Influence of uniaxial and biaxial tension on meso-scale ... · Influence of uniaxial and biaxial tension on meso-scale geometry and strain fields in a woven composite ... state

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Composite Structures 77 (2007) 405–418

Influence of uniaxial and biaxial tension on meso-scale geometryand strain fields in a woven composite

P. Potluri *, V.S. Thammandra

Textile Composites Group, Textiles and Paper School of Materials, University of Manchester, Manchester M60 1QD, United Kingdom

Abstract

Representative volume element (RVE) of a woven composite is often idealised to have identical geometry in the warp and weft direc-tions. While this may be true in the case of consolidating a completely relaxed preform with equal warp and weft densities, majority ofwoven composites have non-ideal RVE geometry. The present paper investigates the influence of membrane stresses – applied during themoulding process – on the mechanical properties of the finished composite. Crimp interchange due to uniaxial stress and tow flatteningdue to biaxial stress have been computed by a fabric compliance model developed by the authors. Compliance model is based on theprinciple of stationary potential energy by taking into consideration tow bending, compression and extensional energies, and the externalwork done by the tensile forces. It has been shown that a uniaxial tensile stress applied to the preform results in an increase in tensilemodulus and a corresponding reduction in the transverse modulus, as a result of reduced crimp in the loading direction with a corre-sponding increase in the transverse direction. Tensile strength and stiffness increase in the loading direction, and the mode of failure ini-tiation changes from ‘knee phenomena’ to tow failure. A biaxial stress applied during the forming stage results in crimp reduction in bothaxial and transverse directions; this leads to an improved tensile modulus and strength of a composite laminate.

RVE of the deformed geometry has been modelled using ABAQUS CAE pre-processor. The elastic moduli were computed using theconcept of equivalent strain energy proposed by Zhang and Harding. The resulting strain fields were compared for various cases of uni-axial and biaxial extension.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Woven composites; Fabric tension; Meso-scale; Tow geometry

1. Introduction

Biaxial woven, non-crimp and braided fabrics are popularreinforcements for moulding composite parts with complexdouble-curvatures. These fabrics are subjected to in-planetensile and shear forces, and out-of-plane compression andbending forces during moulding process. While there is alarge volume of literature on shear [1–4] and compression[5,6], very little is published on the tensile behaviour. Boisseet al. [7], Sagar et al. [8] presented tensile load-deformationbehaviour of plain woven fabrics. Bending is less criticalfor relatively thin 2D fabrics bent to modest curvatures,but may become significant for thicker 3D fabrics. The pres-

0263-8223/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruct.2006.10.005

* Corresponding author. Tel.: +44 161 200 4128; fax: +44 161 955 8128.E-mail address: [email protected] (P. Potluri).

ent paper deals with the load-deformation behaviour ofwoven fabrics subjected to thread-line tensions in uniaxialand biaxial configurations. The main objective here is tostudy the effect of yarn tensions on meso-scale geometry oftextile unit cell, which in turn affects the micro-mechanicalbehaviour of a composite RVE.

1.1. Woven fabric geometry

Woven fabrics are characterised by the interlacement ofwarp and weft yarns orthogonal to each other, resulting incrimp or waviness in both the directions. The crimp has asignificant influence on both the moduli and strength of acomposite structure. A multi-filament glass or carbon yarnis normally modelled with an elliptical (Fig. 1a) or a lentic-ular cross-section. In the case of uniaxial loading, crimp

mailto:[email protected]

Free zone

Contact Zone

h1/2h2/2 d2

d1

Yarn straightening in theloading direction

Yarn cr

impin

g and

jamm

ing in

tran

sver

se

direct

ion

Fig. 1. (a) Free and contact zones in a fabric, (b) crimp interchange.

406 P. Potluri, V.S. Thammandra / Composite Structures 77 (2007) 405–418

value reduces in the loading direction and increases in thetransverse direction. This phenomenon of crimp inter-change continues until yarns in the loading directionbecome straight or the yarns in the transverse directionreach a jammed state (Fig. 1b). In the case of biaxial load-ing with identical loading in two orthogonal directions,there is a slight reduction in the crimp in both directionsdue to yarn flattening. However, in the case of unequalbiaxial loading, crimp interchange does take place in thedirection of higher loading. In this paper, two limit cases,

θ01

1

R1

O

(x,y)X

Y

ac1

bc1

θ

Fig. 2. (a) Elliptical path, (b) tow centre line in conta

uniaxial and equi-biaxial (simply referred to as biaxial inthe rest of the paper), have been considered.

