J- INTEGRAL ESTIMATES FOR STRAIN- HARDENING MATERIALS I N DUCTILE FRACTURE PROBLEMS Satya N . A t l u r i " and M . Nakagalci'k'' School of Engineering Science and Mechanics Georgia I n s t i t u t e of Technology A t l a n t a , Georgia 30332 Abstract A f i n i t e element procedure i s developed t o analyze p l a n e d u c t i l e f r a c t u r e problems i n t h e presence of l a r g e - s c a l e y i e l d i n g n e a r t h e crackt i p . 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 n e a r t h e c r a c k - t i p , corresponding t o t h e s t r a i n - h a r d e n i n g m a t e r i a l model, a r e embedded i n elements n e a r t h e c r a c k - t i p . The developed e l a s t i c - p l a s t i c i n c r e mental f i n i t e element method i s based on a hybrid displacement model. The r e s u l t s f o r t h e J - i n t e g r a l a t v a r i o u s load l e v e l s , f o r s e v e r a l s t a n d a r d f r a c ture- toughness t e s t specimens, a r e compared with a v a i l a b l e experimental r e s u l t s . The i m p l i c a t i o n of t h e r e s u l t s i n predicting i n i t i a t i o n of crack growth, and i t s s t a b i l i t y , i n d u c t i l e m a t e r i a l s i s discussed. Nomenclature JKII J Linear- elastic s t r e s s intensity factors (Mode I and I1 r e s p e c t i v e l y ) A p a t h - i n t e r a l defined by Eshelby (1) Under t h e u s e of a J2-flow t h e o r y o f p l a s t i c i t y , t h e p h y s i c a l s i g n i f i c a n c e of J a s a measure of t h e characteristic crack-tip elastic- plastic stress/ s t r a i n f i e l d i s s t i l l v a l i d . When a J2-flow theory i s used, t h e path- independence of t h e J - i n t e g r a l , even under monotonic l o a d i n g cannot be e s t a b l i s h e d . However, under monotonic l o a d i n g , t h e use of a t o t a l s t r a i n t h e o r y i s j u s t i f i a b l e i n s t u d i e s of s t r e s s / s t r a i n f i e l d s n e a r a s t a t i o n a r y c r a c k bec a u s e , approximately p r o p o r t i o n a l loading o c c u r s . During t h e l o a d i n g of a cracked body from t h e v i r gin s t a t e t o the c r i t i c a l s t a t e , the effective s t r e s s can be expected t o i n c r e a s e everywhere i n t h e r e g i o n . Thus, under monotonic l o a d i n g , approximate p a t h independence of t h e J - i n t e g r a l may be expected even under J2-flow t h e o r y of p l a s t i c i t y . Thus, t h e J - i n t e g r a l d e f i n e d by and Rice (2y P Applied load i n s t a n d a r d f r a c t u r e t e s t specimens such a s t h r e e - p o i n t bend and compact t e n s i o n 6 Load p o i n t displacement a.. 1J Stress tensor de!. 1J Total incremental s t r a i n dcFj Incremental p l a s t i c s t r a i n J2 Second i n v a r i a n t of d e v i a t o r i c s t r e s s tensor Introduction The s t r u c t u r e of t h e dominant s i n g u l a r i t y i n s t r e s s e s and s t r a i n s n e a r a c r a c k - t i p i n p l a n e problems, f o r power law hardening m a t e r i a l s , has ) Rice and Rosenbeen s t u d i e d by H u t ~ h i n s o n ( ~and g , r e r ~ ( ~ )Within . t h e l i m i t a t i o n s of a s m a l l - s t r a i n , J 2 deformation t h e o r y of p l a s t i c i t y , t h e amplitude of t h e above dominant s i n g u l a r i t y , . f o r pure mode problems, i s d i r e c t l y r e l a t e d t o t h e w e l l known Ji n t e g r a l introduced by Rice(2) and e a r l i e r by E s h e l b y ( l ) . Thus t h e J - i n t e g r a l can be taken t o be a convenient measure of t h e s t r e n g t h of singul a r i t y i n such problems. The J - i n t e g r a l i s d e f i n e d by, J =s (wdy - aU, TK a x ds) r where x , y a r e c o o r d i n a t e s normal t o t h e c r a c k front, y being p e r p e n d i c u l a r t @t h e c r a c k s u r f a c e , ds i s a d i f f e r e n t i a l a r c l e n g t h along any contour r , beg i n n i n g along t h e bottom s u r f a c e of t h e crack and +; *+e ending along t h e top s u r f a c e , W i s t h e s t r a i n energy d e n s i t y , TK a r e components of t r a c t i o n on t h e s u r f a c e o f t h e i n t e r i o r body c u t by I-, and uK a r e displacement components. The J - i n t e g r a l i s path- independent i f t h e s t r a i n energy i s a s i n g l e valued f u n c t i o n of t h e s t r a i n s . This i s t h e c a s e i f the material is nonlinearly e l a s t i c , or i n the c a s e of an e l a s t i c - p l a s t i c m a t e r i a l i f a J 2 - d e f o r mation t h e o r y i s used, so long as t h e e f f e c t i v e I n t h e c a s e of an e l a s t i c s t r e s s i s n o t decreased. s o l i d ( l i n e a r o r n o n - l i n e a r ) t h e J - i n t e g r a l can be i n t e r p r e t e d as t h e e n e r g y flow towards t h e crackt i p p e r u n i t o f c r a c k e x t e n s i o n . This energy r e l e a s e i n t e r p r e t a t i o n of J cannot be a p p l i e d t o t h e process of crack e x t e n s i o n i n e l a s t i c - p l a s t i c s o l i d s even w i t h a J2-deformation t h e o r y . However, under t h e use of a J2-deformation t h e o r y , t h e J i n t e g r a l can be i n t e r p r e t e d a s t h e p o t e n t i a l energy d i f f e r e n c e between two i d e n t i c a l l y loaded bodies having neighboring crack s i z e s , i . e . , -aP/al,, where P i s t h e p o t e n t i a l energy and R i s t h e crack length. A s s o c i a t e P r o f e s s o r , Member, AIAA. Post- Doctoral Fellow. 140 where a i . i s t h e s t r e s s a t any p o i n t and d c i j i s the s t r a i n a t t h e p o i n t , may be taken a s a v a r i a b l e s p e c i f y i n g t h e s e v e r i t y of t h e c o n d i t i o n s a t t h e c r a c k - t i p when l a r g e - s c a l e y i e l d i n g i s present. I f t h e magnitudes of t h e l o a d s i s d e s i g n a t e d by The a g e n e r a l i z e d load parameter Q , t h e n J = J(Q). c r i t i c a l g e n e r a l i z e d load Qc a t which crack growth i n i t i a t i o n o c c u r s i n p l a n e , d u c t i l e , f r a c t u r e problems can be s t a t e d by t h e c o n d i t i o n , J = Jc =J(Qc). It i s noted a t t h i s p o i n t t h a t t h i s c r i t e r i o n J=Jc i s a c r i t e r i o n f o r i n i t i a t i o n o f crack growth only, without any s t a t e m e n t of s t a b i l i t y o r i n s t a b i l i t y of such crack growth. I n r e c e n t experiments, Begley and Landes(5) have demonstrated t h e p o t e n t i a l of t h e J - i n t e g r a l as a fracture i n i t i a t i o n c r i t e r i o n i n the large-scale y i e l d i n g range. I n e x p e r i m e n t a l l y determining c r i t i c a l J i n e l a s t i c - p l a s t i c problems, Begley and Landes(5) u t i l i z e d t h e i n t e r p r e t a t i o n o f J a s t h e r a t e of change w i t h r e s p e c t t o t h e crack l e n g t h of t h e a r e a under t h e load v e r s u s t h e load- point d i s p lacement curves of s t a n d a r d f r a c t u r e t e s t s p e c i mens such a s t h e t h r e e - p o i n t bend specimen, and t h e t e n s i o n specimen. Thus t h e o r i g i n a l experimental p r o t o c o l o f Begley and Landes c a l l s f o r testi n g v i r g i n specimens w i t h d i f f e r e n t c r a c k s i z e s . For f r a c t u r e t e s t specimens such as c i t e d above, where t h e only geometric dimension of i n t e r e s t i s the uncracked ligament l e n g t h , r e c e n t l y Rice, P a r i s and Merkle(6) have shown t h a t t h e J - i n t e g r a l can be evaluated from a s i n g l e , e x p e r i m e n t a l l y g e n e r a t e d , load- displacement r e c o r d . This simple e m p i r i c a l formula of Rice e t a 1 (6) f o r e s t i m a t i n g J f o r t h r e e point bend and compact t e n s i o n t e s t specimens a s sumes t h a t t h e crack depth i s a t l e a s t s u f f i c i e n t SO t h a t t h e p l a s t i c y i e l d zone i s confined t o t h e uncracked ligament r e g i o n ahead of t h e c r a c k . Turning t o a n a l y t i c a l e s t i m a t i o n procedures, when s m a l l - s c a l e y i e l d i n g e x i s t s n e a r t h e c r a c k - t i p , J i s simply r e l a t e d t o t h e e l a s t i c s t r e s s - i n t e n s i t y f a c t o r . For l a r g e - s c a l e y i e l d i n g , such simple c a l c u l a t i o n s a r e n o t p o s s i b l e , s i n c e J depends on t h e geometry, a p p l i e d l o a d i n g s , and m a t e r i a l p r o p e r t i e s i n a complicated way. In t h e a n a l y s i s o f simple f r a c t u r e t e s t specimens, a c r u d e , but u s e f u l , approximate procedure f o r e s t i m a t i o n of J has been used by Bucci e t a l ( 7 ) who " e x t r a p o l a t e " from small s c a l e y i e l d i n g range t o f u l l y p l a s t i c range using I r w i n ' s " p l a s t i c i t y c o r r e c t i o n s . " The m a t e r i a l d e s c r i p t i o n used by Bucci e t a l ( 7 ) i s e l a s t i c p e r f e c t l y - p l a s t i c . On t h e o t h e r hand, f o r pure power-law hardening m a t e r i a l s ( e on), u s i n g a deformation t h e o r y of p l a s t i c i t y , simple, approxia t e procedures f o r e s t i m a t i n g J were r e c e n t l y p r e ented by Goldman and Hutchinson(8). - Thus t h e r e c l e a r l y e x i s t s a need f o r a c c u r a t e umerical procedures t o e s t i m a t e J i n a r b i t r a r y plane problems of d u c t i l e f r a c t u r e mechanics, w i t h a r b i t r a r y boundaries, under a r b i t r a r y loading cond i t i o n s , a d when t h e m a t e r i a l p r o p e r t i e s a r e c h a r a c t e r i z e d as e l a s t i c - p l a s t i c w i t h a r b i t r a r y s t r a i n - h a r d e n i n g . Such a numerical procedure should f i r s t be t e s t e d f o r i t s v a l i d i t y by comparison with a v a i l a b l e experimental r e s u l t s i n standard t e s t specimens used i n f r a c t u r e toughness t e s t ing, such a s t h e t h r e e - p o i n t bend specimen, t h e compact t e n s i o n , and t h e c e n t e r cracked specimen. This, t h e n , i s t h e main i n t e n t o f t h i s paper. sponding t o t h e n o n l i n e a r m a t e r i a l model, was embedded i n t h e s e n e a r - t i p elements, whereas t h e 0 v a r i a t i o n was approximated i n each s e c t o r element and solved f o r i n t h e s e n s e of t h e f i n i t e - e l e m e n t method; ( b ) m a i n t a i n i n g c o n t i n u i t y of displacements and t r a c t i o n s , between n e a r - t i p elements with singul a r s t r e s s / s t r a i n assumptions and t h e f a r - f i e l d elements w i t h r e g u l a r s t r e s s / s t r a i n assumptions, through a hybrid displacement f i n i t e element model T h i s procedure i s t h e one used by t h e authors(lO9') i n l i n e a r e l a s t i c f r a c t u r e problems; (c) using a J 2 - f l o w t h e o r y of p l a s t i c i t y and a r b i t r a r y kinemat i c hardening which w i l l a c c u r a t e l y model t h e Bauschinger e f f e c t under f u l l y r e v e r s e d and c y c l i c l o a d i n g ; (d) u s i n g an incremental f i n i t e element s o l u t i o n procedure t h a t w i l l be s u i t a b l e i n t h e l i m i t i n g c a s e of 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 s , a s w e l l as c y c l i c loading s i t u a t i o n s . The " i n i t i a l s t r e s s i t e r a t i o n " approach a s o r i g i n a l l y proposed by Zienckiewicz e t a l ( I 2 ) i s used f o r t h i s purpose; (e) developing a more a c c u r a t e f i n i t e element method f o r incremental a n a l y s i s of e l a s t i c - p l a s t i c problems. The more common approach i n t h e l i t e r a t u r e i s t o u s e " c o n s t a n t - s t r e s s " elements, and based on t h e s t r e s s l e v e l i n t h e element, t h e whole element e i t h e r y i e l d s o r s t a y s e l a s t i c . Thus i n problems, such a s t h e p r e s e n t , where t h e y i e l d zones n e a r t h e c r a c k - t i p p l a y a dominant r o l e i n t h e a n a l y s i s and i t s i n t e r p r e t a t i o n , i n o r d e r t o o b t a i n a reasonably a c c u r a t e d e s c r i p t i o n of t h e y i e l d zone, a v e r y f i n e f i n i t e element mesh i s needed. However, i f h i g h e r - o r d e r elements a r e used, and i f " p l a s t i c i t y - c o r r e c t i o n " i t e r a t i o n s a r e p e r formed a t s e v e r a l p o i n t s w i t h i n t h e element, i t t h e n b e c o m e s p o s s i b l e t o g i v e a smoother d e f i n i t i o n of t h e y i e l d zone. Thus, a p o r t i o n of element, i n t h e p r e s e n t f o r m u l a t i o n , can y i e l d , w h i l e t h e r e s t of t h e element can remain e l a s t i c . The p r e s e n t a n a l y s i s i s a l s o based on a smalldeformation, s m a l l - s t r a i n theory. However, t o p r o p e r l y account f o r c r a c k - t i p b l u n t i n g , a f i n i t e deformation a n a l y s i s i s n e c e s s a r y . (Such an a n a l y s i s w i l l be p r e s e n t e d i n a forthcoming p a p e r . ) The u n i a x i a l s t r e s s - s t r a i n curve o f t h e m a t e r i a l i s considered t o be of Ramberg-Osgood type [ e = (o/E) + Using t h e p r e s e n t method, d e t a i l e d a n a l y s e s were performed f o r t e s t specimens of t h e types three- point- bend, compact t e n s i o n , and c e n t e r cracked t i p . The m a t e r i a l s considered were A533B p r e s s u r e v e s s e l s t e e l and a medium s t r e n g t h N i - C r Mo- V s t e e l f o r which experimental r e s u l t s a r e a v a i l a b l e i n open l i t e r a t u r e , Begley and Landes(5). I n a l l t h e c a s e s , e x c e l l e n t c o r r e l a t i o n of t h e p r e s e n t r e s u l t s w i t h t h e c i t e d experimental r e s u l t s was n o t e d . R e f e r r i n g now t o some d e t a i l s of a n a l y s i s , i n Ref. (8) a c i r c u l a r c o r e element w i t h embedded dominant s i n g u l a r i t y of t h e type derived i n (3,4) i s used n e a r t h e crack t i p , The 0-dependence