Analysis of Woven Fabric Strengths

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    ELSEVIER

    Composites Science and Technology 56 (1996) 311-327@ 1996 Published by Elsevier Science Limited

    Printed in Northern Ireland. All rights reserved0266-3538(95)00114-X 0266.3538/96/$15.00

    ANALYSIS OF WOVEN FABRIC STRENGTHS: PREDICTIONOF FABRIC STRENGTH UNDER UNIAXIAL AND BIAXIALEXTENSIONS

    Ning PanDi vi sion of Texti les and Clot hing, Biological and Agri cultural Engineeri ng Departm ent, Uni versit y of Calif ornia, D avi s,

    California 95616, USAAbstractThis study of w oven fabric strength begins by focusingon the effect s of the crossi ng poi nt s, w here w arp andw eft yarns i nterlace w it h each other t o form a fabri c,t ow ards the ult imate fabric tensil e strength. When afabri c is under uniaxi al or biaxial tension, t heyarn- yarn int eracti ons at the crossing point s are foundt o consi st of t w o component s, a pressure-i ndependentadhesi ve component and a pressure-dependent fri c-t ional one, of w hich the lat t er s proved t o be dominant .On the basis of t he analy sis, the yarn crit ical l ength int he fragment at i on process duri ng fabr ic extensi on isdefi ned and calculat ed. M oreover, by t reati ng t he abricas an assembl age of yar n bundl es, t he di stri but io ndensity funct ion of the strengt hs betw een indi vi dualyar ns in the same bundl e i s prov ided based on t hesingle yarn str ength di stri buti on, and the fractur eprocess of the bundl e can be bet t er descri bed using t hi sfunction.

    Then, by incorporati ng the crit ical yarn l ength int ot he distr ibut ion densit y functi on of the yarn str ength,t he in sit u behavi or of the yarn bundle in a fabric ispredicted. Follow ing the chain of sub-bundle model,t he mean fabr ic str ength and t he fabr ic stress/st rai ncurve are predicted more r eali sticall y for both uniaxi aland biaxial tension cases.

    The predict io ns of the present t heory comparefavorabl y w it h t he measured fabri c strengths bothprevi ously report ed and t ested in the present w ork. Adetai led parametr ic study is also carried out .Keyw ords: w oven fabric strength, uniaxial and biaxialextensions1 INTRODUCTIONAlthough biaxial woven fabrics existed long before theIndustrial Revolution, and fabric manufacturingtechniques have since advanced enormously, ourunderstanding of the mechanical behavior of thefabrics is still very limited.In the past few decades, instead of simply beingused as a clothing material, woven fabric has gainedwider applications in many other areas including some

    load-carrying structural utilizations. Recently, wovenfabrics have been applied as the fiber preform forfiber-reinforced composites. Consequently, it becomesincreasingly desirable to develop a more advancedmechanistic theory governing the fabric responsesunder various loading situations so as to provide apowerful design tool for fabric structures in heavyload-carrying applications.So far, attempts to investigate the mechanicalbehavior of woven fabrics have more or less followedthree paths. The well-known Peirce geometricalmodel has been considered to be the first suchundertaking. By assuming a woven fabric as a highlyidealized geometrical object, Peirce was able todescribe the deformational behavior of the fabricunder external loading. Painter* and Love3 furtherrefined this technique in their analyses of fabricproperties. Yet, because of the level of ignorance ofthe mechanistic load/deformation relationships andexcessively idealized features of the constituent yarnsin the fabric, this technique is very limited in terms ofreliability and applicability.By incorporating the mechanisms of force equilib-rium into the geometrical model and lifting some ofthe over-restrained assumptions, the second analyticalapproach was developed. For example, Hearle et a1.4carried out extensive studies on the mechanicalbehavior of woven fabrics by applying both geometri-cal and mechanistic analysis. Taylo? used amechanical model based loosely on Peirces fabricgeometry to examine the tear strength of a fabric as afunction of the yarn strength and the force required toslip one yarn set over another. Postle et a1.6 appliedthe energy method to determine yarn and fabricdeformation by minimizing the strain-energy function.However, the problem associated with this techniqueis the complicated, often cumbersome, mathematicalequations. Further study of fabric tear strength hasrecently been reported by Scelzo et a1.7-Another approach is based on continuum mechanicsby treating the fabric as a continuum laminate. Kilbywas perhaps among the first to apply this approach tostudy the anisotropic nature of woven fabrics, i.e. thedirectional dependence of the tensile modulus. Also,

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    312 N. Panby treating the fabric as a spatially periodic interlacednetwork of orthogonal yarns and by modeling theindividual yarns as extensible elastica, Warren wasable to determine theoretically the in-plane linearelastic constants of a woven fabric. The drawback ofthis type of analysis lies in the neglect of the structuraldetails and the yarn-yarn interactions in the fabric.This technique therefore is mainly applicable to theprediction of the elastic material constants rather thanthe strength of a fabric.

    A new stimulation to the study of the mechanics offabrics has emerged since the 1980s from researchersin the composites field. Fabrics and other textilestructures as preform for the so-called textilestructural composites have attracted more and moreattention from the composites community. This stemsfrom the need for improvements in intra- andinterlamiar strength and damage tolerance, especiallyin thick-section composites. Textile composites offerthe potential of providing adequate structural integrityas well as shapeability for near-net-shape manufactur-ing. The major theoretical contributors in this areaseem to include mainly Chou and his colleagues, andtheir work is best summarized by Chou,, Ko andPastore,* Pastore and Gowayed,13 Hearle and Du,14and Li et a1.15 Largely, the technique in analysingwoven fabrics of Chou is also based on the laminatetheory of continuum mechanics. By treating a fabric asa laminate with or without inclusion of the yarnundulation (crimping) effect, it was possible toestablish the constitutive equations for various fabrics.Again, this method only leads to the prediction of theelastic material constants as well as fabric thermalproperties, and is unable to tackle the problem offabric strengths which rely primarily on the propertyextremes in the system. Additionally, in all theanalysis on woven fabric-reinforced composites, theyarn-yarn interactions at the fabric interlacing pointshave been ignored. As is demonstrated later in thepresent study, the yarn-yarn interaction at the fabriccrossing points provides a fundamental mechanism offabric strength enhancement, and it will likely play animportant role in reinforcing the composite.Shahpurwala and Schwartz16 attempted to predictthe tensile strength of woven fabrics using derivedstrength distributions of the constituent yarns. Bytreating the fabric as a bundle of yarns, they appliedthe statistical theory of Daniel? to predict thestrength of the fabric. They found that when theinteractions between yarns are ignored, the theoreticalprediction is as low as a factor of 0.67. Theyconcluded that a more realistic prediction can beobtained if the fabric is modeled as a bundle of yarnsegment whose length, termed as effective or criticallength, is much shorter than the original yarn length.By back calculation using the weakest-link scalingbased on the known fabric and yarn strength

