ijcst mechanicalpropertiesoffabric materials for draping...

Mechanical properties of fabricmaterials for draping

simulationZ Wu

Department of Mechanical Engineering Hong Kong Universityof Science and Technology Kowloon Hong Kong

Peoplersquos Republic of China

CK AuSchool of Mechanical and Production Engineering

Nanyang Technological University Singapore

Matthew YuenDepartment of Mechanical Engineering Hong Kong University

of Science and Technology Kowloon Hong KongPeoplersquos Republic of China

Keywords Fabrics Drape Mechanical properties

Abstract Most of the cloth simulation and modelling techniques rely on the energy function of thesystem The geometric deformation is related to the energy function by the fabric materialcharacteristics which are usually difregcult to measure directly This paper discusses how the fabricmaterial properties are related to the measurable mechanical properties of the fabric such as tensilemodulus Poissonrsquos ratio etc These properties are incorporated into a cloth simulator to producedraping results The simulated image and real object are then compared to show the realism

IntroductionCloth modelling has received considerable attention recently The applicationsof this modelling technique are mainly for graphic and engineering purposeBoth visual realism and physical accuracy are of equal interest in these tworegelds A lot of researches about cloth modelling have been conducted whichmainly focuses on developing physically based models for the cloth objectThese models can be classireged into two categories the continuum model anddiscrete model For continuum model the fabric is considered as continuousobject with mass and energies distributed throughout The governingequations are derived from the variational principle For the discrete model theobject is modelled as a collection of point masses with some relations amongeach other A cloth model can be continuous or discrete but the computationalmethods are ultimately discrete (Gibson and Miritich 1997) In addition thesemodels always possess certain physical quantities particularly thedeformation energy which is deregned to afford simulation algorithms

The Emerald Research Register for this journal is available at The current issue and full text archive of this journal is available at

httpwwwemeraldinsightcomresearchregister httpwwwemeraldinsightcom0955-6222htm

IJCST151

56

Received June 2002Accepted October 2002

International Journal of ClothingScience and TechnologyVol 15 No 1 2003pp 56-68

q MCB UP Limited0955-6222DOI 10110809556220310461169

Literature review for cloth modellingThe deformable surface proposed by Teropoulos and Fleischer (1988) is atypical continuum model for the cloth Furthermore various approaches(Chen et al 1999 Teropoulos and Fleischer 1988 Volino et al 1995) arediscussed to deregne the potential energy of a deformable model The regnalequilibrium state of the model is obtained by minimizing the potential energywith respect to the material displacements Finite element method (Eischen andClapp 1996 Tan et al 1999) is also used to regnd an approximation for acontinuous function that satisreges the equilibrium expression

Particle system (Breen et al 1992 1994 Eberhardt et al 1996) is a typicalexample of the discrete model for modelling of cloth The cloth is representedby a set of particles The energies of each particle are deregned The regnalequilibrium shape of the cloth occurs at the minimum energy of the wholeparticle system Mass-spring system is another example of the discrete modelfor cloth object (Howelett 1997 Provot 1995) The cloth is modelled as acollection of point masses connected by springs in a lattice structure This is asigniregcant approximation of the true physics that occurs in the cloth Thecurrent status in cloth modelling and animation research can be found inVolino and Magnenat-Thalmann (2000)

Recently cloth modelling tends to focus on two major areas

(1) Incorporating fabric property into the cloth model to give an accurateand realistic simulation (Breen et al 1992 1994 Eberhardt et al 1996)

(2) Improving the computational efregciency to give a fast simulation(Baraff and Witkin 1998)

Both are crucial in simulating the cloth since the demand of accuracy realismand fast computation is increasing for engineering and graphic animation

ObjectiveIt can be seen that most of the cloth modelling techniques rely on the energyfunction of the system The geometric deformation is related to the energyfunction by the fabric material characteristics which are usually difregcult tomeasure directly

