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1026 PLASTIC PARTS PROCESSING II

PART IV. Cyclic Processes: Extension Dominated

12 COMPRESSION MOLDING

Compression molding uses thermosetting resins to make objects with simple

geometry. (Compare transfer molding, which is used for parts with intricate

geometries.) Sonta

molded a simple rectangular parallelepiped out of sheet

molding compound (SMC). He followed the curing process of this material by

using dynamic dielectrometry in a specially constructed compression mold; the

mold halves had to be electrically insulated from each other. High-speed data

acquisition permitted Sonta to follow these reactions, which were over in 40 to

60 s. Hence, as in casting, this technology may be used to increase the rate of

production by replacing ﬁxed time schedules such as ‘‘cure for t seconds or

minutes,’’ with curing times based on objectively measured data. (Ramanathan

studied this application of dynamic dielectrometry as well as slower reactions

observed with epoxies.)

Logsdon

modeled the compression molding of a cafeteria tray, also made

from SMC. In addition, he did experimental work on the squeezing ﬂow prob-

lem, using a sample of Silly Putty (polydimethylsiloxane) in an Instron 4505. If

the preform is a cylinder, e.g., D

⫽ H, then the squeezing operation causes the

sides of the cylinder to become curved. The shape of this free surface is deter-

mined by the surface tension of the polymer. In addition, this process is a man-

ifestation of the uniform biaxial extensional viscosity of the polymer. Hence,

these data are necessary to accurately relate the compression stroke to the mold-

ﬁlling process.

13 BLOW MOLDING

Constants and Deﬁnitions

⫽ Diameter of the mold cavity

⫽ Diameter of the parison

⫽ Height of the mold cavity

⫽ Radius on the free ﬁlm

⫽ Radius of the mold cavity

⫽ Radius of the parison

⫽ Volume of the ﬁlm

⫽ Volume of the parison

⫽ Wall thickness of the molded object

max

⫽ Maximum wall thickness of the molded object

⫽ Wall thickness of the parison

⫽ Distance down the side wall, measured from the horizontal center plane

Blow molding is used to make a hollow object whose opening is smaller than

the outside dimensions. It begins with the cyclic extrusion of a parison, the

vertical extrudate from a concentric or eccentric annular die. The mold halves

close around the parison, pinching off the bottom. Gas enters the hollow center

of the parison and blows it against the insides of the mold. When the part is

sufﬁciently cooled, the mold is opened and the part removed. This is the usual

method for making bottles, most with molded threads for caps, some with hollow

14 THERMOFORMING 1027

handles, the latter of which may be contiguous (as in milk and juice bottles—

concentric openings) or noncontiguous (as in the microwave reheatable syrup

bottles—eccentric openings). The innovation of the programmable parison

makes it possible to obtain a controllable nonuniform wall thickness, providing

or minimizing squeezability.

Riney

modeled the blow molding of a concentric cylindrical bottle from a

uniformly thick parison. A volume balance, similar to that employed in the

analysis of thermoforming, yields the approximation

W ⫽ W if D ⫽ H (13.1)

冉冊

max PMM

W ⫽ if D ⬍ H (13.2)

max MM

11 2 2 2 1 2

–– –

兹{[ ( H ⫺ R )/R ] ⫹ H }

24 MMM4 M

This thickness occurs at the vertical center of the molded object; the remain-

ing free ﬁlm is like a pointed football. (Note that the parison does not move

sideways as a cylinder, as implied by Middleman

in Problem 3-24.) Riney

has written an Excel spreadsheet to calculate the wall thickness of the blow-

molded object, from the initial contact point to the last (which occurs in the

bottom and top edges). She also attempted to obtain analytical solutions using

Maple but was unsuccessful.

14 THERMOFORMING

Constants and Deﬁnitions

⫽ Dimensionless numerator mold constant

⫽ Dimensional denominator mold constant (length)

⫽ Dimensionless denominator mold constant

⫽ Thickness of the free ﬁlm

⫽ Thickness of the frozen ﬁlm

⫽ Thickness of the frozen ﬁlm at the initial touch

⫽ Thickness of the original preform

⫽ Radius of the free ﬁlm, prior to it touching the mold

⫽ Radius of the free ﬁlm

⫽ Distance from the center of a mold outward toward the side

⫽ Slant height, distance down the wall of the mold

⫽ Slant height of the mold

Thermoforming is used to make relatively thin-walled objects with simple or

complex surfaces, provided undercuts are absent or small. The starting material

is generally a ﬂat sheet of uniform thickness; it may be a homogeneous sheet

produced by extrusion or a multilayered sheet by ‘‘co-extrusion.’’

