Modeling of Elastic, Thermal, and Strength/Failure Analysis of Two-Dimensional Woven Composites—A...

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Levent Onal1

Sabit Adanur2

Department of Textile Engineering,Auburn University,

115 Textile Building,Auburn, AL 36849

Modeling of Elastic, Thermal,and Strength/Failure Analysis ofTwo-Dimensional WovenComposites—A ReviewThe usage of textile structures as a reinforcement for polymer composites became essen-tial in many industrial applications in, for example, the marine and aerospace industriesbecause of their favorable stiffness and strength to weight ratio. Determination of elasticproperties and failure behavior of textile reinforced composites is vital for industrialdesign and engineering applications. This paper aims to present a review of numericaland analytical models for elastic, thermal, and strength/failure analysis of 2D reinforcedwoven composites. Major modeling techniques and approaches are presented. A state ofthe art review of woven fabric composites is presented starting from earlier one-dimensional models to recent three-dimensional models. The intention is not to give adetailed analysis of the mathematical approaches to the models discussed, but rather toinform researchers about the main ideas of previous works. This review article cites 122references. �DOI: 10.1115/1.2375143�

ntroductionPolymer matrix composites continue to offer undeniable prom-

se for meeting future industrial design challenges. Fiber rein-orced composites are widely used in lightweight structural com-osites for automotive, marine, aircraft, and civil engineering.heir favorable mechanical properties and efficient reinforcementaterials with low fabrication cost and easy handling secure their

lace in the industry �1–4�. In terms of structural design, charac-erization of textile composites is getting important. Nevertheless,he anisotropic and heterogeneous nature of composite materialsffects the prediction of their mechanical properties.

Composite mechanics has grown to include a wide range ofontinuum and discrete methods. These methods are used to studynd predict the fiber/matrix composite behavior �5�. The efforts ofarlier researchers were just to simplify the problem by makingany assumptions, which could generalize the structure, but give

omewhat unrealistic results. There could be distinct differencesetween the experimental data and predicted results. The relevantpproaches included various rules of mixture approximations, cy-indrical and hexagonal models, and individual boundary methods6�. However, the cost and labor intensive ways of experimentallyetermining mechanical properties directs the investigations toodeling and other predictive tools. Modeling techniques can pre-

ict fabric weight, volume fraction ratio, weave architecture, yarnrimp �undulation�, and properties of constituents for the mostppropriate cost effective ways.

Studies on composite mechanics progressed from unidirectionalomposites to three-dimensional fabric reinforced composites.arly studies were started in the 1960s. The work of Shaffer �7� isased on the idealized model in which both resin and fiber are inhe form of rectangular bands that fill out the full layer. Hashinnd Shtrickman �8� developed a method for obtaining bounds forhe effective moduli of composite elastic materials by a variationalpproach. Then, by using the variational method, Hashin andosen �9� derived bounds and expressions for the effective elasticoduli of fiber reinforced materials arranged in both hexagonal

1Current Address: Department of Textile Engineering, Erciyes University, 38039ayseri, Turkey.

2To whom correspondence should be addressed.
Transmitted by Assoc. Editor S. Adali.
pplied Mechanics Reviews Copyright © 20

om: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 09

and random arrays. The study of Chen and Cheng �10� is con-cerned with the hexagonal array employing infinite series solu-tions, such as a Fourier method and the sum of squares ofweighted errors in the least square method. Whitney �11� consid-ered filament anisotropy and assumed the domain as transverselyanisotropic with the filament axis as the axis of elastic symmetry.

Woven FabricsWeaving is the interlacing of two constituents that are orthogo-

nal to each other. Warp yarn is that yarn system in woven fabriclying parallel to the selvedges of the cloth. The weft �or filling� isthe yarn system in a woven fabric lying across the width of thecloth perpendicular to warp yarns �12�. Within the woven fabricstructure, effective yarn diameter and yarn length in unit weaveare significant for fabric modeling. Plain, twill, and satin are threefundamental weaves that are usually used in composites �Fig. 1�.Plain weave is a commonly used basic and balanced reinforce-ment for woven composites. The twill weave has a looser inter-lacing and the weave is characterized by a diagonal line. The satinweave has good drapability and smooth surface with minimumthickness �13�.

Woven fabrics that are used as reinforcement in composites canbe classified as two-dimensional �2D� and three-dimensional �3D�structures. In 2D structures, the thickness of the fabric is smallcompared to its in-plane dimensions. Two-dimensional wovenfabrics are generally anisotropic, have poor in-plane shear resis-tance and have less modulus than the fiber materials due to yarnundulation caused by intersecting points of the constituents �14�.Within the composite structure, the effective thermal and thermo-mechanical properties depend on weave style, properties of theconstituent materials, and fiber volume fraction �15�.

In the case of 3D woven structures, out-of-plane properties areas important as in-plane properties, and cannot be neglected as itis possible for 2D structures. Modeling studies on 3D structuresare not in the scope of this review.

In general, mechanical properties of woven fabrics depends onweave parameters such as yarn size, yarn spacing, weave archi-tecture, and laminate parameters such as stacking sequence and
ply angle.
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bjectives and Approach2D structures, particularly woven fabric composites, have been

ecognized to be more suitable than unidirectional composites fortability with the essence of warp and weft yarns. Such constitu-nts offer balanced properties in the fabric plane. Such advantagesave caused an increasing interest in the use of woven fabricomposites for design applications and have attracted attention tohe studies for predicting mechanical properties of woven com-osites. Major modeling techniques have been underlined initially.his paper has been designed in the order of elastic analysis,

hermal analysis, and strength/failure analysis. In addition, studiesn elastic analysis are arranged into two subsections as analyticalethods and numerical methods. This review aimed to gather the

evelopment steps of woven fabric reinforced composites and toresent the researchers’ background information of modeling forurther improvements.

valuation of ModelsDifferent methods have been used to predict the mechanical

roperties of composite materials. The Saint Venant sense andifferent techniques related to the least square method were usedn the early 1960s �7–10�. Elementary laminate theory and nu-

erical methods �particularly finite element and finite differenceethods� are three different modeling options that are available in

he literature.The elementary models give simple forecasts of elastic proper-

ies of woven fabric lamina. Each lamina was modeled as a uni-irectional structure, a network of straight fiber bundles, a seriesf curved beams on elastic foundations, and a homogenous aniso-ropic material �16�.

