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Analytical study of strength and failure behaviour of plain weave fabric composites made of twisted yarns
Composites: Part A 33 (2002) 697±708
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Analytical study of strength and failure behaviour of plain weave fabric composites made of twisted yarns N.K. Naik*, R. Kuchibhotla Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Received 5 December 2000; revised 10 August 2001; accepted 3 January 2002
Abstract A two-dimensional analytical method is presented for the failure behaviour of plain weave fabric composites made of twisted yarns. The studies have been carried out on laminates with different con®gurations under on-axis uni-axial tensile loading. The cross-sectional area of the yarn was taken to be elliptical and the yarn path was taken to be sinusoidal. Different stages of failure are considered in the analysis. It has been observed that there is no signi®cant reduction in tensile strength properties of plain weave fabric composites as a result of twisting of yarns. For E-glass yarns, twisting of yarns up to 58, can facilitate ease of fabrication without signi®cantly compromising the strength properties of the woven fabric composites. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Fabrics/textiles; B. Strength; C. Analytical modelling; C. Damage mechanics
1. Introduction Advanced textile structural composites are ®nding increasing use for many high performance applications during last one decade. Textile techniques such as twodimensional (2D) and three-dimensional (3D) weaving, braiding, knitting and through-the-thickness stitching have assisted in enhancing the performance of textile composite structures. Such composites are characterised by enhanced through-the-thickness elastic and strength properties, impact/fracture resistance, damage tolerance and dimensional stability. Additionally, textile structural composites are associated with near-net-shape and cost effective manufacturing processes. For high in-plane speci®c stiffness and high in-plane speci®c strength applications, 2D woven fabric (WF) composites can be the competitors to laminated composites made of unidirectional (UD) layers. To derive the maximum bene®ts of the textile structural composites, an improved understanding of the detailed structure of the reinforcement with the advances in fabric formation techniques is essential. * Corresponding author. Tel.: 191-22-576-7114; fax: 191-22-572-2602. E-mail address: [email protected] (N.K. Naik). Abbreviations: C1, C2, laminate con®guration-1 and -2; FSI, ®ll shear failure initiation; FTI, ®ll transverse failure initiation; PMBFI, pure matrix block failure initiation; UD, unidirectional; WF, woven fabric; WTC, complete failure of warp; WTI, warp transverse failure initiation; 1D, 2D, 3D, one-, two-, three-dimensional
The textile structural composites are made using the woven, braided, knitted or stitched preforms. The textile preforms are planar or 3D. The special feature of the textile preforms is the signi®cant reinforcement interconnectivity between adjacent planes of reinforcements. This interconnectivity provides additional interface strength to supplement the relatively weak ®bre/resin interface. The interconnectivity is mainly in the plane of the preform for the planar textile preforms. Such materials are known as 2D textile preforms. The 3D textile preforms for the structural composites are fully integrated continuous reinforcement assemblies having multi-axial in-plane and out-of-plane reinforcement orientations. Formation of different textile preforms is an important stage in composites technology. WFs are produced by the process of weaving in which the fabric is formed by interlacing warp and ®ll strands/yarns. Knitted preforms are made by interlooping whereas braided preforms are made by intertwining. One of the important requirements of the reinforcing elements during preform preparation is the lateral cohesion. For this reason, the twisted yarns are used for preparing the textile preforms rather than the straight strands. Twisted yarns are normally used for increasing the lateral cohesion of the ®laments and also for ease of handling. By twisting yarns, possible micro damages within the yarn can be localised, leading to possible increase in the failure strength of the yarn. For this reason, twisted yarns are normally used for making textile preforms, especially for
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Nomenclature a yarn width aij extensional compliance matrix axt, ayt as shown in Fig. 4 Aij extensional stiffness matrix E, G, n elastic properties g inter-yarn gap h maximum yarn thickness hm matrix thickness ht fabric thickness HL lamina thickness l sub-element, element, section dimensions L unit cell dimensions nx, ny, nz number of sub-elements, elements, sections in the x-, y-, z-directions Å Q transformed reduced stiffness matrix R radius of yarn S shear strength Vf ®bre volume fraction x, y, z cartesian coordinates XT, YT tensile strength zxi x; y; i 1±4 yarn shape parameters zyi y; i 1±4 yarn shape parameters a angle of twist for the yarn 1 normal strain 1p pre-strain s normal stress s x load per unit area in x-direction q x; q y local off-axis angle of the undulated yarn t shear stress Superscripts el refers to element nl non-linear o overall properties s quantities of strand sec refers to section sel refers to sub-element y quantities of yarn Subscripts f, F quantities in ®ll direction L longitudinal axis of the ®bre T transverse axis of the ®bre w, W quantities in warp direction 1,2 local coordinates x, y global coordinates
E-glass reinforcements. Twisting of yarns may lead to possible reduction in ultimate failure strain. But twisting of yarns may be unavoidable for engineering reasons. The objective of the present work is to study the failure behaviour of plain weave fabric laminates made of twisted
yarns (Fig. 1). This requires the knowledge of mechanical behaviour of twisted yarns along with the effect of laminate con®guration made of WF layers. There are many research papers on the mechanical behaviour of the WF composites under in-plane on-axis uni-axial tensile loading [1±44]. But these studies are based on WF composites made using straight strands. The mechanical behaviour of straight strands is very similar to the mechanical behaviour of UD composites and is well understood. The mechanical behaviour of twisted yarns is signi®cantly different. This in turn, can effect the mechanical behaviour of WF composites made of twisted yarns. 2. Mechanical behaviour of twisted impregnated yarns Twisting of yarns introduces lateral cohesion leading to ease of handling of the yarns during preform fabrication. Studies are available on the mechanical behaviour of twisted impregnated yarns [45±48]. An analytical method is presented in Ref. [47] for predicting the elastic properties of twisted impregnated yarns made of long unbroken ®laments. In this analysis, varying degree of twist in ®laments at different radii of the yarn and possible migration and microbuckling of ®laments are considered. The effects of twist angle and the extent of migration and microbuckling on the elastic properties and the pre-straining of the yarn are presented. An analytical method is presented for the prediction of longitudinal and transverse tensile strengths of twisted impregnated yarns in Ref. [48]. It has been observed that the transverse tensile strength of twisted impregnated yarns can increase compared to that of corresponding impregnated strands. This is because of the lateral pressure generated during twisting. Effective transverse tensile strength of the twisted impregnated yarns varies as a function of radial position of the yarn (Fig. 2). Near the periphery, there is a marginal increase in the effective transverse tensile strength whereas the increase is signi®cant near the centre of the twisted impregnated yarns. Even though there is reduction in the mean longitudinal tensile strength of the twisted impregnated yarns compared to those of corresponding impregnated strands, the reduction is marginal. Considering possible increase in transverse tensile strength, reduction in longitudinal tensile strength and pre-straining of the ®laments, optimum twist angle should be provided. Further, it has been observed that the variation of the shear strength as a function of radial position is marginal for different twist angles. Shear strength of twisted impregnated yarns can be taken to be equal to the shear strength of the corresponding straight impregnated strands. Variation of effective transverse tensile strength, YTy as a function of radial position at sections is shown in Fig. 2. This ®gure is based on circular cross-sectional area for the twisted yarn. A linear variation of the transverse tensile strength as a function of radial position is assumed in the
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Fig. 1. Idealised representation of 2D orthogonal plain weave fabric composite made using twisted yarns.
present study. Elastic and strength properties of the twisted impregnated yarns are presented in Table 1 for E-glass/ epoxy. The pre-strain, 1 p introduced in the ®laments during twisting of yarns is also presented. As the twist angle increases, pre-strain in the ®laments increases. In the present study, the yarn cross-section is taken as an ellipse with minor axis-to-major axis ratio 0.8 3. Lamina geometry For the evaluation of the mechanical behaviour of plain weave fabric composites, mathematical representa-
tion of lamina geometry is essential. The mathematical shape functions used were either linear, circular or sinusoidal functions [1±4,9,22]. Earlier studies were based on WF composites made of straight strands. For such materials, strand thickness-to-strand width ratio is low. It is in the practical range of 0.05±0.1. Such fabrics can be tightly woven with very small value of inter-strand gap. Inter-strand gap-to-strand width ratio can be in the range of 0.0±0.1. Twisted yarns are characterised by near circular crosssectional area. For typical twisted yarns, yarn thicknessto-yarn width ratio (h/a) can be in the range of 0.6±1.0 with a possible elliptical cross-sectional area. For such
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Fig. 2. Variation of effective transverse tensile strength, Y yT as a function of radial position at sections.
