Spun Yarn Strength as a Function of Gauge Length and Extension rate: A Critical Review

Transcription

1 Voume 4, Issue 2, Fa24 pun Yarn trength as a Function of Gauge Length and Extension rate: A Critica Review Anindya Ghosh,. M. Ishtiaque, R.. Rengasamy, P. Ma, and A. Patnaik Department of Textie Technoogy Indian Institute of Technoogy, Hauz Khas, New Dehi-6, India ABTRACT It has been known for many years that the strength of a spun yarn depends on the two important testing parameters, namey, the gauge ength and rate of extension. There is no doubt that a studies reating to the infuence of gauge ength and extension rate on yarn strength are invauabe both in theory and practice. In this artice, a critica review of various theoretica and practica aspects of the infuence of gauge ength and extension rate on yarn strength has been discussed. Keywords: Yarn trength, Gauge Length, Extension rate, Weak-Link, trength distribution. Introduction The standard measurement of yarn strength is executed at mm gauge ength and 2 ± 3 sec. However, during the post spinning operations, namey warping and beaming, a onger than mm ength of yarn experience stresses. In addition, most recenty, researchers [] have shown that the experimentay determined strength behavior of yarn at short gauge ength is more appropriate to simuate the mechanica behavior of fabric than those measured at ong gauge ength. Therefore the resuts obtained at standard test methods may not correcty refect the tensie performance of the yarns. Thus, measurement of yarn strength ony at mm gauge ength is not sufficient. In fact, it is not aso reaistic to measure yarn strength at a possibe engths. This can be overcome by using the theoretica reationships of strength and strength variabiity between ong and short specimens. Yarns undergo stress and strains during weaving and other operations that are significanty different from those appied in the standard tensie tests. Therefore the studies of tensie testing of spun yarns at various eve of gauge engths and extension rates on yarn strength are very significant and in this paper a critica review of these aspects has been made. 2. pun Yarn trength as a Function of Gauge Length 2. Theoretica Consideration Many researchers attributed that the presence of faw in the yarn eads to ocaization of stress in excess of theoretica strength, whereby the rupture process is initiated. It thus foows that the fa in strength of a materia with increasing test ength is due to the presence of a distribution of faw of wide ranging magnitude, since the probabiity of encountering a arge fata faw increases with test ength. Emai: anindya_textie@yahoo.com Artice Designation: choary Voume 4, Issue 2,Fa 24

2 Peirce [2], after studying strength variabiity of yarns, proposed the chain weak ink theory. His theorem has been based on the foowing assumptions: A Yarn of ength may be considered as a chain of m inks having the same ength, but various resistance of stretch (Figure ). Breaking oads of adjacent inks are independent variabes, i.e., the ink of maximum strength may foow immediatey one of minimum strength. Figure. Yarn as a inks of twisted fiber bunde. A simpe derivation of Peirce s theorem of yarn strength is given as foows. Let be the yarn strength at a gauge ength. is a random variabe. Aso, suppose that, f( ) and f( ) are the probabiity density functions of yarn faiure at a gauge ength and respectivey. F( ) and F( ) are the probabiity distribution function of yarn faiure at a gauge ength and respectivey. Thus, from the theory of probabiity, we have: df( d ) = f( ), () df( d Now, to find out the probabiity of faiure of a chain of m inks from the probabiity of faiure at any oad X appied to a singe ink ) = f( ), (2) of ength, it is obvious from the weakest ink theory that the chain as a whoe fais, if any one of its inks fais. If it is assumed that P ( X) is the probabiity of faiure of a specimen of ength at a oad X, then P ( X) = ( ) F, (3) Thus, the probabiity of non-faiure of a specimen of ength at a oad X = - F( ). Therefore, the probabiity of simutaneous non-faiure of a m inks at a oad X =[ F( ] m ). o, the probabiity of faiure at a gauge ength is expressed as [ F( ] m F( ) ) =, (4) Artice Designation: choary 2 Voume 4, Issue 2,Fa 24

