hysteresis of tensile load strain route of knitted fabrics under extension and recovery

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Textile Research Journal

DOI: 10.1177/0040517507090504 2009; 79; 275 Textile Research Journal

Masaru Matsuo and Tomoko Yamada Processes Estimated by Strain History

Hysteresis of Tensile load � Strain Route of Knitted Fabrics under Extension and Recovery

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Textile Research Journal Article

Textile Research Journal Vol 79(3): 275–284 DOI: 10.1177/0040517507090504 www.trj.sagepub.com © 2009 SAGE PublicationsLos Angeles, London, New Delhi and Singapore

Hysteresis of Tensile load – Strain Route of Knitted Fabrics under Extension and Recovery Processes Estimated by Strain History

Masaru Matsuo1 and Tomoko YamadaDepartment of Textile and Apparel Science, Faculty of Human Life and Environment, Nara Women’s University, Nara 630-8263, Japan

In [1–3], we studied the mechanical anisotropy and thenon-linearity of the several knitted fabrics to study the ori-gin of the fit of the fabrics to the human body. The analysiswas carried out in terms of three categories. One was largedeformation modes [1] under strip biaxial extension (cor-responding to uniaxial extension, with the fixed dimension

in the direction perpendicular to stretching) using the “lin-earizing method” proposed by Kawataba et al. [4, 5].1 Thesecond was the stress relaxation mechanism [2] using plain

Abstract Hysteresis phenomenon of tensileload (stress) under extension and recovery proc-esses were estimated in terms of time dependenceof tensile load by using several kinds of knittedfabrics. The measurements were done under stripbiaxial extension corresponding to uniaxial exten-sion with a fixed dimension in the direction per-pendicular to elongation. The present conceptwas based on tensile load (stress) relaxation inelongation and restricted sides. To attempt mathe-matical evaluation, a simple hypothesis describedby hereditary integral was proposed to explain thehysteresis of tensile load (strain), strain routesunder extension, and recovery processes, in whichthe extension and recovery routes were formu-lated as addition and subtraction of step functionof strain history, respectively. The calculatedcurves were in good agreement with the experi-mental curves in the large deformation region.The theoretical calculation to give the best fit withexperimental curve could be realized by using nadopted as an exponent to describe the “lineariz-ing method” (F = Yen, where F is the tensile loadand e is strain of a fabric) proposed by Kawabataet al. The present evaluation improved on theirproposal, since the present theory can representthe hysteresis routes by using the same value ofparameter n.

Key words extension and recovery processes,hysteresis of tensile load (strain)-strain routes,linearizing method, strain history

1 Corresponding author: e-mail: [email protected]



276 Textile Research Journal 79(3)TRJTRJ

knitted fabrics, in which the fabrics were treated as a two-dimensional visco-elastic body and the corresponding prin-ciple between elasticity and visco-elasticity was used toanalyze stress relaxation. The third was large expansionand contraction [3] of plain knitted fabrics containing poly-urethane filaments on the basis of the changes of structuraland mechanical properties of the cotton yarns as a doubleyarn.

The hysteresis of fabrics has been analyzed by Kawa-bata et al. [4, 5] for the bending and shear processes of fab-rics. In their work, the properties were estimated as astructural body and not as a continuum in terms of the dis-placement of individual yarns and fibers in the fabric byexternal applied load. In spite of a number of establishedworks [6–15] concerning excellent fabric-deformation the-ory based on fabric structure mechanics, any theoreticalmethods of fabrics were out of the framework of theirstructural changes under extension and recovery processes.Certainly, the analysis of hysteresis under extension andrecovery routes is a difficult problem and there is no fun-damental concept in any material science fields.

This paper deals with the tensile load and strain rela-tionship under extension and recovery processes to analyzeconsiderable hysteresis of knitted fabrics according to ourprevious studies [1–3]. To pursue the analysis, the hystere-sis is simply analyzed as a hypothesis associated with stressrelaxation and the estimation is carried out by using hered-itary integrals.

Experimental Details

Three kinds of knitted fabric were used as test specimensand their properties are listed in Table 1 [1–2]. Each fabricwas cut into strips of 10 × 10 cm2. Among the three fabrics,two kinds of plain knitted fabric, which were termed asSpecimen A and Specimen C, were of the same knittedstructure. In Specimen A, polyurethane filaments weremixed as a double yarn together with a cotton yarn, whileSpecimen C was a single yarn fabric of cotton fibers. Speci-men B had a 1 × 1 rib structure produced by mixed yarnsof cotton and polynosic fibers. Using these three samples,the tensile load and strain relationship under extension

and recovery processes was obtained by using a strip biaxialtensile tester (KESG-2-SB1), according to the schematicdiagram in Figure 1.

