In-Mold Shrinkage and Stress Prediction in
Injection Molding
G . TITOMANLIO and K. M. B. JANSEN*
Dipartimento di Ingegneria Chimica ed Alimentare
Universita di Salem0
84084 Fisciano (SA), Italy
In-mold shrinkage may occur for product parts that solidify under low holding
pressure and are not restricted by ribs or flanges. It not only affects the final
product dimensions but in addition may have a large effect on the residual stress
distribution. A simple elastic model is used to study the effect of in-mold shrinkage
on final product dimensions and residual stress distributions. Friction between
polymer surface and mold wall as well as deformation of the mold cavity are taken
into account. The model uses local values for temperature, pressure, and crystallization, which belong to the standard output of most simulation codes.
INTRODUCTION
or predicting product dimensions of injection
molded products, one has to distinguish between
the as-molded dimensions (scope of this paper) and
the post molding behavior (time effects). The latter
may consist of physical aging, post crystallization,
water absorption, or recovery of frozen-in orientation.
Shrinkages in injection molded products are usually
calculated from F'vT data (1-41, which necessarily
leads to isotropic shrinkage predictions. However, it is
well known that in injection molding thickness
shrinkages can be a factor 2 to 10 larger than length
and width shrinkages (2). Therefore, a different approach was developed and the as-molded dimensions
are calculated from the stress change during ejection
(5).This means that boundary condition effects (geometrical constraints, friction effects) can also be included. The effect of molecular orientation, however,
can only be included indirectly [via differences in expansion and crystallization coefficients).
In our first paper (5) this idea was worked out in
more detail using a simple elastic model. The model
was used to calculate final shrinkage and stress distribution in case of free quenching and injection molding with constrained shrinkage in the mold. Thus predicted shrinkages were seen to compare excellently
with measurements on amorphous polystyrene (6).
Since the analysis uses local values for pressure, temperature, and crystallization (or reaction) effects, it
can be coupled with any simulation code furnishing
these data.
F
In the present paper, our theory will be specialized
for the case in which shrinkage occurs before mold
opening. In addition, effects of mold deformation and
friction between polymer and mold wall will be discussed.
BASIC MODEL EQUATIONS
Consider a thin slab that is cooled from the outside.
Let z denote the thickness coordinate, ranging from
zero at the surface to 0 at the midplane, and let x and
y be two mutually perpendicular directions in the
plane of the slab. The partial solidified slab consists of
two solid layers with polymer melt in between. If there
is symmetry with respect to the midplane, the two
solid layers have the same thickness, z,(x, t), and the
molten core has thickness 2 ( 0 - 2,). The solid layers
shrink uniformly in the x and y directions (no thickness dependence) and are treated as being purely
elastic. Although deviatoric stress component in the
melt could be taken into account, they are negligible
with respect to stress components in the solid under
standard processing conditions. Only isotropic
stresses (hydrostatic pressure) are therefore considered in the melt. Further, for simplicity all stresses
and strains are assumed to be equibiaxial (i.e. 0;= cry,,
and sU = eyU)and the modulus of elasticity of the solid
is considered to be constant. If desired, these latter
restrictions can be dropped (see Ref. 5).
At each instant, the stress in thickness direction is
a, = -P(x, t). Applying Hooke's law, the stress along
length direction in the solid can be written as:
To whom correspondence should be addressed. Present address: Dept. of Mechanical Eng.. Twente Unlversity. PO Box 217 Enschede. The Netherlands.
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
Vol. 3s, NO. 15
2041
G. Titomanlio and K.
M.B.Jansen
In these equations the dot indicates differentiation
with respect to time and the bar stands for averaging
over the solidified layer thickness. After integration of
the above equation, the strain change in the solid
layer becomes:
where s,( x, z , t ) is given by
z, t ) stands for the observable strain, and
Here E,(x,
E'(x,
I
t ) = - pP dt
(5)
- ~ P ( xt ,)
where to is an arbitrary starting time,
is the hydrostatic strain with pas the linear compress-
ibility and t,,(x,z , t ) denotes the local instant of solidification. Further, d denotes the sum of the isotropic
shrinkage effects (in particular thermal shrinkage eT =
a[T - T,],
crystallization shrinkage E' = -CJ[ - &,I and
reaction shrinkage 8 = -C,[( - (,I). The subscripts s
in T,,[, and 6, refer to the respective values at the
solid melt interface. Remark that although & A x ,z , t )
depends on z , ,E = d Q d t is independent of the thickness coordinate. The unknown quantity E,(x, z , t ) is
obtained from the force balance, which can be written
as :
solid
S,
dZ
= F,(x,
t ) , F,
= F,
+ Fjr+ D P .
