calculation of residual stresses in injection molded products

7/28/2019 Calculation of Residual Stresses in Injection Molded Products

1/16

Rheologica A cta R h e o l A c t a 3 0 : 2 8 4 - 2 9 9 ( 19 9 1)

Ca l cu l a t i o n o f r e s i d u a l s t re s s e s i n i n jec t i o n m o l d ed p ro d u c t sE P . T . B a a i j e n sPhil ips Research Laboratories , Eindhoven, The Netherlands

Abstract." Both f low- and thermally-indu ced res idual s t resses which arise duringthe in jec t ion m old ing o f am orphous the rm oplas t ic po lym ers a re ca lcu la ted inthe filling an d post-filling stage. To achieve this, a compressible version o f theLeo no v mo del is employed. Tw o techn iques to calculate flow-induced residualstresses are investigated. First, a direct ap pro ach is developed where th e pressurepro blem is form ulat ed using the viscoelastic ma terial mode l. Second, generalizedNewtonian materia l behavior is assumed in formulat ing the pressure problem,and the resulting flow kinematics is used to calculate normal stresses employingthe compress ible Leon ov model . Th e la t ter technique gives comparable resul ts ,while reducing the com putat ion al cos t s ignificant ly.Key words." nje cti on _m olding; viscoelasticity; c_ompressible L_eonovmodel; _flow-indu ced residual s_tresses; _thermally-induced residual stres ses; am orp hou spolym ers; _numerical analysis

N o t a t i o n a n d s y m b o l sA c c o n j u g a t e o f t e n s o r AA a d e v i a t o r i c p a r t o f A( - ) c o l u m n( . ) r t r a n s p o se o f a c o l u m n( - ) m a t r i x( : ) m a t e r i a l t i m e d e r i v a t i v e( ' ) e e l a st i c p a r t o f ( ' )( - ) p p l a s t i c p a r t o f ( ' )Symbol description

g

20

Qv

aTcpJN1

c o e f f ic i e n t o f t h e r m a l e x p a n s i o ns h e a r r a t es p e c i f i c i n t e r n a l e n e r g yh e a t c o n d u c t i o n c o e f f i c i e n tr e l a x a t i o n t i m ev i s c o s i t yr e d u c e d t i m ed e n s i t ys p e c i fi c v o l u m et i m e - t e m p e r a t u r e s h i f t f a c t o rs p e c i f i c h e a t a t c o n s t a n t p r e s s u r ev o l u m e c h a n g e f a c t o rf i r s t n o r m a l s t r e s s d i f f e r e n c e

p p r e s s u r er i n t e r n a l h e a t s o u r c e

g r a d i e n t o p e r a t o r w i t h r e s p e c t t o c u r r e n t c o n -f i g u r a t i o n

~ '0 g r a d i e n t o p e r a t o r w i t h r e s p e c t t o r e f e r e n c ec o n f i g u r a t i o n

h h e a t f l u xv e l o c i t y

p o s i t i o n v e c t o ra C a u c h y s t re s s t e n s o rz e x t r a s t r e ss t e n s o rD r a t e o f s t r a i n t e n s o rF d e f o r m a t i o n t e n s o rB l e ft C a u c h y - G r e e n t e n s o rL v e l o c i t y g r a d i e n t t e n s o r

1 . I n t r o d u c t i o nI n j e c t i o n m o l d i n g i s a c o m m o n l y a p p l i e d p r o c e s s -

i n g t e c h n o l o g y f o r p l a st i cs . I n t h e p a s t d e c a d e , a t t e n -t i o n i n n u m e r i c a l s i m u l a t i o n s o f t h e i n j e c t io n m o l d i n gp r o c e s s h a s b e e n f o c u s e d o n t h e f i l li n g st a g e. T h e k e yi t e m s i n t h e s e c a l c u l a t i o n s a r e t h e p r e d i c t i o n o f


2/16


3/16


4/16

Baaijens, Ca lculation of residual stresses in injection m olde d produ cts 287A r e l a t i o n f o r t h e p r e s s u r e p r e m a i n s t o b e g i v e n .

T h i s i s d o n e i m p l i c i t l y b y t a k i n g a s u i t a b l e r e l a t i o n f o rt h e s p e c i f i c v o l u m e . T h e s o - c a l l e d T a i t e q u a t i o n i s as u c c e s s f u l m o d e l f o r a m o r p h o u s p o l y m e r s ( Z o l l e r ,[10 ] ) and i s g i ven by

v (13, T ) = ( a o + a l ( T - T g )) ( 1 - 0 . 0 8 9 4 I n ( 1 + B ) ) , ( 3.9 )

w h e r e Tg i s t h e p r e s s u r e - d e p e n d e n t g l a s s t r a n s i t i o nt e m p e r a t u r e , i . e . , T g ( p ) = T g ( O ) + s p , a n d B(T )=B 0 exp ( - B 1 T) .Plast ic s t resses." T h e p l a s t i c p a r t o f a i s c h o s e n a s

a p = 2 ~ ' ( T , p ) D d . (3 .10)T i m e - t e m p e r a t u r e s u p e r p o s it i on : T h e r m o r h e o l o g i c a l -l y s i m p l e b e h a v i o r i s a s s u m e d , i m p l y i n g t h a t

~ l k = a T ( T ) V l k O , O k = a T ( T ) O k O , t l ' = a T ( T ) t l ' o ,(3 .11)

w h e r e a T i s t h e s h i f t f a c t o r a n d ~ /k 0 a n d 0 k0 a r e t h ev i s c o s it y a n d r e l a x a t i o n t i m e a t a r e f e r e n c e t e m p e r a -t u r e , s a y T 0. T h e s h i f t f a c t o r i s g o v e r n e d b y t h e W L Fe q u a t i o n i f T>_ Tg, i . e . ,

C~(T- To)l o g a r ( T ) - , ( 3 .1 2 )C 2 + T - T ow h i l e , b e l o w Tg

a T ( T ) = a T ( T g ) . (3 .13)T h e r e a s o n f o r t h i s l a s t c h o i c e i s t h a t b e l o w Tg, r e l ax -a t i o n p r o c e s s es a r e e x t r e m e l y s lo w a n d h e n c e t a k ep l a c e a t a t i m e s c a l e b e y o n d c u r r e n t i n t e r e s t .

Linear i zed model . " I n t h e s o l i d i f i e d s t a t e , d e f o r m a -t i o n s a n d r o t a t i o n s a r e s m a l l . I n t h a t c a s e t h e a b o v em o d e l r e d u c e s t o a l in e a r M a x w e l l m o d e l . H e n c e ,

n , tr e = E z ek , r e k = ~ 2 rl k e -( te - s k ) D a ( s ) d s ,k= 1 o Oko (3 .14)w h e r eq 1(qk = ~ - - d s , q = t , s . (3 .15)o a r ( s ) O k

B y l in e a r i z i n g = v t r ( D ) a n e x p l i c it r e l a t i o n f o r t h ep r e s s u r e p c a n b e d e r i v e d :

tp ( t ) = ~ (/3 T - x t r ( D ) ) d s , (3 .16)o

w h e r ef l = a x , a = - - a n d - .

v K v

(3 .17)3 . 2 T h e r m a l b e h a v io r

T o c a l c u l a t e t h e t e m p e r a t u r e d i s t r i b u t i o n i n t h ep o l y m e r , c o n s t i t u ti v e e q u a t io n s f o r t h e h e a t f l u x a n dt h e s p e c i f i c i n t e r n a l e n e r g y a r e g i v e n .H e a t f l u x : T h e h e a t f l u x v e c t o r / ~ is a s s u m e d t o o b e yF o u r i e r ' s l a w , t h a t i s

h = - ~ T . ( 3 .1 8 )S p e c i f i c i n t e r n a l e n e r g y : I f e l a s t i c e f f e c t s u p o n t h es p e c i f i c i n t e r n a l e n e r g y a r e d i s r e g a r d e d i t c a n b es h o w n t h a t ( S i tt e r s [ 2 ])

T OQ= c p i " - p v t r (D) + - -7 -7= P , (3 .19 )Q - 0 1w h e r e Cp i s t h e s p e c i f i c h e a t a t c o n s t a n t p r e s s u r e .

4. Injection mo lding theory4 . 1 I n t r o d u c t i o n

I n t h is s e c t i o n t h e s e t o f b a l a n c e e q u a t i o n s a n d c o n -s t i t u t i v e e q u a t i o n s o f t h e p r e v i o u s s e c t i o n s a r es i m p l i f i e d c o n s i d e r a b l y b y i n t r o d u c i n g a n u m b e ro f g e o m e t r i c a l a s s u m p t i o n s . O n l y n a r r o w , w e a k l yc u r v e d c a v i t i e s a r e c o n s i d e r e d s u c h t h a t t h e t h i n f i l ma p p r o x i m a t i o n h o l d s ( s e e F i g . 1 ). F o r g e n e r a l i z e dN e w t o n i a n m a t e r i a l b e h a v i o r t h i s p r o c e d u r e i s w e l ldesc r i bed i n , e .g . , S i t t e r s [2 ]. Here a v i scoe l a s t i c m at e-r i a l m o d e l i s u s e d .

