8/13/2019 Bf 02485866

1/9

C T M E C H N I C S I N I C A , V o l . 20 , N o . 6 , D e c e m b e r 2 4T h e C h i n es e S o c i e t y o f T h e o r e ti c a l a n d A p p l i e d M e c h a n i c sC h i n e s e J o u r n a l o f M e c h a n i c s P r e s s , B e i j in g , C h i n aA l l e r t o n P r e s s , I N C . , N e w Y o r k , U . S . A .

ISSN 0 5 6 7 - 7 7 1 8

A M I C R O - M E C H A N I C A L M O D E L O F K N I T T E D F A B R I C A N D I T SA P P L I C A T I O N T O T H E A N A L Y S I S O F B U C K L I N G

U N D E R T E N S I O N I N W A L E D I R E C T I O N :M I C R O - M E C H A N I C A L M O D E L

Z H A N G Y i t o n g ~ X N ) t L I C u i y u ~ N ) X U J i a f u ~ , ~ N )Department of Mechanics, Tian jin U niversity, Tianjin 300072 , China)

A B S T R A C T : T h e t y p i c a l m i c r o - k n i t t i n g s t r u c t u r e o f k n i t t e d f a b r i c , w h i c h m a k e s i t v e r y d i ff e r e n tf r o m w o v en f a b r ic , is d e s c r i b e d . T h e t e n s i l e t e s t s o f k n i t t e d f a b r i c ar e r e p o r t e d . T h e d e f o r m a t i o n s o ft h e m i c r o - k n i t t i n g s t r u c t u r e s a r e c a r ef u l l y s t u d i e d . T h e s t u d y i n d i c a t e s t h a t w h e n a k n i t t e d f a b r i c s h e e ti s s u b j e c t e d t o a t e n s i o n a l o n g w - d i r e c t i o n a n e x t r a c o m p r e s s i v e s t r es s f i el d i n si d e l o o p i n c - d i r e c t i o ni s i n d u c e d . T h e e x t r a s t r e ss f ie l d i s a l s o d e t e r m i n e d t h r o u g h a n a l y s i s . F i n a l l y , a m i c r o - m e c h a n i c a lm o d e l o f k n i t t e d f a b r i c i s p r o p o s e d . T h i s w o r k p a v e s t h e w a y f o r t h e s i m u l a t i o n s o f b u c k l i n g m o d e s o fa k n i t t e d f a b r ic s h e e t a s a r e o b s e r v e d i n e x p e r i m e n t s .K E Y W O R D S : c o n s t i t u t i v e t h e o r y , k n i t t e d f a b r ic , m i c r o - k n i t t i n g s t r u c t u r e , b u c k l i n g , t e n s i le t e s t s

1 I N T R O D U C T I O NF o r a l o n g t i m e t h e w o v e n f a b r i c w a s c o n s i d -

e r e d a s t h e s a m e a s a t h i n o r t h o t r o p i c p l a t e w i t h v e r ys m a l l e l as t ic m o d u l i . A l a r g e n u m b e r o f p a p e r s a b o u tt h e s i m u l a t i o n s o f t h e d e f o r m a t i o n o r d r a p i n g o f w o -v e n f a b r i c w e r e p u b l i s h e d , s e e R e f s . [ l ~ 3 ] a n d t h e r e f -e r e n c es q u o t e d t h e r e , b u t t h e e r r o r s o f s i m u l a t i o n sw e r e n o t v e r y s m a l l . I t i s n o t e d t h a t b a s e d o n t h eo r t h o t r o p i c c o n s t i t u t i v e m o d e l i t i s n o t p o s s i b l e t os i m u l a t e s o m e s p e c i a l b u c k li n g m o d e s o f w o v e n f a b -r i c t h a t h a v e b e e n o b s e r v e d i n e x p e r i m e n t s [4 ]. S h o u l dt h e c o n s t i t u t i v e m o d e l o f w o v e n f a b r i c b e e x a c t l y t h es a m e a s t h a t o f a f a b r ic - r e i n f o r ce d c o m p o s i t e p l a t e ?M o t i v a t e d b y t h e q u e s ti o n , a m i c r o - m e c h a n i c a l m o d e lo f w o v e n f a b r i c is p r o p o s e d a n d , b a s e d o n t h e m o d e l ,a se r i e s o f b u c k l i n g m o d e s o f w o v e n f a b r i c h a v e b e e ns i m u l a t e d s u c c e s s f u l l y [5 ~z] .