(31
where F, and Ffr stand for the force exerted by the
mold wall and the friction force in x direction, respectively. The DP term actually consists of two parts, the
first one being the stretching force exerted by the melt,
(D- Z J P , and the second one follows from the definition of s, in Eq 1 . All forces are expressed per unit
width. The friction force in general depends on the
pressure values along the flowpath. If x = 0 corresponds to the gate and x = L to the free moving end,
one may write for the friction force per unit width
(6)
stands for the strain during free shrinkage, and
(7)
The strain change &I:, is constant along the thickness and gives the local strain change in length direction of all solid layer. The superscript s refers to the
solidified layer. If in the definitions above, to = 0 refers
to the start of solidification, the reference length (and
width) coincide with the mold dimensions in injection
molding. This local shrinkage with respect to the mold
dimensions will be denoted as 8, and S,, thus
S,(x, t ) =.;I&,:
In addition to the local strain change (i.e. the one that
depends on x and y) one may define the global deformations (or shrinkages) as:
(S,)(t) =L
P(x,t ) du
~ r ( P ( x t,) ) ( L- XI,
loL
8, dx, solidified layer
(4)
8, d y .
W
with qfras the friction factor. The brackets (.) imply an
average over the flowpath. The last term of Eq 4 may
serve as a first approximation. Note that Ffr can be
either positive or negative depending on the direction
of movement. The force balance has to be evaluated
after every small time interval dt. We therefore differentiate Eq 3 with respect to time and write:
solidified layer
(8a)
(8b)
Thus the shrinkages (S,) and (8,) are functions of time
only. In case the strain distribution E:, is uniform
along length direction, E&,,S,, and (8,) will only differ
by a constant. A similar argumentation holds for
E;,, 6,.
and (6,). Thickness shrinkage, S,(x, t ) , is
obtained b y integrating E,,(x, z , t ) over both fluid
and solid part:
SJx, t) = D
(I. + 1EL 1'
dz
dz) , solid and fluid
n
from which &,(x,
t ) is readily solved:
-.
&,,(x, t ) = & f i ' + 2042
1-v
'
(FX/zs).
Stress distributions are obtained by substituting
expressions for E,, such as Eqs 5 and 6, in the stress
equation, Eq 2. The stress distribution in the free
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1!3H, VOI.36,NO. 15
In-Mold Shrinkage and Stress Prediction in Injection Molding
quenching case is given as:
E
free
s,,
(x,z , t ) = __
1-
[ E xfree
,
-
EJ]
I
, free quenching
trz
(gal
In injection molding with zero shrinkage in the mold,
the strain before ejection is simply given by ,& = 0,
while during ejection the strain changes by a finite
value. After ejection, strain changes follow a free
shrinkage pattern. As is shown in Ref. 5 this results in
IMO case before ejection
s,
IMO'
(x,Z , t ) = K [ P ,
- P,]
E
+1-v
~
[ZJ(t)-
(9b)
d]
same, but after ejection (9c)
Here K = (1 - 2v)/(1 - v), Ps(x, z ) = ox,t,,) is the local
solidification pressure and the bar stands for averaging over the solidified layer. For further reference also
the expressions for product shrinkage in length and
thickness direction are given:
(lob)
Here t, denotes the moment of ejection, which is assumed to occur after complete solidification.
In practice the mold will deform slightly under the
holding pressure applied. Since deformations are
small, the effects are linear and one may write for the
additional dimensional product thickness
(11)
where E, is the elastic modulus of steel (about 2.1 X
lo5 MPa), Pfl is the pressure at the instant the gate
freezes off and C, is a constant that depends on the
mold geometry. An estimate can be obtained by considering the bending of a plate with thickness s and
span length 1. Then C, is given as (7,8)
( 1 + v)1"
+ 1.2
S
[x - x"1
where x is the dimensionless span length coordinate
(ranging from 0 to l),n = 1 for a clamped plate and n =
2 for a simply supported plate. Typically Cd0.5) is of
order 0.1 m. Another effect of this mold deformation is
that it causes a slower decrease of the actual cavity
pressure. This is one of the reasons for the often observed discrepancy between measured and simulated
cavity pressures.