For the last 20 years, more computer compatible techniquesave been used for mechanics of composites. Finite element meth-ds �FEMs� have attracted more attention because of their adapt-ble strength for computerization. It is a common tool to obtainhe stress-strain properties in a constrained elastic or inelasticolid. The finite element method is easy to apply for solid con-tructions with almost all levels of integrity. Computer adaptabil-ty allows overcoming the complexity of matrices, so that 3D

odels are simpler to generate. Standard application procedure ofEMs to textile reinforced composites can be in the followingrder: �1� defining the textile composite structure in terms of unitells and analyzing the unit cell using FEM and �2� rearranginghe entire geometry depending on the unit cells for predicting

echanical properties.The homogenization method, which is mostly based on math-

matical theories, has become very useful in studying the micro-tructure of composites. The homogenization solutions basicallynvolve two main parts. The homogenization part represents theeriodically heterogeneous material with relatively equal homoge-eous properties. The second part is the disturbance area caused

Fig. 1 X diagrams of some basic weaves

y local irregularities. In the approach using homogenized theory

8 / Vol. 60, JANUARY 2007


by Lene and Leguillon �17�, Lene �18�, and Jonsson �19�, a unitcell problem governing mechanical properties for composites withperiodic microstructure has been originated by employing anasymptotic expansion of the field variables in two length scales.The fundamentals of the homogenization method was publishedby Babuska �20�, and Benssaussen et al. �21�. Homogenizationmodels can be based on analytical and numerical approaches. Re-lationships between the properties of reinforcement and matrixand the final elastic properties of the woven composite can easilybe modeled using an analytical approach. Other homogenizationmodel concepts can be based on laminate theory and energy ap-proach using numerical methods �22,23�. Cox and Flanagan �24�described the basis of the energy approach in a review of appli-cation of analytical methods on textile composites. Several re-searchers have been utilizing the analytical approach to modelfabric waviness �25–27�, in-plane as well as out-of-plane proper-ties �23,24�. Some researchers adapt the analytical models to com-puter software with the help of the finite element method. Thosemodels enable practical ways for structural design and fractureand damage analysis �28,29�.

Mathematical homogenization originated from partial differen-tial equations �20,30–34�. The basis of this method is that a pa-rameter is introduced that describes the fineness of the microstruc-ture. It could be practical to employ the method for nonlinearproblems due to nonobligatory periodic essence.

Analysis of Elastic Properties

Analytical Methods. The analytical studies to predict the elas-tic properties of woven composites started in the early 1970s.Halpin et al. �35� employed the lamination analogy for predictingthe elastic stiffness of two- and three-dimensional composites.The warp and weft yarns in the unit cell of a woven fabric laminawere modeled as angle plied laminates, which were then stackedto form balanced and symmetrical laminates. The theoretical re-sults for woven fabric lamina were shown to be only qualitativelycorrect.

The frame of the analytical models on woven composites wasexpanded by Ishikawa and Chou �26�, who used the laminatetheory to determine mechanical behavior of woven fabric rein-forced composites. They presented three models with differentassumptions for in-plane properties of basic woven structures.These are the mosaic model �3,26,36�, the fiber undulation model�3,37�, and the bridging model �37�. The mosaic model handles afabric composite as an assemblage of asymmetrical cross-plylaminates �Fig. 2�. Whether the pieces of cross-ply laminate are iniso-strain or in iso-stress condition or not, the boundaries of thestiffness predicted from the mosaic model can be assessed. This isa one-dimensional �1D� model, in which strand continuity andstress disturbance of the assemblage are omitted. The isostraincondition depends on the assumption of constant strain state in themid-plane lamina while the isostrain condition considers constantstress. The applicability of classical laminate plate theory is ofprinciple value in this model. The transverse dimension of a com-posite should be significantly smaller than its in-plane direction.

Fig. 2 The schematic of the mosaic model for an eight-harness satin †26‡

Upper bounds were given by parallel model and lower bounds of

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nfinite stiffness constants were given by series model.Fiber continuity, which is not considered in the mosaic model,

as been included in the fiber undulation model. However, theodel still does not include transverse loading effect in the struc-

ure. Elastic and knee behavior of a threadwise strip of a fabricomposite is another approach of the series model. The unit cellFig. 3� idealizes the structure as three parts that are a straightross-ply region, an undulated cross-ply region, and a pure matrixegion. Even though the undulation model is one step ahead of theosaic model, it is applied on the straight regions only. Classical

aminate plate theory is used in undulated regions for each infini-esimal slice of the threadwise strip. The fiber undulation modelas mainly designed for structures with a small repeat unit such

s plain weave. The authors implied that for a single layer ofabric, coupling between extension and bending was present.ower bounds for stiffness were determined when bending wasermitted, whereas upper bounds were obtained when curvatureas required to be zero. This was in agreement with the conclu-

ions of Whitcomb et al. �38�, where they conducted researchsing the finite element method. It was asserted �39� that the fiberndulation model has some similarities with the curved fiberheory �40�.