K n i t t e d f a b r ic c o n s is t s o f m o r e c o m p l i c a t e d m i -c r o s t r u c t u r e s . T h e s i m u l a t i o n s o f d r a p i n g o r b u c k l i n go f k n i t t e d f a b r ic , a n d t h e d e t e r m i n a t i o n o f it s c on -

Rece ived 9 Oc tober 2003 , rev ised I March 2004

s t i t u t i v e m o d e l h a v e n o t b e e n p a i d a d u e a t t e n t i o ni n l i t e r a t u r e . I n t h e p r e s e n t p a p e r w e d e s c r ib e t h et y p i c a l m i c r o - k n i t t i n g s t r u c t u r e s o f k n i t t e d f a b r ic , r e-p o r t t h e r e s u l t s o f t h e t e n s i le t e s t s o f k n i t t e d f a b r ic ,a n a l y z e t h e d e f o r m a t i o n o f t h e m i c r o - k n i t t i n g s t ru c -t u r e , a n d , f i n al ly , p r o p o s e a m i c r o - m e c h a n i c a l m o d e lf o r k n i t t e d f a b r i c .

T h e m o r e d e t a i l s a b o u t r e c e n t a d v a n c e s i n f a b r i cm e c h a n i c s c a n b e f o u n d i n t h e r e v i e w a r t i c l e [8] a n dt h e b o o k [ 9 ] .

2 M I C R O - K N I T T I N G S T R U C T U R E O FK N I T T E D F A B R I C

K n i t t e d f a b r i c is m a d e o f i n t e r l o c k i n g l o o p s o fy a r n or t h r e a d . I t s t y p i c a l m i c r o - k n i t t i n g s t r u c t u r ei s s h o w n i n F i g . l a ) , a n d t h e l o c a l l y e n l a r g e d l o o p sa r e s h o w n in F i g s . l b ) , c ) , d ) a n d e ) . T h e d i r e c -t i o n i n w h i c h t h e l o o p s a r e i n t e r l o c k e d i s c a l l e d t h ew a l e - d i r e e t i o n , o r i n s h o r t h a n d f o r m , w - d i r e c t i o n ; t h ed i r e c t i o n a l o n g w h i c h t h e l o o p s l i n e u p i n a c o u r s e i s

* T h e p ro je c t s u p p o r t e d b y th e N a t io n a l N a tu ra l S c ie n c e F o u n d a t io n o f C h in a 1 0 27 2 07 9 )t E -m a ih y t z h a n g @ t ju . e d u . c n

8/13/2019 Bf 02485866

2/9

624 CT MECH NIC SINICc a l l ed t h e c o u r se- d i r e c t i o n , o r c - d i r ec t i o n . B o t h t h ed i r e c t i o n s a r e sh o w n in F ig . l ( a ) .

2004

a) Typica l mic ro-kni t t ing s t ruc ture ofkni t ted fabr ic d) The locally enlarged two loops sub-jected a shear a long w-direction

(b) The locally enlarged loops sub-jected a tension along w-direction

(c) The locally enlarged loops sub-jected a tension along c-direction

(e) A locally enlarged smallest e lementsubjected a shear

Fig.1 The stru cture schematics of the knit ted fabr icT h e i n t e r lo c k in g l o o p s o f t h r e a d i n k n i t t e d f a b -

r i c p l a y tw o im p o r t a n t r o l e s: f i r st , t h e i n t e rl o c k in gb e tw e e n th e l o o p s c o n n e c t s t h e l o o p s t o f o r m th e f a b -r ic shee t and t r ansfe r loading; and , second, the loop-sh a p e d th r e a d s i n k n i t t e d f a b r i c c a n u n d e r g o l a r g e re x t e n s iv e d e f o r m a t io n t h a n t h e s t r a ig h t t h r e a d s i n w o -v e n f ab r i c . S o m e t e s t s sh o w th a t t h e t e n s i l e r i g id i tyo f k n i t t e d f a b r i c is m u c h l e s s t h a n t h a t o f w o v e n f ab -r i c c o m p o se d o f t h e s a m e th r e a d s . T o sh o w th a t , o n em a y s t r e t c h a sh i rt c o m p o se d o f w o v e n f a b ri c a lo n gw a r p / w e f t d i r ec t io n a n d a T - s h i r t c o m p o s e d o f k n i t -t e d f a b r i c a lo n g w - / c - d i r e c t i o n , r e sp e ct i v e ly . I t w o u ldb e f o u n d t h a t t h e l a t t e r d e f o r m a t i o n i s m u c h m o r eth a n t h e f o r m e r o n e . T h i s i s b e c a u se t h e l o o p - sh a p e dk n i t t i n g s t r u c tu r e s o f k n i t t e d f a b r i c c a n d e f o r m v e r ye x t e n s iv e ly t h r o u g h c h a n g in g o f t h e g e o m e t r i c a l sh a p eo f l o o p s . T h u s k n i t t e d f a b r i c i s m u c h so f t er t h a nw o v e n f a b r i c a n d i s w e l l su i t a b l e f o r u n d e r w e a r .W e a s su m e th a t i n a t y p i c a l m ic r o - k n i t t i n gs t r u c tu r e t h e l o o p s a r e c i r c l e s a n d t h e a d j a c e n t l o o p s

