Kutz M. Handbook of materials selection
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996 PLASTIC PARTS PROCESSING II
1.3 Sources of Momentum
Since force F equals mass m times acceleration a, and acceleration is the time
rate of change of velocity v, and momentum is mass times velocity, then (for
constant mass)
mdv d(mv)
F ⫽ m a ⫽⫽
dt dt
This force per unit area A, which is a stress
, that produces a shear or exten-
sional motion is a flux of momentum, the time rate of momentum per unit
area.
The magnitudes and directions of the fluxes of momentum are given by the
stress tensor
. In component form the first subscript designates the direction in
which the momentum is being transferred, and the second designates the direc-
tion in which the fluid is moving (the velocity).
11 12 13 11 12 13
⬅
⫽
(1.9)
21 22 23 12 22 23
冤冥冤冥
31 32 33 13 23 33
The six different components of this three-by-three symmetric matrix character-
ize the rheological properties of a polymer melt.
In plastic parts processing, the sources of momentum may be classified as
‘‘drag,’’ ‘‘pressure,’’ ‘‘gravitational,’’ or ‘‘centrifugal’’; hence, the processes may
be given those names.
Drag-Flow Processes
Drag-flow processes are those in which a moving object, usually metal, supplies
the momentum that is transferred to the plastic, causing it to move. In extrusion
it is the turning screw; in wire coating it is the moving wire; in calendering it
is the rotating rolls.
Pressure-Flow Processes
Pressure may be a source of momentum that produces the flow, a source of
momentum that opposes the flow, or a sink of momentum reducing the magni-
tude of the flow. Pressure is a ‘‘positive’’ source when an extrudate flows through
a die; pressure is a ‘‘negative’’ source for the melt within the channels of an
extrusion screw; the injection pressure in a molding process is required because
there is a pressure drop as the melt flows through the sprue, runner, gate, and
cavity.
Vacuum, a negative pressure, is an essential operating parameter in thermo-
forming; it is the agent that removes the air between the sheet and the mold,
and the resulting positive pressure gradient across the sheet causes the sheet to
take on the shape of that mold.
Gravitational-Influenced Processes
Some processes, such as vertical fiber spinning, benefit from gravity as a source
of momentum, which helps to reduce the diameter of the fiber. Gravity aids the
sheet movement in thermoforming, if the mold is below the plastic.
2 INTRODUCTION 997
Centrifugal-Based Process
Rotational molding relies on the centrifugal forces developed from two rotating
axes to press the plastic melt against the inside surfaces of the mold.
2 INTRODUCTION
Engineering problems are usually one of two types: If a machine needs to be
specified for a given task, the problem is one of design; if the performance of
an existing machine needs to be characterized, the problem is one of rating. In
the former situation, there are usually more variables to be specified than there
are equations; constraints must be assigned. In the latter situation, the constraints
are fixed; they limit the variation in performance that may be achieved through
changes in the operating parameters.
2.1 Design Problem
In the design problem, some measure of performance must exist and be known.
This constitutes the dependent variable. It is a function of three categories of
variables—the geometric dimensions of the shaping apparatus, the properties of
the plastic, and the operating parameters.
Measure of Performance
In continuous processes, this is usually the rate of production of a given profile.
In extrusion, this is the rate of production of a specified shape; in wire coating,
it is the rate of production of a wire with a specified coating thickness; in
calendering, it is the rate of production of a sheet of specified width and thick-
ness.
In cyclic processes, the measure of performance is usually the number of
parts produced annually, monthly, weekly, or hourly, which is related to the
reciprocal of the cycle time. A component of the cycle time is the processing
time—the time it takes to shape the object; other components of the cycle time
may be the time required to load a preform and unload the product, close and
open a mold, and/or solidify the plastic.
Geometric Factors
The objects whose geometric dimensions may influence the performance of a
process include the barrel, screw, and die in extrusion; the wire and die in wire
coating; the rolls in calendering; etc. In general, these characteristics cannot be
changed during processing. However, there are other distances that can be
changed, such as the gap between the two plates that constitutes a lip die or the
movable core that determines the thickness of the parison in blow molding.
Material Properties
Although the rheological properties of the polymer melt (defined in Section 2.2)
determine its motion, other physical and thermal properties have an influence
on the measure of performance. These include the density
, the heat capacity
C
P
, and the thermal conductivity k.
Operating Parameters
An operating parameter of a process might be the translational or rotational
speed of a moving object (such as the wire in wire coating, the screw in extru-
998 PLASTIC PARTS PROCESSING II
sion, or the roll in calendering). Another such parameter is the pressure in in-
jection molding, blow molding, or film blowing.
