2D Braided Composites a Review for Stiffness Critical Applications

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Composite Structures 85 (2008) 43–58

2D braided composites: A review for stiffness critical applications

Cagri Ayranci, Jason Carey *

Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2G8

Available online 10 October 2007

Abstract

Composite materials offer numerous advantages over conventional engineering metals. Over the years, the use of composite materialshas increased significantly. Braiding is a promising and already very commonly used method to form continuous fiber reinforced com-posite materials. Braided structures are used in a broad range of applications including, but not limited to, medical, aerospace, and auto-motive. This paper reviews studies published in the field of 2D braiding in order to outline advantages and disadvantages of the process,common preform impregnation techniques, and common stiffness critical applications. Furthermore, elastic property prediction modelspublished in the field are presented for the purpose of stiffness critical designs and applications.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Braided composites; 2D braiding; Preform impregnation; Fiber reinforced composites; Elastic constants of braided composites

1. Introduction

Braiding has been used since 1800s to produce textilefabrics. New demands for high production rate manufac-turing of high quality composite materials have focusedattention on braiding. A conventional braiding machinehas fiber carriers moving in a circular pattern [1]. Half ofthe carriers move clockwise, the others counterclockwise,in an intertwining serpentine motion producing a desiredbraid pattern, such as 2-dimensional tubular and flatbraids.

The braiding process competes with other compositematerial or composite preform manufacturing techniquessuch as filament winding, pultrusion, and tape lay-up.The advantages and disadvantages of 2D braiding are dis-cussed in the following sections.

2. Common terminology

Braiding: A composite material preform (Fig. 1a) manu-facturing technique. A braiding machine is used to inter-

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doi:10.1016/j.compstruct.2007.10.004

* Corresponding author. Tel.: +1 780 492 7168.E-mail address: [email protected] (J. Carey).

twine fibers to create desired braid architecture before orduring the impregnation of the fibers.

Braid angle: The angle between the longitudinal direc-tion of the braided preform and the deposited fiber,Fig. 1b.

Volume fraction: Relative amount of one constituent ofthe composite to the remaining constituents.

Unit cell: Smallest repeating element of a braided com-posite, Fig. 1b.

Crossover regions: Regions where intertwining fiber towsare deposited on top of each other in a unit cell.

Undulation region: The region where fiber tows undulatefrom one crossover region to the other, Fig. 1b.

Matrix only region: Remaining parts of the unit cellwhere fiber undulations or fiber crossovers do not exist,Fig. 1b.

3. Braid architecture

Braiding is a composite material preform manufacturingtechnique where a braiding machine deposits continuous,intertwined, fiber tows to create desired reinforcing braidarchitecture before or during the impregnation of the fibers.

There are three commonly used braid architectures:Hercules braid, regular braid, diamond braid. Hercules

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Fig. 1. (a) various braided composites (first three from the left), different preform sizes (last two on the right); (b) braid architecture (i.e., unit cell, braidangle, undulating region, matrix only region).

44 C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58

braid is a braid where each yarn passes over and thenabove three other yarns, where in regular braid each yarncrosses over and below two yarns, and finally if each yarncrosses over and below one other yarn in a repeating man-ner, it is called a diamond braid [2,3]. Adding axial fibersalong the mandrel axis is called a triaxial braid, and itincreases bending and tension strength and also stiffnessof braided composite materials. Triaxial braids need tobe formed/braided on a mandrel due to the geometric nat-ure of the process, whereas it is sometimes possible to pro-duce a biaxial braided preform without the use of amandrel. Tubular triaxial braids resist to radial shrinkage,and flat triaxial braids resist to shrinkage in width undertensile loads. Hence, these preforms are compatible as rein-forcements in pultrusion process [3].

4. Introduction to 2D braids

The most common commercial applications of braidedcomposites are, but not limited to, over-braided fuel lines,braided air ducts, rocket launch tubes, and aircraft struc-tural parts [1]. Other possible applications are catheters,automotive shaft reinforcement, sporting equipment, etc.

Conventional braiding machines produce preformseither vertically or horizontally. Most braiding machinesare said to be Maypole-type machines due to the serpentineor maypole strand carrier path. There are also Rotarybraiders which use two rotating tables. Although they havehigher production rates than Maypole braiders, they cannot produce flat braids. Flat braids must be produced bycarriers following two intersecting serpentine paths; how-

ever the intersecting paths form a single path by removalof the horn-gear. This forces the carriers to reverse theirmotion at the end of the track and form a flat braid insteadof completing a circular track on the machine to form atubular braid [2,3]. Maypole and Rotary tubular braid pre-forms are the same in terms of their architectures [2]. Fibersused to produce braided preforms can be dry or prepreg [1].The braiding process competes well with filament winding,pultrusion, and tape lay-up. Braiding compares favorablyin terms of structural integrity of components, design flex-ibility, damage tolerance, repair ability, and low manufac-turing cost [4]. Braiding advantages are high rate of stranddeposition on the mandrel, ability to produce complexshapes, low capital investment cost [1], and minimal laborcost [3]. The most important braiding process disadvantageis the difficulty in producing low braid angle preforms.

Munro et al. [5] presented a direct comparison of braid-ing to one of its major competitors, filament winding.Advantages and disadvantages of both high productionrate reinforced composite manufacturing techniques werehighlighted with respect to design and manufacturingmethodology and manufacturing aspects. They emphasizedthat it was not possible to determine the better processsince both have similarities, advantages and disadvantagescompared to the other and the selection of the manufactur-ing technique would largely be product dependent [5].

The kinematic analysis of the braiding process has beenstudied since 1950s [6–9]. Du and Popper [7] proposed adetailed time dependent model that predicts the microge-ometry of a fiber preform braided on an axisymmetricmandrel in terms of the relationship between braid angle,

C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58 45

fabric cover factor, yarn volume fraction, convergence zonelength and rate of braid formation. The model also outlineslimits of the braiding process as a result of jamming ofyarns [7].

Early studies showed that the crimp angle and braid angleaffect the strength and stiffness of the braided composites.Phoenix [10] presented experimental findings that verify thatan increase in the crimp angle or the braid angle causesdecrease in the strength of the braided composite [10]. Smithand Swanson [11] investigated the stiffness and strengthproperties of 2D braided carbon/epoxy composites underbiaxial tension and compression loading. Influential factorson stiffness were fiber volume, braid angle, percentage offibers in the braid and axial directions [11].

Braided composites are usually used in applications thatrequire high shear and torsional strength and stiffness. A±45� braid angle was proven suitable for such applications[3,12]. They also offer increased transverse moduli, trans-verse strength, damage tolerance, dimensional stabilityand near net shape manufacturing capabilities [13]. Thetransverse moduli and strength, and dimensional stabilityof braided composites arise from off-longitudinal-axis ori-ented fibers. Damage tolerance results from the lockingmechanism between the intertwined fibers of the braidarchitecture that prevents or limits yarn delamination.Low velocity impact damage tolerance capability of lami-nated composites has long been recognized and methodsof further improving the damage tolerance of the compos-ites have been studied [14]. Braiding is listed as one of themanufacturing techniques to produce aircraft primarystructures at lower cost and with better damage tolerantproperties. Jackson [15] and Kuykendall [16] reported onstudies investigating resin transfer molding (RTM) impreg-nated 2D braided preforms as one manufacturing tech-nique used to produce aircraft primary structures atlower cost and with better damage tolerant properties.They indicated that RTM technique makes possible toachieve up to 60% fiber volume fractions. Thicker partscan be achieved by adding any desired number of braidedlayers; this is an advantage of 2D braiding. Lack ofthrough the thickness tows and long manufacturing timesfor multi-lamina stacking procedures were listed as the dis-advantages of the 2D braids [15,16]. The authors indicatedthat 3D braiding addressed these disadvantages; however,the high cost of the 3D braiding machinery was a majordisadvantage. As an example, for their study, authors indi-cated that the 2D braided components cost 10% less thanthat of the 3D braided components, and hence 2D braidingwas chosen as the manufacturing technique [15].