As shown in Figure 1, the extension is carried out alongone direction and the strain in the transverse direction isrestricted to zero. That is, mode (a) is the extension andreturn processes in the warp direction (X1-axis) and mode(b), in the weft direction (X2-axis). The extension weredone at 8.2%/s and fixed at a standard speed of KESG-2-SB1. Hence the extended speed was fixed to be the same asthe return speed. The compulsory return by automatic con-trol was done immediately after the extension. To obtainaccurate results, the measurements in one direction weretaken five times by changing the fabric samples. The five

Table 1 Characteristics of three kinds of knitted fabric, A, B, and C.

Specimen Fiber Yarn Yarn count (tex)Twist number

(n/cm)Structure

Mass (kg/m2)

A Cotton 93.6%Polyurethane 6.4%

SpunContinuous-filament

1/14.81/ 2.2

6.0 Plain weave 0.215

B Cotton 45%Polynosic 55%

Mixtured spun 1/14.8 6.4 1 × 1Rib 0.106

C Cotton 100% Spun 1/14.8 7.8 Plain weave 0.110

Figure 1 Schematic diagram of strip biaxial strain: mode(a) strain in the warp direction (X1-axis) and mode (b)strain in the weft direction (X2-axis).



Hysteresis of Tensile load – Strain Route of Knitted Fabrics M. Matsuo and T. Yamada 277 TRJ

curves were confirmed to show mostly similar profiles. Theinner stress within the knitted fabric changed from positiveto negative and this tendency was considerable for Speci-men B and Specimen C because of the compulsory returnprocess of the tester. This mechanism will be discussedlater in more detail.

Incidentally, time dependence of tensile load, termed asload relaxation, was measured according to the previousmethod [2]. The instantaneous strain was applied alongone direction and the strain in the transverse direction wasrestricted to zero.

Results and Discussion

Time Dependence of Load by Instantaneous Constant StrainIn the previous paper [2], it may be concluded that whenan instantaneous constant strain along the X1 directionis applied to a knitted fabric at e22 = 0, as shown in mode(a) in Figure 1, the responded stress could be representedas a function of time.

At e22 = 0:

(1)

and at e11 = 0:

(2)

where Yij are tensile load (stress) relaxation stiffness [2].F11 and e11 are tensile load (stress) and strain, respectively,in the warp direction (X1-axis), while F22 and e22 are tensileload (stress) and strain in the weft direction (X2-axis). Forfabrics, the tensile load is normalized by the fabric width,which is used as a special quantity instead of stress,because of the difficulty in measuring the cross-sectionalarea of fabrics.

Figure 2 shows an example of the tensile load relaxationcurve against time along the X1 direction, mode (a), meas-ured for Specimen A. It is seen that the logarithmic curveof tensile load against time can be represented as the sum-mation of two straight lines. Of course, the applied instan-taneous strain must be limited to the regions assuring thelinear relationship of the logarithmic plots of tensile load(stress) versus logarithmic values of strain, as discussed interms of the application limit of the “linearizing method”

in [1, 2]. The similar tendency obtained for Specimen Awas confirmed for the other fabrics (Specimens B and C).Such behavior suggests that the knitted fabrics were com-posed of two kinds of relaxation mechanism; the rapidmode and slow mode. If this is the case, the relaxationmodulus is composed of rapid and slow relaxation mecha-nisms. Then the time dependence of load Yij(t) in equa-tions (1) and (2) can be represented as

Yij(t) = Aij exp(–αij t) + Bij exp(–βij t) (3)

Equation (3) is similar to the tensile load (stress) relax-ation mode of visco-elastic bodies [2]. However, we mustemphasize that equation (3) is a way of representing ten-sile load (stress) and the origin of time dependence isattributed to a number of unknown factors such as plasticdeformation, slippage, and friction between yarns due tothe complicated structure of knitted fabrics. Such a simpleanalysis has been successfully used to analyze tensile load(stress) relaxation mechanism of knitted fabrics [2]. Basedon this concept, the hysteresis of tensile load (stress), thestrain curves under extension, and recovery processes isestimated for the three knitted fabrics. In this paper, the

e11o

F11 t( ) Y11 t( )e11o=

F22 t( ) Y21 t( )e11o=

F11 t( ) Y12 t( )e22o=

F22 t( ) Y22 t( )e22o=

Figure 2 Stress relaxation curves along warp direction,when instantaneous stress as shown in ode (a) in Figureis applied to Specimen A. The curve can be classified intotwo components. Upper side: experimental curve (1) astraight line (2), lower side: (1) to (2) becomes a straightline.




theoretical analysis is performed by using four parameters,Aij, Bij, αij, and βij, obtained experimentally by assuming thetwo relaxation modes of tensile load. The method wasdescribed elsewhere [2] in detail.

