K.W. Garrett and J.E. Bailey, The effect of resin failure strain on the tensile properties of ...... Leif. E. Asp and Lars. A. Berglund*. Div. of Polymer Engineering.
DOCTORAL THESIS DIVISION O F P O L Y M E R
1995:178 D ISSN 0348-8373
ENGINEERING ISRN
HLU-TH-T--178-D--SE
Transverse failure initiation in polymer composites LeifE. Asp
TEKNISKA HÖGSKOLAN I LULEA
JL!H
LULEÅ U N I V E R S I T Y O F T E C H N O L O G Y
1995:178 D
DOCTORAL THESIS
ISSN 0348-8373
Division of Polymer Engineering
Transverse failure initiation in polymer composites
Leif B. Asp
A K A D E M I S K
A V H A N D L I N G
som med vederbörligt tillstånd av Tekniska Fakultetsnämnden vid Tekniska Högskolan i Luleå för avläggande av teknologie doktorsexamen kommer att offentligen försvaras i sal A 117 vid LuTH, den 24 november 1995 kl 10 . Avhandlingen försvaras på engelska. 00
Fakultetsopponent: Dr John Morton, Defence Research Agency, England
TEKNISKA HÖGSKOLAN I LULEA
LULEÅ UNIVERSITY OF TECHNOLOGY
To Mom and Dad
I enjoy to get an idea, not to have Lens P. Ås 1995
i
ABSTRACT T r a n s v e r s e f a i l u r e is o n e o f the m o s t i m p o r t a n t f a i l u r e m o d e s i n p o l y m e r c o m p o s i t e s . T h e p h e n o m e n o n o f t e n causes t h e f i r s t d e v i a t i o n s f r o m n o n l i n e a r l a m i n a t e b e h a v i o r . A l s o , i n pressure vessels a n d pipes, f l u i d leakage t h r o u g h a p a t h o f transverse cracks is o f t e n t h e l i m i t i n g d e s i g n c r i t e r i o n . I n the present w o r k , experimental and theoretical studies focused o n the m i c r o m e c h a n i c a l l e v e l h a v e been c a r r i e d o u t . The objective was to i n v e s t i g a t e transverse failure initiation i n the matrix. The other major m e c h a n i s m of failure initiation, f i b e r / m a t r i x debonding, was n o t considered. The triaxial n a t u r e o f the m a t r i x stress state i n glass f i b e r / e p o x y w a s c o n f i r m e d b y f i n i t e e l e m e n t a n a l y s i s . E x p e r i m e n t a l r e s u l t s f o r g l a s s y epoxies s u b j e c t e d
to
c o m p o s i t e - l i k e stress states d e m o n s t r a t e d large r e d u c t i o n s i n s t r a i n t o f a i l u r e as c o m p a r e d w i t h u n i a x i a l l o a d i n g . The t r i a x i a l stress state is t h e r e f o r e b y itself a sufficient explanation f o r the l o w transverse strain to f a i l u r e i n p o l y m e r composites. Plastic y i e l d i n g i n the m a t r i x w a s d e m o n s t r a t e d n o t to b e t h e cause o f f a i l u r e i n i t i a t i o n . I n s t e a d c a v i t y i n d u c e d c r a c k i n g w a s s u g g e s t e d as a f a i l u r e m e c h a n i s m . A c r i t e r i o n w a s p r o p o s e d b a s e d o n a critical v a l u e f o r the
dilatational energy
density. Comparison
with
e x p e r i m e n t a l results f o r epoxies subjected to a v a r i e t y o f m u l t i a x i a l load-cases s u p p o r t e d the c r i t e r i o n . A d d i t i o n a l s u p p o r t w a s o b t a i n e d f r o m c o m p a r i s o n w i t h e x p e r i m e n t a l results i n the l i t e r a t u r e f o r transverse f a i l u r e o f glass f i b e r / e p o x y at d i f f e r e n t f i b e r contents. A l t h o u g h t h e e p o x y m a t r i x w a s d i f f e r e n t f r o m those i n the p r e s e n t s t u d y , g e n e r a l t r e n d s i n d a t a
were
s u p p o r t e d b y p r e d i c t i o n s based o n the c r i t e r i o n a n d f i n i t e e l e m e n t analysis. T h e r m a l r e s i d u a l stresses w e r e f o u n d t o be i m p o r t a n t f o r h i g h f i b e r contents. Based o n the c r i t e r i o n , a conservative estimate o f c o m p o s i t e s t r a i n t o f a i l u r e w a s o b t a i n e d . T h i s is reasonable since the c r i t e r i o n p r e d i c t s i n i t i a t i o n , n o t f i n a l f a i l u r e . Based o n the m o d e l , e f f e c t s f r o m changes i n c o n s t i t u e n t properties were examined i n a parametric f i n i t e element analysis. Fiber m o d u l u s w a s f o u n d to s t r o n g l y i n f l u e n c e transverse f a i l u r e . I n t r o d u c t i o n o f a t h i r d p h a s e i n t e r p h a s e b e t w e e n f i b e r a n d m a t r i x w a s also i n v e s t i g a t e d . B e n e f i c i a l results o n transverse f a i l u r e s t r a i n caused b y m a t r i x i n i t i a t i o n w a s o b s e r v e d f o r t h i n r u b b e r y interphases.
iii
PREFACE D u r i n g t h e years 1992-95 I h a v e h a d the o p p o r t u n i t y t o w o r k i n the d i v i s i o n o f P o l y m e r E n g i n e e r i n g at the d e p a r t m e n t o f M a t e r i a l s a n d M a n u f a c t u r i n g E n g i n e e r i n g . M y w o r k has been i n the area o f f a i l u r e i n i t i a t i o n i n transversely t e n s i l e l o a d e d u n i d i r e c t i o n a l glass f i b e r / e p o x y c o m p o s i t e s .
Initiation of
t r a n s v e r s e cracks i n a u n i d i r e c t i o n a l c o m p o s i t e is, i n d e e d , a c o m p l e x a n d i n t e r e s t i n g p h e n o m e n o n . T h e i n v e s t i g a t i o n has b e e n r e s t r i c t e d to c o n c e r n m a t r i x i n i t i a t e d transverse f a i l u r e , o n l y .
T h e r e are a n u m b e r o f p e o p l e w h o deserves m y g r a t i t u d e as t h e y h a v e been i m p o r t a n t t o m y thesis w o r k . I sincerely t h a n k m y a d v i s o r P r o f e s s o r Lars B e r g l u n d f o r his i n s p i r a t i o n a n d s u p p o r t . Professor B e r g l u n d creates a m a g i c a t m o s p h e r e i n w h i c h research becomes a p l e a s u r e , a n d I a d d i c t e d . A l l m y colleagues at the d i v i s i o n o f P o l y m e r E n g i n e e r i n g a l o n g w i t h m y f e l l o w postgraduates
are
acknowledged
for helping and
for o f f e r i n g me
their
f r i e n d s h i p s . Special t h a n k s are d u e to Professor Peter G u d m u n d s o n , at the R o y a l I n s t i t u t e o f T e c h n o l o g y , a n d Professor Ramesh Talreja, at Georgia Tech. f o r t h e i r scientific c o n t r i b u t i o n s a n d constant interest i n m y w o r k .
F i n a l l y I w o u l d l i k e t o express m y g r a t i t u d e to m y f i a n c é e Pia f o r a l w a y s s u p p o r t i n g m e a n d f o r p u t t i n g u p w i t h m y absence.
L u l e å , September 1995
Leif A s p
iv
LIST OF PAPERS
T h i s thesis is based o n the f o l l o w i n g papers:
I
L . E . A s p , L . A . B e r g l u n d a n d P. G u d m u n d s o n , Effects o f c o m p o s i t e l i k e stress state o n the f r a c t u r e o f epoxies, Comp. Sci. Techn., 53, (1995), p p . 27-37.
II
L . E . A s p a n d L . A . B e r g l u n d , A b i a x i a l t h e r m o - m e c h a n i c a l d i s k test f o r glassy p o l y m e r s , s u b m i t t e d to Exp.
III
Mech.
L . E . A s p , L . A . B e r g l u n d a n d R. Talreja, A c r i t e r i o n f o r crack i n i t i a t i o n i n glassy p o l y m e r s subjected to a composite-like stress state, s u b m i t t e d to Comp. Sci.
IV
Techn.
L . E . A s p , L . A . B e r g l u n d a n d R. Talreja, P r e d i c t i o n o f m a t r i x i n i t i a t e d transverse f a i l u r e i n p o l y m e r composites, s u b m i t t e d to Comp.
Sci.
Techn.
V
L . E . A s p , L . A . B e r g l u n d a n d R. Talreja, Effects of f i b e r a n d i n t e r p h a s e o n m a t r i x i n i t i a t e d transverse f a i l u r e i n p o l y m e r composites, s u b m i t t e d t o Comp. Sci.
Techn.
v
CONTENTS
page ABSTRACT
i
PREFACE
iii
L I S T O F PAPERS
iv
CONTENTS
v
I N T R O D U C T I O N w i t h a s u m m a r y o f the p a p e r s
1
PAPER I
11
Effects o f c o m p o s i t e - l i k e stress state o n the f r a c t u r e o f epoxies.
PAPER I I
A b i a x i a l t h e r m o - m e c h a n i c a l d i s k test f o r
43
glassy p o l y m e r s .
PAPER I I I
A c r i t e r i o n f o r crack i n i t i a t i o n i n glassy p o l y m e r s
69
subjected to a c o m p o s i t e - l i k e stress state.
PAPER I V
P r e d i c t i o n o f m a t r i x i n i t i a t e d transverse f a i l u r e i n
103
p o l y m e r composites.
PAPER V
Effects of f i b e r a n d interphase o n m a t r i x i n i t i a t e d transverse f a i l u r e i n p o l y m e r composites.
135
ion
Asp;
3
Introduction
INTRODUCTION
General Background C o m p o s i t e m a t e r i a l s u s e d f o r aerospace a p p l i c a t i o n s u s u a l l y c o n s i s t o f c o n t i n u o u s h i g h p e r f o r m a n c e f i b e r s , ie c a r b o n f i b e r s , i n a p o l y m e r m a t r i x , eg a t h e r m o s e t s u c h as e p o x y o r a t h e r m o p l a s t i c s u c h as PEEK. These t y p e s o f c o m p o s i t e m a t e r i a l s are c h a r a c t e r i z e d b y h i g h specific l o n g i t u d i n a l s t i f f n e s s a n d s t r e n g t h . H i g h p e r f o r m a n c e composites are m a n u f a c t u r e d b y s t a c k i n g o f t h i n p r e p r e g plies o f u n i d i r e c t i o n a l f i b e r o r i e n t a t i o n . These plies are stacked at d i f f e r e n t f i b e r d i r e c t i o n s t o a l a m i n a t e t h a t meets r e q u i r e d m e c h a n i c a l p r o p e r t y specifications. D u e t o m a t e r i a l h e t e r o g e n e i t y a n d a n i s o t r o p y , cracks p a r a l l e l to t h e f i b e r s w i l l appear i n plies o r i e n t e d p e r p e n d i c u l a r to the l o a d d i r e c t i o n , e v e n at l o w l o a d levels. I n t r a l a m i n a r transverse c r a c k i n g i n o f f - a x i s plies is one o f the f i r s t 1
2
failure modes ' .
Although
final failure of p o l y m e r composites
with
c o n t i n u o u s f i b e r s u s u a l l y i n v o l v e s f i b e r f r a c t u r e , transverse c r a c k i n g is also o f great i m p o r t a n c e . Transverse cracks r e d u c e l a m i n a t e s t i f f n e s s a n d are also k n o w n t o i n i t i a t e other t y p e s o f d a m a g e such as local d e l a m i n a t i o n a n d f i b e r f r a c t u r e . T h e i n v e s t i g a t i o n b y Spencer a n d H u l l
3
on pressurized
glass
f i b e r / p o l y e s t e r p i p e s p r o v i d e s a n e x a m p l e o f h o w transverse cracks f o r m e a r l y i n t h e d e f o r m a t i o n process o f a c o m p o s i t e structure. Onset o f w e e p a g e d u e to transverse cracks o c c u r r e d at transverse strains o f about 0.2 % w h e r e a s f i n a l f a i l u r e o c c u r r e d m u c h later. The effect o f the m a t r i x o n transverse f a i l u r e is o f interest. I n the l i t e r a t u r e , s t u d i e s h a v e c o m p a r e d t h e s t r a i n to f a i l u r e i n transverse t e n s i o n a n d t h e 1
s t r a i n t o f a i l u r e o f the p u r e m a t r i x l o a d e d i n u n i a x i a l t e n s i o n '
4 - 7
. Uniaxial
m a t r i x s t r a i n to f a i l u r e v a r i e d f r o m 1.5 to 70 %. Transverse s t r a i n t o f a i l u r e s o f c o r r e s p o n d i n g f i b e r composites w e r e d r a m a t i c a l l y smaller a n d v a r i e d o n l y i n t h e r a n g e 0.2 to 0.9 %. T h e r e are m a n y e x p l a n a t i o n s f o r the d i s c r e p a n c y i n s t r a i n to f a i l u r e b e t w e e n the transversely l o a d e d composite a n d the u n i a x i a l l y l o a d e d neat resin. Transverse c o m p o s i t e f a i l u r e m a y i n i t i a t e b y d e b o n d i n g d u e to a w e a k interface, b y presence o f v o i d s or i n regions b e t w e e n f i b e r s i n contact w i t h each other. A l s o , presence o f s t i f f f i b e r s causes a t r i a x i a l stress 8
state "
10
8
as w e l l as stress a n d s t r a i n
and B r o u t m a n
1 2
1 1
m a g n i f i c a t i o n i n the m a t r i x . A g a r w a l
p o i n t e d o u t the state o f stress to be the m o s t i m p o r t a n t f a c t o r
i n f l u e n c i n g i n i t i a t i o n o f f a i l u r e . H i g h l y m a g n i f i e d stresses o r t r i a x i a l stresses m a y i n i t i a t e f a i l u r e i n t h e m a t r i x , e v e n at l o w g l o b a l c o m p o s i t e
loads.