5. Resin impregnation of 2D braided fibers

5.1. Manual impregnation

One of the limiting factors of broader use of compositematerials is from inconsistent mechanical properties due tostress concentrations originating from the voids that occur

in the materials as a result of non-homogeneous impregna-tion of fibers.

During the manufacturing process of the braided com-posites, fiber impregnation is as important as preform pro-duction. Manual impregnation of the preform, such asbrushing or massaging resin into the preform, is the sim-plest and least expensive method but has its limitations[17,18]. In this type of impregnation, to avoid prematurecure, resins with long gel time must be selected. Further-more, product quality depends highly on the skill level ofthe operator applying the resin onto the preform, and thiscan lead to inconsistent mechanical properties. This can beaddressed by using preimpregnated (prepreg) fibers [17,18].Kruesi et al. [19] suggested use of an impregnation ring thatpreimpregnates fibers prior to their deposition onto themandrel. This is done by a controlled amount of resinapplied to the fibers through small pores while they arepassing through the proposed impregnation ring. It wasreported that very low void content, ranging from 3.71%to 1.74%, was achieved. Also high fiber volume fractionsin excess of 60% were achieved [19]. This process may pro-vide consistent specimen fiber volume fraction while alsodecreasing production time.

5.2. Commingled fibers

In some applications thermoplastic (TP) resins may bepreferred over thermosetting resins. One of the reasonsfor using TP resins is to decrease composite manufacturingtime, because TP resins do not need chemical reaction timeas the thermoset resins. Fujita et al. [20] investigated com-mingled and un-commingled yarns as impregnating sys-tems to increase the uniformity of mechanical propertiesof braided composites. In commingled yarns, reinforcingfibers and matrix fibers are commingled together, whilefor un-commingled yarn, the reinforcing fibers and matrixfibers are placed next to each other. Specimens were man-ufactured by compression molding. The commingled yarnspecimens required lower pressures and shorter holdingtimes compared to un-commingled specimens [20]. Addi-tional advantage of thermoplastic resins is the greater frac-ture toughness compared to thermosetting resins [21].

Bechtold et al. [22] modeled the impregnation processfor braided and pultruded tubes. Due to the difficulty inbraiding preimpregnated thermoplastic tapes, powderimpregnated or commingled yarns were used. Braided com-mingled yarns are preheated slightly above the thermoplas-tic melting temperature prior to entering the heatedpultrusion die. The complete melting process of the ther-moplastic and subsequent impregnation of the fibers occursin the heated die, which is followed by a pressurized cool-ing stage through a die for calibration purposes [22].

5.3. Resin transfer molding based processes

Brookstein [17,18] underlined that consistency in fibervolume fractions and hence mechanical properties may also


be achieved by using other automated impregnation tech-niques such as resin transfer molding (RTM) [17,18].RTM creates high fiber volume composites with very lowvoid content. This leads to homogeneous products. Inaddition, near net shape products are possible to produce.Circumferential frames, keel frames, and window framesare some examples of RTM manufactured braided com-posites [23,24].

In RTM, a completed preform is put in a tool or mold.The part and the resin are heated to optimal temperaturefor the resin to have minimal viscosity. Resin is thenapplied to the preform under pressure. Later the necessarycuring procedure for the specific resin is followed [25]. Min-imal machining requirement of these products decrease theend cost. It also avoids the negative effects of machinedcomposite parts, such as stress concentration factors intro-duced at the machined location of the part. Also due to thedamage of matrix in the machined region, environmentaleffects such as moisture and other existing chemicals effectthe fibers, matrix, and the interface and hence this effect thestrength and elastic properties of the machined composites.

Michaeli et al. [26] used RTM to manufacture a braidreinforced tubular composite where the reinforcementwas placed over a flexible tubing and inserted into theRTM mold. The tube was pressurized and resin injected.Good fiber placement and controlled impregnation as wellas good surface finish were achieved.

However, resin permeability through the preform playsa major role in the quality of products manufactured byRTM. Charlebois et al. [27] reported on permeability char-acteristics and mechanical properties of braided fabrics.Authors investigated permeability of 2D biaxial braidedglass fibers at three braid angles ±35�, ±45�, ±50�, andfound that change in braid angle effect the fiber volumefraction and thus permeability. Permeability of ±45� and±50� angles decreased as the fiber volume fraction wasincreased. However, permeability of ±35� angle was notaffected from the fiber volume fraction change [27].

Vacuum assisted resin transfer molding (VARTM) hasalso been used to manufacture braided composites [28].VARTM offers low cost for high volume production, largeand complex shapes capabilities and high fiber volumefractions compared to hand lay up [29]. VARTM processrequires that a dry preform be placed in a mold (or tool),low viscosity resin be transferred to the preform under vac-uum, followed by the resin curing procedure. It is used bymany industries [30]. Some other advantages of VARTMand RTM are their low volatile organic chemical (VOC)emission and good part surface quality production ability[31].

RTM and VARTM provide cost reductions in compos-ite materials compared to using prepregs. Prepreg materialsoffer good toughness to the composites; however, the resinsused have high viscosities that can not be used with theRTM/VARTM techniques. Pederson et al. [28] addressedthis issue and proposed to achieve better toughness usingRTM. For this, the resin system toughening agent that is

used in the prepreg materials had to be manufactured ina fiber form and directly braided into the preform alongwith the reinforcing fibers (without compromising braidstructural integrity). Experimental results demonstratedsimilar mechanical properties between proposed RTMand conventional prepreg autoclave manufactured com-posites [28].

Uozumi et al. [32] proposed a new technique to manu-facture near-net-shaped composites using RTM impreg-nated 2D braiding, followed by a forging process tominimize cost as compared to 3D braiding. ‘‘I”, ‘‘J”,‘‘T”, ‘‘Z” shaped composites are listed as producible.Authors found superior tensile properties with the braidedspecimens compared to equivalent aluminum specimens,suggesting possible aircraft applications for weight savings.Also, the braiding/RTM process was reported to haveapproximately 34% cost savings compared to the hand-lay-up/ autoclave process [32].

6. Applications

Braid reinforced composite materials have a broadrange of industrial applications. Based on the aforemen-tioned advantages, such as the specific strength, these mate-rials are preferred increasingly over the conventionalengineering metals. This section outlines some of the broadapplications of braided composites.

Brookstein [17,18] listed structural columns, rods,shafts, pressure vessels, and plates as some classical appli-cations where braid reinforcement had replaced conven-tional materials. Brookstein suggested, with notheoretical or experimental evidence to support the claims,the structural limits of braided structure. It was stated thatbraided structure could be used for tensile load carryingapplications if the braid angle did not exceed 15�. In thecases of compression loading and thin-wall buckling,delamination could be overcome by the circumferentialreinforcing nature of braided fabrics (if 20% of the fiberplacement was at a ±45� braid angle). Shafts were listedas ideal components manufactured using composite materi-als, where axially placed fibers provide stiffness, and ±45�braid provided torque transmission reinforcement. Heshowed, through modeling, that 54.74� braided pressurevessels are also good candidates for braided compositeapplications [17,18].

