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A model for the in-plane permeability of triaxially braided reinforcements
Composites: Part A 42 (2011) 165–172
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Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
A model for the in-plane permeability of triaxially braided reinforcements A. Endruweit ⇑, A.C. Long Faculty of Engineering (M3) – Division of Materials, Mechanics & Structures, University of Nottingham, University Park, Nottingham NG7 2RD, UK
a r t i c l e
i n f o
Article history: Received 8 June 2010 Received in revised form 1 November 2010 Accepted 2 November 2010
Keywords: A. Fabrics/textiles C. Analytical modelling E. Braiding E. Resin transfer moulding (RTM)
a b s t r a c t For multilayer preforms from three different triaxial carbon fibre braids with bias angles 45°, 60° and 70° at fibre volume fractions of 0.54 and 0.59, the in-plane permeability was characterised experimentally. A finding of high practical relevance is that, at a bias angle of approximately 55°, the principal flow direction switches from the preform 0° direction to the preform 90° direction. For different zones of characteristic yarn arrangement in the braid unit cell, the local permeability was modelled as a function of bias angle, global fibre volume fraction, and geometrical yarn parameters. The global braid permeability was derived from numerical flow simulation based on simplified 2D models of the textile architecture and local permeabilities, taking into account effects of nesting between adjacent layers. Calculated trends for the permeability as a function of the braid angle reproduce the experimental results qualitatively well. Quantitatively, agreement depends on the value of a geometry parameter, which decreases with increasing fibre volume fraction. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Braiding technology allows the automated fabrication of twodimensional (2D) tubular and three-dimensional (3D) fibrous structures with complex geometries at high reproducibility. This study focuses on 2D tubular braids, in particular triaxial, which can be used as preforms in the production of hollow composite components or components with a lightweight core via Liquid Composite Moulding (LCM). The arrangement of interlacing continuous yarns in an oriented pattern results in high stability of the preform and high strength of the finished component. For composites with different types of tubular braided preforms, the mechanical properties are discussed in a variety of studies, for which only a few examples are given here. Stiffness and strength of composites with biaxially braided preforms were determined experimentally, e.g., by Charlebois et al. [1]. Ayranci and Carey [2] reviewed meso- and macro-scale models for stiffness prediction. Smith and Swanson [3] studied the strength of composites with triaxially braided preforms experimentally. Failure in tension and damage development was studied by Ivanov et al. [4] and Littell et al. [5] and related to the braid meso-scale structure. The manufacture of braided preforms was described in detail by Ko et al. [6]. They discussed the dependence of the yarn arrangement, which can be used for prediction of mechanical properties, on the braiding parameters. The impregnation of braided preforms with a liquid resin was investigated by Sayre and Loos [7]. They found good agreement between experimental observations and ⇑ Corresponding author. Tel.: +44 0115 95 14037. E-mail address: [email protected] (A. Endruweit). 1359-835X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2010.11.003
injection simulations in terms of flow patterns and injection times in vacuum assisted resin transfer moulding for stitch-bonded layers of triaxial braids. Information on the permeability of 2D braided preforms, which determines the impregnation behaviour, is sparse. Antonucci et al. [8] monitored the flow front propagation in resin film infusion of two different braids with undisclosed architecture using fibre optic sensors and presented results for the through-thickness permeabilities calculated based on the measured data. Long [9] conducted radial flow experiments on biaxial braids and found that for a ±45° braid, the dependence of the permeability on the fibre volume fraction follows the trend predicted by the Kozeny– Carman relationship [10]. A decrease in permeability along and perpendicular to the mandrel direction with increasing braid angle at constant fibre volume fraction was explained by the reduction of inter-bundle void space. Charlebois et al. [1] characterised the permeability of biaxial braids with fibre angles of ±35°, ±45° and ±50° in unidirectional flow experiments along the mandrel direction. However, all preforms in their study were braided at ±45°, and fibre angles of ±35° and ±50° were adjusted by shearing. This procedure has a significant effect on the yarn spacing, and preforms at a given fibre angle obtained this way are not equivalent to preforms braided at the corresponding angle. The observed decrease in permeability with increasing fibre volume fraction was less strong at a fibre angle of ±35° than at ±45° and ±50°. This suggests a correlation with the packing density, which was highest at ±35°, near the braid locking angle. At constant fibre volume fraction, the permeability had a maximum at a fibre angle of ±45°. This was explained by the influence of yarn spacing and yarn alignment relative to the injection direction.
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In this study, the local in-plane permeability of triaxially braided carbon fibre preforms is modelled as a function of the preform geometry parameters, which are correlated to the meso-scale fibre arrangement. Local permeabilities are mapped onto a 2D model of the braid unit cell, for which impregnating flow can be simulated numerically using commercial software, thus allowing the global braid permeability to be determined. This simplified 2D approach eliminates detailed geometry modelling and solution of the flow problem in 3D, which is typically challenging in terms of meshing capabilities and computing resources. In addition, effects of nesting, which are expected to be significant here, can be easily incorporated in the model. For comparison with the numerical results, the permeability is determined experimentally at different braid angles and fibre volume fractions. 2. Preforms 2.1. Braid architecture Triaxial carbon fibre tubes comprising 0° yarns, oriented along the tube axis, and bias yarns oriented at angles ±b relative to the 0° direction, were braided on a cylindrical mandrel. The bias angle can be adjusted by setting the braid feed, i.e. the ratio between mandrel translational speed and braider rotational velocity. The bias yarns form a regular braid, which has a similar fibre arrangement as a 2 2 twill weave fabric [6]. The 0° yarns, which are embedded between the bias yarns, are straight and show no crimp when the braid is uncompressed. On the other hand, crimp of the bias yarns is increased compared to biaxial braids, because of the presence of the 0° yarns. For characterisation of the braid properties, the tubes were sliced and unrolled to produce flat sheets. The braid architecture is illustrated in the geometry model [11] in Fig. 1. The spacing between the axes of adjacent 0° (axial) yarns, xa, is given by the width of the rectangular flattened braid, W, and the number of yarns, Na. The spacing between the axes of adjacent bias yarns is
xb ¼
2W cos b; Nb
ð1Þ
where Nb is the number of bias yarns. A factor 2 is introduced since Nb refers to the total number of yarns in direction of ±b. As illustrated in Fig. 1 and described in Table 1, the textile unit cell can be decomposed into zones of different local fibre arrangement. As in biaxial braids, adjacent bias yarns make contact if the spacing corresponds to the (uncompressed) yarn width wb0, i.e. if the bias angle is
xβ β
Table 1 Yarn arrangement in different zones of unit cell. Zone
Yarns
1 2+/2 3 4 5+/5 6
None One bias (orientation b or b) Two bias (orientation b and b) One 0° yarn 0° and one bias (orientation b or b) 0° and two bias (orientation b and b)
b0 ¼ arccos
wb0 Nb : 2W
ð2Þ
If b is equal to b0, zones 1, 2, 4 and 5 in Fig. 1 vanish. If the bias angle exceeds b0, the yarns are compressed laterally. Assuming elliptical yarn cross-section, the width wb is reduced at constant yarn thickness tb0, until a value
wb1 ¼
4R2f cf ; tb0 V bmax
ð3Þ
where Rf is the filament radius, and cf is the yarn filament count, is achieved. This corresponds to an angle
! 2R2f cf Nb ; b1 ¼ arccos Wt b0 V bmax
ð4Þ
which is determined by the maximum achievable fibre volume fraction in the yarn, Vbmax. If wb is reduced beyond wb1, Vbmax stays constant while the yarn thickness increases because of (out-of-plane) reordering of filaments, according to
tb ¼
4R2f cf : wb V bmax
ð5Þ
These purely geometrical considerations are based on the assumption that the yarns are parallel and undergo uniform lateral compression. Forces in the yarns, in particular axial tension determined by the braiding process, and constraints on the thickness are not taken into account. Deviations from the described ideal situation may also occur because of crimp, and because twisting may allow partial overlap of adjacent yarns. Here, multilayer preforms from braids with three bias angles, 45°, 60°, 70°, as illustrated in Fig. 2, were characterised experimentally. All braids comprise 72 yarns with filament count cf = 12 K in 0° direction and 144 bias yarns with cf = 6 K. At a value of W of approximately 0.35 m and wb0 as listed in Table 2, the angle b0 is approximately 68° according to Eq. (2). This implies that in the preforms with a bias angle of 70°, inter-yarn voids are closed. Because of lateral compression, the yarn cross-sections are changed and the fibre volume fraction in the yarns is increased. The theoretical superficial densities of the braids,
Nb kb 1 ; S0 ¼ Na ka þ cos b W
ð6Þ
where ka and kb are the linear densities of yarns along the 0° direction and of bias yarns (Table 2), are 395 g/m2, 490 g/m2 and 639 g/ m2. The corresponding values measured by weighing specimens of given dimensions differ by less than 3% from these values. 2.2. Compression and multilayer effects
y xa
x Fig. 1. Unit cell of a triaxial braid (for the example of b = 60°), decomposed into zones of different basic yarn arrangement.
