On the strength distribution of bundles of threads, and its asymptotic approximation

Abstract

The following paper is concerned with the distribution which arises in the theory of breaking strength of what Ve shall tent bundles of threads. A bundle is defined to be a group Of threads of equal length p= side by side with their corresponding ends clamped together so that when extension is applied to the bundle, all its constituent threads extend at an equal rate. It is required to relate the statistical distribution of breaking strength of bundles having a given number of threads of a given length to the known distribution of breaking strength of single threads of a standard length.The problem presents itself naturally in the theory of strength testing of textile materials. The procedure of determining the breaking strength of bundles has similarities with a test widely used in practice for estimating the strength of textile yarns, known as the "hank" test for woollen and worsted yarns, and the "!.ea" test for cotton yarns, The method adopted is to reel a hank of yarn containing a stated number of turns of specified circumference, and to apply the breaking load to the hank by stretching it between two hooks. We have used the term "bundle", however, in preference to "hank" or "lea" for the reason that in the practical test just described the amount of slip at the hooks, although usually small, is indeterminate, so that the conditions of such a test are ill-defined and ambiguous.A preliminary discussion is given : of the physical principles involved., and results are stated-which apply in general to very large bundles whose constitution is indistinguishable from the population of single threads. The work in this section overlaps that of Peirce to a large extent, and many of the results quoted were also given by hiss,The distribution of bundle strength is then obtained, and subsequently attention is concentrated on the case where the load distributes itself equally between the threads. Two peculiar forms of the single thread strength distribution are considered in detail,The second main section of the paper is devoted to a study of the asymptotic behaviour of the bundle strength distribution for large numbers of threads, and a method is developed which reduces the problem to its simplest terms and provides a preliminary solution in the practically important case.In the final section, the method of steepest descents independently confirms some of. the 'results already obtained, and provides a means of studying the asymptotic behaviour in more detail.