北京化工大学：《材料导论》课程阅读材料（金属材料）2D_braided_composites_A_review_for_stiness_critical_applications

Available online at www.sciencedirect.com ScienceDirect COMPOSITE STRUCTURES ELSEVIER Composite Structures 85(208)43-58 www.lsevicr.com/locatc/compstruct 2D braided composites:A review for stiffness critical applications Cagri Ayranci,Jason Carey" Availableonine 10October 2007 Abstract Composite materials offer numerous advantages over conventional engineering metals.Over the years.the use of composite materials of the proce 贴 er Braided composit:2D braiding Preform impregation:Fibr reinfored composites:Elastic constants of braided composites 1.Introduction he fih Braiding has been used since 1800s to produce textile Brad ange:The angle between the longitudinal direc. omposite the braided prefom a the eptdr mate Relative am has fiber carriers moving in a circular pattern Half of e constituent of the composite to the remaining constituents. the carriers move clockwise,the others counterclockwise. Unit cell:Smallest repeating element of a braided com- posite,Fig.Ib brad pattern,such re d The braiding process competes with other composite The reonwere berduate ite preform manufacturing techniques from one crossover region to the other,Fig.Ib. g parts o cussed in the following sections Fig.Ib. 2.Common terminology 3.Braid architecture Braiding:a composite material preform (Fig.la)manu Braiding is a composite material preform manufacturins facturing technique.A braiding machine is used to inter- technique where a braiding machine deposits continuous used braid ) Hercules braid,regular braid,diamond braid.Hercules

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 materials has 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 models published 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 textile fabrics. New demands for high production rate manufac￾turing of high quality composite materials have focused attention on braiding. A conventional braiding machine has fiber carriers moving in a circular pattern [1]. Half of the carriers move clockwise, the others counterclockwise, in an intertwining serpentine motion producing a desired braid pattern, such as 2-dimensional tubular and flat braids. The braiding process competes with other composite material or composite preform manufacturing techniques such 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￾twine fibers to create desired braid architecture before or during 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 of the composite to the remaining constituents. Unit cell: Smallest repeating element of a braided com￾posite, Fig. 1b. Crossover regions: Regions where intertwining fiber tows are deposited on top of each other in a unit cell. Undulation region: The region where fiber tows undulate from one crossover region to the other, Fig. 1b. Matrix only region: Remaining parts of the unit cell where fiber undulations or fiber crossovers do not exist, Fig. 1b. 3. Braid architecture Braiding is a composite material preform manufacturing technique where a braiding machine deposits continuous, intertwined, fiber tows to create desired reinforcing braid architecture before or during the impregnation of the fibers. There are three commonly used braid architectures: Hercules braid, regular braid, diamond braid. Hercules 0263-8223/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2007.10.004 * Corresponding author. Tel.: +1 780 492 7168. E-mail address: jason.carey@ualberta.ca (J. Carey). www.elsevier.com/locate/compstruct Available online at www.sciencedirect.com Composite Structures 85 (2008) 43–58

C.Ayramel.J.CareyI Compaslte Smructures 85(2008)43-58 b Unit Cell Undulating regior sites (first three from the left),diferent preform sizes (last two on the right):(b)braid architecture (ie.,unit cell,braid only region】 and ther path by crosses over and below two yarns,and finally if each yarn motion at the end of the track and form a flat braid instead ove and elow on oth 3car trac the ma ine to form a increases bending and tension strength and also stiffness of braided Triaxial braids need to braiding process compete well with filamen on a man tape ing compare duce a biaxial braided preform without the use of a bility.damage tolerance.repair ability,and low manufac turing cost [4 Braiding advantages are high rate of strand braid on mandr forcements in cost 31.The most in tant hraiding s disadvantage is the producing preforms. 4.Introduction to 2D braids Munro et al. 5]presented a di gto one o ect comp The most common com nd disa h hig rate reinforced composite manufacturing techniques were highlighted with respect to design anc tural er po applic ology a nng asp since both have tages and disadvantages either vertically or horizontally.Most braiding machines compared to the other and the selection of the manufactur- are said to be Ma entin ing tec hnique would argely be proc ct dependent 01 D nd Po higher production rates than Maypole braiders,they can detailed time dependent model that predicts not produce flat braids.Flat braids must be produced by carrers following two intersecting serpentine paths;how mandrel in terms or the rel

braid is a braid where each yarn passes over and then above three other yarns, where in regular braid each yarn crosses over and below two yarns, and finally if each yarn crosses over and below one other yarn in a repeating man￾ner, it is called a diamond braid [2,3]. Adding axial fibers along the mandrel axis is called a triaxial braid, and it increases bending and tension strength and also stiffness of braided composite materials. Triaxial braids need to be 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 a mandrel. Tubular triaxial braids resist to radial shrinkage, and flat triaxial braids resist to shrinkage in width under tensile loads. Hence, these preforms are compatible as rein￾forcements in pultrusion process [3]. 4. Introduction to 2D braids The most common commercial applications of braided composites 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 preforms either vertically or horizontally. Most braiding machines are said to be Maypole-type machines due to the serpentine or maypole strand carrier path. There are also Rotary braiders which use two rotating tables. Although they have higher production rates than Maypole braiders, they can not produce flat braids. Flat braids must be produced by carriers following two intersecting serpentine paths; how￾ever the intersecting paths form a single path by removal of the horn-gear. This forces the carriers to reverse their motion at the end of the track and form a flat braid instead of completing a circular track on the machine to form a tubular braid [2,3]. Maypole and Rotary tubular braid pre￾forms are the same in terms of their architectures [2]. Fibers used 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 favorably in 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 strand deposition on the mandrel, ability to produce complex shapes, low capital investment cost [1], and minimal labor cost [3]. The most important braiding process disadvantage is 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 production rate reinforced composite manufacturing techniques were highlighted with respect to design and manufacturing methodology and manufacturing aspects. They emphasized that it was not possible to determine the better process since both have similarities, advantages and disadvantages compared 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 been studied since 1950s [6–9]. Du and Popper [7] proposed a detailed time dependent model that predicts the microge￾ometry of a fiber preform braided on an axisymmetric mandrel in terms of the relationship between braid angle, 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, braid angle, undulating region, matrix only region). 44 C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58

C.Ayrancl.J.Carey/Composite Structures 85(2008)43-58 45 fabric cover factor,yarn volume fraction,convergence zone in the materials as a result of non-homogeneous impregna- tion of nbers the mng proces Ding the manufacturing process s of the braided com Early studies showed that the crimp angle and braid angle duction. Manual impregnation of the preform.such as affect the strength and stiffness of the braided composites brushing or massaging resin into the preform,is the sim and st expen e metho but has its decrease in the str the praided composite [1o Smith resins with longg and Swanson [1]investigated the stiffness and strength more,product quality depends highly on the skill level of the operator applying ther resin ont and thi me 17 18 the baid re fihe ntage h Kruesi et al [19]suggested use of an impregnation ring that preimpregna es fibers prior to their deposition onto the nand lone by con led am 3.121 They also through the pro posed impregnation ring It wa reported that very low void content,ranging from 3.71% near net henanfactimgcapebites 01.74% volume fraction ented fibers.Damage tolerance results from the locking decreasing production uime. mechanism between the intertwined fibers the brai tha pre yarn 5.2.Commingled fibers nated comosites has ong ben and methods In some applications thermoplastic (TP)resins may be of further thermosetting resins.One of the reasons or using I s is to decr e c as the thermose resins.Fujita et a 20 investigated com properties.Jackson [15]and Kuykendall [16]reported on mingled and un-commingled yarns as impregnating sys. tems to increase the unif s.In used to orms as one manulacturing tec for un-commingled yarn,the reinforcing fibers and matrix They technique makes ssible to s are placed next to each other.Specimens were man um ons afactured by compr sion molding.Th of r parts Lack of imes compared to un-commingled specimens 201 Addi through the thickness tows and long manufacturing times tional advantage of thermoplastic resins is the greater frac were listed as ture toughness comp ared to therm osetting resins [211 Ine au ded tu the high cost of the 3d braiding machinery was a maio braiding preimpregnated thermoplastic powder uthors ind d or commingled】 s were use Braided com 2D ighty above was chosen as the manufacturing technique [15]. pultrusion die.The complete melting process of the ther. moplastic and subsequent impregnation of the fibers occurs 5.Resin impregnation of 2D braided fibers 5.1.Manual impregnation 5.3.Resin transfer molding based processes One from inconfc that orig

