Transient and Dynamic Characterization of Viscoelastic Solids

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7 Transient and Dynamic Characterization of Viscoelastic Solids S. S. S T E R N S T E I N

Downloaded by CORNELL UNIV on October 1, 2016 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0203.ch007

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

the

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

Downloaded by CORNELL UNIV on October 1, 2016 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0203.ch007

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)

(3)

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 *

Downloaded by CORNELL UNIV on October 1, 2016 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0203.ch007

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