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Experimental studies and analysis of the draping of woven fabrics
Composites: Part A 31 (2000) 1409–1420 www.elsevier.com/locate/compositesa
Experimental studies and analysis of the draping of woven fabrics U. Mohammed, C. Lekakou*, M.G. Bader School of Mechanical and Materials Engineering, University of Surrey, Guildford, Surrey GU2 5XH, UK Received 27 August 1999; accepted 13 April 2000
Abstract This study investigates and compares the draping and forming of four types of woven fabrics, namely a loose plain weave (basket weave), a tight plain weave, a satin and a twill weave. The fabrics were draped over a hat mould consisting of a hemispherical dome surrounded by a flat base. The draping of each fabric was examined in terms of wrinkle formation, boundary profile of the draped fabric, distribution of fibre orientation and local shear angles. A theoretical analysis of the experimental results involved the calculation of the distributions of the fibre volume fraction and mechanical properties, in terms of components of the reduced stiffness matrix, from the experimental data of local shear angles. 䉷 2000 Published by Elsevier Science Ltd. Keywords: Draping; A. Fabrics
1. Introduction The draping of fabrics onto surfaces of varying geometry has received much attention over recent years due to the upsurge in the use of fabrics in the manufacturing of three-dimensional composite components. Common problems in the fabricating process of three-dimensional shells of complex curvature include unwanted features such as wrinkling, kinks or tears, the presence of which depends on the geometry of the mould surface and the type of fabric. Furthermore, during draping and forming of a fabric there are local variations of fibre orientation, fibre volume fraction and possibly fabric thickness which are going to affect processability and the mechanical properties of the final product. Finite element numerical techniques have been applied for the simulation of the draping of various types of reinforcements following either a solid mechanics approach [1– 5] or a viscous approach [6]. Regarding this type of computer simulation, there is clearly a need for experimental data of materials properties to be used as input in the finite element code, as well as other data of various draping parameters to be used for validation of the theoretical predictions. Yu et al. [7] compared the formability of different types of woven fabrics in terms of shear rigidity, tensile rigidity * Corresponding author. Tel.: ⫹44-1483-879622; fax: ⫹44-1483876291. E-mail address: [email protected] (C. Lekakou). 1359-835X/00/$ - see front matter 䉷 2000 Published by Elsevier Science Ltd. PII: S1359-835 X( 00)00 080-4
and bending rigidity. Mohammed et al. [8] carried out shear tests for the four fabrics included in the present study and presented a detailed theoretical analysis of their shear behaviour, following a solid mechanics approach. Regarding the measurement of various structural parameters of the draped fabrics, a common method, originating from the metal forming processes (for example [9]) includes the inscription of a grid on the textile before draping and the subsequent study of the grid distortion in the draped textile [10] to predict strain distribution. Laroche and Khanh [11] and Khanh and Liu [12] presented a comparison of their experimental data from the forming of a fabric prepreg over a rounded top cone with numerical predictions from a “fishnet” model. Their predictions of local yarn angle were very close to the experimental data for a plain weave and an 8-harness satin, and also the experimental results from the draping of the two prepregs with the different type of weaves were similarly very close. No details about the experimental procedure of prepreg forming were presented in their paper. Bickerton et al. [13] presented measurements of local yarn angle of a stitched reinforcement draped again over a rounded top cone. The present study considers the draping and forming of dry fabrics in order to simulate the preforming stage in processes such as resin transfer moulding (RTM). The aim of the study is to provide a comprehensive set of experimental data regarding formability of different types of woven fabrics, as well as draping parameters required for the validation of computational and other theoretical simulations of the draping of fabrics. Four different types of
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Table 1 Specifications [14] of the woven fabrics used in this study Weave type
LPW
TPW
TWILL
5HSW
Coding Thickness Areal density (kg/m 2) Ends/10 mm Picks/10 mm
Basket 0.58 0.529 1.1 1.2
Y0212 0.48 0.546 6.7 6.3
Y0185 0.28 0.331 11.8 11.8
Y0227 0.23 0.297 22.4 21.3
woven fabrics were examined: a loose plain weave (basket weave); a tight plain weave; a twill and a 5-harness satin weave. Wrinkling behaviour was investigated in both draping and shear tests. Detailed maps of the measured grid angles after fabric draping were assembled, assuming to represent maps of local angles between crossing fibre yarns. A theoretical analysis of the experimental results was carried out, according to which the local fibre volume fraction and local values of components of the reduced stiffness matrix of a formed fabric laminate were calculated from the experimental data of local shear angle. These data have the potential to be used in the prediction of the permeability of the deformed fabric and also the mechanical properties of the final composite product. 2. Materials Currently, most of the pure and hybrid woven fabrics
often used in textile composites are simple two-dimensional weaves, such as plain, twill and satin weaves, which are identified by the repeating pattern of the interlaced regions in the warp and weft directions. In this study, four E-glass woven fabrics were tested: a loose plain weave (LPW), a tight plain weave (TPW), a twill weave (TWILL), and a 5harness satin weave (5HSW). The fabrics were supplied by Fothergill Engineered Fabrics Ltd and their specifications are presented in Table 1 [14]. The terms “ends” and “picks” refer to the number of warp yarns and weft yarns, respectively. Comparing the two plain weaves, the loose plain weave had an average yarn width of 5 mm and an average gap size between yarns of 2.5 mm whereas the tight plain weave had an average yarn width of 1.3 mm and an average gap size of 0.25 mm. More details about the microstructure of the four fabrics both in unsheared and sheared states are given in another study by Mohammed et al. [8]. 3. Draping of fabrics: equipment and experimental procedures A mould was constructed for draping experiments onto a double curvature geometry, combining a curved and a flat part. The selected mould shape comprised a hemispherical dome surrounded by a flat rim. The mould consisted of two parts: the upper part was made of transparent poly(methyl methacrylate) (PMMA—“perspex”). A rectangular shaped solid block was used of 500 × 500 × 137 mm 3, in which a
Fig. 1. (a) Photograph of the hemispherical mould as used for the draping of woven fabrics. (b) Mould geometry and dimensions.
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Fig. 2. Diagram showing features of the draping process for one fabric quadrant.
hemispherical cavity of 100 mm radius was machined in the central region (see Fig. 1a and b). The upper half of mould was chosen to be transparent so that the draped fabric could be viewed. The lower mould was made of aluminium and consisted of a square shaped flat surface of dimensions 500 × 500 mm 2 with a solid aluminium dome of 97 mm radius at the centre.
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The following procedure was used in the draping experiments. A square piece of fabric of 360 × 360 mm 2 was cut from each of the fabrics listed in Table 1. Then a square grid of 20 × 20 divisions was inscribed onto the cut sample using a thick ink pen, where the ratio of the ink thickness to yarn width was 0.3, 0.8, 0.8 and 2 for the LPW, TPW, TWILL and 5HSW, respectively. The grid was subdivided into four quadrants for easy identification. Four quadrants were also marked to divide the hemispherical mould surface and the surrounding flat rim on the male part. The whole idea was to fix the centre of the fabric grid on the apex of the mould dome and to fit the internal boundaries (lines of symmetry) of the quadrants of the fabric exactly to the corresponding cross-lines of symmetry on the male mould. The reason for this was to formalise the draping procedure of a layer of fabric onto the male mould and achieve satisfactory and accurate results. The next stage of the draping experiment was the shaping process. The shaping process was carried out in two stages in this study with minimal tensioning of the fabric throughout. The first stage, which was the most important and difficult, was to place the fabric on the hemispherical dome and the flat rim of the male (lower) mould, so that each quadrant
Fig. 3. Photographs of fabrics draped over a hemispherical dome surrounded by a flat base: (a) loose plain weave (basket weave); (b) tight plain weave; (c) twill weave; (d) 5-harness satin weave.
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Table 2 Angles of the deformed grid elements for wrinkles in the draped fabrics and locking element angles in picture-frame shear tests [8] Weave
a min (⬚)
a max (⬚)
a locking (⬚)
LPW TPW TWILL 5HSW
No wrinkles 43–44 37 32
No wrinkles 49–56 48 34
29 80 65 64
of the grid sat exactly over the corresponding quadrant of the mould and the cross-lines of symmetry of fabric fitted over the corresponding lines of symmetry printed on the male mould. To facilitate this, a piece of double-sided Sellotape娃 was placed on the apex of the hemisphere and also at locations on each of the internal boundary lines of each quadrant of the mould (on the cross-lines of symmetry), both on the hemispherical part and the flat surface (see Fig. 2). Then, the central position on the grid was marked and the cloth was placed so that this point coincided with the apex. For each quadrant of the grid and the mould, the internal quadrant boundaries of the fabric were fitted manually over the corresponding lines of the mould. Using the Sellotape娃 pieces, the fitted cloth was constrained securely at this position prior to closing the mould. The second stage involved the forming of the fabric onto the mould surface. This was achieved by lowering the upper mould slowly and gradually until the mould was closed and housed the draped fabric.