T h e c o n c e p t o f g e n e r a l i z e d N e w t o n i a n f l o w o n l yh o l d s f o r f u l ly - d e v e l o p e d f lo w s i t u a t i o n s . H o w e v e r ,f u l l y d e v e l o p e d f l o w n e v e r o c c u r s d u r i n g i n j e c t i o nm o l d i n g . Y e t , g e n e r a l i z e d N e w t o n i a n m a t e r i a lb e h a v i o r m a y b e a g o o d a p p r o x i m a t i o n i n c o rer e g i o n s, s u f f i c i e n t l y f a r d e p a r t e d f r o m s u d d e n g e o -m e t r y c h a n g e s ( i. e ., c o n t r a c t io n s ) a n d t h e f l o w f r o n t ,w h i l e th e t e m p e r a t u r e n e e d s t o b e s u f f ic i e n t l y h i g h .


5/16

288 Rh eolog ica Acta, Vo l. 30, No. 3 (1991)l e t A b e a n a r b i t r a r y s e c o n d o r d e r t e n s o r a n d d s o m ev e c t o r , t h e n t h e c o m p o n e n t s o f A , r e s p e c t iv e l y d , w i t hr e s p e c t t o O 1 f o l l o w f r o m A i j = A - ~ . g i , a n da i = d ' g , r e s p e c t i v e l y , w i t h i = 1 , 2 , 3 .

T h e c h a n n e l h e i g h t i s d e n o t e d b y h m , a n d t h e c u r -r e n t p r o d u c t w a l l t h i c k n e s s b y h c .

32 : IOoFig. 1. Def ini t ion o f the local bas is Ot

N e a r t h e s o l i d i f i e d l a y e r s f u l l y - d e v e l o p e d f l o w i su n l i k e l y t o o c c u r , s i n c e r e l a x a t i o n p r o c e s s e s s l o wd o w n c o n s i d e r a b ly a s t h e t e m p e r a t u r e a p p r o a c h e s t h eg l a s s t r a n s i t i o n t e m p e r a t u r e . D u r i n g p o s t - f i l l i n g t h et e m p e r a t u r e d r o p s i n t h e e n t ir e c h a n n e l . S o , i t m a y b ee x p e c t e d t h a t v i s c o e l a s t i c p h e n o m e n a d o a f f e c t t h ef i l l i n g a n d p o s t - f i l l i n g s t a g e s .

I n c i d e n t a l l y , v i s c o e l a s t i c p h e n o m e n a a r e d e f i n i t e l yp r e s e n t a t t h e g a t e w h e r e e n o r m o u s e l o n g a t i o n a l e f -f e c t s t a k e p l a c e . H o w e v e r , e l o n g a t i o n a l f l o w i s n o tc o n s i d e r e d i n t h e l u b r i c a t i o n t h e o r y . T h i s i s a s e r io u ss h o r t c o m i n g o f m o s t m o l d - f i l l i n g a n a l y s i s p r o g r a m s ,b e c a u s e t h e o v e r a l l p r e s s u r e d r o p m a y b e l a r g e l y a f -f e c t e d b y t h e e l o n g a t i o n a l e f f e c t s a t t h e g a t e . T h es o l u t i o n t o t h i s p r o b l e m i s b e y o n d t h e s c o p e o f t h i sp a p e r .

T h e m e c h a n i c a l a n d t h e r m a l d e s c r i p t i o n o f t h es o l i d s t a t e i s q u i t e d i f f e r e n t f r o m t h a t o f t h e m o l t e ns t a te . A s i m p l e a n d e f f e c t i v e m e m b r a n e t y p e o f s t r u c -t u r a l m e c h a n i c s t h e o r y i s u s e d t o d e s c r ib e t h e d e v e l o p -m e n t o f t h e r m a l l y - i n d u c e d r e s i d u a l s t r e s s e s .

I n t h e l i q u i d s t a t e t h e c o m p r e s s i b l e L e o n o v m o d e li s u s e d , w i t h a s e t o f n r e l a x a t i o n t i m e s ( O k ) a n dv i s c o s i t i e s ( r / k ) . I n t h e s o l i d s t a t e t h e l i n e a r i z e d v e r -s i o n , t h e g e n e r a l i z e d M a x w e l l m o d e l i s u s e d , w i t ha n o t h e r s e t o f m r e l a x a t i o n t i m e s ( O k ) a n d v i s c o s it i e s(Ok).P r e l i m i n a r i e s : A t e a c h p o i n t xR o f t h e m i d p l a n e R , al o c a l o r t h o n o r m a l b a s is O 1 = { g l , g z , g 3} c a n b e d e -f i n e d , s u c h t h a t g l a n d g z a r e t a n g e n t i a l t o R , a n d g 3i s n o r m a l t o R , e . g . , g 3 = t ~. A s a g l o b a l o r t h o n o r m a lb a s is O 0 = { g 0 , d , g ~ is d e f i n e d . T o t r a n s f o r m q u a n -t i t ie s w i t h r e s p e c t t o O 0 t o O 1 , t h e r o t a t i o n m a t r i x Qi s d e f i n e d a s Q i j = 5 " g o . L e t Y R i d e n t i f y a p a r t i c l e a tt h e m i d p l a n e R . T h e p o s i t i o n v e c t o r o f a p a r t i c l ea l o n g K t h a t e m a n a t e s f r o m 2 R i s d e n o t e d b y 2 . N o w ,

4 . 2 F l o w - i n d u c e d r e s i d u al s t r es s e sA s s u m p t i o n s :1 ) W i t h r e s p e c t s t o O 1 , it is a s s u m e d t h a t t h e c o n -

t r i b u t i o n o f t h e n o r m a l s t re s se s c a n b e d i s r e g a r d e di n t h e m o m e n t u m e q u a t i o n . T h i s c a n b e m a d ep l a u s i b l e b y n o t i n g t h a t g r a d i e n t s i n t h e t h i c k n e s sd i r e c t i o n a r e f a r s u p e r i o r t o t h e i n - p l a n e g r a d i e n t s .

2 ) T h e p r e s s u r e i s i n d e p e n d e n t o f t h e g 3 d i r e c t i o n .3 ) T h e m a t e r i a l is a s s u m e d t o f l o w i n t o t h e d o m a i nw i t h a f u l l y d e v e l o p e d v e l o c i t y p r o fi l e .

4 ) F o u n t a i n p h e n o m e n a a r e n o t c o n s i d e r e d .5 ) T h e r m a l c o n d u c t i o n t a n g e n ti a l t o t h e m i d p l a n e Ri s d i s r e g a r d e d .

D u e t o t h es e a ss u m p t i o n s, t h e m o m e n t u m e q u a t i o na p p r o x i m a t e d b y ( w i t h r e s p e c t t o O 1 )s

87 p = r , ( 4. i )8x3 ~P = P ( X l , X 2 ) , ( 4 . 2 )

w i t h; [ O o x ( 4 . 3 )T h e a b o v e a p p r o x i m a t i o n i s o f t e n r e f e r re d t o a s t h et h i n f i l m o r l u b r i c a t i o n t h e o r y . E m p l o y i n g t h ev iscoe las t i c m od e l ( 3 . 6 ) , i t f o l low s f r om o" = o" + O 'p ,a e = - p I + Te a n d ( 3 . 1 0 ) t h a t

= r / ' + r e 3~ i3 8 x 3n r /= E " k a f k : 6 ,k = l O k

i = 1,2 . (4.4)W i t h r e s p e c t t o O1, t h e c o n t i n u i t y e q u a t i o n ( 2 . 1 ) i sw r i t t e n a s

~TI" ~- 0V"--~3 - ~) , 1T=~ [V1 V2] . (4 .5)OX3


6/16

Baaijens, Calculat ion of res idual s tresses in inject ion mold ed prod ucts 289F r o m t h i s t h e s o - c a l l e d p r e s s u r e p r o b l e m c a n b e

d e r i v e d ( s e e A p p e n d i x ) :P E G i v e n T ( , t ) , f i n d p ( x l , x 2 , t ) > _ O s u c h t h a t

S + ~O d h_ _ j dx3r ( S V P + ' r ) = - s - ~ - ~ - , p > - 0 , ( 4.6 )

