An elastic-plastic analysis of fatigue crack closure in ... · AN ELASTIC-PLASTIC ANALYSIS OF...

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AN ELASTIC-PLASTIC ANALYSIS OF FATIGUE CRACK CLOSURE IN MODES I AND 11. * ** M. Nakagaki and Satya N. Acluri School of Engineering Science and Mechanics Georgia Institute of Technology Atlanta, Georgia 30332. Abstract In this paper, an efficient elastic-plastic finite element procedure to analyze crack-closure and, its effects on fatigue crack growth under general spectrum loading, is presented. A hybrid- displacement finite element procedure is used to properly treat the stress and strain singularities near the crack-tip; and crack-growth under cyclic loading is simulated by the translation of certain "core" elements, near the crack-tip, in which pro- per stress and strain singularities were embedded. Both pure Mode I and Mode I1 types of cyclic load- ing are considered. In the mode I case, fo~rr types of cyclic loading, viz., constant amplitude block loading, high-to-low block loading, low-to-high block, and a single overload in an otherwise con- stant amplitude block loading are considered; whereas in Mode 11, only a constant amplitude block loading is considered. Detailed results are presented for crack-closure and opening-stresses, crack-surface deformation profiles, etc., in each case. Certain observations, based on the present numerical results, concerning various factors that cause krack growth acceleration or retardation un- der general spectrum loading, are presented and discussed. Introduction: As discussed recently in an expository article by Schi jve [ 11, the mechanism of crack-closure, as first observed experimentally by Elber 1123 , i s gen- erally considered to be a predominant mechanism that contributes to "interaction effects", which cause crack-growth retardation or accoleiation, under variable-amplitude fatigue loading. It is also generally understood [l] that the crack- c l o s u r e phenomenon is cuased by residual plastic deformations remaining in the wake of the advanc- ing crack-tip, as initially postulated by Elber. Analytical models that lend theoretical support to the existence of crack-closure phenomenon in fa- tigue crack-growth, and provide some rationality for the adoption of an effective stress-intensity range, based on closure effects, for the correla- tion of fatigue crack-growth rates, have also been proposed by Budiansky end Hutchinson [3] . As for a more general analysis of extcnding cracks under general block cyclic loading, to obtain crack- closure stresses, crack opening stresses, details of crack-surface deformations, and residual stres- ses in tkie crack-tip region, etc., elastic-plastic finite element analysis were first performed by Newman an1 his colleagues [4,5]. Apart from these analyses, the authors are aware of similar attempts only bv Oh51 and h i s co-workers [6,7]. The stud- C - LC;'~ Ln L4- /J LU~LGLU~Z~U ri~de I case url~y. AL- so, since the crack-growth was simulated in [4-71 - ** K~esearch Engineer, Professor, Member, AIAA by shifting a finite element node (the current crack tip) to an immediately adjacent node, and since constant strain-triangle type finite ele- ments were used to model the cracked structure, a very fine finite element mesh (with the small- est element often being of the order of 10-3 times the crack le2gth) is necessary in the mod- eling procedures of 14-71. Thus the finite ele- ment computations of the type given in [4-71 can be very expensive, especially when cyclic load- ing of arbitrary spectrum are considered. In the present paper, an alternate cost- effective and accurate el; s tic-plastic finite element procedure to analyze fatigue crack clos- ure, and its effects, under general spectrum loading, is presented. In the present procedure, the well-known Hutchinson-Rice-Rosengren [8,9] type strain and stress singularities, for strain- hardening materials, are embedded in specially developed elements near the crack-tip. This elim- inates the need for a very fine mesh near the tip. For instance, the crack-tip elements in the pre- sent procedure are of the order of 10-I of the crack-length, as compared o const nt strain tri- angles of the order of lo-' to lo-' times the crack-length generally used in the procedures of [4-71 . A hybrid-displacement finite element me- thod [ 10,11] is used in developing these special elements. Also in the present procedure, crack- growth is simulated by: (i) the translation of a core of the forementioned special elements by an arbitrary amount in the desired direction, (ii) reinterpolation of requisite data in the new finite element mesh, and (iii) proportional relax- ation of tractions in order to create a new crack surf ace. Since the forementioned special elements near the crack-tip are of circular-sector shape, centered at the crack-tip, crack growth in an arbitrary direction, from the initial-crack axis, under general mi~ed mode cyclic loading, can be modeled. The presently considered cases, of crack-geometries and far-f ield applied loading, result in "small-scale" yielding conditions near the crack-tip. In these cases, a static-conden- sat ion procedure is employed, wherein, the plastic portion of the structure is isolated from the elastic; the stiffness of the former keeps changing whereas that of the latter remain fixed. This condensat ion procedure results in a consid- erable saving of computer time. In the present paper, both Modes I and I1 type cyclic loadings are considered. In the mode I case, four different types of cyclic block loading, viz., constant-amplitude, high-to-low, low-to-hieh. and sinpl~ n.rorl-=A in I~I nth~v- wlse constant anplltude block, are considered. Detailed results are presented for crack-opening Cop~righl@American Instilute of Aeronautic.) and Astronnrlics. lnc. WQ All riuhto *:*.;:-r.. ;?

Transcript of An elastic-plastic analysis of fatigue crack closure in ... · AN ELASTIC-PLASTIC ANALYSIS OF...

Page 1: An elastic-plastic analysis of fatigue crack closure in ... · AN ELASTIC-PLASTIC ANALYSIS OF FATIGUE CRACK CLOSURE IN MODES I AND 11. * ** ... constant amplitude block loading, high-to-low

AN ELASTIC-PLASTIC ANALYSIS OF FATIGUE CRACK CLOSURE IN MODES I AND 11.

* ** M. Nakagaki and S a t y a N . Acluri

School o f Engineering Sc ience and Mechanics Georgia I n s t i t u t e of Technology

At lan ta , Georgia 30332.

Abstract

I n t h i s paper , an e f f i c i e n t e l a s t i c - p l a s t i c f i n i t e element procedure t o analyze crack-closure and, i t s e f f e c t s on f a t i g u e c rack growth under g e n e r a l spectrum load ing , is presen ted . A hybrid- displacement f i n i t e element procedure is used t o p roper ly t r e a t t h e s t r e s s and s t r a i n s i n g u l a r i t i e s nea r t h e c rack- t ip ; and crack-growth under c y c l i c loading i s s imulated by the t r a n s l a t i o n of c e r t a i n "core" elements, near t h e c r a c k - t i p , i n which pro- pe r s t r e s s and s t r a i n s i n g u l a r i t i e s were embedded. Both pure Mode I and Mode I1 types of c y c l i c load- i n g a r e considered. I n t h e mode I c a s e , f o ~ r r types o f c y c l i c loading, v i z . , constant ampl i tude block load ing , high-to-low block loading, low-to-high block, and a s i n g l e over load i n an o the rwise con- s t a n t amplitude block loading a r e cons ide red ; whereas i n Mode 11, only a cons tan t amplitude block loading is considered. De ta i l ed r e s u l t s a r e presented f o r c rack-c losure and open ing-s t r e s ses , c rack-sur face deformat ion p r o f i l e s , e t c . , i n each case . Cer ta in obse rva t ions , based on t h e present numerical r e s u l t s , concerning v a r i o u s f a c t o r s t h a t cause krack growth a c c e l e r a t i o n o r r e t a r d a t i o n un- d e r genera l spectrum loading, a r e p resen ted and d i scussed .

In t roduc t ion :

A s d iscussed r e c e n t l y i n an expos i to ry a r t i c l e by Sch i jve [ 11, t h e mechanism of c rack-c losure , as f i r s t observed exper imental ly by E l b e r 11 23 , i s gen- e r a l l y considered t o be a predominant mechanism t h a t c o n t r i b u t e s t o " i n t e r a c t i o n e f f e c t s " , which cause crack-growth r e t a r d a t i o n o r a c c o l e i a t i o n , under va r i ab le -ampl i tude f a t i g u e load ing . I t i s a l s o genera l ly understood [l] t h a t t h e c rack- c l o s u r e phenomenon i s cuased by r e s i d u a l p l a s t i c deformations remaining i n t h e wake of t h e advanc- ing c rack- t ip , a s i n i t i a l l y p o s t u l a t e d by Elber . Ana ly t i ca l models t h a t lend t h e o r e t i c a l suppor t t o t h e ex i s t ence of c rack-c losure phenomenon i n f a - t i g u e crack-growth, and provide some r a t i o n a l i t y f o r t h e adoption of an e f f e c t i v e s t r e s s - i n t e n s i t y range, based on c l o s u r e e f f e c t s , f o r t h e c o r r e l a - t i o n of f a t i g u e crack-growth r a t e s , have a l s o been proposed by Budiansky end Hutchinson [3] . A s f o r a more general a n a l y s i s of extcnding c r a c k s under g e n e r a l block c y c l i c loading, t o o b t a i n crack- c l o s u r e s t r e s s e s , c r a c k opening s t r e s s e s , d e t a i l s o f crack-surface deformat ions , and r e s i d u a l s t r e s - s e s i n tkie c r a c k - t i p r eg ion , e t c . , e l a s t i c - p l a s t i c f i n i t e element a n a l y s i s were f i r s t performed by Newman an1 h i s co l l eagues [4,5]. Apart from these ana lyses , the a u t h o r s a r e aware of s i m i l a r a t tempts on ly bv Oh51 and h i s co-workers [6,7]. The s tud-

C - LC; '~ Ln L4- / J L U ~ L G L U ~ Z ~ U r i ~ d e I case u r l ~ y . AL- so , s i n c e the crack-growth was s imula ted i n [4-71 - ** K ~ e s e a r c h Engineer , P ro fessor , Member, A I A A

by s h i f t i n g a f i n i t e element node ( t h e c u r r e n t c rack t i p ) t o an immediately ad jacen t node, and s i n c e constant s t r a i n - t r i a n g l e type f i n i t e e l e - ments were used t o model the cracked s t r u c t u r e , a very f i n e f i n i t e element mesh (with t h e small- e s t element o f t e n being of the o rde r of 10-3 t imes the crack l e2g th ) is necessary i n t h e mod- e l i n g procedures of 14-71. Thus t h e f i n i t e e l e - ment computations of the type given i n [4-71 can be very expensive, e s p e c i a l l y when c y c l i c load- i n g o f a r b i t r a r y spectrum a re considered.