In the next section the injection molding process will
be studied in more detail. Special attention will be paid
to cases of non-zero shrinkage before ejection (8, # 0
and/or 8, # 0).This analysis is therefore an extension of
the previous analysis (5).
INJECTION MOLDING WITH SHRINKAGE INSIDE
MOLD
DifFeremt Caws of Shrinkage During Molding
Injection molding is a widely used polymer process
that roughly consists of three stages: filling, packing/
holding, and cooling. In the filling stage, a hot polymer
melt rapidly fills a cold mold with a cavity of the product shape desired. During the holding stage extra material is forced into the cavity in order to compensate
for shrinkage during solidification. Often the pressure
during holding is much higher than during filling.
After a certain time the cavity entrance (or gate)
freezes and no more material can enter the cavity. The
product, however, remains in the mold until it is sufficiently solidified. This last part of the molding process is called the cooling stage. After mold opening
and ejection the product is allowed to cool to room
temperature.
J u s t after the filling stage the injected product may
be regarded as a fluid surrounded by a thin, elastic
shell. Because of the melt pressure, the shell tends to
be stretched in all three directions. As time proceeds,
the shrinkage forces (in length and width direction) in
the growing solidified layers increase, while simultaneously the pressure stretching effect decreases. It is
therefore possible that at some time the shrinkage
forces become larger than the melt stretching force,
causing the product to shrink inside the mold. The
onset of length (or width) shrinkage, written as t: (respectively t;), then follows from the force balance (Eq
3 ) with F , = 0. For simplicity it is assumed in the
following that t: equals t:. Thickness shrinkage on the
other hand, starts at a time value (denoted a s tZ)when
the local pressure vanishes, thus: f i t : ) = 0. Several
shrinkage regimes can be identified, depending on
which of the two times (t: or t:) is larger.
i) IMO-case. If length and thickness shrinkage do
not take place before ejection, the no-shrinkage case
described by E q s 9b and c and 1 Oa and b is recovered.
Note that, since it was assumed that stresses do not
relax in the solid, the instant of complete solidification, denoted as tsl, may be used instead oft, both in
the equations for stresses and shrinkage. This is also
true for the cases listed below.
ii) t: < t,, t: 2 t, (IMx-case).Length shrinkage starts
before ejection while thickness shrinkage does not yet
occur. This can be the case for a material with a
relatively large thermal shrinkage that is molded under low holding pressure or for a part of a product with
a solidified cross section upstream.
iii) t: < t,, t: 2 t, (IMz-case).Thickness shrinkage
before ejection and length shrinkage after ejection.
This case occurs most likely for a product part whose
length shrinkage is hindered by geometrical con-
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST lm, VOl. 3s,
No. 15
2043
G. Titornanllo and K. M. B. Jansen
straints and that is molded under low holding pressures; in any case, local cavity pressure must vanish
before ejection (or before complete solidification).
iua) t: < tr < t, ( I b - c a s e ) . Length shrinkage followed by thickness shrinkage; both before ejection.
Also this case may occur when holding pressure is
low. Length shrinkage starts a s the pressure reduces
to a small but non-zero value. If the pressure then
continues to drop, the length shrinkage is followed by
thickness shrinkage. Remark that after the onset of
both length and thickness shrinkage no external
forces act upon the product. The product then shrinks
according to the free shrinkage curve.
iub) < < t, (IMw-case]. First thickness shrinkage, then length shrinkage; both before ejection. Only
possible in situations where the outer layers are solidified under large pressures and thus contain a large
amount of compressive strain. If the pressure then
suddenly vanishes, thickness shrinkage will start,
while length shrinkage does not yet occur since all
thermal shrinkage is counteracted by the frozen-in
compressive strain.
In the next three paragraphs the shrinkage curves
of each of the cases ti), iii], and iu] will be examined.
Thickness Shrinkage in the Mold ( t c t,. t ,* 2 t,.
IMz-case)
In this paragraph the case is considered in which
the product remains fixed in length and width direction by geometrical constraints, but is allowed to
shrink in thickness direction a s soon a s the pressure
vanishes. The amount of material in the mold remains
constant afterwards.