    distributions, they determined the critical lengthsranging from a high of 18.7 mm to a low of 6 mm,depending on the type of fabrics. Their study provesthat the mechanical behavior of a yarn in a fabricdiffers considerably from the observed ex situ yarnperformance. However, this study fails to establish atheoretical relationship between the critical sub-bundle yarn length and the interactions between yarnsin a tensioned fabric. Instead, they proceeded with amore or less empirical approach. A similar approachhas been adopted by Boyce et a1.18 and Seo et al.19 tomore accurately predict the fabric tensile strength.There have also been studies focusing on themechanical behavior of fabrics under biaxial loading,for instance, as reported by Clulow and Taylor,20Freeston et aL.,*, Reichardt et al.** and Skelton andFreeston. These studies, however, mainly dealt withthe establishment of the constitutive relationships offabrics specifying the elastic performance, and rarelytouched the issue of strength prediction.

    Recently Pan and Yoon24 have proposed ananalytical model to describe the yarn pullout processfrom a woven fabric. This theory is able to predict therelationship of the maximum pullout load and theembedded yarn length in a woven fabric. Themaximum load is shown in the theory to be related tothe yarn-yarn interactions, and the mechanical andgeometrical properties of the fabric and the yarn. Thisapproach is useful in understanding the nature of theyarn interactions in a fabric and the structure-reinforcing mechanism of woven fabrics.The present work is aimed to develop a theoreticalapproach for fabric strength prediction, based on theresults achieved by previous studies. The basicprinciple in this work is quite similar to the modelproposed by Shahpurwala and Schwar@ that a fabricbe treated as a system of chains formed by yarnsub-bundles whose length is equal to the so-calledcritical length. One of the major features in thepresent study is the derivation of the expression forthe critical yarn length, I ,, based on the yarn-yarninteractions and the mechanical and geometricalproperties of the fabric and its constituent yarns. Theyarn-yarn interactions are characterized in terms ofthe adhesive and frictional actions occurring at theinterlacing points where the yarns in warp and weftsystems contact and interact. By incorporating theadhesive and frictional mechanisms into the analysis,we will be able to include the transverse extensionwhich enhances the frictional force, and the yarnsurface topology, which affects both the adhesive andthe frictional actions, into our analysis. Consequentlywe are able to predict the fabric strength and itsvariation at both uniaxial and biaxial extension cases.

    It becomes known that a fabric is an extremelycomplex structure for mechanistic analysis. There existseveral structural levels from fibers to yarns and

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    Anal ysis of w oven fabri c str engths 313

    eventually to the fabric. Each level has its owngeometrical and mechanical variables which control orinfluence to varying degrees the fabric behavior. It willbe a formidable or even impossible task to include allthese variables into a fabric model without simplifica-tion. This simplification is done in the present work inthree ways. First, we approximate a real fabricstructure with an idealized model of more regulargeometry; second, we exclude in our model the factorswith ignorable or only marginal effects on the finalresults; last, for the variables included in our model,we express those unmeasurable variables in terms ofthe measurable ones. Through these approaches, wecan retained the essence of the physics but avoidunnecessary complexity so as to make our modelconcise, more user-friendly, and easy to apply.To begin with, we adopt the following assumptionsin our analysis:

    1.2.

    3.

    4.

    5.

    The tensile strength distribution of the in-dividual yarns is of Weibull form.The yarn jamming effect during fabric extensionis negligible. Evidently, this assumption is moreacceptable when dealing with a fabric with tightstructure. Moreover, in a biaxial loading casewhere extension is exerted on both warp andweft directions, mutually perpendicular to eachother, the yarn jamming effect during fabricstretch is less severe than in a uniaxial loadingcase.Variations, and their changes during fabricextension, of the structural and geometricalparameters of a fabric are negligible.The interactions between yarns in a fabric underextension will not affect the form of the strengthdistribution function of the individual yarns.When a yarn in a fabric breaks, the load it wascarrying is equally shared among the survivingyarns. The effects of stress concentration anddynamic wave propagation are ignored.

    2 FABRIC TENSILE STRENGTH AS ASTOCHASTIC VARIABLEA biaxial woven fabric can be treated as an assemblyof two systems of yarn bundles perpendicularlyinterlaced with each other in a two-dimensionalformat. Therefore, fabric strength is the resultant ofthe yarn strength taking into account the interactionsbetween the two yarn systems.2.1 Weibull statistics of the strength of a single yarnAccording to the above assumption of the Weibulldistribution function of yarn tensile strength, for asingle yarn with length I,, the probability of itsbreaking load being aY can be described by atwo-parameter Weibull function:

    F(a,) = 1 - exp[ - &a,~$7 (1)

    where (Ye s the Weibull scale parameter, and pY is theshape parameter of the yarn. The shape parameter isan indicator of the variation in yarn breaking loads. Ahigher pY value corresponds to a lower variation, andwhen BY--+=0, the variation would approach zero andthe yarn breaking load would be independent of itslength.The mean or the expected value of the yarnbreaking load, (TV an then be calculated as:

    (2)where I is the Gamma function, and the standarddeviation of the breaking load is given by:

    L

    @,=a, r 1+;( ,ii I2 1+$ -1C J (3)YIt should be noted here that we have establishedthese relationships between the yarn tensile strength

    and its Weibull parameters in a generalized manner.The effects on yarn strength due to factors such as thefiber end slippage in a staple yarn, fiber migration, andthe fiber obliquity have all been reflected in the valuesof the Weibull parameters. However, it has beenshown recently by Realff et a1.25 hat there has been ademonstrated failure mechanism change for shortfiber yarns, reflected by the change of the value of theWeibull shape parameter, By, as a function of thegauge length at which the yarn is tested. Since we willuse in our model the Weibull shape parameter testedat the standard gauge length, the effect of thisobservation by Realff et al. will be evaluated in thespecific examples presented towards the end of thepaper.