To overcome the drawbacks of the mosaic and undulation mod-ls, the bridging model was proposed, particularly for satin weave,ecause of the presence of non-interlacing regions. The interlacedegions are separated from one another. The concept of the bridg-ng model is given in Fig. 4, where the unit cell is determined as

hexagonal shape. The unit cell consists of an interlaced regionnd its surrounding areas. Four regions in the model consist oftraight fill threads and can be regarded as pieces of cross-plyaminates. The fifth region has an interlaced structure with anndulated fill thread. This model is a combination of the mosaicnd undulation models with a different approach and is separatedrom the others as being a two-dimensional �2D� model. However,he undulation and yarn continuity are taken into account onlylong the longitudinal direction. It was mentioned that the undu-ation model characterizes the plain weave better than the bridge

odel since there is hardly any straight thread region in the plaineave �37�. The predictions of the elastic moduli of fiber compos-

tes derived from the three analytical models above were com-ared with experimental data �41�.

Naik and Shembekar �42–44� suggested a two-dimensionalrimp model for elastic analysis of two-dimensional plain weave,hich is the extension of Ishikawa and Chou’s one-dimensionalodels �3,26,36,37�. Their approach was published in three parts.amina analysis of plain weave fabric was presented in the first

Fig. 3 Schematic of the fiber undulation model †26‡

art. Part 1 takes different parameters into consideration such as

pplied Mechanics Reviews


yarn thickness, undulated length of yarn, yarn continuity in thewarp and weft directions, actual cross-sectional geometry of yarn,and gap between adjacent yarns. The unit cell in this model con-sisted of individual divided sections �Fig. 5�. Two methods toassemble the discrete sections were proposed: series-parallelmodel �SP� and parallel-series �PS� model. Predictions deducedfrom these models were compared to each other in terms of elasticconstants. Elastic analysis of woven fabric laminated composites

Fig. 4 Schematic of the bridging model †37‡

Fig. 5 The unit cell of plain weave lamina where the points of
subdivision are along the X and Y axes †42‡
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as given in the second part in which consideration was mainlyocused on the stacking sequence of different laminae with respecto each other. Lamina sequence was found to affect the elasticroperties. Overall fiber volume fraction for woven fabric lami-ates was found to be very close to that of unidirectional cross-plyaminate using appropriate yarn undulation selection and laminahickness. Part III contains laminate design that is confined towo-directional orthogonal plain weave fabric composites underniaxial tensile loading. The effect of the gap between adjacentarns and the effect of laminae sequence with different ply num-ers on the elastic constants of woven laminates were studied. Itas found that the optimum gap within the constituents dependsn the fabric structure and material system.

As a further progression for the works �42–44�, Naik andanesh �45� considered the undulation and continuity of the yarn

n the principle constituents directions. The gap between adjacenttrands, different materials, and geometrical properties of warpnd weft yarns including their cross-sections were also taken intoccount. Two “refined models” were presented: the slice arrayodel �SAM� and the element array model �EAM�. Fiber volume

raction and elastic constants were deduced from these idealizedodels. Fibers were accepted as transversely isotropic and their

roperties were evaluated from the composite cylinder assem-lage model �46,47�. The analysis was carried out by dividing thenit cell into several slices. These slices are the idealized form ofctual crimped parts converted into four-layered laminate includ-ng matrix sections. The EAM is proposed to overcome the con-traints of SAM such as mismatching results when an off-axisngle is high. The unit cell is divided into three elements of in-nitesimal thickness instead of four in SAM. Then the elasticonstants of the warp and weft yarns are transformed for the localff-axis angle. The results of the refined models were reported toave been matched well with the experimental procedure.

Jortner �48� suggested a simple model to determine the changesn stiffness due to ply nesting in carbon/carbon plain weave com-osites. In this model sinusoidal lines represented the fill yarns inhe idealization of several elementary stacking sequence of fourloth layers. Dividing the lines into a number of elements, theffective elastic properties were computed using a volume-eighted stiffness averaging method. The handicap of this ap-roach was the simplicity of the idealized fiber architecture, whichimits investigation into localized effects, such as changes on towross-sectional shape, which in turn provides insufficient detail fortrength analysis.

As a further work of �49� on thermoelastic analysis for wovenomposites, Naik �50� found that the crimp angle increased sig-ificantly with increasing yarn size for the same yarn spacing,arn packing density, and overall fiber volume fraction. Shearoduli and Poisson ratio in the plane axis were expressed to be

nsensitive to yarn size.Falzon et al. �51� formulated a technique to analyze the com-

action problem with a pointwise lamination approach using clas-ical laminate theory. Although the model was based on the fiberndulation model developed by Ishikawa and Chou �3,26,36�, it ispplied in both warp and weft directions and has some similaritiesith the one used by Naik and Shembekar �42,43�, with ply nest-

ng being ignored. The proposed analysis was only valid for crimpngles less than 20°. Comparison was performed between the nu-erical results generated from the model and experimental data

nd other analytical methods.Scida et al. �52� generated a micromechanical model calledESOTEX, which is similar to Ganesh and Naik’s �53–55�, and

xtended it to other woven composites while adapting the hybrid-zation principle. The aim of the model was to predict the elasticehavior of composite materials with nonhybrid fundamentaleaves �plain, twill, and satin� and hybrid weave fabrics. The unit

ell for the proposed model is given in Fig. 6. The classical thinaminate theory was employed in which yarn undulations in both
arp and weft directions were taken into consideration. This per-


mitted one to incorporate the geometrical and mechanical param-eters of each constituent �resin, warp and weft yarns� withoutsacrificing the relative translatory motion caused by superposingseveral laminae.

