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

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

  • 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.. ;?

  • 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