N Theoretical Analysis on Tensile Properties of Textured Yarns By Sueo Kawabata* and Tetsuyuki Sasai* *, Members, TMSJ *Department of Polymer Chemistry, Kyoto University, Kyoto * *Faculty of Engineering, Gunma University, Kiryu, Gunma Prefecture Based on the Journal of the Textile Machinery Society of Japan, Transactions, Vol. 29, No. 10, T140-145 (1976) Abstract In this paper, tensile properties of textured yarns will be analysed. First, the property is calculated on the basis of a "independent fiber model" where each fiber contributes independently to the yarn tensile property, i.e., no mechanical interactions among fibers are considered. One of the purpose of this paper is to examine the accuracy of this no-interaction assumption for making distinct the mechanical interaction among fibers. Second, the tensile property of a crimped single fiber is calculated by applying the so-called "strain additive method"~l~ to a helical coil model of a fiber. Then the tensile property of a textured yarn is introduced by summing up the tensile property of each crimped fiber. A good agreement between the calculated and the observed values is obtained over a wide range of the tensile atrain of the yarn by assuming the independent model. No interaction is observed except the small deformation region. The hysteresis behavior observed at the repeated extension is also examined, and it is confirmed that the independent model is still suitable for this case. The experimental results agree well with the calculated results with good accuracy, too. 1. Introduction For analysing the tensile property of a crimped fiber, KondoL2~ considered two models for a crimped fiber : a plane zig-zag form and a three-dimensional helical spring form. He examined the conformity between the calculated and the experimental values of load-extension curves. Kawasaki et al.~37 and Holdaway~4~ also expressed the loadextension curve by applying a helical coil model. It is however difficult to apply their theory to our study just as it is, because they did not take into consideration the elongation of fibers themselves during the extension of the spring model. This neglect leads to considerable error in actual cases, and is not suitable for our precise calculation of the load-extension curve. In this paper, we first apply a large deformation equation equation of a helical coill3,6,7] to calculate the tensile property of a crimped fiber with no fiber elongation. Then we use the so-called "strain additive method"~l~ to introduce the fiber elongation to its load-extension property. The strain additive method is as follows: The tensile deformation of a helical coil is composed of two deformations, coil extension and fiber extension. Both deformations, of course, occur at the same time during the extension process of the helical coil. However, the analysis of it becomes much complicated. Therefore, we calculate first the load-extension curve with no fiber elongation using the equation of large deformation of a helical coil, and combine it with the tensile property of the fiber in a straight form by the following Vol. 24 No,1(1978) method such that both elongations corresponding to the same stretching force are summed up together. Then, we calculate the tensile property of the textured yarn using an "independent fiber model", and will examine whether the mechanical interaction between fibers will exist or not by comparing the calculated results with the experimental ones. 2. Theory 2.1 Yarn model We assume that the tensile property of each crimped fiber contributes independently to the tensile property of a yarn with no interaction among fibers. We call this the "independent fiber model". Following the model, the tensile force of a yarn can be obtained by summing up the forces of all fibers in the yarn as follows; F(A)=j f (2) i_1...(1) where t~11v1v /~] : extension ratio 11L~1V of Vl ~i a yarn,6l111 F(A) : tensile force of the yarn as a function of A f (A): tensile force of a crimped fiber in the yarn as a function of A N: number of fibers in the yarn cross-section. In the case when all fibers in a yarn have the same tensile property, Eq. (1) becomes F(A) =N f (A)...(2) In order to examine the conformity of this independent 13
fiber-model, it is necessary to know the precise tensile property of a crimped fiber. 2.2 Model of a crimped fiber We have assumed two models on the form of a crimped fiber, a helical coil model and a plane-arc-bar model.~6~ Comparing the values calculated on those two models with the experimental results, we found already that values from the helical coil model showed better agreement with the experiment than values from the plane-arc-bar model.~61 Thus we use the helical coil model for the crimped fiber. Fig. 1 shows the helical coil model. The details are given in the next section. 2.3 Tensile force of a crimped fiber We devide the extension deformation of a helical coil model into two deformations. The first is the extension deformation due to the extension of a helical coil while the fiber itself is assumed not to be elongated along the fiber axis. We call this the "coil effect region". Another is the extension deformation due to the elongation of the fiber straightened. We call this the "extension effect region". In our method, the latter region starts when the coil effect region ended just after the coil was straightened. The "critical extension state" is the state when the coil became straight at the end of the coil effect region. It should be noted that these deformations and states do not exist in actual cases, but are only considered as an ideal model of deformation. Then, the effects of these two regions are combined by strain additive method as follows; x(f)=xh(f)+xt(f)...