Parametric sensitivity analysis to maximise auxetic effect of polymeric fibre based helical yarn
Introduction
It has been suggested that helical auxetic yarn (HAY) can be woven into technical auxetic textiles and placed in a composite for body armour and blast mitigation applications [1]. A material that is auxetic exhibits a negative Poisson’s ratio (NPR). This means that the cross-section of the material will become larger when a tensile force is applied in the transverse direction, and smaller when a compressive force is applied in the transverse direction. The opposite is true in conventional materials [2]. Auxetic materials have been found to possess a range of unconventional mechanical properties. These include increased indentation hardness [3], fracture toughness [4], strain energy dissipation [5], and shear toughness [6], [7]. These unusual properties make auxetic structure ideal for use in many applications in many different disciplines, including defence, fashion, medicine and sport [8], [9], [10].
The helical auxetic yarn (HAY) presented itself as the most fitting structure to be chosen for the study; as well as having the greatest potential to maximise the auxetic effect [11], [12], [13], [14], [15], [16], [17], [18], [19]. The HAY has been shown to be capable of achieving a NPR of −6.8 [17], which is the highest published value of any auxetic material to date. An array of HAYs that make up a fabric sheet all exhibiting a NPR would amplify the auxetic effect, thus increasing the desirable enhanced mechanical characteristics. Work published by previous authors up to now, has identified key parameters involved in manipulating the behaviour of the HAY, such as the starting wrap angle [13], [14], [15], [17], tensile moduli of component materials [13], [15], [16], [17], and fibre diameter sizes [13], [14]. However, contradictory claims made by various authors as to which design configuration increases the auxetic effect have made it evident that further investigation is required to determine the optimal design parameters. Whilst previous studies have shown that lowering the starting wrap angle tends to result in a higher maximum NPR [13], [14], [17]; none have identified at which angle that assumption becomes untrue. This study aims to determine the optimal design configuration for maximising the auxetic effect by finding the starting wrap angle which results in the greatest NPR using finite element analysis (FEA). The scope of the study was extended to investigate the effect of changing core and wrap diameter sizes through FEA.
Papers published on HAYs since they were first designed by Hook [7] have studied the design characteristics of the structure and focused on identifying the defining factors that contribute to the auxetic behaviour in the material [13], [14], [15], [16], [17]. This has been done mainly through simulated or experimental tensile testing; or by comparisons of both tests. The structure of the HAY can be defined by various geometrical parameters that all have an effect on its auxeticity to varying degrees. These were the factors considered in the modelling of the yarn. The factors are: wrap diameter, core diameter and starting wrap angle [13]. The material properties that affect a HAY’s performance are: Young’s modulus of both core and wrap; and Poisson’s ratio of core and wrap. Various studies of HAYs [11], [12], [15] showed that the wrap angle has a profound effect on the NPR value. For example, different values for NPR obtained by Miller et al. [11] based on three different HAY designs. The lowest wrap angle of 10° resulted in the largest NPR value of −5.8, indicating that a low starting wrap angle is desired for maximising the auxetic behaviour of the HAY. The results of Sloan et al. [15] study generally agreed with these findings. The HAY with the lowest angle of 13° achieved a maximum NPR of around −2.5. The HAYs with wrap angles of 30° and 38° did not ever have a negative Poisson’s ratio when put in tension. This revealed that the influence of the starting angle of the wrap on NPR is so prominent, that an angle too large will prevent the auxetic effect from taking place at all [15].