Kawabata Evaluation system (Kawabata 1975) is a common equipment tomeasure the mechanical properties of a fabric Breen et al (1994) proposed amethod to derive the energy equations of the cloth model from fabricmeasurement data produced by the Kawabata Evaluation System Firstlya function is determined to approximate the Kawabata plots Then theapproximate function is related to the energy function of the model Lastly theresulting equations are scaled to produce energy values in standard physicalunits Eischen and Bigliani (2000) used a regfth order polynomial to approximatethe Kawabata plot and regtted the approximation into the constitutive equationsto obtain the elastic constants of the fabric

Mechanicalproperties of

fabric materials

57

This paper discusses the characteristics of the fabric material in terms of aset of measurable mechanical properties These properties are incorporated intoa continuum model for draping simulation and the results are then comparedwith the actual situations

Cloth draping simulationFigure 1 shows the pseudo-codes of a cloth draping simulator based onTeropoulos and Fleischer (1988) continuum model The state of a cloth elementis described by its position x and velocity Ccedilx Based on this state the clothdeformation is deregned For instance the metric and curvature tensors are thetypical deformation measurements The behaviours of the cloth arecharacterized by the strain energy due to the deformation Furthermoreinternal forces are induced because of this strain energy Applying thefundamental laws of dynamics such as Newtonrsquos second law of motion on theseelements the acceleration x is obtained which is used to calculate the new stateof the element for the next time step through numerical integration Once thenew state is computed collision detection is performed Basically there are twotypes of collision detection called cloth self-collision and cloth object collisionIf the collision is detected at the new state the new state will be given up

Figure 1The pseudo codes of acloth draping simulator

IJCST151

58

and collision responses are imposed as constraints at that instant Bothcollision detection and response are implemented as separate module in thesimulator for easy software management

Mechanical properties of fabric materialThe mechanical properties of a fabric material are measured by the KawabataEvaluation System under the assumption that fabric is anisotropic andorthotropic in warp and weft directions Three basic tests are performedtensile shear and bending test Figure 2 shows the plotting of the tensile andbending test of a fabric sample with 60 per cent wool and 40 per cent polyester

The mechanical behaviours of the cloth are governed by the strain energyaccumulated due to deformation Two main deformations occur in clothdraping metric deformation due to in-plane forces and curvature deformationdue to out-of-plane forces The strain energy density U is written as

U = 1=2

XlabC abdgldg (1)

where lab is the strain tensor due to deformation and C abgd is the elasticmodulus tensors a b g and d are indices to denote the directions of theprinciple axes

Pis the aggregation of the strain energies due to metric and

curvature deformationThe elastic modulus tensors C abdg are symmetric tensors hence

C abdg = C abgd = C badg = C dgab (2)

When the warp and weft directions coincide with the principle coordinatesystem of the fabric C 1112 = C 1211 = C 1222 = C 2212 = 0 and only regvecomponents of the tensor C 1111 C 1122 C 1212 C 2211 and C 2222 are non-zero

From Hookrsquos law

s11

s22

s12

2

64

3

75 =1

1 2 u1u2

D1 u2D1 0

u1D2 D2 0

0 0 Ds(1 2 u1u2)

2

64

3

75

e11

e22

2e12

2

64

3

75 (3)

t11

t22

t12

2

664

3

775 =1

1 2 m 1m 2

H 1 m 2H 1 0

m 1H 2 H 2 0

0 0 H s(1 2 m 1m 2)

2

664

3

775

k11

k22

2k12

2

664

3

775 (4)

where s and t are the stress due to metric and curvature deformationrespectively e and k are the strain due to metric and curvature deformationrespectively D1 D2 are the tensile modulus in the principal axes Ds are theshear modulus n1 n2 are Poissonrsquos ratio H1 H2 are the bending modulus in


fabric materials

59

Figure 2Tensile and bending testresult from theKawabata EvaluationSystem

IJCST151

60

the principal axes Hs are the twisting modulus m 1 m2 are physical quantitiesanalogue to the Poissonrsquos ratios

n1D2 = n2D1 (5)

and

m 1H 2 = m 2H 1 (6)

