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V I S C O E L A S T I C I T Y P H E N O M E N A I N A M O R P H O U S H I G H P O L Y M E R I C S Y S T E M S

H e r b e r t L e a d e r m a n

I . I n t r o d u c t i o n 1 I I . Creep a n d R e c o v e r y B e h a v i o r of A m o r p h o u s P o l y m e r s 5

1. B u l k a n d Shear R e s p o n s e 5 2. R e t a r d e d E l a s t i c D e f o r m a t i o n in S h e a r : C r o s s - L i n k e d P o l y m e r s 5

3. Effect of T e m p e r a t u r e on R e t a r d e d E l a s t i c i t y : C r o s s - L i n k e d P o l y m e r s . . 8 4. R e t a r d e d E l a s t i c D e f o r m a t i o n in S h e a r : N o n c r o s s - L i n k e d P o l y m e r s . . . 11 I I I . Stress R e l a x a t i o n a n d D y n a m i c R e s p o n s e of A m o r p h o u s P o l y m e r s 15

1. Stress R e l a x a t i o n 15 2. Representation of D y n a m i c R e s p o n s e 17

3. D y n a m i c B e h a v i o r of A m o r p h o u s P o l y m e r s 22 I V . M o d e l s , S p e c t r a , a n d Operator E q u a t i o n s 25

1. T h r e e - E l e m e n t M o d e l : C r o s s - L i n k e d P o l y m e r s 25 2. F i v e - E l e m e n t M o d e l : C r o s s - L i n k e d P o l y m e r s 30 3. S i x - E l e m e n t M o d e l : N o n c r o s s - L i n k e d P o l y m e r s 33

V . S u p e r p o s i t i o n ; C o n t i n u o u s S p e c t r a 37 1. T h e S u p e r p o s i t i o n Principle 37 2. R e l a t i o n between Creep a n d R e l a x a t i o n 40

3. R e t a r d a t i o n a n d R e l a x a t i o n S p e c t r a 42 4. C a l c u l a t i o n of R e t a r d a t i o n a n d R e l a x a t i o n Spectra 45

5. R e l a t i o n s between R e s p o n s e C u r v e s I n v o l v i n g Spectra 50

a. Difference F u n c t i o n s 50 b. V i s c o s i t y , S t e a d y - S t a t e C o m p l i a n c e , a n d R e l a x a t i o n S p e c t r u m 51

V I . M o l e c u l a r Structure a n d L i n e a r Viscoelastic B e h a v i o r 52 1. G l a s s y S t a t e , G l a s s M o d u l u s , G l a s s C o m p l i a n c e 53

2. V o l u m e a n d Shear V i s c o e l a s t i c i t y 55 3. N o n l i n e a r V i s c o e l a s t i c i t y 57

a. R u b b e r l i k e R e g i o n 58 b. G l a s s l i k e B e h a v i o r 58 c. Textile F i b e r s 59 4. L i n e a r Viscoelastic B e h a v i o r a n d C o n s t i t u t i o n of A m o r p h o u s P o l y m e r i c

S y s t e m s 60 Nomenclature 61

I. Introduction

I n o t h e r c h a p t e r s of t h i s a n d t h e p r e c e d i n g v o l u m e , c e r t a i n a s p e c t s of t h e p h e n o m e n o l o g i c a l t h e o r y of l i n e a r v i s c o e l a s t i c b e h a v i o r a r e d i s c u s s e d ( C h a p t e r s 2 a n d 1 1 , V o l u m e I ) , a n d a l s o of t h e b e h a v i o r of s y s t e m s m a n i f e s t -

1

i n g l i n e a r v i s c o e l a s t i c i t y ( C h a p t e r s 2 a n d 4 , V o l u m e I I ) . T h e m a i n p u r p o s e of t h i s c h a p t e r i s t o d e s c r i b e s o m e r h e o l o g i c a l p h e n o m e n a m a n i f e s t e d i n h i g h p o l y m e r i c s y s t e m s , m o r e e s p e c i a l l y l i n e a r v i s c o e l a s t i c i t y i n a m o r p h o u s s y s t e m s , a n d t o s h o w t h e q u a l i t a t i v e a n d q u a n t i t a t i v e r e l a t i o n s a m o n g t h e p h e n o m e n a of l i n e a r v i s c o e l a s t i c i t y r e s u l t i n g f r o m t h e p h e n o m e n o l o g i c a l t h e o r y .

S u p p o s e t h a t w e s t r e t c h a p i e c e of r u b b e r , s p e c i f i c a l l y n a t u r a l r u b b e r v u l c a n i z e d i n t h e u s u a l w a y w i t h o u t fillers. W h e n t h e r u b b e r i s r e l e a s e d , w e find t h a t i t s n a p s b a c k s u b s t a n t i a l l y t o i t s o r i g i n a l l e n g t h . I t w o u l d a p - p e a r t h a t t h e r e l a t i o n s h i p s b e t w e e n t h e a p p l i e d f o r c e s a n d r e s u l t i n g d e f o r m a - t i o n s s h o u l d b e d e a l t w i t h a d e q u a t e l y b y t h e t h e o r y of l a r g e e l a s t i c d e f o r - m a t i o n s , a s d i s c u s s e d i n C h a p t e r 10 of t h e p r e c e d i n g v o l u m e . N o w let u s s u p p o s e t h a t w e b e n d a s t r i p of p o l y s t y r e n e a t r o o m t e m p e r a t u r e ; w e find t h a t t h e m a t e r i a l u n d e r g o e s o n l y a s m a l l d e f o r m a t i o n w i t h o u t r u p t u r e . I f s u c h a s t r i p i s b e n t b u t n o t b r o k e n a n d t h e n r e l e a s e d , i t w i l l a g a i n s e e m t o s n a p b a c k t o i t s o r i g i n a l f o r m . I t w o u l d a p p e a r t h a t t h e r e l a t i o n s b e t w e e n f o r c e s a n d d e f o r m a t i o n s i n s u c h a m a t e r i a l s h o u l d b e a d e q u a t e l y d e a l t w i t h b y t h e c l a s s i c a l t h e o r y of e l a s t i c i t y , s o m e a s p e c t s of w h i c h a r e d i s c u s s e d i n C h a p t e r 2 of t h e p r e c e d i n g v o l u m e .

H o w e v e r , w e find i n c a r e f u l e x p e r i m e n t s t h a t t h e r u b b e r i n t h e first c a s e d o e s n o t q u i t e r e t u r n i m m e d i a t e l y t o i t s o r i g i n a l l e n g t h ; t h e r e i s a s l i g h t r e s i d u a l e l o n g a t i o n , w h i c h g r a d u a l l y r e d u c e s w i t h t i m e , a n d u n d e r c e r t a i n c i r c u m s t a n c e s m a y e v e n t u a l l y d i s a p p e a r c o m p l e t e l y . T h i s effect i s m o r e m a r k e d , t h e g r e a t e r t h e o r i g i n a l e l o n g a t i o n of t h e r u b b e r a n d t h e l o n g e r t h e t i m e t h a t i t w a s h e l d i n t h e s t r e t c h e d c o n d i t i o n . T h e s a m e t y p e of b e h a v i o r i s f o u n d i n t h e c a s e of t h e p o l y s t y r e n e s p e c i m e n ; s u c h a t i m e - d e p e n d e n t e l a s t i c r e c o v e r y m a y b e t h o u g h t of a s a r e t a r d e d e l a s t i c r e c o v e r y . T h i s t i m e effect i s m a n i f e s t n o t o n l y f o l l o w i n g r e m o v a l of l o a d , b u t a l s o w h i l e t h e l o a d i s a c t i n g o n t h e s p e c i m e n . S u p p o s e t h a t a s t r i c t l y c o n s t a n t t r a n s v e r s e l o a d i s a p p l i e d t o t h e p o l y s t y r e n e s t r i p , a n d t h e d e f o r m a t i o n o b s e r v e d f r o m t i m e t o t i m e ; t h i s w i l l b e f o u n d t o i n c r e a s e v e r y g r a d u a l l y , b u t a t a d e c r e a s i n g r a t e ; t h e s a m e i s t r u e f o r m e a s u r e m e n t s of t h e l e n g t h of t h e \ru l c a n i z e d r u b b e r s t r e t c h e d b y a c o n s t a n t f o r c e . T h u s t h e effect of a s u d d e n a p p l i c a - t i o n o r r e m o v a l of l o a d i s o b s e r v e d n o t s i m p l y a s a s t r a i n w h i c h a p p e a r s o r d i s a p p e a r s a t t h e i n s t a n t of l o a d a p p l i c a t i o n o r r e m o v a l , b u t a s a s t r a i n w h i c h a p p e a r s o r d i s a p p e a r s g r a d u a l l y s u b s e q u e n t t o t h i s i n s t a n t . T h i s ef- f e c t w a s o b s e r v e d o v e r a h u n d r e d y e a r s a g o i n s u c h m a t e r i a l s a s g l a s s , m e t a l w i r e s , a n d r u b b e r ; t h e w o r k w a s l a r g e l y e m p i r i c a l u n t i l s o m e p h e n o m e n - o l o g i c a l r e l a t i o n s p r o p o s e d b y B o l t z m a n n i n 1874 w e r e s h o w n t o d e s c r i b e t h e s e t i m e - d e p e n d e n t effects.

