Huang

FINITE EXTENSION OF A VISCOELASTIC MULTIPLE FILAMENT YARNt

N. C.HUANG Departmentof Aerospace and MechanicalEngineering,University of Notre Game, Notre Dame, IN 46556, U.S.A. Meceiwd 2 September 1977: in revised fan

Zl Lkccmber 1977: r&ted

for publication 15 F&wary

1978)

M&ad-This paper is conceme-d with the invest&ion of finite extension of a vkwekstk multipk fikment and sin& kyered yirn subjected to axial forces and twistinftmoments.The &men1 in the yarn is consideredas a Iii viscoekstic sknder curved rod with circukr cross scctkn and heiii uw&ra& Thetbeoryof~kmlerctavedrodsisemployedinUle~whenby(kcmnnrreofrbcfibnentis atromsdtoktlrlidcatlyraullsuchthtchcaossrectioabfthe~psrpssdicPbtotbeuirdtbcyM is approximately@tical. Ill our study, we have ako assumedthat thwe is sii@lIg~~~M!n~~ deformationof the yarn. Geometricalnonlbwuity in introducedby the reswnms sectionofthebhmeot.lnordertoiltustntethetachaiqueiatrodoadhae,iaomwwricrlrarlylh,the extensional retaxationmodulusof the Went is derived from a model of tlwee-ekwnt ‘s&d and Poisson3 ratioof the fikment is regardedas coostant.Exampks are preseotedfor the.extensioos of yarnswith tixedends and yarns with free ends. NOTATION radii of iikments eqn (51) co&kknt of kinetic friction ~~fromthe~teroftbeyuntotkpoiatoftursrryoffllrnwftts sqn 01). FL. Mb) tensik rekxation modulus cqn 00) E(O) componentsof the stress resultantvector

applied axial force eQn(51) pitch of the helix unit vectors constants eqn (51) kngths of yarns lengths of 6kments components of the moment vector applied twistiw moment eqn (51) numberof fikments const8nt force between lUaments eqn (51) radii coordinateof the fikment wn (51) time ns reztangukr cartesian coordinates Ft. l(c) small increments constant flxialstrainoftheyarn axial strainof the iikment hetii an&s principalnormalcmvatures unit vectors eons (6) and (11) Poisson’s ratio torsions pokr anek an&oftwistoftbeyarnperunitkn@b eqn (51)

tThis research is supportedby the National Sciince Fotwktion Grant ENG7645775. Ss VOL

I4 NO. 7-D

519

N. C.HUNG

580

INTRODUCTION

The mechanical behavior of most textile materials exhibits what is known as timesensitivity. For a yarn subjected to an axial load, creep deformation is observed following instantaneous elastic response. In order to analyze the yarn problem in a more realistic manner, the theory of viscoelasticity must be employed. The creep deformation of linearly viscoelastic continuous filament yarn has been investigated by Jones[l]. His analysis is based on the small deformation theory whereby geometrical nonlinearity is ignored. Also in Jones’ study, the yarn material is considered to be incompressible; the normal stresses acting on any yarn element in the transverse directions are assumed to be equal and the shearing stresses are neglected. The finite extension of a linearly viscoelastic two-ply filament yarn has been analyzed by Huang[21 based on the theory- of slender curved rods(31. In Huang’s study, geometrical nonlinearity is introduced as a result of reductions in helical angle and filament cross section. In this paper, we shall study the problem of finite extension of multiple filament linearly viscoelaatic yarns subjected to axial forces and twisting moments at both ends. Similar to the case of the two-ply 5lament yarn, we shall treat the filament as a slender curved rod with circular cross section and helical configuration. We shall assume that the curvature of the filament is sufficiently small such that the cross section of the filament perpendicular to the yarn axis is approximately elliptical. Also, we shall consider that there is siipping between filaments during deformation of the yarn. It is found that the creep deformation of the yarn is governed by two nonlinear integral equations which is solved numerically by a modified Newton’s method. Two probkms have been selected for study, namely, the extension of a yarn with fixed en& and the extension of a yarn with free ends. Effects of initial helical angle and the superposition of a twisting moment on the axial extension of the yarn have also&en included in the study. GEOMETRYOFTHEYARN Let us consider a long linearly viscoelastic n-ply filament yarn as shown in Fii. l(a).

