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Elastic-plastic and viscoelastic behavior of a continuous filament yarn
Int. J. ~wch. Sv/. Pergamon Press. 1974. Vol. 16, pp. 679-687. Printed in Great Britain
ELASTIC-PLASTIC AND VISCOELASTIC BEHAVIOR CONTINUOUS FILAMENT YARN
OF A
NORMAN JONES D e p a r t m e n t of Ocean Engineering, Massachusetts I n s t i t u t e of Technology, Cambridge, Massachusetts 02139, U.S.A.
(Received 12 March 1973, and in revised form 18 January 1974) S u m m a r y - - T h e tensile behavior of continuous filament yarns with fibres which are made from elastic-perfectly plastic or various viscoelastic materials is discussed herein. Theoretical expressions are developed for the m a x i m u m load carrying capacity (limit load) and creep characteristics when the displacements of a y a r n remain small. NOTATION C
COS
1 defined in Fig. 1 r
t 'u
El,
radius defined in Fig. 1 time
1/L
viscous coefficients for linear viscous, Maxwell and Kelvin models, respectively E.,,, Es,, linear elastic constants for linear elastic, Maxwell and Kelvin models, respectively L magnitude of 1 which is defined in Fig. 1 when 8 = a and r = R R outside radius of y a r n T t o t a l axial load on a y a r n OL helix angle of outermost fibres 81" St" 8~ axial a n d transverse strains of a fibre and axial y a r n strain, respectively 8 helix angle defined in Fig. 1 axial and transverse stresses in a fibre ~t" fit (7o uniaxial yield stress packing factor : ratio of the fibre cross-sectional area to the y a r n cross-section F defined b y equation (32)
(')
~/~t ( ) INTRODUCTION
HEAa~LE1 has presented a comprehensive review of the work of many authors who have contributed towards an understanding of the behavior of a highly twisted continuous filament yarn. These yarns are not used frequently in practice but they are particularly convenient for theoretical and experimental investigations. Moreover, the behavior of a continuous filament yarn provides v a l u a b l e i n s i g h t i n t o t h e m e c h a n i c s o f t e x t i l e fibres, y a r n s a n d f a b r i c s , 1 t y r e s ~ and ropes and wire cables. 3 T h e t e n s i l e b e h a v i o r o f a c o n t i n u o u s f i l a m e n t y a r n , w h i c h is m a d e f r o m a l i n e a r e l a s t i c o r a n e l a s t i c - p l a s t i c m a t e r i a l , h a s b e e n e x a m i n e d b y H e a r l e et al.,i, 5 679
680
NORMAN JONES
T r e l o a r a n d H e a r l e 6 a n d v a r i o u s o t h e r a u t h o r s as d i s c u s s e d b y H e a r l e . 1 A p a r t i c u l a r l y s i m p l e e x a c t t h e o r e t i c a l a n a l y s i s is p r e s e n t e d h e r e i n for a cont i n u o u s f i l a m e n t y a r n w h i c h is m a d e f r o m a n e l a s t i c p e r f e c t l y p l a s t i c m a t e r i a l . I t is also r e m a r k e d a n d d e m o n s t r a t e d b y t h e a u t h o r e l s e w h e r e ~ h o w t h e l i m i t t h e o r e m s o f p l a s t i c i t y , w h i c h were d e r i v e d b y D r u c k e r et al., s m a y b e u s e d t o o b t a i n t h e e x a c t l i m i t or collapse load. T h e m a i n p u r p o s e o f t h i s a r t i c l e is t o e x a m i n e t h e t i m e - d e p e n d e n t or creep b e h a v i o r o f a c o n t i n u o u s f i l a m e n t y a r n since n o t h e o r e t i c a l s t u d i e s o n t h i s p h e n o m e n o n a p p e a r to have b e e n reported previously. I n the interests of b r e v i t y , t h e creep, r e c o v e r y a n d r e l a x a t i o n c h a r a c t e r i s t i c s o f y a r n s , 7 w h i c h are l o a d e d a x i a l l y a n d i d e a l i z e d as e i t h e r M a x w e l l or K e l v i n v i s c o e l a s t i c m a t e r i a l s , are o m i t t e d f r o m t h i s article. A n a x i a l l y l o a d e d c o n t i n u o u s f i l a m e n t y a r n , w h i c h is m a d e f r o m e i t h e r l i n e a r v i s c o u s or f o u r - e l e m e n t v i s c o e l a s t i c m a t e r i a l s , is d i s c u s s e d b r i e f l y h e r e i n . CONTINUOUS
FILAMENT
YARN
A continuous filament yarn is sketched in Fig. 1. Each filament or fibre of the yarn follows a helical path which lies on a cylinder with a constant radius r. Thus, a yarn is composed of concentric tubes of filaments and all the filaments of a single tube have the same helix angle 0. R ~r~ _ __
2 f i r
I
,
h
h
FIG. 1. Continuous filament yarn. S i m p l e model I t is straightforward to show 1 when only tensile strains a r e p r e s e n t in t h e filaments that el = ~ cos ~0 (1) provided no change in the yarn diameter occurs during extension and where el( = $l/1) and e~( = 8h/h) a r e t h e f i l a m e n t a n d y a r n s t r a i n s , respectively. The total axial load (T) on a yarn is T = . l : a r 27rr cos 2 0tF dr,
(2)*
where a 1 is the tensile stress in a f i l a m e n t a t r a d i u s r a n d R is t h e r a d i u s o f t h e o u t e r m o s t fibres. ~F is known as a packing factor and is the ratio of the filament cross-sectional a r e a to the corresponding y a r n cross-section. * I t can be shown t h a t the total axial load (T) is accompanied b y a total torsional moment, M = S~ ~I 21rrSsin 0 cos 0LF dr, which acts about the yarn axis. This twisting m o m e n t would be removed when alternating the lay of the filaments.
E l a s t i c - p l a s t i c a n d viscoelastic b e h a v i o r of a c o n t i n u o u s f i l a m e n t y a r n
681
R e f i n e d model T h e s i m p l e m o d e l of a y a r n , w h i c h w a s i n t r o d u c e d in t h e l a s t s e c t i o n , is n o w refined in o r d e r t o c a t e r for t h e d i a m e t r a l c o n t r a c t i o n o f a y a r n a n d t h e influence o f t r a n s v e r s e stresses. T h e d i a m e t e r of a fibre is a s s u m e d t o h e sufficiently s m a l l so t h a t t h e e l e m e n t o f a y a r n , w h i c h is s h o w n i n Fig. 2, is c o m p o s e d of a large n u m b e r of fibres.
•
!
,17
FIG. 2.
dr
E l e m e n t o f a continuous f i l a m e n t y a r n .