2.2 Design Solution Algorithm
The following procedure, or algorithm, gives one a logical approach for obtain-
ing a solution to each processing problem.
●
One should draw a picture of the process, identifying an appropriate ge-
ometry and the associated dimensions.
●
Identify the nonzero components of the velocity and the spatial coordi-
nates of which they are functions.
●
Write and simplify the equation of continuity.
●
Write and simplify the equation(s) of motion.
●
If variations in temperature are to be included, write and simplify the
equation of energy.
●
Combine the above solutions or solve those equations simultaneously to
determine an expression for the measure of performance.
Geometry of Process
Plastic parts processes may be described by using rectangular, cylindrical, bi-
polar, or spherical geometries. In a rectangular coordinate system, the standard
order is R(x,y,z). In a cylindrical coordinate system, the standard order is C(r,
,z),
where 0
⬍
⬍ 2
. The bipolar system is two nonconcentric cylindrical systems
with parallel axes; it is particularly useful in describing the wire coating process
to make household wiring. In a spherical coordinate system, the standard order
is S(r,
,
), where 0 ⬍
⬍
and 0 ⬍
⬍ 2
; these angles correspond to
latitudes and longitudes of Earth, respectively (although the latter are measured
Ⳳ90⬚ north and south and Ⳳ180⬚ east and west).
Velocity Flow Field
Having specified an appropriate geometric coordinate system, one must specify
the directions in which the velocity components are nonzero. If possible, one
also identifies the directional dependencies of these components. These decisions
help to define the process kinematics.
Equation of Continuity
The equation of continuity (EOC) is the differential form of the conservation of
matter. Polymer melts are regarded as being incompressible; this is true except
for injection molding where high pressures are involved. In vector form, the
EOC is
⭸
⫺div(
v) ⫽ (2.1)
⭸t
for the unsteady-state flow of compressible fluids,
2 INTRODUCTION 999
div(
v) ⫽ 0 (2.2)
for the steady-state flow of compressible fluids, and
div(v)
⫽ 0 (2.3)
for the flow of incompressible fluids, whether unsteady or steady state. [Note:
the vector calculus operators divergence (div) and gradient (grad) permit a con-
cise presentation of equations; the complete component forms in several coor-
dinate systems are readily available in a number of texts, including those by
Bird et al.
1,2
]
Kinematics of Flow
The kinematics of the processing flows should be identified because these de-
termine which rheological properties of the polymer melt will influence the
measure of performance.
Rheological Properties of Polymer Melt
There are six rheological properties of a polymer melt that influence the mea-
sures of performance. If the processing flow is dominated by shear, then (i) the
shear viscosity
and the (ii) first ⌿
1
and (iii) second ⌿
2
normal stress functions
of the melt will play important roles in determining the shape of the plastic part.
These properties are defined as follows:
21
⬅⫺ (2.4)
˙
␥
21
⫺
11 22
⌿ ⬅⫺ (2.5)
1
2
(˙
␥
)
21
⫺
22 33
⌿ ⬅⫺ (2.6)
2
2
(˙
␥
)
21
where
0
11 12 13 11 12
⬅
⫽
0 (2.7)
12 22 23 12 22
冤冥冤冥
00
13 23 33 33
It is known from experimental data that ⌿
1
⬎ 0, ⌿
2
⬍ 0, and 兩⌿
1
兩 ⬎ 兩⌿
2
兩.
If the processing flow is dominated by an extensional flow, then either (i) the
uniaxial extensional viscosity
3
, (ii) the planar extensional viscosity
4
, or (iii)
the uniform biaxial extensional viscosity
6
will play an important role in de-
termining the size of the product. (The numerical subscripts are those originally
used by Denson.
