7/28/2019 PB80

1/4

Po lym er Bu l l e t in 2 , 72 7 -7 30 (1980) P o l y m e r B u l l e t i n9 by Springer-Verlag 19~

Model Rheological Equat ions of State in the L inearViscoelasticity of Polymeric CompositesH o r i a P a v e n a n d V i o r i c a D o b r e s c uChe mica l Resea rch Ins t itu te , ICECHIM, SpL Indepe nden te i 202 , Bucha res t , Ruman iaSummary_T h e u t i l i t y o f m o d e l r h e o l o g ic a l e q u a t i o n s o f s t a t e is n o w a w e l l e s t a b li s he d q u e s t i o ni n c a s e o f p o l y m e r i c m a t e r ia l s . U s in g th e v i s c o e la s t ic o p e r a t o r s m e t h o d , w e i n d i c a tethe pos s ib i l i t y to d e r i ve such equa t i ons on the bas is o f mec han ica l mode ls to be usedf o r p o l y m e r i c c o m p o s i t e m a te r ia ls .I n t r o d u c t i o nThe e va lua t i on o f e las t ic and v iscoe las t ic p rope r t i es o f m u l t i phase po l ym er i c m ed ia isa p r o b l e m o f t h e g r e a te s t i n t e re s t an d t h e r e ar e n o w s o m e a u t h o r i t a t i v e m o n o g r a p h s( N I E L S E N 1 9 7 4, M A N S ON a r id S P E R L I N G 1 9 7 6 ) a n d a l o t o f v a l u a b le c r i t i c a lr ev ie w s ( K A R D O S , H A L P I N a nd N I C O L A I S 1 9 75 , L I P A T O V 1 9 77 , K A U S C H1 9 7 7 , D I C K I E 1 9 7 8, S H E N a n d K A W A I 1 9 7 8 ) in t h e f ie l d , w h e r e s i g n i fi c a n ti n fo rma t ion and many re fe rences can be found .I n t h e t h e o r y o f e l a s ti c c o m p o s i t e s a r e w e l l k o n w n t h e s o - - c a ll e d e x t r e m e b o u n d s ,c o r r e s p o n d i n g t o u n i f o r m s tre ss ( Re us s a p p r o x i m a t i o n ) a n d t o u n i f o r m s t ra i n ( V o i g ta p p r o x i m a t i o n ) , w h i c h e s t ab li s h t h e n e ce ss ary c o n d i t i o n s t o b e s a t is f ie d i n m o d e l sc a l c u l a t i o n .T a ka y an a gi an d c o - -w o r k e r s ( T A K A Y A N A G I , H A R I M A a nd I W A T A 1 96 3 )p r o p o s e d m o d e l s f o r h e t e ro g e n e o u s p o l y m e r i c m a t e r ia l s , a nd t h e y a n d a l so a n u m b e ro f o th e r au tho rs repo r ted s ign i f i can t succes i n rep resen t i ng the v i scoe las t icp rope r t i es o f a va r i e ty o f such ma te r i a l s in te rm s o f these mod e ls . I t, i s r e levan t tha ti n these wo rk s , as a f i r s t s tep , by a co r responde nce p r i nc ip le , the resu lt s ob ta ined fo re las t i c componen ts a re ex tended to v i scoe las t i c ones , fo r s imp le pa ra l l e l and se r i esm o d e l s . T h e n , i n o r d e r t o c o r r e l a t e in a r e a so n a b le w a y t h e t h e o r y a n d t h ee x p e r i m e n t a l d a ta , t h e s o - -c a ll e d T a k a y a n a g i m o d e l s a r e p r o p o s e d .T h e m e t h o d t o be p r e s en t e d b e l o w p r o v i d e s r a t h e r t h e c o m p o s i t e t h e o l o g i c a le q u a t i o n o f s t at e ( R E S ) a nd c o n s e q u e n t l y t h e e x p l i c i t f o r m o f th e c o m p o s i t er h e o l o g i c a l p a r a m e t e r s , t h a n t h e c o m p o s i t e c o m p l e x m o d u l u s , as is gi ve n i n t h el i te ra tu re . Na tu ra l l y , the mecha n ica l response o f th e com pos i te , expressed byd i f f e r e n t q u a n t i t i e s , r e s ul ts in a s i m p e w a y , t o o .Theo ry_The l inea r v i scoe las t ic behav o i r o f po l ym er i c ma te r i a l s can be desc r i bed by theo p e r a t o r f o r m o f t h e R ES ( A L F R E Y 1 9 48 , F U L G G E 1 9 76 , P A V E N 1 9 7 8)