Normally, this sheet is warmed from above by radiant heaters; its own weight

causes it to sag. A ‘‘reverse blow’’—air from below—may be used to further

stretch the sheet, the once-again-concave sheet is indexed over a female mold,

1028 PLASTIC PARTS PROCESSING II

its edges are clamped around the mold, and a vacuum is created between the

mold surface and the sheet. This last step draws the sheet against the ‘‘cold’’

mold surface, causing the sheet to solidify.

Among the many variations of thermoforming are drape molding, which is

the use of a male mold, and mechanical assist, the use of a wooden plunger

covered with mole skin to aid the sheet in reaching deep portions of a mold.

The initial heating may be selective by using screens to absorb the radiant heat

above areas that are to remain relatively thick or do not need to stretch very

much.

An inherent characteristic of thermoformed objects is their nonuniform wall

thickness. To the author’s knowledge, the only shape not subject to this result

is the hemisphere. If a circular sheet of thickness h

is clamped above a ‘‘hem-

ispherical mold,’’ the heated free ﬁlm being thermoformed is a portion of a

hollow sphere; this ﬁlm touches the hemispherical mold at a single instant,

which, upon freezing, has a uniform thickness. This thickness, h

, determined

from a volume balance, is h

, as shown below:

–

V ⫽

␲

14 3 3

––

⫽ ()

␲

[R ⫺ (R ⫺ h ) ] (14.1a)

23 ƒ ƒ ƒ

14 3

––

⫽ ()

␲

[3(R ⫺ h )Rh⫹h ]

23 ƒ ƒ ƒƒ ƒ

Since R

⬎⬎ h

, then h

⬎⬎ ; since R

⫽ R

ƒƒ

–

V ⬇ 2␲Rh h ⫽ h (14.1b)

ƒƒ ƒ 20

(Note that a material balance is not required since the densities of the starting

‘‘preform’’ and the resulting thermoformed object are the same.)

For ‘‘all’’ other molds, a volume balance yields the following ordinary dif-

ferential equation:

dh Ah

⫺⫽ (14.2)

dz B

⫺ Cz

as shown by Garrett

and subsequent researchers. The various shapes studied

so far have been recently classiﬁed by Gerstle

as ‘‘symmetric’’ molds in which

there is a single geometry of the ﬁlm frozen each instant, versus ‘‘asymmetric’’

molds in which there are two different geometries of ﬁlm frozen each instant.

In addition, there are ‘‘one-plane’’ molds, which are deﬁned by a single line

(either extended in one direction or rotated around a given axis), versus ‘‘two-

plane’’ molds, which require two such lines. The following outline includes

references to the cited works:

●

Symmetric Molds

(a) One-plane molds: Hemispherical,

71,72

open cylindrical,

71,72

coni-

cal,

70,71,73,74

triangular

70,71,73,74

(b) Two-plane molds: Deep truncated triangular,

70,71,73,74

shallow trun-

cated triangular

70,73,74

14 THERMOFORMING 1029

●

Asymmetric Two-Plane Molds: Deep truncated conical,

shallow trun-

cated conical,

deep cylindrical,

71,72

shallow cylindrical

71,72

Garrett

initiated these studies by developing a Galerkin ﬁnite-element pro-

gram in FORTRAN. Wooldridge

rewrote it for Apple II computers and added

new geometries; she also corrected the solution of Rosenwieg

for the shallow

truncated cone. Passiﬁume

used FORTRAN and added new geometries; she

also demonstrated the need to redeﬁne the ﬁnite-element mesh for positions near

the ‘‘end’’ of the process. Herps

tackled the problem of ‘‘edges’’—locations

where two planes meet; she solved the biharmonic and von Karman equations

for an hemioctahedral mold. Gerstle

developed a spreadsheet model for ther-

moforming, utilizing the Runge–Kutta–Fehlberg ﬁfth-order integration scheme

in Visual Basic.

