Technically, sag is the bowing of a plastic sheet during the heating phase of thermoforming. Practically, it is a major bugbear. Many processing problems, including the following are directly attributed to sag:

  • Nonuniform sheet thickness, even before the sheet contacts the mold.
  • Nonuniform sheet temperature, since the top-center of the sheet moves away from the top heater while the bottom-center moves toward the bottom heater.
  • Localized webbing owing to greater sheet surface area.

Excessive sag has restricted the use of PP homopolymer and prevented the use of many types of nylons.

Mathematically, the shape of the sagging sheet can be modeled in two dimensions as a plate flexing under its own weight, so long as the neutral axis remains within the thickness of the sheet. It has been shown that the extent of sag depends on the width of the sheet, the thickness of the sheet, and the instant elastic modulus of the polymer. When the sheet has sagged substantially, its shape can be modeled in two dimensions as a parabola or catenary. It has been proposed that the instant tensile strength be used instead of the instant elastic modulus. (This is documented in J.L. Throne’s Technology of Thermoforming, Hanser, 1996, described elsewhere.)

A recent review of the problem has yielded a new model, based on linear viscoelasticity. Consider the Maxwell parallel spring-and-dashpot mechanical model, Figure 1.

Maxwell parallel spring-and-dashpot mechanical model

Figure 1

In tension, the spring constant is related to the polymer elastic modulus, and the resistance of the dashpot is related to the polymer extensional viscosity. The equation that describes the Maxwell model is:

Maxwell model equation

The stress is α,  σ is time, ηe is extensional viscosity, E is elastic modulus, and ε is elongation. The Maxwell model response to a constant stress, σ0, is shown in Figure 2.

Maxwell model response to a constant stress

Figure 2

The solution to the Maxwell model for this loading configuration is:
Maxwell model solution

Note that the elongation of the model is linear with time, σ, with the proportionality being the ratio of the stress and the extensional viscosity. The zero-time asymptotic sag is given as the ratio of the applied stress to the instant elastic modulus. For sag, the stress is just the weight of the sheet per unit area, the weight being a function of the density of the sheet and the sheet thickness.

Can this model be verified? Figure 3 [C. 0 Cruz, Jr., “The Sag Process in Modified Polypropylene”, SPE ANTEC Tech. Papers, 40 (1994), pp. 854-858.] shows the time-dependent isothermal sag for unmodified and acrylic-modified polypropylenes. It is apparent that the extent of sag is linear with time.

Effect of Modifier on Isothermal Sag of Polypropylene

Figure 3

Why is this important? Because in resin suppliers’ research laboratories, time-dependent module and elongational viscosities can be measured under very controlled conditions. It appears from here that the simple Maxwell linear viscoelastic model can be used to quickly screen materials to determine temperature-sensitive sag for polymers that are currently commercial, as well as those that are being proposed as lower sag polymers.

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