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MICROCOPY RESOLUTION TEST CHARTI4ATIKWAL BUREAU OF STANDARflS.1963-A

U~FILE COPYARL-STRUC-R.421 AR-04-493

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Cl1 DEPARTMENT OF DEFENCEt- DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORIES

MELBOURNE, VICTORIA

Structures Report 42 I

ON THE APPUCATION OF THESTRAIN ENERGY DENSITY THEORY IN PREDICTING

CRACK RINIATION AND ANGLE OF GROW1-

by

A.K. WONG

DTICfo PbJ Rl e IELECTEApproved for P0l c Relems

Sc((0 C O EAjIh OF AUSTRAUA IN1; ........

SEPTEMBER I1"r

87 7 10 010

AR-004-MN

DEPARTMENT OF DEFENCEDEFENCE SCIENCE AND TECHNOLOGY ORGANISATION

AERONAUTICAL RESEARCH LABORATORIES

SMrUCIUM REPORT 421

OK1 T MU OA2ION OW T93

8maSr nemiy 003mr 7momY IN iininezwoCRACK DNrAIIOt AND ANGLE OP GROWTH

by

A.K. WON

until recently, the one-ommter ingular expressio for streemnear a crack-tip was widely thos* to be sufficiently a alte over areasonable region for any geometry and leafig conditiom. This view hasbeen fast changing due to the growing e I aps that the lakk of higherorder terms can significantly affect the solution. particularly ,,. certainbiaxial loading conditions. In this context. the present paper' exguaines thestrain energy density criterion for fractuM, and the onseqwwas of theassumption of a 1/r energy singularity in the formulation an its application. Itis found that this assumption imposs a rather severe restria" oa the regionfor which the criterion is applicable, and that its application on as arMitrarilyselected 'small' distance from the crack-tip (a procedure which ha beenadopted by many experimentalists and finite element analyts), can lead toerroneous results.

(C) ONOUW'ALTE OF ALU'UA;ZA IWM

POSTAL ADDRESS: Directm, Aeronutical Remefhc Laboraories.P.O. Box 4331. Melboume. Victmia. 3001. Au sia.

rI

Page No.

SUMMARY

NOTATIION

1. INTODUCIION 1

2. SOLM N FOR THE INCND CRACK PROBLEM 2

2.1 The Strain hergy Density Criterion 4

8. THE APPLICATION OF THE S-THEORY IN PREDICINGBRrITLE FAILURE 5

8.1 Predicting the Onset of Failure 5

8.2 Predcting the Direction of Growth a

4. CONCLUSON 9

ACKNOWLEDGEMINT8 10

REFERENCES

FIGURE

Aossion por

7DETRIBU72ON NFTZS GRUilDTIC TAB

I Munnounced 13DOCUMENT CONTROL DATA Just lest t

3,D"sribut io

1vat bi sa/er_!v8 1aili Codesvel/I

V a])lo!

NOTATION

A,C, Fins three coesciests of the ledi" expansion of the strain energdensity fanction

* Half crack ength of specimen

& Half width of specimen

E Elastic modulus

F Futlon of elastic modulus and Poisson's ratio

A Half height of specimen

J J-lntegrl

je Critical J for determining fracture

K1, K Mode-I1 and Mode-I Strews Intensity factors

K1. Critical stres Intensity factor for Mode- frctre

k Ratio of remote lateral load to remote vertical load

r Radial distance from crack-tip

rj Incremental crack growth

rc Critical crack growth increment

ro Radius defaing core region

S Strain energy density factor

S*, 5,, S First three coelcieats of the series expansion of the strain enewg

density factor

S) Strain energy density factor at load incrementj

S, Critical strain energ density factor foe determiaing fracture

s Integration path for the J-integral

T 1Tio vector on s

a Displacement vector on sVe Displacement in the y-dlrectloa at the first corner node behind the crack-tip

W Strain energy density fuction

we Critical strain energy density function

e.# Cartesian coordinate axis system

3 Complex vector &+ i

o Length parameter2Tr

Inclination of the crackrxr Complex fuaction of lad, biaxinity and age of mnclination

a1 Distance of the lest comner node behind the crack-tip to the crack-tipis d,fS, Normal and shear strains

