research article experimental and theoretical research on

13
Research Article Experimental and Theoretical Research on the Stress-Relaxation Behaviors of PTFE Coated Fabrics under Different Temperatures Yingying Zhang, 1 Shanshan Xu, 1 Qilin Zhang, 2 and Yi Zhou 2 1 Jiangsu Key Laboratory of Environmental Impact and Structural Safety in Engineering, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China 2 College of Civil Engineering, Tongji University, Shanghai 200092, China Correspondence should be addressed to Yingying Zhang; [email protected] Received 30 April 2015; Revised 16 May 2015; Accepted 18 May 2015 Academic Editor: Jo˜ ao M. P. Q. Delgado Copyright © 2015 Yingying Zhang et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. As polymer composites, the stress-relaxation behaviors of membrane materials have significant effects on the pattern cutting design, the construction process analysis, and the stiffness degradation of membrane structures in the life cycle. In this paper, PTFE coated fabric is taken as the research object. First, the stress-relaxation behaviors under different temperatures (23 C, 40 C, 50 C, 60 C, and 70 C) are studied, and the variations of main mechanical parameters are got. en, a simple review of several current viscoelastic models is presented. Finally, several common models for the material viscoelasticity are used to compare with the test results. Results show PTFE coated fabric is typically viscoelastic. e stress relaxation is obvious in the initial phase and it decreases with time increasing. e stress decreases significantly and then tends to a stable value. With temperature increasing, the decrease rate of membrane stress decreases and the final stable value increases. is material performs obvious hardening with temperature increasing. Most of the current models can make good prediction on the stress-relaxation behaviors of PTFE coated fabrics under different temperatures. e results can be references for the determination of pattern shrinkage ratio and construction process analysis of membrane structures. 1. Introduction At the beginning of the 1970s, glass fibers coated with PTFE (polytetrafluoroethylene) developed by the NASA were first used in civil engineering. Among the commonly used membrane materials, the glass fibers coated with PTFE are welcomed by the designers for their good durability and stiff- ness [1]. Many famous landmark membrane structures are built with PTFE coated fabrics, for example, membrane roof of EXPO Axis in Shanghai, China (Figure 1), and Shenzhen Baoan Stadium in Guangdong, China (Figure 2). e PTFE coated fabric is obviously nonlinear, anisotropic, and viscoe- lastic plastic under tensile loading. Its mechanical properties are affected by the loading history, as well as the woven struc- ture and coating type. Significant viscoelastic characteristics including creep and stress relaxation can be observed in the material tests [24]. Besides, as a typical polymer composite, the viscoelastic characteristics should be considered in the design and analysis of membrane structures [5, 6]. e overall stiffness of membrane structures is supported by the prestress and the curvature form of membrane surfaces. e process of pattern cutting and the joining of separate parts are carried out under the zero stress state. en, the membrane materials are stretched to the initial state during the forming process and the prestress is the main loading. erefore, the membrane materials should be transformed from the initial state to the zero stress state, which is also called “the cutting pattern design” [1]. Due to the nonlinear and viscoelastic properties of membrane materials, the shrinkage compensation should be considered in the pattern cutting design, as shown in Figure 3. Obviously, the shrinkage ratio is related with the membrane stress state and the improper value will affect the forming of membrane structures [711]. e determination of shrinkage ratio is a key and complex technology and there are few published references about this aspect. In actual engineering, the determination of shrinkage ratio is always got by experience or simple tests and it is always Hindawi Publishing Corporation Advances in Materials Science and Engineering Volume 2015, Article ID 319473, 12 pages http://dx.doi.org/10.1155/2015/319473

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Page 1: Research Article Experimental and Theoretical Research on

Research ArticleExperimental and Theoretical Research on the Stress-RelaxationBehaviors of PTFE Coated Fabrics under Different Temperatures

Yingying Zhang1 Shanshan Xu1 Qilin Zhang2 and Yi Zhou2

1 Jiangsu Key Laboratory of Environmental Impact and Structural Safety in Engineering China University of Mining and TechnologyXuzhou Jiangsu 221116 China2College of Civil Engineering Tongji University Shanghai 200092 China

Correspondence should be addressed to Yingying Zhang zhangyingying85163com

Received 30 April 2015 Revised 16 May 2015 Accepted 18 May 2015

Academic Editor Joao M P Q Delgado

Copyright copy 2015 Yingying Zhang et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As polymer composites the stress-relaxation behaviors ofmembranematerials have significant effects on the pattern cutting designthe construction process analysis and the stiffness degradation of membrane structures in the life cycle In this paper PTFE coatedfabric is taken as the research object First the stress-relaxation behaviors under different temperatures (23∘C 40∘C 50∘C 60∘C and70∘C) are studied and the variations of main mechanical parameters are got Then a simple review of several current viscoelasticmodels is presented Finally several common models for the material viscoelasticity are used to compare with the test resultsResults show PTFE coated fabric is typically viscoelastic The stress relaxation is obvious in the initial phase and it decreases withtime increasing The stress decreases significantly and then tends to a stable value With temperature increasing the decrease rateof membrane stress decreases and the final stable value increases This material performs obvious hardening with temperatureincreasing Most of the current models can make good prediction on the stress-relaxation behaviors of PTFE coated fabrics underdifferent temperatures The results can be references for the determination of pattern shrinkage ratio and construction processanalysis of membrane structures

1 Introduction

At the beginning of the 1970s glass fibers coated withPTFE (polytetrafluoroethylene) developed by theNASAwerefirst used in civil engineering Among the commonly usedmembrane materials the glass fibers coated with PTFE arewelcomed by the designers for their good durability and stiff-ness [1] Many famous landmark membrane structures arebuilt with PTFE coated fabrics for example membrane roofof EXPO Axis in Shanghai China (Figure 1) and ShenzhenBaoan Stadium in Guangdong China (Figure 2) The PTFEcoated fabric is obviously nonlinear anisotropic and viscoe-lastic plastic under tensile loading Its mechanical propertiesare affected by the loading history as well as the woven struc-ture and coating type Significant viscoelastic characteristicsincluding creep and stress relaxation can be observed in thematerial tests [2ndash4] Besides as a typical polymer compositethe viscoelastic characteristics should be considered in thedesign and analysis of membrane structures [5 6]

The overall stiffness of membrane structures is supportedby the prestress and the curvature form of membranesurfaces The process of pattern cutting and the joining ofseparate parts are carried out under the zero stress stateThen the membrane materials are stretched to the initialstate during the forming process and the prestress is themain loading Therefore the membrane materials should betransformed from the initial state to the zero stress statewhich is also called ldquothe cutting pattern designrdquo [1] Dueto the nonlinear and viscoelastic properties of membranematerials the shrinkage compensation should be consideredin the pattern cutting design as shown in Figure 3 Obviouslythe shrinkage ratio is related with the membrane stress stateand the improper value will affect the forming of membranestructures [7ndash11]

Thedetermination of shrinkage ratio is a key and complextechnology and there are few published references about thisaspect In actual engineering the determination of shrinkageratio is always got by experience or simple tests and it is always

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2015 Article ID 319473 12 pageshttpdxdoiorg1011552015319473

2 Advances in Materials Science and Engineering

Figure 1 Membrane roof of Shanghai EXPO Axis

Figure 2 Shenzhen Baoan Stadium China

the secret technology of many companies The Europeandesign guide for tensile surface structures proposed that thelong-term strain curves of membrane material can be gotthrough the biaxial cyclic tests by incorporating the effectsof temperature and time [1] The shrinkage ratio can be gotaccording to the curves of long-term strain The Japaneseresearchers also conducted the biaxial cyclic tests to getthe shrinkage ratio but the corresponding test protocolsare different from that of the European recommendations[12] Some researchers used the viscoelastic models to getthe shrinkage ratio and the parameters in the viscoelasticmodels can be got by uniaxial creep tests [13] Besides afterstretched forming significant stress relaxation will appearin the membrane surface under the interaction of windand snow which may lead to the reduction and redistribu-tion of membrane stress [14ndash16] Therefore the viscoelasticproperties of membrane materials should be considered inthe cutting pattern design and construction process analysis[11]

Nowadays the viscoelastic constitutive models are builtby adding the viscous components to the elastic constitutivemodels Actually plastic strains can be observed under avery low stress state and they perform significant memorycharacteristics subject to the loading history There are twoprincipal methods to get the constitutive relations macro-scopic models and microscopic models The microscopicmodels are built by the properties of yarns coatings andinterfaces [11 17] They can reflect the deformation mecha-nisms of internal structures but there are always too manyunknown parameters They may decrease the application

efficiency and increase the computation complexity There-fore many researches choose the macroscopic mathematicalmodels to describe the viscoelastic properties of coatedfabrics The mathematical models are always composedof elastic components and viscous components such asMaxwell model Kelvin model and generalized viscoelasticmodel [18 19] Besides some researchers conduct the three-component model four-component model multicomponentmodel fractional Maxwell model fractional exponentialmodel and others to describe the viscoelastic behavior ofmaterialsThe studies show that with the prediction accuracyincreases with parameter number increasing but too manyunknown parameters are also not convenient to the engineer-ing application As shown in the existing references the linearviscoelastic models are always used in the current analysisand the prediction accuracy needs to be improved

