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Thermoforming triangular troughs.

INTRODUCTION

The thermoforming process takes uniformly thin polymer sheets and forms them into usable products. The modern thermoforming process has four phases: free forming, plug-assisted forming, constrained forming, and straight thermoforming (1) in which plug assist is never used. Not every phase is required for all thermoforming products. A plug-assisted part goes through a free forming, plug-assisted forming, and constrained-forming series of phases. While a straight thermoformed part consists of the free-forming phase.

Free forming takes place first. Here, a uniformly thin polymer sheet is drawn through a slot. The sheet is initially at thickness h0 with an infinite radius of curvature R. Figure 1 describes the problem geometry. Appendix A explains the geometry more specifically.

[FIGURE 1 OMITTED]

Analysis is restricted to Newtonian fluids, and so our results also apply to nearly Newtonian melts. The triangular trough formed by a prismatic mold is the only shape considered in this analysis. This arises whenever we need to thermoform a straight edge into rigid packaging.

Figure 1 describes the thin sheet behavior as it forms a cylinder. Fig. 2 describes the initial unformed polymer sheet, Fig. 3 depicts the mold geometry, and Fig. 4 shows the variables of the constrained forming case.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

Table 1 summarizes all dimensional variables used in the analysis. Table 2 presents the resulting dimensionless variables.

LITERATURE REVIEW

This problem of thermoforming wedges bears looking into, since there has not been any significant research done on the topic. A review of the relevant literature is discussed here and summarized in Table 3 (1-12).

Relevant literature on thermoforming was taken into account, but the important literature was the mathematical analysis of the thermoforming problem described by Rosenzweig et al. (2) in 1979. First, he looked into the problem of thermoforming cones and truncated cones. Later (3), Rosenszweig returned to the cones problem in 1983 and went on to look at the problem of triangular troughs and truncated triangular troughs. Williams (4) had looked into this issue of triangular troughs in 1970. Neither came up with an analytical solution. Rosenzwieg (3) presents numerical data for a triangular trough, but gives up on the analytical solution. This article strives to further the analytical work of Williams and Rosenzweig, in the manner of Kershner and Giacomin (1), who also derived their motivation from the Williams article of 1970.

MOTIVATION

Thermoforming Is an often used industrial process. It is frequently employed to create hollow packaging, such as reusable food containers. These applications require a quick, cost-effective process. Tooling costs for thermo forming molds are significantly less than those for injection molding (5). Despite being cost-effective, tolerances are still important in these applications. Thermoforming does not have the tight tolerances that injection molding does, and the thickness distribution of part walls is not well understood. Most thermoforming processes are based on previous experience and empirical knowledge, The literature review for this article did not rind significant findings of analysis of the thermoforming process for rectangular prismatic containers (hollow triangular troughs). After the success of thermoforming cones analysis by Kershner and Giacomin [1], the problem of troughs was investigated.

The thermoforming industry needs more analytical methods. Since most of the knowledge is empirical, there is usually a trial-and-error element in the start-up process of a new operation. With appropriate research, this could be avoided, resulting in less waste and a quicker design-to-product timeline. If the thickness distribution could be calculated directly based on operating parameters, much time and effort could be saved, and hence the motivation for this article.

This work then has several overreaching goals. The first goal is to further the work of previous authors in the context of an analytical solution. The end result desired is an analysis of sheet thickness based on speed, which is adimensional, and can be applied to any thermoformed triangular trough. Finally, these results are to be collected into a series of worked examples that can be followed by any practitioner.

BACKGROUND

As stated in the Introduction, thermoforming begins with a uniformly thin polymer sheet. This sheet is heated up in an oven (usually a separate piece of equipment) and then brought over the mold. The sheet is then brought in contact with the mold. Air or pressure assist is applied to form the sheet to the mold. A plug is sometimes employed to assist in an even thickness distribution.

As the melt progresses through the three stages of thermoforming (free forming, plug-assisted forming, and constrained forming) or in straight thermoforming, we will to be able to measure the thickness distribution in the walls. Each of these stages is treated separately in this research. The wall thickness variability of the final-formed part is the main concern with thermoformed parts (5).
TABLE 1. Dimensional variables.

Name Symbol

Initial sheet thickness [h.sub.o]

Radius of curvature R

Triangular trough height H

Incline angle of triangular [beta]
trough

Slit half width [r.sub.o] [equivalent to] H/tan[beta]

Initial sheet width 2[r.sub.o]

Time t

Final thickness h

Contact angle of sheet's [empty set]
edge
Contact length Z

Pressure drop [DELTA]P =[P.sub.i] - [P.sub.o]

Inner pressure [P.sub.i]

Outer pressure [P.sub.o]

Finished part height H

Free-forming interval [t.sub.[empty set]]

Constrained-forming interval [t.sub.[kappa]]

Thickness during constrained [h.sub.[kappa]]
forming


The thermoforming process can be in line with the extrusion process, with the fresh sheets of polmer coming straight to the mold to be formed as they are made, Thermoforming can be done with a one-sided mold and a two-sided mold. This analysis focuses on the shape of the melt as it approaches the bottom of a mold, but can be applied to an air-assisted melt molded upward, since the gravity is neglected.

THERMOFORMING MECHANICS

We use the thin membrane approach for bubble inflation of Baird and Collias [6] for the thermoforming mechanics. Figure 1 describes our moving cylindrical coordinate system, centered inside the trough. Figure 2 describes the initial melt behavior. After applying our physical intuition, we assume constant density, and evaluation the continuity of equation. This can be integrated to find the velocity in the r-direction (evaluated at r = R) to be

[[upsilon].sub.r] = [[absolute value of [R.sup.R]]/r]. (1)
TABLE 2. Dimensionless variables and groups.