* Angular coordinate at crack-tip

A Fitnctio, of Poisson's ratio

V Pewsosna ratio

IF Ratio of the circumfree to the dinkmet of a circl

*rs, op few Normal and shear srse

0. fl W I'e ERlomoephic functions

* 1. INTHODUCTIONThe analysis of the strem field araoad cracks is an elastic wld has been well documented madkt perbaps mochi older than the &Mi now know as Liner Elastic Fracture Mechanics. Oneof the early woek Is due to Westergad III, who as"aei the shaip crack proliem using aI IS IfI ctlom expasion appcoach. ha a lawe paper, Westergased 14 1 xpse $be solusom tothe sam problem bn term of opIe- amalyi Inactions. This. lowe work has since gainedmws meeg -t6a and itfrequestl reeif by the amabse dactvu moanks. It ean beahwam that doe Watergasrilm"h baob to the hlowlag claualal mob for the sitems laidam the tb' of a sutre-crask coatained in han lS. plate lairled by a reot ww Imaltale

sum (sa(y aa l

K, e2 2 )

*sa - - - em -)-

way ~ 2 2o 2* 0"~

where K, is known as $be dtress intensity factor.However, a smal error made is the Weatergaaud solution remwaasindetected for aheost

thirty yeary. By using the move general complex potentias approach ot Muahheliahvili IM.Sih 14 showed that the arbitrary msttig of a particular consant is the Weaterpard soles"oto sero was generally lawaldi. Discussion ans the effects of the error was subsequently lawsup by ft n i Liebowkts . il late, Elis st aW M showed tha the amro was eaqulem

* to the omission ofa aca-slaguhareri am r th sream expresios. Dy -anl.lmlg a m agiteatre-cracbed plate under blzal remote bmade r 1peependleular to the crc) ail he lpl.ato the crack), it was shown that the solution, correct to the sero-th order teem is r, b

*=KI~*

' I coi g 2 2 + Si(lW1b)

Comparing Eqs 1.2 with Eop 1.1, ad aettlg k 0 for malaxial loadlag, it iswe thatthe only dlbfeance is the aen-slagulac sem #is the expression for r,. Of course. as r - 0,

t thibistism bt expected to become aqliglhl . However, kt does adt do so as rapidly as oe wouldMea. malaly because of te square root ect of the singular temn. For the sakajal tase as asexamople, & - 0 ad K, - vuwii, so that hr 9 - (r, Eq. 1.2a becomes

68 9 st - 11]. (1.31

hm the er caused by neglsetlag the noneliaguiag, term to be lms shaa 1% my,. is in required

u6m hW e md -I h dog Vow ~i,. NOA dotht a k he.. Poet (h - )doh mm~h mr/a weaM he.. ewe a=no NV406 MINs lP jewel AM for O. am of

fe~ar, b pbeoedsooe amu mloh pamde deram s v mlog ONe md &adam.. at labm hadpoomlma Gwhimedo with commoid data. h.asuheegmmtpepm,

Ebb at i M I meataed that o hea ohmi -td Imow deadl m $owne mg rataso depaml dgpiem asmd th aef d~t At~h ipplul ha ml Oh. uhbmy *=ma oft oh. inbegml tam dIl. Ih s ek m Okd toh iaxt M decO ml thmbob el to biemest

peklm ofth. mWW d1 Vow& whes wfti Mkb's sbdo mwasn e*hlt aili 1 .Tbli werk cetama oh. liwatga. em o dof 1th eAslem d1th. aem-eblov Wm

im oh. hmudatm al 1Th. S-Otuemy, sod te eeieqmm md IMkoltis I t aplmole am aremuit. h wN he ahewas that em the dino* minima vamWm saw be algamtlym alfectm, mloh. minmm peeclmeri o 11h 5-omy em be .1.1.1.

I. goWTUON FOR TM3 M1CLU4 CRACK PSOULDAlb Mhmoot h dem Wh oh. esmidma 1 oh sam in lor m, we wil Wa to a paiow wbk& a wal boews ads"lm asios. Camsider the balt. plat emodelg a eta cim&al bob SommbeeedthhbiAlhbe&m sod reuhms hIn f 2. NfIawI*Mookb&lbbv at, Oh hlmneqk edsohm wh"s mpo -At mohuo aim

a(s) W,18)rn!(W + ro NI.

