Through-thickness permeability study of orthogonal and angle-interlock woven fabrics
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DOI: 10.1007/s10853-014-8683-4
- Cite this article as:
- Xiao, X., Endruweit, A., Zeng, X. et al. J Mater Sci (2015) 50: 1257. doi:10.1007/s10853-014-8683-4
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Abstract
Three-dimensional (3D) woven textiles, including orthogonal and angle-interlock woven fabrics, exhibit high inter-laminar strength in addition to good in-plane mechanical properties and are particularly suitable for lightweight structural applications. Resin transfer moulding (RTM) is a cost-effective manufacturing process for composites with 3D-woven reinforcement. With increasing preform thickness, the influence of through-thickness permeability on RTM processing of composites becomes increasingly significant. This study proposes an analytical model for prediction of the through-thickness permeability, based on Poiseuille’s law for hydraulic ducts approximating realistic flow channel geometries in woven fabrics. The model is applied to four 3D-woven fabrics and three 2D-woven fabrics. The geometrical parameters of the fabrics were characterized by employing optical microscopy. For validation, the through-thickness permeability was determined experimentally. The equivalent permeability of inter-yarn gaps was found to account for approximately 90 % of the through-thickness permeability for the analysed fabrics. The analytical predictions agree well with the experimental data of the seven fabrics.
Introduction
Because of their high specific stiffness and high specific strength, polymer composites have found use in the aerospace, nautical, automotive and sports equipment industries [1, 2, 3], where they replace other materials, in particular metals. In the aeronautic and automotive industries, lightweight composite structures have become important in the development of sustainable fuel-efficient transport solutions [4]. Demand for cost-effective manufacture of high-performance composite structures with woven textile reinforcements has driven research into liquid composite moulding (LCM) processes. A key research topic is characterization of the reinforcement permeability tensor, which determines the impregnation of the reinforcement with liquid resin in LCM [5, 6, 7, 8]. Quantifying the permeability accurately and reliably remains a major challenge, because resin flow paths within deformable textile reinforcements are inherently geometrically complex and variable.
Here, K_{xx}, K_{yy} and K_{zz} are the principal permeabilities.
Here, M is the number of binder yarns per fabric surface area, k a form factor, n is the filament count of the binder yarns, R_{f} is the filament radius and ψ is the angle between the axis of the binder yarns and the fabric plane. Equation 4 cannot predict K_{zz} directly since the parameter k for a particular fabric needs to be determined from experiments.
In the present study, an analytical model is derived from a generalized Poiseuille’s law for predicting K_{zz} of 3D-woven fabrics based purely on geometrical information on the fabric architectures. For orthogonal and angle-interlock 3D-woven reinforcement fabrics, the model was validated with the experimental permeability data. For comparison, plain and twill weave 2D fabrics were analysed.
Theoretical analysis of \( \varvec{K}_{{\varvec{zz}}} \)
Orthogonal and angle-interlock woven fabrics
Sub-layer of 3D-woven fabrics
Here, the parameter, λ, is related to the yarn height and determines the curvature of the channel geometry. The smaller the value of λ, the sharper the tip of the yarn cross section. The exact flow channel geometry can be obtained from microscopic images of cross sections of warp and weft yarns, where coordinates can be measured using image analysis software and approximated by a second-order polynomial using least-squares analysis [22]. This allows the value of λ in Eq. 15 to be determined directly.
In Eqs. 19 and 20, yarn permeabilities are derived assuming hexagonal fibre arrangement. For a square fibre arrangement, the constants (53 and \( 9\sqrt 6 \)) would need to be replaced with 57 and \( 9\sqrt 2 \).
Experimental study of \( \varvec{K}_{{\varvec{zz}}} \)
The equations for the permeabilities of sub-layers, K_{f}, and entire fabrics, K_{zz}, were applied to four 3D-woven carbon fibre reinforcement fabrics (two orthogonal and two angle-interlock fabrics) and validated based on experimental permeability data.