8/13/2019 Bf 02485866

3/9

Vol.20 No.6 Zhang YT et al.: Micro-mechanical Model of Knitted Fabric 625are tangent to each other on their contact points, forexample, at the points A , B , C 1 , C 2 , etc, as shown inFig.l b). Thus it follows that in the triangle 01 02 03we have 0 l C 1 = 0 3 C 1 and 03A]-0102 where O1,02, and 03 are the centers of three adja cent loops.

Let D, R denote the diameter of the thread andthe mid-radius of the circle loop of the thread, re-spectively. Eviden tly, we have O103 = 2R, O a A =R + D / 2 , 0 2 03 = 2 R - D . It is easy to show thatthe D and R must satisfy the following equati on

p p - x ) p - 2 R ) p - 2 r + D ) - x 2 R + D / 2) 2 = 0 1)4where p is a half of the circumference of triangleO10 203 , a nd p = ~ 4R + x - D); x is the distancebetween two centers of adjacent loops in w-direction,and x = O102 = ~ f f 6 R D + D 2 / 2 , which we refer toas the distance between two adjacent courses.

A solut ion of Eq. 1) is R = k D , where1. 625 7.. . < k < 1 .62 58. --, We take k = 1.626 inthe present paper. The ot her solutions of Eq. 1) arephysically meaningless for this problem.

The knitting elements filled with two pieces ofthreads are called cells and two adjacent cells in c-direction are shown in Fig.l a) in a thick dashed line.The cells are lined up continuous ly in c-direction. Thecells are symmetric about the wale-axis.

The adjacent two cells in w-direction have a com-mon part, e.g., the cell O1 and the cell 02 in Fig.l b)have a common part between the lines which passpoints V and A, respectively. The distance betweentwo points U and V in a cell, denoted by U V , is calledthe effective leng th of the cell, and it is easy to obt ain

U V = 0 1 0 2 = X / 6 R D + D 2 / 2 = 1.970R 2a)U B = 0.543R 2b)BM1 = 0.889R 2c)

If the loops in Fig.l b) are subjected to a ten-sion along w-direction the threads will be stretchedin w-direction, the loops will become slender, and thefibers inside loops will be compressed. So ther e will bea tensile stress field in w-direction in threads. Froma macroscopic viewpoint, the knitted fabric sheet willbe extended in w-direction and at the same time becont racte d in c-direction, i.e. the response of knit tedfabric is similar to the response of a knitted-fabric-reinforced composite plate when it is subjected to atension along w-direction.

If the loops in Fig. l c) are subjec ted a tensionalong c-direction the threads will be stretched in c-direction, the loops will become flat, and the fibers ofthreads will be compressed against each other at theircontact points such as the points G1, G2, HI,//2, C1and C2 etc. So there will be a tensile stress field in c-direction in threads. From a macrosco pic viewpoint,the knitted fabric sheet will be extended in c-directionand, meanwhile, be contr acte d in w-direction, i.e. inthis case the r esponse of knitte d fabric is also simi-lar to the response of a knitted-fabric-reinforced com-posite plate when it is subjected to a tension alongc-direction.

A smallest knitting element in knitted fabric issketched in Fig.l d) in thick solid line and is plottedas an enlargement in Fig.l e). There are four smallestknitti ng elements in Fig. l d). A piece of knit ted fab-ric consists of a large number of the smallest elementscontinuously connected with each other. When a knit-ted fabric sheet is subjected a shear along w-directionthe smallest knitting element will be subjected to ten-sion and compression in two directions perpendicularto each other, respectively, as shown in Fig.l e). Inthis case the response of knitted fabric is also simi-lar to the response of a knitted-fabric-reinforced com-posite plate when it is subjected to a shear along w-direction.

BO1 = 0.457R 2d)0 1 N 1 = R 2e)

For convenience it is assumed that the thickness ofknitted fabric is unity.3 A N A L Y S I S O F D E F O R M A T I O N O F

M I C R O K N I T T I N G S T R U C T U R EWe start with the analysis of the deformation

of loops of knitted fabric when they are subjected toa tension along w-direction, c-direction and a shearalong w-direction, respectively.