A c o n s i d e r a t i o n o f t h e g e o m e t r i c a l r e q u i r e m e n t s for t h i s m o d e l 1 gives ej = e~(cos ~ O - v , sin 2 0),
(3)
where v~ =
dr/r dh/h
(4)
is k n o w n a s t h e y a r n l a t e r a l c o n t r a c t i o n r a t i o . T h e r a d i a l e q u i l i b r i u m e q u a t i o n for t h e y a r n e l e m e n t i l l u s t r a t e d i n Fig. 2 is 1 c3at
4"tr2
7 ~-~ = - W ( o l - a,) w h e n i t is a s s u m e d t h a t t h e s t r e s s e s (at) in t h e t w o t r a n s v e r s e d i r e c t i o n s a r e e q u a l . T h i s equation can be written in the form ~a_...~t= ~a t - ,- at i~u u
(5)
u = l/L
(6)
where
682
NORMAbT JONES
a n d 1e = h 2 + 41r2 r e f r o m Fig. 1. I t is e v i d e n t t h a t t h e t o t a l a x i a l load o n t h e y a r n in t h i s case is*
T =
(alcoseO+a~sineO) 2rrr~Fdr
or
l
T-
o,&o,)o u
~---:-~ A
(7)*
where c
=
cosa
=
h/L.
ELASTIC-PLASTIC
(8)
BEHAVIOR
I n o r d e r t o e x a m i n e t h e e l a s t i c - p l a s t i c b e h a v i o r of a c o n t i n u o u s f i l a m e n t y a r n t it is a s s u m e d t h a t t h e fibres a r e m a d e f r o m a n elastic p e r f e c t l y p l a s t i c m a t e r i a l w i t h a Y o u n g ' s m o d u l u s E I a n d a u n i a x i a l t e n s i l e y i e l d s t r e s s a0. I t is e v i d e n t f r o m e q u a t i o n (3) t h a t t h e l a r g e s t a x i a l s t r a i n occurs a t t h e y a r n c e n t e r . T h e r e f o r e , t h e y i e l d s t r e s s is first e x c e e d e d b y t h o s e fibres w h i c h a r e l o c a t e d a t t h e c e n t e r o f a y a r n . A s t h e a x i a l l o a d is i n c r e a s e d f u r t h e r a p l a s t i c zone d e v e l o p s w i t h i n t h e c e n t r a l p o r t i o n 0 ~
--
-
w h e r e al, a2 a n d aa a r e p r i n c i p a l stresses. E q u a t i o n (9) r e d u c e s t o a 0 = a!--at
(10)
for t h e r e f i n e d m o d e l a n d is i d e n t i c a l t o t h e p r e d i c t i o n of t h e T r e s c a y i e l d c o n d i t i o n . N o w s u b s t i t u t i n g e q u a t i o n (10) i n t o t h e e q u i l i b r i u m e q u a t i o n (5) a n d i n t e g r a t i n g gives at = a0 log, u + a 0 C1
( 11 )
al = ao(l +log~u+Cl),
(12)
and w h e r e C~ is a n a r b i t r a r y c o n s t a n t of i n t e g r a t i o n . T h e s t r e s s e s i n t h e o u t e r elastic zone p ~
and
{v~(1--u eva-l)
(1 + v , ) ( 1 - u en+t) c ~
w h e r e vl is a n a x i a l P o i s s o n ' s r a t i o ( t r a n s v e r s e s t r a i n / a x i a l s t r a i n ) for a t e n s i l e load o n a fibre. E q u a t i o n s (13) a n d (14) a g r e e w i t h t h e c o r r e s p o n d i n g r e s u l t s g i v e n b y Hearle. 1 I n o r d e r t o s i m p l i f y t h e following t h e o r e t i c a l p r e s e n t a t i o n it is a s s u m e d t h a t d e f o r m a t i o n o c c u r s a t c o n s t a n t v o l u m e , i.e. vl = vv = ½. T h u s ,
E f t ' { 3(l+u~)ce (~I = --~-~ ~
l-log, u)
(15)
E1 e~ (3(1 - u 2) c ~ 4-log, u~.
(16)
a n d
at=-
2
(
/
* I t c a n b e s h o w n t h a t t h e t o t a l a x i a l l o a d (T) is a c c o m p a n i e d b y a t o r s i o n a l m o m e n t , M = ~ (a I - a t ) s i n 0 cos 0 2rrr e t F d r , w h i c h a c t s a b o u t t h e y a r n axis. H o w e v e r , M - - - 0 w h e n t h e l a y o f t h e fibres is a l t e r n a t e d . t T h e refined m o d e l o f a y a r n is e m p l o y e d t h r o u g h o u t t h i s article w h i l e t h e r i g i d - p l a s t i c a n d e l a s t i c - p l a s t i c b e h a v i o r a c c o r d i n g t o t h e s i m p l e m o d e l o f a y a r n is d e v e l o p e d elsewhere3
E l a s t i c - p l a s t i c a n d viscoelastic b e h a v i o r of a continuous filament y a r n
683
T h e c o n t i n u i t y of at (equations (11) a n d ( 16 )) b e t w e e n t h e inner plastic a n d o u t e r elastic zones at r = p or u -- u 1 gives C1 -- - E tu 1e~ 2 a\(l°g" o
+~-~u1~3c2( 1 - u ~ ) ) - l o g , u 1
(17)
while t h e inner edge (r = p or u = u~) of t h e o u t e r elastic zone yields plastically w h e n 2
)
\u~
1 = a0
(18)
a c c o r d i n g to e q u a t i o n s (10), (15) a n d (16). T h e t o t a l axial load on a y a r n is
~R2LFrE, e, [
. ,
9c'
_ u : ) + ~ ( 6 c , log,(ujc)+c~_u:)]
(19)
which is o b t a i n e d b y s u b s t i t u t i n g e q u a t i o n s (11), (12) a n d (17) a n d e q u a t i o n s (15) a n d (I6) into e q u a t i o n (7) for t h e regions c<<.u~u I (O<<.r~p) a n d Ul<~U<~ 1 (D<~r<~R), respectively. T h e non-dimensionalized p a r a m e t e r ul, which m a r k s t h e b o u n d a r y b e t w e e n t h e elastic and plastic regions, is g i v e n b y e q u a t i o n (18). I t is e v i d e n t t h a t e q u a t i o n (19) predicts
~'1 + 9c 2 + 3c 2 log, c] T = ~ R ~ ~FEr ~, \ ~ - - -1] - : - ~ - ]
(20)
w h e n t h e entire y a r n is elastic (i.e. p -- 0 or u 1 = c) which agrees w i t h Hcarle. 1 T h e entire y a r n is plastic w h e n p = R or u~ = 1 a n d T -- IrR 2 a0 ~F {3c2 loge(1/c) This is t h e e x a c t collapse or l i m i t load* according to t h e limit t h e o r e m s of plasticity, 8 since a statieMly admissible stress field has b e e n associated w i t h e q u a t i o n (21), while it is also possible to associate a k i n e m a t i c a l l y admissible d i s p l a c e m e n t field. The e x a c t limit load (equation (21)) can be o b t a i n e d w i t h far less effort 7 w h e n t h e filaments of a y a r n are idealized as a rigid-perfectly plastic m a t e r i a l , since t h e limit t h e o r e m s o f p l a s t i c i t y t o g e t h e r w i t h a n u m b e r of corollaries, which were d e v e l o p e d b y D r u c k c r et al., s show t h a t t h e m a x i m u m load of a c o n t i n u u m or s t r u c t u r e is i n d e p e n d e n t of t h e influence of m a t e r i a l elasticity. The v a r i a t i o n of t h e axial load (T) w i t h t h e axial y a r n strain (e~) according to e q u a t i o n (19) is s k e t c h e d in Fig. 3. Plastic collapse of the entire y a r n in t h e refined m o d e l
T ,rrRe~I,cro 9c 2- I 4
-r'
I
r .....
c---c' 3 c'c'c'c'cI'c'~-
Ef£y %
FIG. 3. T o t a l axial load (T) vs t h e y a r n strain (e~) for a y a r n m a d e f r o m a n e l a s t i c - p e r f e c t l y plastic material. * A t w i s t i n g m o m e n t a c c o m p a n i e s this tensile load w h e n all t h e fibres h a v e t h e s a m e lay as discussed in t h e f o o t n o t e associated w i t h e q u a t i o n (7).