2D braiding may be used to manufacture structuralcomponents as well. Kobayashi et al. [33] reported manu-facturing a T-shape braided graphite epoxy composite trussjoint. Authors proposed a different continuous productionmanufacturing method for structural components such asT-shaped trusses. At the end of the process the whole T-shape had two layers of continuous triaxial braiding. Inthis study EPIKOTE 828 epoxy resin with an amine systemhardener (KC1118) was used. Fibers were impregnated in avacuum and an autoclave was used for curing. It wasreported that the braided T-shaped truss joint had higherstrength than a similar cloth tape component [33].


Hamada et al. [34,35] reported a new technique to pro-duce tubular braided products that are more resistant tointerlaminar delamination, also referred to as through-the-thickness toughness. The technique uses a conventional2D braider in a multireciprocal fashion to produce a multi-layer braided laminate. Through-the-thickness fibers weresimultaneously added to the braid through a three tracksystem where the spindles travel from one track to theother creating a three-dimensional structural network ofstrands. It was observed that propagation of interlaminardelamination was impeded during lateral compression testsof said manufactured tubular braided specimens [34,35].

Due to their specific strength and tailorable mechanicalproperties, composites have long been the preferred mate-rials for aviation [15,23,24,32,36]. White [36] reported onmanufacturing, testing, and cost analysis of a Kevlar49�/epoxy blade spar. Ballistic tests were done to evaluatethe structural damage. After complete armor penetration,static retesting of spar section did not show any detectablechanges in the elastic behavior, which was attributed to thebraided fabric delamination resistance. Also, ultrasonic C-scan inspection of the structure was assessed and satisfac-tory results were observed. Finally, the cost evaluation ofthe braided structure revealed 33% savings compared to fil-ament wound glass blade spar [36].

The sports equipment industry highly utilizes the bene-fits offered by braided composite materials. Casale et al.[37] reported on design and fabrication of a braided bicycleframe using Kevlar/graphite braided hybrid preformsimpregnated with Epon 828 epoxy resin and D-230 curingagent. The frame was manufactured by braiding the four-piece frame over a foam core and subsequent joining pro-cess. Five prototype bicycles were produced [37].

Production of braid reinforced laminated wood baseballbats have been reported by Axtell et al. [38,39]. Reversedballoon molding was used to manufacture the bats. Duringthis process an elastomeric tube was inflated and themolded component pushed onto it. The tube was subse-quently deflated to wrap the part for the curing process.Following curing, the tube was again inflated forcing thecured product out.

Neogi et al. [40,41] published their findings on designanalysis and fabrication of a self deployable structural ele-ment, constructed of a foam core, internal bladder, braidedload carrying preform and an outer jacket, which was orig-inally developed to minimize payload volume on spaceshuttle missions. The proposed structure had a minimumvolume at the onset; using a resistance wire embedded inthe foam core as a heat source, the structure expandedand cured. A carbon/epoxy system was chosen for thebraid because of low coefficient of thermal expansion, highlongitudinal and torsional stiffness and interlaminarstrength. As a result of the study, 80% volume savings wereachieved compared to original designs. The authors sug-gested using a triaxial braid structure due to the lower spe-cific stiffness of the final product compared to aluminumstructures. They also listed emergency sailboats, deployable

antennas and tent frames as other possible applications ofdeployable structures [40,41].

Braided composites have also been suggested for usewith structural reinforced concrete components since flex-ural strength and ductility of reinforced concrete memberscan be improved with braided composite jackets [42]. Lifespans of reinforced concrete structures can be improvedby using corrosion resistant and high specific strengthbraided fiber reinforced polymer (FRP) rebars instead ofconventional steel rebars. The non-ductile behavior ofbraid reinforced FRP rebars were also addressed byresearchers: Hampton et al. [43], and Lam et al. [44]reported on hybrid Kevlar-Carbon FRP rebars manufac-tured using a braiding/pultrusion process exhibiting desir-able ductile behavior similar to conventional steel rebars[43,44].

Karbhari et al. [45,46] studied crush performance andenergy absorbing capabilities of braided composites. Braidenergy absorbing capabilities could be eventually used inindustrial applications such as car bumpers. They reportedthat triaxially braided composites increased the energyabsorbing performance of the braided composites [45],and the occurrence of damage prior to onset of crushingaffected crush performance [46].

Braid reinforced composite materials have been exten-sively studied for biomedical applications. Hudgins et al.[47,48] suggested replacing the natural intervertebral discwith a prosthetic intervertebral disc. The proposed dischad a core of elastomeric polymer and a braid reinforcedouter shell. Braided shells proved to provide compressivestrength to the design [47,48]. Moutos et al. [49] reportedtubular braided structures with elastomeric cores that weremanufactured and tested to mimic the properties of ante-rior cruciate ligaments [49]. Reinhardt et al. [50] underlinedthe high numbers of hip replacement surgeries conductedevery year in the world, and the need for a design thatwould have tailorable mechanical properties, enhancedfatigue life, and biocompatibility. Authors proposed adesign that consisted of balsa wood core with six layersof braided carbon preforms manufactured by RTM usinga vinyl ester matrix. The study was designed as a basisfor future studies but early mechanical performance ofthe design were reported to be excellent; however, resin bio-compatibility issues were left for future studies [50].Another example of biomedical application of braidedcomposites, braided carbon/PEEK composite bone plates,were fabricated and tested by Fujihara et al. [51]. Braidedfabric reinforcement was chosen for this work based onbetter in plane properties and out of plane delaminationresistance. Promising results encouraged researchers to fur-ther investigate the effect of the braid angles and platethicknesses on the bending performance of the compositeplates; braid angle was identified as important for thickplates. For example, it was suggested that a 2.6 mm thickplate with a 10� braid angle was suitable for forearm treat-ments [51–53]. Finally in dentistry, braided compositeswere used in dental posts that require varying stiffness


along the shaft. This was obtained by varying the braidangle along the post shaft [54].

Braided tubular products can also be used as cathetersin medical field. Carey et al. published a study about thedesign of fiber reinforced composite catheters [55]. Theyanalyzed the required rigidities of conventional cathetersand set design objectives to achieve these targets by useof braided composites [55].

7. Typical challenges in applications: joining methods

braided and machined holes in 2D braided composites

Composite materials, including braided composites,may be manufactured to near net shape to avoid anypost-manufacturing processes; however, there are alsonumerous applications that require multi-part assemblywith other composite or non-composite components.Assembly may be accomplished through adhesive or poly-meric bonding as well as mechanical joints. Use of adhe-sives involves studying adhesive shear strength, surfacefinish of substrates and coupling agents. For the purposesof this review, mechanical joints were investigated sincethey require holes or other shape openings in the structuresand will have effects on the integrity of the parts.

Composite materials are susceptible to develop stressconcentrations around holes and cutouts. Tsiang et al.[56] and Brookstein [57] compared the effect of integrallyformed braided holes and machined holes on strength ofcylindrical braided composites. Specimens with braidedand machined holes were tested under tensile loads. Inaverage, specimens with braided holes were observed tobear loads that were 1.23 times higher than that ofmachined holes. Observations on specimen failure modeswere presented; however, limited micromechanical discus-sions to explain the observed phenomena were provided.In another set of tensile experiments, load was appliedthrough pins inserted into the braided and machined holes.On average, specimens with braided holes supported 1.8times greater loads than those with machined holes. Thiswas associated to the fiber discontinuity at the machinedholes [56,57].

