Oloffsons Model

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DOI: 10.1177/004051756903900907

1969 39: 844Textile Research JournalI. M. Stuart

Some Developments of Olofsson's Theory of Woven Fabric Structure

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844Literature Cited

1. Ba1cerzyk, E., Calvet, E., Boivinet, P., and Hempel,K., Thermogenese de mouillage de fibres cellulosiques dans l'eau et le methanol, Compt, Rend,256, 3674-3676 (1963).2. Balcerzyk, E., Boivinet, P., Calvet, E., and Hempel,K., Therrnogenese de mouillage des fibres d'alcoolpolyvinylique, Compt, Rend. 256, 4005-4007(1963).3. Ba1cerzyk, E. and Kozlowski, W., MicrocalorimetricInvestigation of the Wetting of Some ProteinFibers by Various Liquids, Prseqlqd Wlokienniczy,21,296-297 (1967).4. Ba1cerzyk, E., Kozlowski, W., and Wlodarski, G.,Thermokinetics of Cellulose Fiber Penetration byPhysically and Chemically Reactive and the Structure of Accessible Regions, Textile Res. J. 39,666677 (196)9.4a. Balcerzyk, E. and Hempel, K., Calorimetric Investigations of Molecular Sieves, To be published.

5. Calvet, E. and Prat, H., "Microcalorimetrie. Applications physico-chimiques et biologiques," Paris,Masson, 1956, pp. 83-84 and 87-88.6. Calvet, E. and Prat, H., ibid., Chapter VIII.7. Crank, j. , "The Mathematics of Diffusion," Oxford,Clarendon Press, 1957.

TEXTILE RESEARCH JOURNAL8. Fraser, R. D. B. and Rogers, G. E., Bromine Allworden reaction, Biochem. Bioplzys. Acta 16, 307316 (1955).9. Frei , R., Synthetische Molekularsiebe fu r selektiveAdsorption, Chem, Rundschau 13,509-510 (1960),10. Hermans, P. H., Staining of Viscose Rayon CrossSections, Textile Res. J. 18, 9-18 (1948).11. Mercer, E. H., The Heterogeneity of the Keratinfibers, Textile Res. J. 23, 388-397 (1953).12. Nikonovich, G. V., Leont'eva, S. A., Byrkhanova,N. D., and Usmanov, Kh. U., The Structure ofthe Surface and of Ultrathin Sections of PolynosicFibers, Khim. Volokna 5,54-59 (1965).13. Novikova, S. A. and Vorob'eva, V. 1., MicroscopicInvestigation of Dyed Cross-sections of ViscoseFibers, Khim. Volokna 3, 51-53 (1964).14. Palil, N. B., Dweltz, N. E., and Radhakrishnan, T.,Studies on Decrystallization of Cotton, TextileRes. J. 35, 517-523 (1965).15. Trotman, E. R., "Dyeing and Chemical Technologyof Textile Fibres," London, Griffin, 1964, pp. 93-99.16. Usmanov, Kh. U. and Nikonovich, G. V., "TheElectron Microscopy of Cellulose," Tashkent. PressAcad. Sci. Uzbek. SSR, 1962, pp. 51-66 (Russ.).17. Woods, H. ]. , Some Problems in the X-Ray Diffraction and Electron Microscopy of Keratin Fibers.J. Polymer Sci. Part C, 20, 37-48 (1967).Manuscript received October 16, 1968.

------- < ! ' ~ - ~ Ijlr;ji>ii> -------Some Developments of Olofsson's Theory

of Woven Fabric StructureI. M. Stuart

Division of Textile Physics, CSIRO, Ryde, Sydney, AustraliaAbstract

The theory of woven fabric structure and yarn shape proposed by B. Olofsson is described and appraised. Some developments of this theory are given. These developments enable more straight-forward application of the theory to be made. Alternat ivebut similar formulas are deduced which are applicable to the case of a parti al ly se tfabric under no tension. Some applicat ions of these formulas are described and theconnection with similar theories of yarn shape used by Peirce is shown.Keywords1. Theories, "'oven Fabric Structure, Yarn Crimp.3. Models (Mathematical), Mathematical Analysis.4. Yarn Tension, Yarn Geometry, Yarn Crimp; Bending Rigidity; Bending Moment,Curvature.5. Crimp Amplitudes, Crimp Radius.

