AMMRC IR 78-15

~ STRESS INTENSITY ESTIMATES FOR APERIPHERALLY CRACKED

~ ‘ SP HERICAL VOID AND A .~~~~~~~

~ HEMISPHERICAL SURFACE PiT

-~~~~~~~~~~ FRANCIS 1. BARA1TA‘

~ MECHANICS OF MATERIA LS DIVISION

March 1978

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To be published in the Journal of the American Ceramic Society.

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Ceramic materials VoidsStress intensity Surface pits

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ABSTRACT

Stress intensity factors are estimated for three-dimensional defectswhich occur in ceramic bodies. The two idealized cases considered are:(4) a spherical void with a circumferential crack at its equator stressedby uniaxial tension at infinit~•and (

~ a hemispherical pit at a free sur-face of a semi-infinite body also stressed by uniaxial tension with a cir-cumferential crack at the semiequator of the pit. These stress intensityfactors, which are given as a function of the crack length L to radius Rof the spherical void or hemispherical pit , are considered as estimatesbecause of the approximate nature of the analysis..

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INTRODUCTIONResearchers in recent years, such as Evans and Tappin,1 Molner and Rice,2

Baratta et al.,3 and Bansal and Duckworth,~ to name a few , have attempted to uti-lize linear elastic fracture mechanics to determine fracture energy or mode I frac-ture toughness (K i~

) of ceramic materials from individual tension or bend specimensby strength and flaw characterization. This attractive approach, if accuratelyaccomplished, is worthwhile pursuing because it will provide in situ ~~ data froma simple specimen configuration. However, it is difficult to accurately assessstress intensity factors associated with even regularly-shaped single inherentflaws in ceramic materials because of the complexity of three-dimensional mathe-inatical analyses. Only a limited number of such solutions are available. Althoughthere are other reasons why stress intensity factors associated with inherent flawsin materials are ill defined, such as irregular shape, defects distribution, etc.,only those single flaws of regular shape which do occur’~ in ceramic materials, seeKirchner et al.5 also, are considered here. Such results, although approximate,and heretofore unavailable, will aid in fracture mechanics applications and provideguidance until more rigorous analyses become available.

Therefore, the object of this paper is to estimate stress intensity factorsfor (a) a spherical void with a circumferential crack at its equator in an infi-nite body under uniaxial tensile siress, as shown in Figure 1, and also include

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Th~~’I. EVANS, A. G., and TAPPIN, G. The Effects of Microwuctuve on the Stress to Propagate Inherent flaws. Proc. Br. Ceram. Soc..

no. 20, 1972, p. 275-297.2. MOLNER , B. K.. and RICE, R. W. Strength Anisotropy in Lead Zircate Thanate Transducer Rings. Am. Ceram. Soc . Bul., V. 52,

no. 6, 1973 . p. 505-509.3. BARAUJI F. I., DRISCOLL , G. W., and KATZ , R. N. The Use of Fra cture Mechanics and Fractography to Define Surface f inish

Requirements for 513N4 In Ceramics for High Performance ApplIcations. 3. 1. Burke , A. F. Gorum, and R. N. Katz , eds., Brook Hill ,Chestnut Hill , Ma ssachu setts , 1974.

4. BANSAL , K. , and DUCKWORTH , W. H. Fracture Stress ns Related to Flaw and Fracture Mirror Sizes. J. Am. Ceram. Soc.. V. 6.not. 74, 1977 , p. 304-310.

S. KIRCHNER. H. P., GROVER. IL. M., and SOTFER. W . A. Characteristics of Flaws at Fracture Origins and Fracture Stress’FbwSize Relations in Vatio~s Ceramics. Mtls. Sd. and Eng., no. 22. 1976. p. 147-156.

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(b) a circumferentially-cracked hemispherical pit at a free surface of a semi-infinite body also stressed by uniaxial tension, as shown in Figure 2. Stressintensity estimates for these two cases will be developed .

