http://trj.sagepub.com/Textile Research Journal

http://trj.sagepub.com/content/72/11/954The online version of this article can be found at:

 DOI: 10.1177/004051750207201104

2002 72: 954Textile Research JournalH. Liu, S.K. Tandon and E.J. Wood

Part I: Wear Model of Cut Pile Carpet  

Published by:

http://www.sagepublications.com

can be found at:Textile Research JournalAdditional services and information for    

  http://trj.sagepub.com/cgi/alertsEmail Alerts:

 

http://trj.sagepub.com/subscriptionsSubscriptions:  

http://www.sagepub.com/journalsReprints.navReprints:  

http://www.sagepub.com/journalsPermissions.navPermissions:  

http://trj.sagepub.com/content/72/11/954.refs.htmlCitations:  

What is This? 

- Nov 1, 2002Version of Record >>

at TEXAS SOUTHERN UNIVERSITY on November 11, 2014trj.sagepub.comDownloaded from at TEXAS SOUTHERN UNIVERSITY on November 11, 2014trj.sagepub.comDownloaded from

954

Probability Fracture Mechanics of Wear in Wool CarpetPart I: Wear Model of Cut Pile Carpet

H. LIU, S. K. TANDON, AND E. J. WOOD

Wool Research Organisation of New Zealand (Inc.), Christchurch, New Zealand

ABSTRACT

Modem probability fracture mechanics forms the foundation of Carnaby’s carpetdurability theory. On the basis of this theory, mathematical models have been developedto predict the wear life of wool cut pile carpet. In this paper, the existing models areupgraded to provide an accurate mathematical expression of Carnaby’s theory, in line withthe current advances in knowledge about the fatigue mechanism of wear in wool carpet,The life distribution function proposed by Camaby is redefined to achieve a direct

measurement of the model parameters. The advanced model gives a better fit to the

experimental observations than the previous one.

Carpet wear is a major concern of customers andmanufacturers, since carpet performance changes withthe loss of fibers [9]. The pile of a wool carpet is wornaway in service through the fatigue process of foot traffic[7]. A number of empirical and analytical models havebeen developed to predict the thickness of wool carpetagainst treads [2, 3, 11 ]. On the basis of modern proba-bility fracture mechanics, Carnaby [ I, 21 proposed thatthe durability of wool carpet is determined by its initialconditions, life distribution of fibers, and fatigue site

distribution along fibers. The basic set-up of a wool

carpet is described by the initial conditions. The life

distribution is the number of broken fibers over the total

number of fibers during wear, or the cumulative proba-bility of fatigue failures. The fatigue site distribution

gives the remaining length of a broken fiber. This theorygives a complete mathematical description of wool car-pet wear.The Weibull distribution is commonly used in fatigue

failures problems [8] and is based on the weakest chain

model (i.e., fatigue failure occurs when the weakest linkbreaks). Wear in wool carpet is a typical example ofWeibull’s model. Studies of the fracture mechanisms inwool carpet wear have revealed that fiber damage is

spread fairly uniformly throughout the depth of the pilein straight walking, and there might be bias under otherwalking forces (e.g., turning wear) [4, 6, 7]. From the

existing experimental data, we can construct life and

fatigue site distributions. Combining the initial condi-

tions, life distribution of fibers, and fatigue site distribu-tion along the fibers into one equation, we can thusderive an analytical model.

A constant contact distance between assumed fatiguepoints along a wool fiber was included in Carnaby’smodel [21, because fibrillation due to interfiber contactwas believed to be the mode of fatigue failure in the1980s [4, 12]. Several hypotheses were proposed in the1990s: Axial compression fatigue, instead of fibrillationfailure due to intercontact, was proposed as the majorcause of fatigue failure in wool carpets [4, 5). The

plucking (by surface shear forces) and subsequent recoilof pile fibers may also be causes. Therefore, Carnaby’smodel is refined in this study to a pure mathematicalexpression of his theory, and the contact distance is

eliminated in line with recent advances in knowledge offatigue mechanisms.The original uniform distribution of Carnaby’s model

[ 1, 2, 11 is extended to a universal fatigue site distribu-tion to fit a wider range of experimental results. In

addition, the life distribution of wool fibers is redefinedas the cumulative probability of the fatigue failures ofsingle fibers, instead of the failures of fiber segments.After these modifications, the life distribution parameterscan be directly measured from experimental data,whereas the parameters in Carnaby’s original modelwere indirectly obtained by calibration.