4. Draping of fabrics: experimental results The ability of fabrics to form over three-dimensional shapes without having to use undue force defines the drapeability of fabrics. One method of comparing different fabrics is based on draping experiments. The shapes of the draped fabrics and the deformation of the grid, inscribed on each fabric prior to deformation, are seen in Fig. 3a–d for the loose plain (basket) weave, the tight plain weave, the twill weave and the satin weave, respectively. Each of the four draped fabrics deformed by angular rotation of the warp and weft yarns at their crossovers to produce a star shaped geometric feature. This deformation mode was observed on the grid inscribed on the fabrics prior to draping. Draping did not destroy the yarn formation and the weave pattern. In the draping processes shown in Fig. 3a–d the initial inscribed grid remained mostly intact after draping. This can be experimentally observed by following the deformations of straight lines drawn on the fabrics prior to draping. These lines covering successive warp yarns or weft yarns became curved during draping but remained mostly continuous. However, some widening of the thickness of grid lines was observed at macro-level in the case of loose plain (basket) weave, containing grid line discontinuities at the micro-level of yarn width, while no widening of any
individual yarns was observed. After closer observation, this indicated fibre slipping for about one yarn width, at maximum, in the case of loose plain weave. This was not evident in the other weaves because of (a) their small yarn width and (b) the tightness of weave which would not have facilitated fibre slipping. The phenomenon of wrinkling in the draping of fabrics is caused by compressive forces induced during forming, resulting in gross buckling deformation through the thickness of the fabric. These compressive forces arise from significant material compression when the warp and weft fibres are sheared beyond the “locking shear angle” necessary to form the doubly curved shapes. The resulting compressive strain which cannot be accommodated by the fabric’s shear process results in the buckling or wrinkling of the fabric. In Fig. 3a the draped basket weave fabric (loose plain weave) is seen to have formed without wrinkling not only over the upper part of the hemisphere, but also the lower part of the hemisphere as well as the flat part. This is because the locking shear angle of the fabric was never reached within the mould design of the drape experiments. On the contrary when the photographs of the draped tight plain weave, twill and satin weave are examined (Fig. 3), it can be noticed that, whereas on the hemispherical surface no wrinkles were formed, wrinkles are visibly present on the flat surface. In the dome region, two simultaneous opposing processes were responsible for the absence of wrinkles. Whilst the fabric underwent shearing to conform to the mould geometry, the upper mould, as it closed, conferred a smoothing force which opposed the compressive forces, so that the effect of the latter was reduced. This is a similar effect as that observed in the diaphragm forming [15,16] of fibre reinforced thermoplastics where the rubber diaphragm imparts membrane tensile stresses. As a result, the combination of stresses on the fabric part over the hemispherical mould surface managed to overcome the wrinkling there. At the transition zone, i.e. near the equator, there was reduced smoothing and the fabric was compressed deeply onto the base of the hemisphere shearing the fibres on the flat surface, in the process. Thus, because of the tightness and density of some weaves, the locking angle was reached. When the fibres could not accommodate the orthogonal compressive forces, they buckled and produced wrinkles which were not further smoothed by opposing forces. A survey was carried out to determine the angles, a , of the deformed elements that fell within the wrinkle areas of the draped fabrics. The angle of an undeformed element (without any wrinkles) is 90⬚. The whole area where wrinkles appeared was considered for this investigation. All the elements angles inside the area were traced on a transparent paper and measured with a protractor. The largest angle of deformed wrinkled elements, angle (a max), corresponds to the least deformed wrinkled element where there are small wrinkles; the lowest angle of deformed wrinkled elements, angle (a min), corresponds to the most deformed wrinkled
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Fig. 4. Boundary profiles of the draped fabrics.
element where wrinkles are more pronounced. The measurements are tabulated in Table 2 for the fabrics which formed wrinkles, i.e. the tight plain weave, the twill weave and the 5-harness satin. From the measured angles, the higher is the element angle containing the wrinkles the more susceptible may be the fabric to produce the shear locking effect. However, the values obtained do not correspond to the exact locking shear angles, which are very difficult to identify with this kind of experiment, especially as one cannot quantify the size of wrinkles. Nevertheless, these values can be used to compare fabrics in terms of wrinkling during draping over a certain mould geometry. Table 2 also presents the locking element angle, a locking, below which shear locking (in terms of the onset of wrinkling) was observed in picture-frame shear tests [8]. In these tests, the fabric was placed in a picture-frame which was extended at its diagonally opposite ends at a constant speed of 5 mm/min. The locking element angle from the pictureframe shear tests is obviously larger than the grid element angles observed in the wrinkled regions after draping for TPW, TWILL and 5HSW, illustrating that shear locking precedes any serious wrinkling observed in the fabrics draped over the hat geometry of this study. Also, a max for all three fabrics with wrinkling after draping are notably smaller than the locking element angles in the pictureframe shear tests. This could be attributed to the smoothing forces that the closing mould exerted on the fabric, smooth-
Fig. 5. Deformed grid element during shear; a is the angle of the deformed grid element and g is the shear angle.
ing out wrinkling, whereas such forces were not present in the picture-frame shear tests. The tight plain weave has the largest angles of wrinkled elements in comparison to the other weaves in both draping and picture-frame shear tests, which indicates that the fibre yarns will lock earlier in shear than for the satin and twill weaves. For the same shear forces generated on the fabrics during the drape experiment, the tight plain weave will have more prominent and a greater number of wrinkles in these areas. The draped fabric with the next size of angles of wrinkled elements is the twill weave. The satin weave is perhaps the best fabric for use in drape applications since it produces the smallest angles and extent of wrinkled elements when compared to the other tight weaves. In the picture-frame shear test, the satin weave seems comparable to the twill weave fabric which might indicate that fibre slipping might also be involved in the draping of the satin weave, apart from fabric shear. The loose plain weave (basket weave) is of course superior to the satin weave in terms of wrinkles, since it formed no wrinkles when draped over the hemispherical dome surrounded by the flat rim in this study. However, it might have other disadvantages, for example in the area of mechanical properties, due to the large gaps between fibre yarns. Fig. 4 presents top viewed, boundary profiles of each fabric draped over the hat mould geometry. The observed profiles are close to each other although the extent of wrinkling differed in the various fabrics. The profile of the draped satin weave seems to have a lower degree of rotational symmetry, relative to the x; y axes through the centre in Fig. 4, than the profiles of the other fabrics. The local fibre orientation is represented by the orientation of the deformed gridlines inscribed on the fabric prior to draping. So, the first task was to measure the angles a of the grid elements (see Fig. 5) of the fabric after draping. Figs. 6–8 present shaded contours interpolated from the map of measurements of the deformed grid elements
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Fig. 6. Shaded contours of the angle, a , of grid elements after the draping of the loose plain weave (basket weave) onto a hemispherical dome surrounded by a flat base.
angle as a function of location on the draped fabric for the loose plain weave (basket weave), tight plain weave and twill weave, respectively. The thus processed angle measurements at the nodes of the deformed grid were derived as an average quadrant set of the corresponding measurements over the four quadrants of the fabric. These contours provide us with the zones of deformation patterns on the entire double curvature hemispherical surface and the surrounding flat surface. Starting from the apex of the dome, one notices that no fabric deformation was recorded and the original 90⬚ grid element angle was retained. Hence, this area can be regarded as a zero deformation zone. The zero deformation zone extends around the mid-lines of the fabric,
forming a diamond or a cross of elaborate shape, depending on the fabric (see Figs. 6–8). At this point, it must be mentioned that the mid-lines of the fabric are lines of symmetry and have also been constrained over the corresponding cross-lines of the mould during the positioning process of draping. Next to the zero deformation zone is the transition zone, within each of the four quarters of the hemisphere. In this zone, the fibres bend and rotate at the yarn cross-overs to form onto the hemisphere. The recorded angles of deformed elements are less than 90⬚. As we move away from the apex toward the edge of the hemispherical dome, there are zones of increasing shear, where the fabric deformation continues to rise. Toward the end of the
Fig. 7. Shaded contours of the angle, a , of grid elements after the draping of the tight plain weave onto a hemispherical dome surrounded by a flat base.
Fig. 8. Shaded contours of the angle, a , of grid elements after the draping of the twill weave onto a hemispherical dome surrounded by a flat base.
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Fig. 9. Shaded contours of the angle, a , of grid elements after the draping of the 5-harness satin weave onto a hemispherical dome surrounded by a flat base.
hemisphere (equator), the fabric deformation reaches its peak and this extends to the region where the hemisphere ends and the flat section begins. In this region, the highest attainable fabric deformation was obtained around the diagonals of the fabric. When one considers the grid elements angles contours for the loose plain weave (basket weave) first (see Fig. 6), it can be seen that the pattern follows the general trends as described above. A dark diamond of 90⬚ element angles is observed around the apex of the hemispherical dome which corresponds to the zero-shear zone. After that, there are about nine notional zones where the grid elements angle ranges between 84–88 and 52–56⬚ within the hemispherical part. At the equator, where the hemispherical dome changes to the flat surface, the fabric deformation is maximum and a grid element angle of 34⬚ was recorded. However, this maximum shear is concentrated only in a small area along the diagonal direction of each fabric quadrant. Outside this zone of maximum deformation, shear starts to decrease at
Fig. 10. Plots of the values of the angle, a , of the grid elements along the diagonal fabric direction as a function of the normalised arc distance L=S from the apex for the four woven fabrics draped over a hemispherical dome surrounded by a flat base.