PROCEDURE

The procedure is one of applying the appropriate stress, compatible with t~iedefect discontinuity, to the crack tip for the known limiting cases of stressintensity and utilizing an interpolation scheme between limits to deterr.uine inter-mediate values. According to Barenblatt,6 the stress intensity can be obtainedby loading the crack faces with the negative of the stresses that would normallyexist on the plane of the crack, if the crack was absent. However, in the analysesto follow, rather than apply the stress distribution to the crack face, only thestress caused by the defect discontinuity will be applied to the crack tip as sug-gested by Cartwright.*

In order to demonstrate the details of the procedure and evaluate the validityof the approach, it will be applied to a problem already solved by more accuratemeans. Consider a crack in a region of high stress concentration, such as a singleradial crack originating from a circular hole in an infinite plate subjected to uni-axial uniform tension at infinity, as shown in Figure 3a. This problem hd.- beensuccessfully solved by Bowie7 and will be used to evaluate the following approach .

We initially consider the crack length L to be very small compared to the holeradius R. Since the hole radius is very large compared to the crack length we canconsider it to be at an edge of a semi-infinite plate, as shown by the equivalentsystem in Figure 3b, but with a stress distribution °~~ r that arises because ofthe hole. The normalized stress intensity ratio K1/a irL for an edge crack in a

_ _ _ _ _ _ -- ~J _L_-__----~- ~/ ‘ ‘ L./ .1

Figure 2. A hemispherica l surface pit with a circumferential crack at its semiequator.

‘CAR TWRIGHT. 0. 1.. University of Southampton, England. Internal PublicatIon.6. BA RFNNLATt . G. I. The Mathematical Theory of Equilibrium Cracks in Brittle Fr ctu re. Advances in Applied Mrcbi~n~c~ . ~~- ~ .

Academic Press , 1962.7. BOWIF , 0. 1. AnalysIs of an Infinite Plate Containing Radial Cracks Originating at the Houndan ’ of an Internal Circul.ir c’.e.

3. Math. Phys .. V. 35. no. II . 1956 , p. 60-71.

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semi-infinite plate stressed in tension at infinity given by Paris and Sih8 isK1/a/~t = 1.12, where a is the remotely applied uniform tensile stress. At theopposite extreme when the crack length is very large compared to the hole radius,i.e., L/R + ~~, the normalized stress intensity ratio Kj/cW~t= 0.707, from Refer-ence 7. It appears that this ratio is dependent upon some function of L/R whichhas limits of 1.12 when L/R = 0 and 0.707 when L/R + w, Thus, intermediate valuesof the normalized stress intensity ratio can be approximated by choosing a functionwhich is well behaved between the above stated limits. The function arbitrarilychosen for this problem as well as those to follow is of the form:

f(L/R) = c - k(tan~~ L/R)m. (1)

Thus, for a circular hole with one radial crack the normalized stress inten-sity ratio is:

K1/a0(r)vGt = f(L/R) = c - k(tan~~ L/R)m, (2)

where o0(r) is the stress distribution caused by the circular hole, provided byTimoshenko and Goodier,9 which is

Oe (r) o[l + (l/2)(R/r)2 + (3/2)(R/r)~ ], r ~ R , (3)

1 H fOj ( r )-

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Figure 3. Circular hole with asingle radial crack.

a. Cracked Circular Hole b. Equivalent Cracked Circular Hole

8. PARIS. P. C., and SIH. G. C. Stress Analysis of Cracks. ASTM-STP 381, 1970, p. 3013.9. TIMOSHENKO, S., and G000IER, .1. N. Theory of Elasticity. McGraw-Hill Book Company. Inc.. 2nd Ed., 1951.

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where r is the radial distance from the center of the hole. Substitution of Equa-tion 3 into Equation 2 and transforming the coordinate axis r by R + L, gives:

K1/o&1 = [c - k(tan~~ L/R)m] [1 + (l/2)(L,R ~ 1) + (3/2)(L,R’ 1). (4)

Equation 4 represents the estimated normalized stress intensity ratiofor a radial crack of length L emanating from a circular hole of radius R. Theparticular constants which appear in Equation 4 are readily determined by knowingc = 1.12 when L/R = 0; k and m are found by allowing f(L/R) = 0.707 when L/R ~~,and f(L/R) = 0.94 when L/R = 3.0 (taken from Reference 7). The latter value waschosen so that the error span on the function would be minimized throughout thetotal range of L/R. Thus, c = 1.12; k was found to be 0.119 and m to be 2.748.

Equation 4 was then utilized to obtain K/avi~t as a function of L/R from 0 toinfinity. It is seen that these results shown in Table 1 are within ±6% when com-pared to the more accurate data of Reference 8. Although the results of this pro-cedure are approximate, it is believed that the errors involved when applied to theproblems represented by Figures 1 and 2 will be of similar magnitude as those givenin Table 1. Thus, a similar approach is used to solve these problems of interest.