Following the study of Brookstein et al., Wang and hisco-workers published contradictory findings [58–60].Authors outlined that, in the previous studies, the overallwall thickness of the tube specimens were not controlleddue to excess resin surrounding the holes. It was suggestedthat these thicker resin rich regions contributed to theincreased the local bearing strength. In Wang and his co-workers’ studies, wall thickness was kept constant. Changein fiber angles in the surrounding regions of the holesresulted in decreased bearing strength. They concluded thatsimilar or greater bearing strengths were found formachined holes as compared to braided holes. Other stud-ies on 3D braided composites support their findings[58–61].

However, Fujita et al. [62] published studies on compar-ison of machined holes versus braided holes on flat braided

bars, and they found results parallel to that of Brooksteinet al. They stated that machined holes have lower bearingstrengths; however, their work concentrated on the effectof hole diameter on bearing strength. Smaller diameterscaused more local disturbance on the orientation of fibersthan larger diameters leading to resin rich regions andlower bearing strengths than their larger counterparts.Results were validated using numerical analysis. They con-cluded that fiber orientations around the holes significantlyaffected the bearing strength and failure mode [62]. Theydid not comment on the issue (i.e. the resin-rich regionssurrounding the holes) raised by Wang et al.

Ohki et al. [63,64], and Nakai et al. [65] evaluated theeffect of machined versus braided holes in end loadedflat-braided specimen with a centralized hole. Specimenswith braided holes had higher strength properties duringboth static and fatigue testing. From microscopic observa-tions, authors conclude that the damage mechanism of themachined holes is related to the fiber–resin interface, whilethe damage mechanism of the braided holes is related tothe reorientation of the continuous fibrous strand pathcaused by the presence of the hole [63–65].

8. Elastic constant predictive models

Mechanical behavior of 2D braided composites can bediscussed in terms of elastic behavior, plastic behavior,and failure behavior such as ultimate strength and failuremechanism [66]. This review, due to the broadness of thetopic, focuses on elastic behaviors of braided composites.Important publications about the remaining two categoriesare listed for bibliographical purposes but not detailed.

Elastic property prediction of 2D braided compositeshas been studied far more than their plastic behavior andfailure behavior. Braided structures are assumed to behavelinearly in the elastic range. In the plastic region, a non-lin-ear behavior is observed which increases the complexity ofthe problem. Nevertheless, a number of studies have beenpublished regarding plasticity behavior and failure charac-teristics of braided composites [67–74]. Other papers thathave dealt with elastic properties, as well as strength andfailure mechanisms, will be covered in detail.

The majority of braid analysis developed to date canfind its origins in earlier woven fabric composite and lam-inated composite analysis; hence, this review also outlinesmajor studies published in these fields to create a basisfor the overall discussion. In this view, braided compositescan be seen as a specific form of woven fabric composites,or textile composites [75].

Some of the models discussed are based on the wellknown Classical Laminate Plate Theory (CLPT). Duringthe discussions of this review, it is assumed that readersare familiar with the well documented CLPT analysis, suchas by Jones [76].

In early 1970s, Halpin et al. [77] developed a model pre-dicting elastic stiffness and thermal expansion properties ofshort fiber composites from a laminate analogy. This lam-


inate analogy was extended to two- and three-dimensionalwoven fabric composites. Authors reported that predictedand experimental results for woven fabric composites werecompared and found to be ‘‘qualitatively correct”.

Whitney and Halpin [78], analyzed laminated aniso-tropic tubes subjected to combined tension or compression,internal pressure, and torque. Authors listed the governingequations as equilibrium, compatibility, strain and curva-ture displacement, and constitutive relations. The analysiswas done using Donnel’s approximations.

Some of the most influential studies that followed werepublished by Ishikawa and Chou [79–81] who proposedand compared three stiffness and strength predictive mod-els that formed the basis to many subsequent textile fabriccomposite models, namely, the ‘‘mosaic”, ‘‘fiber undula-tion”, and ‘‘bridging” models. The models study the small-est repeating unit of the fabric, the unit cell. The propertiesof the unit cell are assumed to be representative of the over-all composites. Mosaic model treats the system as anassemblage of asymmetric cross-ply laminates. The modeluses the Classical Laminate Plate (CLPT) theory as thebasis of the analysis. The model was analyzed using bothiso-strain and iso-stress assumptions to respectively obtainupper and lower bound composite stiffness properties. Thefiber undulation model was developed to validate andimprove the mosaic model. Undulation (crimp) and conti-nuity characteristics of the fibers in woven fabric compos-ites omitted in the mosaic model were considered. Due tophysically occurring matrix only regions, this model alsoallowed the recognition of changes in the overall fiber vol-ume fraction of the unit cell. The undulating fibers,assumed to follow a path described by a sinusoidal func-tion, were used to calculate stiffness matrices of CLPTanalysis. The local stiffness matrices used in the calculationof the CLPT A, B, D matrices were computed as a functionof the local undulation angle (called ‘‘local off-axis angle”

by Ishikawa and Chou). The authors stated that the undu-lation of the fibers reduced the effective stiffness of the com-posite in the longitudinal direction, and that the maximumstrain occurs at the mid-point of the undulating fiber. Thebridging model was developed for satin weave fabrics andis therefore out of the scope of this review [79–81].

Ishikawa and Chou also characterized geometric andmaterial properties of hybrid woven fabrics [82], and investi-gated effects of these fabric parameters on elastic propertiesby using the mosaic model. In this model, due to the hybridnature of the fabric, in-plane and bending moduli (Aij, and Bij

matrices) are no longer uniform in the repeating region.Gaps that may exist between the fibers were neglected anda close mesh configuration was adapted. In this report, Ishik-awa and Chou also investigated the thermal expansion coef-ficients and thermal bending coefficients. Investigation wasconducted using the mosaic model and one-dimensionalfiber undulation model. Agreement was found betweenexperimental and theoretical results [82].

Tsiang et al. [83] investigated the longitudinal and trans-verse mechanical properties of triaxial braided graphite/

epoxy cylinders using a simple micromechanics theorybased model. The braid architecture was modeled as astructure composed of unidirectional-ply and bias-angleply yarns. The brief description of the model provided sta-ted that material properties were calculated by applying theprinciple of superposition to the two sub-layers. Results forthe longitudinal and transverse elastic modulus and Pois-son’s ratio were provided, and were stated to be in reason-able agreement with experimental results.

Yang et al. [84] proposed a predictive model for triaxial-ly braided composites elastic properties. Unlike woven fab-ric models (45� fiber deposition angle), this model, basedon the Ishikawa and Chou’s fabric undulation model[79], assumes 60� fiber deposition angle. The model utilizesthe geometrical characterization of the braid architecturewhere the triaxial fabric composite is treated as an assem-blage of three laminae; bias and longitudinal yarn laminae.The corrugated yarns impregnated with matrix are takeninto account in the initial calculation, and the contributionof the matrix only regions are subsequently consideredusing a Rule of Mixtures prediction. The upper bound iscalculated from a laminate that consists of three laminaestacked together with fibers in the bias braid and longitudi-nal angles, and the lower bound is calculated from the pro-posed model. As a result of the analysis, the stiffness of thenon-orthogonal woven fabrics was determined to bestrongly influenced by the fiber deposition angles. Themodel was not verified experimentally [84].