Vandeurzen et al. performed 3D micromechanical analysis ofwoven composites in two parts �13,56�. After proposing a geomet-ric analysis for woven fabrics �13�, the authors proceeded to use adiscretization procedure with micromechanics to determine ho-mogenized stiffness parameters �56�. A new geometric concept,consisting of a library of 108 rectangular macrocells to build acomplex material unit cell, was generated to predict the most im-portant aspects of composite behavior. Each macrocell was subdi-vided into microcells. Elastic analysis was carried out based onthe mathematical descriptions of macro- and microcells. The newmodel had the capability to predict the stiffness matrix as well asshear modulus accurately.

Stewart et al. �57� investigated the in-plane properties of intra-ply hybrid glass/carbon satin weave composites. Mechanical prop-erties of a single-layer hybrid satin weave were analyzed initiallyand then the analysis was extended to a laminate. The laminateswere considered as symmetrical to eliminate coupling and bend-ing coefficients of the stiffness matrix. It was found that 0°/45°/0°laminate order provides the best in-plane stiffness property whenthe layers in the laminate are identical. These results were ex-panded to the optimization of in-plane and bending propertiessuch as laminate and middle layer thickness �58�.

Chaphalkar and Kelkar �59� developed an analytical modelbased on the classical laminate theory �CLT� to predict the stiff-ness of twill woven composites. Their new approach to CLT con-sidered tow undulations and continuity along the warp and weftdirections. The model has the capability to take into account vari-ous tow cross sections such as oval and rectangular.

Bystrom et al. �30� applied mathematical homogenization topredict elastic properties of fiber reinforced composites. The tech-nique was called reiterated homogenization. The authors adaptedthe homogenization method to woven fabric composites for com-putation of stiffness matrix �23�. Results from the reiterated ho-mogenization technique were compared with the existing semi-analytical models of the previous authors �28,54�. The comparisonwas performed to predict the elastic properties of the tows andalso the macroscopic elastic properties of woven composites. A

Fig. 6 Unit cell of a 2/2 twill fabric where it was divided intothree subsections along the Y axis †52‡

critical discrepancy between the results of semi-analytical models



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nd reiterated homogenization technique was reported.Hoftsee et al. �60� generated a geometric description for plain

eave fabric in the first part of their three papers. Fabric hadndergone in-plane shear deformation during forming. A rela-ively simple geometric analysis was followed and the effects ofber strengthening under tension or compression were presented.he authors used these geometric relations in conjunction with ainamatic draping simulation to determine the deformed fabricrchitecture of a draped component for determining a local stiff-ess matrix in sheared regions �61�. The experimental verificationas performed in the last part of the series, in which they com-ared predictions with measurements, both in the tow and biasirections, from specimens with shear deformations �62�.

Determining the fiber orientation is one of the important issuesn the optimal design of laminated composite structures. Grediac63� focused on the aspect of optimal design in terms of stiffnessroperties of laminated thin woven composites. Elastic propertieshat require in-plane bending and coupling were considered in theesign procedure. A stacking sequence, which is necessary forechanical properties, was obtained from the optimization

rogram.Reissner mixed variational principle is an analytical method for

redicting stress fields of fibrous composites. Roy and Sinh64,65� applied this method to woven fabric composites. In therst part of the series, mathematical formulation of the structureas made including the geometrical analysis. Stress distribution

long the warp and weft of the unit cell was analyzed at a repre-entative volume element �Fig. 7�. The results received from thisechnique were validated with the FE results. According to theomparison, the shear-stress continuity is only achieved with theenalty approach. However, there is no stress continuity at theubregions where the thicknesses are small.

Numerical Methods. Numerical methods are employed forredicting the elastic constants of textile composites because ofhe difficulties in solving some complex equations using analyticalechniques. Among the numerical methods, finite element analysisas been widely used for the analysis of textile reinforced com-osites. Zhang and Harding �66� employed the strain energyquivalence principle using a finite element method for microme-hanical analysis of the elastic constants of a plain weave fabricamina. The fiber undulation model was used for plain weaveabric in only one direction. As a result of the comparison betweenhe numerical and experimental data, it was suggested that thendulation model be extended to two dimensions.

Whitcomb �67� developed an iterative algorithm for global/ocal finite element stress analysis. The form of analysis describedn the article used global and local meshes with appropriateoundary conditions �Fig. 8�. The limitation was due to differ-nces in the stiffness matrix of the global and local models.

The unit cell continuum model �UCCM� presented by Foye

ig. 7 Representative volume element of the plain weave com-osite with six subregions †65‡

68� incorporated a virtual displacement concept in the finite ele-



ment solution. The finite element solution consisted of elementsthat were heterogeneous hexahedra bricks. The heterogeneous so-lution has a limitation, although it is successful in cutting downthe number of elements. In addition, the model needs high com-putational memory and long run time requirements.

Woo and Whitcomb �69� conducted research on the applicationof the finite element method for mechanical evaluation of wovencomposites. They subsequently extended their two-dimensionalmodel to three dimensions �70�. Macroelements were divided intosmaller domains to calculate the stiffness matrix. The nodal dis-placements were used as the boundary conditions for a submodelof the unit cell. Despite great computational savings, discrepan-cies at the boundaries of the global/local domain still remained.

Chapman and Whitcomb �71� studied the stress distributionswithin the unit cell plain weave reinforced composites using con-ventional finite elements. They exploited the symmetry of the unitcell by analyzing a small section of the geometry under appropri-ate periodic boundary conditions. Two different methods wereused specifying the material orientation within the tow. This wasdone by a convergence study to determine the number of ele-ments. The meshes used in the convergence study are given inFig. 9. The predictions of failure stress were not presented explic-itly, despite the use of a modified Tsai Hill criterion.

