1. eng. & app l. sci. Vol. J3 No. I Jan . - Ju ne 20 0.J I SSN 1023 -862

COMP UTATION OF CRITICAL CRACK SIZES FOR CRACK GROWTHPREDICT ION IN THIN PLATES

Abid Ali Khan* and Iqba l Rasool Memon*

ABSTRACT

In this research critical stress int ensity fa ctor K ilo corr esponding to Mode I fractu re has been experi ­mentally evaluated. Thin metal sheets with s ingle and doub le edge crack are used as fe.lt specimens '. The

pra cti cal resu lts are compared wi th estima ted values of KIC

available in the fr acture mechan ics lit erature. The

study also includes th e experimentatio n to compute number of des ign load cycles that wo uld propagate an

exis ting crack to the crit ica l length under cyclic loading.

7 1

College of Aeronautical Eng ineering, National University of Science and Technology, P..IF Academy. Risalpur,l' akistan.

INTRODUCTI ON

The eng inee ring proce dures used by the indus­try to promote structura l integri ty of mach ine partsare continuously evo lving. Early enginee rs used manycut and try practices in their machine designs. Thi sled to the deve lopment of experimentally oriented test sand development procedures. As the enginee ring pro­f,'ss ion deve loped further. analyt ical techni ques werealso introduced to make improvements in design pro­cedures. Sign ifican t changes are still taking place ininstrumentation, ana lysis procedu res, and methods ofhandl ing and analyz ing data . As long as these tech­niques conti nue to evolve, enginee rs have the oppo r­tunity to utili ze these new resource s to make improve­ments in their fatigue design procedures.

Failure of structures due to fatigue or develop­ment of cracks is one of the major phenomenon thatoccur in everyday life. A number of str uctures suchas bridges, aircraft and industrial stru ctures havefailed catastrophicajl y due to initiation and pro pa­gation of cracks. It is the refo re, import ant to reali zethe import ance of crack initi ation and propagation inmetal structu re failur es. Nature of loads on structuresmakes it impossible to eliminate cracks. However, thegrowth of these cracks can be moni to red til l theyreach critica l size. To perform this check crack growthrate is measured at different sizes of cracks .

For structural materials, the tolerable flaw sizesare much larger than any initia l undetected flaws.However for structures subjec ted to fatig ue loadin g

(or stress corrosion cracking) the initial cracks cangrow through out the structure. Th us an over allapproac h to preve nting fracture or fat igue failures inlarge welded structures ass umes that a small flaw ofcertain geometry exists after fabricati on and that thisflaw can either cause brittle fracture or grow by fa­tigue to the critic al size. To ensure that the structuredoes not fail by fracture, the calculated critical cracksize, at design load must be sufficient ly large. and thenumber of cyc le of loading requ ired to grow a sma llcrack to cr itical crack size must be greater tha n thedesign life of the structure.

The crit ica l stress intensit y facto r is a too l,which helps in determin ation of cr itica l value of crac klength s corresponding to the applied nominal stress.Thi s parameter forms basis to predict the structura llife of a component based on the length of crackpresent. Th us the parame ter is of vita l importance infracture mechanics, engi neers and des igners use it todetermine the str uctura l integri ty of their designedstructural parts for machines.

CRACKING PROC ESS

In metals of interest to industr y, fatigue damageand crack growth typically occ ur by the process ofreserve d-slip or reversed plas tic strain 1.2. The amountof slip on a give n cycle relates to the amount of cycli cstrain imposed. Therefore , the co rre latio n of fatigueres istance with strain or plastic stra in is , logical andconsisten t with the , phys ics of th e pro cess . Theamount of slip that develops under the action of given

,

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J. eng. & ap p l. s c i. Vol . 2J No. / .IU I7. - Ju ne 20 1l -!

strain depends on how easy it is fOI' two planes ofatoms to move past eac h other along the slip plane.Thus the component of force applied normal to theslip plane makes slip easier if it is tensil e, by separat­ing the planes , and correspon dingly hard er if it iscompression . for this reason , correlation of fatigueresistance with both strai n and stre ss is logica l, andis more cons istent with the physics of the proce ssthan j ust the strain .

The slip process is responsible for fat igu e cracknuclea tion and the micro- and macro-crack growthprocess . Key differ ences in the slip behav iour dur ingnucleation and growth are in the magn itude of slipand in the volume of material that undergoes slip. Incrack nuc leation, the str ain level is small , but may berather widespr ead compared to crack growth. \ 1 hichconcentrates high strain in the relatively small plasticzone at the crack tip.