Hence the material modulus tensors can be expressed in terms of a set ofmeasurable quantities

C1111G = D1

1 2 u1u2

C1122G = C2211

G = u2D1

1 2 u1u2

C2222G = D2

1 2 u1u2

C1212G = Ds

9gtgtgtgtgtgt=

gtgtgtgtgtgt

(7)

C1111B = H 1

1 2 m 1m 2

C1122B = C2211

B = m 2H 1

1 2 m 1m 2

C2222B = H 2

1 2 m 1m 2

C1212B = H s

9gtgtgtgtgtgtgt=

gtgtgtgtgtgtgt

(8)

where the CG and CB are the elastic modulus tensors due to metric andcurvature deformation respectively

If the cloth is stretched in a direction making an angle y with one of theprincipal axis direction the tensile strain Ey is related to the tensile stress sy as

Dy =sy

ey(9)

Similarly the bending strain ky and the bending moment ty can beexpressed as

Hy =ty

ky(10)

According to the transformation laws of tensor

ey = [cos2y sin2y siny cosy]

e11

e22

2e12

2

664

3

775 (11)


fabric materials

61

s11

s22

s12

2

6664

3

7775 =

sycos2y

sysin2y

2 sycosy siny

2

6664

3

7775 (12)

and

ky = [cos2y sin2y siny cosy]

k11

k22

2k12

2

6664

3

7775 (13)

t11

t22

t12

2

664

3

775 =

t ycos2y

tysin2y

2 t ycosy siny

2

664

3

775 (14)

Derivation of equations (9)-(14) can be found in most common strength ofmaterials literatures (Ryder 1973)

Substituting equations (11)-(14) into equations (9) and (10) and combiningwith equations (3) and (4) yields the tensile modulus and the bending modulusin this direction

1

Dy=

1

D1cos4y +

1

Ds2

n 1

D12

n 2

D2

sup3 acutecos2y sin2y +

1

D2sin4y (15)

1

H y=

1

H 1cos4y +

1

H s2

m 1

H 12

m 2

H 2


1

H 2sin4y (16)

Since the ordfPoisson ratioordm of bending m 1 and m 2 are usually small they can beneglected Taking the angle y = 458 the tensile Poisson ratio and the twistingmodulus are written as

n 1 =1

2D1

1

D1+

1

D2+

1

Ds2

4

D45 8

sup3 acute n 2 = D2

n 1

D1(17)

IJCST151

62

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)

Hence the mechanical properties of modulus D1 D2 Ds H1 H2 and Hs can bemeasured directly by the Kawabata Evaluation System Other parameters suchas Poissonrsquos ratio and twisting modulus are obtained from the tensile shearand bending modulus by using equations (17) and (18)

Simulation examplesA set of examples is used to illustrate the effects on cloth draping simulationwith the incorporation of the mechanical properties of the fabrics

Example 1 A piece of cloth falling on a sphereThe mechanical properties of the fabric are listed in Table I The size of thecloth is 1 pound 1 m with meshing size of 50 pound 50 nodes and the radius of the sphereis 0125 m The animation process is shown in Figure 3

Example 2 Cloth draping over a sphere with various fabric materialsThe bending modulus of the cloth is changed to show the draping effectsThree cases are performed

Case 1 the bending modulus is enlarged by 5 timesCase 2 the original bending modulus is usedCase 3 the bending modulus is reduced to 15 of the original values

The draping behaviour is shown in Figure 4 The more rigid fabric in case 1gives a 4-wrinkle draping mode while the softer fabric generates an 8-wrinkledraping mode

