7 Transient and Dynamic Characterization of Viscoelastic Solids S. S. S T E R N S T E I N
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Rensselaer Polytechnic Institute, Materials Engineering Department, Troy, NY 12181
This chapter reviews the rudimentary aspects of linear viscoelasticity theory as applied to the transient and dynamic mechanical characterization of solids. Various sources of experimental error are discussed, and a de tailed derivation of machine and load cell compliance corrections as applied to dynamic moduli data is given. The Dynastat system and associated computerized data acquisition and processing equipment are described. This instrument provides closed-loop control of either load or displacement covering a frequency range of DC to 200 Hz. A transient stress relaxation curve on a glassy polymer is given and illustrates the ability of the Dynastat to apply a displacement rapidly (15 ms) with out overshoot or ringing. Dynamic data on a very stiff carbon—epoxy laminate (stiffness of 1000 Ν/mm) are presented versus both frequency and temperature. The effects of the matrix glass transition on the storage and loss stiffnesses of the laminate are illustrated.
JALPPLICATIONS OF LINEAR VISCOELASTIC
test m e t h o d s a n d
d a t a to
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s t u d y o f p o l y m e r i c s o l i d s , m e l t s , a n d s o l u t i o n s are w e l l d o c u m e n t e d (1). T h e t h r e e - d i m e n s i o n a l t h e o r y o f v i s c o e l a s t i c i t y is p r e s e n t e d w i t h m a t h e m a t i c a l r i g o r b y C h r i s t e n s e n (2) a n d s o m e e x a m p l e s o f n o n 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 n d t h e o r y are also g i v e n e l s e w h e r e ( 3 , 4 ) . Some examples of complexities introduced b y the simultaneous exis tence of both v o l u m e a n d shear linear viscoelastic processes i n poly m e r i c s o l i d s are f o u n d i n t h e l i t e r a t u r e (5, 6). T h e a d d e d c o m p l e x i t i e s associated w i t h t h r e e - d i m e n s i o n a l effects ( v o l u m e a n d shear p r o c e s s e s ) , a n i s o t r o p y , a n d n o n l i n e a r i t y w i l l b e i g n o r e d i n t h i s c h a p t e r . It 0065-2393/83/0203-0123$07.25/0 © 1983 A m e r i c a n C h e m i c a l Society
Craver; Polymer Characterization Advances in Chemistry; American Chemical Society: Washington, DC, 1983.
124
POLYMER CHARACTERIZATION
is a l s o a s s u m e d i n t h i s c h a p t e r t h a t t h e m a t e r i a l i s s u b j e c t e d t o a s p a t i a l l y u n i f o r m stress a n d s t r a i n . I n its s i m p l e s t f o r m , t h e o n e - d i m e n s i o n a l t h e o r y o f l i n e a r v i s c o e l a s t i c i t y states t h a t t h e s t r e s s σ at t h e p r e s e n t t i m e t i s g i v e n b y . de , - )~j^du
t
= J
a(t)
E (t
/1N
(1)
u
r
—00
w h e r e E i s t h e stress r e l a x a t i o n m o d u l u s , a f u n c t i o n o f t i m e , a n d e(u) is t h e s t r a i n h i s t o r y o v e r a l l p a s t t i m e u ^ t. r
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A n a l t e r n a t i v e f o r m u l a t i o n o f t h e t h e o r y g i v e s t h e s t r a i n at t h e p r e s e n t t i m e e(t) i n t e r m s o f a n a r b i t r a r y l o a d h i s t o r y σ(μ)
e(t)
=
f
*
da - u)—du du
D (t c
J
— 00
where D
c
is t h e c r e e p c o m p l i a n c e . E v a l u a t i n g E q u a t i o n 1 f o r a s t e p
f u n c t i o n i n s t r a i n o c c u r r i n g at t i m e z e r o , t h a t i s e(u) e(u)
(2)
= 0 for u < 0 a n d
= € f o r u > 0, o n e o b t a i n s 0