In nuc leat ion, the slip process occ urs in thehighl y stressed locations. Once cra cking begin s. thedeformation forme rly acc ommodated by the slip dis­tributed in the cr it ica l locati on is taken up by defor­mation at , and ahead of, the crack tip and along thewake of the crac k.

DIFFERENT MODES OF CRACK

To establ ish method s o f stress analysi s forcracks in elast ic solids, it is convenient to define freerelative movements of two crack surfaces. These dis­placement modes represent the local defo rmat ion in

an infin ites ima l ele ment containing a crack front. Thethree mode s are shown in Fig. 1.

(a) Mo de I: Opening mode , it is charac terised bylocal d isplacement that are symmetric with respect tox-yand x-z planes. The two fracture surfaces are dis­placed perpend icularly oppo site directions.

(b) Mode II : Shear mode, where local displacem entsare symmetrical with respect to the x-y plane and skewsymmetric with respect to x-z plane . The two fracturesurfaces slide over eac h other in alterat ion perpen­dicu lar to the line of the crack tip .

(c) Mode III : Tearin g mode . it is assoc iated withlocal displaceme nt that are skew symmetric with re­spect to both x-y and x-z planes . The two fractures

72

I."".';:\' 1023-862

y

y

Fig , I. Diffe rent Modes of Crack SlIIjl'IC'C Displacement.

surfa ces slide over each other in a direct ion that ispara llel to the line of the crack front.

CRACK GROWTH

Crack growth adversely affects the life of struc ­ture and strength of mater ial. If regul ar inspect ionsare carried out to arrest the crac k growth then struc­ture life can be enhanc ed and strength can be re­tained . Differ ent repair schemes are adopted to arre stthe cracks in stru tures, the most succe ssful being

drilling holes at crack tips to avoid further growth.Crack growt h mainly take s place due to one of thefollowing five mechanism s:

(a) Fatigue due to cyclic loading

(b) Stress corros ion due to susta ined loadin g

(c) Creep

.J. eng. & appl. sci. Vol. 23 No. 1 Jan. - JUlie 200./ ISSt\' 1023 -862

(d) Hydrogen induced cracking

(c) Liquid metal induced crac king

materials the rate of fatigue crack growth can be ex­pressed by the Par is equation

C & III = Materia l constant s.

TYPES OF SPECIMEN

EXPERIMENTAL EVALUATION OF KIC

(2)

The critical intensity factor is a parameter ofvita l importance for all service materia ls as far as thefrac ture mechanics is concerned. When KIt: of a ma­terial is known, then vita l prediction can be made onthe life o f a material in service. KIt: is a function ofapplied nomina l stresses, crack size and the geo metryof the spec imen. When KIC' the des ign nomin al stressand obvio usly the crack geomet ry is known, then thecrit ical value of crack length can be figured out fromII. re lation Therefore under the particu lar conditionIe . ,

of st ress when crack length is below the critica l valuethe materi al will not fail under given operating condi­tions.

The basic aim of this research is to estimate KIt:

value of thin metal sheets. The standard test proce-

.. d All. ' dalAs cracks grow 111 size an LJ increases, I dN

becom es very high and fractu re is imminent as Ke orKIt: (defined in sub sequent paragraph) are approached.Different other-? researchers have also obtained goodcorrelations with similar spectrum characteri zation forrandom loadings. In these studies of crack growthunder random loadi ng, stress spect ra were representedby a cont inuous, unimoda l distr ibution, in particularby a Rayleigh distri bution function. The usefulnessof th is type of app roach for load histories describedby other types of distributions is uncertain". Also ,such statistica lly based approac hes lose the loadingsequ ence, but again that is only of concern if se­quence effects cannot be neglected and an attempt totake them into account is thereby necessitated .

daldN = Increase in crack size with number of loadcycles applied

where,

da =C(!J.K)IIIdN

N = Number of cyc les

(I)

(J Appl ied nominal stress

a Crack length (or dept h)!

where,

C = A dimension less quantit y used to account forthe y type of crack loadi ng, and the ratio ofcrack size to specimen or component dimen ­s ions.