Example 3 Cloth draping over a sphere with various fabric materialsTwo pieces of cloth with different fabric materials draping over a sphere isshown in Figure 5 The results are compared with the real images The size ofthe cloth is 06 pound 06 m and the radius of the sphere is 009 m The propertiesare listed in Table II

Example 4 Table cloth draping with various fabric materialsThe same simulation in example 3 is performed by using a cloth with varyingfabric materials The mechanical properties are listed in Table III

Each simulation is then compared with the appearance of real table clothsituations as shown in Figure 6

Example 5 Dresses with various fabric materialsThree typical fabric materials cotton polyester and silk are listed in Table IVThe results are simulated with a mannequin as shown in Figure 7

r kgm2 D1 Nm D2 Nm Ds Nm H1 mNm H2 mNm Hs mNm n1 n2

0125 3657 3627 368 507 441 155 0096 0086

Table IMaterial properties

of cotton and rayon


fabric materials

63

Figure 4Draping behaviours withvarious fabric materials

Figure 3The animation of clothfalling on a sphere

IJCST151

64

Figure 5Different types of cotton

cloth draping over asphere

Materialsample r kgm2 D1 Nm D2 Nm Ds Nm H1 mNm H2 mNm Hs mNm n1 n2

Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical

properties of twodifferent cotton

samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165

Table IIIMaterial properties

of cotton and rayon


fabric materials

65

DiscussionFabric is made from threads in woven or knitted patterns with the structuresresulting from different weaving or knitting techniques Due to the complexityof the fabric microstructure it is necessary and practical to treat the fabric asan engineering material in draping modelling One of the feasible solutions is to

Figure 6Table cloth simulationwith different fabricmaterials


Cotton 0208 3475 2865 191 176 127 27 0215 0177Polyester 0212 3071 1823 467 109 90 231 0264 0169Silk 0076 1388 827 124 081 076 0089 0285 0170

Table IVMaterial propertiesof cotton polyesterand silk

IJCST151

66

assume the fabric to be continuous elastic in both the modelling and theexperimental aspects Hence it is equivalent to modelling the draping of afabric sheet Fabric material exhibits the orthogonal anisotropy behaviourswhich leads to different mechanical properties in the weft and warp directionFigure 8 shows the tensile modulus of cloth The mechanical properties of thefabric are considered in three different directions the weft and warp directionsand an angle of 45 8 from one of the weft direction Once these properties aremeasured and calculated they can be inserted into the cloth model for drapingsimulation

Figure 7Dresses simulation withvarious fabric materials

Figure 8The tensile modulus of

cloth


fabric materials

67

References

Baraff D and Witkin D (1998) ordfLarge steps in cloth simulationordm SIGGRAPHrsquo98 ComputerGraphics Proceedings Annual Conference Series pp 43-54

Breen DE House DH and Getto PH (1992) ordfA physically-based particle model of wovenclothordm Visual Computer Vol 8 No 5-6 pp 264-77

Breen DE House DH and Wozny MJ (1994) ordfPredicting the drape of woven cloth usinginteracting particlesordm Computer Graphics (SIGGRAPHrsquo94 Proceedings) Addison-WesleyReading MA pp 365-72

Chen MX Wu Z Sun QP and Yuen MMF (2003) ordfA wrinkled membrane model for clothdraping with multigrid accelerationordm Journal of Manufacturing Science and Engineering(in press)

Eberhardt B Weber A and StrasserW (1996) ordfA fast macrexible particle-system model for clothdrapingordm IEEE Computer Graphics and Applications pp 52-9

Eischen JW and Bigliani R (2000) ordfContinuum versus particle representationordm in House DHand Breen DE (Eds) Cloth Modeling and Animation AK Peters pp 79-122

Eischen JW and Clapp TG (1996) ordfFinite-element modelling and control of macrexible fabricpartsordm IEEE Computer Graphics and Applications pp 71-80

Gibson SFF and Miritich B (1997) ordfA survey of deformable modeling in computer graphicsordmTechnical report TR-97-19 Mitsubish Electric Research Laboratory