T h i s p h e n o m e n o n w a s k n o w n a m o n g G e r m a n s c i e n t i s t s a s "elastische Nachwirkung" w h i c h m a y b e t r a n s l a t e d , retarded elasticity. T h i s m a y b e

o b s e r v e d i n o t h e r t y p e s of e x p e r i m e n t s . L e t u s s u p p o s e , f o r e x a m p l e , t h a t a c o n s t a n t b e n d i n g d e f o r m a t i o n i s a p p l i e d t o t h e s t r i p o f p o l y s t y r e n e . I f t h e d e f o r m a t i o n i s m a i n t a i n e d , f o r e x a m p l e , b y a t r a n s v e r s e f o r c e , i t i s f o u n d t h a t t h e f o r c e a p p l i e d t o t h e s t r i p i s n o t c o n s t a n t , b u t d i m i n i s h e s s l i g h t l y w i t h t i m e , t h a t i s , stress relaxation t a k e s p l a c e . I f s u b s e q u e n t l y t h e s t r i p i s r e s t o r e d t o i t s o r i g i n a l c o n f i g u r a t i o n , a n d h e l d c l a m p e d i n t h a t p o s i t i o n , i t i s f o u n d a t f i r s t t h a t a s m a l l f o r c e i s r e q u i r e d t o d o t h i s ; t h e f o r c e d e c a y s w i t h t i m e a n d e v e n t u a l l y d i s a p p e a r s . A l t e r n a t i v e l y , if a f t e r s t r e s s r e l a x a - t i o n i s a l l o w e d t o t a k e p l a c e , t h e s p e c i m e n i s t h e n s i m p l y r e l e a s e d , t h e r e w i l l b e a g r a d u a l r e c o v e r y , a s i n t h e e a r l i e r e x p e r i m e n t d e s c r i b e d .

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

W h i l e d e v e l o p m e n t s a l o n g t h e s e l i n e s w e r e t a k i n g p l a c e i n t h e s t u d y of t i m e effects i n m e c h a n i c a l b e h a v i o r , i t w a s o b s e r v e d t h a t s i m i l a r t i m e ef- f e c t s w e r e m a n i f e s t e d i n t h e e l e c t r i c a l b e h a v i o r of d i e l e c t r i c s , a n d c o n s e - q u e n t l y v e r y s i m i l a r p h e n o m e n o l o g i c a l t h e o r i e s w e r e d e v e l o p e d t o d e s c r i b e t i m e effects i n e l a s t i c b e h a v i o r a n d i n d i e l e c t r i c b e h a v i o r . T h e h i s t o r y of t h e s e d e v e l o p m e n t s o v e r a h u n d r e d y e a r s u p t o a b o u t 1940 h a s b e e n re- v i e w e d b y L e a d e r m a n .1 F u r t h e r m o r e , i n t h e field of e l e c t r i c a l e n g i n e e r i n g , s p e c i f i c a l l y i n l i n e a r c i r c u i t t h e o r y , t h e s a m e p r i n c i p l e s w e r e f o u n d t o a p p l y . T h e s e i d e a s f o r m t h e b a s i s of w h a t m a y b e c a l l e d linear response theory.

S o m e a s p e c t s of t h i s t h e o r y , m o r e e s p e c i a l l y a s t h e y a p p l y t o h i g h p o l y m e r i c s y s t e m s , a r e d i s c u s s e d i n C h a p t e r s 2 a n d 11 of t h e p r e c e d i n g v o l u m e .

T i m e effects i n t h e e l a s t i c p r o p e r t i e s of m a t e r i a l s s u c h a s p o l y s t y r e n e a n d v u l c a n i z e d n a t u r a l r u b b e r a t r o o m t e m p e r a t u r e i n s i m p l e c o n s t a n t - s t r e s s o r c o n s t a n t - d e f o r m a t i o n t e s t s a s d e s c r i b e d a b o v e m i g h t b e t h o u g h t of a s a r e l a t i v e l y u n i m p o r t a n t p e r t u r b a t i o n u p o n i d e a l e l a s t i c b e h a v i o r . U n d e r m o r e g e n e r a l c o n d i t i o n s , h o w e v e r , s u c h t i m e effects b e c o m e m o r e i m p o r t - a n t a n d p l a y a d o m i n a n t r o l e i n t h e d e s c r i p t i o n of e l a s t i c b e h a v i o r . S p e - c i f i c a l l y , if t h e r u b b e r i s c o o l e d , o r if t h e p o l y s t y r e n e i s h e a t e d , t h e n t h e s e t i m e effects b e c o m e m o r e p r o n o u n c e d . I n f a c t , if t h e r u b b e r i s c o o l e d t o a s u f f i c i e n t l y l o w t e m p e r a t u r e , s a y b e l o w — 7 0 ° C , i t w i l l b e g l a s s y , l i k e p o l y - s t y r e n e a t r o o m t e m p e r a t u r e i n s u c h c o n s t a n t - l o a d o r c o n s t a n t - d e f o r m a - t i o n t e s t s . S i m i l a r l y , if t h e p o l y s t y r e n e i s h e a t e d t o a b o u t 130° C , i t i s f o u n d t h a t t h e s t r i p w h i c h i s g l a s s y a t r o o m t e m p e r a t u r e b e c o m e s l i m p a n d r u b -

1H . L e a d e r m a n , " E l a s t i c a n d Creep Properties of F i l a m e n t o u s M a t e r i a l s a n d Other H i g h P o l y m e r s . " Textile F o u n d a t i o n , W a s h i n g t o n , D . C , 1943.

b e r l i k e ; w h e n s t r e t c h e d t o a m o d e r a t e e x t e n t i t w i l l s n a p b a c k f a i r l y r a p i d l y a n d , a f t e r t h e d e l a y e d e l a s t i c r e c o v e r y h a s c e a s e d , i t w i l l m a n i f e s t o n l y a s l i g h t p e r m a n e n t d e f o r m a t i o n , w h i c h i s of t h e n a t u r e of flow. T h e a n a l o g y i s b e t t e r if w e c o m p a r e t h e h i g h - t e m p e r a t u r e b e h a v i o r of p o l y s t y r e n e w i t h t h e r o o m - t e m p e r a t u r e b e h a v i o r of a r a w ( u n v u l c a n i z e d ) r u b b e r . S u c h a m a t e r i a l o n b e i n g s t r e t c h e d d o e s n o t r e t u r n e v e n t u a l l y t o i t s o r i g i n a l l e n g t h b u t s h o w s a p e r m a n e n t d e f o r m a t i o n t h e m a g n i t u d e of w h i c h d e p e n d s u p o n t h e e x t e n t a n d d u r a t i o n of t h e s t r e t c h i n g , a s i n t h e c a s e of t h e h e a t e d p o l y - s t y r e n e s t r i p .

I n p r i n c i p l e , o r d i n a r y r a w o r v u l c a n i z e d r u b b e r c a n b e s h o w n t o b e g l a s s - l i k e a t room temperature a l s o ; t h i s h o w e v e r , w o u l d r e q u i r e t h e s t r e t c h i n g e x p e r i m e n t t o b e c o m p l e t e d i n , s a y , a m i c r o s e c o n d . S u c h a n e x p e r i m e n t w o u l d b e d i f f i c u l t t o p e r f o r m . W e c a n , h o w e v e r , c a r r y o u t a d y n a m i c e x p e r i - m e n t i n w h i c h t h e f o r c e o r d e f o r m a t i o n a l t e r n a t e s w i t h a f r e q u e n c y of 1 m e g a c y c l e p e r s e c o n d ; a s w e w i l l see, t h i s c a n b e t h o u g h t of a s e q u i v a l e n t t o a " s t a t i c " e x p e r i m e n t i n w h i c h t h e d e f o r m a t i o n ( o r f o r c e ) i s o b s e r v e d 1 m i c r o s e c o n d a f t e r i n s t a n t a n e o u s a p p l i c a t i o n of f o r c e (or d e f o r m a t i o n ) . I n s u c h a d y n a m i c e x p e r i m e n t t h e r u b b e r i s f o u n d t o e x h i b i t t h e m e c h a n i - c a l b e h a v i o r of a g l a s s y p l a s t i c . T h u s , r o u g h l y s p e a k i n g , a n o n c r o s s - l i n k e d a m o r p h o u s p o l y m e r c a n e x h i b i t t h e b e h a v i o r of a g l a s s y p l a s t i c , o r i d e a l r u b b e r l i k e b e h a v i o r , o r t h e b e h a v i o r of a v i s c o u s l i q u i d , d e p e n d i n g u p o n t h e t e m p e r a t u r e a n d t i m e s c a l e of t h e e x p e r i m e n t . I f t h e m a t e r i a l i s c r o s s - l i n k e d , s u c h a s v u l c a n i z e d r u b b e r or c r o s s - l i n k e d p o l y s t y r e n e , t h e n i n g e n - e r a l i t b e h a v e s u n d e r l i m i t i n g c o n d i t i o n s l i k e a n i d e a l g l a s s y e l a s t i c b o d y o r a n i d e a l r u b b e r l i k e e l a s t i c b o d y . O n e of t h e o b j e c t i v e s of t h i s c h a p t e r i s t o d i s c u s s i n d e t a i l t h e p h e n o m e n a o b s e r v e d u n d e r i n t e r m e d i a t e c o n d i t i o n s , i n c l u d i n g r e l a t i o n s b e t w e e n t h e r e s p o n s e p a t t e r n s of a g i v e n m a t e r i a l i n d i f f e r e n t t y p e s of t e s t .

