An analysis of a fibrous assembly at large deformations has been explored as a means of predicting the compressional behavior, especially the compression hysteresis. The essential principle is to classify all the fiber contact points in the assembly as either slipping or nonslipping, so they can be dealt with separately. The compressional modulus and Poisson's ratio are derived and shown to be dominated by the mechanical properties and the directional distribution of the individual fibers within the assembly. An iterative algorithm, in which the system geometry is updated on successive incre ments, is developed to cope with large and nonlinear deformations. A comparison between the theoretical prediction and the experiment has indicated reasonable agree ment. An improvement could probably be achieved by including the contribution of fiber viscoelasticity to the total hysteresis.

The initial response of the unit fibrous cell to an externally applied shear stress is assumed to involve both the bending of fiber sections and slippage at contact points. The criterion used to determine whether a contacting fiber will bend the contacted fiber section or whether it will slip along it depends on the relative angles of the two fiber sections to the external stresses. The total proportion of slipping and nonslipping contact points is thus derived using the density function of fiber orientations within the unit fibrous cell. The derived modulus values for shear behavior are related to the moduli derived earlier for compression behavior. The symmetry rules for selected idealized orientation distributions ( e.g., random) of the fibers in the unit cell provide a check on the validity of the derivations.The use of continuum mechanics to model the behavior of assemblies of fibers remains restricted by inadequate knowledge of the constitutive properties of the unit fibrous cell. In the last three years since we first raised this issue [4], a number of papers have appeared that have improved our understanding of how the unit cell responds in tension [ 2 ] and compression [ 3,8 ] . This understanding includes a prediction [ 3,8 ] of the various Poisson's ratio terms so that algorithms are now available for estimating many of the tangent compliance terms in the general material properties matrix.While these recent analyses of tension and compression have been able to draw on a long established legacy of published research dating back at least to the theories of Grosberg [ 5 ] and van Wyk ( I 1 ] , respectively, the same does not apply to shear behavior. Indeed while it has long been recognized that shear between layers of fibers in yarns [ I ] or fabrics [ 5 ] does occur, the theoretical treatment of shear as a continuum strain has received only minimal attention. Where it has been modeled, the shear deformation has been treated as frictional slippage between layers of fibers. As such there has been no consideration of the low-strain response of the assembly under shear stress, no elastic strain energy due to shear has been calculated, and only the inelastic mechanisms associated with catastrophic shear failure ( i.e., massive slippage) have been dealt with.It is cert.ainly true that such simple textile deformations as twisting a yarn [9] or bending a fabric [ 5 ] involve almost immediate shear failure, because the ' essentially parallel fibers slide past each other with almost no restraint in accommodating large deformations of the yarn or fabric. The simplest way of treating this problem is to assume a high initial linear shear stiffness [ 5 J ( e.g., equal to the fiber modulus) up to a small critical shear stress. Thereafter, the assembly may be considered to shear via frictional slippage with no increase in shear stress. The work done against friction must be accounted for in the energy equations, but the tangent compliance in shear for initial shear stresses above the threshold level may in fact be rega...

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The compressional mechanism of a random fiber assembly is analyzed by an energy method. An infinitesimal fiber segment, which is bounded by two neighboring contact points, is chosen as the unit bending element. The geometry of this element is characterized by its arc length, curvature, and orientation. The change in bending energy of each fiber segment due to the compression of the assembly is derived in terms of the compressional strain and the Poisson's ratio of the assembly. The summation of each energy contribution is done using a continuous joint probability density function of the length and orientation of the segments.

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The extent to which the methods of continuum mechanics may be used to model the mechanical properties of an idealized unit cell of untwisted but aligned fibers has been explored. An important question is the extent to which the continuum assumptions are applicable to a unit cell of fibers and the restrictions that exist with regard to interpolation of continuum solutions. The symmetry properties of the fiber bundle are used to show that, for small strains and linear elastic behavior, the fiber bundle might best be characterized mechanically as a degenerate square- symmetric homogeneous continuum, which is fully defined by six independent elastic constants. None of the existing yarn models includes all six of the elastic constants needed for a rigorous small-strain linear analysis; however, nearly all deformations of textile interest involve both large strains and nonlinear material deformation, and some of the difficulties involved in representing the constitutive properties of the material are discussed, using a tangent compliance matrix. The novel test methods needed indicate that the fiber bundle has various fundamental properties that have little relation to those of single fibers. The development of better test procedures for the key continuum parameters and consequent improve ments in the modeling of both mechanical and geometrical nonlinearities are seen as important stepping stones to improved modeling of the mechanics of textile structures.

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Received as an invited paperThe same fundamental approach that has previously been found reasonably accurate in predicting the tensile behaviour ofbulky singles yams at small extensions of up to 10% is used here to model yarn torsion. The theoretical analysis is based on discrete-fibre-modelling principles, an energy method, and a 'shortest-path' hypothesis. It is applied to the twisting behaviour of bulky singles yarns which may have non-uniform fibre-packing-densify across their cross-sections. The twisting of a yam both under constant yam length and under constant yarn tension is modelled. The contributions to yam torque due to fibre tension, fibre torsion, and fibre-bending can all be calculated. Comparison of the theoretical models with some experimental data will be presented in Part II. INTRODUCTION 1.1 Significance of Yam TorqueTorque has practical significance for a number of the characteristics of yams, fabrics, and carpets. The magnitude of initial torque and the torsional-deformation and torsional-recovery properties of yams influence the twist distribution in singles yams, the tendency of singles yams to snarl, yam-twist instability, the balance of twist in ply yams, skewness of woven fabrics, spirality of knitted fabrics, tuft integrity of cut-pile carpets (this affects carpet appearance-retention) and effects or defects such as pebble or pucker in fabrics and frise tufts and tip-curl tufts in cut-pile carpets.There are three options for the theoretical treatment of freshly twisted yams: snarl; apply constraint; allow to untwist. We know it is impracticable to apply a significant torque to a singles yam in the absence of an axial tension, and a freshly twisted yam will invariably snarl or twist up on itself unless it is constrained from doing so. This phenomenon of yam twisting up on itself has been called 'torsional buckling' [1]. Accordingly, a singles yam with a straight axis can generally only be regarded as subject to a combined-load case of tension and a restraining torque [2].For example, in the cut-pile-carpet trade, the demand for yams of good twist stability is high, as these are essential for piece-dyeable qualities and a high level of tuft integrity in use. For many cut-pile carpets, it is necessary to have a yam which is straight and free from any tendency to untwist, even at the cut ends. Twist stabilisation or setting is an essential process for all yams to be used in cut-pile carpets. It is known that it is impossible to achieve a high degree of set if the fibres in the yam to be set are not significantly strained [3]: this applies to wool and probably to other fibres used in carpet-pile yams. The strains in individual fibres are translated into a residual yarn torque, and the greater the torsional stress in the