## Abstract

The dynamics of fibre slippage within general non-bonded fibrous assemblies is studied in the situation where the assembly is subjected to general small cyclic loads. Two models are proposed. The first is applicable when the general cyclic loading is complemented by an occasional tugging force on one end of a fibre, which causes it to gradually withdraw from the assembly, such as might occur during the pilling of a textile. The second considers the situation in which the cyclic perturbations act around a constant background load applied to the assembly. The dynamics is reminiscent of self-organized critical behaviour. This model is applied to predict the progressive elongation of a single yarn during weaving.

## 1. Introduction

When non-bonded fibrous assemblies, such as most yarns, woven, non-woven and knitted fabrics in which individual fibres have not been chemically or thermally bonded to one another, are subjected to external loads, their subsequent deformation involves fibre deformation and fibres slipping over their neighbours. The mechanics of assembly deformation, including quasi-static slippage from fibre ends, has been well studied (Hearle *et al*. 1969), but the dynamics of fibre slippage has thus far been neglected; to the author's knowledge, there has been no work published on the dynamics of fibre slippage in cyclically perturbed non-bonded fibrous assemblies. Typically, it has been included by introducing empirical factors without considering the micro-dynamics of the slippage process, for instance in the case of compression see Carnaby & Pan (1989)

However, problems involving fibre slippage are numerous and important in the textile world. Especially, when the load is cyclic, the dynamics of fibre slippage is the determining factor in the long-term response of the assembly. One classic example is the creation of pills on the surface of an assembly, which is reviewed in Ukponmwan *et al*. (1998). Pills consist of partially and fully withdrawn fibres whose exposed ends entangle together, and pilling occurs when the exposed ends are repeatedly tugged while the assembly is repeatedly deformed. Another application, discussed in Weinsdorfer & Wimalaweera (1987) and Blanchonette (1996), where fibre slippage is important, is provided by weaving, where the warp threads, which are always under some tension, are repeatedly pulled, rubbed and twisted in order to create the shed and consolidate the weft into the fabric. During this process the warp threads can slowly elongate and may eventually fail, which is to be avoided. This is a creep process, where the cyclic loads replace the role of applied heat in the traditional engineering-type study of creep. However, because movement of one segment of a fibre can cause other segments of that fibre to move too, the fibrous creep is more complicated than for traditional materials. The progressive change of shape of garments, known as bagging, is due to similar causes. Yet, another example is the progressive compaction of a compressed batt of fibres that is subjected to vibrations (see Carnaby & Pan 1989).

Fibre slippage occurs in other settings too. For example, fibres in fibre-reinforced composites can bridge cracks and thereby inhibit crack growth by reducing the stress concentration at the crack tip (Kelly 1964; Kim & Mai 1998; Matthews *et al*. 2000). When the fibres slip from the surrounding matrix the crack may propagate. A significant number of problems involve cyclic loads (Zhou & Mai 1993; Zhang *et al*. 2001), similar to those considered in this paper. In modelling this phenomenon, it is assumed that the cyclic or gradually increasing load causes the frictional properties at the fibre–matrix interface to degrade slowly through wearing out or smoothing of the fibre surface roughness.

However, this approach is not justified for the fibrous assemblies considered in this paper which are not chemically bonded. When, for whatever reason (part of), a fibre slips through a surrounding fibrous assembly, it immediately comes into contact with new parts of the assembly which hamper further slippage. Thus, in non-bonded fibrous assemblies, there is no simple analogue of the debonding process occurring in fibre-reinforced composites.

Closer analogues may be the ‘reptation’ of polymer molecules through materials (Rouse 1953; de Gennes 1971, 1979; Doi & Edwards 1978, 1988), or sliding charge-density waves (Coppersmith 1987, 1988; Tang *et al*. 1987). In each of these problems, a ‘fibre’ (respectively, a polymer chain and a group of charges) under the influence of an external force slips through a background ‘assembly’ which inhibits its motion. These two examples have bonding properties between the fibre and the assembly, which are similar to those studied in this paper. However, it is difficult to directly translate either of these situations into one that is appropriate to use here.

This paper proposes and solves two models for fibre slippage in cyclically perturbed non-bonded fibrous assemblies.