Howelett P (1997) ordfCloth simulation using mass-spring networksordm MSc dissertationDepartment of Computer Science University of Manchester

Kawabata S (1975) ordfThe standardization and analysis of hand evaluationordm Hand Evaluationand Standardization Committee of the Textile Machinery Society of Japan Osaka

Provot X (1995) ordfDeformation constraints in a mass-spring model to describe rigid clothbehaviorordm Proceedings of Graphics Interface pp 147-54

Ryder GH (1973) Strength of Materials ELBS and Macmillan NY

Tan ST Wong TN Zhao YF and Chen WJ (1999) ordfA constrained regnite element method formodelling cloth deformationordm Visual Computer Vol 15 No 2 pp 90-9

Teropoulos D and Fleischer K (1988) ordfDeformable modelsordm Visual Computer Vol 4 No 6pp 306-31

Volino P and Magnenat-Thalmann N (2000) Virtual Clothing Theory and Practice SpringerBerlin

Volino P Courchesne M and Thalmann NM (1995) ordfVersatile and efregcient techniques forsimulating cloth and other deformable objectsordm SIGGRAPHrsquo95 Computer GraphicsProceedings Annual Conference Series pp 137-44

IJCST151

68

Literature review for cloth modellingThe deformable surface proposed by Teropoulos and Fleischer (1988) is atypical continuum model for the cloth Furthermore various approaches(Chen et al 1999 Teropoulos and Fleischer 1988 Volino et al 1995) arediscussed to deregne the potential energy of a deformable model The regnalequilibrium state of the model is obtained by minimizing the potential energywith respect to the material displacements Finite element method (Eischen andClapp 1996 Tan et al 1999) is also used to regnd an approximation for acontinuous function that satisreges the equilibrium expression

Particle system (Breen et al 1992 1994 Eberhardt et al 1996) is a typicalexample of the discrete model for modelling of cloth The cloth is representedby a set of particles The energies of each particle are deregned The regnalequilibrium shape of the cloth occurs at the minimum energy of the wholeparticle system Mass-spring system is another example of the discrete modelfor cloth object (Howelett 1997 Provot 1995) The cloth is modelled as acollection of point masses connected by springs in a lattice structure This is asigniregcant approximation of the true physics that occurs in the cloth Thecurrent status in cloth modelling and animation research can be found inVolino and Magnenat-Thalmann (2000)

Recently cloth modelling tends to focus on two major areas

(1) Incorporating fabric property into the cloth model to give an accurateand realistic simulation (Breen et al 1992 1994 Eberhardt et al 1996)

(2) Improving the computational efregciency to give a fast simulation(Baraff and Witkin 1998)

Both are crucial in simulating the cloth since the demand of accuracy realismand fast computation is increasing for engineering and graphic animation

ObjectiveIt can be seen that most of the cloth modelling techniques rely on the energyfunction of the system The geometric deformation is related to the energyfunction by the fabric material characteristics which are usually difregcult tomeasure directly

Kawabata Evaluation system (Kawabata 1975) is a common equipment tomeasure the mechanical properties of a fabric Breen et al (1994) proposed amethod to derive the energy equations of the cloth model from fabricmeasurement data produced by the Kawabata Evaluation System Firstlya function is determined to approximate the Kawabata plots Then theapproximate function is related to the energy function of the model Lastly theresulting equations are scaled to produce energy values in standard physicalunits Eischen and Bigliani (2000) used a regfth order polynomial to approximatethe Kawabata plot and regtted the approximation into the constitutive equationsto obtain the elastic constants of the fabric


fabric materials

57




IJCST151

58




U = 1=2

XlabC abdgldg (1)






From Hookrsquos law

s11

s22

s12

2

64

3

75 =1

1 2 u1u2

D1 u2D1 0

u1D2 D2 0

0 0 Ds(1 2 u1u2)