As polymer composites the mechanical properties ofcoated fabrics should be sensitive to temperature and loadingprotocols [5 6] Recently many scholars studied the effect oftemperature on the mechanical properties of polymer mate-rials Minte carried out a series of tests on the materials andconnections and proposed the corresponding temperaturesinfluence factors [20] Zhang et al proposed the temperatureinfluence coefficient of tensile strength for coated fabrics byexperimental researches [21] Wang et al used the regressionmethod to fit the test data under two different temperatures[22] Ambroziak and Kosowski described the test methodto study the effect of temperature on mechanical propertiesof PVC-coated polyester fabric used in tensile membranestructures and proposed two nonlinear models for constitu-tive relations the piecewise linear model and theMurnaghanmodel [23] From above the current researches are mainlyimposed on the variations of tensile strength and breakingstrain under different temperatures However there are fewreferences on the stress relaxation of PTFE coated fabricsunder different temperatures The effects of temperature onthe material viscoelastic behaviors should be consideredwhich may be important for the cutting pattern analysis andconstruction process analysis of membrane structures

First this paper studied the stress relaxation of PTFEmembrane materials under different temperatures by uniax-ial tests Then a simple review of several current viscoelasticmodels is presented Finally several viscoelastic models areused to describe the variation law of relaxation modulus andthe fitting accuracies are compared

2 Materials and Test Setup

In this study the Sheerfill-II manufactured by the Saint Gob-ain Company is taken as the research object The specimensare with the length of 300mm and the width of 50mmand the gauge distance is 200mm The thickness is 08mmThe tests are carried out by the electronic tensile machinewith temperature test box as shown in Figure 4 During theexperiment the specimens should be put in the temperaturebox at least 1 hour before the tests The temperatures are23∘C 40∘C 50∘C 60∘C and 70∘C respectively The initialprestress is 4 kNm and the test time of stress relaxation is72 hours

Advances in Materials Science and Engineering 3

X

x

y

Y

Z

ii

j

j

T1(X Y Z)T2(x y)

Figure 3 Shrinkage compensation from the initial state to the zero stress state

Figure 4 Uniaxial tensile machine with temperature box

3 Current Viscoelastic Constitutive Models

Several common viscoelastic constitutive models are intro-duced and the corresponding expressions for stress relaxationare listed

31 Classic Maxwell Model The classic Maxwell models arecomposed of one spring and one viscous component Atconstant initial stress total strain of elastic component andthe viscous component is shown as follows

120576 = 1205761 + 1205762 (1)

where 1205901 = 1198641205761 is the expressions of the spring componentand 1205902 = 120578 1205762 is the expressions of the viscous componentThe balance equation is 1205901 = 1205902 = 120590 and the deformationcoordination equation is 120576 = 1205761 + 1205762

At a constant strain the differential equation can beobtained the stress-relaxation expression of classic Maxwellmodels is shown as follows

120590 (119905) = 1198641205760119890minus119905120591119894

(2)

where 120591119894

= 120578119894119864119894is the relaxation time of the Maxwell

component 119864 is the elastic modulus and 1205760 is the initial

strain From (2) the decrease rate of stress relaxation isrelated with 120578119864 and the relaxation time 120591

119894is related with

material properties With the viscous value 120578 decreasing therelaxation time decreases and the relaxation rate increases If120578 rarr infin the stress relaxationwill end which is the horizontalsection of the stress-relaxation curves

32 Generalized Viscoelastic Models (Three-ComponentModel Five-Component Model and Seven-ComponentModel) The Maxwell models are composed of spring andviscous componentsThe generalizedMaxwell model is com-posed of several Maxwell models as shown in Figures 5(a)and 5(b) The constitutive relation of Maxwell model is1205761 = 1119864 + 1205901120578 and the constitutive relation of the springmodel is 1205902 = 11986421205762

Based on the balance equation and deformation coordi-nate equation the constitutive relation of three-componentmodel is as follows

120576 = 1205761 =11198641

+

1205901120578

=

11198641

( minus 2) +1120578

(120590 minus 1205902) (3)

By introducing the unit jump function the expression forrelaxation modulus is shown as follows

119884 (119905) = 119864119890+

119899

sum

119894=1119864119894119890minus119905120591119894

(4)

where the 119864119890is the final relaxation modulus and 120591

119894= 120578119894119864119894

is the relaxation time of the ith Maxwell component 1205760 is theinitial strain

33 Fractional Maxwell Model The fractional Maxwell mod-els are mathematically composed of the spring and viscouscomponents The classic Maxwell model is composed ofspring and viscous components [24 25] If we replace thespring and viscous components by two fractional viscoelasticmodels then a fractional Maxwell model is created as shownin Figure 5(c)

The constitutive relations can be described as follows

120590 (119905) +

1198641120591120572

1

1198642120591120573

2

119889120572minus120573

120590 (119905)

119889119905120572minus120573

= 1198641120591120572

1119889120572

120576 (119905)

119889119905120572

(5)

4 Advances in Materials Science and Engineering

120576 = 1205760

1205901 1205902

1205761

1205762

EeE1

1205781

(a) Three-compo-nent model

120576 = 1205760

Ee E1

1205781

E2

1205782 120578n

En

(b) Generalized Maxwell model

(E1 1205911 120572)

(E2 1205912 120573)

(c) FractionalMaxwell model

Figure 5 Several viscoelastic models

The expressions of fractionalMaxwell relaxationmoduluscan be got by the Fourier transform and Mellin inversetransform

If 0 le 120573 lt 120572 lt 1 the relaxation modulus can be got asfollows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(119905 ≪ 120591)

119884 (119905) cong

119864

Γ (1 minus 120572)

(

119905

120591

)

minus120572

(119905 ≫ 120591)

(6)

where 120572 is the attenuation index 120591 is the attenuation char-acteristic time 119864 is the modulus and Γ(119909) is the completeGamma function

34 Fractional Exponential Model Due to the similarityof constitutive relations of elastic and viscoelastic bodiesthe answer of the correspondence principle in viscoelasticproblems can be got according to elastic problems In orderto achieve the relationship between the nonlinear viscoelasticconstitutive relation and the nonlinear elastic constitutiverelationship the viscoelastic constitutive theory of the elastic-ity recovery correspondence principle is proposed by Zhang[26] Then the fractional exponential model is proposed asshown in

119884 (119905) = 119884infin+ (1198840 minus119884

infin) exp minus120573 [(120574 + 120572) 119905]

1minus120572 (7)

where119884infinis the long-term relaxationmodulus119884

0is the tran-

sient relaxation modulus and 120572 120573 and 120574 are the parameters

35 Burgers Model The Burgers model is a combinationmodel composed of a Maxwell model and a Kevin modelIt is always called ldquofour-component modelrdquo due to thefour components in its expression The Burgers model is acommon model that can describe the material viscoelasticbehaviors

The constitutive relation of Maxwell model is 1205761 = 1205761015840

1

+

12057610158401015840

1

= 11198641+ 12059011205781and the constitutive relation of Kevin

model is 1205902 = 1205901015840

+ 12059010158401015840

= 11986421205762 + 1205782 1205762 The expressions of

the balance and deformation coordination conditions are 120590 =

1205901 = 1205902 and 120576 = 1205761 + 1205762Therefore the differential expressions of Burgers model

are got as follows

1198642 120576 + 1205782 120576 =

12057821198641

[ +(

11986411205781

+

11986411205782

+

11986421205782

) +

1198641119864212057811205782

120590]

120590 (0+) = 1198641120576 (0+

)

(0+) = 1198641 120576 (0+) minus 11986421 (

11205781

+

11205782) 120576 (0+)

(8)

The expressions of Burgers model for the material stress-relaxation behaviors are shown in

119884 (119905) =

1198641120572 minus 120573

[(

11986421205782

minus120573) 119890minus120573119905

minus(

11986421205782

minus120572) 119890minus120572119905

] (9)

where 120572 = (1199011+radic1199012

1

minus 41199012)21199012 120573 = (119901

1minusradic1199012

1

minus 41199012)21199012

1199011= 12057811198641+ (1205781

+ 1205782)1198642 and 119901

2= 1205781120578211986411198642

4 Results and Discussions

41 Experimental Results of Stress-Relaxation Tests Thestress-strain curves of PTFE coated fabrics under differenttemperatures are shown in Figure 6 The relaxation moduluscan be got by dividing the stress by the strain and therelaxation curves under different temperatures are shownin Figure 7 The stress-time curves are nonlinear and thereis a linear relationship between the logarithm of relaxationmodulus and time In the first three hours the stress relax-ation has completed 80 of the total relaxation value Inthe following the stress gradually reaches a stable value Thechanging of temperature has few effects on the curves ofstress relaxation and relaxation modulus With temperatureincreasing the rate of stress relaxation is faster and it is easyto reach the stable state The stable value increases with tem-perature increasing which is different from other traditionalmaterials (eg PVC-coated fabrics) The material performsobvious hardening with temperature increasing which isconsistent with the material behaviors in cyclic loading tests[27 28]

