Corotating or Codeforming Models for Thermoforming: Free Forming

Our previous work [J Pol Eng, 32, 245 (2012)] explores the role of viscoelasticity for the simplest relevant problem in thermoforming, the manufacture of cones. In this previous work, we use a differential model employing the corotational derivative [the corotational Maxwell model (CM)] for which we find an analytical solution for the sheet deformation as a function of time. This previous work also identifies the ordinary nonlinear differential equation corresponding to the upper convected Maxwell model (UCM), for which she finds no analytical solution. In this paper, we explore the role of convected derivative by solving this UCM equation numerically by finite difference. We extend the previous work to include sag by incorporating a finite initial sheet curvature. We treat free forming step in thermoforming and find that the convected derivative makes the free forming time unreasonably sensitive to the initial curvature. Whereas, for the CM model, we get a free forming time that is independent of initial sheet curvature, so long as the sheet is nearly flat to begin with. We cast our results into dimensionless plots of thermoforming times versus disk radius of curvature.