Name Symbol Definition

Thickness T h/[h.sub.0]

Radius of curvature [rho] R/[r.sub.0]

Sheet shape [[alpha].sub.0] [h.sub.0]/[r.sub.0]

Free-forming time [theta] [DELTA]Pt/[mu]

Constrained-forming [THETA] [DELTA]Pt/[mu]
time

Melt radius of [[rho].sub.[kappa]] [R.sub.[kappa]]/[r.sub.o]
curvature when
constrained forming
begins

Triangular trough apex [[rho].sub.f] R/[r.sub.o]
sharpness

Free-forming interval [[theta].sub.[phi]] [DELTA]Pt/[mu]

Constrained-forming [[THETA].sub.[kappa]] [DELTA]Pt/[mu]
interval

Total forming time [PI] [[theta].sub.[phi]] +
 [[THETA].sub.[kappa]]

Draw ratio l H/[2r.sub.0] =
 tan[beta]/2

Final melt contact [Z.sub.[kappa]f] [Z.sub.kf]/[r.sub.o]
length

Sheet uniformity [gamma] h([Z.sub.[kappa]f])/
 h([Z.sub.[kappa]] =0)

Plug shape factor [[alpha].sub.[pi]] [R.sub.[pi]]/[r.sub.0]

Melt proximity to [omega] r/[r.sub.o]
centerline

Stroke [sigma] [d.sub.[pi]]/H

Stress [SIGMA] t/[DELTA]P

TABLE 3. Previous work [free forming ([phi]), plug-assisted forming
([pi]), constrained forming (k); liquid ([lambda]), solid ([sigma]).

 Truncated
 Truncated Triangular triangular
 Cylinder Cone cone trough trough

Hart- Smith
and Crisp,
1967

Sheryshev et X
al., 1969

Williams, X X
1970

Williams, X X
1970

Tadmor and X
Gogos, 1979

Rosenzweig, X X
et al.,
1979

Rosenzweig, X X
1983

Rosenzweig,
1983

Throne, X
1979

Pearson, X X X X
1985

Allard et X X
al., 1986

Baird and X X
Collias,
1998

Baird and
Collias,
1998

Osswald and X
Hernandez-
Ortiz, 2006

Kershner and X X
Giacomin,
2007

This X
article

 Tensile Constitutive Nonuniform
 Phases stresses Geometry behavior thickness

Hart- Smith [empty set] X X [sigma] X
and Crisp,
1967

Sheryshev et [kappa] X
al., 1969

Williams, [empty set] X X [sigma]
1970 [pi]

Williams, [kappa], X X [sigma] X
1970 [pi]

Tadmor and [kappa] X
Gogos, 1979

Rosenzweig, [kappa] X
et al.,
1979

Rosenzweig, [kappa] X
1983

Rosenzweig, [kappa] X [sigma]
1983

Throne, [phi], X X [sigma]
1979 [kappa]

Pearson, [kappa] X X [lambda]
1985

Allard et [empty set] X [lambda]
al., 1986 [kappa]

Baird and [kappa] X [lambda]
Collias,
1998

Baird and [kappa] X
Collias,
1998

Osswald and [kappa]
Hernandez-
Ortiz, 2006

Kershner and [empty set] X X [lambda]
Giacomin, [pi]
2007

This [empty set]
article [pi] X X [lambda]

 Speed Reference

Hart- Smith and Crisp, 1967 [7]

Sheryshev et al., 1969 [8]

Williams, 1970 [4]

Williams, 1970 [4]

Tadmor and Gogos, 1979 [9]

Rosenzweig, et al., 1979 [2]

Rosenzweig, 1983 [3]

Rosenzweig, 1983 [3]

Throne, 1979 [10]

Pearson, 1985 [11]

Allard et al., 1986 [12]

Baird and Collias, 1998 [6]

Baird and Collias, 1998 X [6]

Osswald and Hernandez- Ortiz, 2006 X [5]

Kershner and Giacomin, 2007 [1]

This article


The momentum equation in the r-direction is then found, and the rr-component of the stress tensor is evaluated, which allows us to find

[DELTA]P = [[integral].sub.R.sup.[R+h]][[[[tau].sub.[theta][theta]] - [[tau].sub.rr]]/r]dr. (2)

For thin films, we can assume h/ho [much less than] 1, so the argument in the integral in (2) is nearly constant and we can apply the thin film approximation from Bird et al. [13], who first used this in tackling a growing viscoelastic bubble problem. Appendix B discusses this further.

[DELTA]P = ([[tau].sub.[theta][theta]] - [[tau].sub.rr])[h/R]. (3)

Introducing the Newtonian constitutive equation gives

[[tau].sub.[theta][theta]] = [[2[micro]A(t)]/[r.sup.2]] = [[2[micro]R[absolute value of R]]/[R.sup.2]] (4)

and

[[tau].sub.rr] = [[-[mu]A(t)]/[r.sup.2]] = [[[mu]R[absolute value of R]]/[R.sup.2]]. (5)

From these we can see that

[[tau].sub.[theta][theta]] = - 2[[tau].sub.rr], (6)

and we can also see that the inside stress (at r = R) exceeds the outside stress (at r = R+h) by a factor of

[[[tau].sub.[theta][theta].sup.0]/[[tau].sub.[theta][theta].sup.i]] = [[[tau].sub.rr.sup.0]/[[tau].sub.rr.sup.i]] = [(1 + [h/R]).sup.2]. (7)

In evaluating (3) at r = R for a Newtonian fluid, we substitute (4) and (5) into (2) to get

[DELTA]P = [[3[mu][absolute value of R]h]/2[R.sup.2]]. (8)

Adimensionalizing (8) with Table 3 gives

[absolute value of [[d[rho]]/[d[theta]]] = [[[rho].sup.2]/[T[[alpha].sub.0]]]. (9)

Which, in turn, describes the form the initial sheet takes on during thermoforming.