3
)
These properties are defined as follows:
1000 PLASTIC PARTS PROCESSING II
⫺
11 22
⬅⫺ (2.8)
3
˙
11
⫺
11 22
⬅⫺ (2.9)
4
˙
11
⫺
11 33
⬅⫺ (2.10)
6
˙
11
where the constraints on
ii
are as given in Section 1.2 and˙
00
11 12 13 11
⬅
⫽ 0
0 (2.11)
12 22 23 22
冤冥冤冥
00
13 23 33 33
Constitutive Equations for Polymer Melt
The simplest model that relates
21
and
21
is that for a Newtonian fluid, i.e.,˙
␥
Newton’s law:
21
⫽⫺ ⫽ ƒ(˙
␥
) (2.12)
21
˙
␥
21
i.e.,
is a constant. If the fluid requires
21
to exceed an initial yield value,
0
,
after which Newton’s law is applicable, the material is known as a Bingham
plastic
4
:
⫺
21 0
⫽⫺ ⫽ƒ(˙
␥
) (2.13)
B21
˙
␥
21
Although there have been a myriad of equations presented in the literature
(e.g., as cited by Skelland
5
) to describe the shear viscosity of a non-Newtonian
fluid, almost all may be reduced to one of the three following generalized forms:
Generalized Eyring Model. A generalized Eyring model may be written as
⫺
arcsinh(˙
␥
/˙
␥
)
⬁
a
⌫ ⬅⫽ (2.14)
⫺
˙
␥
/˙
␥
0
⬁
a
where
a
is the shear rate at which the viscosity is
a
and ⌫
a
⬅ (
a
⫺
⬁
)/˙
␥
(
0
⫺
⬁
) ⫽ arcsinh(1) ⫽ 0.8814. Since
0
and
⬁
are properties of the liquid,
this model has no adjustable parameters.
With
⬁
⫽ 0, this has been called the Sutterby model
6
(in which the
a
was˙
␥
thought to be arbitrary) or the Eyring-Prandtl model.
7,8
When
⬁
⫽ 0, it is called
the Eyring-Powell model,
9
which is sometimes shown in a pseudo-two-parameter
form. O’Nan
10
investigated, among other models, one from his own thought (a
shear stress version of the Eyring-Powell model):
2 INTRODUCTION 1001
⫺
arcsinh(
/
)
⬁
a
⌫ ⬅⫽ (2.15)
⫺
/
0
⬁
a
where
a
is the shear stress at which the viscosity is
a
and ⌫
a
⬅ (
a
⫺
⬁
)/
(
0
⫺
⬁
) ⫽ arcsinh(1) ⫽ 0.8814; it has no adjustable parameters.
Generalized Yasuda Model. A generalized Yasuda model may be written as
⫺
1
⬁
⌫ ⬅⫽ (2.16)
m (1
⫺
n)/m
⫺
[C ⫹ (˙
␥
/˙
␥
)]
0
⬁
a
This model gives the viscosity as a function of shear rate; C must be unity for
the equation to satisfy the low-shear-rate limit criterion. However, when C
⫽ 0
and
⬁
⫽ 0, this model becomes the power law model, credited to Ostwald
11
and de Waele.
12
This is often used, because of its simplicity, as the lowest-level
non-Newtonian model in polymer processing calculations; it is inherently in-
correct because it predicts an infinite viscosity wherever the shear rate is zero.
(In polymer processing, the shear rate is zero whenever a geometric symmetry
exists, such as along the centerline of a circular conduit or in the center plane
between two parallel plates; it is not a trivial situation that may be ignored.) If
C
⫽ 0 and the power law range is truncated at
0
, then one has the Spriggs
model.
13
If C ⫽ 0 and
0
⫺
⬁
⫽ K, then one has the Sisko model.
14
With C ⫽ 1, the slope of the high shear-rate range of the ln ⌫ vs. ln , or˙
␥
ln(
⫺
⬁
) vs. ln , curve is ⫺(1 ⫺ n), from which n may be determined. [Note˙
␥
that since n is a positive number less than unity, ⫺(1 ⫺ n) cannot be more
negative than
⫺1.] is the shear rate at an arbitrary data point through which˙
␥
a
one wishes the equation to pass; m is determined from the equation
(1
⫺ n)(ln 2)
m ⫽ (2.17)
ln(1/
⌫ )
a
where ⌫
a
⫽ (
a
⫺
⬁
)/(
0
⫺
⬁
) and
a
is the viscosity at
a
.If
⬁
is not˙
␥
known, then the ln(
) vs. ln( ) curve is used to determine n, and
0
/
a
replaces˙
␥
1/⌫
a
in the equation for m. In this case, a convenient arbitrary data point is the
‘‘corner shear rate,’’
c
, defined as the value of at which the
0
line intersects˙
␥
˙
␥
the extrapolation of the high shear-rate line. At this shear rate, the corresponding
value of the viscosity is
c
, and m ⫽ (1 ⫺ n)(ln 2)/ln(
0
/
c
).
Since m is determined by n and the arbitrary point selected, this is a ‘‘two
adjustable-parameter’’ model—the Yasuda model.
