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The shear properties of woven carbon fabric
Composite Structures 47 (1999) 767±779
www.elsevier.com/locate/compstruct
The shear properties of woven carbon fabric M. Nguyen a,*, I. Herszberg a, R. Paton b a
The Sir Lawrence Wackett Centre for Aerospace Design and Technology, RMIT University, G.P.O. Box 2476V, Melbourne, Vic. 3001, Australia Cooperative Research Centre for Advance Composite Structures Ltd. (CRC-ACS), 506 Lorimer Street, Fishermens Bend, Vic. 3207, Australia
b
Abstract This paper presents experimental and theoretical studies of the shearing properties of carbon plain weave fabrics and prepregs. The shearing characteristics of these materials are determined by the use of a picture frame shear rig which is loaded by a mechanical test machine. The shear force/angle curves are presented for the experiments conducted with the various test materials. A proposed shear model based on previous research which idealizes the fabric yarns as beam elements is presented. Using fabric geometric and material parameters, the model predicts the initial slip region of the fabric, as well as the more dominant elastic deformation range. Comparisons of the experimental and theoretical results were conducted to validate the model. A discussion of the ®ndings from the analysis is also given, with particular focus relating to the accuracy, limitations and advantages oered by such a model. Results indicated that the slip model gives modestly accurate predictions, whilst the elastic modulus model showed very good correlation with experimental data. Ó 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Shearing; Fabric model; Slip; Elastic modulus
Nomenclature k a, b, / c d l A, B, C, D a, b, c as dc E F Fc f f1 , f2 h
I eigen value solution eigen value fabric parameter functions shear angle maximum yarn de¯ection height coecient of friction model coecients positions along yarn de¯ection model slip distance acting along contact length eective yarn contact length elastic tensile modulus shear force along shear bar linkage reaction force at position c elemental shear force in an arbitrary direction elemental shear force in 1, 2 direction yarn crimp height
k L M, N, O, P m P1 , P2 R s1 , s2 t t1 , t2 V vab vbc w1 ; w2 X, Y X1 , X2 x1 ; x2 ; . . . ; x8
yarn cross-sectional area moment of inertia spring constant total specimen fabric length coecients of shear model solution internal moments pitch spacing in 1, 2 direction resultant shear frame force tow spacing in 1, 2 direction tensile force elemental tension force in 1, 2 direction yarn normal force yarn de¯ection along length ab Yarn de¯ection along length bc tow widths in 1, 2 direction coordinate axes arbitrary lengths along the X-direction fabric architectural parameters
1. Introduction
*
Corresponding author. E-mail address: [email protected] (M. Nguyen).
The forming behaviour of fabric preforms is of important consideration for the manufacture of composite components. The ability of the fabric material to
0263-8223/99/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 0 ) 0 0 0 5 1 - 9
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conform to various shapes is highly dependent on the fabric architecture, which in turn will in¯uence the resultant mechanical properties of the composite part. During preform forming, two major deformation mechanisms are considered to be important; these being simple shear and shear-slip (refer to Fig. 1). Of these mechanisms in relation to woven fabrics, simple shear has been indicated to be the more dominant behaviour [1]. In undergoing fabric shearing, it was initially suggested by Mack and Taylor [2] that the reinforcement behaves as a pin-jointed mesh, whereby the tows rotate or trellis about the fabric cross-overs. By further assuming that the yarns are inextensible and incompressible, Potter [3], Robertson [4], Van West [5] and Long [6], employed it to map the fabric draping behaviour over shapes of varying pro®les. The outcome of these research eorts was to reinforce the concept that the ability of fabrics to shear is a particularly important forming characteristic. This characteristic enables the manufacture of 2-dimensional preforms into complex 3-dimensional shapes. Further work by McBride [7] showed that during fabric shearing, the spatial positioning of the fabric architecture will vary, such that local increases in the ®bre volume fraction will occur. Investigation conducted by Tam and Gutowski [8] with ®bre-aligned materials identi®ed in-plane and interply shear as important parameters in¯uencing the forming behaviour of aligned ®bre-reinforced composites. Results showed that when the material experiences high angles of shearing, wrinkling of the fabric occurred. Woven fabrics dier from aligned ®bre materials, as trellising or rotation of the yarn is the dominant deformation mechanism. Also the eect of inter-ply shear becomes an in¯uential deformation mechanism, when stacks of fabric plies are required to undergo forming simultaneously [9]. However, when high angles of material shear are induced into woven fabrics, they too exhibit similar behaviour to aligned ®bre materials; whereby wrinkling occurs due to a build-up of in-plane compressive forces. The characterization of the fabric shear properties has predominantly consisted of testing the materials
Fig. 1. Major mechanisms for fabric deformation.
within a shear rig ®xture which resembles a picture frame [10,11]. Results using a variety of glass and carbon fabrics have indicated that the shearing behaviour can be divided into two distinct regions. Before fabric lock-up, whereby low shearing loads are required; and after lock-up, which is characterized by a sudden increase in the load generally accompanied with material wrinkling. Fabric lock-up is described as when high angles of yarn rotation occur such that the fabric architecture becomes ÔjammedÕ, and this jamming eect results in a rapid increase in the shearing force. Further results from these tests have indicated that regardless of the weave type, the fabrics all exhibited similar trends in shear deformation. By considering a plain woven unit cell and assuming that yarn rotation about the region of overlap behaves as a pin-jointed mesh. Prodromou and Chen [12] provided a geometric model to predict the occurrence of fabric lock-up. Recent work by Wang [13], highlighted that this fabric lock-up angle is highly dependent on the type of shear testing method employed. By characterizing the fabric shear behaviour of a material with the bias-extension and simple shear rail test, results showed that there were noticeable dierences between the fabric locking angles obtained. Early attempts to model the plain-weave fabric architecture typically assumed the yarn cross-section was circular and various inferences were made about the amount of yarn crimp [14,15]. Presently it is evident that the yarns can be more aptly described as elliptical, and that variations in the elliptical cross-sectional pro®le exist along the length of the yarn [16]. Though the understanding of the fabric architecture has progressed, there exists a critical need to provide models which accurately describe the shear properties of fabric preforms. Research into modelling the shear properties is scarce and there currently only exist a handful of methods. Initial attempts by Grosberg and Park [17] idealized the yarns as beam structural elements, allowing the ability to predict both the initial and elastic shear modulus of the fabric. Two disadvantages are considered to be associated with this model. Firstly the presentation of the theoretical solution is in a form not particularly suitable for practical use, and secondly, the model does not consider undulation of the yarns. Modelling by Katabawa et al. [18±20] did incorporate yarn undulation, but assumed a linear function of the torsional moments required to induce fabric shearing. The various coecients used to characterize this linear function, could not be de®ned theoretically and hence were experimentally solved. Elaboration of this model by Ye [21], enabled the contact force at the yarn crossovers to be determined, though both the models by Grosberg and Park, and Katabawa et al. showed quite good correlation with experimental results. Validation consisted of comparisons with cloth, polyester and wool
M. Nguyen et al. / Composite Structures 47 (1999) 767±779
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fabrics, which are not suitable reinforcement materials for advanced composites. More recent developments by Sinoimeri and Drean [22] used energy methods to study the fabric shear properties. The introduction of a rheological model allowed tow slippage to be accounted. Unfortunately this model suers from the use of arbitrarily coecients to predict the fabric shearing behaviour, and that there is limited reference to the fabric architectural parameters within the model itself. In this paper, a formulation for the shear properties of plain woven fabrics is presented. Theoretical solutions are generated from the model and are compared with experimental results obtained from a shear rig test ®xture. A discussion on the ®ndings is given, with particular focus on the accuracy, limitations and bene®ts oered by such a model.
A variety of carbon fabrics were tested, with their properties summarized in Table 1. For each material, a minimum of ®ve tests were conducted. The test materials also consisted of two carbon plain weave prepregs. These prepregs were comprised from T300 and T650 yarns and impregnated with Hexcel F593 and Hexcel F584 modi®ed epoxy resin, respectively. An important characteristic of fabric deformation is when the material experiences high shearing angles (c), resulting in the fabric architecture becoming ÔjammedÕ. This jamming is also commonly referred to as fabric locking, the eect of which increases the forces required to deform the fabric and also leads to wrinkling. During this experimental investigation, locking of the fabric is de®ned when wrinkling of the material within the picture frame was observed to occur.
2. Experimental
The fabric test size is 240 mm 240 mm, minus a 25 mm radius quadrant cut from each corner. This quadrant is required so that rotation of the hinges can occur freely. The test material is then placed into the ®xture via a clamping arrangement located along each bar linkage. Care is taken to ensure that the weft and warp tows are positioned parallel to the frame. The shear frame is loaded into an INSTRON 45 kN screw driven testing machine and the brace then removed. To minimize the viscosity behaviour of the prepregs, testing was carried out at a rate of 5 mm minÿ1 . Picture frame testing for a range of dry and impregnated fabrics has typically used shear rates of 10 mm minÿ1 . However, a reduced loading rate of one half this speed is not expected to signi®cantly alter the material shearing characteristics. This speculation is supported with work from Wang [23], who showed that minimal variation of the shear modulus was observed for test speeds ranging from 1 to 10 mm minÿ1 .
2.1. The picture frame experiment Experimental investigation of the shearing behaviour of dry and prepreg fabrics was carried out using a picture-frame apparatus. The aim of this experiment is to determine the relationship between shearing angle and load for dierent weave architectures (i.e., variable pattern and fabric parameters). An illustration of the shear frame apparatus which is essentially a square fourbar linkage is shown in Fig. 2. A removable brace was designed to allow transport and loading of the apparatus so as to minimize fabric shearing prior to testing. Experiments were carried out at ambient temperature, with force and displacement history recorded into a personal computer. This type of testing apparatus has been used successfully to characterize the shear properties of dry fabrics [10,12] and composite laminates [11].
2.2. Procedure
2.3. Test results
Fig. 2. Schematic of the picture frame, with removable brace attached.