Peripherally-Cracked Spherical Void

Coodier10 provides the stress distribution for a spherical defect in an infi-nite medium stressed in tension. Because of the defect three stresses are developed,which are indicated schematically in Figure 4; they are a radial stress 0r actingnormal to the interface; a stress G~ acting in a tangential direction to a meridian(the north-south pole axis of the sphere is considered aligned with the axis of theapplied stress); and another tangential stress 0th normal to both ar and a9. In thefollowing analysis, the maximum tensile stress i~ assumed to cause crack initiation.

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Tab’e 1. COMPARISON OF APPROXIMATE TO EXACTSTRESS INTEN SITY RESULTS FOR A CIRCULAR

HOL E WITH A RADIAL CRACK °~

L/R From Eq. 3 Reference 7 % Di fferenceO 3.36 3.39 -10 .1 2.73 2.73 0 R0.2 2.32 2.30 +10.3 2.04 2.04 0 — Figure 4. Coordinate

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0.6 1 .56 1 .64 ..5 stress state around a

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L5 1.~3 L18 -42.0 1.03 1.06 —33.0 0.94 0.94 05.0 0.86 0.81 +6

10.0 0.78 0.75 +40.708 0.707 0

10. 000DIER,J. N. Concentration of Stress Around Spherical and C)’Thid~*.1 Inclusion: and Rates. Trana. Ant. Soc. N. E., v . ~% ,1933. p. 39.44

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It is indicated in Reference 10 that the maximum tensile stress for a spheri-cal void occurs when 0 = 0, resulting in a “hoop stress” 00, which girdles theequator, It is expected that this tensile stress, if becoming large enough, caninitiate an axisyinmetric crack and extend it in the radial direction as envisionedin Fi gure 1. The stress distribution when 0 = 0 from Reference 10 is given in thefol.iowing equation:

= 0(2(7 5) [(4 - Sv)(R/r)3 + 9(R/r)5] + 1 ), (5)

where ‘v is Poisson’s ratio of the material and r is the radial distance from thecenter of the spherical void (r ~ R).

To obtain an approximation for the stress intensity factor associated withFigure 1, we again initially assume that the crack length L is small compared tothe sphere radius R. Also, as before, we further assume that when the crack issmall, the influence of the spherical void on the crack is negligible and thus canbe considered as an edge crack as shown in Figure 3b, but with a stress distribu-tion a0(r) provided by Equation 5 that arises because of the spherical void. Also ,we shall assume that Equation 1 is applicable as well as the constants (except k)previously given as c = 1.12 and m = 2.748; k is determined by realizing that inthe extreme limit when L/R + ~ the defect geometry approaches a disk crack and thusK1/0,ct+ 2/n, according to Sneddon.11 Therefore , making use of this end limit inEquation 1, which is substituted along with Equation S into Equation 2, results inthe following normalized stress intensity expression:

= [c - k(tan-1 L/R) m] (2(7 - 5v) [(4 - 5~~(L,Rl+ l)~

1 \51 (6)

+ ~~L/R + 1) J + 1

where c = 1.12, m = 2.748, and k is determined to be 0.101.

The resulting calculations using Equation 6, which represents the normalizedstress intensity ratio for a spherical void cracked at its equator and stressedby uniform uniaxial tension at infinity, are shown in Table 2 as a function ifL/R from 0 to infinity and for v = 0.25 and 0.30.

Tab’e 2. STRESS INTENSITY RATIO FOR APeripherally—Cracked Hemispherical Pit PERIPHERALLY-CRACKED SPHERICAL VOIDThe final case considered is a hemispherical

pit at a free surface of a semi-infinite body L/R~ 0.25 0.30with a crack initiated at the semiequator of the 0 2.26 2.30pit and extended radially by a uniaxial uniformtensile stress as shown in Figure 2. As outlined 0.55 L26 1:26previously, we initially assume that L is very 1.00 1.~0 108

small compared to R. We further assume that whenthe crack is small , the influence of the hemispher- 5.00 0:79 0.79ical pit on the crack is negligible and thus can 10.00 0.72 0.72be considered as an edge crack as shown in ______

0.64 0.64

II. SNEDDON. I. N. Th~ Distribution of Stress In the Nelghbo4,ood of a Crack in an Elastic Solid. Proc. Roy. Soc. London. v . A-187,1946.

