A POST FIRST-PLY FAILURE ANALYSIS OF COMPOSITE LAMINATES

A. K. pandey* and J. N. ~ e d d ~ + V i r g i n i a P o l y t e c h n i c I n s t i t u t e and S t a t e U n i v e r s i t y

Blacksburg, V i r g i n i a

A b s t r a c t p l a n e - s t r e s s reduced s t i f f n e s s c o e f f i c i e n t s ( i , j = 1 , 2 , 6 )

A f a i l u r e a n a l y s i s of laminated composite p l a t e s i s p resen ted . F i r s t o r d e r s h e a r de fo rma t ion p l a t e theo ry i s used i n t h e f i n i t e - element a n a l y s i s of t h e laminated p l a t e s under in- p l a n e and t r a s v e r s e load ing . Macromechanical models of f a i l u r e , such a s maximum s t r e s s , maximum s t r a i n , Hoffman, Tsai-Wu, and T s a i - H i l l f a i l u r e c r i t e r i a a r e e v a l u a t e d f o r v a r i o u s l a m i n a t e f a i l u r e . A s t r e n g t h r e d u c t i o n p rocedure is p resen ted f o r p o s t - f i r s t - p l y f a i l u r e a n a l y s i s . F i r s t - p l y f a i l u r e load and p o s t - f i r s t - p l y f a i l u r e load a r e p resen ted f o r a laminated p l a t e wi th ho le and wi thou t h o l e i n t h e c e n t e r . E f f e c t of s t r e n g t h r e d u c t i o n on f a i l u r e sequence is i n v e s t i g a t e d . A l l f a i l u r e c r i t e r i a g i v e a lmost t h e same f i r s t p l y f a i l u r e load f o r a l l l a m i n a t e s under in-plane load ing . T s a i - H i l l c r i t e r i a g i v e s d i f f e r e n t r e s u l t t han t h e o t h e r c r i t e r i a f o r t r a n s v e r s e load ing . Sequence and l o c a t i o n of f a i l u r e depends on t h e amount of s t r e n g t h

shea r s t r e s s s t r e n g t h s of a lamina i n yz , xz and xy p l a n e s , r e s p e c t i v e l y

shea r s t r a i n s t r e n g t h of laminae i n t h e yz, xz and xy p l a n e s , r e s p e c t i v e l y

components of t he compliance ma t r ix

d i sp lacemen t components i n t h e x , y , z d i r e c t i o n s , r e s p e c t i v e l y

nodal v a l u e s of d i sp lacemen t s u,v,w ( i = 1 , 2 , ..., n)

p o s i t i o n c o o r d i n a t e s i n c a r t e s i a n sy s tem

r e d u c t i o n and f a i l u r e c r i t e r i a be ing used. compress ive s t r e s s s t r e n g t h of lamina i n the x, y, z d i r e c t i o n s , r e s p e c t i v e l y Nomenclature

e x t e n s i o n a l , f l e x u r a l - e x t e n s i o n a l and f l e x u r a l s t i f f n e s s e s ( i , j =

1 ,2 ,6 )

t e n s i l e s t r e s s s t r e n g t h of lamina i n t h e x, y, z d i r e c t i o n s , r e s p e c t i v e l y

p l a t e planform dimensions a long t h e x and y d i r e c t i o n s , r e s p e c t i v e l y

compress ive s t r a i n s t r e n g t h of lamina i n t h e x , y , z d i r e c t i o n s , r e s p e c t i v e l y

l a y e r e l a s t i c moduli i n d i r e c t i o n s a long f i b e r s and normal t o them, r e s p e c t i v e l y

t e n s i l e s t r a i n s t r e n g t h of lamina i n t h e x, y , z d i r e c t i o n s , r e s p e c t i v e l y

components of t h e s t r e n g t h t e n s o r s i n f a i l u r e c r i t e r i a column v e c t o r of g e n e r a l i z e d nodal

d i sp lacemen t s l a y e r i n p l a n e and t h i c k n e s s s h e a r moduli c u r v a t u r e s wi th r e s p e c t t o x and y

c o o r d i n a t e s t o t a l t h i c k n e s s of t h e l amina te

s t r a i n components ( i = 1 , 2 , ..., 6 ) t h i c k n e s s of i - t h l a y e r

o r i e n t a t i o n of t h e m-th l a y e r (m = 1 , 2 , ..., L) s h e a r c o r r e c t i o n c o e f f i c i e n t s

a s s o c i a t e d w i t h t h e yz and xz p l a n e s , r e s p e c t i v e l y ( i = 4 , 5 ) s t r e s s components ( i = 1 ,2 , . . , 6 )

element s t i f f n e s s m a t r i x f i n i t e - e l e m e n t i n t e r p o l a t i o n f u n c t i o n s ( i = 1 , 2 , . . . , n )

assembled s t i f f n e s s m a t r i x r o t a t i o n s of a t r a n s v e r s e normal about t h e y and x c o o r d i n a t e s , r e s p e c t i v e l y

s t r e s s coup les and s t r e s s r e s u l t a n t s , r e s p e c t i v e l y ( i =

1 ,2 ,6 )

s h e a r s t r e s s r e s u l t a n t s ( i = 1.2)

* ~ r a d u a t e Research A s s i s t a n t , Department of Eng inee r ing Sc ience and Mechanics.

lifton on C. Garvin P r o f e s s o r , Department of Eng inee r ing Sc ience and Mechanics.

Copyright O American Institute of Aeronautics and Astronautics, Inc., 1987. All rights resewed.

