Thermoforming Cones

Transcription

Thermoforming Cones

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© 2007 Carl Hanser Verlag, München
Wissenschaftlicher
Arbeitskreis der
UniversitätsProfessoren der
Kunststofftechnik
Zeitschrift Kunststofftechnik
Journal of Plastics Technology
archivierte, rezensierte Internetzeitschrift des Wissenschaftlichen Arbeitskreises Kunststofftechnik (WAK)
archival, reviewed online Journal of the Scientific Alliance of Polymer Technology
www.kunststofftech.com; www.plasticseng.com
eingereicht/handed in: 22.10.2006
angenommen/accepted: 26.01.2007
Melissa A. Kershner, Prof. A. Jeffrey Giacomin,
Mechanical Engineering Department, Polymer Engineering Center and Rheology
Research Center, University of Wisconsin, Madison, Wisconsin
Thermoforming Cones
This new analysis for thermoforming cones focuses on the manufacturing process speed. Specifically,
we’ve distinguished between what happens before and after (free versus constrained forming) the
melt touches the conical mold. We derive analytical solutions for the time required for both cases, and
sum them to get the total forming time. We restrict our analysis to the fabrication of cones, the simplest relevant problem in commercial thermoforming. We further confine our analysis to the Newtonian case, adimensionalizing our results at every step. For free forming, one dimensionless group arises, the geometric shape factor (α0), and for constrained, two such factors arise (α and sec β). We also calculate the stresses in the deforming melt, since these govern the residual stresses in the thermoformed part. We then derive an expression for wall uniformity; we find that it just depends on the
mold geometry. Finally, we attack the problem of plug assist, deriving an expression for the improvement in wall uniformity achieved through plug assist. Our analytical solutions provide benchmarks for
numerical analysts to test their code accuracy. Four worked examples illustrate how to reduce the results to engineering practice.
Autor/author
Melissa A. Kershner, Prof. A. Jeffrey Giacomin,
Mechanical Engineering Department,
Polymer Engineering Center and
Rheology Research Center,
University of Wisconsin, Madison,
WI 53706-1608
USA
© Carl Hanser Verlag
E-Mail-Adresse: [email protected]
Webseite: rrc.engr.wisc.edu
Tel.: +1 / 608 / 262-7473
Fax: +1 / 608 / 262-7473
Zeitschrift Kunststofftechnik/Journal of Plastics Technology 3 (2007) 1
M.A. Kershner, A. J. Giacomin
Thermoforming Cones
Thermoforming Cones
M.A. Kershner, A.J. Giacomin,
Mechanical Engineering Department, Polymer Engineering Center and
Rheology Research Center, University of Wisconsin, Madison, Wisconsin
This new analysis for thermoforming cones focuses on the manufacturing process speed. Specifically, we’ve distinguished between what happens before and
after (free versus constrained forming) the melt touches the conical mold. We
derive analytical solutions for the time required for both cases, and sum them to
get the total forming time. We restrict our analysis to the Newtonian fabrication
of cones, the simplest relevant problem in commercial thermoforming.
1 INTRODUCTION
Thermoforming is the mass production of thin non-hollow products from uniformly thick flat sheets. We divide modern commercial thermoforming into four
phases. Firstly, in free forming ( φ ), a thin untouched sheet deforms under an
applied air pressure. In plug assisted forming ( π ), a carefully designed solid
shape then touches and stretches some of the sheet to reshape it. The parts
touching the plug do not stretch. Once released from this plug, the reshaped
sheet again deforms freely till it contacts the mold. The material not yet touching the mold then continues to stretch till the mold is covered and we call this
constrained forming ( κ ). Sometimes, thermoforming is done without the plug
assist and this paper attacks this special case. Tadmor and Gogos [1] call this
straight thermoforming. We solve for both the mold covering speed and the
product thickness profile. Table 1 compares this paper with previous work.
We begin by modeling free forming, where a uniformly thin polymeric film is
formed from a thin flat disk inflated through a round hole into a growing thin
sphere. Williams [4] confirmed this spherical shape experimentally. These thin
spheres transition from lenticular, through hemispherical, to bulbous as illustrated in Figure 1. Figure 2 illustrates the initial condition, a flat disk of thickness h0 , and thus of infinite radius of curvature, R .
We restrict this analysis to Newtonian liquids, so we expect this work will apply
accurately to low molecular weight systems such as polyester which is commonly used to thermoform stiff clear packaging. Our analytical solutions also
provide benchmarks for numerical analysts to test their code accuracy. We further restrict our analysis to the fabrication of cones, the simplest product shape
in commercial thermoforming.