It can be seen from Fig. 1a that the interlacing yarn hastwo regions, contact zone and a free zone. The cross-sec-tional shape of the transverse yarns defines the geometryof a contact zone. The main geometric constraint for main-taining the contact between the interlacing yarns is that thesum of the crimp heights is equal to the sum of the yarnthickness.

h1 þ h2 ¼ b1 þ b2 ð1Þwhere h1, h2 are crimp heights and b1, b2 are yarn thickness,for warp and weft, respectively.

Using suffix ‘1’ to denote the warp yarn or warp direc-tion and suffix ‘2’ for the weft yarn or weft direction, theordinate of the ellipsoidal yarn path is given by (Fig. 2a)

yeiðxÞ ¼bci

aciðaci �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

ci � x2

qÞ; 0 6 x 6 xci ð2Þ

where

aci ¼ðaj þ biÞ

2bci ¼

ðbj þ biÞ2

for i ¼ 1 if j ¼ 2 and

� i ¼ 2 if j ¼ 1

The curvature of yarn path at any point along its length iscalculated using the following equation

q ¼ 1

R¼

d2ydx2

1þ dydx

� �2h i3

2

a2

b2

θ01

θ1

R

x1

p1/2

X

Y

OR

ct region, (c) polynomial geometry in free region.

P. Potluri, V.S. Thammandra / Composite Structures 77 (2007) 405–418 407

The radius of curvature at the end of contact zone can beexpressed as

Ri ¼a2

ciða2ci � x2

ciÞ þ b2cix

2ci

� �32

a4cibci

ð3Þ

where

xci ¼a2

ci sin hiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

ci sin2 hi þ b2ci cos2 hi

q ð4Þ

The horizontal projection of contact width (xci), radius ofcurvature (Ri) and slope of yarn path (hi) at the end of con-tact zone are related by Eqs. (3) and (4).

The yarn path in the free zone is represented by a fifth-order polynomial. The degree of polynomial depends onthe number of geometric boundary conditions. The poly-nomial has to satisfy the following five conditions, takingthe origin as the centre of the yarn length between intersec-tions (Fig. 2c):

dydx ¼ hi at x ¼ pi=2;dydx ¼ hi at x ¼ �pi=2;

q ¼ 0 at x ¼ 0;

q ¼ 1=Ri at x ¼ �pi=2;

q ¼ �1=Ri at x ¼ pi=2;

ð5Þ

The first two conditions enforce the continuity of slope atthe end of contact zone, the third condition enforces thepoint of inflexion to occur at the centre of yarn path, thelast two conditions enforce the continuity of curvature.Since five conditions are available, it is possible to definea fifth-degree polynomial.

ypiðxÞ ¼ c1xþ c2x2 þ c3x3 þ c4x4 þ c5x5

It has been found that the first two conditions do not leadto two independent equations. However, an additionalcondition of the slope at the inflexion point (h0i) has beenimposed to obtain the following condition:

@ypi

@x¼ h0i at x ¼ 0;

On substitution of the above conditions and by solving theresulting simultaneous equations, we can get the constantsc1 to c5 in terms of the variables.

c1i ¼ h0i

c2i ¼ 0

c3i ¼ð1þh2

i Þ32

3piRiþ 8

3hi�h0i

p2i

� ��

c4i ¼ 0

c5i ¼ � 45

ð1þh2i Þ

32

p3i Riþ 16

5ðhi�h0iÞ

p4i

� �

ð6Þ

The spacing, crimp heights and the curvilinear lengths ofyarns in the warp and weft directions are given by the fol-lowing expressions.

xi ¼ 2xci þ pi

hi ¼ 2yeiðxciÞ þ 2ypiðpi=2Þ

li ¼ 2R xci

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ dyei

dx

� �2q

dxþ 2R pi

2

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ dypi

dx

� �2r

dx; i ¼ 1; 2

ð7Þ

1.2. Tensile load-deformation behaviour

The yarn path, consisting of two elliptical sections and apolynomial, can be computed using the principle of statio-nery potential energy. It can be observed from the geomet-ric equations that there are eight variables namely b1, h1,h01, p1, b2, h2, h02 and p2. These variables can be foundby minimizing the total potential energy function of theunit cell. The total energy of the unit cell under biaxialloads is given by

V ¼ �F 1ðx1 � X 1Þ � F 2ðx2 � X 2Þ þ U e þ U b þ U c ð8Þwhere x1, x2 are deformed and X1,X2 are undeformed yarnspacings (hence, dimensions of the unit cell) in the warpand weft directions, respectively. The first two terms repre-sent the potential energy of external loads from the unde-formed configuration and the next three terms representthe strain energy stored in the yarns forming the unit celldue to tensile (Ue), bending (Ub) and compression (Uc).For relatively small tensile forces applied during forming,tensile energy may be neglected for an untwisted glass/car-bon fibre tow.