of t h i s dominant s i n g u l a r term (where r and 0 a r e polar coordinates centered a t the crack- tip) i s determined i n ( 3 , 4 ) from t h e numerical s o l u t i o n of a nonlinear, fourth order, ordinary d i f f e r e n t i a l equation f o r each v a l u e of t h e exponent ' n ' i n t h e m a t e r i a l power-hardening law ( e o n ) . For mixedmode problems t h e d e t e r m i n a t i o n of t h i s 8-dependence i s much more c ~ m p l i c a t e d ( ~ ) .Moreover, t h e use o f J2-deformation t h e o r y of p l a s t i c i t y i n ( 8 ) makes i t i n v a l i d i n s i t u a t i o n s where s i g n i f i c a n t unloading i s observed. Such unloading s i t u a t i o n s are observed i n cases i n v o l v i n g s t a b l e c r a c k - I n t h e following we p r e s e n t d e t a i l e d r e s u l t s and t h e i r comparison with a v a i l a b l e experimental d a t a , f o r t h e c a s e of a compact t e n s i o n specimen made of A533B p r e s s u r e v e s s e l s t e e l . A d i s c u s s i o n of t h e i m p l i c a t i o n of t h e s e r e s u l t s i n p r e d i c t i n g crack- growth i n i t i a t i o n , and i t s s t a b i l i t y , i n d u c t i l e materials i s presented. T h e o r e t i c a l Formulation Hybrid Displacement F i n i t e Element Model For I n cremental E l a s t i c - P l a s t i c S o l u t i o n In view of t h e above d i s c u s s i o n , t h e r e s e a r c h P r e s e n t l y r e p o r t e d has t h e o b j e c t i v e s : ( a ) developing c i r c u l a r s e c t o r shaped embedded s i n g u l a r i t y elements n e a r t h e c r a c k - t i p . The c o r r e c t r dependence of t h e dominant s i n g u l a r s o l u t i o n , c o r r e - The need f o r employing a hybrid- displacement t y p e f i n i t e element model, t o e n f o r c e c o n t i n u i t y 141 of displacements and e q u i l i b r i u m of t r a c t i o n s between n e a r - t i p elements (with s i n g u l a r s t r e s s / s t r a i n s ) and f a r - f i e l d elements (with r e g u l a r polynomial type s t r e s s e s / s t r a i n s ) , h a s been d i s cussed i n detail(lOyll) i n t h e context of problems of l i n e a r e l a s t i c f r a c t u r e mechanics. The same b a s i c t h e o r y has been extended h e r e , t o formulate an incremental procedure f o r e l a s t i c - p l a s t i c analys i s of cracked s t r u c t u r e s . The a n a l y s i s procedure i s based on a smalls t r a i n , small displacement, incremental flow t h e o r y In g e n e r a l , we d e f i n e SN t o be t h e of p l a s t i c i t y . s t a t e of t h e body b e f o r e t h e a d d i t i o n of t h e Nth load- increment, whereas S N + ~i s t h e s t a t e a f t e r t h e Nth l o a d indrement. I n s t a t e SN, t h e s t a t e s of s t r e s s , s t r a i n , and deformation i n t h e e l a s t i c p l a s t i c cracked body a r e presumed t o be known. During t h e p r o c e s s of t h e Nth i n c r e m e n t , t h i s s t a t e i s considered t o be i n a s t a t e of " i n i t i a l s t r e s s . " I n t h e following we t r e a t , as a g e n e r i c c a s e , t h e t r a n s i t i o n of t h e cracked body from a r e f e r e n c e s t a t e SN t o a f u r t h e r deformed s t a t e Swl. I n t h e increment from SN t o Swl, t h e incremental s t r e s s e s w i l l be f i r s t c a l c u l a t e d from t h e l i n e a r e l a s t i c c o n s t i t u t i v e law. I f t h e r e i s p l a s t i c flow i n t h e body, t h i s e l a s t i c e s t i m a t e of incremental s t r e s s e s w i l l be h i g h e r than i n t h e a c t u a l s t r a i n - h a r d e n i n g body. The a c t u a l s t r e s s and s t r a i n increments i n t h e presence o f p l a s t i c i t y w i l l then be obtained by modifying t h e e l a s t i c p r e d i c t i o n s u s i n g t h e " i n i t i a l - s t r e s s i t e r a t i o n " approach i n each i n c r e ment, as o r i g i n a l l y proposed by Zienckiewicz, e t a1(12)e Analogous t o t h e v a r i a t i o n a l p r i n c i p l e governing t h e hybrid- displacement model f o r l i n e a r problems (10,11), t h e incremental v a r i a t i o n a l p r i n c i p l e governing t h e p r e s e n t formulation can be s t a t e d a s 6An = 0 , where, interelement compatibility a p r i o r i . I n elements surrounding t h e c r a c k - t i p , displacements corresponding t o t h e Hutchinson- Rice- Roserigren s i n g u l a r i t i e s , f o r hardening m a t e r i a l s , a r e included, t = f(Au 1J i,j + A U j , i) As,, A vi TLi = independently assumed displacements, a t a V m , which i n h e r e n t l y s a t i s f y i n t e r element c o m p a t i b i l i t y c r i t e r i a , = Lagrange m u l t i p l i e r t o e n f o r c e t h e comp a t i b i l i t y c o n d i t i o n A u i = Avi a t a V m , = l i n e a r e l a s t i c i t y tensor for the Ei jka material, = doij noij t = incremental t o t a l s t r a i n , = E AFi,ATi t r u e s t r e s s increment i n t h e c u r r e n t increment i n t h e p r e s s u r e of p l a s t i c i t y , t~ = - ~= E : ~ ~ ~ A ; ~ ~ A ~ current constitutive relation tensor, r e l a t i n g t o t a l increment of s t r e s s t o t o t a l increment of s t r a i n , and, = p r e s c r i b e d increments of body f o r c e s , and s u r f a c e t r a c t i o n s , r e s p e c t i v e l y , i n t h e c u r r e n t increment. The m a t r i x EEjka i s d e r i v e d from t h e c l a s s i c a l p l a s t i c i t y t h e o r y based on ( a ) Huber-Mises-Hencky y i e l d c o n d i t i o n , (b) D r u c k e r ' s n o r m a l i t y condition f o r incremental p l a s t i c s t r a i n , and (c) Z i e g l e r ' s m o d i f i c a t i o n of P r a g e r ' s kinematic hardening r u l e , The form of EEjka f o r t h e c a s e s of p l a n e s t r a i n an p l a n e s t r e s s , r e s p e c t i v e l y , i s given i n Appendix A The Euler e q u a t i o n s corresponding t o 6Arr(6Aui; 6 T ~ i ; G A v i )= 0 l e a d s t o , (EijkAAckA-Au".) t 1 J > .J+ A F i + [ o q j , j + F?] = O i n Vm (4 An(Aui,Avi,TLi) = M r L {[, ([uyj t - Ao..ij + -12 AsKA fk t 'ijkA1 " i j nui = A V ~ at avm m = l "m r - + AFJ LFY / 1 \ nui) (7; + ATi) dV + a m' T~~ = T; - Avids} + (3) m + f i n i t e element, e n t i r e boundary of t h e mth am' element, - a p o r t i o n of a V m where s u r f a c e t r a c t i o n s m 0 -0 JC nuij i n i t i a l s t r e s s , body f o r c e , and s u r f a c e t r a c t i o n , r e s p e c t i v e l y , i n t h e s t a t e SN - adjustment t o t h e e l a s t i c over-pred i c t i o n of s t r e s s i n each " p l a s t i c i t y c o r r e c t i o n " i t e r a t i o n during t h e curr e n t increment ( " i n i t i a l - s t r e s s " i t e r a t i o n of Zienckiewicz e t a 1 (I2)) . A ui 6 Eq. (5) s t a t e s t h a t t h e t r a c t i o n s d e r i v e d from t h e assumed i n c r e m e n t a l i n t e r i o r displacements match independently assumed boundary t r a c t i o n s ( T L ~ ) . F i n a l l y , Eq. (6) i s t h e statement o f i n t e r e l e m e