T r a n s v e r s e f a i l u r e i n i t i a t e d b y the t r i a x i a l stress state is l i k e l y i n m a t e r i a l s
Asp;
Introduction
4
w i t h a s t r o n g a n d t o u g h f i b e r / m a t r i x i n t e r f a c e . T h i s thesis
concerns
transverse f a i l u r e i n i t i a t e d i n t h e m a t r i x b y presence o f a t r i a x i a l stress state. S e v e r a l s t u d i e s s u g g e s t t h e t r i a x i a l stress state to b e i m p o r t a n t f o r 1 0
1 3
1 4
i n i t i a t i o n o f transverse f a i l u r e i n the m a t r i x at l o w s t r a i n s ' ' . Gaggar a n d B r o u t m a n c a l c u l a t e d a s t r a i n m a g n i f i c a t i o n f r o m the t r i a x i a l stress state i n a 1 0
h o m o g e n e o u s m a t r i x p r o d u c e d b y the i n h i b i t i o n o f Poisson c o n t r a c t i o n . A f a i l u r e c r i t e r i o n based o n d i s t o r t i o n energy t h e o r y w a s chosen. B y use o f t h i s c r i t e r i o n , t h e c a l c u l a t e d s t r a i n t o f a i l u r e o f a d u c t i l e m a t r i x w a s 1.6 % . H o w e v e r , t h e s t r a i n to f a i l u r e f o r a b r i t t l e m a t r i x w a s p r e d i c t e d t o be l a r g e r a n d as h i g h as 3 % . A l s o , t h e analysis o f de K o k et a l .
1 3
is o f interest. F i n i t e
e l e m e n t c a l c u l a t i o n s s h o w h i g h l o c a l strains i n t h e m a t r i x at l o w g l o b a l c o m p o s i t e strains, d u e t o the t r i a x i a l stress state. The v o n M i s e s y i e l d c r i t e r i o n is a p p l i e d a n d l o c a l shear s t r a i n s are s h o w n t o concentrate i n a t h i n b a n d . L o c a l y i e l d i n g occur at l o w g l o b a l strains. The m a t r i x is a s s u m e d t o be i d e a l elasto-plastic. F u r t h e r m o r e , e x p e r i m e n t a l results b y N i c h o l l s
1 4
demonstrate
l o w s t r a i n t o f a i l u r e f o r p o l y m e r s subjected to a b i a x i a l tensile l o a d . I n o r d e r t o i n v e s t i g a t e t h e e f f e c t s f r o m t h e t r i a x i a l stress state, a n u n d e r s t a n d i n g o f t h e stress state i n the m a t r i x o f a t r a n s v e r s e l y l o a d e d c o m p o s i t e is n e e d e d . I n l i t e r a t u r e , stress analyses o n t r a n s v e r s e l y l o a d e d polymer
composites
have
been
performed analytically
8
as 9
well 1 5
1 8
n u m e r i c a l l y , u s i n g f i n i t e d i f f e r e n c e or f i n i t e e l e m e n t m e t h o d s ' " .
as The
stress analyses o f m i c r o m e c h a n i c a l m o d e l s p r o v i d e i n f o r m a t i o n s u c h as; n o r m a l a n d shear stresses, m a x i m u m p r i n c i p a l stress, a n d v o n M i s e s e f f e c t i v e stress. Stress analysis c o m b i n e d w i t h f a i l u r e c r i t e r i a p r e d i c t f a i l u r e i n i t i a t i o n . T h e m a x i m u m p r i n c i p a l stress c r i t e r i o n has b e e n 9
used
for polymer
1 9
c o m p o s i t e s ' . H o w e v e r , since the stress state i n the m a t r i x is h i g h l y t r i a x i a l , the m a x i m u m p r i n c i p a l stress c r i t e r i o n w i l l l e a d to erroneous a n d o p t i m i s t i c p r e d i c t i o n s . T h e v o n M i s e s c r i t e r i o n has also b e e n a p p l i e d t o p o l y m e r 9
1 5
c o m p o s i t e s ' . T h e v o n M i s e s e f f e c t i v e stress based o n the second i n v a r i a n t o f t h e d e v i a t o r i c stress tensor is used to p r e d i c t f a i l u r e i n i t i a t i o n i n r e g i o n s w i t h h i g h shear stresses. A n a l y s i s o f a t r a n s v e r s e l y l o a d e d c o m p o s i t e reveals v a r i a t i o n s i n t h e m a t r i x stress state w i t h p o s i t i o n . Regions w i t h h i g h shear stresses as w e l l as 9
r e g i o n s w i t h h i g h d i l a t a t i o n a l stresses are active i n the p o l y m e r m a t r i x . I t is t h e r e f o r e o f i n t e r e s t t o i n v e s t i g a t e the m e c h a n i c a l b e h a v i o r o f p o l y m e r s s u b j e c t e d t o e i t h e r h i g h d i l a t a t i o n a l or h i g h d i s t o r t i o n a l stresses. H i g h d i s t o r t i o n a l stresses l e a d t o p l a s t i c y i e l d i n g . T h e e f f e c t o f stress state o n 2 0
y i e l d i n g i n glassy p o l y m e r s is w e l l u n d e r s t o o d . For glassy p o l y m e r s , shear
Asp;
d r i v e n y i e l d i n g b u t also c r a z i n g
5
Introduction
2 1
has b e e n e m p h a s i z e d w i t h a t t e n t i o n g i v e n
to the associated i n f l u e n c e o f h y d r o s t a t i c stress. A n u m b e r o f y i e l d a n d craze 2 1
2 4
c r i t e r i a h a v e b e e n p r o p o s e d f o r t h i s p u r p o s e " . T h e present s t u d y w i l l f o c u s o n y i e l d a n d f a i l u r e c r i t e r i a f o r epoxies s u b j e c t e d t o d i f f e r e n t stress states. I n o r d e r to subject glassy p o l y m e r s t o h i g h d i l a t a t i o n a l stresses, m u l t i a x i a l tensile tests are needed. Several m u l t i a x i a l tensile tests h a v e been p r o p o s e d i n 1 4
the l i t e r a t u r e '
2 5 - 2 6
. M ö n c h et a l
2 5
d e v e l o p e d the b i a x i a l t e n s i o n c r u c i f o r m
test m e t h o d w h i c h has been s u c c e s s f u l l y a p p l i e d to m e t a l s a n d composites. H o w e v e r , t h e c r u c i f o r m test is d i f f i c u l t t o p e r f o r m , e s p e c i a l l y f o r b r i t t l e m a t e r i a l s . T h e corners o f the c r u c i f o r m s p e c i m e n act as stress raisers a n d are l i k e l y to i n i t i a t e f r a c t u r e . S u l t a n a n d M c G a r r y
2 6
p e r f o r m e d b i a x i a l tensile tests
o n p r e s s u r i z e d e p o x y tubes. I n t h e i r s t u d y a p r e s s u r i z e d silicone o i l i n s i d e the c y l i n d e r p r o v i d e s the h o o p stress w h i l e a tensile test m a c h i n e applies the axial stress. B o t h i n t h e c r u c i f o r m a n d t h e p r e s s u r i z e d t u b e test, c o m p l i c a t e d e x p e r i m e n t a l set-ups are n e e d e d . A s i m p l e r test m e t h o d w a s suggested b y 1 4
N i c h o l l s . H e a p p l i e d b i a x i a l tensile l o a d i n o r d e r t o investigate the e f f e c t o f the b i a x i a l stress state o n the s t r a i n to f a i l u r e o f neat resins. N i c h o l l s c l a m p e d a s h o r t a n d w i d e s p e c i m e n i n a tensile tester, c r e a t i n g a b i a x i a l stress state. H o w e v e r , t h e stress state is d i f f i c u l t t o a n a l y z e as c l a m p i n g c o n d i t i o n s are c r i t i c a l i n t h i s t y p e o f test. N e v e r t h e l e s s , a l l test m e t h o d s described above s u b j e c t t h e s p e c i m e n s t o b i a x i a l t e n s i l e stress states. T o f u r t h e r e n h a n c e d i l a t a t i o n a l stresses a n d r e d u c e d i s t o r t i o n a l c o n t r i b u t i o n s to the stress f i e l d , t r i a x i a l tensile tests are d e s i r e d . T h e p o k e r - c h i p test m e t h o d subjects the 2 7
2 8
s p e c i m e n t o a t r i a x i a l tensile stress s t a t e ' . I n this test a p o k e r - c h i p s h a p e d s p e c i m e n is b o n d e d b e t w e e n t w o r i g i d c y l i n d r i c a l substrates. L o a d is a p p l i e d i n the d i r e c t i o n o f t h e c y l i n d e r s . A s a consequence, a t r i a x i a l tensile stress state is a c t i v a t e d i n the p o l y m e r specimen. The p o k e r - c h i p test was a p p l i e d t o 2 7
2 8
r u b b e r s i n the late f i f t i e s a n d the s i x t i e s ' . I n the present s t u d y glassy
epoxies
w e r e subjected to an
almost
e q u i t r i a x i a l tensile ( h y d r o s t a t i c tensile) stress state b y t h e p o k e r - c h i p test m e t h o d . Results suggest f a i l u r e t o i n i t i a t e b y c a v i t y - i n d u c e d b r i t t l e f a i l u r e r a t h e r t h a n b y y i e l d i n g . For t h i s reason a f a i l u r e c r i t e r i o n based o n a c r i t i c a l value f o r the dilatational energy density was developed. Predictions made based o n the d i l a t a t i o n a l energy experimental data
9
d e n s i t y c r i t e r i o n are
supported
by
f o r transverse stresses at a n d strains to transverse f a i l u r e
i n i t i a t i o n i n glass f i b e r r e i n f o r c e d epoxies.
Asp;
Introduction
6
Objective of the thesis T h e objective o f the thesis is t o d e t e r m i n e the effects o f the t r i a x i a l stress state i n the m a t r i x o n transverse
f a i l u r e i n i t i a t i o n i n c o n t i n u o u s glass f i b e r
r e i n f o r c e d epoxies.
Summary of the papers T o evaluate the i n f l u e n c e o f t r i a x i a l stresses o n transverse crack i n i t i a t i o n i n t h e m a t r i x o f a p o l y m e r c o m p o s i t e , analysis o f t h e stress state is n e e d e d . I n Paper I a p r e l i m i n a r y stress analysis o f a square ( q u a d r a t i c ) f i b e r d i s t r i b u t i o n is p e r f o r m e d u s i n g F E M ( F i n i t e E l e m e n t M o d e l l i n g ) . The analysis does n o t take r e s i d u a l t h e r m a l stresses i n t o account. A r e g i o n o f h i g h d i l a t a t i o n a l stress is o b s e r v e d at the f i b e r poles, see Paper I . T h i s t r i a x i a l tensile stress state is 2 7
2 8
m i m i c k e d i n the p o k e r - c h i p t e s t ' . T h e p o k e r - c h i p strains to f a i l u r e i n t h e p r i m a r y l o a d i n g d i r e c t i o n w e r e 0.5 to 0.8 % , whereas strains t o f a i l u r e i n t h e u n i a x i a l tests w e r e 1.8 to 7 % . T h e t r i a x i a l stress state i n c o m p o s i t e m a t r i c e s m a y t h e r e f o r e b y itself be a s u f f i c i e n t e x p l a n a t i o n f o r l o w values o f transverse c o m p o s i t e strains t o f a i l u r e . A d d i t i o n a l tests w e r e r e q u i r e d to e v a l u a t e t h e i n f l u e n c e o f d i l a t a t i o n a l stresses o n the s t r e n g t h o f p o l y m e r s . A m e t h o d f o r t e s t i n g glassy p o l y m e r s u n d e r b i a x i a l t e n s i l e l o a d i n g w a s d e v e l o p e d , see Paper I I . I n the test m e t h o d , a d i s k o f e p o x y is b o n d e d b e t w e e n a steel r i n g a n d a steel d i s k . T h e test m e t h o d is d e s i g n e d t o subject the s p e c i m e n t o a b i a x i a l tensile stress state u n d e r c o o l i n g . A n a p p r o x i m a t e a n a l y t i c a l m o d e l w a s d e v e l o p e d f o r stress analysis o f the d i s k u n d e r c o o l i n g . The results f r o m t h i s test i n c o m b i n a t i o n w i t h those f r o m the p o k e r - c h i p test p r o v i d e f a i l u r e d a t a f o r glassy p o l y m e r s subjected t o h i g h d i l a t a t i o n a l stresses. I n P a p e r I I I a d i l a t a t i o n a l e n e r g y d e n s i t y c r i t e r i o n is d e v e l o p e d f o r p r e d i c t i o n of f a i l u r e i n i t i a t i o n i n glassy p o l y m e r s subjected t o a c o m p o s i t e l i k e stress state. I n the analysis o f the p o k e r - c h i p data, a d d i t i o n a l t h e r m a l r e s i d u a l stresses are t a k e n i n t o a c c o u n t . T h i s d i l a t a t i o n a l e n e r g y d e n s i t y c r i t e r i o n is f o u n d to g i v e c r i t i c a l e n e r g y densities f o r a t r i a x i a l c o m p o s i t e - l i k e stress state t h a t correlate w e l l to those o f b i a x i a l tensile l o a d cases. A l s o , as p o l y m e r s are k n o w n to be sensitive to h y d r o s t a t i c pressure, the p o s s i b i l i t y o f y i e l d i n i t i a t e d f a i l u r e is e x a m i n e d . A s a result, y i e l d i n i t i a t e d f a i l u r e i n glassy e p o x i e s subjected t o a c o m p o s i t e - l i k e stress state is r u l e d o u t . T h e r e s u l t s t h e r e f o r e i m p l y t h a t c a v i t y - i n d u c e d b r i t t l e f a i l u r e i n glassy epoxies subjected t o a c o m p o s i t e - l i k e stress state is i n d e e d t a k i n g place a n d can be p r e d i c t e d b y the suggested d i l a t a t i o n a l energy d e n s i t y c r i t e r i o n . N u m e r i c a l analyses o f t h r e e d i f f e r e n t f i b e r d i s t r i b u t i o n g e o m e t r i e s
are
p e r f o r m e d i n Paper I V . The o b j e c t i v e is t o p r e d i c t transverse stress at a n d
Asp;
Introduction
7
s t r a i n t o f a i l u r e i n i t i a t i o n o f u n i d i r e c t i o n a l glass f i b e r / e p o x y ( G F / E P ) c o m p o s i t e s . A l l analyses are r e s t r i c t e d to glass f i b e r c o m p o s i t e s because o f t h e i r i s o t r o p y . A l l analyses i n Paper I V i n c l u d e t h e r m a l r e s i d u a l stresses d u e to differences i n t h e r m a l coefficient of expansion between fiber and matrix. T r a n s v e r s e f a i l u r e i n i t i a t i o n is p r e d i c t e d b y the d i l a t a t i o n a l e n e r g y d e n s i t y a n d t h e v o n Mises y i e l d criteria. I n a l l cases, i n d e p e n d e n t o f f i b e r d i s t r i b u t i o n o r f i b e r v o l u m e f r a c t i o n , the n u m e r i c a l results suggest f a i l u r e to i n i t i a t e d u e t o h i g h d i l a t a t i o n a l e n e r g y d e n s i t y . T h e transverse s t r e n g t h s o f a s q u a r e 9
( q u a d r a t i c ) f i b e r d i s t r i b u t i o n are c o m p a r e d to e x p e r i m e n t a l data b y d e K o k . T h e c o m p a r i s o n s u p p o r t s the a b i l i t y o f the d i l a t a t i o n a l e n e r g y
density
c r i t e r i o n to p r e d i c t transverse
GF/EP
failure initiation i n continuous
composites. Paper V contains m o d e l i n g results based o n FEM-analysis a n d
the
d i l a t a t i o n a l e n e r g y d e n s i t y c r i t e r i o n . The o b j e c t i v e is t o s t u d y e f f e c t s o f c o n s t i t u e n t p r o p e r t i e s o n f a i l u r e i n i t i a t i o n as p r e d i c t e d b y the m o d e l . I t is a s s u m e d t h a t the f i b e r / m a t r i x b o n d r e m a i n s i n t a c t . I n t h e f i r s t p a r t , a p a r a m e t r i c s t u d y is c o n d u c t e d i n o r d e r to d e m o n s t r a t e h o w m e c h a n i c a l p r o p e r t i e s o f the f i b e r s a f f e c t the stress at a n d s t r a i n to f a i l u r e i n i t i a t i o n . S p e c i f i c e x a m p l e s f o r c a r b o n f i b e r a n d glass f i b e r r e i n f o r c e d e p o x y is p r e s e n t e d . I n the second p a r t , a p a r a m e t r i c s t u d y is c o n d u c t e d t o d e m o n s t r a t e h o w m e c h a n i c a l p r o p e r t i e s a n d thickness o f a t h i r d phase i n t e r p h a s e a f f e c t t h e stress at a n d s t r a i n t o f a i l u r e i n i t i a t i o n o f a c o m p o s i t e l o a d e d i n transverse tension. Specific examples of rubber, thermoplastic, a n d
intermediate
m o d u l u s i n t e r p h a s e s are presented. N u m e r i c a l stress analysis b y t h e f i n i t e e l e m e n t m e t h o d is c o n d u c t e d o n a square f i b e r array. T h e v o n M i s e s y i e l d c r i t e r i o n a n d the d i l a t a t i o n a l energy d e n s i t y c r i t e r i o n are a p p l i e d t o locate t h e zones o f y i e l d i n g a n d c a v i t a t i o n - i n d u c e d b r i t t l e f a i l u r e . A l s o , t h e p o s i t i o n o f m a x i m u m r a d i a l stresses at the f i b e r / i n t e r p h a s e a n d i n t e r p h a s e / m a t r i x interfaces are e x a m i n e d . Fiber m o d u l u s is s h o w n t o h a v e a large i n f l u e n c e o n transverse c o m p o s i t e stress at a n d s t r a i n to f a i l u r e i n i t i a t i o n . I n t r o d u c t i o n o f a t h i r d phase interphase b e t w e e n f i b e r a n d m a t r i x is s h o w n to increase stress at a n d s t r a i n t o f a i l u r e i n i t i a t i o n f o r t h i n , l o w m o d u l u s , h i g h Poisson's r a t i o interphase composites.