Geometrical fabric parameters for four 3D woven carbon fibre fabrics
Fabric | Structure | Fibre bundles | Inter-bundle voids | |||
---|---|---|---|---|---|---|
Width (mm) | Height (mm) | Width (mm) | Height (mm) | |||
A_{1} | N’ = 1, | Warp | 4.01 ± 0.19 | 0.41 ± 0.03 | 0.97 ± 0.12 | 0.31 ± 0.06 |
L = 2 mm, | Weft | 3.16 ± 0.19 | 0.38 ± 0.04 | 0.49 ± 0.20 | 0.34 ± 0.04 | |
V_{F} = 0.41, λ = 0.5 | Binder | B_{w} 1.40 ± 0.16 | B_{j} 0.37 ± 0.07 | |||
A_{2} | N’ = 2, | Warp | 4.02 ± 0.21 | 0.42 ± 0.05 | 0.83 ± 0.15 | 0.23 ± 0.05 |
L = 3.5 mm, | Weft | 3.29 ± 0.20 | 0.45 ± 0.07 | 0.42 ± 0.20 | 0.30 ± 0.05 | |
V_{F} = 0.47, λ = 0.5 | Binder | B_{w} 1.48 ± 0.16 | B_{j} 0.35 ± 0.04 | |||
O_{1} | N’ = 1, | Warp | 1.81 ± 0.06 | 0.38 ± 0.02 | 0.31 ± 0.05 | 0.38 ± 0.02 |
L = 5 mm, | Weft | 2.07 ± 0.12 | 0.35 ± 0.03 | 0.32 ± 0.07 | 0.35 ± 0.03 | |
V_{F} = 0.55, λ = 2.5 | Binder | B_{w} 0.62 ± 0.06 | B_{j} 0.15 ± 0.03 | |||
O_{2} | N’ = 1, | Warp | 1.77 ± 0.08 | 0.40 ± 0.03 | 0.29 ± 0.04 | 0.40 ± 0.03 |
L = 4.6 mm, | Weft | 2.06 ± 0.11 | 0.32 ± 0.02 | 0.27 ± 0.07 | 0.32 ± 0.07 | |
V_{F} = 0.59, λ = 2.5 | Binder | B_{w} 0.73 ± 0.17 | B_{j} 0.15 ± 0.06 |
The through-thickness permeability was measured in a saturated uni-directional flow experiments. In a stiff cylindrical flow channel with a liquid inlet at the bottom and a liquid outlet on top (inner diameter 80 mm), fabric specimens are held in position by stiff perforated plates, which allow parallel flow perpendicular to the fabric plane. The distance between the perforated plates is given by the height of spacer rings. Engine oil with known viscosity-temperature characteristics (\( \mu \approx 0.3\;{\text{Pa}} \cdot {\text{s}} \) at 20 °C) was used as a test fluid. The flow rate is set on a gear pump and monitored using a flow meter. Pressure transducers are mounted on both sides of the fabric specimen for measurement of the pressure drop [11]. The value of K_{zz} was calculated according to Eq. 1 with the constant flow rate (laminar flow with small Reynolds numbers) and measured pressure drop. Each test was repeated three times with a fresh sample.
Geometrical fabric parameters for three 2D woven fabrics
Fabric | Structure | R_{f} (μm) | Yarn V_{f} | L (mm) | λ (mm^{−1}) | Yarn spacing | Yarn width | ||
---|---|---|---|---|---|---|---|---|---|
S_{j} (mm) | S_{w} (mm) | D_{j} (mm) | D_{w} (mm) | ||||||
P_{1} | Plain (100 % cotton) | 4.3 | 0.56 | 0.323 | 5.23 | 0.470 | 0.410 | 0.405 | 0.279 |
T_{1} | 2/1 twill (67 % polyester, 33 % cotton) | 5.9 | 0.56 | 0.419 | 3.81 | 0.340 | 0.480 | 0.310 | 0.310 |
T_{2} | 2/2 twill (60 % cotton, 40 % polyester) | 5.7 | 0.56 | 0.610 | 4.10 | 0.342 | 0.446 | 0.313 | 0.380 |
The through-thickness permeability of 2D fabrics was measured according to BS EN ISO 9237:1995. The apparatus for the experiment is an air permeability tester FX 3300. While the fabric is clamped in position, a suction fan forces air to flow perpendicularly through the fabric. The volumetric flow rate is measured and divided by the specimen area to give the velocity of air flow. The pressure drop in the experiment for all fabrics is set to 500 Pa, with an accuracy of at least 2 %. Using the measured velocity, pressure drop and fabric thickness, permeability is calculated according to Darcy’s law.
Results and discussions
The cross sections in Fig. 5b, illustrate the internal geometry of orthogonal and angle-interlock woven fabrics. The warp and weft yarns in the orthogonal 3D-woven fabric are straight and parallel. Binder yarns follow paths through the fabric thickness, fixating warp and weft yarns and generating inter-yarn gaps to form flow channels. In the angle-interlock woven fabric, binder yarns follow paths resembling sine/cosine curves through the layers of warp and weft yarns. The cross section normal to the weft direction shows an offset between layers of weft yarns by half a yarn width, which needs to be considered for definition of the angle-interlock fabric unit cell. The white rectangular frames in the top views of the fabrics illustrate the fabric unit cell areas. Measuring the geometrical dimensions of each fabric unit cell allows the sub-layer permeability, K_{f}, and the permeability of the entire fabric, K_{zz}, to be predicted.