3 By using this reation, frequency distribution of faiure at any gauge ength can be worked out. If the form of the function f( ) foows a norma distribution, it gives: where and are the mean and standard deviation of., () Figure 2. Appication of Peirce s theory to a norma distribution. Curves for various test engths, cacuated from the norma distribution at m =. The reations of Equations and can be substituted in Equation 4, and new distribution is then defined. Figure 2 shows an exampe of this. It wi be noticed that even though we start with a symmetrica norma distributions, the derived distribution at other engths are skew. The distribution of F( ) is thus known in terms of,, and m. Anayzing this expression, and making some mathematica approximation, Peirce obtained Equations giving the mean strength, and standard deviation, for specimen ength. The reations are: = 4. 2 m, (6) = m, (7) The Equations 6 and 7 are known as the Peirce s Equations of yarn strength. The resut of the mathematica approximation is given in the Tabe. Artice Designation: choary 3 Voume 4, Issue 2,Fa 24

4 Tabe I: Experimenta and theoretica vaues obtained from approximate Peirce s Formua Vaue of m Cacuated vaue of From Peirce s formua 4. 2( m ) Cacuate d vaue of From Peirce s formua m Zurek et a [3, 4] modified the Equations 6 and 7 after making some empirica correction, which are given by = m, (8) = m Neckar and Das [] considered yarn strength as a stationary, ergodic, Marcovian, stochastic process, and on the basis of these strength vaues, the response of yarn strength at any gauge ength was simuated. They measured the yarn strength that foows the criteria of a stochastic process i.e. the strength of adjacent yarn sections of a gauge ength was measured in a consecutive manner. They took a ong ength of yarn, which is divided into m equa sections of ength. The strength of the yarn 7, (9) was stochasticay measured at a gauge ength of mm in the Instron tensie tester attached with a specia device for semiautomatic feeding of the yarn in between the jaws of the tensie tester. The autocorreation function was cacuated to anayze the strength vaues of adjacent yarn sections as a function of yarn ength. The correation coefficients of experimentay measured strengths as a function of the distance aong the yarn are shown in Figure 3. Artice Designation: choary 4 Voume 4, Issue 2,Fa 24

5 Figure 3. Autocorreation function of strength of cotton carded rotor-spun yarn (2 tex). Neckar and Das [] aso obtained the empirica reationships concerning the strength and the strength variabiity between ong and short specimens on the basis of simuated strength data for different gauge engths by the statistica regression technique. The approximated reationships for 2 tex count cotton carded rotor-spun yarn were found to be as foows: = m, () = m. 89, () If the probabiity distribution function of faiure at a gauge ength foows a two-parameter Weibu distribution [,6,7], we have: F( ) = exp x r, (2) Then the strength distribution F( ) at any gauge ength is given by F( ) = exp x where x = scae parameter, r = shape r parameters, and x = x m. The vaues of x and r are positive. For data foowing a r, (3) Weibu distribution with scae parameter x and shape parameter r, the mean, and variance are given by = x Γ( + ) r µ, (4), () Artice Designation: choary Voume 4, Issue 2,Fa 24

6 Therefore, the coefficient of variation is CV = µ Γ( + 2 / r ) = { Γ( + / r} 2 / 2, (6) where µ, s 2, and CV are the mean, variance, and coefficient of variation (fraction), respectivey, and Γ (.) is the cassica gamma function. The expression of Γ (n) is given by n Γ( n) = exp( t ) t dt, (7) The scae parameter x is reated to the strength of the faw in a yarn. Typicay the scae parameter is numericay cose to the mean yarn tenacity. It is noticed from the Equation 6 that the coefficient of variation of yarn tenacity depends ony on shape parameter r. Rosen [8] has shown that for.92. CV., CV r or CV / r. In other words, r is an inverse measure of the coefficient of variation. The shape parameter r represents the dispersion of yarn strength per unit ength of yarn. More precisey, shape parameter reated to the Poisson distribution of faws per unit ength of yarn. A greater vaue of r indicates a sma yarn strength variation. When r?, there wi be no variation of yarn strength and the mean yarn strength wi then become independent of gauge ength. In genera, vaues of r between 2 and 4 correspond to britte fiber, whereas a vaue of 2 is appropriate for a ductie materia [9]. If it is assumed with some approximation that the yarn breaking oad is proportiona to thickness, one arrives to the concusion that with sections of shorter or equa ength to that of drafting wave, Peirce s formua does not hod good. Furthermore, Peirce negected the fact that the strength of adjacent fracture engths is reated. The reationship between the strength of adjacent fracture zones arises party because the same ong fiber wi occur in a number of fracture zones, and party because of the non-random irreguarities introduced into the thickness of the yarn by the spinning machines. An improvement on Peirce s theory has been worked out and appied to yarns by pencer-mith []. He suggested quite a different approach to the probem of effects of sampe ength on the vaue of breaking oad. He aso pointed out that the strengths of neighboring fracture zones in yarns are reated to one another. According to him, the average yarn strength at any specimen ength can be cacuated from the average strength, variabiity, and seria correogram of the strength of the fracture zones. pencer-mith has worked out the theory in detai and obtained the reation: = W( m ).F( m)., (8) where is the fracture-zone ength, and respectivey. Artice Designation: choary are the mean vaues for engths and 6 Voume 4, Issue 2,Fa 24