Fig. 3 shows the shear frame picture rig deforming at 0°, 15°, 30° and 47° of shear, and the corresponding position of the rig deformation on the shear force/angle curve. White lines drawn 50 mm in from the edge of the bar linkages, show that there is a uniform state of shearing occurring within this highlighted region. The shear force/angle relationships for the fabrics listed in Table 1 are plotted in Fig. 4. A number of observations are obtained from these results. · PW (1K) fabric exhibited a steeper shear curve compared to the PW (3K) material. This is thought to occur because of the PW (1K) having a combination of a small tow size and compact weave structure. The result is to increase the frictional resistance acting at the fabric cross-overs. Similar results have been noticed
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Table 1 Speci®cations of fabrics used in experimentation Fabric type (code)
Construction (tows cmÿ1 )
Areal weight (g m2 )
PW (1K) PW (3K) 2/2 twill (3K) 2/2 twill (6K) PW-prepreg (T300) PW-prepreg (T650)
6:4 6:5 4:6 4:5 5:2 5:3 4:3 4:4 4:7 4:8 4:8 4:9
90 198 220 375 280 270
Nominal thickness (mm) 0.20 0.24 0.32 0.43 0.30 0.28
Measured locking angle (c, deg)
Calculated locking angle (c, deg)
46 50 72 64 38 34
40 44 ) ) 31 22
Fig. 3. Dierent stages of fabric shear deformation, tested at a nominal rate of 5 mm/min, corresponding to various positions of a PW (3K) shear deformation history at (a) 0°, (b) 15°, (c) 30°, (d) 47°.
Fig. 4. Fabric and prepreg shear force/angle relationships for the material types listed in Table 1.
M. Nguyen et al. / Composite Structures 47 (1999) 767±779
and various discussions relating to these observations have been given [12,24]. · Impregnation of fabric with resin as in the case of prepregs has the eect of increasing the materialÕs shear modulus compared to dry fabrics. Results further indicate that the PW-prepreg (T650) has a much higher shear modulus compared to that of the PWprepreg (T300). This variation between the shearing properties is considered to be due to the dierence in the resin properties. · Despite the weave type, both dry fabrics and the prepregs exhibit similar shear deformation behaviour, which can be described as parabolic in nature. · As twill weave fabrics require low deformation loads, these materials experience fabric lock-up at much higher angles of shear. Consequently they are considered to be more ÔconformableÕ than plain weaves, because of their ability to be contoured to complex shapes before the occurrence of wrinkling. 3. The shear models Grosberg et al. [25] demonstrated that the trends for fabric shearing typically consist of three stages; the initial region, where shear is due to the ¯exural deformation of the tows; the mid-region, where shear resistance becomes dominated by a bearing load type contact; and the last region, where there is a rapid load increase due to ÔjammingÕ of the fabric structure. As the fabric undergoes these deformation phases, the material architecture must change so that the dierent phases can be accommodated. Henceforth, it is necessary to both understand and describe the fabric architecture if a model is to be developed. 3.1. De®nition of fabric architecture To model the shearing property of the plain weave fabrics, an idealized repeating, geometrically symmetrical cell section is depicted in Fig. 5. Though there are dierent ways to how the cell can be represented [16], modelling of fabric shear properties has generally consisted of the following pattern. In determining the fabric deformation characteristics using this unit cell structure, undulation of the yarns is not considered. The implications of this assumption on the accuracy of the theoretical results are discussed further in Section 5. By specifying a set of geometric parameters as shown in Fig. 5 with the material elastic properties, mathematical expressions for the initial slip and elastic modulus regions can be derived. Fig. 6(a) shows a representation of the square shear rig deforming from a state of c 0, through to a shear angle of c. As the fabric material is sheared, the forces acting on the bar linkages can be discretized into shear and tensile
771
Fig. 5. Illustration showing the full repeating and geometrically symmetrical unit cell of the plain weave fabric structure.
force components acting perpendicular and parallel to the fabric tow. This is shown Fig. 6(b). At low angles of shearing (0° ! 2°), if the theory of small angle approximations is utilized [26], then the acting tensile force component will have negligible magnitude and hence can be ignored. Consequently for this deformation range, which is referred to as the initial slip region, the fabric material is considered to be in a state of pure shear (Fig. 6(c)). 3.2. The initial slip region This region of deformation has been extensively dealt with by Grosberg and Park [17], and in obtaining the solution a number of key assumptions were presented, which are as follows: 1. As the initial region of slip exists for only small angles of shear, the model can be idealized to be in a state of pure shear, as shown in Fig. 6(c) where the elemental shear forces can be described as f1 F P1 =L; f2 F P2 =L: 2. The yarn contact lengths (dc ) acting at the crossovers, are the same lengths in both directions. 3. Yarns are inextensible during the shearing process. 4. Deformation of the yarns is considered to be only due to bending. It was necessary to understand the process of how this initial slip model was determined, as these principles provided the fundamentals in the development and improvement of the more important and dominant elastic modulus model. During reformulation of the solution presented by Grosberg and Park, two major modi®cations were incorporated, which were the re-de®nition of the normal force (V) and revision of the solution itself. 3.2.1. The normal force (V) In woven fabrics, the undulation of the yarns will exert pressure at the cross-overs, which in turn induces frictional resistance to shearing. The de®nition of this normal force was initially provided by experimental and theoretical work by Pierce [14]. By considering the
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Fig. 6. (a) Fabric shearing under applied load (R). (b) As the applied load increases, the shearing forces acting along the bar linkages (F) can be resolved into components of tension (t) and shear (f) acting parallel and perpendicular to the tows. (c) For the unique case of the initial slip region as the shear angles are assumed to be small, the fabric can be idealised to be in a state of pure shear.
problem from a structural point of view, Katabawa et al. [20] idealized the yarns as beam elements. These beams whose ends are attached to ®xed wall supports, enable the prediction of the normal contact force of the yarn using material and geometric parameters from the fabric architecture. However, if the beams are rigidly attached to the wall and then de¯ection at the centre occurs by application of a normally applied load, this condition would invalidate a key assumption whereby the yarns are inextensible. To overcome this inconsistency, the yarn undulation behaviour is suggested to be more accurately modeled as a beam, having a pin and roller support, as shown in Fig. 7. Hence the normal force can be approximated as V
48 E I h ; 3 P1;2
1
where P1 or P2 refer to the pitch length in the warp or weft directions, respectively. 3.2.2. Revised solution Using the problem statement, boundary conditions and theory as described by Grosberg and Park, a reformulation of the solution was performed. Comparisons of Grosberg and ParkÕs and the revised solutions indicate that though both are dimensionally correct, there exists a small discrepancy which has a considerable in¯uence on the overall results, this dierence being the
Fig. 7. Idealised bending of a ®bre yarn.
presence of cubic power terms present in the solution provided by Grosberg and Park, compared to square power terms in the revised equation. The extent and impact of these dierences in describing the shearing properties of woven fabrics will be investigated and discussed in a future article. The revised fabric shearing solution, incorporating the normal force function as described in Eq. (1) for the initial slippage region is as follows: " 2 F as c P2 L ÿ dc 1 ÿ 8EI L dc # 2 as 2 P1 L ÿ dc 1 ÿ dc as c : 0 ! 1; dc 2 l V 24dc dacs 3 ÿ 2 dacs i h ii F h h 3 as as 1 P L ÿ d 1 ÿ L ÿ d 1 ÿ P 2 c 1 c dc dc 4L c:0!
as 1: dc
dc is determined from methods as described from Grosberg and Park [17]. 3.3. The elastic modulus region Grosberg et al. [25] elaborated on the shear modelling work by Grosberg and Park [17] by next considering the elastic shear modulus region. Unfortunately the solution provided is not presented in a manner which makes comprehension easy, particularly if it is desirable to understand how fabric parameters in¯uence the material shear characteristics. The complexity of the equations is further exacerbated by the use of Fourier series to
M. Nguyen et al. / Composite Structures 47 (1999) 767±779
express the de¯ected yarn modal shape. To overcome these drawbacks, a model is proposed which uses the same assumptions, boundary conditions and problem statement as initially set up by Grosberg et al., but varies by its approach in determining and de®ning the solution for the elastic shear modulus of plain woven fabrics. 3.3.1. Analysis The analysis still maintains the geometrically symmetrical unit cell, however for this region, the mechanism in which fabric shearing occurs is dierent. As high angles of yarn rotation occur, the contact area between adjacent yarns near the fabric cross-overs will increase. This contact resistance to fabric shearing is now assumed to be more dominant than the frictional restraints acting about the yarn cross-overs. The idealization of this contact resistance to fabric shearing, takes the form of a spring bearing load mechanism. It is further assumed that this mechanism acts through a point located at the end of the cross-over contact length, dc . This concept is illustrated in Fig. 8. Tow rotation for this model deformation state is able to occur via a pin-joint acting about the tow cross-overs. Grosberg et al. argued that this is plausible as there is now constant frictional resistance about the tow overlap region after complete slippage has taken place. Furthermore, as the majority of the resistance to fabric
773
shearing will be dominated by the spring contact force, the contribution of friction in resisting shear is expected to be small to minimal. As the fabric continues to be sheared, the shear forces acting on the bar linkages can be separated into tension and shear components acting parallel and perpendicular to the tows, respectively (Fig. 6(b)). By considering that these force components act on one-half of a yarn (as the unit cell is symmetrical), the problem can be described by a free body diaphragm as seen in Fig. 8(b) and Fig. 9. 3.3.2. Model expression The amount of vertical yarn-de¯ection along the length X1 and X2 can be described as vab A cos k X1 B sin k X1 ÿ f2 =t1 X 1 d X1 : 0 ! P1 ÿ dc =2;
4
vbc C cos k X2 D sin k X2 f2 P1 2d ÿ1 X2 t1 dc dc X2 : 0 ! dc =2; 5 q where k Et1 I , and the elemental shear and tension forces acting on a single tow are described as
Fig. 8. (a) Point bearing loads acting at the edge of the cross-over contact length on a unit cell. (b) Representation in the form of a beam de¯ection diagram, showing a single yarn in a state of shear and tension with support springs.