    2.2 The fabric strength excluding the effects ofyarn-yarn interactionsThe tensile behavior of a fabric would be identical tothat of its constituent yarns if all yarns were uniformin their tensile properties, and if the interactionsbetween the two perpendicular yarn systems werenegligible. Unfortunately, in reality, these two factorsare too significant to be excluded; this complicates theotherwise very straightforward relationship.Let us first consider only one yarn bundle systemwhere NYyarns form a parallel bundle with no contactor interaction between the individual yarns. As thereis more or less variation between the mechanicalbehavior of the individual yarns in the system, this willinevitably lead to a discrepancy between theproperties of the single yarns and of the yarn bundle.Because of the statistical nature of single yarn

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    314 N. Panstrength, the yarn bundle strength will obey astatistical distribution as well. Based on the analysis byDaniels, for a large bundle of high NY value, thedensity distribution function of the bundle strength ahapproaches a normal form:

    1H((Th) = 60, exp

    _ (uh - %I*20; I (4)

    where G is the expected value of the bundle strengthand can be calculated from the yarn parameters as:(TV = (Z,CX,&,-+ixp ( 1L6

    and Oh is the standard deviation of the strength:0; = (Zyayp,)pt[xp( - i)] [ - exP( - i,]NF1

    (f-5)As is well known, the strength of this yarn bundle islower than that of its constituent yarns. This can be

    seen by comparing eqns (2) and (5). The strengthvariation of the yarn bundle is also smaller (dependingamong other factors on its size N,,) than that of thesingle yarn given in eqn (3).

    It is apparent that the tensile modulus of this yarnbundle is taken here to be identical to that of the yarn.To conclude, the strength variation betweenindividual yarns will result in a reduction of both

    mean strength and the strength variation of the yarnbundle formed. Furthermore, if the interactionsbetween yarns in the fabric are ignored, a fabric canbe treated as a yarn bundle system in either of the twodirections, i.e. the longitudinal (the warp) or thetransverse (the weft) direction, and then eqn (5) infact gives the fabric mean strength per yarn. So,according to the above theory, the fabric meanstrength per yarn, as well as its standard deviation,will be lower than that of its constituent yarns, or thefabric is weaker than the yarns owing to the propertyvariation between the yarns. This conclusion isactually in contradiction with the experimental resultswhich have shown that the mean fabric strength peryarn is generally greater than the mean strength of itsconstituent yarns. Obviously, the main cause for thisdiscrepancy lies in the exclusion of the effects of theyarn-yarn interactions in the above analysis. Asshown below, these interactions will reinforce thefabric structure so as to achieve a fabric strengthhigher than that of the yarns. As the yarn-yarninteractions are determined by many statisticalvariables, this leads to a fabric strength of probabilisticor stochastic nature.

    From now on, unless specified otherwise, all theanalyses will be focused on the warp direction in afabric.

    2.3 Yam-yam interactions at interlacing points, thecritical length and the in situ yarn strengthIn order to include the yarn-yarn interactions in ourmodel, we have to take into consideration the secondyarn bundle system, which is in the directionperpendicular to and is interlaced with the above yarnbundle system to form a self-locked planar fibroussystem-a woven fabric.

    The interlacing (or crossing) points are the majorlocations where interactions between yarns in the twobundle systems take place, through which the yarnsform an interlocked structure. Without the interac-tions occurring at the interlacing points, a wovenfabric would be equivalent to a system of two sheets,each made of parallel but isolated yarns; the resultantproperties would be entirely different from those of apractical fabric. In other words, the yarn interaction atthe crossing points is the essential feature for a wovenfabric, and will affect more or less all the fabricproperties. Therefore, these interactions can be usedas an indicator of, or a probe to, the various fabricproperties, and, according to Pan and Yoonz4 can beinvestigated through a single yarn pullout test fromthe fabric.As revealed by the statistical chain of sub-bundletheory first applied to woven fabric by Shahpurwalaand Schwartz,16 it is expected that yarns, once woveninto a fabric system, will behave differently due to theinteractions between yarns in the fabric underextension, and the interactions will inevitably alter theproperties of the yarns. Because of these interactions,the fabric is enhanced, and the concept of chain ofsub-bundle theory can better specify this enhancementeffect. Based on the theory, a fabric can be consideredequivalent to a system made of chains of sub-bundlesof yarns whose length is defined by a critical length I,,We are going to derive a theoretical expression for 1,in terms of the fabric and yarn properties.As stated above, the yarn-yarn interactions in awoven fabric mainly take place at the yarn crossingpoints through yarn-yarn contact. However, thecontact area at an interlacing point between a warpand a weft yarn is only partial, as shown in Fig. 1 of anidealized illustration of a plain weave. If an externaltension ay is applied to the fabric, a force equilibriumcan be established at a differential portion d_xas:

    where, as illustrated in Fig. l(a), t, and wy are theshorter and longer axes of the yarn cross section, ry isthe shear resistance, and L,/2 is the actual contactlength between yarns, for one contact point. Here L,is the circumference of the yarn cross section, i.e.

    L, = 1*517c(t,+ WY) v@Q (8)

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    Analysis of woven fabric strengths 315

    (a)

    (b)Fig. 1. Idealized geometrical features at yarn interlacingpoints of a plain weave fabric: (a) the dimensions of the yarncross-section and spacing; (b) the interlacing angle.

    and rzY s the fabric count (number of yarns per fabriclength).