As discussed in detail, e.g. by Potluri and Sagar [12], compression mechanisms for a single fabric layer are change of crimp by fibre bending, closing of gaps in the fabric structure, and flattening of the yarns. All of these are affected by the yarn packing density, which, in braids, increases with increasing bias angle as implied in
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β = 45°
β = 60°
β = 70°
Fig. 2. Triaxial braids with different bias angles b; field of view 42 mm 43 mm.
the inter-yarn gaps on the braid surfaces are closed, i.e. if b > b0. It depends on the distance xa between the 0° yarns and thus affects all preforms described above equally.
Table 2 Properties of the different yarns in the braids: filament count cf, linear density k (measured by weighing yarns at given length), average yarn width w0 (measured from photographs), average yarn height t0 (measured using a Vernier caliper at minimum applied pressure).
Bias 0°
cf (K)
k (g/m)
w0 (mm)
t0 (mm)
6 12
0.392 0.810
1.85 2.82
0.35 0.43
3. Permeability modelling 3.1. Single layer For each zone in the decomposed unit cell of the braid (Fig. 1), the local through-thickness fibre arrangement is approximated as illustrated in Fig. 3 (one braid layer). The effective local in-plane permeability of a zone can be estimated based on the permeability of the fibre bundles at the given orientations and the equivalent permeabilities of voids [14]. The height of each zone corresponds to the thickness of the braid, which is determined by the sum of the yarn thicknesses. If the braid does not undergo any throughthickness compression,
Eq. (1). An additional effect may occur in triaxial braids with wide spacing of bias yarns. In through-thickness compression, the initially straight 0° yarns may crimp because of forces exerted by the bias yarn on the one side, and fill spaces between bias yarns on the other side, thus contributing to the layer compressibility. In multi-layer specimens, packing density and compression behaviour are determined by nesting between adjacent layers. Nesting generally refers to (partial) filling of inter-yarn voids on the surface of a layer by yarns of an adjacent layer. Depending on the fabric architecture, in particular the yarn packing density in each layer [13], and layer alignment, this effect can be significant. For braids, nesting of the bias yarns on the surfaces is significant only if b < b0. Based on numerical experiments, Lomov et al. [13] concluded that, because of the presence of 0° fibres, nesting is less significant for triaxial braids than for biaxial braids. However, in triaxial braids, significant periodic thickness variations across the unit cell, caused by the inserted 0° yarns, give rise to the additional effect of ‘‘structural’’ nesting. Zones of minimum braid thickness of a layer are partially filled with zones of maximum thickness of an adjacent layer. This effect can occur even if
(tβ+ta)/2
ta/2
t0 ¼ t a0 þ 2t b1 :
ð7Þ
The thickness ta0 of the uncompressed 0° yarns is given in Table 2. If b < b1, the thickness of the bias yarns is tb0 (Table 2), and the fibre volume fraction, Vbb1, is determined from tb0 and wb. Otherwise, Vbb1 is constant at a value of Vbmax, and the bundle thickness increases from tb0 to tb1 according to Eq. (5) because of lateral compression. In through-thickness compression, changes of the yarn crosssection along the yarn axes and potential crimping of the 0° yarns are neglected. The decrease in thickness from the initial value t0 to the compressed value t is assumed to be distributed equally to 0° and bias yarns, i.e.
+/– β
tβ
t
0°
(tβ+ta)/2
1
2
3
–/+ β
tβ
tβ
ta/2
4
5
6
Fig. 3. Idealised through-thickness arrangement of yarns in the different zones of the unit cell (blank space indicates void); height of voids is indicated for each zone.
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t b ¼ t b1
t t0
and t a ¼ ta0
t : t0
ð8Þ
The yarn widths wa and wb are assumed to stay constant. For the compressed 0° yarns, the fibre volume fraction, Vba, is determined in analogy to Eq. (5) from ta and wa0 (Table 2). Similarly, the fibre volume fraction in the bias yarns, Vbb, is calculated from tb and wb. If any of the calculated values is greater than Vbmax, it is replaced by Vbmax, implying a change in yarn cross-sectional shape in compression. Vbmax is assumed to be identical for bias and 0° yarns. The principal permeability values k1i (parallel to its axis) and k2i (perpendicular to its axis) of a fibre bundle (aligned filaments with radius Rf) with fibre volume fraction Vbi, are calculated based on the equations derived by Gebart [15],
k1i ¼
R2f ð1 V bi Þ3 4c1 V 2bi
and k2i ¼ c2
sffiffiffiffiffiffiffiffiffiffiffiffi !5=2 V btheo R2f ; 1 V bi
ð9Þ
where c1, c2 and Vbtheo are geometrical constants. The permeabilities of the bias and 0° yarns can be determined by substituting Vbb and Vba for Vbi. Because of the yarn cross-sectional shapes, voids occur in each layer of the zones illustrated in Fig. 3, even if adjacent bundles are in contact. Since the geometry of the voids is complex, there is no analytical solution for their influence on the layer permeability. It is estimated based on the simplified geometry illustrated in Fig. 4. For each type i of yarn, the total cross-sectional area of voids is represented by rectangular gaps with total width
Dwi ¼ wi
pR2f cfi V bi ti
ð10Þ
:
The void space is assumed to be divided into two gaps with widths Dwi/2 and height ti (Fig. 4). The equivalent permeability of each gap along the yarn is estimated by the first term of the series suggested by Ni et al. [16] for description of ducts with rectangular cross-section, kei ¼ fc
8t 2i ðDwi =2Þp4
2
2
!
Dwi ðcoshðpDwi =ð2t i ÞÞ 1Þ sinh ðpDwi =ð2ti ÞÞ þ : 2 2 ðp=t i Þsinh ðpDwi =ð2t i ÞÞ
Perpendicular to the yarn axis, the influence of the gaps is small. Thus, for each bounding box as in Fig. 4, the effective principal permeability values can be estimated by
K 1i ¼
wi Dwi Dwi wi k2i k1i þ kei and K 2i : wi wi wi Dwi
The in-plane permeability of a zone containing M different material layers, each with thickness ti, is estimated by thicknessweighted averaging:
K¼
M X ti Ki ðDai Þ: t i¼1
ð13Þ
Here, Dai is the angle included by the axis of the yarn in layer i, oriented in the direction ai, and a reference direction, described by a0. Thus, the in-plane permeability tensor of layer i in reference coordinates is 2
Ki ðDai Þ ¼
K 1i cos2 Dai þ K 2i sin Dai
ðK 2i K 1i Þ sin Dai cos Dai
ðK 2i K 1i Þ sin Dai cos Dai
K 1i sin Dai þ K 2i cos2 Dai
:
A layer containing no yarn is represented by a wide gap with height ti and characterised approximately by the isotropic equivalent permeability [16]
Ki ¼
t 2i 12
1 0 0
1
;
ð15Þ
For a gap with dimensions given by the values for w0 and t0 in Table 2, this differs by less than 2% from the permeability derived from the expression in Eq. (11). From thickness-weighted averaging of layer permeabilities as formulated in Eq. (13), the local zone permeabilities in reference co-ordinates with a0 = 0° are determined as
K1yx ¼
K2yx
K3yx
t2 12
0 1
ð16Þ
; 2
K 1b cos2 b þ K 2b sin b
ðK 2b K 1b Þ sin b cos b
ðK 2b K 1b Þ sin b cos b ðt b þ t a Þ3 1 0 ; þ 48t 0 1
K 1b sin b þ K 2b cos2 b
tb ¼ t
þ ta t
K 1b cos2 b þ K 2b sin b
0
0
K 1b sin b þ K 2b cos2 b
t 3a 48t
Δwi/2
!
2
2
2t b ¼ t
K4yx ¼
1 0
1 0 ; 0 1
K 1a
0
0
K 2a
þ
ð17Þ
!
2
t3b 6t
1 0 0
1
;
ti
wi
2
!
ð14Þ
ð11Þ
Truncation of the series was found to induce relative errors of less than 2% for the range of Dwi and ti that is relevant here. The correction factor fc introduced here describes the void geometry and is related to the hydraulic diameter [15], which describes the ratio between the void cross-sectional area and perimeter. Since the ratio of cross-sectional area and perimeter is smaller for a duct enclosed by a rectangle and an ellipse than for a rectangular duct with equivalent cross-sectional area (Fig. 4), fc is smaller than 1. The values of fc decrease with increasing degree of compression, i.e. changing yarn cross-section.
ð12Þ
wi - Δwi
Fig. 4. Yarn cross-section and bounding box; simplification of void geometry.