Advances in Materials Science and Engineering 5

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(a) Warp

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(b) Weft

Figure 6 Stress-relaxation curves of PTFE coated fabrics under different temperatures

23∘C40∘C50∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

minus2 0 2 4 6

60∘C70∘C

lg(ts)

(a) Warp

23∘C40∘C50∘C

60∘C70∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

0 2 4 6

lg(ts)

(b) Weft

Figure 7 Relaxation modulus of PTFE coated fabrics under different temperatures

42 Comparisons of Experimental Results of Stress-RelaxationTests The above constitutive models are used to analyze thestress-relaxation behaviors of PTFE coated fabrics and theprediction accuracy of several models is compared

421 Classic Maxwell Model The classic Maxwell models arefitted by (2) and the fitting results are shown in Figure 8 Theresults show that the classic Maxwell models cannot be usedto analyze the stress-relaxation behaviors of PTFE coatedfabrics

422 Generalized Maxwell Model (Three-Component ModelFive-Component Model and Seven-Component Model) Thegeneralized Maxwell models (three-component model five-component model and seven-component model) are usedto fit the relaxation modulus of PTFE coated fabrics underdifferent temperatures All three models are got by self-defined formulas The fitting results are compared withthe test data and the corresponding prediction accuracy iscompared as shown in Figures 9 10 and 11 respectively Thefitted parameters of three models are listed in Tables 1 2 and3 respectively

6 Advances in Materials Science and Engineering

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

(a) Weft

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

(b) Warp

Figure 8 Prediction results of the classic Maxwell model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 9 Predictions of the generalized Maxwell model (three-component model)

423 Fractional Maxwell Model In order to express thestress-relaxation behaviors directly the stress-relaxationbehavior can be described as follows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(10)

Here if 1198961= 119864120591120573

Γ(1 minus 120573) the above function can be simpli-fied as follows

lg119884 (119905) = lg 1198961 minus120573 lg 119905 (11)

The fractional Maxwell models are fitted by (11) and thefitting results are shown in Figure 12 and Table 4

Advances in Materials Science and Engineering 7

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 10 Predictions of the generalized Maxwell model (five-component model)

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 11 Predictions of the generalized Maxwell model (seven-component model)

424 Fractional ExponentialModel The fractional exponen-tial models are fitted by self-defined formulas and the fittingresults are shown in Figure 13 and Table 5

425 Burgers Model The Burgers model is fitted by self-defined formulas and the fitting results are shown in Figure 14and Table 5

From the above all the above viscoelastic models candescribe the stress-relaxation behaviors of PTFE coated fab-rics under different temperatures However the expressionsof classic Maxwell models are relatively simple comparedwith the other models They have strict guidelines forthe application of the ordinary differential equations indescribing the complex constitutive relations They can onlydescribe the simple trend of stress relaxation and cannot

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 2: Research Article Experimental and Theoretical Research on

2 Advances in Materials Science and Engineering

Figure 1 Membrane roof of Shanghai EXPO Axis

Figure 2 Shenzhen Baoan Stadium China

the secret technology of many companies The Europeandesign guide for tensile surface structures proposed that thelong-term strain curves of membrane material can be gotthrough the biaxial cyclic tests by incorporating the effectsof temperature and time [1] The shrinkage ratio can be gotaccording to the curves of long-term strain The Japaneseresearchers also conducted the biaxial cyclic tests to getthe shrinkage ratio but the corresponding test protocolsare different from that of the European recommendations[12] Some researchers used the viscoelastic models to getthe shrinkage ratio and the parameters in the viscoelasticmodels can be got by uniaxial creep tests [13] Besides afterstretched forming significant stress relaxation will appearin the membrane surface under the interaction of windand snow which may lead to the reduction and redistribu-tion of membrane stress [14ndash16] Therefore the viscoelasticproperties of membrane materials should be considered inthe cutting pattern design and construction process analysis[11]

Nowadays the viscoelastic constitutive models are builtby adding the viscous components to the elastic constitutivemodels Actually plastic strains can be observed under avery low stress state and they perform significant memorycharacteristics subject to the loading history There are twoprincipal methods to get the constitutive relations macro-scopic models and microscopic models The microscopicmodels are built by the properties of yarns coatings andinterfaces [11 17] They can reflect the deformation mecha-nisms of internal structures but there are always too manyunknown parameters They may decrease the application

efficiency and increase the computation complexity There-fore many researches choose the macroscopic mathematicalmodels to describe the viscoelastic properties of coatedfabrics The mathematical models are always composedof elastic components and viscous components such asMaxwell model Kelvin model and generalized viscoelasticmodel [18 19] Besides some researchers conduct the three-component model four-component model multicomponentmodel fractional Maxwell model fractional exponentialmodel and others to describe the viscoelastic behavior ofmaterialsThe studies show that with the prediction accuracyincreases with parameter number increasing but too manyunknown parameters are also not convenient to the engineer-ing application As shown in the existing references the linearviscoelastic models are always used in the current analysisand the prediction accuracy needs to be improved

As polymer composites the mechanical properties ofcoated fabrics should be sensitive to temperature and loadingprotocols [5 6] Recently many scholars studied the effect oftemperature on the mechanical properties of polymer mate-rials Minte carried out a series of tests on the materials andconnections and proposed the corresponding temperaturesinfluence factors [20] Zhang et al proposed the temperatureinfluence coefficient of tensile strength for coated fabrics byexperimental researches [21] Wang et al used the regressionmethod to fit the test data under two different temperatures[22] Ambroziak and Kosowski described the test methodto study the effect of temperature on mechanical propertiesof PVC-coated polyester fabric used in tensile membranestructures and proposed two nonlinear models for constitu-tive relations the piecewise linear model and theMurnaghanmodel [23] From above the current researches are mainlyimposed on the variations of tensile strength and breakingstrain under different temperatures However there are fewreferences on the stress relaxation of PTFE coated fabricsunder different temperatures The effects of temperature onthe material viscoelastic behaviors should be consideredwhich may be important for the cutting pattern analysis andconstruction process analysis of membrane structures

First this paper studied the stress relaxation of PTFEmembrane materials under different temperatures by uniax-ial tests Then a simple review of several current viscoelasticmodels is presented Finally several viscoelastic models areused to describe the variation law of relaxation modulus andthe fitting accuracies are compared

2 Materials and Test Setup

In this study the Sheerfill-II manufactured by the Saint Gob-ain Company is taken as the research object The specimensare with the length of 300mm and the width of 50mmand the gauge distance is 200mm The thickness is 08mmThe tests are carried out by the electronic tensile machinewith temperature test box as shown in Figure 4 During theexperiment the specimens should be put in the temperaturebox at least 1 hour before the tests The temperatures are23∘C 40∘C 50∘C 60∘C and 70∘C respectively The initialprestress is 4 kNm and the test time of stress relaxation is72 hours

Advances in Materials Science and Engineering 3

X

x

y

Y

Z

ii

j

j

T1(X Y Z)T2(x y)

Figure 3 Shrinkage compensation from the initial state to the zero stress state

Figure 4 Uniaxial tensile machine with temperature box

3 Current Viscoelastic Constitutive Models

Several common viscoelastic constitutive models are intro-duced and the corresponding expressions for stress relaxationare listed

31 Classic Maxwell Model The classic Maxwell models arecomposed of one spring and one viscous component Atconstant initial stress total strain of elastic component andthe viscous component is shown as follows

120576 = 1205761 + 1205762 (1)

where 1205901 = 1198641205761 is the expressions of the spring componentand 1205902 = 120578 1205762 is the expressions of the viscous componentThe balance equation is 1205901 = 1205902 = 120590 and the deformationcoordination equation is 120576 = 1205761 + 1205762

At a constant strain the differential equation can beobtained the stress-relaxation expression of classic Maxwellmodels is shown as follows

120590 (119905) = 1198641205760119890minus119905120591119894

(2)

where 120591119894

= 120578119894119864119894is the relaxation time of the Maxwell

component 119864 is the elastic modulus and 1205760 is the initial

strain From (2) the decrease rate of stress relaxation isrelated with 120578119864 and the relaxation time 120591

119894is related with

material properties With the viscous value 120578 decreasing therelaxation time decreases and the relaxation rate increases If120578 rarr infin the stress relaxationwill end which is the horizontalsection of the stress-relaxation curves

32 Generalized Viscoelastic Models (Three-ComponentModel Five-Component Model and Seven-ComponentModel) The Maxwell models are composed of spring andviscous componentsThe generalizedMaxwell model is com-posed of several Maxwell models as shown in Figures 5(a)and 5(b) The constitutive relation of Maxwell model is1205761 = 1119864 + 1205901120578 and the constitutive relation of the springmodel is 1205902 = 11986421205762

Based on the balance equation and deformation coordi-nate equation the constitutive relation of three-componentmodel is as follows

120576 = 1205761 =11198641

+

1205901120578

=

11198641

( minus 2) +1120578

(120590 minus 1205902) (3)

By introducing the unit jump function the expression forrelaxation modulus is shown as follows