Fig. 9. Free body diagram of yarn de¯ection under tension and shear.
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f2
F P2 sin c=2; L
t1
F P2 cos c=2: L
d : 11 P2 =2 Hence the total shear angle of the warp and weft yarns is described as
6
c=2weft
3.3.3. Boundary conditions A, B, C, D in Eqs. (4) and (5) are coecients obtained from the boundary conditions. These boundary conditions are 1. at X2 0; vbc 0; 2. at 2t1 d f1 P1 ÿ ; X2 dc =2; vbc k dc k dc 3. at 2t1 d f1 P1 ÿ ; X1 P1 ÿ dc =2; vab k dc k dc 4. ÿo vab 1 o vbc dc P1 ÿ dc : oX1 oX2 2 2
c c=2warp c=2weft :
12
Results for the expression of the total shear force (F) along a bar linkage as a function of the shear angle (c) are given in Eq. (9). To simplify this solution, it is allowed that P1 P2 due to yarn symmetry and that the co-tangent trigonomic function is re-expressed using polynomial approximation terms. The ®nal solution for the shear force as a function of two unknown variables, shear angle (c) and the spring constant (k) is given in the form of a cubic equation 2
M k; c F 3 N k; c F O k; c F P k; c 0; c : 0 ! 90° ÿ arcsin 2 W =P
13
7
where the coecients of the solution M, N, O, P are in terms of the fabric material and geometric parameters. Note that the solution is theoretically valid up to the state when fabric locking occurs.
If the boundary conditions are used to obtain the coef®cient A, then results will show that the solution of A is non-zero. However the expression of A is also a function of d and hence for Eq. (7) to be satis®ed, d must equal the expression given in Eq. (8).
15x2 15x2 15x5 15x7 5 ÿ 5 ÿ 5 ; 2a5 2a 2b 2b 15x1 15x2 3x2 3x2 15x4 3x5 15x6 3x7 N ÿ 5 ÿ 5 3 3 5 ÿ 3 5 ÿ 3 ; 2a a 2a a 2b 2b b b
The maximum tow de¯ection (d) will occur when X1 0. This will result in Eq (4) being d A d ) A 0:
d
k
h
ÿf2 P1 k dc
f22t1dc
P1 dc
ÿ1
M
i
h i h i cot k 2dc k ÿfk 2 dcP1 f2t2 P1 ÿ dc cot k P12ÿdc ÿ ft12 Pd1c ÿ 2 : k 1 ÿ k2td1c cot k 2dc cot k P12ÿdc ÿ d2c
3.3.4. Solution The maximum yarn shear angle as a result of shearing and tensile deformation along the warp direction can be approximated as
8
3x1 3x2 x2 x2 3x4 x5 3x6 x7 ÿ 3 ÿ x3 3 ÿ 3 ÿ x8 ; 3 a a b b a a b b x1 x1 x4 x6 P ÿ ÿ a b b a Oÿ
i h i h i9 8 h k P1 ÿdc ÿP1 P1 ÿP1 dc k dc 1 ÿ t11 Pd1c ÿ 2 = 2 c 2 P2 c < k k dc 2t1 dc ÿ 1 cot 2 k k dc 2t1 P1 ÿ dc cot sin ÿ ; F L P1 2: 1 k 1 ÿ k2tdc cot k:d2 c cot k: P12ÿdc ÿ dc2 h i h i h i9 8 k P2 ÿdc ÿP2 P2 ÿP2 P2 k dc dc 1 1 < = k ÿ 1 cot k P ÿ d cot ÿ ÿ 2 2 c 2t2 2t2 t2 2 k dc dc 2 k dc dc 2 P1 c sin ; ; L P2 2: k 1 ÿ 2t cot k:dc cot k P2 ÿdc ÿ 2 k1 dc
c=2warp
d : P1 =2
2
2
dc
where 10
As the unit cell is symmetrical, the tow shear angle in the weft direction can also be approximated as
x1 ÿc tan c x2
c/ ; cos c
P2 / ; L k dc x6 ÿ
x5
4P2 / ; L k dc
2 P1 ÿ dc / ; P1 sin c
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2c ; dc cos c 4P2 :/ ; x4 ÿ L2 k dc x3
2dc P1 ÿ 1 /; P1 sin c dc 4 P1 x8 ÿ2 P1 sin c dc
775
Although it is well known that the yarn cross-sectional area varies along the length of the yarn, for simplicity of the analysis, these cross-sectional pro®les are generalized by having a constant elliptical shape. This allows the area moment of inertia (I) to be de®ned based on the yarn geometry. The empirical spring constant values were selected so that the theoretical solution would correlate closely with the experimental trends.
x7
and ®nally where r P2 sin c dc ; a 2 LEI r P2 sin c P1 ÿ dc ; b 2 LEI r P2 sin c : / LEI
4.2. The initial slip model
By de®ning an appropriate spring constant (k) with the fabric material and geometric properties, the shear force (F) can be determined from Eq. (13) for a given value of the shearing angle (c). One drawback of the model from a theoretical aspect is that it is unable to accurately predict shear forces at regions of very low shear. This is because the model assumes that there are always components of shear and tension acting on the fabric tows for the solution to be valid. This does not re¯ect the physical condition in the unique case of when c 0, when the shear force on the frame will be zero. To cater to this discrepancy between experimental and theoretical solution when c 0, the theoretical solution is slightly modi®ed by using a normalized factor of F jc0 . This factor is determined by assigning a spring constant, and evaluating the shear force from Eq. (13) for c 0. The modi®cation of the solution in terms of this normalized factor is now expressed as
Fig. 10(a) and (b) show the corresponding shear force and shear angle relationship for plain weave fabrics of 1K and 3K tow sizes, respectively, for the initial slip region. This region is typically characterized by the initially high rise in the shear modulus followed by levelling of the curve. Results indicate that the model provides modest correlation where initial slippage occurs and the magnitudes of the shear force within this range. Using linear interpolation, Table 3 shows the average predicted shear modulus in comparison with the experimental values for this fabric deformation zone. These results indicate that there is a 44% and 35% dierence between the average experimental and model shear moduli for PW (1K) and PW (3K), respectively. For prepreg fabrics, no calculations for the initial region of fabric slip were performed. It is evident from Fig. 10(c) that prepregs do not exhibit the same initial slip trends as compared to dry fabrics. This dierence is attributed to prepregs having the introduction of resin into the fabric architecture. This resin in¯uence on the material shear properties is discussed later in Section 5.
Fmod F jc:0!p=2 ÿ F jc0 :
4.3. The elastic modulus model
14
Henceforth the resultant force required to deform the trellis frame (R) can be determined as R 2 Fmod cos 45 ÿ c=2:
15
4. Modelling results To investigate the eectiveness of this model, the analysis as discussed in Section 3 was used to model the initial slip region and elastic modulus behaviour, of a range of plain weave dry fabrics and prepregs as detailed in Table 1. 4.1. Material property input The four plain weave fabric con®gurations considered and their corresponding geometric and material properties required for the model validation are given in Table 2. Fabric geometric properties were obtained from observations using a stereo microscopic.
Using the simpli®ed solution as provided by Eq. (13) and the inputs as given by Table 2, the elastic model trends for the dry fabrics and prepregs are compared with experimental results. These comparisons are shown in Figs. 11 and 12. They show that there is generally good agreement with the experimental trends from zero shear, up to the angle of fabric locking. For the shearing range of 0° to approximately 30°, the model underestimates the preform shear modulus, with variations from the experimental results ranging from 2% to 14%. The larger errors within the model generally arise in predicting regions of low shear and are considered to be due to the in¯uence of the normalizing factor. As the shear angle increases, the in¯uence of this normalizing factor decays and hence accuracy is improved. Results also indicate there is a phase where the model goes from underestimating to overestimating the material shear properties. For prepregs, this transitional phase occurs after fabric locking, whereas for fabrics this happens before locking. Why the model behaves like
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Table 2 Main parameters required for model validation Parameter
Units
PW (1K)
PW (3K)
PW-prepreg (T300)
PW-prepreg (T650)
P1 P2 W1 W1 s1 s2 L Yarn thickness E I dc k
(mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (GPa) (m4 ) (mm) (N/m)
3.15 3.04 1.20 1.16 0.41 0.40 240 0.08 210 3.1 ´ 10ÿ16 0.48 13500
4.26 4.32 1.51 1.46 0.61 0.70 240 0.11 220 1.17 ´ 10ÿ16 0.52 5000
4.22 4.30 1.80 1.80 0.41 0.45 240 0.15 220 2.9 ´ 10ÿ16 0.47 35000
4.20 4.15 1.95 1.90 0.25 0.18 240 0.14 230 2.61 ´ 10ÿ16 0.51 65000
Fig. 10. Shear force/angle relationship at the initial slip region for (a) PW (1K) with experimental and model trends, (b) PW (3K) with experimental and model trends and (c) experimental trends for both prepregs.
Table 3 Predicted and experimental shear modulii and angles of shear for the initial fabric slip region Shear modulus (N/deg) PW (1K) PW (3K)
Slip region (deg)
Experiment
Model
Experiment
Model
2.5 1.4
3.6 1.9
0 ! 0:40 0 ! 0:35
0 ! 0:18 0 ! 0:25
this is not fully understood, however overall, the accuracy of the solution is still maintained regardless of where the model transitional phase is situated. As stated in Section 3.3.3, the elastic modulus model is valid up to where fabric locking occurs. Analysis of the modelÕs accuracy beyond the lock angle, shows that there is still reasonably good correlation with the experimental results. This is particularly evident for the prepreg materials and the PW (3K) fabric. These results highlight the possible application of the elastic
modulus model to predict shear properties beyond fabric lock-up. 5. Discussion of results The picture-frame experiment has been successfully used to examine the shearing behaviour of carbon plain weave dry fabrics and prepreg materials at ambient temperature. The ®ndings are summarized in Fig. 4,
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Fig. 11. Comparison of theoretical and experimental results for (a) PW (1K) with k 13500 N/m and (b) PW (3K) with k 5000 N/m.