I n t h e e x a m p l e s of m e c h a n i c a l b e h a v i o r d i s c u s s e d a b o v e , m e c h a n i c a l e n e r g y i s p a r t l y s t o r e d i n a b o d y u n d e r g o i n g d e f o r m a t i o n a n d p a r t l y d i s - s i p a t e d a s h e a t ; s u c h b e h a v i o r i s k n o w n a s viscoelastic behavior. A s p e c i a l c a s e i s w h e n t h i s i s linear; u n d e r t h e s e c i r c u m s t a n c e s t h e b e h a v i o r b e c o m e s a m e n a b l e t o s i m p l e m a t h e m a t i c a l t r e a t m e n t . C h a p t e r s 2 a n d 11 of t h e p r e - c e d i n g v o l u m e d i s c u s s c e r t a i n t h e o r e t i c a l a s p e c t s of l i n e a r v i s c o e l a s t i c i t y ; i n t h i s v o l u m e C h a p t e r s 4 a n d 11 d e a l w i t h a s p e c t s of l i n e a r v i s c o e l a s t i c b e h a v i o r f r o m t h e e x p e r i m e n t a l p o i n t of v i e w .

I n t h e e x a m p l e s p r e v i o u s l y c o n s i d e r e d , t h e p h e n o m e n a a r e d u e t o p h y s i - c a l c h a n g e s ; i n p r i n c i p l e , t h e e x p e r i m e n t s c a n b e r e p e a t e d i n d e f i n i t e l y . H o w e v e r , a t s u f f i c i e n t l y e l e v a t e d t e m p e r a t u r e s it i s p o s s i b l e f o r c h a i n s c i s - s i o n a n d c r o s s - l i n k i n g t o t a k e p l a c e . F o r e x a m p l e , i n a c r o s s - l i n k e d p o l y m e r s u b j e c t t o a c o n s t a n t s t r e s s a t a s u f f i c i e n t l y h i g h t e m p e r a t u r e , c h a i n s c i s s i o n w i l l l e a d t o a g r a d u a l i n c r e a s e i n d e f o r m a t i o n , a n d c r o s s - l i n k i n g w i l l r e s u l t

i n a p e r m a n e n t d e f o r m a t i o n s u b s e q u e n t t o r e m o v a l of s t r e s s . T h e s e t i m e - d e p e n d e n t p h e n o m e n a i n m e c h a n i c a l b e h a v i o r r e s u l t i n g f r o m c h e m i c a l c h a n g e s a r e d e n o t e d b y t h e t e r m chemorheology. T h e s u b s e q u e n t c o n s i d e r a - t i o n s i n t h i s c h a p t e r d o n o t refer t o c h e m o r h e o l o g y , w h i c h i s c o n s i d e r e d i n g r e a t e r d e t a i l i n C h a p t e r 2 . T h e r h e o l o g y o f crystalline p o l y m e r s i s i n g e n - e r a l m o r e c o m p l i c a t e d t h a n t h a t o f a m o r p h o u s p o l y m e r s , a n d i s n o t a s w e l l u n d e r s t o o d ; t h i s s u b j e c t i s n o t t r e a t e d i n t h i s c h a p t e r , b u t i s d i s c u s s e d i n C h a p t e r 7 o f t h i s v o l u m e .

II. C r e e p a n d R e c o v e r y B e h a v i o r o f A m o r p h o u s P o l y m e r s 1 . BU L K A N D SH E A R RE S P O N S E

W h e n a b o d y u n d e r g o e s a d e f o r m a t i o n , t h e n a n o r i g i n a l l y s p h e r i c a l p a r - t i c l e i n g e n e r a l b e c o m e s d e f o r m e d t o a n e l l i p s o i d , a n d s u f f e r s a t r a n s l a t i o n a n d a r o t a t i o n a s w e l l a s a c h a n g e i n v o l u m e . I f w e c o n s i d e r a s m a l l c u b i c a l e l e m e n t o f t h e d e f o r m e d m a t e r i a l , t h e n t h e s t r e s s s y s t e m a c t i n g o n t h e s u r - f a c e s o f t h i s e l e m e n t m a y b e r e s o l v e d i n t o n o r m a l a n d t a n g e n t i a l t r a c t i o n s ; if t h e o r i e n t a t i o n of t h e e l e m e n t i s s u i t a b l y c h o s e n , t h e t h r e e p a i r s o f n o r - m a l t r a c t i o n s c a n b e m a d e t o b e e q u a l . T h e s e n o r m a l t r a c t i o n s , w h i c h a r e n u m e r i c a l l y e q u a l t o t h e " h y d r o s t a t i c s t r e s s / ' a r e r e s p o n s i b l e f o r t h e v o l - u m e c h a n g e , w h i l e t h e t a n g e n t i a l t r a c t i o n s , o r " s h e a r - s t r e s s c o m p o n e n t s / ' a r e r e s p o n s i b l e f o r t h e c h a n g e i n s h a p e . T h e s e c o n s i d e r a t i o n s a r e d e a l t w i t h i n C h a p t e r s 2 a n d 10 o f t h e p r e c e d i n g v o l u m e f o r e q u i l i b r i u m e l a s t i c b e - h a v i o r a n d a l s o i n C h a p t e r 1 1 f o r v i s c o e l a s t i c b e h a v i o r . A m a t e r i a l m a n i - f e s t i n g v i s c o e l a s t i c b e h a v i o r e x h i b i t s t i m e - d e p e n d e n t e l a s t i c i t y i n t h e r e l a t i o n b e t w e e n t h e v o l u m e a n d t h e h y d r o s t a t i c s t r e s s , a s w e l l a s i n t h e r e l a t i o n b e t w e e n s h a p e a n d t h e t a n g e n t i a l t r a c t i o n s , o r s o - c a l l e d " s h e a r s t r e s s e s . " I f w e c o n s i d e r , s a y , a p i e c e o f r u b b e r u n d e r a s i m p l e t e n s i l e f o r c e , t h e n t h e r e e x i s t a h y d r o s t a t i c s t r e s s a s w e l l a s s h e a r s t r e s s e s . U n d e r c e r t a i n c i r c u m s t a n c e s t h e r e s p o n s e t o s h e a r s t r e s s d o m i n a t e s t h e r e s p o n s e t o h y d r o - s t a t i c s t r e s s ; t h e r e i s t h e n a v e r y s i m p l e r e l a t i o n s h i p b e t w e e n t h e v i s c o e l a s - t i c b e h a v i o r i n s i m p l e t e n s i o n ( u n d e r s m a l l e l o n g a t i o n s ) a n d t h e b e h a v i o r i n s h e a r . H o w e v e r , w h e n t h e m a t e r i a l e x h i b i t s g l a s s l i k e b e h a v i o r t h i s i s n o t t r u e , a n d t h e r e s p o n s e t o l o n g i t u d i n a l s t r e s s o r s t r a i n i s n o l o n g e r s i m p l y r e l a t e d t o s h e a r b e h a v i o r . T h e s e a s p e c t s a r e d i s c u s s e d i n S e c t i o n V I o f t h i s c h a p t e r . I n t h i s c h a p t e r w e w i l l d e a l m a i n l y w i t h t h e r e s p o n s e t o s h e a r s t r e s s o r s t r a i n .

2. RE T A R D E D EL A S T I C DE F O R M A T I O N I N SH E A R : CR O S S -LI N K E D PO L Y - M E R S

L e t u s c o n s i d e r t h e v i s c o e l a s t i c b e h a v i o r i n s h e a r of a c r o s s - l i n k e d p o l y - m e r . B y w a y o f e x a m p l e w e w i l l c o n s i d e r a s p e c i m e n o f v u l c a n i z e d r u b b e r , d e f o r m e d i n s i m p l e s h e a r a s s h o w n i n F i g . 1 . W h e n a c o n s t a n t f o r c e F i s

TAN-I.

F F I G . 1

SHEAR STRAIN

1 1 1 1 1

SHEAR STRESS APPLIED

1 1 1 1 1

SHEAR STRESS REMOVED

TIME

F I G . 2 F I G . 1. SIMPLE SHEAR CREEP TEST

F I G . 2 . SHEAR STRAIN AS FUNCTION OF TIME AND TEMPERATURE, VULCANIZED RUBBER

A P P L I E D AS S H O W N , THE SHEAR STRESS σ (OR, TO B E CORRECT, THE SHEAR TRACTION) IS OF COURSE EQUAL TO F D I V I D E D B Y THE TOTAL AREA SHEARED. T H E SHEAR STRAIN 7 I N EITHER OF THE PIECES OF RUBBER IS EQUAL TO THE DISPLACEMENT D I V I D E D B Y

THE THICKNESS OF EITHER PIECE OF RUBBER.

