Effect of Pressure History on Shrinkage and Residual
Stresses-Injection Molding With Constrained
Shrinkage
K. M.B. JANSEN* and G . TITOMANLIO
Dipartimento di Ingegneria Chimica ed Alimentare
Universitadi Salerno
84084 F’isciano (SA),Italy
A simple elastic model for residual stresses and shrinkage of a thin solidifying
product is proposed. The model results in expressions ready to be used. It accounts
for shrinkage anisotropy between in-plane and thickness directions, caused by
different constraints in deformation. The model uses local values for temperature,
pressure, crystallization, and (if present) extent of reaction, which belong to the
standard output of most simulation codes. It is therefore assumed to be valid also
for complex shaped products.
INTRODUCTION
n most polymer processes, the plastic material is
deformed in the desired shape while being in the
molten state. In the next processing step this shape is
rapidly frozen-in, often under pressure. Examples of
such processes are injection molding, compression
molding, and extrusion. During solidification stresses
build-up while, simultaneously, the product dimensions change (freequenching or compression molding)
or are held constant (injection molding). After release
of eventual constraints, final product dimensions may
differ considerably from the initial ones. The above
discussed processing stresses are of importance since
they add up to the mechanical stresses that a product
may experience during use. Knowledge of dimensional
accuracy on the other hand, is indispensable when
molding high precision products.
Present shrinkage theories usually rely on thermodynamic considerations only. Following the PVT-diagram from the glass transition to ambient conditions,
an average value for the final product volume at a
certain location is obtained [Isayev et al. (1).Schmidt
(2). Yang et al. (3)].Those models do not take care of
details in shrinkage development and are obviously
limited to predictions of isotropic shrinkage values. In
reality, high anisotropy in shrinkage is usually observed (2);indeed, thin products are free to shrink in
thickness direction, while in-plane shrinkage is restricted by the layers already solidified. The limitations of those theories can be removed by combining
I
Present address: Dept. of Mechanlcal Eng.. Twente Univ.. The Netherlands
the thermodynamic analysis mentioned above with a
thermomechanical one.
In a similar way, the PVT-diagram can be used to
calculate density distributions in products solidified
under continuously changing pressures. Greener (41
used this idea to predict density distributions in injection molding products.
At the same time, theories were developed for calculating residual stress distributions because of polymer processing. Struik (5)proposed a simple analytical model for predicting free quenching stresses, thus
coupling restricted thermal shrinkage with stress formation. A similar (numerical) formulation is used by
Goslinga (6)and Menges (7);if desired, relaxation effects can be included (5,6, 8-10). An excellent review
on models used for stress formation under atmospheric pressure is given by Isayev ( 1 1).
For processes such as injection molding the pressure effect has to be included in the analysis since
solidification may occur under (high) pressure. This
was first recognized by Titomanlio (12). If shrinkage
during solidification is inhibited, the model simplifies
considerably. Brucato (13)and Jansen (14) showed
that in that case the residual stress distribution is
determined by the pressure profile rather than by
thermal shrinkage. Models lacking the pressure contribution (5-10) are therefore not suitable for predicting stresses in injection molding. Recently more sophisticated viscoelastic models have been developed
that combine viscoelastic relaxation processes with
the effect of a non-constant pressure during solidification [Baaijens (15), Douven (16). Rezayat (17)l. All
models (12-17), however, assume a constant product
length during solidification.
POLYMER ENGINEERING AND SCIENCE. MID-AUGUST 1996, Vol. 3s, NO. 15
2029
K. M. B. Jansen and G . Titornanlio
Fig. 1 . Schematic view of constrained solidifying slab.
This paper shows the link between simple theories
predicting shrinkage (1-3), density distributions (41,
and residual stresses (5-7, 12-14) in polymer products. For all these effects (including shrinkage anisotropy), simple analytical expressions will be obtained.
BASIC THEORY
Consider a thin slab that is cooled from the outside.
Let z denote the thickness coordinate, ranging from
zero at the surface to D in the center, and let x and y
be two mutually perpendicular directions in the plane
of the slab (see Fig. I). The position of the solid-melt
interface will be indicated as z,(t).
The analyses will be carried out under the following
assumptions:
1. Continuity of stress and strain at the solid-melt
interface;
2. Shear stress components can be neglected in the
solidified layer;
3. Uniform deformation of the solidified layer (i.e. the
deformation in x and y direction does not depend
on 2);
4. The normal stress, a,, is independent of z;
5. No out of plane deformation during solidification;
6. The solid polymer is elastic, while the melt is considered unable to withstand relevant tensile stresses;
7. Frozen-in normal (or flow-induced) stresses can be
neglected;
8. Temperature, pressure, position of solid-melt interface, crystallization shrinkage, and reaction
shrinkage are known.
Assumptions 1 through 5 are standard for these
kind of solidification theories 1e.g. Struik (51, Baaijens
(15)]and do not need much discussion. Note that, in
fact, assumption 3 follows directly from assumption 2.
