The Elastic-Plastic Finite Element Alternating Method (EPFEAM)and the prediction of fracture under WFD conditionsin aircraft structuresPart III: Computational predictions of the NIST multiple site damage experimental results

L. Wang, F. W. Brust, S. N. Atluri

Abstract This report provides a summary of the Elastic-Plastic Finite Element Alternating Method (EPFEAM), theT�-integral fracture mechanics parameter, and the use ofboth the tools to predict the residual strength of aircraftpanels with multiple-site damage. Because this report ismeant to be self-contained and each of the three subjects isa considerable research topic in itself, the report is writtenin three parts. Part I, EPFEAM Theory provides a summaryof the elastic-plastic ®nite element alternating method(EPFEAM) and the algorithms for fracture analysis andcrack growth predictions. Part II, Fracture and the T�-Integral Parameter provides a complete description of theT�-integral fracture parameter including a detailed dis-cussion of the theoretical basis of T� and the practicalaspects of its use for fracture predictions. Finally, Part III,Computational Predictions of the NIST Multiple SiteDamage Experimental Results provides a series of pre-dictions of a number of fracture tests performed at theNational Institute of Standards and Technology (NIST).These predictions are then compared with the experi-mental data, thus validating the present model for com-puting the residual strength under wide-spread-fatiguedamage conditions.

The reader that is interested in all the topics can studyall the three self-contained parts, while the reader that isonly interested in the practical aspects of fracture pre-dictions using these methods can read only Part III.

This is the Part III report, ``Computational Predictionsof the NIST Multiple Site Damage Experimental Results''.

1IntroductionThe economic environment in the United States (and theglobal economy for that matter), that has persisted formore than ten years, is focused on controlling and con-taining costs. This attitude, which has been prevalent inindustry for years and now is the focus of the US gov-ernment as well, has led to a concept called life extension.Life extension refers to the concept of continuing to usestructures long beyond the original intended design life.As long as the maintenance costs associated with extend-ing the life of an aging component or process remain lowerthan the huge capital expenditures associated with de-signing and building a new structure, the life extensionphilosophy will continue. Examples of industries which areactively extending the life of processes include the fossiland nuclear power industries which are attempting to useplants far beyond their intended design life. Also, theaerospace industry, both commercial and military, is at-tempting to use aircraft far beyond their initial designlives. Indeed, methods to evaluate the structural integrityso that life extension of the aging commercial airline ¯eetmay continue is the focus of this report.

In order to de®ne the maintenance requirements of anaging aircraft to ensure its' continued structural integrityand safety, advanced analytical tools for life and residualstrength prediction are necessary. The FAA has beencharged with the development of a wide range of tools forassuring the integrity of aging aircraft. These tools include,among many others, advanced and accurate non-destruc-tive inspection methods, and fracture analysis techniques.The latter is the subject of this report. The FAA Center ofExcellence at Georgia Tech has been involved with thedevelopment of advanced computational methods to per-mit the evaluation of structural integrity of aging aircraftin a rapid, accurate, and ef®cient way. Three major topicsof development are discussed in this report: EPFEAM,Fracture, and Predictive methods.

This report is written in three parts. Part I, EPFEAMTheory provides a summary of the elastic-plastic ®niteelement alternating method (EPFEAM) and the algorithmsfor fracture analysis and crack growth predictions. Part II,Fracture and the T�-Integral Parameter provides a com-plete description of the T�-integral fracture parameterincluding a detailed discussion of the theoretical basis ofT� and the practical aspects of its use for fracture pre-dictions. Finally, Part III, Computational Predictions of theNIST Multiple Site Damage Experimental Results providesa series of predictions of a number of fracture tests per-

Originals Computational Mechanics 20 (1997) 199 ± 212 Ó Springer-Verlag 1997

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Communicated by G. Yagwa, 31 October 1996

L. Wang, F. W. Brust, S. N. AtluriFAA Center of Excellence for Computational Modeling of AircraftStructures, Computational Modeling Center,Georgia Institute of Technology, A. French Building, Atlanta,Georgia 30332-0356, USA

Correspondence to: S. N. Atluri

The authors would like to acknowledge the recommendations andparticipation of Mr. Chris Seher and Drs. Paul Tan and CathyBigelow of the FAA Technical Center. The useful discussions withMr. T. Swift and Dr. J. Lincoln over the course of this work arethankfully acknowledged. Dr. Roland deWit of NIST graciouslyprovided computer disks of the tests that he directed at NIST, inaddition to providing insight into the results.

formed at the National Institute of Standards and Tech-nology (NIST). These predictions are then compared withthe experimental data, thus validating the computationalmodel.

This report focuses on the Part III topic in this reportseries, ``Computational Predictions of the NIST MultipleSite Damage Experimental Results''.

2Overview of Part III ``Application predictionsof the NIST multiple site damage experiments''This Part III report of this series provides validation of theEPFEAM method (Part I) and the use of the T�-integral(Part II). The validation is made by comparing predictionsof residual strength using these techniques, to experi-mental data produced by The National Institute of Stan-dards and Technology (NIST). The experimental data wasdeveloped on large panels which contained a large maincrack with Multiple Site Damage (MSD) cracks ahead ofthe main crack. The predictions provided here are truepredictions, i.e. no `fudging' of either the experimentaldata or the analysis predictions are made to make thepredictions look better. Before providing the validationpredictions it is useful to present a short overview ofpractical elastic-plastic fracture mechanics and the currentstate-of-the-art. The reader is referred to the Part I fordetails regarding the EPFEAM method development andimprovements made here, and to the Part II for detailsregarding the T�-integral development.

The practical development of elastic-plastic fracturemechanics in the United States was driven primarily by theneeds of the Nuclear industry which uses very ductilestainless steels for its piping systems. The nuclear regu-latory authorities permit utilities that are attempting tohave a plant certi®ed, to reduce or eliminate expensivepipe whip restraints if the pipe `Leak Before Break' criteriacan be met. This requires elastic-plastic fracture mechan-ics methods which are based on J-integral Tearing theory1.A complete summary of the philosophy behind the phe-nomenological J-Tearing method has been summarized byHutchinson (1983). Hutchinson (1983) also points outexplicitly the limitations of J-Tearing Theory. These limi-tations include; (i) only monotonic loading can be con-sidered and (ii) only small amounts of crack growth can bepermitted. Despite these limitations a very large data baseexists (see Wilkowski et al. 1995) which clearly shows thatconservative predictions can be expected if a lower bound(compact specimen) J-Resistance curve is used to predictmonotonic loading of one single crack to fracture in a pipe.A large ongoing program (Hopper et al. 1992) shows thatJ-theory cannot account for cyclic effects. Most impor-

tantly for aerospace applications, a far ®eld J value has nomeaning for a large crack with MSD ahead of it.