2

64

3

75

e11

e22

2e12

2

64

3

75 (3)

t11

t22

t12

2

664

3

775 =1

1 2 m 1m 2

H 1 m 2H 1 0

m 1H 2 H 2 0

0 0 H s(1 2 m 1m 2)

2

664

3

775

k11

k22

2k12

2

664

3

775 (4)



fabric materials

59


IJCST151

60


n1D2 = n2D1 (5)

and

m 1H 2 = m 2H 1 (6)


C1111G = D1

1 2 u1u2

C1122G = C2211

G = u2D1

1 2 u1u2

C2222G = D2

1 2 u1u2

C1212G = Ds

9gtgtgtgtgtgt=

gtgtgtgtgtgt

(7)

C1111B = H 1

1 2 m 1m 2

C1122B = C2211

B = m 2H 1

1 2 m 1m 2

C2222B = H 2

1 2 m 1m 2

C1212B = H s

9gtgtgtgtgtgtgt=

gtgtgtgtgtgtgt

(8)



Dy =sy

ey(9)


Hy =ty

ky(10)



e11

e22

2e12

2

664

3

775 (11)


fabric materials

61

s11

s22

s12

2

6664

3

7775 =

sycos2y

sysin2y

2 sycosy siny

2

6664

3

7775 (12)

and


k11

k22

2k12

2

6664

3

7775 (13)

t11

t22

t12

2

664

3

775 =

t ycos2y

tysin2y

2 t ycosy siny

2

664

3

775 (14)



1

Dy=

1

D1cos4y +

1

Ds2

n 1

D12

n 2

D2


1

D2sin4y (15)

1

H y=

1

H 1cos4y +

1

H s2

m 1

H 12

m 2

H 2


1

H 2sin4y (16)


n 1 =1

2D1

1

D1+

1

D2+

1

Ds2

4

D45 8

sup3 acute n 2 = D2

n 1

D1(17)

IJCST151

62

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)












0125 3657 3627 368 507 441 155 0096 0086


of cotton and rayon


fabric materials

63



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68




IJCST151

58




U = 1=2

XlabC abdgldg (1)






From Hookrsquos law

s11

s22

s12

2

64

3

75 =1

1 2 u1u2

D1 u2D1 0

u1D2 D2 0

0 0 Ds(1 2 u1u2)

2

64

3

75

e11

e22

2e12

2

64

3

75 (3)

t11

t22

t12

2

664

3

775 =1

1 2 m 1m 2

H 1 m 2H 1 0

m 1H 2 H 2 0

0 0 H s(1 2 m 1m 2)

2

664

3

775

k11

k22

2k12

2

664

3

775 (4)



fabric materials

59


IJCST151

60


n1D2 = n2D1 (5)

and

m 1H 2 = m 2H 1 (6)


C1111G = D1

1 2 u1u2

C1122G = C2211

G = u2D1

1 2 u1u2

C2222G = D2

1 2 u1u2

C1212G = Ds

9gtgtgtgtgtgt=

gtgtgtgtgtgt

(7)

C1111B = H 1

1 2 m 1m 2

C1122B = C2211

B = m 2H 1

1 2 m 1m 2

C2222B = H 2

1 2 m 1m 2

C1212B = H s

9gtgtgtgtgtgtgt=

gtgtgtgtgtgtgt

(8)



Dy =sy

ey(9)


Hy =ty

ky(10)



e11

e22

2e12

2

664

3

775 (11)


fabric materials

61

s11

s22

s12

2

6664

3

7775 =

sycos2y

sysin2y

2 sycosy siny

2

6664

3

7775 (12)

and


k11

k22

2k12

2

6664

3

7775 (13)

t11

t22

t12

2

664

3

775 =

t ycos2y

tysin2y

2 t ycosy siny

2

664

3

775 (14)