119884 (119905) = 119864119890+

119899

sum

119894=1119864119894119890minus119905120591119894

(4)

where the 119864119890is the final relaxation modulus and 120591

119894= 120578119894119864119894

is the relaxation time of the ith Maxwell component 1205760 is theinitial strain

33 Fractional Maxwell Model The fractional Maxwell mod-els are mathematically composed of the spring and viscouscomponents The classic Maxwell model is composed ofspring and viscous components [24 25] If we replace thespring and viscous components by two fractional viscoelasticmodels then a fractional Maxwell model is created as shownin Figure 5(c)

The constitutive relations can be described as follows

120590 (119905) +

1198641120591120572

1

1198642120591120573

2

119889120572minus120573

120590 (119905)

119889119905120572minus120573

= 1198641120591120572

1119889120572

120576 (119905)

119889119905120572

(5)

4 Advances in Materials Science and Engineering

120576 = 1205760

1205901 1205902

1205761

1205762

EeE1

1205781

(a) Three-compo-nent model

120576 = 1205760

Ee E1

1205781

E2

1205782 120578n

En

(b) Generalized Maxwell model

(E1 1205911 120572)

(E2 1205912 120573)

(c) FractionalMaxwell model

Figure 5 Several viscoelastic models

The expressions of fractionalMaxwell relaxationmoduluscan be got by the Fourier transform and Mellin inversetransform

If 0 le 120573 lt 120572 lt 1 the relaxation modulus can be got asfollows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(119905 ≪ 120591)

119884 (119905) cong

119864

Γ (1 minus 120572)

(

119905

120591

)

minus120572

(119905 ≫ 120591)

(6)

where 120572 is the attenuation index 120591 is the attenuation char-acteristic time 119864 is the modulus and Γ(119909) is the completeGamma function

34 Fractional Exponential Model Due to the similarityof constitutive relations of elastic and viscoelastic bodiesthe answer of the correspondence principle in viscoelasticproblems can be got according to elastic problems In orderto achieve the relationship between the nonlinear viscoelasticconstitutive relation and the nonlinear elastic constitutiverelationship the viscoelastic constitutive theory of the elastic-ity recovery correspondence principle is proposed by Zhang[26] Then the fractional exponential model is proposed asshown in

119884 (119905) = 119884infin+ (1198840 minus119884

infin) exp minus120573 [(120574 + 120572) 119905]

1minus120572 (7)

where119884infinis the long-term relaxationmodulus119884

0is the tran-

sient relaxation modulus and 120572 120573 and 120574 are the parameters

35 Burgers Model The Burgers model is a combinationmodel composed of a Maxwell model and a Kevin modelIt is always called ldquofour-component modelrdquo due to thefour components in its expression The Burgers model is acommon model that can describe the material viscoelasticbehaviors

The constitutive relation of Maxwell model is 1205761 = 1205761015840

1

+

12057610158401015840

1

= 11198641+ 12059011205781and the constitutive relation of Kevin

model is 1205902 = 1205901015840

+ 12059010158401015840

= 11986421205762 + 1205782 1205762 The expressions of

the balance and deformation coordination conditions are 120590 =

1205901 = 1205902 and 120576 = 1205761 + 1205762Therefore the differential expressions of Burgers model

are got as follows

1198642 120576 + 1205782 120576 =

12057821198641

[ +(

11986411205781

+

11986411205782

+

11986421205782

) +

1198641119864212057811205782

120590]

120590 (0+) = 1198641120576 (0+

)

(0+) = 1198641 120576 (0+) minus 11986421 (

11205781

+

11205782) 120576 (0+)

(8)

The expressions of Burgers model for the material stress-relaxation behaviors are shown in

119884 (119905) =

1198641120572 minus 120573

[(

11986421205782

minus120573) 119890minus120573119905

minus(

11986421205782

minus120572) 119890minus120572119905

] (9)

where 120572 = (1199011+radic1199012

1

minus 41199012)21199012 120573 = (119901

1minusradic1199012

1

minus 41199012)21199012

1199011= 12057811198641+ (1205781

+ 1205782)1198642 and 119901

2= 1205781120578211986411198642

4 Results and Discussions

41 Experimental Results of Stress-Relaxation Tests Thestress-strain curves of PTFE coated fabrics under differenttemperatures are shown in Figure 6 The relaxation moduluscan be got by dividing the stress by the strain and therelaxation curves under different temperatures are shownin Figure 7 The stress-time curves are nonlinear and thereis a linear relationship between the logarithm of relaxationmodulus and time In the first three hours the stress relax-ation has completed 80 of the total relaxation value Inthe following the stress gradually reaches a stable value Thechanging of temperature has few effects on the curves ofstress relaxation and relaxation modulus With temperatureincreasing the rate of stress relaxation is faster and it is easyto reach the stable state The stable value increases with tem-perature increasing which is different from other traditionalmaterials (eg PVC-coated fabrics) The material performsobvious hardening with temperature increasing which isconsistent with the material behaviors in cyclic loading tests[27 28]

Advances in Materials Science and Engineering 5

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(a) Warp

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(b) Weft

Figure 6 Stress-relaxation curves of PTFE coated fabrics under different temperatures

23∘C40∘C50∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

minus2 0 2 4 6

60∘C70∘C

lg(ts)

(a) Warp

23∘C40∘C50∘C

60∘C70∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

0 2 4 6

lg(ts)

(b) Weft

Figure 7 Relaxation modulus of PTFE coated fabrics under different temperatures

42 Comparisons of Experimental Results of Stress-RelaxationTests The above constitutive models are used to analyze thestress-relaxation behaviors of PTFE coated fabrics and theprediction accuracy of several models is compared

421 Classic Maxwell Model The classic Maxwell models arefitted by (2) and the fitting results are shown in Figure 8 Theresults show that the classic Maxwell models cannot be usedto analyze the stress-relaxation behaviors of PTFE coatedfabrics

422 Generalized Maxwell Model (Three-Component ModelFive-Component Model and Seven-Component Model) Thegeneralized Maxwell models (three-component model five-component model and seven-component model) are usedto fit the relaxation modulus of PTFE coated fabrics underdifferent temperatures All three models are got by self-defined formulas The fitting results are compared withthe test data and the corresponding prediction accuracy iscompared as shown in Figures 9 10 and 11 respectively Thefitted parameters of three models are listed in Tables 1 2 and3 respectively

6 Advances in Materials Science and Engineering

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

(a) Weft

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

(b) Warp

Figure 8 Prediction results of the classic Maxwell model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 9 Predictions of the generalized Maxwell model (three-component model)

423 Fractional Maxwell Model In order to express thestress-relaxation behaviors directly the stress-relaxationbehavior can be described as follows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(10)

Here if 1198961= 119864120591120573

Γ(1 minus 120573) the above function can be simpli-fied as follows

lg119884 (119905) = lg 1198961 minus120573 lg 119905 (11)

The fractional Maxwell models are fitted by (11) and thefitting results are shown in Figure 12 and Table 4

Advances in Materials Science and Engineering 7

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 10 Predictions of the generalized Maxwell model (five-component model)

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 11 Predictions of the generalized Maxwell model (seven-component model)

424 Fractional ExponentialModel The fractional exponen-tial models are fitted by self-defined formulas and the fittingresults are shown in Figure 13 and Table 5

425 Burgers Model The Burgers model is fitted by self-defined formulas and the fitting results are shown in Figure 14and Table 5

From the above all the above viscoelastic models candescribe the stress-relaxation behaviors of PTFE coated fab-rics under different temperatures However the expressionsof classic Maxwell models are relatively simple comparedwith the other models They have strict guidelines forthe application of the ordinary differential equations indescribing the complex constitutive relations They can onlydescribe the simple trend of stress relaxation and cannot

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 3: Research Article Experimental and Theoretical Research on

Advances in Materials Science and Engineering 3

X

x

y

Y

Z

ii

j

j

T1(X Y Z)T2(x y)

Figure 3 Shrinkage compensation from the initial state to the zero stress state

Figure 4 Uniaxial tensile machine with temperature box

3 Current Viscoelastic Constitutive Models

Several common viscoelastic constitutive models are intro-duced and the corresponding expressions for stress relaxationare listed

31 Classic Maxwell Model The classic Maxwell models arecomposed of one spring and one viscous component Atconstant initial stress total strain of elastic component andthe viscous component is shown as follows

120576 = 1205761 + 1205762 (1)

where 1205901 = 1198641205761 is the expressions of the spring componentand 1205902 = 120578 1205762 is the expressions of the viscous componentThe balance equation is 1205901 = 1205902 = 120590 and the deformationcoordination equation is 120576 = 1205761 + 1205762

At a constant strain the differential equation can beobtained the stress-relaxation expression of classic Maxwellmodels is shown as follows

120590 (119905) = 1198641205760119890minus119905120591119894

(2)

where 120591119894

= 120578119894119864119894is the relaxation time of the Maxwell

component 119864 is the elastic modulus and 1205760 is the initial

strain From (2) the decrease rate of stress relaxation isrelated with 120578119864 and the relaxation time 120591

119894is related with

material properties With the viscous value 120578 decreasing therelaxation time decreases and the relaxation rate increases If120578 rarr infin the stress relaxationwill end which is the horizontalsection of the stress-relaxation curves