Practical methods for predicting the residual strength ofthe MSD problem include the plastic zone technique (Swift1985), the crack tip opening angle (CTOA) method(Newman et al. 1993), and the T�-integral method (Pyoet al. 1994a, 1994b; Singh et al. 1994; Brust 1995a). Theplastic zone method predicts that the main crack will linkwith an MSD crack if their plastic zones overlap. Theplastic zones are estimated using an elastic analogy such asan Irwin estimate. However, the Irwin estimate of theplastic zone size is not very accurate, especially for hard-ening materials. Moreover, there is no theoretical justi®-cation for choosing plastic zone overlaps as a link-upcriterion. The CTOA method, which requires a rather de-tailed ®nite element analysis to apply, has been shown tobe able to predict MSD failures. However, the CTOA is®nite element mesh size dependent in general and oftenrequires expensive analyses due to the need for extrememesh re®nement. Moreover, CTOA is a geometric pa-rameter and cannot be used under more general fractureconditions such as cyclic loading and rate dependentfracture. Furthermore, the CTOA for fracture initiationand stable growth is measured in small planar specimens,wherein the crack axis lies in the plane of the specimenand the crack plane is perpendicular to the plane of thespecimen. In this case, the CTOA has a reasonably clearphysical basis. However, for cracks in cylindrical-shelltype fuselage structures, and when such cracks bulge, theCTOA and the plane in which it should be measured areambiguous. Thus, the implementation of a critical CTOAfor fracture as measured in planar laboratory specimens asa criterion for fracture under WFD conditions in curvedfuselage structures is far from simple. The T�-integral is ageneral energy based parameter (see the Part II volume)and has been shown to be applicable under severe oper-ating conditions of cyclic loading and rate dependentfracture (see Part II). Furthermore, T�, being an energyquantity, can be de®ned for cracks in any geometry (in-cluding a shell type fuselage structure) and when cracksbulge (Shenoy et al. 1994).

Perhaps the most appealing aspect of the T�-integralmethod is that its use is completely analogous to classicalJ-Tearing theory. In fact, under the conditions (statedabove) where J is valid, the T� method and the J methodare identical! To use T� a resistance curve is developed.This is usually obtained by performing a generation phaseanalysis of a specimen geometry. This one curve is thenused as the fracture criterion for application phase pre-dictions for any other cracked geometry of interest, in-cluding MSD and cyclic load conditions. Indeed, theanalogy with the established J-Tearing method should beclear. In addition, the T� method combined with the ef®-cient EPFEAM method, and the fact that relatively coarsemeshes may be used has made this method practical forroutine use today.

The next section summarizes the experimental results;and the following section provides predictions and theircomparisons with test data for eight separate tests. It willbe seen that T� provides good predictions of MSD link-upand residual strength.

1 The Europeans rely on the so-called R-6 method to make elastic-plastic fracture predictions. This method essentially interpolatesbetween LEFM and the rigid-plastic (strength of materials) limitload solution. The method limitations include (i) only one largecrack can be considered (ii) displacement predictions cannot bemade (iii) cyclic loading cannot be accounted for, among others.The Asians, particularly the Japanese rely on J-Tearing theory.

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3Summary for experimental resultsThis section provides a short summary of the tests per-formed at NIST. Please see deWit et al. (1995) for moredetails.

Table 1 summarizes the series of tests carried out atNIST. As seen at the top of Table 1, the specimens were 90-inches wide, 150-inches high, 0.04-inch thick, made of2024-T3 aluminum. Because of the large size of thesespecimens, a 400 kip machine with extended grips wasused. These ¯at sheet tests were performed to eliminatecomplications due to curvature and stringers so that goodload and displacement control data could be obtained. Inthis fashion, good basic fracture properties and linkagecriteria were obtained.

The cracks were introduced by using a series of sawsculminating with the sharpest jeweler's cut available hav-ing a tip radius of 0:003 inch2. The MSD cracks were cutfrom 0.22-inch holes that were drilled into the sheets be-fore the saw cuts to simulate cracks emanating from rivetholes as occurs in aircraft fuselage. The grips were spe-cially designed to enforce uniform stress applied to thespecimens and were veri®ed with strain gages at low loadsduring the beginning of the tests. The measured strainswere within plus or minus 10% of the uniform value. Itshould be noted here that after crack growth and extensiveplastic deformation, the uniformity of these stresses willnot be achieved. In addition, anti-buckling guides wererequired so that the specimens did not buckle out-of-planesince the thickness is very small.

The test procedure consisted of pulling the specimen tofracture under displacement control. The displacementwas generally applied at load intervals of 20 to 45 KN. Theentire tests typically lasted from 15 to 20 minutes. Finalfracture occurred with an audible rip. It was also clear that

both tips of the main crack did not grow at the same loads(deWit 1995). This must be kept in mind when reviewingthe results since crack growth was only recorded at onecrack tip.

As seen in Table 1, the ®rst three tests were single centercrack panel tests with different initial crack sizes. TestsMSD-4 through MSD-10 were tests of panels with a multi-site damage (MSD) in front of a main crack (except MSD-6which was a repeat of MSD-3 without the use of anti-buckling guides). Figure 1 de®nes the MSD crack de®ni-tion parameters. It is seen that the MSD tests had variousdifferent main crack sizes and different spacing of theMSD cracks. Figure 2 provides a graphical representationof the crack growth behavior and MSD link-up results asobserved from the tests. The detailed experimental loadsand crack growth response from these tests will be sum-marized in the next section during the discussion ofcomparisons between prediction and test results. Note thatthe experimental failure loads during the link-ups and atthe end of the tests are shown in the next section.