One can also solve this problem without the thin film approximation, and Appendix B shows that this does not yield an analytical solution for [theta]([rho]) either. This general result can, however, be expanded in a Taylor series about infinite R. Appendix C evaluates this, which applies shortly after the thermoforming begins. This solution would apply to films of any thickness, such as those for which h/[h.sub.0] [much less than] 1 does not hold true.

We define a Reynold's number for thermoforming as

Re = [[2[r.sub.0]H[[rho].sub.d][DELTA]P]/[theta]], (10)

where [[rho].sub.d] is the density of the material, and not the adimensional radius of curvature. We can then define our Poiseuille number as

Ps = [[[[rho].sub.d]gH[theta]]/[[DELTA]P]], (11)

where again [[rho].sub.d] is the density of the material. Here, g is the gravitational force on the material. For (10) and (11), the rest of the variables are found in Tables 2 and 3.

FREE-FORMING GEOMETRY

For free forming, the sheet thickness depends on the radius of curvature as

T = [1/[2[rho]arcsin (1/[rho])]], (12)

so that

[dT/[d[rho]]] = [1/[2[[rho].sup.3]arcsin [(1/[rho]).sup.2][square root of (1/[1 - [[rho].sup.2]]]]]][1/[2[[rho].sup.2]arcsin (1/[rho])]], (13)

which is true for both the flat (lenticular) and cylindrical (bulbous) cases. This analysis will concentrate on the lenticular case, which is more relevant to the triangular trough problem.

The contact angle of the edge of the sheet is evaluated from the slope of the melt's edge at [r.sub.0]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The free forming bubble height can be observed by the practitioner. Figure 1 shows this as a. The dimensionless bubble height is then

[H.sub.b] = [rho] - [absolute value of [square root of [[[rho].sup.2] - 1)]; H[greater than or equal to]1, (15)

which is true for the lenticular shape and

[H.sub.b] = [rho] + [absolute value of [square root of [[[rho].sup.2] - 1]; H[less than or equal to]1, (16)

which is true for the bulbous case.

FREE-FORMING RESULTS

Substituting (12) into (9) gives us

|[[d[rho]]/[d[theta]]]| = [[2[[rho].sup.3] arcsin (1/[rho])]/[[alpha].sub.0]] (17)

which may be rearranged to give

|[[d[rho]]/[d([theta]/[[alpha].sub.0])]]| = 2[[rho].sup.3] arcsin (1/[rho]) (18)

or

([theta]/[[alpha].sub.0]) = [[integral].sub.[infinity].sup.[rho]][[d[rho]]/[2[[rho].sup.3]arcsin (1/[rho])]]; [infinity][greater than or equal to][rho][greater than or equal to]1. (19)

which has no analytical solution. Appendix D provides a plot the argument of (19) in log--log coordinates versus [rho], and this plot reveals a straight line of slope 2 for most of the plot. Hence

[[theta]/[[alpha].sub.0]] [congruent to] [[[rho].sup.-2]/2] (20)

for [rho] [much greater than] 2, which means that the practitioner can use this equation for the dimensionless radius of curvature as it descends from infinity all the way down to 2. Appendix D explains this concept clearly. Equation 20 then rear-ranges to give us the free-forming time interval as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Also, the limits of the argument of the integral in (19) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

The dimensionless hoop stress can be found by adimensionalizing (4), and using [[tau].sub.[[theta][theta]] with the definition of stress in Table 2, we get

[SIGMA] = [3/[4[rho]]]|[[d[rho]]/[d[theta]]]|, (24)

so that for free forming

[SIGMA] = [[8[[rho].sup.2] arcsin (1/[rho])]/[3[[alpha].sub.0]]]. (25)

CONSTRAINED-FORMING GEOMETRY

In the constrained case, the melt forms onto the prismatic mold. Constrained forming begins as soon as the melt touches the mold side, which is depicted in Fig. 4.

Using the mold geometry, we can find an equation for the radius of curvature in terms of mold dimensions

[rho] = [[[r.sub.0] tan [beta] - Z sin [beta]]/[[r.sub.0] tan [beta] sin [beta]]], (26)

and we know that constrained forming begins when Z = 0, so that we can define

[[rho].sub.[kappa]][equivalent to][1/[sin[beta]]]. (27)

Substituting this into (21) gives

[[theta].sub.[empty set]] = [[2[[alpha].sub.0]]/[(sin[beta]).sup.2]]. (28)

The sheet thickness for the constrained case depends on the radius of curvature as

T = [1/[[[rho]/[tan [beta]]] + 2[rho] arcsin (1/[rho])]], (29)

so that

[[dT]/[d[rho]]] = [[[[-2]/[[rho][square root of [1 - (1/[[rho].sup.2])] + 2arcsin (1/[rho]) + [1/[tan [beta]]]]/[(2[rho]arcsin (1/[rho]) + [rho]/[tan [beta]]).sup.2]]. (30)

The constrained melt thickness is also related to the contact length (Z in Fig. 1), as defined by Rosenzweig (3),

T = [T.sub.[kappa]][([H - [Z.sub.[kappa]] sin [beta]]/H).sup.[[1/[beta]]tan [beta] - 1]]] (31)

and from geometry, the constrained melt thickness can be compared with the initial sheet thickness by

[Wzh.sub.0] = [Wh.sub.k](z + 2R arcsin (L/[2R])), (32)

which can be adimensionalized to

[T.sub.0] = [T.sub.[kappa]](1 + [[2R]/[Z.sub.[kappa]]] arcsin (1/[rho])). (33)

Combining (31) with (33) gives the thickness profile as

T = [[[(1 - [Z.sub.[kappa][florin]] cos [beta]).sup.[1/[beta]]tan [beta] - 1]]/(1 + [[2[rho]]/[Z.sub.[kappa][florin]]] arcsin (1/[rho]))]] (34)

since [T.sub.0] [equivalent to] 1.