*
Corresponding author ([email protected]).
Zeitschrift Kunststofftechnik 3 (2007) 3
1
Sheryshev et
al. (1969)
X
Williams (1970)
X
Tadmor and
Gogos (1979)
X
Rosenzweig et
al. (1979)
X
Throne (1979)
Allard et al.
(1986)
X
X
X
Pearson (1985) X
X
X
X
X
X
φ ,π
X
X
σ
κ ,π
X
X
σ
X
[3]
X
[4]
[4]
X
X
[1]
κ
X
X
[5]
κ ,π
X
σ
X
[6]
κ
X
X
σ
X
[7]
φ ,κ
X
X
λ
X
[8]
λ
X
[9]
X
[10]
X
κ
X
Osswald and
Hernández –
Ortiz (2006)
X
κ
X
This Paper
X
Table 1:
[2]
κ
Baird and
Collias (1998)
X
Reference
σ
Speed
X
Uniform Thickness
Constitutive Behavior
X
κ
X
Williams (1970) X
Geometry
φ
Tensile Stresses
Hart-Smith and
Crisp (1967)
Phases
Truncated Wedge
Thermoforming Cones
Wedge
Truncated Cone
Cone
Cylinder
φ ,π ,κ
X
X
λ
X
X
Previous work [free forming ( φ ), plug assisted forming ( π ),
constrained forming ( κ ); liquid ( λ ), solid ( σ )]
2
Thermoforming Cones
Figure 1: Thin sphere transitioning
3
Thermoforming Cones
Figure 2: Initial undeformed disk (just before thermoforming)
Table 2 summarizes these and other dimensional variables. Furthermore, Table 3 defines the corresponding dimensionless variables, including thickness
(T), radius of curvature ( ρ ) and height ( H ) of the free forming bubble.
Name
Initial disk thickness
Bubble height
Radius of curvature
Slit radius
Time
Final thickness
Contact angle of sheet’s edge
Contact length
Pressure drop
Inner pressure
Outer pressure
Plug radius
Plug displacement
Cone apex sharpness
Table 2:
Symbol
h0
a
R
L
r0 ≡
tan β
t
h
φ
zk
Δ P ≡ Pi − Po
Pi
Po
Rπ
dπ
Rf
Finished part height
Hp
Free forming interval
Constrained forming interval
Thickness during constrained
forming
tφ
tκ
hκ
Dimensional variables
4
Name
Thickness
Radius of curvature
Bubble height
Free forming time
Constrained forming time
Thermoforming Cones
Symbol Definition
h / h0
T
ρ
R / r0
a / r0
H
ΔPt
θ
12 μ
ΔPt
Θ
12 μ
Free forming interval
θφ
ΔPt
12 μ
ρ = ρκ
Θκ
ΔPt
12 μ
Total forming time
Π
θφ + Θκ
Draw ratio11
l
Constrained forming interval
Cone height
Conical melt length
Thickness during constrained forming
Melt radius of curvature when constrained forming begins
Sheet uniformity
ρ = ρf
L tan β
=
2r0
2
2l
zk / r0
Zκ
Tκ
h / hκ
ρκ
Rκ / r0
ϒ
h( zκ f )
h( zκ = 0)
1 + cos β ⎛ sin 2 β ⎞
⎜
⎟
2α 0 ⎝ 2l cos β ⎠
Cone shape factor
α
Stress
Σ ij
Disk shape
α0
h0 / r0
Plug shape factor
απ
Rπ / r0
ρf
R f / r0
Centerline proximity
ω
r / r0
Stroke
σ
dπ / L
Cone apex sharpness
Table 3:
1− sec β
τ ij
ΔP
Dimensionless variables and groups
5
Thermoforming Cones
2 THERMOFORMING MECHANICS
For the mechanics of the stretching sheet, we follow the thin membrane approach for bubble inflation of Baird and Collias [9]. We employ moving spherical coordinates centered in the bubble.
