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ELASTIC CONSTANTS AND INTERNAL FRICTION
OF REINFORCED COMPOSITES
H. Ledbetter
To cite this version:
H. Ledbetter.
ELASTIC CONSTANTS AND INTERNAL FRICTION OF REINFORCED
COMPOSITES.
Journal
de
Physique
Colloques,
1985,
46
(C10),
pp.C10-573-C10-578.
JOURNAL DE PHYSIQUE
Colloque CIO, supplBment au n o 12, Tome 46, decembre 1985 page C10-573
ELASTIC CONSTANTS AND INTERNAL FRICTION OF REINFORCEDCOMPOSITES H.M. LEDBETTER
Fracture
and
Deformation Division, Center for MaterialsScience , National Bureau of Standards, Boulder, Colorado
80303. U.S.A.
Abstract
- We d e s c r i b e experimental s t u d i e s on the a n i s o t r o p i c e l a s t i c con-
s t a n t s and i n t e r n a l f r i c t i o n of reinforced composites. Reinforcement types include f i b e r and f a b r i c . Studied m a t e r i a l s include boron-aluminum, g l a s s - epoxy, boron-epoxy, graphite-epoxy, and aramid-epoxy. We made most measure- ments with a Marx three-component o s c i l l a t o r a t k i l o h e r t z frequencies. In a l l c a s e s , e l a s t i c - c o n s t a n t d i r e c t i o n dependence f i t r e l a t i o n s h i p s derived f o r homogeneous monocrystals. Usually, e l a s t i c s t i f f n e s s and i n t e r n a l f r i c - t i o n show an inverse r e l a t i o n s h i p . In no case d i d the inclusion-matrix in- t e r f a c e appear t o c o n t r i b u t e s i g n i f i c a n t l y t o i n t e r n a l f r i c t i o n .I - INTRODUCTION
A physical property of a composite m a t e r i a l depends mainly on t h r e e ingredients: matrix property, inclusion property, and phase geometry. Phase geometry includes many v a r i a b l e s : volume f r a c t i o n , inclusion shape, inclusion o r i e n t a t i o n , inclusion s i z e , and inclusion d i s t r i b u t i o n . For some composites, e s p e c i a l l y a t higher tempera- t u r e s , t h e inclusion-matrix i n t e r f a c e a f f e c t s a physical property.
The present study considers two s t r o n g l y r e l a t e d physical p r o p e r t i e s : e l a s t i c con- s t a n t s and i n t e r n a l f r i c t i o n . Especially, we focus on Young modulus, E , and i n t e r - n a l f r i c t i o n
,
Q-', determined in rod-shaped specimens i n a Young-modulus
(extensional-wave) mode.
E l a s t i c constants e n t e r many aspects of composite-material behavior: s t i f f n e s s - weight r a t i o , load-deflection, e l a s t i c i n s t a b i l i t y , thermoelastic s t r e s s , r e s i d u a l s t r e s s , sound-wave v e l o c i t i e s , m a t e r i a l c h a r a c t e r i z a t i o n , r e l a t i o n s h i p t o o t h e r physical p r o p e r t i e s ( f o r example, thermal expansivity and s p e c i f i c h e a t ) , p l a s t i c deformation, t h e o r e t i c a l s t r e n g t h , and nondestructive evaluation.
C10-574 JOURNAL DE PHYSIQUE
I n t e r n a l f r i c t i o n o f c o m p o s i t e s r e l a t e s t o many o f t h e above phenomena. I n a d d i t i o n , i t r e l a t e s e s p e c i a l l y t o s t r u c t u r a l damping and t o d e t e c t i o n o f premoni- t o r y f a i l u r e a s m a n i f e s t e d i n c r a c k i n g and d e l a m i n a t i o n .
I1
- EXPERIMENT
We measured Young modulus and Young-modulus-mode i n t e r n a l f r i c t i o n u s i n g a Marx three-component o s c i l l a t o r i n t h e k i l o h e r t z - f r e q u e n c y r e g i o n / I / . I n t h i s method. t h e Young-modulus v a l u e a r i s e s from t h e specimen r e s o n a n c e f r e q u e n c y and t h e i n t e r n a l f r i c t i o n from t h e half-power w i d t h o f t h e r e s o n a n c e peak. T y p i c a l s p e c i - mens were c y l i n d r i c a l r o d s 5 mm i n d i a m e t e r and 2 t o 10 cm l o n g .
