CHAPTER 9 THE EFFECTS OF GAUGE LENGTH AND STRAIN RATE ON THE TENSILE PROPERTIES OF REGULAR AND AIR JET ROTOR SPUN COTTON YARNS

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1 170 CHAPTER 9 THE EFFECTS OF GAUGE LENGTH AND STRAIN RATE ON THE TENSILE PROPERTIES OF REGULAR AND AIR JET ROTOR SPUN COTTON YARNS 9.1 INTRODUCTION It is the usual practise to test the yarn at a gauge length of 50 cm using Instron Tensile Tester, Uster Dynamometer, Tensorapid or Tensojet for having an idea about the quality of yarns. Since the yarns are subjected to tensile stress at various gauge lengths during spinning and weaving processes, it will be of interest to study the tensile properties at these various lengths. A considerable amount of work has been carried out on the effect of gauge length on ring rotor and air jet rotor yarns by Balasubramanian et al (1985). Pan et al (1997) have investigated the size effects on the mechanical behavior of fibres namely polypropylene, polyester, polyamide and carbon fibres. It was Meredith who did pioneering work on the effect of strain rate on the mechanical properties of fibres. Hussain et al (1990) have examined the effects of gauge length on the mechanical properties of ring and rotor yarns. Realff et al (1991) have carried out studies on the effect of gauge length on the tensile properties of ring and friction spun yarns. Despite numerous studies, very few have examined breaking strain and no one has checked both properties at the same time, let alone together with the initial modulus. Furthermore, although gauge length effects have been examined by many research workers, they have focused mainly on strength.

2 171 Moreover with the increasing awareness of the considerable impact that variations in yarn properties have in determining their mechanical behaviour and structures made from them, theoretical approaches to specify the statistical distribution of the fibre properties have become highly desirable. In addition, as pointed out by some research workers, the effect of gauge length has practical significance. It is a well-known fact that in a yarn structure under extension, fragmentation occurs prior to the failure of the structure. As a result, the yarn will eventually break into much shorter length. Since the properties of the overall structure are derived from the fibers, this much shorter effective fibre length leads to a completely different system behaviour than that predicted, from the starting fibre length. The discrepancy between the two is caused by largely by the gauge length or size effects of the yarns. It may be noted that hardly any work has been carried out on the effects of gauge length on the initial modulus of spun yarns. Also many theoretical models have been developed to predict the initial modulus of twisted continuous filament yarns and for spun yarns made out of different fibres, this type of work is necessary. Prediction of initial modulus of spun yarns is indeed complex, and the findings of this Chapter have shed some light on its implications. This chapter examines the tensile properties of regular and air jet rotor cotton yarn at different gauge lengths and strain rates on their tenacity, elongation and initial modulus. Also, the applicability of the Weibull model to yarns has to be statistically tested and reported.

3 Modeling of Tensile Properties The Weibull distribution was discovered as early as the latter half of the twenties in a discussion relating to the asymmetric distribution of the extreme values in a sample. This distribution is one of the most widely used life time distributions in reliability engineering. It is a versatile distribution that can take on the characteristics of other types of distributions, based on the value of the shape parameters b. This model is used for simulating the long time distributions of the tensile properties. A random variable X possesses a Weibull distribution if it has a probability density function of the form f (h) b h a a b b1 h a (9.1) where a - Scale parameter b - Shape parameter h - Variable a, b, h > 0 The cumulative distribution function is, b h a d.f 1 (9.2) The Weibull distribution function is based on the flaw theory with the assumption that the flaws are distributed at random with a certain density per unit volume. The result is a representation of the material as a series model, or a chain, in which failure depends on the weakest link. The Weibull distribution function is expressed as the probability of fracture which is given by