Ub is computed from the following equation using themoment–curvature (M–j) relationship of the yarn repre-sented by a function fb(j). The bending moment–curvaturerelationship has been obtained using Kawabata bendingtester [9]

Ub ¼Xi¼2

i¼1

Z li

0

Z ji

0

fbiðjÞdj

� �ds ð9Þ

Uc is computed from the following equation using the nor-mal load–compression strain (Fc–d) curve of the yarn rep-resented by a function fc(d). The load–compression curvehas been measured using Kawabata compression tester [9].

U c ¼Xi¼2

i¼1

Z di

0

fciðdÞdd

� �ð10Þ

The energy function given by Eq. (8) should be minimizedsubject to the following non-linear constraint that governsthe geometric constraint of plain weave fabric (Eq. (1)):h1 + h2 = b1 + b2.

The solution of the deformed state essentially involvesminimization of the potential energy function subject tothe geometric constraints involved. For this purpose, it iscomputationally efficient to utilise various algorithmsdeveloped in the field of mathematical programming forconstrained optimisation. In this work, NAG library rou-tine (nag-nlp-sol) has been employed to solve the con-strained minimisation problem [10].


Fig. 3 shows the yarn path when the fabric is in a relaxedstate. It may be noted that for computing relaxed yarnpath, bending energy is the only significant term in Eq.(8). Fig. 4 shows the yarn path under biaxial tension of

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.5 1 1.5 2 2.5

Distance along warp x (mm)

Ord

inat

e o

f ce

ntr

e lin

e y

(mm

)

Fig. 3. Geometry of centre line of yarn path under relaxed state.(a1 = a2 = 1.76 mm, b1 = b2 = 0.36 mm, p1 = p2 = 2.116 mm;h1 = h2 � = 0.0; h01 = h02 = 0/276).

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-1.25 -0.75 -0.25 0.25 0.75 1.25


Ord

inat

e o

f ce

ntr

e lin

e y

(mm

)

Undeformed

BiaxiallyExtended

Fig. 4. Yarn centre line under (equi-)biaxial deformation.

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-1.25 -0.75 -0.25 0.25 0.75 1.25


Ord

inat

e o

f ce

ntr

e lin

e y

(mm

)

warp-deformed

undeformed

weft-deformed

Fig. 5. Yarn geometry due to uniaxial loading.

20 N per yarn applied in both warp and weft directions.There is a slight reduction in crimp in both directionsdue to yarn flattening. Fig. 5 shows the yarn path underuniaxial loading of 20 N/yarn. It can be seen that the yarn(warp) in the loading direction is nearly straight while thetransverse yarn (weft) is highly crimped.

2. Micro-mechanical analysis of woven textile composites

Various analytical techniques have been developed topredict the elastic properties of representative volume ele-ments (RVEs) of textile composites. Bogdanovich and Pas-tore [11] presented a concise review of all these methods.Among these techniques are methods of averaging mechan-ical properties of the constituent materials, property predic-tions based upon the detailed geometric descriptions of thereinforcement, and finite element methods treating matrixand fibre as discrete components. Many models of themechanical response of textile composites have their rootsin the analysis of curved or wavy fibres within single lami-nates. Tarnopol’skii et al. [12] developed early models toaverage the response of laminates containing curved fila-ments. Again Tarnopol’skii et al. [13] were perhaps the firstauthors to present analytical techniques intended specifi-cally for textile reinforced composites. They approachedthe problem as a generalisation of unidirectional compos-ites using the so-called modified matrix method. The con-cept was to reduce the three-dimensional extent of theproblem by combining the fibres in a particular directionwith the matrix material to create an effective medium inthe sense of unidirectional micromechanics. Out of variousmodels based on stiffness averaging, the models put forwardby Ishikawa and Chou [14] were more rational as they takeinto account fibre continuity in the thickness direction andthe fibre undulation, which are distinctive features of textilecomposites. The basic principle in their models is to treat asystem of two orthogonal sets of fibres in a woven compos-ite as a laminate with two laminae placed in 0/90 direction.They proposed three different models, i.e., mosaic model,crimp (fibre undulation) model and bridging model. Themosaic model idealises the fabric composite as an assem-blage of asymmetric cross-ply laminates and the elasticproperties of the composite were found by stiffness (parallelmodel) or compliance (series model) averaging which giveupper and lower bounds for the stiffness constants of thecomposite. The key simplification is the omission of thefibre continuity and undulation (crimp) that exists in anactual fabric. The crimp model incorporates the fibre conti-nuity and undulation in one direction by describing thegeometry of reinforcement using trigonometric functions.The bridging model employs both stiffness and complianceaveraging in two orthogonal directions and is suitable tomodel satin weave composites. Naik and Ganesh [15]extended the models of Ishikawa and Chou by consideringthe fibre undulations in two orthogonal directions usingmore detailed trigonometric functions. Models based on afinite element approach offer to examine the detailed stress