fabrics, the ratio inter-yarn gap-to-yarn width (g/a) would be higher. For the present study, yarn cross-sections are considered to be elliptical with major axis along the width and minor axis along the thickness of the yarn. Sinusoidal functions are considered for the yarn path along the loading direction. Typical requirements for the mathematical shape functions for modelling the geometry of the plain weave fabric lamina are: 1. The yarn cross-section and yarn undulation should be exactly simulated. 2. Possible inter-yarn gap should be considered. 3. The interface contact between the warp and ®ll yarns should be maintained. 4. The undulation angle at a given cross-section of the yarn should be the same. 5. The yarn should be continuous and there should not be any abrupt change in the slope of the yarn along its length. Considering the above aspects, mathematical expressions
have been presented in our earlier work [22] for plain weave fabric lamina made of straight strands. The objective of the present study is to de®ne the geometry of the 2D orthogonal plain weave fabric lamina made of twisted yarns for the prediction of thermo mechanical behaviour. For this, the elliptical cross-section of the yarn and the sinusoidal path of the yarn are used. Hence, the interface contact between the warp and ®ll yarns cannot be maintained throughout. As such, with twisted yarns, there may not be interface contact between the warp and the ®ll yarns throughout. But the actual cross-sectional area is considered. Yarn undulation is exactly simulated along the loading direction whereas only the equivalent area is considered with respect to transverse direction. As can be seen, the yarn cross-sectional area, yarn undulation along the loading direction and realistic interyarn gap are the important requirements for the accurate prediction of thermo mechanical behaviour. Even though there is geometrical inconsistency with respect to the transverse direction, it would not affect the predicted thermo mechanical properties. 3.1. Lamina geometry: elliptical cross-section Representative unit cells of the plain weave fabric lamina made of twisted yarns are shown in Fig. 3. The crosssections with respect to warp and ®ll directions are presented in Fig. 4. The expressions for the shape functions to de®ne the geometrical unit cell are as follows: In the Y±Z plane, i.e. along the warp direction (Fig. 4),
h zy1 y 2 f 2
"
2y 12 af
2 #1=2
zy2 y 2zy1 y zy3 y
y0!^
y0!^
af 2
af 2
hf py cos af 1 gf 2
zy4 y hw 1 zy3 y
Table 1 Elastic and strength properties of twisted impregnated yarns: E-glass/epoxy, Vfy 0:7 (Yarn diameter 1 mm: Shear strength of the yarn, Sy 39:7 MPa: Y yT values are at r=R 0: The quantities in the bracket are at r=R 1)
a (8)
E yL (GPa)
E yT (GPa)
G yLT (GPa)
G yTT (GPa)
n yLT
n yTT
X yT (MPa)
Y yT (MPa)
1 p (%)
0 2 5 10
51.50 51.31 50.32 47.13
17.70 17.70 17.67 17.59
5.85 5.85 5.86 5.90
6.67 6.67 6.67 6.66
0.313 0.313 0.315 0.320
0.327 0.327 0.326 0.322
1397 1395 1385 1348
27.9 (27.9) 28.2 (27.9) 30.6 (27.9) 38.8 (27.9)
± 0.061 0.380 1.543
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Fig. 3. Representative unit cells of the plain weave fabric lamina made of twisted yarns.
In the X±Z plane, i.e. along the ®ll direction (Fig. 4), 2 #1=2 " hw 2x zx1 x; y 2hy1 y 12 aw 2 a x0!^ w 2 2 #1=2 " hw 2x 2hy1 y zx2 x; y 2 12 aw 2 a x0!^ w 2
Fig. 4. Cross-sections of plain weave fabric lamina made of twisted yarns.
hy1 y
hf 2 zy3 y 2
hy2 y
hf 2 zy2 y 2
If zx2 x; y 2 zx3 x; y # 0 then zx2 x; y zx3 x; y: The local undulation angle of the ®ll yarn at any crosssection along its length is given by:
h px 2 hy2 y zx3 x; y 2 w cos aw 1 gw 2 a y0!^ f 2
qf x tan21
hw px 1 hy2 y 2 hf zx4 x; y 2 cos aw 1 gw 2 a y0!^ f 2
and the warp yarn at any cross-section along its length is
d zx x; y dx 3 phw px tan21 sin aw 1 gw 2 aw 1 gw
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given by: d zy y dy 3 phf py 21 sin tan af 1 gf 2 af 1 gf
qw y tan21
4. Laminate geometry A laminate is formed by stacking individual layers one over the other. In multi-directional laminates made of UD layers, a variety of laminate con®gurations can be obtained by varying the orientation angle of individual layers. On the other hand, in the case of WF laminates, different stacking patterns can be obtained even without considering the orientations of individual layers as a variable [22]. One of the typical WF laminate con®gurations is to maintain the same warp and ®ll directions for all layers. This laminate con®guration is similar to UD composite consisting of many layers. In the case of WF laminates with aligned warp and ®ll directions, different laminate con®gurations are obtained by shifting the adjacent layers in such a way that the yarns of one layer are not in-phase with the yarns of the adjacent layers. The shift of one layer with respect to the adjacent layer can be in warp and/or ®ll directions. In an actual laminate, the relative movements of the fabric layers are affected by friction between fabric layers, local departure in yarn perpendicularity, possible variation of number of counts from place to place in the fabric and constraints on the relative lateral movement of the layers
during lamination. Hence, an actual laminate would have scattered zones of different combinations of shift [22]. The idealised cases of laminate on-axis shapes are shown in Fig. 5. In con®guration-1 (C1), there is no relative shift between adjacent layers, i.e. each layer is exactly stacked over the adjacent layer. All the layers are in-phase. Such a con®guration is termed as aligned con®guration. In con®guration-2 (C2), the adjacent layers are shifted with respect to each other by a distance of a 1 g=2 both in the ®ll and the warp directions. In this case adjacent layers are out-of-phase. Such a con®guration is also termed as bridged con®guration. As can be seen from Figs. 3±5, the WFs consist of interlacing regions in which both the warp and ®ll yarns are present and the gap regions where either warp or ®ll yarns are present. In the case of C1, for the WF laminate as a whole, clear interlacing regions and gap regions can be seen. In the case of C2, gap region of one layer is bridged by the interlacing region of the adjacent layer. In an actual WF laminate, the shift of different layers with respect to the adjacent layers would be random. Based on the photo micrographic studies and the predictions obtained using analytical models in the case of WF laminates made of straight strands, it has been shown that the predictions based on C2 con®guration match well with the experimental data [15,23]. 5. Analysis The stress and failure analysis of the geometrical representative unit cell is based on volume averaging method. A geometrical representative unit cell for a typical plain weave fabric lamina made of twisted yarns is shown in Fig. 3.
Fig. 5. Representative unit cells for different laminate con®gurations-for analysis.
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703
Fig. 6. Plain weave fabric lamina made of twisted yarns: unit cell idealised, parallel-series scheme.