1 4

(W + r 10 -400)(2.4)

md ON. us... ON. by mA.

thm htam.eesm1t.aoeAhm smeaulm m vb ml .alea oh. udhm waso 0 <c P/aC 1, t mep he

obwe60 aNods @som" O A solM O

34

*~~~cs Cos-~osD f2?.

2 2

wbee K, and Kr anth. stsu inkemity kctos given by

K, - -(lk) sll~. (2.9)

Flemming FeIDe's law, the in-planme elastic strains are

ewhe!(o

F -,--ILj fo plane strain,

and E ad vawe the elastic modules and Poissom's ratio respectively.Hence, the strain energy density function W for a plane problem is gies by

W - (msts + tri + 26(2.11)

+ "[u + 20., +au,- t aiJ

Substituting Eqs 2.? into Sq. 2.11, we obtain

A-4 B+C

A- j~2K7'(14eas)I(3-cs#J- All+cos)l

+ 41(0 - 4*0+IcomW) +.%(I + #Cos* + acos0)I+SWKKguisofems - 1) +Alen#+1)jl,

2FVIAco c(2+1),sW~

Cm -'-k) ca 2.iF

3

Note that the heackim at the Oo~hplwen ba the Slstrum uf leank.kds to twoadditional tam in W. me df which Miar'dhusFw the other In nmssmgimr.To m hew this aaet t application of the S-Ibee~my, i tdv Ie Wuansube the hubselthdohebsy.

The sa snamg daft altuiem wa ftat Iepe by Sib ~and Ifu appleat an beenuwell deemm-I Ie (use for eimmple MI - lUi). INe hamltin bahsed as the assumpton thutthe Atas mer dafth function now the crackl* poessus a I/r tisguity Thin mW he

dellnedsuh*m

The criterion cosemlag cack iiito a h nl fpwbi hnbsdo h

By thsi 1 Crc W %tesombegn n. diset rthe subeegde si ty betoe

ypthe ub1 2. rcueI mmms hntelclmamu d ece a critical value S.,andtha Sois smteral ammterwhch aracterimes the fracture stregt

orr cetcahed sptecie ue oe odtos as h ietos rc rp

By* kgWA - aTe Eq .4 ofe, eln stalInrnal cod it , a j.. coaetaresl gore rbyt

lagSSt

whenr upo reachhesg 3, Ittal showsp tht heamun oorm irntal sei nermd Indtion bofth pltse dfm of sad c t pmtrial

that W w a u W pact i ouhes r e hnowee, rak e nsito of 0su a 0,flee=Ktead 0 ippoied. 11 acouv es thm phenoetancdton, a hesoldnvaeen xressied sfoh thlat.ing So *(rre to exstn bate orem deata, catummehnc uue oUtsoto

re gurty soh a raphyicad eepreiction Hy pothi &It Shew hAMous the cMoere-n Thart hh conberdiion by rmt he setiond t tepo@o a W c ad that e materia

awhiWE ahu intersec te wth (e.Practc hevtter aeril mot ro unles a 40tellit

Varou Aiel stengths and fracture touighaes. Sih J12l showed that r. rasged krm 0.000Is. (0.015 mms) to 0.01M4 In. (&.3416 mas).

4

. THE APPLICATION OF THE S-THEORY IN PREDICTING BRITTLE FAIL-UEOne advantage of the strain energy density criterion over most other fracture criteria is thatboth the angle of growth and critical load may be predicted by a single parameter. Furthermore,without the restriction of self-simlar growth as required by other criteria such as the criticalenergy release rate approach, the 5 criterion may be applied in complex load systems. Forexample, by means of Snite element modelling, the S distribution around a crack-tip containedin a complex structure under an arbitrary load system may, in theory, be used to predict thefailure load as well as the direction of propagation. However, it will be shown in this sectionthat without sufcient care, this practice can lead to erroneous results.