Δwi/2
ð18Þ
ð19Þ
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K5yx ¼
ta t
K 1a 0
0 K 2a
2
K 1b cos2 b þ K 2b sin b
ðK 2b K 1b Þ sin b cos b
ðK 2b K 1b Þ sin b cos b 1 0 ; þ 12t 0 1
K 1b sin b þ K 2b cos2 b
tb þ t
adjacent layers is xa/2 along the x-axis. This results in a complex overlap pattern of the different zones in the unit cells (Fig. 1) with thirteen property sets as indicated in Table 3. For values of b between 45° and b0, the average layer thickness (Fig. 5) is approximated by
!
2
b 45 ta þ ; t ¼ tb 1 þ b0 45 2
t 3b
ð20Þ K6yx ¼
ta t
K 1a
0
0
K 2a
2tb þ t
which replaces Eq. (7). This includes the following special cases, for which the permeability can be estimated analytically:
2
For b = 45°, structural nesting and bias yarn nesting are combined, and the average layer thickness is estimated as tb + ta/2. Inter-yarn void spaces are assumed to be filled with yarns of adjacent layers, and the permeability is assumed to be uniform across the braid unit cell. It can be estimated from Eq. (21), which applies if two nested layers with average thickness are considered. For b = b0, only structural nesting with an estimated average layer thickness 2tb + ta/2 occurs. Inter-yarn voids disappear, and only zones 3 and 6 remain in each individual layer. In analogy to Eq. (22), the global permeability can be estimated from the resulting strips of 3–6 overlap and 6–6 overlap in parallel (Ky) and in series (Kx). The same applies for b > b0.
!
K 1b cos2 b þ K 2b sin b
0
0
K 1b sin b þ K 2b cos2 b
2
; ð21Þ
where the superscripts refer to the numbers of the respective zones, and the indices y and x refer to the notation in Fig. 1. For the unit cell in Fig. 1, analytical calculation of the global permeability is generally not possible, in particular because of the non-zero off-diagonal elements of the local permeability tensors for zones 2 and 5. However, if b P b0, i.e. the bias yarns are in contact, only zones 3 and 6 remain in the unit cell. For flow along the braid axis (in y-direction, zones 3 and 6 in parallel), the flow rates in the zones add up to the total flow rate, and the rule of mixtures applies to the flow velocities. For flow along the circumference (in x-direction, zones 3 and 6 in series), the pressure differentials in the zones add up to the total pressure differential. The corresponding global permeabilities can be estimated as
Ky ¼
xa wa 3 wa 6 Ky þ K xa xa y
and K x ¼
xa K 3x K 6x
ð23Þ
For all other values of b, the braid permeability can be determined numerically. 4. Results and discussion 4.1. Experimental results
; 6
wa K 3x þ ðxa wa ÞK x
ð22Þ
For the triaxially braided preforms described above, the unsaturated in-plane permeability was measured in radial flow experiments at constant injection pressure. As described in detail elsewhere [17], circular fabric specimens with a radius of 200 mm were placed in an injection tool, which is made from aluminium with an additional stiffening steel structure to minimise deflection. A test fluid, engine oil with given viscosity, was injected through a central circular injection gate with a radius of 5 mm. The
respectively. 3.2. Multiple layers Eqs. (16)–(21) apply for a single braid layer. For more than one layer, nesting is modelled assuming that the shift between
Table 3 Nesting between adjacent layers, n and n + 1: x indicates overlap of different zones in unit cell (Fig. 1). Zones layer n 1 Zones layer n + 1
1 2+ 2 3 4 5+ 5 6
2+
2
3
4
5+
5 x
x x
x x
x
x x x x
x
x x
x
x
x
x
t
t
Fig. 5. Illustration of significant difference between layer thickness without nesting and average layer thickness with nesting.
6 x x x x x x x x
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For the 0°/±45° preforms, the properties are dominated by the 0° yarns, while the contribution of the bias yarns is isotropic. Ky and the ratio of anisotropy Ky/Kx decrease significantly with increasing Vf because of nesting of the bias yarns and closing of inter-yarn void space in compression. For the 0°/±60° and 0°/±70° preforms, the main flow direction is dominated by the bias fibres. Because there is less void space than in the 0°/±45° preforms, the changes in Ky and Kx due to compression are less significant, and Ky/Kx does not change significantly with increasing Vf.
0.8 0.7
Ky
K / 10
-10
m
2
0.6 0.5 0.4 0.3 0.2
Kx
4.2. Numerical results
0.1 0 40
45
50
55
60
65
70
75
β / deg Fig. 6. In-plane permeability values Ky and Kx (along the preform 0° and 90° directions) as functions of the bias angle b at a target fibre volume fraction Vf = 0.54; filled symbols: experimental results, mean values and standard deviations, based on three repeats, are indicated; open symbols: results from model at fc = 0.05 with trend lines.
0.45 0.40
0.30
K / 10
-10
m
2
0.35
Ky
0.25 0.20 0.15
Kx
0.10 0.05 0.00 40
45
50
55
60
65
70
75
β / deg Fig. 7. In-plane permeability values Ky and Kx (along the preform 0° and 90° directions) as functions of the bias angle b at a target fibre volume fraction Vf = 0.59; filled symbols: experimental results, mean values and standard deviations, based on three repeats, are indicated; open symbols: results from model at fc = 0.03 with trend lines.
flow front propagation along three co-planar axes was determined by evaluating the readings from an array of pressure transducers, which allows the principal permeability values and the orientation of the principal permeability axes to be determined from the flow front position as a function of time. For 0°/±45° preforms at six layers, 0°/±60° preforms at five layers and 0°/±70° preforms at four layers, two different target values for the global fibre volume fraction Vf, 0.54 and 0.59, were set by appropriate adjustment of the cavity height h. The measured permeabilities along the preform 0° and 90° directions, Ky and Kx (Fig. 1), are plotted in Figs. 6 and 7. At both fibre volume fractions, Ky decreases continuously with increasing bias angle. Kx shows a maximum at a bias angle of 60°, which results from the competing effects of increased flow in x-direction because of reorientation of the bias yarns (dominating for b < 60°) and decreasing inter-yarn void spaces (dominating for b > 60°). This behaviour is qualitatively similar to that observed for biaxial braids [9]. For b = 45°, Kx is lower than Ky, whereas for 60° and 70°, it is higher. This implies a switch of principal flow directions. Interpolation of the data suggests that, at the given braid parameters, isotropic flow behaviour can be expected at bias angles of approximately 55°.
The in-plane permeability of a triaxial braid can be estimated analytically only for b = 45° and b > b0 as discussed in Section 3.2. For any arbitrary braid angle, it can be determined from virtual injection experiments, i.e. simulation of uni-directional resin injection at constant injection pressure along the x- and y-direction of the braid unit cell (Fig. 1). Here, numerical results were produced for braid angles of 50°, 55°, 60° and 65° in addition to analytical solutions at b = 45° and b = 70°. For calculation of local permeability values as discussed above, the values of w0 and t0 for bias yarns and 0° yarns (Table 2) suggest that the initial fibre volume fraction in the yarns, Vb, is 0.45. The maximum fibre volume fraction in the yarns is estimated as Vbmax = 0.78 based on square packing. Values for the constants in Eq. (9) are given in Table 4. The small difference between Vbmax and Vbtheo was introduced to account for imperfect alignment of filaments and avoid k2i in Eq. (9) being zero, which could cause numerical problems. In analogy to the ‘‘grid average’’ method described by Wong et al. [18], local permeabilities, derived for the overlap pattern specific to each braid angle, were projected onto a 2D mesh as illustrated in Fig. 8. The equations describing unsaturated flow of a viscous liquid through a porous medium considering conservation of the fluid mass were solved based on a non-conforming finite element method [19], which is implemented in the PAM-RTM software (ESI Group). From the average time for the flow front to travel across the unit cell, the principal permeability value in the respective direction was determined from the one-dimensional formulation of Darcy’s law. It could be argued that 3D flow simulation, based on a detailed model of the fibre arrangement and void geometry, should in theory give more accurate results. However, this may be mitigated by the problem of accurately obtaining the actual geometry parameters, which are subject to variability in the braid structure and thus may induce new sources of uncertainty. In addition, accurate 3D modelling of the fibre structure may be limited by the meshing capabilities. Solving the 3D flow problem, which requires implementation of different equations on different domains (frequently used are Stokes equation for flow in inter-yarn voids, Brinkman equation for flow in yarns, as discussed, e.g., by Ngo and Tamma [20]), may require considerable computing resources, and problems of numerical stability may arise [21]. On the other hand, idealised values of the geometrical dimensions required as input for the 2D modelling approach can be determined based on the considerations in Section 2.1. Decomposition of the 2D geometry into zones of different yarn overlap configuration allows effects of nesting to be incorporated for simulation of flow in multi-layer stacks. Solution of the flow problem is simplified by assuming Darcy’s law to describe flow at any
Table 4 Filament radius Rf, geometry constants c1, c2 and Vbtheo (square packing [15]) and maximum achieved yarn fibre volume fraction Vbmax. Rf (mm)
c1
c2
Vbtheo
Vbmax
3.5 103
1.78
0.40
0.785
0.78
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Fig. 8. Finite element mesh of unit cell at b = 60° for flow simulation (with nesting); 1702 triangular finite elements; different colours indicate zones with different overlap configuration.
