Composites: Part B 66 (2014) 368–380

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Composites: Part B

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A micromechanical model of woven structures accounting for yarn–yarncontact based on Hertz theory and energy minimization

http://dx.doi.org/10.1016/j.compositesb.2014.05.0271359-8368/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +33 3 83 59 57 24; fax: +33 3 83 59 55 51.E-mail addresses: frej_ch@yahoo.fr (F. Chaouachi), rahali.yosra@gmail.com

(Y. Rahali), jean-francois.Ganghoffer@univ-lorraine.fr (J.F. Ganghoffer).

F. Chaouachi a, Y. Rahali a, J.F. Ganghoffer b,⇑a Unité de recherche de mécanique des solides, structures et développements technologiques, ESSTT, Université de Tunis, BP56, Bab Mnara 1008, Tunisiab LEMTA, Université de Lorraine, 2, Avenue de la Forêt de Haye, TSA 60604, BP 160, 54518 Vandoeuvre Cedex, France

a r t i c l e i n f o

Article history:Received 16 April 2013Received in revised form 25 April 2014Accepted 26 May 2014Available online 12 June 2014

Keywords:A. Fabrics/textilesB. Mechanical propertiesC. Micro-mechanicsHertzian contact between yarns

a b s t r a c t

The equilibrium shape of plain weave fabric, accounting for yarn to yarn contact and incorporating thetransverse compressibility, is presently investigated, relying on an energetic analysis of the textile. Eachyarn is modeled as an undulated beam at a mesoscopic level, with a shape modeled by a Fourier series. Inorder to get an accurate description of the contact between yarns, we have considered the contact occur-ring on a surface (distributed contact) of finite extent according to Hertz contact theory. This methodallows evaluating the reaction forces between weft and warp from the pressure distribution in the con-tact zone between yarns, and the evolution of the contact area versus the applied tension. The overallresponse of fabric under uniaxial and biaxial tension is computed as the minimum of the total potentialenergy of the set of interwoven yarns, including the work of reaction forces on the contact zones. The cal-culated nonlinear response for two important materials (glass, carbon) is due to the change of the geom-etry, namely the crimp variation and the change of contact area during increased loading. The contactarea increases with the applied tensile force and is comparable to the yarn diameter. Only the contribu-tion of the displacement due to flexion is shown to be sensitive to the distribution of the forces on thecontact surface. A good agreement between the model predictions and tensile measurements of theresponse of plain weave fabric has been obtained.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Considerable attention from composite manufacturing sectorhas been devoted to textile composites and especially woven fab-rics. These structures have many advantages over unidirectionalfiber reinforced composites, such as enhanced dimensional stabil-ity over a large range of temperatures [1], more balanced proper-ties in the fabric plane, and better impact resistance. Themechanical behavior of fabrics is clearly a multi-scale problem.Woven fabrics are structured, hierarchical materials, having threestructural levels [2], called macro-level, meso-level and micro-level, where the first has the largest and the latter has the smallestorder of magnitude. The mechanical behavior of woven fabrics iscomplex due to the intricate interactions of the yarns that consti-tute the fabric ‘‘mesostructure’’. The multi-scale nature of thefabric leads to a complex mechanical behavior. Micro-scaleapproaches take repartition and contact between thousands offibers into account. Some studies have been performed at this scale

[3], but the analysis of the mechanical behavior of the woven unitcell at this scale is very difficult accounting for the number of fibers(some thousands) in a unit cell. In macro-scale approaches, fabricsare modeled as membranes (or shells if bending is taken intoaccount); these approaches are used in forming simulations. Theyfocus on the characterization of fabric behavior without explicitlymodeling the geometric structure of the textile; the fabric is mod-eled as a homogeneous continuum which obeys a specific constitu-tive relation. These continuous models do not include crimp andinterlacement effects, which are important features of the fabricreinforcement behavior.

The analyses at meso-scale consider the geometry of the wovenunit cell but consider the yarns as continuous domains, accountingpossibly for single yarn interactions. A lot of mesostructurallybased analytical models have been developed for the study ofwoven fabrics behaviors. Such analytical models use mathematicalrelations to predict the mechanical response of the fabric and itscomponent yarns in specific modes of deformation. In addition,these models can be used to quantify homogenized materialproperties for use in continuum models. Hearle et al. [4] describeda number of classical analytical fabric models. Kawabata et al.[5–7] presented general theories for uniaxial, biaxial and shear

F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380 369

deformation properties of plain weave fabrics based on a simplifiedmodel representing the structure of the fabric unit cells. The basicgeometry used in this theory is based on one-dimensional stiffnesselements (bars) representing yarns, and the connecting stiffnesselements at the intersection of yarns to model the compressionbetween yarns. Other researchers have developed improved ana-lytical models, including Realff et al. [8], who modify Kawabata’suniaxial model to include more complex behaviors such as yarnflattening and consolidation. Kato et al. [9] proposed an analyticalmodel for predicting the constitutive behavior of a coated fabriccomposite that is based on the pin-joined lattice-type geometry.Recently, Ben Boubaker et al. [10–12] developed a meso-level dis-crete model for plain weave, which was then extended to generalmonolayer textiles in Assidi et al. [13]. For a more detailed descrip-tion of several other analytical models, see [14,15].

Analytical models of the fabric mesostructure can be incorpo-rated into anisotropic continuum formulations to yield models thattrack the fabric mesostructure as the continuum deforms. A num-ber of authors have presented models for woven fabrics with con-tinuum properties calculated from a deforming unit cell. Boisseet al. [16,17] constructed a three-dimensional unit cell like thatof Kawabata for quasi-static and dynamic simulations of compositeforming but did not incorporate the resistance to yarn rotation orthe inertia of uncrimping. Rattensperger et al. [18] follow a similarapproach for modeling fabric-reinforced hydraulic hoses, with thefabric lattice geometry similar to that used by Kato, and adopt aconventional finite element formulation with rebar reinforce-ments. Tanov and Brueggert [19] presented a mesostructurallybased continuum model that includes shear and locking resistancethrough diagonal spar elements within the assumed unit cell net-work. Boisse et al. [20,21] later used a planar unit cell for dynamicsimulations by incorporating shear resistance at the cross-overpoint but did not include cross-locking. Hamila et al. [22] extendedthis approach to a shell finite element which included the resis-tance to bending of the yarns.

