Calculation of the deformation properties of materials made of chemical fibres

Abstract

A comparatively simple repeating element whose relative deformation corresponds to the deformation of the entire sample can be distinguished in many materials made of chemical (including rigid-chain) fibres. A simple program based on the MATLAB mathematics package that allows calculating the relative deformation of each element and consequently also the deformation of the sample is described. The result of calculating the deformation of a repeating element consisting of two circles of different radius twisted to opposite sides is reported as an example of running the program.Fibres and thread made from chemical, including rigid-chain, polymers are used for manufacturing fabrics, knits, and other textile materials for different applications. Since such fibres and thread stretch very little on deformation of the sample, deformation primarily takes place due to bending of the fibres and thread. In most textile materials except for nonwovens, a comparatively simple repeating element whose relative deformation corresponds to the relative deformation of the entire sample can be distinguished. A photomicrograph of one type of knit is shown in Fig. 1a and a photomicrograph of a column separated (prepared) from this knit is shown in Fig. 1b. A comparison of photomicrographs 1a and b shows that the shape and size of the loops changes little when the fibre is free from interacting with the other fibres. This indicates that the fibre used in manufacturing the knit is plastically deformed, as otherwise it would be in the form of a straight line. After calculating the relative deformation of such a repeating element, we obtain the relative deformation of the entire sample.It is convenient to use the two-dimensional nonlinear theory of elasticity for these calculations. We will assume that only concentrated, and not distributed, forces act on the examined element (henceforth the elastic line).Consider a two-dimensional elastic line obtained as a result of the fact that a fibre of arbitrary initial curvature (for example, plastically deformed) is loaded with some finite number of concentrated forces and external moments. We will use the approach proposed by Popov.*The elastic line can always be divided into segments in the general case so that concentrated forces F 0 and F 1 and external bending moments M 0 , M 1 were applied only on ends 0 and 1 (Fig. 2) of the examined line segment (they are taken with consideration of the effect of the cut parts of the fibre on segment 0-1). If there are no distributed forces on segment 0-1, then from the condition of equilibrium of the segment, we have F 0 + F 1 = 0 or F 0 = -F 1 . We stipulate the direction of force F 0 applied at the initial point of segment 0-1 of the elastic line as the basic direction.Let us introduce angle δ read counterclockwise from direction F 0 to axis X (or clockwise from axis X to the direction of force F 0 ) (i.e., the slope of axis X to force vector F 0 ). We will consider the initial curvature of the rod (fibre) (the curvature is a quantity inverse to the radi...