Figure 1 defines r. Assuming constant density, the continuity equation becomes
∂ 2
( r vr ) = 0
∂r
(1)
Integrating gives:
vr =
A(t )
r2
(2)
where A(t ) is a function of time. On the inside surface (at r = R ) the fluid velocity is
vr ( R) = − R&
(3)
for a lenticular film, and for a bulbous one:
vr ( R) = R&
(4)
hence
A(t ) = R& R2
(5)
and from continuity
vr =
R& R2
r2
(6)
Neglecting fluid inertia, the r-component of the equation of motion reduces to
0=
τ +τ
−∂p 1 ∂ 2
− 2
r τ rr ) + θθ φφ
(
∂r r ∂r
r
(7)
where τ ij is the component of the extra stress tensor corresponding to the flux
of x j momentum in the positive xi direction. Hence, τ φφ and τ θθ are negative in
tension. Rewriting (7):
6
Thermoforming Cones
∂π rr τ θθ + τ φφ − 2τ rr
=
∂r
r
where the rr component of the total and extra stress tensors are related by
(8)
π rr = τ rr + p
(9)
where at the inside surface
π rr (R) = − P(R)
(10)
and outside,
π rr (R + h) = − P(R + h)
(11)
Integrating, the equation of motion [(7)] gives
ΔP = ∫
R+ h
R
⎡τ θθ + τ φφ − 2τ rr ⎤
⎢⎣
⎥⎦ dr
r
(12)
For thin films, that is, when:
h
 1
h0
(13)
Bird et al. [12] proposed that the argument for the integral in (12),
(τ θθ + τ φφ − 2τ rr ) / r will be nearly constant. This is because, for a thin film, nei-
ther the stresses (τ θθ + τ φφ − 2τ rr ) , nor the radial position r , will vary much
through the film thickness. Thus, (12) becomes:
ΔP = (τ θθ + τ φφ − 2τ rr )
ΔP = (τ θθ + τ φφ − 2τ rr )
h
R
(14)
We call this the thin film approximation, and Appendix A further explores its importance. We can thus further simplify (14), by specifically evaluating the stresses at r = R :
R
h
R
(15)
For a Newtonian fluid:
τ θθ = τ φφ
2 &
vr −2 μ R R
= −2 μ =
r
r3
(16)
and
7
Thermoforming Cones
2 &
∂vr 4 μ R R
τ rr = −2 μ
=
∂r
r3
Thus the tensile stress inside the stretching film:
τ θθ
i
i
− 2 μR 2 R&
− τ rr
=
=
2
R3
(17)
(18)
exceeds the stress outside:
τ θθ
o
o
− 2 μR 2 R&
− τ rr
=
=
2
(R + h )3
(19)
by the factor:
τ θθ o τ rr o ⎛ h ⎞
=
= ⎜1 + ⎟
τ θθ i τ rr i ⎝ R ⎠
3
(20)
Eliminating the stresses in (15) gives
ΔP =
12 μ R& h
R2
(21)
which reduces to
ΔP ρ 2
dρ
=
12 μTα 0
dt
(22)
which has been adimensionalized using Table 3. Hence,
dρ
ρ2
=
dθ Tα 0
(23)
which describes how a thin disk’s shape evolves during thermoforming (by definition, the dimensionless radius of curvature, ρ , never falls below unity).
3 FREE FORMING GEOMETRY
The lenticular spherical cap thickness depends on its radius of curvature as
8
ρ+
T=
ρ 2 −1
;T ≥
2ρ
Thermoforming Cones
1
2
(24)
so that
dT
=
dρ 2 ρ
1
ρ2 −1
;T >
1
2
(25)
and for the bulbous spherical cap
ρ − ρ 2 −1
T=
2ρ
;T ≤
1
2
(26)
so that
dT
−1
1
=
;T <
dρ 2 ρ ρ 2 − 1
2
(27)
The following discontinuity thus occurs
dT
dρ
=±
T → 12
±
1
4
(28)
We obtain the contact angle of the sheet’s edge (defined in Figure 1)
from the bubble’s slope evaluated at r0 :
φ x =r =
0
π
2
+ a tan
−1
ρ −1
2
(29)
With difficulty, practitioners can sometimes observe the free forming bubble and
measure its height (defined as a in Figure 1).
The dimensionless bubble height depends on its radius of curvature
for the lenticular shape as
H =ρ−
ρ 2 −1 ; H ≥ 1
(30)
and for the bulbous shape
9
H=ρ+
Thermoforming Cones
ρ2 −1 ;H ≤ 1
(31)
4 FREE FORMING RESULTS
Substituting (22) into (21) for T , for a lenticular cap we get
dρ
2ρ 3
=
dθ α ρ + ρ 2 − 1
0
(
)
(32)
Integrating gives
α0 ⎡
⎛1⎞
ρ2 −1 2 ⎤
+ ⎥
θ = ⎢ − arcsin ⎜ ⎟ −
ρ4
ρ ⎦⎥
4 ⎣⎢
⎝ρ⎠
(33)
for bulbous cap growth, and for lenticular cap growth, we substitute (24) into
(21)for T , and integrate
dρ
2ρ 3
=
dθ α ρ − ρ 2 − 1
0
(
)
(34)
to get
α0 ⎡
⎤
⎛1⎞
ρ2 −1 2
−
−
+
4
θ = ⎢ arcsin ⎜ ⎟ +
π
⎥
ρ4
ρ
4 ⎣⎢
⎝ρ⎠
⎥⎦
(35)
Eqs. (33) and (35) are universal, and central to this paper. From these we see
that the shape switch from lenticular to bulbous always occurs at the same dimensionless time when,
θ
ρ =1
>
α0
8
(4 −π )
(36)
where.
