EUROPEAN
POLYMER
JOURNAL
European Polymer Journal 41 (2005) 2511–2517
www.elsevier.com/locate/europolj
Study on residual stresses of thin-walled injection molding
Tong-Hong Wang, Wen-Bin Young
*
Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan 70101, Taiwan, ROC
Received 4 December 2003; received in revised form 6 April 2005; accepted 11 April 2005
Available online 22 June 2005
Abstract
The residual stresses of the thin-walled injection molding are investigated in this study. It was realized that the
behavior of residual stresses in injection molding parts was affected by different process conditions such as melt temperature, mold temperature, packing pressure and filling time. The layer removal method was used to measure the residual stresses at a thin-walled test sample by a milling machine. This simple method was demonstrated to be adequate for
a thin-walled part. Moldings under different conditions were investigated to study the effects of the process conditions
on the residual stresses of a thin-walled product using the elastic and viscoelastic models. The mold temperature was
found to affect the size of the core region and residual stress on the surface layer of a thin-walled part in our studied
range. The packing pressure was insensitive to the residual stresses in the studied high-pressure range. The residual
stresses predicted by the viscoelastic model are about the same level and trend as compared to the experimental
measurement.
2005 Elsevier Ltd. All rights reserved.
Keywords: Injection molding; Thin-walled molding; Residual stress; Layer removal
1. Introduction
Residual stresses are the stresses left inside the molding product under the condition of no external loads. In
the molding process, internal stresses are frozen inside
the mold cavity. After demolding, the residual stresses
will redistribute and cause the part shrinkage and warpage. Possible thermal stresses may still be introduced
into the part after demolding because of further cooling
to the room temperature. The residual stresses commonly discussed in injection molding may include the
*
Corresponding author. Tel.: +886 627 575 7563672; fax:
+886 2389940.
E-mail address: youngwb@mail.ncku.edu.tw (W.-B.
Young).
flow-induced residual stress and thermal induced residual stress. The flow-induced residual stresses include
those caused by polymer chain preferential orientations
and freeze-off packing pressure, and the thermal induced
residual stress is caused by non-uniform cooling of the
molding part.
Measurement of the residual stresses in a molding
part can be performed by either a non-destructive way
or a destructive way. The technology of photo-elasticity
utilizes the effect that the induced stresses inside a material will diffract the incoming light and form an interaction pattern. This pattern can be related to the stress
level and distribution inside the material. Neves and
Pouzada [1] measured the molecule orientation in thickness direction using this technology. Treuting and Read
[2] used the layer removal method to measure the residual stress in each layer of a molding product. The layer
0014-3057/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.eurpolymj.2005.04.019
2512
T.-H. Wang, W.-B. Young / European Polymer Journal 41 (2005) 2511–2517
removal method removed a small layer from the surface
of the part, leading to a measurable deformation. The
deformation shape of the part can be correlated to the
residual stress in the removed layer. Deformation
shapes for removing consecutive layers are measured
until the middle plane of the part. In this way, the distribution of residual stresses through the thickness
direction of a part can be measured. Zoetelief et al. [3]
used the same method for measuring the through thickness residual stresses. Instead of measuring the part
deformation after layer removed, Miller and Ramani
[4] used an attached strain gage to estimate the stress
in the layer removal method. Except the machining
and cutting, Jasen et al. [5] used an excimer laser for
the layer removal to avoid the problems of the machining stress. Turnbull et al. [6] made a comparative assessment of techniques for evaluating residual stress in
polymers. Emphasis was placed on the layer removal
and hole-drilling methods. A more speculative approach, the chemical probe technique, using the sensitivity of the threshold stress for environment stress
cracking to particular chemicals, has been developed
to characterize near-surface stresses. Chien et al. [7] also
had some discussion on the residual stresses for thinwall molded part.
In modeling the development of residual stresses in
injection molded parts, it was realized that the stress
relaxation effect played an important role. Kabanemi
et al. [8] used a thermoviscoelastic model with volume
relaxation to calculate the residual stresses. A finite
element based on plate theory was employed to simulate the injection parts with complex shapes. The thermal and packing pressure variations in injection mold
processes were not included in the model calculation;
instead a simple thermal equilibrium equation was
used. Lee et al. [9] has presented the physical modeling
and basic numerical analysis results of the entire injection molding process, in particular with regard to both
flow-induced and thermally-induced residual stress and
birefringence in an injection molded center-gated disk.