I n the p resen t paper, an a l t e r n a t e c o s t - e f f e c t i v e and a c c u r a t e e l ; s t i c - p l a s t i c f i n i t e element procedure t o analyze f a t i g u e crack c los - u re , and i t s e f f e c t s , under general spectrum loading, is presented. I n the p resen t procedure, t h e well-known Hutchinson-Rice-Rosengren [8 ,9] type s t r a i n and s t r e s s s i n g u l a r i t i e s , f o r s t r a i n - hardening m a t e r i a l s , a r e embedded i n s p e c i a l l y developed elements near t h e c rack- t ip . This elim- i n a t e s the need f o r a very f i n e mesh near t h e t i p . For ins tance , t h e c r a c k - t i p elements i n t h e pre- s e n t procedure a r e of t h e order of 10-I of the crack- length , a s compared o const n t s t r a i n t r i- ang les of t h e o rde r o f lo-' t o lo-' t imes t h e crack- length g e n e r a l l y used i n the procedures of [4-71 . A hybrid-displacement f i n i t e element me- thod [ 10,11] is used i n developing t h e s e s p e c i a l elements. Also i n t h e present procedure, crack- growth is simulated by: ( i ) the t r a n s l a t i o n of a c o r e of t h e forementioned spec ia l elements by an a r b i t r a r y amount i n t h e des i red d i r e c t i o n , ( i i ) r e i n t e r p o l a t i o n of r e q u i s i t e d a t a i n the new f i n i t e element mesh, and ( i i i ) p ropor t iona l re lax- a t i o n of t r a c t i o n s i n o rde r t o c r e a t e a new crack s u r f ace. Since t h e forementioned s p e c i a l elements nea r t h e c rack- t ip a r e of c i r c u l a r - s e c t o r shape, cen te red a t t h e c rack- t ip , crack growth i n an a r b i t r a r y d i r e c t i o n , from the i n i t i a l - c r a c k axis, under general m i ~ e d mode c y c l i c loading, can be modeled. The p r e s e n t l y considered c a s e s , of crack-geometries and f a r - f i e l d app l i ed loading, r e s u l t i n "small-scale" y ie ld ing cond i t ions near t h e c rack- t ip . In t h e s e cases , a s ta t ic-conden- s a t i o n procedure i s employed, wherein, t h e p l a s t i c por t ion of t h e s t r u c t u r e is i s o l a t e d from t h e e l a s t i c ; t h e s t i f f n e s s of the former keeps changing whereas t h a t of t h e l a t t e r remain f ixed. This condensat i o n procedure r e s u l t s i n a consid- e r a b l e saving of computer time.

I n t h e p resen t paper, both Modes I and I1 type c y c l i c loadings a r e considered. I n the mode I case , four d i f f e r e n t types of c y c l i c block loading, v iz . , constant-amplitude, high-to-low, low-to-hieh. and s i n p l ~ n . r o r l - = A i n I ~ I n t h ~ v -

w l s e constant a n p l l t u d e block, a r e considered. De ta i l ed r e s u l t s a r e presented f o r crack-opening

Cop~righl@American Instilute of Aeronautic.) and Astronnrlics. lnc . WQ A l l riuhto *:*.;:-r.. ;?

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9 c ~ t a a e s , L ~ d ~ A - L L V ~ ~ L c b L L e b b t 4 b , ~ l d c K - b i i K L i i C e

deformation p r o f i l e s , and e f f e c t i v e s t r e s s - i n t e n - s i t y f a c t o r ranges f o r f a t i g u e crack growth, i n each case of block loading. Also presented he re - i n a r e d e t a i l e d d i scuss ions , based on t h e ob ta ined numerical r e s u l t s , concerning var ious f a c t o r s t h a t cause crack growth a c c e l e r a t i o n o r r e t a r d a - t i o n and delay e f f e c t s under d i f f e r e n t types of c y c l i c loading. F i n a l l y , t h e r e s u l t s of an i n - v e s t i g a t i o n of a center-cracked panel under ex- t e r n a l pure shear (Mode 11) c y c l i c loading, of conscz ~ ~ p l i t u d e , a r c presented.

Br ief Discussion o f Mathematical D e t a i l s of Analysis Procedure:

Ln t h e present paper an incremental , updated Lagrangean f i n i t e element formulation, f o r f i n i t e - deformation e l a s t o - p l a s t i c i t y , is used. The flow theory o f p l a s t i c i t y t h a t is used i s c h a r a c t e r i z e d by t h e w e l l known ~ u b e r - ~ i s e s - H e n c k y i n i t i a l y i e l d c r i t e r i o n , and a Prager-Ziegler type kinemat ic hardening r u l e . I n the present work, a s men- t ioned e a r l i e r , c i r c u l a r - s e c t o r shaped s i n g u l a r i t y elements ( i n which, a displacement f i e l d t h a t cor- responds t o the forementioned Hutchinson-Rice Rosengren s i n g u l a r f t i e s i n s t r a i n s and s t r e s s e s , fo r s t ra in-hardening e l a s t o - p l a s t i c m a t e r i a l s , a r e embedded) a re used near t h e c rack- t ip , a s shown i n Fig. 1. The above s i n g u l a r elements a r e surrounded by "regular" eight-noded i soparamet r i c q u a d r i l a t e r a l elements, a s a l s o shown i n Fig. 1. Compatibi l i t y of d isplacements , ane r e c i p r o c i t y of t r a c t i o n s , between t h e s e "s ingular" and "regu- l a r " elements a r e enforced through a Lagrangean M u l t i p l i e r technique ( t h u s leading to a ' h y b r i d ' element formulation), a s shown below.

Let y b e - t h e c u r r e n t ( say , i n a s t a t e C ) i Car tes ian S p a t i a l coord ina tes of a p a r t i c l e , l t o

be used a s a reference system f o r the c u r r e n t i n - crement, i e . fromC t o C

N w 1 . . Let sYj be t h e t r u e

(Cauchy) s t r e s s in C Let S i j pa'

(- s ~ ' - T~ ) be the r a t e of Second Piola-Kirch- i j W i j hoff (P-K) s t r e s s r e f e r r e d to CN. I t is noted

t h a t sty1 is the Second P-K S t r e s s i n S t a t e t N)

Cwl a s r e fe r red t o CN. Let Gi r ep resen t t h e r a t e N of deformation from CN, and l e t u - 3bi/aYj.

i , j Also, l e t elements rn = 1, 2 , . . p be the s e c t o r - shaped s i n g u l a r i t y elements, and m = ptl .. M be the surrounding regu la r e lements . I t can be shown t h a t the v a r i a t i o n a l p r i n c i p l e governing: ( i ) t h e cond i t ions of equi l ibr ium i n each element ( s i n g u l a r a s well ' a s r e g u l a r ) , ( i i ) t r a c t i o n boundarv condi- t i o n s f o r each element, where such e x i s t , and ( i i i ) condi t ions of c o m p a t i b i l i t y o f d i s p l a c e - ments and rec ip roc i ty of t r a c t i o n s between s ingu- l a r and regu la r elements, can be s t a t e d a s t h e s t a t i o n a r y condrtion of t h e f u n c t i c n a l :

REPRODUCIBILITY OF THE ORIGINAL PAGE IS POOR - r nc? L - .,* .*I r "[email protected]

In the above, F is the f u n c t i o n a l f o r the group of s i n g u l a r 8

e h m e n t a , whi l e F is t h a t f o r t h oup of reg- u l a r elements. ~ h & , i n t h e a i n p f s r :few. t one assumes: ( i) an a r b i t r a r y displacement f i e l d ui wi th in t h e element nm t o inc lude the proper s t r a i n / stress s f n g u l a r i t i e s as w e l l a s non-singular poly- nomial v a r i a t i o n s ; ( i i ) an independent displacement f i e l d vi a t t h e s ingular-e lement boundary, anm, and (iii) Lagrange M u l t i p l i e r TLi a t the boundary t o en fo rce t h e c o n s t r a i n t ; =+ a t the boundaries of s i n g u l a r elements. Whoriasf i n t h e r egu la r elements, developed through t h e usua l compat ible-displace- rnent method, one assumes a s i n g l e compatible d i s - placement f i e l d Gi. Thus, i t is seen t h a t i f vi a t the boundary o f s i ~ g l u a r element is s o chosen t h a t i t matches wi th vi o f r e g u l a r elements a t t h e i n t e r f a c e s of s i n g u l a r and r e g u l a r elements, t o t a l displacement compa t ib i l i ty is enforced throughout t h e s t r u c t u r e .

~ l s o , i n E q . ( I ) , W i s tbe r a t e p o t e n t i a l f o r t h e r a t e - o f second P-K s t r e s s Sij, such t h a t ,

a W / a i where 6 i s t h e updated Lagrangean i j 7 i j .

s t r a i n r a t e , Ei j=(f) (ui , + u ). This r a t e po- j, i

t e n t i a l h a s been conabwantl:' de r ived from a pos tu- l a t e d r a t e p o t e n t i a l f o r t h e c o r o t a t i o n a l (Jaumanx$ r a t e of Kirchhoff s t r e s s , us ing t h e well-known c l a s s i c a l r a t e t h e o r i e s o f ( f i n i t e deformation), ra te- independent e l a s t o - p l a s t i c i t y , as d iscussed i n d e t a i l e lsewhere [12, 131. These d e t a i l s a r e omitted h e r e due t o space reasons .

The d e t a i l s of assumed v a r i a b l e s hi, , and TLi for t h e s i n g u l a r elements and a r e ornltted he re . However, f o r t h e present purposes , i t is worth no t - ing t h a t each sector-shaped s i n g u l a r element has 3 nodes a long t h e c i rcumference, and 4 nodes along each of the r a d i a l boundary l i n e s . Along t h e c i r - cumferen t i a l boundary, t h e f i e l d v . i s assumed i n the form,

where 9 i s t h e c i r c u m f e r e n t i a l angle ; and %i..a3i

a r e expressed i n terms of t h e r e s p e c t i v e d i sp lace - ments a t t h e 3 nodes a long t h e circumference. I t $an b e seen t h a t the above boundary displacement, v i s compatible with t h a t of t h e surrounding 8 i '

noded i soparamet r i c q u a d r i l a t e r 3 l s . Along-the r a - d i a l boundaries of the s i n g u l a r elements, vi i s assumed a s :

dl.-re r i s t h e r a d i a l d i s t a n c e from t h e c r a c k - t i p ; n is t h e exponent i n t h e m a t e r i a l hardening law (i e , , from t h e un iax ia l s t r e s s - s t r a i n curve, 6 - 0 " ; BP t h e 2 i a s t i c - s t r a i n , and a the s t r e s s ) ; aRd t h e c ~ c f f i c i e n t s bli . . . bli a r e expressed i n terms of tire r e s p e c t i v e nodal d ~ s p l a c e m e n t s a t t h e forementioned 4 nodes.

n.. - - - . . , . . . . L : I S L 1y i t IS I ! u , - - L L : S L -..- . -L - - - - . S L

p r i n c i p l e i n Eq. (1) assumes t h a t the c u r r e n t s t a t e CN is e x a c t l y e q u i l i b r a t e d . I n genera l , such i s not t h e case ; and hence, equi l ibr ium

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corr-ec.t i , t i L L P : 3 t 10118 t l f ' I L ~ ,.rewton- k a p t ~ s o ~ b type have t o be performed a t the end of e a c h increment- a l s o l u t i o n . The d e t a i l s of these i t e r a t i o n s have a l r e a d y been p r e s e n t e d e lsewhere [ 11,121.

