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THE DYNAMIC ELASTIC AND PLASTIC RESPONSE OF FIBER-REINFORCED SOLIDS
H. Kolsky, J. Mosquera
To cite this version:
H. Kolsky, J. Mosquera. THE DYNAMIC ELASTIC AND PLASTIC RESPONSE OF FIBER- REINFORCED SOLIDS. Journal de Physique Colloques, 1985, 46 (C5), pp.C5-565-C5-571.
�10.1051/jphyscol:1985573�. �jpa-00224807�
page C5-565
THE DYNAMIC ELASTIC AND PLASTIC RESPONSE OF FIBER-REINFORCED SOLIDS
H. Kolsky and J.M. ~os~uera':'
Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, U.S.A.
R6sum6 - On d 6 c r i t l a r6ponse mecanique de p o u t r e s c a n t i l e v e r s de composites m 6 t a l l i q u e s s o l l i c i t 6 e s p a r des chargements dynamiques t r a n s v e r s e s . Le comportement o b s e r v e e s t compare aux p r e d i c t i o n s a n a l y t i q u e s .
A b s t r a c t - The mechanical response o f m e t a l l i c , composite c a n t i l e v e r s t o t r a n s v e r s e dynamic l o a d s i s described. The observed b e h a v i o r i s compared t o a n a l y t i c p r e d i c t i o n s .
INTRODUCTION
As a r e s u l t o f t h e l a r g e and growing i n t e r e s t i n t h e p r o p e r t i e s o f f i b e r - r e i n f o r c e d m a t e r i a l s , much e f f o r t has been e x e r t e d d u r i n g t h e l a s t two o r t h r e e decades t o s t u d y t h e i r mechanical b e h a v i o r . The work d e s c r i b e d i n t h i s paper i s concerned w i t h t h e response o f beams o f f i b e r - r e i n f o r c e d s o l i d s when t h e y a r e s u b j e c t e d t o dynamic t r a n s v e r s e l o a d i n g . The e x p e r i m e n t a l o b s e r v a t i o n s f o u n d i n t h i s work a r e compared w i t h t h e a n a l y t i c p r e d i c t i o n s o b t a i n e d f r o m t h e c o n s i d e r a t i o n o f v a r i o u s mathematical models w h i c h have been used t o s i m u l a t e t h e p h y s i c a l b e h a v i o r o f fi b e r - r e i n f o r c e d m a t e r i a l s . Most o f t h e e x p e r i m e n t a l work has been c a r r i e d o u t on composite beams w h i c h were f a b r i c a t e d i n t h e l a b o r a t o r y and w h i c h c o n s i s t e d o f bundles o f t h i n p a r a l l e l s t e e l p i a n o - w i r e s embedded a x i a l l y i n beams o f a l e a d t i n a l l o y .
The work w h i c h has been c a r r i e d o u t on t h e s e beams f a l l s i n t o one of t h r e e
c a t e g o r i e s : ( a ) e l a s t i c deformations, ( b ) p l a s t i c d e f o r m a t i o n s c a r r i e d o u t s l o w l y enough f o r t h e e f f e c t s o f wave p r o p a o a t i o n t o be i g n o r e d , ( c ) r a p i d dynamic l o a d i n g where wave e f f e c t s must b e t a k e n i n t o account. The r e s u l t s o f t h e s e e x p e r i m e n t a l
s t u d i e s a r e d e s c r i b e d below, and a r e i n each case compared w i t h t h e a n a l y t i c a l p r e d i c t i o n s .
The p r i n c i p a l way i n w h i c h f i b e r - r e i n f o r c e d s o l i d s d i f f e r f r o m i s o t r o p i c ones i s t h a t s i n c e i n such m a t e r i a l s t h e e x t e n s i b i l i t y i n t h e f i b e r d i r e c t i o n i s vesy much s m a l l e r t h a n i t i s i n t h e d i r e c t i o n s p e r p e n d i c u l a r t o it, t h e i m p o r t a n c e o f shear d e f o r m a t i o n i s v e r y much g r e a t e r f o r such m a t e r i a l s t h a n i t i s f o r i s o t r o p i c ones.
In
t h e l i m i t , t h e i d e a l f i b e r r e i n f o r c e d s o l i d i s d e f i n e d as one i n which t h e f i b e r s a r e c o m p l e t e l y i n e x t e n s i b l e and t h e m a t r i x m a t e r i a l i s i n c o m p r e s s i b l e . F o r such a m a t e r i a l shear i s t h e o n l y p o s s i b l e mode o f mechanical d e f o r m a t i o n .( a ) E l a s t i c Response
The dynamic e l a s t i c f l e x u r a l response o f beams i s a problem o f some c o m p l e x i t y ( c f . ( 1 ) ) . The s i m p l e s t t h e o r y o f f l e x u r a l wave p r o p a g a t i o n i s an i s o t r o p i c e l a s t i c beam g i v e s t h e e x p r e s s i o n
where cI i s t h e phase v e l o c i t y o f a t r a i n o f s i n u s o i d a l f l e x u r a l waves o f wave-
"NOW at Universidad del Cauca, Popayan, Colombia, S.A.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985573
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l e n g t h A, K i s t h e r a d i u s o f g y r a t i o n o f t h e c r o s s s e c t i o n a b o u t t h e n e u t r a l a x i s , E i s Young's modulus o f t h e m a t e r i a l o f t h e beam and p i s i t s d e n s i t y . E q u a t i o n ( 1 ) i s , however, o n l y approximate and does n o t t a k e i n t o a c c o u n t f o r examnle, t h e r o t a r y i n e r t i a o f s e c t i o n s o f t h e beam, o r t h e d i s t o r t i o n o f t h e c r o s s s e c t i o n b y shear f o r c e . T h i s e q u a t i o n , however, g i v e s r e a s o n a b l y a c c u r a t e v a l u e s f o r cI so l o n g as K/A<<1.
The p r o p a g a t i o n o f f l e x u r a l waves i n a n i s o t r o p i c e l a s t i c s o l i d s i s a more complica- t e d problem, and f o r a g i v e n v a l u e of K/A t h e d e v i a t i o n s f r o m i t a r e f o u n d t o be l a r g e r . The magnitude o f t h e s e d e v i a t i o n s depends on t h e v a l u e o f t h e non-dimen- s i o n a l r a t i o E/G, where E i s Young's modulus a l o n g t h e f i b e r d i r e c t i o n and G i s t h e v a l u e o f t h e shear modulus i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e f i b e r s . S a y i r ( 2 ) has d e r i v e d e x p r e s s i o n s f o r cII, a n improved v a l u e o f t h e v e l o c i t y o f f l e x u r a l waves i n such beams, b y t r e a t i n g t h e e q u a t i o n s o f m o t i o n o f t h e beams by t h e methods o f a s y m p t o t i c a n a l y s i s . He has expressed h i s r e s u l t s f o r beams o f c i r c u l a r c r o s s s e c t i o n i n terms o f a non-dimensional parameter p w h i c h he d e f i n e s by t h e r e l a t i o n
( R i s t h e r a d i u s o f t h e c i r c u l a r c r o s s s e c t i o n ) . F o r p < < l he d e r i v e s t h e e x p r e s s i o n
As p g e t s l a r g e r t h e shear d e f o r m a t i o n s become more and more prominent, and as p- t h e v e l o c i t y o f f l e x u r a l waves approaches t h e shear wave v e l o c i t y c 2 = ( ~ / p ) ' / ' , i n t h e medium. F o r p > > l , S a y i r g i v e s t h e approximate r e l a t i o n
I n o r d e r t o t e s t t h e v a l i d i t y o f r e l a t i o n s ( 3 ) and ( 4 ) experiments were c a r r i e d o u t on t h e dynamics, e l a s t i c , f l e x u r a l response o f composite beams which c o n s i s t e d o f m e t a l w i r e s embedded a x i a l l y i n e i t h e r l e a d t i n a l l o y m a t r i c e s , o r m a t r i c e s o f n a t u r a l r u b b e r . The v a l u e o f E/G f o r t h e m e t a l composite, a, w h i c h c o n s i s t e d o f phosphor bronze w i r e s embedded i n a l e a d - t i n a l l o y m a t r i x , was 3.2 w h i l e one w h i c h had s t e e l p i a n o w i r e r e i n f o r c e m e n t , 8 , had a v a l u e o f E/G o f 4.75.
These two composites l e a d t o v a l u e s o f p < < l and e a u a t i o n ( 3 ) would be expected t o a p p l y t o them. The o t h e r composite c o n s i s t e d o f s t e e l w i r e s embedded i n a m a t r i x o f n a t u r a l rubber, f o r t h i s composite E/G = 7 . 4 ~ 1 0 and t h i s l e d t o v a l u e s o f p>>l, 4 so t h a t e q u a t i o n ( 4 ) would b e expected t o a p p l y .
The e x p e r i m e n t a l arrangement used f o r measuring b o t h E and cII e x p e r i m e n t a l l y i s shown d i a g r a m a t i c a l l y i n F i g . 1. I t may be seen t h a t t h e beam i s f r e e l y suspended a t i t s two ends, and i s e x c i t e d e l e c t r o m a g n e t i c a l l y e i t h e r i n l o n g i t u d i n a l o r
f l e x u r a l o s c i l l a t i o n . From t h e f r e a u e n c i e s o f l o n g i t u d i n a l resonance, E i s obtained, w h i l e from t h e f r e q u e n c i e s o f f l e x k r a l resonance, t h e v a l u e o f cII can b e i n f e r r e d . I n o r d e r t o o b t a i n an e x p e r i m e n t a l v a l u e o f G t h e beam was s e t i n t o t o r s i o n a l o s c i l l a t i o n e l e c t r o m a q n e t i c a l l y and t h e r e s o n a n t f r e q u e n c i e s measured.
A comparison between t h e e x p e r i m e n t a l o b s e r v a t i o n s and t h e p r e d i c t i o n s o f e q u a t i o n ( 3 ) i s shown i n F i g . 2. W h i l e a comparison between t h e r e s u l t s o b t a i n e d w i t h t h e r u b b e r composite and e q u a t i o n ( 4 ) i s shown i n F i g . 3. I t may be seen t h a t i n b o t h cases t h e agreement i s q u i t e good. The e x p e r i m e n t a l d e t a i l s o f t h i s work a r e d e s c r i b e d i n an e a r l i e r paper ( 3 ) .
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(b) Dynamic P l a s t i c Deformation
Now, most commercially a v a i l a b l e f i b e r r e i n f o r c e d s o l i d s have values o f E/G i n the range o f 5-20, thus f o r transverse e l a s t i c deformations o f c i r c u l a r beams, t h e wavelength Amust be v e r y small compared t o the r a d i u s o f beam R f o r p>>l so t h a t shear predominates over extension. Thus f o r such m a t e r i a l s , t h e i d e a l f i b e r - r e i n f o r c e d s o l i d gives a v e r y poor r e p r e s e n t a t i o n o f e l a s t i c deformation.
This, however, does n o t apply t o l a r g e p l a s t i c deformations, where most o f t h e s t r a i n i s i n t h e form o f shear. To show t h i s we c a r r i e d o u t some impact e x p e r i - ments on long c a n t i l e v e r beams o f t h e composite, formed by embedding s t e e l piano w i r e s i n a l e a d - t i n m a t r i x . The set-up i s shown i n Fig. 4(a), a l o n g c a n t i l e v e r specimen i s mounted w i t h i t s c e n t r a l p o r t i o n , B y h e l d l i g h t l y between two clamp jaws, two c a n t i l e v e r s e c t i o n s o f equal lenqth, A and C , p r o t r u d e from each s i d e o f
t h e clamp.
,piver hamer Loading hamner Specimen
F i g u r e 5
F i g u r e 4
The t i p o f t h e clamp A i s h i t by a t r a v e l l i n g hammer which i s a c c e l e r a t e d i n a Hyge' shock t e s t i n g machine. 'The 'Hyge' machine which i s shown s c h e m a t i c a l l y i n Fig. 5, c o n s i s t s e s s e n t i a l l y o f an a i r gun which accelerates a moving c a r r i a g e t o a speed i n t h e range o f 10-30 meters/sec. A 'hammer' i s mounted on t h e c a r r i a g e , and
C5-568 JOURNAL DE PHYSIQUE
the nose o f t h e hammer h i t s t h e specimen under i n v e s t i g a t i o n . When a l o n g beam o f an i s o t r o p i c metal, e.g. aluminum a l l o y was mounted i n t h e clamp and h i t , a s h o r t p l a s t i c hinge was formed a t t h e boundary between s e c t i o n s A and B o f t h e beam, and p l a s t i c deformation was confined t o t h e r e g i o n o f t h i s hinge. When, however, a beam o f composite formed by embeddinq s t e e l w i r e s i n l e a d a l l o y w i r e s was used t h e whole o f s e c t i o n B was sheared p l a s t i c a l l y t h e c a n t i l e v e r s A and C simply r o t a t e d as r i g i d bodies about t h e i r boundaries w i t h
B y
see F i g . 4 ( b ) .I n o r d e r t o analyze t h e deformation, a f o r c e - t i m e p r o f i l e o f the h i s t o r y o f t h e impact i s required, t h i s was obtained from t h e o u t p u t o f e l e c t r i c a l s t r a i n gages mounted on t h e nose o f t h e hammer. A 'Fastax' h i g h speed c i n 6 camera was a l s o used t o produce a c i n e r e c o r d o f t h e impact, and t h e v e l o c i t y o f t h e t r a v e l l i n g hammer immediately p r i o r t o t h e impact was measured b y determining w i t h a microsecond t i m e r t h e i n t e r v a l between two e l e c t r i c a l c o n t r a c t s made by t h e hammer. Unfortun- a t e l y , t h e p l a s t i c response o f the composite i s h i g h l y s t r a i n - r a t e dependent, so t h a t a s e r i e s o f t e s t s over a range o f s t r a i n r a t e s was c a r r i e d o u t i n a standard t e s t i n g machine, t h e r e s u l t s a r e shown i n Fig. 6. These s t r a i n r a t e s a r e lower than those which occurred i n t h e impact, however i t was shown t h a t they c o u l d be w e l l represented by an e m p i r i c a l r e l a t i o n f o r t h e dependence o f mechanical response on r a t e o f s t r a i n p u t forward by Cowper and Symonds ( 4 ) , so t h i s same r e l a t i o n was used t o determine t h e y i e l d s t r e s s a t the s t r a i n r a t e s r e l e v a n t t o t h e impact.
Shear strain ( I ) FS. 1 R.LI. d " n i n c u - ( a M - t i n .Ila,.
F i g u r e 6 F i g u r e 7
To c a r r y o u t t h e a n a l y s i s o f the impact, which i s d i f f i c u l t , t o t r e a t by exact methods, t h e Simp1 i f i e d E l a s t i c P l a s t i c technique developed by Symonds (5) was used. T h i s s p l i t s t h e h i s t o r y o f t h e deformation i n t o t h r e e p a r t s , f i r s t t h e r e i s an e l a s t i c deformation, then a p e r i o d where t h e s t r u c t u r e i s t r e a t e d as r i g i d - p l a s t i c , and f i n a l l y a second e l a s t i c response. The f i r s t period, i . e . t h e i n i t i a l e l a s t i c response was t r e a t e d by t h e method suggested by S t . Venant (6) as formulated by Warburton (7), and the second stage by t h e methods described by Symonds.
F i g u r e 7 shows a comparison between the observed and p r e d i c t e d values o f the f i n a l displacement o f t h e t i p s o f c a n t i l e v e r s A and C . I t may be seen t h a t t h e agreement i s s a t i s f a c t o r y , and t h e work i s f u l l y described i n a forthcoming p u b l i c a t i o n (8).
( c ) Very Rapid P l a s t i c Deformation
The type o f dynamic p l a s t i c deformation described i n t h e previous s e c t i o n i s q u i t e rapid, t h e d u r a t i o n o f the impacts i s o f the o r d e r o f 50-100 m i l l i s e c o n d s . How- ever t h i s d u r a t i o n i s q u i t e long enough f o r p l a s t i c waves t o have t r a v e r s e d t h e l e n g t h o f t h e beams many times, and t h e e f f e c t s o f nave propagation does n o t have t o be considered. With impacts o f v e r y much s h o r t e r d u r a t i o n however, wave propaga- t i o n can become v e r y i m p o r t a n t and Spencer, Jones and t h e i r colleagues have t r e a t e d
Now i t can be seen by examining t h e s t r e s s s t r a i n curves shown i n Fig. 6 t h a t t h e mechanical response o f t h e composites can be reasonably modeled f o r l a r g e p l a s t i c deformations by a r i g i d p l a s t i c approach w i t h l i n e a r s t r a i n hardness.
T h u s , i f t h e transverse f o r c e i s Q and t h e shear s t r a i n i s y = ;)nr/ax,where w i s the transverse displacement, we have
where Q0 i s t h e y i e l d f o r c e i n shear,which i s h i g h l y rate-dependent,and Q1 i s t h e slope o f t h e s t r a i n hardening p o r t i o n o f t h e curve and does n o t appear t o be p a r t i c u l a r l y s t r a i n rate-dependent.
Now i f we consider t h e equation o f motion o f an element o f t h e beam i n transverse motion we g e t
where m i s t h e mass o f u n i t l e n g t h o f t h e beam,and w i s t h e transverse displacement.
From ( 5 )
( 6 ) i s t h e wave equation showing t h a t p l a s t i c shear waves t r a v e l a t the constant v e l o c i t y (Q1/m) 1/2
.
The d u r a t i o n o f t h e impact can be analyzed and i t s d u r a t i o n determined. T h i s gives t h e p o i n t along t h e beam where t h e p l a s t i c wavefront stops when the beam comes t o r e s t . Experiments were c a r r i e d o u t on c a n t i l e v e r beams by detonating small e x p l o s i v e charges i n c o n t a c t w i t h s t e e l p e l l e t s t o produce s h o r t impacts a t t h e i r t i p s . The momentum conveyed was measured w i t h a b a l l i s t i c pendulum. A f t e r t h e exoeriment t h e p o s i t i o n which the p l a s t i c f r o n t had reached c o u l d be seen on t h e beam. These p o s i t i o n s were compared w i t h those p r e d i c t e d t h e o r e t i c a l l y . The r e s u l t s a r e shown i n F i g . 8 and i t can be seen t h a t t h e agreement i s s a t i s f a c t o r y . The f i n a l displacement o f t h e c a n t i l e v e r t i p was a l s o determined and compared w i t h experiments. Here once again s a t i s f a c t o r y agreement was obtained and i s shown i n Fig. 9.
F i n a l l y by using very l i g h t hammers on t h e 'Hyge' machine impacts o f s h o r t e r dura- t i o n could be achieved. These were long enough f o r t h e p l a s t i c wavefront t o reach t h e clamo, and under i d e a l c o n d i t i o n s t h e p l a s t i c wave should be r e f l e c t e d there.
I n f a c t i t was found t h a t w h i l e t h e cin6 r e c o r d showed t h e progress o f t h e p l a s t i c wavefront from t h e t i p t o t h e clamp q u i t e c l e a r l y , n o r e f l e c t e d wave was observed.
Nevertheless,the f i n a l displacement o f t h e t i p was found t o be i n reasonable agreement w i t h t h a t p r e d i c t e d by t h e theory. The r e s u l t s a r e shown i n F i g . 10.
The work i s described i n (13).
JOURNAL DE PHYSIQUE
F i n a l p o s i t i o n o f the d i s c o n t i n u i t y Qt from s t r e s s - s t r a i n r e l a t i o n
-
Q1 from v e l o c i t y measures
Experimental values z
I _
Impulse (dyn-sx1o4)
Theory
-
exper.
I__
rL
Impulse (dyn-sx104)
F i g u r e 8 F i g u r e 9
V e l o c ' t y o f t h e hammer (m-s-1)
F i g u r e 10
Acknowedgements
The authors express t h e i r thanks t o Mrs. Ezoura Fonseca f o r p r e p a r i n g t h e manuscript o f t h i s paper, t o Professor A.C. P i p k i n f o r v a l u a b l e discussions and t o Messrs.
W.F. Cary and J.F. Tracey and G.J. LaBonte, J r . f o r t h e i r t e c h n i c a l help.
The work described here was c a r r i e d o u t under Grant MEA 8200938 o f t h e N a t i o n a l Science Foundation, t h e authors wish t o r e c o r d t h e i r deep a p p r e c i a t i o n t o t h e Foundation f o r i t s generous funding.
References
( 1 ) Kolsky, H. ' S t r e s s Waves i n S o l i d s ' , Clarendon, Oxford (1953).
( 2 ) S a y i r , M. ' I n g . A r c h i v ' 49, 309 (1980)
( 3 ) Kolsky, H. and Mosquera, J.M. 'Mechanics of M a t e r i a l Behavior', E l s e v i e r , Amsterdam, p.201 (1984).
(4) Cowper, G.R. and Symonds, P.S., Brown Univ. Rept. C l l - 2 8 (1957).
( 6 ) St. Venant, B . , J . Ecole Polytechnique P a r i s ,
59,
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