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Fi gure 3b, with the stress distribution a9(r) that arises because of the surfacepit discontinuity. Again , as before, we utilize Equation 2, with c = 1.12 andm = 2.748; but determine k from the end limit when L/R + ~~~. Smith and Alavi12 givethe variation of K1/a/t around the semicircular crack boundary and indicate thatthe maximum value occurs at the junction of the crack with the free surface and is1.21 x 2/r . This results in k 0.140.

The stress distribution a0(r) is provided by Eubanks,13 who has determined the

variation of the circumferential stress along the axis of symmetry as a function ofr at the base of the pit. Equation 2, including the constants previously given,i.e., c = 1.12, m = 2.748, and k = 0.140, and the normalized stress distribution13a8(r)/a given in tabular form will provide the normalized stress intensity ratiofor a peripherally-cracked hemispherical surface pit with v = 0.25. Such results,including a0(r)/a, are given in Table 3 as a function of L/R from 0 to infinity.

RESULTS AND DISCUSSION

The results of the procedure applied to the known stress intensity case ofone radial crack emanating from a circular hole indicated that the error was ±6%.The procedure was then extended to the two cases of interest assuming that engi-neering accuracy would be preserved.

Table 2 presents the normalized stress intensity ratio K1/aV~t as a functionof L/R from 0 to infinity for the circumferentially-cracked spherical void , withPoisson’s ratio being 0.25 and 0.30. This table indicates that Poisson’s ratio,within the range considered, has little effect on the stress intensity ratio.

Table 3 presents K1/o/ t as a function of L/R from 0 to infinity for thecircumferentially-cracked hemispherical surface pit when v = 0.25. It is evidentthat the magnitude of the normalized stress intensity ratio for the peripherally-cracked surface pit is greater than that of the peripherally-cracked sphericalvoid for the same given L/R value.

Restrictions on Application

Table 3. NORMALIZED STRESS INTENSITY RATIO .

FOR A PERIPHERALLY-CRACKED HEMISPHERICAL The application of Equation 6 and Tables 2SURFACE PIT , WITH • 0.25 and 3 has certain restrictions resulting fromL/R c,8 (r) /a K ; /ovGU the implied assumptions given in References 100 2.23 2.50 and 13 when determining the appropriate stress0.15 1.63 1.82 distributions . These restrictions are:0.35 1.29 1.44

a. the material is isotropic and2.00 1.00 0.99 homogeneous;3.00 1 .00 0.935.00 1.00 0.88 b. the defect is very large compared to10.00 1 .00 the grain size of the crystalline

aggregate;

12. SMITH. F. W .. and ALAV I. N. J. Stress intensit y Facto~s for e Pert.Through Circular Sorfece Raw. Proc. Ut In tl . Conf . onPressure Veaael Tech.. Delvt , Holland. 1969.

13. EUBANKS. R. A. Stress Concentration Air to a Hemispherical Pit at a Free Surface. 3. AppI. Mech.. v . 21 , 1954, p. 57.62.

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c. there are n - other defects closer than four diameters away;d. the cracked spherical void can be no closer than four diameters

to the body boundary; ande. the cracked hemispherical pit on the free surface can be no closer

than lout diameters away from any other body boundary.

SUMMARY

1. An approximate method is presented which allows estimates of normalizedstress intensity for three-dimensional defects such as a circumferentially-crackedspherical void and a circumferentially-cracked hemispherical surface pit as a func-tion of crack length to void or pit radius when L/R is between 0 and inf inity.

2. The three-dimensional defects considered in this paper can be thought ofas being two-dimensional at their limits when L/R = 0 and when L/R -

~ infinity.Wit h the appropriately known limiting normalized stress intensity values obtainedfrom the literature, an interpolation scheme was employed to obtain intermediatevalues when L/R was between 0 and infinity. The results given by Equation 6 andTables 2 and 3 are expected to be within engineering accuracy.

3. It was shown that for the spherical void case the variation of Poisson ’sratio between 0.25 and 0.30 had little effect on stress intensity.

4. It was also shown that the stress intensity factor associated with acircumferentially-cracked hemispherical surface pit was greater than that of acircumferentially-cracked spherical void for the same given L/R value.