F i n i t e Element Model l ing o f Crack Growth

The s t e p s i-I t he f i n i t e e lement s i m u l a t i o n of c r a c k growth i n t h e p r e s e n t procedure may be de- s c r i b e d a s : ( i ) g e o m e t r i c a l change i n t h e c r ack s u r f a c e boundary; ( i i ) t r a n s l a t i o n o f t h e crack- t i p s i n g u l a r i t i e s t o t h e advanced crack- t i p ; and ( i i i ) r e l e a s e o f s u r f a c e t r a c t i o n s on t h e newly c r e a t e d c r a c k s u r f a c e .

The change i n t h e c r a c k s u r f a c e boundary is made by t r a n s l a t i h g t h e whole s e t o f c r a c k - t i p c o r e e l emen t s , a s shown i n F i g . 1 , by a r b i t r a r y d i s l ~ n c e ba i n the d i r e c t i o n o f in tended c r a c k e x t e n s i o n ; t h u s t h e new c rack - t i p node whlch i s d e s i g n a t e d by t h e c e n t e r of t h e s ec to r - shaped c o r e e l emen t s need n o t be c o i n c i d e n t w i t h any p r e v i o u s l y e x i s t i n g f i r n i t e element node b e f o r e e x t e n s i o n . Thus even though the f i x e d boundary i n t he u n c r a c ked 1 igament o f t h e s t r u c t u r e is changed, t h e cons t r a i n i n g condi- t i o n o f t he nodes need not be a l t e r e d . Elements immediately a d j a c e n t t o t he c o r e nius t be r e a d j u s t e d t o f i t t o t h e t r a n s l a t e d c o r e . This p roces s of t r a n s l a t i n g the c o r e mesh a l s o moves t h e embedded s i n g u l a r i t y i n the e lements t o t h e new c r a c k t i p a r e a , l e av ing no s i n g u l a r i t i e s b u t l a r g e deforma- t i o n s and s t r a i n s i n t h e wake o f advanced crack- t i p .

A l l t h e 5x5 Gauss ian d a t a p o i n t s i n e a c h o f t he t r a n s l a t e d c o r e e l emen t s (and a l s o t h e 5x5 p o i n t s f o r t h e convent iona 1 e l e m e n t s ) may g e n e r a l l y n o t co in- c i d e w i t h t hose b e f o r e t r a n s l a t i o n , a t which p l a s t i c h i s t o r y d a t a such a s c u r r e n t s t r e s s e s , p l a s t i c s t r a i n s , p l a s t i c a l l y d i s s i p a t e d work, y i e l d s u r f a c e t r a n s l a t i o n , e t c . , a r e a v a i l a b l e . T h e r e f o r e t he d a t a a t p o i n t s i n t h e new mesh a r e e s t i m a t e d by l i n e a r l y i n t e r p o l a t i n g d a t a on f o u r Gaussi-an p o i n t s i n t h e o ld mesh t h a t a r e n e a r e s t t o t h e p o i n t under q u e s t i o n i n t he new mesh. The s imple bu t cumbersome mathemat ica l d e t a i l s of t h i s i n t e r p o l a t i o n and smoothing p roces s a r e omi t t ed he;e f o r t he sake of b r e v i t y . With t h e f i t t e d p l a s t i c d a t a and the new e l emen t geometry, e l emen t s t i f f n e s s m a t r i c e s a r e r e c a l c u l a t e d f o r t h e c o r e e lements a s w e l l a s f o r t h e su r round ing r e a r r a n g e d e lements and t h e globa 1 s t i f f n e s s i s a p p r o p r i a t e l y mod i f i ed . Subsequent e q u i l i b r i u m check i n e r a t i o n s u s i n g t h e new s t i f f n e s s o f t h e s t r u c t u r e c o r r e c t f i t t i n g e r r o r s , i f ar , i n t h e p l a s t i c i t y d a t a i n t he new mesh. A t t he sdme t ime , t he t r a c t i o n s ove r t h e d i s t a n c e AB (ha s s shown i n F i g . 1) a r e i n c r e m e n t a l l y removed, w i t h e q u i l i b r i u m check i t e r a t i o n s be ing used a t each s t e p , t o c r e a t e a new t r a c t i o n - f r e e c r a c k s u r f a c e of

l e n g t h Aa. Tht f i n i t e element s i m u l a t i o n o f c r a c k e x t e n s i o n by t h e d e s i r e d amount, La , is now complet- e d .

Analys is of F ? t i ~ u e Crsck Growth Under Mode I Cycl ic Loadin2 D e s c r i p t i o n of t he Problem

Throughout t h e s e r i e s of t h e p r e s e n t e l a s t i c - p l a s t i c f i n i t e e lement ana lyses of f , t i g u e crack growth under Mode I t ype c y c l i c l o a d i n g , a t h i n r e c t a n g u l a r p l a t e w i t h a c e n t r a l c r a c k and under uni form t e n s i l e s t r e s s e s , i n a d i r e c t i o n normal t o t h e c r a c k - a x i s . a t t h e edpp? of t h e p l a t e , i s con- s i a ~ r e a (See k ~ g . 2 ) . lnca ulmenslons of the p l a t e a r e : h a l f width w = 230mm; and ' l a l f - c r a c k l eng th a = 27.3mm, r e s p e c t i v e l y . The m a t e r i a l is con- syde red t o be a 2024-T3 Aluminum a l l o y , whose

~ ~ r r ~ ~ ~ ~ ~ ~ l l c a l proper t l e a q r e c l rarac ter l , ra oi l : y l e l a stress,^ = 350MNImL; and young's modulus, E = 70,000 MN%'. The m a t e r i a l i s assumed t o be e l a s - t i c - p e r f e c t l y - p l a s t i c . I t is noted t h a t t h e above problem d e f i n i t i o n i s i d e n t i c a l t o t h a t used by Newman [ 5 ] . The p l a t e i s assumed t o be i n a s t a t e of p lane s t r e s s .

Because of t he s y m e r r i e s of geometry, a p r l i e d l o a d i n g , and m a t e r i a l homogeneity, only a q u a r t e r of t he cracked p l a t e i s ana lysed . F i g . 2 shows the f i n i t e elemen& breakdowr t h a t is used. A t o t a l of 6 s ec to r - shaped s i n g u l a r i t y elements nea r t h e c r a c k - t i p , and 43 convent i o n a l quad^ tt it isoparame t r i c e lements a r e employed. Some of t hese isoparame t r i c e l emen t s a r e 6 noded t r i a n g l e s , whi le t h e m a j o r i t y a r e 8 noded q u a d r i l a t e r a l s (See Fig . 2 ) . I t i s s e e n t h a t t he t o t a l number of nodes i n t h e f i n i t e element mesh f o r t h e q u a r t e r - p l a t e is 171, w i th a t o t a l number of deg rees of fniedom of 311.

The r ad ius of t he sec tor -shaped s i n g u l a r i t y e l emen t s is chosen a s D = 2.;tmn; i e . , p / a = -103. While t he c r ack -ex tens ion per cyc l e 9f l o s d i n g , Aa, can be a r b i t r a r y ( i e . , no t r e l a t e d t o t he f i n i t e element mesh s i z e ) i n t h e present a n a l y s i s proced- u r e , i t i s chosen t o be h a = 0 . 1 4 m i n t he present s e r i e s of computa t ions .

Techniques t o Minimize Computational( CPU) Time

F i r s t l y , we n o t e t h a t the n e a r - t i p e lements used p r e s e n t l y a r e of t h e order 10 - I t imes the serni- c r a c k l eng th ; and the t o t a l number r f a l g e b r a i c equa- t i o n s f o r the above problem a r e only 311.

I n the p re sen t procedure , a tangent modulus ( s t i f f n e s s ) approach is used i n each increment and i n each i t e r a t i o n i n t h e r e spec t ive increment . Thus a f a s t e r convergence is obtained i n t h e pro- c e s s of i t e r a t i o n s of e q u i l i b r i u m c o r r e c t i o n , e t c . However, i t is noted t h a t cn ly the s t i f f n e s s m a - t r i c e s of the p l a s t i c po r t i on of t he s t r u c t u r e need t o be changed i n t h e p re sen t tangent modulus ap- p roach , whereas, t hose o f the e l a s t i c po r t ion r e - main f i xed .

For the p r e s e n t l y cons idered l e v e l s of app l i ed f a r - f i e l d tens i on on t h e specimen, on ly "small- s c a l e " y i e l d i n g c o n d i t i o n s p r e v a i l n e a r t h e c r ack - t i p . A t y p i c a l p l a s t i c - z o n e s i z e n e a r t h e c r ack - t i p f o r t he p r e s e n t l y cons idered c l a s s of Mode I probiems i s shown i n F ig . 3 , being superimposed on t h e f i n i t e element mesh. Tt i s seen t h a t t he p l a s - t i c p o r t ion of t h e s t r u c t u r e (des ignated "Region P") is cons ide rab ly s m a l l e r than the e l a s t i c p o r t i o n (des igna t ed "Region F"). Thus only t h e s t i f f n e s s m a t r i x of r eg ion P need t o be changed i n each load- s t e p and each i t e r a t i o n i n the p re sen t procedure . '

However, t he t o t a l number of t imes , s ay N , t h a t t h e combined s t i f f n e s s ma t r ix ( fo r Regions P + E ) must be i nve r t ed i n t h e cou r se of a n a l y z i n g a typ- i c a l problem, s ay t h e c a s e a growing cr .?ck under c o g s t a n t smpl i tude c y c l i c loading , is h = [ (no . of

*Even though t h e g e n e r a l formula t ion p re sen ted i n E q . (1) is f o r f i n i t e deformat ions , t he f i n i t e de- fo rma t ion e f f e c t s i n t he p re sen t c l a s s of problems ..-- - -- I.,. "-,.. c . L - .."-A?....;".,"t ,-,!.I.. 4" the . . i r : - . : c . -

of t h e c r a c k - t i p . Thus, changes t o t he s t i f f n e s s of r e g i o n E due t o f i n i t e deformation e f f e c t s , if any, a r e ignored h e n c e f o r t h .

Page 4: An elastic-plastic analysis of fatigue crack closure in ... · AN ELASTIC-PLASTIC ANALYSIS OF FATIGUE CRACK CLOSURE IN MODES I AND 11. * ** ... constant amplitude block loading, high-to-low

i c e r a t i o n s / e v c l e ) X (number o f load i n c r e m e n t s p e r , y c i e ) XCno. o r load cycles)]. For a t y p i c a l p r o - blem, s ay F! c y c l e s of c o n s t a n t a m p l i t u d e l o a d i n g , a t y p i c a l v a l u e f o r N c a n be N = 4 x 28 x 8 = 896, ( i t . , 4 i t e r s t i o n s p e r e ; c l e , w i t h 28 l o a d i n c r e - m e n t s / c y c l e , e t c . ) . Th - is - a r a t h e r enormous a - mount of computing; and h e n c e a more e c o n o m i c a l way of s o l v i n g the s t i f f n e s s e q u a t i o n s i s manda tory .