n t displacement c o m p a t i b i l i t y t h a t i s e n f o r c e d , i n t h e p r e s e n t method, by means of Lagrange m u l t i p l i e r s , TLi. a r e given, ,Fi,Ti= so Eq. ( 4 ) r e f e r s t o t h e e q u i l i b r i u m of t o t a l I f s t a t e sN was i n t r u e e q u i l i stresses i n SNtl. brium, t h e term ~ $ ' j , j FP i n Eq. ( 4 ) would be equal t o zero. However, due t o t h e i n h e r e n t num e r i c a l e r r o r s i n t h e peicewise l i n e a r incremental s o l u t i o n p r o c e s s , t h e s t a t e SN may n o t be i n t r u e e q u i l i b r i u m . Thus, r e t a i n i n g t h e p r i o r - e q u i l i brium e r r o r term oOj FP = s i 0 i n Eq. ( 4 ) l e a d s t o an e q u i l i i r i u m check i t e r a t i o n , s i m i l a r t o t h e one used by Hofmeister e t a l ( 1 3 ) . I n t h e above, domain of t h e mth at m 9 a + A T. T ~ ~ ( A Aui) v ~ ds a r b i t r a r i l y assumed displacements, i n each element, t h a t need n o t s a t i s f y 142 where $ I S and C Y ' S are undetermined parameters and Aq's are incremental nodal displacements. The functions in the matrices A and R are appropriately &osen, to correspond to singular stresses/strains in elements in the immediate vicinity of the crack tip, but are otherwise arbitrary. The functions in the matrix L are functions at the boundary aVm that uniquely interpolate for Av at aVm in terms of relevant nodal Aq's. The element total strains are derived from the Using the approxelement (Au), as (net) = [W](p). imations ( 8 ) in the functional of Eq. ( 3 ) , we Using the common procedure for "assembling" the finite elements, the functional An is expressed in terms of the nodal displacements of the entire structure, Aq". Thus, An = $ [Aq*][Kl(Aq") - [Ag]{AFl + AQo + AQ'] . (16) From Eq. (16), the final finite-element equations can be derived as, where, [K] is the linear-elastic stiffness matrix of the entire structure, {Aq*) are the nodal displacements of the system, (AF1] are the consistent nodal forces in the current increment, {AQO) are "nodal-force-imbalance'' vector to check the equilibrium of the structure at the beginning of the increment, and {As') are the equivalent nodal forces due to the iterative "plasticity-correction" adjustments associated with the "initial-stressiteration" approach. Thus in the present solution procedure, during each increment, (AQ') a. taken to be zero initially and at the end of the plasticity correction iteration, it should con1:erge to zero. It is noted that in the present approach, the stiffness matrix [K] remains the same in all the increments (and thus needs to be "inverted" only once) and this facilitates the extension of the analysis procedure to study slow stable crack growth or fatigue crack growth situations. (AF") Field Assumptions for a Circular-Sector Shaped "Singular-I:lement" Near the Crack-tip T -0 [L] (T }ds = AS indicated in the previous section the present hybrid-displacement formulation calls for assumed functions, in each element, for (i) element-interior displacement field Aut ; (ii) element-boundary displacement field Avi, and (iii) an equilibrated element boundary-traction field TLi. Such assumptions for elements away from the crack-tip have already been presented(ll). Only the assumption for circular-sector shaped "singular-elements" surrounding the crack-tip are given here. It is first noted that the structure of the dominant singularity, for pure-mode plane problems, and for the nonlinear material model employed, was shown in(3y4) to be, m where, [E] is the matrix of elasticity constants, an$ ( 0 " ) and{Aci"} are representations for cryj and Auij, respectively. Since the undetermined parameters (CY)and (p) are arbitrary and independent for each element, taking variations of An with respect to ( 8 ) and (CY)yields, respectively, for each element, and 0.. 1J Using the above equations to express (CY)and ( p ) , for each element, in terms of the respective (Aq) of each element, the functional An can be expressed entirely in terms of the element nodal displacements, (Aq) , only. Thus, M 7 An(Aq) (i[Aql[Kml(Aq) >, m= 1 = + [Asl{AFl+AQo+AQ') - - 1/ n+l-ciij(8); K r P eij - K,r -n/n+1E ij ( 0 ) and where (r,6) are polar coordinates centered at the crack-tip and n is the exponent in the material hardening law. The geometrical shape of the special crack-tip element is a six-noded circular-sector as shown in Fig. 1. In this "singular-element", the following assumptions have been made. constant where, B 1 Element-interior displacement (singular-element) Aur 143 = Blr+ 2 p 2 S + B 3 r € 2 + p 5r2 2 +p6r 0 + P,, + P12 cos + Bl4 l/n+l cos 'pr P,,s i n - P12 p,, l/n+l 8 2 'pr s i n 'pr - + l/n+l 'prl'wl s i n 'pr 1/*1 0 cos 'pr - p,, e experimental r e s u l t s have been p r e v i o u s l y r e p o r t e d by Begley and Landes(5). The compact t e n s i o n specimen i s of one i n c h t h i c k n e s s , and of i n p l a n e dimensions a s shown i n F i g . 2 . The specimen i s made of A533B p r e s s u r e v e s s e l s t e e l , whose m a t e r i a l property, i . e . , uniaxial s t r e s s - s t r a i n curve, i s shown i n Fig. 3 . This u n i a x i a l s t r e s s - s t r a i n curve, f o r t h e convenience of implementation i n the p r e s e n t numerical procedure has been c l o s e l y approximated by an a p p r o p r i a t e s t r a i n - h a r d e n i n g type Ramberg-Osgood law a s a l s o shown i n F i g . 3 . Even though t h e l o a d i n g of t h e compact t e n s i o n s p e c i men i n t h e a c t u a l experiment i n v o l v e s a p i n - h o l e i n t h e specimen, t h e e f f e c t of t h i s p i n has n o t been simulated i n t h e p r e s e n t numerical procedure; i n s t e a d , a p o i n t l o a d i n g has been assumed. s i n 'prl / n + l + p13 + P,., I./n+l s i n 'pr s i n 'pr e 2 + p13 COS l/n+l e l/n+l (pr For t h e p r e s e n t compact t e n s i o n specimen, t h e i n t e r p r e t a t i o n of t h e J - i n t e g r a l , used by Begley and Landes(5) i n t h e i r experimental procedure, i s given by where, cp i s the g l o b a l angular c o o r d i n a t e , and 8 i s t h e angle measured from t h e e l e m e n t ' s symmetric axis. 6 or J = 0 It has been e s t a b l i s h e d t h e combination of a r e g u l a r displacement f i e l d (PI t h r u 89) and t h e asymptotic displacement f i e l d ( @ l ot h r u p15) allows one t o use "singular-elernents", n e a r t h e c r a c k - t i p , of a s l a r g e a s i z e as a t e n t h of t h e crack l e n g t h , without l o o s i n g numerical accuracy. B 2 Element-Boundary Displacement F i e l d ( s i n g u l a r ) element) For t h e " s i n g u l a r - e l e m e n t " , Avi i s assumed t o be: along AB, AC: of t h e type: alrlfn+'+ along BC: of t h e type: bl + b20 a r 2 + a 3 + b3B 2 r 3 7 8 * 3 r 3e 3 The e m p i r i c a l formula, E q . ( 2 3 ) , o f Rice e t a l , f o r t h e compact t e n s i o n specimen, c o n s i d e r s only t h e e f f e c t of t h e bend ending moment of t h e applied l o a d