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Strain-rate dependence of the elastic modulus of filled and porous epoxy composites
Int. J. Mech. Svi. PergamonPress Ltd. 1967. Vol.9, pp. 605-608. Printedin Great Britain
S T R A I N - R A T E D E P E N D E N C E OF T H E ELASTIC MODULUS OF F I L L E D A N D POROUS E P O X Y COMPOSITES L. J. COHEN* and 0. ISHAI Department of Mechanics, Technion-Israel Institute of Technology, Israel (Received 11 April 1967 and in revised form 15 M a y 1967)
Summary--Compressive tests were carried out on filled and porous epoxy composites at constant temperature for different constant strain levels, covering the volumetric fillercontent and void-ratio ranges up to 42 per cent a n d 66 per cent respectively. The experimental elastic modulus was found to increase with filler content and decrease with void ratio, showing approximate linearity with log strain-rate. Normalized composite moduli as function of filler content and void ratio fall within the limits of the approximate simplified solutions based on cubic models for two decades of strain-rate. NOTATION Cl C~ E0 E~ Ef
volumetric filler content volumetric void ratio elastic modulus, composite elastic modulus, matrix elastic modulus, filler INTRODUCTION
THIn note deals with an experimental study of the strain-rate dependence of Young's modulus for two-phase particulate systems of high modular ratio. It forms part of a more comprehensive study of the elastic properties of polymeric heterogeneous systems. 1, Several theoretical approaches are available for evaluating the elastic moduli of two-phase composite systems. These approaches range, in mathematical complexity, from the variational technique of Hashin a and Hashin and Shtrikraan' to the simplified approaches of Paul 6, Counto6 and Ishai 7, based on models satisfying fundamental physical boundary conditions, b u t none of them covers the problem of strain-rate dependence, and they are confined to available experimental data indicating dependence of the composite modulus on the properties a n d proportions of the constituents. Recently, Uemura and Takayanagis and Hashin °, 10 have tried to evaluate the moduli of a viscoelastic material in terms of the viscoelastic properties and of the proportions of the constituents, using the correspondence principle. Uemura presents experimental data for several composite polymeric systems, which indicate agreement with his approach. Ishai and Cohen x recently obtained agreement between the simplified theory and experiment for the elastic modulus of filled and porous epoxy composites up to a filler content of 52 per cent a n d a void ratio of 70 per cent. I t was decided to examine the validity of such laws at a different level of constant strain-rate. * Also at the Douglas Aircraft Co., Missile and Space System Division, Santa Monica, California. 605
606
L. J. COHE~
and O. ISHAI
T h e i n g r e d i e n t s a n d t e s t i n g t e c h n i q u e were i d e n t i c a l to t h a t of Ref. 1. L o a d was a p p l i e d a t a r a t e of 0 . 1 - 5 0 m m / m i n ( s t r a i n - r a t e 0 . 0 0 2 7 - 1 . 3 5 m i n - 1 ) , e x c e e d i n g t h e m a x i m u m yield level. C o m p r e s s i v e s t r a i n s were d e t e r m i n e d f r o m t h e r e c o r d e d loaddeflection c u r v e s ( I n s t r o n ) u n d e r a l l o w a n c e for a d d i t i o n a l deflections d u e to t h e l o a d i n g system. P l o t s of c o m p o s i t e m o d u l u s i n c o m p r e s s i o n vs. log s t r a i n - r a t e (see Figs. 1 a n d 2) s h o w a r o u g h l y l i n e a r r e l a t i o n s h i p for b o t h filled a n d p o r o u s s y s t e m s . A t a g i v e n s t r a i n - r a t e ,
I
•
1 ~L~
~1
o
I
I
i
1
I
. 'L~--~s--'-T~
'
.
I
,
~1
Cf:42%
i r
I I
"~
I
4
6 8 10"z
2
4
f : G '/.
6 8 10"1 2 STRAINRATE ~ (rain"1)
10°
Fzo. 1. I n f l u e n c e of s t r a i n - r a t e o n elastic m o d u l u s of c o m p o s i t e filled s y s t e m .