As was mentioned above, the onset of thickness
shrinkage is determined by the condition
P(tZ) = 0, onset of thickness shrinkage
(131
Since here 6, (and iyy)
remain zero until ejection, the
length (and width) shrinkages are given by Eq 1O a and
the stress distribution in x direction by Eq 9b. The
thickness shrinkage until ejection is determined by
considering the strains in both solid and fluid (superscripts s andf,respectively):
At the instant of ejection there can be some thickness expansion. This effect was already calculated in
Ref. 5 a s a function of ejection pressure, which is zero
here. Also a n expression for shrinkage after ejection
was obtained in the reference cited. Summing up the
three contributions one obtains:
t > t,
(15)
where & = z,/D. The thickness shrinkage reported
above is, in fact, equal to the thickness shrinkage of
injection molding with hindered length shrinkage in
the mold (Eq 1O b ) , plus the shrinkage term given by Eq
14 above.
The length shrinkage change upon ejection is given
by Eq 35a of Ref. 5 (with P, = 0) and by the length
shrinkage after ejection, ~ 5
The. final length shrinkage in the case examined here, IMz, is therefore given
by the same equation a s that of the IMO case (Eq 1Oa).
Furthermore, the residual stress equations in the IMz
case are identical with those of the IMO case (Eq 9c).
Remark that, in fact, the stress distribution does not
depend on E, (see Eq 2).
Effect of Holding Time on Shrinkage and Stress
Distrtbution in IMz C a s e
In order to study the implication of the above equations, we will consider the molding of a typical polystyrene resin characterized by: as = 1.10-4 K-’,a’ =
2.10-4K-’, E = 4000 MPa, v = 0.35, p’ = 2p” = 1.5 X
MPa-’, C,, = C, = 0, a = lo-’ m2/s, and T, =
100°C. Cooling is assumed to be one-dimensional
with constant wall temperature, T, = 50°C and initial
temperature TL= 250°C. The pressure curve is considered to be a s follows:
P( t 5 t,,) = p o t / tpo linear increase: filling
p(t,o < t
5
tpl) = P,,
constant; holding
P(t,, < t 5 t;, = Pmm(l- c[t - tPJ)
linear decrease; cooling
dZ= ( € J f + 8.’)
P ( t 2 t:) = 0
fluid
Substituting d‘ = 0, EL = 0 and averaging over the
total thickness then results in
where & = z,/D and S p is the thickness shrinkage in
The
free quenching as given by Eq 9a with E,=.‘,.
second term in Eq 14 accounts for the Poisson contraction because of a hindered length shrinkage. The
superscript IMz refers to the injection molding case
with thickness shrinkage only.
2044
(16)
where the constants are taken a s Po = 5 MPa, P,, =
20 MPa, tpo = 1.0 s, and c = 0.4 s-’. Length shrinkage
is prevented by geometrical constraints and ejection
occurs at t, = 10 s. Remember that the shrinkage and
stress equations discussed here are valid for any
(complex) temperature and pressure history. In fact,
the schematic temperature and pressure history chosen here, serve for illustrative purposes only.
The results of the length shrinkage calculations according to Eq 10a are shown in F@.l a for four different values of holding time, defined a s t, = tpl - tp0.
Depending on t,, thickness shrinkage (F@.Ib, E q s 1 3
and 14) starts between 4.5 s and 7.5 S , i.e. before
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
VOl, 36,NO. 15
In-Mold Shrinkage and Stress Prediction in Injection Molding
0.20
I
Isolidification
/
completely solidified
I
0.00 1
-
1
I
I
I
-0.20 -
6
P
E.
I
I
I
-0.40
-
3
I
I
I
I
I
I
-0.60
-0.80
01.' 00
compression
1 .oo
0.50
1.50
0.00
2.00
0.20
0.40
0.60
wall
0.80
1.00
mid
Z lD
at/D2 (-1
Fig. 2. Residual stress distributions corresponding to the
shrinkage curves of Fig. 1 a (Mzcase).
0.20
1
-0.60
tribution equals that of the corresponding IMO case.
The discontinuity at z/D = 0.142 reflects the pressure
jump at the moment of cavity filling, t@. Note that the
part of the stress distribution near z/D = 1 is constant
for all shrinkage curves with < tsl. This corresponds
to stress formation at zero pressure. For the th = 4.0 s
case the flat central part is absent since there solidification was complete before the pressure dropped to
zero.