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four successive zones on the flat section and the grid element angle ranges between 64 and 68⬚ at the corner of the fabric sample. A similar trend can be observed for the tight plain weave fabric in Fig. 7 although the contour pattern of the grid element angles is not identical to that of the loose plain weave. Starting from the apex, again a central zerodeformation zone is observed of a diamond/elaborate cross shape. Thereafter, the transition zone follows where the dome curvature begins and grid element angles in the range of 80–84⬚ were recorded in each fabric quadrant. After this zone, the rest of the zones experience an increase in fabric deformation until shear reaches a maximum value and grid element angles in the range of 52–56⬚ were recorded close to the end of hemispherical dome. At the transition zone to the flat section, grid element angles of 42⬚ were observed. When the grid element angle contours are considered for the twill weave (see Fig. 8), a similar pattern is observed but by no means identical as those of the loose and tight plain weaves. The zero-deformation zone around the apex is of cross shape the sides of which extend along the mid-lines of the fabric indicating minimum shear deformation due to symmetry. Then, moving to the transition zone where the deformation begins the grid element angles range between 81 and 86⬚. From there onwards, there is a continuous increase in the deformation of the fabric in each quadrant with grid element angles in the range of 76–81⬚ up to 45– 50⬚ near the end of the hemisphere. In the transition region where the hemispherical surface ends and the flat section begins, a minimum grid element angle of 34⬚ was recorded. After this zone, the rest of the flat section consists of several zones of deformation. In these zones deformation falls gradually and the deformed grid element angles lie in the range 45–50⬚ until they reach 72–76⬚ at the corner of the fabric tail. Fig. 9 presents the interpolated shaded contours for the satin weave. The general trend of shear deformation is the same as for the other weaves. The zero deformation zone is of an elaborate cross-shape. Some lack of rotational symmetry is observed in the central cross-pattern, which is also observed in the deformed boundary profile of the satin weave in Fig. 4. A minimum grid element angle of 38⬚ is recorded on the diagonal line on the flat part just after the dome edge. After that the grid element angle increases until it reaches 64–69⬚ at the fabric’s tail. Fig. 10 presents the grid element angle, a , data, averaged over the four fabric quadrants, for the loose plain (basket), tight plain, twill and satin weaves along the diagonal direction as a function of the normalised distance L=S from the apex. The normalising factor is 0.5pR where R is the radius of the dome mould. Hence, at L=S 0 is the apex, at L=S 1 is the dome edge, L=S ⬍ 1 is within the dome and L=S ⬎ 1 is on the flat surface surrounding the dome. The displayed error bars correspond to the minimum and maximum variation of
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5. Determination of the fibre volume fraction distribution of the draped fabrics The next step was to calculate local changes in the fibre volume fraction due to the inhomogeneous shear distribution on the draped fabric. Let us consider an undeformed, rectangular fabric element of sides a and b (see Fig. 5). During draping it is considered that the fabric element deforms following pure shear, the length of its sides a and b remains unchanged during shear, there is no compression and the element thickness, t, remains unchanged. In this case, taking into account that the element mass remains the same before and after shear, it can be derived that Fig. 11. Plots of the values of the shear angle, g , along the diagonal fabric direction as a function of the normalised arc distance L=S from the apex for the four woven fabrics draped over a hemispherical dome surrounded by a flat base.
measurements from the average for each quadrant. The four fabrics in Fig. 9 display the same general features but their deformation data are not identical. The grid element angle starts from values close to 90⬚ in the zero-deformation zone at the apex of the dome L=S 0 and decreases to a minimum angle around L=S 1 at the edge of the dome where the maximum shear occurs. Thereafter, deformation starts declining progressively and this continues up to the end of the fabric. From the value of the angles of the grid elements Fig. 10, the local shear angles, g , can be calculated as g p=2 ⫺ a: Fig. 11 presents the corresponding data of shear angle for the loose plain weave (basket weave), tight plain weave, twill and satin weave against the normalised arc distance L=S from the apex along the diagonal direction of each fabric. The shear angle rises from 0 at the apex of the dome to a maximum around L=S 1 and then it falls again on the flat surface along the tails of the fabric. The loose plain weave (basket weave) and the twill weave exhibit the highest shear angle of 56⬚, followed by the satin weave which has a maximum shear angle of 52⬚ and lastly the tight weave with a maximum shear deformation angle of 48⬚. At points where L=S ⬎ 1 the flat section begins and the fabrics experience less deformation resulting in low shear angle values.
Table 3 Calculated initial and maximum (after drape) fibre volume fractions for each draped weave Fabric
Vfo
Vfmax
Vfmax/Vfo
LPW TPW TWILL 5HSW
0.36 0.44 0.46 0.50
0.64 0.66 0.83 0.82
1.77 1.50 1.80 1.64
Vf
Vfo Vfo sin a cos g
1
where Vfo is the fibre volume fraction of the unsheared element and Vf is the fibre volume fraction after shearing. The values of Vfo can be obtained from the expression Vfo
N ra tr
2
where, r a is the areal density of the weave (kg/m 2), N is the number of plies (in this case, N 1), t is the ply thickness (m), and r is the density of fibre glass (kg/m 3). By using Eqs. (1) and (2) and data from Table 1, the distributions of the fibre volume fractions of the sheared grid elements of the draped fabrics were calculated. Table 3 summarises the calculated initial fibre volume fractions, Vfo, of the unsheared weaves used in the analysis and the maximum fibre volume fractions calculated from the shear data of the drape experiments. Fig. 12 shows the curves of the calculated fibre volume fractions of the sheared grid elements along the diagonal direction for the four weaves, as functions of the normalised arc distance L=S from the apex of the dome. The curves start from Vfo at the apex of the dome, the zero-deformation zone, and rise to a maximum around L=S 1; where the maximum shear occurs in the fibre yarn directions. Then the fibre volume fraction falls along the diagonal line of the flat part of the fabric. Compression of weaves results to a maximum fibre volume fraction, around 0.62–0.65 for through thickness compression [17,18], corresponding to the packing limit. Similar values for the packing limit in in-plane compression have been considered in the modelling of the in-plane shearing of these fabrics [8]. Beyond this limit it is reasonable to expect that in in-plane compression wrinkling or ply thickening will occur. The loose plain weave reaches a maximum calculated Vf close to this limit after draping. However, the calculated maximum Vf for the tight plain weave, twill and satin clearly exceed the maximum acceptable Vf and, hence, they exhibit wrinkling after being draped in the hat mould geometry. A thick line around Vf 0:65 [8,17,18] in Fig. 12 represents the plateau of the approximately acceptable maximum Vf. The corresponding cells of unacceptable Vfmax after draping as calculated by relation (1) are bold in
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Fig. 12. Graphs of the calculated fibre volume fraction along the diagonal direction of the four tested fabrics against the normalised arc distance L=S from the apex, where the woven fabrics were draped over a hemispherical mould surrounded by a flat rim. Thick line represents approximate maximum acceptable Vf.
Table 3. It must also be pointed out that after wrinkling the assumptions associated with relation (1) are unlikely to hold and also the angles between gridlines might not represent the local angles between crossing fibre yarns.
6. Determination of the stiffness distribution in draped fabric laminates Changes in the fibre orientation and the fibre volume fraction in the draping of fabrics result in an inhomogeneous distribution of mechanical properties for the composite part. In this section, the distribution of the reduced stiffness matrix components of a fabric laminate will be evaluated on the basis of the laminated plate theory (for example [19]), where the laminate has the form of a hemispherical hat. Each ply of woven fabric is modelled as a combination of two unidirectional layers (see Fig. 13), at 0 and 90⬚ for an unsheared fabric, and at 0 and a degrees for a fabric sheared by an angle g . The effect of crimp and the type of weave are not incorporated in this type of theoretical analysis.
Fig. 13. Representation of a woven fabric by a combination of two plies of unidirectional fibres in unsheared and sheared state. Note that the interfibre spacing also changes on shearing (not shown in this figure).
Each layer of unidirectional fibres in Fig. 13 has mechanical properties depending on the principal directions 1 and 2 (where 1 is the fibre direction and 2 is perpendicular to the fibre direction) and on the fibre volume fraction. The elastic properties of E-glass have been taken [20] as Ef 72 GPa; nf 0:3 and Gf 27:7 GPa; where Ef is the fibre Young’s modulus, n f is the fibre major Poisson’s ratio and Gf is the fibre shear modulus. As an example, the matrix of the laminate was assumed for be an epoxy with the following elastic properties [20]: Em 3:5 GPa; nm 0:35 and Gm 1:3 GPa; where subscript m refers to the matrix. Assuming that the fibre volume fraction in each unidirectional fibre ply changes during shear by the same amount as the overall fibre volume fraction of the fabric, the effect of the change of Vf during shear on the elastic properties of each unidirectional fibre ply is given by the following equations [19], where subscripts f and m refer to the fibre and matrix, respectively: E1 Ef Vf ⫹ Em 1 ⫺ Vf
3
(Rule of mixtures) 1 V 1 ⫺ Vf f ⫹ Ef Em E2
4
n12 nf Vf ⫹ nm 1 ⫺ Vf
5
1 V 1 ⫺ Vf f ⫹ G12 Gf Gm
6
where the void content of the laminate is assumed to be zero Vf ⫹ Vm 1: The influence of the change in fibre orientation during the draping of a fabric required for the calculation of the modulus of the composite product can be ascertained by considering the rectilinear system of coordinates x; y in relation to the system of principal directions (1,2) for each ply in unsheared and sheared state as is illustrated in Fig. 13.
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following relations [19] Q 11 Q11 m4 ⫹ 2 Q12 ⫹ 2Q66 n2 m2 ⫹ Q22 n4 Q 22 Q11 n4 ⫹ 2 Q12 ⫹ 2Q66 n2 m2 ⫹ Q22 m4 Q 12 Q11 ⫹ Q22 ⫺ 4Q66 n2 m2 ⫹ Q12 n4 ⫹ m4 Q 66 Q11 ⫹ Q22 ⫺ 2Q12 ⫺ 2Q66 n2 m2 ⫹ Q66 n4 ⫹ m4 Fig. 14. The effect of the Vf change on E1 and E2, the elastic moduli in the principal directions of a unidirectional E-glass fibre ply in the draping of the loose plain weave (basket weave) over a hemispherical dome surrounded by a flat rim.