Hence, i n order to i m p r o v e f a i l u r e
p r o p e r t i e s , a p p l i c a t i o n o f t h i n r u b b e r i n t e r p h a s e s is s u g g e s t e d .
initiation Finally,
p o s i t i o n a n d m o d e o f f a i l u r e i n i t i a t i o n is f o u n d to d e p e n d s t r o n g l y o n f i b e r a n d i n t e r p h a s e properties.
Asp;
Introduction
8
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25. E. M ö n c h a n d D . Galster, A m e t h o d f o r p r o d u c i n g a d e f i n e d u n i f o r m b i a x i a l tensile stress f i e l d , Brit. J. Appl. Phys., 14, (1963), p p . 810-812. 26. J . N . S u l t a n a n d F.J. M c G a r r y , E f f e c t o f r u b b e r p a r t i c l e size o n d e f o r m a t i o n m e c h a n i s m s i n glassy epoxy, Pol. Eng. Sci., 13, (1973), p p . 29-34. 27. A . N . G e n t a n d P.B. L i n d l e y , I n t e r n a l r u p t u r e o f b o n d e d r u b b e r c y l i n d e r s i n t e n s i o n , Proc Roy Soc ( L o n d o n ) 249A, (1959), p p . 195-205. 28. G . H . L i n d s e y , T r i a x i a l f r a c t u r e studies, J. Appl. 4852.
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Asp; Paper I
13
Effects of composite-like stress state on the fracture of epoxies L e i f E. A s p a n d Lars A . B e r g l u n d * Div. of Polymer Engineering Luleå University of Technology S-971 87 Luleå, Sweden
Peter G u d m u n d s o n Department of Solid Mechanics Royal Institute of Technology S-100 44 Stockholm, Sweden
Abstract T h e s t r a i n t o f a i l u r e o f a t r a n s v e r s e l y l o a d e d c o m p o s i t e is m u c h l o w e r t h a n f o r the p u r e m a t r i x i n u n i a x i a l tension. Several studies o f c o m p o s i t e s suggest t h e t r i a x i a l m a t r i x stress state as one o f t h e e x p l a n a t i o n s . I n o r d e r t o i n v e s t i g a t e t h i s e x p e r i m e n t a l l y , a t r i a x i a l tensile test p r e v i o u s l y u s e d f o r r u b b e r s ( p o k e r - c h i p test) w a s s u c c e s s f u l l y a p p l i e d t o f o u r e p o x i e s i n t h e glassy state. T h e chosen s p e c i m e n g e o m e t r y m i m i c k e d the m o s t severe stress state i n the m a t r i x as d e t e r m i n e d b y f i n i t e element analysis o f a t r a n s v e r s e l y l o a d e d glass f i b e r / e p o x y ( G F / E P ) c o m p o s i t e . T h e p o k e r - c h i p s t r a i n s t o f a i l u r e i n t h e p r i m a r y l o a d i n g d i r e c t i o n w e r e 0.5 t o 0.8 % , w h e r e a s u n i a x i a l s t r a i n s t o f a i l u r e w e r e 1.8 to 7 % . T h e t r i a x i a l stress state i n c o m p o s i t e m a t r i c e s m a y t h e r e f o r e b y itself be a s u f f i c i e n t e x p l a n a t i o n f o r l o w v a l u e s o f transverse c o m p o s i t e strains to f a i l u r e .
1. INTRODUCTION F i n a l f a i l u r e o f p o l y m e r composites w i t h c o n t i n u o u s f i b e r s u s u a l l y i n v o l v e s f i b e r f r a c t u r e . H o w e v e r , transverse c r a c k i n g p a r a l l e l to the f i b e r d i r e c t i o n is also o f great i m p o r t a n c e . Transverse cracks reduce l a m i n a t e s t i f f n e s s a n d are also k n o w n to i n i t i a t e o t h e r types o f damage s u c h as l o c a l d e l a m i n a t i o n a n d f i b e r f r a c t u r e . T h e i n v e s t i g a t i o n b y Spencer a n d H u l l o n p r e s s u r i z e d glass f i b e r / p o l y e s t e r p i p e s p r o v i d e s an e x a m p l e o f h o w transverse cracks f o r m 1
e a r l y i n the d e f o r m a t i o n process of a c o m p o s i t e s t r u c t u r e . O n s e t o f w e e p a g e
To whom correspondence should be addressed.
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d u e to transverse cracks o c c u r e d at transverse strains o f a b o u t 0.2 % w h e r e a s f i n a l f a i l u r e o c c u r e d m u c h later. T h e e f f e c t o f the m a t r i x o n transverse f a i l u r e is o f interest. Several studies h a v e c o m p a r e d t h e s t r a i n to f a i l u r e i n transverse t e n s i o n a n d t h e s t r a i n t o 2
6
failure of the pure matrix loaded i n uniaxial tension " . Uniaxial m a t r i x s t r a i n s t o f a i l u r e v a r i e d f r o m 1.5 to 70 %. T r a n s v e r s e s t r a i n t o f a i l u r e s o f c o r r e s p o n d i n g f i b e r composites w e r e d r a m a t i c a l l y smaller a n d v a r i e d o n l y i n the range 0.2 t o 0.9 % . W e suggest a d i v i s i o n of explanations f o r this p h e n o m e n o n i n t o t w o categories. T h e f i r s t c a t e g o r y o f explanations is b a s e d o n t h e m o r e severe stress state i n t h e c o m p o s i t e m a t r i x o r at t h e f i b e r / m a t r i x i n t e r f a c e as c o m p a r e d w i t h t h e u n i a x i a l p u r e m a t r i x case. T h e n o n - u n i f o r m f i b e r d i s t r i b u t i o n i n c o m m e r c i a l l y processed materials m a g n i f i e s t h i s effect. T h e second c a t e g o r y o f e x p l a n a t i o n s is based o n the existence o f m a t e r i a l f l a w s s u c h as v o i d s o r i n t e r f a c i a l d e b o n d s . T h i s e x p l a n a t i o n c a n also i n c l u d e a c o n s i d e r a t i o n o f the stress state i n a composite. A s a s t a r t i n g p o i n t f o r a n analysis o f transverse f a i l u r e , w e c o n s i d e r t w o m a j o r m e c h a n i s m s f o r f a i l u r e i n i t i a t i o n . O n e is f i b e r / m a t r i x i n t e r f a c i a l d e b o n d i n g . D e b o n d i n g is the f i r s t event a n d f i n a l f a i l u r e occurs b y l i n k i n g o f debonded
sites.
Debonding
d u r i n g transverse
loading
has
been
3
d e m o n s t r a t e d f o r glass f i b e r / p o l y e s t e r . T h e o t h e r m a j o r m e c h a n i s m is f a i l u r e i n i t i a t i o n i n t h e m a t r i x . This m e c h a n i s m is m o s t l i k e l y i n m a t e r i a l s w i t h a s t r o n g a n d t o u g h f i b e r / m a t r i x interface. I t is the m e c h a n i s m to w h i c h the present s t u d y relates. A schematic o f the t w o m e c h a n i s m s is presented i n F i g u r e 1. L e t us c o n s i d e r t h e o r e t i c a l treatments o f t r a n s v e r s e f a i l u r e w i t h t h e exception of studies related to failure initiation b y interfacial d e b o n d i n g . Christensen
and
Rinde
analyzed
the
transverse
strength
using
5
m a c r o m e c h a n i c a l f r a c t u r e m e c h a n i c s . A l t h o u g h t h i s a p p r o a c h has s o m e t h e o r e t i c a l j u s t i f i c a t i o n , the i n h e r e n t f l a w size o f the m a t e r i a l is u s e d as a 7
f i t t i n g p a r a m e t e r . T h e a p p r o a c h is n o t applicable t o t o u g h m a t r i c e s . A n o t h e r p r o b l e m is l a c k o f c o n n e c t i o n b e t w e e n s u c h a m a c r o s c o p i c m o d e l a n d m i c r o m e c h a n i c a l m o d e l s . M i c r o m e c h a n i c a l m o d e l s are d e s i r a b l e since t h e y m a y i m p r o v e o u r u n d e r s t a n d i n g of the p h y s i c a l m e c h a n i s m s i n v o l v e d . Kies considered n o n - u n i f o r m strain d i s t r i b u t i o n i n the m a t r i x u s i n g a 8
square a r r a y o f f i b e r s . T h e s t r a i n concentration f a c t o r is a n average q u a n t i t y f o u n d t o be a f u n c t i o n o f f i b r e v o l u m e f r a c t i o n a n d f i b e r a n d r e s i n m o d u l i . F r o m h i s e q u a t i o n , a n u m b e r f o r the strain m a g n i f i c a t i o n can b e o b t a i n e d as a
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f u n c t i o n of fiber v o l u m e fraction. Chamis developed a related model, w h e r e 9
t h e m a g n i f i c a t i o n i n m a t r i x stress d u e to the f i b e r s is e x p r e s s e d . H i s m o d e l relates t h e t r a n s v e r s e c o m p o s i t e s t r e n g t h t o the u n i a x i a l m a t r i x s t r e n g t h . G a r r e t t a n d Bailey u s e d the t h e o r y b y Kies to e x p l a i n the relative i n s e n s i t i v i t y o f t r a n s v e r s e c o m p o s i t e s t r a i n to f a i l u r e t o u n i a x i a l s t r a i n to f a i l u r e o f t h e 4
m a t r i x . T h e m a j o r l i m i t a t i o n o f s u c h approaches is o b v i o u s f r o m the f i n a l result. I t is i n the f o r m o f one average s t r a i n o r stress m a g n i f i c a t i o n n u m b e r , u n a b l e t o express l o c a l changes i n s t r a i n w i t h p o s i t i o n . M o r e a d v a n c e d stress analyses h a v e also been p e r f o r m e d . T i r o s h et al d e t e r m i n e d the stress d i s t r i b u t i o n a r o u n d a single f i b e r e m b e d d e d i n a linear 1 0
elastic m a t r i x . W i t h t h e center o f the f i b e r as a s t a r t i n g p o i n t , the h i g h e s t stress w a s f o u n d at a d i s t a n c e o f 1.2 t i m e s the f i b e r r a d i u s . T h e stress d i s t r i b u t i o n i n a single f i b e r m a t e r i a l is, h o w e v e r , v e r y d i f f e r e n t as c o m p a r e d w i t h a real composite. Greszczuck developed an approximate analytical elasticity s o l u t i o n f o r the stress d i s t r i b u t i o n a r o u n d an i d e a l i z e d d i s t r i b u t i o n 1 1
of f i b e r s . I n a w i d e l y q u o t e d study, A d a m s and Doner used f i n i t e d i f f e r e n c e analysis i n o r d e r t o solve the p l a n e elasticity p r o b l e m f o r a s q u a r e 1 2
array of circular f i b e r s . The purpose, however, was to calculate
the
t r a n s v e r s e m o d u l u s r a t h e r t h a n to e s t i m a t e consequences f o r t r a n s v e r s e failure. G a g g a r a n d B r o u t m a n calculated a s t r a i n m a g n i f i c a t i o n f r o m the t r i a x i a l stress state i n a h o m o g e n e o u s m a t r i x p r o d u c e d b y the i n h i b i t i o n o f P o i s s o n 1 3
c o n t r a c t i o n . A f a i l u r e c r i t e r i o n based o n d i s t o r t i o n e n e r g y t h e o r y
was
chosen. B y use o f this c r i t e r i o n , the calculated t r i a x i a l s t r a i n t o f a i l u r e o f a d u c t i l e m a t r i x w a s 1.6 %. H o w e v e r , the t r i a x i a l s t r a i n t o f a i l u r e f o r a b r i t t l e m a t r i x w a s p r e d i c t e d to be larger a n d as h i g h as 3 %. For transverse f a i l u r e i n i t i a t i o n i n the m a t r i x , the analysis o f D e K o k et al.
