On the Poisson's ratios of a woven fabric

Abstract

Although it is undeniable that the Poisson's effect on the behavior of a woven fabric is crucial, there have been relatively few papers devoted to this subject. In this study, a mechanical model for a woven fabric made of extensible yarns is developed to calculate the fabric Poisson's ratios. Theoretical results are compared with the available experimental data. A thorough examination on the influences of various mechanical properties of yarns and structural parameters of fabrics on the Poisson's ratios of a woven fabric is given. The prediction of Poisson's ratios in this paper will enable more rigorous studies on such important issues of fabric bending and draping behaviors.

Introduction

The Poisson's ratio is one of the fundamental properties of any engineering materials, and represents important mechanical characteristics for a woven fabric in many applications including in engineering systems which incorporate textile fabrics as structural elements. Such structures include inflatable containers, tires, certain plastic laminated sheets, belting of various kinds, parachutes, sails, mackintoshes, etc. The magnitudes of Poisson's ratios can attain some peculiar values for woven fabrics [1], very different from those for conventional engineering materials, leading to unusual stress–strain relationships.

Literature indicates that many investigators have studied the response of woven fabrics to a planar stress system. Kilby [2] in 1963 put forward a simple trellis model for a woven fabric and analyzed the planar stress–strain relationship, based on the continuum mechanics for an anisotropic elastic lamina; yet the author admitted the deficiency of this theoretical model for not exhibiting the Poisson's ratios. Assuming a relation of yarn curvatures between the released and the stressed states for a woven fabric, Olofsson [3] soon after gave a mathematical analysis of equilibrium conditions, stress–strain relationships in extension and compression, and energy in bending. Grosberg and Kedia [4] analyzed the initial load-extension modulus of a cloth and showed that it depends not only on the bending modulus of the yarn and the geometry it takes up in the cloth, but also on the strain history of the fabric. An excellent summary of the analysis of the mechanical properties of woven fabrics prior to 1969 was included in the monograph by Hearle et al. [5], who derived the Poisson's ratio of a woven fabric assuming that the yarn extension and compression were negligible.

By means of optimal-control theory, de Jong and Postle [1] applied the general energy analysis of fabric mechanics to the woven fabric structure for deformations, where the yarn extension was introduced into the theory. It was noted that the theoretical calculations of the fabric Poisson's ratios based on the assumption of inextensible yarns are in conflict with the experimental results. Next, Huang [6] presented a methodology for analyzing the problem of the finite biaxial extension of a plain woven fabric, including such effects as the initial stresses due to partial setting of yarns, loss in bending stiffness associated with fiber slippage in the yarn and the contact deformation of the yarns at the crimp in the analysis. Numerical solutions were found by using the Runge–Kutta method for the problems of fabric uniaxial extension and biaxial extension with equal stresses. But the fabric Poisson's ratios were not dealt with in the analysis. Leaf and Kandil [7] constructed a simple mechanical model termed the `straight-line' or `saw-tooth' scheme to represent an idealized woven fabric and presented an analysis of the initial load-extension behavior of plain woven fabrics. A closed-form analytical solution was found for the initial Young's modulus and the Poisson's ratio of the fabric, when the yarns were assumed to be inextensible and incompressible.

By modeling the individual yarn as extensible elastica, Warren [8] in 1990 determined the in-plane linear elastic constants of woven fabrics. Results of this theoretical analysis compare favorably with the measured in-plane Young's moduli of woven fabrics. Pan [9] proposed a fabric model as the chain of yarn sub-bundle. The fabric stress–stain curve and the mean fabric strength were predicted for both uniaxial and biaxial tension cases. Using the finite element method, Tarfaoui et al. [10] recently studied theoretically the mechanical behavior of textile structures of two different weave types: plain and twill. However, they admitted that analyzing their numerical results proved to be very hard and thus demanded a study of the stress field in the fabric unit cell.

Insofar as the experimental determination of fabric Poisson's ratios is concerned, Lloyd and Hearle [11] in 1977 examined their suggested method of a uniaxial tensile test. It was found that the method does not provide a reliable technique and that, to overcome the shortcomings inherent in the specimen geometry, a biaxial test method is needed. Bao et al. [12] in 1997 studied the error sources for measuring the apparent Poisson's ratio of textile fabrics by uniaxial tensile testing. In order to correct the experimental error, samples of various dimensions were utilized for their experiments, and some concrete examples to acquire apparent Poisson's ratios of fabrics by the uniaxial tensile test were also shown.

Up to now, it is hard to get accurate Poisson's ratio measures due to lack of reliable experimental techniques for woven fabrics [13]. While the significance of the effects of Poisson's ratios on fabric drape and other behaviors is well recognized, their values were mostly estimated, based on those for ordinary solid materials, for fabric modeling and simulations [14], [15]. Although a few papers dealt with the theoretical prediction of Poisson's ratios for non-wovens [16], [17], there has been little done analytically determining Poisson's ratios for woven fabrics based on the assumption of extensible yarns in fabrics, and discussing the influences of the mechanical properties of yarns and the geometrical parameters of fabrics on the Poisson's values.

This paper tries to fill the need. First, following the approach put forward by Warren [8], a mechanical model for woven fabrics made of extensible yarns is developed to get a closed-form expression for Poisson's ratios. Then, the analytical results are compared with the experimental measurements. Finally, a parametric analysis is given for the effects of various geometric and mechanical parameters on the Poisson's ratios of woven fabrics.