15
If m ⫽ 2, it is the Carreau
model
16
that passes through the ‘‘no longer arbitrary’’ point (
a
,
a
), where ⌫
a
˙
␥
⫽
(1
⫺
n)/2
.Form ⫽ 1, with
⬁
⫽ 0, it is the Cross model
17
that passes through
1
–
2
the point where
a
⫽
0
/2
(1
⫺
n)
. These two models have one adjustable parameter,
n. The Cross model is utilized in ‘‘C-MOLD,’’ a software package for polymer
processing. (C-MOLD is a registered trademark of Mold-Flow, Louisville KY).
1002 PLASTIC PARTS PROCESSING II
Generalized Meter Model. A generalized Meter model may be written as
⫺
1
⬁
⌫ ⬅⫽ (2.18)
m (1
⫺
n)/m
⫺
[C ⫹ (
/
)]
0
⬁
a
This model gives the viscosity as a function of shear stress. C must be unity
for the equation to satisfy the low-shear-stress limit criterion. However, when
C
⫽ 0 and
⬁
⫽ 0, it yields the power law model.
With C
⫽ 1, the slope of the high-shear-stress range of the ln ⌫ vs. ln
,or
ln(
⫺
⬁
) vs. ln
, curve is ⫺(1 ⫺ n), from which n may be determined. [Note
that since n is a positive number less than unity,
⫺(1 ⫺ n) cannot be more
negative than
⫺1.]
a
is the shear stress at an arbitrary data point through which
one wishes the equation to pass; m is determined from the equation
(1
⫺ n)(ln 2)
m ⫽ (2.19)
ln(1/
⌫ )
a
where ⌫
a
⫽ (
a
⫺
⬁
)/(
0
⫺
⬁
) and
a
is the viscosity at
a
.
If
⬁
is not known, then the ln
vs. ln
curve is used to determine n, and
0
/
a
replaces 1/⌫
a
in the equation for m. In this case, a convenient arbitrary
data point is the corner shear stress,
c
, defined as the value of
at which the
0
line intersects the extrapolation of the high-shear-stress line. At this shear
stress, the corresponding value of the viscosity is
c
, and m ⫽ (1 ⫺ n)(ln 2)/
ln(
0
/
c
).
Sometimes this model is written as
⫺
1
⬁
⌫ ⬅⫽ (2.20)
␣
⫺
1
⫺
C ⫹ (
/
)
0
⬁
a
where C must be unity for the equation to satisfy the low-shear-stress limit
criterion. Here,
a
is the value of
for which ⌫ ⫽ or where
⫽ (
0
⫹
⬁
)/
1
–
,
2
2; hence,
a
is frequently referred to as a mean value of
. When
⬁
⫽ 0 and C
⫽ 1, the equation is the Meter model.
18
When
⬁
⫽ 0 and C ⫽ 1, it is the Ellis
model.
19
When
⬁
⫽ 0 and C ⫽ 1, it is the DeHaven model.
20
The first two are
one-parameter models, and the third is an inappropriate two-parameter model.
If C
⫽ 1 and
␣
⫽ 3, one has the Reiner-Philippoff model
21
; this model has
no adjustable parameters.
Brodkey Model. The Brodkey model
22
is based on an analogy with the
chemical kinetics of homogeneous reactions. It is especially versatile in that it
can be used to describe time-dependent materials (using reaction rate constants)
as well as time-independent ones (using reaction equilibrium constants); the
other models included in this section are limited to the latter case. The time-
dependent fluids are known as rheopectic, if the viscosity increases with the
duration of shear, or thixotropic, if the viscosity decreases with the duration of
shear.
2 INTRODUCTION 1003
Denny and Brodkey
23
expressed their model in terms of the fraction of mol-
ecules whose structure is unchanged by shear, F, according to the equation
ƒ(
) ⫺ ƒ(
)
⬁
F ⫽ (2.21)
ƒ(
) ⫺ ƒ(
)
0
⬁
where ƒ(
) ⫽ A
1 / 3.5
. (Other researchers
24
would recommend 3.4, rather than
3.5.) Using U for ‘‘unchanged’’ and C for ‘‘changed,’’ a kinetic equation for the
reaction U
↔ C may be written as
dU dF
mn m n
⫺⫽k⬘U ⫺ k⬘C or ⫺⫽k⬘F ⫺ k⬘(1 ⫺ F) (2.22)
12 12
dt dt
in addition,
p1
k⬘ ⫽ k
(2.23)
llij
where l ⫽ 1, 2 and
ij
is the shear stress. Combining these equations, one has
dF
mp1 np2
⫺⫽kF
⫺ k (1 ⫺ F)
(2.24)
1 ij 2 ij
dt
for time-dependent materials. For time-independent fluids, one sets
⫺dF/dt ⫽
0toget
n
k (1 ⫺ F)
1
p1
⫺
p2 p
⫽ K
⫽ (2.25)
ij ij
m
kF
2
where K ⫽ k
1
/k
2
and p ⫽ p
1
⫺ p
2
. In these equations, the parameters are k
1
, k
2
,
m, n, p
1
, and p
2
,orK, m, n, and p, respectively.