S i n c e t h e p l a s t i c - z o n e i s a s m a l l - s i z e , by a n a p p r o p r i a t e node numbering scheme, t h e s t i f f n e s s m a t r i x o f t h e p l a n t i c zone can be a r r a n g e d , a s i n F i g . 3 . b , s o t h a t i t is a small s u b - s e t of t h e g l o - b a l s t i f f n e s s m a t r i x of t h e s t r u c t u r e (even though t h e p l a s t LC zone s i z e keeps changing w i t h l o a d , a n a p p r o x i m a t e p r e l i m i n a r y a n a l y s i s can be made t o d e t e r m i n e i t s s i z e a t maximum l o a d ) . Then we use a s t a t i c c o n d e n s a t ion p r o c e d u r e t o f i r s t e l i m i n a t e t h e e q u a t i ~ - n s c o r r e s p o n d i n g t o t h e nodes i n t h e e - l a s t i c p o r t i o n , i n t h e v e r y f i r s t l o a d - i n c r e m e n t . T h e r e a f t e r on ly t h e e q u a t i o n s f o r t h e nodes i n Re- g i o n P need t o be o p e r a t e d upon , i n a l l s u b s e q u e n t load s r e p s and i n t e r a t i o n s . Thus, i n t h e example c i t e d above, (N = 8 9 6 ) , i n a l l b u t one o f t h e 896 s o l u t i o n s , t h e number o f e q u a t i o n s b e i n g s o l v e d is r a t h e r v e r y s m a l l , and c o r r e s p o n d t o t h e t o t a l number o f u n c o n s t r a i n e d n o d a l d i s p l a c e m e n t s i n Re- g i o n p. T h i d e n a b l e s t h e p r e s e n t c o m p u t a t i o n s t o be e c o n o m i c a l l y f e a s i b l e .

A l s o , the computer program i s s o a r r a n g e d t h a t t h e d a t a o b t a i n e d from c o m p u t a t i o n u p t o t h e end of a g i v e n s p e c t r u m o f l o a d i n g can b e used as i n - p u t d a t a a t t h e b e g i n n i n g of a d i f f e r e n t s p e c - trum o f l o a d i n g . For i n s t a n c e , a t t h e end o f a c o n s t a n t Hi -ampl i tude c y c l i c l o a d i n g , t h e d a t a is s t o r e d on a d i r e c t a c c e s s permanent d i s c f i l e and used a s i n i t i a l c o n d i t i o n s f o r a low a m p l i t u d e cy - c l i c l o a d spec t rum; i n t h i s p r o c e s s n o t o n l y t h e c a s e o f Hi -ampl i tude l o a d i n g b u t a l s o t h e c a s e of H i to -Lo b l o c k l ~ a d i n g is s o l v e d .

p r e s e n t c y c l i c l o a d i n g c a s e , a t t t ,e i n e t a n ~ t h e d is p lacement ( i n t h e d Erect i o n of a p p l i e d t e n s i o n ) a t one o r more nodes o n t h e c r a c k s u r f a c e becomes n e g a t i v e , f u r t h e r u n l o a d i n g bebopped. The computa- t i o n a l p r o c e d u r e is t h e n s w i t c h e d t o a d i s p l a c e - ment c o n t r o l t y p e , and t h e above n e g a t i v e d i s p l a c e - ments a r e p r e c i s e l y e n f o r c e d t o be z e r o ; t h u s f i n d - Eng t h e p r e c i s e s t r e s s Leve l a t which t h e c l o s u r e c o n s t r a i n t on L h e r e e p e c t i v e node must be c n f o r e a d .

A f t e r t h e c r a c k - c l o s u r e i s d e t e c t e d , t h e res- p e c t i v e n o d e ( s 1 a r e c o n s t r a i n e d t h e r e a f t e r , u n t i l t h e r e s t r a i n i n g f o r c e ( e ) a t t h e n o d e ( s ) jus t be- comes zero and begins t o b e t e n s i l e i n n a t u r e . The c o r r e s p o n d i n g a p p l i e d s tress leve 1 d e f inee t h e c r a c k - o p e n i n g streas. The d e t a i l s o f t h e above p r o c e s s a re d i s c u s s e d and g r a p h i c a l l y i l l u s t r a t - e d e l s e w h e r e [ 12,143.

F i n a l l y , some comments on t h e p r e s e n t l y ob- s e r v e d p a t t e r n s c f c r a c k - c l o s u r e a r e g i v e n , be- f o r e p r o c e e d i n g t o a d i s c u s s i o n of s p e c i f i c c a s e s . In g e n e r a l , c l o s u r e was n o t i c e d t o o c c u r a t t h e node c l o s e s t t o t h e c u r r e n t c r a c k - t i p , as i n d i c a - t ed i n t t , e sequence of u n l o a d i n g s t e p s i n F i g s . 4 a - c . However, if t h e c u r r e n t c r a c k - s u r f a c e p r o - f i l e i s i r r e g u l a r , a s ~n t h e c a s e of Hi-to-Low b l o c k l o a d i n g t o be d i s c u s s e d l a t e r , c r a c k - c l o s u r e rnqy f i r s t occur a t t h e node c l o s e s t t o t h e c r a c k - t i p ; however , i n t h e s u b s e q u e n t u n l o a d i n g s t e p , c l o s u r e may o c c u r a t a node fa r - removed from t h e c r a c k - t i p , a s i n d i c a t e d i n F i g . 4 d . From t h e r e - s u l t s t o be d i s c u s s e d l a t e r , t h i s p a t t e r n o f c r a c k - c l o s u r e a p p e a r s t o c o n t r i b u t e s i g n i f i c a n t l y t o g rowth r e t a r d a t i o n and d e l a y e f f e c t s .

C r i t e r i o n For Crack-Extens ion S t r e s s L e v e l

Moni tor inn of Crack-Closure and Opening i n t h e F i n i t e Element Mqdel

Let us assume, t h a t a t a g i v e n i n s t a n t o f t i m e ( a t a g i v e n p o i n t i n t h e l o a d i n g h i s t o r y ) , t h e l o - c a t i o n of t h e c r a c k - t i p , the l o c a t i o n o f t h e 4 nodes on t h e r a d i a l l i n e (which c o i n c i d e s w i t h t h e c r ack s u r f a c e ) of the s i n g u l a r s e c t o r e l e m e n t , t h e l o c a t i o n s o f a l l o t h e r nodes on t + e c r a c k a x i s , as w e l l t h e c u r r e n t (deformed) p r o f i l e o f t h e c r a c k s u r f a c e , a r e a l l known. We d e n o t e t h e c u r r e n t no& on t h e c r a c k s u r f a c e as "upda ted Lagrangean Nodes". Let u s now assume t h a t t h e c r a c k t i p i s now f u r t h e r e x t e n d e d by ,an a r b i t r a r y a m m n t (Aa) and t h e s t r u c - t u r e is t h e n s u b j e c t e d t o f u r t h e r l o a d i n g . We

- f i r s t n o t e t h a t , i n t h e p r e s e n t deve lopment , the b o u n d a r ) ~ d i s p l a c e m e n t ( i n t h e d i r e c t i o n of t h e ap- p l i e d normal s t r e s s ) a l o n g a r a d i a l l i n e o f a " s i f i g u l a r " s e c t o r e l e m e n t f s of the form g i v e n i n E q . ( 3 ) . Using t h e above e q u a t i o n , and knowing, a ~ r i - or i . the r a d i a l c o o r d i n a ' e s , ( i e . , r , a s measi : * i

from the c u r r e n t c r a c k - t i p ) , of t h e r e s p e c t i v e nodes on t h e r a d i a l l i n e of the s e c t o r e lement i n i t s immedia te ly p rev ious l o c a t i o n , one can compute t h e v a l u e s o f Av a t t h e above mentioned "Updated Lag - r a n g e a n ~ o d e g . " By a d d i n g ( o r s u b s t r a c t i n g , a s t h e c a s e mav b e , ) t h e s e i n c r e m e n t a l d i s p l a c e m e n t s t o t h e rev i n l ~ s l v known - 7 - l x r c . ZT. ., - - ., . ., , . 1 - - . . D L :he c u r r e n t c r a c k s u r f a c e d e f o r m a t i o n p r o f i l e i s made.

In t h e p r e s e n t work , a s t u d y is made t o a r r i v e a t a c r i t e r i o n f o r t h e stress l e v e l , g , a t which f a t i g u e c r a c k growth o c c u r s . In p r i o r e f i t e r a t u r e , t h i s c r a c k - e x t e n s i o n s t r e s s l e v e l was chosen a r b i - t r a r i l y . For i n s t a n c e , i n [ 5 j t h e c r a c k is e x - tended a t t h e maximum a p p l i e d s t r e s s i n e a c h c v c l e even i n a g e n e r a l s p e c t r u m ( f o r i n s t a n c e , h i g h - t o - l o w , l o w - t o - h i g h , e t c . , ) l o a d i n g , where as i n [ 7 ] t h e c r a c k was e x t e n d e d a t t h e a p p l i e d s t r e s s l e v e l a t which the r e s t r a i n i n g n o d a l f o r c e a t t h e new c r a c k - t i p becomes z e r o . In t h e p r e s e n t s t u d y , for i n s t a n c e i n a c o n s t a n t - a m p l i t u d e ( z e r o - t o - t e n - s i o n ) c y c l i c l o a d i n g , i t was found t h a t t h e c r a c k opening and c l o s u r e s t r e s s e s , oo and o r e s p e c t - i v e l y , were s e n s i t i v e t o t h e choBen o F' I n t h e p r e s e n t work , a c r i t e r i o n , oex = 0 qX p ( o

O P max-"od where p is a c o n s t a n t o f p r o p o r t i o n a l i t y , is p o s t - u l a t e d ; and p is o b t a i n e d by c a l i b r a t i o n s u c h t h a t t h e c a l c u l a t e d a c o r r e l a t e d w i t h t ha t a b s e r v e d i n e x p e r i m e n t a l ' 'studies s u c h a s i n [ 2 2 . However, i t i s n o t e d t h a t t h i s c o n s t a n t o f p r o p o r t i o n a l i t y p m a y , t o some e x t e n t , be dependent on t h e numer i - c a l m e t h o d o l o g y employed i n f a t i g u e c r a c k model ing i t s e l f . T h u s t h e above d e s c r i b e d c a l i b r a t i o n may be c o n s i d e r e d a s v a l i d ol l ly i n t h e c o n t e x t o f t h e p a r t i c u l a r methodology employed i n t h e p r e s e n t work. T h r e e d i f f e r e n t t e s t c a s e s , each w i t h a d i f f e r e n t magni tude of c o n s t a n t a m p l i t u d e ( z e r o t o t e n s i o n ) c y c l i c l o a d i n g , were s t u d i e d w i t h . . - - . . , , . . 9 . . . \ h i c. -, . L L V ~ ~ L I 1 Cdl-16 ( kIYe LOX

t h c above mentioned c o n s t a n t o f p ropor t i o n a l i t y , p. The i d e a was t o s e l e c t a I p ' t h a t y i e l d s r e - a u l t s , i n e a c h c a s e , f o r (oo /arna ) t h a t a r e i n b e s t agreement v i t h t h e expe!imenfal r e s u l t s [21