The value of C is that comput ed as if no crackswere present. The K[ factor has units of MPa - m1/2 orksj - in l/2 . Each of above values is also a function oftemperature, part icularly for those structure s exhibit­ing a transition from britt le to ductile behaviour. Workon simulat ion and analysis of crack growth in ducti lemater ia ls is an important sepa rate research area l.".The fatigue cycle is usually described by !JK, which

equals K",~ \.-K"' i'" Here 11.""" and 11.", ,,, are values of theopening mode stress intensity factor 11." calcul atedfrom the- maximum and minimum stress during thefat igue cycle. If the minimum stress in a cycle iscompressive, it is conv eniently taken as zero. Th isis a simplification based on the assumpti on that acrack closes when the load fall s to zero, or below it.There is also a value of !JK below which no growthtypically occurs, called the threshold stress intensity,iJK . It has been shown exp erimen tally that !JK has

Iii

the major influence on the fatigue crack growth. It hasalso been found in one study' that if LJ K is constant,the fatigue crack growth rate is constant. For many

Fatigue crack growth tests arc relativ ely stra ightforward and over the past 30 years have been carr iedout in large numbers. Stress intensity factors are con­venient means of correlating fatigu e crack growthdata , bec ause the stre ss condit ions at the crack tipcan be describ ed by this single parameter. The stressintensity factor is the K

Iterm present in equatio n 1.

This factor depend s primaril y on crack s ize and thestress app lied to a region, but also varie s with differ­ent types of cracks and loadin gs. All K

Ifactor s have

gene ral form such as

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1. eng. & appl. sci. Vol. 23 No. I Jail. - JUlie 200-1

(b) Aluminium test specimen of standard dimensionswith edge notc h.

ll'lOl"'U-O ':rn

1gem

j

ISS'N I0l3-S62

The calculated Kw Values for thin metal plateswith various crack lengths are compared with theknow n values of cr itica l stress intensity facto r givenIn the handouts.

Th in aluminium plates of9 :\14 .75:\ 0,4 ern withsingle and double notch of lengths 0.4, 0.8, 1.2, 1.6

and 2.0 ern at the mid length of the plate as show n inthe Fig. 3 were prepared for the tests. The plates weretested under bending and tensi le loads. 10 ton tensi letest ing mach ine, the loadi ng mecha nism out lined inFig. 4 was used and fract ure load was noted for theplates with differen t crack lengths. Nom inal stresswas then calcul ated for noted value of fractur e load.The geome try facto r was ca lculated using the fract uremechanics for the double edge crack geome try . As­sumi ng the initial crack length as cr itica l. correspond ­ing value of K,c was then calculated for all the platesunder bending and tensile load and tabulated in Tables

I and 2. The calculated values of KI(' for double edgenotc hed plate are tabulated in Table I and for singleedge notched plate are given in Table 2. The resul tsreveal that bending and tensi le stress va lue whichhave been computed expe rimenta lly decrease withincrease in flaw size. Furthermore, the qua nt itat ivedecrease in stress values is re latively smalle r fordouble notches as compared to test pieces with singlenotch and same flaw size.

(4)

(5)

(3)

C =1.12 + 0.43(a lTV)

- 4.79(a nV)2 + 15.46(a I W)3

C =1.12 - 0.23(a / W)

+10.56(a I W)2- 21.74(a I W)3

+30.42(aIW)4

C = I+ 0.256(aI W)

-1 .152(a I W) 2+ 12.2(a / W)3

dure demands the speci men of sta ndard dimensions.Thus the use of thin metal shee ts in standard testmethod is restri cted. Therefore, two types of spec i­men were consid ered for determ ination of K,c

(a) Thin alumin ium plates of reaso nab le dime nsionswith single and doub le edge cracks.

In order to determine K1C

edge notch of di fferentlengths are produced on the thin plates and it isass umed that eac h one is the cr itica l length of thecrac k. Thin alumin ium plates are tested under the twotypes of loadin g, i.e. slow bending load and tens ileload ing. The specimen fracture stress is assumed asthe nominal stress level for a particular test specimen.The geo metry factor expression for sing le and doubleedge crack specimens Me taken from geo metries inFig. 2 and correspo nd ing relationsh ip from equat ions3, 4 and 5 were used for further computa tion.

+---4.75 em----++---4.75 em----+

Fig. 3. Different Test Specimeno

c

c

o

c

o

2.--,

Fig. 2. Geometry Function of COII/II/OII Cases Fig . .J. Tensile and Bending Test Loading Mechanism

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J. el1g. & appl. sc i. Vol. 23 No . I Jan . - June 200-!