5. The restrictions indicated in the previous section must be recognizedwhen applying Equation 6 or utilizing Tables 2 and 3.

AC KNOWLEDGMENT

The author wishes to acknowledge the helpful advice given by David J.Cartwright from the University of Southampton, England, NATO Fellow to the ArmyMaterials and Mechanics Research Center.

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Coamsanaer. Watervlie t Arsena l . Witeryliet , Washington. 0. C. 20546New Yor k 12189 1 AIIM: Mr. B. G. Achhemmer

I ATIN : Dr. I. DavIdson 1 Mr . 6. C. Deuts ch . Code 8W1 Mr. 0. P. kendall National Aeronautics and Space Admlni&tration.Mr. J. F. Throap Marshall Space Flight Center , Huntsville

Commander. U. S. Army Foreign Science and Techno logy Alabama 35812Center . 220 7th Street. N. E. • Charlottesville. I AIIM: P. J. Schwinghae r, (H(~ Di rect or ,Virgin ia 22901 MaP Laboratory

1 AIIM: Mr. Marley . Mil itary Tech 1 Mr. H. A. Wilson , (HA l, Build ing 4612

_

6

So. No. ofC~ipies To Cop i es To

National Aerovu xtl cs and Space Adnini stration , Brown Unive rs i ty, Providence . Rhode I s l a n d 02912Langley Research Center, Hampton , Virginia 23665 I 71511: Prof. 3. 8. R i ce

I AIIM : Mr. H. F. Hardrath , Ma il Stop l8~4 I Prof. a. N . Findley , Di vision of Eng.. Box 0Mr . P . Foye , Ma il Stop 188A I Prof. P . C. Paris

National Aeronautics And Space Administration , Carnegie-Mellon University , Department of MechanicalLewis Research Center, 21DDO Brookpark Road , Engineering. Scisenley Park , Pi ttsburgh .Cleveland, Ohio 44135 Pennsylvania 15213

I AIIM: Mr. S. S. Mansos 1 AIIM: Dr. 3. L. SwedlowDr. 1. E. Srawley , Mail Stop 105-1 1 ~~~~~~ 3. 0. Labahn , Colorado School of Mines,I Mr. V . F. Br~av , Jr. Golden , Colorado 8040 1Mr. H. H. Hirschberg, Head , Fatigue ResearchSection , Mail Stop 49-1 1 Prof . J. Dvorak. Civil Engineering Department, Duke

University, Durham, North Carolina 27706I Mechanical Properties Data Center , BelfourStulen . Inc. , 13917 V . Bay Shore Dr iv e , George Washington Unive rsi ty , School of EngineeringIranerse City . MIchiga n 49684 and Applied Sciences, Washington , 0. C. 20052

1 AIIM: Or. H . Liebow i tzNational Bureau of Standar ds . U. S. Department ofCo,merce . Washingt on, 0. C. 20234 LehIgh University, Bethlehem, Pennsy lvania 18015

1 AlIt: Mr. 3. A. Bennett 1 AIIM: Prof. G. C. SIhProf. P. RobertsI Mr. V. F. Anderson, Atoniics International , P rof. 8. P . WeiCa noga Pa rk , CalIfornia 91303 1 Prof. F. Erodgan

Mi deest Research Institute , 425 Coker Boulevard, Massachusetts Institute of Technology ,Kansas City, MIssou ri 64110 tanbridge . lSassachacetts 021391 AIIM: Mr. G. Gross 1 ArlN : Prof. B. L. Averbach , Materials Center. 13-50821 Mr. A. Hurlich , Convair Din . • General Dynamics 1 Prof. F. A. McClintock. Roam 1-304

Corp. • Ma il Zone 630-01 , P. 0. Boo 80847, I Prof. P. H. PelloaxSan Dingo , Cal ifornia 92138 I Prof . I. H. H. P lan , Dept. of Aeronautics

and Astronaut ics1 Mr. J. G. Kaufma n , Alcoa Research Laboratories , I Prof. A. S. Argon . Room 1-306hew Kensington , Penns yl vania 15068

I Mr. P. N. Randa l l , TRW Systems Group — 0-1/2210, Syracuse University, Department of Chemical

One Space Park, Redondo Beach, Californ ia 90278 Engineering and Metallurgy, 409 L i n k H a l l ,Syracuse , New Yo rk 13210

I Or. L. A. Steigerwald , TRW Metals DIvision , 1 AIIM: Mr. H. W. L ixp . 0. Box 250, Minerva , Ohio 44657 1 Dr. V. Weiss , Metallurg ical Res.