Q 16 Q11 ⫺ Q12 ⫺ 2Q66 nm3 ⫺ Q22 ⫺ Q12 ⫺ 2Q66 n3 m Q 26 Q11 ⫺ Q12 ⫺ 2Q66 n3 m ⫺ Q22 ⫺ Q12 ⫺ 2Q66 nm3 10
According to Fig. 13, the system of coordinates , x; y is aligned with the principal directions (1,2) of the 0⬚ ply both in unsheared and sheared state, whereas the 90⬚ ply changes direction with respect to the x; y system of coordinates after shear. The relationship between in-plane stress and in-plane strain in each ply is expressed by
where m cos a and n sin a: Once the components of the transformed reduced stiffness matrix are calculated for each equivalent unidirectional fibre ply of the woven fabric laminate model, the transformed reduced stiffness matrix of the complete “fabric laminate” under loading in the x; y directions is given by the relation
2
Q ij fabric 0:5 Q ij 0 ⫹ Q ij a
2 Q11 6 7 6 6 s y 7 6 Q 12 4 5 4 s xy Q 16
sx
3
Q 12 Q 22 Q 26
32 3 ex Q 16 76 7 6 7 Q 26 7 5 4 ey 5 gxy Q 66
7
where s is the stress tensor, e is the strain tensor and Q is the transformed reduced stiffness matrix. The stiffness matrix of each unidirectional fibre ply in the woven fabric laminate model is defined as 2
Q11
6 Q ij 6 4 Q12 Q16
Q12
Q16
7 Q26 7 5
Q26
Q66
8
where Q11, Q12, Q22 and Q66 are called the reduced stiffnesses and can be expressed in terms of the elastic properties of a unidirectional fibre ply [19] as: Q11 E11 = 1 ⫺ n12 n21 Q12 n21 E11 = 1 ⫺ n12 n21
which originates from the relation giving the laminate force matrix, N Nlaminate
N Zhn X n1
hn ⫺ 1
s n dz
N Zhn X n1
⫹
hn ⫺ 1
n e0 dz Q
N Zhn X n1
hn ⫺ 1
n kz dz Q 12
3
Q22
11
Q22 E22 = 1 ⫺ n12 n21 Q66 G12
9
Q16 Q61 0 The stiffness components of the 0⬚ ply remain the same in the unsheared and sheared state since the 0⬚ ply is aligned with the x; y system and does not change orientation after shear. The a degree ply is at 90⬚ with respect to the x; y system in the unsheared state and at a degrees with respect to the x; y system in the sheared state. The transformed reduced stiffness matrix components of the a degree ply in an off axisloading test in the x and y directions are given by the
where the laminate consists of N unidirectional fibre plies, n is the trans(hn –hn⫺1) is the thickness of each ply, Q formed reduced stiffness matrix of the n ply, e 0 is the mid-plane strain and [k] is the mid-plane plate curvature. By assuming that the fabric thickness is very small, the second term of the sum in Eq. (12) is not so significant. By also assuming that each of the warp and weft unidirectional fibre plies in a woven fabric model has the same thickness and fibre volume fraction, the definition of the transformed reduced stiffness matrix of the laminate is established and relation (11) is derived. The above analysis and calculation procedure of the major components of the transformed reduced stiffness matrix of a sheared fabric loaded in the x; y directions was applied to the loose plain weave (basket weave) draped over the hemispherical hat mould. This fabric was chosen because it draped without wrinkling and the evaluated values of fibre volume fraction according to relation (1) did not exceed the maximum packing fraction (see Table 3). The weave was considered balanced (see Table 1) and the fabric was considered as in-plane symmetric material. First the effect of the change of Vf due to shear during draping was incorporated in the elastic properties in the principal directions (1,2) of each equivalent unidirectional
U. Mohammed et al. / Composites: Part A 31 (2000) 1409–1420
Fig. 15. The reduced stiffnesses of the 0⬚ unidirectional fibre ply in the draping of the loose plain weave (basket weave) over a hemispherical dome surrounded by a flat rim.
fibre ply, according to Eqs. (3)–(6). Fig. 14 presents the effect of the Vf change on E1 and E2. For the composite component, it is evident that E1 is the most important modulus value. It increases as one moves away from the apex along the diagonal direction of the fabric, it reaches a maximum value (maximum Vf) due to maximum shear at the edge of the hemisphere and then it falls gradually on the flat surface. E2 is low for the equivalent unidirectional Eglass fibre ply where, as mentioned earlier, an epoxy matrix has been considered in this case-study. After the effect of the Vf during shear has been taken into account, the major reduced stiffnesses for a unidirectional E-glass fibre ply can be calculated from relations (9). (Notice that n21 n12 E2 =E1 [19]). These stiffness values are directly applicable to the 0⬚ ply and are presented in Fig. 15. Again it can be seen that Q11 is the most important stiffness component, which corresponds to E1 and follows its trend due to the effect of changing Vf across the draped ply. The transformed reduced stiffnesses of the a degree unidirectional fibre ply were evaluated from Eq. (10) which expresses the effect of the change in fibre orientation. The effect of change in the fibre volume fraction after drape was also taken into account in the values of the reduced stiffnesses [Qij] in the principal directions. Hence Fig. 16 presents the transformed reduced stiffnesses of the a degree unidirectional fibre ply of the loose plain weave fabric (basket weave) draped over the hemispherical hat for loading in the x; y directions (see Fig. 13), incorporating the
Fig. 16. The transformed reduced stiffnesses of the a degree unidirectional fibre ply in the draping of the loose plain weave (basket weave) over a hemispherical dome surrounded by a flat rim. Loading is in the x; y directions (see Fig. 12).
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effects of changing Vf and changing fibre direction during shear. It can be seen that at the apex L=S 0 Q 22 is maximum and is the important reduced stiffness component whereas the other reduced stiffnesses are negligible. At this location a 90⬚ and Q 22 corresponds to the modulus along the orthogonal fibre direction (see Fig. 13). As one moves away from the apex along the diagonal direction of the fabric, the fibre orientation changes due to shear and, as a result Q 22 decreases whereas Q 11 increases reaching a maximum at the edge of the hemisphere (around L=S 1) where the maximum shear occurs: consequently change in fibre orientation takes place. By using Eq. (11) the transformed reduced stiffnesses of the “fabric laminate” were calculated as a combination of the corresponding stiffnesses of the 0 and a degree unidirectional fibre plies, when the loading is applied in the x; y directions (see Fig. 13). Fig. 17 presents the reduced stiffnesses of the loose plain weave (basket weave) fabric along the normalised diagonal distance L=S outwards from the apex when the fabric is draped over the hemispherical hat. These stiffnesses are with respect to loading in the x; y directions (see Fig. 13). 7. Conclusions In this study, there have been presented a set of data on the deformation of fabrics draped over a hemispherical dome surrounded by a flat base. These data may be used to assess fabric drapeability and to validate computer simulations of draping. Four different types of fabric were examined: a loose plain weave, a tight plain weave, a twill and a satin weave. The four types of fabrics presented different drapeability in terms of wrinkling, with the tight plain weave exhibiting the worst drapeability. Differences in the deformation behaviour between the four fabrics resulted in a certain extent of differences in the distribution of local shear angles. The obtained data of local gridline orientation and local shear angle were used to first evaluate the distribution of local fibre volume fractions in each draped fabric. It was found that in regions of high shear, in some of the fabrics, the fibre volume fraction reached the packing limit and relation (1) could not be employed any more. It was thought
Fig. 17. The reduced stiffnesses of the loose plain weave (basket weave) fabric draped over a hemispherical dome surrounded by a flat rim. Loading is in the x; y directions (see Fig. 12).
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that, when this occurs, wrinkling may be expected for the draped fabric. The results of fibre volume fraction and shear angle distributions were used to calculate components of the transformed reduced stiffness matrix of a formed fabric laminate following a laminated plate theory analysis. References [1] Boisse P, Cherouat A, Gelin JC, Sabhi H. Experimental study and finite element simulation of a glass fibre fabric shaping process. Polymer Composites 1995;16(1):83–95. [2] Mohammed U, Lekakou C, Bader MG. Mathematical and experimental studies of the draping of woven fabrics in resin transfer moulding (RTM). In: Visconti IC, editor. Proceedings of ECCM-8, vol. 2. Cambridge: Woodhead Publishers, 1998. p. 683–90. [3] Dong L, Lekakou C, Bader MG. Solid mechanics draping simulations of woven fabrics. ICCM-12, Paris, 1999. [4] Dong L, Lekakou C, Bader MG. Submitted for publication. [5] Dong L, Lekakou C, Bader MG. Submitted for publication. [6] Martin TA, Bhattacharyya D, Collins IF. Bending of fibre-reinforced thermoplastic sheets. Composites Manufacturing 1995;6(3/4): 177–87. [7] Yu JZ, Cai Z, Ko FK. Formability of textile preforms for composite applications. Part 1: characterisation experiments. Composites Manufacturing 1994;5(2):113–22. [8] Mohammed U, Lekakou C, Dong L, Bader MG. Shear deformation and micromechanics of woven fabrics. Composites A 2000; 31A(4):299–308. [9] Tan Z, Melin L, Magnusson C. Application of an image processing technique in strain-measurement in sheet-metal forming. Journal of Materials Processsing Technology 1992;33(3):299–310.
[10] Long AC, Rudd CD, Blagdom M, Smith P. Characterising the processing and performance of aligned reinforcement during preform manufacture. Composites A 1996;27A:247–53. [11] Laroche D, Vu-Khanh T. Forming of woven fabric composites. Journal of Composite Materials 1994;28(18):1825–39. [12] Vu-Khanh T, Liu B. Prediction of fibre rearrangement and thermal expansion behaviour of deformed woven-fabric laminates. Composites Science and Technology 1995;53:183–91. [13] Bickerton S, Simacek P, Guglielmi SE, Advani SG. Investigation of draping and its effects on the mould filling process during manufacturing of a compound curved composite part. Composites A 1997;28(9/10):801–16. [14] Reinforcement fabrics—glass fabrics for plastics reinforcement technical data. Fothergill Engineered Fabrics Ltd, Lancashire, 1997. [15] Delaloye S, Niedermeier M. Optimisation of the diaphragm forming process for continuous fibre-reinforced advanced thermoplastic composites. Composites Manufacturing 1995;6:135–44. [16] Bradaigh CM, McGuinness CB, Pipes RB. Numerical analysis of stresses and deformations in composite materials sheet forming: central indentation of a circular sheet. Composites Manufacturing 1993;4(2):67–83. [17] Saunders RA, Lekakou C, Bader MG. Compression and microstructure of fibre plain woven cloths in the processing of polymer composites. Composites A 1998;29A:443–54. [18] Saunders RA, Lekakou C, Bader MG. Compression in the processing of polymer composites: 1. A mechanical and microstructural study for different glass fabrics and resin. Composites Science and Technology 1999;59:983–93. [19] Powell PC. Engineering with polymers. London: Chapman and Hall, 1983. [20] Naik NK. Woven fabric composites. Lancaster: Technomic, 1994.