Alternative approaches suitable for simulating general fabricdeformations are based on finite elements: either they discretelymodel in three dimensions every yarn in the weave, or theyhomogenize the response of the yarns, typically by treating thefabric as an anisotropic continuum. The discrete approach, usedby Shockey et al. [23–25], Ng et al. [26], Duan et al. [27], Zhanget al. [28], and Talebi et al. [29], among others, has the advantageof capturing all yarn interactions and providing a detailed descrip-tion of all mechanisms of fabric deformation. It is thus valuable forstudying the various modes of deformation of the yarns and theeffects of parameters such as weave style and yarn properties onmacroscopic deformation. However, building these elaborate finiteelement meshes is tedious, and simulating actual structures orevents of significant duration is computationally prohibitive inmost cases. On the other hand, the homogenized, continuumapproach, used by Peng and Cao [30–32], Xue et al. [33,34], andBaseau [35], amongst others, allow greater computational effi-ciency and it can be easily integrated into multi-component sys-tem models. Although this approach can accurately represent themacroscopic response of woven fabrics, it does not explicitly modelthe modes of deformation or interactions between yarns, andtherefore cannot predict all of the necessary yarn-level aspects ofdeformation. Peng and Cao [30,31] applied a dual homogenizationtechnique to predict the mechanical behavior of woven fabricsfrom a sequence of models for fibres. In the first step, a unit cellwas made for the yarn section to estimate the effective elastic con-stants of fibre yarn. Then, another unit cell at the meso-level wasconstructed to represent the periodic structure of the woven fabric.Peng and Cao [32] developed a continuum mechanics-based non-orthogonal constitutive model to characterize the anisotropicmaterial behavior of woven composite fabrics under large

deformation. Moreover, Xue et al. [34] developed an integratedmeso-macro model for the prediction of mechanical properties ofwoven fabrics under large deformations. The same authors previ-ously suggested a non-orthogonal constitutive model for the pre-diction of mechanical properties at the macro scale [33], relyingon experimental measurements to identify the constants of theirmodel. Baseau [35] developed continuum formulations for ‘‘fila-mentary networks’’, appropriate for non-reinforced fabrics. Kinget al. [36] proposed a new approach for the continuum modelingof fabrics by selecting a geometric model for the fabric weave cou-pled with constitutive models for the yarns behavior. The lastauthors used energy minimization method to relate the fabricsstructural configuration to the macroscopic deformation.

A common assumption in the literature is that the contactbetween yarns is punctual [13,37]; this assumption can be consid-ered as valid for structures sufficiently large compared to the areaof contact. In [37], although sliding between yarns is modeled, thecontact is represented using concentrated forces, and not distrib-uted pressures as in a standard contact mechanics approach. How-ever, the contact area becomes comparable to the size of therepresentative unit cell (we can reasonably assume textile to beperiodic, and consisting of the spatial repetition of a certain armor)in the case of textiles (as shown on Fig. 6 in [37]). One accordinglyexpects that the contact between the two sets of yarns will have asignificant effect on the overall mechanical response. It is then theprincipal objective of this contribution to quantitatively assess theeffect of the distribution of contact forces between yarns on themechanical response of plain weave fabric.

The principal goal of this contribution is to extend the previousdiscrete model of textile monolayers based on energy minimiza-tion following Ben Boubaker et al. [10,11] and Assidi et al. [13] tothe consideration of the contact between warp and weft. An ana-lytical approach will be followed in this contribution, and Hertzanalytical contact theory will be employed. The yarns will be mod-eled at an intermediate meso level, so that the fibrous structurewill not be explicitly described. Nevertheless, we shall evaluatethe effective moduli of yarns from the fiber level using some modelof the literature.

This works is organized as follows: in Section 2, the contactmodel between yarns based on Hertz theory is presented, leadingto an expression of the reaction force exerted by the transverseyarn, and the variation of the contact area with load. The potentialenergy of the woven structure is constructed in Section 3, incorpo-rating the work of the reaction forces. An algorithm for the resolu-tion of the set of equations of the model is presented in Section 4.Results of uniaxial simulations evidencing the effect of warp toweft contact interactions are presented (Section 5). Finally, a sum-mary of the main results and perspectives are given (Section 6).

2. Hertzian contact between yarns

We analyze in the sequel the mechanical response of such a tex-tile monolayer; in view the subsequent analysis, it is convenient tosplit the sets of interwoven yarns into two interacting subsystems:the set of warp denoted Xwa and the set of weft denoted Xwe. Eachyarn is modeled as an undulated beam assumed to be flexible andextensible. We further make the assumption that the contactbetween yarns is perfect and does not induce relative sliding.Accordingly, one shall involve energetic principles in combinationwith Hertzian contact mechanics in order to evaluate the equilib-rium shape of the monolayer submitted to a tensile (uniaxial orbiaxial) loading.

We consider plain weave fabric as the assembly of two sets oforthogonal yarns (warp and weft) in an initial configuration. Eachyarn within the woven structure is assimilated to a homogeneous

370 F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380

beam submitted to the contact forces exerted by the transverseyarns (Fig. 2.1).

The main assumptions of the present model are as follows:

– The tensile force remains aligned with the direction of loading.– The yarns are represented as undulated beams endowed with

effective mechanical properties, in line with the previouslymentioned meso-scale analyses. Those effective properties willbe determined from micromechanical models of the literature.

– In order to simplify the calculations, the initial yarn section(before contact occurs) is assumed to be circular (this will how-ever not restrict the generality of the presented methodology).

– The yarns are in perfect contact with no relative sliding.

The vertical deflection of each yarn (modeled as a beam ele-ment) is expressed as a Fourier series with coefficients dependingupon the forces applied to the beam element. Timoshenko beammechanics leads to the following expression of the initial deflec-tion of warp and weft in the absence of external load [11]:

w jo�weðyÞ ¼

XNwa

n¼1

awen; j sin ð j� 1Þpþ n

pyLwe

� �ð1Þ

wko�waðxÞ ¼

XNwe

n¼1

awan;k sin ðk� 1Þpþ n

pxLwa

� �ð2Þ

with wjo�we; wk

o�wa respectively the initial deflections of the warpand weft, Lwa, Lwe respectively the projected length of the yarns inthe x and y directions, n the number of harmonics, Nwa, Nwe respec-

tively the number of warp and weft, awan;k

� �n2½1;Nwe �

; awen; j

� �n2½1;Nwa �

are

the Fourier series coefficients (detailed in Appendix B, see Eqs.(B.4) and (B.5)) and j, k the labeling of warp and weft in the wovenstructure (Fig. 2.1).

Those expressions (Eqs. (B.4) and (B.5)) are valid only for concen-trated (punctual) reaction forces; the goal of this contribution is todevelop more general expressions for the Fourier series expansions

Fig. 2.1. Micromechanical mo

describing the shape of woven monolayers, relying on a more accu-rate and realistic representation of the contact between yarns.