IntroductionD. Olofsson has proposed [4, 5J a model of th e

structure of simplewoven fabrics. From this model,th e shape and various properties related to theshape of th e wave form of the yarn crimp ar e predicted, and a relationship is determined which

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846 TEXTILE RESEARCH JOURNAL

1\1 = (- B1/;) K.

eter 1/; which he calls the form factor and proposesthat

Thus, with this postulated proportionality between the released and in-fabric curvatures, theyarn behaves as an originally straight rod with abending rigidity of (- B1/;). It is this property thatenables the subsequent analysis to proceedsmoothly.

Writing Equation 2 asdOI = (1 + 1/;) deds u ds

and integrating this equation between the pointss = 0, 0 = Om and s = 1/2, 0 = 0, Olofsson obtains

(4a)

(4b)1and {32 = - ~ p2 + Q2B

dOas s = 0,0 = Om and - = 0ds '1and at s = 2' 0 = 0.

He introduces parameters a and {3where tan a = Ijl

where

11 + c = p= F (2E - F), (Sa)~ = 2k (2E - F) (Sb)k = sin Om/2 (Sc)

d281/;B ds2 = - P sin 0+ Q cos 8,

(r /2F = ) 0 (1 - k2 sin2 lp)-ldlp

He obtains a general solut ion of the differentialequation under these boundary conditions, whichgives parameters like x, y, and s as explicit functionsof 0, a, and {3. He proceeds to give the results (ingraphical form) of the calculation from this generalsolution of the ratio hfP and the angle Om as func-tions of the yarn crimp (c = ~ - 1), for a varietyof values of a under different conditions of loadingand both positive and negative values of 1/;. Hedemonstates that, irrespective of the condi tionshused, the dependence of p and Om on c closely fol-lows that obtaining when Q = 0,' so that a = O.The shape of the yarn under these conditions hecalls the "a = 0 geometry." It can be shown thatthis is the form assumed by an elastica under compression (see Fig. 2). The equations describing theover-all parameters of this shape are: .

with boundary conditions,

yarns with the introduction of one parameter. !/t.the form factor.Olofsson proceeds to establish the differentialequation which follows from the equilibrium of theyarn under the action of the forces P and 2Q. Thisis

(3)

(2) = (1 + 1/;) K

where the subscript 0 refers to the released state.This equation enables 1/; to be estimated.To justify such an apparently sweeping assump

tion as that embodied in Equation 2, it might bepointed out that the detailed shape of a crimp-waveform has only a second-order effect on relationshipsderived from the shape. This has been demonstrated by Olofsson [4J, as shall be described below, and is further demonstrated in this paper.Among possible released crimp-wave forms that (a)have an inflection where they cross the fabriccenter plane, (b) have zero slope at the point ofmaximum displacement from the center plane, and(c) have continuous curvature, the selection of theone which fulfills Equation 2 probably has littleeffect on the results obtained therefrom. The assumption is justified because it facilitates the analysis and enables the consideration of partially se t

As will be seen, this assumption facilitates theanalysis of the differential equation describing theequilibrium of the yarn under the forces P and 2Q.It becomes possible to account for a released curvature in the yarn, and thus to consider a partially se tfabric without assuming a form for this releasedcurvature.

When 1/; = 0, K o = K and thus the yarn is se t atthe curvature it exhibits in the fabric.When 1/; = - 1, K = 0, corresponding to a completely unset yarn which would be straight on removal from the fabric. Substituting Equation 2into 1, Olofsson obtains



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SEPTEMBER 1969 847and

{r/2E = [ (1 - k2 sin2 lP)ldlPOlofsson proposes that when questions arise inwhich the shape of the yarn crimp form comes intoconsideration the "a = 0 geometry" be used. I t isshown below how Equations 5 can lead to simplerand more easily applied relationships.

Equation Sa as expansions in powers of th e modulusk (See Byrd and Friedman [IJ)

F = 11"/2 I1 + lk2 + (9/64)k4 + O(k6)}E = 11"/211 -lk2 - (3/64)k4 + O(k6)}

we obtain th e expression for th e crimp e as a function of k

~ c = k l l + (9/16)k2 + O(k.)}

Fig. 2. Forces acting on yarn center line in a = 0and a = r/ 2 conditions.

or k as a function of ek = .yell - ge/16 + o(e2) } (7)

Then, by expressing!: and Om as expansions inppowers of k, and using Equation 7 to express theseexpansions as power expansions in th e crimp c, weobtain

(9)

(8)t 4 . t:P=;;: 'VeIl + (3/16)e + 0(e2) }8m = 2 ~ c ' l 1 - (19/48)c + o(e2) }and

. ~p

where 8m is measured in radians.The following expressions have been found in

similar ways or by differentiating Equations 8, 9,and 5(a).