In a later study, Yang et al. [85] proposed the ‘‘FiberInclination Model” based on a modified CLPT to predictthe elastic properties of three-dimensional textile (wovenand braided) composites. Here, the unit cell used for theanalysis is assumed to be composed of an assemblage ofinclined unidirectional laminae. The idealized unit cellwas described as fiber bundles oriented in four body diag-onal directions. All the yarns in one direction were assumedto form inclined laminae after matrix impregnation. Therest of the analysis was explained as an extension of thefiber undulation model developed by Ishikawa and Chou.In the analysis, contribution of pure matrix regions to thestiffness matrices were neglected (interested reader mayrefer to the original text for the modifications and neces-sary assumptions). Authors recognized and underlined thatthe CLPT ignores the interactions of fiber yarns at theinterlocking points and stated that it is still a convenienttechnique for the analysis. Predictions and experimentalfindings were in good agreement [85].

Whyte [86] proposed an analytical model, the FabricGeometry Model (FGM), to predict the properties ofthree-dimensionally braided structures. FGM is based ona modified CLPT where the unit cell is defined as repeatingvolumes. The stiffness matrix is developed for each yarn inthe unit cell by calculating the stiffness matrix of the equiv-alent unidirectional lamina and transforming it into thestructural coordinate system. The contributions of eachyarn are superimposed with respect to their volumetriccontribution. Authors also suggest calculating the strain


at every new strain level to account for the non-linearbehavior of the materials [86]. Pastore and Gowayed [87]underlined two major disadvantages of the FGM modelpresented earlier. First, the theoretical mathematical deri-vation is not compatible with the basic transverse isotropyused in the model. Secondly, the transformation matricesused for the stiffness calculations are not consistent. Intheir paper, authors address these problems. They com-pared stiffness averaging and compliance averaging tech-niques, and they compared predicted and experimentalresults for triaxially braided, as well as orthogonal glassreinforced composites. The self consistent FGM model,as it was called, was used to predict elastic propertiesresults using both stiffness averaging and compliance aver-aging technique. Authors highlighted that in all cases thestiffness averaging technique provided better predictions.

Ko et al. [2] and Lei et al. [88] presented the Finite CellModel (FCM) in which the unit cell for the structure wasdefined as an assemblage of brick-shaped elements. TheFCM defines the composite as a ‘‘space truss” and henceeach yarn is considered individually. The yarns areassumed to be diagonals of the unit cell and analyzed aspin-jointed two-force truss members, which makes thismodel suitable for finite element analysis.

Soebroto et al. [89] published a design framework forbraided tubular composites. The objective was to fill agap in the field by creating a link between textile preformmanufacturers and structural designers creating designcurves such as effect of braid angle on fabric diameterand transverse speed required for a given braider to achievea certain diameter tubular braided preform. They also usedFGM model by Whyte [86], to predict elastic propertiesand strength of 2D braids. Soebroto et al. theoretical pre-dictions, taken approximately from a graphic, werereported in the range of 5–85�. Experimental verificationwas done for 20� and 70� braid angles and appear to followthe general trend with the models published after them,such as the longitudinal elastic modulus decreases as thebraid angle changes increases [89]. However, for the regionbetween 30� and 60�, their predictions appear nearly linearcompared to other models that have a more curved shape.Again, FGM was originally developed for three-dimen-sionally braided composites and does not include undulat-ing fiber strands. Hence, following the comparison of linearversus curved predictions of the different models, it may beconcluded that for two-dimensionally braided compositesmore sensitivity in the results may be obtained by methodsthat account for fiber undulation.

A woven fabric study was published by Naik and She-mbekar [90–92] as a series of three publications, namely,lamina and laminate analysis and laminate design. Naikand Shembekar indicated that the early elementary lami-nate theory models developed, such as mosaic and undula-tions models by Ishikawa and Chou, were simple but notaccurate because of the one-dimensional nature of thesemodels leading to large discrepancies between predictedand experimental results. Conversely, authors indicated,

numerical models were accurate but complex. To addressearly model concerns, Naik and Shembekar proposed sim-ple but accurate generalized two-dimensional models topredict the elastic properties of woven fabric composites.Their models account for fiber continuity and undulationin both the weft and warp directions, matrix only regionsand cross sectional geometry of the yarns in the unit cell.The Naik and Shembekar two-dimensional model was anextension of the Ishikawa and Chou one-dimensionalmodel. Only non-hybrid two-dimensional plain weave fab-ric lamina was considered. The unit cell was divided intostraight cross ply, undulated cross ply, and matrix onlyregions. The undulating tow paths were modeled usingsinusoidal functions. The elastic constants were calculatedusing a Cylindrical Assemblage Model (CAM) in the prin-cipal material directions. Each infinitesimal region of theunit cell was analyzed using CLPT [90–92].

In the analysis, the unit cell is assumed to be comprisedof sub-sections along and perpendicular to the loadingdirections. Each sub-section is comprised of infinitesimalpieces. A uniform, unidirectional, in-plane load wasassumed to be applied to the woven fabric. The infinitesi-mal sub-sections in the unit cell, which are in series withthe loading direction, were assumed to be under constantstress. On the other hand, the infinitesimal sub-sectionsthat are parallel to the loading axis were assumed to haveconstant strain in their mid-planes [90–92]. Following thisapproach, they created two models: Series Parallel Model(SPM) and Parallel Series Model (PSM). The SPM was cre-ated by assembling all the infinitesimal pieces in series withthe loading direction under iso-stress condition, and thenassembling all the sections along the loading directionunder iso-strain condition. The PSM was created by fol-lowing the same approach in the reverse order. Naik andShembekar stated that the SPM provides the lower boundsof the in-plane stiffness constants, whereas the PSM pro-vides the upper bounds of the stiffness constants. Followingexperimental verification PSM was recommended forwoven fabric composites [90–92]. It should be underlinedthat PSM was developed for woven fabric composites,which can not be generalized to include braided structuresthat may have different angle orientations.

Finally, Naik and Shembekar [92] underlined the superiorproperties and advantages of woven fabric composites tothat of unidirectional composites such as shorter build time,complex shape capability and ease of mold impregnationbecause of the intertwined structure. Authors highlightedthe fact that the elastic behavior of a unidirectional laminaand a thin laminate are the same, whereas this may not benecessarily true for a woven fabric lamina and thin wovenfabric laminate because of the macroscopically heteroge-neous nature of the woven fabric lamina; they also outlinedthe limited number of studies published in this field [92].Authors studied the effect of stacking sequence or shift oflaminae to obtain optimal laminates. Since this is beyondthe scope of this review, interested readers are referred tothe original publication [92].


Masters et al. [93] studied the mechanical properties oftriaxially braided composites both analytically and experi-mentally, which could serve as a database of experimentalresults for comparison purposes with predicted results ofmodels available at the time. The experiments used 2 � 2braided AS4/epoxy resin composite flat panels impreg-nated by RTM. Braid angle, size of the braided yarn andsize of the longitudinal yarn were varied to obtain three dif-ferent architectures. A processing science model was usedto construct the braided unit cell geometry. Mechanicalproperties of the braided composites were predicted usingfour different approaches, namely, laminate, corrected lam-inate, diagonal brick, and finite element model. Laminatemodel was the simplest model where all the tows were trea-ted as unidirectional plies in a symmetric laminate. A cor-rection factor was applied to this model to compensate forthe ignored fiber undulations to create the corrected lami-nate model. Diagonal brick model [94] is an extension ofthe above FCM. The finite element model was based ona previous model proposed by one of the authors of thepaper, Foye R.L., where the unit cell was analyzed as acombination of sub-cells. They found that all model predic-tions were comparable to experimental findings and the dif-ferences between them were not significant; however, finiteelement method predictions were best. Also studied was thesensitivity of experimental measurements to strain gagesizes. Findings concluded that large gage sizes, such asthe 2.54 cm gage length of some extensometers, were pref-erable [93]. Master and Ifju [95] later published a detailedstudy where they outline Moire interferometry, X-ray radi-ography, and surface replication techniques as alternativesto inspecting or testing methods for braided composites[95].