Tensile Load Relaxation Under Extension and Recovery ProcessesThe tensile load-strain curves under extension and recov-ery processes provided largely different routes with consid-erable hysteresis. The large difference is analyzed on thebasis of the concept that the drastic decrease of tensileload is attributed to the tensile load (stress) relaxationmode. If the strain is due to time dependence described asa summation of step functions concerning strain history,tensile load (stress) under extension process can be repre-sented by using the hereditary integral according to theBoltzmann superposition principle [16].

That is, the load (stress) at t may be described as thesummation of load (stress) by step at t′ < t, since the load(stress) added at the arbitrary time must be dependentupon the strain history ε′(t′) at t′ < t. Thus, we have

(4)

where

The tensile load (stress) relaxation modulus for all thepresent knitted fabrics under the extension process wasfound to be represented by two relaxation modes (asshown in Figure 2) and then equation (3) can be given sim-ply by omitting the suffix:

(5)

Substituting equation (5) into equation (4), we have

(6)

where

(7)

In equation (7), εν is the strain rate (speed) and t1 is theelapsing time up to strain ε0. Thus:

(8)

When t reached t1, the strain ε became ε0 and the strainimmediately decreased with the same constant speed εν.Such compulsory return process of the tester occurredbeyond t > t1 and the strain became zero at 2t1. Accord-ingly, we have

or (9)

where

or (10)

The parameters described in equations (9) and (10) can beobtained quantitatively by stress relaxation measurements.

If the decrease of strain is described as a subtraction ofstep functions concerning strain history under compulsoryrecovery process of the tester (2t1 > t > t1), tensile load(stress) of the knitted fabric under the recovery processcan be represented by using the hereditary integral. Thenunder the recovery process at t > t1, we have

(11)

After integration, equation (11) can be rewritten as

(12)

F t( ) ε0Y t( ) Y t t′–( )0+

t

∫dε′dt′--------dt′+=

ε0Y t( ) Y t t′–( )σ t′( )[ ]0+t+= ε t′( )

0+

t

∫dY t t′–( )

dt′-----------------------dt′–

ε t( )Y 0( ) ε t′( )0+

t

∫dY t t′–( )d t t′–( )

-----------------------dt′+=

dY t t′–( ) dt′⁄ dY t t′–( )– d t t′–( )⁄=

Y t( ) A αt–( )exp B βt–( )exp+=

F t( )ε0tt1

------ A B+( )ε0t′t1

-------- αAe α t t ′–( )– βBe β t t ′–( )–+ dt′0

t

∫–=

t1εεv----=

F t( ) 1 e αt––( )Aα---- 1 e βt––( )B

β---+

ε0

t1----=

t 2 εε0----–

t1= t t1– 1 εε0----–

t1=

t1ε0

εν----= t 2t1

εεν----–=

F t( ) A B+( )ε0tt1---

ε0t′t1

-------- α– Ae α t t ′–( )– βBe β t t ′–( )–– dt′0

t1

∫+=

ε0

t1----– t′ t1–( ) ε0+

t1

t

∫ α– Ae α t t′–( )– βBe β t t ′–( )–– dt′–

F t( ) A B+( ) Ae α t t′–( )––A

αt1--------e α t t ′–( )–+

+=

Aαt1--------e αt– Be β t t ′–( )– B

βt1-------e β t t′–( )– B

βt1-------e β t––+–

–

At1---- t 1

α---–

e α t t ′–( )– t11α---–

–

–

Bt1--- t 1

β---–

e β t t′–( )– t11β---–

–

–

2A 1 e α t t ′–( )––

2B 1 e β t t ′–( )––

+ + ε0




Substituting equations (9) and (10) into equation (12), wehave

(13)

where

(14-1)

(14-2)

(14-3)

(14-4)

(14-5)

(14-6)

Figure 3 shows the experimental and theoretical curvesestimated for Specimen A. The measurements were donein accordance with mode (a) in Figure 1. The experimentalcurves (solid curves) show considerable hysteresis underextension and return processes, even in small deformationprocesses but the theoretical curves (dotted curve) showthat slight linearity and the hysteresis of the predictedcurves were not significant. Even so, the theoretical curvesindicate the possibility that the hysteresis can be realizedby hereditary integral under extension and recovery proc-esses.