32 Generalized Viscoelastic Models (Three-ComponentModel Five-Component Model and Seven-ComponentModel) The Maxwell models are composed of spring andviscous componentsThe generalizedMaxwell model is com-posed of several Maxwell models as shown in Figures 5(a)and 5(b) The constitutive relation of Maxwell model is1205761 = 1119864 + 1205901120578 and the constitutive relation of the springmodel is 1205902 = 11986421205762

Based on the balance equation and deformation coordi-nate equation the constitutive relation of three-componentmodel is as follows

120576 = 1205761 =11198641

+

1205901120578

=

11198641

( minus 2) +1120578

(120590 minus 1205902) (3)

By introducing the unit jump function the expression forrelaxation modulus is shown as follows

119884 (119905) = 119864119890+

119899

sum

119894=1119864119894119890minus119905120591119894

(4)

where the 119864119890is the final relaxation modulus and 120591

119894= 120578119894119864119894

is the relaxation time of the ith Maxwell component 1205760 is theinitial strain

33 Fractional Maxwell Model The fractional Maxwell mod-els are mathematically composed of the spring and viscouscomponents The classic Maxwell model is composed ofspring and viscous components [24 25] If we replace thespring and viscous components by two fractional viscoelasticmodels then a fractional Maxwell model is created as shownin Figure 5(c)

The constitutive relations can be described as follows

120590 (119905) +

1198641120591120572

1

1198642120591120573

2

119889120572minus120573

120590 (119905)

119889119905120572minus120573

= 1198641120591120572

1119889120572

120576 (119905)

119889119905120572

(5)

4 Advances in Materials Science and Engineering

120576 = 1205760

1205901 1205902

1205761

1205762

EeE1

1205781

(a) Three-compo-nent model

120576 = 1205760

Ee E1

1205781

E2

1205782 120578n

En

(b) Generalized Maxwell model

(E1 1205911 120572)

(E2 1205912 120573)

(c) FractionalMaxwell model

Figure 5 Several viscoelastic models

The expressions of fractionalMaxwell relaxationmoduluscan be got by the Fourier transform and Mellin inversetransform

If 0 le 120573 lt 120572 lt 1 the relaxation modulus can be got asfollows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(119905 ≪ 120591)

119884 (119905) cong

119864

Γ (1 minus 120572)

(

119905

120591

)

minus120572

(119905 ≫ 120591)

(6)

where 120572 is the attenuation index 120591 is the attenuation char-acteristic time 119864 is the modulus and Γ(119909) is the completeGamma function

34 Fractional Exponential Model Due to the similarityof constitutive relations of elastic and viscoelastic bodiesthe answer of the correspondence principle in viscoelasticproblems can be got according to elastic problems In orderto achieve the relationship between the nonlinear viscoelasticconstitutive relation and the nonlinear elastic constitutiverelationship the viscoelastic constitutive theory of the elastic-ity recovery correspondence principle is proposed by Zhang[26] Then the fractional exponential model is proposed asshown in

119884 (119905) = 119884infin+ (1198840 minus119884

infin) exp minus120573 [(120574 + 120572) 119905]

1minus120572 (7)

where119884infinis the long-term relaxationmodulus119884

0is the tran-

sient relaxation modulus and 120572 120573 and 120574 are the parameters

35 Burgers Model The Burgers model is a combinationmodel composed of a Maxwell model and a Kevin modelIt is always called ldquofour-component modelrdquo due to thefour components in its expression The Burgers model is acommon model that can describe the material viscoelasticbehaviors

The constitutive relation of Maxwell model is 1205761 = 1205761015840

1

+

12057610158401015840

1

= 11198641+ 12059011205781and the constitutive relation of Kevin

model is 1205902 = 1205901015840

+ 12059010158401015840

= 11986421205762 + 1205782 1205762 The expressions of

the balance and deformation coordination conditions are 120590 =

1205901 = 1205902 and 120576 = 1205761 + 1205762Therefore the differential expressions of Burgers model

are got as follows

1198642 120576 + 1205782 120576 =

12057821198641

[ +(

11986411205781

+

11986411205782

+

11986421205782

) +

1198641119864212057811205782

120590]

120590 (0+) = 1198641120576 (0+

)

(0+) = 1198641 120576 (0+) minus 11986421 (

11205781

+

11205782) 120576 (0+)

(8)

The expressions of Burgers model for the material stress-relaxation behaviors are shown in

119884 (119905) =

1198641120572 minus 120573

[(

11986421205782

minus120573) 119890minus120573119905

minus(

11986421205782

minus120572) 119890minus120572119905

] (9)

where 120572 = (1199011+radic1199012

1

minus 41199012)21199012 120573 = (119901

1minusradic1199012

1

minus 41199012)21199012

1199011= 12057811198641+ (1205781

+ 1205782)1198642 and 119901

2= 1205781120578211986411198642

4 Results and Discussions

41 Experimental Results of Stress-Relaxation Tests Thestress-strain curves of PTFE coated fabrics under differenttemperatures are shown in Figure 6 The relaxation moduluscan be got by dividing the stress by the strain and therelaxation curves under different temperatures are shownin Figure 7 The stress-time curves are nonlinear and thereis a linear relationship between the logarithm of relaxationmodulus and time In the first three hours the stress relax-ation has completed 80 of the total relaxation value Inthe following the stress gradually reaches a stable value Thechanging of temperature has few effects on the curves ofstress relaxation and relaxation modulus With temperatureincreasing the rate of stress relaxation is faster and it is easyto reach the stable state The stable value increases with tem-perature increasing which is different from other traditionalmaterials (eg PVC-coated fabrics) The material performsobvious hardening with temperature increasing which isconsistent with the material behaviors in cyclic loading tests[27 28]

Advances in Materials Science and Engineering 5

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(a) Warp

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(b) Weft

Figure 6 Stress-relaxation curves of PTFE coated fabrics under different temperatures

23∘C40∘C50∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

minus2 0 2 4 6

60∘C70∘C

lg(ts)

(a) Warp

23∘C40∘C50∘C

60∘C70∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

0 2 4 6

lg(ts)

(b) Weft

Figure 7 Relaxation modulus of PTFE coated fabrics under different temperatures

42 Comparisons of Experimental Results of Stress-RelaxationTests The above constitutive models are used to analyze thestress-relaxation behaviors of PTFE coated fabrics and theprediction accuracy of several models is compared

421 Classic Maxwell Model The classic Maxwell models arefitted by (2) and the fitting results are shown in Figure 8 Theresults show that the classic Maxwell models cannot be usedto analyze the stress-relaxation behaviors of PTFE coatedfabrics

422 Generalized Maxwell Model (Three-Component ModelFive-Component Model and Seven-Component Model) Thegeneralized Maxwell models (three-component model five-component model and seven-component model) are usedto fit the relaxation modulus of PTFE coated fabrics underdifferent temperatures All three models are got by self-defined formulas The fitting results are compared withthe test data and the corresponding prediction accuracy iscompared as shown in Figures 9 10 and 11 respectively Thefitted parameters of three models are listed in Tables 1 2 and3 respectively

6 Advances in Materials Science and Engineering

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

(a) Weft

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

(b) Warp

Figure 8 Prediction results of the classic Maxwell model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 9 Predictions of the generalized Maxwell model (three-component model)

423 Fractional Maxwell Model In order to express thestress-relaxation behaviors directly the stress-relaxationbehavior can be described as follows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(10)

Here if 1198961= 119864120591120573

Γ(1 minus 120573) the above function can be simpli-fied as follows

lg119884 (119905) = lg 1198961 minus120573 lg 119905 (11)

The fractional Maxwell models are fitted by (11) and thefitting results are shown in Figure 12 and Table 4

Advances in Materials Science and Engineering 7

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 10 Predictions of the generalized Maxwell model (five-component model)

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 11 Predictions of the generalized Maxwell model (seven-component model)

424 Fractional ExponentialModel The fractional exponen-tial models are fitted by self-defined formulas and the fittingresults are shown in Figure 13 and Table 5

425 Burgers Model The Burgers model is fitted by self-defined formulas and the fitting results are shown in Figure 14and Table 5

From the above all the above viscoelastic models candescribe the stress-relaxation behaviors of PTFE coated fab-rics under different temperatures However the expressionsof classic Maxwell models are relatively simple comparedwith the other models They have strict guidelines forthe application of the ordinary differential equations indescribing the complex constitutive relations They can onlydescribe the simple trend of stress relaxation and cannot

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

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CeramicsJournal of

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CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Biomaterials

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 4: Research Article Experimental and Theoretical Research on

4 Advances in Materials Science and Engineering

120576 = 1205760

1205901 1205902

1205761

1205762

EeE1

1205781

(a) Three-compo-nent model

120576 = 1205760

Ee E1

1205781

E2

1205782 120578n

En

(b) Generalized Maxwell model

(E1 1205911 120572)

(E2 1205912 120573)

(c) FractionalMaxwell model

Figure 5 Several viscoelastic models

The expressions of fractionalMaxwell relaxationmoduluscan be got by the Fourier transform and Mellin inversetransform

If 0 le 120573 lt 120572 lt 1 the relaxation modulus can be got asfollows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(119905 ≪ 120591)

119884 (119905) cong

119864

Γ (1 minus 120572)