4Analysis predictionsThis section provides predictions of eight of the NIST testssummarized in Table 1. The tests analyzed are: MSD-1 toMSD-5 and MSD-7 to MSD-9. MSD-6 is a single crack testwith a crack size identical to the MSD-3 test. This test(MSD-6) was performed without buckling guides to ob-serve the differences between a test with and withoutbuckling guides. Since our analyses are two dimensional,we are basically accounting for the buckling guides. Alsotest MSD-10 is a repeat of test MSD-7. Hence, all uniquetests in Table 1 have been modeled. In addition, analyseswere also performed on three of the Foster-Miller tests(Broek 1993) that had a single crack. These latter threeanalyses, which were performed by the FAA Center ofExcellence at Georgia Tech and reported in Pyo et al.(1994a, 1994b), were performed as part of the veri®cationexercise.

The ®rst subsection provides veri®cation of the EP-FEAM method and the computer code as well as obtainingthe T�-Resistance curve that is used for all other predic-tions, and the last subsection provides the test predictions

Table 1. Test matrix for NISTtests (deWit et al. 1995) Test No 2024-T3 aluminum, 0.04 inch thick, 90 inch wide, 150 inch high

Main crack MSD cracks

2a(in)

a(in)

dMSD

(in)sMSD

(in)2aMSD

(in)numberper side

MSD-1 14.0 7.0MSD-2 8.0 4.0MSD-3 20.0 10.0MSD-4 14.0 7.0 7.5 1.0 0.4 3MSD-5 5.6 2.8 3.5 1.5 0.6 3MSD-6 20.0 10.0 (no anti-buckling guides)MSD-7 20.0 10.0 10.5 1.5 0.5 5MSD-8 19.0 9.5 10.5 1.5 0.5 10MSD-9 10.0 5.0 6.5 1.0 0.4 10MSD-10 (r) 20.0 10.0 10.5 1.5 0.5 5

(r) repeat of MSD-7

2 No fatigue pre-cracks were introduced. It has been shown(Thomson, Hoadley and McHatton 1993) that, except in veryductile materials such as stainless steel, a sharp saw cut have adifferent initiation load compared with a fatigue pre- crack. Sincealuminum should be considered moderately ductile, the initiationloads may have been somewhat affected by use of a saw cut.

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using only this single resistance curve as the crack growthcriterion.

4.1EPFEAM code validation and resistance curve calculationA generation phase analysis was performed on the twosingle crack tests MSD-2 and MSD-3. A generation phaseanalysis is de®ned as an analysis of the load versus crackgrowth response of test data using the ®nite element al-ternating method (or ®nite element method), in order to

calculate a resistance curve. The resistance curve could beCTOA but in this case we obtain the T�-Resistance curve.Generation phase analyses of test data to develop a resis-tance curve is usually performed when obtaining the curvedirectly from the test is time consuming, expensive, andimpractical. It should be noted that both CTOA (Newmanet al. 1993) and T� (Okada et al. 1995) resistance curvescan and have been obtained directly from experiments butdoing so is not practical nor necessary.

The load versus crack growth data from NIST tests 2and 3 are plotted in Fig. 3. These data were ®tted with thecurve also illustrated in Fig. 3 (for convenience in per-forming the analyses) and two analyses were performed. Inthe ®rst analysis, the EPFEAM code was used. In the sec-ond analysis, a classical ®nite element method (FEM) wasperformed. The material properties used for the analysisare illustrated in Fig. 4. For the classical FEA crack growthwas modeled using a node release technique, while for theEPFEAM analysis, the method discussed in the Part I re-

Fig. 1. Test description. a Typical MSD test panel ± this caseshows two MSD cracks with the main crack. b Cross sectionshowing test panel crack location and de®nitions of MSD crackspacing

Fig. 2. Experimental results illustrating the MSD link-upbehavior for each of the tests (Taken from deWit et al. 1995)

Fig. 3. Experimental data for MSD tests 2 and 3 (MSD-2 andMSD-3). These tests were analyzed using EPFEAM to evaluate amaterial T�-Resistance curve for use in application phasepredictions

Fig. 4. Elastic and plastic material properties for 2024-T3aluminum used for all analyses

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port was used. The plane stress assumption was used andeight node isoparametric elements were modeled. Eitherhalf or quarter symmetry of the cracked plates were usedfor analyses (a half model is useful to validate the code).

The calculated T�-Resistance curves from this genera-tion phase analysis are illustrated in Fig. 5 for cases MSD-2and MSD-3, respectively. It is clearly seen that predictionsof T� are basically identical between the two methods. Thisis a complete validation of the EPFEAM code since thenumerical evaluation of T� consists of (see Part II) inte-grating stress and strain (and their derivatives) typequantities at many locations along many paths sur-rounding the growing crack tip. Figure 5 illustrates theevaluation along 6 paths ranging in size from � 0:087 inchto � � 0:35 inch (please refer to Part II for a completedescription of the path de®nitions for T�). In addition, T�was calculated along many other paths with identical goodcomparison. For T� to compare so well between the EP-FEAM and classical FEA along many paths is strong proofthat the full ®eld solution using EPFEAM throughout theentire body is calculated very accurately. It should be

noted that EPFEAM was compared with the asymptoticHRR solution (see the Part I report in this series) and withgood comparison. However, the present comparison isstronger proof of the accuracy of EPFEAM since the HRR®eld comparison is only near the crack tip and the effect ofstress and strain derivative terms are not considered whencomparing results to HRR ®elds.

Figure 6 places all of the results on Fig. 5 on a singleplot. It is seen that the T�-resistance curves for the MSD-2(initial crack size 2a � 8-inches) and MSD-3 (initial cracksize 2a � 20-inches) are very close throughout the crackgrowth history. Ideally, the T�-Resistance curves for agiven � would be identical regardless of the crack size. Thedifferences illustrated in Fig. 6 are quite small, well withinthe scatter band of experimental error, and the lot to lotmaterial variability. Indeed, variations of the J-Resistancecurves from specimen to specimen (especially the initia-tion value, JIC) are quite signi®cant (see Rahman and Brust1995). The resistance curves for � � 0:087 were ®tted withan equation which appears in Fig. 6. This curve representsan average of the tests MSD-2 and MSD-3 and it was usedfor all other application phase analyses discussedthroughout the rest of this report. A discussion of themeaning for the different � path size de®nitions and theirphysical signi®cance is fully discussed in the Part II report.