The open cylindrical mold is described by

dh h

⫺⫽ (14.3a)

dz R

since C

⫽ 0; ‘‘open’’ refers to the fact that this hole has no bottom (i.e., the

mold is very deep). The nonuniform wall thickness is given by the analytical

solution

h ⫽ h exp ⫺ (14.3b)

冉冊

ƒ i

as shown by Passaﬁume;

here, the initial touch occurs when R just equals D,

–

where D is the diameter of the hole.

The conical mold is described by Eq. 14.2 with

sin

␤

tan

␤

A ⫽⫺2 B ⫽ HC⫽ sin

␤

(14.4a)

⫺ cos

␤

where

␤

is the angle of the cone measured from the horizontal to the side of

the mold and H is the axial height of the cone; the analytical solution for the

nonuniform wall thickness is

sec

␤

⫺

h ⫽ h 1 ⫺ sin

␤

(14.4b)

冋冉冊册

ƒ i

where

–

h ⫽ h (1 ⫹ cos

␤

) (14.4c)

i 20

The triangular mold is described by Eq. 14.2 with

11 H 1

A ⫽⫺ B ⫽ C ⫽ (14.5a)

␤

tan

␤

sin

␤

tan

␤

tan

␤

where

␤

is the angle of the mold measured from the horizontal to its side and

1030 PLASTIC PARTS PROCESSING II

H is the center-plane height of the mold; the analytical solution for the non-

uniform wall thickness is

(tan

␤

⫺

h ⫽ h 1 ⫺ sin

␤

(14.5b)

冋冉冊册

ƒ i

where

h ⫽ sin

␤

(14.5c)

冉冊

␤

The deep truncated triangular mold is deﬁned by two converging inclined

sides, each forming an angle

␤

with the horizontal, truncated by a third hori-

zontal plane; the adjective deep refers to the fact that the initial touch of the

free ﬁlm with the mold takes place on the sides, rather than the bottom. This

mold is symmetric in that the linear strip laid down on the bottom after it is

touched has the same geometry as the linear strip laid down on the sides. Prior

to the bottom being contacted, the sides of this mold have the nonuniform thick-

ness given by Eq. 14.5b.

After the bottom is touched there are four linear strips being frozen, one on

each side and two on the bottom, each of these moving away from the centerline

of that bottom. Hence, there is contiguity but lack of mathematical continuity

in the nonuniform wall thickness distribution; i.e., the slope of the curve that

relates h to z changes when the bottom is contacted. The bottom is touched at

the instant

H(1

⫹ cos

␤

) W

z ⫽⫺ (14.6a)

sin

␤

where W is the width of the mold at the top. Now,

21 H 1

A ⫽⫺ B ⫽ C ⫽ (14.6b)

␤

tan(

␤

/2) sin(

␤

/2) tan(

␤

/2)] tan(

␤

/2)

and

[2 tan (

␤

/2)]/

␤

⫺

H/sin

␤

⫺ z

h ⫽ h (14.6c)

冉冊

ƒ t

–

W ⫺ H/tan

␤

)

where h

is the h of Eq. 14.5b when z is the z

of Eq. 14.6a.

The shallow truncated triangular mold is deﬁned the same as the deep, but

the adjective shallow indicates that the bottom is touched by the free ﬁlm be-

fore the sides. Using x to indicate the distance outward from the centerline

towards the side, the sides are contacted when

14 THERMOFORMING 1031

WH(1 ⫹ cos

␤

)

x ⫽⫺ (14.7a)

2 sin

␤

For x ⬍ x

␪

iii

h ⫽ (14.7b)

␪

where

h ⫽ (14.7c)

␪

12 2

–

( W ⫺ x) HWH

R ⫽⫹R ⫽⫹ (14.7d)

2H 28H 2

––

W ⫺ xW

␪

⫽ arctan

␪

⫽ arctan (14.7e)

冉冊冉冊

R ⫺ HR⫺ H

F ⫽ exp 冕 (14.7f)

冉冊

␪

The deep truncated conical mold is deﬁned by one plane rotated around an

axis, forming an angle

␤

with the horizontal, truncated by a second horizontal

plane; the adjective deep refers to the fact that the initial touch of the free ﬁlm

with the mold takes place on the side rather than the bottom. This mold is

asymmetric in that the circular strip laid down on the bottom after it is touched

has a different geometry than that laid down on the sides. Prior to the bottom

being contacted, the sides of this mold have the nonuniform thickness given by

Eq. 14.4b.