fabric cover factor, yarn volume fraction, convergence zone length and rate of braid formation. The model also outlines limits of the braiding process as a result of jamming of yarns [7]. Early studies showed that the crimp angle and braid angle affect the strength and stiffness of the braided composites. Phoenix [10] presented experimental findings that verify that an increase in the crimp angle or the braid angle causes decrease in the strength of the braided composite [10]. Smith and Swanson [11] investigated the stiffness and strength properties of 2D braided carbon/epoxy composites under biaxial tension and compression loading. Influential factors on stiffness were fiber volume, braid angle, percentage of fibers in the braid and axial directions [11]. Braided composites are usually used in applications that require 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 stability and near net shape manufacturing capabilities [13]. The transverse moduli and strength, and dimensional stability of braided composites arise from off-longitudinal-axis ori￾ented fibers. Damage tolerance results from the locking mechanism between the intertwined fibers of the braid architecture that prevents or limits yarn delamination. Low velocity impact damage tolerance capability of lami￾nated composites has long been recognized and methods of further improving the damage tolerance of the compos￾ites have been studied [14]. Braiding is listed as one of the manufacturing techniques to produce aircraft primary structures at lower cost and with better damage tolerant properties. Jackson [15] and Kuykendall [16] reported on studies investigating resin transfer molding (RTM) impreg￾nated 2D braided preforms as one manufacturing tech￾nique used to produce aircraft primary structures at lower cost and with better damage tolerant properties. They indicated that RTM technique makes possible to achieve up to 60% fiber volume fractions. Thicker parts can be achieved by adding any desired number of braided layers; this is an advantage of 2D braiding. Lack of through the thickness tows and long manufacturing times for multi-lamina stacking procedures were listed as the dis￾advantages of the 2D braids [15,16]. The authors indicated that 3D braiding addressed these disadvantages; however, the high cost of the 3D braiding machinery was a major disadvantage. As an example, for their study, authors indi￾cated that the 2D braided components cost 10% less than that of the 3D braided components, and hence 2D braiding was 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 composite materials is from inconsistent mechanical properties due to stress 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 as brushing 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 premature cure, resins with long gel time must be selected. Further￾more, product quality depends highly on the skill level of the operator applying the resin onto the preform, and this can lead to inconsistent mechanical properties. This can be addressed by using preimpregnated (prepreg) fibers [17,18]. Kruesi et al. [19] suggested use of an impregnation ring that preimpregnates fibers prior to their deposition onto the mandrel. This is done by a controlled amount of resin applied to the fibers through small pores while they are passing through the proposed impregnation ring. It was reported that very low void content, ranging from 3.71% to 1.74%, was achieved. Also high fiber volume fractions in excess of 60% were achieved [19]. This process may pro￾vide consistent specimen fiber volume fraction while also decreasing production time. 5.2. Commingled fibers In some applications thermoplastic (TP) resins may be preferred over thermosetting resins. One of the reasons for using TP resins is to decrease composite manufacturing time, because TP resins do not need chemical reaction time as the thermoset resins. Fujita et al. [20] investigated com￾mingled and un-commingled yarns as impregnating sys￾tems to increase the uniformity of mechanical properties of braided composites. In commingled yarns, reinforcing fibers and matrix fibers are commingled together, while for un-commingled yarn, the reinforcing fibers and matrix fibers are placed next to each other. Specimens were man￾ufactured by compression molding. The commingled yarn specimens required lower pressures and shorter holding times 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 process for braided and pultruded tubes. Due to the difficulty in braiding preimpregnated thermoplastic tapes, powder impregnated or commingled yarns were used. Braided com￾mingled yarns are preheated slightly above the thermoplas￾tic melting temperature prior to entering the heated pultrusion die. The complete melting process of the ther￾moplastic and subsequent impregnation of the fibers occurs in 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 fiber volume fractions and hence mechanical properties may also C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58 45

46 C.Ayranci,J.Carey/Composite Structures 85(2008)43-58 be achieved by using other automated impregnation tech- used in the prepreg materials had to be manufactured in niques such as resin transfer molding (RTM)[17,18]. a fiber form and directly braided into the preform along RTM creates high fiber volume composites with very low with the reinforcing fibers (without compromising braid void content.This leads to homogeneous products.In structural integrity).Experimental results demonstrated addition,near net shape products are possible to produce. similar mechanical properties between proposed RTM Circumferential frames,keel frames,and window frames and conventional prepreg autoclave manufactured com- are some examples of RTM manufactured braided com- posites [28]. posites [23,24]. Uozumi et al.[32]proposed a new technique to manu- In RTM,a completed preform is put in a tool or mold. facture near-net-shaped composites using RTM impreg- The part and the resin are heated to optimal temperature nated 2D braiding,followed by a forging process to for the resin to have minimal viscosity.Resin is then minimize cost as compared to3 D braiding.“T”,“J' applied to the preform under pressure.Later the necessary “T”,“Z”shaped composites are listed as producible.. curing procedure for the specific resin is followed [25].Min- Authors found superior tensile properties with the braided imal machining requirement of these products decrease the specimens compared to equivalent aluminum specimens, end cost.It also avoids the negative effects of machined suggesting possible aircraft applications for weight savings. composite parts,such as stress concentration factors intro- Also,the braiding/RTM process was reported to have duced at the machined location of the part.Also due to the approximately 34%cost savings compared to the hand- damage of matrix in the machined region,environmental lay-up/autoclave process [32]. effects such as moisture and other existing chemicals effect the fibers,matrix,and the interface and hence this effect the 6.Applications strength and elastic properties of the machined composites. Michaeli et al.[26]used RTM to manufacture a braid Braid reinforced composite materials have a broad reinforced tubular composite where the reinforcement range of industrial applications.Based on the aforemen- was placed over a flexible tubing and inserted into the tioned advantages,such as the specific strength,these mate- RTM mold.The tube was pressurized and resin injected. rials are preferred increasingly over the conventional Good fiber placement and controlled impregnation as well engineering metals.This section outlines some of the broad as good surface finish were achieved. applications of braided composites. However,resin permeability through the preform plays Brookstein [17,18]listed structural columns,rods, a major role in the quality of products manufactured by shafts,pressure vessels,and plates as some classical appli- RTM.Charlebois et al.[27]reported on permeability char- cations where braid reinforcement had replaced conven- acteristics and mechanical properties of braided fabrics. tional materials.Brookstein suggested,with no Authors investigated permeability of 2D biaxial braided theoretical or experimental evidence to support the claims, glass fibers at three braid angles±35°，±45°，±50°，and the structural limits of braided structure.It was stated that found that change in braid angle effect the fiber volume braided structure could be used for tensile load carrying fraction and thus permeability.Permeability of +45 and applications if the braid angle did not exceed 15.In the +50 angles decreased as the fiber volume fraction was cases of compression loading and thin-wall buckling, increased.However,permeability of +35 angle was not delamination could be overcome by the circumferential affected from the fiber volume fraction change [27. reinforcing nature of braided fabrics (if 20%of the fiber Vacuum assisted resin transfer molding(VARTM)has placement was at a +45 braid angle).Shafts were listed also been used to manufacture braided composites [28]. as ideal components manufactured using composite materi- VARTM offers low cost for high volume production,large als,where axially placed fibers provide stiffness,and 145 and complex shapes capabilities and high fiber volume braid provided torque transmission reinforcement.He fractions compared to hand lay up [29].VARTM process showed,through modeling,that 54.74 braided pressure requires that a dry preform be placed in a mold (or tool), vessels are also good candidates for braided composite low viscosity resin be transferred to the preform under vac- applications [17,18]. uum,followed by the resin curing procedure.It is used by 2D braiding may be used to manufacture structural many industries [30].Some other advantages of VARTM components as well.Kobayashi et al.[33]reported manu- and RTM are their low volatile organic chemical (VOC) facturing a T-shape braided graphite epoxy composite truss emission and good part surface quality production ability joint.Authors proposed a different continuous production [31] manufacturing method for structural components such as RTM and VARTM provide cost reductions in compos- T-shaped trusses.At the end of the process the whole T- ite materials compared to using prepregs.Prepreg materials shape had two layers of continuous triaxial braiding.In offer good toughness to the composites;however,the resins this study EPIKOTE 828 epoxy resin with an amine system used have high viscosities that can not be used with the hardener(KC1118)was used.Fibers were impregnated in a RTM/VARTM techniques.Pederson et al.[28]addressed vacuum and an autoclave was used for curing.It was this issue and proposed to achieve better toughness using reported that the braided T-shaped truss joint had higher RTM.For this,the resin system toughening agent that is strength than a similar cloth tape component [33]