W E WILL N O W DISCUSS THE RESPONSE OF THE RUBBER W H E N A CONSTANT SHEAR STRESS Σ0 IS APPLIED TO THE S P E C I M E N AT ZERO T I M E , A N D R E M O V E D AT S O M E S U B - SEQUENT T I M E h , W H I C H M A Y B E OF THE ORDER OF M I N U T E S TO HOURS. I N F I G . 2 IS S H O W N A TYPICAL SHEAR STRAIN B E H A V I O R AS A FUNCTION OF T I M E W H E N THE TEMPERATURE COVERS THE RANGE F R O M , S A Y , 5 0 TO — 7 0 ° C . A T THE HIGHEST TEMPERATURE THERE APPEARS I M M E D I A T E L Y A N ELASTIC RESPONSE W H I C H S T A Y S PRACTICALLY CONSTANT DURING THE T I M E UNDER LOAD A N D DISAPPEARS ALMOST I M - M E D I A T E L Y FOLLOWING REMOVAL OF LOAD. A T A S O M E W H A T LOWER TEMPERATURE, S A Y , 0 ° C , THE S A M E IS SUBSTANTIALLY TRUE, B U T THE DEFORMATION IS GREATER FOR THE S A M E LOAD. T H I S IS I N QUALITATIVE ACCORDANCE W I T H THE PREDICTION OF THE STATISTICAL THEORY OF RUBBERLIKE ELASTICITY FOR SUCH NETWORK S Y S T E M S , AS DISCUSSED I N C H A P T E R 1 1 ( V O L U M E I ) . A T A STILL LOWER TEMPERATURE, FOR E X - A M P L E , AROUND — 4 0 ° C . FOR A NATURAL RUBBER G U M VULCANIZATE, THE E Q U I - L I B R I U M DEFORMATION IS NOT REACHED I M M E D I A T E L Y B U T ONLY AFTER S O M E T I M E ; O N REMOVAL OF LOAD THE DEFORMATION DOES NOT DISAPPEAR I M M E D I A T E L Y B U T A G A I N REQUIRES THE PASSAGE OF T I M E . A T LOWER TEMPERATURES THE A P P R O A C H TO EQUILIBRIUM IS STILL SLOWER, A N D THE RETARDED ELASTIC RECOVERY IS AS INDICATED I N F I G . 2 . FI NA LL Y, AT A SUFFICIENTLY LOW TEMPERATURE, S A Y , — 7 0 ° C , ONLY A SMALL T I M E - I N D E P E N D E N T DEFORMATION IS OBSERVED, W H I C H DISAPPEARS I M M E D I - ATELY O N REMOVAL OF THE LOAD. U N D E R THESE CONDITIONS THE MATERIAL HAS THE S A M E BEHAVIOR AS A GLASSY PLASTIC AT R O O M TEMPERATURE.

L E T U S S U P P O S E THAT THE E X P E R I M E N T B E REPEATED W I T H A DIFFERENT APPLIED LOAD, THAT IS, W I T H A DIFFERENT SHEAR STRESS. I T IS FOUND GENERALLY THAT THE D E -

f o r m a t i o n of t h e s a m p l e a t a n y g i v e n t e m p e r a t u r e a n d t i m e i n s t a n t i s not p r o p o r t i o n a l t o s h e a r s t r e s s , b u t i n c r e a s e s s o m e w h a t f a s t e r t h a n t h e s h e a r s t r e s s ; h e n c e , in general, t h e r e t a r d e d e l a s t i c b e h a v i o r i n s h e a r of v u l c a n i z e d r u b b e r i s n o n l i n e a r .2 H o w e v e r , if t h e s t r e s s i s s m a l l e n o u g h , t h e n t h e b e - h a v i o r i s a p p r o x i m a t e l y l i n e a r . I f w e n e g l e c t t h e d e v i a t i o n f r o m l i n e a r i t y , t h e n , b y m e a n s of t h e g e n e r a l t h e o r y of l i n e a r v i s c o e l a s t i c b e h a v i o r d i s c u s s e d s u b s e q u e n t l y i n t h i s c h a p t e r a n d e l s e w h e r e i n t h i s t r e a t i s e , w e c a n d e r i v e r e l a t i o n s h i p s b e t w e e n t h e r e s p o n s e of a m a t e r i a l ( a t t h e s a m e t e m p e r a t u r e ) i n d i f f e r e n t t y p e s of t e s t .

W i t h t h i s a p p r o x i m a t i o n , w e c a n s p e c i f y t h e r e t a r d e d e l a s t i c b e h a v i o r of a v u l c a n i z e d r u b b e r , o r a n y o t h e r m a t e r i a l m a n i f e s t i n g v i s c o e l a s t i c b e - h a v i o r , i n a c o n s t a n t s h e a r s t r e s s t e s t . L e t y(t) b e t h e s h e a r s t r a i n a t a t i m e t a f t e r a p p l i c a t i o n of a c o n s t a n t s h e a r s t r e s s σ0. T h e n , a t a g i v e n t e m p e r a - t u r e , t h e r a t i o y(t)/a0 i s a p p r o x i m a t e l y i n d e p e n d e n t of σ0 if t h i s i s s u f f i - c i e n t l y s m a l l ; t h i s r a t i o i s c a l l e d t h e shear creep compliance ( a t t i m e t a t t h e s p e c i f i e d t e m p e r a t u r e ) a n d i s d e n o t e d b y J(t). L e t u s n o w c o n s i d e r h o w /(t) v a r i e s w i t h t e m p e r a t u r e f o r s o m e s p e c i f i e d v a l u e of t w h i c h w e w i l l d e n o t e b y tc. F r o m F i g . 2, w e see t h a t t h i s b e h a v i o r i s g i v e n a s s h o w n b y t h e f u l l l i n e i n F i g . 3 , i n w h i c h J(tc) i s p l o t t e d a g a i n s t t e m p e r a t u r e . A t l o w t e m p e r a t u r e s t h i s i s s u b s t a n t i a l l y i n d e p e n d e n t of t e m p e r a t u r e a n d i s of t h e o r d e r of 1 0 ~10 c m .2/ d y n e . A t h i g h t e m p e r a t u r e s t h i s i s of t h e o r d e r of 1 C T6

c m .2/ d y n e a n d d e c r e a s e s s o m e w h a t w i t h i n c r e a s e i n t e m p e r a t u r e . I f tc i s i n c r e a s e d o r r e d u c e d , t h e s t e e p l y r i s i n g p o r t i o n i s s h i f t e d t o t h e l e f t o r t h e r i g h t i n F i g . 3 , a s i n d i c a t e d b y t h e b r o k e n l i n e s . T h e r e c i p r o c a l of J(tc) c o u l d b e u s e d a s a m e a s u r e of t h e ' ' s t i f f n e s s ' ' of r u b b e r a s a f u n c t i o n of t e m p e r a t u r e , b u t a s s e e n f r o m F i g . 3 t h i s s t i f f n e s s d e p e n d s t o s o m e e x t e n t u p o n t h e a r b i t r a r y c h o i c e of tc.

F r o m F i g . 2 a g a i n w e c a n d r a w c o n c l u s i o n s c o n c e r n i n g t h e r e s i d u a l d e - f o r m a t i o n a t a s p e c i f i e d t i m e s u b s e q u e n t t o t h e i n s t a n t of r e m o v a l of s t r e s s , w h i c h t i m e w e w i l l d e n o t e b y tr. A t s u f f i c i e n t l y h i g h a n d l o w t e m p e r a t u r e s t h e r e s i d u a l d e f o r m a t i o n a t t h i s s p e c i f i e d t i m e i s s u b s t a n t i a l l y z e r o ; t h i s d e f o r m a t i o n a t t a i n s a m a x i m u m a t s o m e i n t e r m e d i a t e t e m p e r a t u r e , w h i c h m a y b e a r o u n d — 5 0 ° C . i n t h e r u b b e r s p e c i m e n c o n s i d e r e d a b o v e i n t e s t s of " o r d i n a r y " d u r a t i o n ( m i n u t e s t o h o u r s ) . W i t h c h a n g e i n tr, t h e b e l l s h a p e d p l o t of r e s i d u a l d e f o r m a t i o n a g a i n s t t e m p e r a t u r e h e r e a g a i n s h i f t s a l o n g t h e t e m p e r a t u r e s c a l e .