1

Dy=

1

D1cos4y +

1

Ds2

n 1

D12

n 2

D2


1

D2sin4y (15)

1

H y=

1

H 1cos4y +

1

H s2

m 1

H 12

m 2

H 2


1

H 2sin4y (16)


n 1 =1

2D1

1

D1+

1

D2+

1

Ds2

4

D45 8

sup3 acute n 2 = D2

n 1

D1(17)

IJCST151

62

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)












0125 3657 3627 368 507 441 155 0096 0086


of cotton and rayon


fabric materials

63



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68




U = 1=2

XlabC abdgldg (1)






From Hookrsquos law

s11

s22

s12

2

64

3

75 =1

1 2 u1u2

D1 u2D1 0

u1D2 D2 0

0 0 Ds(1 2 u1u2)

2

64

3

75

e11

e22

2e12

2

64

3

75 (3)

t11

t22

t12

2

664

3

775 =1

1 2 m 1m 2

H 1 m 2H 1 0

m 1H 2 H 2 0

0 0 H s(1 2 m 1m 2)

2

664

3

775

k11

k22

2k12

2

664

3

775 (4)



fabric materials

59


IJCST151

60


n1D2 = n2D1 (5)

and

m 1H 2 = m 2H 1 (6)


C1111G = D1

1 2 u1u2

C1122G = C2211

G = u2D1

1 2 u1u2

C2222G = D2

1 2 u1u2

C1212G = Ds

9gtgtgtgtgtgt=

gtgtgtgtgtgt

(7)

C1111B = H 1

1 2 m 1m 2

C1122B = C2211

B = m 2H 1

1 2 m 1m 2

C2222B = H 2

1 2 m 1m 2

C1212B = H s

9gtgtgtgtgtgtgt=

gtgtgtgtgtgtgt

(8)



Dy =sy

ey(9)


Hy =ty

ky(10)



e11

e22

2e12

2

664

3

775 (11)


fabric materials

61

s11

s22

s12

2

6664

3

7775 =

sycos2y

sysin2y

2 sycosy siny

2

6664

3

7775 (12)

and


k11

k22

2k12

2

6664

3

7775 (13)

t11

t22

t12

2

664

3

775 =

t ycos2y

tysin2y

2 t ycosy siny

2

664

3

775 (14)



1

Dy=

1

D1cos4y +

1

Ds2

n 1

D12

n 2

D2


1

D2sin4y (15)

1

H y=

1

H 1cos4y +

1

H s2

m 1

H 12

m 2

H 2


1

H 2sin4y (16)


n 1 =1

2D1

1

D1+

1

D2+

1

Ds2

4

D45 8

sup3 acute n 2 = D2

n 1

D1(17)

IJCST151

62

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)












0125 3657 3627 368 507 441 155 0096 0086


of cotton and rayon


fabric materials

63



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68


IJCST151

60


n1D2 = n2D1 (5)

and

m 1H 2 = m 2H 1 (6)


C1111G = D1

1 2 u1u2

C1122G = C2211

G = u2D1

1 2 u1u2

C2222G = D2

1 2 u1u2

C1212G = Ds

9gtgtgtgtgtgt=

gtgtgtgtgtgt

(7)

C1111B = H 1

1 2 m 1m 2

C1122B = C2211

B = m 2H 1

1 2 m 1m 2

C2222B = H 2

1 2 m 1m 2

C1212B = H s

9gtgtgtgtgtgtgt=

gtgtgtgtgtgtgt

(8)



Dy =sy

ey(9)


Hy =ty

ky(10)



e11

e22

2e12

2

664

3

775 (11)


fabric materials

61

s11

s22

s12

2

6664

3

7775 =

sycos2y

sysin2y

2 sycosy siny

2

6664

3

7775 (12)

and


k11

k22

2k12

2

6664

3

7775 (13)

t11

t22

t12

2

664

3

775 =

t ycos2y

tysin2y

2 t ycosy siny

2

664

3

775 (14)