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 low void content. This leads to homogeneous products. In addition, near net shape products are possible to produce. Circumferential frames, keel frames, and window frames are 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 temperature for the resin to have minimal viscosity. Resin is then applied to the preform under pressure. Later the necessary curing procedure for the specific resin is followed [25]. Min￾imal machining requirement of these products decrease the end cost. It also avoids the negative effects of machined composite parts, such as stress concentration factors intro￾duced at the machined location of the part. Also due to the damage of matrix in the machined region, environmental effects such as moisture and other existing chemicals effect the fibers, matrix, and the interface and hence this effect the strength and elastic properties of the machined composites. Michaeli et al. [26] used RTM to manufacture a braid reinforced tubular composite where the reinforcement was placed over a flexible tubing and inserted into the RTM mold. The tube was pressurized and resin injected. Good fiber placement and controlled impregnation as well as good surface finish were achieved. However, resin permeability through the preform plays a major role in the quality of products manufactured by RTM. Charlebois et al. [27] reported on permeability char￾acteristics and mechanical properties of braided fabrics. Authors investigated permeability of 2D biaxial braided glass fibers at three braid angles ±35, ±45, ±50, and found that change in braid angle effect the fiber volume fraction and thus permeability. Permeability of ±45 and ±50 angles decreased as the fiber volume fraction was increased. However, permeability of ±35 angle was not affected from the fiber volume fraction change [27]. Vacuum assisted resin transfer molding (VARTM) has also been used to manufacture braided composites [28]. VARTM offers low cost for high volume production, large and complex shapes capabilities and high fiber volume fractions compared to hand lay up [29]. VARTM process requires 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 by many industries [30]. Some other advantages of VARTM and 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 materials offer good toughness to the composites; however, the resins used have high viscosities that can not be used with the RTM/VARTM techniques. Pederson et al. [28] addressed this issue and proposed to achieve better toughness using RTM. For this, the resin system toughening agent that is used in the prepreg materials had to be manufactured in a fiber form and directly braided into the preform along with the reinforcing fibers (without compromising braid structural integrity). Experimental results demonstrated similar mechanical properties between proposed RTM and 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 to minimize cost as compared to 3D braiding. ‘‘I”, ‘‘J”, ‘‘T”, ‘‘Z” shaped composites are listed as producible. Authors found superior tensile properties with the braided specimens compared to equivalent aluminum specimens, suggesting possible aircraft applications for weight savings. Also, the braiding/RTM process was reported to have approximately 34% cost savings compared to the hand￾lay-up/ autoclave process [32]. 6. Applications Braid reinforced composite materials have a broad range of industrial applications. Based on the aforemen￾tioned advantages, such as the specific strength, these mate￾rials are preferred increasingly over the conventional engineering metals. This section outlines some of the broad applications 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 no theoretical or experimental evidence to support the claims, the structural limits of braided structure. It was stated that braided structure could be used for tensile load carrying applications if the braid angle did not exceed 15. In the cases of compression loading and thin-wall buckling, delamination could be overcome by the circumferential reinforcing nature of braided fabrics (if 20% of the fiber placement was at a ±45 braid angle). Shafts were listed as ideal components manufactured using composite materi￾als, where axially placed fibers provide stiffness, and ±45 braid provided torque transmission reinforcement. He showed, through modeling, that 54.74 braided pressure vessels are also good candidates for braided composite applications [17,18]. 2D braiding may be used to manufacture structural components as well. Kobayashi et al. [33] reported manu￾facturing a T-shape braided graphite epoxy composite truss joint. Authors proposed a different continuous production manufacturing method for structural components such as T-shaped trusses. At the end of the process the whole T￾shape had two layers of continuous triaxial braiding. In this study EPIKOTE 828 epoxy resin with an amine system hardener (KC1118) was used. Fibers were impregnated in a vacuum and an autoclave was used for curing. It was reported that the braided T-shaped truss joint had higher strength than a similar cloth tape component [33]. 46 C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58

C.Ayranci,J.Carey/Composite Structures 85 (2008)43-58 47 Hamada et al.[34,35]reported a new technique to pro- antennas and tent frames as other possible applications of duce tubular braided products that are more resistant to deployable structures [40,41]. interlaminar delamination,also referred to as through- Braided composites have also been suggested for use the-thickness toughness.The technique uses a conventional with structural reinforced concrete components since flex- 2D braider in a multireciprocal fashion to produce a multi- ural strength and ductility of reinforced concrete members layer braided laminate.Through-the-thickness fibers were can be improved with braided composite jackets [42].Life simultaneously added to the braid through a three track spans of reinforced concrete structures can be improved system where the spindles travel from one track to the by using corrosion resistant and high specific strength other creating a three-dimensional structural network of braided fiber reinforced polymer(FRP)rebars instead of strands.It was observed that propagation of interlaminar conventional steel rebars.The non-ductile behavior of delamination was impeded during lateral compression tests braid reinforced FRP rebars were also addressed by of said manufactured tubular braided specimens [34,35. researchers:Hampton et al.[43],and Lam et al.[44] Due to their specific strength and tailorable mechanical reported on hybrid Kevlar-Carbon FRP rebars manufac- properties,composites have long been the preferred mate- tured using a braiding/pultrusion process exhibiting desir- rials for aviation [15,23,24,32,36].White [36]reported on able ductile behavior similar to conventional steel rebars manufacturing,testing,and cost analysis of a Kevlar [43.441. 490/epoxy blade spar.Ballistic tests were done to evaluate Karbhari et al.[45,46]studied crush performance and the structural damage.After complete armor penetration. energy absorbing capabilities of braided composites.Braid static retesting of spar section did not show any detectable energy absorbing capabilities could be eventually used in changes in the elastic behavior.which was attributed to the industrial applications such as car bumpers.They reported braided fabric delamination resistance.Also,ultrasonic C- that triaxially braided composites increased the energy scan inspection of the structure was assessed and satisfac- absorbing performance of the braided composites [45], tory results were observed.Finally,the cost evaluation of and the occurrence of damage prior to onset of crushing the braided structure revealed 33%savings compared to fil- affected crush performance [46]. ament wound glass blade spar [361. Braid reinforced composite materials have been exten- The sports equipment industry highly utilizes the bene- sively studied for biomedical applications.Hudgins et al. fits offered by braided composite materials.Casale et al. [47,48]suggested replacing the natural intervertebral disc [37]reported on design and fabrication of a braided bicycle with a prosthetic intervertebral disc.The proposed disc frame using Kevlar/graphite braided hybrid preforms had a core of elastomeric polymer and a braid reinforced impregnated with Epon 828 epoxy resin and D-230 curing outer shell.Braided shells proved to provide compressive agent.The frame was manufactured by braiding the four- strength to the design [47,48].Moutos et al.[49]reported piece frame over a foam core and subsequent joining pro- tubular braided structures with elastomeric cores that were cess.Five prototype bicycles were produced [37]. manufactured and tested to mimic the properties of ante- Production of braid reinforced laminated wood baseball rior cruciate ligaments [49].Reinhardt et al.[50]underlined bats have been reported by Axtell et al.[38,39].Reversed the high numbers of hip replacement surgeries conducted balloon molding was used to manufacture the bats.During every year in the world,and the need for a design that this process an elastomeric tube was inflated and the would have tailorable mechanical properties,enhanced molded component pushed onto it.The tube was subse- fatigue life,and biocompatibility.Authors proposed a quently deflated to wrap the part for the curing process. design that consisted of balsa wood core with six layers Following curing,the tube was again inflated forcing the of braided carbon preforms manufactured by RTM using cured product out. a vinyl ester matrix.The study was designed as a basis Neogi et al.[40,41]published their findings on design for future studies but early mechanical performance of analysis and fabrication of a self deployable structural ele- the design were reported to be excellent;however,resin bio- ment,constructed of a foam core,internal bladder,braided compatibility issues were left for future studies [50. load carrying preform and an outer jacket,which was orig- Another example of biomedical application of braided inally developed to minimize payload volume on space composites,braided carbon/PEEK composite bone plates, shuttle missions.The proposed structure had a minimum were fabricated and tested by Fujihara et al.[51].Braided volume at the onset;using a resistance wire embedded in fabric reinforcement was chosen for this work based on the foam core as a heat source,the structure expanded better in plane properties and out of plane delamination and cured.A carbon/epoxy system was chosen for the resistance.Promising results encouraged researchers to fur- braid because of low coefficient of thermal expansion,high ther investigate the effect of the braid angles and plate longitudinal and torsional stiffness and interlaminar thicknesses on the bending performance of the composite strength.As a result of the study,80%volume savings were plates;braid angle was identified as important for thick achieved compared to original designs.The authors sug- plates.For example,it was suggested that a 2.6 mm thick gested using a triaxial braid structure due to the lower spe- plate with a 10 braid angle was suitable for forearm treat- cific stiffness of the final product compared to aluminum ments [51-53].Finally in dentistry,braided composites structures.They also listed emergency sailboats,deployable were used in dental posts that require varying stiffness

