Optimally discretized finite elements for boundary-layer stresses incomposite laminates

A s t u d y of boundarylayer s t r e s s e s i n f i n i t e w id th composite l a m i n a t e s is p r e s e n t e d . Quasi t h ree -d imens iona l , e ight-node i s o p a r a m e t r i c e l e ments a r e used i n c o n j u n c t i o n wi th a r e c e n t l y developed mesh o p t i m i z a t i o n p rocedure . A c o n s i s t e n t s t r e s s e v a l u a t i o n scheme i s in t roduced f o r more a c c u r a t e s t r e s s d e t e r m i n a t i o n i n a s e v e r e l y d i s t o r t e d mesh. S t r e s s e s n e a r t h e l a m i n a t e edge a r e o b t a i n e d by t h e use o f o p t i m a l l y d i s c r e t i z e d f i n i t e e l emen t s . For i l l u s t r a t i o n , r e s u l t s from an o p t i mized mesh c o n s i s t i n g of twelve e l emen t s a r e shown t o demons t r a t e t h e accuracy and e f f i c i e n c y o f t h e c u r r e n t approach. E x c e l l e n t agreement between t h e p r e s e n t s o l u t i o n and t h e s i n g u l a r l a m i n a t e e l a s t i c i t y s o l u t i o n i s o b t a i n e d . Comparison w i t h o t h e r f i n i t e e lement a n a l y s e s hav ing h i g h l y r e f i n e d , nonop t imized g r i d s i s a l s o r e p o r t e d t o i l l u s t r a t e t h e s u p e r i o r performance of t h e p r e s e n t method. 1. I n t r o d u c t i o n New and a c c e l e r a t e d demands t o employ s t r u c t u r a l composi tes i n e n g i n e e r i n g a p p l i c a t i o n s have encouraged r e s e a r c h e r s t o examine many a s p e c t s of t h e mechanics of composi te m a t e r i a l s . Of p a r t i c u l a r i n t e r e s t h a s been t h e complex behav io r o f s t r e s s e s and r e l a t e d f a i l u r e problems n e a r t h e edge of comp o s i t e l a m i n a t e s . The s o l u t i o n of such a problem r e q u i r e s t h e a n a l y s i s o f a l a y e r e d , a n i s o t r o p i c m a t e r i a l sys t em w i t h geometr ic boundar i e s . Due t o t h e complex n a t u r e of t h e problem, v a r i o u s numerical s t u d i e s have been a t t empted t o determine t h e s : r e s s e s and de fo rma t ion i n compos i t e s , f o r example, Refs . 110. Upon examining a f i n i t e w i d t h l a m i n a t e subj e c t e d t o uniform a x i a l e x t e n s i o n , e a r l y r e s e a r c h e r s no ted t h a t l a r g e s t r e s s g r a d i e n t s were e v i d e n t a s t h e l a m i n a t e edge was approached. Thus, a s ingul a r i t y had been deduced and, s u b s e q u e n t l y , v a r i o u s methods were employed i n an e f f o r t t o o b t a i n an i m proved approximat ion o f t h e s t r e s s e s w i t h i n t h e "boundary l a y e r " r e g i o n of t h e l a m i n a t e edge. Rec e n t l y , t h e e x a c t o r d e r o f t h e edge s t r e s s s ingul a r i t y f o r t h e l a m i n a t e e l a s t i c i t y problem h a s beyo de te rmined by an e i g e n f u n c t i o n expansion method,'' and a s s o c i a t e d edge s t r e s s e s have been o b t a i n e d by a boundary -co l loca t ion t echn ique . l2 Due t o i t s v e r s a t i l i t y and s i m p l i c i t y , t h e f i n i t e e lement method has p l ayed a s i g n i f i c a n t r o l e i n s t u d y i n g e l a s t i c i t y p oblems w i t h s i n g u l a r i t y . It h a s been n o t e d l 3 ¶ 14. l 4 t h a t s i g n i f i c a n t l y improved s o l u t i o n accuracy and convergence can be ach ieved f o r t h i s c l a s s of problems by t h e use of a s i n g u l a r e lement f o r m u l a t i o n o r mesh o p t i m i z a t i o n p rocedure . The s i n g u l a r f i n i t e e lement approach r e q u i r e s knowledge o f t h e e x a c t s t r u c t u r e o f t h e s i n g u l a r e l a s t i c i t y s o l u t i o n and, t h e r e f o r e , t h e f o r m u l a t i o n i s r a t h e r s o p h i s t i c a t e d . 14, The r e c e n t l y developed *Assoc ia t e P r o f e s s o r , Dept. of T h e o r e t i c a l and Appl ied Mechanics. Member AIAA. **Graduate A s s i s t a n t , Dept. of T h e o r e t i c a l and Appl ied Mechanics. mesh o p t i m i z a t i o n p rocedure i n f i n i t e e lement ana l ,y s e s p r o v i d e s an a t t r a c t i v e a l t e r n a t i v e f o r examining t h e s i n g u l a r e l a s t i c i t y problem. T h i s approach h a s t h e advantage of u s i n g s i m p l e , c o n v e n t i o n a l , nons i n g u l a r e l emen t s w i t h much fewer degrees of f r e e dom and , t h u s , s o l v e s t h e problem e f f e c t i v e l y and e f f i c i e n t l y , a s demonstra ted i n Refs . 13 , 16 and 17. I n t h i s pape r , a s t u d y o f t h e boundarylayer s t r e s s e s i n composi te l a m i n a t e s i s p r e s e n t e d by employing a s u i t a b l e mesh o p t i m i z a t i o n scheme i n con junc t ion w i t h a q u a s i three-dimensional f i n i t e e lement ana l y s i s . I n t h e n e x t s e c t i o n , a fo rmula t ion o f t h e quas i t h ree -d imens iona l , e ight-node i s o p a r a m e t r i c element i s p r e s e n t e d . B a s i c theo ry and procedure of mesh o p t i m i z a t i o n f o r t h e problem a r e g iven i n S e c t i o n 3. Discuss ion of a c o n s i s t e n t s t r e s s e v a l u a t i o n scheme f o r de te rmin ing s t r e s s e s a l o n g t h e p l y i n t e r f a c e i n a composi te l a m i n a t e is r e p o r t e d i n S e c t i o n 4 . Num e r i c a l examples a r e shown i n S e c t i o n 5. Comparis o n s w i t h t h e l amina te e l a s t i c i t y s o l u t i o n and prev ious numer ica l a n a l y s e s employing h i g h l y r e f i n e d g r i d s a r e made t o i l l u s t r a t e t h e accuracy and e f f i c i ency of t h e p r e s e n t approach. 2 . F i n i t e Element Formulat ion I n g e n e r a l , t h e complex s t r e s s s t a t e i n a f i n i t e d i m e n s i o n a l composi te l amina te w a r r a n t s t h e use of a th ree -d imens iona l e l emen t . However, s i g n i f i c a n t s i m p l i f i c a t i o n s can be made by c o n s i d e r i n g t h e w e l l known Pipes-Pagano problem,' which consists of a symmetric composi te l amina te under uniform a x i a l e x t e n s i o n , E~ = E ~ (F ig . 1 ) . That i s , t h e d i sp lacemen t s u i n a l a m i n a t e may be w r i t t e n a s