Table 5 Comparison of analytically calculated permeability values at b = 70° with and without nesting; experimental data are listed for reference. Vf
0.54 0.59
With
Without
Experiment
Ky (1010 m2)
Kx (1010 m2)
Ky (1010 m2)
Kx (1010 m2)
Ky (1010 m2)
Kx (1010 m2)
0.274 0.138
0.313 0.162
2.375 2.323
0.209 0.126
0.174 ± 0.038 0.117 ± 0.018
0.288 ± 0.099 0.205 ± 0.030
point in the model. Effective local permeability values for each of the thirteen zones can be calculated from the equations in Section 3. The advantages of using a commercial software package for solution of the flow problem are its availability, robustness and flexibility regarding input mesh data, the latter rendering it convenient for studying different braid configurations. While solution of steady-state flow at complete impregnation of the braid would be sufficient for permeability calculation, the commercial software simulates a complete virtual injection experiment involving multiple solutions at different time steps, thus allowing the flow front propagation to be tracked. However, the computational cost for 2D simulations is generally low. Here, computation times were less than one minute on a standard PC. Despite the simplifications inherent to the ‘‘grid average’’ method, Wong et al. [18] found that results compare favourably with results from more computationally intensive 3D numerical approaches. Analytical (b = 45°, 70°) and numerical results (b = 50°, 55°, 60°, 65°) for Vf = 0.54 at fc = 0.05 and Vf = 0.59 at fc = 0.03 are plotted in Figs. 6 and 7. Because of simplifications in geometrical modelling and uncertainties induced by the numerical solution, e.g. mesh-related issues and accuracy of determining the average flow front arrival time, the individual data points show considerable scatter. Simulations of the same scenario at different mesh densities (for the example of b = 65°, Vf = 0.54, fc = 0.05: 2915 elements and 9976 elements) suggest that the discretisation may result in uncertainties of the permeability of up to 5%. Another source of uncertainty is that approximations in the calculation procedure may induce small differences between the effective Vf and the target value of Vf, which is assumed for evaluation of the simulations. The induced difference between the permeability derived from the simulations and the analytical solution was found to be 7% for b = 45°. However, trend lines fitted to the data reproduce the experimental results qualitatively well. Most importantly, the change in principal flow direction, which was observed in the experimental results and is of highest practical relevance, is predicted at approximately the right values of b (60° and 58°). The values of fc were selected to achieve reasonable quantitative agreement of calculated trends and experimental results for both Ky and Kx at the respective fibre volume fractions. With increasing global Vf, i.e. increasing degree of bundle compression, the cross-sections of fibre bundles as indicated in Fig. 4 are expected to become more rectangular. This implies that the ratio of
cross-sectional area and perimeter of the voids, and thus fc, decreases. However, based on the available data, the dependence of fc on Vf cannot be clearly identified. The correlation of the global permeability with fc reflects that it is determined mainly by a network of inter-yarn gaps. It is to be noted that bundle cross-sectional shape and thus fc depend not only on through-thickness compression, but, at given bundle thickness, also on braid deformation, e.g. by shearing of the bias yarns. To give an impression about the significance of (structural) nesting on the results, analytical results from Eq. (22) for b = 70° with and without nesting at identical fc are compared in Table 5. The results for Ky are significantly higher without nesting than with nesting, because gaps in zones 3 (Fig. 3) facilitate flow along the y-direction. For Kx, the results without nesting are lower than those with nesting, because the level of yarn compression in zones 6, which is related to the average layer thickness and determines the flow along the x-direction, is higher than with nesting. This results in a change of principal flow direction, which is oriented along the y-direction if nesting is not considered.
5. Conclusions For triaxial braids comprising a given number of axial (0°) yarns and twice as many yarns oriented at the braid bias angle (±), the influence of the bias angle on the size of inter-yarn void spaces and yarn cross-sections was described. For analysis of the void space in multilayer preforms, the effect of nesting needs to be considered, which generally refers to partial filling of voids between bias yarns on the surface of a layer by yarns of an adjacent layer. An additional effect of ‘‘structural’’ nesting due to thickness variations in triaxial braids, which depend on the spacing between the 0° yarns, was postulated. For predictive estimation of the braid permeability, the braid unit cell was decomposed into zones of characteristic yarn arrangement. A set of equations was derived allowing description of the local permeability of each zone as a function of bias angle, braid thickness or global fibre volume fraction, and geometrical yarn parameters. A complex yarn overlap pattern, characterising the effect of nesting in multilayer preform, was identified. For three different carbon fibre braids with bias angles 45°, 60° and 70°, the in-plane permeability at multiple layers was charac-
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terised experimentally. Along the 0° direction, the permeability decreases continuously with increasing bias angle. Along the 90° direction, it shows a maximum at a bias angle of 60°. This results in a switch of the principal flow direction from the preform 0° direction for b = 45° to the preform 90° direction for b = 60° and b = 70°. Interpolation suggests isotropic flow behaviour at a bias angle of approximately 55°. The braid permeability was derived from numerical flow simulations based on simplified 2D models of the textile architecture and averaged local permeabilities. Calculated trends for the permeability as a function of the braid angle reproduce the experimental results qualitatively well. The change in principal flow direction was predicted with good accuracy. Quantitative agreement between numerical and experimental results depends on the appropriate choice of a geometry parameter, which is related to the yarn cross-section and decreases with increasing fibre volume fraction. Comparison of results generated with and without consideration of ‘‘structural’’ nesting indicates that this effect has a significant influence on the in-plane permeability, in particular along the preform 0° direction. Acknowledgements This work was funded by the Engineering and Physical Science Research Council (EPSRC) via the Nottingham Innovative Manufacturing Research Centre (EP/E001904/1). References [1] Charlebois KM, Boukhili R, Zebdi O, Trochu F, Gasmi A. Evaluation of the physical and mechanical properties of braided fabrics and their composites. J Reinf Plast Compos 2005;24(14):1539–54. [2] Ayranci C, Carey J. 2D braided composites: a review for stiffness critical applications. Compos Struct 2008;85(1):43–58. [3] Smith LV, Swanson SR. Strength design with 2-D triaxial braid textile composites. Compos Sci Technol 1996;56(3):359–65.
[4] Ivanov DS, Baudry F, Van Den Broucke B, Lomov SV, Xie H, Verpoest I. Failure analysis of triaxial braided composite. Compos Sci Technol 2009;69(9):1372–80. [5] Littell JD, Binienda WK, Roberts GD, Goldberg RK. Characterization of damage in triaxial braid composites under tensile loading. technical report NASA/TM2009-215645. Hanover: NASA Center for AeroSpace Information; 2009. [6] Ko FK, Pastore CM, Head AA. Handbook of industrial braiding. Atkins and Pearce; 1989. [7] Sayre JR, Loos AC. Resin infusion of triaxially braided preforms with throughthe-thickness reinforcement. Polym Compos 2003;24(2):229–36. [8] Antonucci V, Giordano M, Nicolais L, Calabro A, Cusano A, Cutolo A, et al. Resin flow monitoring in resin film infusion process. J Mater Process Tech 2003;143– 144:687–92. [9] Long AC. Process modelling for liquid moulding of braided preforms. Compos Part A – Appl Sci 2001;32(7):941–53. [10] Carman PC. Fluid flow through granular beds. T I Chem Eng – Lond 1937;15:150–6. [11] TexGen (Version 3.3.0). University of Nottingham; 2008. [viewed 02.09.09]. [12] Potluri P, Sagar TV. Compaction modelling of textile preforms for composite structures. Compos Struct 2008;86(1–3):177–85. [13] Lomov SV, Verpoest I, Peeters T, Roose D, Zako M. Nesting in textile laminates: geometrical modelling of the laminate. Compos Sci Technol 2003;63(7):993–1007. [14] Endruweit A, Harper LT, Turner TA, Warrior NA, Long AC. Random discontinuous carbon fiber preforms: experimental permeability characterization and local modelling. Polym Compos 2010;31(4):569–80. [15] Gebart BR. Permeability of unidirectional reinforcements for RTM. J Compos Mater 1992;26(8):1100–33. [16] Ni J, Zhao Y, Lee LJ, Nakamura S. Analysis of two-regional flow in liquid composite molding. Polym Compos 1997;18(2):254–69. [17] Endruweit A, McGregor P, Long AC, Johnson MS. Influence of the fabric architecture on the variations in experimentally determined in-plane permeability values. Compos Sci Technol 2006;66(11–12):1778–92. [18] Wong CC, Long AC, Sherburn M, Robitaille F, Harrison P, Rudd CD. Comparisons of novel and efficient approaches for permeability prediction based on the fabric architecture. Compos Part A – Appl Sci 2006;37(6):847–57. [19] Trochu F, Gauvin R, Gao DM. Numerical analysis of the resin transfer molding process by the finite element method. Adv Polym Technol 1993;12(4):329–42. [20] Ngo ND, Tamma KK. Complex three-dimensional microstructural permeability prediction of porous fibrous media with and without compaction. Int J Numer Meth Eng 2004;60(10):1741–57. [21] Mardal KA, Tai XC, Winther R. A robust finite element method for Darcy– Stokes flow. SIAM J Numer Anal 2002;40(5):1605–31.