~ X ~ d x 3= - J z + J 2 ' g i = t l' , i = 0 , 1 , 2 , ( 4 . 7 )J0 s -_ f T = [ f 1 3 ~ 2 3 ] ,

s i x 3 ~ ' i 3 d x 3 ,i3 J 1 r q s e_ ' i 3 d x 3 +J o S - S - t l '

i = 1 , 2 . ( 4 .8 )

I n t h i s , S i s t h e s o - c a l l e d f l u i d i t y c o e f f i c i e n t , a n ds + a n d s - d e n o t e t h e lo c a t i o n s o f t h e s o l i d if i e d l a y -e r s . I n a d d i t i o n , f r e p r e s e n t s t h e c o n t r i b u t i o n o fs+e l a s t i c e f f e c t s , ~ ~ / Q d x 3 r e p r e s e n t s c o m p r e s s i b i l i ty ,

s -a n d d h / d t a c c o u n t s f o r m o l d e l a s t i c i t y . N o t e t h a t f o rs y m m e t r i c f l o w J 1 = 0 . T h e p r e s s u r e p i s n o t a l l o w e dt o d r o p b e l o w z e r o , b e c a u s e a s s o o n a s p b e c o m e sz e r o t h e m a t e r i a l l o s e s c o n t a c t w i t h t h e m o l d . O b -v i o u s ly , b o u n d a r y c o n d i t io n s n e e d t o b e i n t r o d u c e d i nt h e d e f i n i t i o n o f P E , h o w e v e r , t h e y a r e i d e n t i c a l t ot h e N e w t o n i a n c a s e a n d c a n b e f o u n d i n , e . g . , S i t t e r s[ 2] , t h e y a r e b r i e f l y s u m m a r i z e d h e r e . A t t h e g a t ee i t h e r th e f l o w r a t e o r t h e p r e s s u r e i s p r e s c r i b e d . T h ep r e s s u r e i s a s s u m e d n e g l ig i b l e a t t h e f l o w f r o n t a n d n of l u i d m a y p e n e t r a t e t h e s o l i d w a l l s o f t h e m o l d .

E q u a t i o n ( 4 . 6 ) i s t a k e n a s a s t a r t i n g p o i n t f o r t h ef i n i te e l e m e n t i m p l e m e n t a t i o n . V a r i o u s a s p e c t s o f t h isi m p l e m e n t a t i o n c a n b e f o u n d i n , e . g . , S i t t e r s [ 2 ] .

P r o b l e m P E d e f in e s t h e d i r e c t m e t h o d , i n t h e s e n s et h a t v i s c o e l a s t i c m a t e r i a l b e h a v i o r i s a s s u m e d a t t h eo n s e t o f d e r i v i n g t h e p r e s s u r e p r o b l e m . I n t h e i n d i r e c tm e t h o d P E i s s i m p l i f i e d s u b s t a n t i a l l y b y a s s u m i n gg e n e r a l i z e d N e w t o n i a n m a t e r i a l b e h a v i o r . I n t h a tc a s e , f = Q , a n d t h e v i s c o s i t y r / ' is r e p l a c e d b y t h es t e a d y s t a t e v i s c o s i t y o f t h e i n c o m p r e s s i b l e L e o n o vm o d e l [ 5 ], se e S e c t i o n 4 . 4 , E q . ( 4 . 3 3 ) . T h e r e s u l t i n gf l o w k i n e m a t i c s ( v e l o c i t y a n d s h e a r r a t e f i e ld ) a r e s u p -p l i e d t o t h e v i s c o e l a s t i c c o n s t i t u t i v e e q u a t i o n t oc a l c u l a t e ( a s a p o s t - p r o c e s s i n g s t e p ) f l o w - i n d u c e dr e s i d u a l s t re s se s , g i v i n g r e m a r k a b l y g o o d r e s u l t s, e v e ni n t h e p o s t - f i l l i n g s t a g e .

4 . 3 T h e r m a l l y - i n d u c e d r e s i d u a l s t r e s s e sT h e r m a l l y - i n d u c e d r e s i d u a l s t r e s s e s d e v e l o p i n t h e

s o l i d i f i e d l a y e r s . I n t h i s s e c t i o n t h e l i n e a r i z e d m o d e l( 3 . 1 4 ) - ( 3 . 1 7 ) is r e d u c e d b y t a k in g a n u m b e r o fk i n e m a t ic a n d d y n a m i c a s s u m p t i o n s i n t o a c c o u n t .T h e r e d u c e d c o n s t i t u t i v e e q u a t i o n i s t h e n e m b e d d e di n a m e m b r a n e t y p e o f f i n i t e e l e m e n t .

A s s u m p t i o n s :1 ) D i s p l a c e m e n t s a n d r o t a t i o n s a r e s m a l l .2 ) B e n d i n g is u n i m p o r t a n t . T h i s is r e a s o n a b l e b e c a u s e

t h e i n - p l a n e d e f o r m a t i o n s a r e s m a ll , t h e c h a n n e l i so n l y w e a k l y c u r v e d a n d t h e o u t - o f - p l a n ed i s p l a c e m e n t s a r e s e v e r e ly re s t r i c te d b y t h e m o l d .Y e t , a s s o o n a s t h e p r o d u c t i s e j e c t e d f r o m t h em o l d b e n d i n g is o n e o f t h e k e y p h e n o m e n a .

3 ) T h e d i s p l a c e m e n t f i e ld t~ i s h o m o g e n e o u s i n t h en o r m a l d i r e c t i o n , e . g . ,t~(Y) = f i (~R) (4.9)

S o , i f t h e s o l i d i f i e d l a y e r s a r e s e p a r a t e d b y f l u i d ,t h e y b o t h d i s p la c e i n t h e s a m e m a n n e r : t h e y a r e v i r-t u a l l y c o u p l e d t o e a c h o t h e r .

4 ) D u e t o t h e p r e s e n c e o f t h e m o l d , a l l n o r m a l d i s -p l a c e m e n t s a r e a s s u m e d t o b e s u p p r e s s e d :

~ ( ~ N ) ' f i = 0 . ( 4 . 1 0 )I n f a c t , d u e t o s h r i n k a g e a s m a l l c l e a r a n c e b e t w e e np r o d u c t a n d m o l d m a y d e v e l o p a n d h e n c e s m a l ll a t e r a l d i sp l a c e m e n t s m a y o c c u r . Y e t , t h e s e a r e s u f -f i c i e n tl y sm a l l t o b e d i s r e g a r d e d i n a f i r s t a p p r o x -i m a t i o n .

5) W i th re sp ec t to 0 1, the shea r s t ra ins e l3 and e23 a red i s r e g a r d e d .

6 ) T h e n o r m a l s t re s s 0-33 i s h o m o g e n e o u s i n t h e n o r -m a l d i r e c t i o n .

7 ) A s l o n g a s a 3 3 < 0 t h e m a t e r i a l s t ic k s t o t h e m o l d .8 ) S o l i d i f i c a t i o n t a k e s p l a c e a t t h e g l a s s - t r a n s i t i o n

t e m p e r a t u r e , a n d f l o w - i n d u c e d s tr e ss e s d o n o tr e l a x a f t e r s o l i d i f i c a t i o n . T h e f l o w - i n d u c e d s t r e s su p o n s o l i d i f i c a t i o n i s d e n o t e d r u e .

9 ) M o l d e l a s t i c i t y i s d i s r e g a r d e d , i . e . , d h / d t = 0 .R e d u c e d c o n s t i tu t i v e e q u a t io n : S i n c e d i s p l a c e m e n t s i nt h e s o l i d i f i e d m a t e r i a l a r e s m a l l , t h e l i n e a r i z e d c o n -s t it u ti v e m o d e l ( 3 . 1 4 ) - ( 3 . 1 7 ) c a n b e u se d . A s s u m et h a t a t t = tn t h e s t r e s s s t a t e is k n o w n . C o n s i d e r t h es t a t e t r a n s i t i o n t n ~ t n + l , tn + 1 - tn = zl t . D u r i n g t h i st r a n s i t i o n i t is a s s u m e d t h a t T a n d D a r e c o n s t a n t . L e t


7/16

290 Rhe ologica Acta , Vol . 30, No. 3 (1991)

A T = i " ( t n + O A t a n d g = D A t .t e n s o r a t t = t n + l , a n + l , sa t i s f i e s

a n + l = 6 " + ~ t r ( g ) I + 2 G g d ,

T h e C a u c h y s t r e s s

(4 .11)w i t h

I f a ll t h e m a t e r i a l h a s s o l i d i f i e d i n t h e n o r m a l d i r e c -t i o n , t h e n t h e m a t e r i a l m a y e i t h e r s t ic k t o t h e w a l ls o rm o v e t a n g e n t i a l t o t h e m i d p l a n e R . I f i t s t ic k s to t h ew a l l s t h e n e n = e 22 = 0 a n d ( d u e t o a s s u m p t i o n 3 a n dhm/29 ) ~ a33 d x 3 = 0 ; s o