/77 np -Q, . ", q : ( s )whe re P and Q a re the cha rac te r i s t i c li nea r v i scoe las t ic ope ra to rs , P r and q r ther h e o l o g i c a l p a ra m e t e r s o f t h e b e h a v i o r m o d e l , D e - d r / d r r t h e r - - t h o r d e r t i m ede r i va t i ve , and O 'and ~ the s tress and the s t ra in i n th e un i - -d im ens iona l case .I n o r d e r t o d e r iv e t h e e x p l i c i t f o r m o f t h e c o m p o s i t e R E S, t h e r h e o lo g i c a lpa ram e t res o f wh ich a re exp ressed as fun c t i on s o f the rheo log i ca l pa rame te rs o f

0 1 7 0 - 0 8 3 9 / 8 0 / 0 0 0 2 / 0 7 2 7 / $ 0 1 . 0 0

7/28/2019 PB80

2/4

728

t h e c o mp o n e n t s a nd o f t h e i r v o l u me fr a c t io n s , we w i l l c o n s id e r f i rs t t h e t wobasic--para l le l and series mo dels .The "a " and " b " l i nea r v iscoe las t ic behav io r mode ls a re cha rac te ri zed by the Pa,Qa and P6 , Q6 ope ra to rs , respec t ive ly . Fu r the r , t he c o r respond ing vo lum ef ra c t io n s a r e v a n d vb (v +vb = | , v b = v ,v = i - v ) , s e e F i g . 1 .

; ; IF i g . 1 Para l le l (1 ) and s er ies (2) mo delsThe pa ra l le l coup l ing ._ . The "a " and " b " com ponen ts , rep resen t ing l inea rv iscoe las t ic behav io r mode ls , obe y to ind iv id ua l RES o f the fo rm (1 ) . On the

o the r hand , the de f in i t i on re la t ionsh ip fo r the com pos i te s t ress, (7 " , and s t ra in ,~r areo - = ~ , + ,,b ~ , ~ = ~ = E 6 , ( z )

wh ere (~'~, ~'6and ~,~, ~'6are the ind ivi du al co m po ne nt stresses and strains.Tak ing in to ac coun t the re la t ions (1 ) and (2 ), one ob ta ins the com pos i te RE[~] o" : [ c 1 - ~ q ~ , + ~ Q 6 ] e , (a )

wh ich is s im i la r to eq (1) , i .e . a lso a l inear v iscoe last ic law.The se r ies coup ling .. . The " a " and " b " com pone n ts a re desc ribed by ind iv idu a lRE$ o f the fo rm (1) . H owever , t he de f in i t ion re la t ionsh ips fo r the com pos i te st ressand s t ra in changes no w as fo l lo w s

~ = z ' , ~ G b , ~ ' = ~ , = ~ , (~ )whe re ~ , ~ '~, ~ and ~ , E a, ~6 have the same s ign i f icance as above.The co r respo nd ing com pos i te RE $ is g iven by

[ c l- ~ , )s q , + z , ~ , ] ~ : [q = :q 6] E , (5)wh ich means tha t the rhe o log ica l behav io r o f t he co mp os i te is again o f l i nea rv iscoe last ic type .I t resu l ts tha t , in genera l , the com pos i te RES d i f fe rs f rom those o f thec o mp o n e n t s , n o t o n l y q u a n t i t a t i v e l y b u t a ls o q u a l i t a ti v e l y ; i n t h e c a s e o f s omes imp le fo rms o f t he co mp one n t RES i t i s poss ib le to ge t to qu a l i t a t i v e ly ana logousfo rm s o f t he com pos i te RES. O f cou rse , the resu l t ing com pos i te RES (3 ) and (5 )a re a t leas t quan t i t a t i ve ly d i f f e ren t . An ano the r s ign i f i can t consequence o f t h isapproach concerns the poss ib i l i ty to obta in , as a second s tep, the form of thec o mp o s i t e o p e ra t o r mo d u lu s , a n d t h e c o r re s p o n d e n t s o f t h e f a mo u s " ru l e o fm i x t u re s " , i r r es p e c ti v e t h e " i n ve rs e ru l e o f m i x t u re s " (P A V E N a nd D O B RE S CUi 980).As po in ted above , the T akayana g i app roach appeared as an a t tem p t to express theaverage mechanical response between bounds, by simple paral le land series rules ofmix ing . I t i s s t ra igh t fo rwa rd to ob ta in no w the rhe o log ica l co r responde n t o f t heTakaya nag i mode ls , as i t resul ts by the op e ra to r desc r ip t ion o f t he c om pos i te R E$ .In the case o f m ode l I (parameters,~t, ~ ) one obta ins,on the bas is o f th e re la t ionsh ips(3) and (5 ) and a f te r some m a them at ica l ma n ipu la t ions , the resu l ting RE $ as