Wright et al. [14] observed that the selection of boundary conditions at the end of the yarn have a minimal effect on the yarn behaviour after 10 cycles. A model made up of 10 cycles displayed results within 1.5% accuracy of a model of 50 cycles – showing that the end effects did not have a profound effect on result accuracy at that many cycles. The findings indicated that designing of the HAY model did not require an excessive number of cycles. Wright et al. [14] also claimed that the stiffness of the fibre components were the major influences on the performance of the HAY. They alleged that a HAY with a low stiffness ratio is not fit-for-purpose, and it was shown that a lower core/wrap diameter ratio results in a higher NPR value. The study also highlighted the effect that increasing the Young’s modulus of the wrap had on the NPR value. Study by Sibil and Rawan [13] claimed that by increasing the core/wrap diameter ratio – and therefore lowering the stiffness ratio – the value of NPR can be increased. They showed that a HAY with a stiffness ratio of 0.003 (achieved by virtually reducing the wrap diameter by three orders of magnitude) displayed a higher magnitude of NPR when compared to a HAY with a stiffness ratio of 7.4. These contrasting results made it difficult to predict the ideal design parameters the HAY should have had, but elucidated the need for further study to determine the ideal design configuration to maximise the auxetic effect. To further develop the design of the HAY, it was important to identify the optimum configuration of core and wrap that this previous work had not shown. However, both studies showed that the core/wrap diameter ratio is inversely proportional to the core/wrap stiffness ratio.
For the arrangement of wrap and core to work as intended; the wrap should be composed of material with a relatively high Young’s modulus, and the core should be composed of material with a Young’s modulus lower than that of its wrap counterpart [18]. It is also important to note that the behaviour of HAY fabrics is heavily affected by the inter-yarn contact friction and shear, and in single HAYs this is compounded by the friction and contact between the core and the wrap. However, in this study, the model does not consider contact friction between the core and the wrap, and therefore, the purpose of this study is to optimise the design features of the auxetic behaviour under deformation mechanisms and enhanced properties because of having a negative Poisson’s ratio.
Section snippets
Methodology
The model in this study was analysed under the axial tensile load test conditions. The tensile testing method is the only procedure to date, that has been used to determine the NPR of a HAY [11], [12], [13], [14], [15], [16], [17], [18], [19]. The load was applied until a maximum deformation of 1 mm per cycle of HAY had been reached [17], and the model does not consider contact friction between the core and the wrap.
Analytical analysis
The initial geometrical parameters of each design of HAY were determined using simple trigonometry. Fig. 1 shows the arrangement of wrap and core components for each cycle of HAY form a right-angled triangle, where the length of hypotenuse is equal to the length of the wrap if the wrap was unravelled and laid out straight. The initial length of the core, Lc0 – and therefore the length one cycle of HAY – can be found by calculating the length of the bottom leg of the triangle. The known input
Finite element analysis
HAY’s computational design in this study were modelled using SOLIDWORKS® 3D CAD software. The helical arrangement of the wrap component around the core was simply modelled using the ‘helix’ feature and by extruding the base of a concentric 2D core with a centre point aligned with the centre of the helix. The model is illustrated in Fig. 2(a). Since all the models consisted of the same two components wound together in a helical manner, the design parameters can be easily edited using a
Validation of negative Poisson’s ratio
As can be seen in Fig. 4, the same kind of typical response was being induced by each model for a total deformation of 10 mm. This response was explained by the general pattern of each curve. When strain was first applied, it caused the wrap to tighten itself around the peripheral of the core – resulting in the total diameter shrinking below its initial size. This is shown by the sudden spike at the beginning of the graph, where the Poisson’s ratio of the HAY shot up, then rapidly decreased due
Sensitivity analysis of core Poisson’s ratio
The sensitivity test, whilst not vital to the aims of the study, provided a good means of obtaining more realistic results from the simulations. Had the model used by Sibal and Rawan [13] been used as the base model for all subsequent designs generated in this study, the behaviour and responses of the HAYs would not have been expected to vary by a great deal. This is evident from the similarity in results between Model A1 and Model A2. What it means for this study however, is that the results
Conclusion
Several designs of HAYs were generated so that their auxetic performance characteristics can be identified. The designs were tensile tested through finite element simulations to determine their maximum NPR and the rate of strain at which it was achieved. A maximum NPR of −12.04 was achieved by lowering the wrap starting angle. The findings show that a starting wrap angle of 7° will produce the highest NPR. This should be the design angle for optimised performance of the yarn. Additionally, the
Acknowledgements
This paper presents independent research and this work did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. However, the authors would like to acknowledge advice about the potential application of auxetic structures for oil & gas safety applications by Mr. John Locke, Expro North Sea Ltd, Stirling, UK.
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