After the bottom is touched, there are two circular strips being frozen, one

on the lateral side and one on the bottom; the latter is moving away from the

center point of the bottom. Hence, there is contiguity, but lack of mathematical

continuity, in the nonuniform wall thickness distribution; i.e., the slope of the

curve that relates h to z changes when the bottom is contacted. The bottom is

touched at the instant

H(1

⫹ cos

␤

) W

z ⫽⫺ (14.8)

sin

␤

where W is the diameter of the mold at the top. (The equations for determining

h(z) after the bottom is touched are complicated; see Wooldridge.

)

The shallow truncated conical mold is deﬁned the same as the deep, but the

adjective shallow indicates that the bottom is touched by the free ﬁlm before

the side. Using x to indicate the distance outward from the center point towards

the side, the side is contacted when

1032 PLASTIC PARTS PROCESSING II

1 H(1 ⫹ cos

␤

)

x ⫽ W ⫺ (14.9)

2 sin

␤

[The equations for determining h(z) after the side is touched are complicated;

see Wooldridge.

]

The deep cylindrical mold is deﬁned as a truncated open cylindrical mold,

where the side is contacted before the bottom; hence, prior to the bottom being

touched, the nonuniform wall thickness is given by Eq. 14.3b. After this, the

thickness is given by the solution to Eq. 14.2 with

⫽ (

␲

⫺ 2)R ⫺ (2

␲

⫺ 6)z

B ⫽ 2R (14.10)

⫽ (

␲

⫺ 2)(R ⫺ z) ⫺ 2R

The shallow cylindrical mold is deﬁned as a truncated open cylindrical mold,

where the bottom is contacted before the side is touched; hence, prior to the

side being touched, the nonuniform wall thickness is given by Eq. 14.2 with

⫺ z

A ⫽ [(H ⫺ R)(H ⫹ R ⫹ 2z) ⫺ 3z ] arccot

冉冊

222

B ⫽ R(H ⫺ R ⫺ z ) (14.11)

C ⫽

After this, Eq. 14.10 is applicable.

15 ROTATIONAL MOLDING

Rotational molding is used to make relatively thick-walled hollow objects, where

only the outside of the part has a well-deﬁned geometry; the inside surface

depends on the effectiveness of the spinning process.

An appropriate weight of polymer pellets are placed within the two mold

halves; this weight is approximately the product of the surface area, the required

part thickness, and the density of the solid polymer. The mold is closed, heated,

and rotated on two axes. When sufﬁcient time has elapsed for melting to take

place (determined by solution of the heat transfer problem), the mold is cooled,

the polymer solidiﬁed (another heat transfer problem), and the mold is opened

to remove the part.

A 55-gal barrel, with an integral lid, may be manufactured this way; the solid

that bridges the lid to the barrel is cut, allowing the lid to be separated and

subsequently snapped onto the barrel.

PART V. Cyclic Process: Quiescent

16 CASTING

Casting is a quiescent process used to encapsulate nonpolymer objects. This is

often used for objets d’art, such as water-white paperweights with butterﬂies,

coins, etc., enclosed within. The industrial object studied by Ramanathan

was

REFERENCES 1033

the epoxy encapsulation of small transformers, such that the electrical contacts

were exposed. He found that this process, which lasted about 30 min, could be

monitored by dynamic dielectrometry. This method followed the resistance, ca-

pacitance, and dissipation function of the epoxy resin as it cured, utilizing a

120-Hz AC source. This technique may be used to increase the rate of production

by replacing ﬁxed time schedules such as ‘‘cure for t minutes,’’ with curing

times based on objectively measured data.

17 CONCLUSION

Nearly one-half of this brief discourse on plastic parts processing presented the

underlying principles. The remaining summaries of the individual processes are,

at best, ‘‘introductions’’; each has had several entire books written on them. One

should consult the following Bibliography for sources of further information.

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1034 PLASTIC PARTS PROCESSING II

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