3.1 Predicting the Onset of Failure

Consider the uniaxial problem described previously. Since the direction of growth for this caseis known a priori (namely along the x-axis), substituting 0 = 0*, 0 = 90* , K11 = 0 into Eq. 2.12and multiplying through by r yields an expression for S along the critical direction, viz.,

S K2 (I -A) - Ki(I-A) +02r (3.1)

And since K, = e a,

' [.(I - (I - k)V'ii/+ () (3.2)

ITThe one-term representation, Eq. 2.14, may be obtained by taking the limit of Eq. 3.2 asr/a - 0, giving

S = -( -A). (3.3)

Taking a typical value of v = 0.3, Fig. 4 shows a comparison between the one-term rep-resentatiou and the higher order expression for S . It may be seen that there is significantdiference between Eq. 3.2 and Eq. 3.3 At the relatively small value of r/o = 0.02, the neglectof the higher order terms gives rise to eera of appxocimately 20% in both the plane stressand plane strain results. Indeed, values of r/a ? 0.02 have not been uncommon in many anal-yses using the strain energy density method (e.g., 1101, J131). What is more important in thisexample is that along the critical direction, the one-term expression always over-estimates thecorrect value of S for r/a > 0. Since S. is derived from the one-term expression (Eq. 2.16),the prediction of fracture load using the S-theory by means of finite element modelling, or byphysically monitoring the elastic strains at some small but finite distance from the crack tip,will invariably produce an over-estimate. In the above example, if the strain-energy density Sat ra = 0.02 is monitored, and Hypothesis 2 of the criterion is applied so that fracture loadIs assumed to be reached when S = S., then S(r/o - 01 would be greater than Se by about20% and hence the fracture strength would be over-estimated by 9.5%. The non-conservativenature and the relatively large magnitude of the error involved is certainly undesirable, and itshows how the blind application of the S-theory in design work can be extremely dangerous.

For the S-theory to be applicable, the analysis must be confined to a region for which theone-term expression, Eq. 3.3, is valid. Allowing an error of say 6 %, we get from Eq. 3.2

, 6

-m.- 1M m____ im

( -(A)VTa -r/a :5 0.06. (3.4)

The solution which satifies the condition 0 < r/a 4: I is given by

r< 0.00124. (3)a-

Equation 3.5 represents aa upper limit to the region where the strain energy density analysismay be applied with acceptable accuracy. However, we recall that there exists also a lower limitr. which defines the core region (see §.1). Consequently, the region where the S-theory is validmay be given by

r. : r : 0.00124a. (3.6)

Equation 3.6 implies

ro0.00124(

The restrictions inferred by Eqs 3.6 and 3.7 are severe, and in some cases, may be consideredas impractical. For the high fracture strength 4140 steel (t. = 0.01345 in.) quoted in 112 as anexample, the minimum crack length required for a valid LEFM analysis would be 10.85 in. (275.6mm). Such a condition would be difficult to satisfy, particularly for real life situations wherecracks of much shorter lengths are detected and require analysis. Equation 3.5 on the otherhand can almost always be satisfied in a finite element model in which mesh sizes of the orderof 0.001 a may be readily handled on a modern computer. However, using such a fine meshfor a linear analysis may be difficult to justify as stress intensity factors have been successfullycomputed by other methods with much coarser grids. To highlight this point, a finite elementmodel of a centre-cracked plate (v = 0.3) subjected to uniaxial tensile loading and plane strainconditions was established. Because of symmetry, only one quarter of the plate which has awidth to height ratio b/h = 1, and a crack-length to width ratio a/b = 0.25 was modelled.Figure 5 shows the two mesh schemes adopted for the analysis. The parameters calculatedinclude

1) The stress intensity factor K, given by the expression (141

,FK, = -2v r- (3.8)

where v, and A, are respectively the y-dispacement and distance from the crack tip atthe Arst comer node behind the crack tip, and F is as defined in Eq. 2.10.