I n t r o d u c t i o n

The u s e of composi te m a t e r i a l s i n s t r u c t u r a l a p p l i c a t i o n s i s d i c t a t e d by t h e o u t s t a n d i n g s t r e n g t h s t i f f n e s s , low s p e c i f i c g r a v i t y of f i b e r s , low maintenance c o s t s , and t h e flexibility i n t a i l o r i n g the s t i f f n e s s and s t r e n g t h i n t h e p r e l i m i n a r y d e s i g n s t a g e of comnlex v e h i c l e s t r u c t u r e s . IJse of composi te ~ m t e r i a l s i n advanced underwater and space v e h i c l e can r e s u l t i n s i g n i f i c a n t i n c r e a s e i n pay l o a d , weight r e d u c t i o n , ranqe and speed , m a n e u v e r a h i l i t y , f u e l e f f i c i e n c y and s a f e t y . F u n c t i o n a l r equ i rmen t s and s a f e t y and economic c o n s i d e r a t i o n s r e q r ~ i r e d e s i g n e r s t o use r e l i a b l e and a c c u r a t e but economical method of d e t e r m i n i n g s t r e s s e s and i d e n t i f y i n g f a i l u r e mechanisms.

An a c c u r a t e p r e d i c t i o n of t h e s t r e n g t h and f a i l ~ l r e ( o r r e l i a b i l i t y ) of s t r u c t u r e s made of composi te l a m i n a t e s r e q u i r e s a r e a l i s t i c modeling of t h e m a t e r i a l behavior ( i . e . m a t e r i a l c h a r a c t e r i z a t i o n ) , a c t u a l geometry ( e . ~ . , m a t e r i a l d i s c o n t i n u i t i e s , Teometr ic d i s c o n t i n u i t i e s , e t c . ) , and k i n e m a t i c s of de fo rma t ion ( e . g . , t h i c k n e s s e f f e c t s , i n t e r l a m i n a r s t r e s s d i s t r i b u t i o n s , l a r g e deEormat ion, e t c . ) .

The freedom t o t a i l o r t h e s t i f f n e s s and s t r e n g t h of f i l a m e n t a r y composi tes i n t h e p r e l i a i n a r y d e s i g n s t a g e of complex s t r u c t u r e s demands f u r t h e r a t t e n t i o n t o d e t a i l s wFlich e i t h e r \ave no c o u n t e r p a r t i n m e t a l l i c s t r u c t u r a l d e s i g n , o r a r e cons ide red of secondary impor t ance . I n many r e s p e c t s , a g r e a t e r need e x i s t s f o r r e l i a b l e and a c c u r a t e a n a l v t i c a l p r e d i c t i v e t echn iques For composi te s t r u c t n r e s than f o r me ta l s t r ~ l c t u r e s . The d i f f i c u l t i e s a s s o c i a t e d w i t h a n a l y z i n g composi te s t r u c t ~ ~ r e s a r e magnif ied because of f r e e edges , h o l e s , j o i n t s , s t i f F e n e r s , and hending- s t r e t c h i n ? coup l ing . The problem of c a l c u l a t i n g d e t a i l e d s t r e s s d i s t r i b u t i o n s around d i s c o n t i n u i t i e s f o r u se w i th a n a l y t i c a l f a i l u r e p r e d i c t i o n t echn iques i n g e o m e t r i c a l l y complex s t r u c t u r e s p r e c l u d e s t h e u s e of c l a s s i c a l a n a l y t i c a l methods. Yumerical methods, such a s t h e F i n i t e c lement method p rov ide a l t ~ r n a t i v e p r a c t i c a l Teans of c a l c u l a t i n g s t r e s s d i s t r i b u t i o n s .

The p r e s e n t s tudy uses k i n e m a t i c models t h a t accoun t f o r t r a n s v e r s e s t r e s s e s and c o n s t i t u t i v e models t h a t u t i l i z e a v a i l a b l e macromechanics models t o p r e d i c t s t r e n g t h and f a i l u r e i n l amina ted composi te p l a t e s . The co r r e spond ing compu ta t iona l models a r e developed t o de t e rmine s t r e s s e s and p r e d i c t f i r s t - p l y and p o s t - f i r s t - p l y F a i l u r e i n composi te l a m i n a t e s .

C o n s t i t u t i v e Models f o r F a i l u r e

The d i s c u s s i o n h e r e d e a l s w i t h a review of t h e f a i l u r e c r i t e r i a of composi te l a m i n a t e s and i t s a p p l i c a t i o n i n t h e f i r s t - p l y - f a i l u r e a n a l y s i s of composi te l a m i n a t e s ( ) . F a i l u r e c r i t e r i a Eor composi te m a t e r i a l s i s more d i f f i c u l t t o p o s t u l a t e hecause i t i s more invo lved i n te rms O F s t r u c t u r a l and m a t e r i a l complexi ty i n comparison t o i s o t r o p i c m a t e r i a l s . P h y s i c a l p r o p e r t i e s of t he c o n s t i t u e n t phases can cover ext reme r anges . Ye in fo rced m a t e r i a l s a r e h igh modulus e l a s t i c and b r i t t l e f i b e r s , whereas t h e b i n d e r m a t e r i a l s a r e d u c t i l e an4 v i s c o e l a s t i c . There e x i s t a number of f a i l u r e

c r i t e r i a and a b r i e f d e s c r i p t i o n of t he f a i l u r e c r i t e r i a a s a p p l i e g t o composi te ma te r i . 1s i s 3 p r e s e n t e d hy T s a i ( ) and T s a i and ~ a h n ( ) .