Model DevelopmentThe steps of model derivation follow a logical se-

quence. First, we count the number of broken fibers, thenthe broken-off lengths after one break and k breaks.

Finally, we estimate the remaining total fiber length bytaking away the broken-off segments. Important data,such as pile mass and pile thickness, are calculated fromthe estimated fiber lengths.

at TEXAS SOUTHERN UNIVERSITY on November 11, 2014trj.sagepub.comDownloaded from

955

NUMBER OF BROKEN FIBERS AT TREADS t

Assume the same life distribution F(t) = I - exp[ -a(t - c)t)] (cumulative Weibull function of eachwool fiber, where t = number of treads, a > 0, b > 0.c >_ 0, and F(t) is an average measurement from allfibers and independent of length). Following Carnaby’sargument U. 2], the probabilities that fibers do not breakor break k times at treads t are I - F( t ) and [ F( t ) ] (k= 1, 2, ... ), respectively. Let G be the number offibers per unit area in a cut pile carpet. The unbrokenfibers per unit area are expected to be G[ I - F(t)] (i.e.,the expected value, or the product of probability andtrials). The number of fibers broken at least once is

GF(t).The sum of unbroken fibers and fibers with at least one

break is G[ I - F(t)] + GF(t) = G. The fibers brokenat least twice are among the GF(t) fibers, not the unbro-ken fibers. We may assume that all weak zones have

developed since experiments began, and hence the prob-ability that the GF(t) remaining segments will break isstill F(t). Therefore, there are G[F(t)]’‘ fibers broken atleast twice among the GF(t) remaining fibers. In general,the opportunity for a fiber to break k times is [ F< 1 ) ] ~ , andthe corresponding number of broken fibers is G[ F( t ) ] A ( k= 1,2,...).

This discussion implies that the second breakage oc-curs in the remaining segment of the first break. In fact,the broken-off segments are removed by vacuum clean-ing or other means during carpet use or wear experi-ments. Hence, the probability of a second breakage of apile fiber occurring at the broken-off segment is zero. Inother words, the second breakage of a fiber is located inthe segment remaining in the pile.

REMAINING AND BROKEN-OFF TOTAL FIBER LENGTHAFTER ONE BREAK

Mathematical expectation is defined as the mean of

experiments [ 10]. If x denotes a discrete random variablewith values x,, x,, ... , .r&dquo; and corresponding probabil-ities p,, p,, ... , PII (PI l + P2 +...+ p~t = 1 ), themathematical expectation (or simply the expectation) ofx is denoted by

Interpret this stitement to estimate the remaining totalfiber length as follows: let x, , X 2’ ... , x&dquo; E [0, 1 ]denote the remaining lengths as fractions of the originallengths of n fibers with corresponding probabilities P(43= xj) = Pi (breakage at xj), where j = 1, 2, ... , n and

( p + P2 + - - - + pn = I ). If all fibers have the sameinitial length 1. the mean of the remaining fiber lengths isE(~3) as a fraction of 1. where ~3 is the random variableof broken site distribution as a fraction of the originallength. In other words, the mean remaining and broken-off lengths equal I x E(~3) and x ( l - E( ~z ) ] ,respectively.

’

Define ~, as the random variable of initial lengthdistribution and E( ~, ) _ ! as the mean length of n fibers.Among the n fibers. m.~ fibers have the same lengthI,&dquo;;(~m; = n. i = 1. 2, ... , k). Thus. the remainingtotal length of the n fibers after one fatigue failure is

where nE( ~, ) (or nl) is the original total fiber length.The broken-off length can be calculated from Equation 2as

REMAINING AND BROKEN-0FF TOTAL FIBER LENGTH ATTREADS l .