As was shown by Baaijens (15)flow induced stresses
are typically about 100 times lower in magnitude than
the rest of the residual contributions, thus justifying
2030
assumption 7. Assumption 6, however, seems to be a
rather strong simplification of the complex viscoelastic behavior inherent to polymeric materials. It implies
that in the solid, all relaxation effects are neglected.
The model thus overestimates calculated stress and
deformation values. However, since relaxation times
increase exponentially with decreasing temperature,
and the temperature drops rather quickly while passing Ts,relaxation processes are effectivelyhalted when
cooling through T,. A more elaborated justification is
given by Buckley (10). In addition, Douven (16)
showed for injection molding that full viscoelastic calculations result in a stress distribution similar to that
calculated with a simple elastic model (see Fig. 2).One
thus may conclude that for processes involving relatively high cooling rates, the model used above constitutes a reasonably accurate approximation of the viscoelastic behavior.
All data mentioned in assumption 8 are standard
output of numerical simulation codes. It depends on
the software at hand, which effects are being taken
into account (e.g. convection, dissipation, and crystallization effects on temperature and pressure, or cooling rate on glass transition). In this paper simple temperature and pressure histories will be used for
illustrative purposes only.
Derivation of Stress Equation
The deformation of a solid body can be characterized by its displacements, u, (i = x , y, z).In solid state
mechanics, however, it is more common to use strains
instead of displacements. The axial strain components are given as [see e.g. Landau et al. (IS)]:
The strain component st, can thus be interpreted a s
the local relative displacement of a small dash, with
length Ax: at its reference state. Vice versa, the dis-
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1998, Vol. 36,NO. 15
Effect of Pressure History
and aging. We prefer the hydrostatic strain, 8,to be
written separate from the r‘ summation. The isotropic
shrinkage effects mentioned here are defined as:
y, z , t )
E;I(X,
a d T G a [ T ( x , y. z , t ) - T,] < 0
i = x , y, z
(5a)
Pcr
- Pam
c , , = ____
3Pam
(5b)
-15
0.00 0.20 0.40 0.60 0.80 1.00
mid
wall
zl D
Fig. 2. Comparison between elastic prediction Null line) and
uiscoelastic prediction [dashed line) of residual stresses in
injection molding; after Ref. 16.
placements can be seen a s the integral of the strain
components, starting from a fvred point.
According to assumption 2 we have uw = uyz= a, =
0 and .sxy= cyz =
= 0, where u denotes the stress.
Substituting a, = -P(x, y, t )in Hooke’s law, the stress
equations can be cast in the following form
uxx(x.y, z . t ) = - P ( x , y, tl
+ s x x ( x ,y, z , t )
uyy(x.y. z. t ) = -P(x, y, t ) + s y y ( x y,
, z, t)
U A X ,
y, z , tl
(2)
- P ( x , y. t )
=
where the stress components s, are given as
7.
I3
s,,(x, y, z , t ) = l_v2 [ E * ,
Syy(x.y, Z , tl
=
~
E
1-2
+ VE:,]
[&Ey + v&]
solid part
solid part
(3)
s x x ( x ,y, 2, t ) = 0.
In the fluid all stresses equal -P(x, y, t )and thus s, =
s, = 0. Further:
syy=
E?I(X.
y. z , t ) = E l [ - (41+ PI.
(4)
Here and E: + ep is the true mechanical strain, is the
observable strain (ALIL),
and d, = ei + :E + 6 + . . .
accounts for isotropic shrinkage effects, in particular
those related to thermal expansion and (if present)
crystallization shrinkage and chemical reaction (curing). Other possibilities are viscoelastic volume effects
c -_.__
APR
R-3
p
(5c)
with 5 as the degree of crystallinity, C, as the linear
shrinkage from a n amorphous to a 100%crystalline
material, 5 as the conversion index of any reaction, C,
a s the linear shrinkage due to complete reaction (6 =
l),and p as the compressibility. T,. 6,. and 5, are the
values of T,6, and 5 at the moment of solidification.
Note that since the eJ terms are isotropic here, the
indices x, y, and z can as well be dropped. Further,
expansion (and the corresponding stretching forces) is
defined as positive. Consequently, shrinkage (and
compression forces)will be negative. Since usually the
temperature decreases and both crystallinity and conversion index increase, changes in eT, &, and 8 tend to
be negative (shrinkage). The hydrostatic strain differs
from the other isotropic strains in that it is defined to
be zero for zero pressure instead of at the solidification
pressure.
For further reference we will also give the strain in
thickness direction, derived from Hooke’s law with
a, = -P:
,&
-
+ p) -
1-v
[&
+E;~],
solid part.
(6)
In case of hindered shrinkage in x and y direction,
and E~~ are fixed and Eqs 2 through 4 directly yield
the stress distribution (see injection molding case beE,
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1996, VOI. 36, NO. 15
2031
K . M. B. Jansen and G. ntomanlio
low). In general, however, the strains are different
from zero and have to be determined from the force
balance. Since in the analysis below only the thickness coordinate is of importance, the variables x and y
will be omitted where possible.
