A phenomenological theory of the mechanics of cold drawing

Abstract

When the amount of stretch in a taut fiber varies rapidly with distance z along the fiber axis, the tension T on a cross section normal to the axis is not determined by only the temporal history of the local stretch, but is affected by the variation of stretch with z. Here a study is made of the implications of a constitutive relation giving the tension in a polymeric fiber as a function of the local stretch, λ, the spatial derivative, v_z, of the velocity, and the first two spatial derivatives of the stretch, λ_z and λ_zz. When this relation is placed in the equation for balance of momentum in the axial direction, dynamical equations are obtained whose equilibrium solutions for λ as a function of z can be rendered explicit. The solutions describe necks, bulges, drawing configurations, and periodic striations. The assumption that motions resulting from slow changes in applied tension or length are homotopies formed from these equilibrium solutions is compatible with many of the observed properties of tension-induced necking and cold drawing in polymeric fibers. Properties of non-equilibrium solutions of the dynamical equations are also discussed here. Lyapunov functions appropriate to several types of boundary conditions, including elongation under a dead load, are constructed. It is shown that the presence of viscous stresses has a strong effect on the types of traveling waves that can occur in long fibers. A detailed discussion is given of a class of traveling waves, called "steady draws," that correspond to the continuous drawing processes frequently employed to improve the stiffness and tenacity of synthetic fibers.