4 T E N S I L E T E S T S O F K N I T T E D F A B R I CTensile tests of knit ted fabric were carried out

in Shimad zu Testing Machine recently. The speci-mens were cut off from a piece of knit ted fabric tha tis usually used to make vests. The specimens wereironed and then were laid freely on table for moretha n 24 h before tests. The thickness of specimens is0.245 mm, the width 90 mm and the length 300 ram.The ph otograph s of specimens under tension along w-direction and along c-direction are shown in Figs.2 a)and b), respectively. We can see clearly tha t thespecimen under tension along w-direction buckled and

8/13/2019 Bf 02485866

4/9

8/13/2019 Bf 02485866

5/9

V o l . 2 0 N o . 6 Zhang YT et al.: Micro-mechanical Model of Knitted Fabric 627However, the knitted fabric is very much differ-

ent from the orth otro pic composite plate, e.g. theknitted-fa bric-reinfo rced composite plate. The obvi-ous differences between t hem are as follows. First, theformer possesses a micro-knitting structure whereasthe latter does not (the micro-knitting structure of thelatter is bonded to the matrix); secondly, the stressesin the former are acted on the micro-knitting struc-tures alone whereas the stresses in the latter are actedboth on the micro-knitting structures and the matrix.We should carefully consider the def ormatio ns of themicro-k nitti ng str uctu re and decide if and how Eq. (3)might be modified in order to describe correctly themechanical property of knitted fabric.

Another important reason for constitutiveEq.(3) to be modified is that based on Eq.(3), thebuckling of the kn itte d fabric sheet shown in Fig.2(a)canno t be predicted. In fact, it has been proved tha twhen a sheet compose d of orth otro pic material s issubjected a uniaxial tension along any direction, theout-of-plane buckling like that shown in Fig.2(a) willdefinitely not occur[51.

Careful observation of the deformation of a cellshows that when the knitted fabric sheet is subjectedto a tension in w-direction the fibers inside the cellare subjected to a compression due to the contractionof the cell in c-directi on. Therefore, there is an ex-tra compressive stress field inside the cell that doesnot exist in orth otrop ic materials. This extra stressfield is induced in knitted fabric by the micro-knittingstructures and should be added to the stress field de-termined by Eq.(3) in order to describe correctly thestresses acted on the fibers in knitted fabric.

From Fig.l(c) we also see that when the knittedfabric is subjected to a tension in c-direction, thereare compressive stresses only at the contact points ofloops, such as the points Gx, G2, H1,/-/2, C1 and C2in Fig. l(b) . These compressive stresses exist only atsome points and are not continuous in knitted fab-ric. Thus there is no extra stress field to be addedto Eq.(3) when the knitted fabric is subjected to atension in c-direction.

In the following section we will carefully analyzethe deformation of a cell when it is subjected to atension in w-direction in order to determine the ex-tr a compressive stress field. Since it is not possibleto measure the extra stress field by tests, it is veryimportant to determine the extra stress field throughanalysis.

C E L L S U B J E C T E D T O T E N S I O NA L O N G w D I R E C T I O N

For convenience we first introduce some terms.We call the bo und ar y curve of a cell the shell of thecell as shown in thick line in Fig.4; the points J1 andJ2 the end points of the cell, t he poin ts I1 a nd 12 theinner points of the cell; and the appr oxim atel y circle-shaped area inside the shell, with length and width ofUV and N2N1, respectively, the effective inner spaceof the cell. From Fig. l(b) it is evident tha t t he endpoints of cell O1 is also the inner points of cell O2.

The fibers in a cell, i.e. the shadow par t in a cellshown in Fig.4 or in Fig.l(a), occupy almost all the ef-fective inner space of the cell. To facilitate analysis weassume that the effective inner space of the cell is fillednwith t he fibers of the th reads and t he arcs N1 J1 andN2J2 are replaced by straight lines NxJ1 and N2J2, re-spectively, as shown in Fig.4. Simple calcul ation s give

nN~J~ = N J = 1.716R and N1J1 = N2J2 = 1.644R,thus the relative error of the replacement is abo ut 3 .The angle a shown in Fig.4 is the angle mad e by thelines N1 J1 and I 1 J 1

U

N1

Fig.4 A simplified cell in initialconfiguration

The deformed configuration of the cell issketched in Fig.5(a), with the letters with prime sig-nifying the points in the initial configuration.