(n = 2,3,4,. . .). The yarn consists of a single layer of Blaments which form a cylindrical tube. The cross section of the filament in the undeformed state is considered to be circular with radius u,,. The center line of the undeformed filament is prescribed by the following rectangular Cartesian coordinates: x~=r~cos&,

x2=r0sin&,

x3=&,

(1)

where r. and & are the polar coordinates and &,,is a constant which is related to the length of the filament of one turn of twist measured along the axis of the yarn /is by i

The unit vectors in the tangential, principal normal and binormal directions of the center line of the undeformed 6kment are . Ao=

$(-r. sin hOi + r. cos &f + &&),

k;o=

-(cos hi + sin f$&,

(4)

60--!-(kosin&$-kocos&f+rd) - * respectively, where i: I and k^are unit vectors in the xl, x2 and x&irections and po = (rt +

bz)ln.

Note that the principal normal of the filament is in the radial direction and toward the axis of the yarn. The helical angk of the center line of the undeformed filament is ‘90= tan-’

2.

Finite extension of a viscoclastic

multipk

filament yam

581

(b) Secttan A-A

(a)

Fii. 1. Geometry

of the problem.

Let us define the curvature vector by dE/ds where dE is the iniktitesimal rotation vector of the coordinate axes in i, i; and idirections in a distance ds along the center line of the filament. The principal normal curvature of the center line of the’ undeformed @antent is then the component of the curv&ure in the binormal diction. The torsion of the center line is the component of the curvature in the tangential diiecti0n.t They are

1 . Ko = POstn 4v

=-Los 7o PO

eo

(8)

respectively. The component of the curvature in the principal normal direction is xero. Let us cut the yarn by a plane perpendicular to the axis of the yarn. The cross section is shown in Pii. l(b). We shall follow Phillips and Costello[S] and assume th&thCCUWSltUl-COf tbc filament is su&iently small such that the cross section of each tflament is elliitical. Hence the cross section of the undeformed yarn consists of it ellipses tangent to each other with center of ellipse at (fncos (2iw/n), r&n (2iw/n)) (i = 0, 1,. . . n - 1). We shag also assume that the common tangent of any two kghboriqg ellipses passes through the center of the cross section.sincetbeprincipelnormplofthefilruneatisintberadirrldircctioa,tbelsrrjorpxisoftbc ellipse is 2a&ko and the minor axis is 2a,+ By analytical geometry, we can easily show that -l/2

(

q = a& b1 sin*f - ao2cos’ f

)

.

Equation (9) agrees with the corresponding relation given in [4]. When n = 2, we obtain from quation (9) that ro= 40 which agrees with the result of [2]. The yarn is subjected to an axial tension P and a twisting moment fi in the direction of the original twist of the yarn. In the deformed state, as a result of contact deformation, the cross section of the 6lament is no longer circular. Since the con&uration of 6lament is helical, the analysis of this type of contact problem is extremely d&t&. In the following, we shall neglect the contact deformation and assume that the cross section of the filament in the deformed state remains circular with radius a. The center line of the filament in the deformed state remains helical. Let r and 4 be the polar coordinates of any point on the deformed center line of the t3ament. We have equivaknt equations identical to qns (l)-(9) with tBt s&cript 0 dekted. Hence. K =

tin Love’s textI41. our torsion is refened

f

Sin 8,

+ose,

to as the twist of the filament.

(10)

N. C. HUANG

582

p=

8 = tan-’ f.

(9 + k2p2,

(11)

Also, we have -112

k’sin’t-a2cos2f)

.

(12)

The distance from the center of the cross section of the yarn to the point of tangency of two neighbori~ ellipses is found as (13) The line of contact of the deformed filament is helical with radius d. Next. let us consider a cross section of the deformed filament as shown in Fii. l(c). To determine the central angle 2s between the points of contact, let us first project the circular cross section on the pkne perpendicuhu to the axis of the yarn. The projection is also an ellipse with major axis 2a and minor axis 2uk/p as shown by dotted line in Fii. l(b). Note that the projection of the helical tine of contact on the plane perpendicular to the axis of the yarn is a circk with radius d and center at 0. The angle /3 can be determined from the intersection of the projection of the cross section and the projection of the line of contact. similar approach for the determination of @ has also been employed by Costello and PhiUips[61. It is found that cos B = (p’- [p2(d2 + k2) - a2k2]'"}/(ar).