Derivation of Strain Rates from the Force
Balance
The force balance on the solid layer requires that
the external forces along a certain direction ( x or y)are
equal to the integral over the stress distribution:
A similar balance equation can be written for a,,,. Here
F, is the force per unit width due to the elastic response of the mold wall which, for a planar cavity, can
be negative only. FJr is the force (per unit width) because of friction on the xy-plane. Although the friction
force is distributed on the xy-plane, here it is taken as
concentrated at the plane edges and function of time
only. As a first approximation the total friction force
per unit width can be approximated as:
Ffr(t)=
1
~ J P I y.
X ~tktw
pr(P(t))(L- X I ,
=FJt),
&,(t) =
E( iix+ i p ) F, - I'Fu
+ Z , E , solid part
E
(10)
where the bar stands for the average over thEolidified
layer thickness (see notation). Remark that EiP can be
written as -(1 - 2 v ) P if v is constant. The strain
change after solidification is obtained by integrating
Eq 10 from the moment of solidification of a layer z ,
denoted a s t,,, to time t:
The stress component s,(z, t ) is obtained by integrating over time the stress rate S , given by Eq 9:
(71
where qJr is the friction factor, which usually varies
between 0.1 and 0.5 [Weast et al. (19))and (p(t))
stands for the pressure averaged over the length direction. Note that Frr can be either positive or negative.
For brevity, the sum of the external forces (including
the D o t ) term) will be denoted as Fx(t).With this, the
force balance over the solid part can be written as:
\:sgliddz
Muki and Sternberg (8).Many authors, however, still
use Eq 9with E = E(TIx, y, z , tl, e x , y. z, tl) (e.g. (5-7,
9). In the basic derivation that follows we therefore
allow E to be a function of position and time.
Since E J t ) is not function of the thickness coordinate (assumption 31,it is readily eliminated from Eqs
8 and 9:
F, = Fw,x+ FJr,,+ DP(t).
+ v(&,,- i'y, - i p ) ) d t ,
With this, Eq 10 and
;P
(12)
= -PP we then arrive at
(8)
(13)
Since the force balance has to be evaluated after every
small time step d t , we use Leibnitz's theorem for the
differentiation of an integral to obtain:
where S ; ' = U ~ is the stress distribution without
external forces or pressure contributions (free quenching stresses):
Here s,(z,, t ) is the stress without pressure contribution (Eq 2) at the solid-melt interface; there it is just
zero. The dot denotes differentiation with respect to
time. For S , we write (see Eq 3):
E
s,, = 1 - 2 [&*,,
~
+ v&*,,],
solid part
(9)
Equations similar to Eqs 8 and 9 can be written for the
y direction by interchanging the subscripts x and y.
Note that strictly speaking, elastic materials with temperature (or crystallization or conversion) dependent
modulus do not belong to the class of thermorheologically simple solids, as was already pointed out by
2032
Equation 14 is in fact the analytical representation of
the numerical iteration scheme as derived by Menges
et al. (7) and Eduljee et al. (19).The rest of Eq 13
describes the effect of the melt pressure and other
external forces on the stress distribution.
In the next section the shrinkage curve and stress
distribution will be calculated under the hypothesis
that the modulus can be taken as constant. The expressions then simplify considerably. For further ref-
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
VOI. 36,No. 15
Effect of Pressure History
the strain over both the solid (see Eq 6) and the fluid:
erence we define the engineering shrinkages as:
6,( t)lb =
(k I,'
&dx)
1
,
= r'."
zz -
&?
solidified layer
~
2v
1-v
[&E
- dz]
&pf=
di
0
solid part
( 18a)
fluid part,
( 18b)
resulting in
(191
(15~1
The shrinkages 6, and 6, are thus the measurable
surface shrinkage in length and width direction, while
8, is the total thickness shrinkage. The superscripts s
andfrefer to the solid and the fluid state, respectively.
Remark that, in view of assumption 3, the surface
shrinkage rate is identical with the shrinkage rate of
the rest of the solidified layer. The engineering shrinkages are defined to be zero at the start of solidification
(i.e. at t = 0) and are negative for shrinkage and positive in case of expansion.
Here we used the fact that the average of the [EEd'&, term is proportional to the average of dz and
thus vanishes.
Example 1 :Closed Form Expressions of Stress and
Strain in Free Quenching
If crystallization and reaction effects are absent and
the wall temperature Tw is constant, the temperature
distribution during the initial cooling stage can be
approximated a s (Carslaw et al. (201, p. 97)
I=
a t l D 2 < 0.2 (20)
SPECIAL CASES OF THE BASIC THEORY
From now on we assume that the modulus is constant and that both width (W) and length (L)are much
larger than the thickness (20).Moreover, the stresses
and strains along x and y direction are assumed to be
identical. Under these hypothesis several special
cases will be analyzed. For each case both stress distribution and shrinkage curve will be derived. Furthermore, for the injection molding case, the effect of
ejection will be elucidated. In order to illustrate how
the equations may be used, a few examples will be
given.