CONSTRAINED-FORMING RESULTS

Inserting (29) into (9) gives us

|[[d[rho]]/[d[theta]]]| = [[[rho].sup.3]/[[alpha].sub.0]](1/[tan [beta]] + 2 arcsin (1/[rho])), (35)

which can be rearranged as

|[d[rho]]/[d([theta]/[[alpha].sub.0]])| = [[rho].sup.3] (1/tan[beta] + 2 arcsin (1/[rho])) (36)

or

([theta]/[[alpha].sub.0]) = [[integral].sub.[infinity].sup.[rho]] [[d[rho]]/[[rho].sup.3] (1/tan[beta] + 2 arcsin (1/[rho]))]]; [infinity] [greater than or equal to] [rho] [greater than or equal to] 1. (37)

Equation (37) has no general analytical solution, out Appendix D provides an analytic solution for large radii of curvature. For constrained forming, the equation to be used depends on the value of [beta] (the corner angle of the prismatic mold). Using the large radius of curvature method of Appendix D, we get a different equation for each value of [deta]. Integrating (37) numerically yields both Table 4 and Fig. 5, a universal chart for the constrained forming triangular troughs.
TABLE 4. [theta]/[[alpha].sub.0] values for (37) for various
values of [beta].

 [beta]
[rho] 10 20 30 40 50

1 0.069669 0.11886 0.15857 0.1939 0.22807
2 0.019697 0.03662 0.052312 0.067919 0.084573
3 0.009083 0.017423 0.025611 0.034199 0.043858
4 0.005205 0.010152 0.015167 0.020595 0.026904
5 0.003368 0.00664 0.010024 0.013764 0.018212
10 0.000861 0.001736 0.002682 0.003778 0.005154
20 0.000218 0.000444 0.000695 0.000994 0.001381
50 3.51E-05 7.21 E-05 0.000114 0.000164 0.000231

 [beta]
[rho] 60 70 80

1 0.26392 0.3052 0.35933
2 0.10378 0.12816 0.164
3 0.05564 0.07161 0.097134
4 0.034893 0.046228 0.065541
5 0.023995 0.032488 0.047706
10 0.007067 0.010149 0.016575
20 0.001944 0.002912 0.005196
50 0.000331 0.000512 0.000987


[FIGURE 5 OMITTED]

Initial calculations for Table 4 and Fig. 5 led to inconsistencies. These calculations were done using the Romberg integration technique, In these calculations, it seemed that at values of [deta] around 45[degrees] and at larger values of [rho] caused convergence problems of our solution. This prompted a closer look at those values gives us Table 5 and Fig. 6. Figure 7 then depicts Fig. 6 in log-log coordinates.
TABLE 5. Close up of [theta]/[[alpha].sub.0] values for (37) for various
values of [beta].

 [beta]
[rho] 30 35 40 41 42 43 44

1 0.15857 0.17656 0.19390 0.19732 0.20073 0.20414 0.20755
1.25 0.11377 0.12807 0.14209 0.14488 0.14768 0.15048 0.15328
1.5 0.08488 0.09632 0.10770 0.10998 0.11228 0.11458 0.11689
1.75 0.06567 0.07501 0.08439 0.08628 0.08819 0.09011 0.09204
2 0.05231 0.06006 0.06792 0.06952 0.07113 0.07275 0.07438
2.25 0.04266 0.04919 0.05586 0.05723 0.05860 0.05999 0.06140
2.5 0.03545 0.04103 0.04677 0.04795 0.04914 0.05034 0.05156
2.75 0.02993 0.03475 0.03975 0.04077 0.04181 0.04286 0.04393
3 0.02561 0.02982 0.03420 0.03510 0.03602 0.03695 0.03789
4 0.01517 0.01781 0.02060 0.02118 0.02177 0.02237 0.02298
5 0.01002 0.01183 0.01376 0.01417 0.01458 0.01500 0.01543
10 0.00268 0.00320 0.00378 0.00390 0.00403 0.00415 0.00428
15 0.00122 0.00147 0.00174 0.00179 0.00185 0.00192 0.00198

 [beta]
[rho] 45 46 47 48 49 50

1 0.21095 0.21436 0.21777 0.22119 0.22462 0.22807
1.25 0.15610 0.15892 0.16175 0.(6460 0.16747 0.17035
1.5 0.11921 0.12155 0.12390 0.12628 0.12867 0.13108
1.75 0.09399 0.09595 0.09793 0.09993 0.10195 0.10399
2 0.07604 0.07770 0.07939 0.08110 0.08282 0.08457
2.25 0.06282 0.06425 0.06570 0.06718 0.06867 0.07019
2.5 0.05279 0.05404 0.05531 0.05659 0.05789 0.05922
2.75 0.04501 0.04611 0.04722 0.04835 0.04950 0.05067
3 0.03884 0.03981 0.04080 0.04180 0.04282 0.04386
4 0.02360 0.02423 0.02488 0.02554 0.02622 0.02690
5 0.01587 0.01632 0.01677 0.01724 0.01772 0.01821
10 0.00442 0.00456 0.00470 0.00485 0.00500 0.00515
15 0.00204 0.00211 0.00218 0.00225 0.00232 0.00240

 [beta]
[rho] 55 60

1 0.24559 0.26392
1.25 0.185/6 0.20087
1.5 0.14356 0.15697
1.75 0.11462 0.12617
2 0.09373 0.10378
2.25 0.07816 0.08698
2.5 0.06622 0.07403
2.75 0.05687 0.06383
3 0.04939 0.05564
4 0.03062 0.03489
5 0.02088 0.02400
10 0.00602 0.00707
15 0.00282 0.00334


[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

Again the Romberg technique had issues solving for values of [beta] around 45[degrees] and at larger values of [rho]. This prompted a switch to an alternate numerical integration technique, the adaptive Gauss-Kronrod quadrature. This approach seemed more robust, had no convergence issues, and generated the values found here in Tables 4 and 5. and their resulting plots.