4 173 S 1 exp V o u m (9.3) where V is the volume of the element under consideration, is the applied uniform tension stress, and u, o and m are material constants. If the Weibull function is truly applicable to the material used for small test specimens and for the structural components is truly identical, then u, o and m will indeed be material constants and will not change with volume. Textile yarns are not uniform and, therefore, have different mechanical properties at different gauge lengths. In addition to this, they exhibit varying strengths, due to their internal structure which means that there is no specific strength value to represent their mechanical properties. This leads to the necessity of employing statistical analyses for their safe utilisation in design and manufacturing. One of these analyses is the Weibull distribution, which has recently been used for modeling textile yarn strength results. Weibull distribution has the capability to model experimental data of very different characters. This is one of the reasons for its wide utilisation nowadays. In recent years research papers and books dealing with the historical development and application of this statistical method have been published (e.g., Hallihan 1993, Dodson 1994). In his book, Dodson described estimation, approaches for Weibull distribution parameters. Barbero et al (2000) applied this analysis to model the mechanical properties of composite materials and suggested the Weibull distribution as a practical method in the determination of 90% and 95% reliability values used in composite material mechanics.

5 INITIAL MODULUS OF TWISTED CONTINUOUS FILAMENT YARNS The literature is replete with the experimental and predicted values of initial modulus of twisted continuous filament yarns. The following formulae were given for predicting initial modulus as a function of twist by Hearle et al (1969). Yarn mod ulus cos 2 Fibre mod ulus (9.4) where = helix angle. Later Hearle et al (1969) modified this formula taking into account fibre and yarn Poisson s ratio. 2 Yarn mod ulus c C ln C. (9.5) 2 Fibre mod ulus 4 4 (1 c ) where C = cos. While these above models were applicable for twisted continuous filament yarns made out of any fibre, not much work seems to have been carried out on the prediction of initial modulus for spun yarns made from different fibres. But a very important fact is that a spun yarn behaves like a continuous filament yarn at very short gauge length namely 1 cm and so the model which is used for predicting the initial modulus of twisted continuous filament yarn can be used as a first approximation for spun yarns at a very small gauge length.

6 MATERIALS AND METHODS Materials Table 9.1 gives details of the yarns used in the study. Table 9.1 Yarns used S.No. Description T.P.M 1 Rotor yarn 20Ne (29.52 tex) yarn 20Ne (29.52 tex) Methods tensile tester Tensile tests were carried out on an Instron 6021 universal strength with computerized monitoring. Five different gauge lengths, namely, 100,200,300,400 and 500 mm and three strain rates 10%/min, 15%/min and 20%/min were chosen for the test. For each gauge length 20 tests were carried out and the mean was considered. In all, 30differnt yarn samples were tested as given in Table 9.2.

7 176 Table 9.2 Detail of yarns tested S.No. Gauge Length mm Strain rate %/min Type of yarn TPM Rotor yarn Rotor yarn Rotor yarn yarn yarn yarn Rotor yarn Rotor yarn Rotor yarn yarn yarn yarn Rotor yarn Rotor yarn Rotor yarn yarn yarn yarn Rotor yarn Rotor yarn Rotor yarn yarn yarn yarn Rotor yarn Rotor yarn Rotor yarn yarn yarn yarn 990

8 Modeling of Tensile Properties Weibull distribution Weibull distribution is used to model extreme values such as failure times and fortune strength. Two popular forms of this distribution are two and three parameter Weibull distribution. The (Cumulative) distribution function of the three parameters. Weibull distribution is given as follows (Ghosh 1999), x a F(x; a, b, c) 1exp c b (9.6) a 0, b 0, c 0, where a, b and c are the location, scale and shape parameters respectively. When a = 0 in the equation, the distribution function of the two-parameter Weibull distribution is obtained. The threeparameter Weibull distribution is suitable for situations in which an extreme value cannot take values less than a. In this chapter the two parameter Weibull distribution which can be used in rotor yarn testing will be considered. The distribution function in this case can then be written as follows: c x F(x; a,b,c) 1exp b0, c0 b (9.7) The parameters b and c of the distribution function F(x; b, c) are estimated from observations. The methods usually employed in the estimation of these parameters are methods of linear regression, method of maximum likelihood and methods of moments (Hallinan 1993, Dodson 1994, Taljera 1981). Among these methods, use of linear regression goes back to the days when computers were not available. The linear regression line was fitted manually with the help of Weibull graph papers. Linear regression is still