Whitcomb et al. �38� conducted numerical studies for wovenfabric composites using the FE method. They concluded that theupper bound approximation is more appropriate and is exactlytrue for perfectly symmetric laminates in the case of multilayeredlaminates. This finding was in agreement with Ishikawa and Chou�37�. Another finding of Whitcomb et al. was that stiffness in-creased with the number of layers when eight or more layers wereconsidered.

Fig. 8 „a… Global/local analysis of a sheet involving two dis-tinct finite element meshes. „b… Locally refined global model†67‡.

Mathematical instabilities can appear when there is a large dif-

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erence in the stiffness matrix of the fibers and composites duringhe application of the UCCM �68�. Solving the accuracy problemaused by this drawback was attempted with a modification byowayed et al. �72� using a homogenization approach with their

abric geometry model �73�. This fabric geometry model was un-er the framework of the finite element solution to homogenizehe material at the subelement level. Therefore, a modified UCCM

odel and fabric geometric model were integrated to ensure ac-urate representation of complex fabric preforms. The modifiedechnique was compared with the experimental results for fivearness satin weaves and three-dimensional orthogonal weave.owever, the predicted results only matched well with the experi-ental results for in-plane properties.A method for analyzing the stress-strain state of a loaded textile

omposite material was evaluated by Glaessgen et al. �74� usinghe I-DEAS geometric finite element modeling software. Theirechnique provided a physical basis for comprehending the dis-lacement, strain, stress, and failure parameters of plain weaveomposites. However, they did not present a comparison of theiresults with other models.

Iso-strain, flexural, and finite element models were developedor the investigation of elastic properties and stress distribution oflain weave composites under in-plane tensile loading by Ito andhou �75�. Analysis was performed on various configurations

rom unit cell �iso-phase� to random phase laminate. A two-imensional geometrical model was generated. The iso-strainodel was applied to the unit cell and then it was expanded to

ut-of-phase and random-phase laminates using a laminate model.he analytical results were evaluated by finite element analysisnder boundary conditions determined in a 2D geometric model.

Computational simulations of the shaping process have beenroposed based on mechanical approach and finite element

Fig. 9 Finite element meshes for a

ig. 10 One-quarter elementary mesh of finite element simu-
ation for the unit cell of woven fabric †76‡


method instead of usual geometrical approaches. Boisse et al. �76�analyzed nonlinearities induced by undulation variations of theyarns during deformation from biaxial tests. Finite element simu-lation for the unit cell of woven fabric was formed in the field oflarge strains �Fig. 10�. The deformation energy was calculated asthe sum of the energies of each elementary cell. It was found thatthe deformation of the fabric is low and the influence of the un-dulation variations is more important.

A multi-scale three-dimensional modeling technique that couldbe useful for theoretical and finite element analysis was studied topredict the linear elastic properties for opened-packing woven fab-ric unit cells by Tan et al. �77�. The developed technique involvesmacro-unit-cell-blocks that are divided into several microblocks.Macro- and microblocks were defined for plain, twill, and satinweave individually. The technique was valid for pure tension andshear response. The capability of the modeling technique wasverified using theoretical and finite element analysis for individualweaves. A good correlation was observed among the results.

Tan et al. �78� compared experimental and numerically derivedfindings of elastic constants for carbon/epoxy plain weave com-posites. They presented two models for the elastic constants andfailure strengths. A 3D finite element model for elastic constantswas the sinusoidal yarn model. 3D FEA mesh of the unit cell isgiven in Fig. 11. The other one was an analytical model for failurestrengths named the sinusoidal beam model. The results showedthat failure strengths were closely related to fiber volume fractionof a yarn and the mechanical properties were closely related to theoverall fiber volume fraction of the composites. Only single unitcell geometry was analyzed using the finite element method andthe authors did not use FE analysis to determine failure behavior.

A global/local approach is an effective way to study the stressdistribution of a structure in great detail. Substructuring, hybridtechniques, and exact zooming can be examined with global/localanalysis techniques. Haryadi et al. �79� utilized this technique tocalculate the static response of simply supported composite plateswith small cracks. The Ritz method was employed to compute theresponse of the structure with a global approach. A local approachis taken in a small area in the neighborhood of the cracks, discri-tized using a finite element mesh.

Tabiei and Jiang �80� developed a simple model that is a two-dimensional extension of woven fabric with an interface contain-

t cell of plain weave composite †71‡

Fig. 11 3D FEA mesh for a plain weave unit cell „removing the
matrix from the model… †78‡


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ng nonlinear finite element code. A micromechanical compositeaterial model for woven fabric with nonlinear stress-strain rela-

ions is developed and implemented using ABAQUS for nonlinearnite element structural analysis. The representative unit cell wasivided into many subcells, and uniform stress distribution in eachubcell was assumed to obtain the effective stress-strain relationsFig. 12�. Numerical simulation contained shear nonlinearity. Theeveloped formulation was not validated experimentally.

Jiang et al. �81� employed a homogenization method that en-

Fig. 12 Micro-model devel

bles the constitutive equations for local/global analysis of plain



weave fabric composites. Effective engineering constants for awoven composite were obtained from a novel micromechanicsbased method. A representative unit cell of a plain weave compos-ite was divided into many infinitesimal subcell blocks to obtainthe effective stress-strain relations of each subcell. Therefore,based on the average strains, the global average stresses can bedetermined. Numerical results gathered from global averagestresses were generated for engineering constants and compared toother numerical predictions.

d by Tabiei and Jiang †80‡

The anisotropic form of the linear Hook’s law was studied by

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ehrabadi and Cowin �82�. The maximum number of eigenten-ors for any elastic symmetry was found to be six. Bejan andatresu �83� computed the components of this elasticity tensorith Borland C3.1 language. Eigenvalues and eigenvectors of the

tiffness or compliance matrix were obtained using tensorial pre-entation of Kelvin formulation. The approach was applied tolain weave fabric lamina.