a t t h e n e t s e c t i o n ligament. More r e c e n t l y , Merkle and Corten(14) d e r i v e d an improved formula f o r t h e J - i n t e g r a l , f o r t h e compact t e n s i o n s p e c i m e n , which can again be e v a l u a t e d from a s i n g l e specimen t e s t d a t a f o r t h e load- displacement curve. T h i s improved formula t a k e s i n t o account t h e e f f e c t s of both t h e a x i a l f o r c e and bending moment a t t h e n e t - s e c t i o n li ament. The e m p i r i c a l formula ) be s t a t e d a s , of Merkel and C ~ r t e n ( ? ~can + a g r4 + alor 48 + CYllr4e2 + cyl2' 4e 3 +cy (22) where b i s t h e l e n g t h of t h e uncracked ligament. Thus J has t h e simple d e f i n i t i o n : twice t h e work of deformation d i v i d e d b t h e uncracked ligament l e n g t h . Rice e t a l ( G have hypothesized t h a t t h e E q . (23) i s a p p l i c a b l e when t h e notch- depth i s s u f f i c i e n t l y h i g h so t h a t t h e p l a s t i c i t y encountered i s confined t o t h e uncracked ligament. 4 + c u G3r e + c y r 3 e 2 + u dP P where P i s t h e load and 6 i s t h e load p o i n t d i s placement. The above r e l a t e J t o t h e r a t e of change w i t h r e s p e c t t o c r a c k s i z e , a , of t h e a r e a under t h e load v e r s u s t h e l o a d - p o i n t displacement, P v s . 6 , c u r v e s . Thus i n t h e experimental evaluat i o n of J , t h e s e P v s . 6 curves a r e generated f o r d i f f e r e n t crack s i z e s , a , i n v i r g i n specimens subj e c t t o monotonic l o a d i n g . P(d6crack) $ ( r , e ) = u l r 2 + c u r e2 + ( u 3 r 2 e 2 + c u r2e 3 5 ai 6 crack B3 Element Boundary- Traction F i e l d ( s i n g u l a r element) The t r a c t i o n s T L ~f o r t h e " s i n g u l a r element" a r e assumed from t h e s t r e s s e s derived from t h e Airy s t r e s s - f u n c t i o n , +cy ,' I n specimens such a s t h e compact t e n s i o n , where t h e uncracked ligament i s s u b j e c t p r i n c i p a l l y t o bending but t h e load i s a p p l i e d by concentrated load P , i t has r e c e n t l y been shown by Rice, P a r i s and Merkle(6) t h a t t h e J - i n t e g r a l can be d i r e c t l y e v a l u a t e d from a s i n g l e specimen load- displacement curve, thus The above assumption can be seen t o s a t i s f y t h e i n t e r - e l e m e n t displacement c o m p a t i b i l i t y . 2 s \.a p (-%l6d6 J = 2 n+2/n+l Or 15 I t can be seen t h e assumptions i n s e c t i o n B 1 through B3 correspond t o approximations t o t h e asymptotic s o l u t i o n , 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 given i n E q . ( 1 8 ) . n J = - - Results For purposes of e v a l u a t i o n of t h e p r e s e n t num e r i c a l procedure, i t has been a p p l i e d t o s o l v e t h e c a s e of a compact t e n s i o n specimen, f o r which 2 +-a' b 144 (1-&-a t h e e n t i r e uncracked ligament has y i e l d e d . This uncontained p l a s t i c i t y i n the c a s e of p l a n e s t r e s s i s b e l i e v e d t o have s i g n i f i c a n t e f f e c t s on slow s t a b l e c r a c k growth, i n p l a n e s t r e s s s i t u a t i o n s , which w i l l be s t u d i e d i n a subsequent paper. where, and f u r t h e r , a = c r a c k l e n g t h , c = h a l f t h e n e t ligament width ( s e e F i g . 2 ) , P = applied f o r c e , B = specimen t h i c k n e s s , K = e l a s t i c s t i f f n e s s a t the load p o i n t , A p = p l a s t i c displacement of t h e applied load due t o t h e c r a c k , and b = 2c. I n t h e p r e s e n t numerical f o r m u l a t i o n , s i n c e s t r e s s / s t r a i n d a t a a t each p o i n t and a t each load bend a r e c a l c u l a t e d , t h e J - i n t e g r a l i s e v a l u a t e d numerically u s i n g t h e formula of Eq. (2) d i r e c t l y . This d i r e c t e v a l u a t i o n of J a s a p a t h - i n t e g r a l , has been c a r r i e d o u t i n t h e p r e s e n t s t u d y , along four d i f f e r e n t p a t h s a t each load l e v e l , a s shown i n Fig. 4 . The average of t h e s e f o u r d i f f e r e n t p a t h i n t e g r a l s i s taken t o be t h e v a l i d J a t t h e p a r t i c u l a r load l e v e l . Using t h e numerically e v a l uated P v s . 6 c u r v e , t h e J - i n t e g r a l i s a l s o evaluat e d u s i n g t h e above c i t e d e m p i r i c a l formulae of Rice e t a1(6) and Merkle and Corten(14). F i n a l l y t h e p r e s e n t l y computed J - i n t e g r a l v a l u e s w i l l be compared w i t h t h e experimental r e s u l t s of Begley and Landes (5) . D e t a i l e d R e s u l t s f o r t h e Compact Tension Specimen F i r s t we n o t e t h a t i f , i n t h e assumed f i e l d f u n c t i o n s f o r t h e " s i n g u l a r element" a s d e t a i l e d e a r l i e r , one s e t s n = 1, then t h e assumptions correspond t o an approximation f o r t h e asymptotic solutions for displacements/stresses i n the l i n e a r e l a s t i c c a s e . Thus i n t h e f i r s t increment of t h e p r e s e n t i n c r e m e n t a l s o l u t i o n procedure, s e t t i n g n = 1, t h e l i n e a r e l a s t i c c a s e can be computed. The v a l u e o f t h e normalized s t r e s s i n t e n s i t y f a c t o r , i n the l i n e a r e l a s t i c case, as calculated d i r e c t l y from t h e b u i l t - i n asymptotic s t r e s s / d i s p l a c e m e n t f i e l d i n t h e p r e s e n t procedure, was found t o be KI/P = 6.602. T h i s i s i n e x c e l l e n t agreement w i t h t h a t r e p o r t e d by Bucci e t a l ( 7 ) , f o r s i m i l a r geometry, which i s KI/P = 6.79 . The J - i n t e g r a l , f o r t h i s l i n e a r e l a s t i c c a s e , was a l s o d i r e c t l y computed around f o u r d i f f e r e n t p a t h s as shown i n F i g . 4 . The average v a l u e of J f o r t h e s e f o u r p a t h s , (with a +_ 1%v a r i a t i o n between t h e p a t h s ) was found l b - i n / i n 2 p e r u n i t l o a d , which t o be .1477 X corresponds t o a v a l u e KI/P = 6.880. Thus i t i s found t h a t t h e s t r e s s - i n t e n s i t y f a c t o r e v a l u a t i o n from t h e J - i n t e g r a l i s a l s o h i g h l y a c c u r a t e i n t h e p r e s e n t procedure. F i g . 7 shows t h e r e s u l t s f o r l o a d - l i n e d i s p l a c e ment ( a t p o i n t A i n F i g . 4) v a r i a t i o n w i t h load f o r t h e c a s e s of p l a n e s t r e s s and p l a n e s t r a i n , o b t a i n ed i n t h e p r e s e n t a n a l y s i s , u s i n g t h e RambergOsgood m a t e r i a l behavior approximation a s shown i n F i g . 