Experimental studies and analysis of the draping of woven fabrics U. Mohammed, C. Lekakou*, M.G. Bader School of Mechanical and Materials Engineering, University of Surrey, Guildford, Surrey GU2 5XH, UK Received 27 August 1999; accepted 13 April 2000
Abstract This study investigates and compares the draping and forming of four types of woven fabrics, namely a loose plain weave (basket weave), a tight plain weave, a satin and a twill weave. The fabrics were draped over a hat mould consisting of a hemispherical dome surrounded by a flat base. The draping of each fabric was examined in terms of wrinkle formation, boundary profile of the draped fabric, distribution of fibre orientation and local shear angles. A theoretical analysis of the experimental results involved the calculation of the distributions of the fibre volume fraction and mechanical properties, in terms of components of the reduced stiffness matrix, from the experimental data of local shear angles. 䉷 2000 Published by Elsevier Science Ltd. Keywords: Draping; A. Fabrics
1. Introduction The draping of fabrics onto surfaces of varying geometry has received much attention over recent years due to the upsurge in the use of fabrics in the manufacturing of three-dimensional composite components. Common problems in the fabricating process of three-dimensional shells of complex curvature include unwanted features such as wrinkling, kinks or tears, the presence of which depends on the geometry of the mould surface and the type of fabric. Furthermore, during draping and forming of a fabric there are local variations of fibre orientation, fibre volume fraction and possibly fabric thickness which are going to affect processability and the mechanical properties of the final product. Finite element numerical techniques have been applied for the simulation of the draping of various types of reinforcements following either a solid mechanics approach [1– 5] or a viscous approach [6]. Regarding this type of computer simulation, there is clearly a need for experimental data of materials properties to be used as input in the finite element code, as well as other data of various draping parameters to be used for validation of the theoretical predictions. Yu et al. [7] compared the formability of different types of woven fabrics in terms of shear rigidity, tensile rigidity * Corresponding author. Tel.: ⫹44-1483-879622; fax: ⫹44-1483876291. E-mail address: [email protected] (C. Lekakou). 1359-835X/00/$ - see front matter 䉷 2000 Published by Elsevier Science Ltd. PII: S1359-835 X( 00)00 080-4
and bending rigidity. Mohammed et al. [8] carried out shear tests for the four fabrics included in the present study and presented a detailed theoretical analysis of their shear behaviour, following a solid mechanics approach. Regarding the measurement of various structural parameters of the draped fabrics, a common method, originating from the metal forming processes (for example [9]) includes the inscription of a grid on the textile before draping and the subsequent study of the grid distortion in the draped textile [10] to predict strain distribution. Laroche and Khanh [11] and Khanh and Liu [12] presented a comparison of their experimental data from the forming of a fabric prepreg over a rounded top cone with numerical predictions from a “fishnet” model. Their predictions of local yarn angle were very close to the experimental data for a plain weave and an 8-harness satin, and also the experimental results from the draping of the two prepregs with the different type of weaves were similarly very close. No details about the experimental procedure of prepreg forming were presented in their paper. Bickerton et al. [13] presented measurements of local yarn angle of a stitched reinforcement draped again over a rounded top cone. The present study considers the draping and forming of dry fabrics in order to simulate the preforming stage in processes such as resin transfer moulding (RTM). The aim of the study is to provide a comprehensive set of experimental data regarding formability of different types of woven fabrics, as well as draping parameters required for the validation of computational and other theoretical simulations of the draping of fabrics. Four different types of
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Table 1 Specifications [14] of the woven fabrics used in this study Weave type
LPW
TPW
TWILL
5HSW
Coding Thickness Areal density (kg/m 2) Ends/10 mm Picks/10 mm
Basket 0.58 0.529 1.1 1.2
Y0212 0.48 0.546 6.7 6.3
Y0185 0.28 0.331 11.8 11.8
Y0227 0.23 0.297 22.4 21.3
woven fabrics were examined: a loose plain weave (basket weave); a tight plain weave; a twill and a 5-harness satin weave. Wrinkling behaviour was investigated in both draping and shear tests. Detailed maps of the measured grid angles after fabric draping were assembled, assuming to represent maps of local angles between crossing fibre yarns. A theoretical analysis of the experimental results was carried out, according to which the local fibre volume fraction and local values of components of the reduced stiffness matrix of a formed fabric laminate were calculated from the experimental data of local shear angle. These data have the potential to be used in the prediction of the permeability of the deformed fabric and also the mechanical properties of the final composite product. 2. Materials Currently, most of the pure and hybrid woven fabrics
often used in textile composites are simple two-dimensional weaves, such as plain, twill and satin weaves, which are identified by the repeating pattern of the interlaced regions in the warp and weft directions. In this study, four E-glass woven fabrics were tested: a loose plain weave (LPW), a tight plain weave (TPW), a twill weave (TWILL), and a 5harness satin weave (5HSW). The fabrics were supplied by Fothergill Engineered Fabrics Ltd and their specifications are presented in Table 1 [14]. The terms “ends” and “picks” refer to the number of warp yarns and weft yarns, respectively. Comparing the two plain weaves, the loose plain weave had an average yarn width of 5 mm and an average gap size between yarns of 2.5 mm whereas the tight plain weave had an average yarn width of 1.3 mm and an average gap size of 0.25 mm. More details about the microstructure of the four fabrics both in unsheared and sheared states are given in another study by Mohammed et al. [8]. 3. Draping of fabrics: equipment and experimental procedures A mould was constructed for draping experiments onto a double curvature geometry, combining a curved and a flat part. The selected mould shape comprised a hemispherical dome surrounded by a flat rim. The mould consisted of two parts: the upper part was made of transparent poly(methyl methacrylate) (PMMA—“perspex”). A rectangular shaped solid block was used of 500 × 500 × 137 mm 3, in which a
Fig. 1. (a) Photograph of the hemispherical mould as used for the draping of woven fabrics. (b) Mould geometry and dimensions.
U. Mohammed et al. / Composites: Part A 31 (2000) 1409–1420
Fig. 2. Diagram showing features of the draping process for one fabric quadrant.
hemispherical cavity of 100 mm radius was machined in the central region (see Fig. 1a and b). The upper half of mould was chosen to be transparent so that the draped fabric could be viewed. The lower mould was made of aluminium and consisted of a square shaped flat surface of dimensions 500 × 500 mm 2 with a solid aluminium dome of 97 mm radius at the centre.
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The following procedure was used in the draping experiments. A square piece of fabric of 360 × 360 mm 2 was cut from each of the fabrics listed in Table 1. Then a square grid of 20 × 20 divisions was inscribed onto the cut sample using a thick ink pen, where the ratio of the ink thickness to yarn width was 0.3, 0.8, 0.8 and 2 for the LPW, TPW, TWILL and 5HSW, respectively. The grid was subdivided into four quadrants for easy identification. Four quadrants were also marked to divide the hemispherical mould surface and the surrounding flat rim on the male part. The whole idea was to fix the centre of the fabric grid on the apex of the mould dome and to fit the internal boundaries (lines of symmetry) of the quadrants of the fabric exactly to the corresponding cross-lines of symmetry on the male mould. The reason for this was to formalise the draping procedure of a layer of fabric onto the male mould and achieve satisfactory and accurate results. The next stage of the draping experiment was the shaping process. The shaping process was carried out in two stages in this study with minimal tensioning of the fabric throughout. The first stage, which was the most important and difficult, was to place the fabric on the hemispherical dome and the flat rim of the male (lower) mould, so that each quadrant
Fig. 3. Photographs of fabrics draped over a hemispherical dome surrounded by a flat base: (a) loose plain weave (basket weave); (b) tight plain weave; (c) twill weave; (d) 5-harness satin weave.
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Table 2 Angles of the deformed grid elements for wrinkles in the draped fabrics and locking element angles in picture-frame shear tests [8] Weave
a min (⬚)
a max (⬚)
a locking (⬚)
LPW TPW TWILL 5HSW
No wrinkles 43–44 37 32
No wrinkles 49–56 48 34
29 80 65 64
of the grid sat exactly over the corresponding quadrant of the mould and the cross-lines of symmetry of fabric fitted over the corresponding lines of symmetry printed on the male mould. To facilitate this, a piece of double-sided Sellotape娃 was placed on the apex of the hemisphere and also at locations on each of the internal boundary lines of each quadrant of the mould (on the cross-lines of symmetry), both on the hemispherical part and the flat surface (see Fig. 2). Then, the central position on the grid was marked and the cloth was placed so that this point coincided with the apex. For each quadrant of the grid and the mould, the internal quadrant boundaries of the fabric were fitted manually over the corresponding lines of the mould. Using the Sellotape娃 pieces, the fitted cloth was constrained securely at this position prior to closing the mould. The second stage involved the forming of the fabric onto the mould surface. This was achieved by lowering the upper mould slowly and gradually until the mould was closed and housed the draped fabric.