Because of the numerical integration technique errors, further analysis is required. From Eq. 36 we see that

[[d[rho]]/[d([theta]/[[alpha].sub.0])]] = [[rho].sup.3] ([1/[tan[beta]]] + 2 arcsin (1/[rho])); [[d[rho]]/[d([theta]/[[alpha].sub.0]]] [greater than or equal to] 0, (38)

but

[[d[rho]]/[d([theta]/[[alpha].sub.0]] = - [[rho].sup.3] ([1/[tan[beta]]] + 2 arcsin (1/[rho])); [[d[rho]]/[d([theta]/[[alpha].sub.0])]]] [less than or equal to] 0. (39)

If |x| = y, then y [greater than or equal to] 0 for all values of x. Equations 38 and 39 imply

[[d([theta]/[[alpha].sub.0])]/[d[rho]]] = [1/[[[rho].sup.3](1/tan[beta] + 2 arcsin (1/[rho]))]]; [[d[rho]]/[d ([theta]/[[[alpha].sub.0])] [greater than or equal to] 0 (40)

and

[[[d([theta]/[[alpha].sub.0])]/[d[rho]]] = [-1/[[rho].sup.3] ([1/[tan[beta]]] + 2 arcsin (1/[rho]))]]; [[d[rho]]/[d([theta]/[[alpha].sub.0)]] [greater than or equal to] 0, (41)

so that

[[theta]/[[alpha].sub.0]] = [[integral].sub.[infinity].sup.[rho]] [1/[[[rho].sup.3]([1/tan[beta]] + 2 arcsin (1/[rho]))]]; [[d[rho]]/[d([theta]/[[alpha].sub.0]][greater than or equal to] 0, (42)

but

[[theta]/[[alpha].sub.0]] [[rho].[integeral].[infinity]][-1/[[[rho].sup.3]]([1/tan[beta]] + 2 arcsin (1/[rho]))]]; [[d[rho]]/[d([theta]/[[alpha].sub.0]]] [less than or equal to] 0 (43)

and

[[theta]/[[alpha].sub.0]] = [[rho].[integral].[infinity]][1/[[[rho].sup.3](1/tan[beta] + 2 arcsin (1/[rho]))]]; [rho] [less than or equal to] [[rho].sub.0] (44)

TABLE 5. Close up of [theta]/[x.sub.0] values for (37) for various values of [beta]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (45)

This analysis shows us that a solution should always be possible, and positive, as shown in the tables and plots generated numerically.

Another close look shows the limits of the argument of the integral in (37) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (46)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (47)

In our preceding analysis, all of the integration begins at [rho] = 1. However, the practitioner wants to know the forming time for values 0 < [rho] [less than or equal to]1. As we note in Eq. 37, the integral is not defined for [rho] < 1. Using the Romberg integration technique in Maxima (the open-source version of Macsyma), we are not even able to estimate a solution at small values of [rho], and the same story happened with the adaptive Gauss-Kronrod quadrature. The limit of Eq. 37 as [rho] goes to zero is essentially zero, as division by infinity leads to a solution of zero.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (48)

The inability to solve for an analytical result for the constrained forming case had several mathematical foundations. First of all, Eq. 37 contains an arcsin (1/[rho]), which is undefined for values of 0 & lt; [rho] < 1, because it results in solutions containing only complex numbers. The equation is not defined in the real-number plane. There is a series solution to the arcsin (1/[rho]), but it is also not valid for 0 < [rho] < 1. The Romberg integration technique in Maxima, "quad" and "quadl," and "quadgk" functions in Matlab, the trapezoid rule, and other computational math methods all yielded these same results. In this case, as was found by Rosenzweig, the only estimated solution to this problem is a computer simulation. The proposed future work in Polyflow would be able to estimate Eq. 37 for us.

In addition, the practitioner knows from experience that achieving a very sharp point in a thermoformed triangular trough is difficult. In practice, we call this detailing. To quantify this, Table 3 defines the dimensionless triangular trough edge sharpness as [rho]f. The required thermoforming time to get that sharpness is then

[[THETA].sub.f] = |[THETA]([[rho].sub.f]) - [THETA]([[rho].sub.[kappa]])|, (49)

and the total manufacturing time required for the triangular trough is then

[[THETA].sub.T] = [[theta].sub.[empty set]] + [[THETA].sub.f] = [[theta].sub.[empty set]] + |[THETA]([[rho].sub.f]) - [THETA]([[rho].sub.k])|. (50)

We can find the stresses in the constrained melt following the same method that we used for free forming.

[summation] = [[4[[rho].sup.2]]/[3[[alpha].sub.0]]](1/[tan [beta]] + 2arcsin (1/[rho])). (51)

PLUG ASSIST

A plug is often used in thermoforming to even the wall thickness. Figure 8 describes the variables and plug-assist geometry.