Contents lists available at ScienceDirect
Composites: Part A journal homepage: www.elsevier.com/locate/compositesa
A model for the in-plane permeability of triaxially braided reinforcements A. Endruweit ⇑, A.C. Long Faculty of Engineering (M3) – Division of Materials, Mechanics & Structures, University of Nottingham, University Park, Nottingham NG7 2RD, UK
a r t i c l e
i n f o
Article history: Received 8 June 2010 Received in revised form 1 November 2010 Accepted 2 November 2010
Keywords: A. Fabrics/textiles C. Analytical modelling E. Braiding E. Resin transfer moulding (RTM)
a b s t r a c t For multilayer preforms from three different triaxial carbon fibre braids with bias angles 45°, 60° and 70° at fibre volume fractions of 0.54 and 0.59, the in-plane permeability was characterised experimentally. A finding of high practical relevance is that, at a bias angle of approximately 55°, the principal flow direction switches from the preform 0° direction to the preform 90° direction. For different zones of characteristic yarn arrangement in the braid unit cell, the local permeability was modelled as a function of bias angle, global fibre volume fraction, and geometrical yarn parameters. The global braid permeability was derived from numerical flow simulation based on simplified 2D models of the textile architecture and local permeabilities, taking into account effects of nesting between adjacent layers. Calculated trends for the permeability as a function of the braid angle reproduce the experimental results qualitatively well. Quantitatively, agreement depends on the value of a geometry parameter, which decreases with increasing fibre volume fraction. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Braiding technology allows the automated fabrication of twodimensional (2D) tubular and three-dimensional (3D) fibrous structures with complex geometries at high reproducibility. This study focuses on 2D tubular braids, in particular triaxial, which can be used as preforms in the production of hollow composite components or components with a lightweight core via Liquid Composite Moulding (LCM). The arrangement of interlacing continuous yarns in an oriented pattern results in high stability of the preform and high strength of the finished component. For composites with different types of tubular braided preforms, the mechanical properties are discussed in a variety of studies, for which only a few examples are given here. Stiffness and strength of composites with biaxially braided preforms were determined experimentally, e.g., by Charlebois et al. [1]. Ayranci and Carey [2] reviewed meso- and macro-scale models for stiffness prediction. Smith and Swanson [3] studied the strength of composites with triaxially braided preforms experimentally. Failure in tension and damage development was studied by Ivanov et al. [4] and Littell et al. [5] and related to the braid meso-scale structure. The manufacture of braided preforms was described in detail by Ko et al. [6]. They discussed the dependence of the yarn arrangement, which can be used for prediction of mechanical properties, on the braiding parameters. The impregnation of braided preforms with a liquid resin was investigated by Sayre and Loos [7]. They found good agreement between experimental observations and ⇑ Corresponding author. Tel.: +44 0115 95 14037. E-mail address: [email protected] (A. Endruweit). 1359-835X/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesa.2010.11.003
injection simulations in terms of flow patterns and injection times in vacuum assisted resin transfer moulding for stitch-bonded layers of triaxial braids. Information on the permeability of 2D braided preforms, which determines the impregnation behaviour, is sparse. Antonucci et al. [8] monitored the flow front propagation in resin film infusion of two different braids with undisclosed architecture using fibre optic sensors and presented results for the through-thickness permeabilities calculated based on the measured data. Long [9] conducted radial flow experiments on biaxial braids and found that for a ±45° braid, the dependence of the permeability on the fibre volume fraction follows the trend predicted by the Kozeny– Carman relationship [10]. A decrease in permeability along and perpendicular to the mandrel direction with increasing braid angle at constant fibre volume fraction was explained by the reduction of inter-bundle void space. Charlebois et al. [1] characterised the permeability of biaxial braids with fibre angles of ±35°, ±45° and ±50° in unidirectional flow experiments along the mandrel direction. However, all preforms in their study were braided at ±45°, and fibre angles of ±35° and ±50° were adjusted by shearing. This procedure has a significant effect on the yarn spacing, and preforms at a given fibre angle obtained this way are not equivalent to preforms braided at the corresponding angle. The observed decrease in permeability with increasing fibre volume fraction was less strong at a fibre angle of ±35° than at ±45° and ±50°. This suggests a correlation with the packing density, which was highest at ±35°, near the braid locking angle. At constant fibre volume fraction, the permeability had a maximum at a fibre angle of ±45°. This was explained by the influence of yarn spacing and yarn alignment relative to the injection direction.
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In this study, the local in-plane permeability of triaxially braided carbon fibre preforms is modelled as a function of the preform geometry parameters, which are correlated to the meso-scale fibre arrangement. Local permeabilities are mapped onto a 2D model of the braid unit cell, for which impregnating flow can be simulated numerically using commercial software, thus allowing the global braid permeability to be determined. This simplified 2D approach eliminates detailed geometry modelling and solution of the flow problem in 3D, which is typically challenging in terms of meshing capabilities and computing resources. In addition, effects of nesting, which are expected to be significant here, can be easily incorporated in the model. For comparison with the numerical results, the permeability is determined experimentally at different braid angles and fibre volume fractions. 2. Preforms 2.1. Braid architecture Triaxial carbon fibre tubes comprising 0° yarns, oriented along the tube axis, and bias yarns oriented at angles ±b relative to the 0° direction, were braided on a cylindrical mandrel. The bias angle can be adjusted by setting the braid feed, i.e. the ratio between mandrel translational speed and braider rotational velocity. The bias yarns form a regular braid, which has a similar fibre arrangement as a 2 2 twill weave fabric [6]. The 0° yarns, which are embedded between the bias yarns, are straight and show no crimp when the braid is uncompressed. On the other hand, crimp of the bias yarns is increased compared to biaxial braids, because of the presence of the 0° yarns. For characterisation of the braid properties, the tubes were sliced and unrolled to produce flat sheets. The braid architecture is illustrated in the geometry model [11] in Fig. 1. The spacing between the axes of adjacent 0° (axial) yarns, xa, is given by the width of the rectangular flattened braid, W, and the number of yarns, Na. The spacing between the axes of adjacent bias yarns is
xb ¼
2W cos b; Nb
ð1Þ
where Nb is the number of bias yarns. A factor 2 is introduced since Nb refers to the total number of yarns in direction of ±b. As illustrated in Fig. 1 and described in Table 1, the textile unit cell can be decomposed into zones of different local fibre arrangement. As in biaxial braids, adjacent bias yarns make contact if the spacing corresponds to the (uncompressed) yarn width wb0, i.e. if the bias angle is
xβ β
Table 1 Yarn arrangement in different zones of unit cell. Zone
Yarns
1 2+/2 3 4 5+/5 6
None One bias (orientation b or b) Two bias (orientation b and b) One 0° yarn 0° and one bias (orientation b or b) 0° and two bias (orientation b and b)
b0 ¼ arccos
wb0 Nb : 2W
ð2Þ
If b is equal to b0, zones 1, 2, 4 and 5 in Fig. 1 vanish. If the bias angle exceeds b0, the yarns are compressed laterally. Assuming elliptical yarn cross-section, the width wb is reduced at constant yarn thickness tb0, until a value
wb1 ¼
4R2f cf ; tb0 V bmax
ð3Þ
where Rf is the filament radius, and cf is the yarn filament count, is achieved. This corresponds to an angle
! 2R2f cf Nb ; b1 ¼ arccos Wt b0 V bmax
ð4Þ
which is determined by the maximum achievable fibre volume fraction in the yarn, Vbmax. If wb is reduced beyond wb1, Vbmax stays constant while the yarn thickness increases because of (out-of-plane) reordering of filaments, according to
tb ¼
4R2f cf : wb V bmax
ð5Þ
These purely geometrical considerations are based on the assumption that the yarns are parallel and undergo uniform lateral compression. Forces in the yarns, in particular axial tension determined by the braiding process, and constraints on the thickness are not taken into account. Deviations from the described ideal situation may also occur because of crimp, and because twisting may allow partial overlap of adjacent yarns. Here, multilayer preforms from braids with three bias angles, 45°, 60°, 70°, as illustrated in Fig. 2, were characterised experimentally. All braids comprise 72 yarns with filament count cf = 12 K in 0° direction and 144 bias yarns with cf = 6 K. At a value of W of approximately 0.35 m and wb0 as listed in Table 2, the angle b0 is approximately 68° according to Eq. (2). This implies that in the preforms with a bias angle of 70°, inter-yarn voids are closed. Because of lateral compression, the yarn cross-sections are changed and the fibre volume fraction in the yarns is increased. The theoretical superficial densities of the braids,
Nb kb 1 ; S0 ¼ Na ka þ cos b W
ð6Þ
where ka and kb are the linear densities of yarns along the 0° direction and of bias yarns (Table 2), are 395 g/m2, 490 g/m2 and 639 g/ m2. The corresponding values measured by weighing specimens of given dimensions differ by less than 3% from these values. 2.2. Compression and multilayer effects
y xa
x Fig. 1. Unit cell of a triaxial braid (for the example of b = 60°), decomposed into zones of different basic yarn arrangement.
