Theoretical study on the bending rigidity of filament yarns with an elliptical cross-section using energy method. I. Theoretical modeling

The geometrical model of yarn currently in use is assumed to be a circular cylindrical under the bending deformation, but this assumption is not really true because the cross-section of yarn has to change to an elliptic shape when the compressive force acted on the yarn is considered. Therefore, as the first part of a series of work, the present paper considers the fact that yarn geometry changes from a circular cylindrical to an elliptical cylindrical under an external applied bending moment, and then analyzes the relationship of bending rigidity between constituent filaments and the formed yarn, which is useful in predicting or selecting specified bending property of filaments by the requirements of yarn. In the present paper, the bending and torsional energies stored in filaments are analyzed according to their deformations by the energy method. The explicit formula is then obtained to quantify the bending rigidity relationship between the filament and yarn, in which four characters are featured as the eccentricity of elliptical cross-section of yarn, the ratio of bending to torsional rigidity of filament, the helix angle of the filament on yarn surface and the number of filaments inside the yarn. Moreover, the bending rigidity of filament yarns formulated by an earlier method with a circle shape assumption can also be obtained by the proposed method with the eccentricity being equal to zero. Based on the analytical solution, the earlier method in which the circle shape takes place of the elliptical shape of yarn for simplification can also be evaluated. This will be depicted in the Part II of this series of work, in which the relative error between the ideal and revised model would be analyzed, and the numerical simulations of relationship between filament and yarn would also be made.