I f t h e r e t a r d e d e l a s t i c b e h a v i o r i s l i n e a r , t h e n i t i s n o l o n g e r n e c e s s a r y t o r e s t r i c t t h e e x p e r i m e n t a l m e t h o d t o t h a t s h o w n i n F i g . 1 . F o r e x a m p l e , t h e s p e c i m e n c o u l d h a v e t h e s h a p e of a h o l l o w c i r c u l a r c y l i n d e r , i n w h i c h t h e i n n e r c y l i n d r i c a l s u r f a c e i s e i t h e r d i s p l a c e d a l o n g i t s a x i s3"5 o r r o t a t e d a r o u n d

2 H . LEADERMAN, / . Polymer Sei. 1 6 , 2 6 1 ( 1 9 5 5 ) .

3 M . REINER, "TWELVE LECTURES ON THEORETICAL RHEOLOGY." NORTH HOLLAND, AMSTER- D A M , 1 9 4 9 .

J{tc) J{t)

Temperature log t

F I G . 3 F I G . 4

F I G . 3. Creep compliance a t time tc as function of temperature F I G . 4. Creep compliance of vulcanized rubber

i t s a x i s r e l a t i v e t o t h e o u t e r c y l i n d r i c a l s u r f a c e ,6, 7 a s d i s c u s s e d i n C h a p t e r 2 , V o l u m e I , a n d C h a p t e r 1 1 , V o l u m e I I ; o r i t c o u l d b e i n t h e f o r m of a t h i n d i s k .8 ,9 I n f a c t , t h e r e a r e a l a r g e n u m b e r of p o s s i b i l i t i e s d e p e n d i n g u p o n t h e p h y s i c a l p r o p e r t i e s o f t h e t e s t s p e c i m e n a t r o o m t e m p e r a t u r e a n d u n d e r t h e t e s t c o n d i t i o n s ; m a n y o f t h e s e m e t h o d s a r e s i m i l a r t o m e t h o d s u s e d f o r m e a s u r i n g v i s c o s i t y . T h i s a s p e c t i s c o n s i d e r e d i n t h e c h a p t e r o n e x p e r i m e n t a l m e t h o d s ( C h a p t e r 1 1 , V o l u m e I I ) a n d w i l l n o t b e c o n s i d e r e d f u r t h e r h e r e . W e w i l l a s s u m e t h a t f r o m t h e e x p e r i m e n t a l o b s e r v a t i o n s t h e r e c a n b e d e r i v e d t h e r e s p o n s e i n a s i m p l e s h e a r s t r e s s t e s t , s u c h a s i n t h e a r r a n g e m e n t o f F i g . 1 .

3 . EF F E C T O F TE M P E R A T U R E O N RE T A R D E D EL A S T I C I T Y : CR O S S -LI N K E D PO L Y M E R S

I t i s i n s t r u c t i v e t o p l o t t h e creep c o m p l i a n c e i n F i g . 2 a g a i n s t t h e l o g - a r i t h m o f t i m e a s i n F i g . 4 . T h e effect o f i n c r e a s e i n t e m p e r a t u r e o n creep c o m p l i a n c e c a n b e i n t e r p r e t e d a s a r e d u c t i o n i n t h e e q u i l i b r i u m v a l u e o f t h e creep c o m p l i a n c e , o r equilibrium compliance, Je ; a n d a c h a n g e i n t i m e s c a l e , c o r r e s p o n d i n g t o a l a t e r a l s h i f t a l o n g t h e l o g a r i t h m i c t i m e a x i s . T h u s b y p l o t t i n g f o r e a c h t e m p e r a t u r e J(t)/Je a s o r d i n a t e a g a i n s t l o g a r i t h m o f t i m e , a n d s h i f t i n g t h e c u r v e s h o r i z o n t a l l y o n t h e l o g t a x i s , a s i n g l e c u r v e i s

4 T . G . F o x , Jr., a n d P . J . F l o r y , / . Am. Chem. Soc. 70, 2384 (1948).

5 J . D . F e r r y , L . D . G r a n d i n e , and E . R . F i t z g e r a l d , J. Appl. Phys. 24, 911 (1953).

6 H . L e a d e r m a n , J. Polymer Sei. 13, 371 (1954).

7 H . L e a d e r m a n , R . G . S m i t h , and R . W . Jones, Polymer Sei. 14, 47 (1954).

8 G . J . Dienes, Colloid Sei. 2, 131 (1947).

9 A . v a n R o s s e m a n d H . v a n der M e i j d e n , Rubber Age (Ν. Y.) 23, 438 (1928).

9

NATURAL RUBBER VULCANIZATE

O L

- 2 Ο 2 4 6

LOG t (MIN), - 6 0EC

8 1 0

- 1 2 - 1 0 8 - 6 - 4

-2

LOG t (MIN), 3 1EC

F I G . 5 . NORMALIZED CREEP COMPLIANCE: NATURAL RUBBER AND BUTYL RUBBER VULCAN- IZATES.

obtained. As an example, in Fig. 5 are shown the creep compliance curves obtained in this way for two different vulcanized rubbers;

the abscissa scales represent, respectively, the time scales for — 60° C. and 31° C. For the abscissa scale corresponding to temperature T, the curves in Fig. 5 represent the normalized creep compliance "reduced to temperature If now the ordinate scale is multiplied by the value of J

at the temperature T, then the curves represent the (non-normalized) creep compliance re- duced to temperature T. We see, for example, from Fig. 5 that, for the particular vulcanizates represented there, 1 min. at —60° C. is equivalent to 10~

min. at 31° C, and 10~

min. at the higher temperature is equiva- lent to 10

min. at -60° C.

In Fig. 6 is represented diagrammatically the behavior of a cross-linked amorphous polymer when a constant shear stress σ

is applied from zero time to time ti, and subsequently removed; the ordinate represents values of the shear strain at time t, namely 7(0, divided by σ

. The diagrams repre- sent from top to bottom experiments carried out at successively higher temperatures. At the lowest temperature there is seen to be a substantially instantaneous deformation, which disappears immediately on removal of stress (actually a finite time is required for the propagation of a stress wave through the material). This deformation corresponds to an increase in internal energy, and is associated mainly with van der Waals' forces be- tween polymer chains. The ratio of strain to stress, namely the compli- ance, is of the order of 1CT

cm.

/dyne; this is the order of magnitude of the compliance of organic glasses, both polymeric and nonpolymeric. It is ap- propriate to call this limiting value of compliance corresponding to very low temperatures (or, from Fig. 5, to very short times), the

this will be denoted by J

.

At a higher temperature it is seen from the two middle diagrams in Fig. 6

1 0 F . S . CONANT, G . L . HALL, AND W . J . L Y O N S , A p p l . Phys. 21, 4 9 9 ( 1 9 5 0 ) .

10 HERBERT LEADERMAN

0 ti

Shear stress Shear stress t t applied removed

FIG. 6. Effect of temperature on retarded elasticity: cross-linked polymer

that there occurs the glassy deformation immediately upon application of

stress, and then a retarded elastic deformation; on removal of stress there

occurs an "immediate" elastic recovery corresponding to the glass compli-

ance, and then the retarded elastic recovery. The retarded elasticity is due

to the gradual uncurling or curling up of randomly kinked polymer chains,

and this corresponds principally to a change in entropy (cf. Chapter 11,

Volume I). When the temperature is further increased, as represented, for

example, in the lowest diagram of Fig. 6, then the time scale of the retarded

elastic deformation becomes shortened so much that it is not possible to

observe the glassy deformation corresponding to J

. Since a small but

nevertheless finite time is required for the application of stress, the actual

course of the deformation as a function of time may follow the curve indi-

cated by the broken line in the lowest diagram of Fig. 6. Thus a deformation

observed at an early instant after application of stress may correspond to

some point such as Ζ in Fig. 6. Such an observation, unlike the true limit-

ing value corresponding to J

, has thus no scientific meaning. Under these

c o n d i t i o n s o n l y t h e c r e e p c o m p l i a n c e J(t) i s m e a s u r a b l e ; w e c a n s a y t h a t t h e l i m i t i n g v a l u e of t h i s q u a n t i t y , n a m e l y Jg, i s n o t e x p e r i m e n t a l l y a c c e s - s i b l e i n s u c h o r d i n a r y c r e e p e x p e r i m e n t s e x c e p t a t a s u f f i c i e n t l y l o w t e m - p e r a t u r e .

4. RE T A R D E D EL A S T I C DE F O R M A T I O N I N SH E A R : NO N C R O S S -LI N K E D PO L Y - M E R S

W e w i l l n o w c o n s i d e r t h e r e s p o n s e i n a s i m i l a r c o n s t a n t s h e a r s t r e s s e x - p e r i m e n t of a n o n c r o s s - l i n k e d p o l y m e r . T h e a r r a n g e m e n t c o u l d b e a s s h o w n , f o r e x a m p l e , i n F i g . 7, w h e r e t h e m a t e r i a l f i l l s a n a r r o w g a p b e t w e e n t w o c i r c u l a r c y l i n d e r s . O n e of t h e c y l i n d e r s i s fixed; a c o n s t a n t t o r q u e W c a n b e a p p l i e d t o t h e o t h e r c y l i n d e r . T h e s h e a r s t r e s s i s p r o p o r t i o n a l t o t h e t o r q u e , a n d t h e s h e a r s t r a i n t o t h e a n g l e of r o t a t i o n of t h e " f r e e " c y l i n d e r r e l a t i v e t o t h e fixed c y l i n d e r .