We now consider a piece of knitted fabric whichis subjec ted to a tension along w-direction. Denotethe tension per unit area (measured in the initial con-figuration) by c]. The tensile load acting on the widthof 2R + D of the fabric shee t is equal to q (2R + D)and, from Fig.l(b), the load acts on three pieces ofthreads. Thus th e load acting in w-directi on on onethread is

6 ANAL YSIS OF DEFO RMAT ION OF P = q(2R + D /3 = 0.871 7Rq (4)

8/13/2019 Bf 02485866

6/9

F i g . 5 a )

R2 U R1r \ r r w

/ I :o r L d

F~ F1

g r

T h e p r e s s u r e e x e r t e d b y t h efibers ins ide cel l

F i g . 5 b ) T h e r e s u l t a n t f o r c es o f i n n e rp r e s s u r e

R 2 c R ~~ R 1

1F2 ~ ? F1

P i g . 5 c ) T h e d e c o m p o s e d R t a n d R2

D e n o t e t h e f o r ce s a c t in g o n t h e e n d p o i n t s o f t h e c e l la l o n g t h e t h r e a d d i r e c t io n s b y F 1 a n d F 2 , r es p e c t iv e l y , F i g . 6 a ) T h e e x t r a p r e s s u r e f ie l d i n w - d i r e c t io n

CT MECH NIC SINIC 2 0 0 4t e r m e d t h e e n d f o r c e s o f t h e c e ll , a s s h o w n i n F i g . 5 a ) .I t f o ll o w s t h a t F 1 = F 2 = P / c o s ~ , w h e r e o ~ i s t h ea n g l e m a d e b y t h e l i n e s N ~ J ~ a n d I ~ J ~ a s s h o w n i nF i g . 5 b ) . T h e v a l u e o f c~~ w i ll b e d e t e r m i n e d l a t e r .6 .1 E q u i l i b r i u m o f t h e S h e l l o f t h e C e l l

B e s i d e t h e e n d f o r c e s Fz a n d F . 2 , a c t i n g o n t h ee n d p o i n t s I ~ a n d I ~ o f t h e s h e l l , t h e s h e l l is a l s os u b j e c t e d t o i n n e r p r e s s u r e f r o m t h e f i b er s in s id e t h ec e ll a s s h o w n i n F i g . 5 a ) . T h e r e s u l t a n t f o r c es o f t h ep r e s s u r e , d e n o t e d b y R 1 a n d / ~ a s s h o w n in F i g . 5 b ) ,m u s t p a s s t h e i n n e r p o i n t s J ~ a n d J.~ o f t h e c e ll , a n dm u s t b e e q u a l i n m a g n i t u d e a n d o p p o s i t e i n d i r e c t i o nt o t h e f o r c e s F 1 a n d F 2 , r e s p e c ti v e l y . T h i s i s b e c a u s et h e r e a c t i n g f o r c e s o f R 1 a n d R 2 a r e j u s t t h e e n d f o r c e so f a n o t h e r s h e ll .

T h e s h e l l i s i n e q u i l i b r i u m s t a t e u n d e r t h e a c -t i o n s o f t h e f o r c e s F 1 , F 2 , to g e t h e r w i t h R 1 a n d R 2 .R 1 a n d R 2 c a n b e d e c o m p o s e d i n t o R l w a n d R l c ,

R 2 w a n d R 2 c , r e s p e c t i v e l y , a s s h o w n i n F i g . 5 c ) . W eh a v e

Rlw - R~w = P R lc = R2c = P t a n c~~ 5)

6 .2 D e f o r m a t i o n o f t h e C e l lD u e t o t h e p r i n c i p l e o f a c t i n g a n d r e a c t i n gf o r c e s , t h e r e a r e t w o p r e s s u r e f i e l d s i n s i d e t h e c e l l :

o n e i s t h e p r e s s u r e f i e l d i n c - d i r e c t i o n c o r r e s p o n d i n gt o R l c a n d R 2 c , o n t h e a r e a o f w h o l e e f f e c ti v e i n n e rs p a c e o f t h e c e ll ; a n o t h e r i s t h e p r e s s u r e f i el d i n w -d i r e c t i o n c o r r e s p o n d i n g t o R l w a n d R 2 w , o n l y o n t h ea r e a a b o v e t h e l i n e I [ I ~ i n s i d e t h e c e ll . B o t h p r e s -s u r e f ie l d s a r e i n d u c e d b y t h e m i c r o - k n i t t i n g s t r u c -t u r e s a n d a r e c a l l e d e x t r a p r e s s u r e f i e l d s a s s h o w n i nF i g . 6 a ) a n d F i g . 6 b ) , r e s p e c t i v e l y .

6 2 8

8/13/2019 Bf 02485866

7/9

V o l . 2 0 N o . 6 Zhang YT e t a l. : M ic ro mechan ica l Mode l o f Kn i t ted Fabric

_ _

F i g . 6 ( b ) T h e e x t r a p r e s s u r e f ie ld i n c - d i r e c t io nF r o m t h e m a c r o s c o p i c v i e w p o i n t , t h e c el ls ar e

v e r y s m a l l , t h u s w e c a n a s s u m e t h a t b o t h p r e s s u r ef ie l d s i n s i d e a c e ll a r e h o m o g e n o u s , t h u s t h e m a g -n i t u d e s o f e x t r a p r e s s u r e s i n w - d i r e c t i o n a n d i n c -d i r e c t i o n c o r r e s p o n d i n g t o t h e i n it ia l c o n f i g u r a t i o na r e , r e s p e c t i v e l y ,