(3) where f : tensile force along the coil-axis x(f) : elongation of the crimped fiber along the coilaxis xh(f) : elongation of the coil model as a function of the applied tensile force. The calculation is carried out under the assumption that the fiber is not elongated. XT(f) : elongation of the fiber as a function of the applied tensile force after the fiber is straightened. Fig. 1 Helical coil model. In the coil effect region, the tensile force, fh, can be calculated by following equations~5'6'7~ (see Appendix); f( n d4e ~ 2)= -a-- R~ sin a cost a (\/'+2m(1---)-1) -stn a 1- ~ sin a ~... (4) where m : =G/E G : shear modulus of the fiber E: tensile modulus of the fiber d: diameter of the fiber when the cross-section is assumed circular A : extension ratio of the helical coil. Here, a, R and n are reduced from next equations; sina = o/lc...(5) R=l~ cosa/2,rn...(6) n=lo/(2~rr~tana)...(7) A where to : length of the coil before extension l~ : length of the coil at the critical extension state. It is equal to the length of the fiber under no tension. a : helical angle n : number of turns of the coil R : radius of the coil. In the extension effect region, the tensile property of the straightened fiber is used for obtaining xt(f ). 2.4 Critical extension ratio l~ is defined as the length of the crimped fiber at the critical extension state, and the critical extension ratio A~ is defined as follows :...(8) For the discussion of the tensileproperty of a crimped fiber, it is difficult to define the initial length of the crimped shape because of its uncertainty. However, the elongated state of a crimped fiber can be determined clearly by considering the critical extension state as the base length. Now, we can define A which is the stretch ratio based on the length at the critical extension state such that, A=l/l~...(9) =(lha)/l~/lo =A/A~ 3. Experiment (10) 3.1 Preparation of specimens and experimental apparatus Specimens used in this expriement are polyester falsetwisted textured yarns of 150 d, 48 fil. An apparatus to bring about and stabilize the crimps of yarns is shown schematically in Fig. 2. The specimen is passed for 15 sec with no tension through a pipe to which saturated water vapor at 100 C is supplied from the boiling vessel. After this process, the yarn is dried up for 24 hrs in room condi- 14 Journal of The Textile Machinery Society of Japan
tion. For measuring the tensile properties of yarns and fibers, the precise tensile tester KES-G1 is used. For the torsional experiment of a single fiber to obtain shear modulus G, KES-Y1 is used. Both testers were developed by Kawabata. 3.2 Measurement of tensile properties of fibers Fig. 3 shows the stretch and recovery behavior of a single fiber observed by repeated tensile test. The numbers put near the curves show the order of repeated stretch: 1 indicates the first cycle of the vergin specimen after the vapor treatment, 2 the second cycle and so forth. The fourth stretch and thereafter give almost the same cruves as the third. Thus, the specimen after the third stretch is assumed to be in the stable state. Young's modulus E of the fiber is estimated from the slope of the straight part of the curve for each cycle. G is measured directly by the torsional testing machine under constant tension. The fiber length l~ can be measured by the photographic observation of the crimped fiber. The number of crimps per unit length of the fiber is determined by calculating the auto-correlation function of the projected curve of the crimped fiber as shown later, and is used as the value of n of the helical coil model. The fiber length at the critical state, l~, is presumed, as is illustrated in Fig. 4, from the intetsecting point of the horizontal axis with the extrapolated line of the linear part of the forceextension curve of the crimped fiber. The value of 1, is again confirmed by observing the fiber length on its photographic picture. Those values obtained above are then substituted into Eq. (4) to obtain a force-extension curve of a crimped fiber. Here, we have to call into a question about the fiber to fiber variation of these values, as there is considerably large variation between fibers. One reason for this is probably due to the variation of drawing conditions during spinning; the correlation between E and d is clearly observed as shown in Fig. 5. To make this variation effect on calculated results small, averages of about thirty fibers are used here as the representative values. Especially, we use the average d 4E instead of d ~E in Eq. (4) because of the correlation be- Fig. 4 Estimation of l~ of a crimped fibr. Fig. 2 Apparatus for bringing about and stabilizing crimps. Fig. 3 Repeated extension of a crimped fiber (experimental). Vol. 24 No, 1(1978) Fig. 5 Relation between E and d. Mark x points out the average. 15
Table 1 Characteristic values of crimped fibers. tween E and d. These characteristic values are shown in Table 1 for the first and the third cycles of repeated stretch. 3.3 Comparison between calculated and experimental results The tensile property of a fiber in the coil effect region is calculated by substituting the values shown in Table 1 into Eq. (4). On the other hand, the tensile property of the fiber is estimated from the force-extension curve of the crimped fiber by extrapolating the linear part of the curve, and we obtain the curve of the extension effect region as a fiber property. The complete curve in the tensile prosess is then obtained by combining the calculated values in the coil effect region with the observed values in the extension effect region using the strain additive method shown by Eq. (3). The tensile property of a fiber calculated from Eq. (3) is shown by dotted lines in Fig. 6. Both the calculated property in the coil effect region and that in the extension effect region are also shown seperately in this figure by solid lines. The curve of the third cycle is calculated by using the observed values of n and h from the crimped fiber after the second stretch, and the steretch ratio is defined by l~ of the first cycle. In Fig. 7, the calculated force-extension curve of a crimped fiber is compared with the experimental one. It is found that a good agreement is obtained between them. Because characteristic values vary considerably among fibers as mentioned before, a good agreement can only be obtained by using their averages. The curve of the third cycle slips aside from that of the first cycle by considerable amount as shown in Figs. 6 and 7. This shift is due to the change of the crimp shape by repeated extension, resulting in the change of n and R. Fig. 8 is the trace of crimped fibers projected on a plane after each repeated stretch. The change of shapes can be observed clearly. 