where erf(x) is the well known error function and a
denotes the thermal diffusivity. Performing the differentiations and integrations according to Eqs 16, 17
and 19,one then arrives at [e.g. Goslinga (6)
for Eqs 21
and 221:
d y ( Z ,
t )=
E
~
-v
r*F/ - as"Uz, t ) - T J I ,
ts*
t,,=--
Caae i: Free Quenching
(21)
(221
This is the classical case studied in the inorganic
glass literature and is added for the sake of further
reference. Some excellent reviews on this matter are
given by Struik (5)and Isayev (11).Simplifying Eqs 1 1
and 14 one easily arrives at
E
- ____ [ S,"..( t ) - e i x ( z , t)]:,,, solid part
1-v
22
4ak2
*
[
e-kz- \;;;.ierfc($)]),
i < 0.2 (23)
where
( 16)
Equation 16 corresponds to Struik's cooling stress
equation if eJ is replaced by eT = a(T - T,). For the
shrinkage in thickness direction we have to integrate
and ierfdx) stands for the integral of the error function
[see (20)l.For small times this ierfdx) term in Eq 23 can
be neglected. From the above equations we see that the
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 199s, Vol. 3s, NO. 15
2033
K. M. B. Jansen and G. Titornanlio
surface shrinkage in x (and y) direction increase logarithmically with time, whereas the thickness shrinkage
increases linearly with z, 0.e. as the square root of time).
As has been noted by various authors, the expression of
Eq 22 (and hence also the stresses at the surface) tends
to go to --oo as ts0approaches zero. It is important, however, to realize that t,,, (the moment of solidification of the
surface layer) can never be exactly zero but rather it is
directly related to the heat transfer coefficient, h,or the
Biot number (Bi= hD/h), which may become large but
not infinite. tso can be calculated by inversion of the
surface temperature equation [as given by Carslaw et al.
(201, p. 711:
t o= 1 /(k,Bi)',
--(
v -
3v3
k,=T,Jv l + - c + - c +
2
8
-
...
For small 2 the free shrinkage curve can thus be
approximated by
(25)
and is seen to be proportional to the logarithm of the
heat transfer coefficient.
Clearly, for larger times the error function approximation of the temperature field is no longer valid and
a different equation should be used. An expression for
the free quenching stress distribution valid for larger
times is given by Struik (51,p. 249. The thickness
shrinkage for t -+ m a s derived from Eq 19 simply
becomes
Thus the final thickness shrinkage consists of two
parts related to shrinkage of the fluid part from the
initial temperature, T*, to T, and contraction of the
solid part from T, to the final temperature T,.
In case of instantaneous crystallization at T = T,
(from 5, to 5"). we may directly calculate the surface
shrinkage from E q s 17 and 19 and find for the crystallization contributions:
Note that Eqs 27 and 28 have approximately the same
structure as E q s 22 and 23, i.e. a surface shrinkage
that increases with the logarithm of z,(t) and a thickness shrinkage that increases proportional to z,(t). It
2034
can be shown that the singularity that arises a s t,ItSo
is because of the discontinuous shrinkage at the mo-
ment of crystallization. For instantaneous reaction at
T = T, similar equations may be derived. Note that in
case of crystallization or reaction the temperature distribution will deviate from Eq 20. Hence also E q s 21
through 25 are expected to be different and the true
free quenching stresses and strains should be calculated from E q s 16 through 19. However, E q s 21
through 25 may still be used to obtain a first estimate
terms.
of the ST and
UL
Case ii) Injection Molding with Hindered
Shrinkage During Solidification
During injection molding, shrinkage before ejection
can be prevented by either interaction forces at the
mold wall (friction),large enough pressures, or by geometrical constraints. The injection molding case considered here is that with both zero thickness and
length shrinkage in the mold. The more general case
with non-zero shrinkages will be treated in a subsequent paper. Note, however, that if the shrinkages
occur after complete solidification, the equations that
will be obtained below for residual stresses, final
shrinkage, and final density distribution remain valid.
Substituting EU = EYY= 0 in E q s 12 and 2 results in
the stress distribution before ejection:
Here we used the fact that EJ( z, tsZ)= 0, according to Eq
5. The superscript IMO refers to the injection molding
(1M) case with zero shrinkage. The first term in the
second line of E q 30 reflects the effect of the solidification pressure. The factor j3E/(1 - v) (with j3E = 1 2v) should be evaluated at the moment of solidification, thus at T,,P,. This may be of importance since
during solidification v is known to change rapidly from
0.50 to about 0.30.The second term of E q 30 corresponds to the unilateral compression stress (Landau
et al. (18,p. 14),whereas the last term comprises the
correction for thermal, crystallization, and reaction
shrinkage. Note that at the solid-melt interface ( z = z,)
the stresses always equal -P(t), as expected.