P P t a n c ~P w - - P c ( 6 )B M 1 U VS u b s t i t u t i n g E q s . ( 4 ) , ( 2 c) a n d ( 2 a ) i n t o E q s . ( 6 ) g i v e s

Pp w = 1 .1 25 0 . 9 s o 6 q7 )] r ]

P c = 0 . 5 0 7 6 t a n c ~ R 0 .4 4 2 6 t a n c ~ q

W e n o w c o n s id e r t h e d e f o r m a t i o n o f t h e c el l. T h e r a -d i a l d e f o r m a t i o n o f t h r e a d s u n d e r c o m p r e s s i o n c a n b ed e s c r i b e d b y

d = D e - T p ( 8 )w h e r e d is t h e d e f o r m e d d i a m e t e r o f t h r e a d , r/ i s ac o n s t a n t r e l a t e d t o t h e m a t e r i a l s o f t h r e a d , a n d p i st h e p r e s s u r e w i t h r e s p e c t t o t h e i n i t ia l c o n f i g u r a t i o no f t h r e a d .

T h u s w e h a v er w = U B + B O = U B e - ~ q w + B O

9 )tr c = O 1 N { = O 1 N l e ~q~

w h e r e r w i s t h e r a d i u s i n w - d i r e c t i o n o f t h e d e f o r m e dc e ll a n d r c t h e r a d i u s i n c - d i r e c t i o n o f t h e d e f o r m e dc e l l , a s s h o w n i n F i g . 5 ( b ) .

AS i n ce S O < BtOi M I N I w e t a k e a p p r o x i -

m a t e l yB O , - B 0 1 = 0 . 4 5 7 R

S u b s t i t u t i n g E q s . ( 2 b ) , ( 2 d) a n d t h e a b o v e e q u a t i o ni n t o E q s . ( 9 ) g i v e s

r w : 7 ( C t ) r c r c : R e - ~ p~ ( 1 0 )

6 2 9w h e r e

r (q ) 0 . 5 4 3 R e + 0 . 4 5 7 R e 'J vc ( 1 1 )T h e d e f o r m e d c o n f i g u r a t i o n o f t h e c e l l i s

s k e t c h e d i n F i g . 5 ( a ). W e n o w a s s u m e t h a t t h e c u r v enN f U N . ~ i s a p p r o x i m a t e l y h a l f o f a n e l l ip s e.

T h e s t r e t c h o f t h e s h e l l o f t h e c e l l i n w - d i r e c t i o nd u e t o t h e c h a n g e o f t h e s h a p e o f t h e s h e ll ( o r lo o p ) i sm u c h l a r g e r th a n t h a t d u e t o t h e e x t e n s i o n o f t h r e a d s ,t h u s w e c a n a s s u m e t h a t t h e t o t a l l e n g t h o f t h e s h e llo f t h e c el l ( o r l o o p ) d o e s n o t v a r y d u r i n g d e f o r m a t i o n .I n o t h e r w o r d s , t h e e x t e n s i o n o f t h r e a d i s n e g l e c t e di n t h e m i c r o - m e c h a n i c M a n a l y s i s o f t h e c e ll , b u t i sc o n s i d e r e d i n c o n s t i t u t i v e E q . ( 3 ) . T h u s w e h a v e

N NIU N 1 ,N 1 J , = U N , ( 1 2 )U s i n g t h e a p p r o x i m a t e e q u a t i o n f o r t h e c i r cu m f e r e n c eo f a n e l l ip s e , w e w r i t e E q . ( 1 2 ) a s

+ - +40 . 6 9 2 5 r c

s i n c ~C~ R + 1 . 6 6 4 R = 3 . 2 3 5 R ( 13 )

W i t h t h e a i d o f E q s . ( 1 0 ) , ( 7 ) a n d ( 4 ), 1 3 ) c a nb e w r i t t e n a s

f ( R , r h q , c~ ) = 0 (14 )w h e r e c ~ i s t h e d e f o r m e d c~ a s s h o w n i n F i g . 5 ( b ) .