Table 1Fabric specifications

Specification Warp Weft

Linear density (tex) 600 600Sett 4.724 ends/cm 4.724 picks/cmYarn thickness (mm) 0.36 0.36Yarn width (mm) 1.76 1.76Crimp (%) 1.2 1.6


field throughout the RVE, potentially providing the infor-mation necessary for failure analysis and damage propaga-tion studies. Three-dimensional models employingconventional finite elements have been presented by manyresearchers but notable work was done by Whitcomb [16]and Dasgupta et al. [17]. Typically the yarns are describedas lenticular in cross-section and the yarn path is describedthrough some trigonometric relationship. The yarn wastreated as a transversely isotropic material and the matrixwas treated as an isotropic material. The stress field underin-plane loads was created by imposing appropriate dis-placements consistent with the boundary conditions. Theconstraint forces generated at the faces of RVE were relatedto the external forces and average normal strains and aver-age Poisson’s ratios were determined. The effective Young’smodulus in a particular direction was then calculated usingthe energy balance equation by equating the work done bythe resultant force in that direction to the strain energystored in RVE due to the displacement in the correspondingdirection. The analysis was used to study several differentvariations of plain weaves that illustrate the effect of towwaviness on composite modulii, Poisson’s ratios and inter-nal strain distributions. A more detailed geometric descrip-tion of plain weave fabric geometry for creating FEAmodels was attempted by Kuhn and Charlambides [18] bytaking into account the asymmetry of the cross-sectionand variation of the cross-sectional geometry from contactregion to free region. They developed continuous mathe-matical functions to describe the yarn’s surface. Glaessgenet al. [19] were the first to construct geometry of a preformrelating it to the mechanical properties of yarns and usedthe resultant geometry for finite element modelling of com-posite RVE. They were able to achieve a maximum fibrevolume fraction of 42% compared to 25–30% achieved byall other models which used idealised descriptive geometricfunctions to represent the yarn path. Zhang and Harding[20] used 3D finite element analysis to find the strain energystored in the whole RVE under uniform uniaxial extensionand obtained the effective elastic constant using the princi-ple of equivalent strain energy. The yarn geometry was idea-lised as straight inclined segments and the undulation wasalso considered in only one direction. Nevertheless, the con-

Fig. 6. (a) RVE model generated using ABAQUS CAE pre-processor, (b) Tomaterial orientation. (a = 2.116 mm, bt = 0.36 mm, bm = 0.10 mm).

cept of strain energy equivalence gave a reasonable predic-tion of the elastic properties of a woven fabric composite.

2.1. Creation of tow geometry and representative volume

element (RVE)

The resin impregnated fibre bundle (roving) in a compos-ite is commonly referred as tow. The analytical/FEM mod-els that predict the effective modulii of woven compositesrequire a geometric description of the tow path and hencethe fibre volume. The usual practice in formulating all thesemodels has been to explicitly describe the fibre path usingsome trignometrical functions without making any refer-ence to the mechanical properties of the fibre bundle. Therovings used in textile composites have a finite bendingrigidity and have to obey certain geometric constraints gov-erned by the fabric structure. Hence the explicit geometricdescription of textile reinforcement may not represent theactual orientation or actual volume contained in the com-posite. This may be particularly significant in unbalancedweaves where both the warp and weft spacings are differentor warp and weft yarns of different linear densities are used.The yarn path in the initial state of the fabric has beenobtained by minimizing the bending energy of the yarnfor the given spacings and thickness of the yarn (Fig. 3).The fabric under consideration here is the plain-woven glassfabric woven from 600 Tex roving. Fabric specifications arepresented in Table 1. The composite specimen was formedusing an unsaturated polyester resin matrix.

Fig. 6 shows the 3D FEA model constructed using theABAQUS CAE pre-processor. The yarn cross-sectionwas swept along the yarn path shown in Fig. 6b to produce

w path divided into regions and assigned with local coordinate system for

Fig. 7. (a) Finite element mesh of RVE. (b) Deformed state of RVE (0.5%uniaxial strain).

30

40

50

60

ss (

MP

a)


the yarn shape. Cutting the volume occupied by the yarnsin the solid matrix volume generated the matrix pockets.Copies of the individual parts i.e. yarn and matrix werethen generated and assembled to produce the unit cell ofa plain weave composite representative volume element(RVE).