dρ
dθ
=0
(37)
ρ =1
Figure 3 illustrates this shape switch.
10
Thermoforming Cones
Figure 3: Dimensionless radius of curvature as a function of dimensionless
time.
Adimensionalizing (18) gives the dimensionless hoop stress inside the stretching film:
Σφφ =
−1 d ρ
6 ρ dθ
(38)
For the lenticular melt,
α 0 Σφφ =
(
−ρ 2
3 ρ+
ρ 2 −1
)
(39)
which is negative and peaks at
ρ=
2
(40)
3
where
α 0 Σφφ =
−4 3
≅ −0.257
27
(41)
and for the bulbous melt
11
α 0 Σφφ =
(
−ρ 2
3 ρ−
ρ −1
2
Thermoforming Cones
)
(42)
Figure 4 illustrates this. So both the radius of curvature and the magnitude of
the tensile stress, Σφφ , are initially infinite. In practice, there is always a little
sag [13, 14], so the initial radius of curvature is always finite. Furthermore, unlike a rubber, Σφφ is initially increasing.
Figure 4: Dimensionless stress versus dimensionless radius of curvature
5 CONSTRAINED FORMING GEOMETRY
The constrained melt takes on the shape of the conical mold, as shown in Figure 5. A lenticular melt forms into a lenticular cone, and a bulbous melt forms
into a bulbous cone.
12
Figure 5:
Thermoforming Cones
Dimensional schematic of transitioning cone
π⎞
⎛
( A ≡ zk sin β and B ≡ zk sin ⎜ β − ⎟ )
2⎠
⎝
From geometry and the mass balance we can relate the dimensionless radius
of curvature to the conical melt length [9]
ρ=
2l − Zκ sin β
sin β tan β
(43)
Now free-forming ends when melt first touches the cone, that is, when Z k = 0 .
Hence,
ρκ =
2l
sin β tan β
(44)
Substituting into (33) and (35) give the free-forming intervals, θφ , for the bulbous melt:
13
α0 ⎡
ρκ 2 − 1 2 ⎤
⎛ 1 ⎞
+ ⎥
θφ = ⎢ − arcsin ⎜ ⎟ −
4 ⎢
ρκ 4
ρκ ⎥⎦
⎝ ρκ ⎠
⎣
Thermoforming Cones
(45)
and for the lenticular melt:
θφ =
α0 ⎡
⎛ 1
⎢ arcsin ⎜
4 ⎢
⎝ ρκ
⎣
⎤
ρκ 2 − 1 2
⎞
+
−
− π + 4⎥
⎟
4
ρκ
ρκ
⎥⎦
⎠
(46)
respectively. We can also relate the constrained melt thickness t , to the contact length [1]
Z
T = Tκ ⎛⎜ 1 − κ sin β ⎞⎟
2l
⎝
⎠
sec β −1
(47)
From geometry, we can relate the thickness during constrained forming, hκ to
the initial disk thickness
π L2 h0
4
=
π L2 (1 − cos β )hκ
2 sin2 β
(48)
which adimensionalizes to
Tκ =
T0
( 1 + cos β )
2
(49)
1
( 1 + cos β )
2
(50)
or
Tκ =
since T0 ≡ 1 .
To get the thickness profile, we substitute (50) into (47):
1
⎛ Z
⎞
T = (1 + cos β ) ⎜ 1 − κ sin β ⎟
2
2l
⎝
⎠
sec β −1
(51)
combining (43) and (51)
⎛ sin 2 β ⎞
1
T = (1 + cos β ) ⎜
⎟
2
⎝ 2l cos β ⎠
sec β −1
(52)
14
When β exceeds
π
2
Thermoforming Cones
, the constrained melt’s shape changes from lenticular to
bulbous [1, 4,5,9].
