Through-thickness permeability modelling of woven fabric under out-of-plane deformation
Abstract
When a woven fabric is subject to a normal uniform loading, its properties such as tightness and through-thickness permeability are both altered, which relates to the fabric out-of-plane deformation (OPD) and dynamic permeability (DP). In this article, fabric OPD is analytically modelled through an energy minimisation method, and corresponding fabric DP is established as the function of loading and fabric-deformed structure. The total model shows the permeability a decrease for tight fabric and an increase for loose fabric when the uniform loading increases. This is verified experimentally by fabric OPD, static and dynamic permeabilities. Experimental tests for both permeabilities showed good agreement with the corresponding predictions, indicating the fact that tight fabric becomes denser and loose fabric gets more porous during OPD. A sensitivity study showed that an increase of fabric Young’s modulus or a decrease of fabric test radius both lead to an increase of DP for tight fabric and opposite for loose fabric. The critical fabric porosity and thickness were found for inflexion of fabric DP trend during the OPD, which contributes to the optimum design of interlacing structure applied to protective textiles and composites.
Introduction
Many technical textiles during bulking, such as inflation of airbag and artificial blood vessel, are subject to pressure loading perpendicular to the fabric in-planar. The perpendicular loading easily causes the interlacing structural fabric a deformed out-of-plane curved profile. The transient permeability and the subsequent protective effect and transport efficiency are thereafter varied dynamically during the deformation. Therefore, it is important to model this fabric out-of-plane deformation (OPD) behaviour and the relationship with the fabric dynamic permeability (DP), as the complete understanding of the mechanism from fabric OPD to fabric DP will be desirable for optimum design of such technical textiles and exploration of their new applications.
As known, 2D woven fabric is a flexible, discontinuous and anisotropic sheet, and can be easily deformed by an out-of-plane pressure load, which may involve fabric nonlinear tensile, shear, bending and compression behaviours. Hursa [1] used a fabric unit-cell geometrical model to predict the OPD with its micromechanical tensile, which may contribute to the link of fabric deformation to DP. King et al. [2] proposed a continuum constitutive model for predicting fabric mechanical behaviour in-planar direction. The approach relied on the selection of a geometrical model for the fabric weave, coupled with constitutive models for the yarn behaviours. The structural configuration was related to the macroscopic deformation through an energy minimisation method. This method is useful as it covers all aspects of mechanical properties in the fabric OPD. Lin et al. [3] developed an analytical model for the OPD of a square textile composite by its own weight. Trigonometric functions were used to describe the in-plane and out-of-plane displacements of any point. Energy minimisation approach was employed to analyze the composite tensile, shear, bending and exerted external force. The predictions for the maximum deflections of a few weave composites showed good agreement with corresponding finite element simulations.
Transverse permeability (K_{⊥}) of yarns developed by a few researchers
Researcher | Hexagonal | Quadratic |
---|---|---|
\( K_{\text{h}} = \frac{16}{9\pi \sqrt 6 }\left( {\sqrt {\frac{\pi }{{2\sqrt 3 V_{\text{f}} }}} - 1} \right)^{2.5} \cdot R_{\text{f}}^{2} \) | \( K_{\text{q}} = \frac{16}{9\pi \sqrt 2 }\left( {\sqrt {\frac{\pi }{{4V_{\text{f}} }}} - 1} \right)^{2.5} \cdot R_{\text{f}}^{2} \) | |