ψ-ΕΛ)
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T h i s e x p e r i m e n t d e f i n e s a n i d e a l stress r e l a x a t i o n test. S i m i l a r l y , a s t e p f u n c t i o n i n stress m a y b e a p p l i e d ^ n a m e l y , a(u) = 0 f o r u < 0 a n d σ(η) = σ f o r u > 0, a n d E q u a t i o n 2 m a y b e i n t e g r a t e d to o b t a i n 0
^
= D (i)
(4)
c
T h i s e x p e r i m e n t d e f i n e s a n i d e a l c r e e p test. C l e a r l y t h e h i s t o r i e s t h a t d e f i n e E a n d D are d i f f e r e n t ; h o w e v e r , these f u n c t i o n s are r e l a t e d b y t h e s i m u l t a n e o u s s o l u t i o n o f E q u a t i o n s 1 a n d 2 to o b t a i n r
c
t J E (t r
- u) D (u) c
du
= t
(5)
0
O n l y for a n e l a s t i c s o l i d w h e r e b o t h Ε a n d D are n o t t i m e d e p e n d e n t d o e s o n e o b t a i n ED = 1. T h e c o n s t a n t s t r a i n o r stress h i s t o r i e s u s e d t o d e f i n e t h e m a t e r i a l r e s p o n s e f u n c t i o n s E (t) a n d D (t) are b u t t w o o f t h e m a n y p o s s i b l e histories o f d e f o r m a t i o n . A n o t h e r h i s t o r y c o m m o n l y u s e d to d e f i n e m a t e r i a l r e s p o n s e is a s i n u s o i d a l s t r a i n (or stress) h i s t o r y . T h i s h i s t o r y r
c
Craver; Polymer Characterization Advances in Chemistry; American Chemical Society: Washington, DC, 1983.
7.
STERNSTEiN
Characterization
of Viscoelastic
Solids
125
is c o n v e n i e n t l y r e p r e s e n t e d u s i n g p h a s o r n o t a t i o n as i n A C e l e c t r i c a l n e t w o r k s , t h a t is = € e
e(u)
0
i(ÛU
(6)
w h e r e e is t h e s t r a i n a m p l i t u d e , ω i s t h e a n g u l a r f r e q u e n c y o f t h e s i n e w a v e ( r a d i a n s p e r s e c o n d ) , a n d u i s h i s t o r i c a l t i m e u ^ t. B y u s i n g E q u a t i o n 6 i n E q u a t i o n 1, t h e s t e a d y state r a t i o o f i n s t a n t a n e o u s s t r e s s to i n s t a n t a n e o u s s t r a i n i s s e e n t o c o n s i s t o f b o t h a n i n - p h a s e a n d a n o u t - o f - p h a s e c o m p o n e n t . T h u s , t h e r a t i o c a n b e e x p r e s s e d as a c o m p l e x or d y n a m i c m o d u l u s E *
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0
alt) ~^y=
E* = E' + iE"
(7)
w h e r e E' r e p r e s e n t s t h e r a t i o o f i n - p h a s e stress t o s t r a i n a n d E " r e p r e s e n t s t h e r a t i o o f o u t - o f - p h a s e stress to s t r a i n . T h e o u t - o f - p h a s e stress l e a d s t h e s t r a i n b y 90° a n d i s t h e r e f o r e r e p r e s e n t e d i n E q u a t i o n 7 as the imaginary part o f E * . A n a l t e r n a t i v e p r o c e d u r e f o r d e v e l o p i n g E q u a t i o n 7 is g i v e n as f o l l o w s . C o n s i d e r t h a t t h e m a t e r i a l m a y b e m o d e l e d as a p a r a l l e l c o m b i n a t i o n o f a s p r i n g a n d a d a s h p o t (the s o - c a l l e d K e l v i n - V o i g h t b o d y ) . F o r t h e s p r i n g t h e stress d e v e l o p e d is p r o p o r t i o n a l to t h e s t r a i n (i.e., H o o k e a n e l a s t i c ) a n d c a n b e w r i t t e n as ^ ( e l a s t i c ) = ke
(8)
w h e r e k is a s p r i n g c o n s t a n t . T h e d a s h p o t r e p r e s e n t s a N e w t o n i a n v i s c o u s r e s p o n s e a n d g i v e s r i s e to a stress p r o p o r t i o n a l to s t r a i n r a t e , namely o-(viscous) = ne
(9)
W h e r e e is t h e s t r a i n rate a n d η is a v i s c o s i t y c o n s t a n t . I f t h e s t r a i n i s sinusoidal, then € = € s i n