- hm/2m~ = - P s ( t n ) I + ~ e - ~ t k r ~ ( t n ) hm/2 3

- h m / 2 0 " 3 3 4 ( ~ + 3 t?- f l A T I + T v e , (4.1 2) 0-33 - , (4.1 6)hm/2 3tn +A t 1 I - - d x 3~ A t k = - - d s , (4 .13) - h m / 2 4 G + 3 f f ;t . a r ( s ) O k

(4 .14)

1 tn~ 1 1 tn+lS K d ,f l = - A t tn A t t .m 1 tn f l OkOe_( t .+ , ,_~k) ds ,

O= E Okok = 1 tnw h e r e P s d e n o t e s t h e h y d r o s t a t i c p r e s s u r e i n t h e s o l i ds t a t e, a s o p p o s e d t o t h e p r e s s u r e i n th e m e l t p . L e t a tt = tg s o l i d i f i c a t i o n t a k e p l a c e , t h e n 0 "(tg ) = - p ( t g ) l+ T v e ( t g ) . S o , t h e a b o v e u p d a t i n g s c h e m e f o r 0 " h o l d sf o r t > t g . T h e c o n s t i t u t i v e e q u a t i o n ( 4 . 1 1 ) c a n b es i m p l i f i e d c o n s i d e r a b l y . D u e t o a s s u m p t i o n s 6 a n d 7 ,t h e n o r m a l s t re s s 0 "3 3 c a n b e d e t e r m i n e d b yd i s t i n g u i sh i n g b e t w e e n t h e s i t u a t i o n w h e r e a f l u i d s ti lle x i s t s a n d t h e s i t u a t i o n w h e r e a l l t h e f l u i d h a ss o l i d i f i e d i n t h e x 3 - d i r e c t i o n .

R e m a r k 4 . 1 : T h e c h o i c e 0 - ( tg ) = - p ( t g ) I + r ve (t g ) isc o n t r o v e r s i a l . I t is n o t c l e a r to w h a t e x t e n t f l o w - i n -d u c e d r e s i d u a l s t r es s e s a c t u a l l y c o n t r i b u t e t o t h e s t r e sss t a t e a t t e m p e r a t u r e s s i g n i f i c a n t l y b e l o w t h e g l a s st r a n s it i o n t e m p e r a t u r e . T h e e x i s te n c e o f n o r m a ls t re s s e s i n a p o l y m e r m e l t i s a t t r i b u t e d t o t h e m o l e -c u l e ' s a b i l it y to r e g a i n i ts m o s t p r e f e r r e d , u n o r i e n t e d ,s t a te . T h i s i s o n l y p o s s i b l e a t s u f f i c i e n tl y h i g h t e m p e r -a t u r es . T h e r e f o r e t h e a b o v e c h o i c e o f 0-(tg) i s q u e s -t i o n a b l e , a n d a n a l t e r n a t i v e s e e m s t o b e : a u g ) =- p ( t g ) L T h i s l a s t d e f i n i t i o n i s u s e d i n p r a c t i c a lc a l c u l a t i o n s l a te r o n . H o w e v e r , a s r v e ( t g ) i s v e r ys m a l l c o m p a r e d t o t h e f i n a l s t re s s s t a t e a t r o o m t e m -p e r a t u r e , t h e a c t u a l d i f f e r e n c e b e t w e e n t h e t w o a l t e r -n a t i v e s i s v e r y s m a l l .

I f a f l u i d p h a s e s t il l e x i s ts , t h e n o -3 3 i s o b v i o u s l ye q u a l t o t h e p r e s s u r e i n t h e m e l t

0"33 = - P (4.15 )N o t e t h a t p _ > 0 .

o t h e rw i se 0"33 = 0 .I n t h e a b o v e it is c r u c i a l t o k n o w w h e t h e r o r n o t t h e

m a t e r i a l s t i l l c o n t a c t s t h e m o l d . T h e c o n t a c t c o n d i -t i o n s a r e f o r m u l a t e d a s f o l l o w shm-hc


8/16

Baaijens, Calculat ion of residual s tresses in inject ion mo lded prod ucts 291

T h e d e f i n it i o n o f a a n d b m a y s e em r a t h e r a d h o c , b u tt h e y a p p e a r q u i t e n a t u r a l l y w h e n ( 4 . 1 1 ) i s w r i t t e n i na f o r m l i k e ( 4 . 1 8 ) .M e m b r a n e e l e m e n t : A t a c e r t a i n t i m e , s a y t = t o t h ed i s p l a c e m e n t f i e l d a ( g R , t n ) a n d t h e s t r e s s f i e l do -( g, t n ) a re a s s u m e d t o b e k n o w n . D u r i n g t h e s t a t et r a n s i t i o n t~-~tn+ ~, t h e d i s p l a c e m e n t f i e l d c h a n g e sw i t h A a a n d i t is d e t e r m i n e d b y e m p l o y i n g th e f i n i t ee l e m e n t m e t h o d w i t h b i - l i n e a r q u a d r i l a t e r a l m e m -b r a n e e l e m e n t s .

L e t t h e p h y s ic a l m e m b r a n e d o m a i n b e m a p p e d o nt h e u n i t c u b e b y m e a n s o f t h e l o c a l c o o r d i n a t e s ~ , r/a n d ( s u c h t h a t ~ , t /, ~ e [ - 1 , 11 . T h e g e o m e t r y a n d t h ed i s p l a c e m e n t f i e l d a r e i n t e r p o l a t e d a s

42 = ~ N a ( 4 , r l ) g a , (4 . 24)a = l4a a = ~ N a ( ~ , r l )A a a . ( 4 . 2 5 )

a=t

T h e s t r a i n - d i s p l a c e m e n t m a t r i x , s e e H u g h e s [ 14 ], isd e f i n e d a s

B = I n t . . . n 4 ] , ( 4 . 2 6 )

w i t h

I o ! l- a = B 2 , B i = = Q i j . ( 4 . 2 7 )B , o x i A s s u c h B i s a r r a n g e d t o b e c o m p a t i b l e w i t h e i n t h es e n s e t h a t

e = B _ A u , A u r = [ Au r3 . . . A u r 4 ] , (4 . 28)w h e r e A u a c o n t a i n s t h e c o m p o n e n t s o f A t ~ a t n o d e aw i t h r e s p e c t t o t h e l o c a l b a s i s O 1 .

T h e e l e m e n t s t i f f n e s s m a t r i x K e a n d t h e e l e m e n tf o r c e v e c t o r f e a r e g i v en b y

K ~ = [ K a b ] , f e = ~ a l , a , b = l . . . . . 4 , ( 4 .2 9 )w h e r e

g a b I T T ^O a B-a -MB_bQ bdA , (4 . 30)A

w h e r e A r e p r e s e n t s th e a r e a o f t h e m i d p l a n e o f t h ee l e m e n t . T h e d e f i n i t i o n o f t h e m a t e r i a l m a t r i x 5 7 I a n dt h e i n i ti a l st r es s c o l u m n ~ d e p e n d s o n w h e t h e r o r n o tt h e m a t e r i a l c o n t a c t s t h e m o l d . I f t h e m a t e r i a l c o n -t a c t s t h e m o l d , t h e n

s - h m / 2M = j M d x 3 + j M d x3 ,

- h m / 2 s +s - h m / 2

~ = l ~ d x 3 + I ~ d x 3 ,- h m / 2 s +

(4 . 32)