F I G . 7 . A r r a n g e m e n t for concentric cylinder viscometer

7 W/ * o

Time

F I G . 8 . Creep a n d recovery of noncross-linked polymer : effect of time under stress

L E T U S A S S U M E AS BEFORE THAT A CONSTANT SHEAR STRESS Σ0 IS A P P L I E D AT ZERO T I M E , A N D RELEASED AT A SUBSEQUENT T I M E t\. T H E SHEAR STRAIN AS A FUNCTION OF T I M E M A Y FOLLOW THE CURVE ABC I N F I G . 8 . I F THE SHEAR STRESS IS SMALL E N O U G H , THEN THE SHEAR STRAIN AT A N Y T I M E t IS PROPORTIONAL TO Σ0 ; THE M A - TERIAL THUS MANIFESTS LINEAR VISCOELASTIC BEH AV IOR . A S BEFORE, D E P E N D I N G U P O N THE MECHANICAL PROPERTIES OF THE MATERIAL AT R O O M TEMPERATURE A N D AT THE TEST TEMPERATURE, A S I M P L E SHEAR T Y P E OF A P P A R A T U S ,11 OR ONE OF THE ARRANGE- M E N T S DESCRIBED PREVIOUSLY M A Y B E U S E D ; THE ROTATIONAL VISCOMETER OF F I G . 7 N E E D NOT NECESSARILY POSSESS A NARROW G A P .6' 7 T H E S E CONSIDERATIONS A G A I N C O M E UNDER THE H E A D I N G OF EXPERIMENTAL TECHNIQUES, A N D W E WILL A G A I N A S - S U M E THAT FROM THE EXPERIMENTAL OBSERVATIONS THERE CAN B E DERIVED THE SHEAR STRAIN BEHAVIOR DURING APPLICATION OF A CONSTANT SHEAR STRESS A N D FOL- LOWING REMOVAL OF STRESS.

R E T U R N I N G TO F I G . 8 , IT IS SEEN THAT IF THE VISCOELASTIC B E H A V I O R IS A S S U M E D TO B E LINEAR, THEN W E M A Y PLOT AS ORDINATE THE SHEAR STRAIN AT T I M E t, N A M E L Y y(t), D I V I D E D B Y THE M A G N I T U D E OF THE SHEAR STRESS Σ0 . T H E CURVE ABD W H I C H REPRESENTS THUS THE CREEP C O M P L I A N C E J(t) IS SEEN TO PROCEED NOT TO A N EQUILIBRIUM VALUE B U T TO A LIMITING SLOPE; THE RECIPROCAL OF THIS SLOPE IS CALLED THE viscosity, η. I T IS N O W A S S U M E D THAT THE STRESS IS R E M O V E D AT A T I M E ti SUBSEQUENT TO THE T I M E AT W H I C H THE SLOPE B E C O M E S CONSTANT. T H E R E IS A RETARDED ELASTIC RECOVERY S H O W N B Y THE BROKEN LINE BC> A N D A P E R M A - NENT SET EQUAL TO σ0ίι/η. I F THE STRESS IS R E M O V E D INSTEAD AT ANOTHER T I M E h , CORRESPONDING TO A POINT D O N THE CREEP C O M P L I A N C E CURVE SUBSEQUENT TO THE T I M E AT W H I C H THE SLOPE B E C O M E S CONSTANT, THEN THE P E R M A N E N T SET CORRESPONDS TO A C O M P L I A N C E OF fe/V, THE RECOVERY CURVE DE IS IDENTICAL I N S H A P E TO THE RECOVERY CURVE BC. T H E MATERIAL MANIFESTS flow W H E N UNDER LOAD ( I N ADDITION TO T I M E - D E P E N D E N T ELASTICITY) SINCE THE P E R M A N E N T SET IS PROPORTIONAL TO THE T I M E UNDER SHEAR STRESS. T H E FLOW IS Newtonian SINCE IT IS PROPORTIONAL TO THE SHEAR STRESS. I F THE STRESS IS R E M O V E D AT A POINT F I N THE D I A G R A M CORRESPONDING TO A T I M E £3, W H I C H IS PREVIOUS TO THE ATTAIN- M E N T OF THE STEADY-FLOW STATE, THEN THE SUBSEQUENT P E R M A N E N T SET CORRE- S P O N D S TO A C O M P L I A N C E OF h/ η ; THE RECOVERY CURVE FG IS H O W E V E R NOT OF THE S A M E S H A P E AS THE CURVES BC A N D DE.

W E WILL N O W CONSIDER THE EFFECT OF TEMPERATURE O N THE BEHAVIOR OF A N O N - CROSS-LINKED P O L Y M E R I N SUCH A N E X P E R I M E N T . L E T A Β I N F I G . 9 REPRESENT THE CREEP C O M P L I A N C E J(t) AT A TEMPERATURE T2 U P TO T I M E t\ . I F THE STRESS Σ0 IS R E M O V E D AT THIS INSTANT, THEN THE STRAIN y(t) AT SUBSEQUENT T I M E S D E - CREASES AS S H O W N B Y THE CURVE BC IN THE PLOT OF y(t)/a0 AGAINST T I M E I N F I G . 9 . T H E INTERCEPT OF THE LINEAR PART OF A Β ON THE ORDINATE A X I S , N A M E L Y AXy

IS EQUAL TO THE COMPLIANCE CORRESPONDING TO THE TOTAL ELASTIC RECOVERY. T H E CURVE A DE REPRESENTS THE BEHAVIOR AT A LOWER TEMPERATURE Τ ζ. H E R E THE

11 Κ . E . VAN HOLDE AND J . W . WILLIAMS, / . Polymer Set. 1 1 , 2 4 3 ( 1 9 5 3 ) .

Time

F I G . 9 . Creep a n d recovery of noncross-linked p o l y m e r : effect of temperature

final s l o p e i s l e s s , a n d t h e t i m e t o a t t a i n t h e s t e a d y s t a t e i s g r e a t e r t h a n a t Τ2. T o a first a p p r o x i m a t i o n , t h e i n t e r c e p t of t h e s t r a i g h t p a r t of AD o n t h e o r d i n a t e a x i s i s t h e s a m e a s b e f o r e ; t h e m a g n i t u d e of t h e r e c o v e r y i s a g a i n a p p r o x i m a t e l y a s a t t e m p e r a t u r e T2, b u t t h e r e c o v e r y i s s l o w e r . I f t h e t e m p e r a t u r e i s l o w e n o u g h , t h e r e i s e v i d e n c e of a g l a s s y i n s t a n t a n e o u s c o m p l i a n c e a s i n d i c a t e d b y A Y a n d DZ. F i n a l l y , a t a s u f f i c i e n t l y l o w t e m p e r a t u r e T4, t h e r e o c c u r s o n l y a n i n s t a n t a n e o u s d e f o r m a t i o n c o r r e - s p o n d i n g t o a g l a s s c o m p l i a n c e o f t h e o r d e r of m a g n i t u d e o f 1 ( T10 c m .2/ d y n e a s r e p r e s e n t e d b y AYFG i n F i g . 9 . A t a higher t e m p e r a t u r e Τι, o n t h e o t h e r h a n d , t h e c r e e p c o m p l i a n c e i s a s s h o w n b y t h e c u r v e AH.

T h e s e c u r v e s a d m i t of a s i m p l e i n t e r p r e t a t i o n . T h e c r e e p c o m p l i a n c e c o n - s i s t s of t h r e e c o m p o n e n t s , a g l a s s c o m p l i a n c e , a t i m e - d e p e n d e n t r e t a r d e d - e l a s t i c c o m p l i a n c e , a n d a N e w t o n i a n flow. T h e m a g n i t u d e of t h e t i m e - d e p e n d e n t e l a s t i c c o m p l i a n c e i s a p p r o x i m a t e l y i n d e p e n d e n t of t e m p e r a t u r e b u t t h e r a t e of a p p r o a c h t o t h e s t e a d y s t a t e i s v e r y t e m p e r a t u r e - d e p e n d e n t . A t t h e t e m p e r a t u r e T4 o n l y t h e g l a s s c o m p l i a n c e i s o b s e r v e d f o r t h e t i m e s c a l e of t h e e x p e r i m e n t . A t t h e h i g h e r t e m p e r a t u r e Τ ζ t h e r e i s a l s o o b s e r v e d t h e t i m e - d e p e n d e n t e l a s t i c i t y a n d f l o w ; a t t h e t e m p e r a t u r e Ί\ t h e g l a s s y c o m p l i a n c e c a n n o l o n g e r b e e x p e r i m e n t a l l y o b s e r v e d a n d a t t h e t e m p e r a - t u r e Τι t h e flow d o m i n a t e s a l s o t h e t i m e - d e p e n d e n t e l a s t i c i t y , w h i c h i s n o w v e r y r a p i d . T h e c o m p l i a n c e A Y w e h a v e a l r e a d y c a l l e d t h e g l a s s c o m p l i a n c e Jg ; t h e c o m p l i a n c e A X w e w i l l c a l l t h e steady-state elastic compliance Je ; a n d XY w e w i l l c a l l t h e delayed elastic compliance Jd . T h e s e q u a n t i t i e s a r e o b t a i n a b l e f r o m t h e r e c o v e r y c u r v e s a s i n d i c a t e d i n F i g . 9 .