In injection molding, shrinkage is partly compensated by injecting extra mass during the holding
stage. For the calculation of the (final) thickness
shrinkage it is therefore important to assure that this
extra mass is taken into account in the shrinkage
calculation. In general we may write for the relative
changes of mass, rn, average density, p , and polymer
volume V:
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
Vol. 36, No. 15
Effect of Pressure History
If the volume occupied by the polymer is enforced to
remain constant (first part of the holding stage, A V =
0).the effect of extra mass is an increase in density. If,
on the other hand, also the mass supply ceases ( A m =
0, after gate sealing), the average density must remain
constant and the pressure is enforced to follow the
average (temperature) shrinkage. After the pressure
reaches zero, density starts to increase as the volume
decreases (shrinkage in second part of the holding
stage). The gate sealing time, denoted as tgs,depends
on several parameters, among which there is certainly
the gate geometry. In the time interval between gate
sealing and the onset of shrinkage, tshr.one may write:
upon ejection:
- k)pfpe
9
te
< tsl
(35b1
Here k(t,)stands for the average of the solidification
pressure just before ejection and 8 = z,/D. Note
that SFolEis independent of P,. For t > tsl the mass
inside the cavity is constant and Eq 33 gives the relation
between P, and PSI= P(tsll. The shrinkage in thickness
direction can thus also be written as:
+ ? i x ( t e ) + te 2 t s l ( 3 5 ~ )
From which we derive for the pressure curve
Equations 32 and 33 certainly hold also if tgsis replaced by the time, tSl, of complete solidification.
Effect of Ejection on Shrinkage and Stress
Distribution
Ejection of the product introduces a sudden change
in the boundary condition and it may have a severe
effect on the stress development during further solidification (that is, if the product is not yet completely
solidified before ejection). After ejection there are no
more external forces in a n y of the three directions
and thus C:, = 0 ( i = x, y. z ) . Here and hereafter
the ’ refers to the situation after ejection. If the ejection
occurs at time t,, and the stresses just before ejection
are given as u&z, t,), the average stress change upon
ejection is Air, = C& = -Cu(t,). If we neglect possible
heat effects due to expansion, the change in the e,’
terms is zero and, using Hooke’s law and Eq 2 , we
obtain the following general expression for the expansion upon ejection:
=
PsPe+
[Sxx(t,) + S,,(t,)], solid part
(34b)
fluid part
(34c)
where P, stands for P(t,). Averaging the stress distribution of Eq 30 to calculate
and ij, and substituting this in Eqs 34 and 15 then yields for the expansion
Since the pressure is always positive and the d terms
usually negative (shrinkage), the sign of Eq 35a may
be either positive or negative. If the p; term predominates, the product will expand in x and y directions,
while if the 6’ term predominates, the product contracts upon ejection. For P, = 0 only the &t,) and
& (t,) terms in Eq 35b remain and it is not difficult to
see that in that case the dimensional changes in thickness direction are just of sign opposite to those in
length and width direction.
Flnal Stress Distribution and Total Shrinkage
The total shrinkage after ejection is obviously of
more importance than the dimensional changes during ejection. Mold dimensions or, equivalently, the
dimensions of the first layers that solidify at the mold
surface are taken as a reference situation for shrinkage. Any deformation of the mold cavity will be neglected in this analysis. The total shrinkage is the sum
of three contributions: i) the shrinkage from time 0 to
t, (which is just zero in the case considered here), ii)
the expansion (or shrinkage) upon ejection (Eqs 32
and 34) and iii) the shrinkage after ejection, which is
unconstrained shrinkage (Eqs 17 and 19).This then
results in:
The shrinkages in Eq 36 should be independent of the
moment of ejection if ejection takes place after complete solidification. This is easy to prove for Eq 36a
since for t, > tSl is constant and the free shrinkage
change A S P is equal to A2ix such that the last two
terms of Eq 36a collapse into Z’(t). For the thickness
shrinkage a similar argumentation holds, and using
Eq 35c we seen that also S:Mo(t) is independent oft,
(but does depend on P,, and Zxx(tsl).
Both shrinkage equations, Eqs 36a and b, consist of
a pressure term that is roughly proportional to TP,,
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1996, Vol. 36, No. 15
2035
K. M. B. Jansen and G. Titomanlio
(P,, is the maximum cavity pressure, see Example 2
below) and a (thermal) shrinkage part proportional to
- d ( T , - T,) if we neglect crystallinity and curing. The
relative importance of the two scaling factors tells us if
the shrinkage is mainly pressure or thermally driven.
In order to study this in more detail, we consider the
final shrinkage (as t + w) and scale the equations with
aS(T,- TJ:
aM0
= Np$(tel
+ $(te),
(37a)
where
equations simply reduce to
(40)
where, for constant final temperature, crystallization,
and reaction contributions, the last term tends to vanish. In that case the solution published by Brucato et
al. (13) and Jansen (14) is retained. Measured stress
data are seen to reproduce the shape predicted by Eq
40 fairly well (13, 14, 16). Some sets of measured
stress data (14, 16, 211, however, tend to be a factor 2
or 3 lower in magnitude than those predicted by Eq
40, while others (2) were closely reproduced by this
equation [see (13)l. This is most probably not due to
viscoelastic effects, which are not accounted for in the
model proposed here (see Fig. 2).