4. Draping of fabrics: experimental results The ability of fabrics to form over three-dimensional shapes without having to use undue force defines the drapeability of fabrics. One method of comparing different fabrics is based on draping experiments. The shapes of the draped fabrics and the deformation of the grid, inscribed on each fabric prior to deformation, are seen in Fig. 3a–d for the loose plain (basket) weave, the tight plain weave, the twill weave and the satin weave, respectively. Each of the four draped fabrics deformed by angular rotation of the warp and weft yarns at their crossovers to produce a star shaped geometric feature. This deformation mode was observed on the grid inscribed on the fabrics prior to draping. Draping did not destroy the yarn formation and the weave pattern. In the draping processes shown in Fig. 3a–d the initial inscribed grid remained mostly intact after draping. This can be experimentally observed by following the deformations of straight lines drawn on the fabrics prior to draping. These lines covering successive warp yarns or weft yarns became curved during draping but remained mostly continuous. However, some widening of the thickness of grid lines was observed at macro-level in the case of loose plain (basket) weave, containing grid line discontinuities at the micro-level of yarn width, while no widening of any
individual yarns was observed. After closer observation, this indicated fibre slipping for about one yarn width, at maximum, in the case of loose plain weave. This was not evident in the other weaves because of (a) their small yarn width and (b) the tightness of weave which would not have facilitated fibre slipping. The phenomenon of wrinkling in the draping of fabrics is caused by compressive forces induced during forming, resulting in gross buckling deformation through the thickness of the fabric. These compressive forces arise from significant material compression when the warp and weft fibres are sheared beyond the “locking shear angle” necessary to form the doubly curved shapes. The resulting compressive strain which cannot be accommodated by the fabric’s shear process results in the buckling or wrinkling of the fabric. In Fig. 3a the draped basket weave fabric (loose plain weave) is seen to have formed without wrinkling not only over the upper part of the hemisphere, but also the lower part of the hemisphere as well as the flat part. This is because the locking shear angle of the fabric was never reached within the mould design of the drape experiments. On the contrary when the photographs of the draped tight plain weave, twill and satin weave are examined (Fig. 3), it can be noticed that, whereas on the hemispherical surface no wrinkles were formed, wrinkles are visibly present on the flat surface. In the dome region, two simultaneous opposing processes were responsible for the absence of wrinkles. Whilst the fabric underwent shearing to conform to the mould geometry, the upper mould, as it closed, conferred a smoothing force which opposed the compressive forces, so that the effect of the latter was reduced. This is a similar effect as that observed in the diaphragm forming [15,16] of fibre reinforced thermoplastics where the rubber diaphragm imparts membrane tensile stresses. As a result, the combination of stresses on the fabric part over the hemispherical mould surface managed to overcome the wrinkling there. At the transition zone, i.e. near the equator, there was reduced smoothing and the fabric was compressed deeply onto the base of the hemisphere shearing the fibres on the flat surface, in the process. Thus, because of the tightness and density of some weaves, the locking angle was reached. When the fibres could not accommodate the orthogonal compressive forces, they buckled and produced wrinkles which were not further smoothed by opposing forces. A survey was carried out to determine the angles, a , of the deformed elements that fell within the wrinkle areas of the draped fabrics. The angle of an undeformed element (without any wrinkles) is 90⬚. The whole area where wrinkles appeared was considered for this investigation. All the elements angles inside the area were traced on a transparent paper and measured with a protractor. The largest angle of deformed wrinkled elements, angle (a max), corresponds to the least deformed wrinkled element where there are small wrinkles; the lowest angle of deformed wrinkled elements, angle (a min), corresponds to the most deformed wrinkled
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Fig. 4. Boundary profiles of the draped fabrics.
element where wrinkles are more pronounced. The measurements are tabulated in Table 2 for the fabrics which formed wrinkles, i.e. the tight plain weave, the twill weave and the 5-harness satin. From the measured angles, the higher is the element angle containing the wrinkles the more susceptible may be the fabric to produce the shear locking effect. However, the values obtained do not correspond to the exact locking shear angles, which are very difficult to identify with this kind of experiment, especially as one cannot quantify the size of wrinkles. Nevertheless, these values can be used to compare fabrics in terms of wrinkling during draping over a certain mould geometry. Table 2 also presents the locking element angle, a locking, below which shear locking (in terms of the onset of wrinkling) was observed in picture-frame shear tests [8]. In these tests, the fabric was placed in a picture-frame which was extended at its diagonally opposite ends at a constant speed of 5 mm/min. The locking element angle from the pictureframe shear tests is obviously larger than the grid element angles observed in the wrinkled regions after draping for TPW, TWILL and 5HSW, illustrating that shear locking precedes any serious wrinkling observed in the fabrics draped over the hat geometry of this study. Also, a max for all three fabrics with wrinkling after draping are notably smaller than the locking element angles in the pictureframe shear tests. This could be attributed to the smoothing forces that the closing mould exerted on the fabric, smooth-
Fig. 5. Deformed grid element during shear; a is the angle of the deformed grid element and g is the shear angle.
ing out wrinkling, whereas such forces were not present in the picture-frame shear tests. The tight plain weave has the largest angles of wrinkled elements in comparison to the other weaves in both draping and picture-frame shear tests, which indicates that the fibre yarns will lock earlier in shear than for the satin and twill weaves. For the same shear forces generated on the fabrics during the drape experiment, the tight plain weave will have more prominent and a greater number of wrinkles in these areas. The draped fabric with the next size of angles of wrinkled elements is the twill weave. The satin weave is perhaps the best fabric for use in drape applications since it produces the smallest angles and extent of wrinkled elements when compared to the other tight weaves. In the picture-frame shear test, the satin weave seems comparable to the twill weave fabric which might indicate that fibre slipping might also be involved in the draping of the satin weave, apart from fabric shear. The loose plain weave (basket weave) is of course superior to the satin weave in terms of wrinkles, since it formed no wrinkles when draped over the hemispherical dome surrounded by the flat rim in this study. However, it might have other disadvantages, for example in the area of mechanical properties, due to the large gaps between fibre yarns. Fig. 4 presents top viewed, boundary profiles of each fabric draped over the hat mould geometry. The observed profiles are close to each other although the extent of wrinkling differed in the various fabrics. The profile of the draped satin weave seems to have a lower degree of rotational symmetry, relative to the x; y axes through the centre in Fig. 4, than the profiles of the other fabrics. The local fibre orientation is represented by the orientation of the deformed gridlines inscribed on the fabric prior to draping. So, the first task was to measure the angles a of the grid elements (see Fig. 5) of the fabric after draping. Figs. 6–8 present shaded contours interpolated from the map of measurements of the deformed grid elements
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Fig. 6. Shaded contours of the angle, a , of grid elements after the draping of the loose plain weave (basket weave) onto a hemispherical dome surrounded by a flat base.
angle as a function of location on the draped fabric for the loose plain weave (basket weave), tight plain weave and twill weave, respectively. The thus processed angle measurements at the nodes of the deformed grid were derived as an average quadrant set of the corresponding measurements over the four quadrants of the fabric. These contours provide us with the zones of deformation patterns on the entire double curvature hemispherical surface and the surrounding flat surface. Starting from the apex of the dome, one notices that no fabric deformation was recorded and the original 90⬚ grid element angle was retained. Hence, this area can be regarded as a zero deformation zone. The zero deformation zone extends around the mid-lines of the fabric,
forming a diamond or a cross of elaborate shape, depending on the fabric (see Figs. 6–8). At this point, it must be mentioned that the mid-lines of the fabric are lines of symmetry and have also been constrained over the corresponding cross-lines of the mould during the positioning process of draping. Next to the zero deformation zone is the transition zone, within each of the four quarters of the hemisphere. In this zone, the fibres bend and rotate at the yarn cross-overs to form onto the hemisphere. The recorded angles of deformed elements are less than 90⬚. As we move away from the apex toward the edge of the hemispherical dome, there are zones of increasing shear, where the fabric deformation continues to rise. Toward the end of the
Fig. 7. Shaded contours of the angle, a , of grid elements after the draping of the tight plain weave onto a hemispherical dome surrounded by a flat base.
Fig. 8. Shaded contours of the angle, a , of grid elements after the draping of the twill weave onto a hemispherical dome surrounded by a flat base.
U. Mohammed et al. / Composites: Part A 31 (2000) 1409–1420
Fig. 9. Shaded contours of the angle, a , of grid elements after the draping of the 5-harness satin weave onto a hemispherical dome surrounded by a flat base.
hemisphere (equator), the fabric deformation reaches its peak and this extends to the region where the hemisphere ends and the flat section begins. In this region, the highest attainable fabric deformation was obtained around the diagonals of the fabric. When one considers the grid elements angles contours for the loose plain weave (basket weave) first (see Fig. 6), it can be seen that the pattern follows the general trends as described above. A dark diamond of 90⬚ element angles is observed around the apex of the hemispherical dome which corresponds to the zero-shear zone. After that, there are about nine notional zones where the grid elements angle ranges between 84–88 and 52–56⬚ within the hemispherical part. At the equator, where the hemispherical dome changes to the flat surface, the fabric deformation is maximum and a grid element angle of 34⬚ was recorded. However, this maximum shear is concentrated only in a small area along the diagonal direction of each fabric quadrant. Outside this zone of maximum deformation, shear starts to decrease at
Fig. 10. Plots of the values of the angle, a , of the grid elements along the diagonal fabric direction as a function of the normalised arc distance L=S from the apex for the four woven fabrics draped over a hemispherical dome surrounded by a flat base.