w h i l e o t h e r w i s e t h e i n t e g r a t i o n t a k e s p l a c e o v e r t h ee n t i r e m a t e r i a l t h ic k n e s s , e . g . , f r o m - h m / 2 t o h m / 2 .R e m a r k 4 . 2 : F o r m a l l y , t h e i n t e g r at i o n f r o m - h m / 2t o h m / 2 i s i n c o n s i s t e n t w i t h a s s u m p t i o n s m a d eb e f o r e . T h e l i n e a r i z e d m o d e l i s i n t e n d e d t o t h e a p -p l i e d i n t h e s o l i d s t a t e o n l y , w h e r e a s i n t h e m o l t e ns t a t e t h e n o n - l i n e a r m o d e l s h o u l d b e a p p l i e d . H o w -e v e r , i n p r a c t i c a l s i t u a t i o n s d e f o r m a t i o n s a n d d e f o r -m a t i o n r a t e s a r e s m a l l a s s o o n a s t h e m a t e r i a l l o s e sc o n t a c t w i t h t h e m o l d , s i n c e i n t h a t c a s e m o s t o f t h em a t e r i a l h a s s o l i d i f i ed a l r e a d y . T h a t m e a n s t h a t i n th em e l t t h e e x t r a - st r e ss c o n t r i b u t i o n t o t h e C a u c h y s t re s st e n s o r i s n e g li g i bl e , n o m a t t e r w h a t m o d e l i s a p p l i e d .N o t e t h a t i n th e m e l t r i g id i t y is v e r y l o w , a n d h i g hs t r a i n r a t e s a r e n e e d e d t o c a u s e s u b s t a n t i a l s t r e s s d e -v e l o p m e n t . T h e s h r i n k a g e o f t h e m e l t, h o w e v e r , n e e d st o b e a c c o u n t e d f o r ; i n t e g r a t i n g o v e r t h e e n t i r e h e i g h ti s m e r e l y a w a y t o d o t h i s .R e m a r k 4 . 3 : N o t e t h a t a s l o n g a s t h e m a t e r i a l s t i c k st o t h e w a l l , o n + l c a n b e c a l c u l a t e d d i r e c t l y f r o m( 4 .1 8 ) a s e = Q d u e t o a s s u m p t i o n 7 . I f f r i c t i o n a ls l id i n g b e t w e e n p o l y m e r a n d m o l d i s a l l o w e d , t h i s n ol o n g e r h o l d s .R e m a r k 4 . 4 : T h e c o n s t r a i n t A tT . t i = 0 c a n b e t a k e n i n -t o a c c o u n t b y a p e n a l t y f u n c t i o n m e t h o d . T h a t i s, t h ef o l l o w i n g t e r m i s a d d e d t o t h e s t i f f n e s s m a t r i x(4 . 30) : I e N a _ n n r N b d A , w h e r e . n r = [ ~q .e f l . g2

Af i" e ~ l , a n d e a s u f f i c i e n t l y l a r g e p e n a l t y p a r a m e t e r .

4 . 4 T e m p e r a t u r e e q u a t io nW i t h u s e o f ( 3 . 1 8 ) a n d ( 3 . 1 9 ) , t h e e n e r g y e q u a t i o n

( 2 . 3 ) i s w r i t t e n a sQ C p f F = ~ . ~ T + z : D a T 0 9 .Q 8 f ; p f lea S T T ^Q _ a B a g d A , (4 . 31) (4 . 33)A


9/16

292 Rhe ologica Acta, Vo l. 30, No. 3 (1991)T o t a k e d i s s i p a t i o n i n t o a c c o u n t w i t h o u t h a v i n g t od e a l w i t h v i s c o e l a s t i c p h e n o m e n a , z i n ( 4 . 9 ) i s t a k e nt o r e p r e s e n t t h e g e n e r a l i z e d N e w t o n i a n b e h a v i o r o ft h e m a t e r i a l . H e n c e z = 2 0 0 ) , T ) D a, w her e ?) i s thes h ea r r a te 0 ) = ~ ) . T he v is c o si ty 0 is ob -t a i n e d f r o m t h e s t e a d y s t a t e b e h a v i o r o f t h e i n c o m -p r e s s i b l e L e o n o v m o d e l [ 5 ] a t s i m p l e s h e a r

/700) , T) = r / ' + ~ 2 r /kk = l l + x k 'x k = ]/(1 + 4~)202) . (4.33)

F u r t h e r , t h e c o n d u c t i o n a l o n g t h e c h a n n e l i s d i s -r e g a r d e d d u e t o t h e t h i n n e s s o f t h e c a v i t y c o m p a r e dt o i t s l e n g t h . H e n c e w i t h r e s p e c t t o 0 2 , t h e t e m p e r a -t u r e p r o b l e m t o b e s o l v e d i sPT Gi ven 17 ( , t ) and p ( x l , x z , t ), f i n d T ( , t ) s u c ht h a t

+ 0 ))2 _ (4.3 4)0 x3 \ 0 x 3 , / O T p "D u r i n g t h e f i l l i n g s t a g e t h e c o n t a c t t e m p e r a t u r e i sp r e s c r i b e d a t t h e p r o d u c t m o l d i n t e r f a c e , t h e r e a f t e rt h e m o l d t e m p e r a t u r e i s p r e s c r i b e d a s a b o u n d a r yc o n d i t i o n . O b v i o u s l y , a m u c h b e t t e r w a y t o o b t a i n t h et e m p e r a t u r e b o u n d a r y c o n d i t i o n s is t o s i m u l t a n e o u s l yc a l c u l a te t h e t e m p e r a t u r e d i s t r i b u ti o n i n th e m o l d a n dp r o d u c t t h r o u g h o u t t h e p r o c e s s , y e t th i s is b e y o n d t h es c o p e o f t h i s p a p e r .

5. Computational aspectsI n a l l c a s e s t h e i m p l i c i t E u l e r s c h e m e i s u s e d f o r

t e m p o r a l d i s c r e t i z a t i o n . A t e a c h t i m e s t e p , f i r s t t h et e m p e r a t u r e a n d p r e s s u r e e q u a t i o n a r e s o l v e d i n -d e p e n d e n t l y , w h e r e c o u p l i n g i s e n f o r c e d b y t h ei t e r a t i v e s c h e m e . T h a t i s , t h e s e q u e n c e o f p r o b l e m s t ob e s ol ve d at e ac h t i m e st ep is : P T - - , P E ~ P T ~ P E - - , . . .u n t i l c o nv e r g e n c e . T h e p r e s s u r e p r o b l e m P E is s o lv e db y e m p l o y i n g t h e f i n it e e le m e n t m e t h o d ( F E M ) w i t hl i n e a r e l e m e n t s . T h e n o n - l i n e a r i t y o f t h e r e s u l t i n g s e to f e q u a t i o n s i s d e a l t w i t h i n th i s s e c t io n . T h e t e m p e r a -t u r e p r o b l e m P T i s s o l v e d w i t h a f i n i t e d i f f e r e n c e( F D ) s c h e m e w h e r e t h e d i f f e r e n t i a l g r i d is c e n t e r e d a te a c h e l e m e n t . F o u r c o m p u t a t i o n a l a s p e c t s a r e d e a l tw i t h i n m o r e d e t a i l : t h e s o l u t i o n o f t h e p r e s s u r e p r o -b l e m P E , t h e m e t h o d o f c h a r ac t e r is t ic s t o h a n d l e t h e

m a t e r i a l d e r i v a t i v e s , t h e c a l c u l a t i o n o f t h e s h e a r r a t ei n c a s e o f v i s c o e l a s t i c m a t e r i a l b e h a v i o r , a n d t h ec a l c u l a t i o n o f Bek.S o l u t i o n o f t h e p r e s s u r e p r o b l e m : T h e s y s te m o f E q s .( 4 .6 ) i s h ig h l y n o n - l i n e a r a n d i s s o l v e d w i t h a t w o - s t e pp r o c e d u r e . A f u l l y i m p l i c i t s c h e m e is u s e d i n t h e t i m ed o m a i n .S t e p 1 . A n i n i t i a l e s t i m a t e f o r t h e p r e s s u r e p , a n da s s o c i a t e d p r o p e r t i e s s u c h a s v e l o c i t y a n d s h e a r r a t ef i e l d s , a r e f o u n d b y s o l v i n g f o r g e n e r a l i z e d N e w t o -n i a n m a t e r i a l b e h a v i o r . T h e v i s c o si ty is t a k e n a s t h es t e a d y s t a t e v i s c o s i t y o f t h e i n c o m p r e s s i b l e L e o n o vm o d e l . T h e n P E r e d u c e s t oP V G i v e n T ( , t ) , f i n d p ( x~ , x2 , t ) >_0 s u c h t h a t

S + .7 r ( S g p ) = - ~ ~ d x 3 - d h , p > _ O , (5 .1)s 0 d ts+~ = _ ~ + j 2 , Z = f Xi@dx3, i = 0 , 1 , 2 . ( 5 . 2 )

J 0 s - / 7E q u a t i o n ( 5 . 1 ) i s s o l v e d w i t h a P i c a r d i t e r a t i o ns c h e m e , a s s u m i n g d h / d t = 0

f d x 3 f g i + 1 + V T ( s i ~ T p i +, , _ V /= - - ~ O k O d x 3 1 0 i ( 5 . 3 )

, V / ': 2 i ' +

=2 j i ' I - ~ j 0 , 1 , 2 , ( 5 . 4 )

w h e r e t h e s u p e r s c r i p t s i a n d i + 1 r e s p e c ti v e l y r e f e r t ot h e p r e v i o u s o r t h e c u r r e n t i t e r a t i o n .S t e p 2 . G i v e n a n e s t i m a t e o f t h e p r e ss u r e , s h e a r r a t ef i e l d , e t c . , b y s t e p 1 , P E i s f in a l l y s o lv e d w i t h t h eN e w t o n i t e r a t i o n p r o c e s s . W i t h i n t h e f i n i t e e l e m e n tc o n t e x t , t h e e l e m e n t s t i f f n e s s m a t r i c e s a r e d e t e r m i n e db y n u m e r i c a l d i f f e r e n t i a t i o n .M e t h o d o f c h a r a c t e r i s t i c s : ( P i r o n n e a u [ 1 1 ] a n dM o r t o n e t a l . [ 1 2 ] ) . C o n s i d e r t h e t i m e i n t e r v a lt e [ t n , t n + l] , a n d l e t A t = t n + l - t n . T h e t i m ed e r i v a t i v e p i s a p p r o x i m a t e d b y

p _ p ( 2 , tn+ 1) - p ( S n , tn ) , (5 .5)A t

wh ere ~n des i gn a t es t he pos i t i on a t t i m e t n o f t he ma -t e r i a l p a r t i c l e c u r r e n t l y l o c a t e d a t 2 . T h e m a t e r i a l t i m ed e r i v a t i v e s o f T a n d Bee a r e t r e a t e d l i k e w i s e .