Example 2: Molding of a Thin Strip
The dimensionless number, Np, is just the ratio between the two scaling factors and will be called Pressure Number. Note that for t, 2 tSl. the first term of
both 8; and sf: may be replaced by ZJdtsl) and Psi,
respectively, (Eqs 32 and 33).In the example below we
will graph the 8 and 8 curves as a function of the
ejection time. First, however, expressions for the residual stress distribution in injection molding will be
given.
The stress distribution after ejection regards a part
of the sample solidified before ejection (0 5 z Iz,) and
a part formed after ejection ( z > 2,). For the first part,
the shrinkage between tSzand t is given by: cgolis= 0
+ SiMoIo+ S F / $ while for the second part the shrinkage is just Syl:,,. Substituting this in Eq 9 then
results in:
The injection molding of a thin, plastic strip is considered as an example. As material and operating parameters the following values are considered: as =
l.10-4K-1,d=2aS,E=4000MPa,u=0.35,~=
a=
m2/s, T, = 100 "C, C,, = C, = 0, D = 1 mm,
and Th = 250 "C,T , = T, = 50 "C. We assume a simple
one dimensional cooling with constant wall temperature and a pressure that increases linearly with time
during filling, Rt) = Pot/t, (0It < tN);is constant
during holding, f i t ) = P,, (tpoIt It,,); and decreases
exponentially afterwards,
The pressure curve with Pm, = 6 0 MPa, Po = P,,/6,
t, = 1.0 s, and tpl = 3.0 s is shown in Fig. 3.
With the data of the pressure curve (in particular t@,
tpl, a, and Pm,/Po) we now can-construct the $(t,)
curve used in Eq 37a. For the $ ( t J plot, 2 is the
governing parameter. These curves are represented
in Fig. 4 by thick lines. Note that the thermal shrink70 r
a
60
c
-
50
-
i
I
D
Q
E.
2!
lo
2!
a
with
30
1-v
1-u'
-
20 10
1 -2v
K = - - PE -
-
a t T,, P,.
It is not difficult to prove that the integral over both
parts of this stress distribution vanishes, as it should.
For t, 2 tSl (ejection after complete solidification) the
2036
40-
t [SI
Fig. 3. Cavity pressure curve taken as a basis for the calculations whose results are reported in Figs. 4 through 6.
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
VOI. 36,No. 15
Eflect of Pressure History
2
1
0
Fig. 4 . Dimensionless in-plane
Z x 0
thermal and pressure shrinkage
plots (thick lines) and their combi- t c o
nations resulting in normalized
length shrinkages according to Eq
-1
37a (dashed lines). N p Is pressure
number as defied in Eq 38 and t
Is the dimensionless ejection time.
-2
0.00
0.20
0.40
0.60
0.80
1.oo
.
I
te
age is always negative, while the pressure shrinkage
is positive (or zero). The pressure plot $ ( t e ) may have
a maximum if Psis decreasing at t,. The thin dashed
lines represent the plots of the final (scaled) length
shrinkage, qMo,
for several values of the pressure
number. From these plots it is evident that after
complete solidification
= 0.66 here), the moment of ejection has no further influence on the final
shrinkages. Furthermore, for a pressure number of
1.8 the two terms in Eq 37a balance and the final
length shrinkage becomes approximately zero (assumed that t, 2 tSl). The pressure number in our
example equals 0.9,thus in order to obtain zero
length shrinkage we should mold with a maximum
pressure of 120 MPa, which is rather high but still
feasible.
The thickness shrinkage plots are given in Fig. 5.
The g(te) curve is not only governed by the solidification temperature but also by the ratio d/d.
In particular for small values of t, this will give rise to a large
and negative shrinkage, mainly related to the melt.
For the limit t,lO_this shrinkage is bounded and given
by Eq 26a. The @(t,)plot resembles the pressure
curve since the P , terms are larger than the Psterm.
This is partly because of the effect of the fluid expansion (i.e. the ratio p’/p”).Remark further that, in
(5,
4
2
0
Fig. 5. Dimensionless thickness
thermal and pressure shrinkage
plots (thick lines) and their combinations resulting in normalized
thickness shrinkages according to
Eq 37b (dashed lines). N p is pressure-number as d e m in Eq 38
and t is the dimensionless ejection
time.
0
2
z
6-2
-4
-6
0.00
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
0.20
0.40
VOI.3s, NO. 15
0.60
0.80
1 .oo
2037
K. M . B. Jansen and G . ntornanlio
contrast with the &Mo plots, the @ curves do depend directly on the pressure at the moment of
ejection, P, (or on P,, if t, 2 t s l ) .E q s 37 and 38 in fact
quantify the anisotropy in length and thickness
shrinkage. In particular in our example for t, > t,, if
N p = 0.5, boths:!"
and8iMoare negaiive (-0.727 and
-0.109, respectively); if Np= 1.O gMo is negative
(-0.455) and6iMopositive (0.0257).If Np = 2.0, however, both 6'":
and @'" are positive (0.091 and
0.295, respectively).