-0.80
0.00
Injection Moldiug rith Length Shrinkage in Hold
[t,*< te 5 t:* mx-came)
Isolidification
/
completely solidified
0.00
-0.20
-0.40
0.50
1 .oo
1.50
2.00
at/D2 [-I
Fig. I . Shrinkage plots for lMzcase as afunction of dimensionless time. a: Length shrinkage; b: Thickness shrinkage
(Eqs 14 and 15). Numbers correspond to dttferent holding
times and dots refer t o m & shrinkage values. Process and
material parameters are given in text. Dots refer t o m
shrinkage values.
length shrinkage, which is enforced to be zero until t,.
The change in length shrinkage upon ejection is accompanied by a thickness expansion of -0.3 to 0.4%
due to Poisson's effect (Fig.Ib).It is clear from Fig. la
that the longer the holding time, the less the final
length shrinkage is. The final length shrinkage values
(shown a s dots) vary between -0.478% and -0.406%.
Compared to thickness shrinkage the variation of final
length shrinkage with holding time is, however, small.
There the final shrinkage values vary between
-0.510% and -0.092%. As was already mentioned in
Ref. 5, this clearly shows that thickness and length
shrinkage are not closely related (aswould follow from
a PVT analysis), but vary more or less independently.
In fact, depending on the details of the pressure history and material parameters, the final thickness
shrinkage in the IMz case can be either negative or
positive.
The residual stress distribution for the IMz case is
shown in Q. 2. A s mentioned above this stress disPOLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
Here it is assumed that no thickness shrinkage occurs until ejection, while shrinkage in length direction
may occur before ejection. The onset of length shrinkage (denoted as t:) follows from the force balance, Eq
3. Before the onset of shrinkage the sum of F, and
DP(t)is larger than the integral over ,
s and the difference is absorbed by the mold wall (F, < 0). At the
onset of length shrinkage F, vanishes. Since, at that
moment the stress distribution is still given by
,'::s
the condition for the onset of shrinkage becomes:
The superscript IMx stands for the injection molding
case with shrinkage in length direction. Remind that
the contribution of F, vanishes in the I M x case. The
absolute bars are used to emphasize that the friction
force is positive since it tends to oppose the direction
of movement (which is negative here). Equation 17 is
a n implicit relation that has to be solved by varying t:
until the left hand side equals zero.
A first estimate for determining if shrinkage effects
are important for final properties (i.e. if shrinkage
starts before complete solidification) can be obtained
by assuming that F, = 0 and considering the scaling
factors of the melt stretching force and the isotropic
shrinkage effects. The isotropic shrinkage force (due
VOI. 3s, NO. 15
2045
G . Titornanlw a n d K . M . B. Jansen
to thermal, crystallization, and reaction shrinkage) is
proportional to E(a[T, - T,] + C , , r + C,("), while the
melt stretching force scales with the maximum cavity
pressure, P,,. In order to be consistent with our previous definition (see Ref. 5) we introduce p = ( 1 2 v ) / E and write for the ratio between the scaling factors (the pressure number):
for the IMO case where the product can either expand
or shrink along x at ejection (see Ref. 5).
The change in thickness shrinkage, 6, upon ejection
follows from Eqs 8 c and 20b and c, while the thickness
shrinkage after ejection is just the free thickness
shrinkage. Thus:
A small pressure number then indicates a possible
shrinkage before complete solidification, while if Np >>
1 it is unlikely that length shrinkage occurs in the
mold. The effect of friction corresponds to an increase
of the pressure number and thus reduces the likeliness of length shrinkage. It should be noted that Eq 18
serves only as a rough indication of the onset of
shrinkage, since it neglects all details of the pressure
history.
Shrinkage Calculations
Next, we will calculate the length shrinkage as a
function of time. As before, the shrinkage curve can be
divided in intervals. In the first interval (0 5 t 5 t:) the
shrinkage (rate)is zero, while after the onset of shrinkage (t ? t:) Eq 5 must be used:
E IxM X lo=o,
t:
osttt:
( 19a)
&:""Ii
+ € P ( t ) + __
- g ( t ) ) t,
E
= (E::(t)
t ? t:
From Eq 22b we see that in the I M x case considered
here, the product always expands in thickness direction upon ejection. Note that the equation only depends on the fluid compressibility p' if ejected before
complete solidification. With the shrinkage curves obtained here one can easily calculate the stress distributions for the injection molding case with partial
shrinkage.