[FIGURE 8 OMITTED]

This article has focused on unassisted thermoforming, which produces the least even wall thickness, and thus qualifies as the worst case scenario. The sheet thickness uniformity without plug assist is thus

[??][equivalent to][[h([z.sub.[kappa]f])/[h([z.sub.[kappa]] = 0)]] = [[T([Z.sub.[kappa]f])]/[T([Z.sub.[kappa]] = 0)]], (52)

where [Z.sub.kf] is the final contact length after the desired radius has been reached for the triangular trough's edge. From the geometry, we get

[Z.sub.[kappa]f] = tan [beta](1/[sin [beta]] - [[rho].sub.f]). (53)

Combining (52) with (34) gives the developing dimensionless sheet uniformity

[??] = [[(1 - [Z.sub.[kappa]f]cos [beta]).sup.[1/[beta]]tan [beta] - 1]/(1 + [[2[rho]]/[Z.sub.[kappa]f]]arcsin (1/[rho]))]. (54)

Substituting (53) into (54) gives

[??] = [[([[rho].sub.f]sin [beta]).sup.[1/[beta]]tan [beta] - 1]/(1 + [[2[[rho].sub.f]]/[(sec [beta] - [[rho].sub.f]tan [beta])]]arcsin (1/[[rho].sub.f]))], (55)

which is the final dimensionless uniformity for a sample without plug assist.

Equation 55 dictate's how severely plug assist is needed.

Using only the geometry, Williams [5] derives the following thickness profile for a thermoformed trough using plug assist

[T.sub.[pi]] = [1/[square root of [1 + [([d.sub.[pi]]/[[r.sub.0] - [R.sub.[pi]]]).sup.2]]]], (56)

and he then verified this experimentally. The right side of (56) adimensionalizes as

[T.sub.[pi]][equivalent to][1/[square root of [1 + [([2l[sigma]]/[1 - [[alpha].sub.[pi]]]).sup.2]]]], (57)

where [sigma] is the dimensionless plug displacement (practitioners call this the stroke), and we call [[alpha].sub.[pi]] as the plug shape factor. Because [omega], is the melt proximity to the centerline, does not arise in (57), the plug assist of the triangular trough is a homogenous deformation, with its uniform thickness given by [T.sub.[pi]]. In principle, the trough thickness problem could be nearly eliminated by equating r with the amount of plug assist

[T.sub.[pi]][approximately equal to][??] (58)

In practice, however, the plug normally runs into the mold before this amount of plug assist can be realized. To prevent this collision, the stroke must satisfy the following geometric inequality

[sigma]< [[alpha].sub.0]/2l[(1 - [[alpha].sub.[pi]])/[[alpha].sub.0] - 1]. (59)

Since, in practice

[d.sub.[pi]]/[h.sub.0][much greater than]1. (60)

That is, stroke vastly surpasses sheet thickness, so Eq. (59) simplifies to

[sigma] < 1 - [[alpha].sub.[pi]]/2l tan [beta]. (61)

Pressure

A plastics engineer wants to manufacture a triangular trough more efficiently. She knows the current part shape, but wants to cut down the processing time. To decrease the total forming time to 1 sec, how much total pressure must be applied? This part has an angle of [beta] = 70[degrees], a height of H = 0.5m, a dimensionless sharpness of [p.sub.f] = 10, and it is thermoformed from a polymer sheet of uniform thickness [h.sub.0] = 1.5 mm. Using Table 2, she can find [r.sub.0] [equivalent to] H/tan [beta] = 0.129, and from Table 3 we then find [[alpha].sub.0] [equivalent to] [h.sub.0]/[r.sub.0] = 0.012. From Table 4, the engineer knows that she has a constrained-forming time [[theta].sub.f] = 0.225. She also calculates a dimensionless free-forming time from (2/) of [[theta].sub.[phi]] = 2.4 X [10.sup.-4]. The total adimensional forming time is then [[theta].sup.T] = 0.22524. The polymer being used for the part is nearly Newtonian, and has a viscosity of [micro] = 3.10 X [10.sup.6] Pa.s. The required pressure drop is then [delta]P = [[theta].sub.T][mu]/t = 101.3 psi.

Process Speed and Melt Stress

An engineer wants to manufacture a triangular trough with an angle of [beta] = 60[degrees] and H = 0.5 m from a polymer sheet of uniform thickness /z0 = 1.5 mm. The polymer is nearly Newtonian with a viscosity of [micro] = 3.10 X [10.sup.6] Pa.s. The triangular trough apex is to be blunt, with a high radius of [R.sub.f] = 0.575 m. The external gage pressure is 91.3 psi, and the vacuum pressure is 14.1 psi. What is the total forming time, and what is the stress frozen into the trough's thin edge?

Using (27), we see that [[rho].sub.k] = 1.15 (the dimensionless radius of curvature) when constrained forming begins. Using Table 2, we find [r.sub.0] [equivalent to] H/tan [beta] = 0.289, and from Table 3 we then find [[alpha].sub.0] [equivalent to] [h.sub.0]/[r.sub.0], = 0.005. Substituting these values into (27) gives a free-forming time of [[theta].sub.[phi]] = 0.008. Since the final radius is known, a dimensionless radius can be found from [[phi].sub.f] [equivalent to] [R.sub.f]/[r.sub.0] = 1.99. This is nearly 2, and using Table 4, then gives us 0( = 0.10378, for a total of [[theta].sub.T] [congruent to] 0,1118.

To find the actual processing time, we first add up the pressures to get the pressure drop [delta]P = 105.4, and then we use Table 3 to get t = [[theta].sub.T][mu]/[delta]P = 0.475 s. The melt stress is assumed to be the final stress frozen into the thin edge of the triangular trough, which, from (51), is [summation] = 64.6 MPa.