Lomov et al. �84� developed a mathematical model of the in-ernal geometry of 2D and 3D woven fabrics that is used as a unitell geometry preprocessor for meso-mechanical models of com-osite materials with minimum energy principle. The developedoftware CETKA-KUL allows easy manipulation of fabric andarn data and visualization tools. A textile geometry preprocessorhat provides a link between meso-mechanical models of compos-tes has the capability to construct 2D and 3D-weave structures forhe development of the unit cell cross-section and for computationf the unit cell porosity. Fabric thickness and fiber content of 3Deave can be easily predicted from the input yarn data. They

xpanded their study to contain knitted structures in the model asell �85�. In terms of hierarchy for textile structures, fiber is mod-

led in the yarn and yarn is modeled in the fabric. Active forcescompressive and bending� within the fabric, generated by thenterface of adjacent yarns, were included in the geometric model.eneral ideas of the hierarchical approach are illustrated by mod-

ls of internal geometry of multilayered woven fabrics and weft-nitted fabric topography.

Huang developed the so-called “bridging model” to predict thelastic, inelastic, and ultimate strength behavior of unidirectionalbrous composites �86–88�. The developed models were success-ully applied to non-crimp multiaxial warp knitted fabrics �89� asell as woven and braided fabrics �90�. The most appealing pointf the model is that there is an explicit correlation between stresstates originated from the constituent fiber/matrix material and theverall applied load on the composite. Elliptical yarn cross-ection and sinusoidal undulation of meshed yarns were combinedith the geometric model to describe the fabric geometry with orithout interyarn gaps. Huang’s micromechanical model �bridg-

ng model� does not need any iteration if the overall applied loads in a plane.

Carvelli and Poggi �91� applied homogenization theory for pe-iodic media to unidirectional composites and to woven fabricomposites using a three-dimensional finite element model. Thewo-step numerical model predicted the stiffness and strength ofoven fabric laminates as well as other properties such as lamina

hickness, yarn orientation, and fiber volume fraction. Druker-ragor’s criterion and smeared crack model were adopted as theonstitutive model for the matrix and the fibers. The numericalodel was validated in the elastic field with the available analyti-

al models in the literature and with experiments for nonlinearehavior of a fabric composite.

A new three-dimensional FE method was suggested by Kim

Fig. 13 Voxel mesh refinement: „a… uniform rstrained selective refinement †92‡

nd Swan �92�. This new technique was called the “voxel”



method, which means a 3D pixel; it originated from the field ofimage processing. An automated method was offered to producelocally refined meshes, which describe complex 3D geometrieswith high accuracy level �Fig. 13�. The technique is particularlydesigned to refine hexahedral elements around a tow matrix inter-face. Due to the newly formed nodes on existing boundaries, theuse of kinematic constraints is required to keep the continuity ofthe displacement field. Although the authors suggested the usageof novel technique for geometric refinement, they did not discussthe contribution of it on stress-strain results.

Thermal AnalysisRayleigh �93� and Maxwell �94� were among the first research-

ers to investigate thermal conductivity of multiphase materials. Aninitial attempt directed towards fibrous composites was made bySpringer and Tsai �95�. In 1967 they introduced a model for thethermal conductivity of UD fibrous composites for fibers arrangedin square and hexagonal arrays. In 1981, Han and Costner �96�developed a model using a finite difference method for unidirec-tional and laminated composites. Ishikawa and Chou used theirprevious analytical models �26,36,37� to examine in-plane thermalexpansion and thermal bending coefficients of plain and satinweave fabrics �97�.

Kabelka �98� proposed a method to evaluate the elastic andthermal properties of plain weave fabric composites. Sinusoidalshape functions were used to model the undulation in both warpand weft directions in his 2D model. The actual strand cross-sectional geometry was not considered. Once the properties ofcrimped warp and weft yarns were assessed under the constantstress condition, then the classical laminate theory was used toestimate the overall properties.

Naik and Ganesh predicted the thermal expansion coefficientfor three idealized laminate arrangements �99�. The experimentalvalues were validated with the analytical predictions. The effect ofthe fabric geometry on the plain weave fabric laminates was ana-lyzed and the significance of fabric geometrical parameters on thethermo-elastic behavior of woven fabric laminates was explained.

Thermoelastic properties of 2D orthogonal plain weave fabriclamina were studied by Naik and Ganesh �49� using the closedform analytical method. Yarn crimp, continuity in warp and filldirections, actual yarn cross-section and weave geometry, fibervolume fraction, and possible gap between the two adjacent yarnswere considered in the analysis. The results were obtained usingan element array model with parallel-series combination �45�.

Sankar and Marrey �100� studied the stress gradient effects inthin textile composite plates and beams by performing finite ele-ment analysis on the unit cell. They computed flexural rigidity andcoefficients of thermal expansion and showed that the beam/platestiffness and thermal properties cannot be predicted from theequivalent thermoelastic constants of the material and beam/plate

ement, „b… selective refinement, and „c… con-
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Gowayed et al. �101� developed a model for predicting thehermal conductivity constants of 2D and 3D woven fabric com-osites under steady-state heat transfer conditions. Geometricharacterization of the textile structures were followed by a FEA.icrolevel homogenization for hexahedral brick elements is given

n Fig. 14. The so-called Graphical Integrated Numerical AnalysisGINA�, which is a computer algorithm, was employed to performhe theoretical analysis in the study. The theoretical results werexperimentally verified. The authors investigated the effect of fi-er type and fiber volume fraction for thermal conductivity oflain weave and 3D woven composites in their further study102�.