Hamada et al. [34,35] reported a new technique to pro￾duce tubular braided products that are more resistant to interlaminar delamination, also referred to as through￾the-thickness toughness. The technique uses a conventional 2D braider in a multireciprocal fashion to produce a multi￾layer braided laminate. Through-the-thickness fibers were simultaneously added to the braid through a three track system where the spindles travel from one track to the other creating a three-dimensional structural network of strands. It was observed that propagation of interlaminar delamination was impeded during lateral compression tests of said manufactured tubular braided specimens [34,35]. Due to their specific strength and tailorable mechanical properties, composites have long been the preferred mate￾rials for aviation [15,23,24,32,36]. White [36] reported on manufacturing, testing, and cost analysis of a Kevlar 49/epoxy blade spar. Ballistic tests were done to evaluate the structural damage. After complete armor penetration, static retesting of spar section did not show any detectable changes in the elastic behavior, which was attributed to the braided fabric delamination resistance. Also, ultrasonic C￾scan inspection of the structure was assessed and satisfac￾tory results were observed. Finally, the cost evaluation of the 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 bicycle frame using Kevlar/graphite braided hybrid preforms impregnated with Epon 828 epoxy resin and D-230 curing agent. 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 baseball bats have been reported by Axtell et al. [38,39]. Reversed balloon molding was used to manufacture the bats. During this process an elastomeric tube was inflated and the molded 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 the cured product out. Neogi et al. [40,41] published their findings on design analysis and fabrication of a self deployable structural ele￾ment, constructed of a foam core, internal bladder, braided load carrying preform and an outer jacket, which was orig￾inally developed to minimize payload volume on space shuttle missions. The proposed structure had a minimum volume at the onset; using a resistance wire embedded in the foam core as a heat source, the structure expanded and cured. A carbon/epoxy system was chosen for the braid because of low coefficient of thermal expansion, high longitudinal and torsional stiffness and interlaminar strength. As a result of the study, 80% volume savings were achieved 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 aluminum structures. They also listed emergency sailboats, deployable antennas and tent frames as other possible applications of deployable structures [40,41]. Braided composites have also been suggested for use with structural reinforced concrete components since flex￾ural strength and ductility of reinforced concrete members can be improved with braided composite jackets [42]. Life spans of reinforced concrete structures can be improved by using corrosion resistant and high specific strength braided fiber reinforced polymer (FRP) rebars instead of conventional steel rebars. The non-ductile behavior of braid reinforced FRP rebars were also addressed by researchers: 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 and energy absorbing capabilities of braided composites. Braid energy absorbing capabilities could be eventually used in industrial applications such as car bumpers. They reported that triaxially braided composites increased the energy absorbing performance of the braided composites [45], and the occurrence of damage prior to onset of crushing affected 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 disc with a prosthetic intervertebral disc. The proposed disc had a core of elastomeric polymer and a braid reinforced outer shell. Braided shells proved to provide compressive strength to the design [47,48]. Moutos et al. [49] reported tubular braided structures with elastomeric cores that were manufactured and tested to mimic the properties of ante￾rior cruciate ligaments [49]. Reinhardt et al. [50] underlined the high numbers of hip replacement surgeries conducted every year in the world, and the need for a design that would have tailorable mechanical properties, enhanced fatigue life, and biocompatibility. Authors proposed a design that consisted of balsa wood core with six layers of braided carbon preforms manufactured by RTM using a vinyl ester matrix. The study was designed as a basis for future studies but early mechanical performance of the design were reported to be excellent; however, resin bio￾compatibility issues were left for future studies [50]. Another example of biomedical application of braided composites, braided carbon/PEEK composite bone plates, were fabricated and tested by Fujihara et al. [51]. Braided fabric reinforcement was chosen for this work based on better in plane properties and out of plane delamination resistance. Promising results encouraged researchers to fur￾ther investigate the effect of the braid angles and plate thicknesses on the bending performance of the composite plates; braid angle was identified as important for thick plates. For example, it was suggested that a 2.6 mm thick plate with a 10 braid angle was suitable for forearm treat￾ments [51–53]. Finally in dentistry, braided composites were used in dental posts that require varying stiffness C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58 47

an bu post sha holes ha in medical field.Carey et al.published a study about the of hole diameter on bearing strength.Smaller diameters design of fiber reinforced composite catheters [55]They used more local disturbance on the orientation of fibers analyzed the require than larger resin rich regions anc of braided composites [55] rical analysis.The chdedhathberoicntaionsaroundthchoiesSienicamly Icomposite Composite materials,including braided composites. Ohki et al.[63.64]and Nakai et al.[65]evaluated the may be manufactured to near net shape toavoid any pos wev there are al cen s had highe Assembly may be accomplished through adhesive or poly tions. bonding as well as mechanical joints.Us e of adh s related to lamage of the strand path of this review.mechanical ioints were investigated since caused by the presence of the hole 163-651 8.Elastic constant predictive modek centrations around holes and cutouts.Tsiang et al Mechanical behavior of 2D braided com sites can be plastic behavior. such as t te strength and fa mnosite Important public cations about the remaining two categorie bear were me high than that are liste or bibliographical purpos not detaile men fa compo has sions to explain the observed phenomena were provided failure behavior.Braided structures are assumed to behave In anothe set or tensile exp ments, linearly in the elastic range.In the plastic egion,a non-lir through pin s th complexity of times greater loads than tho with machined holes.This was associated to the fiber discontinuity at the machined papers that holes[56,57列 e dealt propertie as well s strength anc ng th udy of Br tein et ang fail to date Authors outlined that,in the previous studies.the overal find its origins inarlier woven fabric composite and lam- ens ed composit It wa s to increased the local bearing strength.In Wang and his co can he seen as a specific form of woven fabric comnosites workers'studies.wall thickness was kept constant.Change or textile composites [75]. e models dis sed are base eng ey co the dise f thi machined holes as compared to braided holes.Other stud are familiar with the well documented CLPT analysis,such as by Jones 16 Halpin al. 77]devel ped a model pr Fuiita et al 1621 nublished studies short fiber composites from a laminate analogy.This lam-