One i n p o r t a n t a r e a of c o n c e n t r a t i o n , b e s i d e s t h e f a i l u r e l o a d , i s modes of f a i l u r e . 1,aminated composi te may f a i l by f i b e r y i e l d i n g , m a t r i x y i e l d i n g , Eiber breakage, d e l a m i n a t i o n of l a y e r o r by f r a c t u r e . F i r s t t h r e e f a i l u r e modes depend on the c o n s t i t u e n t ' s s t r e n g t h p r o p e r t i e s , whereas d e l a m i n a t i o n i s b a s i c a l l y due t o improper s t a c k i n g sequence of laminae . F r a c t u r e i s caused by the n r e - e x i s t i n g v o i d s and c r a c k s i n t h e c o n s t i t u e n t m a t e r i a l . i l a c roscop ic f a i l t ~ r e c r i t e r i a , urhich a r e d i s c ~ ~ s s e d h e r e , a r e b a s e l on t h e t e n s i l e , compress ive and s h e a r s t r e n g t h of t h e i n d i v i d u a l l a n i n a .

: tos t popu la r f a i l u r e c r i t e r i a , a s d i s c n s s e d by ~ o n i ( ~ ) , a r e maxinum s t r e s s c r i t e r i o n , maximum s t r a i n c r i t e r i o n and q u a d r a t i c polynomial c r i t e r i a such a s t he Tsai-Wu, Cha.nis, Hoffman and H i l l c r i t e r i a . The maximum s t r e s s c r i t e r i o n and inexinuln s t r a i n c r i t e r i o n a r e very s imp le and a r e c a l l e d independent f a i l u r e mode c r i t e r i a . Maximun s t r e s s and s t r a i n c r i t e r i a have some physic11 b a s i s whereas t h e polynomial c r i t e r i a a r e mathemat ica l i n n a t u r e .

Tensor polyno~nia r e p r e s e n t e d by:

c r i t e r i a of f a i l u r e i s

Terms a s s o c i a t e d wi th aq, 0 5 , and 06 ( i . e . F4 , F5 and F6) a r e t aken t o be z e r o , s i n c e shea r s t r e n g t h s a r e t he same f o r p o s i t i v e and n e g a t i v e s h e a r s t r e s s . I t i s a l s o assumed he re t h a t t h e r e i s no i n t e r a c t i o n between s h e a r s t r e s s e s and normal s t r e s s e s , t h u s F16, FZ6 e t c . become ze ro .

Var ious F a i l u r e c r i t e r i a used i n e n g i n e e r i n ? p roEess ion can he exp res sed i n t h e form oE equa t ion (1 ) . Yaximu~n s t r e s s , maxinun s t r a i n , Tsai-Xu, Tsai-IIiLl and l l o f fnan c r i t e r i a a r e used h e r e f o r f a i l u r e p r e d i c t i o n . D e t I i l s of t h e s e c r i t e r i a a r e p re sen ted i n Re fe rence ( 1 ) . I t should he noted h e r e t h a t a l t h o u g h maxinum s t r e s s c r i t e r i o n and maximum s t r a i n c r i t e r i o n a r e independent mode c r i t e r i a , i t can be exp res sed i n polynomial form.

The f a i l u r e c r i t e r i a a r e used he re i n con junc i o n wi th t h e f i r s t - o r d e r s h e a r de fo rma t ion theory( ') t o p r e d i c t f a i l u r e ( f i r s t - p l y f a i l u r e and p o s t - f i r s t - p l y f a i l u r e ) i n laminated composi te p l a t e s .

F i n i t e Element Yodel

V a r i a t i o n a l F o r m u l a t i o n

The p l a t e under c o n s i d e r a t i o n i s composed of a f i n i t e number of o r t h o t r o p i c l a y e r s of uni form t h i c k n e s s , w i th p r i n c i p a l axes of e l a s t i c i t y o r i e n t e d a r h i t r a r i l y w i th r e s p e c t t o t he p l a t e axes . The x and y -coord ina t e s of t h e p l a t e a r e t aken i n t h e midplane ( S Z ) of t h e p l a t e . The d i sp l acemen t f i e l d of t h e f i r s t - o r d e r t h e o r y i s of

t h e form

l le re u l , 9, u3 a r e t h e d i s p l a c e m e n t i n t h e x , y , z d i r e c t i o n s , r e s p e c t i v e l y ; u , v , w a r e t h e a s s o c i a t e d m i d p l a n e d i s p l a c e m e n t s ; and y and 3 a r e t h e r o t a t i o n s a 5 9 u t t h e y and x axes: Y

r e s p e c t i v e l y . The s t r a i n s i n t h e p l a t e s c a n be e x p r e s s e d i n t h e form

Vote t h a t t h e t r a n s v e r s e s h e a r s t r a i n s c 4 and E

a r e c o n s t n n t t h r o u g h t h e t h i c k n e s s . 5

I f one p l a n e of e l a s t t c symmetry p a r a l l e l t o t h e p l a n e of e a c h l a y e r e x i s t s , t h e c o n s t i t v t i v e e q u a t i o n s For t h e l m i n a t e can be w r i t t e n i n t h e Form

- The Ai j , qi j, D i . ( i , j = 1 , 2 , 6 ) , and A ( i , j = 4 , 5 ) a r e t h e i n p l a n e , b e n d i n g - i n p l a n e :dupl ing , b e n d i n g o r t w i s t i n g . and t h i c k n e s s - s h e a r -.

s t i f f n e s s e s , r e s p e c t i v e l y :

and Ni, Yi and q i a r e t h e s t r e s s r e s u l t a n t s ,

Here u . ( i = 1 ,2 , ..., 6) d e n o t e t h e s t r e s s c o m p o n i n t s i n t h e l a m i n a t e c o o r d i n a t e s ( u l = u x , u 2 = uY,u4 = u Z y , u 5 = u and u 6 and

XZ

u ) . H e r e z , d e n o t e s t h e d i s t a n c e From t h e mid- p'fzne t o t h e lower s u r f a c e of t h e n r t h l a y e r , and ki a r e t h e s h e a r c o r r e c t i o n c o e f f i c i e n t s .