The first considers the response of a fibre of finite length as its end is occasionally tugged by a small force while the assembly is repeatedly deformed. Without the latter deformation, a simple model does not lead to progressive withdrawal of the fibre, but it is proposed that the deformation promotes stress relaxation within the fibre and the assembly, and this is shown to result in fibre withdrawal.

The second model considers the response of a finite-length fibre whose end is pulled with a constant force when the background assembly is cyclically perturbed in an arbitrary way. The perturbations have the effect of randomly strengthening or weakening some of the fibre–assembly contact points and this allows progressive movement of the fibre through the assembly. Numerical simulations of the ensuing complicated dynamics suggest that the model might be self-ordered critical as defined in Bak *et al*. (1988) and Bak (1997).

The aim of this paper is to present the theoretical groundwork on which later applications can be based. Here only two applications—pilling and gradual yarn elongation—are sketched. Pilling is normally considered to be a fabric phenomenon, and it is only recently that pill formation on the surface of a single yarn has been studied experimentally, without the extra complication of that yarn being part of a fabric. It is debatable whether the fabric or the yarn construction has a greater effect on pilling propensity. In this paper, the experimental data on yarn pilling are used only to demonstrate agreement between experiment and theory, and the issue of fabric effects is left for future research.

## 2. Occasionally tugging the fibre

The fibre–assembly system is modelled by two straight extensible coaxial cylindrical sleeves of material of equal initial length which are coupled by friction. There is a continuum of contact points between the inner sleeve (the fibre) and the outer sleeve (the assembly).

A cyclic tugging force is applied to this system. In the pull phase of the load cycle, a force *F* is applied to one end of the fibre and the other end of the assembly. This causes both the fibre and assembly to extend, and as they do, the fibre will withdraw a small distance. This is depicted in figure 1. The withdrawn part of the fibre experiences the full force *F*. Inside the assembly, there is progressive transfer of tension from fibre to assembly until the ‘fully gripped region’ is reached, where the fibre and assembly strains are equal. The same comments apply for the right-hand end of the system.

The external force is then reduced to zero, and the fibre moves a small way back into the assembly until the system reaches an equilibrium.

If the same force is applied for a second time, the fibre is pulled out to the same extent as in the first pull and retracts to the same extent. Consequently, if the external force is repeatedly cycled between *F* and zero, there is no progressive withdrawal of the fibre from the assembly: the middle sections of the fibre–assembly system remain ‘stuck’ together.

However, in most realistic situations, the assembly will be continually subjected to cyclic perturbations. For instance, pills (tangled balls of partially and fully withdrawn fibres) often form on apparel textiles when the garment is being continually rubbed, stretched and twisted during wear or washing, while tugging forces may act on a particular fibre only occasionally. This is modelled by allowing the stress concentrations in the fibre–assembly system to dissipate at the end of the tugging-force cycle.

### (a) Basic assumptions and notation

Fibre parameters are labelled with the subscript ‘f’, while parameters pertaining to the assembly have a subscript ‘a’.

The assembly is considered to be a continuous medium and the direction along which the fibre lies is denoted by *z*. Both the fibre and the assembly are homogeneous and isotropic. The fibre is always straight and its natural length is *L*_{f}. The assembly can provide a frictional drag of Newtons on each small section d*z* of the fibre. Therefore, is a force per unit length, and as parts of the fibre within the assembly are stretched, the total frictional drag increases.

It is assumed that the external forces are increased and decreased very slowly, so that the dynamics of the situation during the times of changing external load is unimportant. For the same reason, it is assumed that the frictional damping proportional to the velocity of the fibre is large.

Only the linear region of the mechanics is explored. Poisson's ratios are assumed to be zero. It is assumed that the properties of the fibre and the assembly, such as their elasticity and cylindrical radii, are unaffected by their elongation and the amount of withdrawal. The *zz* component of the stress tensor is the only one of interest: all shearing is ignored.

Denote the elasticities of the fibre and the assembly by *E*_{f} and *E*_{a}, respectively. This means specifically that when under some tension, their strain (dimensionless ratio of change in length to natural length) is the quotient of their internal tension and elasticity: in other words, ‘elasticity’ is the product of the usual Young modulus and the cross-sectional area.

The results will be expanded to first order in *F*/*E*_{f} or *F*/*E*_{a}. It is further assumed that , otherwise the fibre is in danger of withdrawing in one tug. Indeed, in many assemblies made of natural fibres such as wool, , is substantially greater than the breaking load of a fibre, so if *F* approached this limit the fibres would snap, rather than progressively slip.