Trough Sharpness

A plastics manufacturing engineer is designing a sharp triangular trough ([p.sub.f] [much less than] 1) with [beta] = 80[degrees] and H = 0.5 m from a nearly Newtonian ([micro]. = 3.10 X [10.sup.6] Pa.s) polymer sheet of uniform thickness [h.sup.0] = 1.0 mm. The manufacturing engineer applies an external gage pressure of 90 psi and a vacuum of 14.1 psi. The mold has an infinitely pointed tip (which the melt will, of course, never reach). A forming lime of 10 sec is used. What is the resulting apex radius?

[delta]P is found to be 104.1 psi by summing the applied and vacuum air pressures. From Table 3 we get [[THETA].sub.t] = APt/[micro] = 3.36 X [10.sup.-4]. Using (27) we find [[rho].sub.k] = 1/sin [beta] = 1.02. Using Table 2 we can see [[r.sub.0] [equivalent to] H/tan/[beta] = 0.088, and from Table 3 we can find [[alpha].sub.0] [equivalent to] [h.sub.0]/[r.sub.0] = 0.011. Using this value in (28) gives [[theta].sub.[phi]] = 2[alpha]0/[(sin [beta]).sub.2] = 0.023. Plugging this and [[THETA].sub.T] into (50) gives [[theta].sub.f] = |[[theta].sub.T] - [[theta].sub.[phi]],| = 0.023. To solve for the dimensionless radius of curvature, we must divide this by [[alpha].sub.0] which gives 2.09. This value is then looked up for the angle of 80[degrees] in Fig. 4, but it is not found. The approximation does not hold for this value, since ([p.sub.f] [much less than] 1). At [[rho].sub.f]t = 1 we see that [theta]/[alpha] = 0 in the plot and chart, which means that the Taylor expansion should be used. Using just the first term we get

[theta] [congruent to] [[alpha].sub.0] tan [beta] log [rho]/8, (62)

and rearranging gives

[rho] = exp (8[theta]/[[alpha.sub.0] tan [beta]). (63)

Using (63) gives us [rho] = 19.1. Using Table 3 we then get [R.sub.f] = [p.sub.f][r.sub.0] = 1.68 mm.

Stress Calculations Based on Forming Time

The practitioner would like the residual stresses in a part to be as low as possible. This is usually done by raising the temperature of the sheet, which lowers the visco-elastic properties. Our practitioner would like to know how much residual stress is in their part based on the forming time. By substituting Eq. 20 into Eq. 25, we can get a function of residual stresses in the free-forming melt based on processing speed, which gives

[summation] = 4/3 [theta] arcsin [square root of]2[theta]/[[alpha].sub.0] (64)

Using our part from Example 8.3, we know that [[THETA].sub.T] = 3.36 X [10.sup.-4] and [[alpha].sub.0] = 0.011, so we can calculate our dimensionless stress and get [summation] = 0.0064.

A similar process can be done for the constrained-forming case, but of course is different for each value of p, and so Tables 4 and 5 and Figs. 4 and 5 must be used.

CONCLUSIONS

This analysis applies the method of Kershner and Gia-comin (1) to thermoforming triangular troughs and focuses on processing speed. Though triangular troughs are second only to thermoforming cones in simplicity, the troughts problem yields no analytical solutions for forming time. Instead, we combine a series expansion with a numerical solution to provide practitioners with a way to estimate thermoforming time, trough edge sharpness, and frozen-in stresses. These solutions are estimated analytically for the free-forming condition; however, the constrained-forming condition does not supply sufficient analytic solutions in the region of concern. These parameters would have to be determined through numerical techniques are simulations and compare them to these analytical and numerical integration results.

ACKNOWLEDGEMENTS

The Place on Corporation is recognized for its sustaining sponsorship of the Rheology Research Center. The author thanks Ms. Melissa Kershner of the Placon Corporation for her help, Mr. Marty Stephenson of the Pall Corp. for his proofreading the suggestions, Professor Bird of the University of Wisconsin-Madison, Chemical Engineering Department for his invaluable proofreading and guidance, and Mr. Erick Butzlaff for his math expertise.

APPENDIX A: GEOMETRY

Figure 1 describes the geometry of the thermoformed triangular trough and Fig. 2 the initial sheet geometry. We employ a cylindrical coordinate system centered on the trough's midplane. The origin starts above the trough, then approaches the trough, and then enters the trough.

Some basic geometric relations follow. The dimensional radius of curvature is defined as

R [equivalent to] H - Z sin [beta]/tan [beta] sin [beta], (A1)

where H is the overall trough depth and can be found as

H = L/2 tan [beta] = [r.sub.0] tan [beta]. (A2)

Z is defined as the mold length that the melt has touched in constrained forming, which can be found by solving (A1) for Z and substituting into (A2) for H. The resulting definition is

Z = R sin [beta] tan [beta] + [r.sub.0] tan [beta]/sin [beta]. (A3))

These definitions are used throughout this article.