Table I: Results for Double Edge Notched Plate

ISSN 1023-862

S Flaw Geometry Bending Tensile K1C

Tensile K1C

BendingNo Size Factor C Stress ks i Stress ks i ks i -Y i n ksi"in

I 0.1572 1.1 3 3.59 40.56 1.6 10 ~ 8. 1 9

2 0.3150 1.14 3.07 33.46 1.940 21 .22

3 0.4725 1.1 7 2.24 28.74 1.800 23.\1

4 0.6300 1.31 1.31 17.58 1.360 18.28

5 I 0.7876 1.61 0.67 8.68 0.958 12.40

ASPECTS OF LIFE PREDICTION

In order to predict crack growth for differenttypes of loadin gs, sev eral approaches severa l ap­proaches are possib le. None of the appr oaches canbe said to be clearly superior at this time since theamount of testing done to date has not been substan­tial enoug h to indic ate which appr oach might be pre­ferred".

The question of when to use a crack init iationanalysis or crack growt h ana lysis or a comb ination ofthe two remains an outstanding issue to be resolvedon case to case basis. However, it is recommendedthat in applications where inspecti on indicates theexistence of a material or geometr ic notch, crackgrowth ana lysis is appropriate in estimating remaini nglife. Likewise, is s ituations where significant defectsexist and are conside red acceptable or are undetect-

able as the part enters service, a crack growth analy­sis is app ropriate .

Crack growth anal ysis may also be relevantduring design/analysis for mater ials selection and com­ponent sizing. Such is the case where s ignificantdefect s are considered accep table or may be unde­tectable . "Safe" life in these cases requires either nogrowth or slow stable crack grow th to crack size s thatare stable - that is well below critical size at limit loadconditions. In all crack growth analy ses, care shouldbe taken to ensure the validity of procedure adopted .

CRAC K GROWTH ANALYSIS

Crack growth life is expre ssed as the number ofcycles to grow a fatigue crack over a certain length .The important parameters used in the crack growthana lysis concept are stress range (Lla) and the stress

Tab le 2: Results for S ing le Edge Notched Plate

S Flm\' Geome try Bendi ng Tensile K1C

Tens ile K,c Bend ing

No Size Factor C St ress ksi Stress ksi I,:si -Yi n ksi -Yin.I 0.1572 1.162 4.12 45.81 21.12 1.90

2 0.3150 1.300 3.74 43.94 32.05 2.73

3 0.4725 1.51 0 3.14 4J .09 42.65 3.26

4 0.6300 1.800 2.84 "l ...... J" 47.47 4.06.).) ._.)

5 0.7876 2.230 2.40 27.84 55.Q9 4.75

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J. eng. & app l. sc i. Vol. 23 No. I Jan. - Ju ne J()() .f

rati o (R) . The st ress range and stress ra ti o is

given by

6 (J =O'lll ax - O'I11 ;n

R == O"min

0" 111 ax

(6)

(7)

IS,':;N 1023-862

load cyc les applied for eac h interval of I nun wasnoted till the time the crack reached to a critical valueof 20.5 nun from an initial length of 12.5 nun . Theresult s are tabulated in Ta ble 3 . The resul ts revealthat threshold stress intensity has major influenc e onfatig ue growth. It is also evident that da/dN the rateof fatigue crack growth is directly proportional toiniti al cra ck size.

The loca l s tre ss at the crac k t ip can bedescri bed in terms of the st ress intensity factor Kgiven as

(10

6K = C60' Jra) -

K .III I II

1/2=CO'min ( l U I ) (8)

(9)

(10)

CONC LUS ION

The study and the experimenta l work done inth is researc h concluded that the cr it ical stress inten­sity factor is the key parameter to predi ct the servicelife of the structura l members under the known load­ing condi tions. The crack growth rate is measured byusing equ ation no I . The parameters C and m in thisequation must be ca lculated using the graphica l ap­proach , since they are the materia l de pend ent param­

eters.

Th e crack gro wth rate is given by equation I

(Paris Law). The materi al se lect ed for this ana lysiswas Aluminium 7075 -1'6 . The maximum initial flawsize a was taken as 12.5 mm and maximum critical flaw

o

ar size was determined to be 20.5 mm . The specimenwas subjected to constant amplitude load cycle of

