and extract just the informa-tion related to the thermoelastic effect, signal processingis required. The thermoelastic signal is correlated with areference signal of the same frequency as the load cycleapplied. This signal is normally taken from the functiongenerator that feeds the actuator in the servo-controlledtest machine or a strain gauge bonded on the compo-nent. The thermoelastic signal from the infrared camerais sampled at the peak and valley of the reference signalto isolate the thermoelastic data associated with the cyclicload from the background signal. This is often describedas differential thermography. As a result of the correlationprocess a signal proportional to the thermoelastic effectinduced in the component due to the cyclic loading isobtained.

S P E C I M E N P R E PA R AT I O N A N D C A L I B R AT I O NP R O C E S S

In order to increase the surface emissivity of the com-ponent, it is normally sprayed with a matt black paint.Moreover, calibration is required to translate the out-put voltage from the thermoelastic equipment intostresses. The thermoelastic calibration process consists ofdetermining the stress at a point in the image where the

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574 F. A . D IAZ et al.

stress is already known for a given load on the struc-ture. There are different calibration methods2 but in thecase of the reported experiments the calibration proce-dure was carried out using a strain gauge rosette. A gauge(Tokyo Sokki Kenkyujo Co. Ltd, type FCA-2-11, 2 mm,120 ± 0.5) consisting of two orthogonal strain gaugeswas bonded to the specimen in a region of uniform stresswhere the thermoelastic signal was constant (Fig. 1b).From the gauge the first stress invariant at the point wasobtained:

�(σx + σy) = �(σ1 + σ2) = E1 − ν

�(εx + εy). (3)

If the thermoelastic signal is evaluated at the same pointwhere the gauge is located, the calibration constant, A,can be obtained easily by combining Eqs (1) and (3):

A =E

1−ν�(εx + εy)

S. (4)

Once the calibration constant was determined and imagescaptured, it was possible to change from thermoelasticunits to stresses and process the thermoelastic images tocalculate the stress intensity factor.

T S A A P P L I E D T O F R A C T U R E M E C H A N I C S

In recent years, TSA has experienced significant devel-opments in terms of analysing cracked structures. Thesehave been, in part, due to the advent of systems based oninfrared detector arrays, making it possible to reduce thetime for image capture. Previous systems, like SPATE,3

were based on single point detectors with a motor drivenmirror that allowed the detector to scan the surface of thespecimen in a point-by-point manner. This way of scan-ning takes a relatively long time to produce a thermoe-lastic image and during this period the crack could grow,making it difficult to analyse the image. With new sys-tems based on staring array detectors, like DeltaTherm,4

it is possible to reduce the image acquisition time to a fewseconds. Consequently, such systems become highly at-tractive in the study of fatigue crack growth and damageaccumulation in structural components.

Over the last 20 years, different approaches for deter-mining the stress intensity factor from thermoelastic datahave been proposed. Stanley and Chan5 in 1986, devel-oped a method that related the thermoelastic signal tothe first two terms of Westergaard’s equation for the elas-tic stresses in the vicinity of a crack tip under mode Iand mode II loading. In 1993, Stanley and Dulieu-Smith6

presented an approach based on the fact that the isopachiccontour at the crack tip region generally takes the formof a cardioid curve centred on the crack tip. In 1996,Lesniak et al.7 developed a technique that operated by fit-

ting Williams’ stress field equations to thermoelastic datausing a least squares optimization method. In 1997, Linet al.8 reported a method that calculated the stress inten-sity factor from thermoelastic data based on a combina-tion of the equilibrium and compatibility equations withthe J-integral concept. In 1997, Tomlinson et al.9 deriveda technique based on Muskhelishvili’s stress field equa-tions. This technique was used with success in previouswork undertaken with photoelasticity by Nurse and Pat-terson.10 The procedure is based on the multipoint-over-deterministic method (MPODM) developed by Sanfordand Dally.11

Since Elber12,13 first identified the crack closure effect,much research has been carried out on its measurementand interpretation using indirect compliance techniques.Although thermoelasticity has a great potential for fa-tigue crack analysis, little work has been published relatedto quantifying the crack closure effect, probably due tothe novel character of this experimental technique. Ful-ton et al.14 reported some experiments with TSA where apossible effect of crack closure during the fatigue tests wasobserved. They tested both real cracks and spark-erodedslots and observed evidence of crack closure when a fatiguecrack was studied at different R-ratios. Further work wasperformed by Batchelor et al.15 on single edge notchedaluminium panels. In this case crack closure was studiedby measuring the effect on the stress intensity factor of re-moving the wake of the fatigue crack. The results showeda measurable reduction in the closure when the wake ofthe crack was removed. Dıaz et al.16 performed fatiguetests on single edge notched steel specimens. In those ex-periments, the differences between the experimental andtheoretical stress intensity factor ranges decreased as theR-ratio was increased.

P H A S E M A P

Thermoelastic information is normally presented as a vec-tor where the modulus is proportional to the change intemperature experienced by the specimen due to the ther-moelastic effect and the phase denotes the angular shiftbetween the thermoelastic and the reference signals. Themagnitude of the phase is normally constant unless adia-batic conditions are not achieved.

However, there are two phenomena that lead to a lackof adiabatic conditions and hence a change in phase: heatgeneration due to plastic work and the presence of highstress gradients. Both are conditions that occur near thecrack tip region. Consequently, in these circumstancesheat conduction starts taking place blurring the data atthose regions, making direct observation of the crack tipfrom a thermoelastic image difficult. Nevertheless, thethermoelastic data are rich in information and work hasbeen conducted17 to develop a methodology based on the

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MEASURING STRESS INTENSITY FACTORS 575

Fig. 2 Typical phase map for a 6 mmfatigue crack. Illustration of the phase shiftoccurring at the crack tip as a result of theloss of adiabatic conditions (dimensions inmm).

Fig. 3 Typical phase profile along a fatiguecrack and the ligament. Detail of the phaseshift occurring at the crack tip due to theloss of adiabatic conditions.

phase map to estimate the crack tip location. This method-ology does not provide an exact location for the crack tipbut helps in finding an approximate location required forcalculating the SIF. This will be described in next section.

For image processing purposes, the reference and thethermoelastic signals are set in-phase. Consequently, thephase map should be zero in all those points where adi-abatic conditions are achieved. However, near the cracktip, plasticity and high stress gradients lead to a loss ofadiabatic conditions. This loss of adiabaticity can be eas-

ily identified in the phase map (Fig. 2). In addition, furtherinformation about the crack processes can be extracted byobserving the phase profile along the crack and the liga-ment (Fig. 3). Looking at the right side of (Fig. 3), thereis a region where the phase takes a constant value equalto zero. This means that adiabatic conditions are achievedat all those points (region A). Approaching the crack tip,the phase starts taking positive values (region B). Thispositive phase value indicates a loss of adiabaticity due tomonotonic plasticity and presence of high stress gradients.At region C, the phase changes from positive to negative

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576 F. A . D IAZ et al.

and takes negative values until point O is reached. Thisregion, where the phase value turns negative is attributedto the presence of reverse plasticity as the sign of the phaseis now inverted with respect to region A. From the pointO onwards the phase starts changing continuously frompositive to negative. Such behaviour is attributed in part tocontact between the crack faces and also the backgroundreflection coming from inside of the crack mouth as itis opening. Thus, based on previous analysis, point O isadopted as an initial estimate of the crack tip. This loca-tion is not necessarily the real crack tip but as mentionedbefore it is adopted as an initial estimation required for themethod employed to calculate the SIF from thermoelasticdata. This phase pattern was observed in all the thermoe-lastic tests performed and it agrees with other publishedexperimental results18 using thermocouples to quantifythe heat generation due to local plasticity. Nevertheless,although no conclusive results can be reported, furtherwork is underway to link the area of the phase shift to thesize of the plastic zone.

D E S C R I P T I O N O F T H E A L G O R I T H MF O R S I F C A L C U L AT I O N

Previous methods for SIF calculation based on thermoe-lasticity combine thermoelastic information with mathe-matical models describing the sum of the principal stressesaround the crack tip. Those mathematical models arefunctions that depend on the crack length, the stress in-tensity factor and the coordinates of points surroundingthe crack tip. In all the cases, the coordinates of thosepoints must be referred to the crack tip. Thermoelas-tic images can be used to obtain the stress sum of a setof data points surrounding the crack tip. Subsequently,the coordinates of those data points and their values canbe substituted into the mathematical model to formu-late a system of equations to calculate the stress intensity

Fig. 4 Schematic illustration of the data point selection used to reconstruct the stress field and calculate the stress intensity factors fromthermoelastic images.

factor7,9,10. However, the crack tip coordinates must beknown in order to refer the coordinates of the selected datapoint in the thermoelastic images to the crack tip. Never-theless, finding the real crack tip location by visual inspec-tion of a thermoelastic images is not an easy task, and anerror in locating the crack tip is always likely to occur. Theerror introduced when locating the crack tip can stronglyaffect the K calculation. However, in all the previousmethods for calculating the SIF using the thermoelasticdata, the crack tip was either not an issue5, or its locationhad to be assumed by the operator, which constitutes a ma-jor source of error and uncertainty as thermoelastic dataclose to the crack tip are blurred due to the lack of adiabaticconditions.

This aspect has been accounted in a novel algorithm19

based on previous work done by Tomlinson et al.9 in ther-moelasticity and Nurse et al.10 in photoelasticity. The newmethodology not only calculates the SIF but also incor-porates a search routine to find the crack tip coordinates.The algorithm acts by fitting a mathematical expressiondescribing the distribution of the sum of principal stressesat the crack tip to experimental data collected at the cracktip region. The mathematical model is based on Muskhe-lishvili’s approach,20 where the in-plane stresses are de-scribed by two analytical functions of a complex variable.Moreover, the stress equations are expressed in form ofFourier series in complex form where the coefficients ofeach term are complex variables that allow different statesto be modelled. For the calculation of the SIF from ther-moelastic images, an initial estimation for the crack tip isrequired and from that the algorithm starts calculating thereal crack tip coordinates. The initial crack tip estimationis obtained by looking at the phase profile along the crack.This initial location for crack tip is assumed to be at pointO in Fig. 3, as previously described. Subsequently, data arecollected from a set of points in the region surroundingthe crack tip (Fig. 4). In this step, it is necessary to make

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MEASURING STRESS INTENSITY FACTORS 577

Fig. 5 Graph showing the linearrelationship between the vertical distancefrom the crack tip and (Smax)−2 employed inStanley’s methodology for the calculation ofthe stress intensity factor fromthermoelastic data.

sure that points are being collected from the region wherethe validity of the mathematical model is assured. For thecase of a crack with small scale of yielding, three regionscan be identified; a region of large strain and plasticitywhere the model is not valid, as it is based on linear elas-tic fracture mechanics, a region dominated by the cracktip behaviour where linear elastic conditions are achievedand consequently the model is able to describe the stressfield, and a region far away from the crack tip where themodel is not appropriate. Data must be collected fromthe region where the model is valid. To identify such a re-gion Stanley’s methodology5 has been employed. Stanley’smethod combines Eq. (1) with a mathematical expressiondescribed the crack stress sum derived from Westergaard’smodel:

�(σx + σy) = �(σ1 + σ2) = AS = 2�KI√2πr

cos(

θ

2

)

− 2�KII√2πr

sin(

θ

2

). (5)

For the case of pure mode I cracks, Stanley observed that amaximum thermoelastic signal, Smax, along any line paral-lel to the crack, occurred at a 60◦ angle with respect to thecrack. Taking this into account, expression 5 can be rear-ranged into a linear equation relating the vertical distancefrom the crack to any parallel line with the inverse squareof the maximum thermoelastic signal along that particularline, 1/S2

max:

y =(

3√

3�K 2I

4π A2

)1

S2max

. (6)

If the previous relation for a real thermoelastic image isplotted (Fig. 5), three different regions can be identified:

1 Region A. In this region no linear behaviour is observedbecause there is a loss of adiabatic conditions due to highstress gradients and crack tip plasticity.

2 Region B. In this region a clear linear behaviour is ob-served. This can be employed to defined the region of va-lidity of the model as the same behaviour as described byEq. (6) is observed.

3 Region C. In this region there is a deviation from the linearbehaviour observed in region B, which indicates inappro-priate use of this mathematical model.

Based on previous analyses, the region of validity for themodel is identified in (Fig. 5) as the portion of the graphwhere a linear relationship is observed (Region B). Conse-quently, data are collected from the corresponding regionof the thermoelastic image where such a linear relation-ship in Stanley’s plot exists. The collected points are usedto fit the stress field equation described by Muskhelishvili’smodel and to reconstruct the stress field around the cracktip. From the resultant fitting equation the SIF can beinferred.

This new methodology also makes it possible to includethe crack tip location as a variable to be optimized intothe fitting process. The algorithm searches for the cracktip location that minimizes the error for the fitted expres-sion. As a result, the crack tip location obtained from eachprocessed image is calculated, reducing the sources of un-certainty in the SIF calculation. Moreover, the crack tiplocation inferred from each image can be employed to

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578 F. A . D IAZ et al.

monitor the fatigue crack path. The numerical methodemployed in the optimization process is based on a com-bination of genetic algorithms21 and the downhill simplexmethod.22 This numerical approach is more robust thanother methods as it is not based on the derivative of anerror function.

E X P E R I M E N TA L W O R K

The purpose of the experiments was to determine whetherthe TSA technique was sensitive to crack closure andwhether the magnitude of the closure could be quantified.The material used was a ferritic structural steel, BS 1501490LT50. Specimens were cut from a plate manufacturedfrom two 40-mm-thick plates welded together by a multi-pass submerged arc welding process along their edges. Adouble ‘V’ preparation of the plate edges was used andthe welding (Fig. 1) was performed in 16 passes on the topface of the plate and 7 passes from the reverse side. Thefatigue specimens were single-edge notched tensile bars,12 mm thick over a 40 mm gauge length (Fig. 1). A 4 mmslit was spark eroded on the weld centre line from the topweld face, from which the fatigue crack was grown.

Fig. 6 Typical thermoelastic imagescaptures with DT 1500.

The specimen was previously prepared by spraying oneof its faces with a matt black paint (type no 496-782, RSComponents Ltd., UK) to increase the black body radi-ation from the metal surface and have a more uniformemissivity. Additionally, two rosette strain gauges (TokyoSokki Kenkyujo Co., Ltd., type FCA-2-11, 2 mm, 120 ±0.5 ) were bonded to the back face of the specimen, onein the weld region and the other in the parent material,far away from the crack to ensure uniform stresses at thatpoint, as shown in Fig. 1. The strain gauges were used forcalibration and to confirm the Young’s modulus and Pois-son’s ratio values provided by the plate manufacturer. Therosette located at the weld region was employed at thebeginning of the fatigue test to calculated the calibrationconstant A at the weld region. This value was comparedwith the A value obtained from the rosette located in theparent material. Additionally, during the fatigue test, therosette strain gauge located at the weld part of the spec-imen was used to compute any change in the specimencompliance when the crack was approaching the straingauge.

The fatigue machine used for testing was an ESH servo-hydraulic machine with a 100 kN load range and two pin

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MEASURING STRESS INTENSITY FACTORS 579

-10

0

10

20

30

40

50

60

70

4 8 12 16 20 24

Crack length (mm)

SIF

rang

e(M

Pa

m0.

5 )

TheoryExperimental KI, R = 0.13Experimental KI, R = 0.3Experimental KI, R = 0.6Experimental KII, R = 0.13Experimental KII, R = 0.3Experimental KII, R = 0.6

Fig. 7 Experimental and theoretical DKI versus crack length for tests performed with welded specimens at different R-ratios.

jointed grips. Constant amplitude tensile cyclic loads wereapplied along the longitudinal axis at a frequency of 12 Hz.The frequency was selected in order to ensure that adi-abatic conditions were achieved. Experiments were con-ducted at R-ratios ranging from 0.1 to 0.58. Thermoe-lastic images were captured whilst the fatigue crack wasgrown. Images were collected throughout the tests at dif-ferent stages of growth. An example is shown in Fig. 6.The crack growth was also monitored from the reverseside of the specimen using a microscope. Subsequently,the thermoelastic images were processed and mode I andmode II stress intensity factors evaluated at different cracklengths (Fig. 7). In all the fatigue tests conducted, theload was adjusted to give the same nominal �K for eachR-ratio.

VA L I D AT I O N O F T H E R E S U LT S

The results for the stress intensity factor range derivedfrom thermoelastic data, for the case of R = 0.1, werecompared with the effective SIF range obtained by tak-ing into account the change in compliance detected at thecrack tip as the crack starts closing prematurely. To de-tect the compliance change, the longitudinally orientedstrain gauge in the rosette bonded at the weld part ofthe specimen was used (Fig. 1). As the crack approachedthe rosette (2 mm distance between the crack tip and theedge of the rosette) strain data from the strain gauge aswell as load data from the load cell were simultaneouslycollected over a whole load cycle. The data collection pro-cess was aided using PCI data acquisition card (National

Fig. 8 Estimation of the opening load from the change incompliance detected in the load versus displacement plot.

Instruments model PCI 6052E) commanded from a PC(Viglen-Pentium 200 MHz) using Labview software ver-sion 6.0. As a result, both signals previously captured wereplotted on a graph of load versus strain making it possi-ble to clearly detect the change in compliance as the crackstarts opening (Fig. 8). The opening load was estimated asthe point where the slope of the upper portion of the load-ing curve starts deviating from linearity23 (Fig. 8). Subse-quently, this value was employed to calculate the Kopen

according to the expression:24

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580 F. A . D IAZ et al.

Kopen = Cσopen√

πa,

C = 1.12 − 0.231( a

W

)+ 10.55

( aW

)2− 21.72

( aW

)3

+ 30.39( a

W

)4, (7)

where σ open, is the stress corresponding at the openingload, a, is the crack length and W , is the specimen width.

The effective SIF range was calculated according to thefollowing expression:

�Keff = Kmax − Kopen. (8)

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5

Image

No

rmal

ized

SIF

ran

ge

Theoretical (neglecting closure)

Effective SIF from compliance

Effective SIF from TSA

Fig. 9 Comparison of DKI obtained from TSA for several imagescaptured at the same crack length (20.15 mm) and DKI inferred byestimating the Kopen from the change in compliance

Fig. 10 Residual stress profile along theweld.

Simultaneously, a set of crack tip thermoelastic imageswas captured from the same specimen on the oppositeface of the specimen. This set of images was subse-quently processed and the SIF range calculated. The in-ferred SIF for several thermoelastic images correspond-ing to the same crack length (20.15 mm) were comparedwith the value of the SIF range previously calculatedusing the compliance change. Results are presented inFig. 9.

As previously mentioned, the specimen tested had a weldat the central part. Although the weld process was carriedout in high quality conditions, the presence of substan-tial residual stresses due to the welding process was in-evitable. The residual stress profile through the weld wasexperimentally determined by an industrial collaborator(Fig. 10). The method employed consisted of bondingtwo strain gauges in longitudinal and transversal direc-tions respectively at one side of the sample. The resid-ual stress profile was obtained from the readings col-lected from the strain gauges as layers of material wereremoved.

Because of the high value of the residual stress ob-served from the residual stress analysis it was thoughtthat they could influence the actual crack tip stressfield and consequently the value of the SIF. To studyhow the residual stresses affected the final value of thestress intensity factor, the contribution of the resid-ual stresses on the SIF was estimated using a weightfunction.25

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MEASURING STRESS INTENSITY FACTORS 581

-50

0

50

100

150

200

4 9 14 19 24

Crack length (mm)

SIF

(MP

am

0.5)

Kmax+Kres

Kmin+Kres

Kopen from TSA

KresLinear behaviour

Fig. 11 Resultant Kmax, Kmin, Kopen and K res for and R-ratio of 0.6 considering the influence of the residual stresses due to welding in thecalculation of K .

KI =∫ a

0S0(x)m(x, a) dx,

m(x, a) =

[1 + m1

(1 − x

a

)+ m2

(1 − x

a

)2]

√2π (a − x)

m1 = 0.6147 + 17.1844( a

W

)2+ 8.7822

( aW

)6

m2 = 0.2502 + 3.2889( a

W

)2+ 70.0444

( aW

)6

(9)

where S0 is the experimentally determined residual stressprofile along the crack line, m(x, a) is the weight function,m1 and m2 are weight function coefficients, a is the cracklength and W is the specimen width.

Consequently, the actual maximum and minimum SIFwas the resultant of superimposing the SIF calculated as-sociated to the presence of the residual stresses (Fig. 11)and the nominal SIF at the maximum and minimum loads.To account for the presence of crack closure, the Kopen

was calculated from the SIF range values previously in-ferred from the processing of thermoelastic images. Sub-sequently, �K eff was calculated by subtracting the Kopen

from the maximum K taking into account the influenceof residual stresses. Finally, the resultant values for �K eff

were compared those inferred from TSA and those calcu-lated using the nominal stress range. Results are presentedin Fig. 12.

D I S C U S S I O N

The stress intensity factor ranges determined, for differentfatigue tests conducted with TSA are presented in Fig. 7.

The continuous line represents the mode I stress intensityfactor calculated directly from the applied load and a suit-able calibration function.26 The �K I values obtained fromthe thermoelastic images are clearly lower than the the-oretical values. The difference decreases at high R-ratio,which is typical of crack closure and consistent with thework of Fulton et al.14 During the initial stages of thecrack growth, closure should be absent and the theoreti-cal and experimental results are very close to each other.The difference increases as the crack wake becomes fullydeveloped. Mode II stress intensity factors were also cal-culated from the thermoelastic data and were found to bevery close to 0. This was expected because the tests wereset up to be under pure mode I conditions.

