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Theoretical Considerations

When a thermoplastic sheet is softened by heat and then pressure is applied to one side so as to generate a free surface, the shape so formed has a uniform thickness. Hence a simple volume balance will provide the thickness of the shape produced in this forming operation.

In most thermoforming processes, a relatively cold mold is used to produce the desired shape. The effect of this is a molding which has a large variation in thickness because the sheet freezes off at whatever thickness it has been stretched to when it touches the mold.

Consider the thermoforming of a plastic sheet of thickness, ho, into a conical mold as shown in the figure below.

Analysis of thermoforming.

At the shown instant of time t, the plastic is in contact with the mold for a distance S, and the remainder of the sheet is in the form of a spherical dome of radius R and thickness h.

From the geometry of the mold the radius is given by:

R = H - S sin / sin tan

The surface area A of the spherical bubble is given by:

A = 2 R2 (1 - cos )

The change in thickness during an infinitesimal time dt may be estimated by assuming that the volume remains constant:

Substituting for R, the above equation can be reduced to:

dh/h = [2 - sin tan / (1 - cos )] . sin dS / (H - S sin)

This equation can be integrated with the boundary condition that h = h₁ at S = 0. Hence, the thickness h at a distance S along the side of eh conical mold is given by:

h = h1 ( H - S sin / H ) sec - 1

Now, we seek an expression for h1 in terms of ho. We again consider the same boundary condition as above. At the point when the softened sheet first enters the mold it forms part of a spherical bubble which does not touch the sides of the cone. The volume balance is therefore:

(D2 / 4) ho = 2(D/2)2 (1-cos) h1 /(sin)2

Hence,

h1 = ho (sin)2 / 2 (1-cos)

Substituting for h1 in the expression for h, we get:

h = ho (sin)2 / 2 (1-cos) .[H - S sin / H ] sec - 1

Or

h/ho = [(1+cos)/2] [H - S sin / H ] sec - 1

This equation may also be used to calculate the wall thickness distribution in deep truncated cone shapes but note that its derivation is only valid up to the point when the spherical bubble touches the center of the base. Thereafter, the analysis involves a volume balance with a freezing-off on the base and sides of the cone.

Source: Crawford, Plastics Engineering.