S o l v i n g E q . ( 1 4 ) f o r c d g i v e s( 1 5 )

w h e r e 0 < c ~ < a r c s i n R - D / 2 _ 2 4 . 5 9 0 .- J 1 NL e t A w a n d A c d e n o t e t h e s t r e t c h e s o f t h e c e ll i n

w - d i r e c t i o n a n d c - d i re c t io n , r e s p e c t i v e l y , w e h a v eA w - r w + O l A t o - - r c ( 1 6 )

R + 0 1 A 0 1 NN o t i n g t h a t

O ~ A = K J ~ = K N s c o t a = 0 . 6 9 2 5 r c c o t aw e r e w r i t e E q s . ( 1 6 ) a s

Ac = --rc (1 7)Rr w + 0 . 6 9 2 5 c o t o ~ r cA w = R + 1 . 5 1 3 R = / 3 ( q ) A c ( 1 8 )

w h e r e / 3 (@ = 3 @ t ) + 0 . 6 9 2 5 c o t c ~2 . 5 1 3 ( 1 9 )

8/13/2019 Bf 02485866

8/9

6 3 0 ACTA MECHANICA SINICA 2 46 . 3 T h e E x t r a C o m p r e s s i v e S t r e s s F i e ld s

C o r r e s p o n d i n g t o t h e e x t r a p r e s s u r e f i el d i n s i d et h e c e l l i n c - d i r e c t i o n , t h e r e i s a C a u c h y s t r e s s f i e ldi n s i d e t h e c e l l P t a n aa ~ x t _ U V ( 20 )T h i s s t r e s s f ie l d i s a c t e d o n t h e w h o l e a r e a o f t h e e f -f e c t i v e i n n e r s p a c e o f t h e c el l, s o t h a t i t i s c o n t i n u o u so n t h e w h o l e k n i t t e d f a b r ic s h e e t a n d s h o u l d b e a d d e dt o t h e c o n s t i t u t i v e E q s . ( 3 ) .

S i n c e t h e c e l l i s v e r y sm a l l , i t i s a s su m e d t h a tt h e s t r e t c h e s A w a n d A c a r e h o m o g e n o u s i n s i d e a c e l l .T h u s i t f ol lo w s t h a t

U V = A,~UV 2 1 )S u b s t i t u t i n g E q . ( 2 a ) i n t o E q . ( 2 1 ) a n d t h e n i n t oE q . ( 2 0 ) l e a d s t o

a~x _ P t a n a _ 0 . 4 4 2 5 q t a n a (2 2)1 . 9 7 0 A w R A ~C o r r e s p o n d i n g t o t h e e x t r a p r e s s u r e f ie l d i n s i d e

t h e c e l l i n w - d i r e c t i o n , t h e r e i s a l so a C a u c h y s t r e s sf i e ld ins ide the ce l l , bu t i t i s l oca l i ns ide the ce l l , o r ,f r o m t h e m a c r o s c o p i c v i e w p o i n t , i t i s n o t c o n t i n u o u so n t h e k n i t t e d f a b r i c s h e e t . W e d o n o t a d d i t t o t h ec o n s t i t u t i v e e q u a t i o n s .

T h e C a u c h y s t r e s s i n k n i t t e d f a b r i c i n w -d i r e c t i o n c o r r e s p o n d i n g t o t h e t e n s i o n i s

3 P ( ]- - 2 3 )A b o v e e q u a t i o n c a n a l s o b e o b t a i n e d f r o m t h e

b o u n d a r y c o n d i t i o nSw = q (24)

w h e r e S w i s a c o m p o n e n t o f t h e n o m i n M s t r e s s te n s o r .S u b s t i t u t i n g

Sw = Ar (25 )i n t o E q . ( 2 4 ) g i v e s E q . ( 2 3 ) .

M a k i n g u s e o f E q s . ( 2 3 ) , ( 2 2) a n d ( 1 8) l e ad s t ot a n acr~x t = 0 4 4 2 5 ~ - a w ( 2 6 )

x t ~c Tex tO w =i n a l l y , su b s t i t u t i n g a c A w r

~--W Tw i n t o E q . ( 2 6 ) a n d E q . ( 1 8 ) , l e f t w i t hs i n g we a reT = q)Tw 27)

w h e r e= 0 . 4 4 2 5 t a n 2 8 )

7 T H E M I C R O - C O N S T I T U T I V E E Q U A -T I O N S O F K N I T T E D F A B R I C

W e r e w r i t e E q . ( 3 ) a sD i i D i 2 0D~I D~2 0

0 0 C66E c

Ecw(29)

A d d i n g t h e e x t r a c o m p r e s s i v e s t r e s s fi e l d T ~ X tt ot h e a b o v e c o n s t i t u t i v e e q u a t i o n g i v e s t h e m i c r o -c o n s t i t u t i v e e q u a t i o n s o f k n i t t e d f a b r i c

T c = T 2 - T : x t =Tow T~

D i i D i 2 0D~I D22 0

0 2D66E c

Ecw(30)

w h e r eD21 = D~a - ~ ( q )D n D22 D~2 - ~ (q)D12 (31)