Stress Distributionf o r C a s e of Length Shrinkage
Inside the Mold
The stress distribution before ejection is obtained
by substituting the shrinkage equation, Eq 19, in the
stress equation, Eq 2. After rearranging one then obtains
t:
(19b)
Remark that Eq 19b remains valid during and after
ejection. At ejection the last two terms of Eq 19b instantaneously change to zero and the product afterwards shrinks according to the free shrinkage curve.
The strain change upon ejection can be obtained by
substituting API: = P(te)=P, and Ag = -Fim(t,)/z, in Eq
19b. An alternative is to use Hooke's law with ASn =
SJt,) and A3= = -P,:
E rite
I
G
E,,I~
1-v
Ps P e - 7 S d t e ) solid part
= pSP,
+
2v
S,(t,)
= PfPe
Here z: = z,(t:). In a similar way the stress distribution
after ejection can be derived:
(20a)
solid part
(20b)
fluid part
(20c)
Using Eq 20a and writing z, = z,(t,), we may construct
the rest of the IMx-shrinkage curve:
~ I MxX It,
xc
= PsPe -
1-v
7
F i M x ( t , ) / z ,5 0
t = tk
(21a)
For t: < t,, Pxm= DP, + IFfA and since D / z , 2 1 it is not
difficult to show that the right hand side of Eq 21a is
always negative (or zero) causing product shrinkage
upon ejection. This is in contrast to previous findings
2046
where sEoand s? are given by E q 9. Note that the
stress curve is continuous at zz, but discontinuous
at z, (in case of premature ejection only). This discontinuity is given by
discontinuity at z,
(25)
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1996, VOI. 36,NO. 15
In-Mold Shrinkage and Stress Prediction in Injection Molding
If the product is ejected before the (calculated)onset of
shrinkage, shrinkage is enforced to start at the ejection time (i.e. t: = t,). In that case Eq 24 reduces to Eq
9c (if t, 2 tsl) or Eq 39 of Ref. 5 (if t, < tsl). Eq 24 can
thus be considered a s a generalization of the limiting
cases i) and i f ) of the reference cited. Further, since
it can be proven that the integral of s r ' between 0
and z , equals F:*(t), a s it should be. Note that Eq 24
is in fact the analytical form of the numerical iteration scheme presented by Titomanlio et al. (9).
Effect of Holding Time in Case of Length Shrinkage
in Mold
Here we consider a molding process similar to that
of the previous example but with a smaller pressure
decay constant, c = 0.2 s - I . In this way it is ensured
0.20 1
1
I
I
Lsolidification
1
I
completely solidified
0.00
-0.20
-0.40
-0.60
0.00
1 .oo
0.50
1.50
2.00
at/D2 [-I
that no thickness shrinkage occurs before complete
solidification. Friction at the mold surface is assumed
to be absent, thus predicted length shrinkage effects
will be maximal.
By applying Eq I 7 it follows that the onset of length
shrinkage occurs between t: = 5.31 s and 7.04s for
holding times between 1.O s and 4.0 s. For the shortest three holding times shrinkage starts before complete solidification (=6.60s),while for t, = 4 s it starts
after complete solidification.The length and thickness
shrinkage curves are drawn in Figs. 3a and b, respectively. The length shrinkage curves show a change in
slope at the instant the pressure reaches zero (onset of
thickness shrinkage at 7, 8, 9, and 10 s,respectively).
Ejection has no effect on length or thickness shrinkage since both pressure and F, are already zero for all
four holding times (see Eqs 20a and 21b). Final length
shrinkages vary between -0.510 and -0.390% for t h
between 1 s and 4 s, respectively, while corresponding
thickness dimensions lie between -0.137 and
-0.288%.Compared to the case considered in Fig. 1
the effect of pressure on final thickness shrinkage
value is reduced considerably. This is mainly due to
the smaller pressure decay in the example discussed
here.
The residual stress distributions, obtained by substituting the length shrinkage curves of Fig. 3a in Eqs
I and 2, are shown in Fig. 4. Note that although the
length shrinkage at complete solidification of case th =
2.0 s is small, its effect on the residual stress distribution is considerable (change in slope at z / D = 0.66).