(

119905

120591

)

minus120572

(119905 ≫ 120591)

(6)

where 120572 is the attenuation index 120591 is the attenuation char-acteristic time 119864 is the modulus and Γ(119909) is the completeGamma function

34 Fractional Exponential Model Due to the similarityof constitutive relations of elastic and viscoelastic bodiesthe answer of the correspondence principle in viscoelasticproblems can be got according to elastic problems In orderto achieve the relationship between the nonlinear viscoelasticconstitutive relation and the nonlinear elastic constitutiverelationship the viscoelastic constitutive theory of the elastic-ity recovery correspondence principle is proposed by Zhang[26] Then the fractional exponential model is proposed asshown in

119884 (119905) = 119884infin+ (1198840 minus119884

infin) exp minus120573 [(120574 + 120572) 119905]

1minus120572 (7)

where119884infinis the long-term relaxationmodulus119884

0is the tran-

sient relaxation modulus and 120572 120573 and 120574 are the parameters

35 Burgers Model The Burgers model is a combinationmodel composed of a Maxwell model and a Kevin modelIt is always called ldquofour-component modelrdquo due to thefour components in its expression The Burgers model is acommon model that can describe the material viscoelasticbehaviors

The constitutive relation of Maxwell model is 1205761 = 1205761015840

1

+

12057610158401015840

1

= 11198641+ 12059011205781and the constitutive relation of Kevin

model is 1205902 = 1205901015840

+ 12059010158401015840

= 11986421205762 + 1205782 1205762 The expressions of

the balance and deformation coordination conditions are 120590 =

1205901 = 1205902 and 120576 = 1205761 + 1205762Therefore the differential expressions of Burgers model

are got as follows

1198642 120576 + 1205782 120576 =

12057821198641

[ +(

11986411205781

+

11986411205782

+

11986421205782

) +

1198641119864212057811205782

120590]

120590 (0+) = 1198641120576 (0+

)

(0+) = 1198641 120576 (0+) minus 11986421 (

11205781

+

11205782) 120576 (0+)

(8)

The expressions of Burgers model for the material stress-relaxation behaviors are shown in

119884 (119905) =

1198641120572 minus 120573

[(

11986421205782

minus120573) 119890minus120573119905

minus(

11986421205782

minus120572) 119890minus120572119905

] (9)

where 120572 = (1199011+radic1199012

1

minus 41199012)21199012 120573 = (119901

1minusradic1199012

1

minus 41199012)21199012

1199011= 12057811198641+ (1205781

+ 1205782)1198642 and 119901

2= 1205781120578211986411198642

4 Results and Discussions

41 Experimental Results of Stress-Relaxation Tests Thestress-strain curves of PTFE coated fabrics under differenttemperatures are shown in Figure 6 The relaxation moduluscan be got by dividing the stress by the strain and therelaxation curves under different temperatures are shownin Figure 7 The stress-time curves are nonlinear and thereis a linear relationship between the logarithm of relaxationmodulus and time In the first three hours the stress relax-ation has completed 80 of the total relaxation value Inthe following the stress gradually reaches a stable value Thechanging of temperature has few effects on the curves ofstress relaxation and relaxation modulus With temperatureincreasing the rate of stress relaxation is faster and it is easyto reach the stable state The stable value increases with tem-perature increasing which is different from other traditionalmaterials (eg PVC-coated fabrics) The material performsobvious hardening with temperature increasing which isconsistent with the material behaviors in cyclic loading tests[27 28]

Advances in Materials Science and Engineering 5

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(a) Warp

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(b) Weft

Figure 6 Stress-relaxation curves of PTFE coated fabrics under different temperatures

23∘C40∘C50∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

minus2 0 2 4 6

60∘C70∘C

lg(ts)

(a) Warp

23∘C40∘C50∘C

60∘C70∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

0 2 4 6

lg(ts)

(b) Weft

Figure 7 Relaxation modulus of PTFE coated fabrics under different temperatures

42 Comparisons of Experimental Results of Stress-RelaxationTests The above constitutive models are used to analyze thestress-relaxation behaviors of PTFE coated fabrics and theprediction accuracy of several models is compared

421 Classic Maxwell Model The classic Maxwell models arefitted by (2) and the fitting results are shown in Figure 8 Theresults show that the classic Maxwell models cannot be usedto analyze the stress-relaxation behaviors of PTFE coatedfabrics

422 Generalized Maxwell Model (Three-Component ModelFive-Component Model and Seven-Component Model) Thegeneralized Maxwell models (three-component model five-component model and seven-component model) are usedto fit the relaxation modulus of PTFE coated fabrics underdifferent temperatures All three models are got by self-defined formulas The fitting results are compared withthe test data and the corresponding prediction accuracy iscompared as shown in Figures 9 10 and 11 respectively Thefitted parameters of three models are listed in Tables 1 2 and3 respectively

6 Advances in Materials Science and Engineering

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

(a) Weft

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

(b) Warp

Figure 8 Prediction results of the classic Maxwell model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 9 Predictions of the generalized Maxwell model (three-component model)

423 Fractional Maxwell Model In order to express thestress-relaxation behaviors directly the stress-relaxationbehavior can be described as follows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(10)

Here if 1198961= 119864120591120573

Γ(1 minus 120573) the above function can be simpli-fied as follows

lg119884 (119905) = lg 1198961 minus120573 lg 119905 (11)

The fractional Maxwell models are fitted by (11) and thefitting results are shown in Figure 12 and Table 4

Advances in Materials Science and Engineering 7

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 10 Predictions of the generalized Maxwell model (five-component model)

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 11 Predictions of the generalized Maxwell model (seven-component model)

424 Fractional ExponentialModel The fractional exponen-tial models are fitted by self-defined formulas and the fittingresults are shown in Figure 13 and Table 5

425 Burgers Model The Burgers model is fitted by self-defined formulas and the fitting results are shown in Figure 14and Table 5

From the above all the above viscoelastic models candescribe the stress-relaxation behaviors of PTFE coated fab-rics under different temperatures However the expressionsof classic Maxwell models are relatively simple comparedwith the other models They have strict guidelines forthe application of the ordinary differential equations indescribing the complex constitutive relations They can onlydescribe the simple trend of stress relaxation and cannot

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

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CeramicsJournal of

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CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 5: Research Article Experimental and Theoretical Research on

Advances in Materials Science and Engineering 5

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(a) Warp

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C

60∘C70∘C

(b) Weft

Figure 6 Stress-relaxation curves of PTFE coated fabrics under different temperatures

23∘C40∘C50∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

minus2 0 2 4 6

60∘C70∘C

lg(ts)

(a) Warp

23∘C40∘C50∘C

60∘C70∘C

Rela

xatio

n m

odul

us (M

Pa)

110

100

90

80

70

60

50

0 2 4 6

lg(ts)

(b) Weft

Figure 7 Relaxation modulus of PTFE coated fabrics under different temperatures

42 Comparisons of Experimental Results of Stress-RelaxationTests The above constitutive models are used to analyze thestress-relaxation behaviors of PTFE coated fabrics and theprediction accuracy of several models is compared

421 Classic Maxwell Model The classic Maxwell models arefitted by (2) and the fitting results are shown in Figure 8 Theresults show that the classic Maxwell models cannot be usedto analyze the stress-relaxation behaviors of PTFE coatedfabrics

422 Generalized Maxwell Model (Three-Component ModelFive-Component Model and Seven-Component Model) Thegeneralized Maxwell models (three-component model five-component model and seven-component model) are usedto fit the relaxation modulus of PTFE coated fabrics underdifferent temperatures All three models are got by self-defined formulas The fitting results are compared withthe test data and the corresponding prediction accuracy iscompared as shown in Figures 9 10 and 11 respectively Thefitted parameters of three models are listed in Tables 1 2 and3 respectively

6 Advances in Materials Science and Engineering

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

(a) Weft

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

(b) Warp

Figure 8 Prediction results of the classic Maxwell model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 9 Predictions of the generalized Maxwell model (three-component model)

423 Fractional Maxwell Model In order to express thestress-relaxation behaviors directly the stress-relaxationbehavior can be described as follows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(10)

Here if 1198961= 119864120591120573

Γ(1 minus 120573) the above function can be simpli-fied as follows

lg119884 (119905) = lg 1198961 minus120573 lg 119905 (11)

The fractional Maxwell models are fitted by (11) and thefitting results are shown in Figure 12 and Table 4

Advances in Materials Science and Engineering 7

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 10 Predictions of the generalized Maxwell model (five-component model)

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 11 Predictions of the generalized Maxwell model (seven-component model)

424 Fractional ExponentialModel The fractional exponen-tial models are fitted by self-defined formulas and the fittingresults are shown in Figure 13 and Table 5

425 Burgers Model The Burgers model is fitted by self-defined formulas and the fitting results are shown in Figure 14and Table 5

From the above all the above viscoelastic models candescribe the stress-relaxation behaviors of PTFE coated fab-rics under different temperatures However the expressionsof classic Maxwell models are relatively simple comparedwith the other models They have strict guidelines forthe application of the ordinary differential equations indescribing the complex constitutive relations They can onlydescribe the simple trend of stress relaxation and cannot