2.2. Specification of tow properties

The elastic properties of resin-impregnated tow weredetermined using the equations of micro-mechanics [21].The tow is considered as a transversely isotropic materialand hence it requires the specification of six elastic con-stants, i.e. E11, E22, m12, m23, G12, G23 with reference to anorthogonal material coordinate system (Fig. 6b). Thereare different approaches to the prediction of these con-stants, some based on an elementary mechanics approachi.e. stiffness averaging (rule of mixtures), compliance aver-aging, and others based on advanced approaches i.e. strainenergy approach, elasticity solutions and charts developedfrom FEA. While stiffness averaging gives sufficiently accu-rate values for E11 and m11, some refinement is necessary topredict E22 and G12. G23 is taken as equal to G12 and m23 isfound by considering the isotropic behaviour in plane 23[22].

E11 and m12 can be found by the well-known rule of mix-tures given by

E11 ¼ EfV f þ EmV m ð11Þm12 ¼ mf12V f þ mmV m ð12Þ

E22 can be found by the equation based on the method ofaveraging sub regions due to Chamis [21]

E22

Em

¼ 1�ffiffiffiffiffiffiV f

p� �þ

ffiffiffiffiffiffiV f

p

1�ffiffiffiffiffiffiV f

p1� Em

Ef2

� �24

35 ð13Þ

G12

Gm

¼ 1�ffiffiffiffiffiffiV f

p� �þ

ffiffiffiffiffiffiV f

p

1�ffiffiffiffiffiffiV f

p1� Gm

Gf

� �24

35 ð14Þ

The glass fibre is assumed to be isotropic and hence Ef2 istaken equal to Ef. The properties of the fibre and matrixmaterial are shown in Table 2. The volume fraction wasfound using the following equation

V f ¼qy

qfAy

ð15Þ

where qy is the linear density of yarn, i.e., weight of yarnper unit length, qf is the fibre density and Ay is the areaof cross-section of yarn.

Table 2Properties of E-glass, polyester

Property E-glass (N/mm2) Polyester (N/mm2)

E 72,500 2500G 29,713 925m 0.22 0.35

2.3. Specification of displacement field and solution of model

The stress–strain state corresponding to a uniform dis-placement (strain) along a material principal axis is pre-dicted here. The effective modulus of elasticity of thecomposite is then determined using the concept of equiva-lent strain energy following Zhang and Harding [20]. For

0

10

20

0 0.002 0.004 0.006 0.008

Strain

Str

e

Fig. 8. Experimental stress–strain curve of composite specimen.

Fig. 9. Schematic diagram describing yarn flattening in the case of biaxial extension. (a) Initial state; (b) after applying biaxial tension.

Table 3Properties of E-glass/epoxy tow after biaxial extension of preform

Elasticconstant

Unloaded state(Vf = 0.49)

Biaxially loaded state(Vf = 0.58)

E11 37,310 43,520E22 = E33 8389 10,500G12 = G13 3133 3942G23 3133 3942m12 = m13 0.28 0.27m23 0.33 0.33


example, applying a uniform strain e11 along the warpdirection

U c ¼1

2

Zv

Ec11e211 dv ¼ 1

2Ec11e

211V ð16Þ

where Ec11 is the required homogeneous equivalent modu-lus of the composite and V is the volume of the unit cell(RVE). The strain energy of RVE (Uc) is equal to thesum of the strain energy stored in the warp yarn, weft yarnand the matrix pocket.

Therefore, we have

U c ¼ Uwarp þ Uweft þ U matrix

and

Ec11 ¼2ðU warp þ U weft þ U matrixÞ

e211V

ð17Þ

The software allows output of the strain energy stored ineach component and the volume of each component.Fig. 7(a) and (b) show the FEM mesh of the RVE andthe stress field corresponding to a uniaxial strain of 0.5%in the X-direction. The nodes on the plane x = 0 are con-strained in the X-direction (1) while a displacement of0.01058 mm which corresponds to 0.5% strain was appliedto all the nodes lying in the plane x = a thus specifying auniform displacement field along one principal direction.

Fig. 10. (a) FEM mesh of RVE (a = 2.136; bm = 0.1; bt = 0.302; after biaxiastrain.

The solution of the model yields a stress state correspond-ing to the prescribed displacement as shown in Fig. 7(b).

The strain energy stored in the whole model (RVE) forthis displacement was found to be 0.4905 N-mm and thevolume of RVE was found to be 4.1328 mm3. Substitutionof these values in Eq. (17) gives the effective E11 value ofthe composite to be equal to 9495 N/mm2.