A review paper that utilizes experimental results to com-pare stiffness predictive models available at the time waspublished by Falzon et al. in 1993, [96]. Authors catego-rized the models into three types, namely, the elementarymodels such as fabric geometry model (FGM); the lami-nate theory models such as ‘‘fiber undulation model” and‘‘mosaic model”; and, finally numerical models. Authorsstated that the elementary models are unsuitable forstrength calculation; the laminate models are unable to pre-dict out-of plane elastic properties; while finite elementmodels are complex [96]. Although these observations weretrue at the time, in subsequent years, improvements weremade to these models to address these concerns.

Redman and Douglas [97] proposed a simple analyticalmodel to determine the elastic properties of triaxiallybraided composites. The model utilizes a unique combina-tion of Rule of Mixtures prediction and CLPT. Unlikemany of the previous models presented, the Redman andDouglas model, due to this unique modeling approachcombination, does not require the use of a unit cell. Thelength between neighboring fibers was assumed to be bigenough to neglect the effect of undulating fibers. The triax-ial braid is considered to have three separate plies that allcoexist in the same space. Each ply is assumed to have a

thickness equal to the full braid layer and assumed to haveprimary fiber tows and effective matrix material. Effectivematrix material is assumed to be composed of two second-ary fiber tows and matrix material, and is analyzed usingCLPT as a symmetric laminate. This model may be a goodalternative to obtain fast preliminary design results prior toa detailed analysis [97].

Following Masters et al. [93], Naik et al. [98] conductedan analytical and experimental study on the effects ofbraiding parameters on 2D triaxially braided composites.Braiding parameters were listed as braid angle, yarn sizeand axial yarn content. A Repeating Unit Cell (RUC)was isolated and used for the analysis. Each yarn in theRUC was discretely modeled and sliced. The three-dimen-sional effective stiffness of the RUC was calculated using avolume averaging technique under iso-strain assumption.Although the analysis was conducted in three dimensionswith respect to the XYZ global coordinate axis, the pre-dicted elastic properties mainly showed sensitivity to braid-ing parameter in the longitudinal and transverse directions.The elastic properties in the thickness direction were muchless sensitive to changes in braid angle or percent axial yarncontent. This may be the underlying reason to why manybraiding models following this study analyzed braids onlyin the axial and transverse direction, such as Carey et al.[75]. Stiffness properties were not affected by yarn sizes,but were affected by braid angle and axial yarn content.Increasing the braid angle increased transverse and shearelastic moduli, but decreased longitudinal elastic modulus.It was also reported that the out of plane elastic and shearmoduli were insensitive to these parameters [98].

Following Naik et al., Naik [99] published a study toextend on the previous work. He implemented the analysisin a program code called TEXCAD used for braided aswell as other textile composites. The work was alsoextended to predict strength of woven and braided compos-ites [99–101].

Raju and Wang [102] reported a detailed study aboutclassical laminate theory models for woven fabric compos-ites derived from, but not limit by the simplification of, theIshikawa and Chou models [79]. They first identified arepeating unit in the woven fabric composite, which wasfurther divided into unit cells. This geometrical character-ization was done for plain weave, 5- and 8-harness satinweave structures; this review covers only the plain weavecase. A uniform membrane strain and curvature areassumed at the midplane of the unit cell. The unit cellwas divided into four regions, each subsequently dividedinto four sub-regions composed of undulating and non-undulating regions. As was the case for Ishikawa andChou’s fiber undulation model, Raju and Wang’s modelaccounts for the undulating fibers; however, they use amore accurate geometry to characterizes undulating fibersin the fill and warp directions than its predecessor. Theundulating fibers are assumed to follow a sinusoidal shapefunction as with the model by Naik and Shembekar [90].CLPT stiffness matrices A, B, and D of the unit cells are


calculated as follows. First the A, B, and D matrices of allfour sub-regions are calculated by integrating the Stiffnessmatrix, Q, over the volume of each sub-region. Then, theseare summed over each sub-region to obtain A, B, and D

matrices for each region in the unit cell. Finally, all fourregions are summed to obtain unit cell A, B, and D matri-ces. Authors state that the integrals involving the undulat-ing strand stiffness were calculated numerically withoutspecifying the method selected. In the model, coefficientsof thermal expansions were also obtained. Predicted resultswere compared to many other available models; mostmatched favorably [102]. It should be noted that, as men-tioned, the study was conducted for woven fabrics; hence,fill and warp strands were always perpendicular.

Gowayed et al. [103] proposed a finite element model topredict the elastic properties of textile composites. Thismodel addresses the short comings of the Unit Cell Contin-uum Model (UCCM) by Foye [104]. The UCCM utilizes aunit cell that is divided into subcells. Displacements of thesubcells are calculated using Virtual Work, and summed tocalculate the total displacement. However, Gowayed et al.suggested that the UCCM does not clearly differentiatebetween fiber and matrix material properties in the analy-sis. They state that if the difference between the propertiesof fiber and matrix materials is large, such as for the case offiber reinforced composites, the solution becomes inaccu-rate. Authors suggested correcting this by using the UCCMalong with Whyte’s FGM model [86], where compositematerial fibers and matrix constituents are treated sepa-rately and their contribution to the global stiffness matrixare calculated through superimposing each contributionwith respect to their relative volume fraction. The modelwas verified experimentally [103].

Nakai et al. [105], Hamada et al. [106], and Nakai et al.[107] attempted to use the unit cell predictions to designand predict behavior of braided cylinders upon loadingusing numerical analysis. The analysis was comprised ofa micro analysis, modeling individual resins and fibers asstraight lines, and of a macro analysis, which combinedthe micro models, to form structural elements. They alsostudied the influence of braiding structure on torsionalproperties of braided composite tubes.

Naik and Ganesh [108] studied two-dimensional orthog-onal plain weave fabric laminae through a thermoelasticanalysis. The authors claimed that most of the modelsdeveloped until then, [90,109,110], do not consider theactual strand geometry and cross section; hence, the fibervolume fraction was not included in the models. The fewmodels that included these were complex. Consequently,they outlined a two-dimensional closed form analyticalmethod which takes into consideration the strand undula-tion and continuity in fill and warp directions, strand crosssection, fiber volume fraction, and possible gaps betweentwo adjacent strands. In their model, a unit cell that is com-posed of three layers, fill and warp fibers and matrix, isused. Strand cross section, strand cross sectional shape inwoven form and the strand undulations are defined by

shape functions. Authors compared both circular and sinu-soidal functions for strand undulations and concluded thatthe sinusoidal functions offered better predictions. As manybefore, Naik and Ganesh defined the unit cell of the com-posite as an asymmetric cross-ply laminate. This laminateis assumed to consist of one pure matrix and two unidirec-tional laminae. The thermoelastic properties of woven fab-ric lamina were calculated using CLPT under theassumptions that CLPT is applicable to a unit cell andthe bending deformations of a unit cell are constrainedby the surrounding unit cells. The undulating angle of thestrands is assumed to vary linearly [108]. In their study,12 material systems with different strand and weave geom-etries were analyzed. The results were compared to a previ-ous model by the same author and experimental data. Theyconcluded that the proposed model provides acceptableand quick results. The effect of the ratio of strand thicknessto strand width on elastic constants was also investigated inthe study and the results are provided in a graphical form.They also suggested that the twist of the strand along thefiber undulation direction should be investigated; however,later, Carey et al. [75], calculated this to be negligible in 2Dbraided structures [108].