Before providing predictions of the tests listed in Table 1the re-analysis of some of the Foster-Miller tests (Broek1993) is discussed next. These experiments consisted of¯at panel tests of specimens much smaller than the NISTtests. Three single crack tests and three MSD tests wereanalyzed. Predictions of these tests were made using theT�-integral and reported in References (Pyo et al. 1994a,1994b). Good comparison between prediction and analysiswas observed. Here the three single crack tests were re-analyzed using the T�-Resistance curve equation in Fig. 6.Note that the resistance curve used for these analyses wasobtained from NIST material and not the Foster-Millermaterial.

Fig. 5. T�-Resistance curves calculated from tests MSD-2 (top)and MSD-3 (bottom). Both the EPFEAM (designated as ALT here)and conventional elastic plastic ®nite element calculations (FEM)were made. � represents the size of the path

Fig. 6. The results of MSD-2 and MSD-3 for both EPFEAM (ALT)and classical FEM. Note that T� for this generation phase analysisis very close between the two specimen

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The predictions are application phase analyses. Thetests were analyzed using a displacement control conditionand the T�-Resistance curve is followed. The analysis iscontrolled by the driving force value of T� being forced tofollow the Resistance Curve throughout the crack growthhistory. As such, it is a true prediction. Both the load andthe crack growth are predicted. Figure 7 compares thepredictions for three tests with different initial crack sizes.Good comparison is observed.

The `steps' in Fig. 7 are present due to the nature of thepredictive analysis. Consider a crack growth step. An in-crement of displacement is applied. When the drivingforce value TD

�� � reaches the material resistance value forthe current crack size TR

�� � then the displacement is heldconstant while an increment of crack growth is permitted.If, after the increment of growth, TD

� � TR�, then the crack

must grow more. So again, the displacement is held con-stant and an increment of crack growth is modeled. Assuch, the horizontal `steps' in Fig. 7 represent the amountof crack growth for a given displacement. Note also that,during a growth step with the displacement held constant,the load, which is predicted, drops slightly. If after anincrement of crack growth, TD

� < TR� then the growth of

the crack is stopped. Hence, an additional increment ofdisplacement can be applied. This application phase pre-dictive methodology is illustrated graphically in Fig. 8. Weemphasize that the analyses in Fig. 7, as well as all analysesshown in the next section (which presents MSD analyses),provide predictions of both load and crack growth usingonly one T�-Resistance Curve.

4.2Predictions of NIST experiments ± load versus crack growthThe predictions of the NIST experiments were performedin an identical fashion as described above. However, forthe MSD cases, all cracks were permitted to grow. Thismeans that the main crack and all MSD cracks were forced

to follow their own T�-resistance curve. In general, as themain crack approaches the nearest MSD crack, the smallercrack with the tip closest to the main crack begins growingto meet the main crack. Near the end of the analysis asfailure is predicted, several of the MSD cracks begingrowing. Having many cracks growing at once with eachfollowing their own resistance curve is quite natural withEPFEAM. While simultaneous multiple crack growth canbe modeled using conventional ®nite element methods,doing so is quite cumbersome.

The ®rst prediction shown here is for the single crackcase (MSD-1, see Table 1) that has not been analyzed asyet. Again, the analysis was forced to follow the T�-Re-sistance curve. Recall that the T�-Resistance curve wasobtained from generation phase analyses of cases MSD-2and MSD-3 and the average of these two curves (see Figs. 5and 6) is considered. Again, the T� curve for � � 0:087 wasused for all predictions. However, the T�-Resistance curvefor all of the other values of � shown in Fig. 5 could havebeen used and the results would be identical as discussedin the Part II report in this series.

Figure 9, which again shows the T�-Resistance curveequation that was used at the top of the ®gure as deter-mined from the MSD-2 and -3 generation phase analyses,shows the analysis prediction compared with the data.Notice that this case is for a crack �2a � 14-inches� be-tween the MSD-2 �2a � 8-inches� and MSD-3�2a � 20-inches�. Again, the steps in the analysis resultsare caused by the predictive algorithm used, as was de-scribed in the previous section and illustrated in Fig. 8. Asseen in Fig. 9, both loads and crack growth predictionscompare well with the experimental data.

Fig. 7. Predicted loads and crack growth from application phaseanalyses of Foster-Miller single crack panels P1, P2, P3. The T�-Resistance curve from the NIST data was used

Fig. 8. EPFEAM crack growth algorithm based on a T� fracturecriterion

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4.2.1Cases MSD-4 and MSD-5The experiments labeled MSD-4 and MSD-5 in Table 1 areseen to contain three MSD cracks on each side of the maincrack. The main crack size and the spacing for the MSDcracks are illustrated schematically in Fig. 2. Figure 2 isdrawn to scale so that the main crack size and spacing canbe directly observed. As seen, the main crack for MSD-4 islarger than MSD-5 while the MSD cracks are smaller andspaced closer together. Note that Fig. 2 represents the rightside of the specimens from the centerline of the CenterCracked Panels (i.e., centerline of the main large crack) tothe edge of the specimen. As such the half width is seen inFig. 2 to be 45 inches. The ligaments that failed simulta-neously are also illustrated (gray shading) as observedfrom the experimental results.

Figure 10 illustrates the ®nite element meshes used forMSD-4 and MSD-5. For MSD-4 and MSD-5 half symmetrywas used. This means that the top half of the specimen wasmodeled and both the left and right half sections of thecracks were considered. Figure 10 illustrates a blowup ofleft main crack and corresponding MSD cracks for thesetwo cases. These mesh details are to scale and the numberson the scales represent real distances in inches. Also, theMSD cracks drawn at the bottom of each mesh and theiroccurrence in the mesh are also illustrated. Of course, aquarter model rather than a half model could have beenused (and was used for cases MSD-7, -8, and -9). However,modeling each crack tip separately provides an excellentcheck on the computational methodology. More impor-tantly, the versatility of the EPFEAM method combinedwith the T�-integral fracture methodology can be consid-ered. For instance, by introducing an initial perturbation,the effect of uneven crack growth rates for each crack tipcan be studied. (For instance, the right half main crack tipcould be made slightly larger than the left crack, and theeffect of uneven right and left side crack tip growth and thecorresponding non-symmetric crack growth (which wasobserved experimentally) can be accounted for very easily.