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for 2024-T3 Aluminum a l l o y , which is t h e m a t e r i a l s imu la t ed i n a n a l y s i s . The r e s u l t s , f o r ins .ance . f o r t h e c a s e (omax/u .40) and (I? = umin/umaxrOj

Y S a r e summarized ae f o 1 lows :

S i m i l a r r e s u l t s were o b t a i n e d W f o r t h e c a s e s , (a /a ) = .229 and

(urnax /o = .314. From t h e s e ?%$eeY~ets of res- i t wag observed t h a t ~ 4 . 6 2 v i e l d a r e s u l t s f o r (3 /a 1 t h a t a r e i n b e e t agreement w i t h e x p r i - meRtaTa&servat ions , which i n d i c a t e t h a t [o /bma , a t s t e a d y s t a t e i s about 0.56. I t is hypotf$size3 t h a t t he above c o n s t a n t ~ 1 0 . 6 2 may be used through- o u t t h e r e s t of t h e a n a l y s i s , i e . , f o r c a s e s of g e n e r a l spec t rum l o a d i n g . We a l s o n o t e t h a t when t h e load l e v e l o d u r i n g any c y c l e is f i r s t d e t e r - mined, t he nurnbegQ(and s i z e ) of load s t e p s between t h i s a a n d o , i n t h e r e e p e c t i v e c y c l e , is s o adjus tgf l t h a t TI% pre-chosen l e v e l of o [ = a +

- ) ] c o i n c i d e s w i th one of thgxload-oP ! i k % ? n t : o ~ n t h e c y c l e . We now d i s c u s s t h e r e - s u l t s of a n a l y s i s o f e a c h of t he l o a d i n g c a s e s .

p

1 . 0

.85

,62

.4q

Constant Amplitude Zero- to-Tens ion C y c l i c Loading

( i ) The r e s u l t s f o r a f o r t h e c a s e of (omax/BS) = 0 . 4 and R = ) = 0 , a r e shown

('m i n lamax

Level led - o f f

115 MP

94 I I

79 I I

58 11

i n F ig . 5 , f o r 8 c y c l e s of l oad ing . I t is observed t h a t o r eaches a " s t eady s t a t e " v a l u e of 0.56~ a f t e r O P t he 4 t h o r 5 t h c y c l e . I t is a l s o notedmaX t h a t t h i s v a l u e f o r o /amax(=O. 56) is i n reasonable acco rd w i t h experimenP%l r e s u l t s [2] f o r t he same m a t e r i a l , a 2024 T3 Aluminum a l l o y . Knowing 0 i n each c y c l e , we d e f i n e t h e e f f e c t i v e s t r e s s - i8Pens i t f a c t o r a s :

bKeff = C1 &(ao + NAa) (amax- uop ) where C 1

i s t h e f i n i t e s i z e c o r r e c t i o n f a c t o r f o r t he p re - s e n t c r a c k geometry (which was found t o be C - = 1 1.017 from a f i n i t e element l i n e a r a n a l y s i s of t he c r a c k wi th a = a ; and t h e r e a f t e r assumed t o he c o n s t a n t ) ; N is ehe number of c y c l e s and Aa is the c r a c k growth pe r c y c l e . I t is thus s e e n t h a t bKeff l e v e l s o d i a f t e r a few c y c l e s t o a s t e a d y s t a t e v a l u e .

[o /omax] a t s t e a d y s t a t e OP

. 8 2

. 6 7

.56

.4 1

( i i ) F i g . 6 shows the c r ack s u r f a c e deforma- t i on p r o f i l e s , f o r i n s t a n c e , a t v a r i o u s s t a g e s of unloading d u r i n g t h e 8 t h cyc l e of t ! ~ e p r e s e n t con- s t a n t - a m p l i t u d e ( R = o ) c y c l i c l oad ing . The l a r g e b l u n t i n g a t t he i n i t i a l c r a c k - t i p l o c a t i o n ( a = ao) is observed t o remain permanently. The s u r f a c e

. of t h e extended c r a c k is observed t o be f a i r l y smooth. During t h e 8 t h c y c l e , i t is observed t h a t p r e c i s e c r a c k - c l o s u r e occu r s on ly o v e r t h e a r e a h a ( i e . , on ly a t t h e previnlts I n r a f i n n - F +-hn - - c l - -

L A P neae ) even upon f u i i u n i o a d ~ n g ; however, 1 t . is a l s o noted t h a t a t t h i s s t a g e , t h e o p e n i n ) of t he

c l o s e d node is v e r y s m a l l (See F P ~ 6 ) . 4 c r a c k f a c e s between t h e p o i n t s a and t h . p r e c i s e l y

Low-'To-ti inh Block Load i n q

A two l e v e l b lock load ing , from low t o h igh , w i t h omex i n t he h i g h e r l e v e l being 1.273 t imes the

i n t h e lower l e v e l , is cons ide red . The z%fmum s t r e s s i n t h e lower l e v e l is taken such t h a (O ( u s ) = -314. A s m e n t i o n e d e a r l i e r , t te d a ~ P a p Y h e en8 of 4 c y c l e s of low l e v e l b lock load- i ng (See Fig . 7) is recovered from a cons tant-ampli- t ude t e s t c a s e , w i t h t h e corresponding stress l e v e l . The fo l lowing r e s u l t s were obta ined:

( i) The v a r i a t i o n of crack-opening s t r e s s , a , a s the c y c l i c l oad ing p rog res se s , is shown i n F@. ( 7 ) . It can be s e e n t h a t immediately a f t e r t h e s t e p up i n t h e l e v e l of appl ied stress, a d e c r e a s e s by about 33% of i t s s t eady s t a t e vaP8e co r r e spond ing t o t h e lower Level of b lock loading . Subsequent t o t h i s , 0, inc reases mono t o n i c a ? l y t o a s t e a d y s t a t e v a l u e cgrresponding t o t h e h ighe r l e v e l of block load ing , wi th :n about 5 c y c l e s . P r i o r t o t h i s s t a b i l i z a t i o n , ~ K ~ f d d e f i n e d a s be- f o r e ) i n t he h i g h e r l e v e l of block load ing remains c o n s i d e r a b l y h i g h e r t han t h e s teady s t a t e va lue co r r e spond ing to t h i s s t r e s s l e v e l ; t hus i n d i c a t i n g growth a c c e l e r a t i o n fo l lowing the load s t e p - u p .

( i i ) The r ep re sen ' a t i ve crack s u r f a c e p r ~ f i l e s f o r i n s t a n c e , a t v a r i m s s t a g e s of unloading a t

t h e end of t he 8 t h c y c l e (high s t r e s s ) of t h e cu r - r e n t low- to-high l e v e l b lock loading , a r e shown i n F i g . ( 8 ) . From t h i s F igu re , i t can be s een t h a t t h e s t e p - u p i n t he l e v e l of loading cause8 a. b lun t - i n g of t he c r a c k - t i p (ie., a t the l o c a t i on x / a = 1.02 i n Fig. 8 , when t h e s t ep -up i n l oad ing occurs i n t he p re sen t f i n i t e element s i m u l a t i o n ) . Even d u r i n g the unloading a t the end o f t he p re sen t two l e v e l b lock load ing , a s seen from Fig . (8), t he c r a c k - c l o s u r e occu r s only over the a r e a La ( i e . , on ly a t the previous l o c a t i o n of t he c r a c k - t i p node ) .

Hi~h-To-Low Block Loading

Af t e r 8 consecu t ive cyc l e s of a h igh l e v e l b lock loading ( t h e d a t a a t which poin t is recov- e r e d from the cvrrespond ing cons tant ampl i tude t e s t c a s e ) , t he a is reduced by 21.4% and 8 more cy- c l e s of thismFgduced l e v e 1 block load ing were con- s i d e r e d . The magnitude of t h e app l i ed s t r e s s i n t h e h i g h - l e v e l b lock was such tha t (urnax) -

' b - o = 0.40 . The fo l lowing r e s u l t s were o b k ~ l n e d .

y s

( i ) The v a r i a t i o n of t he crack-opening s t r e s s , a s t he loading p r o g r e s s e s , is shown i n Fig . ( 9 ) .

gpIt is seen t h a t i nmed ia t e ly a f t e r t h e step-down i n t he load l e v e l , no ab rup t dec rease iw as was t h e c a s e i n Low- to-High loading , occu r s igp;he p r e s e n t High- to-Low b lock loading c a s e . A f t e r t he l o a d - l e v e l s t e p d o w n , t a y e d a t a b o u t 0 . 7 0 ( o m a x ) b r Y witF.in t he number o cyclar of low- l e v e l Load cons idered p r e s e n t i y , I t may be p o s s i b l e t ha t , as f u r t h e r number of low-level load c y c l e s a r e con- s i d e r e d andthe c r a c k - t i p grows f u r t h e r and even- t u a l l y s u r p a s s e s t h e p l a s t i c zone c r e a t e d d u r i n g t h e h i g h l e v e l b lock l o a d i n g , t he o d e c r e a s e s t o a b a s e l i n e va lue co r r e spond ing tgP the lower l e v e l b lock loading . However, l i n i i 4a t ions of cornouter f:!nds FrPcl~d9d th" nn~=*hii:ty nf -

.. . t - - - -- L,

l a r g e r number of load c y c l e s a t the low l e v e l . I t

is a l s o seen t h a t a f t e r t h e l oad - l eve l s t e p down, A K f f remains remarkably lower than its base l i n e v a f u e correspond tng t o the low- lcve 1 block loading ;

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.,, ,, L C .. r . r C j p r e s e u c e o t a c o n s i d e r a b l e r e +arcla; L;,I of gr3wtt:. but n c d e l a j .

( i i ) F ig . (10) shows the crack s u r f a c e p r o f i l m du r ing va r ious s t a g e s of unloading i n one of t h e low l e v e l cyc l e s of the p r e s e n t High to-Low block loading . I t i s seen t h a t a t t he stage of unloading i n d i c a t e d by point ' R ' i n F i g . ( l o ) , t h e c r a c k clos- e s o n l y a t the previous c r a c k - t i p ( c l n s u r e a r e a = a Fu r the r unloading, r ep re sen ted by p o i n t C , causes another nade away from rhe c u r r e n t c r a c k - t i p t o c l o s e , a s seen i n F ig . 10. The a r e a of c r ack - c l o s u r e thus i nc reases as t h e unloading p r o g r e s s e s .