3. Also shown, f o r comparison, a r e t h e r e s u l t s f o r both p l a n e s t r a i n and p l a n e s t r e s s c a s e s based on l i n e a r - e l a s t i c a n a l y s i s a s w e l l a s t h e e m p i r i c a l r e s u l t s of Bucci, P a r i s , Landes, and Rice ( 7 ) based on 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 behavior- assumption. I t can be seen t h a t , when t h e specimen y i e l d s , t h e r e i s a s u b s t a n t i a l d i f f e r e n c e between t h e v a l u e s of t h e l o a d - p o i n t displacement ( a t t h e same l o a d ) f o r t h e c a s e s of p l a n e s t r e s s and p l a n e s t r a i n . Moreover, i n t h e p l a n e s t r a i n c a s e , t h e r e i s a good agreement of t h e p r e s e n t r e s u l t s w i t h t h o s e of Bucci, e t a l ( 7 ) , w h i l e such a comparison i s r a t h e r poor i n t h e c a s e of p l a n e stress. F i g . 8 shows t h e v a r i a t i o n of t h e v a l u e of t h e J - i n t e g r a l w i t h t h e l o a d - l i n e displacement f o r t h e p l a n e s t r a i n c a s e . The J - i n t e g r a l was c a l c u l a t e d , u s i n g Eq. ( 2 ) , over f o u r d i f f e r e n t p a t h s which a r e shown i n F i g . 4 . The average v a l u e of J f o r t h e s e f o u r p a t h s (with a +_ 2% v a r i a t i o n between t h e p a t h s ) i s p l o t t e d i n Fig. 8. A l s o shown f o r comp a r i s o n purposes a r e , ( i ) e s t i m a t i o n of J - i n t e g r a l based on l i n e a r - e l a s t i c t h e o r y ( e s s e n t i a l l y a " s m a l l - s c a l e y i e l d i n g " approximation), ( i i ) e s t i mation o f J from t h e computed load- displacement curve, of Fig. 7 , u s i n g t h e e m p i r i c a l formula of Rice e t a 1 ( 6 ) , i . e . , E q . ( 2 3 ) , and, ( i i i ) estimat i o n J from t h e computed load- displacement curve u s i n g t h e e m p i r i c a l formula of Merkle and Corten(l4) i . e . , Eq. ( 2 4 ) . The c r i t i c a l v a l u e of J l c , e x p e r i m e n t a l l y o b t a i n e d by Begley and Landes f o r t h e p r e s e n t one- inch t h i c k specimen, a s w e l l a s G l c obt a i n e d by Bucci e t a l ( 7 ) u s i n g a 1 2 i n c h t h i c k specimen a r e a l s o shown. I t can be seen from Fig. 8 , t h a t t h e p r e s e n t d i r e c t computation of J agrees w e l l w i t h t h e e m p i r i c a l formula of Merkle and C o r t e n ( l 4 ) r a t h e r than w i t h t h a t of Rice e t a1(6) a s can be expected, becuase t h e former e m p i r i c a l r e l a t i o n accounts f o r b o t h t h e a x i a l f o r c e and bending moment e f f e c t s i n t h e compact t e n s i o n specimen. The s u b s t a n t i a l d i f f e r e n c e between t h e .023 i n ) m d t h e experimental computed Jlc( a t 6 c Jlc obtained i n (5) s u g g e s t s t h a t t h e p l a n e s t r a i n assumption i s v e r y poor f o r t h e p r e s e n t one i n c h t h i c k specimen. Y To understand t h e i n f l u e n c e of t h e assumption of t h e p l a n e s t r e s s o r a l t e r n a t i v e l y t h e p l a n e s t r a i n s t a t e i n t h e one- inch t h i c k specimen shown i n Fig. 2 , both c a s e s were t r e a t e d i n t h e p r e s e n t elastic- plastic analysis. Fig. 9 shows t h e v a r i a t i o n of t h e average v a l u e of t h e d i r e c t l y computed J - i n t e g r a l ( o v e r f o u r p a t h s w i t h a +_ 2% v a r i a t i o n between t h e p a t h s ) w i t h t h e l o a d - l i n e displacement f o r t h e p l a n e s t r e s s c a s e . Once a g a i n , f o r comparison purposes, t h e e s t i m a t i o n of J , based on ( i ) l i n e a r - e l a s t i c t h e o r y , ( i i ) t h e computed load- displacement curve and u s i n g t h e e m p i r i c a l formula of Rice e t a l , and ( i i i ) t h e computed load- displacement curve and usi n g t h e r e l a t i o n given by Merkle and Corten, a r e shown. I t can be seen t h a t t h e e m p i r i c a l formula of Merkel and Corten(14) p r o v i d e s an e x c e l l e n t agreement w i t h t h e p r e s e n t d i r e c t computation of J where as t h e formula of Rice e t a 1 p r o v i d e s an u n d e r e s t i m a t i o n by about 10%. The c l o s e n e s s of t h e p r e s e n t p l a n e s t r e s s r e s u l t f o r J ( a t 6-.023in) Fig. 4 shows t h e f i n i t e element model of t h e compact t e n s i o n specimen t h a t i s used, w i t h 30 elements, 93 nodes and 170 degrees of freedom. Fig. 5 shows t h e p l a s t i c zones i n t h e specimen a t v a r i o u s l o a d l e v e l s , when t h e c o n d i t i o n s of p l a n e S t r a i n a r e invoked. F i g . 6 , l i k e w i s e , shows t h e y i e l d zones i n t h e specimen where p l a n e s t r e s s cond i t i o n s a r e assumed. It 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 t h e c a s e of p l a n e s t r a i n t h e uncracked l i g ament, immediately ahead of t h e c r a c k - t i p does n o t Yield even a t a load of P = 22.5 k i p s ; where a s i n t h e c a s e of p l a n e s t r e s s , a t P = 20.8 k i p s , almost 145 and t h e experimental J of Begley and Landes(5) s u g g e s t s t h a t , f o r t h e p r e s e n t specimen of one inch thickness, plane s t r e s s conditions prevail a t t h e c r a c k - t i p . The experimental r e s u l t f o r Jlc f o r t h e p r e s e n t specimen'; obtained i n (5) i s s t i l l lower by about 14% t h a t t h e p r e s e n t p l a n e s t r e s s r e s u l t . I t i s f e l t t h a t some cause of t h i s may be a t t r i b u t e d t o t h e p o s s i b l e presence of s t a b l e c r a c k growth, i n p l a n e s t r e s s s i t u a t i o n s , p r i o r t o f r a c t u r i n g i n specimens i n t h e experiments. The numeri c a l s i m u l a t i o n of s t a b l e c r a c k growth i s p r e s e n t l y being c a r r i e d o u t by t h e a u t h o r s and w i l l be r e p o r t e d i n a subsequent p a p e r . F i g s . 10 and 11 show t h e c r a c k - s u r f a c e deformat i o n p r o f i l e s , a t v a r i o u s load l e v e l s , f o r t h e c a s e s of p l a n e s t r a i n and p l a n e s t r e s s , r e s p e c t i v e l y . I n both c a s e s , t h e b l u n t i n g o f t h e c r a c k - t i p can be n o t i c e d a t load l e v e l s g r e a t e r than approximately 1 5 k i p s . To a s s e s t h e e f f e c t of t h i s b l u n t i n g on t h e s t r e s s / s t r a i n s t a t e s n e a r t h e c r a c k - t i p r e q u i r e s a f i n i t e deformation a n a l y s i s , which i s c u r r e n t l y being c a r r i e d o u t and w i l l be r e p o r t e d It i s a l s o n o t i c e d t h a t i n a subsequent paper. f o r t h e most p a r t , t h e c r a c k s u r f a c e deformation p r o f i l e i s n e a r l y l i n e a r away from t h e immediate v i c i n i t y o f t h e c r a c k t i p . The d i s t r i b u t i o n o f t h e e f f e c t i v e s t r a i n ahead of t h e c r a c k - t i p , i n t h e uncracked ligament, f o r t h e c a s e s of p l a n e s t r a i n and p l a n e s t r e s s a r e shown i n F i g s . 1 2 and 13, r e s p e c t i v e l y . The s i n g u l a r n a t u r e of t h e s t r a i n a t t h e c r a c k - t i p i s immediately n o t i c e d , and i t i s the manifestation of the b u i l t - i n s i n g u l a r i t i e s i n t h e p r e s e n t f i n i t e element procedure. F i g s . 