along the shaft. This was obtained by varying the braid angle along the post shaft [54]. Braided tubular products can also be used as catheters in medical field. Carey et al. published a study about the design of fiber reinforced composite catheters [55]. They analyzed the required rigidities of conventional catheters and set design objectives to achieve these targets by use of 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 any post-manufacturing processes; however, there are also numerous applications that require multi-part assembly with 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, surface finish of substrates and coupling agents. For the purposes of this review, mechanical joints were investigated since they require holes or other shape openings in the structures and will have effects on the integrity of the parts. Composite materials are susceptible to develop stress concentrations around holes and cutouts. Tsiang et al. [56] and Brookstein [57] compared the effect of integrally formed braided holes and machined holes on strength of cylindrical braided composites. Specimens with braided and machined holes were tested under tensile loads. In average, specimens with braided holes were observed to bear loads that were 1.23 times higher than that of machined holes. Observations on specimen failure modes were presented; however, limited micromechanical discus￾sions to explain the observed phenomena were provided. In another set of tensile experiments, load was applied through pins inserted into the braided and machined holes. On average, specimens with braided holes supported 1.8 times greater loads than those with machined holes. This was associated to the fiber discontinuity at the machined holes [56,57]. Following the study of Brookstein et al., Wang and his co-workers published contradictory findings [58–60]. Authors outlined that, in the previous studies, the overall wall thickness of the tube specimens were not controlled due to excess resin surrounding the holes. It was suggested that these thicker resin rich regions contributed to the increased the local bearing strength. In Wang and his co￾workers’ studies, wall thickness was kept constant. Change in fiber angles in the surrounding regions of the holes resulted in decreased bearing strength. They concluded that similar or greater bearing strengths were found for machined 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 Brookstein et al. They stated that machined holes have lower bearing strengths; however, their work concentrated on the effect of hole diameter on bearing strength. Smaller diameters caused more local disturbance on the orientation of fibers than larger diameters leading to resin rich regions and lower bearing strengths than their larger counterparts. Results were validated using numerical analysis. They con￾cluded that fiber orientations around the holes significantly affected the bearing strength and failure mode [62]. They did not comment on the issue (i.e. the resin-rich regions surrounding the holes) raised by Wang et al. Ohki et al. [63,64], and Nakai et al. [65] evaluated the effect of machined versus braided holes in end loaded flat-braided specimen with a centralized hole. Specimens with braided holes had higher strength properties during both static and fatigue testing. From microscopic observa￾tions, authors conclude that the damage mechanism of the machined holes is related to the fiber–resin interface, while the damage mechanism of the braided holes is related to the reorientation of the continuous fibrous strand path caused by the presence of the hole [63–65]. 8. Elastic constant predictive models Mechanical behavior of 2D braided composites can be discussed in terms of elastic behavior, plastic behavior, and failure behavior such as ultimate strength and failure mechanism [66]. This review, due to the broadness of the topic, focuses on elastic behaviors of braided composites. Important publications about the remaining two categories are listed for bibliographical purposes but not detailed. Elastic property prediction of 2D braided composites has been studied far more than their plastic behavior and failure behavior. Braided structures are assumed to behave linearly in the elastic range. In the plastic region, a non-lin￾ear behavior is observed which increases the complexity of the problem. Nevertheless, a number of studies have been published regarding plasticity behavior and failure charac￾teristics of braided composites [67–74]. Other papers that have dealt with elastic properties, as well as strength and failure mechanisms, will be covered in detail. The majority of braid analysis developed to date can find its origins in earlier woven fabric composite and lam￾inated composite analysis; hence, this review also outlines major studies published in these fields to create a basis for the overall discussion. In this view, braided composites can be seen as a specific form of woven fabric composites, or textile composites [75]. Some of the models discussed are based on the well known Classical Laminate Plate Theory (CLPT). During the discussions of this review, it is assumed that readers are familiar with the well documented CLPT analysis, such as by Jones [76]. In early 1970s, Halpin et al. [77] developed a model pre￾dicting elastic stiffness and thermal expansion properties of short fiber composites from a laminate analogy. This lam- 48 C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58

C.Ayranel.J.Careyl Composite Siructures 85(2008)43-58 49 inate analogy was extended to two-and three-dimensiona cylinders using a simple micromechanics theory dmP心Soven fa es wer of t ply and Whitney and Halpin 78 analyzed laminated aniso tropic tubes subjected to combined tension or compression principle of superposition to the two sub-layers.Results for nternal pressur torque.A se elast us and Poi ment with ex ental to was done using Donnel's approximations Yang et al []proposed a predictive model for triaxial a followed were L th and st the nd C dulatio els that formed the basis to many subsequent textile fabric 79assumes 60fiber deposition angle.The model utilize namely,the undula haracterization of the braid architecture tion and e mod dy the sma cell.Th ite is treated as an ass of the unit cellan ssumed to he ve of the over. ated with matrix are taker all composites. Mosaic model treats the system as an The mode of Mi regions are sub quently consid zed a ing bon so-strain and iso-stress assumptions to respectively obtair stacked together with fibers in the bias braid and longitudi upper and ower bound compo al angles, be und to n s a result of the analys nuity char acteristics of the fibers in woven fabric comp ced by the fiber de sition angles. The tes omitted in the mo was not verified experimentally [84] e In a la Yang e al.[85 ume fraction of the unit cel The undulating he elastic erties of three.dimensional textile assumed to follow a path described by a sinusoidal func and braided)composites.Here.the unit cell used for the were used to CLP analysis is assumed to be compose 0 of the unidir of the local undulation angle (called"local offaxis angle onal directions.All the yarns in one direction were a and Chou).The authors state that the undu of the con analysis n exten th strain occurs at the mid-point of the undulating fiber. The In the analysis contribution of pure matrix re s to the bridging model was developed for satin weave fabrics and stiffness matrices were neglected (interested reader may this view [79 eter to the riginal text for the of h rid u en fab c6i gated effects of these fabric parameters on elastic pro ertie interlocking points and stated that it is still a convenient by using the m emodel.In this m the analysis.Predictions and experimenta nature g mo ngs g000 nt [ the Fabri Gaps that may exist between were Geomeury Model (FGM.to predict the aclose mesh configuration was adapted.In this report,Ishik three-dimensionally braided structures.FGM is based on thermal exp sion coe modified CLP where the uni ed as epeating an the stiftne s matrix of the fiber undulation model agreement was found betweer lent unidirectional lamina and transforming it into the experimental and theoretical results [82] tructural coordinate The contrib tions of each

inate analogy was extended to two- and three-dimensional woven fabric composites. Authors reported that predicted and experimental results for woven fabric composites were compared 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 governing equations as equilibrium, compatibility, strain and curva￾ture displacement, and constitutive relations. The analysis was done using Donnel’s approximations. Some of the most influential studies that followed were published by Ishikawa and Chou [79–81] who proposed and compared three stiffness and strength predictive mod￾els that formed the basis to many subsequent textile fabric composite 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 properties of the unit cell are assumed to be representative of the over￾all composites. Mosaic model treats the system as an assemblage of asymmetric cross-ply laminates. The model uses the Classical Laminate Plate (CLPT) theory as the basis of the analysis. The model was analyzed using both iso-strain and iso-stress assumptions to respectively obtain upper and lower bound composite stiffness properties. The fiber undulation model was developed to validate and improve 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 to physically occurring matrix only regions, this model also allowed 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 CLPT analysis. The local stiffness matrices used in the calculation of the CLPT A, B, D matrices were computed as a function of 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 maximum strain occurs at the mid-point of the undulating fiber. The bridging model was developed for satin weave fabrics and is therefore out of the scope of this review [79–81]. Ishikawa and Chou also characterized geometric and material properties of hybrid woven fabrics[82], and investi￾gated effects of these fabric parameters on elastic properties by using the mosaic model. In this model, due to the hybrid nature of the fabric, in-plane and bending moduli (Aij, andBij matrices) are no longer uniform in the repeating region. Gaps that may exist between the fibers were neglected and a close mesh configuration was adapted. In this report, Ishik￾awa and Chou also investigated the thermal expansion coef- ficients and thermal bending coefficients. Investigation was conducted using the mosaic model and one-dimensional fiber undulation model. Agreement was found between experimental 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 theory based model. The braid architecture was modeled as a structure composed of unidirectional-ply and bias-angle ply yarns. The brief description of the model provided sta￾ted that material properties were calculated by applying the principle of superposition to the two sub-layers. Results for the 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, based on the Ishikawa and Chou’s fabric undulation model [79], assumes 60 fiber deposition angle. The model utilizes the geometrical characterization of the braid architecture where 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 taken into account in the initial calculation, and the contribution of the matrix only regions are subsequently considered using a Rule of Mixtures prediction. The upper bound is calculated from a laminate that consists of three laminae stacked 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 the non-orthogonal woven fabrics was determined to be strongly influenced by the fiber deposition angles. The model was not verified experimentally [84]. In a later study, Yang et al. [85] proposed the ‘‘Fiber Inclination Model” based on a modified CLPT to predict the elastic properties of three-dimensional textile (woven and braided) composites. Here, the unit cell used for the analysis is assumed to be composed of an assemblage of inclined unidirectional laminae. The idealized unit cell was described as fiber bundles oriented in four body diag￾onal directions. All the yarns in one direction were assumed to form inclined laminae after matrix impregnation. The rest of the analysis was explained as an extension of the fiber undulation model developed by Ishikawa and Chou. In the analysis, contribution of pure matrix regions to the stiffness matrices were neglected (interested reader may refer to the original text for the modifications and neces￾sary assumptions). Authors recognized and underlined that the CLPT ignores the interactions of fiber yarns at the interlocking points and stated that it is still a convenient technique for the analysis. Predictions and experimental findings were in good agreement [85]. Whyte [86] proposed an analytical model, the Fabric Geometry Model (FGM), to predict the properties of three-dimensionally braided structures. FGM is based on a modified CLPT where the unit cell is defined as repeating volumes. The stiffness matrix is developed for each yarn in the unit cell by calculating the stiffness matrix of the equiv￾alent unidirectional lamina and transforming it into the structural coordinate system. The contributions of each yarn are superimposed with respect to their volumetric contribution. Authors also suggest calculating the strain C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58 49