10/16

Baaijens, Calculation of residual stresses in injection molded products 293

Calculation of the shear rate: To calculate/~ek, theshear rate must be known. Equation (A.2) constitutesa non-linear relation for the shear rate. It is solvedpointwise with a secant-method, such that Bek andOv/Ox3 are calculated simultaneously.Calculation of Be# The k'th mode unimodular elasticleft Cauchy-Green strain tensor Bek is calculatedfrom (3.8). A variable-order variable-time stepbackward difference scheme is used to integrate (3.8)over a certain time interval. During each time inter-val, say tn~tn+l, L is assumed to be constant and theinitial value of [ ~ e k is l ~ e k ( S n , tn)"6 . E x a m p l e

As an example, polycarbonate (PC) is injected intoa cavity of 80 x 50 2 mm (length, width, height), seeFig. 2. Along A a line gate is assumed. The materialproperties of PC in the case where a viscoelastic con-stitutive model is used are given in Table 1.

L i n e g a t e

Fig. 2. Sketch of the cavity, strip of 80.50.2 mm

Table 1. Parameters for PCParameters WLF equation:TO= 200 oCC 1 = -4.2 17C2= 94.95 CThermal properties:~ p = 1.5 J/gK0 . 2 7 1 0 - 3 J/s mm KVisco-elastic properties:v/' = 0.003 0 MPas

0 1 = l 0 - 1 S0 2 = 1 0 - 2 S0 = 10 -3 s0 1 = 1 0 - 1 0 S

Tait parameters:s = 0.51 C/MPasolida o = 868a 1 = 0.22B o = 395.4B a = 2.609 10 -3

th = 9.74 1 0 - 3 MPasr/2 = 6.7 5 1 0 - 3 MPast/3 = 1. 25 1 0 - 3 MPas01 = 894 10 -1 MPas

melt868 mm3/g0.577 mm3/gK316.1 MPa4.078 10 -3 C -I

Processing conditions PC: The material is injected atan average velocity of 1 20mm /s at 320C. Thisvelocity is maintained at A until the holding pressureof 50 MP a is reached. The holding pressure is main-tained at A until the elapsed time exceeds 4.0 s. At thistime the gate is assumed to freeze off. Clearly, an ac-curate model of gate freeze off is not available yet,and it is unlikely that the pressure remains at 50 MP ain reality. The proper modeling of the behavior in andnear the gate is still an open question. As can be seenfrom the results later on, the period of time that thegate remains open determines, to a large extent, theamount of residual birefringence in the product. Themold has a tempera ture of 80 C.Results for PC: Calculations are performed with boththe viscoelastic and the viscous model. Figure 3 showsthe calcula ted pressure hist ory at x~ = 8 mm andx~ = 56 mm for the viscoelastic case. They virtuallycoincide with the results of the viscous model, whichare not shown. During the filling of the mold, for t[0, tf= 0.67] s, the pressure g radu ally rises. The n ashort compression stage follows where the pressurerapidly increases to 50 MPa. Hereafter, due to cool-ing, the pressure slowly drops. After solidification ofthe gate the pressure drops further until the productis ejected from the mold.

Figures 4 to 8 show the evolution o f the first normalstress difference N 1, def ined as N 1 = r~l - r e3, for theviscoelastic (direct) case. N~ is shown at 5 times: atthe end of the filling stage tf = 0.67 s, at the end ofthe compression stage t c = 0.71 s, and at t = 2.7, 3.2and 4.7 s. During the compression stage most of theflow-induced normal stresses relax because the flowrate is low and the temperatur e at the core of the cavi-

P r e s s u r e (MPa)6O

5 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 0 .......... ~ - - x = 8 m m ..............................

30 . . . .

2 4 6 8 10T i m e ( s )

Fig. 3. Numerical pressure trace for PC, viscoelastic materi-al model


11/16

2 9 4 R h e o l o g i c a A c t a , V o l . 3 0 , N o . 3 ( 1 9 9 1 )

o

' ~ 1.0' m r ~ e l n g t h r a m ) ~ i s t a n c e , d p b n e r ~ r ~ )

F i g . 4 . N 1 a t e n d o f f i l l i n g s t a g e , t = 0 . 6 7 s

" g ' L ~

O

0

n n e l le n n m ~ . o . o ^ , -, m i d l o n e~ " tram) i s t d n t ~ " pFi g . 7 . N ~ a t t = 3 .2 1 s

, ' 5 ~~o

O

0

' l a r ~ n e i e n g t h ( m r -q ) ~ i s to n c e m 'd p lG n ~F i g . 5 . N ~ a t e n d o f c o m p r e s s i o n s t a g e

~ ~ 1.0f la n n e l l e n ~ + k ~ o .o , ~ d p l a n e (ram)~ " k r n m ) O i s t a n c e " '

Fi g . 8 . N 1 a t t = 4 .7 s

' g ~ n . ,

OQo- 0r n e l l e n a t h f ~ _ ' , 0 . 0 . _ ~ r ~ i d p l a r ~o " ' ~ u m ) p i s to u ~ - e

Fi g . 6 . N 1 a t t = 2 .7 s

N 1M P a ).12 Ga t er e e z e - o f f ~0.1 .................................................................................................................................................................

0 .08

0 . 0 8

0,0 2 ~ .. .. .. .. .. .. .. .. .. .. .. .. .. . ...........................................................

0 - -0.5 1 1 .5 2 2 .5 3 3 .5 4 4 .5 5T i m e ( s )

F i g . 9 . N , h i s t o r y a t x = 8 r a m , a t a d i s t a n c e o f 0 . 3 5 m mf r o m t h e m i d p l a n e


12/16

Baai jens , Calcu lat ion of res idual s tresses in inject ion mo lded prod ucts 295

t y is s ti ll q u i t e h i g h , c o m p a r e F i g s . 4 a n d 5 . T h i s d o e sn o t a p p l y to r e g i o n s c l o s e to t h e w a l ls , b e c a u s e t h e r et h e t e m p e r a t u r e h a s d r o p p e d b e l o w Tg a n d r e l a x a t i o nh a s v i r t u a l l y s t o p p e d . D u r i n g t h e p o s t - f i l l i n g s t a g es h e a r r a t e s a re s e v e r a l o r d e r s o f m a g n i t u d e s m a l l e rt h a n i n t h e f i ll in g s t a g e . H o w e v e r , d u e t o t h e d e c r e a s -i n g t e m p e r a t u r e a s t i m e p r o c e e d s , s m a l l s h e a r r a t e sm a y s t il l i n t r o d u c e c o n s i d e r a b l e n o r m a l s t r e s se s , a s i sc l e a r l y d e m o n s t r a t e d i n F ig s . 5 t o 8 . F i g u r e 8 s h o w st h e f i n a l N 1 d i s t r ib u t i o n b e c a u s e a t t h a t t i m e a l l m a -t e r i a l h a s s o l i d i f ie d . A f t e r a b o u t t = 2 . 5 s t h e f i r s t n o r -m a l s t re s s d i f f e r e n c e a l m o s t e x p l o d e s t o a v e r y h i g hv a l u e , i n d i c a t i n g t h a t p a c k i n g t i m e s s h o u l d b e li m i t e di f m i n i m a l r e s i d u a l b i r e f r i n g e n c e i s r e q u i r e d . T h i s i sc l e a r l y s h o w n i n F i g . 9 , w h i c h s h o w s t h e N l - h i s t o r ya t x = 8 m m a t a d i s ta n c e o f 0 .3 5 m m f r o m t h e m i d -p l a n e .

