Sag in thermoforming

In thermoforming, flat sheets of plastics are transformed into complex shapes by first softening the sheet, and then by shaping it. While softening, the sheet can extend under its own weight, making it concave. Called sag, this affects how each element of the sheet surface views the heater bank. This can worsen the variations in the heat flux distribution across the sheet. It can also cause the heat flux over and under each element on the sheet to differ. The resulting temperature gradients, either across the sheet or through its thickness, can complicate processability. To suppress sag, practitioners stretch the sheet laterally, using cambered transfer rails to keep the sheet taut. In this article, we model sag analytically, using transport phenomena in cylindrical coordinates, for a thin wide rectangular Newtonian isothermal sheet. We uncover a universal dimensionless relation between sag and time, and a useful dimensionless group that we call sagability. We find that the middle of the sagging sheet, unsuppressed, descends, to leading order, with the cube root of time, and our experiments confirm this. Also, we discover that at a particular time the sag increases rapidly without bound, and we call this phenomenon sag runaway. POLYM. ENG. SCI., 50:2060–2068, 2010. © 2010 Society of Plastics Engineers