Byun et al. [111] proposed a novel braiding and pultru-sion manufacturing technique during which the fiber towsare preimpregnated and subsequently braided on a Teflonmandrel. Impregnated preforms are cured in a curing dieand cut into pieces. Authors proposed an analytical modelfor elastic properties of braided products that first calcu-lates the effective compliance matrix of a yarn based onits length then uses this information to obtain the effectivestiffness of the composite by averaging the stiffness con-stants of the axial yarn, braided yarn, and matrix as func-tions of their volume fraction in the composite. The modeldoes not allow for open-mesh braid configuration. Limitedexperimental data was provided [111].

A three-dimensional tow inclination model was pro-posed by Branch et al. [112] to calculate elastic constantsof two-dimensional textile and three-dimensional braidedcomposites. The global constitutive equation of the com-posite material is derived using an iso-strain approach forthe unit cell and averaging all tow segments and matrixwithin the unit cell [112].

Tsai et al. [31] used a CLPT-based model to predict stiff-ness and strength of braided tubular composites. Theyintroduced two models, bridge and crimp, that are similarto those of Ishikawa and Chou [79]. The experimental andpredicted values were generally not in god agreement; how-ever, better agreement was found with the crimp model.

Robitaille et al. [113] stated the importance of realisti-cally characterizing preform geometry for use in predictivemodels. They proposed a method to describe preforminterlacing geometrical patterns by a series of vectors.Authors indicate that the geometries can be used in predic-tive models as well as permeability studies of preforms toobtain optimal impregnation of fibers. In a subsequentstudy [114], the group underlined the difficulty in character-


izing the complete structures for such purposes; hence, theypresented an algorithm to generate geometric characteriza-tion of unit cells for textile and composite materials. Possi-ble useful applications were calculation of localpermeability values and local stress distributions in thesematerials. Their examples are mostly aimed towards per-meability calculations.

Wong et al. [115] investigated permeability models andproposed two new numerical models, namely, grid averageand stream surface methods. Grid average simplifies theunit cell domain to a rectangular grid in the longitudinaland transverse directions plane from which the permeabil-ity tensor components are calculated. In the Stream Sur-face model the unit cell domain is first divided into basicvolumes that consisted of open channels and porous towsfrom which permeability can also be calculated.

Aggarwal et al. [116] proposed an analytical model fortheir braided dental post and bone plate calculations. Itwas stated that many current models for braided compos-ites ignored unit cell inter-yarn gap for which they pro-posed a micromechanical model. Here undulating fibersand yarn cross sections were considered. The geometricalcharacterization was based on a unit cell, authors calledRepeating Unit Cell (RUC). Each sub-cell in the RUC istreated as assemblage of spatially oriented unidirectionallaminates of transversally isotropic properties. The stiffnessof the RUC was calculated using a modified CLPT. Theyassumed that the CLPT is applicable in infinitesimal sub-cells generated within the RUC. The subcells consisted oftwo braiding yarns and a single matrix lamina. Experimen-tal results were used to validate the model. Only longitudi-nal elastic modulus results were provided; good agreementbetween predicted and experimental results was found.

In another work, Aggarwal et al. [117] proposed an ana-lytical model based on a repeating unit cell (RUC)approach for in-plane elastic constants of two-dimensionalbraided composites. The geometric model considers yarnundulations and inter-yarn gap using a sinusoidal shapefunctions. Under iso-stress and iso-strain assumptionsengineering constants of each sub-cell is calculated andaveraged over the RUC volume. The paper considers flatbraids. Both upper and lower braiding yarns follow thesame undulation path but have different cross sectionalarea shapes. Yarns in the unit cell were calculated as anassemblage of small straight yarns. This approach was alsofollowed in the local undulating yarn segments. They per-formed a sensitivity analysis of the yarn thickness/yarnwidth (t/a) ratio and concluded that in-plane elastic con-stants decrease slightly as the ratio increases. Intern yarngap was found to affect the RUC volume fraction and con-sequently mechanical performance. Also, changes in yarnaspect ratio affect the undulation and therefore mechanicalperformance [117].

Byun [118] proposed a detailed model to predict geomet-rical characteristics, undulation yarn angle, fiber volumefraction and three-dimensional engineering constants of2D braided composites. Byun underlined the simplicity of

the calculation procedures compared to lamination theory[93] and yarn-discretization [98] models. First, a geometricmodel of the triaxial braid is developed; yarn shape param-eters measured from photomicrographs are used to charac-terize yarn geometric relationships within the unit cell,which lead to the prediction of the undulation angle, fibervolume fraction and elastic constants. The resin impreg-nated yarns are modeled as unidirectional composite rods.The effective compliance matrix of the unidirectional com-posite is found through simple transformation through theundulation angle and subsequent averaging of the trans-formed compliance matrix. The author stated that aftertransformation the specific geometry of the yarn is nolonger significant and it can be treated as layers of ortho-tropic materials. It is assumed that, once loaded, each layerundergoes iso-strain. The effective stiffness of the compositeis found by averaging the stiffness of each layer based onvolume. The stiffness matrix is inverted to get the compli-ance matrix, from which engineering constants of the triax-ially braided composite are obtained. The model wasexperimentally verified: predicted and experimental fibervolume constants were in good agreement; however, theundulation angle was under-predicted. In the conclusionsauthor suggested more experiments were needed to supportthe model predictions.

Harte and Fleck [119,71] studied the necking and tensilebehavior of braided tubes. Elastic moduli of which werepredicted using laminate plate theory. The mechanics ofneck propagation was investigated; the authors concludedthat braided structures can be very effective in energyabsorbing applications because they deform in tension atconstant stress for large extensional strains. Failure mech-anisms of braided composites under compression and tor-sion were also investigated.

Huang [66,120,121,72] underlined that, for woven andbraided fabrics, many of the available models were devel-oped for elastic behavior (i.e. small displacements) andthere are very few models for plastic behavior and strengthpredictions. Authors proposed a micromechanical model,the bridging model, for woven and braided fabrics capableof determining elastic, plastic and ultimate strength behav-ior of fiber composites under any arbitrary load condition.Concisely, in the model the overall applied load on thecomposite is explicitly correlated with the stress statesdeveloped in the fiber and matrix constituents. Huang[66,120,121,72] divided the woven/braided composite intoa repeating unit cell (RUC) further divided into four sub-elements consisting of two yarns and pure matrix regions.Each sub-element component, assigned a local coordinatesystem, can be locally treated as a unidirectional compos-ite. Relative coordinate transformations are provided withrespect to the global axes. Yarn undulations are defined bysinusoidal functions. Following the iso-strain assumption,an average stiffness/compliance of the sub-element is deter-mined. Based on the iso-stress assumption, the overall stiff-ness/compliance matrix of the unit cell is obtained usingthe contributions of each sub-element. To be more compre-


hensive on the approach, Huang uses a ‘‘bridging matrix”,to correlate the volume averaged stress increments in thefiber and matrix of the representative volume element. Thismatrix represents the load carrying contribution of one ofthe constituents in the composite with respect to the otherconstituent (i.e. contribution of fiber with respect tomatrix). The model utilizes this relationship in the calcula-tion of the volume averaged stress relationship. The bridg-ing matrix is populated differently when finding elastic orplastic response, or ultimate tensile strength. The resultscompared favorably to the experimental studies and othermodels available in the literature. Here, Huang also studiedthe effect of gap-ratio of braided fabrics on the predictedproperties via a parametric study [66,120,121,72]. Huang’smodel is correlated with experimental data and results. Dif-ferences between experimental and predicted results are lessthan 13% [122,123].