I11
- RESULTS
Table 1 g i v e s a m b i e n t - t e m p e r a t u r e r e s u l t s f o r s e v e r a l f i b e r - r e i n f o r c e d and c l o t h - r e i n f o r c e d c o m p o s i t e s . F i g u r e 1 shows a l o g - l o g p l o t o f most o f t h e s e r e s u l t s . F i g u r e 2 shows t h e a n g u l a r v a r i a t i o n o f E and Q-l f o r a u n i a x i a l boron-f i b e r - r e i n f o r c e d aluminum-matrix c o m p o s i t e . For t h e w a r p - f i l l p l a n e , F i g . 3 shows a s i m i l a r d i a g r a m f o r a g l a s s - f i b e r - c l o t h - r e i n f o r c e d epoxy-matrix c o m p o s i t e . F i g u r e 4 shows a s i m i l a r diagram f o r t h e Young modulus o f a g r a p h i t e - f i b e r - c l o t h - r e i n f o r c e d epoxy-matrix composite.
I V
-
DISCUSSIONF i g u r e 1 shows an a p p r o x i m a t e l y h y p e r b o l i c r e l a t i o n s h i p between Young modulus and i n t e r n a l f r i c t i o n . I f we assume a r e l a t i o n s h i p
E" Q - ~ = C = c o n s t a n t ( 1 )
t h e n w i t h E i n u n i t s o f 1011 ~ / and Q-l i n u n i t s o f lo-', m ~ a l e a s t - s q u a r e s . f i t g i v e s n = 0.80 and C = 14.3. Although t h i s p r e l i m i n a r y e m p i r c a l E-Q-I r e l a t i o n s h i p r e q u i r e s f u r t h e r s t u d y , i t s u g g e s t s a u s e f u l g u i d e l i n e f o r u n d e r s t a n d i n g and o p t i m i z i n g t h e s e two p h y s i c a l p r o p e r t i e s i n f i b e r - r e i n f o r c e d c o m p o s i t e s .
For a t r a n s v e r s e - i s o t r o p i c - s y m m e t r y m a t e r i a l , w i t h t h e u n i q u e a x i s a l o n g x3, t h e Young modulus i s
where S . . d e n o t e t h e Voigt e l a s t i c c o m p l i a n c e s , Y d e n o t e s t h e a n g l e between t h e 1 J
u n i q u e a x i s and t h e specimen a x i s , and t h e prime d e n o t e s r o t a t i o n away from t h e x3 a x i s . Composites c o n t a i n i n g p a r a l l e l f i b e r s d i s t r i b u t e d e i t h e r randomly o r t r i a n g u - l a r l y i n t h e t r a n s v e r s e p l a n e s h o u l d e x h i b i t t r a n s v e r s e - i s o t r o p i c symmetry.
F i g u r e 2 shows t h a t measurements on a boron-aluminum c o m p o s i t e f i t t h e p r e d i c t i o n s o f Eq. ( 2 ) . A l s o , F i g 2 shows a n i n v e r s e r e l a t i o n s h i p between t h e d i r e c t i o n a l v a r i a t i o n s o f E a n d Q-: : t h e minimum i n E c o r r e s p o n d s t o t h e maximum i n Q-I
,
and v i c e v e r s a . T h i s e x t e n d s t h e g e n e r a l r e s u l t shown i n F i 1 f o r p r i n c i p a l d i r e c - t i o n s . P r e s e n t l y , we do n o t u n d e r s t a n d t h e i r r e g u l a r Q-'.(Y) b e h a v i o r . We specu- l a t e t h a t t h i s may a r i s e from mode c o u p l i n g r e l a t e d t o t h e c o m p o s i t e ' s l a m i n a t e d s t r u c t u r e .For a n o r t h o t r o p i c - s y m m e t r y m a t e r i a l , w i t h p r i n c i p a l a x e s a l o n g x l , x 2 , x 3 , t h e Young modulus ( i n t h e xl-x2 p l a n e , f o r example) is
where t h e a i r e p r e s e n t t h e d i r e c t i o n c o s i n e s between t h e specimen a x i s and t h e p r i n c i p a l a x i s . Laminated c l o t h - r e i n f o r c e d c o m p o s i t e s w i t h 90° a n g l e s between warp- d i r e c t i o n and f i l l - d i r e c t i o n f i b e r s s h o u l d e x h i b i t o r t h o t r o p i c macroscopic symmetry. For g l a s s - e p o x y , F i g . 3 confirms t h a t t h e measurements f i t Eq. ( 3 ) . T h i s f i g u r e a l s o shows Q-'
,
which, l i k e t h e boron-aluminum c a s e , r e l a t e s i n v e r s e l y t o E.modulus behaves s i m i l a r l y t o the glass-epoxy c a s e shown i n Fig. 3. Our preliminary measurements show t h a t Q-' f o r graphite-cloth-epoxy does
not
vary a s expected with angle.V
-
CONCLUSIONSFrom t h i s s t u d y , t h e r e a r i s e s e v e r a l conclusions:
1 . Composites r e i n f o r c e d with u n i d i r e c t i o n a l f i b e r s e x h i b i t t r a n s v e r s e - i s o t r o p i c symmetry. Composites r e i n f o r c e d with c l o t h laminae e x h i b i t o r t h o t r o p i c symmetry. One can describe both c a s e s using s t a n d a r d r e l a t i o n s h i p s derived f o r a n i s o t r o p i c monocrystals.