The residual stress distribution in a product ejected
after complete solidification is represented by the solid
line in Fig. 6 (t, = 10 s, tsl = 6.60 s and using E q 40).
The stress distribution consists of a small tensile zone
near the surface, a compressive minimum, and a tensile zone in the center. Since here R(t,), Ps,,
CT(t,) and
8(tsl) are 32.6 MPa, 24.7 MPa, -0.363%. and
-0.182%, respectively, we seen from Eqs 35a and c )
that upon ejection the product shrinks 0.118% in
length direction and expands 0.136% in thickness
direction. Further, (using Eqs 36 or Figs 4 and 5) the
final product length is predicted to be 0.255% smaller
than the cavity length, whereas the product thickness
will be practically identical with that of the cavity.
The dashed line in Fig. 6 shows the stress profile in
case of premature ejection (t, = 5 s). In that case a
large discontinuity shows up at z, = 0.4710). causing
the surface stresses to reduce with a factor of -2.5.
The molded object will thus consist of several layers
with rather large differences between their average
stress levels, which may cause premature product
failure. Note that since large tensile stresses at the
surface favor the sensitiveness to stress cracking, it is
(at least regarding this aspect) not always disadvantageous to eject before complete solidification.
-a
RELATION BETWEEN STRAIN AND DENSITY
The density at any moment is related to the strain,
since
and thus
where po is a reference density and to corresponds to
the reference situation, for instance To, Po = 0.
The changes in E & Z , t ) between ts. and t for the free
quenching and IMO cases are given by Eqs I 7 and 36a
in combination with Eq 6 , while for the strain change
between the reference state and the moment of solidification we may write:
In absence of crystallization and curing effects and
assuming a constant modulus we may sum the E?
terms given by E q s I 7 and 6 and insert this in Eq 41.
After some algebraic manipulations this then yields
for the free quenching case:
pfree(Z, t -+a)= po
while for the injection molding case (with ejection after
complete solidification and using E q s 6 and 36a we
10
Q
Fig. 6. Example of residual stress
distributions in an injection
molded product with hindered
shrinkage inside the mold. dE is
in-plane residual stress, z and D
denote distance from wall and
half thickness, respectively. Solid
line: ejection after 10 s [Eq 40);
Dashed line: ejection after 5 s
[Eqs 39).
2
0
I
6
-10
-20
I
0.00
I
I
0.20
0.40
I
.
I
0.60
I
0.80
1.oo
mid
wall
ZlD
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 199e, Vd. 36,NO. 15
Effect of Pressure History
find:
where the final temperature distribution, T", is assumed to be constant and P(t + m) = Po = 0. An
alternative (and often less laborious) way of calculat~ o in Eq 41, is by using:
ing the ~ q l lterm
which can be proved with Eqs 3,4,and 6 . Substituting
either Eq 16 or Eq 40 for the s, = syy terms, then
readily results in Eqs 43 and 44, respectively. Thus
the density distribution is either directly related to the
sum of the three strain components or to the sum of
the stress distributions.
One should be aware that the usual experimental
method for determining the density distribution (microtoming the product and using a density gradient
column) does not measure the true density distribution in the stressed object but rather the density after
stress relieve. Thus density distributions measured by
this method (see for instance Greener) can not be
associated to elastic effects as given by Eq 44 but
rather to other densification effects as aging or densification under pressure. Note that these effects are
usually about one order of magnitude smaller than
the elastic effects (compare Greeners pseudo compressibility with p" in Eq 44.
DISCUSSION AND CONCLUSIONS
Assuming a simple elastic behavior in the solid,
general analytical expressions for stress distributions
and shrinkage curves were derived, including effects
of pressure, external forces, crystallization, and reaction. These equations were specialized for free
quenching and injection molding with hindered
shrinkage in the mold. In these limiting cases the
expressions for residual stress distributions reduce to
previously published results. Simple expressions for
shrinkage curves and density distributions, however,
are scarcely found in literature and the equations presented here may be a welcome tool for a first approximation.
The final length shrinkage of a molded product is
shown to depend on a pressure term and a (thermal)
shrinkage term. The ratio of the scaling factors of
these terms, the Pressure Number, was used to study
the relative importance of pressure and thermal
shrinkage effects. It was shown that for Pressure
Numbers of order unity these effects cancel, resulting
in a product with very small length shrinkage after
molding. It has been pointed out that if the starting
point for thickness shrinkage calculation is taken before gate sealing, the effect of extra mass entering the
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1-
cavity should be taken into account. Thickness
shrinkage, in any case, is not related to length shrinkage. Plots of normalized length and thickness shrinkages in injection molded products are calculated assuming a simplified pressure history.
The effect of premature ejection on both final product shrinkage and residual stress distribution is discussed. It was shown that this causes a discontinuity
in the residual stress profile, which may severely hamper proper product performance.