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 6: Research Article Experimental and Theoretical Research on

6 Advances in Materials Science and Engineering

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

(a) Weft

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Stre

ss (k

Nm

)

Time (s)

4

3

2

0 100000 200000 300000

(b) Warp

Figure 8 Prediction results of the classic Maxwell model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 9 Predictions of the generalized Maxwell model (three-component model)

423 Fractional Maxwell Model In order to express thestress-relaxation behaviors directly the stress-relaxationbehavior can be described as follows

119884 (119905) cong

119864

Γ (1 minus 120573)

(

119905

120591

)

minus120573

(10)

Here if 1198961= 119864120591120573

Γ(1 minus 120573) the above function can be simpli-fied as follows

lg119884 (119905) = lg 1198961 minus120573 lg 119905 (11)

The fractional Maxwell models are fitted by (11) and thefitting results are shown in Figure 12 and Table 4

Advances in Materials Science and Engineering 7

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 10 Predictions of the generalized Maxwell model (five-component model)

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 11 Predictions of the generalized Maxwell model (seven-component model)

424 Fractional ExponentialModel The fractional exponen-tial models are fitted by self-defined formulas and the fittingresults are shown in Figure 13 and Table 5

425 Burgers Model The Burgers model is fitted by self-defined formulas and the fitting results are shown in Figure 14and Table 5

From the above all the above viscoelastic models candescribe the stress-relaxation behaviors of PTFE coated fab-rics under different temperatures However the expressionsof classic Maxwell models are relatively simple comparedwith the other models They have strict guidelines forthe application of the ordinary differential equations indescribing the complex constitutive relations They can onlydescribe the simple trend of stress relaxation and cannot

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 7: Research Article Experimental and Theoretical Research on

Advances in Materials Science and Engineering 7

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 10 Predictions of the generalized Maxwell model (five-component model)

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 11 Predictions of the generalized Maxwell model (seven-component model)

424 Fractional ExponentialModel The fractional exponen-tial models are fitted by self-defined formulas and the fittingresults are shown in Figure 13 and Table 5

425 Burgers Model The Burgers model is fitted by self-defined formulas and the fitting results are shown in Figure 14and Table 5

From the above all the above viscoelastic models candescribe the stress-relaxation behaviors of PTFE coated fab-rics under different temperatures However the expressionsof classic Maxwell models are relatively simple comparedwith the other models They have strict guidelines forthe application of the ordinary differential equations indescribing the complex constitutive relations They can onlydescribe the simple trend of stress relaxation and cannot

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 8: Research Article Experimental and Theoretical Research on

8 Advances in Materials Science and Engineering

Table 1 Parameter results of the generalized Maxwell model (three-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5683 5354 5794 5733 5892 5975 5433 5363 5778 61111198641 1443 1169 1109 1251 983 1394 1150 1241 1116 10831198642 1321 969 1187 1190 1237 1383 1320 973 1012 8721198643 1674 1232 1139 1074 1103 1831 1402 1144 1331 123311205911 005 563119864 minus 2 743119864 minus 2 437119864 minus 5 313119864 minus 5 362119864 minus 5 343119864 minus 5 219119864 minus 3 0081 778119864 minus 2

11205912 378119864 minus 5 416119864 minus 5 198119864 minus 3 237119864 minus 3 229119864 minus 3 127119864 minus 3 553119864 minus 2 448119864 minus 5 542119864 minus 5 104119864 minus 4

11205913 175119864 minus 3 205119864 minus 3 403119864 minus 5 0087 0082 0041 186119864 minus 3 0059 376119864 minus 3 368119864 minus 3

Table 2 Parameter results of the generalized Maxwell model (five-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5466 5281 5656 5692 5866 5928 5249 5263 5259 60861198641 313 249 260 233 217 287 326 261 274 2071198642 1152 629 794 743 759 1427 902 892 859 8001198643 874 725 727 1055 711 894 1004 591 811 8131198644 871 796 814 668 658 707 759 676 1182 6271198645 1221 864 820 731 892 1170 814 844 906 62411205911 021 022 034 043 044 024 018 021 041 03111205912 0019 296119864 minus 4 0002 00045 0047 0024 00019 0022 282119864 minus 8 004011205913 131119864 minus 4 196119864 minus 5 202119864 minus 4 299119864 minus 5 00049 364119864 minus 4 0018 209119864 minus 4 620119864 minus 5 0005511205914 10119864 minus 5 00028 145119864 minus 5 487119864 minus 4 736119864 minus 4 00026 157119864 minus 4 170119864 minus 5 0023 711119864 minus 4

11205915 00017 0024 0029 0043 242119864 minus 5 250119864 minus 5 986119864 minus 6 00022 00019 579119864 minus 5

give a more accurate description This aspect can also beseen in the application of Burgers model The Burgers modelcan reflect the stress-relaxation behaviors of PTFE coatedfabrics because it is composed of the Maxwell model andthe Kevin model To a certain extent it should reflectboth the stress-relaxation behavior and the creep behaviorHowever as we know the Kevin model can only reflectthe material creep behaviors and the Maxwell model canonly reflect the material stress-relaxation behaviors Whenpredicting the stress-relaxation behaviors the Burgers modelcan be degenerated into the classicMaxwellmodelThereforeit cannot make good prediction for the material stress-relaxation behaviors For the generalized Maxwell modelsthe equations are composed of classic Maxwell models andspring componentsMore parametersmake it easy to describethe stress-relaxation behaviors accurately Besides the finalmodulus is given which is the stable modulus when thetime is the infinity Therefore it can reflect the ldquomemoryrdquocharacteristics of coated fabrics and can predict the stress-relaxation behaviors The fractional Maxwell models andthe fractional exponential models can solve the predictionlimitation of ordinary differential equationsThey can achievethe interpolation calculation between elastic behaviors andviscous behaviors Therefore the fractional Maxwell modelsand the fractional exponential models are more suitable fordescribing the stress-relaxation behaviors of PTFE coatedfabrics

5 Conclusions

(1) The PTFE coated fabrics are typically viscoelastic Thestress decreases obviously in the initial state and it hascompleted 80 of the total relaxation in the first threehours The decrease rate decreases with time increasing andfinally the stress gradually reaches a stable value The stress-time curves are nonlinear and there is a linear relationshipbetween the logarithm of relaxationmodulus and timeThereare no significant differences between the behaviors of warpand weft

(2) The changing of temperature has few effects on thecurves of stress relaxation and relaxation modulus Withtemperature increasing the rate of stress relaxation is fasterand it is easy to reach the stable value With temperatureincreasing the relaxation modulus increases and the finalstable value increases This is consistent with the behaviorsunder cyclic loading in previous references which may berelated with the properties of glass fibers

(3)The classic Maxwell model cannot make good predic-tion for the material stress-relaxation behaviors due to fewerparameters From the expressions it can be seen that theBurgers model can reflect the stress-relaxation behaviors ofPTFE coated fabrics because the Burgers model is composedof the Maxwell model and the Kevin model However whenusing the Burgers model to predict the stress-relaxationbehaviors its expression is very close to the classic Maxwell

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 9: Research Article Experimental and Theoretical Research on

Advances in Materials Science and Engineering 9

Table 3 Parameter results of the generalized Maxwell model (seven-component model)

Warp Weft23∘C 40∘C 50∘C 60∘C 70∘C 23∘C 40∘C 50∘C 60∘C 70∘C

119864119890

5363 5380 5547 5219 3541 3356 3313 5113 3371 33271198641 177 211 146 224 219 287 205 141 274 1951198642 680 760 535 716 1124 882 558 610 807 6411198643 494 357 677 578 763 1424 649 564 839 5861198644 951 646 597 710 1148 691 698 656 856 7811198645 845 762 551 648 649 1113 2184 523 853 0721198646 717 673 526 842 721 1301 721 502 906 7681198647 823 959119864 minus 8 631 395 936 1358 860 549 1182 80011205911 037 026 069 045 044 024 029 037 041 03311205912 0060 00048 010 00053 minus11119864 minus 7 388119864 minus 4 136119864 minus 5 00025 624119864 minus 5 883119864 minus 4

11205913 398119864 minus 4 419119864 minus 7 761119864 minus 6 minus16119864 minus 7 0046 0024 128119864 minus 4 0066 421119864 minus 7 768119864 minus 5

11205914 00019 459119864 minus 5 00025 0048 minus11119864 minus 7 00027 00011 0013 412119864 minus 7 668119864 minus 8

11205915 0012 0034 637119864 minus 5 633119864 minus 4 701119864 minus 4 286119864 minus 5 392119864 minus 7 583119864 minus 6 407119864 minus 7 247119864 minus 5

11205916 711119864 minus 5 602119864 minus 4 407119864 minus 4 201119864 minus 5 00048 170119864 minus 7 0047 483119864 minus 4 00019 004411205917 649119864 minus 6 minus593 0016 717119864 minus 5 222119864 minus 5 172119864 minus 7 00073 561119864 minus 5 0023 00065

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

minus2 0 2 4 6

lg(120590

(kN

m))

lg(ts)

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

06

05

04

03

0 2 4 6

lg(120590

(kN

m))

lg(ts)