!i
~2
%
:%
=
6
8 t0 "2
2
lo-1 6 TRAIN RATE E (min -1)
~%
s 8 #
FIG. 2. I n f l u e n c e of s t r a i n - r a t e o n elastic m o d u l u s of c o m p o s i t e p o r o u s s y s t e m .
t h e m o d u l u s s h o w e d t h e e x p e c t e d g e n e r a l t r e n d of i n c r e a s i n g w i t h C t a n d d e c r e a s i n g w i t h C~. T h e m o d u l a r v a l u e s were n o r m a l i z e d w i t h r e s p e c t t o t h e i r m a t r i x c o u n t e r p a r t s a t t h e s a m e s t r a i n - r a t e , a n d p l o t t e d as f u n c t i o n of C I a n d C~ i n Figs. 3 a n d 4. T h e a p p r o x i m a t e s o l u t i o n s of P a u P a n d I s h a i ~, u s i n g t h e t w o - p h a s e m o d e l o f a c u b i c i n c l u s i o n e m b e d d e d w i t h i n a c u b i c m a t r i x , a r e also i n c l u d e d . C o m p a r i s o n o f t h e e x p e r i m e n t a l d a t a w i t h t h e t h e o r e t i c a l curves, see Figs. 3 a n d 4, r e v e a l s t h e following c h a r a c t e r i s t i c s :
S t r a i n - r a t e d e p e n d e n c e of t h e elastic m o d u l u s of e p o x y c o m p o s i t e s
607
(1) T h e n o r m a l i z e d m o d u l a r v a l u e s of t h e c o m p o s i t e s y s t e m s for t w o d e c a d e s of s t r a i n r a t e a r e well w i t h i n t h e n a r r o w b a n d defined b y t h e t w o a p p r o x i m a t e solutions, u p t o C I = 42 p e r c e n t a n d C, = 66 p e r c e n t . (2) F o r t h e t w o - d e c a d e i n t e r v a l , t h e effect of s t r a i n - r a t e o n t h e n o r m a l i z e d c o m p o s i t e m o d u l u s w a s a p p r o x i m a t e l y 17 p e r c e n t for t h e filled s y s t e m a n d 8 p e r c e n t for t h e p o r o u s one. T h e f i n d i n g s of t h e a b o v e series of t e s t s o n e p o x y c o m p o s i t e s a t r o o m t e m p e r a t u r e i n d i c a t e a n i n c r e a s e w i t h s t r a i n - r a t e in t h e elastic m o d u l u s in c o m p r e s s i o n , for b o t h filled
• E=0.0027 (rain"~) o 0.027 o 0,135 @ 0.2?0
,# v
uJ
ref, 5 ~j
//
b /
o~
1 0
20
(eq 16)
40
60
~ O L L I I ~ RLLER CONTENT Cf
80
(%)
FzG. 3. I n f l u e n c e o f filler c o n t e n t o n m o d u l a r r a t i o o f filled c o m p o s i t e t o m a t r i x .
•"~10 ~ , ~ i!
8~
strain rate • E = 0~0Z7 (rain-1) o 0~27
0~
~
~
J re[ 1
I
1
I
CONTENT Cv ('1.) Fro. 4. I n f l u e n c e o f v o i d r a t i o on m o d u l a r r a t i o o f porous c o m p o s i t e t o m a t r i x .
~ a d p o r o u s s y s t e m s . T h e n o r m a l i z e d v a l u e s of t h e c o m p o s i t e m o d u l i w i t h r e s p e c t t o t h e i r m a t r i x c o u n t e r p a r t s (for t h e s a m e r a t e ) fall well w i t h i n t h e b o u n d s defined b y t w o simplified a p p r o a c h e s for t h e r a n g e of t w o decades of s t r a i n - r a t e s .
L. J. COHEN and 0. ISHAI
608
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
0. L. Z. Z. B. U. 0. S. Z. Z.
ISHAI and L. J. COHEN, Int. J. Med. Sci. 9, 637 (1967). J. COHEN and 0. ISHAI, to be published. HASHIN, J. a&. Mech. 29, 143 (1962). HASHIN and S. SHTRIKMAN, J. mech. phye. sokL9 11, 127 (1963). PAUL, Trans. Am. inst. mech. Engrs 218, 36 (1960). COUNTO, Mag. Concr. Rea. 16, 129 (1964). ISHAI, Mug. Concr. Res. 17, 198 (1965). UEMURA and M. TAKAYANAGI, J. appl. polym. Sci. 10, 113 (1966). HASHIN, J. appl. Mech. 630 (1965). HASHIN, AIAA JousnaZ4, 1411 (1966).