1 4
is o f interest. F i n i t e e l e m e n t calculations s h o w h i g h l o c a l strains i n the
m a t r i x at l o w g l o b a l c o m p o s i t e strains. The v o n Mises y i e l d c r i t e r i o n is t h e n a p p l i e d a n d local shear strains are s h o w n to concentrate i n a t h i n b a n d . L o c a l y i e l d i n g occur at l o w g l o b a l strains. The m a t r i x is assumed t o be ideal elastoplastic. S t i l l , n o n e o f the e x i s t i n g m i c r o m e c h a n i c a l theories a n d / o r e x p e r i m e n t a l s t u d i e s are able to s a t i s f a c t o r i l y e x p l a i n the r o l e of the m a t r i x i n transverse c o m p o s i t e f a i l u r e a n d f u r t h e r w o r k is needed. A s p a r t o f t h e m o t i v a t i o n f o r t h e p r e s e n t s t u d y , t r a n s v e r s e f a i l u r e is a s s u m e d t o i n i t i a t e i n the m a t r i x . E x p e r i m e n t a l s t u d i e s s u p p o r t the p o s s i b i l i t y o f such a m e c h a n i s m . Several f r a c t o g r a p h i c i n v e s t i g a t i o n s r e p o r t o n f r a c t u r e surfaces w h e r e the f i b e r s are
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c o v e r e d b y m a t r i x m a t e r i a l , see e.g. Bascom et a l . ( C F / e p o x y ) (CF/PEEK)
1 6
1 5
and Purslow
. I n C F / P E E K , d e b o n d i n g appears to be u n u s u a l d u r i n g
t r a n s v e r s e c r a c k i n g , see F i g u r e 2 w h e r e a crack e v e n p r o p a g a t e d t h r o u g h some c a r b o n
fibers.
Several s t u d i e s suggest the t r i a x i a l m a t r i x stress state t o be i m p o r t a n t f o r 1 3
1 4
i n i t i a t i o n o f transverse f a i l u r e i n the m a t r i x at l o w s t r a i n s ' . T h e e f f e c t o f t r i a x i a l stress states o n p o l y m e r f a i l u r e is t h e r e f o r e o f interest. B i a x i a l 1 7
p o l y m e r tests h a v e been r e p o r t e d t o result i n l o w p o l y m e r s t r a i n t o f a i l u r e . I n t r a n s v e r s e c o m p o s i t e tests, t h e p o s s i b i l i t y o f c r a c k i n i t i a t i o n f r o m a m a t e r i a l f l a w c a n n o t be disregarded. For t h i s reason i t w a s desirable t o f i n d a test m e t h o d w h e r e a p u r e p o l y m e r can be subjected t o a t r i a x i a l stress state. T h e p o k e r - c h i p test is such a m e t h o d a n d w a s a p p l i e d to r u b b e r s b y G e n t and L i n d l e y
1 8
1 9
and by Lindsey .
T h e o b j e c t i v e is to investigate i f the p o k e r - c h i p test is a p p l i c a b l e t o glassy epoxies a n d , i f so, c o m p a r e u n i a x i a l tensile d a t a w i t h t h o s e f r o m t r i a x i a l poker-chip experiments.
P r o v i d e d f a i l u r e i n i t i a t i o n i n t h e m a t r i x is
c o n s i d e r e d , t h e i m p o r t a n c e o f the t r i a x i a l stress state i n the m a t r i x m a y t h e n be e s t i m a t e d .
2. EXPERIMENTAL 2.1. Materials The chemical structures of the material components f o r the f o u r e p o x y systems are p r e s e n t e d i n Figures 3 a n d 4. I n three e p o x y systems, p r e s e n t e d i n F i g u r e 3, the e p o x y c o m p o n e n t is D G E B A , ( D E R 332, D o w C h e m C o ) . E a c h s y s t e m has a d i f f e r e n t c u r i n g agent: (i) D E T A ( D E H 20, D o w C h e m C o ) , (ii) M H P A , ( H Y 917, C i b a G e i g y ) a n d a m e t h y l i m i d a z o l e ( M I ) accelerator, ( D Y 070, C i b a - G e i g y ) , (iii) A P T A , (Jeffamine T-403, Texaco C h e m C o ) . I n F i g u r e 4 the f o u r t h s y s t e m is presented, T G D D M ( M Y 720, C i b a G e i g y ) c u r e d b y D D S , ( H T 976, C i b a - G e i g y ) . T G D D M is a n a r o m a t i c e p o x y a n d D D S is a n a r o m a t i c amine.
2.2. Casting procedure T h e f o u r d i f f e r e n t systems w e r e c a r e f u l l y m i x e d b y h a n d , v a c u u m w a s a p p l i e d t o t h e m i x t u r e s ten m i n u t e s b e f o r e casting. T h e m i x t u r e s w e r e t h e n p o u r e d i n t o a f l u o r o p o l y m e r coated a l u m i n i u m m o l d . M a t e r i a l c o m p o s i t i o n s a n d cure schedules are presented i n Table I .
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2.3. Specimen fabrication and testing methods The cast plates w e r e r e m o v e d f r o m the m o l d a n d m a c h i n e d to the s p e c i m e n d i m e n s i o n s r e q u i r e d f o r m e c h a n i c a l t e s t i n g . T h e specimens d e s i g n e d f o r u n i a x i a l t e s t i n g w e r e m i l l e d t o the d i m e n s i o n s suggested b y A S T M D 6 3 8 M 8 1 , t y p e I I , i n a c o m p u t e r c o n t r o l l e d m i l l i n g m a c h i n e . The strains i n u n i a x i a l tests w e r e m e a s u r e d b y a 50 m m gauge l e n g t h extensometer. A m i n i m u m o f seven specimens o f each m a t e r i a l w a s used. 1 8
1 9
For the m u l t i a x i a l p o k e r - c h i p m e t h o d ' , a t h i n c i r c u l a r s p e c i m e n w a s b o n d e d b e t w e e n t w o a l u m i n i u m r o d s a n d l o a d e d to f a i l u r e , see F i g u r e 5. T h e s p e c i m e n s w e r e f i r s t c u t f r o m t h e p l a t e t o squares 30 m m
x
30 m m a n d
b o n d e d t o t h e a l u m i n i u m r o d s b y 7 3 M OST e p o x y a d h e s i v e f i l m f r o m A m e r i c a n C y a n a m i d C o . P r i o r to b o n d i n g , the a l u m i n i u m w a s d e g r e a s e d a n d etched i n c h r o m i c a c i d . T h e e p o x y surfaces to be b o n d e d w e r e g r o u n d 20
a n d t h e n d e g r e a s e d b y t h e use o f a c e t o n e .
For t h e s t r o n g e s t
epoxy,
D G E B A / A P T A , the e p o x y itself w a s u s e d as an adhesive since the a d h e s i v e f i l m f a i l e d p r e m a t u r e l y . A f t e r b o n d i n g , t h e specimens w e r e g r o u n d a n d p o l i s h e d i n t o c i r c u l a r shape. The d i a m e t e r a n d thickness w e r e 30 a n d 4 m m r e s p e c t i v e l y (aspect r a t i o 7.5). The a l u m i n i u m r o d s h a d a l e n g t h o f 100 m m . Joints i n the f r e e ends o f the r o d s w e r e connected t o g r i p p i n g extensions i n a Dartec test m a c h i n e . T h e strains i n the m u l t i a x i a l tests w e r e m e a s u r e d o v e r a distance o f 50 m m b y a sensitive extensometer ( f u l l r a n g e o f d i s p l a c e m e n t ± 0 . 5 m m ) . T h e s t r a i n to f a i l u r e o f the specimen w a s calculated b y s u b t r a c t i o n o f strains i n a l u m i n i u m r o d s . S t r a i n rates f o r u n i a x i a l tests a n d p o k e r - c h i p tests w e r e 1 % p e r m i n u t e a n d 0.2 % p e r m i n u t e respectively. A l l tests w e r e p e r f o r m e d at a m b i e n t c o n d i t i o n s . For the u n i a x i a l tests, Y o u n g ' s m o d u l u s E w a s d e t e r m i n e d as t h e secant m o d u l u s at 0.9 % s t r a i n . F o r the p o k e r - c h i p tests, t h e a p p a r e n t
tensile
m o d u l u s E a w a s calculated as the secant m o d u l u s at 0.1 % s t r a i n . T h e p o k e r c h i p data i n T a b l e I I I are based o n 30 specimens o u t o f 65. Specimens w i t h f a i l u r e at t h e s p e c i m e n / s u b s t r a t e
interface were discarded. Data f r o m a
m i n i m u m o f 7 specimens o f each m a t e r i a l are r e p o r t e d . For D G E B A / A P T A , 3 o u t o f 7 specimens f a i l e d at the interface. Since these specimens h a d h i g h e r strengths t h a n those w i t h t r u e m a t e r i a l f a i l u r e , the data w a s used.
2.4. Fractography Fracture surfaces w e r e s t u d i e d v i s u a l l y and i n a scanning
electronic
m i c r o s c o p e ( S E M ) . T h e f r a c t u r e surfaces w e r e c o a t e d w i t h c a r b o n / g o l d
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( C / A u ) i n a Blazers S C D 050 s p u t t e r coater. The S E M w a s a C A M S C A N S H 8 0 D , o p e r a t i n g at 30 k V accelerating voltage.
3. THEORETICAL ANALYSIS 3.1. FEM-analysis of transversely loaded composites The F i n i t e E l e m e n t M e t h o d ( F E M ) was used to determine the stress state i n a t r a n s v e r s e l y l o a d e d c o m p o s i t e . T h e c o m m e r c i a l l y available A B A Q U S s y s t e m was used f o r the F E M analysis. The generated
cell c o n t a i n e d a f i b e r
s u r r o u n d e d b y r e s i n a n d w a s a u n i t e l e m e n t i n a square a r r a y o f f i b e r s . E l e m e n t s o f general p l a i n s t r a i n w e r e used to generate the mesh. P e r i o d i c i t y w a s t a k e n i n t o a c c o u n t b y c o u p l i n g o f node-displacements. For nodes o n t h e sides p e r p e n d i c u l a r t o the l o a d i n g d i r e c t i o n , the d i f f e r e n c e d i s p l a c e m e n t o f each o p p o s i t e n o d e - p a i r w a s e q u a l to a n i n i t i a l d i s p l a c e m e n t f a c t o r . T h e r e m a i n i n g t w o sides w e r e k e p t s t r a i g h t a n d free to move. T h e Y o u n g ' s m o d u l u s o f t h e glass f i b e r i n the m o d e l w a s E f = 7 6 G P a , Poisson's r a t i o w a s V f = 0 . 2 . F o r t h e r e s i n E = 3 . 0 GPa a n d V = 0 . 3 4 . T h e m
m
c o m p o s i t e m a t e r i a l i n t h e m o d e l h a d a f i b e r v o l u m e f r a c t i o n o f 50.2 % . C a l c u l a t i o n s r e s u l t e d i n a c o m p o s i t e Poisson's r a t i o o f 0.31, close t o t h e 1 2
r e s u l t s p r e s e n t e d b y A d a m s a n d D o n e r . A t r i a x i a l stress state acts i n t h e m a t r i x o f t h e c o m p o s i t e . The r a t i o o f the stress c o m p o n e n t m a g n i t u d e s is a p p r o x i m a t e l y 1:1:2 (x:y:z), w h e r e z, the largest stress c o m p o n e n t , is i n t h e l o a d i n g d i r e c t i o n . T h i s r a t i o v a r i e s w i t h p o s i t i o n i n the c o m p o s i t e . T h e p r e s e n t e d results r e f e r t o a m a t e r i a l v o l u m e i n the v i c i n i t y o f the f i b e r / m a t r i x i n t e r f a c e at the f i b e r center-line. W i t h i n t h i s v o l u m e , the stress m a g n i f i c a t i o n is a p p r o x i m a t e l y - ^ - = 1 . 8 , w h i c h is close to the results o b t a i n e d b y A d a m s 1 2
a n d D o n e r . T h e l o c a l stress i n the z - d i r e c t i o n at the p o s i t i o n (x:y:z) is h e r e ve
d e n o t e d a a n d the g l o b a l average stress i n the z-direction is 0"° 2
3.2. Triaxial test I d e a l l y , w e w o u l d l i k e a t r i a x i a l test w h i c h m i m i c s the m a t r i x stress state d e t e r m i n e d i n the p r e v i o u s section. A n analysis o f the p o k e r - c h i p load-case i n F i g u r e 5 is t h e r e f o r e p e r f o r m e d . The p o k e r - c h i p test s p e c i m e n is a c i r c u l a r d i s k w i t h l a r g e d i a m e t e r . I t is b o n d e d t o r i g i d substrates a n d t e s t e d i n monotonic 1 8
t e n s i o n . The test
m e t h o d has
previously been
used
for
1 9
r u b b e r s ' . I n the analysis o f reference 18, the specimen w a s a s s u m e d t o b e i n f i n i t e l y t h i n . T h i s i m p l i e s v a n i s h i n g i n - p l a n e strains a n d a n o n - z e r o
Asp; Paper I
19
h o m o g e n e o u s s t r a i n i n t h e z - d i r e c t i o n . W i t h this c o n f i g u r a t i o n the stress f i e l d becomes
E(l-v)
T h i s a n a l y s i s is n o t s u f f i c i e n t f o r a f i n i t e r a t i o b e t w e e n t h i c k n e s s
and
d i a m e t e r . T h i s p r o b l e m w a s addressed b y L i n d s e y , Schapery et a l . f o r t h e purpose of polyurethane
21
1 9
t e s t s . F o r p o l y u r e t h a n e , the stress state w a s
almost p u r e l y h y d r o s t a t i c , 1:1:1 (a :o"y:o" ). T h i s w a s d u e to the Poisson's r a t i o x
2
o f close t o 0.5 a n d the aspect r a t i o (specimen d i a m e t e r to thickness) w h i c h w a s larger t h a n 10. A m o r e d e t a i l e d analysis o f the p o k e r - c h i p m e t h o d w a s also p r e s e n t e d b y L i n d s e y , Schapery et a l
2 1
. F i g u r e 5 s h o w s a circular d i s k o f n o r m a l i z e d r a d i u s
(a) w i t h its axis i n the z - d i r e c t i o n , a n d faces z = ± l . N o r m a l i z a t i o n is w i t h respect to the h a l f thickness ( t / 2 ) o f the disk. N o t i c e that the h a l f thickness is taken as u n i t y , t h u s the aspect r a t i o is equal to the n o r m a l i z e d d i s k r a d i u s (a). The d i s k is a s s u m e d t o be l o a d e d b y i n c r e a s i n g the thickness b y 2e, see F i g u r e 5. T h e f o l l o w i n g e q u a t i o n s w e r e assumed to describe the n o r m a l i z e d r a d i a l a n d a x i a l displacements u a n d w
u=
respectively;
2
-{l-z )-g(r)
W = £-Z w h e r e z is n o r m a l i z e d t h i c k n e s s a n d g ( r ) is a n u n p r e s c r i b e d
(3)
function of
n o r m a l i z e d r a d i a l p o s i t i o n . N o t i c e t h a t b o t h z a n d r are n o r m a l i z e d w i t h respect t o t h e h a l f thickness o f the d i s k . T h e strains c o r r e s p o n d i n g t o these displacements are
20
Asp; Paper I
du
~~dr u £e r dw C
(4 a-d)
£
z ~!z~
=
€
du Yn
^ = 2 (r)-Z or
+
"dl
8
L i n d s e y , Schapery et al. d e t e r m i n e d the f u n c t i o n g(r), f r o m the c o n d i t i o n t h a t the z- i n t e g r a t e d e q u i l i b r i u m e q u a t i o n f o r the r a d i a l d i r e c t i o n is to v a n i s h , a n d u s e d t h e e q u a t i o n s (4a-d) to calculate t h e average stresses t h r o u g h t h e thickness o f t h e s p e c i m e n . T h e analytical s o l u t i o n s of the stress c o m p o n e n t s are s o m e w h a t c o m p l i c a t e d . A s i m p l i f i e d f o r m of the a n a l y t i c a l s o l u t i o n s w a s 2 1
also d e r i v e d . T h e stress c o m p o n e n t s are i n s i m p l i f i e d f o r m
h ^3(1--2v)
3v Ee
Ee
(1+v)
3(1 - 2 v )
h/ „
W (i 3
(5)
-2v)
rV3(l-2v) 1 +-—--^1+— Ee (1+v) 2/ «V (l-2v)
Ee
(6)
3
0
3v
IB..