The v i r t u a l work s t a t e m e n t ( t . e . t h e p r i n c i ? l e of v i r t u a l d i s p l a c e m e n t s ) i s g i v e n by

w h e r e i n X i , Yi and q i a r e g i v e n i n t e r m s of t h e g e n e r a l i z e d d i s p l a c e m e n t s by E q . ( 3 ) , and V , 'In and N n s , and 'in and Xns a r e t h e s h e a r f o r c e , normal and t a n g e n t i a l i n - p l a n e f o r c e s , and normal and t w i s t i n g h e n d i n g moments d e f i n e d on t h e boundary T , r e s p e c t i v e l y :

;J = n n N f 2 n n N + n n N n x x l x y 6 y y 2

'1 = n n M + 2 n n X 1 + n n M 11 x x l x y 6 y y 2

where n = ( n , n ) is t \ e u n i t v e c t o r normal t o t h e h o u n d a r y r . X ~ h g v a r i a t i o n a l f o r m u l a t i o n i n d i c a t e s t h a t t h e e s s e n t i a l ( i . e . , g e o l n e t r i c ) and n a t u r a l boundary c o n d i t i o n s o f t h e problem a r e g i v e n by:

ESSENTI4L s p e c i f y : u,, u s , W, y n , a s NATURAL s p e c i f y : N n , N n s , V , ?f ny lqns ( 8)

w h e r e i n un and u s , f o r example , der ln te t \ e normal and t a n g e n t i a l components of t h e i n p l a n e

(4) d i s p l a c e m e n t v e c t o r , u = ( u , v ) . - F i n i t e Element X o d e l

C o n s i d e r a f i n i t e - e l e l l e n t a n a l o g , 2 , of t h e m i d p l a n e of t h e p l a t e , 9. Over a typica!? e l e m e n t , R of t h e mesh Q , e a c h g e n e r a l i z e d d i s p l a c e m e g t 3 is i q t e r p o p a t e d s p a t i a l l y by a n e x p r e s s i o n of t h e form,

where U i i s t h e v a l u e of U a t node i , @ i s t h e i

f i n i t e - e l e m e n t i n t e r p o l a t i o n f u n c t i o n a t node i, and r i s t h e number of nodes i n t h e e l emen t . Fo r s a k e of s i m p l i c i t y , we u s e t h e same i n t e r p o l a t i o n f u n c t i o n f o r each of t h e g e n e r a l i z e d d i s p l a c e m e n t s , (u ,v ,w,$ ,$y) .

S u b s t i t u t i n g Eq. (9 ) i n t o Eq. ( 8 ) , we o b t a i n t h e f o l l o w i n g e q u a t i o n f o r a t y p i c a l e lement

Here { A ) i s t h e column v e c t o r of t h e nodal v a l u e s of t h e g e n e r a l i z e d d i s p l a c e m e n t s , [ K ] i s t h e m a t r i x of t h e s t i f f n e s s c o e f f i c i e n t s , and { F } i s t h e column v e c t o r c o n t a i n i n g t h e boundary and t r a n s v e r s e f o r c e c o n t r i b u t i o n s . The e l emen t s of [ K ] a r e g iven i n t h e Appendix.

The assembled f i n i t e - e l e m e n t e q u a t i o n s a r e of t h e form

D where [K ] is t h e assembled s t i f f n e s s m a t r i x .

F i r s t - P l y and P o s t - F i r s t - P l y F a i l u r e P r e d i c t i o n

The f i n i t e - e l e m e n t procedure d e s c r i b e d i n t h e p reced ing s e c t i o n can be used t o de t e rmine t h e s t r e s s e s ( i n t h e l a m i n a t e c o o r d i n a t e s ) a t any p o i n t of t h e l a m i n a t e . I n g e n e r a l , t h e l a m i n a t e c o o r d i n a t e s do no t c o i n c i d e w i t h t h e m a t e r i a l p r i n c i p a l axes of t h e i n d i v i d u a l p l i e s . S ince t h e f a i l u r e c r i t e r i a d e s c r i b e d e a r l i e r r e q u i r e t h e s t r e s s e s and s t r a i n s w i t h r e s p e c t t o t h e m a t e r i a l c o o r d i n a t e s of each l amina , a t r a n s f o r m a t i o n of t h e s t r e s s e s and s t r a i n s from l a m i n a t e c o o r d i n a t e s t o t h e lamina m a t e r i a l c o o r d i n a t e s is r e q u i r e d . The s t r e s s e s i n m a t e r i a l c o o r d i n a t e s can be de t e rmined i n one of t h e two ways: ( i ) compute s t r a i n s i n t h e l a m i n a t e c o o r d i n a t e s , t r a n s f o r m them t o lamina c o o r d i n a t e s and use t h e lamina c o n s t i t u t i v e e q u a t i o n s t o compute s t r e s s e s ; ( i i ) compute s t r a i n s i n t h e l a m i n a t e c o o r d i n a t e s , compute s t r e s s e s u s i n g t h e lamina c o n s t i t u t i v e e q u a t i o n s w i t h s t i f f n e s s e s r e f e r r e d t o t h e l a m i n a t e c o o r d i n a t e s (Qi .) , and then t r a n s f o r m l a m i n a t e s t r e s s e s t o lamina s t r e s s e s . Both a r e ma thema t i ca l ly e q u i v a l e n t . The second method i s used i n t h e p r e s e n t s t u d y . I f 8, d e n o t e s t h e l a m i n a t i o n a n g l e of t h e m-th l a y e r , t hen t h e

s t r e s s e s a!') i n i t s m a t e r i a l c o o r d i n a t e s can be

o b t a i n e d from t h e l a m i n a t e s t r e s s e s o a a X ' y ' xy'

e t c . u s i n g t h e t r a n s f o r m a t i o n :