### (b) The first tug

Start with an unstressed fibre sitting completely within the unstressed assembly.

After applying the external load, the left-hand end of the fibre slips out of the assembly, dragging the left-hand portion of the assembly slightly to the left. The tension in this part of the fibre reduces with *z* because of friction, until the critical tension of is reached where the fibre and assembly strains are equal. Segments of the fibre and assembly to the right-hand side of this point will not be slipping, but will have expanded by the same rate and by the same amount, and friction will no longer be relevant. At equilibrium, the tension distribution in the system is as depicted in figure 2*a*.

Parameterize the length along the fibre by *s*_{f}, with 0≤*s*_{f}≤*L*_{f}. Before the external tension is applied the fibre may be embedded into space with the function(2.1)In general, the fibre's tension is related to its embedding by(2.2)and in the regions where friction is relevant(2.3)which may be used to derive the embedding function. Demanding continuity of tension and the embedding yields(2.4)where the values of *s*_{f} at the junctions are(2.5)The solution only makes sense if *s*_{2}≤*s*_{3},(2.6)Assume that friction is large enough, or the external force is small enough, so this is the case.

Finally, the distance between the centre of mass of the fibre and of the assembly is(2.7)All results have been expanded to first order.

### (c) The first release

Studying frictional systems is complicated, because there are many equilibrium configurations, and the final result depends on the path taken. In this instance, after release, the tension distributions in the fibre and assembly are shown in figure 2*b*. The parts of the fibre and assembly that were at their critical tensions remain together when the external force is reduced to zero. Using the same technique as above, the distance between the centre of mass of the fibre and of the assembly may be shown to be(2.8)

### (d) Perturbations causing stress relaxation

Although cycling the force between *F* and zero does not lead to any progressive withdrawal (which may be deduced from the fact that parts of the fibre and assembly at critical tension remain stuck to each other throughout the entire pull–release cycle), in many realistic situations, a fibre's end is only occasionally tugged, but the assembly is continually being deformed. This process will continually change the frictional characteristics of the fibre-to-assembly contact points and will tend to remove the local stress concentrations resulting from a tug-and-release, allowing the fibre and assembly to relax so that they return to their original lengths.

If, for instance, *E*_{a}=*E*_{f}, then, because of symmetry, the centres of mass of the fibre and assembly will (on average) remain fixed during the relaxation process (which essentially reduces the friction to zero) and so the distance moved is given by equation (2.8),(2.9)

Similarly, after the *n*th cycle, the withdrawn length is(2.10)The continuum version(2.11)where *ω* is the number of cycles per time-step, is more easily solved.

### (e) Comparison with pilling

Pills form on the surfaces of yarns when they are cyclically rubbed. Trained specialists can estimate, somewhat subjectively, the ‘rate of pilling’ by recording the amount of time taken for the pills' appearance to reach a given point. The results of such an experiment are shown in figure 3, where the rate of pilling was measured as a function of the yarn pitch (J. Lappage 2004, personal communication). Fibres in a yarn wind helically around its axis and the number of complete revolutions per millimetre along the axis that a fibre experiences is the reciprocal of the yarn pitch.

It is believed that fibres resist being pulled from a yarn primarily because of the fibre entanglement and the capstan effect. The latter causes an exponential decay of tension in the fibre due to its helical nature. Disregarding entanglement, because it is difficult to measure experimentally and is likely to be similar for each of the yarns tested, the important parameter is the effective frictional parameter(2.12)Here *r* and *c* are, respectively, the radius and pitch of the helix (so the fibre curvature is for instance).

To leading order, the rate of fibre withdrawal, which is correlated with the rate of pilling, *R*, is dominated by the small-*Δ* regime where the withdrawal rate is small. The formulae in §2*d* then suggest that(2.13)Two fits are shown in figure 3 for *r*=0.3 mm, from which it is evident that the predictions and experiment agree fairly well.

It is known that fabrics, and presumably yarns, with virtually all protruding fibre ends removed (e.g. by the finishing treatment known as singeing) can still become pilled during use. This scenario would be similar to setting *F*∼0 at the start of the process, with *F* initially growing with withdrawn length. It would be interesting to analyse this case when experimental data becomes available.