Dasgupta et al. �103� presented the effective thermal andhermo-mechanical properties of plain weave fiber reinforcedomposite laminates in a micromechanical analysis of the periodicnit cell using the finite element method and thermal resistanceetworks. A review of the two-scale asymptotic homogenizationcheme was included in the beginning and also used in the theo-etical work. Appropriate boundary conditions were defined in thenit cell by the use of three-dimensional finite element analysisor determination of a complete set of orthotropic properties. Itas found that resin conductivity is the dominant constituent in

he thermal conduction properties of laminates.Sankar and Murrey developed the selective averaging method

104� for preferential reinforcements in contrast to Naik andanesh �53–55�. They overcame the problem using more realistic

sostrain and isostress assumptions. The model could predict theffective three-dimensional elastic constants and effective coeffi-ients of thermal expansions for a textile composite. A unit cell isivided along one edge into slices �mesoscale�. Each slice is againubdivided into elements �microscale�. The stiffness constantsere determined from homogenized medium for both isostress

nd isostrain conditions. As a different approach, structural stiff-ess properties of plates ��A�, �B�, and �D� matrices defined inef. �105�� were computed using a selective averaging method

nstead of from the continuum stiffness properties such as Young’sodulus, shear modulus, etc. A comparison was done betweennite-element-based micromechanical methods and analytical so-

utions. However, significant limitation exists when these modelsre applied to an inelastic deformation.

trength/Failure AnalysisIshikawa and Chou employed the maximum strain failure cri-

erion to determine a point of initial failure in woven composites37�. Complex strain and stress fields around the failed regionere neglected in the study.Nonlinearity caused by the shear forces acting on the unidirec-

ional laminates received the attention of early researchers106–109�. Ishikawa and Chou examined the nonlinear behavior

Fig. 14 Micro-level homogenization for hexaing method was employed †101‡

f woven fabric composites �110�. Three types of nonlinearity



including the shear deformation of weft yarns, the extensionaldeformation of the pure matrix region, and transverse cracking ofwarp regions were considered in the technique. Fiber undulationand the bridging models were applied to analyze the nonlinearelastic nature of the fabric composites. It was found that the sheardeformation of weft yarns contributes more to the composite ma-terial nonlinearity than the extensional deformation of pure matrixregions. The geometrical repeating length has an incremental ef-fect on both knee stress and strain for the bridge model applica-tion.

Karayaka and Kurath �28� modeled the elastic properties andfailure behavior of a textile composite using the homogenizationmethod. Their approach was directed to the determination of 3Dstresses in the interlacing region of the composite. The three-dimensional unit cell was considered to be brought together fromidealized subcells. The representative unit cell was homogenizedas a combination of unidirectional and interlaced regions that con-sisted of flat matrix pockets as well as in-plane and interlaced flattow regions. The randomly shifted fabric layers were taken intoaccount in the model. This method particularly gives accurate re-sults for practical applications.

Dow and Rammath �111� developed a strength analysis modelusing the finite element method. Sequential failure of the matrixand fiber was assumed. They proposed that after the matrix fails,the contribution of matrix to the composite strength is reduced.The stresses in the yarn were broken into constituent fiber stress inits local coordinate system and matrix stresses. Failure was pre-dicted based on the average stresses in the fiber and matrix. Toreach a correct verdict, cracks and yarn bonding in the compositeshould be considered for strength analysis and stress distributions.

Naik and Ganesh predicted the shear moduli for three idealizedlaminate arrangements �112�. They proposed angled off-axis ten-sion tests for better shear characterization of laminated compos-ites. Although they predicted the shear failure for a given fabricarchitecture, they failed to investigate the effect of changes infiber architecture on the failure strength or sequence of failure.

Ganesh and Naik �53–55� evaluated the failure behavior ofplain weave fabric laminates under on-axis uniaxial tensile load-ing in three parts. Part 1 contained mathematical expressions fordefining the geometry of the 2D orthogonal plain weave fabriclaminates. An analytical model was developed using geometry forprediction of thermo-mechanical behavior. The actual strandcross-sectional geometry, possible gap between adjacent strands,strand undulation, and continuity along both the warp and weftdirections were considered in the model. Part II focused on ananalytical model to predict the stress-strain history up to the ulti-mate failure of two-dimensional orthogonal plain weave fabriclaminates under on-axis uniaxial static tensile loading consideringall the intermediate stages of failure such as warp yarn transverse

ral brick elements in which stiffness averag-
hed
failure, weft yarn shear transverse failure, and pure matrix block

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ailure. Material and geometrical nonlinearities were consideredor predicting the stress strain behavior. A good correlation wasbtained between the experimental data and the model. The effectf fabric geometry on the failure behavior of 2D orthogonal plaineave fabric laminate was studied in Part III. The analyticalodel proposed by Ganesh and Naik is valid for multidirectional

einforcement in composites.Naik expanded the methods in Refs. �49,112� to predict the

ompressive strength of plain weave fabric composites �113�. Thisodel depends on curved beam/arch analysis. According to the

valuation of different failure modes in woven composites, shearailure of the warp yarn or compression failure caused by bendingtresses proceeded to the ultimate failure of the laminate.

Kurashiki et al. �114� conducted research for the case evaluatedy Naik and Shembekar �44� using FE analysis. They evaluatedtress-strain response considering damage development for plaineave composites with and without phase shifts between layers.he authors concluded that the change in Young’s modulus due tohase shift was small. This and what was reported by Naik andhembekar were in contrast to the findings of Woo and Whitcomb115�, who reported an increase in Young’s modulus from3 to 29 GPa as the phase shifts between two layers.