6 CONSTRAINED FORMING RESULTS
Substituting (52) into (23)
dρ
= αρ 3−sec β
dθ
(53)
where the cone shape factor is
1 + cos β
α≡
2α 0
⎛ sin 2 β ⎞
⎜
⎟
⎝ 2l cos β ⎠
1− sec β
(54)
where Table 3 defines the dimensionless cone height, 2l (Figure 5 defines the
cone height, L ). When β <
π
2
, a lenticular cap progresses down the cone.
Hence,
dρ
π
= αρ 3− sec β ; β <
2
dΘ
(55)
Integrating gives
Θ=
ρ sec β − 2
π
π
; >β ≠
α ( sec β − 2 ) 2
3
(56)
and
Θ=
log(ρ )
α
;β =
π
(57)
3
For thermoforming with β >
π
2
, a bulbous cap progresses down the cone. Hen-
ce,
dρ
π
= −αρ 3− sec β ; β >
2
dΘ
(58)
and integrating gives
15
Θ=
Thermoforming Cones
− ρ secβ − 2
π 4π
;β > ≠
α ( sec β − 2 )
2
3
(59)
and
Θ=
−log(ρ )
α
;β =
4π
3
(60)
Linearizing (56) gives
ln (α ( sec β − 2 ) )
⎛
⎞
1
π
ln ρ = ⎜
;β ≠
⎟ ln Θ +
3
( sec β − 2 )
⎝ sec β − 2 ⎠
(61)
and linearizing (59):
ln (α ( 2 − sec β ) )
⎛
⎞
1
4π
ln ρ = ⎜
;β ≠
⎟ ln Θ +
3
( secβ − 2 )
⎝ secβ − 2 ⎠
(62)
Hence, zero radius of curvature takes an infinite period of constrained forming.
In other words, the melt will never reach the cone apex, and Zκ never reaches
1. Thus, Eqn. (61) and (62) explain why sharp corners are difficult to thermoform. The thermoforming of sharp edges or corners is called detailing.
Letting ρ f be the desired dimensionless apex sharpness, we then get:
Θ f = Θ ( ρ f ) − Θ ( ρκ )
(63)
for the required constrained forming interval. Thus, the total manufacturing time
is
θT = θφ + Θ f
= θφ + Θ ( ρ f ) − Θ ( ρκ )
(64)
For the melt, the stress is
−αρ 1−sec β
π
;β <
Σφφ =
6
2
(65)
which linearizes to
⎛Σ
ln ⎜ φφ
⎝ α
⎞
⎛ −1 ⎞
⎟ = (1 − sec β ) ln ρ + ln ⎜ ⎟
⎠
⎝ 6 ⎠
(66)
16
Thermoforming Cones
and
Σφφ =
αρ 1−sec β
6
;β >
π
2
(67)
which linearizes to
⎛Σ ⎞
⎛1⎞
ln ⎜ φφ ⎟ = (1 − sec β ) ln ρ + ln ⎜ ⎟
⎝ α ⎠
⎝6⎠
(68)
7 PLUG ASSIST
In thermoforming, plug assist is often used to even out wall thickness profiles.
Figure 6 illustrates the variables for plug-assisted thermoforming.
Figure 6: Plug assist
Our work focused on unassisted thermoforming, and thus derives the worst case for wall thickness variation. We define the final wall thickness uniformity as:
ϒ≡
h( zκ f )
h( zκ = 0)
=
T ( Zκ f )
T ( Zκ = 0)
(69)
where zκ f is the final contact length, corresponding to the final desired radius at
the cone tip:
Z κf =
2l
− ρ f tan β
sin β
(70)
Combining (51) with (69) gives:
17
⎛ Zκ f sin β ⎞
ϒ ≡ ⎜1−
⎟
2l
⎝
⎠
Thermoforming Cones
sec β −1
(71)
and substituting (70) into this yields:
⎛ ρ f sin β tan β ⎞
ϒ≡⎜
⎟
2l
⎝
⎠
sec β −1
(72)
and in the limit, for an infinitely sharp cone tip, we get:
1
ϒ ≡ ⎛⎜ 1 − sin β ⎞⎟
⎝ 2l
⎠
sec β −1
(73)
which is the worst case for the cone thickness uniformity. Equations (71) and
(72) dictate just how much plug assist is required to even out the cone wall
thickness distribution. Equation (73) gives the upper bound for this plug assist
requirement.