The number of fibers broken at least once is GF( t ).with the mean remaining and broken-off lengths being Ix E(~3) and x ( 1 - E( ~; ) ] = I - I ~ x E( ~, ),respectively, where I = E( ~, ) is the mean of the originallengths and E(43) is the mean of the remaining lengths asa fraction of E( ~, ). The fibers broken at least twice areamong the GF( t ) fibers. The mean remaining and bro-ken-off lengths of the G( F( t ) ] fibers are denoted by Ix E(~~) x E(f3) and I x E(f3) x (I - E(f3) ] = [I Ix E(~~)] - [I x E(43)] x E(f3). respectively.The loss of total fiber length is expected to be GF( t )I

x (I - E(f3)] at the first break. G [ F( t ) ] ~ ( x E( ~ ~ ) Ix [ I - E( ~ ~ ) J at the second break, and

GI[F(t)]~.[E(~3)]’~-~ X (1 - E(t,) at the kth break.

Taking away the broken-off segments from the originaltotal length GI, the remaining total fiber length L~. (km/m‘’) at treads t is approximately

Equation 4 meets the initial and final conditions (i.e.. L~= Gl when t = c at the beginning of walking and L,.= 0 when t = x).

at TEXAS SOUTHERN UNIVERSITY on November 11, 2014trj.sagepub.comDownloaded from

956

Similarly, the losses of fiber total length (Lei’ km/m2)in cut pile carpet can be denoted by

Particularly, we have E(~3) = 0.5 when fatigue sitesfollow a uniform distribution. Thus,

where L,. = remaining total fiber length in a cut pilewool carpet (km/m’‘), G = number of pile fibers per unitarea in a cut pile wool carpet (/M2),l = initial mean fiberlength (km), and L,, = initial total fiber length (km/m2).

PREDICTIONS OF: PILE MASS AND THICKNESS

Let texf = pile mass per unit length of pile fibers

(g/km), m,. = remaining pile mass per unit area in a cutpile wool carpet (g/M2), /n~ = loss of pile mass per unitarea in a cut pile wool carpet (g/m2), and m~, = initial

pile mass per unit area (g/nr). The remaining pile massand loss of pile mass of a cut pile carpet can then beestimated as

The pile thickness (hue, cm) is the pile height under’

compression and may be converted from pile mass bymultiplying it by a specific volume vp (CM3/g), namely h~.= m~. X vp [ 11 ]. The loss of pile thickness in the initial

. stage of wear is primarily due to flattening and compres-sion [ l, 2]. The subsequent loss of thickness in the wearstage is mainly attributed to the loss of broken-off seg-ments. Hence, the specific volume in the initial stage

(vpo, cm3/g, assume t < 1000) differs from that in thewear stage (vvnx, cm;/g). After some manipulation, thepile thickness he at t treads can be expressed as

The loss of pile thickness (h,,, cm) is given by

where h,. = remaining thickness of a cut pile wool carpet(cm), h&dquo; = m&dquo; X v~x X 10-’~ initial pile thickness(cm), vpx = limiting pile specific volume in the wearstage (cm- g), and vpo = pile specific volume in theinitial stage (CM3/g).

Validation and Simulation

It is important that a mathematical model is validatedto build confidence for further prediction. The validationprocess examines the consistency between the model andthe physical phenomenon and the accuracy of simulationoutcomes compared with experimental data.Camaby’s theory [1, 2] is an accurate mathematical

description of wool carpet wear on the basis of modemprobability fracture mechanics, as discussed in the intro-duction. Following his argument, our model is con-

structed on a solid theoretical and experimental basis.Hence, this validation focuses on the statistical accuracyof the predictions compared with our earlier observations(Tandon et al. [ 11 ] ).