Ee
(1+v)
(7) 3(1-2v) j
/ [aV3F2v)]J 0
w h e r e a, r, a n d z are n o r m a l i z e d d i s k r a d i u s , r a d i a l p o s i t i o n , a n d p o s i t i o n i n thickness d i r e c t i o n respectively. A l l parameters were n o r m a l i z e d w i t h respect t o t h e h a l f thickness o f the d i s k . E is Y o u n g ' s m o d u l u s , v Poisson's r a t i o , I a n d \ are m o d i f i e d Bessel f u n c t i o n s , (07, o"ø, o 0
z
) are n o r m a l stresses
i n r-, 6-, a n d z - d i r e c t i o n s i n a c y l i n d r i c a l c o o r d i n a t e s y s t e m , x
rz
is the shear
stress i n the r / z - d i r e c t i o n . T h e stress d i s t r i b u t i o n a c c o r d i n g t o equations (5-6) at z=0 is p r e s e n t e d i n F i g u r e 6. E q u a t i o n s (5-7) are s i m p l i f i e d . H o w e v e r , L i n d s e y , S c h a p e r y et a l .
2 1
c o n c l u d e d these a p p r o x i m a t i o n s to be s u f f i c i e n t l y accurate i n the c e n t r a l r e g i o n w h e r e r / a < 0 . 6 . T h e analysis close to the edge is less accurate. F E M analyses b y A d a m s et a l . h a v e s h o w n stress concentrations t o be present at 22
t h e i n t e r f a c e ( z = ± l ) c o r n e r s . Since adhesive f a i l u r e is o f n o interest f o r o u r
21
Asp; Paper I
p u r p o s e , w e n e e d t o establish cohesive f a i l u r e i n i t i a t i o n i n the central r e g i o n , a w a y f r o m the interface. T h e d i f f e r e n c e b e t w e e n the p o k e r - c h i p a n d the related b u t t j o i n t test is t h a t the aspect r a t i o o f the p o k e r - c h i p specimen is m u c h smaller. The p u r p o s e is t o test the p o l y m e r m a t e r i a l rather t h a n the adhesive b o n d strength. F o r each choice o f aspect r a t i o o f the p o k e r - c h i p s p e c i m e n , the stress d i s t r i b u t i o n w i l l be a f u n c t i o n o f the n o r m a l i z e d r a d i a l p o s i t i o n r. F i g u r e 6 s h o w s the stress d i s t r i b u t i o n i n the specimen to be f a i r l y h o m o g e n e o u s f r o m the center to a p p r o x i m a t e l y 60% o f the r a d i a l distance ( r / a = 0 . 6 ) . The z-stress is a l m o s t t w i c e as l a r g e as the other t w o . N o t e t h a t o is n o r m a l i z e d w i t h z
respect to Y o u n g ' s m o d u l u s £ . The average o f the n o r m a l i z e d o
z
therefore
becomes larger t h a n one since E is smaller t h a n the a p p a r e n t m o d u l u s o f the s p e c i m e n , Ea , see e q u a t i o n (8). T h e shear stress is zero at z=0. I t has its m a x i m u m v a l u e at the i n t e r f a c e , z = ± l , b u t i t is s i g n i f i c a n t l y l o w e r t h a n the n o r m a l stress. T h e aspect r a t i o o f e p o x y p o k e r - c h i p specimens w a s chosen as 7.5 w h i c h i n d u c e s a r a t i o o f t h e stresses x:y:z o f close to 1:1:2 i n t h e c e n t r a l r e g i o n . H e n c e , the stress state is s i m i l a r t o t h a t l o c a l l y i n the m a t r i x o f a transversely l o a d e d G F / E P c o m p o s i t e . T h e r m a l stresses generated d u r i n g c o o l - d o w n o f the e p o x y a f t e r b o n d i n g to the a l u m i n i u m substrates at elevated temperature, are n o t i n c l u d e d i n the analysis. T h e i r m a g n i t u d e is estimated to be s i m i l a r i n t h e p o k e r - c h i p s p e c i m e n as i n the c o m p o s i t e . H o w e v e r , i f a q u a n t i t a t i v e f a i l u r e c r i t e r i o n is seeked, t h e r m a l stresses h a v e to be i n c l u d e d . They d o n o t c o n t r i b u t e to t h e a p p a r e n t stress at f a i l u r e , t h e i r e f f e c t is t o increase t h e stresses i n t h e p l a n e
a n d change the p r o p o r t i o n s between
the stress
components. F r o m the p o k e r - c h i p test, the a p p a r e n t m o d u l u s Ea m a y be d e t e r m i n e d as
2n\o
rdr z
a
(8) A
e
m
2
s
w h e r e 0" A is the average stress over the b o n d e d surface a n d e is the a p p l i e d Z
z
s t r a i n i n the z - d i r e c t i o n . N o t i c e t h a t o i n e q u a t i o n (8) is i n its u n s i m p l i f i e d z
f o r m e x p r e s s e d b y L i n d s e y , Schapery et a l e q u a t i o n (8) d i f f e r e n t l y , thus:
2 1
. T h i s can be u s e d t o w r i t e
Asp; Paper I
E
_
A
E
3v
1-
1 + v 3 ( 1 - 2 v)
22
2/,(WM) aVM/ (aVF) 0
2/,(oVM|
(9)
WM/ (WM) 0
1+1+ v a4MI (a-M) 0
1+
l-2v a-fMlJya^M)
where M = — 2 (1-v)
T h e m o d u l u s E can be o b t a i n e d e x p e r i m e n t a l l y f r o m a u n i a x i a l test a n d E A f r o m the p o k e r - c h i p test. E q u a t i o n (9) can t h e n be u s e d to calculate Poisson's ratio.
4. RESULTS AND DISCUSSION 4.1. Uniaxial test A l t h o u g h t h e p r i m a r y i n t e r e s t is i n results f r o m t r i a x i a l tests, r e f e r e n c e results f r o m u n i a x i a l tests are needed. U n i a x i a l tensile test d a t a f o r f o u r epoxies are p r e s e n t e d i n T a b l e I I a n d F i g u r e 7. Strains t o f a i l u r e are i n t h e range 1.8 to 7.0 % . The epoxies c u r e d b y a l i p h a t i c amines s h o w 6 to 7 % s t r a i n to f a i l u r e . B e f o r e f a i l u r e , D G E B A / A P T A s h o w e d s i g n i f i c a n t l o c a l i z e d y i e l d i n g i n the f o r m o f n e c k i n g . For a f e w cases, o u r data d i f f e r f r o m results p r e v i o u s l y r e p o r t e d i n the l i t e r a t u r e , see Table I I , p r o b a b l y d u e to d i f f e r e n c e s i n s p e c i m e n g e o m e t r y , s t r a i n rate a n d / o r s p e c i m e n p r e p a r a t i o n . The behavior
of the
f o u r epoxies
is i n a g r e e m e n t
with
current
u n d e r s t a n d i n g of the effect of molecular structure o n u n i a x i a l tensile response. A s expected f r o m its densely cross-linked n e t w o r k , T G D D M / D D S has l o w s t r a i n t o f a i l u r e , 1.8 % , l o w G i c
a
n
d h i g h T g , see Table I I . T h e
Y o u n g ' s m o d u l u s is the h i g h e s t (3.8 GPa) f o r t h i s system. The h i g h m o d u l u s of densely crosslinked aromatic thermosets secondary bonds
is d u e t o h i g h d e n s i t y o f
f r o m efficient p a c k i n g of the aromatic
2 3
segments .
D G E B A / D E T A has l o w c r o s s l i n k density a n d l o w T g , see Table I I . I t also has t h e l o w e s t m o d u l u s o f t h e i n v e s t i g a t e d s y s t e m s , 2.1 GPa, i n d i c a t i n g inefficient molecular p a c k i n g
2 3
a n d p o s s i b l y s o m e viscoelastic effects. T h e
Asp; Paper I
23
2 4
f r a c t u r e t e x t u r e s are i n a g r e e m e n t w i t h results b y M o r g a n a n d O ' N e a l . D G E B A / D E T A fails after significant d e f o r m a t i o n whereas
TGDDM/DDS
s h o w s b r i t t l e f r a c t u r e at l o w strain. The observed
d i f f e r e n c e s i n u n i a x i a l b e h a v i o r create i n t e r e s t i n a
c o m p a r i s o n b e t w e e n the i n v e s t i g a t e d epoxies i n t r i a x i a l l o a d i n g .
4.2. Triaxial test T h e g e o m e t r y o f the p o k e r - c h i p s p e c i m e n w a s chosen i n o r d e r t o m i m i c the m a t r i x stress state i n a G F / E P c o m p o s i t e subjected to transverse l o a d i n g , as e x p l a i n e d p r e v i o u s l y . T h e p o k e r - c h i p test w a s u s e d f o r n a t u r a l r u b b e r b y Gent a n d L i n d l e y
1 8
1 9
and for polyurethane rubbers b y L i n d s e y . H o w e v e r ,
c o m p a r e d w i t h r u b b e r s , p o l y m e r s i n the glassy state h a v e s i g n i f i c a n t l y h i g h e r s t i f f n e s s a n d s t r e n g t h . H i g h e r stress-levels w e r e t h e r e f o r e expected, a c c o m p a n i e d b y r i s k f o r b o n d f a i l u r e at the i n t e r f a c e b e t w e e n the s p e c i m e n a n d t h e m e t a l substrate. A f t e r some e x p e r i m e n t a t i o n , t h e test p r o c e d u r e described i n the e x p e r i m e n t a l section w a s f o u n d to g i v e satisfactory results. The e x p e r i m e n t a l data are presented i n Table I I I ( w i t h u n i a x i a l data) a n d F i g u r e 8. A l l f o u r epoxies e x h i b i t e d a l i n e a r a p p a r e n t stress-strain r e l a t i o n u n t i l f a i l u r e , as is c o m m o n w i t h transverse c o m p o s i t e data. I t is i m p o r t a n t t o n o t i c e t h a t t h e a p p a r e n t stress is d i f f e r e n t f r o m the t r u e stress, see
the
p r e v i o u s t h e o r e t i c a l discussion. A d r a m a t i c r e d u c t i o n i n s t r e n g t h a n d s t r a i n to f a i l u r e is o b s e r v e d as c o m p a r e d w i t h t y p i c a l u n i a x i a l tensile data. L i n d s e y f o u n d a n increase i n s t r e n g t h a n d a decrease i n s t r a i n t o f a i l u r e o f a r u b b e r , 19
as t h e n u m b e r o f stress-axes i n c r e a s e s . T h e s i g n i f i c a n t d i f f e r e n c e i n Poisson's r a t i o b e t w e e n r u b b e r s a n d epoxies (0.5-0.3) is a n i m p o r t a n t f a c t o r . For t h i s reason, q u a n t i t a t i v e comparisons b e t w e e n e p o x y - r u b b e r data are n o t very meaningful. A s a n i l l u s t r a t i o n o f the stress state e f f e c t o n neat epoxies, the a p p a r e n t t r i a x i a l stress-strain c u r v e a n d u n i a x i a l test data are presented i n F i g u r e 9 f o r DGEBA/MHPA. T h e e p o x i e s c a n be d i v i d e d i n t o t w o g r o u p s w i t h r e s p e c t mechanical
behavior.
Epoxies
cured
with
aliphatic curing
to t h e i r agents,
D G E B A / D E T A a n d D G E B A / A P T A , s h o w s i m i l a r stress-strain b e h a v i o r w i t h s t r a i n s to f a i l u r e a r o u n d 0.8%. E p o x i e s c u r e d w i t h a r o m a t i c a n d c y c l o a l i p h a t i c c u r i n g agents, D G E B A / M H P A a n d T G D D M / D D S , also s h o w e d a l m o s t i d e n t i c a l stress-strain b e h a v i o r . These epoxies f a i l e d at strains o f 0.5 % . T h e l o w e r s t r a i n to f a i l u r e o f m o r e b r i t t l e epoxies is n o t u n e x p e c t e d f o r a c o m p o s i t e - l i k e stress state. D a t a o n transverse s t r a i n t o f a i l u r e i n G F / E P
Asp; Paper I
generally
show
matrices
with
low
24
fracture
toughness,
such
as
D G E B A / M H P A a n d T G D D M / D D S (see T a b l e I I ) , to h a v e l o w e r s t r a i n t o f a i l u r e t h a n t o u g h e r matrices. T y p i c a l d a t a f o r transverse s t r a i n to f a i l u r e i n G F / E P f a l l i n the range 0.30.8 % i f t h e f i b e r v o l u m e f r a c t i o n is 0.5-0.6. T h e s i m i l a r i t y i n stress states b e t w e e n t h e p o k e r - c h i p s p e c i m e n a n d t h e m o s t severely stressed p o s i t i o n i n the m a t r i x is a p r o b a b l e e x p l a n a t i o n f o r the fact t h a t p o k e r - c h i p data (0.5-0.8 % ) correlate w i t h c o m p o s i t e data f o r this case. P o k e r - c h i p d a t a c a n b e u s e d i n E q u a t i o n (1) i n o r d e r t o e s t i m a t e t h e Poisson's r a t i o s o f t h e f o u r epoxies. Poisson's ratios b e t w e e n 0.28 a n d 0.36 r e s u l t f r o m s u c h c a l c u l a t i o n s i f the a p p a r e n t m o d u l u s is d e t e r m i n e d as t h e secant m o d u l u s at 0.1 % s t r a i n f r o m the p o k e r - c h i p test data. A m o r e c a r e f u l e v a l u a t i o n is n o t m e a n i n g f u l since t h e c a l c u l a t e d Poisson's r a t i o is v e r y sensitive t o s m a l l v a r i a t i o n s i n the measured m o d u l i . I t is s t i l l e n c o u r a g i n g t o f i n d c a l c u l a t e d v a l u e s i n agreement w i t h e p o x y data w h i c h w e r e i n the range 0.32-0.35.