\( K_{\text{h}} = 0.07R_{\text{f}}^{2} \cdot \left( {\frac{{1 - \sqrt {\frac{{V_{\text{f}} }}{0.907}} }}{{\frac{{V_{\text{f}} }}{0.907}}}} \right)^{2.5} \) | \( K_{\text{q}} = 0.12R_{\text{f}}^{2} \cdot \left( {\frac{{1 - \sqrt {\frac{{V_{\text{f}} }}{0.785}} }}{{\frac{{V_{\text{f}} }}{0.785}}}} \right)^{2.5} \) | |
\( K_{\text{h}} = \frac{{R_{\text{f}}^{2} }}{3\sqrt 3 }\frac{{(1 - l^{2} )^{2} }}{{l^{3} }} \cdot \left( {3l\frac{{\arctan \left( {\sqrt {\frac{1 + l}{1 - l}} } \right)}}{{\sqrt {1 - l^{2} } }} + \frac{{l^{2} }}{2} + 1} \right)^{ - 1} \)\( l^{2} = \frac{2\sqrt 3 (1 - \varPhi )}{\pi } \) | \( K_{\text{q}} = \frac{{R_{\text{f}}^{2} }}{3}\frac{{(1 - l^{2} )^{2} }}{{l^{3} }} \cdot \left( {3l\frac{{\arctan \left( {\sqrt {\frac{1 + l}{1 - l}} } \right)}}{{\sqrt {1 - l^{2} } }} + \frac{{l^{2} }}{2} + 1} \right)^{ - 1} \)\( l^{2} = \frac{4(1 - \varPhi )}{\pi } \) | |
\( K = R_{\text{f}}^{2} \cdot \frac{{\pi \left( {1 - V_{\text{f}}^{*} } \right)\left( {1 - \sqrt {V_{\text{f}}^{*} } } \right)^{2} }}{{24\left( {V_{\text{f}}^{*} } \right)^{1.5} }} \)V_{f}^{*} = 2.22(V_{f})^{2} − 1.22V_{f} + 0.56 (V_{f} ≥ 0.5) V_{f}^{*} = V_{f} (V_{f} < 0.5) |
This paper takes into account the fabric deformation under out-of-plane uniform loading (OPUL), and the effect on the fabric geometric parameters. The fabric DP is thereafter calculated based on the varying geometrical factors. Two typical fabrics (tight and loose) are used to make the experimental verifications. Results and discussions are given, following with the conclusions finally.
Analytical modelling
Fabric deformation under OPUL
Fabric behaviour under OPUL is modelled through assuming an originally flat, stress-free circular fabric sample with axisymmetric deformation. Polar coordinates are used in this particular deflection case.
Permeability of the deformed fabrics
All yarns are assumed with the same ɛ value in the deformation.
Tight fabric permeability
Loose fabric permeability
Experimental verification
Fabric deformation model
The experimental verification to the model of fabric OPD contains two aspects: the w′ and corresponding w. Here a novel experimental device is invented to validate the deformation model.
Design of the fabric deflection tester
The device is designed to produce a vacuum pressure up to 100 kPa. A steel ball with diameter of 4 mm is used to determine the place of w′. A ruler is placed on the top plate across a diameter parallel to the fabric warp, weft and 45° of warp/weft directions, respectively. A vernier caliper is placed on the ruler perpendicularly and movable to determine the displacement of the fabric deflection. Each fabric deflection under a certain pressure load was repeated five times for the three directions with a fresh sample. Average fabric deflections for the repeats were given with standard deviations.
Experimental materials
Fabric specifications in average values before OPUL (±SD)
Fabric | Composition and structure | R_{f} mm | Yarn V_{f} | L mm | α | C mm | D mm | ||
---|---|---|---|---|---|---|---|---|---|
Warp | Weft | Warp | Weft | ||||||
A_{1} | 100 % Nylon plain | 0.34 (±0.01) | 0.68 (±0.01) | 0.34 (±0.01) | 20° | 0.53 (±0.02) | 0.45 (±0.01) | 0.45 (±0.01) | 0.52 (±0.01) |
Fabric | Composition and structure | Ave λ | L mm | α | C mm | D mm | |||
---|---|---|---|---|---|---|---|---|---|
Warp | Weft | Warp | Weft | ||||||
U_{2} | 65 % PET/35 % cotton plain | 12.96 (±0.32) | 0.32 ( ±0.01) | 14° | 0.22 (±0.03) | 0.33 (±0.01) | 0.17 (±0.03) | 0.19 (±0.03) |
An attempt was made to measure v values of the two fabrics using Digital Image Correlation (DIC) equipment according to Hursa’s approach [16], however, the results showed both larger than 1 which is not considered physically realistic. In the next section, a few v values will be used to assess sensitivity.