From the geometry of the deformation, and in cylindrical coordinates, Williams
derived the following for the thickness profile caused by plug assist
T≡
1
⎛ dπ
1 + ⎜ r ln r0
⎜
Rπ
⎝
⎞
⎟
⎟
⎠
(74)
2
and verified this experimentally. This adimensionalizes to
T≡
1
2
1 + ⎛⎜
ω
1
ln
⎜
⎝ σ l απ
⎞
⎟
⎟
⎠
(75)
2
where σ is the dimensionless plug displacement (normally called stroke), α π is
the plug shape factor and ω is the center line proximity. So during plug assist,
the deforming melt’s thinnest part is near the plug’s edge where:
Tmin =
1
1 + ⎛⎜
απ
1
ln
⎜
⎝ σ l απ
2
⎞
⎟
⎟
⎠
2
(76)
18
Thermoforming Cones
and its thickest, near the mold rim, where:
Tmax =
1
2
1 + ⎛⎜
1
1
ln
⎜
⎝ σ l απ
⎞
⎟
⎟
⎠
(77)
2
In principle, we would like to match the severity of the cone thickness problem,
ϒ , with the amount of plug assistance near the rim, Tmax . In practice, however,
the plug normally runs into the cone before this amount of plug assistance can
be realized. To prevent this, the stroke must satisfy this geometric inequality:
σ<
α 0 ⎡ ( 1 − απ ) tan β ⎤
− 1⎥
α0
2l ⎢⎣
⎦
(78)
For a right cylinder, where β =
π
, the plug can never run into the mold; Throne
2
has outlined an approach for this special case [6].
8 WORKED EXAMPLES
8.1 Process speed and melt stress
A plastics engineer wants to manufacture a safety cone with β = 75.5° and
L = 31cm from a disk of uniform thickness h0 = 1.51 mm and from a nearly
Newtonian melt with μ = 3.11 × 106 Pa ⋅ s . She desires a blunt cone of apex radius, R f = 25.4 mm . She employs an external gage pressure of 91.3 psi and a
vacuum of 14.1 psi . Calculate the total forming time, and estimate the stress
frozen into the safety cone, both near its rim and into its blunt tip.
Using Table 3, we calculate l ≡ ( tan β ) / 2 = 1.93 . Substituting into (44) gives a
dimensionless radius of curvature of ρκ = 1.03 when constrained forming begins. Substituting this and α 0 ≡ h0 / r0 = 0.0188 (from Table 3) into (45) then gives
a dimensionless free forming time of θφ = 1.85 × 10 −3 . Combining ρ ≡ R / r0 from
Table 3 with r0 ≡ L / tan β from Table 3 gives the dimensionless radius of curvature of the blunt cone tip ρ f ≡ R f / r0 = 0.312 . Substituting θφ , ρκ ,
ρ f ≡ R f / r0 = 0.312 and
α ≡ (sin 2 β / 2α 0 (1 − cos β )) ( sin 2 β / 2l cos β )
1− sec β
= 36.4
into (64) gives a total dimensionless forming time of θT = 0.015 .
19
Thermoforming Cones
Summing the vacuum and the applied air pressures gives ΔP = 105.4 psi . Using this and Table 3, we find the total-forming time of tT = 12 μ ΘT / ΔP = 0.770 s .
We then substitute ρκ , and α = 36.4 , into (65) to obtain the dimensionless
stress near the rim of Σφφ
rim
= −6.65 , which is tensile. We expect most of this to
freeze into the rim. Using Table 3, we find that this corresponds to a tensile
stress at the rim of Σφφ ΔP = τ φφ
= −5.53 MPa , which is also tensile.
rim
rim
Substituting ρ f , and α into (65) gives the dimensionless stress near the blunt
tip of Σφφ
tip
= −253.7 , which is tensile. We expect most of this to freeze into the
tip. Using Table 3, we find that this corresponds to a tensile stress at the rim
of Σφφ ΔP = τ φφ = −211 MPa .
tip
tip
8.2 Pressure difference and part height
A plastics engineer wants to process the same safety cone as in Example a.,
but his process economics require a forming time to fall below 1.2 seconds.
Find the required ΔP , finished part height, and estimate the stress frozen into
the safety cone, both near its rim and into its blunt tip.
Using θT = 0.015 from Example a., for the required applied pressure difference
we find ΔP = 12ΘT μ / t = 67.7 psi . From the cone geometry, for the finished part
height we get:
Rf
⎛ L
H p = sin β ⎜
− R f tan β −
tan β
⎝ sin β
⎞
⎟ + Rf
⎠
(79)
which give 0.24 m.
8.3 Cone sharpness
A plastics engineer wants to manufacture a pointy cone ( ρ f  1) with β = 75.5°
and L = 0.31m from a disk of nearly Newtonian melt with μ = 3.11 × 106 Pa ⋅ s and
uniform thickness h0 = 1.51 mm . She employs an external gage pressure of
91.3 psi and a vacuum of 14.1psi . Her mold has an infinitely sharp cone tip, and
she employs a long forming time, tT = 10 s . Calculate the resulting apex radius.