A computational micromechanical model was developedby Ivanov and Tabiei [124] to predict elastic properties ofwoven fabric composites. The model is based on a micro-mechanical approach and homogenization technique. It isclaimed that due to the efficiency of the model it is suitablefor large scale finite element analysis. Similar to other mod-els, a unit cell of the composite is divided into four sub-cellswith respect to its fill and warp yarns. Direction of theyarns in each cell is characterized by the braid and undula-tion angles. The homogenization technique used was sum-marized in three steps by the authors: first, partitioning theconstituent stiffness matrices by choosing iso-strain andiso-stress components; second, calculating the interimmatrices; and, finally, calculating the partitions of the effec-tive stiffness matrix [124].

Yan and Van Hoa [125] developed a macrostructuremodel to predict the mechanical behavior of 2D triaxiallybraided composites. Authors used the elastic deformationenergy of a unit cell to calculate the effective stiffness ofthe braided composites. They used this model to predictelastic properties and to conduct a parametric study[126]. The elastic property predictions were compared toresults of Master et al. [93]. In their parametric study, theyseparated independent parameters (yarn and compositegeometrical parameters, and constitutive material con-stants) that affect the analysis of braided composites. Theseparameters can be used for guidance in designs using triax-ial braided composite structures.

Tabiei and Yi [127] compared several numerical analysismethods and proposed a new one to predict the elasticproperties of woven fabric composites. Their model is asimplified version of the earlier ‘‘method of cells” forwoven composites by Tabiei et al. [128]. The authorsclaimed that the new method is more computationally effi-cient and requires less memory than the previous methodsthat were too complex and required high numbers of calcu-lations; thus addressing one of the major disadvantages ofanalyzing braided composites using numerical analysis.

Quek et al. [129] proposed an analytical model for theeffective elastic stiffness of a 2D triaxially flat braided com-

posite capable of investigating the effect of imperfectionson stiffness. A Representative Unit Cell (RUC) comprisedof two braid tows, one axial tow and one matrix layer isdeveloped for the braid geometry. The model uses a Con-centric Cylinder Model (CCM) to predict, with respect tolocal coordinate axes of the fibers, the elastic constants ofthe tows in the composite. The contribution of the undulat-ing fibers, which affect the stiffness in the ply direction, iscalculated by averaging transformed local fiber stiffnessover one complete undulation cycle, called wavelength inthe paper. Finally, the stiffness in the ply directions is trans-formed to the global coordinate system. Stiffness contribu-tions of each ply are assembled together as a function oftheir volume fraction within the RUC to predict overallRUC elastic constants. The predicted results are comparedto experimental and finite element model results; results arein agreement. Based on their results, the authors underlinethe advantages of using the proposed model in terms oftime and computational memory savings compared tofinite element models, which they state should only be usedif ultimate strength is required. Analytical models are pref-erable for elastic properties. Finally, a parametric studyrevealed that braid fiber plies in the RUC have the largesteffect on elastic properties.

Carey et al. [75] proposed a model to predict elastic con-stants of 2D-diamond-braided fiber composites. The modelis a generalization of the model developed by Raju andWang in which the geometry of a braid unit cell is analyzedby dividing the unit cell into 13 regions. These regions arecategorized as overlapping strands, strand undulation, andmatrix only regions. Model was capable of limiting thephysically possible braid angles for a unit cell based onstrand geometry. In the model, unidirectional lamina elas-tic constants are found using micromechanical models. Thelongitudinal modulus and major Poisson’s ratio are calcu-lated using Rule of Mixture predictions. Transverse and in-plane shear moduli are calculated with Halpin–Tsai equa-tions, while out-of-plane shear modulus and Poisson’s ratiowere calculated using stress partitioning parameter and amethod proposed by Ko, respectively [75]. The macromodel is based on a modified CLPT where a volumeweighted stiffness matrix is calculated using the 13 regions.Stiffness matrices of undulation regions are calculatedusing the Gauss–Legendre numerical iteration. Stiffnessmatrices are subsequently transformed to the loading direc-tion axis. Comparison of the predicted results to results ofother models and experimental findings were in god agree-ment. Later they performed a sensitivity analysis of theeffect of constituent elastic constants on braid elastic con-stants. The model is found to be mainly sensitive to longi-tudinal fiber elastic modulus, and matrix elastic and shearmoduli. Authors later used the findings of the model todesign a braided composite catheter through calculationsof axial, flexural and torsional rigidities of braided compos-ites [75,130,55].

In another publication, Carey et al. [131], claimed that,although accurate and promising, available elastic property


predictive models for woven/braided structures arelengthy. They proposed a regression based model simplerthan other available models, for use in the preliminarystages of design with braid/woven composites. The geomet-rical characterization of the braid unit cell was done in ageneralized manner that compensates for different braidsangles and open-mesh braids. Using previously developedanalytical models, elastic constants of braids and laminatespossessing the same angle-ply geometry are calculated. Aunit cell fiber volume fraction was determined. Normalizedelastic modulus values are plotted with respect to unit cellfiber volume fractions. A linear relationship was observedbetween braid and laminate longitudinal and transverseelastic moduli values and fiber volume fractions. Thismodel is underlined to be a very promising pre-design toolfor such composites [131].

Recently, finite element models used for predicting engi-neering properties of 2 � 2 braided composites were devel-oped by Tang et al. [132], Goyal et al. [133], and Goyal andWhitcomb [134]. Tang et al. and Goyal et al. studied theeffect of waviness ratio, a relation between the thicknessof lamina with an undulating yarn and the undulationlength, and the braid angle on the elastic properties ofbraided 2 � 2 braided composites. Transverse propertieswere found to be more sensitive to these parameters. Onthe other hand, the out-of-plane modulus was found tobe almost insensitive. Later, stress concentrations of2 � 2 braided composites were investigated by researchersfrom the same group, Goyal and Whitcomb [134].

Potlori and Mannan [135], and Potlori et al. [136] usedfinite element analysis to determine the mechanics ofnon-orthogonal structures such as braided structures[135]. Flexural and torsional behaviors of biaxial and triax-ial braided composite structure were also investigated.Flexural and torsional rigidities, calculated using a modi-fied CLPT, analysis were in good agreement with experi-mental findings [136].

Very recently, Lomov et al. [137] published a detailedwork on finite element analysis (FEA) of textile compositesthat provides an algorithm of the necessary steps to textilecomposites FEA [137].

Ayranci and Carey [138] indicated that almost none ofthe models used for predicting tubular braided structureelastic constants consider tube curvature in the geometricdefinition of the unit cells. Authors modified the analyticalmodels developed by Raju and Wang, [102], and Careyet al. [75] to compensate for the curvature in the unit cell[138].

9. Conclusion

In this report, braiding technique used for compositematerials manufacturing was reviewed. Advantages anddisadvantages of 2D and 3D braiding were outlined. Resinimpregnation of fibers and different methods to achieveproper impregnation were listed. A broad range of applica-tions of 2D braided composite materials were listed to

show the nearly endless possibilities and advantages of2D braids, which reinforce the notion of braiding as astrong alternative to other types of composite manufactur-ing techniques available. A large number of analyticalmodels used to predict the elastic properties of braidedcomposites were outlined and discussed.

2D braided composite materials offer numerous advan-tages over the conventional materials. The recent improve-ments in their fabrication techniques and understanding oftheir mechanical behaviors through predictive models con-tribute to the increasing popularity of these materials.

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