T h e s e r e su l ts c a n b e c o m p a r e d w i t h c a l c u la t i o n s o b -t a i n e d w i t h t h e i nd i r e c t m e t h o d , w h e r e th e k i n e m a t i c so f t h e g e n e r a li z e d N e w t o n i a n m o d e l i s u s e d t oc a l c u l a t e f l o w - i n d u c e d r e s i d u a l s t r e s s e s w i t h t h e c o m -p r e s s ib l e L e o n o v m o d e l . F i g u r e 10 c o m p a r e s N 1 o b -t a i n e d w i t h b o t h t e c h n i q u e s a t x = 8 m m . Q u a l i t a ti v e -l y , th e r e s u l t s a r e in v e r y g o o d a g r e e m e n t , w h i l e t h ec o m p u t a t i o n t i m e o f t h e i n d i re c t m e t h o d is le ss th a n1 / 1 0 o f th e d i r e c t m e t h o d . T h i s im p l i e s th a t t h e m o r ec o s t - e f f e c t i v e , i n d i r e c t, m e t h o d c a n b e u s e d a s a n i n -d i c a t o r f o r m o l e c u l a r o r i e n ta t i o n , a n d h e n c e a s am e t h o d t o a s s i g n a n i s o t r o p i c m a t e r i a l p r o p e r t i e s t ot h e m o d e l .

F i g u r e s 1 1 t o 1 9 s h o w t h e e v o l u t i o n o f t h e t h e r m a l -l y - i n d u c e d o ' n . I n t h e f i n a l t i m e s t e p t h e m a t e r i a l i sf o r c e d t o c o o l d o w n t o t h e f in a l u n i f o r m t e m p e r a t u r eo f 2 0 C . T h i s e x p l a in s t h e w i d e d i f fe r e n c e b e t w e e nF i g s . 1 8 a n d 1 9 . N o t e t h a t t h e s t r e s s e s a r e p o s i t i v e a tt h e e d g e , n e g a t i v e n e a r t h e e d g e , a n d p o s i t i v e in t h e

N1 (MPa)0.1

0.08

0.06

0.04

0.02

0-1 -0.5 0 0.5Distance midplane (mm)

F i g . 10 . C om par i son o f N 1 a t x I = 8 mm

- - D i r ec t. . . . Indirect

6)

CQ?1.0

u u l d l e n ~ + l . . , , o . o , . ~ ; d n l a n e r a m )~ ' " k r ~ r n ) D i s t d n c e " " ~Fig. 11. Thermal ly induced s t ress a t t = 0.67 s

l i b

7 5 . 0 . ~ 1 .oM o u l d J s l g t h ' " ~ i s t a n c e m i d p J a n e ( r ~ r ~ )Fig. 12. Thermal ly induced s t ress a t t = 0,7 s

Q

75.0Y o u t d I s ! 1.0D i s t o n c

Fig. 13. Thermal ly induced s t ress a t t = 2.7 s


13/16


14/16

Baaijens , Calculat ion of res idual s t resses in inject ion mold ed produ cts 297c o r e . T h i s s t r e s s d i s t r i b u t i o n i s m a r k e d l y d i f f e r e n tf r o m w h a t i s f o u n d i n f r e e q u e n c h e x p e r i m e n t s ( s e eW i m b e r g e r - F r i e d l a n d H e n d r i k s [ 1 5 ] ) . T h e d i f f e r e n c ei s d u e t o t h e p r e s s u r e e v o l u t i o n i n t h e m o l t e n p a r t o ft h e p r o d u c t : s o l i d i f i c a t i o n t a k e s p l a c e a t e l e v a te dp r e s s u r e s .T h e e f f e c t o f m o l d e l a s ti c i ty is s h o w n i n F i g . 2 0 .T h e m o l d e l a s t i c i t y i s m o d e l e d v e r y c r u d e l y b y

dh dh dpd t d p d t (6 . 1 )

w h e r e d h / d p = 0 . 4 g m / M P a i s c h o s e n . A s i s se e nf r o m F i g . 20 , t h is s m a l l a m o u n t o f m o l d e l a s ti c i ty h a sa s i g n i f i c a n t i n f l u e n c e o n t h e p r e s s u r e h i s t o r y i n t h em o l d a n d , t h e r e f o r e , o n t h e f i n al t h e r m a l ly - i n d u c e ds t r e ss s t a t e , a s s h o w n i n F i g . 2 1 . N o t e , h o w e v e r , t h a tt h e t h e r m a l l y - i n d u c e d s t r e s s t h e o r y d o e s n o t i n c l u d em o l d e l a s t i c it y , s o Fi g . 21 m u s t b e h a n d l e d w i t h c a r e .E s s e n t i a ll y , a s s u m p t i o n 9 o f S e c t i o n 4 . 3 i m p l i e s t h a t ~ -

hm/2 ~33 dx 3 = 0 . T h i s a s s u m p t i o n m u s t b e a l t e r e d t o /d "- hm/2f o r m a l l y i n c l u d e m o l d e l a s ti c i ty in t o t h e t h e r m a l l y - i n -d u c e d s tr e ss t h e o r y . T h e c u r r e n t c a l c u l a t i o n o n l y i n - ~ ~ _e l u d e s t h e e f f e c t s o f t h e c h a n g e i n p r e s s u r e h i s t o r y . ~)

F i n a l l y , F i g . 2 2 s h o w s p a r t o f t h e t e m p e r a t u r eh i s t o r y o f a p o i n t a t x~ = 8 m m a t t h e m i d p l a n e . I tc l e a r l y d e m o n s t r a t e s t h e e f f e c t o f h e a t d u e t o c o m - UJo~p r e s s i o n , a p h e n o m e n o n t h a t i s n o t n e g l ig i b l e. ~ o.? -7 . C o n c l u s i o n s

T h e e v o l u t i o n o f b o t h f l o w - a n d t h e r m a l l y i n d u c eds t r e s s e s d u r i n g i n j e c t i o n m o l d i n g i n b o t h t h e f i l l i n ga n d t h e p o s t - f i l l i n g s t a g e i s i n v e s t i g a t e d n u m e r i c a l l y .

A c o m p r e s s ib l e v e r si o n o f t h e L e o n o v m o d e l w a sd e v e l o p e d a n d a p p l i e d .

T h e c a l c u l a t io n s s h o w c l e a r ly t h a t a s u b s t a n t i a lp o r t i o n o f t h e f l o w - i n d u c e d r e s i d u a l s t r e s s e s a r i s ed u r i n g t h e p o s t - f i l l i n g s t a g e . T h i s i s i n a g r e e m e n t w i t he x p e r i m e n t a l d a t a [ 13 ] . T h e i n d i r e c t a p p r o a c h g i ve s ag o o d q u a l i t a t i v e a n d a r e a s o n a b l e q u a n t i t a t i v e p r e d i c -t i o n o f f l o w - i n d u c e d r e s i d u a l s t r e s s e s w h e n c o m p a r e dw i t h t h e d i r e c t , v is c o e l as t i c a p p r o a c h . I n t h i s i n d i r e c tm e t h o d t h e p r e s s u r e p r o b l e m i s d e r i v e d w i t hg e n e r a l i z e d N e w t o n i a n m a t e r i a l b e h a v i o r , a n d t h er e s u l t i n g k i n e m a t i c s i s s u p p l i e d t o t h e v i s c o e l a s t i cc o n s t i t u t i v e e q u a t i o n . T h i s a p p r o a c h i s a v a l u a b l et o o l t o g i v e a q u i c k a n d f a i r l y a c c u r a t e i n d i c a t i o n o ft h e m o l e c u l a r o r i e n t a t i o n i n a n i n j e c t i o n m o l d e dp r o d u c t ( p r o v i d e d t h a t o n e a s s u m e s t h a t f l o w - i n d u c e ds t re s se s a r e a m e a s u r e o f m o l e c u l a r o r i e n t a t i o n ) .

/~ . 0

' v t o U d e n g th ~ ) o ~ i s t a n c m i d p l a n e ' " "Fig. 21. Therm ally induced s tress a t t = 8.7 s in case of moldelasticity

Pressure (MPa)6 0 330

5 0 3 2 5

4 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 O

3O

2 0 3 1 5

1 0 3 1 0

00 2 4 6 8 10Time (s)

- - S er i e s 1 - - S ed es 2

Fig. 20. Pressure his tory a t x = 8 mm with and witho utmold e las t ic i ty

Temperature C)

3 0 5 ...................................................................................................................................................................