The specimens, which had all cracks machined with a saw,did not have `perfect' crack lengths and the correspondingMSD separation (illustrated in Table 1) certainly have astatistical scatter associated with their distances. Theperturbations associated with the left and right side cracktips growing at different rates can have an important effecton failure loads. This should be kept in mind when ob-serving the following predictions since the left and rightcracks did indeed grow at different rates in the experi-ments. The experiments tracked the left side crack only.Finally, it is important to note that in real aircraft fuse-lages, the same effect will occur and can have an importanteffect on residual strength. This is especially true sincestringers and frames are present. This `crack tip pertur-bation' effect is not studied here, but should be consideredin future work since it has not been studied to date in theliterature.

The application phase predictions for MSD-4 and MSD-5 are shown in Figs. 11 and 12, respectively. In these ®g-ures (and all other similar ®gures shown subsequently) theabscissa represents crack growth and the ordinate repre-sents load as predicted from T� fracture methodology. The

Fig. 9. Predicted loads and crack growth compared with exper-imental results from application phase analyses of NIST singlecrack test panel MSD-1

Fig. 10. Finite element meshes used for a MSD-4 and b MSD-5.These meshes represent a blowup of the left crack tip region

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predicted results are for the right crack tip. The right andleft side cracks grew symmetrically since no initial cracksize or other perturbation sources such as slight statisticaldifferences in the T�-Resistance curves within the speci-men are considered in the analyses here. The main cracktip ends at the axis origin in Figs. 11 and 12. The MSDcracks (drawn to scale) can be observed at the top of these®gures. As such, the main crack and MSD link-up points,and corresponding simultaneous ligament failure predic-tions can be observed. It is seen that the MSD-4 predic-tions (Fig. 11) are excellent. Both the link-up loads,maximum load, and crack growth predictions are almostidentical with the experimental values. Figure 12 likewiseshows good predictions for MSD-5 except that the thirdlink-up load is predicted to be identical to the second link-up load. Experimentally, a slight increase in load was ob-served (after the third link-up) at about 4:5 inches of crack

growth. However, these predictions are considered excel-lent given the many sources of statistical variability forthese experiments. This variability includes: material lotvariability of material properties such as stress-strain re-sponse and T�-Resistance curves, slight errors in specimenfabrication which led to the experimentally observed crackgrowth rate differences between the left and right sidecrack tips, etc. The sources of statistical variability arediscussed further in the discussion section.

4.2.2Case MSD-7The application phase prediction for MSD-7 is shown inFig. 13. This test had ®ve MSD cracks ahead of both tips ofthe main crack. The crack spacing and crack sizes areillustrated in Fig. 2 and at the top of Fig. 13. As seen inFig. 2, all of the MSD cracks joined together after thesecond link-up (gray shaded regions ± Fig. 2). Excellentpredictions of both link-up loads, link-up instability, andcrack growth behavior are seen in Fig. 13. Note that MSD-10 test data is also presented here. The MSD-7 and MSD-10tests were identical. Some of the statistical variability inresults can be seen here with the test results for MSD-10higher than those for MSD-7. The analyses for this case, aswell as for MSD-8 and MSD-9 (discussed next), used aquarter model. As such, only the right crack was modeled.

4.2.3Cases MSD-8 and MSD-9These tests had ten MSD cracks ahead of both main cracks.Figure 2 illustrates the crack con®gurations, sizes, andMSD separation. It is seen that test MSD-8 had a largermain crack, larger MSD spacing, and larger MSD separa-tion compared with test results for MSD-9.

The application phase predictions for MSD-8 and MSD-9 are shown in Figs. 14 and 15, respectively. The MSDspacing is illustrated at the bottom of these ®gures. Again,the right main crack tip was modeled and its' tip is at theorigin of Figs. 14 and 15, with the MSD spacing ahead of

Fig. 11. Load and crack growth predictions for MSD-4 comparedwith experimental results

Fig. 12. Load and crack growth predictions for MSD-5 comparedwith experimental results

Fig. 13. Load and crack growth predictions for MSD-7 comparedwith experimental results

206

the main cracks. The initiation load prediction for MSD-8(Fig. 14) is quite good. The predicted load at ®rst link-upis slightly higher than the test data. In addition, the pre-dicted load at second link-up is higher than the test data.After the second link-up, the MSD cracks all linked to-gether as seen in Fig. 14 after about 2:5 inches of crackgrowth. This link-up instability was predicted by the EP-FEAM model quite well.

At this point it is interesting to observe the results afterthe unstable MSD link-up. This unstable link-up occurswhen the main crack links with all of the MSD cracksbetween crack growth of about 2.5 to 14 inches. The ex-periments were performed via displacement control (aswere the analyses). When the MSD instability occurredbetween crack growth of 2.5 to 14 inches the crack wasobserved to `pop' in an unstable fashion, i.e., the crackpropagated in a dynamic unstable fashion as all of the

MSD cracks linked together. Since the displacement washeld constant while the crack jumped almost 12 inches, theload must drop. This is seen in Fig. 14 where the experi-mental load is about 20 Kips after the crack jump at acrack extension value of about 15 inches. The experimentaldisplacement is then increased until the ®nal load of about40 Kips is reached. The predictions, which are all quasi-static, cannot account for this dynamic crack jump atpresent. There are two important events that occurredexperimentally here, and which certainly would occur in areal aircraft. These are: (i) The inertial effects of the crackjump and the greatly increased crack speed and corre-sponding strain-rates near the growing crack tips reducethe size of the plastic zone and lead to a more `brittle type'crack growth compared with quasi-static growth, and(ii) because the displacement was held constant during thecrack jump, the load decreased, and as a result, an unloadreload cycle occurred. It is well known that such unload/reload effects reduce the load carrying capacity of stablegrowing cracks. This latter effect can be accounted for withthe current model (except it may be necessary to imple-ment a more appropriate cyclic plasticity constitutive lawsince we use isotropic hardening for these analyses). Boththe dynamic crack jump effect and the cyclic effect shouldbe considered in future work. It should be emphasized thatboth effects are present in real aircraft since, due to thestringers, frames, and tear straps, displacement controlMSD `crack jumps' will occur in service. Note from Fig. 14that the predicted load slowly decreases at the predicted`crack instability' since we use a quasi-static analysis.