Based on similarity, unit cell for analysis is identi®ed. The unit cell is discretised into sections perpendicular to the loading direction. Loading is assumed to be along the ®ll direction. The sections are further discretised into elements (Fig. 6). The elements are further discretised into subelements. It may be noted that an element consists of three types of regions: longitudinal (®ll) yarn, transverse (warp) yarn and pure matrix. As has been explained earlier, effective transverse tensile strength of the twisted impregnated yarn varies as a function of radial position of the yarn. To take this aspect into account, the longitudinal and transverse yarn regions are further subdivided into sub-elements. Subelements are obtained by discretising the longitudinal and transverse yarn regions along the thickness directions with the same longitudinal and transverse dimensions. It may be noted that the longitudinal and the transverse sub-element properties are derived from the properties of the twisted impregnated yarns. Based on the discretisation, length, width and thickness of the sub-elements are known. Matrix sub-element/region properties are taken to be those of bulk matrix. Based on these, the element properties are derived as explained later. 5.1. Stress and failure analysis The discretisation scheme is shown in Fig. 6. The strain for the elements in the loading direction is the
same as that for the corresponding section by the iso-strain assumption at the section level. el sec 1sel x 1x 1 x
In the transverse direction, the assumption is made that all the elements have the same strain as the corresponding section. Also, the transverse strain for a section is the same as the average transverse strain for the unit cell. el sec 1sel y 1y 1 y 1y
The extensional stiffness matrix for each sub-element is equal to its transformed reduced stiffness matrix. h i sel Asel Q ij The element and section extensional stiffness matrices are obtained as: nz h h i i 1 X Asel Ael ij ij lz Lz n1 h
ny h i i 1 X Asec Ael ij ij ly Ly n1
The extensional compliance matrix of the section is, h i h i21 Asec asec ij ij The extensional compliance matrix of the unit cell/WF
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laminate is, aij
nx h i 1 X l asec Lx n1 ij x
The transverse strain of the unit cell/WF laminate is,
1y a21 s x Here, s x is load per unit area. The longitudinal strain of the section is,
1sec x
sec s x 2 Asec 12 1y sec A11
The longitudinal strain of the unit cell/WF laminate is,
1x
nx 1 X 1sec l Lx n1 x x
The sub-element stress is, sel sel sel s xsel Asel 11 1x 1 A12 1y
Fig. 7. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 08; linear analysis.
The ®ll sub-element stresses in the local co-ordinates:
s 1sel cos2 qf xs xsel
5.1.1. Non-linear analysis Because of the undulation of the yarns, each sub-element is off-axis with respect to loading direction. Considering this aspect, possible non-linear behaviour of each sub-element is considered as follows [49]:
s 2sel sin2 qf xs xsel sel tsel 12 sin qf xcos qf xs x
The above expressions are based on linear analysis. The stresses obtained from the above equations in each sub-element are compared with the permissible stress values. If the induced stress in any sub-element exceeds the permissible limit, that particular sub-element is treated as failed and does not contribute to further load sharing. Generally, it is observed that the failure takes place in a section with longitudinal yarns at an angle with respect to the loading direction. Hence, in such sub-elements, the early modes of failure would be shear failure and transverse failure. The ultimate failure would take place when the ®bres break. Hence ®bre strain/twisted yarn strain/section strain was monitored during loading. The effective permissible twisted yarn strain would be the ®bre strain minus pre-strain due to twisting. When the section strain/ ®bre strain exceeds the permissible strain, the section is supposed to have failed indicating the failure of unit cell/ WF laminate.
1nl x Sxx u xs x u x 1
1 2 cos 4u x2 S5555 s x u x3 64
For E-glass/epoxy, S5555 37 £ 1029 MPa23 at Vfs 0:46: And, Sxx is the compliance element. sel sec It may be noted that s x s xsel ; 1nl and x 1x 1x u x qf x: The sub-element/section strains in local coordinates, 2 nl 1nl 1 cos qf x1x
This value of induced strain is compared with the permissible strain of the twisted yarn/section. 6. Results and discussion Predicted strength properties of the plain weave fabric
Table 2 Predicted strength properties of plain weave fabric laminates made of twisted yarns: E-glass/epoxy, Vfo 0:44; h=a 0:8; g=a 0:2 (linear analysis)
a (8)
0 2 5 10
WTI (MPa)
WTC (MPa)
FTI(FSI) (MPa)
PMBFI (MPa)
XT (MPa)
1 x (%)
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
15 15 15 15
22 22 21 19
47 48 49 52
29 29 30 33
25(32) 25(32) 25(31) 23(32)
29(31) 29(31) 27(31) 25(31)
119 119 119 119
120 120 120 121
227 224 209 155
262 258 240 178
3.73 3.68 3.44 2.53
3.85 3.80 3.55 2.60
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Fig. 8. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 28; linear analysis.
Fig. 10. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 108; linear analysis.
composites made of twisted yarns using linear analysis are presented in Table 2 and in Figs. 7±12. Failure strength and strain data are presented for both C1 and C2 cases with different twist angles. Different stages of failure like warp transverse failure initiation (WTI), complete failure of warp (WTC), ®ll transverse failure initiation (FTI), ®ll shear failure initiation (FSI), pure matrix block failure initiation (PMBFI) and the ®bre breakage, i.e. the ultimate tensile failure (XT) have been monitored. In Figs. 7±10,
WTI, PMBFI and XT have been indicated by `a', `d' and `e', respectively. It can be seen that warp transverse failure initiation is the ®rst mode of failure. Because of the higher undulation angle, ®ll transverse failure and ®ll shear failure also take place early. The next failure is the failure of
Fig. 9. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 58; linear analysis.
Fig. 11. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, con®guration C1, linear analysis.
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Table 3 Predicted strength properties of plain weave fabric laminates made of twisted yarns: E-glass/epoxy, Vfo 0:44; h=a 0:8; g=a 0:2 (non-linear analysis)
a (8)
0 2 5 10
WTI (MPa)
WTC (MPa)
FTI(FSI) (MPa)
PMBFI (MPa)
1 x (%)
XT (MPa)
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
15 15 15 15
22 22 21 19
47 48 49 52
29 29 30 33
25(32) 25(32) 25(31) 23(32)
29(31) 28(31) 27(31) 26(31)
118.4 118.4 118.6 ±
120 120 120 ±
131.2 131.0 129.4 118.6
121.2 120.6 120.4 113.2
1.91 1.91 1.85 ±
2.03 2.02 2.03 ±
pure matrix block. Failure of the pure matrix block would lead to the disintegration of the composite. Effectively, this can be treated as the failure of the WF composite. But the ®bres can still be intact. The ®bres break when the ultimate strain exceeds the permissible strain limit inside the section. This is the ultimate failure of the WF composite. In Table 2, 1 x is the failure strain of the plain weave fabric laminates made of twisted yarns during tensile loading along warp or ®ll directions. As the angle of twist of the yarn increases, the ultimate failure strength and the ultimate failure strain decrease. This is because of the higher pre-strain within the yarn as the twist angle increases. This can be seen from Table 1. The ultimate strength and ultimate strain predictions for C2 are higher than for C1. This is because of the bridging effect in the case of C2. The weaker gap regions of one layer are
bridged by the stronger interlacing regions of the adjacent layer for C2. For twist angles of the yarn, say up to 58, the variation in ultimate strength and ultimate strain values is not signi®cant compared to those of corresponding straight strand. Since twisting of E-glass yarns is an engineering requirement for facilitating ease of preform fabrication, the yarns can be twisted optimally without reducing the strength values signi®cantly. Because of higher h/a ratio in the case of plain weave fabric composites made of twisted yarns, the strength properties are lower compared to the WF composites with lower h/a ratio and cross-ply laminates [23]. From Figs. 11 and 12, it can be seen that the stress±strain behaviour is nearly identical for different angles of twist. Predicted strength properties using the non-linear analysis are presented in Table 3. As expected the ultimate strength and ultimate strain are lower in this case compared to the predictions based on linear analysis. This is because of higher strain state within the sections. The difference in behaviour between the predictions using linear analysis and non-linear analysis is seen only after pure matrix block failure. As indicated earlier, failure of pure matrix block leads to the disintegration of WF composites. Effectively, the corresponding stress can be taken to be practical strength value for the WF composites. From this point of view, both linear and non-linear analysis give identical results. 7. Conclusions A 2D analytical method is presented for the failure behaviour of plain weave fabric composites made of twisted yarns. The studies have been carried out under uni-axial on-axis tensile loading. Twisting of yarns, especially for E-glass composites, is an engineering requirement for facilitating ease of preform fabrication. Based on this, the following conclusions are derived:
Fig. 12. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, con®guration C2, linear analysis.
1. The twisted yarn thickness-to-width ratio would be higher compared to the straight strand thickness-to-width ratio. 2. The variation in tensile strength properties for plain weave fabric composites as a result of twisting of yarns is not signi®cant up to an optimum angle of twist.