Fig. 12. Comparison of theoretical and experimental results for (a) PW-prepreg (T650) k 65000 N/m and (b) PW-prepreg (T300) with k 35000 N/m.
which shows that the shear modulus of the materials is lowest for twill weave fabrics and highest for prepregs. Because of the ability of twill weaves to undergo high angles of shear, they are better suited for forming applications, as wrinkles are induced at a later stage during the manufacturing process. Previous works on the characterization of the shear properties for both dry and impregnated fabrics of varying architectures have exhibited similar results to that presented in this paper [10,12]. For instance, experimental values of the fabric locking angles, agreed well with work conducted by Breuer [10], who reported that the critical fabric shear angle ranged around 40° for carbon plain weaves and 60° for carbon twill weaves.
Though Breuer does not describe how the lock angles were de®ned during testing, results from this experimental work bear similarities to results as presented by Breuer. It is hence assumed that a similar criterion, whereby locking is de®ned as when wrinkles occur during shear testing, was probably used. In attempting to theoretically predict the locking angle, Prodromou and Chen [12] provided a geometric model based on the plain weave unit cell to de®ne a point after which fabric lock-up occurs. This model is described as h arcsin
W ; P =2
16
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where h is the fabric lock angle, W the thickness of the yarn and P the pitch of the unit cell as described in Fig. 5. Estimations of the fabric locking angle using this formulation showed good correlation with experimental results (Table 1). A limitation in the use of beam theory to predict the shear response of fabric preforms is the inability of the model to cater for yarn crimping. Katabawa et al. [20] showed that the in¯uence of the yarn crimp angle (though generally small), when combined with shear will cause fabric deformation as a result of the bending and torsional moments created. The eect of not considering yarn torsion for the initial slip model, may be one plausible explanation as to why there exist large discrepancies between the theoretical and experimental results. However this does not exclude practical reasons for such dierences, which could include incorrect material handling or slight misalignment of the fabric during loading within the picture-frame shear rig. Having considered the in¯uence of yarn crimping and reasons as to why the theoretical and experimental results dier for the initial slip region. It is worth noting that the angles and shear forces associated with this region, are of extremely small magnitudes when compared to the total range of the fabric deformation. Hence, it is expected that there may be practical limitations with the utilization of such a model. Nevertheless this initial slip model is of signi®cant importance, as it provides important fundamentals on which the elastic modulus model is based. Results from the elastic modulus model show that there is good correlation with experimental data with an acceptable level in the margin of error. The accuracy of the correlation within this fabric deformation range also validates the assumption that the eects of yarn crimping can be considered to be negligible. The signi®cance of this deformation region compared to the initial slip region is apparent, as it represents a large proportion of the preform shear properties up till fabric lock-up. Generally, the shear properties of the preform after fabric lock-up has occurred are not required, as wrinkles within the material will already be present. It was noticed that there were diculties in determining the eective contact length (dc ) used in both the initial slip and elastic modulus models, when applying the methodology prescribed by Grosberg and Park [17]. During the evaluation of this length, it was realized that the models are highly sensitive to values of yarn thickness. Yarn thickness is an important parameter, because of its relationship to the yarn cross-sectional area moment of inertia via a cubic power term. Consequently small variations in the measured tow dimensions can be hugely exacerbated, which in turn will directly in¯uence estimations of not only the contact length, but the solution itself.
The use of a dry fabric model to predict the shear behaviour of visco-elastic prepregs, suggests that the deformation mechanisms acting within prepregs are similar to that of dry fabrics. The introduction of resin into the fabric structure is considered to have two eects on the mechanisms of shear deformation. Firstly, it will introduce greater frictional constraints acting at the yarn cross-overs, and secondly, the elastic stiness of the dry fabric yarns will be increased. The overall eect is to require higher shearing forces for the same degree of fabric shear. Consequently as shown in Fig. 10(c), prepregs do not exhibit the initial slip trends and so no slip predictions were performed. Rather the entire prepreg deformation region up to fabric locking is described by the elastic modulus model. Currently the model relies on de®ning the spring constant (k), so that model generation is as accurate as possible. As this value is presently empirical, further modelling is required so that a theoretical description can be obtained. In comparing the current model with existing solutions, one common theme is that all have incorporated various forms of empirical and semi-empirical coecients. The uniqueness of the solution presented is that it not only combines all the unknown variables into a single entity (k), but this factor also represents the physical mechanics required for fabric shear deformation. Furthermore, the breakdown of the solution into a cubic function, whose coecients form parameters relating to the fabric geometric and material properties, enables the application of the solution to be relatively straightforward. 6. Conclusions and summary From the current work the following conclusions were obtained: · The picture frame test experiment is able to adequately characterize the shear deformation behaviour of both dry fabrics and prepregs. · The region in which initial fabric slip occurs contains very low magnitudes of shear force and shear angle in comparison with the entire range of fabric deformation. · Overestimation of the shear modulus within the initial slip region is thought to occur because the model ignores yarn undulation. This eect would need to be incorporated to improve the accuracy of the solution. · Beam bending theory is a satisfactory analytical method which can be employed to predict the elastic modulus of prepregs and dry fabrics. Due to the accuracy of the solutions obtained, this validates the assumption that the in¯uence of yarn crimping can be ignored during modelling of the elastic modulus.
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· The de®nition of the spring constant (k) is an important factor in determining the accuracy of the solution. Currently this is an arbitrary value in which further research needs to be undertaken to characterize it from a theoretical aspect. Finally, the present formulation easily models variation of the plain weave fabric architecture. This makes it a useful tool to understand and quantify how changes in the ®bre architecture during manufacturing in¯uence fabric shear properties. References [1] Long AC, Rudd CD, Blagdon M, Smith P. Characterizing the processing and performance of aligned reinforcements during preform manufacture. Composites: Part A 1996;27A:253±547. [2] Mack C, Taylor HM. The ®tting of woven cloth to surfaces. J Text Inst 1956;47:477±89. [3] Potter KD. The in¯uence of accurate stretch data for reinforcements on the production of complex structural mouldings. Composite 1979;11:161±7. [4] Robertson RE, Hsiue ES, Yeh GSY. Continuous ®ber rearrangements during the moulding of ®ber composites. Polym Compos 1984;53:191±7. [5] Van West BP, Pipes RB, Keefe M. A simulation of the draping of bidirectional fabrics over arbitrary surfaces. J Text Inst 1990;4:448±60. [6] Long AC, Rudd CD, Blagdon M, Smith P. Characterising the processing and performance of aligned reinforcements during preform manufacturing. Composites: Part A 1996;27A:247±53. [7] McBride TM, Chen J. Unit-cell geometry in plain-weave fabrics during shear deformations. Compos Sci Technol 1997;57:345±51. [8] Tam AS, Gutowski TG. The kinematics for forming ideal aligned ®bre composites into complex shapes. Compos Manuf 1990;1(4):219±28. [9] Nguyen M. Investigations of the single diaphragm forming technique. Cooperative Research Centre for Advanced Composite Structures (CRC-ACS) Draft report, 1999. [10] Breuer U, Neitzel M, Ketzer V. Reinicke. Deep drawing of fabricreinforced thermoplastics: wrinkle formation and their reduction. Polym Compos 1996;17(4):643±7.
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Br [11] McGuinness GB, O adaigh CM. Characterization of thermoplastics composite melts in rhombus-shear: the picture frame experiment. Composites: Part A 1998;19A:115±32. [12] Prodromou AG, Chen J. On the relationship between shear angle and wrinkling of textile composite preforms. Composites: Part A 1997;28A:491±503. [13] Wang J, Page JR, Paton R. Experimental investigation of the draping properties of reinforced fabrics. Compos Sci Technol 1998;58:269±80. [14] Pierce FT. The geometry of cloth structure. J Text Inst 1937;45± 95. [15] Olofsson B. A general model of fabric as a geometric-mechanical structure. J Text Inst 1964:541±537. [16] Kuhn JL, Charalambides PG. Modelling of plain weave fabric composite geometry. J Compos Mater 1999;33(3):188±220. [17] Grosberg P, Park BJ. The mechanical properties of woven fabrics, Part V: The initial modulus and the frictional restraint in shearing of plain weave fabrics. Text Res J 1966:420±431. [18] Kawabata S, Niwa M, Kawai H. The ®nite deformation theory of plain-weave fabrics Part I: the biaxial-deformation theory. J Text Inst 1973;64(2):21±46. [19] Kawabata S, Niwa M, Kawai H. The ®nite deformation theory of plain-weave fabrics Part II: the uniaxial-deformation theory. J Text Inst 1973;64(2):47±61. [20] Kawabata S, Niwa M, Kawai H. The ®nite deformation theory of plain-weave fabrics Part III: the shear-deformation theory. J Text Inst 1973;64(2):62±85. [21] Ye L, Daghyani HR. Characteristics of woven ®bre fabric reinforced composites in forming process. Composites: Part A 1997;28A:869±74. [22] Sinoimeri A, Drean JY. A study of the mechanical behaviour of the plain-weave structure by using energy methods: fabric shear. J Text Inst 1996;87(1):120±9. [23] Wang J, Paton R, Page JR. Forming properties of thermoset fabric prepregs at room temperature and elevated temperature. Cooperative Research Centre for Advanced Composites (CRCACS) technical memorandum, 1998. [24] Golden K, Rogers TG, Spencer AJM. Forming kinematics of continuous ®bre reinforced laminates. Compos Manuf 1991;2(3):167±278. [25] Grosberg P, Leaf AV, Park BJ. The mechanical properties of woven fabrics, Part VI: the elastic shear modulus of plain-weave fabrics. Text Res J 1968:1085±1100. [26] James Stewart. Calculus. 2nd ed. Brooks/Cole, 1991.