2) The J-integral given by the numerical integration of the expression 1151

S= Wdu - T. On d., (3.9)

where T and u are the traction force and displacement vectors along the path # respectively.The integration path chosen in this case is shown in Fig. 5. It may be shown that J ispath-independent for an elastic material and is related to the stress intensity factor by thefollowing expression

K, = V3F. (3.10)

6

3) The strain energy density factor 5(= rW) at nodal points along the critical direction(0 = 0"), where W is calculated from the computed stresses (Eq. 2.11). To estimate thestre intensity factor from the strain energy density factor, the conventional expressionwhich neglects the higher oeder terms is used 1121

K, 2sES= ( + -)(, - ,)

In the following presentation, all values of KI and S have been normalised by ut/i andw'a/E respectivel. The computed strain energy density along the critical direction (0 = 0")for Mesh-I is shown in Fig. 6. An immediate observation is the systematic scatter of the resultsabout some mean value. The strain energy density factor appears to be over-estimated at thecorner nodes, and under-estimated a mid-side nodes. Whilst the average deviation of S fromthe mean amounts to appraimately .T %, the scatter of the computed stresses would be onlyapprdmately 2.8 % and is therfore considered acceptable. U the assumption of a constant S ismade over the reion say 0 < r/a f_ 0.04, then a best fit to the datagives the result S = 0.259T,which when used in Eq. &1, gives K, = I.1 This compares with K, = 1.950 obtained usingthe analytical solution given in Rooke and Cartwright 1161, and therefore represents a 9.2 %under-estimation.

The computational results for the refined mesh are shown in Fig. T. Again, a similarsystematic scatter is apparent, and the assumption of a constant S in the range 0 < r/a :5 0.04gives S = 0.2B27, or from Eq. 3.11, K, - 1.848 which is in error by 6.2 %. However, thedownward trend of the S-distribution as predicted by the analytical solution is clearly evidentin this case, and compares well with the plot of Eq. 3.1 (where KI is taken to be 1.950). Thissuggests that some sort of curve fitting to the correct form

S=S..-Fl!+S (3.12)

would provide a more accurate prediction of strain energy singularity. This was indeed foundto be the case. Applying a least squares It for the S data of Mesh-2 to Eq. 3.12, it was foundthat S. = 0.3082. Substituting S. for S in Eq. 3.11 yields K, = 1.930, giving an error of only1 %.

Table I summarises the various methods of computing the stress intensity factor and theirerrors with respect to the analytical solution obtained in 116].

K, K,(v,,Al) K,(J) K,(S=onat.) KI(S.)

Ref. 1161Mesh-1 Mesh-2 Mesh-I Mesh-2 Mesh-I Mesh-2 Mesh-2

1.O0 1.892 1.902 1.894 1.896 1.771 1.848 1.930

% Error 3.0 2.5 2.9 2.8 9.2 5.2 1.0

Table 1. Stress Intensity Factors Calculated by Various Methods

7

It may be seen from Table 1 that the stress intensity factor is predicted accurately byboth the crack opening and the 1-integral even for Mesh-I. The evaluation of K, from Showever required the much finer mesh to achieve an accuracy to within 5 %. For a moreaccurate prediction, a least squares fit of the correct form was required. Whilst this curvefitting technique proved to be useful in the case of Mesh-2, it would be unreasonable to applythis to the data of Mesh-I where a poor correlation to the true solution is expected.

It is also interesting to point out that whilst the J-integral involves the integration of astrain energy term, the problems associated with the S-theory do not appear, as it has beenshown by Eftis et a1171 that the truncation of the higher order terms has no effect on J andEq. 3.10. Another feature of J is that the integration process tends to nullify the systematicscatter of the numerical data and was therefore able to predict K, accurately without theapplication of any best fit technique. In practice, it is most likely that J is used in a non-linear analysis where stress intensity factors are not defined. Hence, J is not generally used forcomputing K,, but instead, it is calculated and compared directly to a critical value J, (which,like K,, and Sc, is a material parameter) to determine whether failure is to occur. The aboveresults merely show that a finite elements approach to the J-integral can be applied with someconfidence. However, it must be remembc red that the restriction of self-similar growth must beobserved, and that its usefulness when extended to true elasto-plastic materials is yet uncertain0i7J.