Fig. 8 shows the experimental stress–strain curve of thecomposite specimen and the average slope of the curve inthe strain range 0–0.5% is taken as the modulus of the spec-imen. The modulus decreases beyond a strain of 0.35%,probably due to the visco-elastic nature of the polyestermatrix. However, the initial modulus of 9100 N/mm2 ishowever very close to the predicted modulus of 9495 N/mm2.

l extension). (b) Deformed state of RVE corresponding to 0.5% uniaxial


3. Analysis of stress field corresponding to biaxial fabric

strain

The objective of this analysis is to study the changes inmagnitude of the stresses due to the change in geometryof the preform. The fabric deformed state correspondingto a load of 20 N each in the warp and weft directions is

Table 4Comparison of stress and strain magnitudes in RVEs of compositea (biaxial e

Component Unloaded Biax

S11 304.00 329S22 24.50 22.0S33 110.50 77.7e11 7.382e � 03 8.66e22 6.567e � 04 8.48e33 1.285e � 02 8.70S11 (matrix) 49.51 24.9e11 (matrix) 1.216e � 02 6.62Effective in-plane modulus E11 = E33 11,794 12,1

a The stress and modulii are reported in MPa (N/mm2).

Fig. 11. Longitudinal strain field (e11) in tow parallel to the direction of imposbiaxially loaded preform.

obtained using the energy model described in Section 1.2,and Fig. 4 shows the geometry of the yarn path. The appli-cation of equal biaxial loads in the case of a balanced plain-woven fabric results in equal loss of crimp and hence equalincrease in spacing in both the directions. The reduction ofyarn thickness due to the inter-yarn compressive forceleads to a further loss of crimp and hence the yarns would

xtension of preform)

ially loaded (tow Vf = 0.49) Biaxially loaded (tow Vf = 0.58)

.70 413.305 28.799 106.402e � 03 9.307e � 037e � 05 2.209e � 045e � 03 9.496e � 033 28.607e � 03 6.808e � 0320 13,973

ed displacement: (a) composite with unloaded preform; (b) composite with


tend to become flatter compared to the case where therewas no yarn compression. Fig. 9 shows the schematic dia-gram describing the effect of yarn flattening during thebiaxial fabric strain applied during processing.

It has been computed from the energy model that thereis an increase in the spacing of the yarns from 2.116 mm to2.136 mm and a decrease in the yarn thickness from0.360 mm to 0.302 mm. Hence the cross-section areadecreases from 0.491 mm2 to 0.413 mm2 and hence thereis an increase in the volume fraction from 0.49 to 0.58(+18.36%). Since the tow properties directly depend uponthe volume fraction, the increase needs to be accountedfor and Table 3 shows the tow properties correspondingto both the unloaded and biaxially strained state. Theunloaded as well as the biaxially strained states refer tothe geometry of a dry fabric.

Fig. 10 shows the FEM mesh corresponding to the com-posite made with biaxially strained fabric. The RVE of thiscomposite is subjected to a uniaxial strain of 0.5%. Themesh size is kept the same as in the case of RVE corre-sponding to the unloaded state (Fig. 7) so as to have anobjective comparison of the stress field.

Table 4 shows the comparison of the stress componentsof the composite produced with an unloaded and a biaxi-ally strained fabric, both corresponding to a uniaxial strainof 0.5%.

The results of the composite with a biaxially strainedfabric are shown separately for original and modified towproperties. The stress components S11, S22 and S33 corre-spond to the local coordinate system shown in Fig. 6. S11

Fig. 12. Transverse strain field (e33) in tow perpendicular to the direction of imwith biaxially loaded preform.

refers to the stress in the tow along the fibre direction,S33 refers to the stress perpendicular to the fibre directionand S22 refers to the stress in the out of plane direction.It can be seen that there is an increase of stress magnitudeS11 in the biaxially strained state, which is accompanied bya decrease in stress magnitude S33. This is mainly due to thereduction of crimp, i.e., fibre waviness, which occurs whenthe fabric is biaxially strained. As the tow wavinessdecreases, the stress in the tow increases for the same aver-age strain but the transfer of stress to the yarn in the trans-verse direction would become less and hence the maximumtransverse stress and strain would decrease with thedecrease of tow waviness. This prediction also corroboratesthe findings of Whitcomb [16]. Figs. 11–13 show the strainfield in different components of composites made withunloaded and biaxially strained fabrics.

A careful examination (Fig. 11) shows that the maxi-mum longitudinal stress in tows parallel to the directionof the imposed displacement occurs at the middle of theedge where the cross-sectional area is least (the regionmarked in figure). The increase of strain (e11) (inFig. 11b) is due to the change in tow orientation after thebiaxial fabric strain and also due to the reduction in thick-ness at this region. It is observed that the maximum trans-verse strain (e33) occurs in tows perpendicular to thedirection of the imposed displacement. Fig. 12 shows thetransverse strain magnitude in the tows perpendicular tothe direction of the imposed displacement. It can be seenthat there is a reduction of transverse strain by as muchas 32% in the case of a composite with a biaxially strained

posed displacement: (a) composite with unloaded preform; (b) composite

Failure of Transverse yarn

Fig. 14. Schematic diagram describing the knee phenomenon in fabriccomposites.

weft

warpupper

lower

Fig. 15. Effect of increase of crimp of weft yarn on warp yarn’s cross-section.


fabric. This is mainly due to the reduction in inclination ofthe tow path at the centre. The higher the inclination of thelongitudinal tows (in the direction of imposed displace-ment), the higher the transfer of stress to the perpendiculartows. It should be noted that transverse stress and trans-verse strain are critical in tows, which have low breakstrains in the transverse direction compared to the longitu-dinal direction (Fig. 14).