Aspect ratios, α, of rectangular gaps (width/length) and corresponding values of conversion friction factor, c
Fabric | α | c |
---|---|---|
P_{1} | 0.496 | 0.973 |
T_{1} | 0.176 | 1.220 |
T_{2} | 0.439 | 1.001 |
A_{1} | 0.878 | 0.893 |
A_{2} | 0.900 | 0.892 |
O_{1} | 0.857 | 0.894 |
O_{2} | 0.827 | 0.897 |
Additional values | 0.100 | 1.323 |
0.167 | 1.231 | |
0.250 | 1.140 | |
0.400 | 1.023 | |
0.500 | 0.972 | |
0.750 | 0.905 | |
1.000 | 0.889 |
Comparison of the predicted yarn and inter-yarn gap permeabilities for 2D woven fabrics
Fabric | \( \varPhi \) (%) | K_{y}/10^{−13} m^{2} | K_{g}/10^{−10} m^{2} | K_{f}/10^{−12} m^{2} | \( \left( {100 \% - \varPhi } \right)K_{\text{y}} /K_{\text{f}} \) (%) | \( \varPhi K_{\text{g}} /K_{\text{f}} \) (%) |
---|---|---|---|---|---|---|
P_{1} | 3.93 | 3.00 | 3.99 | 15.97 | 1.80 | 98.20 |
T_{1} | 1.64 | 4.98 | 2.19 | 4.08 | 12.01 | 87.99 |
T_{2} | 1.08 | 3.13 | 2.63 | 3.14 | 9.86 | 90.14 |
The predicted values of K_{zz} for orthogonal and angle-interlock 3D-woven fabrics were based on Eqs. 8, 12, 18 and 20. The geometrical parameters and fabric specifications for the prediction are taken from Table 1. Fabric ‘A’ shows relatively wide gaps between adjacent parallel yarns and high K_{zz} values owing to the small values of \( V_{\text{F}} \) and high values of K_{i}. For V_{F} = 0.41 (A_{1}), the predicted value of K_{zz}, 28.9 × 10^{−12} m^{2}, is similar to the measured average value, 23.5 × 10^{−12} m^{2}, indicating a relative difference of 23.3 %. For \( V_{\text{F}} \) = 0.47 (A_{2}), the prediction shows very good agreement with experimental data. Comparisons for fabric ‘O’ give similar result. For V_{F} = 0.55 (O_{1}), the measured K_{zz} is 10.3 × 10^{−12} m^{2}, whereas the prediction is 12.9 × 10^{−12} m^{2}. All predicted permeabilities for the 3D-woven fabrics lie within the range defined by the standard deviations of the experimental data. As expected, Fig. 6 shows that K_{zz} decreases with the increasing V_{F} due to the reduction of overall gap space in the fabric.
Conclusions
The though-thickness permeability of orthogonal and angle-interlock 3D-woven fabrics was studied analytically. It is determined by the height and through-thickness permeability of each sub-layer, the number of sub-layers, and the entire fabric thickness. The through-thickness permeability of each sub-layer depends on the yarn permeability in the flow direction, the equivalent permeability of inter-yarn gaps and the areal coverage of the fabric. The yarn permeability was modelled by combining axial and transverse permeabilities based on the local yarn crimp angle. The equivalent gap permeability was modelled based on conversion of the actual gap cross section to a circular cross section and varying the cross section through the fabric thickness according to measured yarn cross-sectional profiles. For seven woven fabrics of different architectures, geometrical fabric parameters were characterized in detail by optical microscopy. Calculation of yarn permeability, equivalent gap permeability and fabric permeability shows that the equivalent gap permeability dominates the fabric permeability, even if the areal coverage of inter-yarn gaps is only around 1 %. Comparison shows close agreement for each sample of predicted and measured values of the through-thickness permeability of orthogonal and angle-interlock woven fabrics, indicating good accuracy of the permeability models. Studies on the sensitivity of the fabric through-thickness permeability to variation of the geometrical parameters and extension of the theoretical analysis to 3D-woven fabrics with different architectures are ongoing.
Acknowledgements
The work was supported in part by the projects: RGC No.: 5158/13E and NSFC funding Grant No. 51373147 and Project code: JC201104210132A.