To understand the e f f e c t s of t he f e a t u r e s of c r a c k - c l o s u r e as i n the presen t c a s e , t h e problem was seanalysed wi th the c o n s t r a i n t of c l o s u r e be- ing removed on the node ( a s d i s cus sed above) f a r away from the c r a c k - t i p , but l eav ing t h e c l o s u r e - c o n s t r a i n t on the node c l o s e s i t o the c r a c k - t i p . The corresponding changes i n o a r e i n d i c a t e d by a broken l i n e in F ig . 9. ~ h e s g ' r e s u l t s i n d i c a t e t h e i n f l u e n c e of p rope r ly imposing the c l o s u r e - c o n s t r a i n t s on nodes even f a r away from t h e c r a c k - t i p : when the considered load ing , a s t h e p r e s e n t H i g h - to-Low case, causes such a t y p e of c r a c k - c l o su re .

S i n n l e Over Load

The case of a s i n g l e over load a f t e r 4 c y c l e s of a c o n s t a n t ampli tude b lock load ing , fol lowed by f u r - t h e r c y c l e s of cons t an t ampl i tude ( equa l i n magnitude t o t h a t before over load) was cons idered . The fo l low- ing t h r e @ caaes were c o n s i d e y d . _ -

I "

I OPrnax - Oop. basq -e . max- base- =op. base

I n the above, a is he maximum a p p l i e d max. base

s t r e s s p r i o r t o o r a f t e r ove r load ; 'over load

is the over load s t r e s s ; is t h e maximup c a l c u l a t e d

Oop. max va lue f o r c rack-opening s t r e s s a f t e r ove r load ; and o is the base l i n e opening s t r e s s f o r an 0 t h -

o base e & i s e cons tant -ampl i tude c y c l i c load a t l e v e l and a l l t he se s t r e s s e s a r e i l l u s t r a t e d 'max.baseY

i n F i g . (11) . The obta ined r e s u l t s a r e d i s c u s s e d b e l o w :

( i ) The v a r i a t i o n s of o dur ing the load cy- c l i n g , f o r t he t h r e e d i f f e r & ? r a t i o s o f s t r e s 3 - ove r load , a r e indica ted i n F igs . ( I ].a, b , c ) r e s p e c t - i v e l y . I n a l l t h r ee over load cases an ab rup t de - c l ease i n o (which r e l a t i v e decrease becomes more predominantoPs the oyer load s t r e s s - r a t i o i n c r e a s e s ) i s no t i ced i rmedia te ly a f t e r the s i n g l e over load a p p l i c a t i o n . Af ter t h i s , i n a l l the t h r e e c a s e s . 3 i n c r e a s e s a g a i ~ t o r cach a peak va lue a

"P op-max b e f o r e l e v e l l i n g o f f a s t e a d y - s t a t e v a l u e . T h i s r e l a t i v e va lues of 0 i n c r e a s e s as t h e ave r load

opmax s t ress r a t i 3 i n c r e a s e s . Also as t he over load s t r e s - r a t i o i n c r e a s e s , t he l a t t e r is the oczurance o f t h i s .r r-- - - .. - . C , , I ' . - - * . - - . . 4.. . . *. .. 1 . 4

-- - st r e s s - i a t Lo occurs i n t he 4 t h cy -

O f 1 ' 2 7 3 ' * O Y T ~ ; f o r over load r a t i o c l e a f t e r overload ( F l g . o f 1.455 t h i s occurs i n t he 8 t h c y c l e a f t e r over- load (Fig. l l b ) ; wh i l e f o r t h e ca se of ove r load -

r a t i o 2 . 0 , ~ is s t i l l i n c r e a s i n g (Fig. l l c ) . T h i s imp l i e s t h a f P t h e h ighe r t h e over load r a t i o is , t h 0 . m

remarkable both t he r e t a r d a t i o n and d e l a y e f f e c t s are, This conc lus ion is i n acco rd wi th t h e obrerva- t i o n o f Schijve [ I ] , bared on t h e expe r imed ta l re- s u l t ~ of Arkem [15!,

( i f ) It is seen t h a t hKeff expe r i ences a sud - den jump innmediately a f t e r t h e ove r load , and t h e n d e - creases below i t s base l i n e l e v e l cor responding :o a cons t an t - ampl i tude c y c l i n g a t a l e v e l ; thus i n d i c a t i n g the p re sence of r%%&@?on and de - lay e f f e c t a ( t h e s e terms are used here in the same sense a s d e f i n e d by Barnard, e t . a 1 [16])in t h i s b i n - g l e ove r load case . It is also noted t h a t t he quan- t i t a t i v e e f f e c t s of r e t a r d a t i o n and d e l a y depend on t h e ove r load r a t i o .

( g i i ) The curve depicting the v a r i a t i o n of the ra t io ( cro - cf ) vi!~?%e over load s?~bk~ : ) ac ::ya8%6 %$PZ&I us ing t h e above d is - cussed 3 d a t a p o i n t s , is shown i n Fig. 12. By ex- t r a p o l a t i o n , t h e t h r e s h o l d v a l u e of t he over load r a t i o , a t which r e t a r d a t i o n e f f e c t s come i n t o p l ay , is s e e n t o be about 1.10. In c o n t r a s t , Bernard e t . a1 161 r e p o r t a t h r e sho ld r ~ e r l o a d ra- t io of 1.3 - 1.4 based on a s e r i e s of exper iments 0,: t h e m a t e r i a l Ducol W30B whose y i e l d s t r e n g t h is 366 MP (, comparable t o t h e p r e s e n t l y cons ide red o J ~ O M P , ) . It i s noted however, t h a t t he prar- ex!! a n a l y s i s is based on a plane-stresct assumption, wh i l e Bernard e t . a l [16] n o t e t h e dependence o f t h e e x p e r i m e n t a l l y determined t h r e s h o l d va lue on t h e specimen th i ckness .

I.

( i v ) The crack- l i n e d e f orrnat ion prof i l e s a t v a r i o u s s t a g e s of un load ing a t the end of t h e con- s i d e r e d number of cyc l e8 a r e shown i n F ig . 13 f o r t he case of over load r a t i o of 2 , whi le s i m i l a r re- s u l t # were noted f o r t h e o t h e r two o v e r l o a d - r a t i o c a s e s a l s o . I t i s s een t h a t t h e a p p l i c a t i o n of t he s i n g l e ove r load t o t h e specimen ( a t t he i n s t a n t when a/no = 1.02 i n F ig . 13 )causes a l a r g e ( p l a s t i c ) b l u n t i n g which is r e t a i n e d i n t h e c r a c k - s u r f a c e pro- f i l e even a s t h e crack advances i n f u r t h e r c y c l i c loading . When the specimen is f u l l y unloaded, a t t h e end of t he c y c l e i l l u s t r a t e d i n F ig . 13 , a lmost t he whole s u r f a c e a r e a ahead of t h e p rev ious ly mrn- t i oned Locat ion of b l u n t i n g is n o t i c e d t o c l o s e , w h i l e t h e c r a c k s u r f a c e a r e a behind t h i s b l u n t i n g l o c a t i o n is seen never t o c l o a e .

/

Analvs is of a Center-Cracked Specimen under Pure Mode 11 Cyc l i c Load'ing

A c e n t e r cracked s q u a r e p l a t e under a c o n s t a n t ampl i tude c y c l i c loading of pure s h e a r , which i s uni formly d i s t r i b u t e d a t t h e edges of t h e p l a t e , i s ana lysed . Plane s t r e s s c o n d i t i o n s are assumed. The m a t e r i a l i s cons ide red t o be 2024-T3 Aluminum a l l o y , ( ~ a m e a s i n t he Pure Mode I case d i scus sed e a r l i e r ) . The dimensions of t h e p l a t e a r e : L=W= 140mm; a = 40mm. The maximum ampli tude of t h e u n i f o r m l ~ d i s t r i b u t e d s h e a r is taken t o b e r = 80MP ( T ~ x/o , = . 2 3 ) . In t he p re sen t probTg%, the &omef!ry b f t he p l a t e w i t h t he c r ack i s sym- m e t r i c a l about both t h e x and y a x i s (See F i g . 1 4 ) , and t h e e x t e r n a l loading i s an t i - symmetr ic w i t h . C-.l.- - . , . . , . . I . L A II d i i ~ j d.t ia .

A s e a r l i e r , the p r e s e n t m a t e r i a l is modeled a s an e l a s t i c - p e r f e c t - p l a s t i c m a t e r i a l . We no te a l s o t h a t t h e p r e s e n t l y cons ide red m a t e r i a l has t he

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same p r s p e r t i e s i n t e n s i o n as i n c o m p r e s s i o n . T I ~ ~ I S , t h e d i s p l a c e m e n t f i e l d h a s t!le f o l l o w i n g a n i i s y m - m e t r i c p r o p e r t i e s :

where u f a t h e d i s p l a c e m e n t i n x c l i r e c t i o n , e t c . ~ u r t h e r l i t is n o t e d t h a t t h e s e d i s p l a c e m e n t s may be d i s c o n t i n u o u s a t t h e c r a c k s u r f a c e , - a < x < a

0' T h u s , ir . t h e f i n i t e e l e m e n t m o d e l i n g , o n l y a q u a r t e r o f the p l a t e is modeled ( a s show1 i n F i g , 1 4 ) w i t h t h e d i s ~ l a c m e n t boundary c o n d i l i o r s : u = 0 a l o n g y = 0, i n t h e u n c r a c k e d l igament o n l y ; a8d u = 0

Y a t nodes a l o n g x = 0. The Linear e l a s t i c r e s u l t s , b a s e d on t h e f i r s t l o a d increntent , 'rom t h e p r e s e n t f i n i t e e lement a n a l y s i s i n d i c a t e :

K I = 0.075; K I I = 3.777 6

which compare f a v o r a b l y w i t h t h e f o l l owing r e s u l t s ( o b t a i n e d by u s i n g t h e f i n i t e - s i z e c o r r e c t i o n f a c - t o r s ) o f Bowie and P e a l [ I 7 2 f o r a n I d e n t i c a l p ro- blem:

KI = 0.0 K I I = 3.899 7

The f a c t t h a t K f O i n t h e p r e s e n t f i n i t e e lement I a n a l y s i s is t h e r e s u l t o f i n h e r e n t n u l r ~ e r i c a l e r r o r s ,

s u c h a s round-of f and t r u n c a t i o n , i n t h e f i n i t e e l e m e n t a n a l y s i s .