For the th = 1.0 s case the central part of the stress
distribution ( z / D2 0.52) shows a n even larger effect.
Length shrinkage in the mold thus increases the
amount of tensile stresses in the central zone, thereby
reducing the tensile stresses at the product surface.
This effect will be larger the smaller the pressure number (Eq 10).
Mixed Case (t:, t: < t,. IMxz and I M a - c a s e )
0.20
Lsolidification j
I
I
0.10
0.00
I
completely solidified
Here it is assumed that geometrical constraints are
absent and both length and thickness shrinkage are
10
I
I
I
I
-am
-0.10
-0.20
3
x
-0.30
I
D
tsi
1-
-0.40
0.00
0.50
1 .oo
1.50
2.00
at/D2 [-I
Fig. 3. Shrinkage plots for IMwcase as afunction of dimensionless time. a: Length shrinkage (Eqs 19 and 21 ); b: Thickness shrinkage (Eq 22).Numbers correspond to diierent holding times. Process and material parameters given in text. Dots
refer toftnal shrinkage values.
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
0.00
0.20
0.40
0.60
wall
0.80
1.00
mid
ZlD
Fig. 4 . Residual stress distributions corresponding to the
shrinkage curves of Fig. 3a (LMxcase).
VOI.36, NO. 15
2047
G. Titornanlio and K . M. B. Jansen
possible. One than has to distinguish between the
subcase that length shrinkage occurs before thickness shrinkage (t: < t:) and vice versa (t: < t:). The
latter is only possible if the product is first solidifled
under relatively high pressure ( N p >> l),followed by a
quick pressure drop (with backflow). For the length
shrinkage curves in the two subcases the following
equations hold:
Here the superscripts IMxz and I M z x are used to distinguish these cases from the previous ones. Note that
both Eqs 26 are special cases of Eq 19 and that after
the onset of both length and thickness shrinkage,
shrinkage equals the free shrinkage curve. The instant of ejection therefore has no influence on the
shrinkage.
By substituting Eqs 26 and 8 = 0 in the equations
for .sZZ and &, one obtains for the thickness shrinkages:
if
t; < t:.
For the subcase t: < t:, the residual stress distribution is given by Eqs 24a and b if z 5 z,(t:) and by the
free quenching stresses if z > z,(t:). In fact, since g(t >
t:) = 0 and F,(t,I = 0, Eq 24 may also be used for times
larger than
In the other subcase, t: < t:, a similar
reasoning holds and the stress distribution is given by
Eqs 23 or 24 with all g(t) and F, terms equated to zero.
Summarizing, one can say that the equations for
length shrinkage (and those for residual stresses) of
the IMx case do not change if thickness shrinkage
occurs before ejection. The equations for thickness
shrinkage, however, are different from both those of
the I M x and IMz case and are given by Eqs 27.
c.
EFFECT OF FRICTION
In order to obtain an idea of the magnitude of the
friction effect, we measured the coefficients of static
friction by tilting a system consisting of a steel bar,
test specimen, and dead weight, until sliding took
place. qJris then given by the tangent of the tilting
angle. As can be seen in Table 1 , the measured friction
coefficients range from 0.12 to 0.18. These results are
considerably lower than most literature values. Most
probably this is due to the fact that we used a steel bar
Table 1. Measured Static Friction Coefficientsfor PolymerSteel Combinations. The Steel (DIN-X38CrMoVlsl) has a
Surface Roughness of R, = 0.14 f 0.02.
2048
Materials
llfr [-I
PMMA-steel
PS-steel
PC-steel
PE-steel
0.15 t 0.01
0.18 -t 0.01
0.13 2 0.01
0.12 2 0.01
with a rather smooth surface finish, similar to that of
the mold cavity.
An estimate of the order of magnitude of the different terms in the force balance, Eq 3, can be obtained
by taking qJr= 0.15, L = 0.1 m, D =
m, and a
cavity pressure of 20 MPa. The friction force per unit
width (Eq 4 )forx = 0 then becomes Ffr= 3 X lo5N/m.
The DP(t) term in the force balance is 2 X lo4N/m at
maximum, which is a n order of magnitude smaller
than the friction force. As a first estimate of the integral over the stresses in the solid layer, the maximum
thermal shrinkage stresses are taken: &(T, - T J D /
(1 - 4.With the data used in the previous examples
this becomes --3 X lo4N/m, which again is a n order
of magnitude smaller than the friction term at x = 0.