2. Usually, but not always, Q-' decreases when E i n c r e a s e s . This i n v e r s e r e l a t i o n - s h i p tends t o hold f o r t h r e e s i t u a t i o n s : a s i n g l e composite where i n c l u s i o n volume f r a c t i o n changes, among various composites, and within a s i n g l e composite versus d i r e c t i o n .
3. A t ambient temperatures, i n a l l composites s t u d i e d t o d a t e ( i n c l u d i n g o t h e r s not r e p o r t e d h e r e ) , we found no i n t e r f a c e c o n t r i b u t i o n t o i n t e r n a l f r i c t i o n .
ACKNOWLEDGMENT
Our composites s t u d i e s received support from s e v e r a l sponsors, e s p e c i a l l y DARPA and t h e DOE O f f i c e of Fusion Energy. M. W. Austin made most of t h e measurements.
REFERENCE
JOURNAL DE PHYSIQUE
TABLE 1. Young modulus and i n t e r n a l f r i c t i o n of s e v e r a l f i b r o u s composites a t ambient temperature.
Fiber I n t e r n a l First-harmonic
o r i e n t a t i o n , Dynamic Young's f r i c t i o frequency,
No. M a t e r i a l degrees modulus, GPa kHz
1 Boron- 0 2 aluminum 90 6 Glass-cloth- epoxy 1 woof 7 Glass-cloth- epoxy 2 woof 29.6 t 0.2 64.9 i 7.5 40 Glass-cloth- epoxy 3 woof 8 Glass-cloth- warp 29.4 i 1.0 68.7
+
6.6 4040 9 epoxy 4 f i l l 26.3 i 1.0 100.1 i 7.8 45-90 10 normal 14.0~ 228.6 55 1 1 Class-cloth- warp 31.4 i 0.4 114.5+
13.9 40-70 12 epoxy 5 f i l l 27.7 221.6 i 85.3 40-70 13 norma 1 15.6 i 0.6 406.5 60 Graphite- epoxy 1 14 Graphite- 15 epoxy 2 16 20 Graphite- epoxy 4 21 Graphite- 0 130. i 1---
60 epoxy 5 Graphite-cloth warp 73.7 27 4 50 epoxy f i l l 64.7 320 50 normal 13.0 1584 50 Graphite- 0 epoxy 1 90 22 Aramid- 0 epoxyFig. 1 - Young modulus v e r s u s i n t e r n a l f r i c t i o n f o r f i b e r - r e i n f o r c e d composite m a t e r i a l s i n Table 1
.
;
I I 1 I " I , I 1 1 1 1 1 1 1 1 I 1 1 1 1 I l l ( I I l 1 l l l L-
-
\-
-
\
-
-
Fig. 2 - D i r e c t i o n a l v a r i a t i o n o f Young modulus and i n t e r n a l f r i c t i o n f o r a u n i a x i a l - b o r o n - f i b e r aluminum-matrix composite. F i b e r s l i e a l o n g x j a x i s . The c u r v e r e p r e s e n t s Eq. ( 2 ) .
1
-
-
r
-
-
-
-
-
-
-
-
-
-
0.1
-
Transverse
r
-
-
eCrosspiy
-
-
-
@Perpendicular to laminae
\-
\-
-
+Aluminum, annealed, high-purity
\JOURNAL
DE
PHYSIQUEWaro
Fig. 3 - D i r e c t i o n a l v a r i a t i o n of Young modulus and i n t e r n a l f r i c t i o n f o r a g l a s s - f i b e r - c l o t h epoxy-matrix laminated composite. F i b e r s l i e i n warp and f i l l d i r e c t i o n s . The curve r e p r e s e n t s E q . ( 3 ) .
F i g . 4