In that work [I I], we tested six specimens of a 100%wool cut pile carpet with initial conditions of G = 7

X 107 (/M2), tex = 1.3 (g/km), vpx = 5.0 (cm~/g),initial pile thickness h&dquo; = 0.57 (cm) at t = 1000 treads,and h9000 = 0.1$ cm at 9000 treads. The average pilethickness was recorded every 200 treads from 1000 to

9200 treads. Using interpolation techniques, the vari-

ables of Weibull distribution are obtained as a = 1.329

X 10-3 , b = 0.797, and c = 1000 when a uniformfatigue site distribution is assumed. The simulation val-ues closely match the observations (Figure 1 ), and thetwo groups of data are strongly correlated, R‘’ = 0.946. This

FIGURE I. Comparison of simulation predictions and observations(data from Tandon et al. III)). >.

at TEXAS SOUTHERN UNIVERSITY on November 11, 2014trj.sagepub.comDownloaded from

957

result shows a closer fit than the model before modificationbased on the two-constant Weibull function (a, b) (R’‘ 2= 0.885, [ 11 ]), and similar to the previous model based onthe three-constant Weibull function (a, b, c) [ 11 ].An arbitrary fatigue site distribution is taken into ac-

count in this model. Unfortunately, the value of E( ~j ) isnot available, apart from the uniform fatigue site distri-bution. Hence, a scenario simulation is given to demon-strate the approach to constructing arbitrary fatigue sitedistributions from experimental data. We may assume afatigue site distribution x xj with bias to the top of thefibers. Construct a piecewise density function as

where .r = the ratio of the remaining length to the

original length of a pile fiber from the backing, g( x)means all criteria of probability distribution, and f ±x.rg(x)dx = 80 (i.e., 80% of mean original length).

Using the same carpet and wear data as given byTandon et al. ( 11 ], the parameters of the life distributionfor E( ~~ ) = 0.8, are estimated as a = 8.766 X t0’Bb = 0.629, and r - 1000. The simulated result is

essentially the same in terms of accuracy (R‘’ - 0.944for E(~~) = 0.8 compared with R2 = 0.946 for E(~~)- 0.5 ).When two wool carpets have the same life distribution

and different fatigue site distributions, the one with thehigher E( ~; ) value should be more durable. Assignmodel parameters as a = 1.329 X 10-;, b = 0.797,c = 1000, ho = 0.68 (cm) at t = 1000 treads, and

E(~~) = 0.5 for uniform distribution and 0.8 for biaseddistribution. The reduction of pile thickness is slower inthe carpet with a bias in fatigue sites toward the top of thepile fibers (Figure 2).

Having gained confidence in the validity of the model,scenario simulations of other parameters, such as the

remaining pile mass, loss of pile mass, remaining totalfiber length, and loss of fiber length, can be conducted byassuming a uniform fatigue site distribution. Given a= 1.329 x 10-3 , b = 0.797, c = 1000, mo = 808g/nr at t = 1000 treads and E(~~) = 0.5, the loss ofpile mass is shown in Figure 3.

Conclusions

Wear in wool carpet is one of the most complicatedstochastic processes, in that the state of each event de-

pends on the states of all previous events. The wear

FIGURE 2. Two sets of pile thicknesses obtained from the same lifedistribution and different fatigue site distributions.

FIGURE 3. Simulation of the loss of pik mass per unit area

process can be described by the mathematical theoryproposed by Carnaby in the 1980s. This study extendsCarnaby’s early models to a universal vision, allowingarbitrary initial conditions, life distribution of single fi-

bers, and fatigue site distribution along a fiber. Thevalidation shows that this model agrees with the exper-imental observations and is more accurate than the pre-vious ones in terms of prediction. Further experimentaland theoretical studies are needed to determine the model

parameters in terms of identifiable properties of woolfibers, yams. carpet construction parameters, and walk-

ing forces in order to identify areas for improving fiberproperties, carpet constructions, and test methods.

ACKNOWLEDGMENTS

We thank the New Zealand Foundation for Research.Science and Technology for funding this research, Dr.Garth Carnaby and Prof. John W. S. Hearle for their

contributions to the mathematics and mechanics.

Literature Cited

1. Carnaby, G. A.. Theoretical Prediction of the Durability ofWool Carpets Under Laboratory and Use Conditions,

at TEXAS SOUTHERN UNIVERSITY on November 11, 2014trj.sagepub.comDownloaded from

958

WRONZ Communication C91, Wool Research Organisa-tion of New Zealand, 1984.