1

Dy=

1

D1cos4y +

1

Ds2

n 1

D12

n 2

D2


1

D2sin4y (15)

1

H y=

1

H 1cos4y +

1

H s2

m 1

H 12

m 2

H 2


1

H 2sin4y (16)


n 1 =1

2D1

1

D1+

1

D2+

1

Ds2

4

D45 8

sup3 acute n 2 = D2

n 1

D1(17)

IJCST151

62

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)












0125 3657 3627 368 507 441 155 0096 0086


of cotton and rayon


fabric materials

63



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68


n1D2 = n2D1 (5)

and

m 1H 2 = m 2H 1 (6)


C1111G = D1

1 2 u1u2

C1122G = C2211

G = u2D1

1 2 u1u2

C2222G = D2

1 2 u1u2

C1212G = Ds

9gtgtgtgtgtgt=

gtgtgtgtgtgt

(7)

C1111B = H 1

1 2 m 1m 2

C1122B = C2211

B = m 2H 1

1 2 m 1m 2

C2222B = H 2

1 2 m 1m 2

C1212B = H s

9gtgtgtgtgtgtgt=

gtgtgtgtgtgtgt

(8)



Dy =sy

ey(9)


Hy =ty

ky(10)



e11

e22

2e12

2

664

3

775 (11)


fabric materials

61

s11

s22

s12

2

6664

3

7775 =

sycos2y

sysin2y

2 sycosy siny

2

6664

3

7775 (12)

and


k11

k22

2k12

2

6664

3

7775 (13)

t11

t22

t12

2

664

3

775 =

t ycos2y

tysin2y

2 t ycosy siny

2

664

3

775 (14)



1

Dy=

1

D1cos4y +

1

Ds2

n 1

D12

n 2

D2


1

D2sin4y (15)

1

H y=

1

H 1cos4y +

1

H s2

m 1

H 12

m 2

H 2


1

H 2sin4y (16)


n 1 =1

2D1

1

D1+

1

D2+

1

Ds2

4

D45 8

sup3 acute n 2 = D2

n 1

D1(17)

IJCST151

62

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)












0125 3657 3627 368 507 441 155 0096 0086


of cotton and rayon


fabric materials

63



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68

s11

s22

s12

2

6664

3

7775 =

sycos2y

sysin2y

2 sycosy siny

2

6664

3

7775 (12)

and


k11

k22

2k12

2

6664

3

7775 (13)

t11

t22

t12

2

664

3

775 =

t ycos2y

tysin2y

2 t ycosy siny

2

664

3

775 (14)



1

Dy=

1

D1cos4y +

1

Ds2

n 1

D12

n 2

D2


1

D2sin4y (15)

1

H y=

1

H 1cos4y +

1

H s2

m 1

H 12

m 2

H 2


1

H 2sin4y (16)


n 1 =1

2D1

1

D1+

1

D2+

1

Ds2

4

D45 8

sup3 acute n 2 = D2

n 1

D1(17)

IJCST151

62

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)












0125 3657 3627 368 507 441 155 0096 0086


of cotton and rayon


fabric materials

63



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68

H s =4

H 45 82

1

H 12

1

H 2

sup3 acute 2 1

(18)












0125 3657 3627 368 507 441 155 0096 0086


of cotton and rayon


fabric materials

63



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68



IJCST151

64




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68




Cotton 1 0218 3475 2865 191 877 614 1142 0215 0177Cotton 2 023 2405 5315 396 116 107 31 0185 0165

Table IIMechanical


samples


Cotton 0208 3475 2865 191 176 127 27 02150177

Rayon 0129 1847 3644 152 44 23 071 01850165


of cotton and rayon


fabric materials

65






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68






IJCST151

66




cloth


fabric materials

67

References

















IJCST151

68




cloth


fabric materials

67

References

















IJCST151

68

References

















IJCST151

68

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