684
NORMAN JONES
Occurs w h e n eu = 2ao/E1(3c 2 - 1), while p l a s t i c y i e l d i n g first occurs a t e~ = (rg/Es. A n i n n e r p l a s t i c zone is s u r r o u n d e d b y a n o u t e r elastic r e g i o n w i t h i n t h e s t r a i n r a n g e % / E s ~
VISCOUS
BEHAVIOR
I-Iooke's law for a l i n e a r elastic isotropic m a t e r i a l m a y b e w r i t t e n (e.g. D r u c k e r ) 9 E vE a,.j = -]--~v eii + (1 + v ) ( 1 - - 2 v ) e ~ S i ~ ,
(22)*
w h i l e t h e c o n s t i t u t i v e e q u a t i o n for a l i n e a r viscous i s o t r o p i c m a t e r i a l is
C ai~ = ~
mC
(23)
eij -~ (1 + m) (1 - 2m) ekk $i¢,
w h e r e (°) is a t i m e d e r i v a t i v e , C is a viscous coefficient a n d m is P o i s s o n ' s ratio. I f a l i n e a r viscous m a t e r i a l is i n c o m p r e s s i b l e (m = ½) as is c u s t o m a r i l y a s s u m e d , t h e n i t c a n b e s h o w n 9 that
ei¢ = 3Si¢/2C.
(24)t
I t is clear t h a t e q u a t i o n (23) c a n b e o b t a i n e d f r o m e q u a t i o n (22) b y r e p l a c i n g s t r a i n s b y s t r a i n r a t e s a n d s u b s t i t u t i n g C a n d m for E a n d v, r e s p e c t i v e l y . T h u s , t h e b e h a v i o r of a y a r n w i t h fibres w h i c h a r e m a d e f r o m a l i n e a r viscous m a t e r i a l c a n b e o b t a i n e d d i r e c t l y f r o m a s o l u t i o n for a l i n e a r elastic y a r n p r o v i d e d t h e a p p r o p r i a t e s u b s t i t u t i o n s a r e m a d e . E q u a t i o n (20) p r e d i c t s t h a t t h e a x i a l s t r a i n r a t e in a y a r n w h i c h is s u b j e c t e d to a n a x i a l l o a d T is
T
(1+9c~+3c21og~c~ -1,
~ = rrR ~ ~CI [ ~
_--T-ZT~] 1
(25)
w h i l e f r o m e q u a t i o n (3) ( w i t h vv = ½) t h e s t r a i n r a t e i n a fibre is
~ 13c 2
~, = y ( ~ - ~ - 1).
(26)
I t is s t r a i g h t f o r w a r d t o s h o w t h a t t h e a x i a l s t r e s s i n a fibre m a y b e w r i t t e n
T
[3(l+u~)(c~/u~)-2(l+log,
at = rrR 2 ~F \ 1 7 ~
1-i-~Tc2 ( log-~)~
u)~ ]
(27)
w h i c h is i d e n t i c a l t o t h e c o r r e s p o n d i n g l i n e a r elastic case. T h e t o t a l s t r a i n a c q u i r e d b y a y a r n d u r i n g a t i m e i n t e r v a l r is 6~ =
VISCOELASTIC
I, dr.
(28)
BEHAVIOR
A M a x w e l l viscoelastic m a t e r i a l m a y b e r e p r e s e n t e d as a l i n e a r elastic s p r i n g in series w i t h a l i n e a r viscous e l e m e n t . T h e r e s p o n s e of t h i s a r r a n g e m e n t is clearly g i v e n b y t h e
* a~j and 6~ are the stress and strain tensors, respectively, where t h e s u b s c r i p t s i a n d j h a v e t h e r a n g e 1-3 ; ~t~ is t h e K r o n e c k e r d e l t a . T h e s u m m a t i o n c o n v e n t i o n is u s e d so t h a t ~kk ~ 611 ~ g22 "~ 633"
t S~j = ais - o~k ~ j / 3 is k n o w n as a s t r e s s d e v i a t o r .
E l a s t i c - p l a s t i c a n d viscoelastic b e h a v i o r of a continuous filament y a r n
685
s u p e r p o s i t i o n of t h e responses of t h e two i n d i v i d u a l elements, or l+v
~j = ~
1 - 2 v ~ k ~ s _ ~ 3S~j
~iJ÷ Es~
3
(29)*
2Ot~"
A K e l v i n t viscoelastic m a t e r i a l m a y be r e p r e s e n t e d as a linear elastic spring which is in parallel w i t h a linear viscous element. The strains across t h e two e l e m e n t s are equal in this idealization. I t can be shown 9 t h a t t h e corresponding c o n s t i t u t i v e relation m a y be written
a~j = ~Esk e~ + ~Csk~ + ~ . ~i~.
(30)$
I t is n o w s t r a i g h t f o r w a r d w i t h t h e aid of t h e g e o m e t r i c a l r e l a t i o n (3), e q u i l i b r i u m e q u a t i o n s (5) a n d (7) a n d t h e c o n s t i t u t i v e relations (22), (24), (29) a n d (30) to e x a m i n e t h e b e h a v i o r of y a r n s w i t h fibres w h i c h are c o n s t r u c t e d f r o m Maxwell, K e l v i n , t h r e e - e l e m e n t or four-element viscoelastic m a t e r i a l s or o t h e r idealizations which h a v e been studied b y various authors, a, ~0 T h e loading, unloading, creep and r e l a x a t i o n response of y a r n s w i t h fibres which are m a d e f r o m either Maxell or K e l v i n viscoelastic m a t e r i a l s h a v e been r e p o r t e d elsewhere3 The four-element viscoelastic idealization is f o r m e d b y placing t h e Maxwell a n d K e l v i n e l e m e n t s in series as indicated in Fig. 4. I t can be shown for t h e E
fk
Eim
5m
~
ii
It-_ Cfk
Fro. 4. Four-element viscoelastic idealization.
p a r t i c u l a r case of a u n i f o r m axial load T, w h i c h is applied at t i m e t = 0 as shown in Fig. 5, t h a t t h e axial strain of a y a r n w i t h fibres obeying t h e four-element viscoelastic idealization is e~ = 7
\Es.~
Cs.