7/28/2019 PB80

3/4

729

{ @ [P ~ q , , . . - c ~ -~ . ~ .~ , c ~ , , - ~ ~ . , , ) J } ~ == {~, IP~, +~,c P ~, - ,~ ~,)j}~ , (8 )and ana logous ly= { A , [ P , A I + ( 1 - , 1 , ) ( ~ , - P , ~ i ) ] ) ~ , ( ? )

i n the case o f mo de l I I ( pa rame te rs ~a ,~), see F ig . 2 . The w e l l kn ow n equ iva lencec o n d i t i o n o f t h e t w o m o d e l s r e su l ts n o w i m m e d i a t e l y t o b e A~ = v ( l +v - A ~ ) ' I , i fPaQ 6 ~ PbQ ~ , v b e in g t h e v o l u m e f r a c t io n o f t h e " b " c o m p o n e n t i n th e " a "m a t r i x c o m p o n e n t .

F ig . 2 The Tak ayanag i mod e ls I( 1 ) and I I ( 2 ) .I t a p p e a r ~w o r t h w h i l e t o w r i t e n o w e x p l i c i t l y , o n t h e b as is o f e q u a t i o n s ( 3 ),( 5 ) - - ( 7 ) t h e f o r m o f t h e r h e o l o g ic a l p a r a m e t e rs i n t h e ca se o f a t y p i c a l c o m p o s i t ema te r ia l o f e las t i c (H oo ke ) / v i scoe las ti c s tanda rd l i nea r (Zene r ) t ype . Thei n d i v i d u a l c o m p o n e n t s R E S a r e g i ve n b y

~ = ~ . o E , ~p , . + ~ . o ) ~ = ( f z . ~ z z . , D ~ e , ~8 ~wh er nd " " " "q ,~ a Pz~ ' p - - ' qz ~ ' qz , ; a re the H (Ho oke ) and Z (Zene r )M , r a : , / 9r h e o l o g lc a l p a r a m e t e r s o f t h e c o m p o n e n t m o d e l s , r e s p e c ti v e ly . I n t h e c a s e o f t h ebas i c~pa ra l l e l and se ries m ode ls resu l ts[ ~ ; ~ . D ) r ~ ) p . o f ~ , + ~ z o ] ( 1 ~ ) p , , f ~ , + r e , , ] : 1 ~ . c 9 )

{ [ ( I - ~ ) ~ . o + ~ , f ~ o ] + [ ( 1 - ~ ) f . , , , + ~ ' P z . , Z ~ . o ] : J ~ = i " / o )= (~o ~z o " ~ . o ~z, 0 ) ~ ,respec t i ve l y . The rheo log i ca l pa rame te rs o f the com pos i te RES, ap pea r ing asf a c t o r s o f t h e D o p e r a t o r s a re c l e a r ly f u n c t i o n s o f t h e r h e o l o g i c a l p a r a m e t e r s of" H " a n d " Z " m o d e ls . I n t h e c a s e j u s t c o n s id e re d t h e t w o R E S e x p r i m eq u a n t i t a t i v e l y d i f f e r e n t , b u t q u a l i t a t i v e l y a n a lo g o u s r h e o l o g ic a l b e h a vi o rs .T h e r h e o l o g i c a l c o r r e s p o n d e n t o f t h e T a k a y a n a g i a p p r o a c h r e s u lt s as f o l l o w s : i nt h e c a se o f a H o o k e / Z e n e r c o m p o s i t e w e h a veI I [ l - u - ~ " ~ A ; ~ e , .o + U - ~ ) ~ e ~ , I D ] ~ ' =Z ~ , ]={[( f ' / J }~z. ,~,~io+J i :~,~.z, ]+[C l - / J )~z. f iZ ~o'=/ i~. l i iO becom ~es}e (11)a nd f o r a Z e n e r / H o o k e c o m p o s i t e t h e c o m p o s i t e R E 5{{ [1 - C l - ~,)~ ] ~ ~ Z , ,o + c1 -~, )~ p,~$ e , . o } + { [ 1 - C - ~ ) ~ l p ,. .o Z , : ., ++ P , . , ~ i ,, 0 ) + 2 ( 1 - ~ ) ~ ? , . . o p i , , f ~ o } O + { [ I - ( 1 - ~ ) ~ ] ? , , , ~ . , . , +