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3. E-glass yarns can be twisted up to 58 so as to facilitate ease of preform fabrication without compromising on the tensile strength properties of plain weave fabric composites signi®cantly. Acknowledgements This work was supported by the Structures Panel, Aeronautics Research and Development Board and ISROIIT(B) Space Technology Cell. References [1] Ishikawa T, Chou TW. One-dimensional micromechanical analysis of woven fabric composites. AIAA J 1983;21:1714±21. [2] Chou TW, Ishikawa T. Analysis and modeling of two-dimensional fabric composites. In: Chou TW, Ko FK, editors. Textile structural composites. Amsterdam: Elsevier, 1989. p. 209±63. [3] Whitcomb JD. Three-dimensional stress analysis of plain weave composites. In: O'Brien TK, editor. Composite materials: fatigue and fracture, ASTM STP 1110, vol. 3. Philadelphia, PA: American Society for Testing and Materials, 1991. p. 417±38. [4] Naik NK, Shembekar PS. Elastic analysis of woven fabric composites. Report No. IITB/AE/ARDB/STR/659/91/01, Bombay: IIT, 1991. [5] Naik NK, Ganesh VK. Prediction of on-axes elastic properties of plain weave fabric composites. Compos Sci Technol 1992;45:135±52. [6] Naik NK, Shembekar PS. Elastic behavior of woven fabric composites: IÐlamina analysis. J Compos Mater 1992;26:2196±225. [7] Shembekar PS, Naik NK. Elastic behavior of woven fabric composites: IIÐlaminate analysis. J Compos Mater 1992;26:2226±46. [8] Naik NK, Shembekar PS. Elastic behavior of woven fabric composites: IIIÐlaminate design. J Compos Mater 1992;26:2522±41. [9] Dasgupta A, Agarwal RK. Orthotropic thermal conductivity of plain weave fabric composites using a homogenization technique. J Compos Mater 1992;26:2736±58. [10] Naik NK, Ganesh VK. Prediction of thermal expansion coef®cients of plain weave fabric composites. Compos Struct 1993;26:139±54. [11] Naik NK. Woven fabric composites. Lancaster, PA: Technomic Publishing Co., Inc, 1994. [12] Hahn HT, Pandey R. Micromechanics model for thermoelastic properties of plain weave fabric composites. J Engng Mater Technol 1994;116:517±23. [13] Raju IS, Wang JT. Classical laminate theory models for woven fabric composites. J Compos Technol Res 1994;16:289±303. [14] Dasgupta A, Bhandarkar SM. Effective thermo mechanical behavior of plain weave fabric-reinforced composites using homogenization theory. J Engng Mater Technol 1994;116:99±105. [15] Ganesh VK, Naik NK. Thermal expansion coef®cients of plain weave fabric laminates. Compos Sci Technol 1994;51:387±408. [16] Naik NK, Ganesh VK. Failure behavior of plain weave fabric laminates under in-plane shear loading. J Compos Technol Res 1994;16:3±20. [17] Shkoller S, Hegemier G. Homogenization of plain weave composites using two-scale convergence. Int J Solids Struct 1995;32:783±94. [18] Ganesh VK, Naik NK. Failure behaviour of plain weave fabric laminates under in-plane shear loading: effect of fabric geometry. Compos Struct 1995;30:179±92. [19] Naik NK, Ganesh VK. Analytical method for plain weave fabric composites. Composites 1995;26:281±9. [20] Naik RA. Failure analysis of woven and braided fabric-reinforced composites. J Compos Mater 1995;29:2334±63. [21] Whitcomb J, Srirengan K. Effect of various approximations on predicted progressive failure in plain weave composites. Compos Struct 1996;34:13±20.
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[22] Ganesh VK, Naik NK. Failure behavior of plain weave fabric laminates under on-axis uni-axial tensile loading: IÐlaminate geometry. J Compos Mater 1996;30:1748±78. [23] Naik NK, Ganesh VK. Failure behavior of plain weave fabric laminates under on-axis uni-axial tensile loading: IIÐanalytical predictions. J Compos Mater 1996;30:1779±822. [24] Ganesh VK, Naik NK. Failure behavior of plain weave fabric laminates under on-axis uni-axial tensile loading: IIIÐeffect of fabric geometry. J Compos Mater 1996;30:1823±56. [25] Ito M, Chou TW. Elastic moduli and stress ®eld of plain weave composites under tensile loading. Compos Sci Technol 1997;57:787±800. [26] Nayfeh AH, Kress GR. Non-linear constitutive model for plain weave composites. Compos Part B: Engng 1997;28B:627±34. [27] Naik NK, Ganesh VK. Thermo-mechanical behaviour of plain weave fabric composites: experimental investigations. J Mater Sci 1997;32:267±77. [28] Ganesh VK, Naik NK. (1/245)8 Off-axis tension test for shear characterization of plain weave fabric composites. J Compos Technol Res 1997;19:77±85. [29] Ganesh VK, Ramakrishna S, Teoh SH, Naik NK. Micromechanical design of textile composites. Mater Des 1997;18:175±81. [30] Srirengan K, Whitcomb JD. Finite element based degradation model for composites with transverse matrix cracks. J Thermoplast Compos Mater 1998;11:113±23. [31] Scida D, Aboura Z, Benzeggagh ML, Bocherens E. Prediction of the elastic behaviour of hybrid and non-hybrid woven composites. Compos Sci Technol 1998;57:1727±40. [32] Ito M, Chou TW. Analytical and experimental study of strength and failure behavior of plain weave composites. J Compos Mater 1998;32:2±30. [33] Kuhn JL, Charalambides PG. Elastic response of porous matrix plain weave fabric composites: part IÐmodeling. J Compos Mater 1998;32:1426±71. [34] Kuhn JL, Charalambides PG. Elastic response of porous matrix plain weave fabric composites: part IIÐresults. J Compos Mater 1998;32:1472±507. [35] Nguyen M, Herszberg I, Paton R. The shear properties of woven carbon fabric. Compos Struct 1999;47:767±79. [36] Tan P, Tong L, Steven GP. Micromechanics models for the elastic constants and failure strengths of plain weave composites. Compos Struct 1999;47:797±804. [37] Bejan L, Poterasu VF. Woven composite material design by orthotropic compliance averaging method using mathematica. Comput Meth Appl Mech Engng 1999;179:53±65. [38] Scida D, Aboura Z, Benzeggagh ML, Bocherens E. A micromechanics model for 3D elasticity and failure of woven-®bre composite materials. Compos Sci Technol 1999;59:505±17. [39] Kuhn JL, Haan SI, Charalambides PG. Semi-analytical method for the calculation of the elastic micro-®elds in plain weave fabric composites subjected to in-plane loading. J Compos Mater 1999;33:221±66. [40] Thom H. Finite element modeling of plain weave composites. J Compos Mater 1999;33:1491±510. [41] Gao F, Boniface L, Ogin SL, Smith PA, Greaves RP. Damage accumulation in woven fabric CFRP laminates under tensile loading: part I, observations of damage accumulation. Compos Sci Technol 1999;59:123±36. [42] Tabiei A, Jiang Y, Yi W. Novel micromechanics-based woven fabric composite constitutive model with material non-linear behavior. AIAA J 2000;38:1437±43. [43] Bystrom J, Jekabsons N, Varna J. An evaluation of different models for prediction of elastic properties of woven composites. Compos Part B: Engng 2000;31:7±20. [44] Nguyen-Hoa H, Vu-Khanh T. Prediction of failure in unsaturated polyester reinforced by plain weave glass fabric. J Compos Mater 2000;34:379±97.
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N.K. Naik, R. Kuchibhotla / Composites: Part A 33 (2002) 697±708
[45] Hearle JWS, Grosberg P, Backer S. Structural mechanics of ®bers, yarns, and fabrics. New York: Wiley, 1969. [46] Byun JH, Chou TW. Effect of yarn twist on the elastic properties of composites. In: Poursartip A, Street K, editors. Proceedings of the ICCM-12, vol. IV. Cambridge, England: Woodhead Publishing Company, 1995. p. 293±99.