(b) Weft

Figure 12 Prediction of the fractional Maxwell model

Table 4 Parameter results of the fractional Maxwell model

Temperature Warp Weft1198961 120573 1198961 120573

23∘C 40214 004896 39919 00479840∘C 39836 004176 39996 00451250∘C 39389 003832 39811 00418360∘C 39452 003903 39033 00399070∘C 39006 003639 39312 003751

model Therefore it cannot make good prediction of stress-relaxation behaviors of PTFE coated fabrics under differenttemperatures

(4) For the generalized Maxwell models all three modelsincluding three-component model five-component modeland seven-component model can make good predictionsfor the material stress-relaxation behaviors Among themthe prediction accuracy of the seven-component model isthe best which indicates that with equation parameterincreasing the prediction accuracy of fitting results increases

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 10: Research Article Experimental and Theoretical Research on

10 Advances in Materials Science and Engineering

Table 5 Parameter results of the Burgers model

Warp Weft120572 120573 1205782 1198641 1198642 120572 120573 1205782 1198641 1198642

23∘C 00048 121119864 minus 6 1724 9333 0058 00074 142119864 minus 6 1389 8063 007240∘C 00062 960119864 minus 7 1471 8026 0067 00059 105119864 minus 6 1538 8441 006450∘C 00054 111119864 minus 6 1563 8841 0064 108119864 minus 6 00062 1493 8497 006760∘C 111119864 minus 6 00052 1587 8936 0062 863119864 minus 7 00081 minus1299 8988 minus0077

70∘C 00066 886119864 minus 7 1424 8031 0071 646119864 minus 7 00071 1370 9048 0072

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 13 Prediction of the fractional exponential model

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa)

100

90

80

70

60

50

(a) Warp

23∘C40∘C50∘C60∘C70∘C

23∘C40∘C50∘C60∘C70∘C

Time (s)0 100000 200000 300000

Rela

xatio

n m

odul

us (M

Pa) 100

110

90

80

70

60

50

(b) Weft

Figure 14 Prediction of the Burgers model

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 11: Research Article Experimental and Theoretical Research on

Advances in Materials Science and Engineering 11

(5) The fractional Maxwell model and the fractionalexponential model are all built by self-defined formulas andthe fitting results are good They can make good predictionof the final relaxation modulus and make worse predictionof the initial phase of the stress relaxation Further researchshould be imposed on proposing a more accurate model todescribe the whole phase of the stress relaxation

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by Covertex Membranes (Shanghai)Co Ltd National Natural Science Foundation of China(Grant no 51308532) the Fundamental Research Funds forthe Central Universities (Grant no 2015QNA57) and aProject Funded by the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutions

References

[1] B Forster and M Mollaert European Design Guide for TensileSurface Structures TensiNet 2004

[2] B N Bridgens P D Gosling and M J S Birchall ldquoMembranematerial behavior concepts practice amp developmentsrdquo TheStructural Engineer vol 82 pp 21ndash27 2004

[3] B Bridgens and M Birchall ldquoForm and function the signif-icance of material properties in the design of tensile fabricstructuresrdquo Engineering Structures vol 44 pp 1ndash12 2012

[4] A Ambroziak and P Kłosowski ldquoMechanical properties forpreliminary design of structuresmade fromPVC coated fabricrdquoConstruction and Building Materials vol 50 no 1 pp 74ndash812014

[5] Y Li and M Wu ldquoUniaxial creep property and viscoelasticndashplastic modelling of ethylene tetrafluoroethylene (ETFE) foilrdquoMechanics of Time-DependentMaterials vol 19 no 1 pp 21ndash342015

[6] W-R Yu M S Kim and J S Lee ldquoModeling of anisotropiccreep behavior of coated textile membranesrdquo Fibers and Poly-mers vol 7 no 2 pp 123ndash128 2006

[7] H Minami C Yamamoto S Segawa and Y Kono ldquoA methodfor measurement of stress-strain curves for elastic analysis onmembrane on the state after stress relaxation or creeprdquo ResearchReport on Membrane Structures The Membrane StructuresAssociation of Japan Tokyo Japan 1997

[8] S Kato T Yoshino T Ono et al ldquoStress-deformation of mem-brane structures based on the fabric lattice model consideringvisco-inelasticitymdashcomparison between the experimental andanalytical resultsrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1997

[9] B Maurin and R Motro ldquoCutting pattern of fabric membraneswith the stress composition methodrdquo International Journal ofSpace Structures vol 14 no 2 pp 121ndash129 1999

[10] H Tsubota A Yoshida and Y Kurokawa ldquoTheoretical analysisof actual initial equilibrium state for membrane structuresbased on cutting patternrdquo in Proceedings of the IASS Symposium

pp 656ndash674 International Association for Shell and SpatialStructures Karnataka India 1988

[11] S Kato and T Yoshino ldquoSimulation for introducing tensionsinto curved membranes considering both of the cutting patternmethod and visco-elasto-plastic characteristics of the fabricsrdquoin Proceedings of the IASS Symposium pp 110ndash111 InternationalAssociation for Shell and Spatial Structures Nagoya Japan2001

[12] T Kotake M Kikushima and K Nishikawa ldquoA study oncompensation of membrane material in consideration of fab-ric characteristicrdquo Research Report on Membrane StructuresMembrane Structures Association of Japan Tokyo Japan 1996

[13] S Ishitoku K Tanaka K Takahashi and N Ataka ldquoDetermin-ing mechanical properties of membranes by uniaxial testingmethodrdquo Research Report on Membrane Structuresrsquo90 TheMembrane Structures Association of Japan Tokyo Japan 1990

[14] J Fujiwara M Ohsaki and K Uetani ldquoCutting pattern designofmembrane structures considering viscoelasticity of materialrdquoin Proceedings of the IASS Symposium on Theory Design andRealization of Shell and Spatial Structures pp 112ndash113 Inter-national Association for Shell and Spatial Structures NagoyaJapan October 2001

[15] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report on Membrane Structures TheMembrane Structures Association of Japan Tokyo Japan 2006

[16] H Nakajima M Saitoh and A Okada ldquoStructure analyticalstudy on stress relaxation in tensile membrane structures-structural principle on tensile membrane structure with spring-strut systemrdquo Research Report onMembrane Structures Mem-brane Structures Association of Japan Tokyo Japan 2007

[17] S Kato H Minami T Yoshino and T Namita ldquoVisco-inelasticconstitutive equations for fabric membranes based on fabriclattice model-simulations for creep and relaxation comparedwith experimentsrdquo Research Report on Membrane StructuresThe Membrane Structures Association of Japan Tokyo Japan1996

[18] H Minami ldquoA multi-step linear approximation method fornonlinear analysis of stress and deformation of coated plain-weave fabricrdquo Journal of Textile Engineering vol 52 no 5 pp189ndash195 2006

[19] J Argyris I S Doltsinis and V D da Silva ldquoConstitutive mod-elling and computation of non-linear viscoelastic solids PartII application to orthotropic PVC-coated fabricsrdquo ComputerMethods in Applied Mechanics and Engineering vol 98 no 2pp 159ndash226 1992

[20] J Minte Das mechanische Verhalten von Verbindungenbeschichteter Chemiefasergewebe [Dissertation] RWTHAachenUniversity 1981

[21] Y Zhang Q Zhang C Zhou and Y Zhou ldquoMechanical prop-erties of PTFE coated fabricsrdquo Journal of Reinforced Plastics andComposites vol 29 no 24 pp 3624ndash3630 2010

[22] H P Wang X W Chen X J Li L T Ma and B Su ldquoTensilestrength distribution of glass fiber reinforced composites atdifferent temperaturesrdquoMaterials Engineering no 7 pp 76ndash782008

[23] A Ambroziak and P Kosowski ldquoInfluence of thermal effectson mechanical properties of PVDF-coated fabricrdquo Journal ofReinforced Plastics and Composites vol 33 no 7 pp 663ndash6732014

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 12: Research Article Experimental and Theoretical Research on

12 Advances in Materials Science and Engineering

[24] H Schiessel R Metzler A Blumen and T F NonnenmacherldquoGeneralized viscoelastic models their fractional equationswith solutionsrdquo Journal of Physics A General Physics vol 28 no23 pp 6567ndash6584 1995

[25] R M Christensen and L B Freund ldquoTheory of viscoelasticityrdquoJournal of Applied Mechanics vol 38 no 3 p 720 1971

[26] W M Zhang ldquoPractical expressions of relaxation modulusand creep compliancerdquo Natural Science Journal of XiangtanUniversity vol 21 no 3 pp 26ndash28 1999

[27] R L Taylor K S Pister and G L Goudreau ldquoThermome-chanical analysis of viscoelastic solidsrdquo International Journal forNumerical Methods in Engineering vol 2 no 1 pp 45ndash59 1970

[28] Y Y Zhang Q L Zhang K Lei and B L Kuai ldquoExperimentalanalysis of tensile behaviors of polytetrafluoroethylene-coatedfabrics subjected to monotonous and cyclic loadingrdquo TextileResearch Journal vol 84 no 3 pp 231ndash245 2014

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 13: Research Article Experimental and Theoretical Research on

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials