Determination of the stress-strain state of the filament yarn

The solution that is obtained using the equation of the curved axis is in the form of an infinite series and practical application is associated with great mathematical difficulties. In addition, the differential equation of the curved axis is obtained for pure bending, i.e. The rod is bent due to the moments acting in the longitudinal sections, and there are no lateral forces. A bending of a flexible thread occurs under the influence of its own weight, which in many cases is directed precisely in the transverse direction of the rod or has a component directed perpendicular to the axis of the flexible thread. All this circumstance makes it necessary to create a new method for calculating the strength and rigidity of flexible threads. In the present paper, for the first time, this problem is considered not in Eulerian a but in Lagrangian variables. in variables not connected with a deformed body, but with an undeformed body. Exact analytical solutions are obtained, and after obtaining the solution in Lagrangian variables it is possible to go over to Euler variables. Moreover, the exact analytical solution obtained is much simpler than existing solutions and the application in practice presents no difficulties.