Summing the vacuum and the applied air pressures gives ΔP = 105.4 psi . Using this and Table 3, we find the total-dimensionless forming time of
ΘT = ΔPtT / 12 μ = 0.194 . Using Table 3, we calculate l ≡ ( tan β ) / 2 = 1.93 , and
20
Thermoforming Cones
substituting into (44) gives a dimensionless radius of curvature when constrained forming begins of ρκ = 1.03 . Substituting this and α 0 ≡ h0 / r0 = 0.0188
(from Table 3) into (45) then gives a dimensionless free forming time of
θφ = 1.85 × 10 −3 . Combining ρκ , ΘT and θφ into (64), and solving for the dimensionless apex radius gives:
ρ f = (α ( sec β − 2 ) (θT − θφ ) + ρκ
1
sec β − 2 sec β − 2
)
(80)
Using this and Table 3 we get ρ f = 3.89 and thus find the cone tip sharpness to
be R f = ρ f r0 = 31.2 cm .
8.4 Plug assist
A plastics engineer wants to process the same safety cone as in Example a.,
but in this case, she wants a more uniform wall thickness distribution. For this
she employs plug assist, specifically using a right cylindrical plug
with Rπ = r0 / 2 = 4.01 cm with a plug displacement of 13.8 cm. Estimate the improvement in wall thickness uniformity?
Combining ρ ≡ R / r0 from Table 3 with r0 ≡ L / tan β from Table 3 gives the di-
mensionless radius of curvature of the blunt cone tip ρ f ≡ ( R f tan β ) / L = 0.312 .
Since Rπ = r0 / 2 , the dimensionless cone shape factor is απ ≡ Rπ / r0 = 1 / 2 . Using Table 3, we calculate l ≡ ( tan β ) / 2 = 1.93 , and substituting this and ρ f into
(72) gives the sheet uniformity, ϒ = 0.0279 . This is the uniformity that would be
obtained without plug assist.
Using Table 3 , we calculate a stroke of σ ≡ dπ / L = 0.446 . Substituting into (77)
gives Tmax = 0.373 . This means that the melt cone’s rim will begin free forming
at a thickness that is 62.7% of the disk’s initial thickness. We thus expect the
plug assist to improve the uniformity to:
ϒπ ≅
ϒ
= 0.0748
Tmax
(81)
which corresponds to roughly a three-fold improvement in cone wall thickness
uniformity.
21
Thermoforming Cones
9 CONCLUSION
This new analysis for thermoforming cones focuses on the manufacturing process speed. Specifically, we’ve distinguished between what happens before
(free forming) and after (constrained forming) the melt touches the conical mold.
We derive the time required for both cases, and sum them to get the total forming time. We’ve restricted our analysis to the Newtonian case and adimensionalized our results at every step. For free forming, one dimensionless geometric
shape factor arises (α 0 ) , and for constrained, two arise ( α and sec β ). We also
calculate the stresses in the deforming melt, since these govern the residual
stresses in the thermoformed part. We then derive an expression for wall uniformity; we find that it just depends on the mold geometry. Finally, we attack
the problem of plug assist, deriving an expression for the improvement in wall
uniformity achieved through plug assist.
10 ACKNOWLEDGEMENT
The authors are indebted to Dr. Zhongbao Chen of the University of Wisconsin
and to Dr. Martin J. Stephenson of the Placon Corporation for their invaluable
advice. We further acknowledge Professor R. Byron Bird for his help with the
thin film approximation. We thank the Placon Corporation of Madison, Wisconsin and Plastic Ingenuity, Inc. of Cross Plains, Wisconsin for their financial support through their memberships in the Industrial Consortium of the Center for
Advanced Polymer and Composite Engineering at the University of Wisconsin.
The Placon Corporation is also recognized for its sustaining sponsorship of the
Rheology Research Center.