The lamina ( i . e . , m a t e r i a l c o a r d i n a t e ) s t r e s s e s o r s t r a i n s a r e t hen used i n a chosen f a i l u r e c r i t e r i o n t o check i f each F i n i t e e lement and each lamina has f a i l e d . I f t h e f a i l u r e c r i t e r i o n i s s a t i s f i e d i n a p l y of an e l e a e n t , t hen t h e i n d i v i d u a l c o n t r i b u t i o n s , c a l l e d f a i l u r e i n d i c e s , oE each s t r e s s component t o t h e t e n s o r polynomial a r e computed. The f a i l u r e i n d i c e s can he used t o i n e r p r e t t h e mode of f a i l u r e . For example, i f t h e

c o n t r i b u t i o n s of t h e a t rn) t o t h e t e n s o r polynomial

i s l a r g e r t han t h e c o n t r i b u t i o n s of o t h e r s t r e s s components, t hen t h e f a i l u r e i n t h a t p l v i s assumed t o occur due t o t h e t e n s i o n o r compress ion

i n 1 - d i r e c t i o n depending on the s i g n of a i m ) . The

f a i l u r e i n d i c e s a r e used t o reduce t h e s t i f f n e s s e s i n t h e p o s t - f i r s t - p l y f a i l u r e a n a l y s i s .

A flow c h a r t oE t h e compu ta t iona l p rocedure f o r p o s t - f i r s t - p l y f a i l u r e p r e d i c t i o n i s g iven i n F i g u r e 1. For a g i v e n l a m i n a t e , one s t a r t s w i th any a r b l t r a r y l o a d i n g and de t e rmines t h e d i s p l a c e m e n t s a t each node. From t h i s d i sp l acemen t , s t r e s s e s and s t r a i n s a r e de termined i n each e lement i n t h e m a t e r i a l p r i n c i p a l d i r e c t i o n a s e x p l a i n e d above. Each p o i n t , w h e r ~ v a l u e s of s t r a i n s and s t r e s s e s a r e known, i s t e s t e d f o r f a i l u r e f o r any g iven f a i l u r e c r i t e r i o n . I f t h e f a i l u r e c r i t e r i o n i s not e x a c t l y s a t i s f led then a p p l i e d load i s e i t h e r i n c r e a s e d o r dec reased depending on t h e value of f a i l u r e i ndex ( F I ) , which a r e c a l c u l a t e d when f a i l u r e c r i t e r i o n i s a p p l i e d a t any p o i n t . T h i s p r o c e s s i s i t e r a t i v e and g i v e s a load w i t h one o r more l o c a t i o n where f i r s t - p l y f a i l u r e o c c u r s . I n t h e p r e s e n t c a s e r e l a t i o n between f a i l u r e load and d i sp l acemen t is l i n e a r so E i n i t e e lement e q u a t i o n s a r e so lved on ly once .

Once f i r s t - p l y f a i l u r e load and l o c a t i o n ( s ) a r e de t e rmined , a n a l y s i s f o r p o s t - f i r s t - p l y f a i l u r e is performed. The e l a s t i c c o n s t a n t s of t h e f a i l e d p l y ( o r p l i e s ) a r e reduced a c c o r d i n g t o t h e scheme e x p l a i n e d he re . C o n t r i b u t i o n (u i ) of each s t r e s s component t o t h e f a i l u r e index i s de t e rmined . S i n c e t o t a l f a i l u r e index a t f a i l u r e is equa l t o one so ( 1 - n ) i s m u l t i p l i e d t o t h e

i e l a s t i c c o n s t a n t . Fo r example, i f a h i s t h e

f a i l u r e index c o n t r i b u t i o n from o ( m ) l t hen 6

w i l l be reduced t o G12( - w ) . A r e d u c t i o n pa rame te r , R has been i n t r o a u c e d . T h i s i s m u l t i p l i e d t o t h e reduced e l a s t i c c o n s t a n t s . T h i s a l l o w s one t o r educe t h e s t i f f n e s s of any fa iLed

2 2 p l y by more amount t han d e s c r i b e d i n t h e above = u cos a + a s i n e + 2a s i n e C O S ~ ,

1 m y m xY m scheme. A f t e r r e d u c t i o n i s done, one de t e rmines t h e f a i l u r e a t some o t h e r p o i n t u s i n g t h e i t e r a t i o n t echn ique . T h i s t ime f a i l u r e load i s u s u a l l y d i f f e r e n t because e l a s t i c c o n s t a n t s a r e reduced i n e a r l i e r s t e p s . T h i s p roces s i s r e p e a t e d u n t i l a l l c r i t i c a l p l i e s have f a i l e d .

a(m) = - s i n e + a C O S ~ 4 xz m yz m

D i s c u s s i o n of R e s u l t s

a (m) = - a c a s e + o s i n e m 4 xz m y z F i r s t - p l y and p o s t - f i r s t - p l y f a i l u r e a n a l y s i s

of l amina ted composi te p l a t e s a r e p r e s e n t e d . Composite m a t e r i a l p r o p e r t i e s of T300/5208

a(m) = - a s i n e C O S ~ + a s i n e cos3' 6 x m m y m graph i t e / epoxy a r e g iven i n Tab le 1 .

t h e nodal d i sp l acemen t s S e t : P =I' and IIP = increment

on o

Displacement f o r P = Displacement on

f o r Po x(Pon/po ) I

CALL STRESS t o Compute S t r e s s e s ach p l y of eache lemen nd f a i l u r e i n d e x , FIN

I F FIN. LT, 1 Pon=Pon+DP - S e t POUP= Po,

IF(NCOUNT.EQ.1) Podn= 0.0

IF (NCOUNT. NE. 1 ) P =P - DP Compute new DP ar$'fbnon i

I F DP < EPS ---+ Reduce e l a s t i c w n s t a n t s

o f f a i l e d p l i e s

%up-%dn DP= --

FACTOR Pon=Podn +DP

EPS=Error

Fig . 1 A f low c h a r t of t h e compu ta t iona l procedure

Table 1 Matcrial Propcrtics of 1'300 5208 Graphite Ilpoxy Compoaitc Materials

Propcrty Value

v23

x, xc Y, = %, Y,. = ZC K S = T ply thickness h,

19.2*106 psi 1.56*lO0 psi

I O b psi 0.81* 10'' psi 0.49' 10"si 0.24 0.49 219.5'10' psi 246 O* 103 psi 6.35'10' psi 23.8*103 psi 9.80* 103 psi 12.6'10' psi 0.005 iqply

has 39 l i n e a r q u a d r i l a t e r a l e l emen t s and 54 nodes. A uniform I n p l a n e edge load ( i n x d i r e c t i o n ) i s used.