,
(31)§
where i~ = 1+9c ~
3c 2logic ~ l_c------V-
(32)
which is o b t a i n e d f r o m e q u a t i o n s (3), (5), (7), (29) a n d (30) since T has t h e s a m e v a l u e in t h e Maxwell elastic a n d viscous elements a n d K e l v i n model, while t h e t o t a l strain is t h e s u m of those in t h e Maxwell elastic a n d viscous e l e m e n t s a n d t h e K e l v i n model, q, 9, 10 T h e c o n s t a n t s of i n t e g r a t i o n are d e t e r m i n e d f r o m t h e r e q u i r e m e n t s t h a t at -- 0 a t u = 1 for all t a n d e~ = 0 at t = 0 for t h e Maxwell viscous e l e m e n t a n d K e l v i n model. W h e n EI~ -~ ¢¢, t h e n t h e first two t e r m s in e q u a t i o n (31) r e m a i n a n d predict t h e response of an axially loaded y a m w i t h fibres w h i c h are idealized as a Maxwell m a t e r i a l . T h e first t e r m is t h e t i m e - i n d e p e n d e n t initial elastic response related to EI~, while t h e second t e r m increases linearly w i t h time, is irreversible, a n d is c o n t r i b u t e d b y t h e viscous element. * The first t w o t e r m s are g i v e n b y t h e t i m e d e r i v a t i v e of t h e elastic strain which is o b t a i n e d b y i n v e r t i n g e q u a t i o n (22). T h e last t e r m is identical to e q u a t i o n (24). t This m o d e l is s o m e t i m e s n a m e d a f t e r Voigt. $ The elastic a n d viscous e l e m e n t s are a s s u m e d to be incompressible a n d isotropic. § I t is a s s u m e d t h r o u g h o u t this article t h a t t h e axial load T is applied fast enough to a v o i d significant viscous d e f o r m a t i o n (e.g. rise t i m e of T,~ C,s~/E1~) a n d y e t slow e n o u g h to a v o i d inertia effects.
686
NORMAN J o ~ E s
f Elm -->~ and Csm -->~, t h e n the last t e r m in e q u a t i o n (31) remains and gives the b e h a v i o u r of a y a r n which is m a d e from a K e l v i n material. I t is clear in this case t h a t ~y is initially zero and a p p r o a c h e s the elastic v a l u e when t >>C1k/Elk. ]
T
o
z"
t
-EfkZ"
6y~R2EfmF~
I + Efm T Jr Efm (I-e cfk Cfr n Efk ~,
)
~ ~ ~ _ _Efk r
i
~'x'Efrn6ecfk 1 Ef m --
fL~m; o
z"
t
FIG. 5. Creep and r e c o v e r y characteristics of a y a r n m a d e from a four-element viscoelastic idealization. The v a r i a t i o n of ev w i t h t i m e according to e q u a t i o n (31) for t h e four-element m o d e l is s k e t c h e d in Fig. 5 t o g e t h e r w i t h t h e u n l o a d i n g or r e c o v e r y characteristics which can be found in a s t r a i g h t f o r w a r d m a n n e r f r o m t h e basic e q u a t i o n s g i v e n in this article. I t is e v i d e n t t h a t t h e spring Elm i m m e d i a t e l y recovers on u n l o a d i n g a t t = ~, while the K e l v i n c o m p o n e n t s (E1~, Clk ) r e t u r n e v e n t u a l l y to t h e i r original s t a t e leaving only t h e irreversible strain a c c u m u l a t e d in the viscous e l e m e n t Csm. CONCLUSIONS
The
continuousfilamentyarnwhich
simple analytical studies of a are presented herein are an attempt to incorporate the more realistic properties of many yarns into the theoretical models of their behavior. As remarked in the Introduction, a continuous filament yarn is not used widely in practice but it is a particularly useful tool for revealing the mechanics of textile fibres, yarns and fabrics, tyres, ropes and wire cables. The rigid-plastic prediction of equation (21) provides a rigorously correct maximum axial load carrying capacity when the yarn material is timeindependent, does not exhibit material strain hardening and the displacements remain infinitesimal. Equation (19) predicts the magnitude of the yarn strain (or displacement) which is associated with any axial load up to the maximum value.
Elastic-plastic and viscoelastic behavior of a continuous filament yarn
687
I t is i n d i c a t e d in Fig. 5 a n d Figs. 6 - 8 o f ref. (7) h o w c o n t i n u o u s filament y a r n s creep, r e c o v e r a n d r e l a x w h e n s u b j e c t e d t o t h e static l o a d h i s t o r y which is s h o w n in Fig. 5. T h e t i m e - d e p e n d e n t characteristics of t h e y a r n s in ref. (7) a n d herein are e x a m i n e d b y a s s u m i n g t h a t t h e fibres are m a d e f r o m m a t e r i a l s w h i c h e x h i b i t linear viscous, Maxwell, K e l v i n or f o u r - e l e m e n t m o d e l viscoelastic properties. Various simplifications h a v e b e e n i n t r o d u c e d into the calculations including t h e a s s u m p t i o n s of isotropy, i n c o m p r e s s i b i l i t y a n d l i n e a r i t y of t h e m a t e r i a l s a n d t h e a s s u m p t i o n t h a t t h e response involves infinitesimal d i s p l a c e m e n t s a n d strains. Clearly, it is i m p o r t a n t for a designer to assess w h e t h e r these simplifications are r e a s o n a b l e for t h e p r o b l e m a t h a n d . I n t h e a p p r o p r i a t e cases, these simple predictions should be sufficient to enable a n e s t i m a t e of t h e i m p o r t a n c e of viscoelastic characteristics to be m a d e . H o w e v e r , it w o u l d be s t r a i g h t f o r w a r d to e x t e n d these m e t h o d s in order to describe t h e response arising f r o m m o r e c o m p l i c a t e d loading histories a n d of y a r n s which are m a d e f r o m fibres e x h i b i t i n g t i m e - d e p e n d e n t characteristics which do n o t c o n f o r m to t h e simple models e x a m i n e d herein. Acknowledgements--The author is indebted to the Engineering Mechanics Program of the
National Science Foundation for their support of this work under Contract No. G K 20189X. The author wishes to record his appreciation to Professor S. Backer of the Department of Mechanical Engineering for introducing him to the fascinating field of textile technology and to Dean A. H. Keil and Professor I. Dyer for their interest and encouragement in this project. REFERENCES 1. J. W. S. HEARLE, P. GROSBERGand S. BACKER,Structural Mechanics of Fibres, Yarns and Fabrics, Vol. 1, Chapter 4. Wiley, l~ew York (1969). 2. S. BACKER, Tire Cord Structure and Properties, Nat. Bur. of Standards, Monograph on Mechanics of Pneumatic Tires, p. 63 (1971). 3. C. W. BERT and R. A. S~Er~, Wire and Wire Products 37, 621-624, 769, 770, 772, 816 (1962). 4. J. W. S. HEARLE, J. Textile Inst. 49, T389 (1958). 5. J. W. S. HEARLE, H. M. A. E. EL-BEHERY and V. M. THAKUR, J. Textile Inst. 52, T197 (1961). 6. L. R. G. TRELOARand J. W. S. HEARLE, J. Textile Inst. 53, T446 (1962). 7. N. JOI~ES, Department of Ocean Eng., Report 73-4. M.I.T., Cambridge, Massachusetts (1973). 8. D. C. DRUCKER, W. PRAGER and H. J. GREE~BERG, Quart. appl. Math. 9, 381 (1952). 9. D. C. DRVCKER, Introduction to Mechanics of Deformable Solids. McGraw-Hill, New York (1967). 10. D. R. BLAND, The Theory of Linear Viscoelasticity. Pergamon Press, London (1960).