[47] Naik NK, Madhavan V. Twisted impregnated yarns: elastic properties. J Strain Anal Engng Des 2000;35:83±91. [48] Naik NK, Singh MN. Twisted impregnated yarns: transverse tensile strength. J Strain Anal Engng Des 2001;36:347±58. [49] Hahn HT, Tsai SW. Non-linear elastic behaviour of unidirectional composite laminae. J Compos Mater 1973;7:102±18.
www.elsevier.com/locate/compositesa
Analytical study of strength and failure behaviour of plain weave fabric composites made of twisted yarns N.K. Naik*, R. Kuchibhotla Department of Aerospace Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India Received 5 December 2000; revised 10 August 2001; accepted 3 January 2002
Abstract A two-dimensional analytical method is presented for the failure behaviour of plain weave fabric composites made of twisted yarns. The studies have been carried out on laminates with different con®gurations under on-axis uni-axial tensile loading. The cross-sectional area of the yarn was taken to be elliptical and the yarn path was taken to be sinusoidal. Different stages of failure are considered in the analysis. It has been observed that there is no signi®cant reduction in tensile strength properties of plain weave fabric composites as a result of twisting of yarns. For E-glass yarns, twisting of yarns up to 58, can facilitate ease of fabrication without signi®cantly compromising the strength properties of the woven fabric composites. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Fabrics/textiles; B. Strength; C. Analytical modelling; C. Damage mechanics
1. Introduction Advanced textile structural composites are ®nding increasing use for many high performance applications during last one decade. Textile techniques such as twodimensional (2D) and three-dimensional (3D) weaving, braiding, knitting and through-the-thickness stitching have assisted in enhancing the performance of textile composite structures. Such composites are characterised by enhanced through-the-thickness elastic and strength properties, impact/fracture resistance, damage tolerance and dimensional stability. Additionally, textile structural composites are associated with near-net-shape and cost effective manufacturing processes. For high in-plane speci®c stiffness and high in-plane speci®c strength applications, 2D woven fabric (WF) composites can be the competitors to laminated composites made of unidirectional (UD) layers. To derive the maximum bene®ts of the textile structural composites, an improved understanding of the detailed structure of the reinforcement with the advances in fabric formation techniques is essential. * Corresponding author. Tel.: 191-22-576-7114; fax: 191-22-572-2602. E-mail address: [email protected] (N.K. Naik). Abbreviations: C1, C2, laminate con®guration-1 and -2; FSI, ®ll shear failure initiation; FTI, ®ll transverse failure initiation; PMBFI, pure matrix block failure initiation; UD, unidirectional; WF, woven fabric; WTC, complete failure of warp; WTI, warp transverse failure initiation; 1D, 2D, 3D, one-, two-, three-dimensional
The textile structural composites are made using the woven, braided, knitted or stitched preforms. The textile preforms are planar or 3D. The special feature of the textile preforms is the signi®cant reinforcement interconnectivity between adjacent planes of reinforcements. This interconnectivity provides additional interface strength to supplement the relatively weak ®bre/resin interface. The interconnectivity is mainly in the plane of the preform for the planar textile preforms. Such materials are known as 2D textile preforms. The 3D textile preforms for the structural composites are fully integrated continuous reinforcement assemblies having multi-axial in-plane and out-of-plane reinforcement orientations. Formation of different textile preforms is an important stage in composites technology. WFs are produced by the process of weaving in which the fabric is formed by interlacing warp and ®ll strands/yarns. Knitted preforms are made by interlooping whereas braided preforms are made by intertwining. One of the important requirements of the reinforcing elements during preform preparation is the lateral cohesion. For this reason, the twisted yarns are used for preparing the textile preforms rather than the straight strands. Twisted yarns are normally used for increasing the lateral cohesion of the ®laments and also for ease of handling. By twisting yarns, possible micro damages within the yarn can be localised, leading to possible increase in the failure strength of the yarn. For this reason, twisted yarns are normally used for making textile preforms, especially for
1359-835X/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 1359-835 X(02)00 012-X
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Nomenclature a yarn width aij extensional compliance matrix axt, ayt as shown in Fig. 4 Aij extensional stiffness matrix E, G, n elastic properties g inter-yarn gap h maximum yarn thickness hm matrix thickness ht fabric thickness HL lamina thickness l sub-element, element, section dimensions L unit cell dimensions nx, ny, nz number of sub-elements, elements, sections in the x-, y-, z-directions Å Q transformed reduced stiffness matrix R radius of yarn S shear strength Vf ®bre volume fraction x, y, z cartesian coordinates XT, YT tensile strength zxi x; y; i 1±4 yarn shape parameters zyi y; i 1±4 yarn shape parameters a angle of twist for the yarn 1 normal strain 1p pre-strain s normal stress s x load per unit area in x-direction q x; q y local off-axis angle of the undulated yarn t shear stress Superscripts el refers to element nl non-linear o overall properties s quantities of strand sec refers to section sel refers to sub-element y quantities of yarn Subscripts f, F quantities in ®ll direction L longitudinal axis of the ®bre T transverse axis of the ®bre w, W quantities in warp direction 1,2 local coordinates x, y global coordinates
E-glass reinforcements. Twisting of yarns may lead to possible reduction in ultimate failure strain. But twisting of yarns may be unavoidable for engineering reasons. The objective of the present work is to study the failure behaviour of plain weave fabric laminates made of twisted
yarns (Fig. 1). This requires the knowledge of mechanical behaviour of twisted yarns along with the effect of laminate con®guration made of WF layers. There are many research papers on the mechanical behaviour of the WF composites under in-plane on-axis uni-axial tensile loading [1±44]. But these studies are based on WF composites made using straight strands. The mechanical behaviour of straight strands is very similar to the mechanical behaviour of UD composites and is well understood. The mechanical behaviour of twisted yarns is signi®cantly different. This in turn, can effect the mechanical behaviour of WF composites made of twisted yarns. 2. Mechanical behaviour of twisted impregnated yarns Twisting of yarns introduces lateral cohesion leading to ease of handling of the yarns during preform fabrication. Studies are available on the mechanical behaviour of twisted impregnated yarns [45±48]. An analytical method is presented in Ref. [47] for predicting the elastic properties of twisted impregnated yarns made of long unbroken ®laments. In this analysis, varying degree of twist in ®laments at different radii of the yarn and possible migration and microbuckling of ®laments are considered. The effects of twist angle and the extent of migration and microbuckling on the elastic properties and the pre-straining of the yarn are presented. An analytical method is presented for the prediction of longitudinal and transverse tensile strengths of twisted impregnated yarns in Ref. [48]. It has been observed that the transverse tensile strength of twisted impregnated yarns can increase compared to that of corresponding impregnated strands. This is because of the lateral pressure generated during twisting. Effective transverse tensile strength of the twisted impregnated yarns varies as a function of radial position of the yarn (Fig. 2). Near the periphery, there is a marginal increase in the effective transverse tensile strength whereas the increase is signi®cant near the centre of the twisted impregnated yarns. Even though there is reduction in the mean longitudinal tensile strength of the twisted impregnated yarns compared to those of corresponding impregnated strands, the reduction is marginal. Considering possible increase in transverse tensile strength, reduction in longitudinal tensile strength and pre-straining of the ®laments, optimum twist angle should be provided. Further, it has been observed that the variation of the shear strength as a function of radial position is marginal for different twist angles. Shear strength of twisted impregnated yarns can be taken to be equal to the shear strength of the corresponding straight impregnated strands. Variation of effective transverse tensile strength, YTy as a function of radial position at sections is shown in Fig. 2. This ®gure is based on circular cross-sectional area for the twisted yarn. A linear variation of the transverse tensile strength as a function of radial position is assumed in the
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699
Fig. 1. Idealised representation of 2D orthogonal plain weave fabric composite made using twisted yarns.
present study. Elastic and strength properties of the twisted impregnated yarns are presented in Table 1 for E-glass/ epoxy. The pre-strain, 1 p introduced in the ®laments during twisting of yarns is also presented. As the twist angle increases, pre-strain in the ®laments increases. In the present study, the yarn cross-section is taken as an ellipse with minor axis-to-major axis ratio 0.8 3. Lamina geometry For the evaluation of the mechanical behaviour of plain weave fabric composites, mathematical representa-
tion of lamina geometry is essential. The mathematical shape functions used were either linear, circular or sinusoidal functions [1±4,9,22]. Earlier studies were based on WF composites made of straight strands. For such materials, strand thickness-to-strand width ratio is low. It is in the practical range of 0.05±0.1. Such fabrics can be tightly woven with very small value of inter-strand gap. Inter-strand gap-to-strand width ratio can be in the range of 0.0±0.1. Twisted yarns are characterised by near circular crosssectional area. For typical twisted yarns, yarn thicknessto-yarn width ratio (h/a) can be in the range of 0.6±1.0 with a possible elliptical cross-sectional area. For such
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Fig. 2. Variation of effective transverse tensile strength, Y yT as a function of radial position at sections.