    If uy is increased to the current in situ tensilebreaking load of the yarn, it follows that the yarncritical length I, is given as:

    orL-3 Zyn,l, = uY2 (9)

    2~,_-cYL,% =YIt is implied in this equation that the original yarn,

    once embedded in the fabric matrix made of otheryarns, has to be treated as a chain of statisticallyindependent yarn segments of length I, whose value isdetermined by the yarn in situ strength, yarn size andthe interactions between yarns as specified in theequation.

    Furthermore, as indicated by Henstenburg andPhoenix,26 the actual fragment lengths are not aconstant and vary in the range ZJ2 to I,. This problemcan be solved by replacing 1, with the mean fragmentlength 3lJ4 if a uniform distribution of this length isassumed. For simplicity however, we still use eqn (10)for the present study.

    More importantly, the fragmentation process hasrevealed that during the extension of the fabric, theyarns are stretched segment by segment throughfabric. So, because of the strong strength-lengthdependence of yarns, the strengths of these yarnsegments will become higher owing to their shorterlengths.Also, during the fabric extension, by definition anyyarn fragment with length longer than 1, is still able tobreak somewhere along its center section as its stressexceeds its current in situ strength, a,,. Therefore, themean length before yarns break into lengths 1, will be41,/3. This will be the length by which the value of Usfor the new yarn fragment is determined. Keeping thisin mind and combining eqs (2) and (10) gives:

    The in situ yarn strength, fly, can then be determinedfrom eqn (2) replacing the original yarn length, I,, bythe critical yarn length, I,, to reflect the yarn-fabricinteractions.

    To accommodate other weave structures besides theplain weave, the above equation can be rewritten as:I, = [&(Z oy)-d(l + $)l(+Py (12)

    where C, represents the length of the yarn-yarncontact area. In the case of plain weave, we have asabove C, = L,/2.2.4 The components of the shear resistance zr atyam contact areaEven with a fabric made of yarns whose weight is solight as to be negligible, a certain force is still requiredto pull a yarn out of the fabric by overcoming theresistances between the yarn contact points. Yet themagnitude of this pullout force will increasesignificantly when certain pressure is applied at thecontact points. In other words, there exist twocomponents of the shear traction ry which resists anyattempt of relative yarn movement at the contact area.Of the two components, one is pressure-independentadhesive force termed zy2, and the other ispressure-related frictional force r,,i governed by thefriction law as explained above. That is

    ry = Tyl + 7y2 (13)The pressure can either be applied directly to thefabric surface, or be generated from tensile loadexerted uniaxially or biaxially to cause a tightening

    effect to the fabric so as to increase the pressure at theyarn contact points.Assume the fabric is under a biaxial loading casewhere the load uL is applied in the warp direction andthe load CT in the weft direction as seen in Fig. 2(a).

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    316 N. Pan

    -*-+-*-*,-*,-+-*,-*-*

    PFig. 2. Various forces on a plain weave fabric. (a) Biaxialloading on the fabric; (b) cross-section along a weft yarn; (c)

    forces at an interlacing point.These external loads will create a pressure, P, t eachcontact point shown in Fig. 2(b) and (c) of across-section of the fabric.The compression force, P, s exerted on a contactpoint of two crossing yarns with circumference L, tocause a frictional force pP where /_L s the frictionalcoefficient. This frictional force is balanced by theshear force r,,% hrough:

    ry1 c, = rQ (14)So the frictional component of the shear resistance is:

    PPTyl =r (15)The resulting normal compression force P as shown

    in Fig. 2(c) can be calculated as:P = 2~~ sin & + 2crL sin 8,) (16)

    where 0, (i = T, L) are the interlacing angles at thelongitudinal and transverse directions, respectively,and can be expressed from Fig. l(b) as:8 = arctan = arctan(4t,n,) (17)

    Since our ultimate goal is to determine the fabrictensile strength in one, say the warp, direction whenthe fabric is under a biaxial loading situation, theinvolvement of the warp stress cr, in eqn (16) bringsconsiderable difficulty for further analysis. To simplifythe result, we need to study the ratios &CT,_ andsin B&in 0,. It is not hard to conceive that when weincrease crL, the value of 8, will decrease, and asimilar connection exists between a?- and 0,.Supposing an identical geometry in both directions ofthe fabric, we can therefore assume a relationship:

    orCT sin oL-= (IS)gL sin C!&

    UT sin & = ffL sin & 09)So eqn (16) becomes:

    P = 4uT sin I& (20)It has to be pointed out, however, in the case of

    uniaxial extension where UT = 0, the value of P willnot be zero because of the contribution from theexisting force aL> 0. Yet, as to be proved later, thiscontribution will be reduced as the fabric is onlytightened in one direction with the load applied, andthere is an absence of the lateral extension @r

    Combining eqs (15), (17) and (20) yields the firstshear resistance component:

    ZYl = F uT sin[arctan(4tyny)] (21)YThe second component of ry can be derived basedon the analysis of Pan and Yoonz4 on the yarn pulloutbehavior from a fabric. When a yarn is pulled out

    from a woven fabric, the maximum adhesive surfaceforce pm at one contact point was derived as:22 wpm=--21P

    tanh 2pwy

    where r, is defined as the elastic shear strength of thecontact area, and p is a factor reflecting thegeometrical and mechanical properties of the yarn:

    $_ YdtY nEY (23)where G,/E, is the ratio of the longitudinal shearmodulus and tensile modulus of the yarn. Similarly,this force is balanced by the second shear resistancecomponent ry2 as: L2-32 2 -PmNote that the between yarn distance by = l/n ,.