3.4 Comparison between calculated and experimental values Using the independent fiber bundle model for textured yarns, we can introduce the force-extension property of a yarn by summing up the calculated force of each crimped fiber under given stretch ratio. This idea is proved sound by Fig. 7 Calculated and experimental curves of the tensile property of a crimped fiber. Fig. 6 Calculated tensile property of a crimped fiber. (Thick solid lines give the calculated properties of the coil effect region. Fine solid lines are of the extension effect region. Dotted lines are obtained from Eq. (3) using those two curves.) 16 Fig. 8 Plane projection of fibers after repeated stretch. Journal of The Textile Machinery Society of Japan
pictures of a yarn stretched as shown in Fig. 10(b). Fig. 10 (a) shows the yarn appearance at A=0.473 of specimen 1 in Table 1, and its tension is extremely low. However, (b) is the appearance in the critical extension state having ~=1.000. In both states, fibers look like independent each other. (a) and (b) in Fig. 9 show these two states respectively. The calculated ourve is shown in Fig. 11 together with the experimental curve. The figure shows that the observed forces at given stretch ratios are a little higher than the calculated in the low extension region. This is probably due to the error caused by the independent fiber bundle model; the crimped fibers contact each other complicatedly and entangle in the low extension state. However once the entwining is released by the first stretch, a good agreement between the experimental and the calculated values can be observed. 3.5 Hysteresis behavior of yarns also apply the fiber. the properties and the structural parameters of The good agreement between calculated and experimental Fig. 11 Calculated and experimental curves of the tensile properties results of force-extension behavior of textured yarns is observed in the stretch process. The similar agreement is observed in the recovering process, and to the analysis of of a textured yarn. which we again use the independent fiber bundle model and 4. Conclusions Fig. 9 Repeated extension of a textured yarn (experimental). We have found first that the tensile property of a crimped fiber can be calculated precisely by using a helical coil model and the strain additive method. Secondly the tensile property of a textured yarn can be calculated by using an independent fiber bundle model. However it is very difficult to obtain the values most adequate to the mechanical properties and to the structural parameters of component fibers because of their large variation. It necessitates us o use average values of those fibers. We also have to use d 4E instead of d4e in Eq. (4), as there is a correlation bweteen d and E. These two cases will result in the error about 5 %. The tensile property of a yarn is then obtained from Eq. (2) based on the assumption of the independent fiber bundle model. The calculated result agrees well with the experimental one. We have also discussed the repeated stretch, which is the origin of both the configuration change of the fiber and the number of turns of helical coil per unit length. If we introduce those changes into calculation, the value obtained agrees well with the experimental value. Fig. 10 Appearance of yarns when stretched. =1.000. (a) A =0.473, (b) Acknowledgement The authors feel grateful to Professor H. Kawai, Kyoto University, for his useful suggestions. They also thank to Associate Professor M. Niwa, Nara Women's University, for her valuable discussion and helpful support. (Most of the experiments in this paper was done in Kyoto University from May, 1970 to February, 1971, by one of the authors, T. Sasai.) Vol. 24 No.1(1978) 17
References [1] S. Kawabata; J. Text. Mac., Soc. Japan, 23, T30 (1970) [2] T. Kondo; Memoirs of Training Institute for Engineering Teachers, Kyoto University, 2, 16 (1966) [3] K. Kawasaki and S. Murata; J. Soc. Fiber Sci. Tech., Japan, 15, 814 (1959) K. Kawasaki; "General Discussions on Textured Yarns", Text. Mac. Soc. Japan (1966) [4] H. W. Holdaway; J. Text. Inst., 47, T586 (1956) [5] S. Kawabata; Proceeding of the 5th Summer Seminar, Soc. Fiber Sci. Tech., Japan (1972) [6] S. Kawabata, M. Niwa and S. Nakabayashi; J. Text. Mac. Soc. Japan, 28, T153 (1975) [7] S. Kawabata and T. Sasai; J. Text. Mac. Soc. Japan, 28, T157 (1975) [8] S. Kawabata, M. Niwa and T. Mamiya; J. Text. Mac. Soc. Japan, 29, T119 (1976) [9] A. E. H. Love; "A Treatise on the Mathematical Theory of Elasticity", First ed., Cambridge University Press, and The Macmillan Co. Dover Publ., New York (1944). Appendix (1892), Fourth ed., The basis of the large deformation theory of a thin lod is expressed in detail in the classical work by A. E. H. Love.E9] The tensile force F of a helical coil, which is obtained by applying his basic equations written on page 397 to a helical coil is as follows; cosai sinai cosai - sina cosa ri ri r _B sinai cosai (cos2ai - cos2a ri ri r where C and B are the torsional and the bending rigidities, a and r the helical angle and the radius of the helical coil before deformation, ai and ri those values after deformation, respectively. By modifying this equation to use more easily, we can get Eq. (6). 18 Journal of The Textile Machinery Society of Japan