fabrics, the ratio inter-yarn gap-to-yarn width (g/a) would be higher. For the present study, yarn cross-sections are considered to be elliptical with major axis along the width and minor axis along the thickness of the yarn. Sinusoidal functions are considered for the yarn path along the loading direction. Typical requirements for the mathematical shape functions for modelling the geometry of the plain weave fabric lamina are: 1. The yarn cross-section and yarn undulation should be exactly simulated. 2. Possible inter-yarn gap should be considered. 3. The interface contact between the warp and ®ll yarns should be maintained. 4. The undulation angle at a given cross-section of the yarn should be the same. 5. The yarn should be continuous and there should not be any abrupt change in the slope of the yarn along its length. Considering the above aspects, mathematical expressions
have been presented in our earlier work [22] for plain weave fabric lamina made of straight strands. The objective of the present study is to de®ne the geometry of the 2D orthogonal plain weave fabric lamina made of twisted yarns for the prediction of thermo mechanical behaviour. For this, the elliptical cross-section of the yarn and the sinusoidal path of the yarn are used. Hence, the interface contact between the warp and ®ll yarns cannot be maintained throughout. As such, with twisted yarns, there may not be interface contact between the warp and the ®ll yarns throughout. But the actual cross-sectional area is considered. Yarn undulation is exactly simulated along the loading direction whereas only the equivalent area is considered with respect to transverse direction. As can be seen, the yarn cross-sectional area, yarn undulation along the loading direction and realistic interyarn gap are the important requirements for the accurate prediction of thermo mechanical behaviour. Even though there is geometrical inconsistency with respect to the transverse direction, it would not affect the predicted thermo mechanical properties. 3.1. Lamina geometry: elliptical cross-section Representative unit cells of the plain weave fabric lamina made of twisted yarns are shown in Fig. 3. The crosssections with respect to warp and ®ll directions are presented in Fig. 4. The expressions for the shape functions to de®ne the geometrical unit cell are as follows: In the Y±Z plane, i.e. along the warp direction (Fig. 4),
h zy1 y 2 f 2
"
2y 12 af
2 #1=2
zy2 y 2zy1 y zy3 y
y0!^
y0!^
af 2
af 2
hf py cos af 1 gf 2
zy4 y hw 1 zy3 y
Table 1 Elastic and strength properties of twisted impregnated yarns: E-glass/epoxy, Vfy 0:7 (Yarn diameter 1 mm: Shear strength of the yarn, Sy 39:7 MPa: Y yT values are at r=R 0: The quantities in the bracket are at r=R 1)
a (8)
E yL (GPa)
E yT (GPa)
G yLT (GPa)
G yTT (GPa)
n yLT
n yTT
X yT (MPa)
Y yT (MPa)
1 p (%)
0 2 5 10
51.50 51.31 50.32 47.13
17.70 17.70 17.67 17.59
5.85 5.85 5.86 5.90
6.67 6.67 6.67 6.66
0.313 0.313 0.315 0.320
0.327 0.327 0.326 0.322
1397 1395 1385 1348
27.9 (27.9) 28.2 (27.9) 30.6 (27.9) 38.8 (27.9)
± 0.061 0.380 1.543
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701
Fig. 3. Representative unit cells of the plain weave fabric lamina made of twisted yarns.
In the X±Z plane, i.e. along the ®ll direction (Fig. 4), 2 #1=2 " hw 2x zx1 x; y 2hy1 y 12 aw 2 a x0!^ w 2 2 #1=2 " hw 2x 2hy1 y zx2 x; y 2 12 aw 2 a x0!^ w 2
Fig. 4. Cross-sections of plain weave fabric lamina made of twisted yarns.
hy1 y
hf 2 zy3 y 2
hy2 y
hf 2 zy2 y 2
If zx2 x; y 2 zx3 x; y # 0 then zx2 x; y zx3 x; y: The local undulation angle of the ®ll yarn at any crosssection along its length is given by:
h px 2 hy2 y zx3 x; y 2 w cos aw 1 gw 2 a y0!^ f 2
qf x tan21
hw px 1 hy2 y 2 hf zx4 x; y 2 cos aw 1 gw 2 a y0!^ f 2
and the warp yarn at any cross-section along its length is
d zx x; y dx 3 phw px tan21 sin aw 1 gw 2 aw 1 gw
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given by: d zy y dy 3 phf py 21 sin tan af 1 gf 2 af 1 gf
qw y tan21
4. Laminate geometry A laminate is formed by stacking individual layers one over the other. In multi-directional laminates made of UD layers, a variety of laminate con®gurations can be obtained by varying the orientation angle of individual layers. On the other hand, in the case of WF laminates, different stacking patterns can be obtained even without considering the orientations of individual layers as a variable [22]. One of the typical WF laminate con®gurations is to maintain the same warp and ®ll directions for all layers. This laminate con®guration is similar to UD composite consisting of many layers. In the case of WF laminates with aligned warp and ®ll directions, different laminate con®gurations are obtained by shifting the adjacent layers in such a way that the yarns of one layer are not in-phase with the yarns of the adjacent layers. The shift of one layer with respect to the adjacent layer can be in warp and/or ®ll directions. In an actual laminate, the relative movements of the fabric layers are affected by friction between fabric layers, local departure in yarn perpendicularity, possible variation of number of counts from place to place in the fabric and constraints on the relative lateral movement of the layers
during lamination. Hence, an actual laminate would have scattered zones of different combinations of shift [22]. The idealised cases of laminate on-axis shapes are shown in Fig. 5. In con®guration-1 (C1), there is no relative shift between adjacent layers, i.e. each layer is exactly stacked over the adjacent layer. All the layers are in-phase. Such a con®guration is termed as aligned con®guration. In con®guration-2 (C2), the adjacent layers are shifted with respect to each other by a distance of a 1 g=2 both in the ®ll and the warp directions. In this case adjacent layers are out-of-phase. Such a con®guration is also termed as bridged con®guration. As can be seen from Figs. 3±5, the WFs consist of interlacing regions in which both the warp and ®ll yarns are present and the gap regions where either warp or ®ll yarns are present. In the case of C1, for the WF laminate as a whole, clear interlacing regions and gap regions can be seen. In the case of C2, gap region of one layer is bridged by the interlacing region of the adjacent layer. In an actual WF laminate, the shift of different layers with respect to the adjacent layers would be random. Based on the photo micrographic studies and the predictions obtained using analytical models in the case of WF laminates made of straight strands, it has been shown that the predictions based on C2 con®guration match well with the experimental data [15,23]. 5. Analysis The stress and failure analysis of the geometrical representative unit cell is based on volume averaging method. A geometrical representative unit cell for a typical plain weave fabric lamina made of twisted yarns is shown in Fig. 3.
Fig. 5. Representative unit cells for different laminate con®gurations-for analysis.
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703
Fig. 6. Plain weave fabric lamina made of twisted yarns: unit cell idealised, parallel-series scheme.