' t - I

1 - - I Quadrant modeled

?

used f o r t h e p o 5 t - f i r s t - p l y f a i l u r e a n a l y s i s U = O

of composi te l a m i n a t e s .

Laminates w i t h a c e n t r a l h o l e a r e ana lyzed f o r i n -p l ane load ing . O the r l a m i n a t e s ( i . e . , l a m i n a t e s w i t h no h o l e i n t h e c e n t e r ) a r e ana lyzed f o r t r a n s v e r s e l oad ing . Composite l amina ted v = o p l a t e s w i t h v a r i o u s s t a c k i n g sequence a r e ana lyzed . Laye r s a r e numbered s t a r t i n g from

(b) Computational domain, f i n i t e element mesh and boundary bottom t o top and f i r s t a n g l e i n t h e l a m i n a t e cond i t i ons used f o r the model n o t a t i o n scheme d e n o t e s t h e o r i e n t a t i o n of t h e f i r s t l a y e r .

--I

Laminate w i t h a C e n t r a l Hole r i g . 2 Geometry, compu ta t iona l domain and f i n i t e

e lement model of a r e c t a n g u l a r l amina ted F i g u r e 2 c o n t a i n s t h e geometry , l o a d i n g and

p l a t e w i t h a c e n t r a l h o l e and s u b j e c t e d t o compu ta t iona l domain For a l a m i n a t e w i t h a c e n t r a l

un i fo rmly d i s t r i b u t e d edge load . h o l e . A mes'l of l i n e a r q u a d r i l a t e r a l e l emen t s i s a l s o shown i n t h e f i g u r e . Q u a r t e r of t h e p l a t e

- - - 9" -, +

v -

w

(a ) Geometry and load ing

' t

F i r s t - p l y f a i l u r e l o a d s p r e d i c t e d by v a r i o u s c r i t e r i a a r e shown i n F i g u r e 3. I t i s observed t h a t maximum s t r e s s , Hoffman and Tsai-Wu c r i t e r i a p r e d i c t a lmos t t h e same load hu t maximum s t r a i n and Tsai -Tl i l l c r i t e r i a p r e d i c t d i f f e r e n t l oad . A l l f a i l u r e l o c a t i o n s a r e t h e same f o r v a r i o u s t h e o r i e s . F a i l u r e o c c u r s i n e lement numher 11 and i n t o p and bottom p l y . Th i s r e s u l t i n d i c a t e s t h a t maximum s t r e s s , maximum s t r a i n , Hoffman, Tsai-Wu and T s a i - H i l l c r i t e r i a a r e e s s e n t i a l l y t h e same f o r i n p l a n e load ing .

1,00(

F P F L

( l b s

90C

0

FPFL = F i r s t - P l y F a i l u r e Load

Xax . S t r e s s :

F a i l u r e C r i t e r i a

F i g . 3 F i r s t - p l y f a i l . u r e l o a d f o r composi te lami- n a t e (45"/ -45"/45") w i t h a c e n t r a l h o l e under i n p l a n e l o a d i n g .

P o s t - f i r s t - p l y f a i l u r e a n a l y s i s r e s u l t s a r e p r e s e n t e d i n T a b l e 2 and T a b l e 3. D i f f e r e n t v a l u e s of r e d u c t i o n pa rame te r R a r e t aken and t h e i r e f f e c t on p o s t - f i r s t - p l y f a i l u r e l o a d s and l o c a t i o n s a r e i n v e s t i g a t e d . Here r e s u l t s f o r R = 1 . 0 and R = 0.1 a r e p r e s e n t e d . A t s t e p number one, T a b l e s 2 and 3 have t h e same f a i l u r e load and l o c a t i o n f o r a l l t h e c r i t e r i a but f a i l u r e l o c a t i o n and load changes a f t e r two more r e d u c t i o n s .

F a i l u r e sequence i s shown i n F i g u r e 4 . The l a m i n a t e sequence i n t h i s c a s e i s [ 4 5 / - 4 5 / 4 5 ] . Hoffman's c r i t e r i a was used f o r o b t a i n i n g t h i s f a i l u r e sequence . When r e d u c t i o n pa rame te r i s 0 .01 , a l l e l e n e n t s t r a n s v e r s e t o t h e h o l e i n y d i r e c t i o n f a i l f i r s t . A r e d u c t i o n pa rame te r of 0.01 i n d i c a t e s t h a t s t i f f n e s s of t h e f a i l e d p l y i s s e t t o a ve ry s m a l l number. S e l e c t i o n of p r o p e r v a l u e of X i s impor t an t f o r a c c u r a t e p o s t - f i r s t - p l y f a i l u r e a n a l y s i s . Note t h a t f a i l u r e load d e c r e a s e s Eor subsequent r e d u c t i o n s a t t h e same l o c a t i o n . The f a i l u r e load i n c r e a s e s when t h e f a i l e d e l emen t s a r e f a r t h e r from t h e c r i t i c a l e l emen t 11.