Since the yarns are described as cylindrical beams, one firstanalyses the Hertzian contact between orthogonal cylinders, inwhich case it is known that the contact area is an ellipse [38].The total reaction force is supposed to be known, and one deter-mines its spatial distribution from Hertz theory as well as the con-tact surface. From Hertz theory [38], the pressure distribution onthe elliptic zone of contact (Fig. 2.2) writes:

P0ðx; yÞ ¼ pm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

a2 �y2

b2

sð3Þ

with the maximum pressure therein:

pm ¼3N

2pabInvolving the normal effort acting on the contact N, and the majorand minor radii of the contact ellipse, respectively quantities a, b,expressing versus the normal effort as

a ¼ m1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3pNðk1 þ k2Þ

2 C1 þ C 01 þ C2 þ C 02� �3

s; b ¼ m2

m1a ð4Þ

The coefficients m1, m2 therein are expressed in [40] through a fic-tive angle u depending upon the curvature radius C (the scalarsC1; C01; C2; C02) and the angle h between the planes of principal cur-vatures representing the relative orientation of both yarns (exposedin Appendix A), viz.

u¼arccos

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1�C 01� �2þ C2�C 02

� �2þ2 C1�C 01� �

C2�C02� �

cosð2hÞq

C1þC01þC2þC 02

0@

1Að5Þ

The coefficients k1, k2 in (6) are dependent on the Poisson’s ratio ofboth yarns, coefficients m1; m2 and the tensile moduli E1, E2 (exposedin Appendix A), as

del of plain weave fabric.

Fig. 2.2. Distribution of the contact pressure [16].

F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380 371

k1 ¼1� m2

1

pE1; k2 ¼

1� m22

pE2ð6Þ

In the initial state (before any external load), the contact force Nis due to the yarns being interwoven, whereas a different contact(interaction) force shall result from the additional effect of anapplied tensile force P. We use in the sequel two different nota-tions for those two reactions, namely N1 when there is no appliedload (P = 0), and N2 when P – 0. The contact force in the absence ofexternal load reflects the sole interactions between weft and warp;when an external load is applied, there is a modification of the con-tact forces. Accordingly, introducing those notations, the previouscoefficients express as in (6), but with the normal effort beinginstead N2. From [10,12], the contact forces exerted by the trans-verse yarns (weft) on the warp are given by:

8ð j; kÞ 2 ½1 . . . Nwe; 1 . . . Nwa�; N1 ¼p4

2BwaN3

we

L3wa

w j;kso�wa

N2 ¼ �p4

2BweN3

wa

L3we

1þ awe

N2wa

!"w j;k

so�we �w j;kso�wa

� �þw j;k

s�wa

þ C2 1� 1�dwa;k

comp

C1

!K2Lwac

K1Lwec

264

375þ dwa;k

comp

375

ð7Þ

where Nwa, Nwe are respectively the number of warp and weft,Bwa; Bwe therein denote the flexural rigidities of the warp and weftrespectively, Lwa; Lwe respectively the projected length of the yarnsin the x and y directions, C1, K1 (resp. C2, K2Þ are the constants ofKawabata of the warp (resp. the weft), Lwa

c (resp. Lwec Þ the curvilinear

length of a warp (resp. a weft) yarn portion defined within a halfperiod, dwa;k

comp represent the deformation under compression of the

warp, w j;kso�wa; w j;k

s�wa; wj;kso�we the amplitudes of the undulations

(before and after the application of an external load) of the warpand the weft yarns at the interlacing point (ck, Fig. 2.3) respectively.

The ratio awe ¼ PwePwe

cris introduced, with Pwe

cr ¼ p2Bwe

L2we

the criticalcompressive buckling loads of the weft.

The expressions of the abscissa of the contact points for the setof warp are parameterized by the index p within a contact zonelabeled by the index k, such that

8k 2 ½1;Nwa�; 8p 2 ½1;NR� ! ck ¼ð2k� 1ÞLwe

2Nwa;

ckp ¼ ck þ2a p� NRþ1

2

� �� �NR

ð8Þ

NR being the number of discretization points in the contact zone.The continuous distribution of the pressure on the contact sur-

face is discretized by dividing this surface into several portions d;one then replaces the continuous contact pressure by a discretedistribution whereby one integrates the pressure in all those zonesto determine the discretized contact force; this gives the following

expressions of the reaction forces Rj;kpwa=we

or Rj;kpwe=wa

in the initial state

(when no external load is applied), considering all contact points:

8p 2 ½1;NR�;R j;kp ¼

ZS

P0ðx; yÞdS

with

R j;kp ¼

Z ck�aþ2aNR

p

� �ck�aþ2a

NRðp�1Þ

� �

�Z ckþb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ðx�ck Þ

2

a2

q� �

ck�b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ðx�ck Þ

2

a2

q� � 3N1

2pab

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðx� ckÞ2

a2 � ðy� ckÞ2

b2

s0@

1Adydx

ð9Þ

a, b are the major and minor radii of the contact ellipse.Note that in the case of a uniaxial or biaxial load, the contact

forces and the contact area keep the same expression by changingthe normal force N1 into N2 according to (7), accounting for theimposed loading.

From the considered assumptions, the coefficients of the Fourierseries (1) and (2) providing the equilibrium shape of the yarns areexpressed in Appendix B, see Eqs. (B.9) and (B.10)).

3. Energetic model and equilibrium shape of plain weave fabric

The work of reaction forces exerted by a single transverse yarn(weft with index j) on a warp labeled with index k is computedfrom the previous expression of the reaction forces (9) as

WR j;kp¼XNR

p¼1

Z w j;ks�wa

w j;kso�wa

ð�1Þð jþ1ÞR j;kp we=wadw j

!ð10Þ

where w j;kso�wa; wj;k

s�wa denote the initial and actual yarn deflections atthe corresponding contact point respectively. Hence, the total workof the reaction force for a given warp writes:

Wkcontact ¼

XNwe

j¼1

XNR

p¼1

Z w j;ks�wa

w j;kso�wa

ð�1Þð jþ1ÞR j;kp we=wadw j

! !ð11Þ

The work of the traction effort Pwa exerted on the warp is given by

Wktraction ¼ PwaDuk

wa ¼ Pwa Duk; fwa þ Duk;ex

wa

� �ð12Þ

The total displacement Duwa therein has been decomposedadditively into a flexural component Du f

wa and an extensional com-ponent Duex

wa; the developed expressions of this work are exposedin Appendix C, see Eq. (C.1).

The potential energy of a single warp is then constructed as thedifference between the internally stored mechanical energy(accounting for yarn extension, flexion and compression)

Ukwa ¼ Uk

f þ Ukex þ Uk

comp ð13Þ

(the calculation of this expression is exposed in Appendix C, Eq.(C.2)), and the work of external forces, consisting of the sum of

Fig. 2.3. Distribution of the reaction forces.

372 F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380

the work of the applied load Wktraction and of the reaction forces

Wkcontact , viz.

Vk ¼ Ukwa �Wk

traction �Wkcontact ð14Þ

The total potential energy of the whole plain weave structure isthen the sum over all warp yarns

wko�waðxÞ ¼

2L3wa

p4Bwa

XNR

p¼1

Rj;kpwe=wa

XNwe

n¼1

XNwe2

m¼1

1n4

sin npNwe

2m� 32

� �þ 2 p�ðNRþ1Þ

2

� �anp

NR :Lwa

� �

� sin npNwe

2m� 12

� �þ 2 p�ðNRþ1Þ

2

� �anp

NR :Lwe

� �26664

37775

0BBB@

1CCCA sin ðk� 1Þpþ npx

Lwa

� �

V ¼XNwa

k¼1

Vk ð15Þ

The equilibrium shape of the fabric is then obtained from theset of configuration parameters fak;ukg that achieve a minimumof V. The simulation of tensile test of plain weave fabric relies onthe algorithm described in the next section.