Therefore, the friction term is the governing term in
the force balance as long a s the pressure remains
larger than a few MPa.
The importance of mold friction for the resulting
stress distribution can also be demonstrated by considering the pressure curve already discussed in the
I M x case with t, = 1 s. In Fig. 5a the length shrinkage
curves are plotted with qJ4L - x ) / D a s a parameter.
The full line is the case without friction and corresponds to the full line in Fig. 3a. All other shrinkage
curves start later. At t = 7 s the pressure becomes
zero, causing a change in the slopes of the shrinkage
curves. All thickness shrinkage curves are identical,
since at the onset of thickness shrinkage (7.0 s ) , the
product is no longer constrained (t: < t:) and will
shrink freely. Corresponding residual stress distributions are depicted in Fig. 6. The effect of friction is to
decrease both the compressive minimum and tensile
stresses in the center and, on the other hand, increase
the tensile stresses at the surface. For the zero friction
case the surface stresses become negative (compressive) as was already seen in Fig. 4. For friction values
larger than 5. shrinkage starts after complete solidification (Fig.5a) and has no more effect on the residual
stresses. The stress distribution therefore becomes
that of the IMO case. For longer holding times this
occurs for even smaller friction values (frictionparameter equal to 0.6, if th = 2.0 s). Since injection molded
products usually have a rather large L / D , the parameter V,~(L- x ) / D may become much larger than the
critical value 5 (0.6).Increasing holding time or holding pressure will reduce the critical friction value even
more. This means that under normal injection molding conditions, length shrinkage in the mold may effectively be prevented by friction at the mold surface (if
not already prevented by geometrical constraints).
CONCLUSIONS
Our theory for predicting final dimensions and
stress distributions of polymer products was extended
to include the effects of shrinkage in the mold in thickness or length direction. The different subcases that
were considered were: only thickness shrinkage: only
length shrinkage; first length shrinkage then thickness shrinkage, and, vice versa, first thickness
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
Yo/. 36,NO. 15
In-Mold Shrinkage and Stress Prediction in Injection Molding
Q
a
0.00
1 .oo
0.50
0.00
2.00
1.50
0.20
0.40
0.60
0.80
1.00
mid
wall
z/D
at/D2 [-]
0.10
length shrinkage was studied. From this analysis it
followed that friction can be dominant and prevent
length shrinkage until pressure drops to a few MPa.
0.00
ACKNOWLEDGMENTS
0.20
solidification
completely solidified
This work was supported by the Italian Research
Council CNR (grant no. 93.03247) and by the Bright
Euram project no. ERBBRE-2CT933027.The authors
acknowledge CNR and the European Commission for
their financial support.
-0.10
-0.20
-0.30
-0.40~r~
t*i
I
t,
,
I
I
I
I
,
,
,
REFERENCES
#
Fig. 5. Shrinkage plots for IMx case for different values of
q,,(L - x ) / D(frictionpararnetc?r).
a:Length shrinkage; b: Thickness shrinkage. Holding tirne t, = 1.0 s, other conditions
identical to those of Figs. 3 and 4. Dots refer toJnal shrinkage
values.
shrinkage, followed by length shrinkage. For all nontrivial cases both expressions for final length shrinkage, thickness shrinkage, and stress distribution were
given.
It was seen that in-mobd length shrinkage may have
a severe effect on both residual stress distribution and
final product length. Furi:her, the effect of friction between polymer and mold wall on the occurrence of
Germany (1987).
3. S.Y. Yang and M. Y. Hon, Tenth Annual PPS Meeting,
Akron, Ohio (1994).
4. J. S. Yu and D. M. Kaylon, Polyrn. Eng. Sci., 31, 153
(1991).
5. K. M. B.Jansen and G. Titomanlio, Polyrn. Eng. Sci., this
issue.
6. K. M. B. Jansen, et al., European PPS Meeting, Stuttgart,
pp. 2.2(1995).
7. S . Timoshenko, Strength of Materials 1, Van Nostrand,
London (1955).
8. G. Menges and P. Mohren, How to Make lnjection Molds,
2nd Ed., Hanser, Munich (1993).
9. G. Titomanlio and V. Brucato, Intern. Polyrn. Process., 1.
55 (1987).
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1998, Vol. 36,No. 15
Revision received October 1995
2049