In Fig . (15a) t h e r e s u l t s f o r t h e d i s p l a c e m e n t s u a t t h e upper and lower c r a c k s u r f a c e s , uU and uL

Y Y Y r e s p e c t i v e l y , a r e p l o t t e d f o r t h e l i n e a r e l a s t i c c a s e . These n u m e r i c a l r e s u l t s f o r uU and u a a r e i- d e n t i c a l i n t h e l i n e a r e l a s t i c case . ' T h i s ' e q u a l i t y o f uU and u 4 is n o t i c e d a s t h e l o a d i n g c o n t i n u e s i n t h e F i r s t c y c l e ( c r a c k b e i n g s t a t i o n a r y ) and when t h e p l a s t i c zone s i z e i s s i g n i f i c a n t a t r = 70 MPa ( s e e F i g . 16) f o r t h e s h a p e of t h e p l a s t i c i t y zone a t 7 = 70 MPa . For l a c k of any o t h e r c r i t e r i a , t h e c r a c k was e x t e n d e d , i n t h e p r e s e n t p r o c e d u r e , a t 70MPa , ( t h e maximum a p p l i e d s t r e s s b e i n g 80 HPaX i n t h e d i r e c t i o n o f t h e i n i t i a l c r a c k i e . , i n the x - d i r e c t i o n . I t is s e e n from F i g . ( 1 5 a ) t h a t s i g - n i f i c a n t i n c r e a s e i n u is brought a b o u t by t h e pro- c e s s o f c r a c k e x t c n s i o X and hence t h e a t t e n d a n t pie t i c u n l o a d i n g ; however , a g a i n , uU and ue a r e a lmos t i d e n t i c a l ( t o t h e 4 t h s i g n i f i c a n t d i g i t r . Thus i t is s e e n t h a t th rough t h e a l l s t a g e s of l o a d i n g , c r a c k e x t e n s i o n and p l a s t i c - u n l o a d i n g , and f u r t h e r l o a d i n g ( t o 8 MPa i n t h i s c a s e ) a f t e r c r a c k e x t e n - s i o n , t h e upper and l o w e r c r a c k f a c e s e x p e r i e n c e i- d e n t i c a l d i s p l a c e m e n t s i n t h e y d i r e c t i o n , i e . , p e r p e n d i c u l a r t o t h e i n i t i a l c r a c k a x i s . On t h e o t h e r h a n d , t h e d i s p l a c e m e n t s u a t t h e u p p e r and lower s u r f a c e s o f t h e c r a c k , u L ' znd o a a r e p l o t t e d i n F i g . (15b) f o r t h e c a s e s *of l o a B i n g when l i n - e a r - e l a s t i c c o n d i t i o n s p r e v i a l (7 = 3 1 . 1 MPa); when a p p r e c i a b l e p l a s t i c i t y d e v e l o p s a t t h e c r a c k - t i p (T = 70 MPa), when t h e c r a c k is e x t e n d e d (and hence t h e r e i s p l a s t i c l o a d i n g ) a t T = 7 0 MPa, and when t h e l o a d is f u r t h e r i n c r e a s e d ro 80 MPa a f t e r c r a c k e x t e n s i o n . I t is s e e n t h a t t h e m a g n i t u d e s o f uU and u A a r e n e a r l y i d e n t i c a l ( t o t h e 4 t h s i g n i f i ? s n t d i g i t x ) , bu t w i t h o p p o s i t e s i g n , i n a l l t h e above c a s e s . However i t is i n t e r e s t i n g t o o b s e r v e t h a t t h e change ( a s a r a t i o o f t h e r e s p e c t i v e v a l u e ' --: --. . - - 1 % < L - C ^-A . y. A,, ,, ,,,,., _ ' .CC- . . "L" . . , 4 C , .L - - .,.e.. # ,. - TqLh>t

a b o u t by t h e p r o c e s s o f r r a c k e x t e n s i o n (and hence p l a s t i c u n l o a d i n g ) i n u i s much more pronounced t h a n i n u . Y

I t i s i n t e r e s t i n g t o n o t e t h a t i n the l i n e a r

e l a s t i r c a s e the i r n r l - - s t i r f a c e d i s ~ l a r e r n ~ ~ ~ s .I: - ~ d uk ( s e e F i g . 1 5 a ) e x h i b i t a lmost a l i n e a r v a r i a r ion fYom t h e c r a c k - t i p ; t h u s i n d i c a t i n g a l a c k of any r a ( i n p a r t i c u l l r , ,,F t y p e f o r l i n e a r e l a s t i c i t y i component i n u f o r t h e l i n e a r e l a s t i c c a s e . On tile o t h e r hand: f o r t h e l i n e a r e l a s t i c c a s e , t i , ? t a n g e n t i a l d i s p l a c e m e n t s uU and l l a (See F i g . 1 5 5 ) d o e x h i b i t a f i b e h a v i o r n a a r t h e x c r a c k - t i p , f o r t h e p r e s e n t Mode I 1 problem. Also, i t i s s e e n from F i g . (15) t h a t , a s p l a s t i c i t y d e v e l o p s , t h e tangen r i a l d i s p l a c e m e n t s uU and ue e x h i b i t a P (a $ j v a r i a t i o n near t h e c:ack-t i.5. However, f o r t i le p u r e Mode 11 c a s e , e v e n i n che presence of p l a s t i c - i t y , t h e a n a l y s e s o f Hutch inson [8] and Rice and Rasengren [ 9 : , i n d i c a t e t h a t t h e r e may n o t be a P (a < %, type v a r i a t i o n i n uv n e a r t h e c r a c k - t i p . gu t t h e p r e s e n t r e s u l t s f o r u i n t h e p r e s e n c e of p l a s t i c i t y , F i g . 1 6 , a r e s e e n Y t o c o n t a i n s u c h an P(CI< 1 / 2 ) tvpe v a r i a t i o n n e a r t h e c r a c k - t i p . How- e v e r , i t should be n o t e d t h a t t h e a n g u l a r v a r i a t i o r , o f t h e s i n g u l a r i t y f u n c t i o n s , G i ( d ) , a s embedded i n t h e p r e s e n t s e c t o r e l e m e n t s a r e b e i n g approximated a s q u a d r a t i c po lynomia ls i n each s e c t o r e l e m e n t . The f a c t t h a t < 1/21 type v a r i a t i o n s i n rl a r e n u m e r i c a l l y o b t a i n e d a l o n g t h e r a d i a l l i n e of Y t h e s e c t o r element l y i n g on t h e c r a c k s u r f a c e , aa i n Fig. 15a s u g g e s t s t h a t t h e above a n g u l a r v a r i a t i o n s a r e n o t being s o l v e d h i g h l y a c c u r a t e l y i n t h e p r e - s e n t numer ica l method. However, i t a p p e a r s t h a t t h e s e numer ica l e r r o r s a r e i d e n t i c a l a t 6 = + n a s w e l l a s a t d = -n, s o t h a t u$ = u$ a s i n F - , 1 5 1 ~ F i n a l l y t h e c r a c k s u r f ace d i s p l a c e m e n t s (uu,uy) and (uudj a t t h e end of the f i r s t c y c l e a f lo%diXg ( i e .

X' X when t h e a p p l i e d stress is brought back t o z e r o ) a r e a l s o i n d i c a t e d i n F i g s . ( l S a ) and (15b) r e e p e c t i v e l v .

Once a g a i n i t i s s e e n t h a t even a f t e r comple te un- l o a d i n g , uU and ua a r e i d e n t i c a l i n magni tude and d i r e c t ion,'where 8s uU x a r e i d e n t i c a l i n magnitude b u t o p p o s i t e i n d i r e c t i o n . Thus f o r t h e p r e s e n t m a t e r i a l , w i t h i d e n t i c a l p r o p e r t i e s i n t e n s i o n a s i n compress ion , i t i s s e e n t h a t i n a l l c a s e s o f p u r e - s h e a r type e x t e r n a l load 'ng , t h e upper and lower s u r f a c e s of t h e c r a c k move t o g e t h e r i n the d i r e c t i o n p e r p e n d i c u l a r t o t h e i n i t i a l c r . 4 ~ ~ - a x i s , w h e r e a s they s l i d e p a s t one a n o t h e r i n t h e d i r e c t - i o n of t h e c r a c k - a x i s . r h u s , i t a p p e a r s f o r t h e s e t y p e s of m a t e r i a l s t h e phenomenon of c r a c k - c l o s u r e , a s observed e x p e r i m e n t a l l y and a s a n a l y s e d p r e s e n t - l y i n Mode I t y p e l o a d i n g c o n d i t i o n s , d o e s n o t occur i n p u r e Mode 11 t y p e c y c l i c l o a d i n g . However t h e p r e s e n t e x p e r i e n c e i n d i c a t e s t h a t c r a c k - c l o s u r e may o c c u r even i n p u r e Mode I 1 c y c l i c l o a d i n g i f t h e m a t e r i a l has d i f f e r e n t p r o p e r t i e s i n u n i a x i a l t e n - s i o n and compress ion . C o n s i d e r a t i o n of s u c h m a t e r - i a l s is n o t pursued i n tile p r e s e n t work. F i n a l l y , t h e computed s h a p e s and s i z e s o f t h e p l a s t i c zone n e a r t h e c r a c k - t i p a t v a r i o u s s t a g e s o f p u r e s h e a r l o a d i n g a r e i n d i c a t e d i n F i g . 16.

Summary and Conclusions:

I f t h e c rack-growth r a t e , d a , d , i n Mode 1 f a t i g ~ l e l o a d i n g , can be assumed t o r e r e l a t e d the e f f e c t i v e s t r e s s - i n t e n s i t y r a n g e , A K e f f , t h e p r e -

s e n t r e s u l t s i n d i c a t e t h a t : ( i ) growth r e t a r d a t ion o c c u r s i n h igh t o low m d s i n g l e o v e r l o a d c a s e s , and (ii) s i g n i f i c a n t f l e l a y e f f e c t s p r i o r t o r e t a r d a - t i o n o c c u r i n t h e c a s e of a s i n g l e - o v e r l o a d - - and

,,=LC n P ' ..\ 1 - - . - d IU ~ C C c l L u d A u u C L L C L L I V G C . - L.L. .L- I \

dominant a s t h e o v e r l o a d r a t i o i n c r e a s e s . From t h e

c r a c k s u r f a c e d e f ormat i o n p r o f i l e s shown i n F i g s . 8 and 13, i t is s e e n t h a t a c o n s i d e r a b l e c r a c k s u r - f a c e b l u n t i n g o c c u r s a t the i n s t a n t when t h e a p p l i d l o a d is s t e p p e d up. T h i s b l u n t i n g a t t h e ~ n s t a n t

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of l o a d - s t e p up a l t e r s t h e p o s s i b l e p a t t e r n of c r a c k c l o s u r e d u r i n g s u b s e q u e n t load c y c l e s , and is s e e n t o be r e s p o n s i b l e f o r the "c'elay" e f f e c t 8 s u c h a s t h e d e l a y e d r e t a r d a t i o n i n t h e s i n g l e o v e r l o a d c a s e , and t h e de layed t r a n s i t i o n of opening stress l e v e l s f rom the base l i n e v a l u e s f o r lower a m p l i - tude b l o c k l o a d i n g t o t h e h i g h e r b a s e - l i n e v a l u e f o r t h e h i g h e r ampl i tude b l o c k l o e d i n g , i n t h e Low- t o - h i g h c a s e . The p h y s i c a l mechanism behind t h e s e e f f e c t s is f u r t h e r d e t a i l e d i n [12 ,14] . F i n a l l y , the phenomenon of c r a c k . c l o s u r e was n o t o b s e r v e d i n t h e p r e s e n t numer ica l model ing of a t h i n c e n t e r - c racked s h e e t (of an e l a s t i c - p l a s t i c m a t e r i a l w i t h i d e n t i c a l ~ r n p e r t i e e i n unfaxfal t e n s i o n a s i n corn p r e s s i o n ) , s u b j e c t a pure s h e a r (Mode 11) c y c l i c Loading of c o n s t a n t a m p l i t u d e . Thus, i t a p p e a r s t h a t i n t h e s t u d y of t h e inore g e n e r a l problem of f a t i g u e crack-growth under mixed-mode l o a d i n g , t h a t c a u s e s s m a l l s c a l e y i e l d i n g 9 n e a r t h e c r a c k - t i p , c r a c k - c l o s u r e e f f e c t s need b e c o n s i d e r e d i n t h e case mode I component on ly . However, t h e p r e s e n t e x p e r - i e n c e i n d i c a t e s t h a t c r a c k - c l o s u r e may o c c u r e v e n i n p u r e Mode I 1 i f t h e m a t e r i a l h a s d i f f e r e n t p r o - p e r t ies i n u n i a x i a l t e n s i o n and compress i o n .

