On the cold drawing of polymers
The paper treats the nonhomogeneous deformation of a slender bar of a viscoelastic material, a polymer. The authors considers the range of conditions under which a neck is formed in the bar, although it at first may both lengthen and deepen, and eventually growths in length without further change of depth. Such a motion is called cold drawing, if it occurs without application of heat. The main assumption made in the paper concerns the form of the dependence of the total tensile force T (in the z-direction, say) on the field of the axial stretch ratio \(\lambda\). The tension T equals the integral over the cross-section of the z-component of the Cauchy stress tensor. The necking as well as the drawing are related to nonhomogeneous stretch fields, hence the assumption about the fading spatial (not temporal) memory of the material of the bar (slender enought to be called fiber) made by the author, seems to be relevant. Then the retardation approximation method developed by the author in the early 60's and 70's [Arch. Ration. Mech. Anal. 6, 355-377 (1960; Zbl 0097.164); 43, 1-23 (1971; Zbl 0325.46042)] leads to a nonlinear relation in which the first and the second derivatives of \(\lambda\) with respect to z take place additionally to the stretch \(\lambda\). Viscosity (dissipative) effects are introduced through a nonlinear dependence of T on the velocity gradient. Due to the assumed forms of the material functions appearing in the derived force-stretch relation, the problem of determining of static equilibrium configuration leads to an autonomous second order ordinary differential equation, the solutions of which split into three classes. They describe: fully developed draw, a neck or a bulge, and periodic striation. It turns out that the derivation of the characteristic quantities in the solutions (e.g. stretches corresponding to a neck or a bulge) is based in the T-\(\lambda\) diagram on the equal areas condition, very well known from the Maxwell construction for phase change problem. Basing on some conjecture about the behaviour of solutions of the equation, the case of a very slow motion called by the author a gradual change of configuration has been considered. Taking into account the dynamical equation of motion the particular solutions have been searched in the form of travelling waves. For several types of boundary conditions the corresponding Lyapunov functions are constructed in the paper. Due to the three classes of the equilibrium solutions also three forms of the particular dynamic solutions are observed, namely: steady draws, solitary waves and periodic waves. The paper is of a value not only for mathematicians.
autonomous second order ordinary differential equation
scientific article; zbMATH DE number 4033217