4. Algorithm for the numerical resolution of the set ofequations

The input data are:

– The yarn geometrical parameters for the considered textilemonolayer, here plain weave.

– The mechanical properties of the warp and weft (tensile moduliand transverse compression parameters).

– The tensile load P.

The results include:

– The evolution of the contact surface between yarns versus theapplied tension.

– The total displacement and the components due to extensionand flexion.

– The contact forces along the yarn.

The methodology relies on the following steps:

– Determination of the initial shape of the yarn and of the initialcontact surface.

– Computation of the contact force N1 in the absence of externalforce.

– Computation of the initial yarn shape from the deflectionamplitude.

– Computation of the reaction forces due to contact at initialstate, thus the initial yarn shape.

– Evaluation of the curvilinear length Ls�wa of the warp [39],expression

Ls�wa ¼Z Lwa

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ dyoðxÞ

dx

� �2s

dx:

– Determination of the flexural and extensional rigidities,quantities

Cf ¼ Bwa

D; Cex ¼ EAwa

D

D is the curvilinear distance between two constitutive nodes(Fig. 2.1).– Initialization of the yarn amplitudes:

8p 2 ½1 . . . NR�;8ði; jÞ 2 ½1 . . . Nwe; 1 . . . Nwa�; wp;iso wa

¼ ð�1Þiþ1Apwa; wp; j

so we ¼ ð�1Þ jApwe

with

Apwa¼ Ao�wa cos

4pa NRþ12 � p

� �NR � Lwa

!;

Apwe¼ Ao�we cos

4pa NRþ12 � p

� �NR � Lwe

!

F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380 373

And

Table 1Mechan

Fibre

TypeNumFilamYarnLongBendPoissPoissTran

Ao�wa ¼Dwa

2; Ao�we ¼

Dwe

2

– Increase of the tensile load along the warp by increments dP,such that Pwa½k� ¼ k � dP, k = 1 . . . ,n, with n the number of calcu-lation steps.

– Expression of the tensile load along the weft as a fraction of thewarp tensile load, Pwe ¼ aPwa½k�; the coefficient a therein is var-ied to simulate the transition from a uniaxial to a biaxial loading(it is indicative of the degree of biaxiality).

– Determination of the total potential energy of the plain weavestructure according to (15), assembling therein all contributionsevaluated in Eqs. (10)–(14).

– Minimization of the potential energy and determination of theequilibrium due to an applied uniaxial or biaxial tension, lead-ing to the tensile response of fabric.

– Evaluation of the reaction forces and of the contact surface.

5. Numerical results and discussion

Since we do not focus on a specific type of yarn, but aim atdeveloping a predictive model able to account for different typesof yarns, computations will be done for two yarns, namely glassand carbon.

We adopt the following choice of the geometrical and mechan-ical parameters, chosen as in [41,42], for the purpose of comparingwith a situation of a finite area incorporating an evolution of thecontact based on Hertz theory.

The geometrical parameters of discretization are selected as:

Lwa ¼ Lwe ¼ 0:022 m,Nwa ¼ Nwe ¼ 16,n0=period ¼ 12,NR ¼ 15.

The mechanical properties of the chosen fibers within the mul-tifilament yarns are [41]:

Young’s modulus: Ef glass ¼ 80:109 Pa and Ef carbon ¼ 238:109 Pa.The yarn behavior viewed as a bundle of filaments is related in a

complicated manner to the arrangement of the fibers and to theirindividual mechanical properties. The measurement of multifila-ment properties in relation to the organization and properties ofthe inherent fibers is a very challenging task; especially, the yarncompressive response is quite complex (nonlinear and non-regu-lar) due to the reorganization of the fibers within the yarn [47].For this reason, we rather rely on analytical models, consideringespecially the analysis given by Alagirusamy and Das [48], whostates that in the unstrained state, the modulus of the twisted yarnand constituent fiber are related by Eq. (16), and Curiskis and Car-naby [46] who express the yarn effective mechanical propertiesfrom the geometrical and micromechanical parameters inherentto the fibers within the yarn (by Eqs. (17) and (18)), as follows:

ical properties of the chosen fibers [41].

type E-Glass Carbon

Multifilament Multifilamentber of filaments 1200 6000ent diameter (lm) 16 7diameter (mm) 0.58 0.57

itudinal tensile moduli (GPa) 1.9 5.66ing stiffness (N mm2) 0.31 0.16on’s ratio mTL 0.77on’s ratio mLT 0.8 0.9 0.8 0.9sverse tensile moduli (GPa) 1.82 1.62 5.44 4.84

The longitudinal tensile moduli is given by EL ¼ Eyarn ¼ Ef �cos2ðqÞ ð16Þ

The Poisson’ ratio expresses as mTL ¼8sin2ðqÞ

p2ð2� sin2ðqÞÞð17Þ

The transverse tensile moduli is given by ET ¼mTL

mLTEL ð18Þ

In the present work, the helical angle of the fibers is chosen asq ¼ 30�. The longitudinal moduli of the bundle of fibers buildingthe yarn are evaluated based on the formula (16)–(18) and reportedin Table 1. In order to calculate the transverse modulus based on(18), we consider Poisson’s ratio mLT as a varying parameter, select-ing the values 0.8 and 0.9 (those values satisfy the conditionEL > ET ). From the previous microstructural fiber parameters, weobtain the mechanical effective properties of the two types of yarnsconsidered in this work given in Table 1.

The two sets of Kawabata constants K corresponding to the twovalues of Poisson’s ratio mLT are then determined for each type ofyarn by using Kawabata’s model:

d ¼ C 1� e�KFL

� �ð19Þ

) dddF¼ CK

Le�

KFL

Especially; for F ¼ 0;dddF¼ CK

L: ð20Þ

In the general case; it holds that ET ¼re¼

FSDlL

! DlF¼ L

ET Sð21Þ

Combining previous relations lead to an identification of the Kawa-bata constant K: consideration of relations (20) and (21) leads toCKL ¼ L

ES hence K ¼ L2

ET :S:C, with L the contact length, S the contact area

and C a constant therein. Using this procedure delivers the follow-ing parameters of the Kawabata compressibility model for the dif-ferent yarns:

For the carbon yarn:

Case1: C = 0.3 � 10�6 and K = 0.78 � 10�3.Case2: C = 0.3 � 10�6 and K = 0.876 � 10�3.

For the glass yarn:

Case1: C = 0.3 � 10�6 and K = 2.32 � 10�3.Case2: C = 0.3 � 10�6 and K = 2.61 � 10�3.

The calculated moduli are reported in Table 1. The bendingmoduli for the carbon and glass yarns are given in [41].