Based on similarity, unit cell for analysis is identi®ed. The unit cell is discretised into sections perpendicular to the loading direction. Loading is assumed to be along the ®ll direction. The sections are further discretised into elements (Fig. 6). The elements are further discretised into subelements. It may be noted that an element consists of three types of regions: longitudinal (®ll) yarn, transverse (warp) yarn and pure matrix. As has been explained earlier, effective transverse tensile strength of the twisted impregnated yarn varies as a function of radial position of the yarn. To take this aspect into account, the longitudinal and transverse yarn regions are further subdivided into sub-elements. Subelements are obtained by discretising the longitudinal and transverse yarn regions along the thickness directions with the same longitudinal and transverse dimensions. It may be noted that the longitudinal and the transverse sub-element properties are derived from the properties of the twisted impregnated yarns. Based on the discretisation, length, width and thickness of the sub-elements are known. Matrix sub-element/region properties are taken to be those of bulk matrix. Based on these, the element properties are derived as explained later. 5.1. Stress and failure analysis The discretisation scheme is shown in Fig. 6. The strain for the elements in the loading direction is the
same as that for the corresponding section by the iso-strain assumption at the section level. el sec 1sel x 1x 1 x
In the transverse direction, the assumption is made that all the elements have the same strain as the corresponding section. Also, the transverse strain for a section is the same as the average transverse strain for the unit cell. el sec 1sel y 1y 1 y 1y
The extensional stiffness matrix for each sub-element is equal to its transformed reduced stiffness matrix. h i sel Asel Q ij The element and section extensional stiffness matrices are obtained as: nz h h i i 1 X Asel Ael ij ij lz Lz n1 h
ny h i i 1 X Asec Ael ij ij ly Ly n1
The extensional compliance matrix of the section is, h i h i21 Asec asec ij ij The extensional compliance matrix of the unit cell/WF
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laminate is, aij
nx h i 1 X l asec Lx n1 ij x
The transverse strain of the unit cell/WF laminate is,
1y a21 s x Here, s x is load per unit area. The longitudinal strain of the section is,
1sec x
sec s x 2 Asec 12 1y sec A11
The longitudinal strain of the unit cell/WF laminate is,
1x
nx 1 X 1sec l Lx n1 x x
The sub-element stress is, sel sel sel s xsel Asel 11 1x 1 A12 1y
Fig. 7. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 08; linear analysis.
The ®ll sub-element stresses in the local co-ordinates:
s 1sel cos2 qf xs xsel
5.1.1. Non-linear analysis Because of the undulation of the yarns, each sub-element is off-axis with respect to loading direction. Considering this aspect, possible non-linear behaviour of each sub-element is considered as follows [49]:
s 2sel sin2 qf xs xsel sel tsel 12 sin qf xcos qf xs x
The above expressions are based on linear analysis. The stresses obtained from the above equations in each sub-element are compared with the permissible stress values. If the induced stress in any sub-element exceeds the permissible limit, that particular sub-element is treated as failed and does not contribute to further load sharing. Generally, it is observed that the failure takes place in a section with longitudinal yarns at an angle with respect to the loading direction. Hence, in such sub-elements, the early modes of failure would be shear failure and transverse failure. The ultimate failure would take place when the ®bres break. Hence ®bre strain/twisted yarn strain/section strain was monitored during loading. The effective permissible twisted yarn strain would be the ®bre strain minus pre-strain due to twisting. When the section strain/ ®bre strain exceeds the permissible strain, the section is supposed to have failed indicating the failure of unit cell/ WF laminate.
1nl x Sxx u xs x u x 1
1 2 cos 4u x2 S5555 s x u x3 64
For E-glass/epoxy, S5555 37 £ 1029 MPa23 at Vfs 0:46: And, Sxx is the compliance element. sel sec It may be noted that s x s xsel ; 1nl and x 1x 1x u x qf x: The sub-element/section strains in local coordinates, 2 nl 1nl 1 cos qf x1x
This value of induced strain is compared with the permissible strain of the twisted yarn/section. 6. Results and discussion Predicted strength properties of the plain weave fabric
Table 2 Predicted strength properties of plain weave fabric laminates made of twisted yarns: E-glass/epoxy, Vfo 0:44; h=a 0:8; g=a 0:2 (linear analysis)
a (8)
0 2 5 10
WTI (MPa)
WTC (MPa)
FTI(FSI) (MPa)
PMBFI (MPa)
XT (MPa)
1 x (%)
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
15 15 15 15
22 22 21 19
47 48 49 52
29 29 30 33
25(32) 25(32) 25(31) 23(32)
29(31) 29(31) 27(31) 25(31)
119 119 119 119
120 120 120 121
227 224 209 155
262 258 240 178
3.73 3.68 3.44 2.53
3.85 3.80 3.55 2.60
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Fig. 8. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 28; linear analysis.
Fig. 10. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 108; linear analysis.
composites made of twisted yarns using linear analysis are presented in Table 2 and in Figs. 7±12. Failure strength and strain data are presented for both C1 and C2 cases with different twist angles. Different stages of failure like warp transverse failure initiation (WTI), complete failure of warp (WTC), ®ll transverse failure initiation (FTI), ®ll shear failure initiation (FSI), pure matrix block failure initiation (PMBFI) and the ®bre breakage, i.e. the ultimate tensile failure (XT) have been monitored. In Figs. 7±10,
WTI, PMBFI and XT have been indicated by `a', `d' and `e', respectively. It can be seen that warp transverse failure initiation is the ®rst mode of failure. Because of the higher undulation angle, ®ll transverse failure and ®ll shear failure also take place early. The next failure is the failure of
Fig. 9. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, a 58; linear analysis.
Fig. 11. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, con®guration C1, linear analysis.
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Table 3 Predicted strength properties of plain weave fabric laminates made of twisted yarns: E-glass/epoxy, Vfo 0:44; h=a 0:8; g=a 0:2 (non-linear analysis)
a (8)
0 2 5 10
WTI (MPa)
WTC (MPa)
FTI(FSI) (MPa)
PMBFI (MPa)
1 x (%)
XT (MPa)
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
C1
C2
15 15 15 15
22 22 21 19
47 48 49 52
29 29 30 33
25(32) 25(32) 25(31) 23(32)
29(31) 28(31) 27(31) 26(31)
118.4 118.4 118.6 ±
120 120 120 ±
131.2 131.0 129.4 118.6
121.2 120.6 120.4 113.2
1.91 1.91 1.85 ±
2.03 2.02 2.03 ±
pure matrix block. Failure of the pure matrix block would lead to the disintegration of the composite. Effectively, this can be treated as the failure of the WF composite. But the ®bres can still be intact. The ®bres break when the ultimate strain exceeds the permissible strain limit inside the section. This is the ultimate failure of the WF composite. In Table 2, 1 x is the failure strain of the plain weave fabric laminates made of twisted yarns during tensile loading along warp or ®ll directions. As the angle of twist of the yarn increases, the ultimate failure strength and the ultimate failure strain decrease. This is because of the higher pre-strain within the yarn as the twist angle increases. This can be seen from Table 1. The ultimate strength and ultimate strain predictions for C2 are higher than for C1. This is because of the bridging effect in the case of C2. The weaker gap regions of one layer are
bridged by the stronger interlacing regions of the adjacent layer for C2. For twist angles of the yarn, say up to 58, the variation in ultimate strength and ultimate strain values is not signi®cant compared to those of corresponding straight strand. Since twisting of E-glass yarns is an engineering requirement for facilitating ease of preform fabrication, the yarns can be twisted optimally without reducing the strength values signi®cantly. Because of higher h/a ratio in the case of plain weave fabric composites made of twisted yarns, the strength properties are lower compared to the WF composites with lower h/a ratio and cross-ply laminates [23]. From Figs. 11 and 12, it can be seen that the stress±strain behaviour is nearly identical for different angles of twist. Predicted strength properties using the non-linear analysis are presented in Table 3. As expected the ultimate strength and ultimate strain are lower in this case compared to the predictions based on linear analysis. This is because of higher strain state within the sections. The difference in behaviour between the predictions using linear analysis and non-linear analysis is seen only after pure matrix block failure. As indicated earlier, failure of pure matrix block leads to the disintegration of WF composites. Effectively, the corresponding stress can be taken to be practical strength value for the WF composites. From this point of view, both linear and non-linear analysis give identical results. 7. Conclusions A 2D analytical method is presented for the failure behaviour of plain weave fabric composites made of twisted yarns. The studies have been carried out under uni-axial on-axis tensile loading. Twisting of yarns, especially for E-glass composites, is an engineering requirement for facilitating ease of preform fabrication. Based on this, the following conclusions are derived:
Fig. 12. Stress±strain behaviour for plain weave fabric laminates made of twisted yarns: E-glass/epoxy, con®guration C2, linear analysis.
1. The twisted yarn thickness-to-width ratio would be higher compared to the straight strand thickness-to-width ratio. 2. The variation in tensile strength properties for plain weave fabric composites as a result of twisting of yarns is not signi®cant up to an optimum angle of twist.