9 178 common among the research workers and will be used for parameter estimation in this chapter. Software programmes with statistical abilities such as MS Excel, SPSS and Microcal origin have replaced the Weibull graph papers Method of linear regression This method is based on transforming Equation (9.8) into c x 1F(x:b,c) exp and taking the double logarithm of both sides. b Hence, a linear regression model in the form y = mx + r is obtained. 1 ln ln c ln (x) c ln(b) 1F(x:b, c) (9.8) Weibull distribution was applied to coir fibres by Kulkarni et al (1983) for studying their strength at different gauge lengths. Use of two parameter Weibull treatment has shown the large scatter observed in the fracture strength of coir fibres ranging in length from m to m and in diameter from m to m. It was found that the strength distribution at any length (and for any diameter fibre) can be represented satisfactorily by unimodal Weibull distributions. The plot on logarithmic coordinates of mean fracture strength against test length for a m diameter coir fibre shows a linear relationship between m and m; below m, the strength is relatively independent of length. Apparently, below a length of m the flaws are evenly distributed and the mean distance between gross flaws is of the order of m. The increase in Weibull parameter, m, with the increase in diameter shows that the flaw distribution is more uniform in fibres of large diameter.

10 179 It has been stated that three-parameter Weibull analysis needs laborious computer analysis (Yasuki Kasai and Makoto Saito 1979, Schneider and Polozotto 1979). Based on the weak-link theory proposed by Weibull (1951) the failure statistics of a uniformally stressed sample of length L, made of flawsensitive material can be represented by the empirical equation P exp L f o m (9.9) where P is the probability of survival at a stress of f, o is the normalising stress at which the stress is that at which P = and m is the Weibull parameter. Equation (9.10) can be rewritten as: 1 ln ln m / n m / n ln L f 0 P (9.10) Equation (9.10) indicates that the plot of 1 ln ln P against ln f will give a straight line for the experimental data obtained at a test length of L, if Weibull distribution is applicable. The two parameters m and o can also be determined from the above equation. The quantity P is estimated from a group of N samples, by noting the number of samples, n, that have fractured at a stress f, or less. Thus n P (1 F) 1 (9.11) N where F is the probability of failure at stress f.

11 RESULTS AND DISCUSSION Influence of Strain Rate and Gauge Length on Stress-strain Curves Figures 9.1, 9.2 and 9.3 depict the stress-strain curves of regular rotor spun and air jet rotor spun yarns at various strain rates, at gauge lengths of 100, 200, 300, 400 and 500 mm. Figure %/min Strain rate: Stress-strain curve

12 181 Figure %/min Strain rate: Stress-strain curve Figure %/min Strain rate: Stress-strain curve

13 Gauge Length Effect on Tenacity and the Strain Rate Influence The results for regular rotor and air jet rotor yarns yielded some interesting results in terms of gauge length. Tenacity decreased as the gauge length increased. This is a well known effect which is due to the existence of weak places. Elongation decreased as the gauge length increased. Initial modulus also followed the same trend. The rate of fall in initial modulus with increase in gauge length is different in different gauge lengths. While there is a 100% drop from 100mm to 200mm, the drop from 200 to 300mm is 40%. This rate decreases with increase in gauge length. Figures 9.4, 9.5 and 9.6 illustrate the effect of gauge length on the tenacity of rotor and air jet rotor spun yarns. This is striking contrast to the findings of Pan et al (1997) who have reported a decrease for the fibres with decrease in gauge length Tenacity gf/tex Gauge length (100mm) Rotor Figure %/min Strain rate: Tenacity - Rotor and air jet rotor yarns

14 183 Tenacity gf/tex) Gauge length (100mm) Rotor Figure %/min Strain rate: Tenacity - Rotor and air jet rotor yarns Tenacity( gf/tex) Rotor Gauge length (100mm) Figure %/min Strain rate: Tenacity - Rotor and air jet rotor yarns