S T R A I N - R A T E D E P E N D E N C E OF T H E ELASTIC MODULUS OF F I L L E D A N D POROUS E P O X Y COMPOSITES L. J. COHEN* and 0. ISHAI Department of Mechanics, Technion-Israel Institute of Technology, Israel (Received 11 April 1967 and in revised form 15 M a y 1967)
Summary--Compressive tests were carried out on filled and porous epoxy composites at constant temperature for different constant strain levels, covering the volumetric fillercontent and void-ratio ranges up to 42 per cent a n d 66 per cent respectively. The experimental elastic modulus was found to increase with filler content and decrease with void ratio, showing approximate linearity with log strain-rate. Normalized composite moduli as function of filler content and void ratio fall within the limits of the approximate simplified solutions based on cubic models for two decades of strain-rate. NOTATION Cl C~ E0 E~ Ef
volumetric filler content volumetric void ratio elastic modulus, composite elastic modulus, matrix elastic modulus, filler INTRODUCTION
THIn note deals with an experimental study of the strain-rate dependence of Young's modulus for two-phase particulate systems of high modular ratio. It forms part of a more comprehensive study of the elastic properties of polymeric heterogeneous systems. 1, Several theoretical approaches are available for evaluating the elastic moduli of two-phase composite systems. These approaches range, in mathematical complexity, from the variational technique of Hashin a and Hashin and Shtrikraan' to the simplified approaches of Paul 6, Counto6 and Ishai 7, based on models satisfying fundamental physical boundary conditions, b u t none of them covers the problem of strain-rate dependence, and they are confined to available experimental data indicating dependence of the composite modulus on the properties a n d proportions of the constituents. Recently, Uemura and Takayanagis and Hashin °, 10 have tried to evaluate the moduli of a viscoelastic material in terms of the viscoelastic properties and of the proportions of the constituents, using the correspondence principle. Uemura presents experimental data for several composite polymeric systems, which indicate agreement with his approach. Ishai and Cohen x recently obtained agreement between the simplified theory and experiment for the elastic modulus of filled and porous epoxy composites up to a filler content of 52 per cent a n d a void ratio of 70 per cent. I t was decided to examine the validity of such laws at a different level of constant strain-rate. * Also at the Douglas Aircraft Co., Missile and Space System Division, Santa Monica, California. 605
606
L. J. COHE~
and O. ISHAI
T h e i n g r e d i e n t s a n d t e s t i n g t e c h n i q u e were i d e n t i c a l to t h a t of Ref. 1. L o a d was a p p l i e d a t a r a t e of 0 . 1 - 5 0 m m / m i n ( s t r a i n - r a t e 0 . 0 0 2 7 - 1 . 3 5 m i n - 1 ) , e x c e e d i n g t h e m a x i m u m yield level. C o m p r e s s i v e s t r a i n s were d e t e r m i n e d f r o m t h e r e c o r d e d loaddeflection c u r v e s ( I n s t r o n ) u n d e r a l l o w a n c e for a d d i t i o n a l deflections d u e to t h e l o a d i n g system. P l o t s of c o m p o s i t e m o d u l u s i n c o m p r e s s i o n vs. log s t r a i n - r a t e (see Figs. 1 a n d 2) s h o w a r o u g h l y l i n e a r r e l a t i o n s h i p for b o t h filled a n d p o r o u s s y s t e m s . A t a g i v e n s t r a i n - r a t e ,
I
•
1 ~L~
~1
o
I
I
i
1
I
. 'L~--~s--'-T~
'
.
I
,
~1
Cf:42%
i r
I I
"~
I
4
6 8 10"z
2
4
f : G '/.
6 8 10"1 2 STRAINRATE ~ (rain"1)
10°
Fzo. 1. I n f l u e n c e of s t r a i n - r a t e o n elastic m o d u l u s of c o m p o s i t e filled s y s t e m .