## 3. Constant external force during assembly deformation

This second model considers the response of a straight fibre as it is pulled by one of its ends with a constant force through a background of assembly fibres which is modelled using a frictional force field. In contrast to the first model, the cyclic perturbation is modelled here by randomly perturbing the friction along the length of the fibre.

In later applications, such as the one presented in §3*c*, the constant force may arise physically via the action of other fibres on this particular fibre. At this simple level, its origin is ignored, however.

The fibre rapidly moves a little way in the assembly under the influence of the external force until it gets stuck due to some particular arrangement of friction along its length. The perturbation then slightly changes the frictional force field and this might let (parts of) the fibre move a little further within the assembly until it gets stuck once again, and so on.

The movement is jerky, not smooth. The dynamics of the fibre system is complex, and the author has not been able to solve the model analytically.

A convenient quantity describing the slippage is the long-time average of the velocity of the fibre's centre of mass, . Interestingly, computer simulations of a discretized version of the model suggest that the probability distribution for the short-time average centre-of-mass velocity is a power law.

### (a) Notation and two examples

The notation used is similar to that of §2: the fibre is described by parameters *E*_{f} and *L*_{f}; and the external constant force is *F*>0. The frictional force, which models the inextensible assembly, acting on length d*z*_{f} is a random function of *z* with a Gaussian distribution of mean and variance .

Physical constraints suggest and to avoid the strange case of a negative frictional force on the fibre, also assume that .

The perturbing influence, *ν*, acting on a unit length of the assembly is modelled by Gaussian white noise with ‘strength’ parameterized by the variance *Ω*, with dimension force^{2} length^{−1} time^{−1}, so that its probability distribution is(3.1)

#### (i) Example 1

Direct numerical simulations require the fibre to be discretized into a number of nodes connected by elastic springs. The assembly is similarly discretized into stationary segments of arbitrary fixed length, *L*_{i}, each with a different friction-per-unit-length *ϕ*_{i} associated with it. Indeed, since real fibrous assemblies are naturally discretized (fibre–assembly contact points occur only a discrete places along the fibre's length due to natural fibre crimp), this description may be more appropriate than the continuum version.

The external force acts on the right-hand node and the nodes move until they are at equilibrium in a marginally stable state. It is advantageous numerically to use near-critical viscous damping to ensure rapid convergence. The damping has little effect on the final results. The maximum frictional force experienced by any fibre node is a linear combination of the product of local *ϕ*_{i} and the length of inter-node material that they act upon.

The perturbation acts on each assembly segment only when the fibre has stopped, and afterwards the fibre may move (more-or-less instantaneously) to a new equilibrium configuration. This essentially provides a discretization of time into units of *τ*: the period of the perturbation. (The parameters *τ* and *L*_{i} simply set the time and length-scales in the problem and may be set to unity in a computer simulation.) The perturbation acts on each assembly segment according to the rule(3.2)where *ν* is a different Gaussian random number for each segment.

Replacement rule (3.2) describes a discrete-time Langevin dynamics, with the factor renormalizing the result so that the variance of *ϕ*_{i} remains unchanged after perturbation. Expanding in small *τ* yields a more conventional presentation,(3.3)while taking the *τ*→0 limit yields the Fokker–Plank equation for the probability distribution, , of *ϕ*_{i}:(3.4)It is now clear that the mean is and the variance as required.

#### (ii) Example 2

If the fibre is inextensible, the maximum frictional force is normally distributed with mean and variance . When the fibre is stationary , the perturbing influence causes the frictional force on the fibre to evolve according to the simple Fokker–Plank equation:(3.5)Here is the probability distribution for the frictional force. The right-hand side of this equation may be derived from the continuum limit of the Langevin presentation above, or by fixing the coefficients in the usual Fokker–Plank equation by demanding that has the required Gaussian form.

The dynamics of moving and perturbing is similar to a Brownian particle in a central force field: when it is to the left-hand side of some point , the fibre is stationary, while when it is to the right-hand side of that point, the fibre is moving.

### (b) Results of numerical simulations

The computer simulation suggests that the fibre system rapidly tends towards a state in which its tension decays linearly to zero along its length. In this state its extended length is(3.6)The fibre reptates through the assembly from one marginally stable state to another in a way that is characteristic of a self-organized system (Bak 1997): the slight movement of one node often has very little effect, but sometimes it causes an ‘avalanche’ of subsequent movement of other nodes.