Labs. • Bldg. 0-61 Dr. George R . Irwin . Department of MechanicalEng ineering, Uni vers i ty of Maryl and , 1 Prof. E. P. Par ker, Department of Materials ScienceCollege Park , Maryland 20742 and Engineering, University of California ,

Berkeley , California 947001 Mr. V. A. Van der Sloys, Researc h Cente r ,Babcock and Wilcox , Alli ance , Ohio 44601 1 Prof. V . Goldsmith , Department of Mechanical

I Mr. B. H. Wundt , 2346 Shirl Lane. Schenecta dy , Lnglneering, University of California.Berkeley , California 94720New Yor k 12309University of Califo rnia , Los Alanos Scienti fic

• Battelle CollaTibas Laboratories. 505 King Avenue , Laboratory, Los Alamos , New Mexico 87544Coluntus , Ohio 43201 ATIN: Dr. 8. Karpp1 ATIN : Mr. 3. Campbell1 Dr. G. I. Hahn 1 Prof. A. 3. McEvlIy , Metallurgy Department U-136 ,

P. G. Hoagland, Metal Science Group University of Connecticut, Storrs , Connecticut 06268Dr. E. Ryblcki 1 Prof. 0. Drucker , Dean of School of Engineering,

General Electric Coeipany. Schenectady, University of IllinoIn , Champaign , Illinois 61820New York 12010

1 AIIM : Mr. A. 3. Brothers, Material s and University of Illi noi s , Urbana, Illinois 618011 AIIM: Prof . H. I. Corten. Department of TheoreticalProcesses Laboratory and Appl ied Mechanics . 212 Talbot Lab.

Genera l Electric C ompany, Schenectady, 1 Pro f . I. J. Dolan , Department of TheoreticalNew Yo rk 12309 and Applied Mechanics

1 AIIN: Mr. Ii. F. Baeckner . Large Steam Turbine 1 Prof . 3. Morrow. 321 Talbot Lab.Generator Department 1 Mr. 6. N. Sinclair , Department of Theoretica l

Mr. S. Yukawa , Meta llurgy Unit and Applied MacpanicsMr. C. C. Zwicky, Jr. 1 Prof. 8. 1. Ste phens , Materials Engineering

General Electr ic Company, Knoll s Atomic Power Div ision University of Iowa , Iowa City,Laboratory, P. 0. Bon 1072, Schenectady, Iowa 52~42New York 12301

1 AIIM: Mr . F . 3. Mehr lnger 1 Prof. D. K. Felbeck , Department of MechanicalEngineering, University of Michigan1 Dr. L. F. Coffin , Room lC4 l-K l , Corp . R&D, 2 046 East Engineering, Ann Arbor , Mich Igan 48109

General Electric Company, P. 0. Box 8.Sc henecta dy. Mew York 12301 1 Dr. 14. L. Williams , Dean of Engineering.

240 Benedum H a l l , University of Pittsburgh,United States Steel Corporation . Monr oeville , Pittsbur gh, Pe nnsylvania 15260Pennsylvania 15146

1 AIIM: Mr. S. P. Novak 1 Prof. A. kobayashi , Department of Me chan icalDr. A. K. Shoemaker, Research Laboratory, Engineering, FU-lO , University of Washington,

Mall S top 78 Seattle , Washington 98195

Westin ghouse Electric Company, Bett is Atomic Power State University of Mew York at Stony Brook ,Laboratory , P . 0. Box 109, West Mifflin , Stony Brook. New York 11790Pennsylvania 15122 1 AlIt: Prof. Fu-Pen Chiang , Department of Mechanics

• 1 AI IM : M r. H. L. Parrish Director , Army Materials and Mechanics ResearchWesting house Research and Development Center , Center , Watertown, Massachusetts 021721310 Beala h Road, Pittsburg h , PennBy lvania 15235 2 AIIM: ORXiAl-Pt.

1 ATTN : Mr. E. I. Wessel 1 DR XP~~-AG-M0Mr. H. 3. Manjoine 1 Author

1 Or. Alan S. Ieteletan, Failure Anal ysis Ass o ci ate a .Suite 4, 11777 Mistlssipp i Avenue,Los Angeles , California 90025

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