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four successive zones on the flat section and the grid element angle ranges between 64 and 68⬚ at the corner of the fabric sample. A similar trend can be observed for the tight plain weave fabric in Fig. 7 although the contour pattern of the grid element angles is not identical to that of the loose plain weave. Starting from the apex, again a central zerodeformation zone is observed of a diamond/elaborate cross shape. Thereafter, the transition zone follows where the dome curvature begins and grid element angles in the range of 80–84⬚ were recorded in each fabric quadrant. After this zone, the rest of the zones experience an increase in fabric deformation until shear reaches a maximum value and grid element angles in the range of 52–56⬚ were recorded close to the end of hemispherical dome. At the transition zone to the flat section, grid element angles of 42⬚ were observed. When the grid element angle contours are considered for the twill weave (see Fig. 8), a similar pattern is observed but by no means identical as those of the loose and tight plain weaves. The zero-deformation zone around the apex is of cross shape the sides of which extend along the mid-lines of the fabric indicating minimum shear deformation due to symmetry. Then, moving to the transition zone where the deformation begins the grid element angles range between 81 and 86⬚. From there onwards, there is a continuous increase in the deformation of the fabric in each quadrant with grid element angles in the range of 76–81⬚ up to 45– 50⬚ near the end of the hemisphere. In the transition region where the hemispherical surface ends and the flat section begins, a minimum grid element angle of 34⬚ was recorded. After this zone, the rest of the flat section consists of several zones of deformation. In these zones deformation falls gradually and the deformed grid element angles lie in the range 45–50⬚ until they reach 72–76⬚ at the corner of the fabric tail. Fig. 9 presents the interpolated shaded contours for the satin weave. The general trend of shear deformation is the same as for the other weaves. The zero deformation zone is of an elaborate cross-shape. Some lack of rotational symmetry is observed in the central cross-pattern, which is also observed in the deformed boundary profile of the satin weave in Fig. 4. A minimum grid element angle of 38⬚ is recorded on the diagonal line on the flat part just after the dome edge. After that the grid element angle increases until it reaches 64–69⬚ at the fabric’s tail. Fig. 10 presents the grid element angle, a , data, averaged over the four fabric quadrants, for the loose plain (basket), tight plain, twill and satin weaves along the diagonal direction as a function of the normalised distance L=S from the apex. The normalising factor is 0.5pR where R is the radius of the dome mould. Hence, at L=S 0 is the apex, at L=S 1 is the dome edge, L=S ⬍ 1 is within the dome and L=S ⬎ 1 is on the flat surface surrounding the dome. The displayed error bars correspond to the minimum and maximum variation of
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5. Determination of the fibre volume fraction distribution of the draped fabrics The next step was to calculate local changes in the fibre volume fraction due to the inhomogeneous shear distribution on the draped fabric. Let us consider an undeformed, rectangular fabric element of sides a and b (see Fig. 5). During draping it is considered that the fabric element deforms following pure shear, the length of its sides a and b remains unchanged during shear, there is no compression and the element thickness, t, remains unchanged. In this case, taking into account that the element mass remains the same before and after shear, it can be derived that Fig. 11. Plots of the values of the shear angle, g , along the diagonal fabric direction as a function of the normalised arc distance L=S from the apex for the four woven fabrics draped over a hemispherical dome surrounded by a flat base.
measurements from the average for each quadrant. The four fabrics in Fig. 9 display the same general features but their deformation data are not identical. The grid element angle starts from values close to 90⬚ in the zero-deformation zone at the apex of the dome L=S 0 and decreases to a minimum angle around L=S 1 at the edge of the dome where the maximum shear occurs. Thereafter, deformation starts declining progressively and this continues up to the end of the fabric. From the value of the angles of the grid elements Fig. 10, the local shear angles, g , can be calculated as g p=2 ⫺ a: Fig. 11 presents the corresponding data of shear angle for the loose plain weave (basket weave), tight plain weave, twill and satin weave against the normalised arc distance L=S from the apex along the diagonal direction of each fabric. The shear angle rises from 0 at the apex of the dome to a maximum around L=S 1 and then it falls again on the flat surface along the tails of the fabric. The loose plain weave (basket weave) and the twill weave exhibit the highest shear angle of 56⬚, followed by the satin weave which has a maximum shear angle of 52⬚ and lastly the tight weave with a maximum shear deformation angle of 48⬚. At points where L=S ⬎ 1 the flat section begins and the fabrics experience less deformation resulting in low shear angle values.
Table 3 Calculated initial and maximum (after drape) fibre volume fractions for each draped weave Fabric
Vfo
Vfmax
Vfmax/Vfo
LPW TPW TWILL 5HSW
0.36 0.44 0.46 0.50
0.64 0.66 0.83 0.82
1.77 1.50 1.80 1.64
Vf
Vfo Vfo sin a cos g
1
where Vfo is the fibre volume fraction of the unsheared element and Vf is the fibre volume fraction after shearing. The values of Vfo can be obtained from the expression Vfo
N ra tr
2
where, r a is the areal density of the weave (kg/m 2), N is the number of plies (in this case, N 1), t is the ply thickness (m), and r is the density of fibre glass (kg/m 3). By using Eqs. (1) and (2) and data from Table 1, the distributions of the fibre volume fractions of the sheared grid elements of the draped fabrics were calculated. Table 3 summarises the calculated initial fibre volume fractions, Vfo, of the unsheared weaves used in the analysis and the maximum fibre volume fractions calculated from the shear data of the drape experiments. Fig. 12 shows the curves of the calculated fibre volume fractions of the sheared grid elements along the diagonal direction for the four weaves, as functions of the normalised arc distance L=S from the apex of the dome. The curves start from Vfo at the apex of the dome, the zero-deformation zone, and rise to a maximum around L=S 1; where the maximum shear occurs in the fibre yarn directions. Then the fibre volume fraction falls along the diagonal line of the flat part of the fabric. Compression of weaves results to a maximum fibre volume fraction, around 0.62–0.65 for through thickness compression [17,18], corresponding to the packing limit. Similar values for the packing limit in in-plane compression have been considered in the modelling of the in-plane shearing of these fabrics [8]. Beyond this limit it is reasonable to expect that in in-plane compression wrinkling or ply thickening will occur. The loose plain weave reaches a maximum calculated Vf close to this limit after draping. However, the calculated maximum Vf for the tight plain weave, twill and satin clearly exceed the maximum acceptable Vf and, hence, they exhibit wrinkling after being draped in the hat mould geometry. A thick line around Vf 0:65 [8,17,18] in Fig. 12 represents the plateau of the approximately acceptable maximum Vf. The corresponding cells of unacceptable Vfmax after draping as calculated by relation (1) are bold in
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Fig. 12. Graphs of the calculated fibre volume fraction along the diagonal direction of the four tested fabrics against the normalised arc distance L=S from the apex, where the woven fabrics were draped over a hemispherical mould surrounded by a flat rim. Thick line represents approximate maximum acceptable Vf.
Table 3. It must also be pointed out that after wrinkling the assumptions associated with relation (1) are unlikely to hold and also the angles between gridlines might not represent the local angles between crossing fibre yarns.
6. Determination of the stiffness distribution in draped fabric laminates Changes in the fibre orientation and the fibre volume fraction in the draping of fabrics result in an inhomogeneous distribution of mechanical properties for the composite part. In this section, the distribution of the reduced stiffness matrix components of a fabric laminate will be evaluated on the basis of the laminated plate theory (for example [19]), where the laminate has the form of a hemispherical hat. Each ply of woven fabric is modelled as a combination of two unidirectional layers (see Fig. 13), at 0 and 90⬚ for an unsheared fabric, and at 0 and a degrees for a fabric sheared by an angle g . The effect of crimp and the type of weave are not incorporated in this type of theoretical analysis.
Fig. 13. Representation of a woven fabric by a combination of two plies of unidirectional fibres in unsheared and sheared state. Note that the interfibre spacing also changes on shearing (not shown in this figure).
Each layer of unidirectional fibres in Fig. 13 has mechanical properties depending on the principal directions 1 and 2 (where 1 is the fibre direction and 2 is perpendicular to the fibre direction) and on the fibre volume fraction. The elastic properties of E-glass have been taken [20] as Ef 72 GPa; nf 0:3 and Gf 27:7 GPa; where Ef is the fibre Young’s modulus, n f is the fibre major Poisson’s ratio and Gf is the fibre shear modulus. As an example, the matrix of the laminate was assumed for be an epoxy with the following elastic properties [20]: Em 3:5 GPa; nm 0:35 and Gm 1:3 GPa; where subscript m refers to the matrix. Assuming that the fibre volume fraction in each unidirectional fibre ply changes during shear by the same amount as the overall fibre volume fraction of the fabric, the effect of the change of Vf during shear on the elastic properties of each unidirectional fibre ply is given by the following equations [19], where subscripts f and m refer to the fibre and matrix, respectively: E1 Ef Vf ⫹ Em 1 ⫺ Vf
3
(Rule of mixtures) 1 V 1 ⫺ Vf f ⫹ Ef Em E2
4
n12 nf Vf ⫹ nm 1 ⫺ Vf
5
1 V 1 ⫺ Vf f ⫹ G12 Gf Gm
6
where the void content of the laminate is assumed to be zero Vf ⫹ Vm 1: The influence of the change in fibre orientation during the draping of a fabric required for the calculation of the modulus of the composite product can be ascertained by considering the rectilinear system of coordinates x; y in relation to the system of principal directions (1,2) for each ply in unsheared and sheared state as is illustrated in Fig. 13.