3 0 00 . 5 1Time (s)

Fig. 22. Effect of heat due to compress ion

1.5


15/16

298 Rhe ologica Acta , Vol . 30, No. 3 (1991)

A s y e t , t h e e f f e ct o f t he m o l e c u l a r o r i e n t a t io n u p o nt h e m a t e r i a l p r o p e r t i e s h a s n o t b e e n t a k e n i n t o a c -c o u n t . T h i s is m a i n l y d u e t o a l a c k o f e x p e r i m e n t a ld a t a . B u t , i n d e e d , t h e i n t e n t i o n i s t o u s e t h e f l o w - i n -d u c e d s t r e s s e s a s a m e a s u r e o f o r i e n t a t i o n s u c h t h a ta n i s o t r o p i c p h e n o m e n a c a n b e i n cl u d e d i n th e c a l c u la -t i o n o f t h e r m a l l y - i n d u c e d r e s i d u a l s t r es s e s. T h i s i s t h et o p i c o f o n g o i n g i n v e s t i g a t i o n s .

I t i s s h o w n t h a t a s m a l l a m o u n t o f m o l d e l a s t ic i t yh a s a m a r k e d e f f e c t o n t h e p r e s s u r e h i s t o r y i n t h em o l d . T h i s is t r u e e v e n t h o u g h t h e m o d e l o f m o l de l a s t i c i t y t h a t i s a p p l i e d i s v e r y c r u d e . A l l c u r r e n t l ya v a i l a b l e c o m m e r c i a l a n a l y s i s c o d e s a s s u m e a r i g i dm o l d . P a r t i c u l a r l y a t l o c a t i o n s i n t h e m o l d a l o n g t h ep l a n e t h a t s e p a r a t e s t w o m o l d h a l f s , m o l d e l a s t ic i t y i sa s i g n if i ca n t p h e n o m e n o n . I t is n o t u n c o m m o n t h a tm o l d e d p a r t s a r e t h i c k e r t h a n t h e i n i t i a l c a v i t yt h i c k n e s s . T y p i c a l l y , a n a l y s is c o d e s t e n d t o u n d e r -p r e d i c t t h e p r e s s u r e in th e m o l d a t s u c h l o c a t i o n s , d u et o t h e d i s r e g a r d i n g o f m o l d e l a s ti c i ty , a n d h e n c e f o r t hu n d e r p r e d i c t t h e p r o d u c t t h ic k n e s s . T h e r e f o r e ,b e s i d e s as t h e r m a l a n a l y s i s o f th e m o l d , a m e c h a n i c a la n a l y s i s o f t h e m o l d i s a l s o r e q u i r e d . T h u s , t h e e n t i r es y s te m , i . e. , m o l d a n d p r o d u c t , s h o u l d b e a n a l y z e dt h e r m o m e c h a n i c a l l y .

Appendix: Der ivat ion o f the pressure problemS u b s t i t u t i o n o f ( 4 . 4 ) i n t o ( 4 . 1 ) g i v e s

( f l t O l~ ..k ~ e X ~ , ~ e T = [ 2 .~ 3 T ~ 3 ] . ( A . 1 )V p : ~)x----~ ~) x3 /S t e p 1 . I n t e g r a t i n g ( A . I ) w i t h r e s p e c t t o x 3 y i e l d s

( no t e t ha t p = p (X l , x2))O v _ 1 ( x 3 V p + c _ r e ) , ( A . 2 )

0x3 r / 'w h e r e c i s a n i n t e g r a t i o n c o n s t a n t c o l u m n .S t e p 2 . T h e v e l o c i t y v is f o u n d b y i n t e g r a t i n g ( A . 2 )f r o m s - t o x 3

X3 0 v !3 1v = f - -- =d x3 = - - ( x 3 V P + c - r e ) d x 3 . ( A . 3 )" s - 8 X 3 - r l ' ~

T h i s h o l d s b e c a u s e a t x 3 = s - v = Q . B e c a u s e v = Q a tx3 = s + a s w e l l , i t f o l l o ws t ha t

_ J 1 1 s + Zeg = J o V ' P + J o s ~ --~ d x 3 " ( A . 4 )

S t e p 3 . I n t e g r a t i n g t h e v e l o c i t y w i t h r e s p e c t t ox 3 o n c e m o r e g i v e s ( a f t e r r e p e a t e d u s e o f~ f g , x d x : f g - ~ f , x g d x ) .s+ ( ( T~

s - ~ J o / / J o s - r l d X 3s + X 3 T e+ J_ T a x 3 . ( A . 5 )

S t e p 4 . I n t e g r a t i o n o f th e c o n t i n u i t y r e l a t i o n ( 4 .5 )f r o m s - t o s + , t h e u s e o f ( A . 5 ) a n d t h e r e c o g n i t i o nt h a t

= s ( 0 v__ dx 3 = v3 (s +) - v3 (s - ) ( A . 6 )hd t s '- O x3g i v e s t h e f i n a l r e s u l t .

References1. Bosho uwers G , We rf J v d (1988) Inject -3, a s imulat ioncode fo r the f il ling s tage of the inject ion mou lding p ro-ces s o f t he r mopl as t i c s . P h D T hes i s , E i ndhoven Uni vT echnol ogy2. S i tters C ( 988) N um er ical s imu lat ion of inject ionmoul d i ng . P h D T hes i s , E i ndhoven Uni v T echno l ogy3 . M anz i one L T ( 1988) Appl i ca t i on o f Com put e r A i dedDes i gn i n i n j ec t i on mol d i ng . Hanse r P ub l , NY, US A4. I s ayev AI , H i ebe r C A ( 1980) T ow ar d a v i s coel a s ti cmode l i ng o f t he i n j ec t i on mol d i ng o f po l ymer s . Rheo lA c t a 1 9 : 1 6 8 - 1 8 25 . L eon ov AI ( 976) Noneq u i l i b r ium t he r mod ynam i csand r heo l ogy o f v i s coe l as t ic po l ymer medi a . Rheo l Ac t a15 :85 - 9 86. S t ickfor th J ( 986) The ra t ional mechanics and ther -modynami cs o f po l ymer i c f l u i ds based upon t he con-cep t o f a va r i ab l e r e f e r ence s t a t e . Rheo l Ac t a25:447 - 4587. S imo JC (1987 ) On a ful ly three dim ension al f ini tes t r a i n v i scoe l a s t i c damage mode l : f o r mul a t i on andc o m p u t a t i o n a l a s p e c t s . C o m p M e t h A p p l M e c h E n g

60:153 - 1738 . Upady ay RK , I s ayev AI , S hen S F ( 1981) T r ans i en tshea r f l ow behav i our accor d i ng t o t he L eonov mode l .R h e o l A c t a 2 0 : 4 4 3 - 4 5 79. Leo nov AI ( 987) On a c lass of const i tut ive equat ionsf o r v i s coe la s t ic li qu ids . J N on- New t on i an F l u i d M ech1 5 : 1 - 5 910 . Z o l l e r P A ( 1982) A s t udy o f t he p r es sur e - vo l ume- t em-pe r a t u r e r e l a t i onsh i ps o f f our r e l a t ed amor phouspo l ymer s : po l yca r bona t e , po l ya r y l a t e , phenoxy andp o l y s u lf o n e . P o l y m S ci 2 0 : 1 4 5 3 - 1 4 6 411 . P i r onneau O ( 19 82) On t he t r ansp or t - d i f f us i on


16/16

Baai jens , Calcu lat ion of res idual s tresses in inject ion mo lded prod ucts 299

a l gor i t hm and i t s app l i ca t i on t o t he Nav i e r - S t okese q u a t i o n . N u m e r . M a t h 3 8 : 3 0 9 - 3 3 212. M orto n KW , Pr ies t ly A, Sul i E (1986) S tabi l i ty analys i so f t h e L a g r a n g e - G a l e r k i n m e t h o d w i t h n o n - e x a c t i n -t e g r a t i o n . O x f o r d U n i v e r s i t y C o m p u t i n g L a b o r a t o r y ,R e p o r t 8 6 / 1 413 . W i m ber ge r - F r i ed l R , J anesch i tz - Kr i eg l H ( 1989)Res i dua l b i r e fr i ngence in C om pac t D i scs , submi t t ed f o rpub l i ca t i on14 . Hu ghes T JR ( 1987) T he f i n it e e l ement me t hod . P r en-t i ce - Ha l l I n t e r na t i ona l E d i t i ons , E ng l ewood C l i f f s ,New Je r sey

15. W i m ber ge r - F r i ed l R , Hendr i ks RD HM ( 1989) T he mea-sur ement and ca l cu l a ti on o f b i r e f r i ngence in quenchedpol yca r bona t e spec i mens . P o l yme r 30 :1143 - 1149(Received September 9, 1990;Au t hor s ' addr es s: accep t ed S ep t ember t 3 , 1990)

P r of . d r . i r . F .P .T . Baa i j ensP h i l i ps Resea r ch L abor a t o r i e sP . O . B o x 8 0 00 05600 JA E i ndhovenT he Ne t he r l ands

calculation of residual stresses in injection molded products

Documents