As discussed in detail, e.g. by Potluri and Sagar [12], compression mechanisms for a single fabric layer are change of crimp by fibre bending, closing of gaps in the fabric structure, and flattening of the yarns. All of these are affected by the yarn packing density, which, in braids, increases with increasing bias angle as implied in
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β = 45°
β = 60°
β = 70°
Fig. 2. Triaxial braids with different bias angles b; field of view 42 mm 43 mm.
the inter-yarn gaps on the braid surfaces are closed, i.e. if b > b0. It depends on the distance xa between the 0° yarns and thus affects all preforms described above equally.
Table 2 Properties of the different yarns in the braids: filament count cf, linear density k (measured by weighing yarns at given length), average yarn width w0 (measured from photographs), average yarn height t0 (measured using a Vernier caliper at minimum applied pressure).
Bias 0°
cf (K)
k (g/m)
w0 (mm)
t0 (mm)
6 12
0.392 0.810
1.85 2.82
0.35 0.43
3. Permeability modelling 3.1. Single layer For each zone in the decomposed unit cell of the braid (Fig. 1), the local through-thickness fibre arrangement is approximated as illustrated in Fig. 3 (one braid layer). The effective local in-plane permeability of a zone can be estimated based on the permeability of the fibre bundles at the given orientations and the equivalent permeabilities of voids [14]. The height of each zone corresponds to the thickness of the braid, which is determined by the sum of the yarn thicknesses. If the braid does not undergo any throughthickness compression,
Eq. (1). An additional effect may occur in triaxial braids with wide spacing of bias yarns. In through-thickness compression, the initially straight 0° yarns may crimp because of forces exerted by the bias yarn on the one side, and fill spaces between bias yarns on the other side, thus contributing to the layer compressibility. In multi-layer specimens, packing density and compression behaviour are determined by nesting between adjacent layers. Nesting generally refers to (partial) filling of inter-yarn voids on the surface of a layer by yarns of an adjacent layer. Depending on the fabric architecture, in particular the yarn packing density in each layer [13], and layer alignment, this effect can be significant. For braids, nesting of the bias yarns on the surfaces is significant only if b < b0. Based on numerical experiments, Lomov et al. [13] concluded that, because of the presence of 0° fibres, nesting is less significant for triaxial braids than for biaxial braids. However, in triaxial braids, significant periodic thickness variations across the unit cell, caused by the inserted 0° yarns, give rise to the additional effect of ‘‘structural’’ nesting. Zones of minimum braid thickness of a layer are partially filled with zones of maximum thickness of an adjacent layer. This effect can occur even if
(tβ+ta)/2
ta/2
t0 ¼ t a0 þ 2t b1 :
ð7Þ
The thickness ta0 of the uncompressed 0° yarns is given in Table 2. If b < b1, the thickness of the bias yarns is tb0 (Table 2), and the fibre volume fraction, Vbb1, is determined from tb0 and wb. Otherwise, Vbb1 is constant at a value of Vbmax, and the bundle thickness increases from tb0 to tb1 according to Eq. (5) because of lateral compression. In through-thickness compression, changes of the yarn crosssection along the yarn axes and potential crimping of the 0° yarns are neglected. The decrease in thickness from the initial value t0 to the compressed value t is assumed to be distributed equally to 0° and bias yarns, i.e.
+/– β
tβ
t
0°
(tβ+ta)/2
1
2
3
–/+ β
tβ
tβ
ta/2
4
5
6
Fig. 3. Idealised through-thickness arrangement of yarns in the different zones of the unit cell (blank space indicates void); height of voids is indicated for each zone.
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t b ¼ t b1
t t0
and t a ¼ ta0
t : t0
ð8Þ
The yarn widths wa and wb are assumed to stay constant. For the compressed 0° yarns, the fibre volume fraction, Vba, is determined in analogy to Eq. (5) from ta and wa0 (Table 2). Similarly, the fibre volume fraction in the bias yarns, Vbb, is calculated from tb and wb. If any of the calculated values is greater than Vbmax, it is replaced by Vbmax, implying a change in yarn cross-sectional shape in compression. Vbmax is assumed to be identical for bias and 0° yarns. The principal permeability values k1i (parallel to its axis) and k2i (perpendicular to its axis) of a fibre bundle (aligned filaments with radius Rf) with fibre volume fraction Vbi, are calculated based on the equations derived by Gebart [15],
k1i ¼
R2f ð1 V bi Þ3 4c1 V 2bi
and k2i ¼ c2
sffiffiffiffiffiffiffiffiffiffiffiffi !5=2 V btheo R2f ; 1 V bi
ð9Þ
where c1, c2 and Vbtheo are geometrical constants. The permeabilities of the bias and 0° yarns can be determined by substituting Vbb and Vba for Vbi. Because of the yarn cross-sectional shapes, voids occur in each layer of the zones illustrated in Fig. 3, even if adjacent bundles are in contact. Since the geometry of the voids is complex, there is no analytical solution for their influence on the layer permeability. It is estimated based on the simplified geometry illustrated in Fig. 4. For each type i of yarn, the total cross-sectional area of voids is represented by rectangular gaps with total width
Dwi ¼ wi
pR2f cfi V bi ti
ð10Þ
:
The void space is assumed to be divided into two gaps with widths Dwi/2 and height ti (Fig. 4). The equivalent permeability of each gap along the yarn is estimated by the first term of the series suggested by Ni et al. [16] for description of ducts with rectangular cross-section, kei ¼ fc
8t 2i ðDwi =2Þp4
2
2
!
Dwi ðcoshðpDwi =ð2t i ÞÞ 1Þ sinh ðpDwi =ð2ti ÞÞ þ : 2 2 ðp=t i Þsinh ðpDwi =ð2t i ÞÞ
Perpendicular to the yarn axis, the influence of the gaps is small. Thus, for each bounding box as in Fig. 4, the effective principal permeability values can be estimated by
K 1i ¼
wi Dwi Dwi wi k2i k1i þ kei and K 2i : wi wi wi Dwi
The in-plane permeability of a zone containing M different material layers, each with thickness ti, is estimated by thicknessweighted averaging:
K¼
M X ti Ki ðDai Þ: t i¼1
ð13Þ
Here, Dai is the angle included by the axis of the yarn in layer i, oriented in the direction ai, and a reference direction, described by a0. Thus, the in-plane permeability tensor of layer i in reference coordinates is 2
Ki ðDai Þ ¼
K 1i cos2 Dai þ K 2i sin Dai
ðK 2i K 1i Þ sin Dai cos Dai
ðK 2i K 1i Þ sin Dai cos Dai
K 1i sin Dai þ K 2i cos2 Dai
:
A layer containing no yarn is represented by a wide gap with height ti and characterised approximately by the isotropic equivalent permeability [16]
Ki ¼
t 2i 12
1 0 0
1
;
ð15Þ
For a gap with dimensions given by the values for w0 and t0 in Table 2, this differs by less than 2% from the permeability derived from the expression in Eq. (11). From thickness-weighted averaging of layer permeabilities as formulated in Eq. (13), the local zone permeabilities in reference co-ordinates with a0 = 0° are determined as
K1yx ¼
K2yx
K3yx
t2 12
0 1
ð16Þ
; 2
K 1b cos2 b þ K 2b sin b
ðK 2b K 1b Þ sin b cos b
ðK 2b K 1b Þ sin b cos b ðt b þ t a Þ3 1 0 ; þ 48t 0 1
K 1b sin b þ K 2b cos2 b
tb ¼ t
þ ta t
K 1b cos2 b þ K 2b sin b
0
0
K 1b sin b þ K 2b cos2 b
t 3a 48t
Δwi/2
!