T h e c r e e p c o m p l i a n c e c u r v e of a n o n c r o s s - l i n k e d a m o r p h o u s p o l y m e r a t a s p e c i f i e d t e m p e r a t u r e h a s t h e g e n e r a l f o r m a s s h o w n d i a g r a m m a t i c a l l y i n F i g . 10, w h e r e J(t) a n d J(t) — t/η a r e p l o t t e d a g a i n s t l o g t. A t v e r y s m a l l v a l u e s o f t t h e r e i s a p l a t e a u i n t h e c u r v e s c o r r e s p o n d i n g t o t h e g l a s s c o m p l i -

log*

F I G . 1 0 . Creep compliance curve of noncross-linked polymer

a n c e Jg of t h e o r d e r of 1 0 ~10 c m .2/ d y n e ; t h e r e i s a s e c o n d p l a t e a u c o r r e s p o n d - i n g t o a c o m p l i a n c e of t h e o r d e r of 1 0- 6 c m2/ d y n e , a s i n a v u l c a n i z e d r u b b e r . F i n a l l y t h e r e o c c u r s t h e s t e e p l y r i s i n g p a r t of t h e J(f) c u r v e o n t h e l o g a r i t h - m i c t i m e p l o t , a s i n d i c a t e d b y t h e b r o k e n c u r v e s i n F i g . 10. T h e q u a n t i t y J(t) — t/η r e p r e s e n t s t h e elastic p a r t of t h e c r e e p c o m p l i a n c e ; a s w i l l b e s h o w n l a t e r , i t r e p r e s e n t s a l s o t h e r e c o v e r y w h e n t h e s t r e s s i s r e m o v e d a f t e r t h e s t e a d y s t a t e i s r e a c h e d i n a c o n s t a n t - s t r e s s e x p e r i m e n t . T h e c u r v e of J(t) — t/η i s a s r e p r e s e n t e d b y t h e c o n t i n u o u s c u r v e i n F i g . 10. T h e r e a s e c o n d r e t a r d e d e l a s t i c p r o c e s s c a n b e o b s e r v e d , l e a d i n g t o a p l a t e a u c o r - r e s p o n d i n g t o a c o m p l i a n c e of t h e o r d e r of 1 0 ~5 c m .2/ d y n e . A t a g i v e n t e m - p e r a t u r e , t h e effect of d e c r e a s e of m o l e c u l a r w e i g h t i s t o m o v e t h e " s l o w "

r e t a r d e d e l a s t i c p r o c e s s t o t h e l e f t , w h i l e t h e " f a s t " r e t a r d e d e l a s t i c p r o c e s s , r e p r e s e n t i n g t h e t r a n s i t i o n f r o m g l a s s l i k e t o r u b b e r l i k e b e h a v i o r , d o e s n o t c h a n g e . C h a n g e i n m o l e c u l a r - w e i g h t d i s t r i b u t i o n c h a n g e s t h e h e i g h t of t h e l a s t - n a m e d p l a t e a u ; t h e o t h e r p l a t e a u s a r e s u b s t a n t i a l l y i n d e p e n d e n t of m o l e c u l a r w e i g h t . T h e s e f a c t o r s w i l l b e c o n s i d e r e d i n m o r e d e t a i l s u b s e - q u e n t l y .

T h e c r e e p c o m p l i a n c e c u r v e f o r a n a m o r p h o u s c r o s s - l i n k e d p o l y m e r c h a n g e s s i g n i f i c a n t l y o v e r s o m e s i x d e c a d e s of l o g a r i t h m i c t i m e a s c a n b e s e e n f r o m F i g . 5. F o r a n a m o r p h o u s n o n c r o s s - l i n k e d p o l y m e r , t h e c o r r e - s p o n d i n g r a n g e f o r J(t) — t/η i s of t h e o r d e r of t e n d e c a d e s , a s i n d i c a t e d i n F i g . 2 5 . I n a c r e e p t e s t t h e p r a c t i c a l r a t i o of t h e l o n g e s t t o t h e s h o r t e s t t i m e of o b s e r v a t i o n i s of t h e o r d e r of 1 03 t o 1 04, c o r r e s p o n d i n g t o t h r e e t o f o u r d e c a d e s o n t h e l o g a r i t h m i c t i m e s c a l e . T h u s i t i s n o t p r a c t i c a b l e t o o b - s e r v e t h e w h o l e c o u r s e of t h e e l a s t i c p a r t of t h e c r e e p c o m p l i a n c e f u n c t i o n b y m e a n s of a n e x p e r i m e n t a t o n e t e m p e r a t u r e . I n o r d e r t o o b t a i n t h e w h o l e c u r v e , i t i s n e c e s s a r y t o c a r r y o u t e x p e r i m e n t s o v e r a r a n g e of t e m p e r a t u r e , a n d i t i s a s s u m e d t h a t t h e p r i n c i p a l effect of c h a n g e of t e m p e r a t u r e i s t o c h a n g e t h e t i m e s c a l e of t h e J(t) a n d J(t) — t/η c u r v e s . T h e r e d u c e d c r e e p c o m p l i a n c e c u r v e f o r a s p e c i f i e d t e m p e r a t u r e i s t h e n o b t a i n e d e s s e n t i a l l y

b y s h i f t i n g c u r v e s o b t a i n e d a t h i g h e r a n d l o w e r t e m p e r a t u r e s a l o n g t h e a x i s o f l o g a r i t h m i c t i m e , u n t i l a l l t h e c u r v e s o v e r l a p t o g i v e a s i n g l e c o n - t i n u o u s c u r v e . T h u s t h e c r e e p c o m p l i a n c e J(t), o r t h e e l a s t i c p a r t o f t h e c r e e p c o m p l i a n c e J(t) — t/η w h i c h i s o b s e r v e d a t t i m e t a t a t e m p e r a t u r e T , w o u l d b e o b s e r v e d a t a t i m e t/aT i f t h e e x p e r i m e n t w e r e c a r r i e d o u t a t t h e r e f e r e n c e t e m p e r a t u r e . T h e t e m p e r a t u r e f u n c t i o n aT i s c a l l e d t h e shift- ing factor; t h i s i s l e s s t h a n u n i t y f o r t e m p e r a t u r e s a b o v e t h e r e f e r e n c e t e m - p e r a t u r e , a n d g r e a t e r t h a n u n i t y f o r t e m p e r a t u r e s b e l o w t h i s . O n t h e o r e t i - c a l g r o u n d s , t h i s p r o c e d u r e s h o u l d b e s l i g h t l y m o d i f i e d a s d i s c u s s e d i n C h a p t e r 1 1 . G e n e r a l l y s p e a k i n g , aT v a r i e s w i t h t e m p e r a t u r e i n t h e s a m e w a y a s t h e v i s c o s i t y .7 , 11 A s w i l l b e s h o w n l a t e r , a s i m i l a r b e h a v i o r i s o b - s e r v e d i n o t h e r t y p e s o f t e s t s o n n o n c r o s s - l i n k e d a m o r p h o u s p o l y m e r s .

III. Stress R e l a x a t i o n a n d D y n a m i c R e s p o n s e o f A m o r p h o u s P o l y m e r s 1 . S T R E S S R E L A X A T I O N

I t h a s p r e v i o u s l y b e e n m e n t i o n e d t h a t v u l c a n i z e d r u b b e r a n d p o l y s t y r e n e a t r o o m t e m p e r a t u r e m a n i f e s t a n o t h e r a s p e c t o f v i s c o e l a s t i c b e h a v i o r , n a m e l y , s t r e s s r e l a x a t i o n : w h e n a g i v e n d e f o r m a t i o n i s a p p l i e d a n d h e l d c o n s t a n t , t h e s t r e s s i s n o t c o n s t a n t b u t d e c r e a s e s g r a d u a l l y w i t h t i m e . A s w i t h r e t a r d e d - e l a s t i c i t y p h e n o m e n a d i s c u s s e d i n t h e p r e v i o u s s e c t i o n , s t r e s s - r e l a x a t i o n p h e n o m e n a i n v u l c a n i z e d r u b b e r a n d p o l y s t y r e n e b e c o m e m o r e m a r k e d a t t e m p e r a t u r e s b e l o w a n d a b o v e r o o m t e m p e r a t u r e , r e s p e c t i v e l y .

S u p p o s e t h a t a c o n s t a n t s h e a r s t r a i n b e a p p l i e d a t z e r o t i m e t o a n a m o r - p h o u s p o l y m e r . I f t h e t e m p e r a t u r e i s s u f f i c i e n t l y l o w , t h e s t r e s s s u b s e - q u e n t l y r e m a i n s c o n s t a n t d u r i n g t h e e x p e r i m e n t ; i f t h e s t r a i n i s s m a l l e n o u g h t h e s t r e s s i s p r o p o r t i o n a l t o t h e s t r a i n . I f t h e e x p e r i m e n t i s r e p e a t e d a t a h i g h e r t e m p e r a t u r e , i t m a y b e f o u n d t h a t t h e s t r e s s , i n s t e a d o f r e m a i n - i n g c o n s t a n t a f t e r t h e i n s t a n t o f a p p l i c a t i o n o f s t r a i n , d e c r e a s e s w i t h t i m e . A g a i n , i f t h e s t r a i n i s s m a l l e n o u g h , t h e s t r e s s a(t) a t a n y t i m e t a f t e r a p p l i - c a t i o n o f t h e s t r a i n 7 0 i s p r o p o r t i o n a l t o t h e m a g n i t u d e o f t h e s t r a i n ; h e n c e w e m a y c o n s i d e r σ(0/το a s a m e a s u r e o f t h e s t r e s s - r e l a x a t i o n b e h a v i o r i n s h e a r a t t h e p a r t i c u l a r t e m p e r a t u r e . T h i s r a t i o i s c a l l e d t h e (shear) relaxa- tion modulusj a n d i s d e n o t e d b y G(t). T h e s t r e s s r e l a x a t i o n b e h a v i o r o f a n a m o r p h o u s p o l y m e r a t d i f f e r e n t t e m p e r a t u r e s i s r e p r e s e n t e d d i a g r a m m a t i - c a l l y i n F i g . 1 1 , i n w h i c h t h e r e l a x a t i o n m o d u l u s G{t) i s p l o t t e d a g a i n s t t h e l o g a r i t h m o f t h e t i m e f r o m t h e i n s t a n t o f a p p l i c a t i o n o f a c o n s t a n t s h e a r s t r a i n . T h e d i a g r a m r e p r e s e n t s t h e b e h a v i o r o f a cross-linked p o l y m e r f o r a g i v e n r a n g e o f v a l u e s o f t i m e t. T h e c u r v e s l a b e l l e d T\ a n d T2 r e p r e s e n t t h e b e h a v i o r a t t w o d i f f e r e n t t e m p e r a t u r e s a s d e s c r i b e d a b o v e . A t s u c c e s s i v e l y h i g h e r t e m p e r a t u r e s , c u r v e s s u c h a s T3 a n d T4 m a y b e o b t a i n e d ; t h e s t r e s s r e l a x e s t o a n e q u i l i b r i u m v a l u e i n t h e c a s e o f a c r o s s - l i n k e d a m o r p h o u s p o l y m e r a s s h o w n i n t h e d i a g r a m , o r t o z e r o , i n t h e c a s e o f a noncross-linked