Acknowledntmen t s : This work was s u p p o r t e d i n p a r t s by N A S A g r a n t No. NSG-1351 and by AFOSR c o n t r a c t No. F49620 78-C-0085. These s u p p o r t s a r e g r a t e f u l l y acknowledged.

R e f e r e n c e s :

e n i n g b l a t e r i a l , " J o u r n a l o f Mechanics and Phva& o f S o l i d s , Vol . 16 , 1968 , pp. 1-12 .

[ l o ) A t l u r i , S . N . m d Nakagaki , M . , " J - I n t e g r a l E s t i m a t e s f o r S t r a i n - H a r d e n i n g M a t e r i a l s i n Duct i n e F r a c t u r e Problems , I t A I A A J o u r n a l , Vol. 1 5 , No. 7 , 1977. pp. 923-931 .

[ l l ] A t l u r i , S . N. and Nakagaki , M. and Chen, W . t i . , "Fiac t u r e Ana l y s is Under Large S c a l e Y i e ld Fng : A F i n i t e Defarmat i o n Embedded S i n g u l a r i t y , E l a s t i c - p l a s t i c I n c r e m e n t a l Finite Element S o l u t i o n i n Flaw Growth and F r a c t u r e " . STP 631, American S o c i e t y f o r T e s t i n g and M a t e r i a l s , 1 9 7 7 , pp. 43-61.

[I21 Nakagaki , M . , and A t l u r i , S . N . , " E l a s t i c - P l a s t i c F i n i t e Element A n a l y s e s of F a t i g u e Crack Growth i n Mode I and Mode I 1 Condi t ions" , NASA -CR- 158987, Nov. 78, 82 p e g e s .

!3] At l u r i , S . N . , "On Some New G e n e r a l and Comple- menta ry Energy Theorems f o r t h e Rate Problems i n F i n i t e S t r a i n , C l a e s i c a l E l a s t o - P l a s t i c i t y " , G e o r g i a I n s t i t u t e of T e c h n o l o g y , Report GIT- ESM-SA-78 1 0 , Aug. 1978. ( I n rev iew f o r pub- l i c a t ion i n J . Mechanics and P h y s i c s of S o l i d s ) .

f 141 N a k a g a k ~ , M . , and A t l u r i , S . N . , " F a t i g u e Crack C l o s u r e and Delay E f f e c t s Under Mode I 9pec t rum l o a d i n g : An E f f i c i e n t E l a s t i c - P l a s t -

S c h i jve , J . , "Four L e c t u r e s on F a t i g u e Crack i c P r o c e d u r e " t o a p p e a r i n Proced . of 3rd I n t . Growth, " engineer in^ F r a c t u r e Mechanics , Vol . TI, Conf. on Mech. Beh. o f M a t e r i a l s , Univ.of Qamtri%e No. 1, 1979 , pp. 165-223. U. K . , S e p t . 1979 ( 9 p a g e s ) .

E l b e r , W . , "Fa t igue Crack C l o s u r e Under C y c l i c [15] Arkema, W. J . , r e s u l t s q u o t e d i n J . S c h i j v e , T e n s i o n , " Engineer ing F r a c t u r e Mechanics , Vol. "Obeervat i o n s on t h e P r e d i c t i o n of F a t i g u e 2 , No. 1, J u l y 1970, pp. 37-45. Crack Growth Propaga t i o n Under V a r i a b l e - Amp1 i -

t u d e Loading", ASTM STP 595, p. 3 , 1976. Budiansky , B. and H u t c h i n s o n , J . W . , " A n a l y s i s o f C l o s u r e i n F a t i g u e Crack Growth". J o u r n a l L16: Bernard , P. J . , L i n d l e y , T. C . , and R i c : l a r d s , of Applied Mechanic8 Vol . 45. KO. 2 , J u n e 1978, C . E . , "Mechanisms of O v e r l o a d R e t a r d a t ior, p p . 267-276. During b a t i g u e Crack P r o p a g a t i o n " , F a t i w e

Crack Growth Under Spec t rum Loads, ASTM S T P 595, Newman, J . C. J r , and Armen, Har ry , J r . , 1976, p p 7 8 - 9 7 . " E l a s t i c - p l a s t i c A n a l y s i s of a P r o p a g a t i n g Crack Under Cyc l ic Loading," , A I A A p a p e r No. [17] Bowie, 0. L. and N e a l , P. M . , "A Note on t h e 74-366, p r e s e n t e d a t the A I A A ~ A S M E ~ S A E 1 5 t h C e n t r a l Crack i n a U n i f o r m e l l y S t r e s s e d S t r i p " . S t r u c t u r e s , S t r u c t u r a l Dynamics, and ater rials C o n f e r e n c e , Las Vega, Nev. . 17-19 A p r i l 1974.

Newman, J . C., J r . , " A F i n i t e - E l e m e n t A n a l y s i s o f F a t i g u e Crack Closure" , Mechanics o f Crack Growth. ASTM STp 590, American S o c i e t y f o r - T e s t i n g and M a t e r i a l s , 1976, pp. 281-301.

O h j i , K . , Ogura, K . , and Okubo, Y . , " C y c l i c A n a l y s i s o f a P r o p a g a t i n g Crack and i t s C o r r e - lat ion with ~ a ' t igue Crack Growth ," Engineer ing . F r a c t u r e Mechanics, 1975, Vol . 7 , pp. 457-464 .

Ogura , K . and O h j i , K . , "FEM A n a l y s i s o f Crack Closure and Delay E f f e c t i A l F a t i g u e Crack Growth Under V a r i a b l e Amplitude Loading. " E n n i n e e r i n p F r a c t u r e Mechanics , 1977, Val . 9 , pp. 473-480.

H u t c h i n s o n , J . W . , " S i n g u l a r Behaviour a t t h e - 3 -6 - .V..".. -. LL:- u I G L I U L ~ F LLQI-P. A , . 4 ~ i a i C i E i ? L i ~ g i . ; ; l ; C i - i d i " .

J o u r n a l of Mechanics and P h y s i c s o f S o l i d s , 1968, Vol. 16 pp. 13 31.

. - In t e r - z t i o n a l . J o u r n a l of F r a c t u r e Mechanics , Vol. 2 , Nov. 1970, pp. 181-182 .

R i c e , J . ?. and Rosengren, G . F. , "Plane S t r a i n Dcformatio? Near a Crack T i p i n a Power Low Hard-

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P:AII. 'N OF DETAIL 'A' IN FIG. 2

MESH ANL NODES BEFORE

MESH AND NODES AFTER TRANSIATION

FIG 1. SCHEMATIC REPRESENTAT ION OF TRANSIATION OF SINGLUR ELPIENTS.

/---- DETAIL 'A'

F T C . 2 . FINTTE nmwr Monn w 4 CENTF~ ruArYLn SPECTMEN ~ E R UNIAXIAL CYCLIC LOADING ( S XNG IJLAR SECTOR ELEMENTS SHOWN WITHIN DETAIL ' A ' ) .

L P PIASTIC ZONE

FIG 30. REPRESENTATIVE SIZE OF YIELD ZONE AT Omx.

FIG 3b. S C U M T I C REPRESENTATION OF INCRPIENTAL EQUATIONS IN THE PRESENCE OF YIELDING.

CLOSED

FIG 4. REPRESENTATIVE FATTERNS OF CRACK CLOSTJRE ; ( 48 - c) : CRACK CLOSES ONLY AT NODES W S E S T TO CRACK TTP. bd. PQACY 17.OSTmF n r r r m c r r rn

AT NODES AWAY FROM CRACK TIP.

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8 C#A1;7( CLOSURE

O CRACK OPENTNG

FIG 5 CRACK CLOSURE AND CRACK OPENING STRESSES IN [XIH!?I'AVT AKPLITUnE (LO) mnIc LUADING.

C1 .96 .98 1. 1.02 1.04 X/ao

FIG 8. CRACK LINE PROFILE DURING UNLOADING IN LOW- TO-HIGH BLOCK LOADING.

4

$ 3 0

V) V)

8 *

; < 0

LYCLE FIG 9 . CHhLX CLOSURE CRACK OPENING STRESSES IN HIGH-m-UU BLOCK

LOADING.

( b ) X I ro

F I G 6a.CRAa LINF PROFILE DURlNG UNLOADING I N CONSTAKI. AWPLITLRIE LOADING: 6b ZNSET 'A' I N F I G 6. I S K 4 G N I F I E D AND SHOW.

0 CRACK OPENING

CRACK CLOSUP-E

L'YCLt F I G 7 '-7(ACN CLOSURE AND CRACK OPENTNG STRESSES IN W - T O - H I G H BMO(

LCHDINC.

AI e

a: CRACK L I N E PROFILE DURING UNLOADING I N HIGH- TO-LOW BLOCK LOADING; b: INSET ' A ' I N F I G 10a. I S MAGNIFIED AND SHOWN.

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FIG 11. CRACK CLOSURE AM) CRACK OPENING STRESS FOR THREE DIFFERENT CASES OF A SINGLE OVER 1A3AI)

'op.mx ' 'op.bars

Omex. bare - 'op. bare

FIG 12. EFFECT OF OVERLOAD-STRESS RATIO ON uop'mx' oap'baee *mx. bare- 'op,bare'

FIG 15, a NORMAL D I S I ~ A C P I E N T FROFILE5 OF THE L!P% ANP LOh'Ffi SURFACES OF THE CRACK: b: TANGENTIAL DISPIACEKK'I PROFILES OF UPPER AND UlWER C IFACES OF THE CRACK.