We then record in Fig. 5.1 the evolution of the compressive dis-placement which is evaluated by Kawabata model [43] versus theload Pwa (defined as a punctual force applied at the end points ofeach warp yarn), for the two previous sets of parameters. Theresponse is more rigid when parameter K decreases. The induced

Fig. 5.1. Variations of the compressive displacement versus the applied force Pwafor different sets of Kawabata constants.

Fig. 5.2. Variation of the contact area versus the applied force Pwa for different setsof Kawabata constants: C = 0.3 � 10�6.

Fig. 5.3. Variation of the reaction force versus the applied force Pwa .

374 F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380

evolution of the contact area and reaction forces is shown inFigs. 5.2 and 5.3 respectively: the contact area increases rapidlywith the loading in a first stage, and the slope of this evolution thendecreases with ongoing loading; this evolution will be explained inthe sequel by a deeper analysis of the deformation mechanisms ofthe fabric. It appears that the compression parameters have a weakeffect on both the reaction force and contact area.

From a numerical point of view, we have tested different dis-cretizations of the contact zone (in case of a finite contact area);the 15 discretization points adopted in all subsequent simulations

Fig. 5.4. Variations of the flexural displacements for two kinds of reinforcement:effect of the contact forces distribution (15R).

are sufficient, since we checked that no more variations of theoverall response are observed for a finer discretization.

For the two yarns, the contact area is important in comparisonto the yarn diameter: the major axis of the large diameter of thecontact area for the glass yarn is evaluated to 0.3 mm, a value sim-ilar to the warp diameter, here 0.58 mm. We notice that for thesame load Pwa, the contact area is greater for the glass yarn in com-parison to the carbon yarn (the same thing for the reaction forceevaluated in Eq. (9)), due to the greater value of the tensilestiffness.

We further analyze the deformation mechanisms, in terms ofthe yarn flexion and extension for the two types of yarns; the var-iation of the displacement in flexion is illustrated in Fig. 5.4 for apunctual contact (referred to as 1R) and a distributed contact(referred to as 15R); the flexural component of the displacementevolves according to a J-shape type response in both cases (1Rand 15R), thereby following the general trend observed in the lit-erature [16]. Fig. 5.4 further shows that the flexural componentof the displacement is sensitive to the distribution of contactforces: this is due to the fact that the reaction force increases inthe first stage of the overall response (a consequence of thedecrease of the yarn crimp, which is an easy deformation mecha-nism), before saturating in the later stage of the response.

We illustrate in Fig. 5.5 the variation of the component of thedisplacement in extension for the two yarns and with the two setsof Kawabata constants. We firstly notice that the component of thedisplacement in extension varies linearly versus the applied forcePwa, and it is further completely insensitive to the compression.This means that yarn compression mostly acts on the flexuralbehavior of the tested fabric. The influence of the yarn compress-ibility on the overall response of composite fibrous reinforcementis well-known in the literature [44]; the present analysis gives fur-ther insight into the mechanisms responsible for this effect at theyarn scale.

We observe on Fig. 5.6 an effect of the way the load distributionis modeled in the contact zone on the overall response of the fabric,for several types of reinforcement materials: the response is morerigid for a punctual contact. This behavior can be explained adopt-ing an energetic point of view: the work of the tensile force is thesame in both contact representations; the total work of the reac-tion forces as well as the internal mechanically stored energy arefound to be greater for punctual contact, thus leading to a stiffermechanical response. Whatever the type of yarns, the two familiesof curves converge to a common response independently of theway the contact is described (punctual or distributed); this canbe explained by the fact that the flexural contribution of the totaldisplacement decreases, while the extensional displacement isinsensitive to the contact forces distribution. The same also holdsfor the compression behavior.

Fig. 5.5. Variations of the extensional displacements for two kinds of reinforcementand for different sets of Kawabata constant. Punctual contact model (1R) anddistributed contact (15R).

Fig. 5.6. Variations of the displacement Duwa for different kind of reinforcement:effect of the contact forces distribution (two sets of Kawabata constant). Cases ofpunctual contact (1R) and distributed contact (15R).

Table 2Characteristics of the plain weave tested samples.

Yarns Weft Warp

Material PET PETType Monofilament MultifilamentNumber of filaments / �210Diameter (mm) 0.45 0.403Average Young’s modulus (MPa) 8260 7940Bending stiffness (N mm2) 1.66 � 10�5 2.59 � 10�6

Kawabata constants C = 0.29 � 10�4;K = 0.32 � 10�5

C = 0.56 � 10�4;K = 0.44 � 10�4

FabricSurface weight (g/m2) 440Density in warp 14 yarns/cmDensity in weft 8.5 yarns/cmCrimp (%) 16.93%Shrinkage (%) 0.30%

Fig. 5.7. Comparison between the micromechanical model and experimentaltensile tests of plain weave samples along the warp direction [45].

F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380 375

In order to provide a first assessment of our predictive modelwith experimental results [45], uniaxial tensile tests of plain weavefabric made of PET yarns have been performed in [45]; the weft is amonofilament, whereas the warp is a multifilament of PET. Thecharacterization of the tensile behavior is performed according tothe standard EN IS13934-1 on the traction machine MTS/20M.The tests have been carried out on samples of initial length of200–50 mm width with a constant elongation rate equal to20 mm/min; the parameters of the tested samples are reportedin Table 2. The compression constants given in Table 2 are deter-mined from Kawabata’s model and the compression measure-ments reporting in Appendix D the measured responses of thetested samples are then compared to the response obtained byour predictive model.

Fig. 5.7 shows the evolution of the tensile force versus the rela-tive elongation (%) in the warp direction, limited to the initial

response under small (macroscopic) strains; a satisfactory agree-ment between the tests and the simulation is obtained.

6. Conclusion

A micromechanical model of plain weave fabric has been devel-oped, based on Hertz theory to model the mechanical contactbetween the constitutive yarns, and on an energy minimizationprinciple to derive the equilibrium shape of the textile under uni-axial tensile loading. Such a micromechanical approach based on arepresentation of each yarn as an undulated beam at a mesoscopiclevel represents a good compromise between the numerical costand the accuracy in the geometrical and mechanical descriptions.A relatively simple method of analysis can accordingly beemployed to predict the tensile response of plain weave with anaccurate description of the contact between yarns. The methodol-ogy can be easily extended to other types of monolayers [13].

As a main novel aspect of this contribution, the finite nature ofthe contact area between warp and weft has been accounted for,resulting in an extended contact (elliptical contact) due to a distri-bution of the contact force over the contact zone. It has beenshown that the compression parameters have a weak effect onboth the reaction force and contact area. In the case of a distributedcontact, there is an evolution of the contact area during loading,due to the size of the contact area being of the same order of mag-nitude as the yarn diameter. The contact area rapidly increaseswith the loading in a first stage, and the slope of this evolution thendecreases with ongoing loading. Nevertheless, it appears that onlythe net effect of contact forces (through their resultant, the reac-tion force) – and not the way the reaction force is distributed onthe contact zone – is important, when considering the overall ten-sile response of fabric.