Fig. 15 illustrates the prediction for MSD-9. Note thatthe initial link-up led to a rapid crack jump (Figs. 2 and15). The rapid crack jump was also predicted via themodel. The maximum load is somewhat underpredictedcompared with the experimental results, but this is con-sidered to be well within the experimental scatter. Thesame effect of the crack jump and unload/reload effect(discussed above regarding Fig. 14) is observed here afterabout 13 inches of crack growth.

4.3Predictions of NIST experiments ± plastic zones and stressesIt was seen in the previous section that the load versuscrack growth predictions based on the EPFEAM methodand the T�-integral fracture theory provided good pre-dictions of the phenomena of MSD crack growth, link-up,and unstable fracture. As such, the tools presented abovecan be readily used to make residual strength predictionsof aircraft fuselage MSD cracking. To complete this sec-tion, we provide some examples which illustrate that EP-FEAM not only provides crack growth and fracturepredictions, but the full ®eld displacements, stresses,strains, etc. are likewise accurately calculated. For thepurpose of this presentation, results for plastic zones andstresses for case MSD-4 are illustrated. Recall that MSD-4had three MSD cracks ahead of each main crack tip. Theresults for all of the other cases are similar to these.

4.3.1Plastic zonesRecall from Table 1 and Figs. 2 and 11 that MSD-4 had amain half crack initial length of 7 inches. Figure 16 illus-

Fig. 14. Load and crack growth predictions for MSD-8 comparedwith experimental results

Fig. 15. Load and crack growth predictions for MSD-9 comparedwith experimental results

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trates the development of the equivalent plastic strainsahead of the main crack and at the MSD cracks. In Fig. 16we focus on the left crack tip and the associated MSD aheadof it. The dimensions along the ordinate and abscissa

represent real dimensions in inches. As such, the plasticzones are real predictions and are not distorted. Figure 16aillustrates the plastic zone contours near the crack tip re-gions just after the main crack has begun growing.

Fig. 16a±d. Plastic zone contour plots for MSD-4 at differentcrack growth locations. The main crack is growing from the leftand the initial half main crack size is 7 inches. The ordinate andabscissa represent real coordinate dimensions in inches

Fig. 17a±d. Stress contour plots �rx� for MSD-4 at different crackgrowth locations. The main crack is growing from the left and theinitial half main crack size is 7 inches. The ordinate and abscissarepresent real coordinate dimensions in inches

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Figure 16a shows the tip of the main crack (atÿ7:2 inches).Observe that the plastic zone size at the main tip, is aboutthe order of the MSD crack lengths. The correspondingload at this point is about 45 Kips as obtained from Fig. 11for 0:2 inches of crack growth. Note that the minimumplastic strain contour plotted here is 0.002 (i.e., the .2%offset strain). Note that the presence of the MSD crackahead of the main crack `stretches' the plastic zone in thecrack growth direction as compared with the usual planestress plastic zone shape for a single crack and that theentire net section between the ®rst MSD crack and themain crack is plastic. Observe also from Fig. 16a that theplastic zone at the left tip of the ®rst MSD crack is be-coming larger, and that the net section between the left tipof the ®rst MSD crack and the right tip of the second MSDcrack is approaching full plasticity. Note also that there aresmall zones of plasticity already developing at the tips of allcracks at this point just after main crack initiation.

Figure 16b illustrates the plastic zone when the maincrack is 8 inches long. This represents a point after themain crack and ®rst MSD crack have linked and the cur-rent main crack is half way through the ligament betweenthe ®rst and second MSD cracks. The residual plasticity atthe ligament between the main crack and the ®rst MSDcrack (which is now a region of the current main crack tipsince the main crack has grown through the ®rst MSDcrack ligament) is seen to be very similar to that seen inFig. 16a. It is also important to observe that the net liga-ment between the current main crack tip (at 8-inches) andthe second MSD crack is fully plastic. Moreover, the liga-ment between the second and third MSD crack is fullyplastic. From Fig. 11 the load is about 49 Kips for crackgrowth of 1-inch and the link-up between the current maincrack and the second MSD crack occurs at a crack growthof 1:3 inches. The plastic zone link-up criteria (Swift 1985)would predict that all cracks have linked at this point sincelinkage is presumed to occur when the net ligament be-comes plastic. Ignoring the fact that the prediction ofplastic zones with the link-up criterion is made using asimple Irwin estimate (based on elastic considerations)while the present plastic zones of Fig. 16 are real predic-tions, it is clear that the plastic zone link-up method is notaccurate. From Fig. 11 it is clear that much more crackgrowth occurs before all cracks link. Moreover, fromFig. 11 it is clear that much more residual strength canoccur before the maximum load is reached.

Figure 16c shows the plastic zone pro®les when themain crack is 9 inches long, i.e., total crack growth of2 inches has occurred. At this point the main crack haslinked with both the ®rst and second MSD cracks and iscurrently halfway through the ligament between the sec-ond and last MSD crack. As seen from Fig. 11, the currentload is about 50 Kip at this point. Again, the plastic zonesalong the main crack faces at the locations where the ®rsttwo MSD crack ligaments were located are similar in sizeand shape to the zone sites before linkage occurred (seeFigs. 16a and 16b). Note that the ligament ahead of the leftside of the last MSD crack is beginning to attain a signi-®cant plastic zone.

Finally, Fig. 16d shows the plastic zone pro®les after allMSD cracks have linked and the current main crack has

grown 0:2 inches ahead of the left tip of the last MSD crack.At this point the main crack has grown a total of 2:9 inchesand the load (from Fig. 11) is currently about 54 Kips. Asseen in Fig. 11, at this point there is still a signi®cantamount of residual strength left in this specimen.