APPENDIX B: THIN FILM APPROXIMATION

The thin film approximation simplifies the analysis of the equation of motion. Eliminating the stresses in (2)

[DELTA]P = [[integral].sub.R.sup.(R + h)][[[tau].sub.[theta][theta] - [[tau].sub.rr]/r]dr = [[integral].sub.R.sup.(R + h)][ - 3[[tau].sub.rr]/r]dr = [[integral].sub.R.sup.(R + h)][ - 3[micro]A(t)/[r.sup.2]/r]dr, (B1)

which simplifies to

[DELTA]P = [[integral].sub.R.sup.(R + h)][3[micro]A(t)/[r.sup.3]dr = [3[micro]A(t)/2[r.sup.2]|.sub.R.sup.(R + h)] = 3[micro].sub.R.sup.[??]/2[1/R - R/[(R + h).sup.2]. (B2)

Using Table 3, (B2) can be adimensionalized to

d[rho]/d[theta] = 1/1/[rho] - [rho]/[([rho] + [[alpha].sub.0]T).sup.2], (B3)

where T is the film thickness as it depends on radius of curvature. Substituting (12) into (B3), we have the solution of a free-forming film

d[rho]/d[theta] = [rho]/1 - 1/[(1 + [[alpha].sub.0]/2[[rho].sup.2][sin.sup. - 1](1/[rho])).sup.2] (B4)

For constrained forming, we insert (29) into (B3) and find

d[rho]/d[theta] = 1/1/[rho] - [rho]/[([rho] + [[alpha].sub.0]/[rho]/tan [beta] + 2[rho] arcsin (1/[rho])).sup.2]. (B5)

Though this is more accurate than (8) for free forming, (B4) and (B5) do not lead to differential equations having analytical solutions.

APPENDIX C: THICK OR THIN FILM--GENERAL SOLUTION

Equation B4 cannot be solved analytically. However, a truncated Taylor series expansion about infinite radius of curvature can be used to approximate the solution. Upon taking the right side of (B4), we take the inverse and expand into a Taylor series to find

d[theta] = - [[alpha].sub.0]/[[rho].sup.2] - [[alpha].sub.0.sup.2]/4[[rho].sup.3] + [[alpha].sub.0]/6[[rho].sup.4] + [[alpha].sub.0.sup.2]12[[rho].sup.5] + 17[[alpha].sub.0]/360[[rho].sup.6] + [[alpha].sub.0.sup.2]/60[[rho].sup.7] + 367[[alpha].sub.0]/15120[[rho].sup.8] + 31[[alpha].sub.0.sup.2]/37880[[rho].sup.9] + ... (C1)

which can be integrated to find

[theta] = [[alpha].sub.0]/[rho]] + [[[alpha].sub.0.sup.2]/[8[[rho].sup.2]] - [[[alpha].sub.0]/[18[[rho].sup.3]] - [[[alpha].sub.0.sup.2]/[48[[rho].sup.4]]] - [[17[[alpha].sub.0]]/[1800[[rho].sup.5]]] - [[[alpha].sub.0.sup.2]/[360[[rho.sup.6]]] - [[[367[[alpha].sub.0]/[104850[[rho].sup.7]] - [[[31[[alpha].sub.0.sup.2]]/[30240[[rho].sup.8]]] - ... (C2)

We repeat this for constrained forming, with the added complexity of the [beta] dependence, the angle of the side of the prismatic mold. The inverse of the right side of (B5) then expands in a Taylor series, again about infinite radius of curvature, to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (C3)

which integrates to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (C4)

APPENDIX D: LARGE RADIUS OF CURVATURE METHOD-FOR FREE FORMING

In general, an analytical solution cannot be found for [[theta]([rho])]. However, for large radii of curvature, we proceed as follows. We first define

g(x) [equivalent to] [1/[2[[rho].sup.3] arcsin (1/[rho])] (D1)

as the argument of the integral in (19). Substituting [rho] = [e.sup.x] into (D1) gives

logg = log ([e.sup.-3x]/[2 arcsin ([e.sup.-x])]). (D2)

Plotting (D2) gives Fig. 9. A straight line of slope -2 is found, except near the g-axis. One can describe this line with (20).

[FIGURE 9 OMITTED]

REFERENCES

(1.) MA. Kershner and A.J. Giacomin, J. Plast. TechnoL, 3, 1 (2007).

(2.) N. Rosenzweig, M. Narkis, and Z. Tadmor, Polym. Eng. Sci., 19, 946 (1979).

(3.) N. Rosenzweig, SPE Tech. Pap., 29, 478 (1983).

(4.) J.G. Williams, J. Strain Anal, 5, 49 (1970).

(5.) T.A. Osswald and J.P. Hernandez-Ortiz, Polymer Processing - Modeling and Simulation, Hanser, Munich (2006).

(6.) D.G. Baird and D.I. Collias, Polymer Processing Principles and Design, Butterworth-Heinemann, Boston (1995); Wiley, New York (1998).

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(9.) Z. Tadmor and C.G. Gogos, Principles of Polymer Processing, Wiley, New York (1979).

(10.) J.L. Throne, Plastics Process Engineering, Marcel Dekker, New York (1979).

(11.) J.R.A. Pearson, Mechanics of Polymer Processing, Elsevier Applied Science Publishers Ltd., London (1985).

(12.) R. Allard, J.M. Charrier, A. Ghosh, M. Marangou, M.E. Ryan, S. Shrivastava, and R.J. Wu, Polym. Eng., 6, 363 (1986).

(13.) R.B. Bird, R.C. Armstrong, and O. Hassager, Eds., Dynamics of Polymeric Liquids, Vol. 1; Fluid Mechanics, 2nd ed., Wiley, New York (1987).

Katie Lynn Lieg, A. Jeffrey Giacomin

Mechanical Engineering Department, Polymer Engineering Center, Rheology Research Center at the University of Wisconsin-Madison, Madison, Wisconsin 53706

Correspondence to: Katie Lynn Lieg; e-mail: kllieg@wisc.edu

This work partially constitutes the Masters of Science Thesis of Katie Lynn Lieg at the University of Wisconsin-Madison, in the Mechanical Engineering Department. Parts of this research were presented at ANTEC 2008 and the Society of Rheology Annual Meeting in 2007.

Contract grant sponsor: Plastic Ingenuity, Inc., Cross Plains; Wisconsin and Placon Corporation of Madison, Wisconsin.

DOI 10. 1002/pen.21239

Published online in Wiley InterScience (www.interscience.wiley.com).
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