4.3. Fracture mechanisms in triaxial test I n o r d e r t o f u r t h e r v e r i f y the accuracy o f the test m e t h o d , crack i n i t i a t i o n at the m e t a l / p o l y m e r i n t e r f a c e needs to be e x c l u d e d . Fractographic studies m a y be u s e d t o c o n f i r m c r a c k i n i t i a t i o n i n the i n t e r i o r r a t h e r t h a n at e i t h e r t h e edge o r t h e s u b s t r a t e / s p e c i m e n interface. T y p i c a l f r a c t u r e surfaces o f t h e f o u r epoxies are presented i n F i g u r e 10. F r a c t u r e surfaces o f the t w o epoxies w i t h the h i g h e s t f r a c t u r e toughnesses (see T a b l e I I ) are p r e s e n t e d
o n the l e f t h a n d side, D G E B A / D E T A
and
D G E B A / A P T A ( a l i p h a t i c c u r i n g agents). For these m a t e r i a l s , a l l r e p o r t e d data are f o r f a i l u r e i n i t i a t i o n i n the central, i n t e r i o r r e g i o n of the specimens as c o n c l u d e d f r o m f r a c t o g r a p h y studies. Characteristic "river" m a r k i n g s o n the fracture surfaces
1 6
c o n f i r m e d crack n u c l e a t i o n a n d g r o w t h w i t h i n t h e
s p e c i m e n , see F i g u r e 1 1 . T o b e a b s o l u t e l y sure, f o r D G E B A / D E T A a t o t a l o f 23 s p e c i m e n s w e r e t e s t e d w h e r e c e n t r a l , i n t e r i o r f a i l u r e i n i t i a t i o n w a s confirmed. T w o d i f f e r e n t t y p e s o f f r a c t u r e surfaces w e r e f o u n d i n D G E B A / D E T A . O n e s h o w e d cracks t o coalesce f r o m several i n i t i a t i o n p o i n t s . T h e o t h e r s h o w e d t h e d e v e l o p m e n t o f one single crack f r o m one i n i t i a t i o n p o i n t , as i n DGEBA/APTA. T h e r e l a t e d b u t t j o i n t tests h a v e m u c h h i g h e r s p e c i m e n aspect ratios a n d h e r e f a i l u r e c o m m o n l y i n i t i a t e s at t h e edge o f t h e
specimen/substrate
Asp; Paper I
25
i n t e r f a c e . A d a m s et a l . r e p o r t e d b u t t j o i n t s o f b r i t t l e e p o x y systems to s h o w a d h e s i v e f a i l u r e at t h e i n t e r f a c e c o r n e r s w h e r e a s b u t t j o i n t s m a d e o f 2 5
plasticised e p o x y f a i l e d a w a y f r o m the c i r c u m f e r e n c e . They
observed
f a i l u r e i n i t i a t i o n at o r close t o the interface f o r b o t h materials. I n a p r e v i o u s p a p e r t h e y s h o w e d t h a t n o stress c o n c e n t r a t i o n s are t o be expected at t h e 2 2
i n t e r f a c e i n t h e c e n t r a l parts o f the b u t t j o i n t s . W i t h a t o u g h adhesive, t h e 2 5
s e n s i t i v i t y t o stress concentrations is l o w e r e d . I n the present i n v e s t i g a t i o n , a t o u g h a d h e s i v e f i l m w a s t h e r e f o r e u s e d t o b o n d the p o k e r - c h i p specimens t o the substrate. I n h i s e a r l y p o k e r - c h i p s t u d y , L i n d s e y presents p o l y u r e t h a n e f r a c t u r e 1 9
surfaces m u c h l i k e those o f D G E B A / D E T A . I n b o t h materials, t h e c e n t r a l p a r t o f t h e s u r f a c e has a large n u m b e r o f n u c l e a t i o n p o i n t s a n d crack g r o w t h is p r e s u m a b l y s l o w . O u t s i d e the coarser central r e g i o n , n o n u c l e a t i o n p o i n t s are present, as expected f o r the f i n a l stages o f crack g r o w t h . Parabolic m a r k s i n d i c a t e a n i n i t i a l crack g r o w t h d i r e c t i o n f r o m the center r a d i a l l y t o w a r d s t h e edge. G e n t a n d L i n d l e y observed i n t e r n a l cavities at stresses as l o w as 15% o f 1 8
the f i n a l s t r e n g t h . T h e cavities g r e w u n t i l t h e y f i l l e d the entire thickness o f the specimen. L i n d s e y
1 9
studied failure mechanisms t h r o u g h transparent
substrates d u r i n g l o a d i n g o f p o l y u r e t h a n e r u b b e r s . H i s results s h o w c a v i t y i n i t i a t i o n i n t h e v i c i n i t y o f the c e n t r a l r e g i o n . T h e cavities g r o w t o f i l l t h e entire thickness o f the specimen. A t t h i s stage t w o sharp cracks appear at t h e e x t r e m i t i e s o f a b u b b l e a n d p r o p a g a t e p e r p e n d i c u l a r to the d i r e c t i o n o f m a x i m u m principal load. F o r t h e t w o epoxies o n the r i g h t h a n d side i n F i g u r e 10, D G E B A / M H P A a n d T G D D M / D D S (aromatic a n d c y c l o a l i p h a t i c c u r i n g agents), the f r a c t u r e s u r f a c e s are v e r y r o u g h . I n s e v e r a l s p e c i m e n s , i n i t i a t i o n p o i n t s
were
i d e n t i f i e d close t o , b u t n o t at the interface, see D G E B A / M H P A i n F i g u r e 10. T h e crack a p p e a r e d to r a p i d l y h a v e g r o w n at a s m a l l angle w i t h respect t o the z - d i r e c t i o n a n d t h e n to h a v e f o l l o w e d the interface. H o w e v e r , f o r m o s t D G E B A / M H P A a n d T G D D M / D D S specimens, i n i t i a t i o n p o i n t s c o u l d n o t be l o c a t e d at a l l . T h e e v i d e n c e f o r crack i n i t i a t i o n i n t h e i n t e r i o r o f these m a t e r i a l s is t h e r e f o r e n o t conclusive. The d i f f i c u l t y to locate i n i t i a t i o n p o i n t s is
related
to
the
low fracture
toughness
of D G E B A / M H P A
and
T G D D M / D D S , see Table I I . S m a l l sizes o f m i r r o r - a n d s m o o t h zones i n 2 6
glassy m a t e r i a l s h a v e been c o r r e l a t e d w i t h l o w f r a c t u r e t o u g h n e s s . F o r cases w h e r e w e c o u l d locate these zones, t h e i r size w a s m u c h smaller t h a n f o r the o t h e r epoxies.
Asp; Paper I
26
F r a c t o g r a p h y d e m o n s t r a t e s c r a c k i n i t i a t i o n t o take place i n t h e c e n t r a l , i n t e r i o r r e g i o n o f t h e specimens f o r D G E B A / D E T A a n d D G E B A / A P T A . F o r b r i t t l e epoxies, T G D D M / D D S a n d D G E B A / M H P A , several specimens s h o w i n i t i a t i o n p o i n t s i n t h e c e n t r a l , i n t e r i o r r e g i o n b u t t h e e v i d e n c e is n o t conclusive.
5. CONCLUSIONS P r e v i o u s s t u d i e s suggest the t r i a x i a l m a t r i x stress state to be i m p o r t a n t f o r 1 3
1 4
t h e l o w v a l u e s o f transverse c o m p o s i t e strains t o f a i l u r e ' . H o w e v e r , i n s u f f i c i e n t e x p e r i m e n t a l data are a v a i l a b l e to v e r i f y t h i s . A t r i a x i a l tensile test p r e v i o u s l y u s e d f o r r u b b e r s ( p o k e r - c h i p test) w a s t h e r e f o r e a p p l i e d to f o u r e p o x i e s i n t h e glassy state a n d p r o v e d s u c c e s s f u l . T h e s p e c i m e n g e o m e t r y m i m i c s the m o s t severe stress state i n t h e m a t r i x . This stress state is chosen o n t h e basis of f i n i t e e l e m e n t analysis o f a transversely l o a d e d G F / E P c o m p o s i t e (square a r r a y o f f i b e r s , f i b e r v o l u m e f r a c t i o n 0.5). T h e t r i a x i a l strains t o f a i l u r e i n the p r i m a r y l o a d i n g d i r e c t i o n are 0.5 t o 0.8 % w h e r e a s u n i a x i a l strains t o f a i l u r e are i n the range 1.8 to 7 % . T h e t r i a x i a l stress state i n c o m p o s i t e m a t r i c e s m a y t h e r e f o r e b y itself be a s u f f i c i e n t e x p l a n a t i o n f o r the l o w transverse c o m p o s i t e strains to f a i l u r e . F i b e r / m a t r i x d e b o n d i n g , p r e e x i s t i n g m a t e r i a l f l a w s a n d n o n - u n i f o r m f i b e r d i s t r i b u t i o n are l i k e l y to f u r t h e r r e d u c e i n i t i a t i o n v a l u e s o f transverse s t r a i n to f a i l u r e . A l t h o u g h n u m e r o u s a t t e m p t s are available i n the l i t e r a t u r e , u n i a x i a l m a t r i x s t r a i n to f a i l u r e c a n n o t be expected to g e n e r a l l y correlate w i t h i n i t i a t i o n v a l u e s o f transverse s t r a i n t o f a i l u r e . Differences i n stress state a n d the g e o m e t r y o f the p r o b l e m w i l l r e s u l t i n vastly d i f f e r e n t f a i l u r e mechanisms.
Acknowledgements M r . J o h n n y G r a h n is g r a t e f u l l y a c k n o w l e d g e d f o r h i s S E M - w o r k . T h e s t u d y w a s p a r t i a l l y f i n a n c e d b y the S w e d i s h N a t i o n a l Board f o r Technical D e v e l o p m e n t ( N U T E K ) and T h e S w e d i s h I n s t i t u t e o f C o m p o s i t e s ( S I C O M P ) .
27
Asp; Paper I
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Aerospace
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Applied to Brittle Materials, A S T M STP 678, e d . S.W. F r e i m a n . A m e r i c a n Society f o r T e s t i n g a n d M a t e r i a l s , P h i l a d e l p h i a , P A , (1980), p p . 136-150. 27. H . Z h a n g a n d L . A . B e r g l u n d , D e f o r m a t i o n a n d f r a c t u r e o f glass b e a d / C T B N - r u b b e r / e p o x y composites, Pol Eng. Sci., 33, (1993), p p . 100-107. 28. G . L u b i n , Handbook Y o r k , (1982).
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TABLES
T a b l e I . C u r e schedule a n d m a t e r i a l c o m p o s i t i o n . Resin system Cure Post-cure DGEBA/DETA DGEBA/MHPA DGEBA/AFTA TGDDM/DDS
2 h / 2 0 °C 2 h / 1 1 0 °C 16 h / 6 0 °C 4 h / 1 5 0 °C
Material Composition (percent by weight) 11.9 % 95,4%,(0.5% MI)
24 h / 1 0 2 ° C 6 h / 140 ° C l h /
44.8% 44%
200°C
T a b l e I I . E x p e r i m e n t a l results a n d s t a n d a r d d e v i a t i o n s f r o m u n i a x i a l tensile 2 4
tests. o"u is the n o m i n a l u l t i m a t e stress. L i t e r a t u r e data: DGEBA/MHPA MATERIAL
2 7
2 8
, DGEBA/APTA "
DGEBA/DETA
3 0
and T G D D M / D D S
DGEBA/MHPA
3 4
DGEBA/DETA - , 2 4
DGEBA/APTA
'
3 1
"
3 3
.
TGDDM/DDS
w E(GPa) present E(GPa) [literature]
2.07±0.15
2.92±0.12
2.93+0.13
3.77±0.07
.
3.08
3.24
4.28
O (MPa) present
69.0+5.4
85.9+3.8
73.1±1.2*
59.9+12.7
O (MPa) [literature]
82
84
73
42-65
e (%) present
7.00±1.50
6.50+1.00
6.14±0.53
1.77+0.44
e (%) [literature]
14
3.1
4.8
1.34
130±20
110
300
69-95
107
150
u
u
u
u
2
Gic O/m ) [literature] Tg(°C) [literature]
93 *Yield stress
240
T a b l e I I I . Fracture data a n d s t a n d a r d d e v i a t i o n s f r o m t r i a x i a l a n d u n i a x i a l tests. Poker-chip Uniaxial Uniaxial Poker-chip Resin system strain to failure strain to failure strength strength (MPa) (MPa) (%) (%) (i) DGEBA/DETA 69.0+5,4 29.1+4.5 7.00+1.5 0.85±0.1 (ii) DGEBA/MHPA 85.9+3.8 26.9±5.7 6.50+1.0 0.57+0.2 (iii) DGEB A / APTA 73.1+1.2* 32.0+2.2 6.14+0.5 0.79+0.1 (iv) TGDDM/DDS 59.9±12 26.6+7.7 1.77+0.4 0.55+0.2 * Y i e l d stress
Asp; Paper I
31
FIGURE CAPTIONS
F i g u r e 1. S c h e m a t i c o f i n i t i a t i o n m e c h a n i s m s f o r t r a n s v e r s e c r a c k s a) Transverse crack i n i t i a t i o n i n the m a t r i x , b) Transverse crack i n i t i a t i o n at the f i b e r / m a t r i x interface. F i g u r e 2. S E M - m i c r o g r a p h o f c a r b o n f i b e r / P E E K cross-section i n cross-ply l a m i n a t e . L o a d a p p l i e d i n the h o r i z o n t a l d i r e c t i o n . M i c r o g r a p h o b t a i n e d b y P r o f R. Talreja, G e o r g i a Inst of T e c h n o l o g y , U S A a n d c o w o r k e r s . Figure 3. C h e m i c a l structures of the c o m p o n e n t s i n the three epoxies based o n D G E B A . T h e e p o x i e s are D G E B A / D E T A , D G E B A / M H P A , and D G E B A / A P T A . I n (iii), X + Y + Z = 5 . 3 . Figure 4. C h e m i c a l structures of the c o m p o n e n t s i n T G D D M / D D S . F i g u r e 5. Schematic o f the p o k e r - c h i p test set-up a n d the c o o r d i n a t e system f o r t h e stress analysis. T h e 8-direction is p e r p e n d i c u l a r to the r - a n d zdirections F i g u r e 6. C a l c u l a t e d n o r m a l stress, o a n d i n - p l a n e stresses, 0" a n d ae f o r D G E B A / D E T A as a f u n c t i o n of r a d i a l p o s i t i o n r / a , a c c o r d i n g to equations (5) a n d (6), aspect r a t i o ( d i a m e t e r / t h i c k n e s s ) o f 7.5. Z /
R
F i g u r e 7. U n i a x i a l stress-strain curves f o r f o u r epoxies. F i g u r e 8. A p p a r e n t n o m i n a l stress v e r s u s s t r a i n i n t h e z - d i r e c t i o n f o r f o u r p o k e r - c h i p epoxies subjected to a composite-like t r i a x i a l stress state. F i g u r e 9. Stress-strain curves f o r D G E B A / M H P A i n u n i a x i a l a n d t r i a x i a l loading. F i g u r e 10. P h o t o g r a p h s o f t y p i c a l f r a c t u r e s u r f a c e s f o r f o u r p o k e r - c h i p specimens o f d i f f e r e n t epoxies. F i g u r e 11. S E M - m i c r o g r a p h s of f r a c t u r e surfaces f o r a) D G E B A / D E T A a n d b) DGEBA/APTA.