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following relations [19] Q 11 Q11 m4 ⫹ 2 Q12 ⫹ 2Q66 n2 m2 ⫹ Q22 n4 Q 22 Q11 n4 ⫹ 2 Q12 ⫹ 2Q66 n2 m2 ⫹ Q22 m4 Q 12 Q11 ⫹ Q22 ⫺ 4Q66 n2 m2 ⫹ Q12 n4 ⫹ m4 Q 66 Q11 ⫹ Q22 ⫺ 2Q12 ⫺ 2Q66 n2 m2 ⫹ Q66 n4 ⫹ m4 Fig. 14. The effect of the Vf change on E1 and E2, the elastic moduli in the principal directions of a unidirectional E-glass fibre ply in the draping of the loose plain weave (basket weave) over a hemispherical dome surrounded by a flat rim.
Q 16 Q11 ⫺ Q12 ⫺ 2Q66 nm3 ⫺ Q22 ⫺ Q12 ⫺ 2Q66 n3 m Q 26 Q11 ⫺ Q12 ⫺ 2Q66 n3 m ⫺ Q22 ⫺ Q12 ⫺ 2Q66 nm3 10
According to Fig. 13, the system of coordinates , x; y is aligned with the principal directions (1,2) of the 0⬚ ply both in unsheared and sheared state, whereas the 90⬚ ply changes direction with respect to the x; y system of coordinates after shear. The relationship between in-plane stress and in-plane strain in each ply is expressed by
where m cos a and n sin a: Once the components of the transformed reduced stiffness matrix are calculated for each equivalent unidirectional fibre ply of the woven fabric laminate model, the transformed reduced stiffness matrix of the complete “fabric laminate” under loading in the x; y directions is given by the relation
2
Q ij fabric 0:5 Q ij 0 ⫹ Q ij a
2 Q11 6 7 6 6 s y 7 6 Q 12 4 5 4 s xy Q 16
sx
3
Q 12 Q 22 Q 26
32 3 ex Q 16 76 7 6 7 Q 26 7 5 4 ey 5 gxy Q 66
7
where s is the stress tensor, e is the strain tensor and Q is the transformed reduced stiffness matrix. The stiffness matrix of each unidirectional fibre ply in the woven fabric laminate model is defined as 2
Q11
6 Q ij 6 4 Q12 Q16
Q12
Q16
7 Q26 7 5
Q26
Q66
8
where Q11, Q12, Q22 and Q66 are called the reduced stiffnesses and can be expressed in terms of the elastic properties of a unidirectional fibre ply [19] as: Q11 E11 = 1 ⫺ n12 n21 Q12 n21 E11 = 1 ⫺ n12 n21
which originates from the relation giving the laminate force matrix, N Nlaminate
N Zhn X n1
hn ⫺ 1
s n dz
N Zhn X n1
⫹
hn ⫺ 1
n e0 dz Q
N Zhn X n1
hn ⫺ 1
n kz dz Q 12
3
Q22
11
Q22 E22 = 1 ⫺ n12 n21 Q66 G12
9
Q16 Q61 0 The stiffness components of the 0⬚ ply remain the same in the unsheared and sheared state since the 0⬚ ply is aligned with the x; y system and does not change orientation after shear. The a degree ply is at 90⬚ with respect to the x; y system in the unsheared state and at a degrees with respect to the x; y system in the sheared state. The transformed reduced stiffness matrix components of the a degree ply in an off axisloading test in the x and y directions are given by the
where the laminate consists of N unidirectional fibre plies, n is the trans(hn –hn⫺1) is the thickness of each ply, Q formed reduced stiffness matrix of the n ply, e 0 is the mid-plane strain and [k] is the mid-plane plate curvature. By assuming that the fabric thickness is very small, the second term of the sum in Eq. (12) is not so significant. By also assuming that each of the warp and weft unidirectional fibre plies in a woven fabric model has the same thickness and fibre volume fraction, the definition of the transformed reduced stiffness matrix of the laminate is established and relation (11) is derived. The above analysis and calculation procedure of the major components of the transformed reduced stiffness matrix of a sheared fabric loaded in the x; y directions was applied to the loose plain weave (basket weave) draped over the hemispherical hat mould. This fabric was chosen because it draped without wrinkling and the evaluated values of fibre volume fraction according to relation (1) did not exceed the maximum packing fraction (see Table 3). The weave was considered balanced (see Table 1) and the fabric was considered as in-plane symmetric material. First the effect of the change of Vf due to shear during draping was incorporated in the elastic properties in the principal directions (1,2) of each equivalent unidirectional
U. Mohammed et al. / Composites: Part A 31 (2000) 1409–1420
Fig. 15. The reduced stiffnesses of the 0⬚ unidirectional fibre ply in the draping of the loose plain weave (basket weave) over a hemispherical dome surrounded by a flat rim.
fibre ply, according to Eqs. (3)–(6). Fig. 14 presents the effect of the Vf change on E1 and E2. For the composite component, it is evident that E1 is the most important modulus value. It increases as one moves away from the apex along the diagonal direction of the fabric, it reaches a maximum value (maximum Vf) due to maximum shear at the edge of the hemisphere and then it falls gradually on the flat surface. E2 is low for the equivalent unidirectional Eglass fibre ply where, as mentioned earlier, an epoxy matrix has been considered in this case-study. After the effect of the Vf during shear has been taken into account, the major reduced stiffnesses for a unidirectional E-glass fibre ply can be calculated from relations (9). (Notice that n21 n12 E2 =E1 [19]). These stiffness values are directly applicable to the 0⬚ ply and are presented in Fig. 15. Again it can be seen that Q11 is the most important stiffness component, which corresponds to E1 and follows its trend due to the effect of changing Vf across the draped ply. The transformed reduced stiffnesses of the a degree unidirectional fibre ply were evaluated from Eq. (10) which expresses the effect of the change in fibre orientation. The effect of change in the fibre volume fraction after drape was also taken into account in the values of the reduced stiffnesses [Qij] in the principal directions. Hence Fig. 16 presents the transformed reduced stiffnesses of the a degree unidirectional fibre ply of the loose plain weave fabric (basket weave) draped over the hemispherical hat for loading in the x; y directions (see Fig. 13), incorporating the
Fig. 16. The transformed reduced stiffnesses of the a degree unidirectional fibre ply in the draping of the loose plain weave (basket weave) over a hemispherical dome surrounded by a flat rim. Loading is in the x; y directions (see Fig. 12).
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effects of changing Vf and changing fibre direction during shear. It can be seen that at the apex L=S 0 Q 22 is maximum and is the important reduced stiffness component whereas the other reduced stiffnesses are negligible. At this location a 90⬚ and Q 22 corresponds to the modulus along the orthogonal fibre direction (see Fig. 13). As one moves away from the apex along the diagonal direction of the fabric, the fibre orientation changes due to shear and, as a result Q 22 decreases whereas Q 11 increases reaching a maximum at the edge of the hemisphere (around L=S 1) where the maximum shear occurs: consequently change in fibre orientation takes place. By using Eq. (11) the transformed reduced stiffnesses of the “fabric laminate” were calculated as a combination of the corresponding stiffnesses of the 0 and a degree unidirectional fibre plies, when the loading is applied in the x; y directions (see Fig. 13). Fig. 17 presents the reduced stiffnesses of the loose plain weave (basket weave) fabric along the normalised diagonal distance L=S outwards from the apex when the fabric is draped over the hemispherical hat. These stiffnesses are with respect to loading in the x; y directions (see Fig. 13). 7. Conclusions In this study, there have been presented a set of data on the deformation of fabrics draped over a hemispherical dome surrounded by a flat base. These data may be used to assess fabric drapeability and to validate computer simulations of draping. Four different types of fabric were examined: a loose plain weave, a tight plain weave, a twill and a satin weave. The four types of fabrics presented different drapeability in terms of wrinkling, with the tight plain weave exhibiting the worst drapeability. Differences in the deformation behaviour between the four fabrics resulted in a certain extent of differences in the distribution of local shear angles. The obtained data of local gridline orientation and local shear angle were used to first evaluate the distribution of local fibre volume fractions in each draped fabric. It was found that in regions of high shear, in some of the fabrics, the fibre volume fraction reached the packing limit and relation (1) could not be employed any more. It was thought
Fig. 17. The reduced stiffnesses of the loose plain weave (basket weave) fabric draped over a hemispherical dome surrounded by a flat rim. Loading is in the x; y directions (see Fig. 12).
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that, when this occurs, wrinkling may be expected for the draped fabric. The results of fibre volume fraction and shear angle distributions were used to calculate components of the transformed reduced stiffness matrix of a formed fabric laminate following a laminated plate theory analysis. References [1] Boisse P, Cherouat A, Gelin JC, Sabhi H. Experimental study and finite element simulation of a glass fibre fabric shaping process. Polymer Composites 1995;16(1):83–95. [2] Mohammed U, Lekakou C, Bader MG. Mathematical and experimental studies of the draping of woven fabrics in resin transfer moulding (RTM). In: Visconti IC, editor. Proceedings of ECCM-8, vol. 2. Cambridge: Woodhead Publishers, 1998. p. 683–90. [3] Dong L, Lekakou C, Bader MG. Solid mechanics draping simulations of woven fabrics. ICCM-12, Paris, 1999. [4] Dong L, Lekakou C, Bader MG. Submitted for publication. [5] Dong L, Lekakou C, Bader MG. Submitted for publication. [6] Martin TA, Bhattacharyya D, Collins IF. Bending of fibre-reinforced thermoplastic sheets. Composites Manufacturing 1995;6(3/4): 177–87. [7] Yu JZ, Cai Z, Ko FK. Formability of textile preforms for composite applications. Part 1: characterisation experiments. Composites Manufacturing 1994;5(2):113–22. [8] Mohammed U, Lekakou C, Dong L, Bader MG. Shear deformation and micromechanics of woven fabrics. Composites A 2000; 31A(4):299–308. [9] Tan Z, Melin L, Magnusson C. Application of an image processing technique in strain-measurement in sheet-metal forming. Journal of Materials Processsing Technology 1992;33(3):299–310.
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