The final result of Olofsson's theory to be considered here concerns the equation he obtains whichrelates the forces P and 2Q, the shape and the formfactor for general a.This is

(6)dPI 11" . t :dh = - -2 'Vel 1 + (7/16)e + o(e2) }I GODdant (10)

where is given by Equation 4b above, y; is the formfactor, and

dl\ 11" .,-dh = -2 'Vctl - (9/16)c + o(c2) Ip con.tantdP Idl = 1+ c+0 (d2)h constant

(11)

(12)

and

a = 11"/2 GeometryIn some studies (for example , of hygral expansion), th e loading conditions on the yarn are P = 0,

2Q ~ O. I t is of interest then to determine theform of Equation 6 above, the force equation, forthis condition. We have then a = 11"/2, B = QB

where k = sin [ ~ ( 8 m - ~ . a ) ]and

ka sin 4>m = sin [!(0 - : : ~ ; 'a)]Developments of Olofsson's Theory

The relationships given by Equations 5 describing the "a = 0 geometry" are inconvenient to useas they stand. However, by a simple procedure ,equations can be obtained from them which aremuch more useful. Thus, expressing F and E in

dOm 11" . t :dh/l = 2" 'Vel 1 + (5/4)c + o(e2) } (13)

(l4a)



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848 TEXTILE RESEARCH JOURNALwhere

andex =kr/2 = sin [r/4 + 8m12]. sin r/4sin cPm = sin [r/4 + 8ml 2J

(14b)

(14c)

are Equations 14 above and1+ c = !- = 1',;..p -V2 sin 8m

h 1',;.. - 2B,;..P= -V2 sin e:(17a)

(17b)

(20)

Equations 14come from those following Equation6 when (a) sgn Q = 1, sqnP = - 1, ex = r/2,(Olofsson's Case 2) or (b) from Olofsson's Case 3,which, when ex = r/2, is mechanically ident icalwith Olofsson's Case 2.Tabulating these functions, start ing with givenvalues of 8m , we obtain the numerical relationship(see Appendix I)

1',;..2 = 3.87 -Ve(l + 0.94c) (15)so that the force Equation 6 reduces to

Q = 15.49 B lfC (1 + 0.94c) 11 - l ~ : ; ~ o l (16)Here, Equation 3 has been used to express v as afunction of I, 10 (8m) , and (8m)0.Equations 15 and 16 are dependent on th e yarn

crimp shape given by "ex = 1f12 geometry." Thisgeometry is determined by the loading on th eyarn, as shown in Figure 2. Further relationshipssimilar to Equations 8 and 9 can be obtained by acombination of analytic and numerical methodsfrom the equations describing this shape. These

The identity of these equations with Olofsson's[4J Equations 36 and 37 when sgn P = - 1, sg nQ = 1 with ex = r/2 follows from th e identity

2k cos 8m = -V2 sin 8m ,a consequence of Equations 14b and 14c.The relationships found are

h _,-p = 1.29 ""e(l + 0.21e) (18)8m = 1.94.,je(1 - 0.41e) (19)

By differentiation of equations for the crimp1 + e = and Equation 18,we obtain

dP I -dh = - 1.55 "c(1 + 0.31e)I eon.tantTable I compares the defining equations and

the numerical expressions for : and Ip h I -eon. t .for the "ex = 0" geometry and the "ex = r/2"geometry.

TABLE I. Defining Equations and Relationshipsa == 0 Geometry a ... 11'/2 Geometry

I1+ c ... - ... F/(2E - F)Ph- ... 2k/(2E - F)Pk == sin 9./2

Defining Equations1 + c = !_... F....P sin em

h F.m - E"mp== ~ 2 s i n 8..k ... sin (11'/4 +9./ 2)ksinfJ.. = sin 11'/4

e; ... 1.94 ' \ ~ ( 1 - 0.41c)dPIh . .. - 1.55 *(1 + 0.31c)

l-eolltlt.