Pm," 1590 KgI' and Pm" = 2 140 Kgf. The number of

The se lection of an initial crack size is in someways the most uncertain aspect of a crack growth lifeevaluati on. If initia l defects are like ly to be presen tfrom the time of fabrication, then the analyst mustdecide on an app ropriate size ba sed on knowledgefrom inspect ions or exp eri ence with the particularcompo nent and manufacturing proce ss bein g cons id­

ered. If the initial size is taken as that predicted to

,

T ab le 3: Fatigue Cra ck Growt h Rate Ca lcu la t ions

S N ao a r a Geometer t. N t.K dald N'N ll\'~

(10- 3 Ill) (10-3 Ill) (10-3 m) Factor C (cycles) (Mpavin) (10-6 m/c) (cycl es)

I 12.5 13.5 13 1.39 1060 12.64 0.94 1060

2 13.5 14.5 14 1.43 860 13.87 1.16 1920

3 14.5 15.5 15 1.47 700 14.36 1.42 2620

4 15.5 16.5 16 1.52 570 15.33 1.75 3190

5 16.5 17.5 17 1.57 470 16.33 2.12 3660

6 17.5 18.5 18 1.63 380 17.44 2.36 4040

7 18.5 19.5 19 1.69 310 18.58 3.22 4350

8 19.5 20.5 20 1.75 260 19.74 3.84 4610

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J eng. & appl. sci. Vo l. 23 No. I J an . - Jun e 200 -1

exist after crack initiation life has been expended, thena size of about two to three mrn could be used . Thatsize co rresponds app rox imate ly to the size of cracksthat exist at fracture in the small lab specimens usedto generate the input data (£-N or a -N) for crackinitiation life pred iction methods .

FUTURE DIRECTIONS

More sophisticated approaches by Dowling " andSocie " have been propo sed for defin ing the transi­tiona l crack size be twee n initiation and propagation,and the analyst may wish to investigate their use.

REFERENCES

I. Leese, G. and S ocie, D., 19 89. Mult iax ial

Fatigue : Analysis and Exp erim ent s . SAE

AE- /-I.

5

6.

7.

8.

ISSN 10 23- 86 2

Barsom, .J. M , 1973. Fati g ue Crack Growth

Un der Variabl e Amplitude Loading in ASTiII

,151-4-B St eel. Flaw Growth and Fra ctur e

Toug hness Testing, ASTiII STP 53 6, pI ';;.

Smith, S. 11. , 1966. Random Loading Fatigue

Cr ack Growth Behavi our of S om e Aluminium

and Titanium Alloy s. S tr uctura l Fatigue in A ir­

craft, ASTM S TP 404 , P 74

S wanson, S. H. , Cicci , F. an d Hoppe, IV , 196 7.

Cra ck Pr opagation in Clad 707 9- T6 Alumin ium

Alloy Sheet Under Constant and Random Am­

plitude Fat igue Loading. Fatigue Crack Pr opa­

gation, AST/v/ STP 415, p . 312.

Fa tigu e Life Predi ction in SA E Fatigue D es ign

Handbook AE - 22. 1997 . Fat igue Design and

Eva lua tio n Comm ittee; Soc ie ty of Automotive

En gineer s, Inc .

2.

3.

-I.

Dabell, B. J, 11)83. Digital Te chniques In Fa­

tigu e. S o ciety of Environ m e nt al Eng in e ers ,

Fat ig ue Group.

Gullerud, A. an d Dodds, R., 2000. Simulation

of D uctile Crack Gro wth us ing Co mputa tional

Cell : Numer ical Aspects. Engineering Fra c­

ture Me ch an ics Vol . 66 No. I, p. 6-1 -92 .

Roy, A. and Dodds, R., 2001. Simulation of

Du ctil e Crack Growt h in Alum in ium Pann els

us ing 3D S ill/ace Cohesive Elements. lnterna­

tiona! Journal of Fractur e Vol. 110, P

21 --/5 .

9. Shen, IV. , Sob oyejo, IV. O. and Soboyejo,

A. B. 0. , 200-/. An Inves t ig at ion on Fatig ue and

Dwell Fatigue Cr ack Growth in Ti.6AI-2Sn--/Zn­

2Mo-0.ISi. Mechanics of Materials Vol. 36 N o.

I - 2, p. 17-1-/0.

10 . Dowling, N. E., 19 79. No tc hed Member Fatig ue

Life Pr edic ti ons Com bining Crack Im itati ons

and Pr op ag ation . Fatigue of Engineering Ma­

terials and Structures. Vol 2, p . 129.

II . Socie, D. F., Morrow, J and Ch en , IV , 19 79. A

Pr ocedur e fo r EST im atin g th e Tot al Fatigu e Li f e

of Notched and Cracked members . Engineer ing

Fracture Mechanics. Vol. II, p . 851 .

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