It was well reported that the breaking strain is firstreached at this point and progressive failure of the towstarts. This is known as the knee phenomenon [23]. Forthe biaxially strained case with modified tow propertiesthat take into account the increase in the volume fraction,the longitudinal stress is increased significantly due to thehigher longitudinal modulus of the tow and there isdecrease of transverse stress when compared to compositewith unloaded preform. Fig. 12 shows the stress field inthe matrix. The reasons as given for transverse stress canalso be attributed to the reduction of stress in the matrixcorresponding to the biaxially loaded preform. In theunstrained preform, the higher inclination of the tow-pathcreates narrow matrix pockets and leads to stress concen-tration along the ridgeline. The effect of equal biaxial fabric

Fig. 13. Strain field in matrix of composite with: (a) unloaded preform; (b) biaxially loaded preform.

Table 5Properties of E-glass/epoxy tow after uniaxial extension of preform

Elastic constant Warp tow (Vf = 0.64) Weft tow (Vf = 0.51)

E11 47,660 38,690E22 = E33 12,432 8803G12 = G13 4673 3290G23 4673 3290m12 = m13 0.26 0.28m23 0.33 0.34

Table 6Effective modulus of RVE for different casesa

Preform E (MPa)

Unloaded 14,559Uniaxially loaded (weft) 3200Uniaxially loaded (warp) 20,075Biaxially loaded 17,044

a Matrix pockets removed.


strain leads to a decrease of stress in the matrix material.The effective in-plane modulus has been computed follow-ing Zhang’s strain energy equivalence approach. It can beseen that the effect of the change in crimp after biaxialdeformation does not seem to significantly affect the effec-tive modulus of the composite as it increases by only 2.7%,but the increase of the modulus is quite significant(+18.48%) if the modified tow properties are used in thecomputation.

4. Analysis of stress field corresponding to uniaxial fabric

strain

The effect of uniaxial extension on the dry fabric pre-form is to increase the fabric strain, i.e., the yarn spacingin the direction of loading while simultaneously decreasingthe yarn spacing in the other direction. While the yarn pathin the loaded direction would become straight, the wavi-ness (crimp) of the yarn path in the other directionincreases. This particular phenomenon is called ‘crimpexchange’ and has already been explained in Section 1.1.As there would be significant change in the fibre orienta-tion after uniaxial strain compared to biaxial strain, itwould be interesting to study the change of the effectivemodulii of composite.

The glass fabric used in the present study is subjected toa load of 20 N per yarn in the warp direction. The changeof yarn path due to this uniaxial fabric strain is computedusing the energy model as shown in Fig. 5. It can be seen

Fig. 16. (a) RVE of a uniaxially strained fabric; (b) RVE applied with 0.5% straweft tow, showing the reduction in warp tow cross-section area due to simplifi

that the yarn path in the warp direction is nearly straightwhile that in the weft direction shows higher crimp.

The part of warp cross section that is in contact with theweft yarn tends to become more rounded as it followsthe weft yarn path. On the other hand, the lower half ofthe cross-section that is not in contact tends to retain itsoriginal shape. Hence, the cross-section now becomesunsymmetrical in the contact region and gradually changesto a symmetrical one at the middle and is again reversed at

in along the transverse tows (without resin pockets); (c) cross-section alongcation.


the contact region of the next cross over point (Fig. 15).The tow geometry needs to be constructed with a varyingcross-sectional area [18]. Due to the limitations of ABA-QUS pre-processor, it has not been possible to create towgeometry with cross section varying along the length. Asa result, it is assumed here that the cross-section is symmet-rical and some simplifications are made to create the geom-etry of RVE corresponding to the preform subjected to theuniaxial fabric strain (Fig. 16). While the warp tows arestraight with zero crimp, the weft tows have relativelyhigher crimp. It can be seen that the warp cross-section fol-lows the weft yarn path up to half the thickness and is con-structed to be symmetrical with respect to the centre, whileretaining the original thickness. The area of the resultingcross-section of the warp yarn is now reduced to0.37 mm2 from the original 0.491 mm2. It may be notedthat this area reduction is mainly due to the simplificationmade to the geometry, and not due to the uniaxial strain.Since the fibre volume is the same, the tow properties havebeen adjusted to take into account the increase in fibre vol-ume fraction. Table 5 shows the properties for the warpand weft tows.