0 C.Ayranci,J.Carey/Composite Structures 85(2008)43-58 at every new strain level to account for the non-linear numerical models were accurate but complex.To address behavior of the materials [86].Pastore and Gowayed [87] early model concerns,Naik and Shembekar proposed sim- underlined two major disadvantages of the FGM model ple but accurate generalized two-dimensional models to presented earlier.First,the theoretical mathematical deri- predict the elastic properties of woven fabric composites. vation is not compatible with the basic transverse isotropy Their models account for fiber continuity and undulation used in the model.Secondly,the transformation matrices in both the weft and warp directions,matrix only regions used for the stiffness calculations are not consistent.In and cross sectional geometry of the yarns in the unit cell their paper,authors address these problems.They com- The Naik and Shembekar two-dimensional model was an pared stiffness averaging and compliance averaging tech- extension of the Ishikawa and Chou one-dimensional niques,and they compared predicted and experimental model.Only non-hybrid two-dimensional plain weave fab- results for triaxially braided,as well as orthogonal glass ric lamina was considered.The unit cell was divided into reinforced composites.The self consistent FGM model. straight cross ply,undulated cross ply,and matrix only as it was called,was used to predict elastic properties regions.The undulating tow paths were modeled using results using both stiffness averaging and compliance aver- sinusoidal functions.The elastic constants were calculated aging technique.Authors highlighted that in all cases the using a Cylindrical Assemblage Model(CAM)in the prin- stiffness averaging technique provided better predictions. cipal material directions.Each infinitesimal region of the Ko et al.[2]and Lei et al.[88]presented the Finite Cell unit cell was analyzed using CLPT [90-92]. Model(FCM)in which the unit cell for the structure was In the analysis,the unit cell is assumed to be comprised defined as an assemblage of brick-shaped elements.The of sub-sections along and perpendicular to the loading FCM defines the composite as a "space truss"and hence directions.Each sub-section is comprised of infinitesimal each yarn is considered individually.The yarns are pieces.A uniform,unidirectional,in-plane load was assumed to be diagonals of the unit cell and analyzed as assumed to be applied to the woven fabric.The infinitesi- pin-jointed two-force truss members,which makes this mal sub-sections in the unit cell,which are in series with model suitable for finite element analysis. the loading direction,were assumed to be under constant Soebroto et al.[89]published a design framework for stress.On the other hand,the infinitesimal sub-sections braided tubular composites.The objective was to fill a that are parallel to the loading axis were assumed to have gap in the field by creating a link between textile preform constant strain in their mid-planes [90-92].Following this manufacturers and structural designers creating design approach,they created two models:Series Parallel Model curves such as effect of braid angle on fabric diameter (SPM)and Parallel Series Model(PSM).The SPM was cre- and transverse speed required for a given braider to achieve ated by assembling all the infinitesimal pieces in series with a certain diameter tubular braided preform.They also used the loading direction under iso-stress condition,and then FGM model by Whyte [86],to predict elastic properties assembling all the sections along the loading direction and strength of 2D braids.Soebroto et al.theoretical pre- under iso-strain condition.The PSM was created by fol- dictions,taken approximately from a graphic,were lowing the same approach in the reverse order.Naik and reported in the range of 5-85.Experimental verification Shembekar stated that the SPM provides the lower bounds was done for 20 and 70 braid angles and appear to follow of the in-plane stiffness constants,whereas the PSM pro- the general trend with the models published after them, vides the upper bounds of the stiffness constants.Following such as the longitudinal elastic modulus decreases as the experimental verification PSM was recommended for braid angle changes increases [89].However,for the region woven fabric composites [90-92].It should be underlined between30°and6O°，their predictions appear nearly linear that PSM was developed for woven fabric composites, compared to other models that have a more curved shape. which can not be generalized to include braided structures Again,FGM was originally developed for three-dimen- that may have different angle orientations. sionally braided composites and does not include undulat- Finally,Naik and Shembekar [92]underlined the superior ing fiber strands.Hence,following the comparison of linear properties and advantages of woven fabric composites to versus curved predictions of the different models,it may be that of unidirectional composites such as shorter build time, concluded that for two-dimensionally braided composites complex shape capability and ease of mold impregnation more sensitivity in the results may be obtained by methods because of the intertwined structure.Authors highlighted that account for fiber undulation. the fact that the elastic behavior of a unidirectional lamina A woven fabric study was published by Naik and She- and a thin laminate are the same,whereas this may not be mbekar [90-92]as a series of three publications,namely, necessarily true for a woven fabric lamina and thin woven lamina and laminate analysis and laminate design.Naik fabric laminate because of the macroscopically heteroge- and Shembekar indicated that the early elementary lami- neous nature of the woven fabric lamina;they also outlined nate theory models developed,such as mosaic and undula- the limited number of studies published in this field [92]. tions models by Ishikawa and Chou,were simple but not Authors studied the effect of stacking sequence or shift of accurate because of the one-dimensional nature of these laminae to obtain optimal laminates.Since this is beyond models leading to large discrepancies between predicted the scope of this review,interested readers are referred to and experimental results.Conversely,authors indicated, the original publication [92]

at every new strain level to account for the non-linear behavior of the materials [86]. Pastore and Gowayed [87] underlined two major disadvantages of the FGM model presented earlier. First, the theoretical mathematical deri￾vation is not compatible with the basic transverse isotropy used in the model. Secondly, the transformation matrices used for the stiffness calculations are not consistent. In their paper, authors address these problems. They com￾pared stiffness averaging and compliance averaging tech￾niques, and they compared predicted and experimental results for triaxially braided, as well as orthogonal glass reinforced composites. The self consistent FGM model, as it was called, was used to predict elastic properties results using both stiffness averaging and compliance aver￾aging technique. Authors highlighted that in all cases the stiffness averaging technique provided better predictions. Ko et al. [2] and Lei et al. [88] presented the Finite Cell Model (FCM) in which the unit cell for the structure was defined as an assemblage of brick-shaped elements. The FCM defines the composite as a ‘‘space truss” and hence each yarn is considered individually. The yarns are assumed to be diagonals of the unit cell and analyzed as pin-jointed two-force truss members, which makes this model suitable for finite element analysis. Soebroto et al. [89] published a design framework for braided tubular composites. The objective was to fill a gap in the field by creating a link between textile preform manufacturers and structural designers creating design curves such as effect of braid angle on fabric diameter and transverse speed required for a given braider to achieve a certain diameter tubular braided preform. They also used FGM model by Whyte [86], to predict elastic properties and strength of 2D braids. Soebroto et al. theoretical pre￾dictions, taken approximately from a graphic, were reported in the range of 5–85. Experimental verification was done for 20 and 70 braid angles and appear to follow the general trend with the models published after them, such as the longitudinal elastic modulus decreases as the braid angle changes increases [89]. However, for the region between 30 and 60, their predictions appear nearly linear compared 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 linear versus curved predictions of the different models, it may be concluded that for two-dimensionally braided composites more sensitivity in the results may be obtained by methods that 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. Naik and 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 not accurate because of the one-dimensional nature of these models leading to large discrepancies between predicted and experimental results. Conversely, authors indicated, numerical models were accurate but complex. To address early model concerns, Naik and Shembekar proposed sim￾ple but accurate generalized two-dimensional models to predict the elastic properties of woven fabric composites. Their models account for fiber continuity and undulation in both the weft and warp directions, matrix only regions and cross sectional geometry of the yarns in the unit cell. The Naik and Shembekar two-dimensional model was an extension of the Ishikawa and Chou one-dimensional model. Only non-hybrid two-dimensional plain weave fab￾ric lamina was considered. The unit cell was divided into straight cross ply, undulated cross ply, and matrix only regions. The undulating tow paths were modeled using sinusoidal functions. The elastic constants were calculated using a Cylindrical Assemblage Model (CAM) in the prin￾cipal material directions. Each infinitesimal region of the unit cell was analyzed using CLPT [90–92]. In the analysis, the unit cell is assumed to be comprised of sub-sections along and perpendicular to the loading directions. Each sub-section is comprised of infinitesimal pieces. A uniform, unidirectional, in-plane load was assumed to be applied to the woven fabric. The infinitesi￾mal sub-sections in the unit cell, which are in series with the loading direction, were assumed to be under constant stress. On the other hand, the infinitesimal sub-sections that are parallel to the loading axis were assumed to have constant strain in their mid-planes [90–92]. Following this approach, 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 with the loading direction under iso-stress condition, and then assembling all the sections along the loading direction under iso-strain condition. The PSM was created by fol￾lowing the same approach in the reverse order. Naik and Shembekar stated that the SPM provides the lower bounds of the in-plane stiffness constants, whereas the PSM pro￾vides the upper bounds of the stiffness constants. Following experimental verification PSM was recommended for woven fabric composites [90–92]. It should be underlined that PSM was developed for woven fabric composites, which can not be generalized to include braided structures that may have different angle orientations. Finally, Naik and Shembekar[92] underlined the superior properties and advantages of woven fabric composites to that of unidirectional composites such as shorter build time, complex shape capability and ease of mold impregnation because of the intertwined structure. Authors highlighted the fact that the elastic behavior of a unidirectional lamina and a thin laminate are the same, whereas this may not be necessarily true for a woven fabric lamina and thin woven fabric laminate because of the macroscopically heteroge￾neous nature of the woven fabric lamina; they also outlined the limited number of studies published in this field [92]. Authors studied the effect of stacking sequence or shift of laminae to obtain optimal laminates. Since this is beyond the scope of this review, interested readers are referred to the original publication [92]. 50 C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58