ELASTIC-PLASTIC AND VISCOELASTIC BEHAVIOR CONTINUOUS FILAMENT YARN
OF A
NORMAN JONES D e p a r t m e n t of Ocean Engineering, Massachusetts I n s t i t u t e of Technology, Cambridge, Massachusetts 02139, U.S.A.
(Received 12 March 1973, and in revised form 18 January 1974) S u m m a r y - - T h e tensile behavior of continuous filament yarns with fibres which are made from elastic-perfectly plastic or various viscoelastic materials is discussed herein. Theoretical expressions are developed for the m a x i m u m load carrying capacity (limit load) and creep characteristics when the displacements of a y a r n remain small. NOTATION C
COS
1 defined in Fig. 1 r
t 'u
El,
radius defined in Fig. 1 time
1/L
viscous coefficients for linear viscous, Maxwell and Kelvin models, respectively E.,,, Es,, linear elastic constants for linear elastic, Maxwell and Kelvin models, respectively L magnitude of 1 which is defined in Fig. 1 when 8 = a and r = R R outside radius of y a r n T t o t a l axial load on a y a r n OL helix angle of outermost fibres 81" St" 8~ axial a n d transverse strains of a fibre and axial y a r n strain, respectively 8 helix angle defined in Fig. 1 axial and transverse stresses in a fibre ~t" fit (7o uniaxial yield stress packing factor : ratio of the fibre cross-sectional area to the y a r n cross-section F defined b y equation (32)
(')
~/~t ( ) INTRODUCTION
HEAa~LE1 has presented a comprehensive review of the work of many authors who have contributed towards an understanding of the behavior of a highly twisted continuous filament yarn. These yarns are not used frequently in practice but they are particularly convenient for theoretical and experimental investigations. Moreover, the behavior of a continuous filament yarn provides v a l u a b l e i n s i g h t i n t o t h e m e c h a n i c s o f t e x t i l e fibres, y a r n s a n d f a b r i c s , 1 t y r e s ~ and ropes and wire cables. 3 T h e t e n s i l e b e h a v i o r o f a c o n t i n u o u s f i l a m e n t y a r n , w h i c h is m a d e f r o m a l i n e a r e l a s t i c o r a n e l a s t i c - p l a s t i c m a t e r i a l , h a s b e e n e x a m i n e d b y H e a r l e et al.,i, 5 679
680
NORMAN JONES
T r e l o a r a n d H e a r l e 6 a n d v a r i o u s o t h e r a u t h o r s as d i s c u s s e d b y H e a r l e . 1 A p a r t i c u l a r l y s i m p l e e x a c t t h e o r e t i c a l a n a l y s i s is p r e s e n t e d h e r e i n for a cont i n u o u s f i l a m e n t y a r n w h i c h is m a d e f r o m a n e l a s t i c p e r f e c t l y p l a s t i c m a t e r i a l . I t is also r e m a r k e d a n d d e m o n s t r a t e d b y t h e a u t h o r e l s e w h e r e ~ h o w t h e l i m i t t h e o r e m s o f p l a s t i c i t y , w h i c h were d e r i v e d b y D r u c k e r et al., s m a y b e u s e d t o o b t a i n t h e e x a c t l i m i t or collapse load. T h e m a i n p u r p o s e o f t h i s a r t i c l e is t o e x a m i n e t h e t i m e - d e p e n d e n t or creep b e h a v i o r o f a c o n t i n u o u s f i l a m e n t y a r n since n o t h e o r e t i c a l s t u d i e s o n t h i s p h e n o m e n o n a p p e a r to have b e e n reported previously. I n the interests of b r e v i t y , t h e creep, r e c o v e r y a n d r e l a x a t i o n c h a r a c t e r i s t i c s o f y a r n s , 7 w h i c h are l o a d e d a x i a l l y a n d i d e a l i z e d as e i t h e r M a x w e l l or K e l v i n v i s c o e l a s t i c m a t e r i a l s , are o m i t t e d f r o m t h i s article. A n a x i a l l y l o a d e d c o n t i n u o u s f i l a m e n t y a r n , w h i c h is m a d e f r o m e i t h e r l i n e a r v i s c o u s or f o u r - e l e m e n t v i s c o e l a s t i c m a t e r i a l s , is d i s c u s s e d b r i e f l y h e r e i n . CONTINUOUS
FILAMENT
YARN
A continuous filament yarn is sketched in Fig. 1. Each filament or fibre of the yarn follows a helical path which lies on a cylinder with a constant radius r. Thus, a yarn is composed of concentric tubes of filaments and all the filaments of a single tube have the same helix angle 0. R ~r~ _ __
2 f i r
I
,
h
h
FIG. 1. Continuous filament yarn. S i m p l e model I t is straightforward to show 1 when only tensile strains a r e p r e s e n t in t h e filaments that el = ~ cos ~0 (1) provided no change in the yarn diameter occurs during extension and where el( = $l/1) and e~( = 8h/h) a r e t h e f i l a m e n t a n d y a r n s t r a i n s , respectively. The total axial load (T) on a yarn is T = . l : a r 27rr cos 2 0tF dr,
(2)*
where a 1 is the tensile stress in a f i l a m e n t a t r a d i u s r a n d R is t h e r a d i u s o f t h e o u t e r m o s t fibres. ~F is known as a packing factor and is the ratio of the filament cross-sectional a r e a to the corresponding y a r n cross-section. * I t can be shown t h a t the total axial load (T) is accompanied b y a total torsional moment, M = S~ ~I 21rrSsin 0 cos 0LF dr, which acts about the yarn axis. This twisting m o m e n t would be removed when alternating the lay of the filaments.
E l a s t i c - p l a s t i c a n d viscoelastic b e h a v i o r of a c o n t i n u o u s f i l a m e n t y a r n
681
R e f i n e d model T h e s i m p l e m o d e l of a y a r n , w h i c h w a s i n t r o d u c e d in t h e l a s t s e c t i o n , is n o w refined in o r d e r t o c a t e r for t h e d i a m e t r a l c o n t r a c t i o n o f a y a r n a n d t h e influence o f t r a n s v e r s e stresses. T h e d i a m e t e r of a fibre is a s s u m e d t o h e sufficiently s m a l l so t h a t t h e e l e m e n t o f a y a r n , w h i c h is s h o w n i n Fig. 2, is c o m p o s e d of a large n u m b e r of fibres.
•
!
,17
FIG. 2.
dr
E l e m e n t o f a continuous f i l a m e n t y a r n .
A c o n s i d e r a t i o n o f t h e g e o m e t r i c a l r e q u i r e m e n t s for t h i s m o d e l 1 gives ej = e~(cos ~ O - v , sin 2 0),
(3)
where v~ =
dr/r dh/h
(4)
is k n o w n a s t h e y a r n l a t e r a l c o n t r a c t i o n r a t i o . T h e r a d i a l e q u i l i b r i u m e q u a t i o n for t h e y a r n e l e m e n t i l l u s t r a t e d i n Fig. 2 is 1 c3at
4"tr2
7 ~-~ = - W ( o l - a,) w h e n i t is a s s u m e d t h a t t h e s t r e s s e s (at) in t h e t w o t r a n s v e r s e d i r e c t i o n s a r e e q u a l . T h i s equation can be written in the form ~a_...~t= ~a t - ,- at i~u u
(5)
u = l/L
(6)
where
682
NORMAbT JONES
a n d 1e = h 2 + 41r2 r e f r o m Fig. 1. I t is e v i d e n t t h a t t h e t o t a l a x i a l load o n t h e y a r n in t h i s case is*
T =
(alcoseO+a~sineO) 2rrr~Fdr
or
l
T-
o,&o,)o u
~---:-~ A
(7)*
where c
=
cosa
=
h/L.
ELASTIC-PLASTIC
(8)
BEHAVIOR
I n o r d e r t o e x a m i n e t h e e l a s t i c - p l a s t i c b e h a v i o r of a c o n t i n u o u s f i l a m e n t y a r n t it is a s s u m e d t h a t t h e fibres a r e m a d e f r o m a n elastic p e r f e c t l y p l a s t i c m a t e r i a l w i t h a Y o u n g ' s m o d u l u s E I a n d a u n i a x i a l t e n s i l e y i e l d s t r e s s a0. I t is e v i d e n t f r o m e q u a t i o n (3) t h a t t h e l a r g e s t a x i a l s t r a i n occurs a t t h e y a r n c e n t e r . T h e r e f o r e , t h e y i e l d s t r e s s is first e x c e e d e d b y t h o s e fibres w h i c h a r e l o c a t e d a t t h e c e n t e r o f a y a r n . A s t h e a x i a l l o a d is i n c r e a s e d f u r t h e r a p l a s t i c zone d e v e l o p s w i t h i n t h e c e n t r a l p o r t i o n 0 ~
--
-
w h e r e al, a2 a n d aa a r e p r i n c i p a l stresses. E q u a t i o n (9) r e d u c e s t o a 0 = a!--at
(10)
for t h e r e f i n e d m o d e l a n d is i d e n t i c a l t o t h e p r e d i c t i o n of t h e T r e s c a y i e l d c o n d i t i o n . N o w s u b s t i t u t i n g e q u a t i o n (10) i n t o t h e e q u i l i b r i u m e q u a t i o n (5) a n d i n t e g r a t i n g gives at = a0 log, u + a 0 C1
( 11 )
al = ao(l +log~u+Cl),
(12)
and w h e r e C~ is a n a r b i t r a r y c o n s t a n t of i n t e g r a t i o n . T h e s t r e s s e s i n t h e o u t e r elastic zone p ~
and
{v~(1--u eva-l)
(1 + v , ) ( 1 - u en+t) c ~
w h e r e vl is a n a x i a l P o i s s o n ' s r a t i o ( t r a n s v e r s e s t r a i n / a x i a l s t r a i n ) for a t e n s i l e load o n a fibre. E q u a t i o n s (13) a n d (14) a g r e e w i t h t h e c o r r e s p o n d i n g r e s u l t s g i v e n b y Hearle. 1 I n o r d e r t o s i m p l i f y t h e following t h e o r e t i c a l p r e s e n t a t i o n it is a s s u m e d t h a t d e f o r m a t i o n o c c u r s a t c o n s t a n t v o l u m e , i.e. vl = vv = ½. T h u s ,
E f t ' { 3(l+u~)ce (~I = --~-~ ~
l-log, u)
(15)
E1 e~ (3(1 - u 2) c ~ 4-log, u~.
(16)
a n d
at=-
2
(
/
* I t c a n b e s h o w n t h a t t h e t o t a l a x i a l l o a d (T) is a c c o m p a n i e d b y a t o r s i o n a l m o m e n t , M = ~ (a I - a t ) s i n 0 cos 0 2rrr e t F d r , w h i c h a c t s a b o u t t h e y a r n axis. H o w e v e r , M - - - 0 w h e n t h e l a y o f t h e fibres is a l t e r n a t e d . t T h e refined m o d e l o f a y a r n is e m p l o y e d t h r o u g h o u t t h i s article w h i l e t h e r i g i d - p l a s t i c a n d e l a s t i c - p l a s t i c b e h a v i o r a c c o r d i n g t o t h e s i m p l e m o d e l o f a y a r n is d e v e l o p e d elsewhere3
E l a s t i c - p l a s t i c a n d viscoelastic b e h a v i o r of a continuous filament y a r n
683
T h e c o n t i n u i t y of at (equations (11) a n d ( 16 )) b e t w e e n t h e inner plastic a n d o u t e r elastic zones at r = p or u -- u 1 gives C1 -- - E tu 1e~ 2 a\(l°g" o
+~-~u1~3c2( 1 - u ~ ) ) - l o g , u 1
(17)
while t h e inner edge (r = p or u = u~) of t h e o u t e r elastic zone yields plastically w h e n 2
)
\u~
1 = a0
(18)
a c c o r d i n g to e q u a t i o n s (10), (15) a n d (16). T h e t o t a l axial load on a y a r n is
~R2LFrE, e, [
. ,
9c'
_ u : ) + ~ ( 6 c , log,(ujc)+c~_u:)]
(19)
which is o b t a i n e d b y s u b s t i t u t i n g e q u a t i o n s (11), (12) a n d (17) a n d e q u a t i o n s (15) a n d (I6) into e q u a t i o n (7) for t h e regions c<<.u~u I (O<<.r~p) a n d Ul<~U<~ 1 (D<~r<~R), respectively. T h e non-dimensionalized p a r a m e t e r ul, which m a r k s t h e b o u n d a r y b e t w e e n t h e elastic and plastic regions, is g i v e n b y e q u a t i o n (18). I t is e v i d e n t t h a t e q u a t i o n (19) predicts
~'1 + 9c 2 + 3c 2 log, c] T = ~ R ~ ~FEr ~, \ ~ - - -1] - : - ~ - ]
(20)
w h e n t h e entire y a r n is elastic (i.e. p -- 0 or u 1 = c) which agrees w i t h Hcarle. 1 T h e entire y a r n is plastic w h e n p = R or u~ = 1 a n d T -- IrR 2 a0 ~F {3c2 loge(1/c) This is t h e e x a c t collapse or l i m i t load* according to t h e limit t h e o r e m s of plasticity, 8 since a statieMly admissible stress field has b e e n associated w i t h e q u a t i o n (21), while it is also possible to associate a k i n e m a t i c a l l y admissible d i s p l a c e m e n t field. The e x a c t limit load (equation (21)) can be o b t a i n e d w i t h far less effort 7 w h e n t h e filaments of a y a r n are idealized as a rigid-perfectly plastic m a t e r i a l , since t h e limit t h e o r e m s o f p l a s t i c i t y t o g e t h e r w i t h a n u m b e r of corollaries, which were d e v e l o p e d b y D r u c k c r et al., s show t h a t t h e m a x i m u m load of a c o n t i n u u m or s t r u c t u r e is i n d e p e n d e n t of t h e influence of m a t e r i a l elasticity. The v a r i a t i o n of t h e axial load (T) w i t h t h e axial y a r n strain (e~) according to e q u a t i o n (19) is s k e t c h e d in Fig. 3. Plastic collapse of the entire y a r n in t h e refined m o d e l
T ,rrRe~I,cro 9c 2- I 4
-r'
I
r .....
c---c' 3 c'c'c'c'cI'c'~-
Ef£y %
FIG. 3. T o t a l axial load (T) vs t h e y a r n strain (e~) for a y a r n m a d e f r o m a n e l a s t i c - p e r f e c t l y plastic material. * A t w i s t i n g m o m e n t a c c o m p a n i e s this tensile load w h e n all t h e fibres h a v e t h e s a m e lay as discussed in t h e f o o t n o t e associated w i t h e q u a t i o n (7).
684
NORMAN JONES
Occurs w h e n eu = 2ao/E1(3c 2 - 1), while p l a s t i c y i e l d i n g first occurs a t e~ = (rg/Es. A n i n n e r p l a s t i c zone is s u r r o u n d e d b y a n o u t e r elastic r e g i o n w i t h i n t h e s t r a i n r a n g e % / E s ~
VISCOUS
BEHAVIOR
I-Iooke's law for a l i n e a r elastic isotropic m a t e r i a l m a y b e w r i t t e n (e.g. D r u c k e r ) 9 E vE a,.j = -]--~v eii + (1 + v ) ( 1 - - 2 v ) e ~ S i ~ ,
(22)*
w h i l e t h e c o n s t i t u t i v e e q u a t i o n for a l i n e a r viscous i s o t r o p i c m a t e r i a l is
C ai~ = ~
mC
(23)
eij -~ (1 + m) (1 - 2m) ekk $i¢,
w h e r e (°) is a t i m e d e r i v a t i v e , C is a viscous coefficient a n d m is P o i s s o n ' s ratio. I f a l i n e a r viscous m a t e r i a l is i n c o m p r e s s i b l e (m = ½) as is c u s t o m a r i l y a s s u m e d , t h e n i t c a n b e s h o w n 9 that
ei¢ = 3Si¢/2C.
(24)t
I t is clear t h a t e q u a t i o n (23) c a n b e o b t a i n e d f r o m e q u a t i o n (22) b y r e p l a c i n g s t r a i n s b y s t r a i n r a t e s a n d s u b s t i t u t i n g C a n d m for E a n d v, r e s p e c t i v e l y . T h u s , t h e b e h a v i o r of a y a r n w i t h fibres w h i c h a r e m a d e f r o m a l i n e a r viscous m a t e r i a l c a n b e o b t a i n e d d i r e c t l y f r o m a s o l u t i o n for a l i n e a r elastic y a r n p r o v i d e d t h e a p p r o p r i a t e s u b s t i t u t i o n s a r e m a d e . E q u a t i o n (20) p r e d i c t s t h a t t h e a x i a l s t r a i n r a t e in a y a r n w h i c h is s u b j e c t e d to a n a x i a l l o a d T is
T
(1+9c~+3c21og~c~ -1,
~ = rrR ~ ~CI [ ~
_--T-ZT~] 1
(25)
w h i l e f r o m e q u a t i o n (3) ( w i t h vv = ½) t h e s t r a i n r a t e i n a fibre is
~ 13c 2
~, = y ( ~ - ~ - 1).
(26)
I t is s t r a i g h t f o r w a r d t o s h o w t h a t t h e a x i a l s t r e s s i n a fibre m a y b e w r i t t e n
T
[3(l+u~)(c~/u~)-2(l+log,
at = rrR 2 ~F \ 1 7 ~
1-i-~Tc2 ( log-~)~
u)~ ]
(27)
w h i c h is i d e n t i c a l t o t h e c o r r e s p o n d i n g l i n e a r elastic case. T h e t o t a l s t r a i n a c q u i r e d b y a y a r n d u r i n g a t i m e i n t e r v a l r is 6~ =
VISCOELASTIC
I, dr.
(28)
BEHAVIOR
A M a x w e l l viscoelastic m a t e r i a l m a y b e r e p r e s e n t e d as a l i n e a r elastic s p r i n g in series w i t h a l i n e a r viscous e l e m e n t . T h e r e s p o n s e of t h i s a r r a n g e m e n t is clearly g i v e n b y t h e
* a~j and 6~ are the stress and strain tensors, respectively, where t h e s u b s c r i p t s i a n d j h a v e t h e r a n g e 1-3 ; ~t~ is t h e K r o n e c k e r d e l t a . T h e s u m m a t i o n c o n v e n t i o n is u s e d so t h a t ~kk ~ 611 ~ g22 "~ 633"
t S~j = ais - o~k ~ j / 3 is k n o w n as a s t r e s s d e v i a t o r .
E l a s t i c - p l a s t i c a n d viscoelastic b e h a v i o r of a continuous filament y a r n
685
s u p e r p o s i t i o n of t h e responses of t h e two i n d i v i d u a l elements, or l+v
~j = ~
1 - 2 v ~ k ~ s _ ~ 3S~j
~iJ÷ Es~
3
(29)*
2Ot~"
A K e l v i n t viscoelastic m a t e r i a l m a y be r e p r e s e n t e d as a linear elastic spring which is in parallel w i t h a linear viscous element. The strains across t h e two e l e m e n t s are equal in this idealization. I t can be shown 9 t h a t t h e corresponding c o n s t i t u t i v e relation m a y be written
a~j = ~Esk e~ + ~Csk~ + ~ . ~i~.
(30)$
I t is n o w s t r a i g h t f o r w a r d w i t h t h e aid of t h e g e o m e t r i c a l r e l a t i o n (3), e q u i l i b r i u m e q u a t i o n s (5) a n d (7) a n d t h e c o n s t i t u t i v e relations (22), (24), (29) a n d (30) to e x a m i n e t h e b e h a v i o r of y a r n s w i t h fibres w h i c h are c o n s t r u c t e d f r o m Maxwell, K e l v i n , t h r e e - e l e m e n t or four-element viscoelastic m a t e r i a l s or o t h e r idealizations which h a v e been studied b y various authors, a, ~0 T h e loading, unloading, creep and r e l a x a t i o n response of y a r n s w i t h fibres which are m a d e f r o m either Maxell or K e l v i n viscoelastic m a t e r i a l s h a v e been r e p o r t e d elsewhere3 The four-element viscoelastic idealization is f o r m e d b y placing t h e Maxwell a n d K e l v i n e l e m e n t s in series as indicated in Fig. 4. I t can be shown for t h e E
fk
Eim
5m
~
ii
It-_ Cfk
Fro. 4. Four-element viscoelastic idealization.
p a r t i c u l a r case of a u n i f o r m axial load T, w h i c h is applied at t i m e t = 0 as shown in Fig. 5, t h a t t h e axial strain of a y a r n w i t h fibres obeying t h e four-element viscoelastic idealization is e~ = 7
\Es.~
Cs.
,
(31)§
where i~ = 1+9c ~
3c 2logic ~ l_c------V-
(32)
which is o b t a i n e d f r o m e q u a t i o n s (3), (5), (7), (29) a n d (30) since T has t h e s a m e v a l u e in t h e Maxwell elastic a n d viscous elements a n d K e l v i n model, while t h e t o t a l strain is t h e s u m of those in t h e Maxwell elastic a n d viscous e l e m e n t s a n d t h e K e l v i n model, q, 9, 10 T h e c o n s t a n t s of i n t e g r a t i o n are d e t e r m i n e d f r o m t h e r e q u i r e m e n t s t h a t at -- 0 a t u = 1 for all t a n d e~ = 0 at t = 0 for t h e Maxwell viscous e l e m e n t a n d K e l v i n model. W h e n EI~ -~ ¢¢, t h e n t h e first two t e r m s in e q u a t i o n (31) r e m a i n a n d predict t h e response of an axially loaded y a m w i t h fibres w h i c h are idealized as a Maxwell m a t e r i a l . T h e first t e r m is t h e t i m e - i n d e p e n d e n t initial elastic response related to EI~, while t h e second t e r m increases linearly w i t h time, is irreversible, a n d is c o n t r i b u t e d b y t h e viscous element. * The first t w o t e r m s are g i v e n b y t h e t i m e d e r i v a t i v e of t h e elastic strain which is o b t a i n e d b y i n v e r t i n g e q u a t i o n (22). T h e last t e r m is identical to e q u a t i o n (24). t This m o d e l is s o m e t i m e s n a m e d a f t e r Voigt. $ The elastic a n d viscous e l e m e n t s are a s s u m e d to be incompressible a n d isotropic. § I t is a s s u m e d t h r o u g h o u t this article t h a t t h e axial load T is applied fast enough to a v o i d significant viscous d e f o r m a t i o n (e.g. rise t i m e of T,~ C,s~/E1~) a n d y e t slow e n o u g h to a v o i d inertia effects.
686
NORMAN J o ~ E s
f Elm -->~ and Csm -->~, t h e n the last t e r m in e q u a t i o n (31) remains and gives the b e h a v i o u r of a y a r n which is m a d e from a K e l v i n material. I t is clear in this case t h a t ~y is initially zero and a p p r o a c h e s the elastic v a l u e when t >>C1k/Elk. ]
T
o
z"
t
-EfkZ"
6y~R2EfmF~
I + Efm T Jr Efm (I-e cfk Cfr n Efk ~,
)
~ ~ ~ _ _Efk r
i
~'x'Efrn6ecfk 1 Ef m --
fL~m; o
z"
t
FIG. 5. Creep and r e c o v e r y characteristics of a y a r n m a d e from a four-element viscoelastic idealization. The v a r i a t i o n of ev w i t h t i m e according to e q u a t i o n (31) for t h e four-element m o d e l is s k e t c h e d in Fig. 5 t o g e t h e r w i t h t h e u n l o a d i n g or r e c o v e r y characteristics which can be found in a s t r a i g h t f o r w a r d m a n n e r f r o m t h e basic e q u a t i o n s g i v e n in this article. I t is e v i d e n t t h a t t h e spring Elm i m m e d i a t e l y recovers on u n l o a d i n g a t t = ~, while the K e l v i n c o m p o n e n t s (E1~, Clk ) r e t u r n e v e n t u a l l y to t h e i r original s t a t e leaving only t h e irreversible strain a c c u m u l a t e d in the viscous e l e m e n t Csm. CONCLUSIONS
The
continuousfilamentyarnwhich
simple analytical studies of a are presented herein are an attempt to incorporate the more realistic properties of many yarns into the theoretical models of their behavior. As remarked in the Introduction, a continuous filament yarn is not used widely in practice but it is a particularly useful tool for revealing the mechanics of textile fibres, yarns and fabrics, tyres, ropes and wire cables. The rigid-plastic prediction of equation (21) provides a rigorously correct maximum axial load carrying capacity when the yarn material is timeindependent, does not exhibit material strain hardening and the displacements remain infinitesimal. Equation (19) predicts the magnitude of the yarn strain (or displacement) which is associated with any axial load up to the maximum value.
Elastic-plastic and viscoelastic behavior of a continuous filament yarn
687
I t is i n d i c a t e d in Fig. 5 a n d Figs. 6 - 8 o f ref. (7) h o w c o n t i n u o u s filament y a r n s creep, r e c o v e r a n d r e l a x w h e n s u b j e c t e d t o t h e static l o a d h i s t o r y which is s h o w n in Fig. 5. T h e t i m e - d e p e n d e n t characteristics of t h e y a r n s in ref. (7) a n d herein are e x a m i n e d b y a s s u m i n g t h a t t h e fibres are m a d e f r o m m a t e r i a l s w h i c h e x h i b i t linear viscous, Maxwell, K e l v i n or f o u r - e l e m e n t m o d e l viscoelastic properties. Various simplifications h a v e b e e n i n t r o d u c e d into the calculations including t h e a s s u m p t i o n s of isotropy, i n c o m p r e s s i b i l i t y a n d l i n e a r i t y of t h e m a t e r i a l s a n d t h e a s s u m p t i o n t h a t t h e response involves infinitesimal d i s p l a c e m e n t s a n d strains. Clearly, it is i m p o r t a n t for a designer to assess w h e t h e r these simplifications are r e a s o n a b l e for t h e p r o b l e m a t h a n d . I n t h e a p p r o p r i a t e cases, these simple predictions should be sufficient to enable a n e s t i m a t e of t h e i m p o r t a n c e of viscoelastic characteristics to be m a d e . H o w e v e r , it w o u l d be s t r a i g h t f o r w a r d to e x t e n d these m e t h o d s in order to describe t h e response arising f r o m m o r e c o m p l i c a t e d loading histories a n d of y a r n s which are m a d e f r o m fibres e x h i b i t i n g t i m e - d e p e n d e n t characteristics which do n o t c o n f o r m to t h e simple models e x a m i n e d herein. Acknowledgements--The author is indebted to the Engineering Mechanics Program of the
National Science Foundation for their support of this work under Contract No. G K 20189X. The author wishes to record his appreciation to Professor S. Backer of the Department of Mechanical Engineering for introducing him to the fascinating field of textile technology and to Dean A. H. Keil and Professor I. Dyer for their interest and encouragement in this project. REFERENCES 1. J. W. S. HEARLE, P. GROSBERGand S. BACKER,Structural Mechanics of Fibres, Yarns and Fabrics, Vol. 1, Chapter 4. Wiley, l~ew York (1969). 2. S. BACKER, Tire Cord Structure and Properties, Nat. Bur. of Standards, Monograph on Mechanics of Pneumatic Tires, p. 63 (1971). 3. C. W. BERT and R. A. S~Er~, Wire and Wire Products 37, 621-624, 769, 770, 772, 816 (1962). 4. J. W. S. HEARLE, J. Textile Inst. 49, T389 (1958). 5. J. W. S. HEARLE, H. M. A. E. EL-BEHERY and V. M. THAKUR, J. Textile Inst. 52, T197 (1961). 6. L. R. G. TRELOARand J. W. S. HEARLE, J. Textile Inst. 53, T446 (1962). 7. N. JOI~ES, Department of Ocean Eng., Report 73-4. M.I.T., Cambridge, Massachusetts (1973). 8. D. C. DRUCKER, W. PRAGER and H. J. GREE~BERG, Quart. appl. Math. 9, 381 (1952). 9. D. C. DRVCKER, Introduction to Mechanics of Deformable Solids. McGraw-Hill, New York (1967). 10. D. R. BLAND, The Theory of Linear Viscoelasticity. Pergamon Press, London (1960).