Since the volume of the matrix pockets would be differ-ent in an RVE corresponding to the uniaxially loaded pre-form from that of an unloaded preform, it has been

Fig. 17. (a) Longitudinal strain field (e33) in tow parallel to the direction of impthe direction of imposed displacement.

proposed to analyse the effective moduli and stress fieldin the RVE considering only the tows. The effective moduliin the warp and weft directions are now computed usingthe superposition method proposed by Whitcomb [16] asthere are no matrix pockets in the RVE. The procedureinvolves applying a uniform strain successively in the longi-tudinal and transverse directions and computing therestraint forces. The modulus is then computed using thework-energy theorem. Table 6 shows the effective in-planemoduli for different cases.

It can be seen, in a composite made with uniaxiallystrained fabric (in the warp direction), the modulusdecreases by as much as 77% in the weft direction andincreases by 40% in the warp direction due to the uniaxialfabric strain in warp direction. Table 6 shows the stress andstrain magnitudes in RVEs with the application of 0.5%strain along the weft direction. Fig. 17 show the strain con-tours in the longitudinal and transverse tows, respectively.A look at Table 7 shows that there is a decrease in longitu-dinal stress and strain (in the case of RVE corresponding tothe uniaxially strained fabric) mainly due to the increase oftow waviness. However there is a decrease of transversestress and strain with an increase of tow waviness, whichis contrary to the expected behaviour. This is mainly dueto the simplification made in the construction of the tow

osed displacement. (b) Transverse strain field (e33) in tow perpendicular to

Table 7Comparison of stress and strain magnitudes in RVEs of composite withunloaded and uniaxially loaded preforms

Component Unloaded Uniaxially loaded

S11 489.00 153.00S22 142.90 24.51S33 128.80 29.24e11 1.197e � 02 3.873e � 03e22 7.087e � 04 1.783e � 03e33 1.429e � 02 2.738e � 03e11 (matrix) 1.216e � 02 6.627e � 03


geometry that is dictated by the limited features of the soft-ware. It can be seen from Fig. 16c that the warp cross-sec-tion is reduced and hence the transverse and longitudinalstiffness is increased owing to the increased volume frac-tion. Moreover the warp cross-section does not have fullcontact with the weft tow-path up to the centre and hencethe transfer of stress from the longitudinal to the transversetow is less. Hence more detailed tow architecture is neces-sary to carefully predict the stress and strain field in thecase of an RVE corresponding to a uniaxially strained fab-ric (future versions of ABAQUS may allow us to varycross-sectional shape along the tow path).

5. Discussion

This paper aimed to study the influence of fabric ten-sions applied during processing/consolidation on themechanical properties of a finished composite. Anenergy-based fabric compliance model has been presentedto predict the change in yarn geometry due to uniaxialand bi-axial loads. In the case of an equi-biaxial loading,there is a slight reduction in the yarn thickness and a cor-responding reduction in crimp in both warp and weft direc-tions. Equi-biaxial loading also leads to an increase in fibrevolume fraction in a tow. In the case of uniaxial loading,yarns in the loading direction become relatively straightwhile the crimp increases in the transverse yarns due tocrimp interchange; this crimp interchange has a significanteffect on the composite mechanical properties.

FEM based micro-mechanical models for three cases ofwoven composites have been presented: unstrained, biaxi-ally strained and uniaxially strained fabrics. In the caseof composites made with biaxially strained fabrics, thereis a slight reduction in crimp, tow thickness, laminate thick-ness and a corresponding increase in tow volume fraction.Tensile modulus increased by 19% in comparison to a com-posite made with unstrained fabric. A more significant fac-tor is the reduction in transverse strain in tows. The towsare weak in the transverse direction and are prone to failuredue to ‘knee phenomena.’ Hence, with a biaxial loadingduring the forming stage, the knee phenomena can be post-poned until a much higher load.

Uniaxial fabric strain leads to straightening of the towsin the loading direction with a corresponding increase intow waviness in the transverse direction. This exacerbates

the knee effect when the composite laminate is loaded inthe transverse direction. Hence, composites made withuniaxially strained fabrics may be suitable for componentswith a significant loading in the axial direction. Due to thelimitations of the ABAQUS pre-processor, a number ofsimplifications have been made to the RVE of a compositemade with uniaxially strained preform. Hence, FE analysisof these composites may be used only for a qualitativeargument. In future, a more sophisticated pre-processorwill be used to create tows with varying cross-sectionalshapes along its length.

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