Theoretical and Experimental Analyses of Poisson Ratios for Plain－Woven Fabrics

CHEN Jian－wen(陈建稳)，CHEN Wu－jun(陈务军)

1 Space Structures Research Center(SSRC)，Shanghai Jiao Tong University，Shanghai 200030，China

2 School of Science，Nanjing University of Science and Technology，Nanjing 210094，China

Theoretical and Experimental Analyses of Poisson Ratios for Plain－Woven Fabrics

CHEN Jian－wen(陈建稳)1，2*，CHEN Wu－jun(陈务军)1

1 Space Structures Research Center(SSRC)，Shanghai Jiao Tong University，Shanghai 200030，China

2 School of Science，Nanjing University of Science and Technology，Nanjing 210094，China

A theoretical model with extensible yarns for plain-woven fabrics is developed to determine the calculation of Poisson ratios．The stress ratio(warp:weft)，as one of parameters corresponding to Poisson ratio variations，isintroduced to complementthe theoretical model．To evaluate the reliability of the theoretical analysis，a series of biaxial tensile tests of a plain-woven fabric with nine stress ratios are conducted carefully，and the theoretical results are compared with the experimentally measured values．The effects of other influencing factors，including geometric and mechanical parameters of yarns，on Poisson ratios are analyzed thoroughly．This solution method could be applied withoutdifficulty to estimations of Poisson ratios and realistic designs for plain-woven fabrics．

Poisson ratios;plain-woven fabrics;mechanical properties; stress ratios;tensile testing

Introduction

Woven fabricsare used in state－of－the－artstructures including architectural structures，inflatable containers，certain plastic laminated sheets， parachutes， airship structures，etc．［1－7］The estimations of the biaxial elastic constants for woven fabrics are of paramount importance for the realistic designs and analyses of above fabric structures［7］．Especially Poisson ratio is one of the crucial properties of woven fabrics and reveals important mechanical characteristics for a woven fabric［8］．And variation of Poisson ratio may cause uncommon stress－strain relationships， dissimilarto those fornormal materials［9］．

Peirce［10］firstly analyzed the relationships between various parameters of woven fabrics，using a geometrical model without consideration of forces applied．Following from the work of Peirce，several researchers have developed models based on the Peirce model．These include Leaf and Anandjiwala［11］，Jong and Postle［12］，Testa et al．［13］，Huang［14］，and Sun et al．［15］．Applying the optimal－controltheory，Jong and Postle［12］introduced the yarn extension to their analyses of deformations．The work indicated that Poisson ratios calculated on the assumption of inextensible yarn were inconsistent with the experimental results．Huang［14］offered a methodology to analyze the problem of the biaxial extension of a plain－woven fabric;however，Poisson ratios of fabrics were not dealt with in the analysis．Leaf and Kandil［16］presented an analysis of the initial load－extension behavior of plain－woven fabrics，and found an analytical solution for the initial elastic modulus and Poisson ratio of an idealized model of which the yarns were assumed to be inextensible and incompressible．

To attain more reliable elastic constants of woven fabrics，a number of researchers have developed models that also include the yarn extension mechanism．Warren［17］determined the inplane linear elastic moduli of woven fabric which was assumed to be a spatially periodic interlaced network of orthogonal yarns． The results of this theoretical analysis were in good agreement with the measured in－plane elastic moduli．Sun and Pan［8］developed a mechanicalmodelfora woven fabric with extensible yarns to calculate the fabric Poisson ratios．The influences of some mechanical properties of yarns and structural parameters of fabrics on Poisson ratios were analyzed．In addition，the experimental determination is another method to obtain the Poisson ratios of fabrics．Lloyd and Hearle［18］analyzed the limitations of a uniaxial tensile test in calculation of Poisson ratios and pointed out that a biaxial test method was required to conquer the limitations of the uniaxial test．Quaglini et al．［19］developed an experimental protocol for mechanical characterization of a plain－woven fabric．The elastic constants including Poisson ratios were determined by fitting uniaxial stress－strain curves along warp，weft and 45°directions．More recent summaries have been presented by Warren［17］and Sun et al．［8］