F i g u r e 1. S c h e m a t i c o f i n i t i a t i o n m e c h a n i s m s f o r t r a n s v e r s e c r a c k s a) T r a n s v e r s e c r a c k i n i t i a t i o n i n the m a t r i x , b ) Transverse crack i n i t i a t i o n at t h e f i b e r / m a t r i x interface.
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F i g u r e 2. S E M - m i c r o g r a p h of c a r b o n f i b e r / P E E K cross-section i n cross-ply l a m i n a t e . L o a d a p p l i e d i n the h o r i z o n t a l d i r e c t i o n . M i c r o g r a p h o b t a i n e d b y P r o f R. T a l r e j a , Georgia I n s t of Technology, U S A a n d c o w o r k e r s .
34
Asp; Paper I
Diglycidyl ether of bisphenol A, DGEBA
C K
-CH-CH-O^Q)-
V
C - Q k ) - C H - C H - C H
r.w_
2
O
(i) Diethylene triamine, DETA
H N - C H - C H - N H - C H 2 C H NH 2
2
2
2
2
(ii) Methyltetra hydrophtalic anhydride, MHPA
(iii) Polyoxy propyleneamine, APTA
CH [OCh1 CH(CH3)] NH 2
2
x
Cht3CH CCH[OCH CH(CI-yLNH 2
2
CH [OCH CH(CK )] NH 2
2
3
z
2
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Figure 3. C h e m i c a l structures o f the c o m p o n e n t s i n the three epoxies based o n D G E B A . T h e e p o x i e s are D G E B A / D E T A , DGEBA/MHPA, and D G E B A / A P T A . I n (iii), X+Y+Z=5.3.
Asp; Paper I
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Tetraglycidyl 4,4' diaminodiphenyl methane epoxy, TGDDM
C H \
2
- C H - C H / /
\
=
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2
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\
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y
X
2
C H - C H - C H 2
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o 4,4' Diaminodiphenyl sulfone, DDS
F i g u r e 4. C h e m i c a l structures o f the c o m p o n e n t s i n T G D D M / D D S .
0 2
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APPLIED LOAD
RIGID GRIPS
r,
u
F i g u r e 5. Schematic o f the p o k e r - c h i p test set-up a n d t h e c o o r d i n a t e s y s t e m f o r t h e stress analysis. T h e 9 - d i r e c t i o n is p e r p e n d i c u l a r t o the r - a n d zdirections
37
Asp; Paper I
F i g u r e 6. C a l c u l a t e d n o r m a l stress, 0" a n d in-plane stresses, 0" a n d a e f o r D G E B A / D E T A as a f u n c t i o n o f r a d i a l p o s i t i o n r / a , a c c o r d i n g t o equations (5) a n d (6), aspect r a t i o ( d i a m e t e r / t h i c k n e s s ) o f 7.5. Z/
R
Asp; Paper I
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100
0
2
4
6
Strain, £ (%) F i g u r e 7. U n i a x i a l stress-strain curves f o r f o u r epoxies.
8
39
Asp; Paper I
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O O
A
A
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06
0.0
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0.6
0.8
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Strain, £ (%) F i g u r e 8. A p p a r e n t n o m i n a l stress versus s t r a i n i n the z - d i r e c t i o n f o r f o u r p o k e r - c h i p epoxies subjected t o a composite-like t r i a x i a l stress state.
40
Asp; Paper I
100
0
2
4
6
8
Strain,£ (%) F i g u r e 9. Stress-strain c u r v e s f o r D G E B A / M H P A i n u n i a x i a l a n d t r i a x i a l loading.
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F i g u r e 10. P h o t o g r a p h s o f t y p i c a l f r a c t u r e surfaces f o r f o u r p o k e r - c h i p specimens o f d i f f e r e n t epoxies.
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F i g u r e 11. S E M - m i c r o g r a p h s of f r a c t u r e surfaces f o r a) D G E B A / D E T A a n d b ) DGEBA/APTA.
Pa
P e r /;
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A biaxial thermo-mechanical disk test for glassy polymers
L e i f . E. A s p a n d Lars. A . B e r g l u n d * Div. of Polymer Engineering Luleå University of Technology S-971 87 Luleå, Sweden
Abstract F a i l u r e c r i t e r i a f o r p o l y m e r s need to i n c l u d e effects f r o m the stress state. F o r t h i s reason, b i a x i a l test results are o f interest. H o w e v e r , b i a x i a l test m e t h o d s u s u a l l y r e q u i r e expensive e q u i p m e n t . I n the presented test m e t h o d , a d i s k o f e p o x y is b o n d e d b e t w e e n a steel r i n g a n d a steel disk. The t e m p e r a t u r e is t h e n l o w e r e d u n t i l f r a c t u r e is o b s e r v e d . E x p e r i m e n t s w e r e p e r f o r m e d o n t h r e e d i f f e r e n t glassy e p o x y p o l y m e r s . The b i a x i a l stress state w a s a n a l y z e d b y f i n i t e element analysis a n d b y a n a p p r o x i m a t e a n a l y t i c a l m o d e l . E x p e r i m e n t a l o b s e r v a t i o n s s u p p o r t t h e a b i l i t y o f the m e t h o d t o p r o v i d e m a t e r i a l p r o p e r t y data. A n a p p r o x i m a t e a n a l y t i c a l m o d e l w a s f o u n d s u f f i c i e n t l y accurate f o r stress analysis a n d d e t e r m i n a t i o n o f the stress state at f a i l u r e .
1. INTRODUCTION The development of i m p r o v e d p o l y m e r composite materials
requires
u n d e r s t a n d i n g o f the r o l e o f m a t e r i a l c o n s t i t u e n t s d u r i n g f a i l u r e processes. L o a d i n g o f the m a t e r i a l i n the w e a k d i r e c t i o n transverse to the f i b e r d i r e c t i o n is c o m m o n l y associated w i t h f r a c t u r e at v e r y l o w stresses a n d strains. For t h i s l o a d i n g case, the p o l y m e r m a t r i x is subjected to a t r i a x i a l stress state at t h e 1
5
microlevel " . 5
I n recent w o r k , the h i g h y t r i a x i a l n a t u r e o f the stress state i n t r a n s v e r s e l y loaded
glass f i b e r / e p o x y w a s
evaluated
b y finite element
analysis.
E x p e r i m e n t a l results f r o m t r i a x i a l tests o n epoxies demonstrated s i g n i f i c a n t l y l o w e r e d s t r a i n - t o - f a i l u r e u n d e r these c o n d i t i o n s as c o m p a r e d w i t h u n i a x i a l 5
r e s u l t s . A c r i t e r i o n f o r f a i l u r e i n i t i a t i o n u n d e r a r b i t r a r y stress state t h e r e f o r e n e e d s t o be d e v e l o p e d . I n t h i s c o n t e x t , t h e n e e d f o r m u l t i a x i a l tests w i t h d i f f e r e n t stress states is a p p a r e n t . A n y p r o p o s e d c r i t e r i o n c o u l d t h e n be c o m p a r e d w i t h e x p e r i m e n t a l data f r o m a v a r i e t y of l o a d i n g cases.
To whom correspondence should be addressed.
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3
6
8
Several b i a x i a l tensile tests h a v e b e e n p r o p o s e d i n l i t e r a t u r e ' " . M ö n c h et al
6
d e v e l o p e d the b i a x i a l t e n s i o n c r u c i f o r m test m e t h o d w h i c h has b e e n
s u c c e s s f u l l y a p p l i e d to metals a n d composites. H o w e v e r , the c r u c i f o r m test is d i f f i c u l t t o a p p l y to b r i t t l e m a t e r i a l s . The corners o f the c r u c i f o r m s p e c i m e n act as stress raisers a n d are l i k e l y t o i n i t i a t e f r a c t u r e . S u l t a n a n d M c G a r r y
8
p e r f o r m e d b i a x i a l tensile tests o n p r e s s u r i z e d e p o x y tubes. I n t h e i r s t u d y a p r e s s u r i z e d s i l i c o n e o i l i n s i d e the c y l i n d e r p r o v i d e s the h o o p stress w h i l e a tensile test m a c h i n e a p p l i e s the a x i a l stress. B o t h i n the c r u c i f o r m a n d the p r e s s u r i z e d t u b e test c o m p l i c a t e d e x p e r i m e n t a l set-ups are needed. A s i m p l e r 3
test m e t h o d w a s suggested b y N i c h o l l s . H e a p p l i e d b i a x i a l tensile l o a d to i n v e s t i g a t e the e f f e c t o f b i a x i a l stress o n the s t r a i n t o f a i l u r e o f neat resins. N i c h o l l s c l a m p e d a s h o r t a n d w i d e s p e c i m e n i n a tensile tester, c l a i m i n g a b i a x i a l stress state to be active. H o w e v e r , the stress state is d i f f i c u l t t o analyze as c l a m p i n g c o n d i t i o n s are c r i t i c a l i n t h i s t y p e o f test. I n t h e p r e s e n t s t u d y a m e t h o d f o r t e s t i n g glassy p o l y m e r s u n d e r b i a x i a l tensile l o a d i n g is d e v e l o p e d . F i n i t e element analysis is p e r f o r m e d i n o r d e r t o d e t e r m i n e t h e stress state at t h e m i c r o l e v e l a n d t o locate sites o f stress concentrations. A l s o , a n a p p r o x i m a t e a n a l y t i c a l m o d e l is i n v e s t i g a t e d i n o r d e r to evaluate i f the stress state c a n be estimated b y a s i m p l i f i e d analysis.
2. EXPERIMENTAL 2.1. Materials T h r e e e p o x y systems w e r e tested. I n t w o o f t h e e p o x y systems, the e p o x y c o m p o n e n t is D G E B A , d i g l y c i d y l ether o f b i s p h e n o l ( D E R 332, D o w C h e m C o ) . T h e D G E B A is c u r e d b y t w o d i f f e r e n t c u r i n g agent: (i)
DET A,
d i e t h y l e n e t r i a m i n e ( D E H 20, D o w C h e m C o ) a n d (ii) A P T A ,
polyoxy
p r o p y l e n e a m i n e , ( J e f f a m i n e T-403, Texaco C h e m C o ) . T h e t h i r d
system
consists o f , (iii) t e t r a g l y c i d y l 4,4' d i a m i n o d i p h e n y l m e t h a n e e p o x y , T G D D M ( M Y 720, C i b a G e i g y ) c u r e d b y 4,4' d i a m i n o d i p h e n y l s u l p h o n e , D D S , ( H T 976, C i b a - G e i g y ) . D G E B A a n d T G D D M are a r o m a t i c epoxies. D E T A a n d A P T A are a l i p h a t i c amines w h e r e a s D D S is a n a r o m a t i c amine.
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2.2. Casting procedure The three d i f f e r e n t systems w e r e c a r e f u l l y m i x e d b y h a n d , v a c u u m w a s a p p l i e d t o t h e m i x t u r e s t e n m i n u t e s b e f o r e casting. T h e m i x t u r e s w e r e t h e n p o u r e d i n t o a f l u o r o p o l y m e r coated a l u m i n i u m m o l d . M a t e r i a l c o m p o s i t i o n s a n d cure schedules are presented i n Table I .
2.3. Specimen fabrication and test methods T h e cast plates w e r e r e m o v e d f r o m the m o l d a n d m a c h i n e d to the s p e c i m e n d i m e n s i o n s r e q u i r e d f o r m e c h a n i c a l t e s t i n g . T h e specimens f o r the t h e r m a l c o n s t r a i n t tests w e r e c u t b y a w a t e r jet cutter. T h e s p e c i m e n d i a m e t e r w a s 60 m m w i t h a t h i c k n e s s o f 2 m m . T h e e p o x y d i s k s w e r e b o n d e d t o steel a d h e r e n t s i n a g u i d e d f i x t u r e t o ensure a l i g n m e n t o f the adherents. The disks w e r e b o n d e d t o the a d h e r e n t s b y one o f the e p o x y resins, D G E B A / D E T A . T h e e p o x y a d h e s i v e w a s c u r e d at 110 ° C u n d e r a l o a d o f 10 k g . P r i o r t o b o n d i n g , t h e steel a d h e r e n t s a n d e p o x y p l a t e s w e r e g r o u n d a n d
then
degreased w i t h acetone. The t w o steel adherents w e r e o f d i f f e r e n t design. The u p p e r a d h e r e n t w a s a steel r i n g w i t h outer a n d i n n e r r a d i u s o f 30 m m a n d 15 m m , r e s p e c t i v e l y . T h e l o w e r a d h e r e n t w a s a d i s k w i t h a r a d i u s o f 30 m m . B o t h a d h e r e n t s w e r e 7 m m t h i c k . T h e use o f a steel r i n g f a c i l i t a t e s c r a c k o b s e r v a t i o n . A s c h e m a t i c o f t h e test g e o m e t r y a n d c o o r d i n a t e s y s t e m is p r e s e n t e d i n F i g u r e 1. Specimens designed f o r uniaxial testing were m i l l e d to the dimensions suggested b y A S T M D638M-81, type I , i n a computer controlled m i l l i n g m a c h i n e . T h e strains i n u n i a x i a l tests w e r e m e a s u r e d b y s t r a i n gauges, t y p e EP-08-125AD-120, m a n u f a c t u r e d b y M e a s u r e m e n t G r o u p Inc. Poisson's r a t i o a n d Y o u n g ' s m o d u l u s at a m b i e n t c o n d i t i o n s w e r e m e a s u r e d i n a n I n s t r o n test machine. T e m p e r a t u r e d e p e n d e n c i e s o f Y o u n g ' s m o d u l u s f o r the epoxies w e r e m e a s u r e d b y D M T A tests, i n a D y n a m i c a l M e c h a n i c a l T h e r m a l A n a l y s e r M K I I I f r o m R h e o m e t r i c Scientific L t d . Tests w e r e p e r f o r m e d o n 30 m m l o n g X
2
c a n t i l e v e r b e a m specimens w i t h cross sectional areas o f 2 2 m m . The D M T A specimens w e r e c u t i n a d i a m o n d w h e e l cutter. T e m p e r a t u r e dependencies o f the t h e r m a l e x p a n s i o n c o e f f i c i e n t s f o r the epoxies w e r e m e a s u r e d f o r a f r e e e x p a n d i n g p l a t e d o w n t o - 1 6 0 ° C . The strains at f r e e e x p a n s i o n w e r e m e a s u r e d b y s t r a i n gauges, t y p e CEA-13-062UT-350 a n d CEA-06-240UZ-120, m a n u f a c t u r e d b y M e a s u r e m e n t G r o u p Inc. The
temperature
d u r i n g cooling
thermometer, A E A by
Automatic
was
measured
Systems
with
Laboratories,
a
digital
using
a
t h e r m o c o u p l e , Pt 100, w i t h a n accuracy o f ± 0 . 1 ° C m e a s u r i n g d o w n to - 2 0 0 ° C .