W e r e m a r k t h a t t h e e x t r a s t r e s s fi el d e x i s t s o n l yi n s i d e t h e c e l l , o r f r o m t h e m a c r o s c o p i c v i e w p o i n t ,o n l y i n s id e t h e k n i t t e d f a b r i c . T h e e x t r a s t r e s s d o esn o t t a k e p a r t i n t h e s a t i s f a c t io n o f b o u n d a r y c o n d i -t i o n s. T h u s t h e s t re s s b o u n d a r y c o n d i t i o n s t a k e th ef o r m , a n a l o g o u s t o t h a t f o r w o v e n f a b r i c in R e f .[ 4]

TOBNB = TA A, B = 1, 2 (32)w h e r e T A i s t h e t r a c t i o n a p p l i e d t o t h e b o u n d a r y w i t hu n i t o u t w a r d n o r m M NA; T~ i s t h e s e c o n d P i o l a -K i r c h h o f f s t r e s s w i t h t h e e x t r a s t r es s t a k e n o u t .

W h e n t h e k n i t t e d f a b r i c is o n l y s u b j e c t e d a te n -s io n i n w - d ir e c t io n d e n o t e d b y q t h e b o u n d a r y c o n d i-t i o n b e c o m e s

Tw = q/ w 33)T h i s i s e q u i v a l e n t t o E q . ( 2 4 ) .8 C O N C L U S I O N A N D R E M A R K S

B y b o t h t e s t s o f k n i t t e d f a b r i c a n d a n a l y s i s o fm i c r o - k n i t t i n g s t r u c t u r e s o f k n i t t e d f a b r i c , a m i c r oc o n s t i t u t i v e E q . ( 3 0 ) f o r i t is p r o p o s e d .

T h e t e s t s o f k n i t t e d f a b r i c sh o w t h a t w h e n ak n i t t e d f a b r i c s h e e t i s s u b j e c t e d t o a t e n s i o n i n w -d i r e c t i o n t h e s h e e t c e r t M n l y b u c k l es . I t h a s b e e np r o v e d t h a t b a s e d o i1 t h e o r t h o t r o p i c c o n s t i t u t i v e

8/13/2019 Bf 02485866

9/9

Vol.20 No.6Eq. 3) the buckling cannot occur [51. Based on the mi-cro constit utive Eq. 30) the buckling of knit ted fabricas shown in Fig.2 a) is analyzed. It is shown thatonly a flexural buckling mode can occur for a knittedfabric sheet. The predictions are in good agreementwith the experiments. The details are repo rted in thesequel of this paper Part 2).

The tests also show that when a knitted fabricsheet is subjected to a tension in c-direction the sheetdoes not buckle. This can also be predict ed by a buck-ling analysis based on the micro constitutive Eq. 30)and following the same pr ocedur e as in Ref.[5]. Thedetails are reported in another article.R E F E R E N E S

1 Kim JH. Fabric mechanics analysis using large defor-mation orthotropic shell theory. [Ph D Thesis]. Deptof Mechanical and Aerospace Engineering, North Car-olina State University, 1991

2 Kang T J, Yu WR. Drape simulation of woven fabricby using the Finite-element method. J o f t h e T e x t i l eI n s t i t u t e 1995, 86 4): 635N648

3 Chen B J, Govindaraj M. A PhisicaIy-based model of

Zhang YT et al.: Micro-mechanical Model of Knitted Fabric 631fabric drape using flexible shell theory. T e x t i l e R e -s e a r c h J o u r n a l 1995, 65 2): 324~330

4 Zhang YT, Fu YB. A micromechanical model of wovenfabric and its application to the analysis of bucklingunder uniaxial tension. Par t 1: the micromechanicalmodel. I n t e r Y E n 9 S c i 2000, 38 17): 1895~1906

5 Zhang YT, Fu YB. A micromechanical model of wovenfabric and its application to the analysis of bucklingunder uniaxial tension. Part 2: Buckling analysis. I n -t e r Y E n g S c i 2001, 39 1): 1~13

6 Zhang YT. Xu JF. Buckling analysis of woven fabricunder simple shear along any direction. T e x t i l e Re-s e a r c h J o u r n a l 2002, 72 2): 147~152

7 Zhang YT, Xu JF. The buckling analysis of wovenfabric subjected to uniaxial tension along any di-rection. A p p l i e d M a t h e m a t i c s a n d M e c h a n i c s 2002,2 3 5 ) : 5 9 7 ~ 6 0 5

8 Zhang YT. The recent advances in fabric mechanics.A d v a n c e s i n M e c h a n i c s 2003, 33 2): 217~226 in Chi-nese

9 Zhang YT. Modern Fabric Mechanics and Theory onStability. Beijing: Peking University Press, 2003 inChinese)

10 Hu JL, Zhang YT. The KES shear test for fabrics.T e x t i l e R e s J o u r n a l 1997, 67 9): 654N664