It is surprising in Fig. 7 that there remains a substan-tial difference between the experimental and theoreticalstress intensities at high R-ratio as contact between thecrack faces should be absent. However, the welded plateswere subjected to substantial internal residual stresses, ofthe order of the yield stress, which may have affected thevalue of the stress intensity factor. Moreover, in Fig. 7the deviation of the SIF range from thermoelastic datawith respect to the SIF range calculated from the appliedload starts to occur at a crack length of approximately 10mm. This length in the residual stress profile presentedin Fig. 10 corresponds to the location where compres-sive residual stresses start appearing. On the other hand,whilst the residual stress field should not affect �K I, it willchange the mean stress and hence the minimum stress in-tensity factor at the crack tip, which could be a possiblereason for the deviation observed in Fig. 7.

c© 2004 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 27, 571–583

582 F. A . D IAZ et al.

Fig. 12 Comparison of the variation of theSIF with the crack length for three differentcases; (1) SIF calculated considering onlythe nominal load range. (2) SIF calculatedconsidering the contribution of the residualstresses in the calculation of K using aweight function. (3) SIF experimentallyobtained from thermoelastic data.

SIF calculated from applied load

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1 10 100

Stress intensity factor (MPa. m1/2)

Cra

ck g

row

th r

ate

(m/c

ycle

s)

Growth equation

R=0.6

R=0.3

da/dN=1.15e-14 ∆K2.76

SIF calculated from thermoelastic data

1.00E-09

1.00E-08

1.00E-07

1.00E-06

1 10 100

Stress intensity factor (MPa . m1/2)

Cra

ck g

row

th r

ate

(m/c

ycle

)

da/dN=8.49e-16 ∆Keff3.71

Fig. 13 Comparison of the fatigue crack growth rate against the SIF range for fatigue tests performed at different R-ratios. (a) SIF rangecalculated from the applied load. (b) SIF range obtained directly from thermoelastic images.

In Fig. 9 the SIF ranges inferred from several thermoelas-tic images for the same crack length are compared to thosecalculated by taking into account the change in compli-ance as the crack starts closing, both results match almostperfectly.

Finally, the residual stress profile along the weld, throughthe plate thickness, was obtained and its contribution tothe stress intensity factor was estimated using a weightfunction (Fig. 11) for the case of R = 0.6. From the resul-tant Kmax and Kmin the effective SIF range was inferred.To calculate the variation of Kopen with the crack length,thermoelastic results were used as it was previously found

to provide very close values of Kopen to those obtained bythe compliance method. Finally, the effective SIF rangeinferred using this methodology was compared with theSIF range obtained derived from thermoelastic images(Fig. 12). Both results show a very good agreement.

Results related to fatigue life prediction are also pre-sented in Fig. 13. In Fig. 13a, the crack growth rate isplotted against the SIF range inferred from the appliedload range, while in Fig. 13b, the crack growth rate isplotted against the SIF range inferred from thermoelasticimages. From this graph it is possible to observe that TSAprovides a more realistic information about the fatigue

c© 2004 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 27, 571–583

MEASURING STRESS INTENSITY FACTORS 583

life of the component as the technique is able to quantifyphenomena like the presence of residual stresses or crackclosure and give a direct measurement of �K eff.

C O N C L U S I O N S

In this paper thermoelastic stress analysis has been pre-sented as a technique with an enormous potential in termsof fatigue crack analysis. Fatigue tests at different R-ratioson welded ferritic steel plates specimens have been re-ported. Stress intensity factors of the growing cracks havebeen evaluated using thermoelasticity. To validate theseresults two different means of validation have been em-ployed: compliance measurements and a numerical anal-ysis based on weight functions. Both approaches pre-dict similar results to those obtained using TSA. Con-sequently, the effect on the mode I stress intensity factorof both crack closure and compressive residual stressesahead of the crack can be experimentally identified usingTSA.

Acknowledgements

The authors gratefully acknowledge the contribution ofSerco Assurance plc, MR Goldthorpe and Associates, andthe Health and Safety Executive for their assistance.

Other parts of this work were supported by an EPSRCgrant (GR/M57712/01) for which the authors are alsograteful. Prof. M. N. James is thanked for his invaluablehelp and support during the compliance measurements.Mr T. Richards and Mr B. Lord are also thanked for theircontinual technical support.

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c© 2004 Blackwell Publishing Ltd. Fatigue Fract Engng Mater Struct 27, 571–583