3.2 Predicting the Direction of Growth

Having seen how the neglected higher order term can affect the prediction of crack initiation bythe S-theory, its effect on the prediction of the direction of crack extension is now examined. Intesting the maximum normal tensile stress theory for predicting the direction of crack extension,Williams and Ewing 1181 conducted experiments on PMMA (polymethylmethacrylate) sheetspecimens which contain a central inclined slit crack and subjected to uniaxial tensile loads.It was found, particularly for steep crack angles (0 -* 0), that the angle of growth deviatedfrom the predicted results presented by Erdogan and Sih [191. The authors attributed thediscrepancies to the fact that only a one-term expansion was used in the expression for thestresses, and showed that by selecting a critical parameter a = f = 0.1(r/a = 0.005), andincluding higher order terms in the expansion, a better fit of experimental data with theory isachieved. In a subsequent discussion, Erdogan and Sih 120] showed that, unlike the maximumnormal tensile stress criterion, the strain energy density criterion provides a better fit to theexperimental data and is relatively insensitive to the parameter a. To investigate this pointfurther, Eq. 2.12 is differentiated with respect to 0 to form

8S 8A aB80 ~ (3.13)

82S 82A 8 2

B80-2 802T + 802 r

and from Hypothesis I of the criterion, the direction of crack propagation is determined when

85 8 $= 0, and - > 0. (3.14)

Assuming v = 0.33 and for plane strain conditions, the solutions to Eq. 3.14 over the range0* < 0 " for s = 0, 0.1 and 0.2 are presented in Fig. 8 together with the scatter band of theresults in 1181. It may be seen that the one-term representation of the S-theory (equivalent tothe case a = 0) fitted the experimental data better than the one-term approach to the maximumstress criterion. It is also noted that the solutions for both a = 0.1 and 0.2 fall within or near theexperimental scatter band. However, there is some evideece that the use of a = 0 for 0 < 45"and a = 0.2 for 0 > 45* can result in a more accurate prediction. This has the implication thatthe core radius r0 may in fact not be a material property. The question of whether r. may be

8i

considered as a material constant has previously been addressed by Ch--g 1211. Chang foundthat in order to obtain reasonable agreement between the experimental data of 1181. 1221. 1231and the strain energy density theory, various values of r./a (ranging from 0.006 to 0.15) had tobe used despite the fact that al data were presented for the same material (PMMA) and forsimilarly sized cracks. He therefore concluded that r./a (and hence r.) 'can hardly be justifiedas a material parameter in the S-theory'.

Chang 1211 also discussed, at length, the dilemmas which may arise in applying the S-theory when no relative minimum, or alternately, when more than one relative minimum in Sexist in the solution. Swedlow 1241 showed that for uniaxial loading configurations, the choiceof the global minimum leads to incorrect predictions. Swedlow then proposed the additionalrequirement that the Sm, which governs fracture must be associated with a tensile hoop stress.On the other hand, Sih and Madenci [101 assert that it is the maximum of all S,.'& whichfirst reaches the material threshold and is therefore the critical factor. For the inclined crackproblem considered, two local minima are found for all 0 except for f = 90", and indeed, it isthe maximum of the two at any given r which corresponds to the results presented in Fig. 8.However, an interesting situation can arise when other loading conditions are considered. As anexample, under a biaxial tension-compression loading system with k = -1, and for P = 60, thepaths of the two minima are as shown in Fig. 9. The corresponding plots of S along the pathsdenoted by i and ii are presented in Fig. 10. It is clear from the plots that the determinationof the maximum of the minima would depend on the selection of r/a.

It has been seen in the uniaxial case that although the use of the S-theory to predictcrack initiation requires an extremely small value of r/a, the restriction was much less severefor predicting the propagation angle as reasonable predictions may be achieved for the range0 < P/a < 0.02. However, the above example shows that this may not be valid in general as thechoice of r/a appears to be crucial in determining whether the crack is to grow in direction i orii. Perhaps this is also evidence for the possibility that the S-theory alone may be insufficientin determining the direction of crack propagation in general, and a modification such as thatproposed by Swedlow [241 may be in order.

4. CONCLUSION

It has been shown that the usual assumption of a 1/r energy singularity is valid only withinan extremely small regime around the crack tip. For the centre-cracked plate considered, thisregion is typically of the order of r/a < 10

-3. As a consequence, the application of the

S-theory at some distance outside this region may result in substantial errors in the predictionof crack initiation. For a finite element analysis, an extrapolation technique using the moreaccurate form is proposed and found to be useful. However, this technique still requires arelatively fine computational grid, and a guideline for maximum mesh size, optimum number ofdata points and the valid extrapolation domain for arbitrary crack and loading configurationshas yet to be established.