As known to all，the Poisson ratios significantly influence the fabric drape and other behaviors．However，the reliable and accurate Poisson ratios for fabrics are difficult to get，as a result of shortage of reliable experimental techniques for fabrics［8］．What's more，Poisson ratios，as reported byGalliot and Luchsinger［20］，vary with the stress ratios and other in situ loading conditions．Poisson ratios used in fabric modeling and structural analysis were mainly estimated on the basis of those for isotropic materials［19］．So far，few studies have analytically and experimentally determined the Poisson ratios for woven fabrics under different biaxial stress ratios．And there is an urgent need to analyze the effects of different stress ratios and other mechanical and geometrical parameters of fabrics on Poisson ratios．

This paper trying to fill the need is organized as follows．Firstly，according to the method proposed by Warren［17］，a theoretical model with extensible yarns for plain－woven fabrics is established to determine an analytical expression of the Poisson ratios．As one of affecting factors，the stress ratios between warp and weft directions are introduced into the theoretical analysis of Poisson ratios．To the best knowledge of the authors，few researches have revealed the effects of the stress ratios on Poisson ratios．Secondly，to evaluate the reliability of the theoretical analysis，a series of biaxial tensile tests of a plain－woven fabric with nine stress ratios are conducted carefully，and the theoretical results are validated by comparing them with the experimentally measured values．Lastly，the effects of influencing factors including stress ratios，geometric and mechanical parameters on Poisson ratios of plain－woven fabrics are analyzed thoroughly．

1 Theoretical Analysis of Fabrics under Biaxial Load

The geometry of the woven fabric and each yarn under consideration here is shown in Fig．1［8，17］．With reference toFig．1(b)，the usual geometrical weave parameters of pick spacing p，yarn length l，and crimp height h are represented in terms of the geometric parameters R and φ0by

where R is the radius of yarn undulation and φ0is the crimp angle．

Fig．1 Geometry of the woven fabric:(a)schematic of woven yarn interlace，(b)cross－section of weave in either x(warp)or y (weft)direction，and(c)forces of model for each yarn

In view of the geometry shown in Fig．1，Eqs．(1)lead to the following relations:

where the subscripts x and y indicate warp and weft yarns，respectively．

The Poisson ratios for a plain－woven fabric are established as follows［8］，

where

And

where I and Arare the moment of inertia and the area for the yarn cross－section，respectively．

Through analysis，the Poisson ratios ofa fabric are determined by the interaction between the warp and the weft yarns，and can be expressed as functions of the structural and mechanical parameters of the system． This exclusive characteristic of a plain－woven fabric is different from a typical continuum．However，their mechanical implications are quite similar．

In order to analyze the effect of stress ratios on the Poisson ratios，a pin－joined truss model proposed by Kawabata et al．［21－22］(shown in Fig．2) was introduced．The model is obtained from a smallestunitofan actualfabric， by approximating the curved yarns with straight rods passing through two consecutive crossover points．As shown in Fig．2(b)，one rod passes through points O1O2，with an angle φ0of the horizontal(x，y)plane;and the axial forces Txand Tyact through the centroids of the cross－sections of yarns．The forces fxand fycan be represented in terms of the axial forces Txand Tyby

Fig．2 Analytical model for stress ratios:(a)the applied forces and geometric angles and(b)geometric relationship

Since equilibrium requires the transverse contact forces to be the same for both warp and weft yarns，there is

And it could be concluded that

Then using Eqs．(2)，we have

The relationship between φ0iand φ0ias shown in Fig．2(b) is given by

where，1 and 2 denote the warp and the weft directions of fabric，respectively．

According to(9)and(10)，the following relations can be determined:

Introducing the stress ratio n and the radius ratio m，

Eq．(11)can be written in the form

Formula(1)determines cos(φ0x/2)and cos(φ0y/2)as

Substituting the relation of Eqs．(14)into Eq．(13)，the radius ratio m can be obtained:

Equations(15)and(12)will be used in the next section to approximately obtain the Poisson ratios(Eq．(3))of fabric under different stress ratios．

2 Biaxial Testing

To assess the validity of a predictive model it is essential to have comprehensive test data with which to compare the model output．Biaxial tensile tests of a cruciform specimen(Fig．3 (b))with slit arms have been carried out using a new biaxial testing machine equipped with two orthogonal independent loading axes(Fig．3(a))．This biaxial testing equipment used was designed and commissioned by our Space Structures Research Center of Shanghai Jiao Tong University．Hydraulic power and series of valves which provide power for the tester are parametrically controlled to realize any spectra．A unique feature of this testing equipment is its feed－back of force which is automatically adjusted by the loading spectrum，and this unique feature makes the loading more precise．A more detailed report can be found in Chen et al．［22］