7 W (m) is a statistica function, tabuated by Tippett [] for vaues of m, and F (m) is the seria correation function, which is expressed as m F(m) = [ m( m ) 2( m ) r 2( m 2) r... 2( m n) r... ] n 2r m where r n = correation coefficient for the strengths of fracture zones., (9) In this expression, W (m) is a numerica factor, F (m) is a factor taking account of the correation of strengths of neighboring zones, and brings in the variabiity. The product W (m). F (m) repaces 4. 2( m ) in Peirce s expression. An adequate theory of weak ink effect has not yet been worked out, though Peirce s theory is a usefu approximation, and pencer-mith s reation is open to criticism on the grounds that it must be based on experimenta resuts for the fracture-zone ength. Apart from the fact that this ength is not known and may be very i defined, it is very ikey that, when jaws are camped on the specimens at a distance apart equa to the estimated fracture-zone ength, the nature of the break wi be different from that at much shorter or much onger engths. When the jaws are cosed together, they wi restrain deformation of the fiber, and the distribution of the strain, giving rise eventuay to rupture, wi be different. The effect of changes in the mechanism of breakage cannot be incuded in any statistica theory, and it seems ikey that different reations woud appy for engths much greater than, and much ess than, the fracture-zone ength. The variations for engths near the fracturezone woud depend on the particuar properties of the fiber. If m =, then from Peirce s Equation 6 Artice Designation: choary = 4., (2) 2 where breaking oad of infinity ong sampe. Therefore, the breaking oad of yarn at any gauge ength 7 = m, (2) If is the mean breaking oad and standard deviation of breaking oads of sampes of mm ength, then And after some transformations, above Equation becomes = , (22) = = b, (23) 4.2s where b is a factor which depends on the characteristics of the materia. A simiar to that of Equation 23, Weibu [2] introduced the reationship of the mean breaking oad of yarn at gauge ength as foows Voume 4, Issue 2,Fa 24

8 = b, (24) For = the Equations 23 and 24, of course, ose sense. Consequenty, the Equation 24 was modified by introduction of an additiona component [2]. Thus the Equation 24 becomes where a is a constant. = a + b, (2) The Equation 2 was worked out as an effect of theoretica considerations. The Equation 2 is aso identica with ippe s [3] expression, which is given as = + b where = breaking oad of a yarn at a gauge ength of =., (26) ome workers [4-6] have attempted empirica approaches to understand the nature of strength variation with test ength. They found the foowing ogarithmic, exponentia, and power aw reationships reating yarn tenacity ( ) and gauge ength () as foows = c + c 2 og, (27) = where c, c 2,.., and c 6 are constants. c 4 c e 3, (28) c 6 = c, (29) A power aw Equation gives rise to singuarities at extreme vaues of, i.e.; when =, tenacity becomes infinity, and when =, tenacity becomes zero, neither of which is actuay true. Hussain et a [7] proposed that yarn tenacity was a modified power aw function of gauge ength to avoid these singuarities: = C + a ( + x d ), (3) where C is the imiting minimum vaue of tenacity, a is the difference between the tenacity at gauge ength (-d) cm and the imiting tenacities, x is a constant. 2.2 Experimenta tudies of the reationship between the Yarn trength and Gauge Length Hussain et a [7] found a significant difference in the gauge ength effect on the Artice Designation: choary 8 strength of ring and rotor spun yarns. The ength effect, which they expressed as a ratio between the tenacity of a given gauge and that of a cm ength, showed no significant difference between ring versus rotor spun yarns at reativey short engths. But the differences were statisticay significant at Voume 4, Issue 2,Fa 24