It has also been shown that the restriction on r/a is somewhat less severe for predictingthe crack growth direction, and that for the uniaxial load case, reasonable agreement betweenexperimental data and predicted results may be achieved over a relatively large range of r/a.Unfortunately, this may not be taken as a general rule as illustrated by the biaxial load examplewhere a dilemma in choosing the correct S.j. may arise when r/a is arbitrarily selected.

In closure, it should be emphasised that, despite the problems revealed by the currentwork, the strain energy density factor should not be disregarded as a useful parameter. Thereis no doubt that the S-theory works well under certain conditions, and its potential in handlingmixed mode fractures is particularly valuable. The close relationship between S and the stressintensity factors (Eq. 2.12), and the reasonable agreement between predicted and existingexperimental data on propagation angles, tend to support this. On the other hand, limitationsto the theory must be identified and realised. Questions such as whether or not r. is a valid

9

t t

_ * m'.~mm,= a s nn im tms n u

materWa parameter, ot what; sh..Id be dome when multiple minima in S aes, have, in thepresent author's opinion, yet to be positively resolvedi.

ACKNOWLED3GEBMENTSThis woek was carvie out mpart of a Commoswenith Advisory Aeroawtlcal Research CouncilCo-operative project os fracture. The asthor wishes to thank Dr. Rhys Jons lb his helpucomments.

10

REFERENCESIII H. M. Westergaard, Stresses at a Crack, Size of the Crack sand the Beading of Reinforced

Concrete, Proc. American Concrete Inst itute., 30 (1934), 93-102.121 H. M. Westergaard, Bearing Pressures and Cracks, 7as. ASMIE, Journal of App liedl Me-

chanics 6 11939), A49-.63.131 N. 1. Muskhelishvili. Some Basic Problems of Mathematical Theory of Elasticity, P. No-

ordboff Ltd.. Groningen, Holland (1953).141 G. C. Sib, On the Westergaaid Method of Crack Analysis, International Journal of Frar.

tuft, 2 (1968), 62".30.161 J. Eftis ad H. Litbowitz, On the Modified Westergaard Equations for Certain Plane Crack

Problems, International Journal of Fracture, 8 (1972), 380-302.161 J. Eftis, N. Subrainonian sand H. Liebowit, Crack Border Stress and Displacement Equa-

tions Revisited. Engineering Fracture Mechanics, 9 (1977), 189-210.171 J. Eftis, N. Subramonian and H. Liebowitz, Biaxial Load Elfects on the Crack Border

Elastic Strain Energy and Strain Energy Rate, Engineering Fracture Mechanics, 9 (19771.753-764.

181 G. C. Sib, A Special Theory of Crack Propagation, Methods of Analysis and Solu tions toCrack Problems 1, ed. G. C. Sib, Wolters-Nocedhoff (1972).

191 G. C. Sib and B. Macdonald, Fracture Mechanics Applied to Engineering Problems - StrainEnergy Density Fracture Criterion, Engineering Fracture Mechanics, 6 (1974), 361-386.

1101 G. C. Sib and E. Madenci, Crack Growth Resistance Characterised by the Strain EnergyDensity Function, Engineering Fracture Mechanics, 18 (1963), 1160-171.

1111 N. N. Au, Application of the Se-Theory to Structural Design, Theoretical and AppliedFracture Mechanics, 4 (1965), 1-11.

1121 G. C. Sib, Ex~perimental Fracture Mechanics: Strain Energy Density Criterion, Experi-mental Evaluation of Stress Concentration and Intensity Factors 7, ed. G. C. Sib, Wolters-Noordhoff (1972).

1131 G. C. Sib and Da-Yu Tzou, Three Dimensional Transverse Fatique Crack Growth in RailHead, Theoretical and Applied Fracture Mechanics, 1, (1984), 103-115.

1141 G. C. Sih and H. Liebowitz, Mathematical Theories of Brittle Fracture, Fracture, AnAdvanced Tratise 2, ad. H. Liebowitz, Academic Press (1968), 67-190.

111 J. R. Rice, A Path Independent Integral and the Approximate Analysis of Strain Concen-tration by Notches and Cracks, Journal of Applied Mechanics, 34 (1968), 371-384.

1161 D. P. Rooke and D. J. Cartwright, Compendium of Stress Intensity Factors, HillingdonPress, Uxbridge, M*iddx. (1976).