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x
T h e d i m e n s i o n s o f t h e t h e r m o c o u p l e w e r e 1 0 2 m m . D u r i n g t h e test t h e t h e r m o c o u p l e w a s p l a c e d o n the free e p o x y surface o f the specimen. The specimen was placed on a perforated cardboard cylinder i n an i n s u l a t e d b o x at r o o m t e m p e r a t u r e , see F i g u r e 2. L i q u i d n i t r o g e n w a s p o u r e d i n t o a n i n s u l a t e d t e f l o n f u n n e l w h i c h e n d e d at t h e b o t t o m o f the b o x . T h e c o o l i n g rate f r o m r o o m t e m p e r a t u r e w a s a p p r o x i m a t e l y 2 ° C p e r m i n u t e . H o w e v e r , t h e c o o l i n g rate i n the b e g i n n i n g w a s h i g h e r t h a n at the e n d , c o o l i n g rate at f a i l u r e was a p p r o x i m a t e l y 1 ° C p e r m i n u t e . The s p e c i m e n w a s o b s e r v e d t h r o u g h a w i n d o w at the t o p o f t h e test c h a m b e r , see F i g u r e 2. T o t a l l y 18 specimens w e r e tested, 6 o f each m a t e r i a l .
3. STRESS ANALYSIS T h e p o l y m e r disks are b o n d e d to steel adherents a n d t h e r m a l l y l o a d e d as t h e t e m p e r a t u r e is l o w e r e d . T h i s t e m p e r a t u r e decrease w i l l cause b o t h t h e p o l y m e r d i s k a n d t h e steel adherents t o c o n t r a c t . D u e t o the d i f f e r e n c e i n t h e r m a l e x p a n s i o n coefficients, the p o l y m e r d i s k is c o n s t r a i n e d b y the steel a d h e r e n t s so t h a t a b i a x i a l tensile stress state is generated i n the p o l y m e r . T h e c o o r d i n a t e s y s t e m is d e f i n e d i n F i g u r e 1, T h e center o f the e p o x y s p e c i m e n is the o r i g i n of the c o o r d i n a t e system.
3.1. FEM-analysis T h e f i n i t e element m e t h o d ( F E M ) w a s used t o d e t e r m i n e the stress state i n a t h e r m a l l y loaded, constrained epoxy disk. The commercially available A N S Y S ® s y s t e m w a s used f o r this analysis. I n the e p o x y d i s k a n d its a d h e r e n t steel substrates a t h r e e - d i m e n s i o n a l stress state is present. H o w e v e r , m o d e l s o f a x i s y m m e t r i c 3-D structures such as the present one can be represented i n 9
e q u i v a l e n t 2 - D f o r m . The ANSYS® p r o g r a m p r o v i d e s elements f o r t h i s t y p e of analysis. I n the present
investigation, the eight-node axisymmetric
h a r m o n i c e l e m e n t , P L A N E 8 3 , w a s used. T h e P L A N E 8 3 e l e m e n t assumes l i n e a r elastic m a t e r i a l . The m e s h is s h o w n i n F i g u r e 3. A t o t a l n u m b e r o f 5000 e l e m e n t s w e r e i n c l u d e d i n t h e m e s h . T h e e p o x y d i s k is a s s u m e d t o b e p e r f e c t l y b o n d e d to the steel adherents. N o d e s o n the s y m m e t r i axis, r = 0 m m , w e r e r e s t r a i n e d i n the r a d i a l d i r e c t i o n . N o d e s o n the l o w e r edge o f the steel d i s k , Z--8 m m , w e r e restrained i n the z - d i r e c t i o n . T o a v o i d p r o b l e m s d u e t o stress a v e r a g i n g f o r d i s s i m i l a r m a t e r i a l s , a d h e r e n t a n d e p o x y stresses w e r e e v a l u a t e d w i t h i n selected elements.
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C a l c u l a t i o n s w e r e p e r f o r m e d f o r one o f the e p o x i e s ,
DGEBA/DETA.
M a t e r i a l d a t a u s e d i n n u m e r i c a l as w e l l as i n analytical analyses are presented i n Table I I .
3.2. Analytical approximation The a p p r o x i m a t e a n a l y t i c a l analysis is based o n the a s s u m p t i o n t h a t the steel a d h e r e n t s are r i g i d i n c o m p a r i s o n to the e p o x y disks. The steel adherents are t h e r e f o r e f r e e t o contract or e x p a n d a c c o r d i n g to the change i n t e m p e r a t u r e . T h i s w i l l s i m p l i f y t h e analysis. O n l y the d i f f e r e n c e i n t h e r m a l e x p a n s i o n c o e f f i c i e n t b e t w e e n t h e e p o x y a n d the steel m u l t i p l i e d b y t h e c h a n g e i n t e m p e r a t u r e are r e q u i r e d f o r c a l c u l a t i o n of the strain i n t h e e p o x y d i s k . T h e a n a l y t i c a l analysis does n o t consider a n y edge effects. T h e stresses i n the e p o x y d i s k w i l l t h e r e f o r e be i n d e p e n d e n t o n the z-coordinate. T h u s , the biharmonic equation
4
V * = 0
(1)
is s o l v e d i n p o l a r c o o r d i n a t e s f o r A i r y ' s stress f u n c t i o n 4>, w h e r e O is a f u n c t i o n o f r a d i a l p o s i t i o n , 4>=3>(r), o n l y . A i r y ' s stress f u n c t i o n is expressed as
2
® = Alogr
2
+ Br logr+Cr +D
(2)
w h e r e A, B, C, a n d D are constants. The constants A a n d B equals zero as the s o l u t i o n m u s t be f i n i t e o n the s y m m e t r i axis. The stress c o m p o n e n t s c a n be w r i t t e n as
-If*^ ° ~ r dr r
2
+
1 d _ 1 d® r 3d ~ r dr 2
2
drKrde
w h e r e t h e i n d i c e s r, 6, a n d z indicate r a d i a l , tangential, a n d n o r m a l d i r e c t i o n s , r e s p e c t i v e l y . A c c o r d i n g to Equations (2) a n d (3), the t w o i n - p l a n e stresses o> a n d ag are i d e n t i c a l a n d the in-plane shear stress z g is zero t h r o u g h o u t the r
disk. T h e stresses are calculated f o r the p l a n e stress s i t u a t i o n , a c c o r d i n g t o
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o",
(4)
£ a n d eø are i d e n t i c a l a n d e q u a l t o r
a.'steel
(5)
epoxy
w h e r e ct teel a n d cCepoxy are the t h e r m a l e x p a n s i o n c o e f f i c i e n t s f o r steel a n d s
e p o x y , respectively, a n d AT is the t e m p e r a t u r e change.
3.3. Temperature dependence of polymer properties T h e elastic p r o p e r t i e s a n d the t h e r m a l e x p a n s i o n c o e f f i c i e n t o f epoxies i n t h e i r 1 0
1 2
glassy state d e p e n d o n t e m p e r a t u r e " . I n t h e d e s c r i b e d analyses,
the
d e p e n d e n c i e s , E(T),
see
cc poxy(T), e
a n d v(T),
are o f d i r e c t i m p o r t a n c e ,
E q u a t i o n s (4) a n d (5). The c o n s i d e r e d t e m p e r a t u r e d e p e n d e n c e o f strains to f a i l u r e o r i g i n a t e s f r o m t h e t h e r m a l e x p a n s i o n c o e f f i c i e n t s o n l y . T h i s is a p p a r e n t f r o m E q u a t i o n (5) a n d the t e m p e r a t u r e d e p e n d e n c e c a n b e t a k e n i n t o account b y i n t e g r a t i o n o f E q u a t i o n (5).
(6)
T h e e f f e c t o n u l t i m a t e stresses caused b y the t e m p e r a t u r e d e p e n d e n c e o f elastic p r o p e r t i e s a n d t h e r m a l e x p a n s i o n c o e f f i c i e n t can be t a k e n i n t o account. T h e l o a d i n g s i t u a t i o n is s u c h t h a t t h e d i f f e r e n c e i n t h e r m a l e x p a n s i o n c o e f f i c i e n t b e t w e e n steel a n d the glassy p o l y m e r leads t o t h e r m a l l y i n d u c e d strains i n t h e glassy p o l y m e r . For a n elastic m a t e r i a l , the stress state at f a i l u r e can t h e r e f o r e be o b t a i n e d f r o m the strains expressed i n E q u a t i o n (6) t h r o u g h k n o w l e d g e o f o n l y the m o d u l i at the f a i l u r e t e m p e r a t u r e . M e a s u r e m e n t o f Poisson's r a t i o at t h e t e m p e r a t u r e o f f a i l u r e requires a n a d d i t i o n a l separate e x p e r i m e n t . H o w e v e r , a c c o r d i n g t o E q u a t i o n (4) a s m a l l change i n Poisson's r a t i o has a m i n o r e f f e c t o n the stresses i n the e p o x y d i s k . Poisson's r a t i o d a t a f r o m the glassy p o l y m e r at r o o m t e m p e r a t u r e c o u l d t h e r e f o r e be used. A s b o t h n u m e r i c a l a n d a n a l y t i c a l analyses
consider linear elastic
m a t e r i a l s , t h e t e m p e r a t u r e d e p e n d e n c i e s o f elastic p o l y m e r p r o p e r t i e s a n d t h e r m a l e x p a n s i o n c o e f f i c i e n t h a v e the same i n f l u e n c e o n t h e results. Since
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t h e p r e s e n t p a p e r m e r e l y i n v e s t i g a t e s t h e test m e t h o d itself, e f f e c t s f r o m temperature dependence of p o l y m e r properties
are
excluded
unless
s p e c i f i c a l l y stated. T h i s is i n o r d e r t o f a c i l i t a t e c o m p a r i s o n of p a r a m e t e r s calculated b y d i f f e r e n t t h e o r e t i c a l m e t h o d s .
4. RESULTS AND DISCUSSION B i a x i a l t e s t i n g o f e p o x i e s w a s c a r r i e d o u t b y c o o l i n g specimens o f t h e geometrical arrangement presented
i n F i g u r e 1. F r a c t u r e w a s
observed
t h r o u g h t h e w i n d o w o n t o p o f the test chamber, see F i g u r e 2. E x p e r i m e n t a l results f o r t h r e e epoxies are presented i n Table I I I . T h e u l t i m a t e stress o
u
is
t h e t r u e stress at f a i l u r e , t a k i n g t h e t e m p e r a t u r e d e p e n d e n c e o f Y o u n g ' s m o d u l u s a n d t h e r m a l e x p a n s i o n c o e f f i c i e n t i n t o account. The stress state is b i a x i a l w i t h e q u a l m a g n i t u d e o f r a d i a l a n d t a n g e n t i a l stress. a
u
a n d e are i n u
the ranges 54 - 73 M P a a n d 0.6 - 1 % respectively f o r the three epoxies. These d a t a seem p h y s i c a l l y r e a s o n a b l e since t h e y are i n b e t w e e n the v a l u e s f o r 5
u n i a x i a l a n d t r i a x i a l l o a d i n g o f the same e p o x i e s . M e a s u r e d c h a n g e , AT,
temperature
w a s b e t w e e n -219 a n d -269 ° C . T h e scatter i n o w a s u
low
c o m p a r e d t o t h e p r e v i o u s p o k e r c h i p test results f o r the same m a t e r i a l s 5
subjected t o a t r i a x i a l stress state . A l l three e p o x y systems w e r e s u c c e s s f u l l y tested b y the t h e r m o - m e c h a n i c a l d i s k m e t h o d , 16 o u t o f t o t a l l y 18 specimens failed d u r i n g cooling. The f r a c t u r e o f a s p e c i m e n w a s o b s e r v e d t h r o u g h the w i n d o w o n t o p o f the test c h a m b e r , see F i g u r e 2. I n a l l cases, the cracks w e r e observed t o g r o w at h i g h speed t h r o u g h the e n t i r e thickness, a n d i n the c e n t r a l regions o f t h e s p e c i m e n . T h e cracks w e r e a l w a y s p e r p e n d i c u l a r to t h e d i s k p l a n e . A t t h e t i m e o f f a i l u r e , a l o u d p o p p i n g s o u n d w a s h e a r d . N o i n i t i a t i o n areas w e r e detected i n situ, neither c o u l d the p o s i t i o n of the i n i t i a t i o n r e g i o n be d e t e r m i n e d conclusively. T y p i c a l features of f r a c t u r e d specimens
are
p r e s e n t e d i n F i g u r e 4. A l l specimens s h o w e d h i g h l y b r a n c h e d cracks, e i t h e r i n the center o r close t o the i n n e r edge o f the steel r i n g , r=15 mm, see F i g u r e 4. T h e b r a n c h i n g p o i n t s are i n t e r p r e t e d as p o i n t s w h e r e h i g h speed cracks 1 3
t r a n s f o r m i n t o t w o or m o r e cracks w i t h l o w e r s p e e d . The stress analysis w i l l be used t o discuss the l o c a t i o n o f crack i n i t i a t i o n .
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4.1. FEM-analysis A n u m e r i c a l stress analysis b y F E M w a s p e r f o r m e d f o r o n e o f t h e epoxies, D G E B A / D E T A , f o r the experimentally investigated temperature
change
(AT=-269 ° C ) . T h e element solutions s h o w h o m o g e n o u s stresses i n the central p a r t o f t h e e p o x y d i s k , r