To give the best fit between experimental and theoreti-cal curves, the modified “linearizing method” is used inthis paper. Namely, as discussed elsewhere [1, 17, 18], theload-strain curve of woven and knitted fabrics showed asimilarly shaped nonlinear curve regardless of the magni-tude of extension. It has been found empirically that thetensile curves can be approximated as [1, 17, 18].In this paper, the same concept is adopted as in the presentsystem. If this is the case, equations (13) and (14) may berewritten as

(15)

(15-1)

(15-2)

(15-3)

(15-4)

(15-5)

(15-6)

Figures 4(a), (b), and (c) show tensile load-strain curves forSpecimen A, in which θ is the tilting angle from the X1direction shown in Figure 1 and modes (a) and (b) in Fig-ure 1 correspond to θ = 0 and 90°, respectively. The calcu-lations were carried out up to ca. 50 N/m, corresponding tosmall deformation and up to ca. 250 N/m, corresponding tolarge deformation. The theoretical curves are in goodagreement with the experimental curves in the elongationand restricted sides, when a suitable value of n is selected.But there is no statistical rule for n-value to give the best fit.The same tendency can be observed for Specimen B elon-gated up to 250 N/cm, as shown in Figure 5.

The negative load is thought to be due to the differencebetween the inner load (stress) relaxation speed within theknitted fabrics and the standard return speed of 8.2%/s ofthe tester (KESG-2-SB1). Namely the return speed of thetester is faster than the load (stress) relaxation within theknitted fabric. The negative load, however, cannot be esti-mated experimentally by a strip biaxial tensile tester(KESG-2-SB1). Even so, it was confirmed that in the laterstages of the recovery process, the excess shrinkage of theknitted fabrics by the faster return speed of the tester makeswaves perpendicular to the return direction of the tester,indicating occurrence of compression under the recoveryprocess. The best fitting value of n at large deformationbecomes larger than that at small deformation. There isobviously considerable plastic deformation with no elasticproperty leading to the slippage of filaments in the knitted

F t( ) F=

KK1– KK2 KK3– KK4 KK5 KK6+ + + +( )ε0=

KK1Aεv

ε0---------

2ε0

εv-------- 1

α---–

A εε0----

–=

KK2 A 1εν

αε0---------–

αε0–

εv------------ 1 ε

ε0----

–

exp=

KK3Bεv

ε0--------

2ε0

εv-------- 1

β---–

B εε0----

–=

KK4 B 1εv

βε0--------–

βε0–

εv----------- 1 ε

ε0----

–

exp=

KK5 2A 1αε0–εv

------------ 1 εε0----

–

exp–

=

KK6 2B 1βε0–εv

----------- 1 εε0----

–

exp–

=

F Yen=

F ε( ) ( KKK1– KKK2 KKK3–+=

KKK4 KKK5 KKK6)ε0+ + +

KKK1Aεv

ε0---------

2ε0

εv-------- 1

α---–

A εε0----

n

–=

KKK2 A 1εν

αε0---------–

αε0–

εv------------ 1 ε

ε0----

n

–

exp=

KKK3Bεv

ε0--------

2ε0

εv-------- 1

β---–

B εε0----

n

–=

KKK4 B 1εv

βε0--------–

βε0–

εv----------- 1 ε

ε0----

n

–

exp=

KKK5 2A 1αε0–εv

------------ 1 εε0----

n

–

exp–

=

KKK6 2B 1βε0–εv

----------- 1 εε0----

n

–

exp–

=




fabric. Incidentally, the negative load of the theoretical curvefor Specimen B due to the compression under recovery proc-ess was shown to be zero.

Figures 6 shows tensile load-strain curves observed forSpecimen C elongated up to 250 N/cm along θ = 0, 45, and90°, in which the negative tensile load of the theoreticalcurve was shown as zero to compare with the correspondingexperimental curve. Surprisingly, for Specimen C, the cal-culated curves in the extension and recovery processes gavethe best fit with the experimental curves, when the calcula-tions were done by using n = 2.8. The value of n in the elon-gation side was also equal to those in the restricted sides.This means that plain knitted fabric (with no polyurethane

filament) takes a simple systematic deformation mode con-cerning yarn friction and compression and then the hypothe-sis represented by hereditary integral satisfies the hysteresisbehavior under extension and recovery processes.

Returning to Figure 5, the best fit did not realize at thesame value of n for Specimen B with 1 × 1 rib structureproduced by mixed yarns of cotton and polynosic fibersand the range of n to give the best fit was 1.8 to 3 except forthe curves at 90°. This means that the region of the fittingvalues is narrower than that for Specimen A with poly-urethane filaments mixed as a double yarn together with acotton yarn, as shown in Figure 4. The values of n underthe recovery process are mostly higher than those under

Figure 3 Experimental (solid) andtheoretical (dotted) tensile load-strain curves in the small andlarge deformation regions esti-mated for Specimen A. The meas-urement was done in accordancewith mode (a) in Figure 1 and theo-retical calculation was carried outby using equation (13).