2. Carnaby, G. A., The Mechanics of Carpet Processes, in"Proc. 7th International Wool Textile Research Confer-

ence, Tokyo," vol. 3, 1985, pp. 327-335.3. Cusick, G. E., and Dawber, S. R. K., Loss of Thickness of

Carpets in Floor Trials, J. Textile Inst. 55, T531-536(1964).

4. Harvey, A. E., Hearle, J. W. S., Jones, T. J., and Noone, P.,A Study of Fatigue Types and Distribution in a ControlledDamaged Wool Carpet, Part II: Turning-Walk Worn Car-pet, WRONZ Report R216, Wool Research Organisationof New Zealand, 1999.

5. Hearle, J. W. S., Lomas, B., and Cooke, W. D., "Atlas of

Fibre Fracture and Damage to Textiles", Woodhead Pub-lication Ltd., Cambridge, England, 1998.

6. Hearle, J. W. S.. Lomas, B., Goodman, P. J., and Camaby.G. A., Microscopic Examination of Worn Carpets, Part I:

Comparison of the Modes of Fibre Failure in Floor Trials.WRONZ Communication C79, Wool Research Organisa-tion of New Zealand, 1983.

7. Hearle, J. W. S.. Mukhopadyay, S. K., and Sengonul, A., AStudy of Fatigue Types and Distribution in a ControlledDamaged Wool Carpet, Part I: Straight-Walk Worn Carpet.WRONZ Report R212, Wool Research Organisation ofNew Zealand, 1997.

8. Li, H. S., and Zhou, C. F., "Engineering Fracture Mechan-ics," Dalian Institute of Science and Technology, China,1990.

9. McFarlane, I. D., and Ross, D. A., Fibre Damage andCarpet Performance, WRONZ Communication C43, WoolResearch Organisation of New Zealand, 1976.

10. Spiegel, M. R., "Theory and Problems of Statistics,"McGraw-Hill, NY, 1961.

11. Tandon, S. K., Camaby. G. A., and Wood, E. J., Evaluationof a Formula for Predicting Carpet Wear, in "Proc. 8thInternational Wool Textile Research Conference,

Christchurch, New Zealand," vol. V, 1990, pp. 429-438.

12. van Wyk, C. M., Note on the Compressibility of Wool, J.Textile Inst. 37, T285-292 (1946).

,Nemuo-riln r-rc-eire·cl Atigiist 17. 200J: accepted March h l3. ’(>I)’-

Part II: Wear Model of Loop Pile Carpet

H. LIU, S. K. TANDON, AND E. J. WOOD

Wool Research Organisation of New Zealand (Inc.), Christchurch, New Zealand

ABSTRACT

A two-stage wear mechanism in a loop pile wool carpet is confirmed by current experi-mental studies. A loop pile wear model is then developed on the basis of the two-stage wearmechanism. This model meets the initial and final conditions of carpet wear and shows that a

loop pile construction of similar pile density has a longer life than the cut pile construction, asproved by observations. The sensitivity tests show that the delay of second-stage wear isinsignificant to the wear life of a loop pile wool carpet, whereas the influence of the shapeparameters b and B of the life distributions is remarkable.

A loop pile carpet is more durable than a cut pilecarpet, since initiall~y there is a double connection of endsto the pile backing in a loop pile. It therefore requires twobreaks to completely break off a segment from a fiber inloop pile. The wear of loop pile fibers may be dividedinto two stages. At the first stage, a fiber breaks into two

parts near the middle of the loop. Each part has oneconnection to the pile backing, similar to the fibers in cutpile. Further breakage in either of the two parts leads tothe loss of fiber mass. We regard the two parts of a looppile fiber after the first breakage as two cut-pile fibers. A

model of loop pile wear can then be developed in a

similar way to the cut pile wear model [9].

Model DevelopmentThe number of first broken fibers over the initial total

number of loop pile fibers is defined as F, ( t ) - I- exp[-u(t - c)’’] (Weibull distribution, t = numberof treads, a > 0. b > 0. c ~ 0) to present the first-stagefatigue mechanism of loop pile fibers in wool carpet.After enough fibers break down from their initial states,

at TEXAS SOUTHERN UNIVERSITY on November 11, 2014trj.sagepub.comDownloaded from