7/28/2019 PB80

4/4

730

+ [ ( I - ~ ) ~ t , + ~ P z , Z ~ o ~ , , , ] O } E , (/ 2)respec t ive ly . I t is obv iou s tha t the more e labo ra ted fo rm o f th is app roach resu lt s ina more comp l ica ted dependence o f the c om pos i te rheo log ica l pa ramete rs on therheo log ica l pa ramete rs and the vo lum e f rac t ions o f the comp onen ts . I t i sin te res ting to no te the s ign i f i can t theo log ica l d i f f e rence be tween the two phys ica l l yd is t inc t cases represented by the c onsidered poss ib i l i t ies o f m ix in g two g ivencom pon ents, one be ing e lastic and th e o the r v iscoe lastic.The o pp or tun i t y o f us ing the above p resen ted app roach is apparen t in the case o fp o l y m e r i c c o mp o s i t e s wh e n t h e mo rp h o lo g y is n o t c l e a r l y d e f i n e d , e s p e c ia l lywhen th is m orph o log y va ries w i th com pos i t ion . M oreove r , t he m e thod is su i tab leto p rov ide d i rec t l y the mechan ica l response o f the com pos i te , co r respond ing tod i f f e re nt typ es o f s t ress (or s t ra in) h is tor ies. Of course, i t is necessary to dev e lopfu r the r the capab i l i t ies o f th is app roach bo th toward s the theo log ica l t rans la t iono f the ex is t ing da ta in the l i t e ra tu re on the m ix tu r e ru les and also toward s theo b t a i n i n g th e rh e o lo g ic a l i n f o rm a t i o n i n t h e c as e o f mo re c o m p l i c a te d s i t u a t io n srep resen ted b y hy b r id po ly m er ic compos i tes .

ReferencesNIE LS EN ,L .E . : Mechan ica l P rope r ties o f P o lymers and Com pos i tes , New Y ork :Marce l Dekk er | 974MA N S O N , J . A . a n d S P E RL I N G , L . H . : P o l y me r B le nd s an d Co mp o s it e s, Ne w Y o rk :P lenum 1976K A R DO S , J . L . , HA L P I N , J . C . a nd N I CO L A I S , L . i n Th e o re t i c a l Rh e o lo g y ,HU TTO N, J . F . , P E A R S O N , J .R . A . an d W A L T E R S , K . ( ed s.) , L o n d o n : A p p I .S c i .Pubis. 1975L IP AT O V,Y u . : Adv . Po l . Sc i. , 22 ,1 ( | 977 )K A U S CH , H . H . : A n g e w. M a k ro m o l . Ch em., 6 0 / 6 | , 1 39 (1 97 7 )DIC KIE , R.A . in Po lym er Blends, PAU L, D.R. and NEW MA N,S . (eds), Ne w Yo rk :Acade mic P ress 1978SH EN,M . and KA W AI ,H . : J .Am.Chem .Soc . , 24 , ! (1978)T A K A Y A N A G I , M . , H A R I M A , H . a n d I W A T A , Y . : M e m . Fa c . E ng . K y u s hu U n i v. ,23, I (1963)AL FR EY ,T . : Mechan ica l Behav io r o f H igh Po lymers , Lon don " In te rsc ience 1948FLUG GE,W . : V iscoe las t i c i t y , Be r l in , He ide lbe rg , New Yo rk : Sp r inge r 1976PAVEN, H. : Mater . p last . , 15 ,163 (1978)P A V E N , H . a n d DO B RE S CU, V . : Re v. Ch im . , 1 9 8 0 , to b e p u b l is h e d

R e c e i v e d M a r c h 2 4 / R e v i s e d a n d a c c e p t e d M a y 1 4, 1 9 8 o

Responsi ble for the text: The Editors (see inside title page ofthis issue). For advertisements: L. Siegel, Kurf~rs tenda ~n 237,D-IO O0 Berli n 15, Tel. (03 0) 882 1031, Telex 01-85411,S p r i n g e r - V e r l a g B e r l i n H e i d e l b e r g N e w Y o r kP r i n t e d i n G e r m a n y b y B e l t z O f f se t d r u c k, H e m s b a c h / B e r g s t r a B eO b y S p r i n g e r - V e r l a g B e r l i n H e i d e l b e r g 1 9 8 0