4.3.2StressesFigure 17 illustrates rx at four different crack growthlengths for MSD-4. The x-component of stress is along thecrack growth direction. The comments regarding the ori-entation of the cracks, etc. made regarding Fig. 16 applyhere (and also for Figs. 17 and 18). Again, we focus on theleft crack tip, and therefore, for the MSD-4 case, the x-coordinate of ÿ7 represents the half main crack tip. Thethree MSD crack lengths and locations appear at the bot-tom of Fig. 17. For Fig. 17a the main crack has grown0:2 inches through the ligament between the main and ®rstMSD crack and at that point the ligament is only 0:1 incheslong. This ligament can be identi®ed by the red color (highrx) above the x � ÿ7:2 and ÿ7:3 in Fig. 17a. In addition,the ligament between the ®rst and second MSD cracksexperiences a large stress state. Also, observe the `cyclical'stress pattern that develops along the crack extension di-rection caused by MSD. Figure 17b shows the rx stressesafter the main (half) crack has grown to 8:0 inches. Like-wise, Figs. 17c and 17d show this stress component formain (half) crack lengths of 9.0 and 9:9 inches, respec-tively.

By observing the progression of rx as the main crackgrows in Figs. 17a through 17d, the following trends arenoted: (i) A large zone of compressive stress developsabove the main crack. Despite the fact that these ®guresshow a blowup of the region in front of the left crack tip, itshould be clear that this compressive zone is large andextends above and along the entire main crack. Indeed,these stresses are the reason that buckling guides are re-quired for the experiments. (ii) A compressive `wake' zoneof larger magnitude stresses above the MSD crack liga-ments is seen. This may be seen from the dark blue colorabove the ligaments between the second and third MSDcracks (i.e. above the ÿ8:0 and ÿ9:0 coordinates). Thislarge wake compression zone is well known to exist for thesingle crack growth problem (see Brust et al. 1985, Brustet al. 1986a for instance) and can result in signi®cant re-sidual stresses after removal of the load. This zone con-tributes to the `unload/reload' phenomena where a cycle ofunloading followed by reloading reduces the load carryingcapacity of the cracked structure (Brust et al. 1986a).

Figure 18a through 18d show the shear stress compo-nent rxy

ÿ �at these same four crack growth locations as

discussed for Figs. 16 and 17. The locations of the currentmain crack tip as well as the MSD cracks can be observedby focusing on the large (red contour) stresses. It is in-teresting to observe that the zone size of the large shearstresses at the main crack tip is rather small when MSD ispresent (Figs. 17a to 17c) while it is much larger when allcracks have linked up and only one large crack remains(Fig. 17d). However, the zone size of the large shearstresses for the MSD cracks (Figs. 17a to 17c) is quite largecompared to the MSD crack lengths.

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Figure 19a through 19d show the same type of con-tour plots for ry, where the y coordinate direction isperpendicular to the crack. Again, the crack tip locations

may be easily identi®ed by the large (red) stress con-tours and the net section ligament stresses are quitelarge. The large zone of dark blue color above the main

Fig. 18a±d. Shear stress contour plots �rxy:� for MSD-4 at dif-ferent crack growth locations. The main crack is growing fromthe left and the initial half main crack size is 7 inches. The ordinateand abscissa represent real coordinate dimensions in inches

Fig. 19a±d. Stress contour plots �ry� for MSD-4 at different crackgrowth locations. The main crack is growing from the left and theinitial half main crack size is 7 inches. The ordinate and abscissarepresent real coordinate dimensions in inches

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crack in these ®gures is the traction free stresses abovethe cracks.

5ConclusionsThis report represents the third in a three part series. Thepurpose of the studies was to: (i) develop an ef®cientmethodology to permit simple analysis of the multi-site-damage problem (Part I), (ii) to revisit the underlyingtheory behind the T� integral fracture parameter and toillustrate the practical calculation of T� (Part II), and (iii)to provide veri®cation of the methodologies developed inParts I and II by providing direct predictions of tests andcomparing results (Part III). The Part I report is entitledEPFEAM Theory and Part II is called Fracture and the T�-Integral Parameter. The present report represents Part III.

This effort used the EPFEAM method detailed in Part Iand the T� fracture method detailed in Part II to providepredictions and corresponding direct comparisons withtest data developed at NIST for FAA on large panels withMSD. In all, NIST performed ten experiments on widepanels, of which, eight were unique and two were repeats.All eight unique tests were analyzed here. In addition,three tests performed by Foster-Miller were reanalyzed(they were originally analyzed in Pyo et al. 1994a, 1994b)as part of the veri®cation process. Of the eight NIST tests,three were single crack tests and ®ve had MSD cracksahead of a main crack. The three Foster-Miller tests con-sidered here were single center crack panel tests.

The predictions were performed as follows:

· First, the tensile stress-strain properties are obtained forthe material (2024-T3 aluminum). This curve is used tode®ne the elastic and plastic material properties.

· Second, a T� material resistance curve was obtained bymodeling two of the single crack NIST tests. In such ageneration phase analysis, the experimental load versuscrack growth record from a single crack test is modeledand the result is the T�-Resistance curve. This curve isthen assumed to be an intrinsic material property that isonly a function of crack growth3 and is used to predictthe failure of all other tests.

· Finally, all the analyses are performed using the ElasticPlastic Finite Element Alternating Method (EPFEAM)using the stress strain properties of the ®rst step. Thedriving force value of T� (i.e., T�D) is forced to follow theT�-Resistance curve (i.e., T�R) from the second step. Assuch, since the analyses were performed via displace-ment control, the predictions consisted of loads andcrack growth. These were then directly compared withexperimental data.

We emphasize that these predictions are simple to obtainwith this newly developed methodology. Moreover, allpredictions were made using the simple three step proce-dure described above and no `fudging' of any results wasmade.

Many of the predictions compared exactly with theexperimental data and at worst, a difference of 15% be-

tween analysis and experiment was obtained. These resultsare considered excellent given the material and test sta-tistical variability.

6DiscussionThe following discussion is provided regarding the po-tential statistical variation of results. In Rahman and Brust(1995) (and many references provided therein) a proba-bilistic investigation of elastic-plastic fracture based on J-Tearing theory was provided. T�-Integral fracture meth-odology should be considered as a general methodologyappropriate for cyclic loading, creep, MSD problems, etc.,with J-Tearing theory as a subset appropriate under cer-tain circumstances. Indeed, with both J and T�, the simplematerial stress strain properties are required. The onlydifference is the resistance curve. However, it is importantto note that JIc � T� at crack initiation and the differencesoccur only after crack growth. As such, the statisticalvariations in (i) stress-strain response and (ii) the T� re-sistance curve will affect the statistical variability of theresults. Let us consider the typical variation of these twoproperties.