Deformed fabric permeability measurement
Results and discussions
Fabric deformation model
Maximum displacement
With a fixed ν value, the predicted w′ is proportional to the cubic root of the OPUL (P) according to Eq. 23a. The ‘EXPT’ dots in Fig. 8 show a nonlinear relationship of w′ and P which is close to the cubic root relationship in the prediction. The graphs also show that a smaller ν value can obtain a higher prediction of w′, and the interval between \( \nu = 0.2\;\; {\text{and}}\; 0.3 \) is much less than that of 0.3 and 0.4, showing the relationship of w′ and ν values is nonlinear. The comparisons show the ν value for Fabric A_{1} is close to 0.3 while Fabric U_{2} is close to 0.2. In the graph, the w′ value for Fabric A_{1} is smaller than that for Fabric U_{2} at the same loading. The reason is a smaller stiffness value of Fabric U_{2}.
Deflection profile
Permeability model of deformed fabric
Tight fabric A_{1}
Loose fabric U_{2}
Table 2 offers initial values of C and D, which can be transferred into R_{g} and R_{y} by Eqs. 7a and 7b. Due to the OPUL, the fabric deflection leads to an increase in the fabric surface area. Yarns tensioning causes an increase in R_{g} and a decrease in L and λ.
Sensitivity study
Figure 16 also shows the relationship by changing a when other parameters are fixed. An increase of a will increase the w′ value of the deformed fabric (Eq. 23a) and \( \varepsilon \) value (Eqs. 24, 25), influencing the final K value. Figure 16a shows that K is decreased with increasing a as V_{f} is increased for tight fabric. The difference of K values between \( a = 41\; {\text{mm}} \) and \( a = 51 \;{\text{mm}} \) is smaller than that of \( a = 31\; {\text{mm}} \) and \( a = 41 \;{\text{mm}}, \)indicating a lower effect of increased a on decreasing the K value. Figure 16b shows K of loose fabric is increased as its a value increases. The reason might be that R_{g} is getting larger relatively as the fabric is deformed more at a larger a value. The difference of K values between a = 31 mm and \( a = 41\; {\text{mm}} \) is smaller than that of \( a = 41\; {\text{mm}} \) and \( a = 51 \;{\text{mm}}, \)indicating that an increase of a will cause the fabric K to increase further.
Conclusions
Three analytical models were proposed for predicting the OPD of woven fabric under perpendicular uniform loading, and corresponding through-thickness permeability of tight weave and loose weave, respectively. The whole models in this paper contribute to the mechanism understanding the effect of external factors on the fabric dynamic permeability, and assists with optimum design of technical textiles in the OPD working environment.
In the modelling of fabric OPD under a uniform load, an energy-based approach was utilised to predict bending energy, strain energy and external energy. The fabric was assumed to behave like a thin film based on the membrane large deformation theory. Minimisation energy of the system was used to derive the relationship of the maximum displacement and the loading. Fabric-deflected shape was characterised by the displacement and a cosine function of the fabric radius. The model for predicting the permeability was based on the accurate prediction of the fabric deformation. Also, it relied on the accurate prediction of the static permeability (Eqs. 1, 5, 6, 7, 8). The assumption was that the yarn width was invariable during the deformation. Fabric thickness was reduced with the same rate of yarn height. Tight fabric permeability was predicted through the increased yarn fibre volume fraction and crimp angle due to the decreased yarn height. Loose fabric permeability was predicted by the increased gap radius due to the enlarged fabric area caused by fabric deflection.
Three kinds of experiments were used to verify the analytical predictions. Fabric OPD was measured by a fabric deflection tester, with loading applied by a vacuum pump. Fabric static permeability was determined by a Shirley air permeability tester while fabric dynamic permeability was tested by a dynamic permeability tester. The predictions for the fabric deflection configurations (tight and loose fabrics) agree with the experimental measurements very well. The deflection causes the yarn fibre volume fraction and the crimp angle to increase, resulting in the permeability of tight fabric to decrease (Fig. 12). In contrast, the deflection leads to the gap radius to increase, obtaining an increased permeability of loose fabric (Fig. 14). The permeability predictions for the deformed fabrics agree with the experimental values well. Sensitivity studies first, investigate the critical fabric porosity and the critical fabric thickness where the increase or decrease of fabric permeability occurs during the fabric deformation. Second the fabric properties, such as Young’s modulus, affect the fabric deformation. An increase of modulus leads to the increase of tight fabric permeability and a decrease of loose fabric permeability when the fabric is under the same pressure load.
Notes
Acknowledgements
The authors would like to thank Airbags International Ltd. for providing experimental materials, Leeds University and UK Unilever Resources Centre for undertaking the experimental tests.
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