The geometrical models generally describe the fabric structuren the undeformed state where fabric is exposed to a state ofn-plane shear deformation in the composite mold. McBride andhen �116� modeled the plain weave fabric by considering the

hear deformation on fabric. The unit cell was represented by a setf four sinusoidal curves, which permitted shear deformation prioro buckling. It was mentioned that both yarn height and yarnpacing were independent from the shear deformation level.

Three-dimensional finite element analysis by Woo andhitcomb �70� for mechanical evaluation of woven composites

lso had failure analysis. The failure behavior of plain weave tex-ile composites was observed. It was concluded that failure behav-or was sensitive to the fiber bundle waviness. Although theoundary region was shallow and was limited to the outermostayer, the stress level was much higher inside than near the freeurface.

Vandeurzen et al. expanded their micromechanical analysis toodel the stress fields �117�. The new micro-mechanical model

ccounted for fabric geometry, yarn interactions, and matrix dis-ribution. A route that contains a multilevel decomposition schemend a multi-step homogenization procedure was followed tochieve a straightforward analytical stress model for woven fabricomposites. This technique was used �118� to predict the onset ofailure using a modified Tsai Hill failure criterion and the effect ofamage using a stiffness reduction method for failed microcells.s a result of a parametric analysis, the authors concluded that theethod was sensitive to fabric geometry.Ito and Chou studied the elastic properties and stress distribu-

ion of plain weave composites under in-plane tensile loading us-ng the FE model. The tension/bending coupling effect is impor-ant for single lamina composite. The iso-strain model wasroposed to predict Young’s modulus and stress distribution oflain weave composite. The stacking configuration of the iso-train model is given in Fig. 15. The flexural model was suggestedor assessing the effect of local bending of yarns. The authorsxtended their 2D geometrical model to a third dimension andnalyzed the plain fabric to develop a novel failure mechanism119�. They applied the maximum stress theory. An iso-strainodel was employed to predict elastic properties, stress distribu-

ions, and strengths under various laminate stacking. The authorsresented the significance of the waviness ratio on the tensileehavior of composites. The drawback of their analysis is thencapability of the failure mode for the woven fabric composite.

Scida et al. �120� developed a model to predict the failuretrength under tensile load in woven composites. They redefinedhe former stiffness model by evaluating the transformed stiffness

atrix.



Hoftsee et al. observed the experimentally determined ultimatetensile strength in the tow direction in addition to their elasticanalysis �60–62� and found out that the strength analysis was notsignificantly affected by the presence of shear deformation �121�.

Naik et al. �122� developed an analytical model for the com-pressive behavior of plain weave epoxy composite under on-axisuniaxial loading. The contributions of warp, weft, and matrix onthe compressive behavior were considered. Curved beams on elas-tic foundation approach were adopted for the analysis. The au-thors determined the stress state throughout the composite as wellas the failure mechanism under in-plane compressive loading.However, the analytical model was only set up for plain weavefabric composites with idealized aligned and bridged configura-tions. The theoretical findings were compared with experimentalresults, the latter being higher than the former one.

Concluding RemarksPractical applications of woven fabric composite materials are

increasing in the area of advanced composite technology. Predict-ing the mechanical properties of woven composites using analyti-cal and numerical methods presents a cost effective tool that re-searchers have been studying for three decades. The complexity ofthe structure makes it difficult to model the textile composite ar-chitecture using analytical methods. Hence, numerical methodslike the finite element method have been employed in variouscases for elastic, thermal, and strength/failure analysis. Eachmodel generally starts with the definition of a unit cell, which isdescribed mathematically using analytical techniques.

In contrast to analytical methods where they are formulated forspecific weave styles, numerical methods �in particular, the finiteelement method� are capable of being more easily adapted to dif-ferent geometries. The models developed to predict the mechanicsof composite materials vary depending on their capabilities togive accurate results. The specifications of individual models aregenerally strong in estimating one characteristic of composite me-chanics. This review attempted to present a helpful tool to under-standing the modeling nature and the chronology of the evaluationof textile composite mechanics.

Despite all the efforts to analyze textile composites in morerealistic ways, there are still discrepancies between experimentaland predicted results. The tendency of the studies is generally infavor of computer based approaches using both finite elementtechniques and analytical models to present more accurate andreasonable results. The efforts toward analysis of deformed rein-forcements have been suggesting better approximations. There-fore, it is necessary to improve the current approximations to a

Fig. 15 Lamina stacking configurations, iso-phase laminate,and random-phase laminate composites †119‡

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cknowledgmentWe are grateful for the funding provided for this work by the

SF CAREER program and Auburn University Peaks of Excel-ence Transportation Pinnacle. The input of Dr. Peter Schwartz isppreciated.

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Dr. Levent Onal was born in Malatya, Turkey in March 1975. He graduated from Istanbul TechnicalUniversity with a B.S. degree in Textile Engineering in June 1998. He received an M.S. degree in TextileEngineering and Science from Istanbul Technical University in February 2000. He graduated with a Ph.D.degree from Auburn University, Department of Polymer and Fiber Engineering in May 2003. He is currentlyworking as assistant professor at Erciyes University, Department of Textile Engineering in Kayseri, Turkey.

Dr. Sabit Adanur is a professor in the Department of Polymer and Fiber Engineering at Auburn University,in Auburn, Alabama. Dr. Adanur has a B.S. degree in Mechanical Engineering from Istanbul TechnicalUniversity, an M.S. degree in Textile Engineering and Science, and a Ph.D. degree in Fiber and PolymerScience, both from North Carolina State University. He worked at Asten Forming Fabrics, Inc., in Appleton,WI for three years as the Product and Process Development Manager. He joined Auburn University in 1992as assistant professor. He has authored three books and more than 60 article length publications.

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