R e c t a n g u l a r Lamina te s Under Rending

Var ious r e c t a n g u l a r composi te l a m i n a t e s a r e ana lyzed and t h e r e s u l t s a r e p r e s e n t e d f o r t r a n s v e r s e l o a d i n g . Only un i fo rmly d i s t r i b u t e d l o a d i n g i s c o n s i d e r e d he re .

( a ) Reduc t i on parameter , R=1.0

( b ) Reduc t i on p a r a n e t e r , R = 0 . 3

( c ) Redduc t i on par f?metcr , R=0.01

F ig . 4 P o s t - f i r s t - p l y f a i l u r e sequence i n a l a m i n a t e (45" l -45" /45" ) w i t h A c e n t r a l h o l e and under i n p l a n e edge load ( t h e numbers i n d i c a t e t h e sequence of f a i l u r e ) .

F i g . 5 Geometry, boundary c o n d i t i o n s and f i n i t e e lement mesh f o r a s imp ly suppor t ed r e c t - a n g u l a r composi te l a m i n a t e under un i fo rmly d i s t r i b u t e d t r a n s v e r s e l o a d .

'l'ablc 2 Cornpanson of Post-l,irst-I'll I:ailure I n a d and Incation for a 1451-451451., Composite Plate with I Iole undcr lnplanc lnadmg for l>iff'crent 1;ailure Criteria. ( R = 1.0)

St~lincss Reduction I n a d and Location at Failure (Ibs)

Stcp l u n i l x r Maxm. Stress Maxm. Strain Hoflinan Tsai-Wu Tsai-Mill

8 10X6.0 1143.0 1085.0 1094.0 1068.0 (20:2 ) (20:2 ) (20:2 ) (192 ) (20.2 )

lurnhcr in parcnthc\is indicate clcmcnt numbcr and pl] numbers whcrc fhilurc has occurred.

Tahlc 3 Comparison of' I'ost-First-I'ly 1;ailurc Inad and I ~ c a t i o n for a 1451-45/45] ,. Composite Plate with I lolc undcr Inplanc Znading for D~ft'ercnt Failure Criteria. ( R = 0.1)

Stiftncss lieduction Load and Location at Failure (lbs)

Stcp \urnbcr Maxm. Strcss Maxm. Strain Hoffman Tsai-Wu Tsai-Ilill

I 908.3 1009.0 906.9 919.0 1007.0 (I 1:1,3)* (I 1:1,3) (I l:1,3) (11:1,3) (11:1,3)

2 fA9.9 757.1 6.17.5 670.0 713.0 (11:2 ) (1 1:2 ) (11:2 ) (11:2 ) (11:2 )

3 884.6 990.9 883.0 897.1 979.5 (19:1,3) (19:1,3) (l9:1,3) (19:1,3) (19:1,3)

4 506.7 571.2 504.0 522.6 532.2 (19:2 ) (19:2 ) (19:2 ) (192 (192 )

5 780.3 869.9 779.0 790.6 858.9 (26.1,3) (26: 1,3) (26: 1,3) (26: 1,3) (26: 1,3)

6 482.1 546.4 480.3 496.8 507.8 (26:2 ) (26:2 ) (26:2 ) (19:2 ) (26:2 )

7 877.9 842.9 877.0 886.1 967.7 (33: 1 ,3) (1 l : l J ) (83: 1,3) (33: 1,3) (33: 1,3)

8 " 363.8 6.18.6 362.0 369.3 38 1 .O (33:2 ) (33:2 ) (33:2 ) (19:2 ) (33:2 )

lu rnhcr in parcnthcsi\ indicate clcmcnt numbcr and ply numbers where failure has occured. -. At this stcp stiffness of. the critical clement reaches 10 percent of the original stiffness, except in case of maxm. strain criteria.

Geometry , c o m p u t a t i o n a l domain and b o u n d a r y c o n d i t i o n s of a s i m p l y s u p p o r t e d r e c t a n g u l a r c o m p o s i t e l a m i n a t e [45" / -45"] a r e shown i n F i g u r e 5 . F i r s t - p l y f a i l u r e l o a d f o r v a r i o u s f a i l u r e c r i t e r i a a r e p l o t t e d i n F i g u r e 6 . T o t a l t h i c k n e s s of t h e l a m i n a t e i n t h i s c a s e is 0 . 1 i n c h e s . T s a i - H i 1 1 c r i t e r i o n p r e d i c t s l o w e r f i r s t - p l y - f a i l u r e l o a d . F a i l u r e l o c a t i o n is t h e same f o r a l l t h e c r i t e r i a and i t o c c u r s i n e l e m e n t number 1, l a y e r numher 2 and 1st G a u s s p o i n t .

P o s t - f i r s t - p l y f a i l u r e l o a d f o r a c o m p o s i t e [45 / -45 /45/ -45IT l a m i n a t e i s p l o t t e d v e r s u s t h e numher of r e d u c t i o n s t e p s i n F i g u r e 7 . T o t a l t h i c k n e s s of t h e l a m i n a t e Is 0 .02 i n c h e s and g e o m e t r y , and boundary c o n d i t i o n s a r e t h e same a s shown i n F i g u r e 5. P o s t - f i r s t - p l y f a i l u r e l o a d i n c r e a s e s f i r s t and t h e n i t d e c r e a s e s w i t h t h e r e d u c t i o n of e l a s t i c c o n s t a n t s .