If geometrical constraints are absent and pressure
is small with respect to isotropic shrinkage effects,
shrinkage may start before ejection. The effect of
shrinkage before ejection on final stress distribution
and total shrinkage can be included in the analysis
and will be the subject of a forthcoming paper.
ACKNOWLEDGMENTS
This work was supported by the Italian Research
Council CNR (no. 93.03247.CT03) and by Bright Euram project ERBBRE2CT933027. The authors acknowledge CNR and the European Commission for
their financial support.
SYMBOLS AND NOTATION
a
Bi
D
E
F,
= Thermal diffusivity [m2/sl.
= hD/h, Biot number.
= Half thickness Iml.
= Modulus of elasticity [Pal.
= Total force per unit width [N/ml.
q r = Friction force [N/m].
F, = Reaction force of mould wall IN/ml.
h = Heat transfer coefficient lW/m2KI.
k = erf'(TS).
L = Length of slab [ml.
N p = Pressure number, Eq 33.
P ( t l = Melt pressure [Pal.
P, = P(t,); pressure just before ejection.
s, = Stress component of Eq 3 [Pal.
i - = (T- Tw)/(Tln- T J ; dim.less temperature.
t = at/D2;dimensionless time.
t s z = Moment of solidification of layer z.
tso = Idem of layer 0 (surface layer) [sl.
t S l = Idem of layer 1 (center of slab) Is].
t, = Moment of ejection Is].
t' = Moment just after ejection Is].
k h r = Onset of shrinkage [sl.
u = Displacement Iml.
w = Width of slab [ml.
2, = Solidified layer thickness [ml.
2, = Solid layer at moment of ejection [ml.
2 = z/D; dimensionless thickness.
Greek Symbols
a = Linear thermal expansion [K-'I.
p = (1 - 2 v ) / E , compressibility IMPa-.'].
at)
d(t)
=
Linear shrinkage.
= Local strain rate.
E*
= E - C@, true mechanical strain.
EJ = Strain contribution ofj(=T, c, R ) .
Vol. 36,NO. 15
2039
K . M.B. Jansen and G. Titomanlw
A = Thermal conductivity [W/mKl.
v = Poisson constant.
cr, = Stress [Pal.
Super-and Subscripts
free = Free shrinkage case.
IM = General injection molding case.
IMO = IM with constrained shrinkage.
IMO’ = Injection molding after ejection.
T = Thermal contribution.
cr = Crystallinity contribution.
e = Ejection.
f = Fluid part.
in = Initial.
R = Reaction.
s = Solidification, solid part.
w = Wall.
Notation
= s z f l z , t)dz/z,.
f
=
@/at.
REFERENCES
1. A. I. Isayev and T. Hariharan, Polym. Eng. Sci., 25, 271
(1985).
2. Th. W. Schmidt, PhD thesis (in German), RWTH Aachen,
Germany (1987).
3. S. Y. Yang and M. Y. Hon, Tenth Annual PPS Meeting,
Akron, Ohio (1994).
4. J. Greener, Polym. Eng. Sci.. 26, 534 (1986).
5. L. C . E. Struik, lnternal Stresses, Dimensional Stabilities
and Molecular Orientations in Plastics. Wiley and Sons,
Brisbane (1990).
6. J. Goslinga, PhD thesis. Twente University, The Netherlands ( 1980).
7. G. Menges et al., SPE AhTEC Tech. Papers, 26. 300
(1980).
8. R. Muki and E. Sternberg, J. Appl. Mech., 193 (June
1961).
9. R. F. Eduljee, J. W. Gillespie, and R. L. McCullough,
Polym. Eng. Sci., 34, 500 (1994).
10. C.P. Buckley. Rheol. Acta, 27, 224 (1988).
11. A. I. Isayev and D. L. Crouthamel, Polym. Plast. Technol.
Eng., 22, 177 (1984).
12. G. Titomanlio et al.,Intern. Polym. Process., 1.55 (1987).
13. V. Brucato, S. Piccarolo, and G. Titomanlio, 2nd lnt.
Conf. on Engineering Materials, 597, Bologna-Modena,
Italy (1988).
14. K. M. B. Jansen, Intern. Polymer Processing, IX, 82
(1994).
15. F. P. T.Baaijens, Rheol. Acta, 30, 284 (1991).
16. L. F. A. Douven, PhD thesis, Eindhoven Technical University, The Netherlands ( 1991).
17. M. Rezayat and T. 0. Stafford, Polym. Eng. Sci., 31.393
(1991).
18. L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 3rd
Ed., Pergamon Press, Oxford (1986).
19. R. C. Weast et al., eds.. Handbook of Chemistry and
Physics, 65th Ed., pp. F-16, CRC Press, Boca Raton, Fla.
20. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in
Solids, 2nd Ed., pp. 71,97, Oxford Science Publications,
Clarendon Press, Oxford (1988).
21. C. H. V. Hastenberg et al., Polym. Eng. Sci., 32, 506
(1992).
Revision received July 1995
POLYMER ENGINEERING AND SCIENCE, MID-AUGUST 1!396, Vol. 36,NO. 15