On the Rheology of Cold Drawing . 11 . Viscoelastic Materials

In this second paper in a series on cold drawing of polymeric fibers and films, asymptotic formulae are derived for the tension in thin bodies of nonlinearly viscoelastic materials subject to inhomogeneous stretching. The formulae, which are valid to within an error of fourth order in sample thickness, give the tension in fibers and films as functionals of the histories of the local stretch and its first two spatial gradients. It is shown that the theory obtained by taking the formulae to be exact is consistent with thermodynamical principles. Material response for various types of histories is discussed, and the theory is employed to calculate the creep response of fibers under static loads. Numerical solutions of the evolution equation for the stretch field in creep show that for an appropriate class of materials with slowly fading memory there is a range of applied loads for which an initially homogeneous deformation evolves into a well defined neck whose edges advance a t high speed along the fiber and in so doing transform moderately stretched material into highly stretched (i.e., drawn) material. The calculated fiber profiles and the predicted dynamics of neck formation and growth are in good accord with familiar observations. A description is given of the circumstances under which one can use instantaneous ( i.e., high speed) response functions to calculate fiber profiles in rapidly growing necks.