The central square of the specimen is 160 mm wide．Each cruciform arm is loaded independently by two clamps mounted on a loading car．The loading car of each arm is equipped with a load cell rated 100 kN to measure the load applied to the specimen．Considering the relationship between the applied load and the stress at the center of the cruciform specimen，a reduction factor of 0．94 is applied to predicting the stress where the strain is measured．The strains are measured by the use of two needle extensometers placed in the warp and the weft directions and bolted on the test specimen using small diameter screws，and the strain gage length is set to be 28 mm．Chen et al．［23］showed that small holes in the specimen did not introduce any unacceptable errors．

Fig．3 Experimental set－up:(a)biaxial testing machine and(b) dimensions(in mm)and orientation of test specimens

The biaxial test protocol for this work was adapted from that of the standard of MSAJ［24］．As shown in Fig．4，in order to remove residual strains and avoid high initial levels of creep，the load profile explores various stress ratios with repeated load cycles and a nonzero pre－stress which appears in a pre－stressing stage(20 min)and at every change of stress ratio．This nonzero pre－stress is set 2．0 kN/m(about 5．0%of the ultimatestrength);this value of pre－stress is higher than that of the standard of MSAJ in which it is set 0 kN/m．For each stress ratio，namely 0∶1，1∶3，1∶2，2∶3，1∶1，3∶2，2∶1，3∶1，and 1∶0，three cycles must be applied and at least three specimens must be tested．For this work，elastic constants have been calculated based on the principle described in the standard of MSAJ［24］．

Fig．4 Biaxial test protocol:(a)radial load regime and(b) part of load history for each specimen

3 Comparison with Experiments and Parametric Studies

Table 1 shows the specification of the samples tested．As the cross－section of the yarn is often not an ideal circular section，the equal area method is used to determine the yarn diameters from the measured values．Yarn dimensions and crimp characteristics have been determined using measurements of fabric cross－section images acquired using a digital camera．Measurements from multiple images were taken and averaged to give typical dimensions for the fabric．The comparisons of the theoretical predictions with the experimental results are listed in Table 2．In general，the analytical calculations are in a reasonable agreement with the measurements．

Table 1 Physical parameters of the plain woven fabric tested

Table 2 Poisson ratios for theoretical model and experimental data

3．1 Stress ratios

Figure 5 illustrates the effect of warp to weft stress ratio (σxx/σyy)on the Poisson ratios of the experimental fabric，for which the parameters are listed in Table 1．In general，the comparison between the results of model developed here and experiment appears to be quite good．It can be seen that with the increase of the stress ratio changing from 0．0 to 3．0，the Poisson ratio vxyfirst increases，then decreases after reaching a maximum of about 0．40，while the Poisson ratio vyxonly decrease steadily and arrives at 0．15 where stress ratio equals 3．0．Likewise，the experimental results，especially of the second and third cycles，experience similar variation trends，except for some slight differences．In addition，according to Fig．5，the experimental values of the first cycle are bigger and fluctuate more strongly than those of other two cycles．This is because that，with loading cycle increasing，the linearity of material response becomes more obvious，and after these cycles the tensile properties of the yarns tend to be stable．

Fig．5 Experimental and model results for variation of Poisson ratios with stress ratios(warp:weft):(a)vxyand(b)vyx(the numbers 1，2，and 3 denote the cycle numbers)

As shown in Fig．5，in general，the analytical calculations are in a reasonable agreement with the measurements，and from Eqs．(3)，Poisson ratios for fabrics are also determined by the properties of yarns and structural geometry of fabrics．Thus the effects of these influencing factors on Poisson ratios of the theoretical model are investigated by parametric studies in the following sections．

3．2 Yarn elastic modulus ratios

Figure 6 demonstrates the effect of elastic modulus ratio(ExS/EyS)between warp and weft yarns on the Poisson ratio of the theoretical model，for which the yarn diameter dyequals 0．180 mm，the crimp height hx(=hy)equals 0．260 mm，and the radius of yarn undulation Rxis 0．560 mm．On the whole，for vxy，it shows that with the increase of the elastic modulus ratio (ExS/EyS)，the Poisson ratio increases，and with the increase of the stress ratio n，the Poisson ratio increases as well．However，for vyx，the results are reversed．These trends were not in line with the previously discussed model［16］of which the yarns were assumed to be inextensible; the Poisson ratios ofthat inextensible yarns model were not affected by the variation of yarn elastic modulus ratios．