LOG Ί

F I G . 1 1 . STRESS RELAXATION: EFFECT OF TEMPERATURE

A M O R P H O U S P O L Y M E R . H E R E A G A I N THE PRINCIPAL EFFECT OF CHANGE I N T E M P E R A - TURE IS TO CHANGE THE T I M E SCALE OF THE E X P E R I M E N T . T H U S I N ORDER TO OBTAIN THE STRESS-RELAXATION CURVE AT A SPECIFIED TEMPERATURE, S A Y T% I N F I G . 1 1 , FOR T I M E S LONGER A N D SHORTER THAN THE PRACTICAL RANGE OF OBSERVATION, THE OBSERVED RELAXATION M O D U L U S CURVES (OR SLIGHTLY M O D I F I E D CURVES AS DISCUSSED I N C H A P T E R 1 1 ) , O B T A I N E D AT HIGHER A N D LOWER TEMPERATURES, RESPECTIVELY, ARE SHIFTED ALONG THE A X I S OF LOGARITHMIC T I M E AS I N THE CASE OF THE CREEP C O M P L I A N C E CURVES DISCUSSED PREVIOUSLY. T H I S TREATMENT OF STRESS-RELAXATION DATA IS CONSIDERED I N M O R E DETAIL I N C H A P T E R 2 . F O R A G I V E N P O L Y M E R , THE S A M E VALUES OF aT SHOULD B E OBTAINED AS FROM CREEP C O M P L I A N C E M E A S U R E - M E N T S .

T H E RELAXATION M O D U L U S CURVE I N SHEAR FOR A M O R P H O U S P O L Y M E R S REDUCED TO A STANDARD TEMPERATURE A N D PLOTTED AGAINST THE LOGARITHM OF THE T I M E IS AS INDICATED D I A G R A M M A T I C A L L Y I N F I G . 1 2 ; THE FULL CURVE REPRESENTS THE B E - HAVIOR OF A CROSS-LINKED P O L Y M E R . F O R LARGE VALUES OF t THE B E H A V I O R OF A NONCROSS-LINKED P O L Y M E R DIFFERS FROM THAT OF A CROSS-LINKED P O L Y M E R , A N D THE B E H A V I O R OF THE FORMER IS S H O W N B Y THE BROKEN CURVE. T H E L I M I T I N G VALUE OF THE RELAXATION M O D U L U S FOR V E R Y SMALL VALUES OF

t

glass modulus

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equilibrium modulus G

J

.

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decay modu-

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T H E RELAXATION M O D U L U S CURVE FOR A NONCROSS-LINKED P O L Y M E R IS SEEN TO POSSESS A N INTERMEDIATE PLATEAU; FOR LONGER VALUES OF T I M E THE STRESS D E C A Y S TO ZERO. T H E PORTION TO THE RIGHT OF THE PLATEAU IS MOLECULAR-WEIGHT D E P E N D - ENT, A N D SHIFTS TO THE RIGHT I N F I G . 1 2 IF THE MOLECULAR W E I G H T INCREASES.12

T H I S IS DISCUSSED I N GREATER DETAIL I N C H A P T E R 2 .

T H E RELAXATION M O D U L U S FUNCTION G(t) IS SEEN TO B E A P P R O X I M A T E L Y THE

12 Α . V . TOBOLSKY AND J . R . MCLOUGHLIN, Polymer Sei. 8 , 5 4 3 ( 1 9 5 2 ) .

Git)

NON-CROSSLINKED

CROSSLINKED

LOG Ί

F I G . 1 2 . RELAXATION MODULUS CURVE, CROSS-LINKED AND NONCROSS-LINKED POLYMER

r e c i p r o c a l of t h e c r e e p c o m p l i a n c e f u n c t i o n J(t); t h e e x a c t r e l a t i o n s h i p i s g i v e n i n S e c t i o n V . I n p r i n c i p l e , t h e c r e e p c o m p l i a n c e c a n b e c o m p u t e d f r o m t h e r e l a x a t i o n m o d u l u s , a n d i n p a r t i c u l a r t w o i m p o r t a n t l i m i t i n g v a l u e s a s s o c i a t e d w i t h t h e c r e e p c o m p l i a n c e c u r v e , n a m e l y t h e s t e a d y - s t a t e c o m - p l i a n c e Je a n d t h e v i s c o s i t y 77, c a n b e c a l c u l a t e d f r o m t h e r e d u c e d G(t) c u r v e . T h i s i s d i s c u s s e d i n S e c t i o n V of t h i s c h a p t e r a n d a l s o i n C h a p t e r 2.

A m e a s u r e of t h e " s t i f f n e s s " of a m a t e r i a l a t a n y t e m p e r a t u r e w h i c h i s o f t e n u s e d i s t h e r e l a x a t i o n m o d u l u s a t a s p e c i f i e d t i m e tc a f t e r a p p l i c a t i o n of s t r e s s . T h e v a l u e of G{tc) m e a s u r e d a s a f u n c t i o n of t e m p e r a t u r e w o u l d b e e x p e c t e d t o f a l l o v e r a s m a l l r a n g e of t e m p e r a t u r e f r o m t h e o r d e r of 1 010

d y n e s / c m .2 t o zero ( f o r a n o n c r o s s - l i n k e d p o l y m e r ) o r t o a b o u t 1 06 d y n e s / c m .2 ( f o r a c r o s s - l i n k e d p o l y m e r ) . T h e a c t u a l c u r v e w o u l d of c o u r s e d e p e n d u p o n t h e c h o i c e of tc ( c o m p a r e F i g . 3 ) . F o r a g i v e n v a l u e of tc a n d t e m p e r a - t u r e , G(tc) a n d J(tc) w o u l d b e a p p r o x i m a t e l y r e c i p r o c a l s of e a c h o t h e r .

A s w i t h t h e c a s e of t h e c r e e p t e s t p r e v i o u s l y c o n s i d e r e d , t h e g l a s s m o d u - l u s i s n o t e x p e r i m e n t a l l y a c c e s s i b l e w h e n t h e t i m e s c a l e of s t r e s s r e l a x a t i o n i s s m a l l c o m p a r e d t o t h e t i m e of a p p l i c a t i o n of t h e s t r a i n . I n t h e u p p e r p a r t o f F i g . 13 i s r e p r e s e n t e d d i a g r a m m a t i c a l l y t h e c u r v e of yoG(t) p l o t t e d a g a i n s t t, a n d a l s o , b y t h e b r o k e n c u r v e , t h e v a l u e s of σ(ί) w h i c h w o u l d b e o b t a i n e d w h e n t h e s t r a i n 7 0 i s a p p l i e d a t a f i n i t e r a t e , a s i n d i c a t e d i n t h e l o w e r p a r t o f F i g . 1 3 . I t i s s e e n t h a t u n d e r t h e s e c o n d i t i o n s a n o b s e r v a t i o n of s t r e s s a t s o m e i n s t a n t f o l l o w i n g a p p l i c a t i o n of s t r a i n b e a r s n o r e l a t i o n t o t h e l i m i t - i n g v a l u e of s t r e s s c o r r e s p o n d i n g t o yoGg .

2. R E P R E S E N T A T I O N O F D Y N A M I C R E S P O N S E

S o f a r w e h a v e b e e n c o n s i d e r i n g w h a t m a y b e c a l l e d step-function experiments. T h e s t r e s s ( o r s t r a i n ) i s zero u p t o a g i v e n i n s t a n t , a n d t h e n c h a n g e s d i s c o n t i n u o u s l y t o a f i n i t e v a l u e ; a l t e r n a t i v e l y , t h e s t r e s s ( o r s t r a i n ) m a y c h a n g e d i s c o n t i n u o u s l y f r o m o n e f i n i t e v a l u e t o a n o t h e r finite v a l u e o r t o z e r o . W e w i l l n o w c o n s i d e r t h e r e s p o n s e of a n a m o r p h o u s p o l y m e r i n