2
2
2t b ¼ t
K4yx ¼
1 0
1 0 ; 0 1
K 1a
0
0
K 2a
þ
ð17Þ
!
2
t3b 6t
1 0 0
1
;
ti
wi
2
!
ð14Þ
ð11Þ
Truncation of the series was found to induce relative errors of less than 2% for the range of Dwi and ti that is relevant here. The correction factor fc introduced here describes the void geometry and is related to the hydraulic diameter [15], which describes the ratio between the void cross-sectional area and perimeter. Since the ratio of cross-sectional area and perimeter is smaller for a duct enclosed by a rectangle and an ellipse than for a rectangular duct with equivalent cross-sectional area (Fig. 4), fc is smaller than 1. The values of fc decrease with increasing degree of compression, i.e. changing yarn cross-section.
ð12Þ
wi - Δwi
Fig. 4. Yarn cross-section and bounding box; simplification of void geometry.
Δwi/2
ð18Þ
ð19Þ
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K5yx ¼
ta t
K 1a 0
0 K 2a
2
K 1b cos2 b þ K 2b sin b
ðK 2b K 1b Þ sin b cos b
ðK 2b K 1b Þ sin b cos b 1 0 ; þ 12t 0 1
K 1b sin b þ K 2b cos2 b
tb þ t
adjacent layers is xa/2 along the x-axis. This results in a complex overlap pattern of the different zones in the unit cells (Fig. 1) with thirteen property sets as indicated in Table 3. For values of b between 45° and b0, the average layer thickness (Fig. 5) is approximated by
!
2
b 45 ta þ ; t ¼ tb 1 þ b0 45 2
t 3b
ð20Þ K6yx ¼
ta t
K 1a
0
0
K 2a
2tb þ t
which replaces Eq. (7). This includes the following special cases, for which the permeability can be estimated analytically:
2
For b = 45°, structural nesting and bias yarn nesting are combined, and the average layer thickness is estimated as tb + ta/2. Inter-yarn void spaces are assumed to be filled with yarns of adjacent layers, and the permeability is assumed to be uniform across the braid unit cell. It can be estimated from Eq. (21), which applies if two nested layers with average thickness are considered. For b = b0, only structural nesting with an estimated average layer thickness 2tb + ta/2 occurs. Inter-yarn voids disappear, and only zones 3 and 6 remain in each individual layer. In analogy to Eq. (22), the global permeability can be estimated from the resulting strips of 3–6 overlap and 6–6 overlap in parallel (Ky) and in series (Kx). The same applies for b > b0.
!
K 1b cos2 b þ K 2b sin b
0
0
K 1b sin b þ K 2b cos2 b
2
; ð21Þ
where the superscripts refer to the numbers of the respective zones, and the indices y and x refer to the notation in Fig. 1. For the unit cell in Fig. 1, analytical calculation of the global permeability is generally not possible, in particular because of the non-zero off-diagonal elements of the local permeability tensors for zones 2 and 5. However, if b P b0, i.e. the bias yarns are in contact, only zones 3 and 6 remain in the unit cell. For flow along the braid axis (in y-direction, zones 3 and 6 in parallel), the flow rates in the zones add up to the total flow rate, and the rule of mixtures applies to the flow velocities. For flow along the circumference (in x-direction, zones 3 and 6 in series), the pressure differentials in the zones add up to the total pressure differential. The corresponding global permeabilities can be estimated as
Ky ¼
xa wa 3 wa 6 Ky þ K xa xa y
and K x ¼
xa K 3x K 6x
ð23Þ
For all other values of b, the braid permeability can be determined numerically. 4. Results and discussion 4.1. Experimental results
; 6
wa K 3x þ ðxa wa ÞK x
ð22Þ
For the triaxially braided preforms described above, the unsaturated in-plane permeability was measured in radial flow experiments at constant injection pressure. As described in detail elsewhere [17], circular fabric specimens with a radius of 200 mm were placed in an injection tool, which is made from aluminium with an additional stiffening steel structure to minimise deflection. A test fluid, engine oil with given viscosity, was injected through a central circular injection gate with a radius of 5 mm. The
respectively. 3.2. Multiple layers Eqs. (16)–(21) apply for a single braid layer. For more than one layer, nesting is modelled assuming that the shift between
Table 3 Nesting between adjacent layers, n and n + 1: x indicates overlap of different zones in unit cell (Fig. 1). Zones layer n 1 Zones layer n + 1
1 2+ 2 3 4 5+ 5 6
2+
2
3
4
5+
5 x
x x
x x
x
x x x x
x
x x
x
x
x
x
t
t
Fig. 5. Illustration of significant difference between layer thickness without nesting and average layer thickness with nesting.
6 x x x x x x x x
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For the 0°/±45° preforms, the properties are dominated by the 0° yarns, while the contribution of the bias yarns is isotropic. Ky and the ratio of anisotropy Ky/Kx decrease significantly with increasing Vf because of nesting of the bias yarns and closing of inter-yarn void space in compression. For the 0°/±60° and 0°/±70° preforms, the main flow direction is dominated by the bias fibres. Because there is less void space than in the 0°/±45° preforms, the changes in Ky and Kx due to compression are less significant, and Ky/Kx does not change significantly with increasing Vf.
0.8 0.7
Ky
K / 10
-10
m
2
0.6 0.5 0.4 0.3 0.2
Kx
4.2. Numerical results
0.1 0 40
45
50
55
60
65
70
75
β / deg Fig. 6. In-plane permeability values Ky and Kx (along the preform 0° and 90° directions) as functions of the bias angle b at a target fibre volume fraction Vf = 0.54; filled symbols: experimental results, mean values and standard deviations, based on three repeats, are indicated; open symbols: results from model at fc = 0.05 with trend lines.
0.45 0.40
0.30
K / 10
-10
m
2
0.35
Ky
0.25 0.20 0.15
Kx
0.10 0.05 0.00 40
45
50
55
60
65
70
75
β / deg Fig. 7. In-plane permeability values Ky and Kx (along the preform 0° and 90° directions) as functions of the bias angle b at a target fibre volume fraction Vf = 0.59; filled symbols: experimental results, mean values and standard deviations, based on three repeats, are indicated; open symbols: results from model at fc = 0.03 with trend lines.
flow front propagation along three co-planar axes was determined by evaluating the readings from an array of pressure transducers, which allows the principal permeability values and the orientation of the principal permeability axes to be determined from the flow front position as a function of time. For 0°/±45° preforms at six layers, 0°/±60° preforms at five layers and 0°/±70° preforms at four layers, two different target values for the global fibre volume fraction Vf, 0.54 and 0.59, were set by appropriate adjustment of the cavity height h. The measured permeabilities along the preform 0° and 90° directions, Ky and Kx (Fig. 1), are plotted in Figs. 6 and 7. At both fibre volume fractions, Ky decreases continuously with increasing bias angle. Kx shows a maximum at a bias angle of 60°, which results from the competing effects of increased flow in x-direction because of reorientation of the bias yarns (dominating for b < 60°) and decreasing inter-yarn void spaces (dominating for b > 60°). This behaviour is qualitatively similar to that observed for biaxial braids [9]. For b = 45°, Kx is lower than Ky, whereas for 60° and 70°, it is higher. This implies a switch of principal flow directions. Interpolation of the data suggests that, at the given braid parameters, isotropic flow behaviour can be expected at bias angles of approximately 55°.
The in-plane permeability of a triaxial braid can be estimated analytically only for b = 45° and b > b0 as discussed in Section 3.2. For any arbitrary braid angle, it can be determined from virtual injection experiments, i.e. simulation of uni-directional resin injection at constant injection pressure along the x- and y-direction of the braid unit cell (Fig. 1). Here, numerical results were produced for braid angles of 50°, 55°, 60° and 65° in addition to analytical solutions at b = 45° and b = 70°. For calculation of local permeability values as discussed above, the values of w0 and t0 for bias yarns and 0° yarns (Table 2) suggest that the initial fibre volume fraction in the yarns, Vb, is 0.45. The maximum fibre volume fraction in the yarns is estimated as Vbmax = 0.78 based on square packing. Values for the constants in Eq. (9) are given in Table 4. The small difference between Vbmax and Vbtheo was introduced to account for imperfect alignment of filaments and avoid k2i in Eq. (9) being zero, which could cause numerical problems. In analogy to the ‘‘grid average’’ method described by Wong et al. [18], local permeabilities, derived for the overlap pattern specific to each braid angle, were projected onto a 2D mesh as illustrated in Fig. 8. The equations describing unsaturated flow of a viscous liquid through a porous medium considering conservation of the fluid mass were solved based on a non-conforming finite element method [19], which is implemented in the PAM-RTM software (ESI Group). From the average time for the flow front to travel across the unit cell, the principal permeability value in the respective direction was determined from the one-dimensional formulation of Darcy’s law. It could be argued that 3D flow simulation, based on a detailed model of the fibre arrangement and void geometry, should in theory give more accurate results. However, this may be mitigated by the problem of accurately obtaining the actual geometry parameters, which are subject to variability in the braid structure and thus may induce new sources of uncertainty. In addition, accurate 3D modelling of the fibre structure may be limited by the meshing capabilities. Solving the 3D flow problem, which requires implementation of different equations on different domains (frequently used are Stokes equation for flow in inter-yarn voids, Brinkman equation for flow in yarns, as discussed, e.g., by Ngo and Tamma [20]), may require considerable computing resources, and problems of numerical stability may arise [21]. On the other hand, idealised values of the geometrical dimensions required as input for the 2D modelling approach can be determined based on the considerations in Section 2.1. Decomposition of the 2D geometry into zones of different yarn overlap configuration allows effects of nesting to be incorporated for simulation of flow in multi-layer stacks. Solution of the flow problem is simplified by assuming Darcy’s law to describe flow at any
Table 4 Filament radius Rf, geometry constants c1, c2 and Vbtheo (square packing [15]) and maximum achieved yarn fibre volume fraction Vbmax. Rf (mm)
c1
c2
Vbtheo
Vbmax
3.5 103
1.78
0.40
0.785
0.78
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Fig. 8. Finite element mesh of unit cell at b = 60° for flow simulation (with nesting); 1702 triangular finite elements; different colours indicate zones with different overlap configuration.