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    Anal ysis of w oven fabri c strengths 317

    orz 4,& w ,Y2 = - tanh 2pw, = 22 wL -Z-J tanh 2pw,CYP (25)YP

    independent of loading situation and directions.As shown above, zy2 value is determined by theyarn mechanical properties G,/E, and the geometricalproperty or the ellipticity of the yarn cross-sectionalarea wy l t y .2.5 The fabric strength as a stochastic variableIn view of the preceding analysis, the strength in thewarp direction of a practical fabric a, has also to betreated as a probabilistic or stochastic variable and itsmean value can be obtained from:

    and the standard deviation:or_ = v@,(I,) (27)

    where V, is the overall fiber volume fraction of thefabric, and E and Or, are the mean yarn bundlestrength and its standard deviation which can becalculated from eqns (5) and (6), except that the yarnlength I, in the equations has to be replaced by thecritical length I, defined in eqn (12). This step iscrucial so that the effect of yarn-yarn interactions canbe included. For convenience, in the followingsections, we will use the same format such as &,(Z,) todesignate a strength b and the length I, at which thestrength is determined.Then the distribution density function of the fabricstrength in the warp direction can be expressed fromeqn (4) as:

    3 THE STRESS/STRAIN CURVE OF AFABRICSuppose the single yarns used for this study are oflinear mechanical behavior prior to failure. The yarnproperties such as the tensile modulus, yarn surfaceand cross-sections as well as fabric parameters areprovided in Table 1.

    For a fabric of length 1, under a given externalstrain l high enough to cause yarn breakage, theyarns in the fabric will not fail at the same timebecause of the variations between the yarn strengths.Instead, they will break gradually according to theirstrength distribution over a certain range of the stressEyeL. Therefore, the fabric tensile stress CT, can beexpressed as:

    (+L = ~L(W&Y~L (29)

    Table 1. The assumed property values for calculationProperty Typicalvalue Unit

    Fiber density, ptFabric fiber volume fraction, V,Fabric width, w,Yam tensile modulus, E,Thread shape parameters, p,Thread scale parameter, c+Yarn cross-section ellipticity, w,/r,Inter-yarn frictional coefficient, pFabric count, nyNumber of yarns in fabric, NY

    1.30 g/cm30.52541.1 (!I%4.05.0 GPa-PY/mm1.40.33.9 mm-100.0

    where E, is the yarn tensile modulus, V, is the overallfiber volume fraction of the fabric, and YL(l,) is thefraction of the number of warp yarns that are notbroken yet at the current stress level EyeL and are stillcarrying the load. This surviving yarn ratio can becalculated according to Pan:27

    (30)where H(v,,) is the distribution density function of theyarn bundle strength defined in eqn (4).4 CALCULATION AND DISCUSSIONIt has to be pointed out that the main focus of thepresent study is to develop a probabilistic model topredict tensile fabric strength. Therefore, when weapply the existing theories on the fragmentationprocess in our analysis, we only use the most widelyaccepted, instead of the most advanced, theoreticalresults so as to avoid distractions and additionalcomplicities. Our model could, of course, be improvedby using it in conjunction with the most recenttheories.4.1 The yam cross-sectional dimensionsIn the present case, the parameters describing theyarn cross-section including the yarn width wy andthickness ty are the ones that have no convenient andreliable way to be measured. However, they can bereplaced with others which are easier to measure orare more familiar in practice.It is rational to assume:

    WY-, - al (31)kYor wy = aI t , (32)

    The value of a, has been assumed to range from 1 to 2based on our experimental observations.There also exists a relationship:t, + wy = D, (33)

    where D, is the nominal or equivalent diameter of the

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    318 N. Panyarns with the assumption of circular yarn cross-sections which can be calculated as: 4

    0, (mm) = 2 Jtex

    V&JO5 x 10where tex is the yarn number, pf is the fiber density ing/cm3, and Vfy s the fiber volume fraction of the yarnwhich is different from, usually greater than, theoverall fiber volume fraction of the fabric, V,, whichappeared in eqn (29), because of the existence of theinter-yarn spaces in a fabric. We can solve from aboveequations the yarn thickness:

    2 dtex

    ty (mm) = 7rpfv,lo51 +ar x 10 (35)Unless stated otherwise, for fabrics of different fiber

    types used in this study, by using pf = 1.30 g/cm3 foracetate fiber, 1.38 g/cm3 for PET and 1.51 g/cm3 forcotton fibers, and assuming identical values al = 1.4,and V, = O-628, we thus calculated their t, values.4.2 Determination of the yam-yarn adhesionThe shear resistance at a yarn-yarn crossing point hasbeen proved to be:

    ry = r,1 + ry2 = F (+T sin[arctan(4t&)]Y

    + 2GWy- tanh2pw,CYP (36)The frictional component rYl is dependent on the

    external pressure, whereas the adhesive componentzY2 s entirely determined by the surface geometry andproperties of the yarns, and is hence an intrinsicsystem property irrelevant to the external loadings.Furthermore, the force rY2, or the elastic shearstrength of the contact area r,, can be determinedtheoretically.

    Assume there is no external load or pressureexerted at the yarn-yarn contact area so that ryl = 0and we have the shear traction as:

    22 wZY= ryz - --LJ tanh 2pw,,GPIf we then pull a yarn out of the fabric, the criticalyarn length at which the yarn will break within the

    fabric rather than be pulled out is still 1, defined in eqn(12), but at the condition rY= ry2. The forceequilibrium on the yarn at breaking becomes:2y2CynyEc $1,) = (lc~y&-8:exp C )$ (38)Y

    Since all other variables can be considered as given,

    the value of ry2 or r, can be obtained by solving theabove equation. It is found during the calculation,however, that the magnitudes of both ry2 and r, areextremely small. Therefore, their effects on thefollowing calculations are in fact negligible. In otherwords, between the two shear resistance components,the frictional component is far more dominant thanthe adhesive one when external load exists. Tosimplify the process, we use in the followingcalculations:

    2, = Tyl (39)This shear force r,, is in fact a reflection of the

    shear bonding strength of the yarn-yarn contactpoint, and is related to yarn surface frictionalbehavior, and the transverse tensile load c+- asindicated in eqn (21). Figure 3 depicts the relationshipbetween ryl and the variables involved. For easycomparison with experimental results later, thetransverse tensile stress a, is expressed as N perthread, and the unit for r,,r follows as N per contactpoint. Here the yarn number tex = 40.0 and inter-yarnfrictional coefficient p = 0.3 are used in the figure.As indicated in eqn (21), the value of rYl increaseslinearly with the transverse tensile load UT. A flatteryarn cross-section, represented by a higher ellipticityof the yarn cross-sectional area, WY/t,, will result in alower shear force ryl, when the thread count nY andthe fiber volume fraction of the yarn Vfy are given asshown in Fig. 3(a). The effects of both Vfy and ny, onthe other hand, can be seen in Fig. 3(b) where CT andWY/t,, are given. It is indicated in this figure that ahigher Vfy or ny, i.e. more fibers in a yarn or moreyarns per unit fabric length, will lead to a higher zylvalue. However, n,, has a more substantial effect on rylwhen V, is greater.In summary, at a given level of the transverse loada,, for a less flat yarn, a denser fabric will result in ahigher shear traction rYl, or a stronger bond at thecontact point between yarns.4.3 The critical yam length 1,Because of the mutual restraints between the warpand weft yarns, these yarns in a tensioned fabric willeach behave as a series of sub-bundle systems made oflinks of length I, whose value is determined from eqn(11) by the mechanical and geometrical properties ofthe yarn and the fabric system. A relative scale 1,/l, isused here, where L, = 152*4mm is the original yarn(fabric) length. Note that the effect of the lengthdiscrepancy between a fabric and a yarn from thefabric, which is caused by yarn crimping within thefabric, is neglected here, since it has been dealt with indetail elsewhere.16

    The ratio 1,/Z, is an indicator of the interactionsbetween yarns during fabric extension, and a smaller1,/Z, value reveals a greater reinforcing effect due to

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    Anal ysis of w oven fabri c str engths 319

    TYl(Nllength)l - 75.1. 5.

    1, 25.1,

    0. 75.0. 5.

    0. 2 0. 4 0. 6 008

    0. 25.

    0. 2 0. 4 0. 6 0. 8 l ' N' yar n)TY

    (N/length)

    0. 125.

    0. 1.

    0. 075.

    0. 05,

    0. 025.

    I, ,,.I.,..,..~. I, ,1.2 3 4 5G! ! i ! LFig. 3. Frictional component, z,, of the shear force at ainterlacing point: (a) zYl versus transverse tension, uTT, atthree levels of elliptic&y WY/t, yarn cross-section; (b) zYlversus fabric count, n,,, at three levels of fiber volumefraction, V,, of yarn.

    the yarn-yarn interactions in a tensioned fabric. Thekey factor for the determination of 1,/l, is the shearbonding strength zYat the yarn interlacing points.

    Figure 4 illustrates the relationship between 1,/Z,and the transverse load aT at different levels of theyarn Weibull parameters. In general, the ratiodecreases at increasing transverse load a,. Thisrelationship is also influenced by the values of theWeibull parameters of the yarn. For example, a

    I 0. 2 0. 4 0. 6 0. 8 l CN%mFig. 4. Effects of variables on 1,/l, ratio: (a) 1,/l, versustransverse tension, uTT,at three levels of shape parameter,/3,,, of yarn strength; (b) 1,/Z, versus transverse tension, a,, atthree levels of scale parameter, q,, of yarn strength.

    smaller value of the shape parameter pY of the yarnstrength, meaning a thread with greater strengthvariation, will lessen the reinforcing effect, leading toa higher 1,/l, value as seen in Fig. 4(a), whereas in Fig.4(b) a greater scale parameter (Ye will intensify theinteractions and enhancement resulting in a smaller1,/l, value.On the other hand, it is clear that the mean fabricstrength per yarn would be greater than the meanyarn strength only when the ratio 1,/l, < 1; this is infact the critical condition for the fragmentation

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    320 N. Panprocess to occur. It in turn defines the critical valuesfor all other variables involved in order for thereinforcing mechanism due to yarn-yarn interactionsat the interlacing points to function. For instance, thiscritical condition

    1,I (40)can be used to define the minimum value of the

    transverse tensile load gT in order to yield a stronger

    0.008,

    0.006

    0.004.

    0.002~I ,,,I,,,,,,,, ,,,,,,,,,,,100 200 300 400 5OOZ&

    @I OTqG0.0070, 006.0, 005.0. 004,0. 003.0. 002.0. 001.

    I L nY2 3 4 5 6( ummFig. 5. The critical value aTlaY ratio: (a) ~~/a;(&)versus original yarn length, I,, at three levels of eliipticityw ,/ t , of yarn cross-section; (b) o,/a,,(l,) versus fabric count,nY,g three levels of fiber volume fraction, V, , of yarn; (c)gT/uY(lY) ersus shape parameter, & of yarn at three levelsof inter-yam frictional coefficient j.~.

    5 10 15 2OPYFig. 5. (Continued.)

    fabric. From this condition we can find the solution inrelative scale c+T/(

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    Analysis of woven fabric strengths 321yarns. On the other hand, once the bundles areinterlaced with each other to form a fabric, upontensioning of the fabric, the interactions at theinterlacing points between yarns will considerablyalter the yarn behavior from its ex situ state, and as aresult, the so-called fragmentation process will occurwhich will increase the mean strength of the fabric.The ultimate value of the fabric strength thus becomesa result of these two competing factors. Since eqns (2)and (5) already provide the results showing theinfluences of the first factor, we need to design amethod where only the second factor, namely theeffects of yarn-yarn interactions in a tensioned fabric,remains to be effective so as to show its effects. Oneconvenient way to do this is to examine the ratio- - -aL(ZJ/aL(I,) where a&) and z(Z,) are the fabricstrengths with and without inclusion of the fragmenta-tion effects; both of them are calculated from eqn(26), except that the critical yarn length I, and theoriginal yarn length 1, are used, respectively. Sinceeqn (26) is derived from the result of the meanstrength for yarn bundles, the effect of the strengthvariations between individual yarns is thus excluded in

    _- -the ratio G&,)/uL(~,).This fabric strength ratio is then plotted against thelateral extension o, in Fig. 6. It can be readily provedthat:

    (42)The effects of the Weibull scale parameter of the

    yarn strength (Yeon the ratio 1,/Z, is already illustratedin Fig. 4(b) and is therefore excluded in the followingfigures. We have to retain, however, the shapeparameter & in our discussion because of itsinvolvement in the expression (l,/Z&.First of all, it is seen from all the curves in Fig. 6that the value K(l,)/

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    322

    (a)

    (cl083 0.4 005 0.6 0.7 0,8 0,9 CTT~(N/Y-)

    1.57 - 2 = 1.0

    1.45. A@

    T = .01. 4, 7 = 3.0

    1.35.