The tensile response of plain weave fabric has been simulatedfor two types of yarns under uniaxial tensile loading, and the sep-arate contribution of yarn flexion and yarn extension has been ana-lyzed. It has been shown that the component of displacement inextension is insensitive to the yarn compression and to the distri-bution of forces on the contact surface, contrary to the flexuralcomponent. The representation of contact (punctual or distributed)is important when crimp changes occur, but it does not affect theoverall response when yarn extension is the main deformationmechanism.

A satisfactory agreement between the predictions of the devel-oped micromechanical model and tensile measurements over plainweave fabric made of PET yarns has been obtained. The proposedmethodology will be extended to textile multilayers such as thickitnerlocks used as preforms in future contributions.

Appendix A. Dimension of the contact surface

For a contact between two cylinders (Fig. A.1), the contact areais an ellipse (Fig. A.2); Hertz proposes the following spatial distri-bution of the contact pressure

P0ðx; yÞ ¼ pm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� x2

a2 �y2

b2

sðA:1Þ

with the maximum pressure therein given by

pm ¼3N

2pab

N the resultant of the normal forces acting on the contact zone,and a; b the two radii of the contact ellipse, given by

a ¼ m

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3pNðk1 þ k2Þ

2 C1 þ C 01 þ C2 þ C 02� �3

s; b ¼ n

ma ðA:2Þ

Fig. A.1. Contact between cylindrical two yarns.

Fig. A.2. Ellipsoidal contact area between two cylindrical yarns.

Fig. B.1. Beam submitted to a punctual vertical force.

376 F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380

This expression involves the following parameters: the coefficientsn, m [40] versus the angle

u¼arccos

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1�C 01� �2þðC2�C 02Þ

2þ2 C1�C 01� �

C2�C 02� �

cosð2hÞq

C1þC 01þC2þC 02

0@

1A

ðA:3Þ

C1, C01 are the maximal and minimal curvatures of solid 1 in theplanes P01 and P001 respectively; C2; C02 are the maximal and minimalcurvatures of solid 2 in the planes P02; P002 respectively, h is the angle

between P01 and P02, and k1 ¼1�m2

1pE1

; k2 ¼1�m2

2pE2

depend on the elastic

properties of both cylinders. We denote E1, E2 the tensile moduliof both yarns (Fig. A.1), and m1; m2 their Poisson’s ratio.

Appendix B. Expressions of the Fourier coefficients

The vertical deflection of each yarn (modeled as a beam ele-ment) is expressed as a Fourier series with coefficients dependingupon the forces applied to this beam element.

Fig. B.2. Distribution of the reaction forces exerted by the

We first consider the case of a single concentrated force (F). Theequilibrium shape of an elastic beam of length L (Fig. B.1) is givenby Timoshenko beam theory [40] as:

wðxÞ ¼XN

n¼1

ansinnpx

L

� �ðB:1Þ

with the Fourier coefficients (before any external load) given by

an ¼2FL3

Bp4

1n4 sin

npcL

� �ðB:2Þ

B: the flexural rigidity of the beam.

N: the number of harmonics.

In the case of an external load (axial or biaxial), the Fourier coef-ficients are given by:

an ¼2FL3

Bp4

1

n2 n2 þ PPcr

� � sinnpc

L

� �

with Pcr ¼ p2BL2 the critical compressive buckling loads of the beam.

transverse yarns (weft) on the warp (labeled with k).

Fig. B.3. Spatial distribution of contact forces on a warp yarn.

F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380 377

The plain weave structure is supposed to be periodical along thex direction; the warp and weft are modeled as elastic beams, sub-jected to punctual forces applied to the top of each crimp (Fig. B.2),distributed in a periodical manner. The Fourier coefficients expressin a more detailed manner from Eq. (B.2) as:

awan;k ¼

2Fwe=waL3wa

Bwap4

1n4 sin

npc1

Lwa

� �� 2Fwe=waL3

wa

Bwap4

1n4

� sinnpc2

Lwa

� �þ 2Fwe=waL3

wa

Bwap4

1n4 sin

npc3

Lwa

� �

� 2Fwe=waL3wa

Bwap4

1n4 sin

npc4

Lwa

� �ðB:3Þ

with

c1 ¼Lwa

8; c2 ¼

3Lwa

8; c3 ¼

5Lwa

8; c4 ¼

7Lwa

8

awan;k ¼

2Fwe=waL3wa

Bwap4

1n4 sin

npLwa

Lwa

8

� �� sin

npLwa

3Lwa

8

� �

þ sinnpLwa

5Lwa

8

� �� sin

npLwa

7Lwa

8

� �

awan;k ¼

2Fwe=waL3wa

Bwap4

1n4 sin

np4

12

� �� sin

np4

32

� �

þ sinnp4

52

� �� sin

np4

72

� �

awan;k ¼

2Fwe=waL3wa

Bwap4

1n4 sin

npNwe

12

� �� sin

npNwe

32

� �

þ sinnpNwe

52

� �� sin

npNwe

72

� �

8n 2 ½1::Nwe�; awan;k ¼

2Fwe=waL3wa

Bwap4

XNwe2

m¼1

1n4 sin

npNwe

2m� 32

� �� �

� sinnpNwe

2m� 12

� �� �

ðB:4Þ

For the case of warp on weft interaction, it holds 8n 2 ½1 . . . Nwa�:

awen; j ¼

2Fwa=weL3we

Bwep4

XNwa2

m¼1

1n4 sin

npNwa

2m�32

� �� ��sin

npNwa

2m�12

� �� �

ðB:5Þ

Hence, the equilibrium states associated to the weft and warp yarns(before any external load) are expressed by:

wko�waðxÞ ¼

2Fwe=waL3wa

Bwap4

XNwe

n¼1

XNwe2

m¼1

1n4 sin

npNwe

2m� 32

� �� ��

� sinnpNwe

2m� 12

� �� ��: sin

npxLwa

� �ðB:6Þ

w jo�weðyÞ ¼

2Fwa=weL3we

Bwep4

XNwa

n¼1

XNwa2

m¼1

1n4 sin

npNwa

2m� 32

� �� ��

� sinnpNwa

2m� 12

� �� ��: sin

npyLwe

� �ðB:7Þ

Nwa, Nwe are respectively the number of warp and weft, Bwa; Bwe

respectively the flexural rigidities of the warp and weft, Lwa, Lwe

respectively the projected length of the yarns in the x and y direc-tions, m the number of considered periods and Fwa=we; Fwe=wa thereaction forces.