C.Ayranci,J.Carey/Composite Structures 85 (2008)43-58 51 Masters et al.[93]studied the mechanical properties of thickness equal to the full braid layer and assumed to have triaxially braided composites both analytically and experi- primary fiber tows and effective matrix material.Effective mentally,which could serve as a database of experimental matrix material is assumed to be composed of two second- results for comparison purposes with predicted results of ary fiber tows and matrix material,and is analyzed using models available at the time.The experiments used 2 x 2 CLPT as a symmetric laminate.This model may be a good braided AS4/epoxy resin composite flat panels impreg- alternative to obtain fast preliminary design results prior to nated by RTM.Braid angle,size of the braided yarn and a detailed analysis [97]. size of the longitudinal yarn were varied to obtain three dif- Following Masters et al.[93],Naik et al.[98]conducted ferent architectures.A processing science model was used an analytical and experimental study on the effects of to construct the braided unit cell geometry.Mechanical braiding parameters on 2D triaxially braided composites. properties of the braided composites were predicted using Braiding parameters were listed as braid angle,yarn size four different approaches,namely,laminate,corrected lam- and axial yarn content.A Repeating Unit Cell (RUC) inate,diagonal brick,and finite element model.Laminate was isolated and used for the analysis.Each yarn in the model was the simplest model where all the tows were trea- RUC was discretely modeled and sliced.The three-dimen- ted as unidirectional plies in a symmetric laminate.A cor- sional effective stiffness of the RUC was calculated using a rection factor was applied to this model to compensate for volume averaging technique under iso-strain assumption. the ignored fiber undulations to create the corrected lami- Although the analysis was conducted in three dimensions nate model.Diagonal brick model [94]is an extension of with respect to the XYZ global coordinate axis,the pre- the above FCM.The finite element model was based on dicted elastic properties mainly showed sensitivity to braid- a previous model proposed by one of the authors of the ing parameter in the longitudinal and transverse directions. paper,Foye R.L.,where the unit cell was analyzed as a The elastic properties in the thickness direction were much combination of sub-cells.They found that all model predic- less sensitive to changes in braid angle or percent axial yarn tions were comparable to experimental findings and the dif- content.This may be the underlying reason to why many ferences between them were not significant;however,finite braiding models following this study analyzed braids only element method predictions were best.Also studied was the in the axial and transverse direction,such as Carey et al. sensitivity of experimental measurements to strain gage [75].Stiffness properties were not affected by yarn sizes, sizes.Findings concluded that large gage sizes,such as but were affected by braid angle and axial yarn content. the 2.54 cm gage length of some extensometers,were pref- Increasing the braid angle increased transverse and shear erable [93].Master and Ifju [95]later published a detailed elastic moduli,but decreased longitudinal elastic modulus. study where they outline Moire interferometry.X-ray radi- It was also reported that the out of plane elastic and shear ography,and surface replication techniques as alternatives moduli were insensitive to these parameters [98]. to inspecting or testing methods for braided composites Following Naik et al.,Naik [99]published a study to [951 extend on the previous work.He implemented the analysis A review paper that utilizes experimental results to com- in a program code called TEXCAD used for braided as pare stiffness predictive models available at the time was well as other textile composites.The work was also published by Falzon et al.in 1993,[96].Authors catego- extended to predict strength of woven and braided compos- rized the models into three types,namely,the elementary ites[99-101]. models such as fabric geometry model (FGM);the lami- Raju and Wang [102]reported a detailed study about nate theory models such as "fiber undulation model"and classical laminate theory models for woven fabric compos- "mosaic model";and,finally numerical models.Authors ites derived from,but not limit by the simplification of,the stated that the elementary models are unsuitable for Ishikawa and Chou models [79].They first identified a strength calculation;the laminate models are unable to pre- repeating unit in the woven fabric composite,which was dict out-of plane elastic properties;while finite element further divided into unit cells.This geometrical character- models are complex [96].Although these observations were ization was done for plain weave,5-and 8-harness satin true at the time,in subsequent years,improvements were weave structures;this review covers only the plain weave made to these models to address these concerns. case.A uniform membrane strain and curvature are Redman and Douglas [97]proposed a simple analytical assumed at the midplane of the unit cell.The unit cell model to determine the elastic properties of triaxially was divided into four regions,each subsequently divided braided composites.The model utilizes a unique combina- into four sub-regions composed of undulating and non- tion of Rule of Mixtures prediction and CLPT.Unlike undulating regions.As was the case for Ishikawa and many of the previous models presented,the Redman and Chou's fiber undulation model,Raju and Wang's model Douglas model,due to this unique modeling approach accounts for the undulating fibers;however,they use a combination,does not require the use of a unit cell.The more accurate geometry to characterizes undulating fibers length between neighboring fibers was assumed to be big in the fill and warp directions than its predecessor.The enough to neglect the effect of undulating fibers.The triax- undulating fibers are assumed to follow a sinusoidal shape ial braid is considered to have three separate plies that all function as with the model by Naik and Shembekar [901. coexist in the same space.Each ply is assumed to have a CLPT stiffness matrices A,B,and D of the unit cells are

Masters et al. [93] studied the mechanical properties of triaxially braided composites both analytically and experi￾mentally, which could serve as a database of experimental results for comparison purposes with predicted results of models available at the time. The experiments used 2 2 braided AS4/epoxy resin composite flat panels impreg￾nated by RTM. Braid angle, size of the braided yarn and size of the longitudinal yarn were varied to obtain three dif￾ferent architectures. A processing science model was used to construct the braided unit cell geometry. Mechanical properties of the braided composites were predicted using four different approaches, namely, laminate, corrected lam￾inate, diagonal brick, and finite element model. Laminate model 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 for the ignored fiber undulations to create the corrected lami￾nate model. Diagonal brick model [94] is an extension of the above FCM. The finite element model was based on a previous model proposed by one of the authors of the paper, Foye R.L., where the unit cell was analyzed as a combination 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, finite element method predictions were best. Also studied was the sensitivity of experimental measurements to strain gage sizes. Findings concluded that large gage sizes, such as the 2.54 cm gage length of some extensometers, were pref￾erable [93]. Master and Ifju [95] later published a detailed study where they outline Moire interferometry, X-ray radi￾ography, and surface replication techniques as alternatives to 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 was published by Falzon et al. in 1993, [96]. Authors catego￾rized the models into three types, namely, the elementary models such as fabric geometry model (FGM); the lami￾nate theory models such as ‘‘fiber undulation model” and ‘‘mosaic model”; and, finally numerical models. Authors stated that the elementary models are unsuitable for strength calculation; the laminate models are unable to pre￾dict out-of plane elastic properties; while finite element models are complex [96]. Although these observations were true at the time, in subsequent years, improvements were made to these models to address these concerns. Redman and Douglas [97] proposed a simple analytical model to determine the elastic properties of triaxially braided composites. The model utilizes a unique combina￾tion of Rule of Mixtures prediction and CLPT. Unlike many of the previous models presented, the Redman and Douglas model, due to this unique modeling approach combination, does not require the use of a unit cell. The length between neighboring fibers was assumed to be big enough to neglect the effect of undulating fibers. The triax￾ial braid is considered to have three separate plies that all coexist in the same space. Each ply is assumed to have a thickness equal to the full braid layer and assumed to have primary fiber tows and effective matrix material. Effective matrix material is assumed to be composed of two second￾ary fiber tows and matrix material, and is analyzed using CLPT as a symmetric laminate. This model may be a good alternative to obtain fast preliminary design results prior to a detailed analysis [97]. Following Masters et al. [93], Naik et al. [98] conducted an analytical and experimental study on the effects of braiding parameters on 2D triaxially braided composites. Braiding parameters were listed as braid angle, yarn size and axial yarn content. A Repeating Unit Cell (RUC) was isolated and used for the analysis. Each yarn in the RUC was discretely modeled and sliced. The three-dimen￾sional effective stiffness of the RUC was calculated using a volume averaging technique under iso-strain assumption. Although the analysis was conducted in three dimensions with 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 much less sensitive to changes in braid angle or percent axial yarn content. This may be the underlying reason to why many braiding models following this study analyzed braids only in 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 shear elastic moduli, but decreased longitudinal elastic modulus. It was also reported that the out of plane elastic and shear moduli were insensitive to these parameters [98]. Following Naik et al., Naik [99] published a study to extend on the previous work. He implemented the analysis in a program code called TEXCAD used for braided as well as other textile composites. The work was also extended to predict strength of woven and braided compos￾ites [99–101]. Raju and Wang [102] reported a detailed study about classical laminate theory models for woven fabric compos￾ites derived from, but not limit by the simplification of, the Ishikawa and Chou models [79]. They first identified a repeating unit in the woven fabric composite, which was further divided into unit cells. This geometrical character￾ization was done for plain weave, 5- and 8-harness satin weave structures; this review covers only the plain weave case. A uniform membrane strain and curvature are assumed at the midplane of the unit cell. The unit cell was divided into four regions, each subsequently divided into four sub-regions composed of undulating and non￾undulating regions. As was the case for Ishikawa and Chou’s fiber undulation model, Raju and Wang’s model accounts for the undulating fibers; however, they use a more accurate geometry to characterizes undulating fibers in the fill and warp directions than its predecessor. The undulating fibers are assumed to follow a sinusoidal shape function as with the model by Naik and Shembekar [90]. CLPT stiffness matrices A, B, and D of the unit cells are C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58 51