(14)

Thus after d is calculated by eqn (13). fl can be determined by eqn (14). Note that for the two-ply filament yams, n = 2, d = j3 = 0. Hence the filaments are in contact to each other at the center line of the yearn. The same con&&n has also been drawn in (21. FORMULATION OFTHEPROBLEM In the following, we shall denote the components of any quantity in the tangential, principal normal and binormal directions by subscripts A, p and Y respectively. Each filament will be treated as a long sknder curved rod. The components of the stress resultant acting on the cross section of the l&unent are denoted by F,, F,, and F, and the components of moment acting on the cross section are denoted by M,, M,,and M, The components ,of the distributed force per unit length of the filament are pA, p,, and p. and the components of the distribution moment per unit length are rn*, m, and m, Along the line of contact, there exist the normal distributed force P and tangential distributed forces Q and R as shown in Fii. l(c). Hence PA=O*

pp

=-2pcosB.

pv=2Qcos@

(15)

and mA -2Qa,

m, = 2Ra sin @, m. = 0.

(16)

If we treat the filament as a one-dimensional sknder body, we can obtain the equations of equilibrium of all forces and moments acting on an element of the filament from the theory of slender curved rods [31. Since the yarn is considered to be long, all derivatives of the stress results and moments with respect to the arc length of the filament must vanish. The equations of equilibrium can be written as

F&-PA =O.

(17)

With eqns (15) and (16). eqns (17)-(22)become

In the fol~~wj~g.we shaff consider that during deformation of the yarn, tkre is sl~~pi~~ between filaments. Let us denote the coefficientaf kinetic friction between fikmcnts by c. We have

Here eqn 125)can be rewritten as

Let us set the local coordinates in the tangential,,principal normal and binarmal directions. Under the assumption of small strains, the Clebsch-Basset momenticurva&tre relations far elastic slender curved rodsf4] are

where EI is the bending stiffness, CJ is tk torsional stars and CTis P~~ss~~“sratio, ~~~~~ C29)is derived frrcptbat the c ~f~~~~~ normal direction is zero for both the undeformed and defamed fkments, In the viwFctutic case, in view of the absence of data, we shaI1assume Poisson’s ratio to br: constant and treat Young’s mudufus E in eqns (281 and (30) as an int@ operation ~~ to &e c&r~spunde~ee principle in v~s~~Iasti~it~.In the follQ~~, we ShaIluse the logon

where T is tk time. In the v~s~~lasti~ case tqns CJ81and ~~~ in (32)

where E(T) is the axial tensile relaxation modulus of the filament. Equa&ion(29) still holds. By

N.

584

C.

WANG

(24),(27),(32) and (331,we obtain

eqns (HI),

F, =

a(p2+cak tan 8)”

+ cr;F, tan @ .

$

(34)

The contact force between filament can be found from eqn {24)as P =-

#_F,7).

2c:#

(35)

Ia the elastic case, 8s a result of Poisson’s effect, the normal S&Z&Iin the t~sve~e direction of the bunt is --aI=

aa

-CT-.

Fn wa E

(36)

By IMDMof the correspo&nce principle, the tangential component of stressresultant in the filament in the viscoehksticcase is E = -$

E*da.

(371

The overaNequilibrium of the internalforces and the appkd force P re@res that [3,5].

SimiIarly,the overall equiiihrium in moments requires

Witheqns (32) and (33),we can rewrite eqn (39) as

~s~~veiy*

The u~~fu~ed and defo~ed length of the fil&rne~tare 10= zAi&j,

1=2u&

(4%

respectiveiy, Hence the axisUstrain of the yarn is (431 and the axial

strain of the giant

is

Finite extension of a viscoclastic

585

multiple filament yam

As a result of Poisson’s effect. we have

a

--

1=-m*.

(45)

00

After elimination of lI and M/MO from eqns (43)-(4% we obtain

The angle of twist of the yarn per unit length is

,=(M--Mo)2c~ Lo

(A& )’

(47)

koMo

After elimination of eA and M/MOfrom eqns (44), (45) and (47). we obtain

Hence

if the ends of the yarn are unconstrained, elongation of the yarn wili be coupled by a twist. On the other hand, if the ends of the yarn are constrained such that fl= 0, we have an additional constrained condition from eqn (48). It is (49) In this case, if a is known, p can be determined from eqn (49) and k can be found from the relation derived from eqns (11) and (12).