When the previous contact model is refined to account for a dis-tribution of contact forces on a finite size contact (Fig. B.3), the pre-vious Fourier series coefficients become

awan;k ¼

2F1we=waL3wa

Bwap4

1n4 sin

npc11

Lwa

� �þ 2F2we=waL3

wa

Bwap4

1n4

� sinnpc1

Lwa

� �þ 2F3we=waL3

wa

Bwap4

1n4 sin

npc12

Lwa

� �

� 2F1we=waL3wa

Bwap4

1n4 sin

npc21

Lwa

� �� 2F2we=waL3

wa

Bwap4

1n4

� sinnpc2

Lwa

� �� 2F3we=waL3

wa

Bwap4

1n4 sin

npc22

Lwa

� �ðB:8Þ

with

c1 ¼Lwa

4; c2 ¼

3Lwa

4; c11 ¼ c1 �

2a3; c21 ¼ c2 �

2a3;

c12 ¼ c1 þ2a3; c22 ¼ c2 þ

2a3

the abscissa of the contact points, and a the major radii of the con-tact ellipse.

awan;k ¼

2F1we=waL3wa

Bwap4

1n4 sin

npc11

Lwa

� �� sin

npc21

Lwa

� � þ 2F2we=waL3

wa

Bwap4

� 1n4 sin

npc1

Lwa

� �� sin

npc2

Lwa

� � þ 2F3we=waL3

wa

Bwap4

� 1n4 sin

npc12

Lwa

� �� sin

npc22

Lwa

� �

awan;k ¼

2F1we=waL3wa

Bwap4

1n4 sin

npc11

Lwa

� �� sin

npc21

Lwa

� � þ 2F2we=waL3

wa

Bwap4

� 1n4 sin

npc1

Lwa

� �� sin

npc2

Lwa

� � þ 2F3we=waL3

wa

Bwap4

� 1n4 sin

npc12

Lwa

� �� sin

npc22

Lwa

� �

awan;k¼

2F1we=waL3wa

Bwap4

1n4 sin

npLwa

Lwa

4�2a

3

� �� ��sin

npLwa

3Lwa

4�2a

3

� �� �

þ2F2we=waL3wa

Bwap4

1n4 sin

npLwa

Lwa

4

� ��sin

npLwa

3Lwa

4

� �

þ2F3we=waL3wa

Bwap4

1n4 sin

npLwa

Lwa

4þ2a

3

� �� ��sin

npLwa

3Lwa

4þ2a

3

� �� �

378 F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380

awan;k ¼

2F1we=waL3wa

Bwap4

1n4 sin

np2

12� 2anp

3Lwa

� �� sin

np2

32� 2anp

3Lwa

� �

þ 2F2we=waL3wa

Bwap4

1n4 sin

np2

12

� �� sin

np2

32

� �

þ 2F3we=waL3wa

Bwap4

1n4 sin

np2

12þ 2anp

3Lwa

� �� sin

np2

32þ 2anp

3Lwa

� �

awan;k ¼

2F1we=waL3wa

Bwap4

XNwe2

m¼1

1n4 sin

npNwe

2m� 32

� �� 2anp

3Lwa

� �

� sinnpNwe

2m� 12

� �� 2anp

3Lwa

� �

þ 2F2we=waL3wa

Bwap4

XNwe2

m¼1

1n4 sin

npNwe

2m� 32

� �� �

� sinnpNwe

2m� 12

� �� �

þ 2F3we=waL3wa

Bwap4

XNwe2

m¼1

1n4 sin

npNwe

2m� 32

� �þ 2anp

3Lwa

� �

� sinnpNwe

2m� 12

� �þ 2anp

3Lwa

� �

Wktraction ¼ Pk

wa

Xnp

j¼1

Xð n04�1ð Þþ ðj�1Þ:ndpð ÞÞ

i¼ 1þ ðj�1Þ:ndpð Þð ÞD cos wk

x;i

� �� cos wk

x;o;i

� �� �0@

1Aþ

D� ðNR�1ÞaNR

� �cos wk

x; n04þ ðj�1Þ:ndpð Þð Þ

� �� cos wk

x;o; n04þ ðj�1Þ:ndpð Þð Þ

� �� �� �þ

Xð NR�1þn04ð Þþ ðj�1Þ:ndpð ÞÞ

i¼ n04þ1ð Þþ ðj�1Þ:ndpð Þð Þ

2aNR

cos wkx;i

� �� cos wk

x;o;i

� �� �0B@

1CAþ

D� ðNR�1ÞaNR

� �cos wk

x; NRþn04ð Þþ ðj�1Þ:ndpð Þð Þ

� �� cos wk

x;o; NRþn04ð Þþ ðj�1Þ:ndpð Þð Þ

� �� �� �þ

Xð NR�2þ 34n0ð Þð Þþ ðj�1Þ:ndpð ÞÞ

i¼ NRþ1þn04ð Þþ ðj�1Þ:ndpð Þð Þ

D cos wkx;i

� �� cos wk

x;o;i

� �� �0B@

1CAþ

D� ðNR�1ÞaNR

� �cos wk

x; NR�1þ 34n0ð Þð Þþ ðj�1Þ:ndpð Þð Þ

� �� cos wk

x;o; NR�1þ 34n0ð Þð Þþ ðj�1Þ:ndpð Þð Þ

� �� �� �þ

Xððndp�n04 Þþððj�1Þ:ndpÞÞ

i¼ NRþ 34n0ð Þð Þþ ðj�1Þ:ndpð Þð Þ

2aNR

cos wkx;i

� �� cos wk

x;o;i

� �� �0B@

1CAþ

D� ðNR�1ÞaNR

� �cos wk

x; ndpþ1�n04ð Þþ ðj�1Þ:ndpð Þð Þ

� �� cos wk

x;o; ndpþ1�n04ð Þþ ðj�1Þ:ndpð Þð Þ

� �� �� �þ

Xðndpþððj�1Þ:ndpÞÞ

i¼ððndpþ2�n04 Þþððj�1Þ:ndpÞÞ

D cos wkx;i

� �� cosðwk

x;o;i� �0

@1A

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

þ ukNdþ1

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

awan;k ¼

2L3wa

Bwap4

X3

p¼1

Fpwe=wa

XN2

m¼1

1n4

sin npNwe

2m� 32

� �þ 2 p�2ð Þanp

3Lwa

� �

� sin npNwe

2m� 12

� �þ 2 p�2ð Þanp

3Lwa

� �2664

3775

0BB@

1CCA

In the general case (for an arbitrary number of reactionforces): it holds the folloying expression of the Fourier seriescoefficients

awan;k¼

2L3wa

Bwap4

XNR

p¼1

Fpwe=wa

XNwe2

m¼1

1n4

sin npNwe

2m�32

� �þ2 p�ðNRþ1Þ

2

� �anp

NRLwa

� �

�sin npNwe

2m�12

� �þ2 p�ðNRþ1Þ

2

� �anp

NRLwa

� �266664

377775

0BBBB@

1CCCCA

ðB:9Þ

and

awen;j ¼

2L3we

Bwep4

XNR

p¼1

Fpwa=we

XNwa2

m¼1

1n4

sin npNwað2m�3

2Þþ2 p�ðNRþ1Þ

2

� �anp

NRLwe

� �

�sin npNwa

2m�12

� �þ2 p�ðNRþ1Þ

2

� �anp

NRLwe

� �266664

377775

0BBBB@

1CCCCA

ðB:10Þ

with NR the number of discretization points in the contact zone.