T a b l e s 4 and 5 p r e s e n t t h e p o s t - f i r s t - p l y f a i l u r e r e s u l t s f o r [45 / -45IT and [C/90IT l a m i n a t e s r e s p e c t i v e l y . Geometry and mesh a r e t h e same a s shown i n F i g u r e 5 . Boundary c o n d i t i o n s f o r [45/-451 l a m i n a t e s a r e shown i n F i g u r e 5 . Boundary c o n x i t i o n s f o r [ 0 / 9 0 I T l a m i n a t e a r e d i f f e r e n t f rom [45/ -45IT l a m i n a t e , i n t h a t u i s r e p l a c e d by v and v i s r e p l a c e d by u. T h i c k n e s s of e a c h l a y e r i s 0 . 0 5 i n c h e s f o r t h e p r e s e n t l a m i n a t e s . T h e s e t a b l e s g i v e f a i l u r e l o a d s and l o c a t i o n s f o r v a r i o u s f a i l u r e c r t i e r i a . P o s t - f i r s t - p l y f a i l u r e l o a d and l o c a t i o n i n t h e p r e s e n t c a s e a r e a l m o s t t h e same f o r a l l c r i t e r i a e x c e p t t h e maximum s t r a i n and H i l l ' s c r i t e r i a . V a l u e of R c h a n g e s t h e f a i l u r e l o a d and l o c a t i o n s u b s t a n t i a l l y .

1 1

10

FPFL ( p s i

9

0

FPFL = F i r s t - P l u r e Load

Max. Max. Hoffman Tsa i - Tsa i - S t r e s s S t r a i n Wu H i l l

F a i l u r e C r i t e r i a

F ig . 6 F i r s t - p l v f a i l u r e l o a d r o r composi te lami- n a t e (45'1-45') under un i fo rmly d i s t r i b u t e d t r a n s v e r s e l o a d .

0.251 I I

0 2 4 6 8 10 12 14 16 1 8 20 22 24

Number o f r e d u c t i o n steps

Fig . 7 P o s t - f i r s t - p l y f a i l u r e l o a d v e r s u s reduc- t i o n s t e p s f o r a r e c t a n g u l a r composi te

E a i l u r e p a t h ( i . e . , i n p o s t - f i r s t - p l y fri i l u r e a n a l y s i s ) .

.Acknoirledgement s The a u t h o r s a r e g r a t e f u l t o t he Arny Research O f f i c e For t he suppor t d u r i n g t h e i n v e s t i g a t i o n . I t i s a g r e a t p l e a s u r e t o acknowledge the t y p i n g of t i i s manuscr ip t by I l rs . Vanessa YcCoy.

Re fe rences

1 . Reddy, J. Y. and A. K. Pandey, "A F i r s t - P l y - F a i l u r e Ana lys i s of Composite Lamina te s , " Computers & S t r u c t u r e s , t o appea r .

?. T s a i , S. W., "A survey of macroscopic f a i l u r e c r i t e r i a f o r composi te m a t e r i a l s , " T e c h n i c a l Repor t , AFIJAL-TR-84-4025, Wright -Pat t e r s o n AFS, Dayton, 0F1, 1984.

3. T s a i , S . V. and Hahn, H . T . , " F a i l u r e a n a l y s i s of composi te m a t e r i a l s , " I n e l a s t i c Rehavior of Composite P l a t e r i a l s , Winter Annual ' l e e t i n g of ASME 1975.

!+. S o n i , S. R . , "4 new look a t commonly used f a i l ~ l r e t h e o r i e s i n composi te l a m i n a t e s , " 24 t h AIAA/ASME/ASCF S t r u c t u r e s , S t r u c t u r a l Dynamics and : l a t e r i a l s Conference , P roceed ings , p. 171, 1983.

5 . T s a i , S. W . , and W I I , E . Y . , "A g e n e r a l t heo ry of s t r e n g t h f o r a n i s o t r o p i c m a t e r i a l s , " J o u r n a l of Composite Y a t e r i a l s , Vol. 5 , -- pp. 58-80, 1371.

6 . Reddy, . J . N . , Energy and V a r i a t i o n a l Nethods i n Appl ied Xechan ic s , Wiley, N e w York, 1984,

Appendix

r l e ~ n e n t s of S t i f f n e s s X a t r i x

Equat ion (11) has t h e form

Sunmary and Recommendations

A f i n i t e e lement compu t s t i ona l p rocedure based on t h e f i r s t - o r d e r s h e a r de fo rma t ion t h e o r y and v a r i o u s F a i l u r e c r i t e r i a i s developed t o p r e d i c t f i r s t - p l y and p o s t - E i r s t - p l y f a i l d r e of composi te l a m i n a t e s s u 5 j e c t e d t o i n p l a n e o r t r a n s v e r s e l oads . A l l f a i l u r e c r i t e r i a cons ide red [ ~ I { A } = { F } i n t he p r e s e n t s tudy e s s e n t i a l l y ag ree i n t h e p r e d i c t i o n of t he F i r s t - p l y f a i l u r e l o a d s and l o c a t i o n s . The r e d u c t i o n p rocedure has where [ K ] i s l i n e a r s t i f f n e s s m a t r i x . The s i g n i f i c a n t e f f e c t on the E a i l u r e l o a d s and

e l emen t s xaP a r e d e f i n e d i n t h e Eollowing. F i r s t , i j

795

'Iablc 4 I'ost-I ust-I'I! I:ailurc Recults I or Anti-S>mmctric Anglc-1'1) 1 1 ' 4 5 1 ., Composite 1'latc ~ n d o r I ransvcrse I ~ ) a d i n g

Stiftnc\\ Ilcductiori Imad and I x)cation at I ailuro (psi)

Stcp \unibcr 'Llaurr?. Strccc .Llaxm. Strain I Iofiman '1, sd-\\'u , . 'I'sai-I Iill

I ahlc 5 I'ost-1:irst-Ply 1:adurc llesults I:or Cross-1'1) 10;90] , Compos~tc Plate undcr ' I rancvcrsc I ,(jading

\ t ~ f t r i c ~ \ I<ctluct~ori Load and Ix)cat~on at I'ailuru (psi) ---

Step Zurr~l>cr L1;ixrn. Strcss Vaxm Strain I loftman I 521-M'u l'sni-l lill

I< = 1 .o

aB All other [K ] are zero.