Finally, 8 can be found from eqn (11). METHOD OF SOLUTION Let us

denote E(0) by E,J and introduce the following dimensionless quantities:

Em = EmEo,

& = Mao,

Fo= r0la0,

PO=pola0, 6 = ah

f= k/00,

(51) For given values of n, eo, the initial geometry of the yam is &term&d equations:

&=csf

from the following

l/2

(cotzeo+cos ) 2s

‘;;

,

(52)

Under given values of c, a, f and A and given function &‘I’), the viacoelostic deformation of

Finiteextensionof a viscocksticmultipktikmcntyarn

587

computation, we shall employ Lee and Roger’s method181to evaluate the hereditary scheme, we first divide t into N intervals with fl = 0’ and T. We have

In our

integral numerically. In our numerical tN+l =

where SN+I=E(rN+l)[f(fl)-fOl-~~1+E(rN+I-rN)lf(rN)

that when time increases step by step, at N which have been determined in the previous time f=fN+lr&+I invoivesf(ti)fori=1,2,... steps. The only unknown in eqn (73) is f(fN+,). By using eqn (73). an integral equation of Volterra’s type can be reduced to an algebraic equation. In the following, we shall analyze the problem of finite extension of yarns with two types of end conditions. and fO is the value of f in the undeformed state. Note

Yam with jixed ends

When the ends of the yarn are fixed, o = 0. The fixed-end twisting moment is determined by eqn (69) for the instantaneous elastic response and by eqn (63) for the viscoelastic response. For each time step, we assume a value for d and compute P and k by

and (75) The values of ?, & @,f&and f. are determined respectively by eqns (55). (57)-(a). The correct value of d must fuBll the condition that the value of g as determined by eqn (68)or (62)is zero. We shall use a modified Newton’s iterative method for t&edetermination of ‘i In this procedure weselectthreevaluesofa’asgivenby&,&-Aandg~+A,whereAisasmaUnumbcr,and compute values of g which are denoted by 81, g2 and 83. The derivative dg/dG at d - 4 can be approximated by the following central difference equations:

9 g’=dd I ali, =$b-ah Hence, according to Newton’s iterative form-&, the new value of d in the iteration would be di+,=isi-

&,=6*-

8

24L h-82'

m

Our iterative procedure continues until the absolute value of g is smaller than a certain prescribed value. It is noted that in our problem, t&erate of conis fast. Computations are first carried out for go= lP, n ‘4, ~“0.7, cmO.2 and ar0.45. Zn Fw. 2, the creep curves c,(r) are shown by solid lines for various values of fi The finite jumps at r~Oinslictfctbc~e~~res~nseoftbey9nr.At~=oD,JI~cmvQIrpporch asymptoticatly the value corresponding to the delayed elastiq mod&s B(m)- 0.7. The &cd end

N, C. HUNG

3-88

twisti~ moments 6(t) are aiso shown by dotted Iines. It is noted that the fixed end twisting moment is nearly constant for this case. In order to show the geometrical nonline~ity, a( 11is plotted against f’in FQ. 3. The Ql) curve as determined by the linear theory is also shown by a dotted line for comparison. Note that the nonlinearity becomes evident when J is large. The helical angl_eof the filament @(1)is plotted against fin Fig. 3. As we may expect, 6( 1)decreases with irmeasing f. In Fig. 4, the dependenceof eY(l),fA{l) and G(l) on n are shown for j = 0.02. When n increases, both q,(l) and f&(i) decrease. The value of fi(l) increases with n because the radial distance F(l) increases with n. The dependence of & (1) and rii(l) on the initial helical angle & is shown in Fig. 5 for n = 4 and r’ = 0.02. When 4 = 0, all &unents are straight. Hence, fk ( 1) = O.OOf and S(l) = 0. It is found that both fA(l) and G(l) increase with do.

0

1 2

c

1

1

I

1

I

3

4

5 n

6

'z

8

Finite extension of a viscoclasticrnu~~k.~l~ent

yarn

589

When the ends of the yarn are free, Si = 0 and of 0. In this case, we have two unknowns Z and k’to be determined. Again, we sh& employ a modii Newton’s iterative method in our computation. Fit, we try out a set of values a’= 4 and R= & and compute g and k according to eqns (68) and (69) for the ins~ntaneous elastic response and eqns (62) and (63) for the viscoetastic response. They are denoted by (g,, trl). Next, we select four sets of values for 6 and k’in the ne~~~~ of 6=iii and I--&. They are (&-A,&), (~~+A,~)~ f&&-S), 14, & + 6). WC have four additional sets of cakuiated vaIues of g and k. They are (gz,Itl),

(83,h3),(gr, h,) and (gs. tr$)respectively. The partial derivatives of g and h with respect to ii and 1 as evaluated at 0’= di and f = & can be approximated by the following central difference equations:

The new set of a’ and k”can be determined by the folIo~ng equations given in Newton’s iterative

method for two variabIes:

where

The iterative procedure conti~~s until the value of a*+ Rzis smaller than 8 certain prescribed value. It is found that the rate of convergence is ofso fast for the use of two variabks. The creep curves, c,,(t), are shown by solid lines in Fig. 6 for 6J0 = lS”, II = 4,~ = 0.7, c = 0.2, Q = 0.45 and diierent values of f In comparison with the creep curves shown in Fw. 2, it is seen~additionai~strrrinofthtysrnconbeinrroducediftbte~oftheyornisrfloued totwist.ItisfoubdtBatwhantbeeadsofEBeyemis~,tbs:axist.extatsionof~yarnis accompanied by an untwist of’the yarn. The an&s of untwist per unit Ien@ of the yarn, - it, be also shown in Fii. 6 by dotted lines. Theesect of surest of a twisting moment on the elongatian of the yarn is sbown in Fig. 7. It is noted that when rii increases,tbcbclicalangk increases and consequently e,(f) decreases. When 1 is m!licidy bKgC,e,(l) can tamcame negative. The fihunent stress fA(1)and the contact force p(f) are a&o plotted against rii in Fii. 7. It is found rhat f,( 1)is essentia@ygoverned by f and the effect of the additional twisting mmmt on fi(l) is ins~ffi~t.

The contact force p(l) increases slightly with 6.

DISCUSSION (1) The finite extension of elastic wire cables has been inves&ated by Costello and Ph~lips[9]~ Their problem corresponds to our problem with T = 0. The concl&ons drawn in their study are very much similar to what we have obtained here for the vkoelastic probkm.

(2) The mechanical behavior of a yarn is governed chietiy by three factors--time, temperature and humidity. In this paper, our investigation is focused on the time-dependent characteristics of the yam and ignoredthe eifects of temperature and hi, Hence our problem is restricted to the case of isothermal en~on~nt with constant humid&y. Shot&l the tempera~ ture and humidity also be time dependent the-relaxation modulus used in our analysis must be mod&d to inch&e the effect of temperature and humidity. (3) Our analysis is based on a model where the deformed cross section of tbe fllamcnt remains circular. In reality, the contract of filaments are of Hertz’s type in both undeformed

N. C. HUANG

590

0

1

0

I

0.5

I

1.0

I

1

I

I

I.5

2.0

2.5

3.0

m

t Fig.6. c,(Oand -ru(r)curvesfor &= IS”. 12 = O.(fret cuds). n = 4. 7 = 0.7. c = 0.2, p = 0.45 and various values of f.

Fig.7. e.~I~./~(l~,o(l)andp(l)xl0vs~ccurvesfor the case of free ends with 6,,= 15”. j=O.O?. y = 0.7, c = 0.2 and Q = 0.45.

n =p.

and deformed states. Additional geometrical nonlinearity can be introduced by the Hertz’s contact deformation of the filament. The result given in this paper is valid if the width of contact is small in compatjson with the diameter of the filament. (4) It is noted that the actual cross sections of filaments are not perfectly circular. Also their distribution is not entirely uniform. To analyze a yarn with nonuniform filaments, statistical theory must be employed. However, our study would still provide a general feature of the finite deformation of viscoeiastic multiple filament yarns.

REFERENCES I. N. JOTS. Elastic-pUic and viscoelastic behavior of a cootittuous filament yarn. hr. 1. Me& Sri. I& 679-687 (1974). 2. N. C. Htuttt#. Ott iittite e~tenrioa of a visco&atic two-ply fRantent yam. TextileRes. I. 48.61-67 (1978). 3. N. C. Huang.Theoriesof elastic skttder curved rods, J. A@. Math. whys. (ZAMP). U l-19 (1973). 4. A. E. H. Love. A Tnatise on the Mathematical Tkwy of Elasticity. Chap. 18. p. 397. Dover, New York. 5. J. W. Pbiltip and G. A. Cost&Jo. Contact stress in twisted wire cables. 1. Engng Mech. Du. Pnx. American Sot. Ciuil Ertgrz. 99, E&G 331-34ltl973). 6. G. A. Costello attd J. W. Phillips. A more erect theory of twisted wire cables. /. Engng h&h. WV. Proc. American SW. Civil Engrs, IU. EMS IO%-1099 (1974). 7. W. FlRgge. Viscocfasticity. Chap. 2. pp. 16-17. Blaisdell. Massachusetts (1%7). 8. E. H. Lee and T. G. Rogers, !Sohttiin of viscoelastic stress analysis probkms using measured creep or relaxation functions. 1. Appt. Mech. 33, 127-133 (1%3). 9. G. A. Costello and J. W. Phillips. Effective modulus of twisted wire cables. /. Engng Mech. Dir.. Proc. American SIX-. Cicil Enqs. 102. EMI, 171-181 (1976).