Appendix C. Expressions of the work of the traction effort andof the deformation energy

The work of the applied load Pwa exerted on the warp is given by

Wktraction ¼ PwaDuk

wa ¼ Pwa Duk; fwa þ Duk;ex

wa

� �ðC:1Þ

with ndp the number of discrete elements on a wave,np ¼ Nwe=2; Nd ¼ ndp� np, n0 ¼ ndp� ððNR � 1Þ � 2Þ, NR the num-ber of discretization points in the contact zone and D the curvilineardistance between two constitutive nodes (Fig. 2.1). The index 0refers to the initial value.

The expression of the internal deformation energy is given by

Ukwa ¼ Uk

f þ Ukex þ Uk

comp ðC:2Þ

with

Ukex ¼

Xnp

j¼1

Xððn04�1Þþððj�1Þ�ndpÞÞ

i¼ð1þððj�1Þ�ndpÞÞ

12 Ck

ei ukiþ1 � uk

i

� �2 þ 12 Ck

e1 i ukðn04þ1Þþ ðj�1Þ�ndpð Þð Þ � uk

n04þððj�1Þ�ndpÞð Þ

� �2

þXððNR�1Þþn04þððj�1Þ�ndpÞÞ

i¼ ðn04þ1Þþððj�1Þ�ndpÞð Þ12 Ck

e2 i ukiþ1 � uk

i

� �2 þ 12 Ck

e1 i ukðNRþ1Þþn0

4þ ðj�1Þ�ndpð Þð Þ � ukNRþn0

4þððj�1Þ�ndpÞð Þ

� �2

þXððNR�2Þþ3�n04 þððj�1Þ�ndpÞÞ

i¼ ðNRþ1Þþn04þððj�1Þ�ndpÞð Þ

12 Ck

ei ukiþ1 � uk

i

� �2 þ 12 Ck

e1 i ukNRþ3�n0

4 þ ðj�1Þ�ndpð Þð Þ � ukðNR�1Þþ3�n0

4 þððj�1Þ�ndpÞð Þ

� �2

þXðndp�n0

4þððj�1Þ�ndpÞÞ

i¼ NRþ3�n04 þ ðj�1Þ�ndpð Þð Þ

12 Ck

e2 i ukiþ1 � uk

i

� �2 þ 12 Ck

e1 i ukndpþ2�n0

4þ ðj�1Þ�ndpð Þð Þ � ukndpþ1�n0

4þððj�1Þ�ndpÞð Þ

� �2

þXðndpþððj�1Þ�ndpÞÞ

i¼ ndpþ2�n04þððj�1Þ�ndpÞð Þ

12 Ck

ei ukiþ1 � uk

i

� �2

266666666666666666666666666664

377777777777777777777777777775

ðC:3Þ

UkF ¼

Xnp

j¼1

Xððn04�1Þþððj�1Þ�ndpÞÞ

i¼ 1þ ðj�1Þ�ndpð Þð Þ

12 Ck

fi wkx;iþ1 � wk

x;i

� �2þ 1

2 Ckf1 i wk

x; ðn04þ1Þþððj�1Þ�ndpÞð Þ � wkx; n0

4þ ðj�1Þ�ndpð Þð Þ� �2

þXððNR�1Þþn04þððj�1Þ�ndpÞÞ

i¼ ðn04þ1Þþððj�1Þ�ndpÞð Þ12 Ck

f2 i wkx;iþ1 � wk

x;i

� �2þ 1

2 Ckf1 i wk

x; ðNRþ1Þþn04þ ðj�1Þ�ndpð Þð Þ � wk

x; NRþn04þ ðj�1Þ�ndpð Þð Þ

� �2

þXððNR�2Þþ3�n04 þððj�1Þ�ndpÞÞ

i¼ ðNRþ1Þþn04þððj�1Þ�ndpÞð Þ

12 Ck

fi wkx;iþ1 � wk

x;i

� �2þ 1

2 Ckf1 i wk

x; NRþ3�n04 þ ðj�1Þ�ndpð Þð Þ � wk

x; ðNR�1Þþ3�n04 þððj�1Þ�ndpÞð Þ

� �2

þXðndp�n0

4þððj�1Þ�ndpÞÞ

i¼ NRþ3�n04 þððj�1Þ�ndpÞð Þ

12 Ck

f2 i wkx;iþ1 � wk

x;i

� �2þ 1

2 Ckf1 i wk

x; ndpþ2�n04þððj�1Þ�ndpÞð Þ � wk

x; ndpþ1�n04þ ðj�1Þ�ndpð Þð Þ

� �2

þXðndp�1þððj�1Þ�ndpÞÞ

i¼ ndpþ2�n04þððj�1Þ�ndpÞð Þ

12 Ck

fi wkx;iþ1 � wk

x;i

� �2þXnp�1

i¼1

12 Ck

fi wkx; ðndp:iÞþ1ð Þ � wk

x; ndp:ið Þ

� �2

266666666666666666666666666664

377777777777777777777777777775

ðC:4Þ

Fig. D.1. Sectional view of the transverse compression device.

F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380 379

The coefficients Ce, Ce1, Ce2, Cf , Cf 1, Cf 2 therein are respectively theyarn rigidities in extension and flexion, expressed versus the yarntensile and flexural rigidities by

Ce ¼EAwa

D; Ce1 ¼

EAwa

D� ðNR�1Þ�aNR

; Ce2 ¼EAwa

2�aNR

; Cf ¼EIwa

D; Cf1

¼ Bwa

D� ðNR�1Þ�aNR

; Cf2¼ Bwa

2�aNR

and

Ukcomp ¼

Lwas

K1C1 � dwa;k

comp

� �ln 1�

dwa;kcomp

C1

!þ dwa;k

comp

" #ðC:5Þ

Lwas is the curvilinear length of a warp yarn; C1, K1 are the Kawabata

constants of the warp, and dwa;kcomp represent the deformation under

compression of the warp.

Appendix D. Transverse compression behavior

Fig. D.1 shows the experimental transverse compression devicedeveloped at LPMT [45]. This device is composed of three parallelplates: the lower plate is attached to the fixed portion of the trac-tion machine. The intermediate plate is connected to the mobilecrossbar and guided in translation. The translational movementof the crossbar ensures the compression of the yarn against theconical glass plate stuck in the upper plate.

Fig. D.2. Transverse compression response of the weft yarns [45].

Fig. D.3. Transverse compression response of the warp yarns [45].

380 F. Chaouachi et al. / Composites: Part B 66 (2014) 368–380

Furthermore, two displacement sensors, with measurementrange �1 mm, directly in contact with the glass plate, used to mea-sure the relative movement between the two compression plates.A force sensor, placed under the tested yarn, measures the com-pressive force exerted on the sample. The adopted compressionspeed is 1 mm/min. The experimental results of the compressionmeasurements are exposed in Figs. D.2 and D.3. The compressionconstants given in Table 2 are determined from Kawabata model(see Eq. (19)), the non-linear least squares method and the com-pression measurements.

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