!i
~2
%
:%
=
6
8 t0 "2
2
lo-1 6 TRAIN RATE E (min -1)
~%
s 8 #
FIG. 2. I n f l u e n c e of s t r a i n - r a t e o n elastic m o d u l u s of c o m p o s i t e p o r o u s s y s t e m .
t h e m o d u l u s s h o w e d t h e e x p e c t e d g e n e r a l t r e n d of i n c r e a s i n g w i t h C t a n d d e c r e a s i n g w i t h C~. T h e m o d u l a r v a l u e s were n o r m a l i z e d w i t h r e s p e c t t o t h e i r m a t r i x c o u n t e r p a r t s a t t h e s a m e s t r a i n - r a t e , a n d p l o t t e d as f u n c t i o n of C I a n d C~ i n Figs. 3 a n d 4. T h e a p p r o x i m a t e s o l u t i o n s of P a u P a n d I s h a i ~, u s i n g t h e t w o - p h a s e m o d e l o f a c u b i c i n c l u s i o n e m b e d d e d w i t h i n a c u b i c m a t r i x , a r e also i n c l u d e d . C o m p a r i s o n o f t h e e x p e r i m e n t a l d a t a w i t h t h e t h e o r e t i c a l curves, see Figs. 3 a n d 4, r e v e a l s t h e following c h a r a c t e r i s t i c s :
S t r a i n - r a t e d e p e n d e n c e of t h e elastic m o d u l u s of e p o x y c o m p o s i t e s
607
(1) T h e n o r m a l i z e d m o d u l a r v a l u e s of t h e c o m p o s i t e s y s t e m s for t w o d e c a d e s of s t r a i n r a t e a r e well w i t h i n t h e n a r r o w b a n d defined b y t h e t w o a p p r o x i m a t e solutions, u p t o C I = 42 p e r c e n t a n d C, = 66 p e r c e n t . (2) F o r t h e t w o - d e c a d e i n t e r v a l , t h e effect of s t r a i n - r a t e o n t h e n o r m a l i z e d c o m p o s i t e m o d u l u s w a s a p p r o x i m a t e l y 17 p e r c e n t for t h e filled s y s t e m a n d 8 p e r c e n t for t h e p o r o u s one. T h e f i n d i n g s of t h e a b o v e series of t e s t s o n e p o x y c o m p o s i t e s a t r o o m t e m p e r a t u r e i n d i c a t e a n i n c r e a s e w i t h s t r a i n - r a t e in t h e elastic m o d u l u s in c o m p r e s s i o n , for b o t h filled
• E=0.0027 (rain"~) o 0.027 o 0,135 @ 0.2?0
,# v
uJ
ref, 5 ~j
//
b /
o~
1 0
20
(eq 16)
40
60
~ O L L I I ~ RLLER CONTENT Cf
80
(%)
FzG. 3. I n f l u e n c e o f filler c o n t e n t o n m o d u l a r r a t i o o f filled c o m p o s i t e t o m a t r i x .
•"~10 ~ , ~ i!
8~
strain rate • E = 0~0Z7 (rain-1) o 0~27
0~
~
~
J re[ 1
I
1
I
CONTENT Cv ('1.) Fro. 4. I n f l u e n c e o f v o i d r a t i o on m o d u l a r r a t i o o f porous c o m p o s i t e t o m a t r i x .
~ a d p o r o u s s y s t e m s . T h e n o r m a l i z e d v a l u e s of t h e c o m p o s i t e m o d u l i w i t h r e s p e c t t o t h e i r m a t r i x c o u n t e r p a r t s (for t h e s a m e r a t e ) fall well w i t h i n t h e b o u n d s defined b y t w o simplified a p p r o a c h e s for t h e r a n g e of t w o decades of s t r a i n - r a t e s .
L. J. COHEN and 0. ISHAI
608
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
0. L. Z. Z. B. U. 0. S. Z. Z.
ISHAI and L. J. COHEN, Int. J. Med. Sci. 9, 637 (1967). J. COHEN and 0. ISHAI, to be published. HASHIN, J. a&. Mech. 29, 143 (1962). HASHIN and S. SHTRIKMAN, J. mech. phye. sokL9 11, 127 (1963). PAUL, Trans. Am. inst. mech. Engrs 218, 36 (1960). COUNTO, Mag. Concr. Rea. 16, 129 (1964). ISHAI, Mug. Concr. Res. 17, 198 (1965). UEMURA and M. TAKAYANAGI, J. appl. polym. Sci. 10, 113 (1966). HASHIN, J. appl. Mech. 630 (1965). HASHIN, AIAA JousnaZ4, 1411 (1966).