Rearranging this model into the normalized form used previously, we have
⫽ [ƒ(
)/A]
3.5
; hence
3.5 3.5
ƒ(
) ⫺ ƒ(
)
⬁
⌫ ⫽ (2.26a)
3.5 3.5
ƒ(
) ⫺ ƒ(
)
0
⬁
or more simply
3.5 3.5
ƒ ⫺ ƒ
⬁
⌫ ⫽ (2.26b)
3.5 3.5
ƒ ⫺ ƒ
0
⬁
This can be expressed as
1004 PLASTIC PARTS PROCESSING II
3.5 3.5
[(ƒ ⫺ ƒ)F ⫹ ƒ] ⫺ ƒ
0
⬁⬁ ⬁
⌫ ⫽ (2.27)
3.5 3.5
ƒ ⫺ ƒ
0
⬁
If
⬁
⫽ 0, then ⌫ ⫽ F
3.5
. Here F is the solution of aF
b
⫹ F ⫺ 1 ⫽ 0, where
for time-independent materials
p 1/n
a ⫽ (K
) (2.28)
ij
and for time-dependent materials
1/n
k
1
p1
⫺
p2
a ⫽
(2.29)
冉冊
ij
k
2
m
b ⫽ (2.30)
n
As explained by Brodkey,
22
m and n are small integers, as expected in chemical
kinetics. If m
⫽ n/2, then
22
F ⫽ [⫺a ⫹ 兹a ⫹ 4/2] (2.31a)
if m
⫽ n, then
1
F ⫽ (2.31b)
a
⫹ 1
if m
⫽ 2n, then
⫺1 ⫹ 兹1 ⫹ 4a
F ⫽ (2.31c)
2a
(The negative roots of these radicals are physically impossible.)
Corotational Jeffreys Model. The above equations say nothing about the
normal stress functions. To do that one must use models based on either the
corotational or codeformational time derivatives; the former is given by
DT ⭸T 1
⫽⫹v䡠grad T ⫹ (
䡠T ⫺ T䡠
) (2.32)
Dt ⭸t 2
where the tensor T may be either
or , each of which is symmetric.˙
␥
The corotational Jeffreys model
25
expands the generalized Newtonian model
by adding the corotational time derivatives of both the stress tensor and the rate
of strain tensor. The adjustable parameters,
1
and
2
, each have the units of
time (i.e., seconds); hence, they may be thought of as stress and rate of strain
time constants, respectively, of the melt:
2 INTRODUCTION 1005
D
D ˙
␥
⫹
⫽⫺
˙
␥
⫹
(2.33)
冉冊
102
Dt Dt
This model predicts the shear properties to be
2
(1 ⫹
˙
␥
)
01221
⫽ (2.34)
22
1 ⫹
˙
␥
121
2
(
⫺
)
01 2
⌿ ⫽ (2.35)
1
22
1 ⫹
˙
␥
121
(
⫺
)
01 2
⌿ ⫽⫺ (2.36)
2
22
1 ⫹ ˙
␥
121
Note that, if
1
⬎
2
, this model does predict that the first normal stress function
(NSF) is positive and the second NSF is negative; it also predicts that
兩⌿
1
兩 ⬎
兩⌿
2
兩. However, the prediction that ⫺⌿
2
/⌿
1
⫽ 0.5 is contrary to observations;
experimental values of this ratio are not constant. This is a characteristic of each
corotational model, and is one disadvantage of such a formulation.
This model predicts the extensional properties to be
⫽ 3
(2.37)
30
⫽ 4
(2.38)
40
⫽ 6
(2.39)
60
These results reveal further deficiencies of this corotational model; it is no better
than the Newtonian model for any extensional viscosity.
Criminale–Ericksen–Filbrey (CEF) Model. In conrast, the CEF model
26
gives the absolutely correct expressions for the shear properties, yet defies util-
itarian value!
11
D ˙
␥
⫽⫺
˙
␥
⫺ ⌿ ⫹ ⌿ ˙
␥
䡠 ˙
␥
⫹ ⌿ (2.40)
冉冊
12 1
22Dt
The results of the analysis for shear properties are
⫽
(2.41)
⌿ ⫽ ⌿ (2.42)
11
⌿ ⫽ ⌿ (2.43)
22
There are no parameters in the model to permit the curve fitting of experimental