On the cold drawing of polymers
type: Item
instance of: scholarly article
title: On the cold drawing of polymers
zbMATH Open document ID: 0634.73030
DOI: 10.1016/0898-1221(85)90137-3
author: Bernard D. Coleman
publication date: 1985-01-01T00:00:00Z
review text: 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 p…
Mathematics Subject Classification ID: 74D10
Mathematics Subject Classification ID: 74G99
Mathematics Subject Classification ID: 74H99
Mathematics Subject Classification ID: 82D60
Mathematics Subject Classification ID: 74D99
Mathematics Subject Classification ID: 74B99
Mathematics Subject Classification ID: 74K10
zbMATH DE Number: 4033217
zbMATH Keywords: nonhomogeneous deformation
zbMATH Keywords: slender bar
zbMATH Keywords: viscoelastic material
zbMATH Keywords: polymer
zbMATH Keywords: cold drawing
zbMATH Keywords: dependence
zbMATH Keywords: total tensile force
zbMATH Keywords: field of the axial stretch ratio
zbMATH Keywords: necking
zbMATH Keywords: nonhomogeneous stretch fields
zbMATH Keywords: fading spatial (not temporal) memory
zbMATH Keywords: fiber
zbMATH Keywords: retardation approximation method
zbMATH Keywords: Viscosity (dissipative) effects
zbMATH Keywords: force-stretch relation
zbMATH Keywords: static equilibrium configuration
zbMATH Keywords: autonomous second order ordinary differential equation
zbMATH Keywords: bulge
zbMATH Keywords: periodic striation
zbMATH Keywords: very slow motion
zbMATH Keywords: gradual change of configuration
zbMATH Keywords: dynamical equation of motion
zbMATH Keywords: travelling waves
zbMATH Keywords: Lyapunov functions
zbMATH Keywords: steady draws, solitary waves
zbMATH Keywords: periodic waves
reviewed by: Witold Kosiński
MaRDI profile type: Publication
full work available at URL: 0898-1221(85)90137-3
OpenAlex ID: W2061964504
cites work: Equilibrium of bars
cites work: Necking and drawing in polymeric fibers under tension
cites work: Q3321549
cites work: An approximation theorem for functionals, with applications in continuum mechanics
cites work: On retardation theorems
cites work: On certain steady flows of general fluids
cites work: Kinematical concepts with applications in the mechanics and thermodynamics of incompressible viscoelastic fluids
cites work: Steady Extension of Incompressible Simple Fluids
cites work: Admissibility criteria for propagating phase boundaries in a van der Waals fluid
cites work: The viscosity-capillarity criterion for shocks and phase transitions
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 4033217