Fig．6 Variation of Poisson ratios with yarn elastic modulus ratio (warp:weft)

According to Fig．6，under different stress ratios，the elastic modulus ratio has considerably different effect on Poisson ratio．More specifically，for example，when the elastic modulus ratio is less than 1，the effect of the stress ratio on the Poisson ratio vxyis complex．While ExS/EySis greater than 1，growth of the stress ratio promotes significantly the Poisson ratio，even beyond 1．0 at some segment of the curve where stress ratio n equals 2．

3．3 Yarn diameter ratio

The effect of the yarn diameter ratio(dxS/dyS)on the Poisson ratio is shown in Fig．7 with equal yarn elastic modulus ratio(ExS=EyS)and equal crimp height(hx=hy)，while allowing the pick spacing ratio change from 2/3 to 3/2．It indicates that with the increase of the yarn diameter ratio changing from 0．3 to 3．3，the Poisson ratio vxyfirst increases severely，and then decreases smoothly afterreaching a maximum，while the Poisson ratio vyxonly decreases rapidly and arrives at 0．1 where stress ratio equals 1．8 approximately．

Fig．7 Variation of Poisson ratios with yarn diameter ratios (warp:weft)

In general，for vxy，it shows that with the increase of the pick spacing ratio， the Poisson ratio increases． More specifically，for example，when the yarn diameter ratio is less than 0．8，the effect of pick spacing ratio on the Poisson ratio vxyis complex．While the yarn diameter ratio is greater than 0．8，increase of pick spacing ratio promotes significantly the Poisson ratio vxy．

3．4 Pick spacing ratio

Please see Fig．8 which presents the influence of the pick spacing ratio(Px/Py)on the Poisson ratio of a woven fabric model．In this model，both the crimp height ratio and yarn diameter ratio equal 1．0，i．e．，hx=hyand dx=dy．With the increase of the pick spacing ratio，the Poisson ratio vxyfirst ascends markedly，and then descends moderately after arriving at a maximum;while the Poisson ratio vyxonly experiences a downward trend．As illustrated by Fig．8 the yarn elastic modulus ratio affects the Poisson ratio vxymore significantly than vyx．More precisely，for Poisson ratio vxy，when the yarn elastic modulus ratio changes from 1/2 to 2/1，the maximum of Poisson ratio even reaches 1．5 from 0．5．It seems that mechanical parametersalso can impactthe Poisson ratio significantly even overweight the structural parameters，which appears inconsistent with results of the earlier study［8］．This result therefore indicates that either mechanical parameters or geometric parameters may play a bigger role in determining Poisson ratios．

Fig．8 Variation of Poisson ratios with pick spacing ratios (warp:weft)

4 Conclusions

As the theoretical model bases on the linear theory，it is particularly relevant to fabrics whose yarns exhibit more linear stress－strain relation．Considering the uncertainties of the yarn elastic modulus and the level of yarn flattening inevitably affecting the geometric parameters of yarns，the theoretical model tends to givepredictions thatare parallelto the experimental data from corresponding tests．To well verify the theoretical model，future work will include more experimental comparisons for other kinds of plain－woven fabrics．Moreover，the cyclic loading may change the crimp level of warp and weft yarns，which affects the biaxial tensile characteristics and elastic constants of the woven fabrics．Therefore an important question for future studies is to determine the effects of the cyclic loading on the geometric parameters of yarns．These effects of cyclic loading should be considered by the theoretical analysis to obtain accurate Poisson ratios of plain－woven fabrics．The influences ofstress ratios，various mechanical properties ofyarns，and structural parameters of fabrics on the Poisson ratios of the woven fabrics are studied．Our current findings expand prior work．This study offers an understanding of the variability of Poisson ratio and provides a guideline for the design of a woven fabric．

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TS151

A

1672－5220(2015)03－0351－06

date:2013－10－21

s:National Natural Science Foundations of China(Nos．51278299);Natural Science Foundation of the Jiangsu Province，China (No．BK 20150775)

* Correspondence should be addressed to CHEN Jian－wen，E－mail:jianwench@yeah．net