www.elsevier.com/locate/compstruct
The shear properties of woven carbon fabric M. Nguyen a,*, I. Herszberg a, R. Paton b a
The Sir Lawrence Wackett Centre for Aerospace Design and Technology, RMIT University, G.P.O. Box 2476V, Melbourne, Vic. 3001, Australia Cooperative Research Centre for Advance Composite Structures Ltd. (CRC-ACS), 506 Lorimer Street, Fishermens Bend, Vic. 3207, Australia
b
Abstract This paper presents experimental and theoretical studies of the shearing properties of carbon plain weave fabrics and prepregs. The shearing characteristics of these materials are determined by the use of a picture frame shear rig which is loaded by a mechanical test machine. The shear force/angle curves are presented for the experiments conducted with the various test materials. A proposed shear model based on previous research which idealizes the fabric yarns as beam elements is presented. Using fabric geometric and material parameters, the model predicts the initial slip region of the fabric, as well as the more dominant elastic deformation range. Comparisons of the experimental and theoretical results were conducted to validate the model. A discussion of the ®ndings from the analysis is also given, with particular focus relating to the accuracy, limitations and advantages oered by such a model. Results indicated that the slip model gives modestly accurate predictions, whilst the elastic modulus model showed very good correlation with experimental data. Ó 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: Shearing; Fabric model; Slip; Elastic modulus
Nomenclature k a, b, / c d l A, B, C, D a, b, c as dc E F Fc f f1 , f2 h
I eigen value solution eigen value fabric parameter functions shear angle maximum yarn de¯ection height coecient of friction model coecients positions along yarn de¯ection model slip distance acting along contact length eective yarn contact length elastic tensile modulus shear force along shear bar linkage reaction force at position c elemental shear force in an arbitrary direction elemental shear force in 1, 2 direction yarn crimp height
k L M, N, O, P m P1 , P2 R s1 , s2 t t1 , t2 V vab vbc w1 ; w2 X, Y X1 , X2 x1 ; x2 ; . . . ; x8
yarn cross-sectional area moment of inertia spring constant total specimen fabric length coecients of shear model solution internal moments pitch spacing in 1, 2 direction resultant shear frame force tow spacing in 1, 2 direction tensile force elemental tension force in 1, 2 direction yarn normal force yarn de¯ection along length ab Yarn de¯ection along length bc tow widths in 1, 2 direction coordinate axes arbitrary lengths along the X-direction fabric architectural parameters
1. Introduction
*
Corresponding author. E-mail address: [email protected] (M. Nguyen).
The forming behaviour of fabric preforms is of important consideration for the manufacture of composite components. The ability of the fabric material to
0263-8223/99/$ - see front matter Ó 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 0 ) 0 0 0 5 1 - 9
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conform to various shapes is highly dependent on the fabric architecture, which in turn will in¯uence the resultant mechanical properties of the composite part. During preform forming, two major deformation mechanisms are considered to be important; these being simple shear and shear-slip (refer to Fig. 1). Of these mechanisms in relation to woven fabrics, simple shear has been indicated to be the more dominant behaviour [1]. In undergoing fabric shearing, it was initially suggested by Mack and Taylor [2] that the reinforcement behaves as a pin-jointed mesh, whereby the tows rotate or trellis about the fabric cross-overs. By further assuming that the yarns are inextensible and incompressible, Potter [3], Robertson [4], Van West [5] and Long [6], employed it to map the fabric draping behaviour over shapes of varying pro®les. The outcome of these research eorts was to reinforce the concept that the ability of fabrics to shear is a particularly important forming characteristic. This characteristic enables the manufacture of 2-dimensional preforms into complex 3-dimensional shapes. Further work by McBride [7] showed that during fabric shearing, the spatial positioning of the fabric architecture will vary, such that local increases in the ®bre volume fraction will occur. Investigation conducted by Tam and Gutowski [8] with ®bre-aligned materials identi®ed in-plane and interply shear as important parameters in¯uencing the forming behaviour of aligned ®bre-reinforced composites. Results showed that when the material experiences high angles of shearing, wrinkling of the fabric occurred. Woven fabrics dier from aligned ®bre materials, as trellising or rotation of the yarn is the dominant deformation mechanism. Also the eect of inter-ply shear becomes an in¯uential deformation mechanism, when stacks of fabric plies are required to undergo forming simultaneously [9]. However, when high angles of material shear are induced into woven fabrics, they too exhibit similar behaviour to aligned ®bre materials; whereby wrinkling occurs due to a build-up of in-plane compressive forces. The characterization of the fabric shear properties has predominantly consisted of testing the materials
Fig. 1. Major mechanisms for fabric deformation.
within a shear rig ®xture which resembles a picture frame [10,11]. Results using a variety of glass and carbon fabrics have indicated that the shearing behaviour can be divided into two distinct regions. Before fabric lock-up, whereby low shearing loads are required; and after lock-up, which is characterized by a sudden increase in the load generally accompanied with material wrinkling. Fabric lock-up is described as when high angles of yarn rotation occur such that the fabric architecture becomes ÔjammedÕ, and this jamming eect results in a rapid increase in the shearing force. Further results from these tests have indicated that regardless of the weave type, the fabrics all exhibited similar trends in shear deformation. By considering a plain woven unit cell and assuming that yarn rotation about the region of overlap behaves as a pin-jointed mesh. Prodromou and Chen [12] provided a geometric model to predict the occurrence of fabric lock-up. Recent work by Wang [13], highlighted that this fabric lock-up angle is highly dependent on the type of shear testing method employed. By characterizing the fabric shear behaviour of a material with the bias-extension and simple shear rail test, results showed that there were noticeable dierences between the fabric locking angles obtained. Early attempts to model the plain-weave fabric architecture typically assumed the yarn cross-section was circular and various inferences were made about the amount of yarn crimp [14,15]. Presently it is evident that the yarns can be more aptly described as elliptical, and that variations in the elliptical cross-sectional pro®le exist along the length of the yarn [16]. Though the understanding of the fabric architecture has progressed, there exists a critical need to provide models which accurately describe the shear properties of fabric preforms. Research into modelling the shear properties is scarce and there currently only exist a handful of methods. Initial attempts by Grosberg and Park [17] idealized the yarns as beam structural elements, allowing the ability to predict both the initial and elastic shear modulus of the fabric. Two disadvantages are considered to be associated with this model. Firstly the presentation of the theoretical solution is in a form not particularly suitable for practical use, and secondly, the model does not consider undulation of the yarns. Modelling by Katabawa et al. [18±20] did incorporate yarn undulation, but assumed a linear function of the torsional moments required to induce fabric shearing. The various coecients used to characterize this linear function, could not be de®ned theoretically and hence were experimentally solved. Elaboration of this model by Ye [21], enabled the contact force at the yarn crossovers to be determined, though both the models by Grosberg and Park, and Katabawa et al. showed quite good correlation with experimental results. Validation consisted of comparisons with cloth, polyester and wool
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fabrics, which are not suitable reinforcement materials for advanced composites. More recent developments by Sinoimeri and Drean [22] used energy methods to study the fabric shear properties. The introduction of a rheological model allowed tow slippage to be accounted. Unfortunately this model suers from the use of arbitrarily coecients to predict the fabric shearing behaviour, and that there is limited reference to the fabric architectural parameters within the model itself. In this paper, a formulation for the shear properties of plain woven fabrics is presented. Theoretical solutions are generated from the model and are compared with experimental results obtained from a shear rig test ®xture. A discussion on the ®ndings is given, with particular focus on the accuracy, limitations and bene®ts oered by such a model.
A variety of carbon fabrics were tested, with their properties summarized in Table 1. For each material, a minimum of ®ve tests were conducted. The test materials also consisted of two carbon plain weave prepregs. These prepregs were comprised from T300 and T650 yarns and impregnated with Hexcel F593 and Hexcel F584 modi®ed epoxy resin, respectively. An important characteristic of fabric deformation is when the material experiences high shearing angles (c), resulting in the fabric architecture becoming ÔjammedÕ. This jamming is also commonly referred to as fabric locking, the eect of which increases the forces required to deform the fabric and also leads to wrinkling. During this experimental investigation, locking of the fabric is de®ned when wrinkling of the material within the picture frame was observed to occur.
2. Experimental
The fabric test size is 240 mm 240 mm, minus a 25 mm radius quadrant cut from each corner. This quadrant is required so that rotation of the hinges can occur freely. The test material is then placed into the ®xture via a clamping arrangement located along each bar linkage. Care is taken to ensure that the weft and warp tows are positioned parallel to the frame. The shear frame is loaded into an INSTRON 45 kN screw driven testing machine and the brace then removed. To minimize the viscosity behaviour of the prepregs, testing was carried out at a rate of 5 mm minÿ1 . Picture frame testing for a range of dry and impregnated fabrics has typically used shear rates of 10 mm minÿ1 . However, a reduced loading rate of one half this speed is not expected to signi®cantly alter the material shearing characteristics. This speculation is supported with work from Wang [23], who showed that minimal variation of the shear modulus was observed for test speeds ranging from 1 to 10 mm minÿ1 .
2.1. The picture frame experiment Experimental investigation of the shearing behaviour of dry and prepreg fabrics was carried out using a picture-frame apparatus. The aim of this experiment is to determine the relationship between shearing angle and load for dierent weave architectures (i.e., variable pattern and fabric parameters). An illustration of the shear frame apparatus which is essentially a square fourbar linkage is shown in Fig. 2. A removable brace was designed to allow transport and loading of the apparatus so as to minimize fabric shearing prior to testing. Experiments were carried out at ambient temperature, with force and displacement history recorded into a personal computer. This type of testing apparatus has been used successfully to characterize the shear properties of dry fabrics [10,12] and composite laminates [11].
2.2. Procedure
2.3. Test results
Fig. 2. Schematic of the picture frame, with removable brace attached.