11 APPENDIX A: THIN FILM APPROXIMATION
Here we explore the virtue of the thin film approximation. Eliminating the stresses in equation (15) gives
ΔP = ∫
2 μ R2 R& + 2 μ R2 R& − 8 μ R2 R&
r4
dr = ∫
−4 μ R2 R&
r4
dr
(82)
Thus,
⎡
dr
1
1 ⎤
= 12 μ R2 R& ⎢
− 3⎥
3
4
R
r
⎣⎢ ( R + h ) R ⎥⎦
ΔP = −4 μ R2 R&
∫
R+ h
(83)
22
Thermoforming Cones
Using Table 3, this adimensionalizes to:
3
dρ ⎡ ρ ( ρ + Tα 0 ) ⎤
=⎢
⎥
dθ ⎢⎣ − ( ρ + Tα 0 )3 + ρ 3 ⎥⎦
(84)
Though this is more accurate than (23), neither substituting (24) or (26) into (84)
for free forming, nor (52) into (84) for constrained forming, leads to differential
equations having analytical solutions. This is why thermoforming analysis relies
so heavily on the thin film approximation.
12 REFERENCES
[1]
Tadmor, Z.,
Gogos, G.G.
Principles of Polymer Processing
John Wiley & Sons, Inc., New York (1979)
[2]
Hart-Smith, L.J.,
Crisp, J.D.C.
Int. J. Eng. Sci., 5, 1(1967)
[3]
Sheryshev, M.A.,
Zhogolev, I.V.,
Salazkin, K.A.
Soviet Plast. , 11, 30 (1969)
[4]
Williams, J.G.
J. Strain Analysis, 5, 49 (1970)
[5]
Rosenzweig, N.,
Narkis, M.,
Tadmor, Z.
Polymer Engineering and Science, 19, 946 (1979)
[6]
Throne, J.L.
Plastics Process Engineering
Marcel Dekker, New York (1979)
[7]
Pearson, J.R.A.
Mechanics of Polymer Processing
Elsevier Applied Science Publishers Ltd., London
(1985)
[8]
Allard, R.,
Charrier, J.-M.,
Ghosh ,A.,
Marangou, M.,
Ryan, M.E.,
Shrivastava, S.,
Wu, R.
J. Polym. Eng., 6, 363 (1986)
[9]
Baird, D.G.,
Collias, D.I.
Polymer Processing Principles and Design
Butterworth-Heinemann, Boston (1995); Wiley &
Sons, New York (1998)
23
Thermoforming Cones
[10] Osswald, T.A.,
Hernández-Ortiz, J.P.
Polymer Processing - Modeling and Simulation
Hanser Publishers, Munich (2006)
[11] Strong, A.B.
Plastics Materials and Processing
3rd ed., Prentice Hall, Upper Saddle River, New
Jersey (2006)
[12] Bird, R.B.,
Armstrong, R.C.,
Hassager, O.
Dynamics of Polymeric Liquids
Vol. 1: Fluid Mechanics, 2nd ed., Wiley & Sons,
New York (1987); see Eq. (8.4-15)
[13] Stephenson, M.J.,
Dargush, G.F.,
Ryan, M.E.
Polymer Engineering and Science, 39, 2199
(1999)
[14] Stephenson, M.J.
An Experimental and Theoretical Study of Sheet
Sag in the Thermoforming Process
PhD Thesis, State University of New York,
Buffalo, NY (August 1997).
Keywords:
englisch:
thermoforming cones, thermoforming mechanics
Contact:
Autoren:
Prof. A. Jeffrey Giacomin
Herausgeber:
Prof. em. Dr.-Ing. Dr. h.c. Gottfried W. Ehrenstein,
Prof. Dr. Tim Osswald
Erscheinungsdatum:
Mai/Juni 2007
24
Herausgeber/Editor:
Europa/Europe
Prof. Dr.-Ing. Dr. h.c. G. W. Ehrenstein, verantwortlich
Lehrstuhl für Kunststofftechnik
Universität Erlangen-Nürnberg
Am Weichselgarten 9
91058 Erlangen
Deutschland
Phone:
+49/(0)9131/85 - 29703
Fax.:
+49/(0)9131/85 - 29709
Thermoforming Cones
Amerika/The Americas
Prof. Dr. Tim A. Osswald, responsible
Polymer Engineering Center, Director
University of Wisconsin-Madison
1513 University Avenue
Madison, WI 53706
USA
Phone:
+1/608 263 9538
Fax.:
+1/608 265 2316
Verlag/Publisher:
Carl-Hanser-Verlag
Jürgen Harth
Ltg. Online-Services & E-Commerce,
Fachbuchanzeigen und Elektronische Lizenzen
Kolbergerstrasse 22
81679 Muenchen
Tel.: 089/99 830 - 300
Fax: 089/99 830 - 156
E-mail: [email protected]
Beirat/Editorial Board:
Professoren des Wissenschaftlichen Arbeitskreises Kunststofftechnik/
Professors of the Scientific Alliance of Polymer Technology
25

Thermoforming Cones

Transcription

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