In Rahman and Brust (1995) a statistical sampling ofNuclear Grade 304 Stainless Steel revealed that the Ram-berg-Osgood plasticity exponent varied between less thanthree and ®ve, with a signi®cant coef®cient of variation. Inaddition, JIc varied between 750 and 2200 KJ/m2, againwith a large coef®cient of variation. These data were ob-tained from eighteen specimens. Using these inputs as wellas a crack size distribution (again obtained from a statis-tical sampling), a probabilistic elastic plastic analysis ofpipe fracture was made using Monte-Carlo as well as Firstand Second Order Reliability Methods (FORM/SORM).The results showed that the cumulative probability ofcrack initiation moment (for through wall cracked pipesubjected to bending) ranged from 0.7 (MN-m) for prob-abilities of 10ÿ8 to 2 (MN-m) with probability of 1. Sincethese are for crack initiation, T� and J methods are equal,and hence, this represents a probabilistic T� calculationfor crack initiation! For the NIST tests, the crack sizevariations and positions are much less than those con-sidered. However, a similar analysis where only materialproperties were varied on a Nuclear Grade Carbon steelalso revealed a signi®cant variation in crack initiation load(see Kurth and Brust 1990). Moreover, the statisticalvariation of both stress strain properties and the T�-Re-sistance curve for 2024-T3 aluminum is not known by usand the statistical variations may be less than for NuclearGrade 304 Stainless Steel4.

Hence, it must be concluded that our worst prediction,which led to a 15% difference from the experiment, is quitegood given the potential statistical variation in experi-mental results.

3 Of course, T� is also a function of temperature and materialtype.

4 Note that the manufacturing processes for Nuclear Grade pipematerials are strictly controlled. As such, since aerospacealuminum fabrication processes are likewise well controlled, wewould not be surprised to ®nd signi®cant variations in propertiesfor 2024-T3.

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ReferencesBroek, D. (1993): The Effects of Multi-site Damage on the ArrestCapability of Aircraft Fuselage Structures. FractuREearch Tech-nical Report No. 9302Brust, F. W.; Nishioka, T.; Atluri, S. N.; Nakagaki, M. (1985):Further Studies on Elastic-Plastic Stable Fracture Utilizing T�Integral. Engineering Fracture Mechanics 22, 1079±1103Brust, F. W.; McGowan, J. J.; Atluri, S. N. (1986a): A CombinedNumerical/Experimental Study of Ductile Crack Growth After aLarge Unloading Using T�, J, and CTOA Criteria. EngineeringFracture Mechanics Vol. 23, No. 3, pp. 537±550Brust, F. W. (1995a): Investigations of High Temperature Damageand Crack Growth Under Variable Load Histories. Intl J of Solidsand Structures (In Press)deWit, R. et al. (1995): Fracture Testing of Large-Scale Thin-SheetAluminum Alloy. NIST Report 5661, Prepared for FAA, May, 1995deWit, R. (1995): Private Communication, June and JulyHopper, A. T. et al. (1992): International Piping Integrity Re-search Group (IPIRG) Program. NUREG/CR-4340 ®nal report.Also, eight semiannual reports, 1985±1989Hutchinson, J. W. (1983): Fundamentals of the PhenomenologicalTheory of Nonlinear Fracture Mechanics. Journal of AppliedMechanics, Transactions of ASME 50, 1042±1051Kurth, R. E.; Brust, F. W. (1990): A Probabilistic Elastic-PlasticFracture Mechanics Analysis of Through-Wall Crack TubularMembers. Proc. Offshore Mechanics and Arctic EngineeringConference, Feb. 19±22, 1990, Pub. by ASMENewman, J. C. Jr.; Dawicke, D. S.; Sutton, M. A.; Bigelow, C. A.(1993): A fracture criterion for wide spread cracking in thin sheetaluminum alloys. Int. Committee on Aeronautical Fatigue. 17thSymposium.Okada, H.; Suzuki Y.; Ma, L.; Lam, P. W.; Pyo, C. R.; Atluri S. N.;Kobayashi, A. S.; Tan, P. (1995): Plane Stress Crack Growth andT� Integral ± An Experimental-Numerical Analysis. to be pre-

sented at the ICES-95, Hawaii, August. Also to appear in theproceedingsPyo, C. R.; Okada, H.; Atluri S. N. (1994a): Residual StrengthPrediction for Aircraft Panels with Multiple Site Damage, Usingthe EPFEAM for Stable Crack Growth Analysis. FAA Center ofExcellence Report by Georgia Institute of Technology, NovemberPyo, C. R.; Okada, H.; Atluri S. N. (1994b): An Elastic-PlasticFinite Element Alternating Method for Analyzing Wide-SpreadFatigue Damage in Aircraft Structures. FAA Center of ExcellenceReport by Georgia Institute of Technology, NovemberRahman, S.; Brust, F. W. (1995): Probabilistic Elastic-PlasticFracture Analysis of Cracked Pipes With CircumferentialThrough-Wall Flaws. Proc. 1995 ASME PVP Conference, Hono-lulu, Hawaii, JulyShenoy, V. B.; Potyondy, D. O.; Atluri, S. N. (1994): A Method-ology for Computing Nonlinear Fracture Parameters for a BulgingCrack in a Pressurized Aircraft Fuselage. FAA Center of Excel-lence Report, MarchSingh, R.; Park, J. H.; Atluri, S. N. (1994): Residual Life andStrength Estimates of Aircraft Structural Components with MSD/MED. In Harris, C. E (ed): Proceedings of the FAA/NASA Inter-national Symposium on Advanced Structural Integrity Methodsfor Airframe Durability and Damage Tolerance, pp. 771±784, SeptSwift, T. (1985): The In¯uence of Slow Stable Growth and NetSection Yielding on the Residual strength of Stiffened Structure.Presented to the 13th International Committee on AeronauticalFatigue, Pisa, ItalyThomson, D.; Hoadley, D.; McHatton, J. (1993): Load tests of ¯atand curved panels with multiple cracks. Final report for FAAtechnical center, Foster-Miller, Inc.Wilkowski, G. M. et al. (1995): Short Cracks in Piping and PipingWelds Research Program. NUREG/CR-4299 ®nal report (inPress). Also, eight semiannual reports, 1990±1995

Part I and II of this report can be found in volume 19, issue 5,pp 356±379

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