9 ong (7 cms) engths. The extent of decrease is greater for ring spun yarns than for OE yarns, indicating that the rotor yarns are more uniform with respect to the ring yarns. Tabe II: Range of faiure zone size for different gauge engths. Yarn system Ring spun Ring spun Ring spun Air jet spun Air jet spun Air jet spun Gauge ength, mm < <2 Faiure zone size*, mm < *The ength of the faiure zone to be the ength of the region of reduced cross section of one of the faied ends Reaff et a [] proposed that the mechanism of faiure might aso change due to a decrease in the test ength. They observed the different range of faiure zone size for ring-spun and air-jet-spun yarns for different gauge engths (Tabe II). According to their observation, as compared to the air-jet spun yarns, ring spun yarns yied higher strength, many broken fibers and a sma faiure zone size at onger gauge engths. But, at gauge engths we beow the fiber stape ength, air-jet spun yarn shows more strength than ring spun yarn because the difference in surface heix ange (θ), since θ > for ring spun yarn and θ for the core fibers of air-jet yarn. Whie comparing the infuence of gauge ength on yarn faiure for ring spun and open-end spun yarn, they found that the ring spun yarns fais by fiber breakage at both ong and short gauge engths. But the open-end yarns show a change in breakage mechanism from a fiber sippage dominant faiure at ong gauge ength (27 mm) to a fiber breakage dominant faiure at short gauge engths (2.7 mm and < 2 mm). Oxenham et.a [8] compared the effect of gauge ength on the strength of ring spun and open end friction spun yarns and found that the strength of the ring spun yarns shows a sharp drop as the gauge ength Artice Designation: choary 9 increases from mm to 4 mm (which is approximatey the fiber ength). The strength of the friction spun yarns aso drops sharpy as gauge ength increases from mm to 2 mm (which is amost equa to the fiber extent in this yarn). For gauge ength greater than 4 mm, the strength of ring spun yarns appears to be fairy constant whereas the strength of the friction spun yarn continues to reduce as gauge ength increases, refecting the discontinuities in the yarn formation zone in friction spinning. Using a Komogorov-mirnov goodness-offit at a significance eve a., Reaf et a [] obtained that the tenacity data for ring, air-jet, and open-end spun yarns were fitted to a two-parameters Weibu distribution. From the Weibu distribution parameters of yarn tenacity at a particuar gauge ength and using the cassica weakest-ink scaing theorem of Peirce [2], they tried to predict the strength response at other gauge engths. They found that none of the spun yarns considered in their studies stricty foow the weakest ink theory when a Weibu strength distribution is assumed. For a the yarns, they observed that there was a noteworthy change in the Weibu shape parameter as a function of gauge ength, indicating a greater variabiity in strength with decreasing gauge ength. Moreover, the Voume 4, Issue 2,Fa 24