Figure 4 Tensile load-strain curvesin the extended and recovery sidesalong (a) θ = 0, (b) 45, and (c) 90°directions measured for SpecimenA, in which experimental and theo-retical curves are shown as solidand dotted curves, respectively.Theoretical calculation was doneby equation (15).




Figure 5 Tensile load-strain curvesin the extended and restricted sidesalong θ = 0, 45, and 90° directionsmeasured for Specimen B, in whichexperimental and theoretical curvesare shown as solid and dottedcurves, respectively. Theoretical cal-culation was done by equation (15).




the extension process for Specimens A and B. The differ-ence is thought to be due to an increase of plastic deforma-

tion under extension process. Hence the load correspondingto the strain under the recovery process becomes much

Figure 6 Tensile load-strain curvesin the extended and restricted sidesalong θ = 0, 45, and 90° directionsmeasured for Specimen C, in whichexperimental and theoretical curvesare shown as solid and dottedcurves, respectively. Theoretical cal-culation was done by equation (15).




lower than the load at the same strain under the extensionprocess. That is, the value of n becomes larger to realizethe drastic decrease in load by small recovery strain. Evenso, the present hypothesis represented by the hereditaryintegral was useful to explain the hysteresis phenomenonof knitted fabrics under extended and recovery processes,since the theoretical curve gave the good fit to the experi-mental curve in the elongation and restricted sides.

The further merit of the present hypothesis gave thebetter fitting curve up to high extension in comparisonwith the simple “linearizing method” discussed elsewhere[1, 17, 18]. Namely, the calculated curves represented by asimple formulation , were in good agreementwith the experimental ones in the small deformationregion, but the deviation became considerable withincreasing strain. In this paper, the four parameters, Aij,Bij, αij, and βij, among five parameters can be obtainedexperimentally and only one parameter, n, is determinedby parameter fitting.

ConclusionWe have analyzed tensile load-strain route showing hys-teresis at elongation and restricted sides by using a new“linearizing method” in terms of tensile load (stress)relaxation as a function of time. In the present theory, thestrains under extension and recovery processes wereassumed to be described as an addition and subtraction ofstep functions concerning strain history under extensionand recovery processes, respectively. Accordingly, tensileload was represented by using the hereditary integral onthe basis of the Boltmann superposition principle. Theexperiment was done by strip biaxial extension correspond-ing to uniaxial extension with the fixed dimension in thedirection perpendicular to stretching. The measurementswere done in the weft (0°), warp (90°), and shear (45°)directions. When the numerical calculation was done byusing n = 2.8, the theoretical curves gave the best fit withthe experimental ones for Specimen C with the simple sys-tematic knitted structure in three directions (θ = 0, 45, and90°) and the best fit could be realized in elongation andrestricted sides. On the other hand, the good fit betweentheoretical and experimental curves for Specimens A andB with complicated knitted structures could be realized upto a large deformation region by selecting a suitable valueof n. This means that in comparison with the simple “line-arizing method” by Kawabata et al., the present “lineariz-ing method” provided a very good fit up to the largedeformation region. This indicated that the present hypoth-esis represented by the hereditary integral plays an impor-tant role in explaining the hysteresis phenomenon of knittedfabrics under extended and recovery processes. However,the proposed concept is, at present, only a hypothesis andfurther studies must be taken into consideration.

Literature Cited

1. Yamada, T., Ito, N., and Matsuo, M., Mechanical Properties ofKnitted Fabrics under Uniaxial and Strip Biaxial Extension asEstimated by a Linearizing Method, Text. Res. J., 73, 985(2003).

2. Matsuo, M., Yamada, T., and Ito, N., Stress Relaxation behav-ior of Knitted Fabrics under Uniaxial and Strip Excitation asEstimated by Corresponding Principle between Elastic andVisco-Elastic Bodies. Text. Res. J., 76, 465 (2006).

3. Yamada, T., and Matsuo, M., Effect of a Polyurethane Fila-ment on Mechanical Properties of Plain Stitch Fabrics. Text,Res. J. (in press).

4. Kawabata, S., “Nonlinear Mechanics of Woven and KnittedMaterials Textile Structural Composites”, Chou, T.-W., andKo, F. K. eds, Elsevier, 1090, 67–116.

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F Yen=



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