52 calculated as follows.First the A.B.and D matrices of all r su are calcula by idal f are summed over each sub-region to obt ain A B and D before.Naik and Ganesh defined the unit cell of the com matrices for each region in the unit cell.Finally,all fou posite as an asymmetric cross-ply laminate.This laminate egions ares that the integrals involving the undulat ric lamina CLPT under the specifying the method selected.In the model,coefficients of thermal Eremcedrem5 the bending deformation unit cell are constrained mos u ioned.the study was conducted for woven fabrics:hence cular analyzed.The e element modelt ous mode ame autho the short o the Unit Cell Contin and quck results.The efect of the ratio of strand thicknes uum Model (UCCM)by Foye [4].The UCCM utilizes nit that nents of the the study and the results are provide culate the total displac suggested that the UCCM does not clearly differentiate later,Carey et al.[75],calculated this to be negligible in 2D between fiber and matrix material properties n the analy braided structure es[108 They te that if th the prope pultru fiber reinforced composites.the solutio mes ina are pre gnated and s ty braided on a Teflon mandrel. regnated preforms are cured in a curing die 86 composite into piece tie rately and their contribution to the slobal stifess matri are calculated through superimposing each contribution its length then uses this information to obtain the effective with respectto lume The mode of the omposite by averaging the yarn, and tal [1061 and Naka 1 [107]attempted to use the unit cell predictions to design does not allow for open-mesh braid configuration.Limited and predic avior cal ana braided ylinders upon loading experimental data was provided [11I] using num thre mo was pre 11 straight lines.and of a macro analysis which combined of two-dim sional textile and three-dimensional braided the micro models,to form structural elements.They also composites.The global constitutive equation of the com studied the of braiding structure on torsiona posite ma 1a1 Gane OgTgidla-dinmensionalorih a4 nd matr1》 onal plain weave fabric laminae through a thermoelastic Tsai et al.31]used a CLPT-based model to predict stiff- . The that most of the model ness and strength braided tubular composites.They 9,110 th 91 Tmp.tha are predicted valucs were generally not in god agreement:how. models that included these were complex.Consequently ever,better agreement was found with the crimp model closed form the and c ity in fill and w interlacing geometrical patterns by a series of vectors two adja nat is com Authors ir and warp fibers an models as well as perme of pref woven form and the strand undulations are defined by

calculated as follows. First the A, B, and D matrices of all four sub-regions are calculated by integrating the Stiffness matrix, Q, over the volume of each sub-region. Then, these are summed over each sub-region to obtain A, B, and D matrices for each region in the unit cell. Finally, all four regions 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 without specifying the method selected. In the model, coefficients of thermal expansions were also obtained. Predicted results were compared to many other available models; most matched 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 to predict the elastic properties of textile composites. This model addresses the short comings of the Unit Cell Contin￾uum Model (UCCM) by Foye [104]. The UCCM utilizes a unit cell that is divided into subcells. Displacements of the subcells are calculated using Virtual Work, and summed to calculate the total displacement. However, Gowayed et al. suggested that the UCCM does not clearly differentiate between fiber and matrix material properties in the analy￾sis. They state that if the difference between the properties of fiber and matrix materials is large, such as for the case of fiber reinforced composites, the solution becomes inaccu￾rate. Authors suggested correcting this by using the UCCM along with Whyte’s FGM model [86], where composite material fibers and matrix constituents are treated sepa￾rately and their contribution to the global stiffness matrix are calculated through superimposing each contribution with respect to their relative volume fraction. The model was verified experimentally [103]. Nakai et al. [105], Hamada et al. [106], and Nakai et al. [107] attempted to use the unit cell predictions to design and predict behavior of braided cylinders upon loading using numerical analysis. The analysis was comprised of a micro analysis, modeling individual resins and fibers as straight lines, and of a macro analysis, which combined the micro models, to form structural elements. They also studied the influence of braiding structure on torsional properties of braided composite tubes. Naik and Ganesh [108] studied two-dimensional orthog￾onal plain weave fabric laminae through a thermoelastic analysis. The authors claimed that most of the models developed until then, [90,109,110], do not consider the actual strand geometry and cross section; hence, the fiber volume fraction was not included in the models. The few models that included these were complex. Consequently, they outlined a two-dimensional closed form analytical method which takes into consideration the strand undula￾tion and continuity in fill and warp directions, strand cross section, fiber volume fraction, and possible gaps between two adjacent strands. In their model, a unit cell that is com￾posed of three layers, fill and warp fibers and matrix, is used. Strand cross section, strand cross sectional shape in woven form and the strand undulations are defined by shape functions. Authors compared both circular and sinu￾soidal functions for strand undulations and concluded that the sinusoidal functions offered better predictions. As many before, Naik and Ganesh defined the unit cell of the com￾posite as an asymmetric cross-ply laminate. This laminate is 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 the assumptions that CLPT is applicable to a unit cell and the bending deformations of a unit cell are constrained by the surrounding unit cells. The undulating angle of the strands 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. They concluded that the proposed model provides acceptable and quick results. The effect of the ratio of strand thickness to strand width on elastic constants was also investigated in the study and the results are provided in a graphical form. They also suggested that the twist of the strand along the fiber undulation direction should be investigated; however, later, Carey et al. [75], calculated this to be negligible in 2D braided structures [108]. Byun et al. [111] proposed a novel braiding and pultru￾sion manufacturing technique during which the fiber tows are preimpregnated and subsequently braided on a Teflon mandrel. Impregnated preforms are cured in a curing die and cut into pieces. Authors proposed an analytical model for elastic properties of braided products that first calcu￾lates the effective compliance matrix of a yarn based on its length then uses this information to obtain the effective stiffness 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 model does not allow for open-mesh braid configuration. Limited experimental data was provided [111]. A three-dimensional tow inclination model was pro￾posed by Branch et al. [112] to calculate elastic constants of two-dimensional textile and three-dimensional braided composites. The global constitutive equation of the com￾posite material is derived using an iso-strain approach for the unit cell and averaging all tow segments and matrix within the unit cell [112]. Tsai et al. [31] used a CLPT-based model to predict stiff- ness and strength of braided tubular composites. They introduced two models, bridge and crimp, that are similar to those of Ishikawa and Chou [79]. The experimental and predicted 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 predictive models. They proposed a method to describe preform interlacing 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 to obtain optimal impregnation of fibers. In a subsequent study [114], the group underlined the difficulty in character- 52 C. Ayranci, J. Carey / Composite Structures 85 (2008) 43–58