Fig. 3 shows the shear frame picture rig deforming at 0°, 15°, 30° and 47° of shear, and the corresponding position of the rig deformation on the shear force/angle curve. White lines drawn 50 mm in from the edge of the bar linkages, show that there is a uniform state of shearing occurring within this highlighted region. The shear force/angle relationships for the fabrics listed in Table 1 are plotted in Fig. 4. A number of observations are obtained from these results. · PW (1K) fabric exhibited a steeper shear curve compared to the PW (3K) material. This is thought to occur because of the PW (1K) having a combination of a small tow size and compact weave structure. The result is to increase the frictional resistance acting at the fabric cross-overs. Similar results have been noticed
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Table 1 Speci®cations of fabrics used in experimentation Fabric type (code)
Construction (tows cmÿ1 )
Areal weight (g m2 )
PW (1K) PW (3K) 2/2 twill (3K) 2/2 twill (6K) PW-prepreg (T300) PW-prepreg (T650)
6:4 6:5 4:6 4:5 5:2 5:3 4:3 4:4 4:7 4:8 4:8 4:9
90 198 220 375 280 270
Nominal thickness (mm) 0.20 0.24 0.32 0.43 0.30 0.28
Measured locking angle (c, deg)
Calculated locking angle (c, deg)
46 50 72 64 38 34
40 44 ) ) 31 22
Fig. 3. Dierent stages of fabric shear deformation, tested at a nominal rate of 5 mm/min, corresponding to various positions of a PW (3K) shear deformation history at (a) 0°, (b) 15°, (c) 30°, (d) 47°.
Fig. 4. Fabric and prepreg shear force/angle relationships for the material types listed in Table 1.
M. Nguyen et al. / Composite Structures 47 (1999) 767±779
and various discussions relating to these observations have been given [12,24]. · Impregnation of fabric with resin as in the case of prepregs has the eect of increasing the materialÕs shear modulus compared to dry fabrics. Results further indicate that the PW-prepreg (T650) has a much higher shear modulus compared to that of the PWprepreg (T300). This variation between the shearing properties is considered to be due to the dierence in the resin properties. · Despite the weave type, both dry fabrics and the prepregs exhibit similar shear deformation behaviour, which can be described as parabolic in nature. · As twill weave fabrics require low deformation loads, these materials experience fabric lock-up at much higher angles of shear. Consequently they are considered to be more ÔconformableÕ than plain weaves, because of their ability to be contoured to complex shapes before the occurrence of wrinkling. 3. The shear models Grosberg et al. [25] demonstrated that the trends for fabric shearing typically consist of three stages; the initial region, where shear is due to the ¯exural deformation of the tows; the mid-region, where shear resistance becomes dominated by a bearing load type contact; and the last region, where there is a rapid load increase due to ÔjammingÕ of the fabric structure. As the fabric undergoes these deformation phases, the material architecture must change so that the dierent phases can be accommodated. Henceforth, it is necessary to both understand and describe the fabric architecture if a model is to be developed. 3.1. De®nition of fabric architecture To model the shearing property of the plain weave fabrics, an idealized repeating, geometrically symmetrical cell section is depicted in Fig. 5. Though there are dierent ways to how the cell can be represented [16], modelling of fabric shear properties has generally consisted of the following pattern. In determining the fabric deformation characteristics using this unit cell structure, undulation of the yarns is not considered. The implications of this assumption on the accuracy of the theoretical results are discussed further in Section 5. By specifying a set of geometric parameters as shown in Fig. 5 with the material elastic properties, mathematical expressions for the initial slip and elastic modulus regions can be derived. Fig. 6(a) shows a representation of the square shear rig deforming from a state of c 0, through to a shear angle of c. As the fabric material is sheared, the forces acting on the bar linkages can be discretized into shear and tensile
771
Fig. 5. Illustration showing the full repeating and geometrically symmetrical unit cell of the plain weave fabric structure.
force components acting perpendicular and parallel to the fabric tow. This is shown Fig. 6(b). At low angles of shearing (0° ! 2°), if the theory of small angle approximations is utilized [26], then the acting tensile force component will have negligible magnitude and hence can be ignored. Consequently for this deformation range, which is referred to as the initial slip region, the fabric material is considered to be in a state of pure shear (Fig. 6(c)). 3.2. The initial slip region This region of deformation has been extensively dealt with by Grosberg and Park [17], and in obtaining the solution a number of key assumptions were presented, which are as follows: 1. As the initial region of slip exists for only small angles of shear, the model can be idealized to be in a state of pure shear, as shown in Fig. 6(c) where the elemental shear forces can be described as f1 F P1 =L; f2 F P2 =L: 2. The yarn contact lengths (dc ) acting at the crossovers, are the same lengths in both directions. 3. Yarns are inextensible during the shearing process. 4. Deformation of the yarns is considered to be only due to bending. It was necessary to understand the process of how this initial slip model was determined, as these principles provided the fundamentals in the development and improvement of the more important and dominant elastic modulus model. During reformulation of the solution presented by Grosberg and Park, two major modi®cations were incorporated, which were the re-de®nition of the normal force (V) and revision of the solution itself. 3.2.1. The normal force (V) In woven fabrics, the undulation of the yarns will exert pressure at the cross-overs, which in turn induces frictional resistance to shearing. The de®nition of this normal force was initially provided by experimental and theoretical work by Pierce [14]. By considering the
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Fig. 6. (a) Fabric shearing under applied load (R). (b) As the applied load increases, the shearing forces acting along the bar linkages (F) can be resolved into components of tension (t) and shear (f) acting parallel and perpendicular to the tows. (c) For the unique case of the initial slip region as the shear angles are assumed to be small, the fabric can be idealised to be in a state of pure shear.
problem from a structural point of view, Katabawa et al. [20] idealized the yarns as beam elements. These beams whose ends are attached to ®xed wall supports, enable the prediction of the normal contact force of the yarn using material and geometric parameters from the fabric architecture. However, if the beams are rigidly attached to the wall and then de¯ection at the centre occurs by application of a normally applied load, this condition would invalidate a key assumption whereby the yarns are inextensible. To overcome this inconsistency, the yarn undulation behaviour is suggested to be more accurately modeled as a beam, having a pin and roller support, as shown in Fig. 7. Hence the normal force can be approximated as V
48 E I h ; 3 P1;2
1
where P1 or P2 refer to the pitch length in the warp or weft directions, respectively. 3.2.2. Revised solution Using the problem statement, boundary conditions and theory as described by Grosberg and Park, a reformulation of the solution was performed. Comparisons of Grosberg and ParkÕs and the revised solutions indicate that though both are dimensionally correct, there exists a small discrepancy which has a considerable in¯uence on the overall results, this dierence being the
Fig. 7. Idealised bending of a ®bre yarn.
presence of cubic power terms present in the solution provided by Grosberg and Park, compared to square power terms in the revised equation. The extent and impact of these dierences in describing the shearing properties of woven fabrics will be investigated and discussed in a future article. The revised fabric shearing solution, incorporating the normal force function as described in Eq. (1) for the initial slippage region is as follows: " 2 F as c P2 L ÿ dc 1 ÿ 8EI L dc # 2 as 2 P1 L ÿ dc 1 ÿ dc as c : 0 ! 1; dc 2 l V 24dc dacs 3 ÿ 2 dacs i h ii F h h 3 as as 1 P L ÿ d 1 ÿ L ÿ d 1 ÿ P 2 c 1 c dc dc 4L c:0!
as 1: dc
dc is determined from methods as described from Grosberg and Park [17]. 3.3. The elastic modulus region Grosberg et al. [25] elaborated on the shear modelling work by Grosberg and Park [17] by next considering the elastic shear modulus region. Unfortunately the solution provided is not presented in a manner which makes comprehension easy, particularly if it is desirable to understand how fabric parameters in¯uence the material shear characteristics. The complexity of the equations is further exacerbated by the use of Fourier series to
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express the de¯ected yarn modal shape. To overcome these drawbacks, a model is proposed which uses the same assumptions, boundary conditions and problem statement as initially set up by Grosberg et al., but varies by its approach in determining and de®ning the solution for the elastic shear modulus of plain woven fabrics. 3.3.1. Analysis The analysis still maintains the geometrically symmetrical unit cell, however for this region, the mechanism in which fabric shearing occurs is dierent. As high angles of yarn rotation occur, the contact area between adjacent yarns near the fabric cross-overs will increase. This contact resistance to fabric shearing is now assumed to be more dominant than the frictional restraints acting about the yarn cross-overs. The idealization of this contact resistance to fabric shearing, takes the form of a spring bearing load mechanism. It is further assumed that this mechanism acts through a point located at the end of the cross-over contact length, dc . This concept is illustrated in Fig. 8. Tow rotation for this model deformation state is able to occur via a pin-joint acting about the tow cross-overs. Grosberg et al. argued that this is plausible as there is now constant frictional resistance about the tow overlap region after complete slippage has taken place. Furthermore, as the majority of the resistance to fabric
773
shearing will be dominated by the spring contact force, the contribution of friction in resisting shear is expected to be small to minimal. As the fabric continues to be sheared, the shear forces acting on the bar linkages can be separated into tension and shear components acting parallel and perpendicular to the tows, respectively (Fig. 6(b)). By considering that these force components act on one-half of a yarn (as the unit cell is symmetrical), the problem can be described by a free body diaphragm as seen in Fig. 8(b) and Fig. 9. 3.3.2. Model expression The amount of vertical yarn-de¯ection along the length X1 and X2 can be described as vab A cos k X1 B sin k X1 ÿ f2 =t1 X 1 d X1 : 0 ! P1 ÿ dc =2;
4
vbc C cos k X2 D sin k X2 f2 P1 2d ÿ1 X2 t1 dc dc X2 : 0 ! dc =2; 5 q where k Et1 I , and the elemental shear and tension forces acting on a single tow are described as
Fig. 8. (a) Point bearing loads acting at the edge of the cross-over contact length on a unit cell. (b) Representation in the form of a beam de¯ection diagram, showing a single yarn in a state of shear and tension with support springs.
Fig. 9. Free body diagram of yarn de¯ection under tension and shear.