10 Weibu scae parameter, mean tenacity, and standard deviation were found to be increased with decreasing gauge ength in a manner not coincident with the weakest-ink scaing theories. From these deviations they concuded that the mechanism of yarn faiure changes in going from ong to short gauge engths. 3. pun Yarn trength as a Function of Rate of Extension Time to break a yarn specimen decreases with the increase of extension rate. Between the time to break and extension rate there is foowing reationship Artice Designation: choary t = E.. V. 6, (3) where E = breaking eongation of yarn (%), = test ength in mm, t = time to break the specimen in seconds, V = extension rate in mm/min. The rate of strain during tensie testing infuences yarn tenacity. Rapid straining of yarn resuts in a higher breaking oad. Midgey and Peirce [9] were the first to study the effect of strain rate on yarn tenacity and showed that the breaking oad of a 36 s sake cotton ring spun yarn was inversey proportiona to the ogarithm of the time to break the yarn. This reationship t t was approximatey vaid over a range of times from / second to a month. Meridith [2] tested yarns over a miionfod range of rates of extension and found that the reation between yarn breaking oad and rate of extension was approximatey inear (actuay sighty concave to the breaking oad-axis) for most fibers. He estabished the foowing empirica Equation for breaking times ranging between a second and an hour 2 F - F 2 = k F og, (32) where F is breaking oad at time t, F 2 is breaking oad in a time t 2, and k is the strength-time coefficient. thus concuded that maximum tenacity The strength-time co-efficient is the gradient occurs not at the maximum rate of extension of the average sope of the ines obtained as observed by Midgey and Pierce [9] but when the breaking oads are potted against at the optimum extension rate. They the ogarithm of the time to break. He attributed these resuts to the foowing observed that the strength of cotton yarn facts. As the rate of extension increases, the decreases by approximatey 9% for a -fod percentage of rupture fibers increases, increase in time to break and the vaue of k resuting in a higher breaking strength i.e., a is cose to.9. He aso stated that the same greater number of fibers are contributing to formua appies to constant-rate-of-oading the breaking oad. At sti arger extension and constant-rate-of extension tests. rates (when yarn tenacity decreased), they Baasubramanian and ahotra [2] faied to proposed that the short time avaiabe may observe a steady increase in tenacity with not be sufficient for the reaignment of increasing rate of extension. They found that fibers, this factor coud therefore cause a tenacity reaches a peak vaue around a strain drop in tenacity of individua fibers which is rate of 2 cm/min and thereafter decines more than what coud be offset by the graduay (Figure 4). This behavior was increase in tenacity due to a higher found to be true for both ring and rotor percentage of fiber rupture. yarns, spun from three different cotton varieties, at three twist eves. The authors Voume 4, Issue 2,Fa 24

11 Figure 4. Effect of strain rate on ring yarn tenacity. Kaushik et a [22] found that as the rate of extension increased, yarn tenacity increased reached a maximum, and then decreased or remained constant for both ring and rotor spun yarn. Deuca and Thibodeaux [23] were the pioneers to show anayticay how ow or high speed testing affects yarn tenacity. They studied yarn spun from UDA Acaa cotton at testing speeds ranging from. to m/min. They found that as the rate of extension increased, the yarn tenacity increased ineary with the ogarithm of the rate of extension from. to m/min. At 2 m/min, yarn tenacity increased sighty, reached a maximum, and then at m/min, it decreased. Chattopadhyay [24] showed that with an increase in the strain rate the tenacity initiay increased up to mm/s for both ring-spun and air-jet-spun yarns and then foowed by a sharp reduction. Oxenham et a [2] compared the tenacity and eongation of different bended yarns tested at Tensojet (4 m/min) and Tensorapid ( m/min). They found that the tenacity vaues of ring, rotor, and air-jet spun yarns tested at Tensojet are higher than those from the Tensorapid. However, in case of air-jet yarns the tenacity vaues measured in Tensojet and Tensorapid showed the east difference than those in ring and rotor spun yarns. Aso, the difference between the Tensojet and the Tensorapid is not significant for / poy/cot bend. They aso found that the yarn tenacity for % cotton and / poy/cot bend the yarn tenacity shows a continuous increase with the ogarithm of the testing speed in both Tensorapid and Tensojet. Guati and Turner [26] found a cose reationship between percent fiber rupture and yarn strength. According to their study, the correation coefficient between the percent fiber rupture and yarn strength were.94,.97, and.99 for 2 s, 3 s, and 4 s count ring yarns. ingh and engupta [27] have shown that the increase in tenacity with the increase in strain rate is directy attributabe to the increasing incidence of fiber rupture (Tabe III). Artice Designation: choary Voume 4, Issue 2,Fa 24