Table 5 Comparison of analytically calculated permeability values at b = 70° with and without nesting; experimental data are listed for reference. Vf
0.54 0.59
With
Without
Experiment
Ky (1010 m2)
Kx (1010 m2)
Ky (1010 m2)
Kx (1010 m2)
Ky (1010 m2)
Kx (1010 m2)
0.274 0.138
0.313 0.162
2.375 2.323
0.209 0.126
0.174 ± 0.038 0.117 ± 0.018
0.288 ± 0.099 0.205 ± 0.030
point in the model. Effective local permeability values for each of the thirteen zones can be calculated from the equations in Section 3. The advantages of using a commercial software package for solution of the flow problem are its availability, robustness and flexibility regarding input mesh data, the latter rendering it convenient for studying different braid configurations. While solution of steady-state flow at complete impregnation of the braid would be sufficient for permeability calculation, the commercial software simulates a complete virtual injection experiment involving multiple solutions at different time steps, thus allowing the flow front propagation to be tracked. However, the computational cost for 2D simulations is generally low. Here, computation times were less than one minute on a standard PC. Despite the simplifications inherent to the ‘‘grid average’’ method, Wong et al. [18] found that results compare favourably with results from more computationally intensive 3D numerical approaches. Analytical (b = 45°, 70°) and numerical results (b = 50°, 55°, 60°, 65°) for Vf = 0.54 at fc = 0.05 and Vf = 0.59 at fc = 0.03 are plotted in Figs. 6 and 7. Because of simplifications in geometrical modelling and uncertainties induced by the numerical solution, e.g. mesh-related issues and accuracy of determining the average flow front arrival time, the individual data points show considerable scatter. Simulations of the same scenario at different mesh densities (for the example of b = 65°, Vf = 0.54, fc = 0.05: 2915 elements and 9976 elements) suggest that the discretisation may result in uncertainties of the permeability of up to 5%. Another source of uncertainty is that approximations in the calculation procedure may induce small differences between the effective Vf and the target value of Vf, which is assumed for evaluation of the simulations. The induced difference between the permeability derived from the simulations and the analytical solution was found to be 7% for b = 45°. However, trend lines fitted to the data reproduce the experimental results qualitatively well. Most importantly, the change in principal flow direction, which was observed in the experimental results and is of highest practical relevance, is predicted at approximately the right values of b (60° and 58°). The values of fc were selected to achieve reasonable quantitative agreement of calculated trends and experimental results for both Ky and Kx at the respective fibre volume fractions. With increasing global Vf, i.e. increasing degree of bundle compression, the cross-sections of fibre bundles as indicated in Fig. 4 are expected to become more rectangular. This implies that the ratio of
cross-sectional area and perimeter of the voids, and thus fc, decreases. However, based on the available data, the dependence of fc on Vf cannot be clearly identified. The correlation of the global permeability with fc reflects that it is determined mainly by a network of inter-yarn gaps. It is to be noted that bundle cross-sectional shape and thus fc depend not only on through-thickness compression, but, at given bundle thickness, also on braid deformation, e.g. by shearing of the bias yarns. To give an impression about the significance of (structural) nesting on the results, analytical results from Eq. (22) for b = 70° with and without nesting at identical fc are compared in Table 5. The results for Ky are significantly higher without nesting than with nesting, because gaps in zones 3 (Fig. 3) facilitate flow along the y-direction. For Kx, the results without nesting are lower than those with nesting, because the level of yarn compression in zones 6, which is related to the average layer thickness and determines the flow along the x-direction, is higher than with nesting. This results in a change of principal flow direction, which is oriented along the y-direction if nesting is not considered.
5. Conclusions For triaxial braids comprising a given number of axial (0°) yarns and twice as many yarns oriented at the braid bias angle (±), the influence of the bias angle on the size of inter-yarn void spaces and yarn cross-sections was described. For analysis of the void space in multilayer preforms, the effect of nesting needs to be considered, which generally refers to partial filling of voids between bias yarns on the surface of a layer by yarns of an adjacent layer. An additional effect of ‘‘structural’’ nesting due to thickness variations in triaxial braids, which depend on the spacing between the 0° yarns, was postulated. For predictive estimation of the braid permeability, the braid unit cell was decomposed into zones of characteristic yarn arrangement. A set of equations was derived allowing description of the local permeability of each zone as a function of bias angle, braid thickness or global fibre volume fraction, and geometrical yarn parameters. A complex yarn overlap pattern, characterising the effect of nesting in multilayer preform, was identified. For three different carbon fibre braids with bias angles 45°, 60° and 70°, the in-plane permeability at multiple layers was charac-
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terised experimentally. Along the 0° direction, the permeability decreases continuously with increasing bias angle. Along the 90° direction, it shows a maximum at a bias angle of 60°. This results in a switch of the principal flow direction from the preform 0° direction for b = 45° to the preform 90° direction for b = 60° and b = 70°. Interpolation suggests isotropic flow behaviour at a bias angle of approximately 55°. The braid permeability was derived from numerical flow simulations based on simplified 2D models of the textile architecture and averaged local permeabilities. Calculated trends for the permeability as a function of the braid angle reproduce the experimental results qualitatively well. The change in principal flow direction was predicted with good accuracy. Quantitative agreement between numerical and experimental results depends on the appropriate choice of a geometry parameter, which is related to the yarn cross-section and decreases with increasing fibre volume fraction. Comparison of results generated with and without consideration of ‘‘structural’’ nesting indicates that this effect has a significant influence on the in-plane permeability, in particular along the preform 0° direction. Acknowledgements This work was funded by the Engineering and Physical Science Research Council (EPSRC) via the Nottingham Innovative Manufacturing Research Centre (EP/E001904/1). References [1] Charlebois KM, Boukhili R, Zebdi O, Trochu F, Gasmi A. Evaluation of the physical and mechanical properties of braided fabrics and their composites. J Reinf Plast Compos 2005;24(14):1539–54. [2] Ayranci C, Carey J. 2D braided composites: a review for stiffness critical applications. Compos Struct 2008;85(1):43–58. [3] Smith LV, Swanson SR. Strength design with 2-D triaxial braid textile composites. Compos Sci Technol 1996;56(3):359–65.
[4] Ivanov DS, Baudry F, Van Den Broucke B, Lomov SV, Xie H, Verpoest I. Failure analysis of triaxial braided composite. Compos Sci Technol 2009;69(9):1372–80. [5] Littell JD, Binienda WK, Roberts GD, Goldberg RK. Characterization of damage in triaxial braid composites under tensile loading. technical report NASA/TM2009-215645. Hanover: NASA Center for AeroSpace Information; 2009. [6] Ko FK, Pastore CM, Head AA. Handbook of industrial braiding. Atkins and Pearce; 1989. [7] Sayre JR, Loos AC. Resin infusion of triaxially braided preforms with throughthe-thickness reinforcement. Polym Compos 2003;24(2):229–36. [8] Antonucci V, Giordano M, Nicolais L, Calabro A, Cusano A, Cutolo A, et al. Resin flow monitoring in resin film infusion process. J Mater Process Tech 2003;143– 144:687–92. [9] Long AC. Process modelling for liquid moulding of braided preforms. Compos Part A – Appl Sci 2001;32(7):941–53. [10] Carman PC. Fluid flow through granular beds. T I Chem Eng – Lond 1937;15:150–6. [11] TexGen (Version 3.3.0). University of Nottingham; 2008.