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M. Nguyen et al. / Composite Structures 47 (1999) 767±779
f2
F P2 sin c=2; L
t1
F P2 cos c=2: L
d : 11 P2 =2 Hence the total shear angle of the warp and weft yarns is described as
6
c=2weft
3.3.3. Boundary conditions A, B, C, D in Eqs. (4) and (5) are coecients obtained from the boundary conditions. These boundary conditions are 1. at X2 0; vbc 0; 2. at 2t1 d f1 P1 ÿ ; X2 dc =2; vbc k dc k dc 3. at 2t1 d f1 P1 ÿ ; X1 P1 ÿ dc =2; vab k dc k dc 4. ÿo vab 1 o vbc dc P1 ÿ dc : oX1 oX2 2 2
c c=2warp c=2weft :
12
Results for the expression of the total shear force (F) along a bar linkage as a function of the shear angle (c) are given in Eq. (9). To simplify this solution, it is allowed that P1 P2 due to yarn symmetry and that the co-tangent trigonomic function is re-expressed using polynomial approximation terms. The ®nal solution for the shear force as a function of two unknown variables, shear angle (c) and the spring constant (k) is given in the form of a cubic equation 2
M k; c F 3 N k; c F O k; c F P k; c 0; c : 0 ! 90° ÿ arcsin 2 W =P
13
7
where the coecients of the solution M, N, O, P are in terms of the fabric material and geometric parameters. Note that the solution is theoretically valid up to the state when fabric locking occurs.
If the boundary conditions are used to obtain the coef®cient A, then results will show that the solution of A is non-zero. However the expression of A is also a function of d and hence for Eq. (7) to be satis®ed, d must equal the expression given in Eq. (8).
15x2 15x2 15x5 15x7 5 ÿ 5 ÿ 5 ; 2a5 2a 2b 2b 15x1 15x2 3x2 3x2 15x4 3x5 15x6 3x7 N ÿ 5 ÿ 5 3 3 5 ÿ 3 5 ÿ 3 ; 2a a 2a a 2b 2b b b
The maximum tow de¯ection (d) will occur when X1 0. This will result in Eq (4) being d A d ) A 0:
d
k
h
ÿf2 P1 k dc
f22t1dc
P1 dc
ÿ1
M
i
h i h i cot k 2dc k ÿfk 2 dcP1 f2t2 P1 ÿ dc cot k P12ÿdc ÿ ft12 Pd1c ÿ 2 : k 1 ÿ k2td1c cot k 2dc cot k P12ÿdc ÿ d2c
3.3.4. Solution The maximum yarn shear angle as a result of shearing and tensile deformation along the warp direction can be approximated as
8
3x1 3x2 x2 x2 3x4 x5 3x6 x7 ÿ 3 ÿ x3 3 ÿ 3 ÿ x8 ; 3 a a b b a a b b x1 x1 x4 x6 P ÿ ÿ a b b a Oÿ
i h i h i9 8 h k P1 ÿdc ÿP1 P1 ÿP1 dc k dc 1 ÿ t11 Pd1c ÿ 2 = 2 c 2 P2 c < k k dc 2t1 dc ÿ 1 cot 2 k k dc 2t1 P1 ÿ dc cot sin ÿ ; F L P1 2: 1 k 1 ÿ k2tdc cot k:d2 c cot k: P12ÿdc ÿ dc2 h i h i h i9 8 k P2 ÿdc ÿP2 P2 ÿP2 P2 k dc dc 1 1 < = k ÿ 1 cot k P ÿ d cot ÿ ÿ 2 2 c 2t2 2t2 t2 2 k dc dc 2 k dc dc 2 P1 c sin ; ; L P2 2: k 1 ÿ 2t cot k:dc cot k P2 ÿdc ÿ 2 k1 dc
c=2warp
d : P1 =2
2
2
dc
where 10
As the unit cell is symmetrical, the tow shear angle in the weft direction can also be approximated as
x1 ÿc tan c x2
c/ ; cos c
P2 / ; L k dc x6 ÿ
x5
4P2 / ; L k dc
2 P1 ÿ dc / ; P1 sin c
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M. Nguyen et al. / Composite Structures 47 (1999) 767±779
2c ; dc cos c 4P2 :/ ; x4 ÿ L2 k dc x3
2dc P1 ÿ 1 /; P1 sin c dc 4 P1 x8 ÿ2 P1 sin c dc
775
Although it is well known that the yarn cross-sectional area varies along the length of the yarn, for simplicity of the analysis, these cross-sectional pro®les are generalized by having a constant elliptical shape. This allows the area moment of inertia (I) to be de®ned based on the yarn geometry. The empirical spring constant values were selected so that the theoretical solution would correlate closely with the experimental trends.
x7
and ®nally where r P2 sin c dc ; a 2 LEI r P2 sin c P1 ÿ dc ; b 2 LEI r P2 sin c : / LEI
4.2. The initial slip model
By de®ning an appropriate spring constant (k) with the fabric material and geometric properties, the shear force (F) can be determined from Eq. (13) for a given value of the shearing angle (c). One drawback of the model from a theoretical aspect is that it is unable to accurately predict shear forces at regions of very low shear. This is because the model assumes that there are always components of shear and tension acting on the fabric tows for the solution to be valid. This does not re¯ect the physical condition in the unique case of when c 0, when the shear force on the frame will be zero. To cater to this discrepancy between experimental and theoretical solution when c 0, the theoretical solution is slightly modi®ed by using a normalized factor of F jc0 . This factor is determined by assigning a spring constant, and evaluating the shear force from Eq. (13) for c 0. The modi®cation of the solution in terms of this normalized factor is now expressed as
Fig. 10(a) and (b) show the corresponding shear force and shear angle relationship for plain weave fabrics of 1K and 3K tow sizes, respectively, for the initial slip region. This region is typically characterized by the initially high rise in the shear modulus followed by levelling of the curve. Results indicate that the model provides modest correlation where initial slippage occurs and the magnitudes of the shear force within this range. Using linear interpolation, Table 3 shows the average predicted shear modulus in comparison with the experimental values for this fabric deformation zone. These results indicate that there is a 44% and 35% dierence between the average experimental and model shear moduli for PW (1K) and PW (3K), respectively. For prepreg fabrics, no calculations for the initial region of fabric slip were performed. It is evident from Fig. 10(c) that prepregs do not exhibit the same initial slip trends as compared to dry fabrics. This dierence is attributed to prepregs having the introduction of resin into the fabric architecture. This resin in¯uence on the material shear properties is discussed later in Section 5.
Fmod F jc:0!p=2 ÿ F jc0 :
4.3. The elastic modulus model
14
Henceforth the resultant force required to deform the trellis frame (R) can be determined as R 2 Fmod cos 45 ÿ c=2:
15
4. Modelling results To investigate the eectiveness of this model, the analysis as discussed in Section 3 was used to model the initial slip region and elastic modulus behaviour, of a range of plain weave dry fabrics and prepregs as detailed in Table 1. 4.1. Material property input The four plain weave fabric con®gurations considered and their corresponding geometric and material properties required for the model validation are given in Table 2. Fabric geometric properties were obtained from observations using a stereo microscopic.
Using the simpli®ed solution as provided by Eq. (13) and the inputs as given by Table 2, the elastic model trends for the dry fabrics and prepregs are compared with experimental results. These comparisons are shown in Figs. 11 and 12. They show that there is generally good agreement with the experimental trends from zero shear, up to the angle of fabric locking. For the shearing range of 0° to approximately 30°, the model underestimates the preform shear modulus, with variations from the experimental results ranging from 2% to 14%. The larger errors within the model generally arise in predicting regions of low shear and are considered to be due to the in¯uence of the normalizing factor. As the shear angle increases, the in¯uence of this normalizing factor decays and hence accuracy is improved. Results also indicate there is a phase where the model goes from underestimating to overestimating the material shear properties. For prepregs, this transitional phase occurs after fabric locking, whereas for fabrics this happens before locking. Why the model behaves like
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M. Nguyen et al. / Composite Structures 47 (1999) 767±779
Table 2 Main parameters required for model validation Parameter
Units
PW (1K)
PW (3K)
PW-prepreg (T300)
PW-prepreg (T650)
P1 P2 W1 W1 s1 s2 L Yarn thickness E I dc k
(mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (GPa) (m4 ) (mm) (N/m)
3.15 3.04 1.20 1.16 0.41 0.40 240 0.08 210 3.1 ´ 10ÿ16 0.48 13500
4.26 4.32 1.51 1.46 0.61 0.70 240 0.11 220 1.17 ´ 10ÿ16 0.52 5000
4.22 4.30 1.80 1.80 0.41 0.45 240 0.15 220 2.9 ´ 10ÿ16 0.47 35000
4.20 4.15 1.95 1.90 0.25 0.18 240 0.14 230 2.61 ´ 10ÿ16 0.51 65000
Fig. 10. Shear force/angle relationship at the initial slip region for (a) PW (1K) with experimental and model trends, (b) PW (3K) with experimental and model trends and (c) experimental trends for both prepregs.
Table 3 Predicted and experimental shear modulii and angles of shear for the initial fabric slip region Shear modulus (N/deg) PW (1K) PW (3K)
Slip region (deg)
Experiment
Model
Experiment
Model
2.5 1.4
3.6 1.9
0 ! 0:40 0 ! 0:35
0 ! 0:18 0 ! 0:25
this is not fully understood, however overall, the accuracy of the solution is still maintained regardless of where the model transitional phase is situated. As stated in Section 3.3.3, the elastic modulus model is valid up to where fabric locking occurs. Analysis of the modelÕs accuracy beyond the lock angle, shows that there is still reasonably good correlation with the experimental results. This is particularly evident for the prepreg materials and the PW (3K) fabric. These results highlight the possible application of the elastic
modulus model to predict shear properties beyond fabric lock-up. 5. Discussion of results The picture-frame experiment has been successfully used to examine the shearing behaviour of carbon plain weave dry fabrics and prepreg materials at ambient temperature. The ®ndings are summarized in Fig. 4,
M. Nguyen et al. / Composite Structures 47 (1999) 767±779
777
Fig. 11. Comparison of theoretical and experimental results for (a) PW (1K) with k 13500 N/m and (b) PW (3K) with k 5000 N/m.