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3. E-glass yarns can be twisted up to 58 so as to facilitate ease of preform fabrication without compromising on the tensile strength properties of plain weave fabric composites signi®cantly. Acknowledgements This work was supported by the Structures Panel, Aeronautics Research and Development Board and ISROIIT(B) Space Technology Cell. References [1] Ishikawa T, Chou TW. One-dimensional micromechanical analysis of woven fabric composites. AIAA J 1983;21:1714±21. [2] Chou TW, Ishikawa T. Analysis and modeling of two-dimensional fabric composites. In: Chou TW, Ko FK, editors. Textile structural composites. Amsterdam: Elsevier, 1989. p. 209±63. [3] Whitcomb JD. Three-dimensional stress analysis of plain weave composites. In: O'Brien TK, editor. Composite materials: fatigue and fracture, ASTM STP 1110, vol. 3. Philadelphia, PA: American Society for Testing and Materials, 1991. p. 417±38. [4] Naik NK, Shembekar PS. Elastic analysis of woven fabric composites. Report No. IITB/AE/ARDB/STR/659/91/01, Bombay: IIT, 1991. [5] Naik NK, Ganesh VK. Prediction of on-axes elastic properties of plain weave fabric composites. Compos Sci Technol 1992;45:135±52. [6] Naik NK, Shembekar PS. Elastic behavior of woven fabric composites: IÐlamina analysis. J Compos Mater 1992;26:2196±225. [7] Shembekar PS, Naik NK. Elastic behavior of woven fabric composites: IIÐlaminate analysis. J Compos Mater 1992;26:2226±46. [8] Naik NK, Shembekar PS. Elastic behavior of woven fabric composites: IIIÐlaminate design. J Compos Mater 1992;26:2522±41. [9] Dasgupta A, Agarwal RK. Orthotropic thermal conductivity of plain weave fabric composites using a homogenization technique. J Compos Mater 1992;26:2736±58. [10] Naik NK, Ganesh VK. Prediction of thermal expansion coef®cients of plain weave fabric composites. Compos Struct 1993;26:139±54. [11] Naik NK. Woven fabric composites. Lancaster, PA: Technomic Publishing Co., Inc, 1994. [12] Hahn HT, Pandey R. Micromechanics model for thermoelastic properties of plain weave fabric composites. J Engng Mater Technol 1994;116:517±23. [13] Raju IS, Wang JT. Classical laminate theory models for woven fabric composites. J Compos Technol Res 1994;16:289±303. [14] Dasgupta A, Bhandarkar SM. Effective thermo mechanical behavior of plain weave fabric-reinforced composites using homogenization theory. J Engng Mater Technol 1994;116:99±105. [15] Ganesh VK, Naik NK. Thermal expansion coef®cients of plain weave fabric laminates. Compos Sci Technol 1994;51:387±408. [16] Naik NK, Ganesh VK. Failure behavior of plain weave fabric laminates under in-plane shear loading. J Compos Technol Res 1994;16:3±20. [17] Shkoller S, Hegemier G. Homogenization of plain weave composites using two-scale convergence. Int J Solids Struct 1995;32:783±94. [18] Ganesh VK, Naik NK. Failure behaviour of plain weave fabric laminates under in-plane shear loading: effect of fabric geometry. Compos Struct 1995;30:179±92. [19] Naik NK, Ganesh VK. Analytical method for plain weave fabric composites. Composites 1995;26:281±9. [20] Naik RA. Failure analysis of woven and braided fabric-reinforced composites. J Compos Mater 1995;29:2334±63. [21] Whitcomb J, Srirengan K. Effect of various approximations on predicted progressive failure in plain weave composites. Compos Struct 1996;34:13±20.
707
[22] Ganesh VK, Naik NK. Failure behavior of plain weave fabric laminates under on-axis uni-axial tensile loading: IÐlaminate geometry. J Compos Mater 1996;30:1748±78. [23] Naik NK, Ganesh VK. Failure behavior of plain weave fabric laminates under on-axis uni-axial tensile loading: IIÐanalytical predictions. J Compos Mater 1996;30:1779±822. [24] Ganesh VK, Naik NK. Failure behavior of plain weave fabric laminates under on-axis uni-axial tensile loading: IIIÐeffect of fabric geometry. J Compos Mater 1996;30:1823±56. [25] Ito M, Chou TW. Elastic moduli and stress ®eld of plain weave composites under tensile loading. Compos Sci Technol 1997;57:787±800. [26] Nayfeh AH, Kress GR. Non-linear constitutive model for plain weave composites. Compos Part B: Engng 1997;28B:627±34. [27] Naik NK, Ganesh VK. Thermo-mechanical behaviour of plain weave fabric composites: experimental investigations. J Mater Sci 1997;32:267±77. [28] Ganesh VK, Naik NK. (1/245)8 Off-axis tension test for shear characterization of plain weave fabric composites. J Compos Technol Res 1997;19:77±85. [29] Ganesh VK, Ramakrishna S, Teoh SH, Naik NK. Micromechanical design of textile composites. Mater Des 1997;18:175±81. [30] Srirengan K, Whitcomb JD. Finite element based degradation model for composites with transverse matrix cracks. J Thermoplast Compos Mater 1998;11:113±23. [31] Scida D, Aboura Z, Benzeggagh ML, Bocherens E. Prediction of the elastic behaviour of hybrid and non-hybrid woven composites. Compos Sci Technol 1998;57:1727±40. [32] Ito M, Chou TW. Analytical and experimental study of strength and failure behavior of plain weave composites. J Compos Mater 1998;32:2±30. [33] Kuhn JL, Charalambides PG. Elastic response of porous matrix plain weave fabric composites: part IÐmodeling. J Compos Mater 1998;32:1426±71. [34] Kuhn JL, Charalambides PG. Elastic response of porous matrix plain weave fabric composites: part IIÐresults. J Compos Mater 1998;32:1472±507. [35] Nguyen M, Herszberg I, Paton R. The shear properties of woven carbon fabric. Compos Struct 1999;47:767±79. [36] Tan P, Tong L, Steven GP. Micromechanics models for the elastic constants and failure strengths of plain weave composites. Compos Struct 1999;47:797±804. [37] Bejan L, Poterasu VF. Woven composite material design by orthotropic compliance averaging method using mathematica. Comput Meth Appl Mech Engng 1999;179:53±65. [38] Scida D, Aboura Z, Benzeggagh ML, Bocherens E. A micromechanics model for 3D elasticity and failure of woven-®bre composite materials. Compos Sci Technol 1999;59:505±17. [39] Kuhn JL, Haan SI, Charalambides PG. Semi-analytical method for the calculation of the elastic micro-®elds in plain weave fabric composites subjected to in-plane loading. J Compos Mater 1999;33:221±66. [40] Thom H. Finite element modeling of plain weave composites. J Compos Mater 1999;33:1491±510. [41] Gao F, Boniface L, Ogin SL, Smith PA, Greaves RP. Damage accumulation in woven fabric CFRP laminates under tensile loading: part I, observations of damage accumulation. Compos Sci Technol 1999;59:123±36. [42] Tabiei A, Jiang Y, Yi W. Novel micromechanics-based woven fabric composite constitutive model with material non-linear behavior. AIAA J 2000;38:1437±43. [43] Bystrom J, Jekabsons N, Varna J. An evaluation of different models for prediction of elastic properties of woven composites. Compos Part B: Engng 2000;31:7±20. [44] Nguyen-Hoa H, Vu-Khanh T. Prediction of failure in unsaturated polyester reinforced by plain weave glass fabric. J Compos Mater 2000;34:379±97.
708
N.K. Naik, R. Kuchibhotla / Composites: Part A 33 (2002) 697±708
[45] Hearle JWS, Grosberg P, Backer S. Structural mechanics of ®bers, yarns, and fabrics. New York: Wiley, 1969. [46] Byun JH, Chou TW. Effect of yarn twist on the elastic properties of composites. In: Poursartip A, Street K, editors. Proceedings of the ICCM-12, vol. IV. Cambridge, England: Woodhead Publishing Company, 1995. p. 293±99.
[47] Naik NK, Madhavan V. Twisted impregnated yarns: elastic properties. J Strain Anal Engng Des 2000;35:83±91. [48] Naik NK, Singh MN. Twisted impregnated yarns: transverse tensile strength. J Strain Anal Engng Des 2001;36:347±58. [49] Hahn HT, Tsai SW. Non-linear elastic behaviour of unidirectional composite laminae. J Compos Mater 1973;7:102±18.