    I GT0. 3 0. 4 0. 5 0. 6 0. 7 0. 8 0. 9 1(N/yam)

    N. Pan

    (b)

    (4

    1.55.1.5.

    1.45,1.4.

    1.35.1.3.

    1.25.1.2

    I 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1oL( L)cJ ,c y)

    OT(Nf~am)

    1.5.

    1.45

    1.4.

    1.35.

    1.3.

    OT 0.3 0.4 0.5 0. 6 0.7 0. 8 0. 9 l (~/ ~arn)Fig. 6. The strength ratio a,(l,)/a,(l,) versus transverse tension, v~, at three levels of: (a) shape parameter, & of yarnstrength, (b) fabric count n,; (c) ellipticity, w,/ty, of yarn cross-section; (d) inter-yarn frictional coefficient p; (e) fiber volumefraction, I$,, of yarn; (f) yarn number (tex).

    4. 7Comparisons between the predicted and tested (1) in this analysis. Yet the conversion between thefabric tensile strengths two can be easily accomplished using the relationship:First, we compare the predictions of the fabric meanstrength using eqn (26), to the results reported byShahpurwala and Schwartz.16 The fabric samples usedby them16 are introduced in Table 2 with their tyvalues calculated from eqn (35), and the correspond-ing Weibull parameters of the yarn strength arereproduced in Table 3. Note that the scale parameter,designated as LY,,~ n Table 3 was defined slightlydifferently in Ref. 16 from the one, ay2, used in eqn

    According to Ref. 16, the yarn length I, = 152.4 mm.So the calculated (yy2 values are also listed in Table 3.It is worth noticing that the physical meaning of CY,,~sthe stress level at which 63.2% of the yarns will break.Since there is usually +,I > $Z,), meaning no more

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    Analysis of w oven fabric strengths 323

    (e> clL( doL( y)1.5.

    1.475.1.45,

    1.425.1.4,

    1.375,1.35.

    1.325,

    Cf) 6,(cJL(y)

    1.45,

    1.4.

    1.35.

    1.3.

    I 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1~~~~~,3 0.4 0.5 0.6 0Fig. 6. (Continued .)

    than 63.2% of the yarns will fail at the stress levelequal to the mean yarn strength

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    324 N. PanTable 2. Fabrics and their parameters from Ref. 16 and t,values

    Property Fabric count, n,, Yarn size(thread/mm) (tex) (&mm)1. Cotton plainWarpWeft2. Cotton twill

    311WarpWeft3. Cotton satin411WarpWeft4. Polyester plainWarpWeft

    4.5 27 7.921.7 42 9.88

    4.4 39 9.522.1 35 9.01

    3.9 32 8.622.0 39 9.523.9 5 3.572.5 20 7.13

    since the results are expressed as N per yarn, the fibervolume fraction of the fabric no longer plays a role inthe calculation and should be set as unity. Theuncorrected results in Table 4 were calculated basedon the model of Daniels and the corrected resultswere based on a modified model of Smith andMcCartney.29

    The data in Table 4 from Ref. 16 are the fabricstrengths under uniaxial loading situation, i.e. whenUT = 0. As we already learned that because of thecontribution to the yarn-yarn normal compressionforce P from the longitudinal external force ur_ asshown in eqn (16) the frictional shear resistancezyl > 0 even when the transverse tension a, = 0.However, in the case of uniaxial loading, thecontribution of uL towards zyl will be reduced becausewhen the lateral tension UT is not present, the fabricis not as tight as a biaxial tensioning case, so thatthe effective normal compression force due to a,will be discounted by a factor A. It is difficult to

    Table 3. Yarn Weibull parameters for fabrics in Table 2Parameter P, q1, N (Y,,~,NmPy/rnrn

    1. Cotton plainWarp 10.08 5.35 2.99 x lo-Weft 10.33 8.38 1.91 x lo-l22. Cotton twill 3/lWarp 6.15 7.43 2.89 x 10m8Weft 8.75 7.19 2.09 x lo-3. Cotton satin 4/lWarp 10.25 5.31 2.43 x lo-Weft 9-88 6.57 5.49 x lo-l14. Polyester plainWarp 19.02 1.77 1.26 x lo-Weft 10.64 5.65 6.54 x 10-l

    determine the value of A theoretically. An estimatedvalue based on an empirical method can be obtained.For brevity, A = O-3 is selected for this study. Also, fornon-plain weave fabric number 2 and 3 in the table,the corresponding P values from eqns (43)-(45) areused.

    To further verify our theory, we have selectedanother five fabrics with various fiber type, weavestructures and Weibull strength parameters deter-mined similarly to Ref. 16 and listed in Table 5. Theyarn thickness t,, and predictions of the fabric strengthas well as its standard deviation are again calculatedusing the present theory.

    It is shown clearly in both Tables 4 and 5 that thepresent predictions of both fabric strength and itsvariation are in reasonable agreement with the testedresults. In Table 4, our predictions are closer to themeasured actual values than the predictions byShahpurwala and Schwartz16 using either of their twomodels. Moreover, the general trend also shows thatour predictions tend to exceed the measured strengthvalues. We suspect that this may be due partly to theequal-load-sharing assumption adopted in the presentanalysis, and a less uniform load-sharing rule betweenthe still surviving yarns will yield lower or morerealistic results.