Fig. 12. Comparison of theoretical and experimental results for (a) PW-prepreg (T650) k 65000 N/m and (b) PW-prepreg (T300) with k 35000 N/m.
which shows that the shear modulus of the materials is lowest for twill weave fabrics and highest for prepregs. Because of the ability of twill weaves to undergo high angles of shear, they are better suited for forming applications, as wrinkles are induced at a later stage during the manufacturing process. Previous works on the characterization of the shear properties for both dry and impregnated fabrics of varying architectures have exhibited similar results to that presented in this paper [10,12]. For instance, experimental values of the fabric locking angles, agreed well with work conducted by Breuer [10], who reported that the critical fabric shear angle ranged around 40° for carbon plain weaves and 60° for carbon twill weaves.
Though Breuer does not describe how the lock angles were de®ned during testing, results from this experimental work bear similarities to results as presented by Breuer. It is hence assumed that a similar criterion, whereby locking is de®ned as when wrinkles occur during shear testing, was probably used. In attempting to theoretically predict the locking angle, Prodromou and Chen [12] provided a geometric model based on the plain weave unit cell to de®ne a point after which fabric lock-up occurs. This model is described as h arcsin
W ; P =2
16
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M. Nguyen et al. / Composite Structures 47 (1999) 767±779
where h is the fabric lock angle, W the thickness of the yarn and P the pitch of the unit cell as described in Fig. 5. Estimations of the fabric locking angle using this formulation showed good correlation with experimental results (Table 1). A limitation in the use of beam theory to predict the shear response of fabric preforms is the inability of the model to cater for yarn crimping. Katabawa et al. [20] showed that the in¯uence of the yarn crimp angle (though generally small), when combined with shear will cause fabric deformation as a result of the bending and torsional moments created. The eect of not considering yarn torsion for the initial slip model, may be one plausible explanation as to why there exist large discrepancies between the theoretical and experimental results. However this does not exclude practical reasons for such dierences, which could include incorrect material handling or slight misalignment of the fabric during loading within the picture-frame shear rig. Having considered the in¯uence of yarn crimping and reasons as to why the theoretical and experimental results dier for the initial slip region. It is worth noting that the angles and shear forces associated with this region, are of extremely small magnitudes when compared to the total range of the fabric deformation. Hence, it is expected that there may be practical limitations with the utilization of such a model. Nevertheless this initial slip model is of signi®cant importance, as it provides important fundamentals on which the elastic modulus model is based. Results from the elastic modulus model show that there is good correlation with experimental data with an acceptable level in the margin of error. The accuracy of the correlation within this fabric deformation range also validates the assumption that the eects of yarn crimping can be considered to be negligible. The signi®cance of this deformation region compared to the initial slip region is apparent, as it represents a large proportion of the preform shear properties up till fabric lock-up. Generally, the shear properties of the preform after fabric lock-up has occurred are not required, as wrinkles within the material will already be present. It was noticed that there were diculties in determining the eective contact length (dc ) used in both the initial slip and elastic modulus models, when applying the methodology prescribed by Grosberg and Park [17]. During the evaluation of this length, it was realized that the models are highly sensitive to values of yarn thickness. Yarn thickness is an important parameter, because of its relationship to the yarn cross-sectional area moment of inertia via a cubic power term. Consequently small variations in the measured tow dimensions can be hugely exacerbated, which in turn will directly in¯uence estimations of not only the contact length, but the solution itself.
The use of a dry fabric model to predict the shear behaviour of visco-elastic prepregs, suggests that the deformation mechanisms acting within prepregs are similar to that of dry fabrics. The introduction of resin into the fabric structure is considered to have two eects on the mechanisms of shear deformation. Firstly, it will introduce greater frictional constraints acting at the yarn cross-overs, and secondly, the elastic stiness of the dry fabric yarns will be increased. The overall eect is to require higher shearing forces for the same degree of fabric shear. Consequently as shown in Fig. 10(c), prepregs do not exhibit the initial slip trends and so no slip predictions were performed. Rather the entire prepreg deformation region up to fabric locking is described by the elastic modulus model. Currently the model relies on de®ning the spring constant (k), so that model generation is as accurate as possible. As this value is presently empirical, further modelling is required so that a theoretical description can be obtained. In comparing the current model with existing solutions, one common theme is that all have incorporated various forms of empirical and semi-empirical coecients. The uniqueness of the solution presented is that it not only combines all the unknown variables into a single entity (k), but this factor also represents the physical mechanics required for fabric shear deformation. Furthermore, the breakdown of the solution into a cubic function, whose coecients form parameters relating to the fabric geometric and material properties, enables the application of the solution to be relatively straightforward. 6. Conclusions and summary From the current work the following conclusions were obtained: · The picture frame test experiment is able to adequately characterize the shear deformation behaviour of both dry fabrics and prepregs. · The region in which initial fabric slip occurs contains very low magnitudes of shear force and shear angle in comparison with the entire range of fabric deformation. · Overestimation of the shear modulus within the initial slip region is thought to occur because the model ignores yarn undulation. This eect would need to be incorporated to improve the accuracy of the solution. · Beam bending theory is a satisfactory analytical method which can be employed to predict the elastic modulus of prepregs and dry fabrics. Due to the accuracy of the solutions obtained, this validates the assumption that the in¯uence of yarn crimping can be ignored during modelling of the elastic modulus.
M. Nguyen et al. / Composite Structures 47 (1999) 767±779
· The de®nition of the spring constant (k) is an important factor in determining the accuracy of the solution. Currently this is an arbitrary value in which further research needs to be undertaken to characterize it from a theoretical aspect. Finally, the present formulation easily models variation of the plain weave fabric architecture. This makes it a useful tool to understand and quantify how changes in the ®bre architecture during manufacturing in¯uence fabric shear properties. References [1] Long AC, Rudd CD, Blagdon M, Smith P. Characterizing the processing and performance of aligned reinforcements during preform manufacture. Composites: Part A 1996;27A:253±547. [2] Mack C, Taylor HM. The ®tting of woven cloth to surfaces. J Text Inst 1956;47:477±89. [3] Potter KD. The in¯uence of accurate stretch data for reinforcements on the production of complex structural mouldings. Composite 1979;11:161±7. [4] Robertson RE, Hsiue ES, Yeh GSY. Continuous ®ber rearrangements during the moulding of ®ber composites. Polym Compos 1984;53:191±7. [5] Van West BP, Pipes RB, Keefe M. A simulation of the draping of bidirectional fabrics over arbitrary surfaces. J Text Inst 1990;4:448±60. [6] Long AC, Rudd CD, Blagdon M, Smith P. Characterising the processing and performance of aligned reinforcements during preform manufacturing. Composites: Part A 1996;27A:247±53. [7] McBride TM, Chen J. Unit-cell geometry in plain-weave fabrics during shear deformations. Compos Sci Technol 1997;57:345±51. [8] Tam AS, Gutowski TG. The kinematics for forming ideal aligned ®bre composites into complex shapes. Compos Manuf 1990;1(4):219±28. [9] Nguyen M. Investigations of the single diaphragm forming technique. Cooperative Research Centre for Advanced Composite Structures (CRC-ACS) Draft report, 1999. [10] Breuer U, Neitzel M, Ketzer V. Reinicke. Deep drawing of fabricreinforced thermoplastics: wrinkle formation and their reduction. Polym Compos 1996;17(4):643±7.
779
Br [11] McGuinness GB, O adaigh CM. Characterization of thermoplastics composite melts in rhombus-shear: the picture frame experiment. Composites: Part A 1998;19A:115±32. [12] Prodromou AG, Chen J. On the relationship between shear angle and wrinkling of textile composite preforms. Composites: Part A 1997;28A:491±503. [13] Wang J, Page JR, Paton R. Experimental investigation of the draping properties of reinforced fabrics. Compos Sci Technol 1998;58:269±80. [14] Pierce FT. The geometry of cloth structure. J Text Inst 1937;45± 95. [15] Olofsson B. A general model of fabric as a geometric-mechanical structure. J Text Inst 1964:541±537. [16] Kuhn JL, Charalambides PG. Modelling of plain weave fabric composite geometry. J Compos Mater 1999;33(3):188±220. [17] Grosberg P, Park BJ. The mechanical properties of woven fabrics, Part V: The initial modulus and the frictional restraint in shearing of plain weave fabrics. Text Res J 1966:420±431. [18] Kawabata S, Niwa M, Kawai H. The ®nite deformation theory of plain-weave fabrics Part I: the biaxial-deformation theory. J Text Inst 1973;64(2):21±46. [19] Kawabata S, Niwa M, Kawai H. The ®nite deformation theory of plain-weave fabrics Part II: the uniaxial-deformation theory. J Text Inst 1973;64(2):47±61. [20] Kawabata S, Niwa M, Kawai H. The ®nite deformation theory of plain-weave fabrics Part III: the shear-deformation theory. J Text Inst 1973;64(2):62±85. [21] Ye L, Daghyani HR. Characteristics of woven ®bre fabric reinforced composites in forming process. Composites: Part A 1997;28A:869±74. [22] Sinoimeri A, Drean JY. A study of the mechanical behaviour of the plain-weave structure by using energy methods: fabric shear. J Text Inst 1996;87(1):120±9. [23] Wang J, Paton R, Page JR. Forming properties of thermoset fabric prepregs at room temperature and elevated temperature. Cooperative Research Centre for Advanced Composites (CRCACS) technical memorandum, 1998. [24] Golden K, Rogers TG, Spencer AJM. Forming kinematics of continuous ®bre reinforced laminates. Compos Manuf 1991;2(3):167±278. [25] Grosberg P, Leaf AV, Park BJ. The mechanical properties of woven fabrics, Part VI: the elastic shear modulus of plain-weave fabrics. Text Res J 1968:1085±1100. [26] James Stewart. Calculus. 2nd ed. Brooks/Cole, 1991.