15 Gauge Length Effect on Initial Modulus The drop in initial modulus with increase in gauge length has some consequences. The initial modulus can provide directives with respect to the reactions of a yarn immediately after it has been put under the load, which can be of considerable importance depending on the particular application of the yarn whether it be weaving or knitting. Thus it is a valuable parameter, which will provide directives with respect to manufacturing problems. The air jet rotor yarns exhibit a higher modulus compared to a conventional rotor spun at all gauge lengths. With reference to the increase in strain rate at a different gauge length, the values of initial modulus are almost constant. Figures 9.7, 9.8 and 9.9 illustrate the effect of gauge length on the initial modulus at different strain rates Initial modulus (gf/tex) Gauge length (100mm) Rotor Figure %/min Strain rate: Initial modulus - Rotor and air jet rotor yarns

16 Initial modulus (gf/tex) Rotor Gauge length (100mm) Figure %/min Strain rate: Initial modulus - Rotor and air jet rotor yarns 350 Initial modulus (gf/tex) Rotor Gauge length (100mm) Figure %/min Strain rate: Initial modulus - Rotor and air jet rotor yarns

17 Test of the Weibull Model Using Yarn Data Two parameter Weibull plots for the tenacity and initial modulus of rotor and air jet rotor spun yarns at different gauge lengths and at a strain rate of 10% / min, 15% / min and 20% / min are shown in Figures 9.10, 9.11 and Using the same procedure developed by Realff et al (1991), the tenacity data for regular rotor and air jet rotor yarns at different gauge lengths were found to fit a two parameter Weibull distribution. If the strength distribution F l0 (x) at a given gauge length lo, follows a two parameter Weibull distribution, we have F l0 (x) = 1-exp [-(x/x 0 ) г ] (9.12) where Xo and г are the scale parameter and shape parameter respectively. by Then the strength distribution F l (x) at any gauge length l is given F l (x) = 1-exp [-(x/x l ) г ] (9.13) where l = ml o and X l =X o m l/r The parameters Xo and г are positive constants. The mean and coefficient of variation for the data following a two-parameter Weibull distribution are given by 1 x0 1 ; r (9.14)

18 x0 T1 T 1 r r (9.15) where μ and 2 are the mean and variance respectively. The scale parameter is numerically close to the mean yarn tenacity. It is noticed from equation (9.15) that the coefficient of variation of yarn tenacity depends only on shape parameter. The Weibull parameters of both tenacity and initial modulus were estimated using data tested at gauge length in mm and the results are provided in Table 9.3. To verify whether both the strength and breaking strain of yarns obeyed the Weibull model, the Kolmogorav goodness of fit test was used, following the procedures given bellow: 1. All the experimental data are ranked in an ascending sequence. 2. Calculate the sample statistical distribution using (i 0.5)/n where n is the total sample number. 3. Calculate the theoretical distribution according to equation using the estimated Weibull parameters. 4. Compare the corresponding pairs from 2 and 3 and find the maximum difference between them denoted by d n ; 5. The critical values of dn are calculated; for significant level =0.05, d nc = 1.36/n 1/2 6. If d n d nc the statistical model is considered a good representation of the data distribution. The decrease in initial modulus is due to a decrease in stress and strain. It is an important point to note that, for a given material, the yarn modulus is not an independent statistical variable anymore once the strength and breaking strain of the material are given.

19 188 Values of d n for both tenacity and strain are of the two types of yarns are calculated and listed in Table 9.3. The critical value d nc =0.608 and by comparing it with d n it is concluded that the Weibull model is an adequate representation of the data distribution of tenacities of the two types of yarns. Table 9.3 Test for tenacity - Strain rate 10%/min G.Length Yarn type (g/tex) d n d nc 100 mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn Table 9.4 Test for tenacity Strain rate 15%/min G.Length Yarn type (g/tex) d n d nc 100 mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn

20 189 Table 9.5 Test for tenacity - Strain rate 20%/min G.Length Yarn type (g/tex) d n d nc 100 mm 200 mm 300 mm 400 mm 500 mm Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn Table 9.6 Test for initial modulus - Strain rate 10%/min G.Length Yarn type (g/tex) d n d nc 100 mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn

21 190 Table 9.7 Test for initial modulus - Strain rate 15%/min G.Length Yarn type (g/tex) d n d nc 100 mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn mm Rotor Yarn Air Jet Rotor Yarn Table 9.8 Test for initial modulus Strain rate 20%/min G.Length Yarn type (g/tex) d n d nc 100 mm 200 mm 300 mm 400 mm 500 mm Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn Rotor Yarn Air Jet Rotor Yarn

22 191 Alpha value Gauge length mm Rotor Alpha value Gauge length (mm) Rotor Figure %/min Strain rate: Weibull model - Tenacity, Initial modulus for Rotor and air jet rotor yarns

23 Alpha Rotor Gauge length (mm) Alpha Rotor Gauge length (mm) Figure %/min Strain rate: Weibull model - Tenacity, Initial modulus for Rotor and air jet rotor yarns

24 Alpha Rotor Gauge length (mm) Alpha Rotor Gauge length(mm) Figure %/min Strain rate: Weibull model - Tenacity, Initial modulus for Rotor and air jet rotor yarns Figures 9.13, 9.14 and 9.15 show the plot of the parameters normalised by their corresponding maximum values. These also show the effect of gauge length succinctly.

25 Normalised value Rotor elongation(%) elongation(%)" Rotor tenacity tenacity Rotor I M IM Gauge length (mm) Figure 9.13 Normalised 10%/min 1.2 Normalised value Rotor tenacity Jet rotor tenacity Rotor elongation Jet rotor elongation Rotor IM Jet rotor IM Gauge length (mm) Figure 9.14 Normalised 15%/min

26 195 Normalised value Rotor tenacity tenacity Rotor elongation elongation Rotor IM IM Gauge length (mm) Figure 9.15 Normalised 20%/min 9.5 CONCLUSION 1. Tenacity of the rotor and air jet rotor yarns are largely influenced by the gauge length and strain rate. 2. Yarn tenacity decreases with increase in gauge length and increases with strain rate. The elongation follows the same trend. 3. The air jet spun yarn yielded a higher tenacity compared to conventional rotor spun yarns. 4. Initial modulus decreases with increase in gauge length and is insensitive to strain rate. Compared to tenacity and elongation of yarns, the fall in initial modulus with an increase in gauge length is quite significant. spun yarns are characterized by higher initial modulus compared to regular rotor spun yarns in comparison with the rotor spun yarns. 5. An important conclusion is that for any yarn, the initial modulus is not an independent statistical variable anymore once the strength and breaking strain of the yarn are given.

27 Weibull model is an adequate representation of the data distribution of both tenacity and strain of yarns. 7. Consequently the models, which are used for predicting the initial modulus of twisted yarn, have to be modified in view of the vast changes that are noticed by the experimental values of initial modulus. The Weibull model is a good theory to specify the statistical distributions of both strength and elongation of any yarn irrespective of the fact that these yarns do not satisfy the definition of a classic fibre originally adopted as assumption in deriving the Weibull model. The implication of gauge length effect on initial modulus is profound. At first sight, it reveals that the increases in both strength and breaking strain of a yarn with a smaller gauge length are the culmination of every fibre in the yarn. Further more, the reason that the modulus of a yarn changes is due to the fact that the gauge length influences the strength and breaking strain of yarn by different amounts. More importantly because of this gauge length effect on yarn modulus of yarns whether they are continuous or spun need to be modified because they have been assumed that the modulus of the yarn is length independent. It should be pointed out that the implications of this size effect on the initial modulus of a material are very significant. First, it reveals that the increases in both the strength and breaking strain of a material with smaller size are the results accumulated from every point of the material. Also the effects influence strength and breaking strain of the material to a different degree causing the modulus of the yarn to change. It has been shown that Weibull model is a good theory to specify the statistical distribution of both tenacity and initial modulus.