14 and 1 5 , r e s p e c t i v e l y , show t h e d i s t r i b u t i o n of t h e e f f e c t i v e stress ahead o f t h e c r a c k - t i p i n t h e uncracked ligament. Once a g a i n t h e s i n g u l a r i t y i n s t r e s s e s , even though weaker than i n s t r a i n s , can be n o t i c e d n e a r t h e c r a c k - t i p , and i s again a m a n i f e s t a t i o n of t h e b u i l t - i n s i n g u l a r i t i e s i n t h e p r e s e n t procedure. It 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 t h e c a s e of p l a n e s t r a i n , t h e e f f e c t i v e s t r e s s , i n t h e immediate v i c i n i t y of t h e c r a c k - t i p , dec r e a s e s d u r i n g t h e i n i t i a l increments of l o a d i n g , but i n c r e a s e s monotonically f o r t h e most p a r t of t h e loading. T h i s s l i g h t i n i t i a l unloading n e a r t h e c r a c k - t i p d i d n o t appear t o have a f f e c t e d t h e results significantly. Discussion and Conclusions I n c o n t r a s t t o t h e common t h e o r e t i c a l d e f i n i t i o n s of p l a n e stress and p l a n e s t r a i n , c h a r a c t e r i z i n g t h e s t r e s s and s t r a i n s t a t e s throughout t h e body, i n f r a c t u r e mechanics terminology, " p l a n e s t r e s s f r a c t u r e " and " plane s t r a i n f r a c t u r e " a r e g e n e r a l l y meant t o d e s c r i b e s t r e s s and s t r a i n c o n d i t i o n s In w i t h i n t h e p l a s t i c zone a t t h e c r a c k - t i p ( l 5 ) . f l a t t e s t specimens w i t h t h r o u g h - t h e - t h i c k n e s s c r a c k s , such a s t h e compact t e n s i o n specimen t r e a t e d here, the r a t i o of the crack- tip p l a s t i c zone s i z e t o specimen t h i c k n e s s becomes t h e c r i t e r i o n o f "plane s t r e s s " v e r s u s t h e "plane s t r a i n ' ' c o n d i t i o n s , The p l a s t i c zone s i z e s f o r t h e p r e s e n t compact t e n s i o n specimen, a s seen from F i g s . 6 and 5 , a r e comparable t o t h e specimen t h i c k n e s s of one i n c h . Hence t h e c o n d i t i o n s of p l a n e s t r e s s can be concluded t o e x i s t n e a r t h e c r a c k - t i p . T h i s i s a l s o seen from t h e c l o s e agreement o f t h e p r e s e n t plane s t r e s s r e s u l t s and t h e experimental r e s u l t s of (5) f o r t h e one i n c h t h i c k specimen. I t has a l s o been e s t a b l i s h e d t h a t f o r compact t e n s i o n t e s t specimens, t h e e m p i r i c a l r e l a t i o n given by Merkle and C o r t e n ( l 4 ) i s much more a c c u r a t e t h a n t h a t given by Rice e t a l ( 6 ) t o e v a l u a t e J from a s i n g l e specimen t e s t d a t a f o r t h e load v s . displacement behavior. The c l o s e agreement o f t h e p r e s e n t J - i n t e g r a l r e s u l t s w i t h t h e experimental r e s u l t s f o r t h e p r e s e n t compact t e n s i o n specimen a s w e l l a s o t h e r t e s t specimens (which a r e n o t r e p o r t e d h e r e due t o s p a c e l i m i t a t i o n s ) i n d i c a t e t h a t t h e p r e s e n t numeri c a l procedure can be used t o a c c u r a t e l y complete t h e J - i n t e g r a l i n p l a n e problems i n v o l v i n g a r b i t r a r y domains, a r b i t r a r y l o a d i n g t h a t causes l a r g e s c a l e y i e l d i n g n e a r t h e c r a c k , and a r b i t r a r y s t r a i n hardening m a t e r i a l s . Comparing t h i s J i n an a r b i t r a r y problem w i t h t h e e x p e r i m e n t a l l y determined Jc (which then becomes a m a t e r i a l p r o p e r t y ) governi n g t h e o n s e t o f c r a c k growth, t h e s i t u a t i o n can t h u s be a s s e s s e d whether t h e crack i n t h e given c a s e i s on t h e verge of growing o r n o t . The r e l a t i v e p a t h independence ( w i t h i n +_ 2% v a r i a t i o n ) of t h e computed J - i n t e g r a l i n d i c a t e s t h a t i t i s indeed a v a l i d parameter t o be used i n a ductile fracture i n i t i a t i o n criterion. As F i n a l l y , t o s t u d y t h e e f f e c t s of t h e mathematis t a t e d e a r l i e r , t h e c r i t e r i o n of c r i t i c a l Jc i s c a l modeling of t h e u n i a x i a l stress s t r a i n curve, o n l y a c r i t e r i o n f o r i n i t i a t i o n of crack growth an a l t e r n a t e Ramberg-Osgood l a w , c l o s e l y approxiw i t h o u t any statement of s t a b i l i t y o r i n s t a b i l i t y mating an e l a s t i c - p e r f e c t - p l a s t i c behavior was of such a c r a c k growth. T y p i c a l l y , c r a c k s do n o t used. T h i s curve was taken t o be e = o/E+ ( U / B ~ ) ~ a b r u p t l y begin t o propagate i n e l a s t i c - p l a s t i c f o r ci 2 u y and E = o / E f o r ci < oy; w i t h n = 50, solids. Instead it i s usual that the i n i t i a t i o n B = 8 . 1 X l o 4 p s i , ciy = 70 X lo3 p s i , E = 29.2 X of c r a c k e x t e n s i o n i s followed by s t a b l e growth lo6 p s i . Thus, t h i s p a r t i c u l a r modeling c o r r e under a continuous i n c r a s e of t h e a p p l i e d load o r sponds almost t o p e r f e c t p l a s t i c behavior a f t e r a t l e a s t of t h e l o a d p o i n t displacement. Ultimateci = 70 x lo3 psi: No s i g n i f i c a n t d i f f e r e n c e s i n l y , t h e required increase f o r continuing quasit h e main r e s u l t , namely t h a t f o r t h e J - i n t e g r a l , s t a t i c c r a c k advance f a l l s t o z e r o , and u n s t a b l e w a s n o t i c e d , even though t h e r e s u l t s d i f f e r e d propagation f o l l o w s . Sometimes t h e s t a b l e growth phase i s s o u n n o t i c e a b l e t h a t i n i t i a t i o n and prosomewhat, i n d e t a i l f o r s t r e s s e s , a s compared with t h e r e s u l t s u s i n g t h e s t r a i n - h a r d e n i n g type Ramberg p a g a t i o n a r e e s s e n t i a l l y c o i n c i d e n t . This i s Osgood approximation f o r t h e m a t e r i a l behavior, a s o f t e n t h e c a s e f o r "plane s t r a i n f r a c t u r e " under shown i n F i g . 3 . The d i f f e r e n c e s between t h e s m a l l - s c a l e y i e l d c o n d i t i o n s . However, t h e ext e n t s of s t a b l e c r a c k growth i n "plane s t r e s s " r e s u l t s u s i n g t h e two d i f f e r e n t types of Rambergs i t u a t i o n s , i n t h e s h e e t s , can be s u b s t a n t i a l as Osgood approximation t o t h e s t r e s s - s t r a i n l a w can observed by Brock(16), Link and Muntz(17), and be seen i n F i g s . 7, 8 and 9 . Bergkvist and Andersson(18). Thus i t i s f e l t t h a t The e x p e r i m e n t a l l y determined crack-mouth opena r e a l i s t i c a n a l y t i c a l framework of f r a c t u r e must i n g ( a t p o i n t B i n F i g . 4 ) of .07 i n . corresponds i n c l u d e n o t only models f o r i n i t i a t i o n of crack t o t h e crack s u r f a c e deformation a t t h e load l i n e growth, but a l s o models f o r subsequent s t a b l e ( p o i n t A i n F i g . 