    Still, there are other potential factors which mayinfluence our predictions. One of them is theaforementioned interconnection between the gaugelength and the Weibull shape parameter pY of staplespun yarns revealed by Realff et a1.25 According tothem, the failure mechanism of a yarn made of shortfibers will change when the testing gauge length isbelow the fiber length, and at such a short gaugelength, the yarn becomes unusually stronger, reflectedby a reduced shape parameter & because more fibersbreak instead of being pulled out. If this principle isapplicable here, the shape parameters determined atstandard gauge length and employed in our model willunderestimate the strengths of the fabrics made ofshort fiber yarns. However, from the data in bothTables 4 and 5, we do not see a noticeable trend ofunderestimation, nor do we see a clear distinctionbetween the predictions for fabrics made of short fiberand continuous filament yarns. This may suggest thatwe cannot treat the critical yarn length in eqn (ll), bywhich our fabric strength predictions are calculated, asthe equivalent gauge length in comparison to theactual testing gauge length on a tensile tester. Thisfailure mechanism change observed on a strengthtester may not dictate the yarn failure process in thefabrics used here.Additionally, it has been reported3 that for somematerials, the Weibull law of eqn (1) mayoverestimate the strength for shorter gauge lengthwhile underestimating the strength for longer length.It again seems not the case for the present data in the

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    Anal ysis of w oven fabri c str engths 325Table 4. Fabric strength and its SD value (in parentheses) tested and predicted (N/yam)

    Uncorrected Corrected Present model Tested1. Cotton plainWarpWeft2. Cotton twill 3/lWarpWeft3. Cotton satin 4/lWarpWeft4. Polyester plainWarpWeft

    3.86 (O-12)6.06 (0.29)4.70 (0.19)5.00 (0.24)3.84 (0.13)4.71 (0.21)1.43 (0.03)4.15 (0.17)

    3.93 (0.11)6.30 (0.26)4.82 (0.18)5.19 (0.22)3.92 (0.12)4.88 (0.19)1.46 (0.03)4.27 (0.15)

    6.12 (0.17)8.56 (0.39)9.14 (0.36)7.22 (0.34)

    4.97 (0.17)8.88 (0.45)7.10 (0.18)6.44 (0.36)5.09 (0.20)5.75 (0.21)1.78 (0.07)4.91 (0.34)

    5.72 (0.18)6.47 (0.29)l-80 (0.04)5.87 (0.21)

    two tables. So our results may in fact confirm thevalidity of using the Weibull function to specify theyarn strength.Another potential factor which may influence ourprediction is the effect of fabric weave structures. Yet,

    by comparing fabrics of different structures in Tables 4and 5, we fail to see a pattern of error associated withthe weave types. This seems to indicate that thepressure and weave type relationships developed ineqns (43)-(45) work reasonably well in our case.

    Table 5. Results for a new set of fabricsFabric*

    1, 100% PET-F 2, 100% acetate-F 3, 100% acetate-F 4, 100% cotton-S 5, 100% cotton-Splain plain satin plain plainThread count, ny (threads/mm)

    3.94 3.783.94 2.21Yarn number (tex)3.28 3.894.92 32.27Yarn thickness, t, (lo- mm)2.89 3.233.54 9.33Yarn shape parameter, p,20.33 45.7831.53 14.50Yarn scale parameter, q, (NmBy/mm)7.29 x lo-l5 144 x lo-2.63 x 10mzl 5.68 x 10-lYarn strength (SD) (N/thread), tested3.79 (0.25) 1.42 (0.03)3.76 (0.13) 2.93 (0.24)Fabric strength (SD) (N/thread), tested4.40 (0.06) 2.09 (0.02)3.78 (0.12) 3.87 (0.13)

    WarpWeft

    WarpWeft

    WarpWeft

    WarpWeft

    WarpWeft

    WarpWeft

    WarpWeft

    WarpWeft

    4.41 2.68 2.682.68 2.21 2.21

    3.99 39.63 36.9131.41 41.35 44.07

    3.28 9.60 9.269.21 9.80 10.12

    12.75 11.54 14.5712.84 37.77 19.54

    1.24 x 1O-5 1.82 x lo- 5.42 x lo-3.63 x 1O-9 4.11 x 1o-21 1.05 x lo-l3

    1.54 (0.15) 3.90 (0.57) 3.67 (0.41)2.92 (0.26) 3.03 (0.10) 3.50 (0.24)

    1.84 (0.03) 3-06 (0.19) 3.14 (0.14)4.04 (0.05) 2-84 (O-10) 3.64 (0.10)Fabric strength (SD) (N/thread), predicted

    3.92 (0.09) 1.46 (0.02) 1.69 (0.05) 3.88 (0.14) 3.693.82 (0.07) (0.11) (0.11)3.18 3.17 (0.11) 2.99 (0.06) 3.59 (0.11)a F, filament yam; S, staple yarn.

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    Anal ysis of w oven fabri c strengt hs 32725. Realff, M. L., Seo, M., Boyce, M., Schwartz, P. &Backer, S., Mechanical properties of fabric woven fromyams produced by different spinning technology: Yarnfailure as a function of gauge length. Textile Res. J., 61(1991) 517.26. Henstenburg, R. B. & Phoenix, S. L., Inter-facial shearstrength studies using the single-filament-composite test.Part 11: A probability model and Monte Carlo

    simulation. Polym. Comp., 10 1989) 389.27. Pan, N., A detailed examination of the translation

    efficiency of fiber strength into composite strength. J.Rein5 Plast. Comp. (in press).28. Hatch, K. L., Textile Science. West Publishing, St Paul,MN, 1993, p. 126.29. Smith, R. L. & McCartney, L. N., Statistical theory ofthe strength of fiber bundles. J. Appl . M ed., 105 1983)601.

    30. Watson, A. S. & Smith, R. L., An examination ofstatistical theories for fibrous materials in the light ofexperimental data. J. M at er. Sci ., 20 (1985) 8260.