Load–extension properties of glass plain knitted technical fabrics: Part I: some additions to the extension of the circular ring problem to make it applicable for a knitted loop head curve

Abstract

Pierce’s well-known solution to the problem of the load–extension behavior of a circular ring has been used as one of the measurement methods of bending rigidities of textile yarns. This paper provides some additions to Pierce’s solution to make it applicable for a knitted loop head curve. It is found that the load applied to a ring is equal to the bending rigidity multiplied by the difference between the squares of the major and minor curvatures of the extended ring. This equality gives way to apply the circular ring problem for obtaining the load–extension properties of a knitted loop head. It is also shown that, for the loop head curve, the elastica curve can be replaced by an elliptical curve for up to 30% of extension, as it is easier to apply.

Load–extension properties of glass plain knitted technical fabrics – Part II: extensions in the course-wise and wale-wise directions

“…In Part I, the elastic curve and elliptical curve were also discussed and compared for the circular ring problem. 11 It was seen that the calculated loads were very close to each other up to 30% of extension when both the curves, namely the elastic and elliptical curves, were applied during the calculation of the extensions. Therefore, here in Part II, the elliptical curve was chosen for the loop head extension problem for simplicity reasons.…”

“…These stages were attributed to the fabric, yarn and fiber extensions, respectively. To be able to apply Hooke’s Law and the Poisson ratio to the knitted samples, the average widths of the samples were calculated by assuming that the side edges of the samples constituted parabolic curves. A theoretical study was carried out for the first stage of extensions and contractions. The following methods were applied in the theoretical work: (i) the model of the fabric was assumed to be as given by Kurbak 1 ; (ii) the frictional restraints were considered based on the work given by Kurbak; 12 (iii) for load–extension properties of the loop arms, the inflexional elastica theory was used as explained by Kurbak; 12 (iv) the load–extension properties of the loop heads were calculated by applying the circular ring problem given in Part I, 11 in which the elastic curves were replaced by elliptical curves. Some simple equations that are suitable to be used in engineering software were found. These simple equations are as follows:…”

“…In Part I of this series of papers, 11 the problem of the extension of a circular ring was discussed and it was argued that the circular ring problem could be applied to the loop head. In the circular ring problem, the applied force does not change along the ring, as shown in Figure 10.…”

“…Equation (30), which was obtained and given in Part I, 11 can be applied as where T c is the force applied in the course-wise direction, Fsc is the equivalent tension in the course-wise direction at the relaxed state (this will be explained later), B is the bending rigidity of the yarn, ρB is the radius of the curvature at point B in Figure 11 and ρA is the radius of the curvature at point A in Figure 11.

Figure 11.The radii of curvatures of the extended circular ring.

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“…The present work aims at compensating for the three objections given above by considering the three-dimensional loop model given by Kurbak 1 (see Figure 1) and applying the extension of a circular ring problem given in Part I. 11 The purpose is to predict the load–extension properties of a fabric in the course-wise and wale-wise directions in order to expand the existing literature by obtaining the contractions in the lateral directions, and to include the effects of frictional resistance based on frictional resistances of relaxed fabrics given by Kurbak. 12

Figure 1.(a) Plain knitted loop model (showing also the parameters α1 and η).

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