12 Tabe III: Effect of strain rate on percentage broken and sipped fibers. Yarn Percentage of trength strength fiber breaking contribution due (gm/tex) (%) to fiber breaking trainrates (cm/min) (gm/tex) trength contribution due to friction of sipping fibers (gm/tex) Broken fibre (%) Log (extension rate) in mm/min Ring yarn Rotor yarn Air-jet yarn OE-friction yarn Figure. Effect of extension rate on percentage of broken fibers. Ghosh et a [28] found that yarn tenacity increases continuousy with the extension rates for a spinning systems. The increase in tenacity with the increase in extension rate is due to the consequent increase in the proportion of fiber breakage, as depicted in Figure. The effect of impact oading at high strain rate is responsibe for more fiber breakage. On the contrary, at sow strain rate the mechanism of yarn faiure is sippage dominated, as more time is avaiabe for yarn to cause rupture. 4. Concusions The foregoing discussion gives an overview of the various theoretica and experimenta aspects of the infuence of gauge ength and extension rate that Artice Designation: choary 2 have been reported so far in the iterature since the interest of this topic made a beginning. The yarns representing different spinning technoogies have aso been concerned in this artice. Finay, an inference may be drawn that the discussions made in this artice is usefu for the textie researchers as a too for further research in the area of yarn strength. Literature cited. Reaff, M. L., eo, M., Boyce, M. C., chwartz, P., and Backer,., Text. Res. J. 6, 7-3 (99). 2. Peirce, F. T., J. Text. Inst. 7, T3- T368 (926). 3. Zurek, W., Mainowski, L., and Potka, E., Proceedings of The Voume 4, Issue 2,Fa 24

13 Technica University Lodz, Vo. 33, pp , Zurek, W., Fredrych, I., and Zakrzewski,., Text. Res. J. 4, (987).. Neckar, B., and Das, D., Paper Presented to the eventh Asian Textie Conference, New Dehi, Weibu, W., J. App. Mech. 8, 293 (9). 7. Thoman, D. R., and Bain, L. J., Technometrics,, 8-8 (969). 8. Rosen, B. W., In Mechanics of Composite Materias: Recent Advances, Pergamon Press, Oxford, Pan, N., Hua, T., and Qui, Y., Textie Res. J. 7, (2).. pencer-mith, J. L., J. Text. Inst. 38, P27-P272 (947).. Tippett, L. H. C., Biometrika, 7, 364 (92). 2. Weibu, W., Ing. Vedensk. Akad. Handingar, No.. 3. ippe, A., Faserf. U. Textit, 9, 63 (98). 4. Grant, J. N., and Morier, O. E., Text. Res. J. 8, (948).. Piay, K. P. R., Proceeding of eventh Tech. Conf. ATIRA, BTRA, and ITRA, PP. -3, Kapadia, D. F., J. Text. Inst. 26, T42-T266 (93). 7. Hussain, G. F. G., Nachane, R. P., Krishna Iyer, K. R., and rinathan, B., Text. Res. J. 6, (99). 8. Oxenham, W., Zhu, R. Y., and Leaf, G. A. V., J. Text. Inst., 83, (992). 9. Midgey, E., and Peirce, F. T., J. Text. Inst., 7, T33-T34 (926). 2. Meridith, R., J. Text. Inst. 4, T (9) 2. Baasubramanian, P., and ahotra, K. R., Text. Res. J., 74-7 (98). 22. Kausik, R. C. D., ahotra, K. R., and Tyagi, G.K., Text. Res. J. 9, 97- (989). 23. Deuca, L. B., and Thibodeaux, D. P., Text. Res. J. 62, 7-84 (992). 24. Chattopadhyay, R., J. Text. Inst, 9, (999). 2. Oxenham, W., and Lee, E. K., Proceedings of Word Congress on Textie in the Mienium, pp , Huddersfied, Guati, A. N., and Turner, A. J., J. Textie Inst. 2, T6-T82 (93). 27. ingh, V. P., and engupta, Textie Res. J. 47, (977). 28. Ghosh, A., Ishtiaque,., M., and Rengasamy, R.., Proceeding of st Internationa Conference Batic Textie & Leather, pp. -6, Kaunas, Lithuania, 23. Artice Designation: choary 3 Voume 4, Issue 2,Fa 24