[171 A. K. Wong and R. Jones, A Numerical Study of Two Integral Type Elaato-Plastic FractureParameters Under Cyclic Loading, forthcoming in Engineering Fracture Mechanics.

[181 J. G. Williams and P. D. Ewing, Fracture under Complex Stress - The Angled CrackProblem, International Journal of Fracture, & (1972), 441-446.

[191 F. Erdogen and G. C. Sib, On the Crack Extension in Plates Under Plane Loading andTransverse Shear, Trans. ASME, Journal of Basic Engineering, 86D (1963), 51"-27.

(201 F. Erdogen and G. C. Sib, Discussion on 'Fracture Under Complex Stress-The AngledCrack Problem', International Journal of Fracture, 10 (1974), 261-265.

1211 K. J. Chang, A Further Examination on the Application of the Strain Energy DensityTheory to the Angled Crack Problem, Trans. ASME, Journal of Applied Mechanics, 49(1982), 377-382.

[221 P. D. Ewing and J. G. Williams, The Fracture of Spherical Shells Under Pressure sadCircular Tuabes with Angled Cracks in Torsion, International Journal ofFracture, 10 (1974),537-644.

1231 H. C. Wu, R. F. Yao and M. C. Yip, Experimental Investigation of the Applied EllipticNotch Problem in Tension, Vrans. ASMIE, Journal of Applied Mechanics, 44 (1977), 466-461.

1241 J. L. Swedlow, Criteria for Growth of the Angled Crack, f~racks and Fracture, ASTMW STP601, American Society for Testing and Materials (1976), 506-521.

11

Olt/-ft-

Faigura A Unnia Loaded Flat-Crark (.eometry

hee

Figure 2. Diaxiafy Loaded Isrliaed-craek ;eoguelry

12

we

wr

Figure 3. Strai Exerg Density Dissbto Abieg se critia Diretum.

0.3

~ 0.3

em *A$ 41M MO GMS 5.10

Figure 4. Comperiomal .i On-Tomm and the Higbo,-Orde, Rd pr eeeusamm of 5

13

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Figur S. Fnite lemenModl o 0 Cet a/ls p Pa0.04

0.0

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r/a

Figure T. Computed S (Mesh-2)

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400

60

so

40

30 Experinental Stter 11iProdktomo Accordiag to the

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- Predictioss Accodieg to the S-ThryIO

10

0 10 20 30 40 so 60 70 60 s o

A.

Figpr S. Prediction af Crack Extenion Dfetioe for the Ineised Crack Probk-m Under Usi-

axal Loading

11

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k = -1

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Figure 9. Paths of Minim.um Strain Energ Density Factor for Biaxial Load Example

0.4

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0.1

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Figurr 10, S-Disluuisg Along Pats i sed ii of Fig. 9

14

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AR-004-493 7 ARL-STRUC-R-421 SEPTEMBER 1986 DST 86/012

bIEAPPLICATION OF THE STRAIN a.dret1ENERGY DENS1TY THEORY IN PREDICTING UNCLASS 1CRACK IN7ITTON AND ANGLE OF GROWTH b tite - absract .hoRf

__ _____________ U U 24______-9. lDngruding Iattuctions

A.K. WONG

10. Corporate Auhor and 11. Xibority (os ppropriate)

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fIb. Citation for olfset paces (ie casu~al aa-eonoint) noy be (mrestricte E4. eawiptions- .CS)ruFracture (mechanics) 1313Strain energy methods 1100Crack initiation

Until recently, the one-parameter singular expression for stresses near acrack-tip was widely thought to be sufficiently accurate over a reasonable region forany geometry and loading conditions. This view has been fast changing due to thegrowing evidence that the inclusion of higher order terms can significantly affect thesolution, particularly under certain biaxial loading conditions. In this context, thepresent paper examines the strain energy density criterion for fracture, and theconsequences of the assumption of a 1/r energy singularity in the formulation on itsapplication. It is found that this assumption imposes a rather severe restriction on theregion for which the criterion is applicable, and that its application on an arbitrarilyselected 'small' distance from the crack-tip (a procedure which has been adopted bymany experimentalists and finite element analysts), can lead to erroneous results.

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