Relationshipsh -- == 1.29 ~ c ( 1 +0.21e)ph- ... 1.27 +0.1ge)p

e.. == 2*(1 - 0.40e)dPIh . .. - 1.57 *(1 + 0.44c)'-oonllt.



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(22)

SEPTEMBER 1969Comparison of the numerical relationships confirms Olofsson's claim that the parameters relating

the shape of the crimp wave (such as hlp and 8m) ,expressed as functions of c, are fairly independentof ex and are well approximated by their values for.a = O. Between ex = 0 and ex = 1T'12, there is adifference of the order of only 2% in the value ofhiP and 3% in the value of Om. This is a strongindication that the shape of the crimp wave formhas only a second order effect on relationship derived from that shape.

Substituting either Equation 9 or Equation 19into Equation 16 for Q, neglecting the factor of cin these expressions, and assuming that t = to(that is, that there is no change in curvilinear lengthwhen the yarn is removed from the fabric), weobtain

Q = 15.49 B;C (1 + 0.94c) 11 - ~ I (21)This equation enables the value of the interyarnforce Q to be calculated for a partially set conditionof a fabric under zero tension.

Some ApplicationsSome relationships are obtained from the equalityof the interyarn forces Ql and Q2' as calculated by

Equation 16 for either warp or weft values. Writ-ing Equation 16 for both warp (suffix 1) and weft(suffix 2), weobtain

_ 15.49 B 1 ~ ( 1 + 0.94 C1) 1.1. IQ l - / 12 '1'1Q _ 15.49 B 2 ~ ( 1 +0.94 C2) 1.1. I2 - 122 '1'2

Expressing 1as a function of c and P,l = p(1 + c)

and equatingQI and Q2'we obtain.w;. 1 + 1.06 C2 PI2 B 2 '/;21 + 1.06 Cl' -{C; = P22 B 1 '/;1

This can be compared with the approximateequation found by Peirce [6J who considered whatis effectively the "ex = 1T'/2 geometry" with initiallys traight yarns , so that '/;1 = !/t2 = - 1. Peirce'sequation is

849In Equation 22, B 1 and B 2 are t he warp and weft

yarn bending rigidities. The measurable quantityin bending r igidi ty is the bending rigidity of unitwidth of fabric (EI). We now assume that thisrigidity is the sum of individual yarn rigidities. 1

11B 1 = (EI) 1PI P2= (Elh ' (1 + Cl) 'P2

I h f 11.. d d thI ere, t e actor - 1S mtro uce because ePIyarn rigidity is a curvilinear one and that contribut-ing to the fabric rigidity would a be rectilinear one,So, substituting in Equation 22 for B 1 and B 2, weobtain an expression for the rat io of the weft formfactor to the warp form factor.

' 1 ~ 2 1 1 1 + 0.06 Cl P2 3 (EI) 1 (23)'I' = -vc;'1 + 0.06 C2' P13' (Elh

DiscussionI t has been shown how the equations describing

Olofsson's "ex = 0 geometry" can be reduced tosimpler approximate forms sui table for both the-oretical and numerical investigations.An alternative system of relationships has been

presented which corresponds to an "ex = -rr/2 geometry." This system corresponds to the case whenthe fabric is under no tension and should be moreappropriate for application to fabrics relaxed undertensionless conditions. I t has been shown how thiscase corresponds to that studied by Peirce [3J (withthe addition, however, of the form factor !/t).Olofsson's theory with ex = 1T'/2 improves on that ofPeirce because of the introduction of the formfactory !/t, which allows the yarns to have a partialset. Relationships between parameters such ash/ ] and Om are independent of the loading level {3and the form factor v. These parameters (/3 and'"] enter into consideration only in expressions likeEquations 6 and 21 which relate the loading to theshape of the yarn wave form.1That this is not generally so has been established byGrosberg [3 J who has found that the fabric bending stiffness

may exceed the sum of the crimped yarn bending stiffness byup to 100%.



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850Appendix I

This appendix gives the outline of the calculationsby which Equations 18, 19, and 2, were determined

TEXTILE RESEARCH JOURNAL(Table AI). The elliptic functions F(f/ and E(q,}were found by interpolation in tables from Byrdand Friedman [1].

TABLE AI. Out line of Ca lcu la tionshlp ! [ Fe(

Oloffsons Model

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