4) of about .023" a s seen i n c r a c k growth,and e s p e c i a l l y f o r i t s t e r m i n a l l o s s F i g . 11). of s t a b i l i t y . Such a f i n i t e element modelling of * 146 Assumed Displacement Hybrid F i n i t e Element Procedure," AIAA J o u r n a l , Vol. 13, No. 6 , pp. 734-740, 1975. s t a b l e c r a c k growth, i n v o l v i n g t h e t r a n s l a t i o n of e n t i r e " s i n g u l a r " n e a r - t i p elements, i s c u r r e n t l y being c a r r i e d o u t and t h e r e s u l t s w i l l be r e p o r t e d in a subsequent p a p e r . Ackhowledgements This work was i n i t i a t e d under t h e support of t h e U . S . A i r Foce O f f i c e of S c i e n t i f i c Research under g r a n t , 74-2667. I n r e c e n t months, t h e a u t h o r s ' work i n t h e b a s i c f i n i t e element methods, such as used h e r e , has been supported by t h e National cience Foundation, under g r a n t ENG 74-21346. The authors g r a t e f u l l y acknowledge t h e s e s u p p o r t s , and the encouragement given by D r . W . J . Walker i n particular. References 1. Eshelby, J. D . , "The Continuum Theory of L a t t i c e D e f e c t s , " S o l i d S t a t e P h y s i c s , Vol. 111, Academic P r e s s , 1956. 2 . Rice, J . R . , "A Path Independent I n t e g r a l and t h e Approximate Analysis of S t r a i n Concentrat i o n by Motches and Cracks,'' J o u r n a l of Applied Mechanics, Trans. ASME, June 1968, pp. 379-386. 1 2 . Zienckiewicz, 0 . C . , Valliappan, S . , and King, I . P . , " E l a s t o - P l a s t i c S o l u t i o n s of Engineering P r o b l e m s , ' I n i t i a l S t r e s s ' F i n i t e Element Approach ,'I I n t e r n a t i o n a l J o u r n a l of Numeri c a 1 Methods i n Engineering, Vol. 1, pp. 75-100, 1969. 13. Hofmeister, L . D . , Greenbaum, G . A . , and Evensen, D . A . , "Large S t r a i n , E l a s t o - P l a s t i c F i n i t e Element Analysis," Proceedings AIAAIASMEI SAE 1 1 t h SDM Conference, Denver, Colorado, A p r i l 1970. 14. Merkle, J . G . , and Cortan, H. T . , "A J - I n t e g r a l Analysis f o r t h e Compact Specimen, Considering Axial Force a s Well a s Bending E f f e c t s , " J o u r n a l of P r e s s u r e Vessel Technology, Nov. 1974, pp. 286-292. 15. Tada, H . , P a r i s , P . C . , and I r w i n , G. R . , "The S t r e s s Analysis of Cracks Handbook," Del Res e a r c h Corporation, Hellertown, P a . , 1973. 16. Broek, D . , I n t e r n a t i o n a l J o u r n a l of F r a c t u r e Mechanics, Vol. 4 , p . 19, 1968. 3. Hutchinson, J . W . , " S i n g u l a r Behavior a t t h e End of a T e n s i l e Crack i n a Hardening Material': J o u r n a l o f Mechanics and Physics of S o l i d s , Vol. 16, 1968, pp. 13-31. 17. Link, F . , and Munti, D . , M a t e r i a l Priif, V o l . 1 3 , p. 407, 1971. 4. Rice, J . R . , and Rosengren, G. F . , " Plane S t r a i n Deformation Near a Crack Tip i n a Power Law Hardening M a t e r i a l , " J o u r n a l of Mechanics and Physics of S o l i d s , Vol. 1 6 , 1968, pp.1-12. 18. B e r g k v i s t , and Andersson, H . , " P l a s t i c Deformation a t a S t a b l y Growing C r a c k - t i p , " I n t e r n a t i o n a l J o u r n a l of F r a c t u r e Mechanics, Vol. 8 , pp. 139, 156, 1972. 5. Begley, J. A. and Landes, J . D . , "The J - I n t e g r a l a s a F r a c t u r e C r i t e r i o n , ' ' F r a c t u r e Tough, ASTM STP 514, 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 , 1972, pp. 1-20. Appendix Incremental E l a s t i c - P l a s t i c C o n s t i t u t i v e Law Using t h e J 2 - t h e o r y o f Huber-Mises-Hencky, the yield c r i t e r i a is written as, 6. Rice, J. R . , P a r i s , P. C . , and Merkle, J . G . , "Some F u r t h e r R e s u l t s of J - I n t e g r a l Analysis and Estimates," Progress i n Flaw Growth and F r a c t u r e Toughness T e s t i n g , ASTM STP 536, American S o c i e t y of T e s t i n g and M a t e r i a l s , 1973, pp. 231-245. 3 " f(J2) = - a a 2 ij ij - ay* = 0 ; where I (5 7 . Bucci, R. J . , P a r i s , P. C . , Landes, J . D . , and Rice, J . R . , " J - I n t e g r a l Estimation Procedures:' F r a c t u r e Toughness, ASTM STP 514, American S c o e i t y of T e s t i n g and M a t e r i a l s , 1972,pp .4070. ij = U ij 1 6 3 KX i j --u (A. 1) The f l o w - r u l e dcp = dh ( a f / a a . .) i s used. The ij 11 P r a g e r - Z i e g l e r hardening r u l e w i t h t h e subsequent yield surface representation, 8 . Goldman, N . L . , and Hutchinson, J . W . , " F u l l y P l a s t i c Crack Problems: The Center-Cracked S t r i p Under Plane S t r a i n , " I n t e r n a t i o n a l J o u r n a l of S o l i d s and S t r u c t u r e s , Vol. 11, No. 5 , pp. 575-593, 1975. f(oij i s used. that - wij) = 0 ; hij = +(aij - cyij) (A.2) For i n f i n i t e s i m a l increments, i t i s seen 9 . Shih, G . F . , " Small- Scale Yielding Analysis of Mixed Mode Plane S t r a i n Crack Problems," Harvard U n i v e r s i t y Report DEAPS-1, 1973. Using E q s . (A.l) t o (A.3), t h e c l a s s i c a l t h e o r y of p l a s t i c i t y y i e l d s , a f t e r some m a n i p u l a t i o n s , t h e following r e l a t i o n between incremental s t r e s s e s and incremental t o t a l s t r a i n s : 10. A t l u r i , S . N . , Kobayashi, A . S . , and Nakagaki, M . , "An Assumed Displacement Hybrid F i n i t e Element Model f o r L i n e a r F r a c t u r e Mechanics,'' I n t e r n a t i o n a l J o u r n a l of F r a c t u r e , Vol. 11, N O . 2 , pp. 257,-271, 1975. doij = Eijka t 11. A t l u r i , S . N . , Kobayashi,,A. S . , and Nakagaki, M., where Eijka t " F r a c t u r e Mechanics A p p l i c a t i o n of an 147 ds t kR (A. 4) i s r e p r e s e n t e d i n m a t r i x form f o r p l a n e s t r a i n and p l a n e s t r e s s c a s e s , r e s p e c t i v e l y , below. (i.) Plane S t r a i n Case (1-V)E where s = E Y FIG I , NOMENCLATUR 'SINGULAR' ELEMENT. sp -I L = 2.w" -----------I FIG 2. COM?ACT TENTION TEST SPECII.!EN CCNFI2JURCTIOS. The parameter C above corresponds t o t h e s l o p e of t h e u n i a x i a l s t r e s s - s t r a i n curve and t h e parameters cyx, c r y , and cyxy a r e o b t a i n e d from (A. 2) and (A. 3 ) . ( i i ) Plane S t r e s s Case dox do Y = [EK-Sl/S] 2 dex t - [(S1S3)/SI = [vEi'-(S + [vE*-(SlS2)/S] de t Y "y:, S IS] dcX t + [ E * - S 22/ S ] d e yt tl'=F,O 12) R -.6!72 k ; ~ 5 PSI ~ - 7 O U 3 0 PSI I Rut S T H E - S T R M UJRVE 20 I 9 . .01 .02 .03 .05 .O't STROIN FIG 3. UNIAXIAL STRESS STRAIN CURVE FOR pi5338 148 , .Ob E STEEL. .07 I a Q 0 J IO 0 FIG 5. PI ASTlC YlCLD ZONES AT VARIOUS LOAD LEVELS STRAIM-HARDEN'NG) I DEPLACEMENT AT WAD LINE PLANE STRAIN, FIG 6 . ;\I:, ( b (IN ) 6 CV3VES FC? COY?ACT TENSION SPECIMEN STRA!N CO:\'DiTICN 1. 3003 / ' DSPLACEXNT AT LOAD LINE FIG G . PLASTC YI!:I.D ZO!IES AT VA!1IOJS LO43 L N E L S !PLAICT STRCSS, S i R~~IN-H/~l<DCPIINO 1. ( IN 1 FIG 9. J VS. 6 CURVES FGR COMPb,CT TENS;ON SPECI!:IEN (PLPNE STRESS 149 CON3!T!CN ;, FI5 14. EFFFCTIVE 51 R iS DISTRIBIITICN IN ?HE UrlCRACKED LlGAhlEN ( PLLNC STRESS, Ii/Y!CEI\'ING 150 1.