For an inextensible fibre, dimensional and geometrical considerations suggest the average centre-of-mass velocity, , should be(3.7)where *h* is an arbitrary function. For an extensible fibre, this velocity is well fit by(3.8)The fit is shown in figure 4.

In generating figure 4, it was necessary to use at least 50 000 perturbations, because the probability distribution of distance moved each cycle is strongly peaked around zero but has an important power-law tail. This suggests that the model is self-organized critical (Tang *et al*. 1987). For example, for the parameter values used in figure 5, the frequency, *k*(*d*), of moving a distance *d*, is(3.9)where the errors are at the one-sigma level. Similar plots occur for other parameter values.

Power-law distributions are given prime importance in the theory of self-organized critical systems (Bak 1997), yet such systems also include fractals. It is difficult to see where fractals could appear in the present model. This, and how the power-law exponent depends on the physical parameters and the model construction, are being investigated by the author.

### (c) Comparison with warp-thread elongation

An experimental rig detailed in Brorens *et al*. (1990) that records the gradual increase in length of a taut yarn, which is cyclically perturbed by twisting and additional loading, produces a typical output shown in figure 6.

To model this process requires a number of additional assumptions about the behaviour of the yarn, and a very simple model is presented here to illustrate how equation (3.8) might be used in practice.

Each fibre is subjected to an external force and a perturbation in a similar way to the one discussed previously. The tension must reduce to zero at both ends of the fibre, but imagining each fibre cut roughly in half yields a situation in which one end of each fibre experiences a constant external force. Assume that all fibres are identical and that *L*_{ext} and *Ω* are all independent of the longitudinal strain of the yarn—which, for a yarn of length *L* and initial length *L*_{0} is defined as . Assume also that friction is due to the capstan force with and scaling in the same way:(3.10)where *r* and *c* are the helical radius and pitch of a fibre wrapping around the outside of the yarn. Similar assumptions have been made in §2*e*. Finally, assume that the yarn elongates with constant volume, so that and , and that .

In terms of the parameter(3.11)equation (3.8) predicts that the rate of change of strain should be(3.12)for some unknown *A* and *B* to be determined from experiment.

Due to natural irregularities, only a small length of the yarn actually elongates, and, unfortunately, the strain data are in terms of the total length of yarn, rather than the strain of the elongating region, which means that *e* may be arbitrarily normalized. By choosing appropriate *A* and *B*, equation (3.12) can be easily made to fit the experimental results.

## 4. Conclusions

Two models of fibre slippage within a non-bonded cyclically perturbed fibrous assembly have been proposed. The first attempts to describe the situation where fibres are only tugged by an external force occasionally compared with the frequency of the perturbation. The amount of movement per tug is given in equation (2.8). The second attempts to model the situation in which fibres are under a constant background load while the assembly is subjected to perturbations. The long-time averaged fibre velocity is given in equation (3.8). Interestingly, the movement consists of a series of ‘avalanches’ and is governed by a power law in a manner highly reminiscent of a self-organized critical system.

Two examples of how this theoretical framework could be applied to experimental situations were sketched. These were pilling as a function of yarn twist, and the gradual elongation of a cyclically perturbed taut yarn. More detailed and diverse applications are currently being investigated by J. W. S. Hearle and A. Wilkins.

The models described in this paper treat the fibres as straight, although in reality they may be following pseudo-helical paths within tubes. Another situation which is being investigated is one in which the fibres have excess lengths present as buckled segments. Perturbations could then cause local flattening of the buckled segments, which would lead to relative fibre movement. An alternative to the micromechanical analysis in this paper is to attempt a statistical mechanics treatment similar to the one discussed in Brujić *et al*. (2003) applied to powders subject to perturbations.

## Acknowledgments

The author thanks John Hearle for general guidance and comments regarding the manuscript. For discussions on the mechanisms of fibre slippage and perturbations, the author wishes to thank Jim Lappage, Malcolm Miao, John Hearle, Clive Marsh, Surinder Tandon and Harry Liu. Financial support was through the FRST postdoctoral fellowship WROX0010.

## Footnotes

- Received March 31, 2004.
- Accepted September 15, 2005.

- © 2005 The Royal Society