Kamal et al. [10] used a three-dimensional numerical
simulation to predict internal stresses in injection
molded components. The warpage of injection–
compression–molded optical media, such as compact
discs and digital video discs, due to asymmetric cooling during production was predicted by Fan et al.
[11].
This study used the layer removal method to measure
the residual stresses in a thin-walled injection molding
part. A simple machine cutting method was used for
the layer removal and demonstrated to be adequate in
this case. The relative importance of processing parameters on the resulting level of residual stresses was also
investigated by numerical simulations using a viscoelastic model. Comparisons with the results from experiments were also discussed.
2. Layer removal stress
After Demolding, the surface traction and moment
will vanish in a flat molding part as shown in Fig. 1.
For the surface in the x-direction, we have the conditions of
Z z0
Z z0
rx ðzÞdz ¼
rx ðzÞz dz ¼ 0
ð1Þ
z0
z0
where z0 is the half thickness. The z-axis is assumed to be
in the thickness direction and x-axis is along the flow
direction during the injection. When a specified thickness of layer is removed from the surface as shown in
Fig. 2, the total force and moment across the crosssection are no longer balanced. The resulting net unit
length force and moment in the x-direction can be expressed as
Z z1
F x ðz1 Þ ¼
rx ðzÞdz
ð2Þ
z0
Z z1
h
z0 z1 i
rx ðzÞ z þ
dz
ð3Þ
M x ðz1 Þ ¼
2
z0
where z1 is the left thickness above the original middle
plane of the part. Notice that, in calculation of the moment, the new middle plane is shifted down to (z0 z1)/2
after some layers removed from the surface. The net unit
z
y
z0
x
-z0
Fig. 1. The coordinate system and surface tractions on crosssections.
z
y
z1
x
-z0
Fig. 2. Element showing layer to be removed.
T.-H. Wang, W.-B. Young / European Polymer Journal 41 (2005) 2511–2517
2513
L
z1+z0
Fig. 4. The finite element model of the strip part.
2z0
Fig. 3. Determination of the curvature of a deformed bar.
length force and moment in the y-direction have the
same forms as Eqs. (2) and (3) except the subscript
changing from x to y. The bending moment of a plate
can be expressed by the bending curvatures [12].
Mx ¼ Eðz0 þ z1 Þ3
½jx ðz1 Þ þ mjy ðz1 Þ
12ð1 m2 Þ
ð4Þ
where jx and jy are the curvatures in the x- and ydirections, E is the youngÕs modulus, and m is PoisonÕs
ratio. Substituting Eq. (4) into (3), one can derive the
residual stress in the x-direction
E
djy ðz1 Þ
2 djx ðz1 Þ
rx ðz1 Þ ¼ þ
z
Þ
þ
m
ðz
0
1
6ð1 m2 Þ
dz1
dz1
þ4ðz0 þ z1 Þ½jx ðz1 Þ þ mjy ðz1 Þ
Z z0
2
½jx ðz1 Þ þ mjy ðz1 Þdz
ð5Þ
z0
The residual stress in the y-direction can be derived by
changing the subscript from x to y. For the case of a part
with equal stresses and curvatures in x- and y-directions,
one has the form for the residual stress as
E
djx ðz1 Þ
rx ðz1 Þ ¼ þ 4ðz0 þ z1 Þjx ðz1 Þ
ðz0 þ z1 Þ2
6ð1 mÞ
dz1
Z z0
jx ðz1 Þdz
ð6Þ
2
z0
model of the part and the node place that will be used
to show the residual stress distribution in the later study.
Material used for the study was ABS (PA-756S from
Chi-Mei, Taiwan). Tables 1–3 list some material constants which will be used in the process simulations.
An ultra high accuracy laser displacement meter (LC2400A, Keyence Corp.) was used to measure the deformation of the mid-point location ion the layer removal
method. It could measure the displacement with accuracy up to 0.1 lm. Samples were molded under the conditions of 0.28 s injection time, 210 C melt temperature,
60 C mold temperature, and 168 MPa packing pressure. The packing time was 0.5 s and cooling time was
4 s.
The samples were cut to the size of 100 · 30 · 1 mm3
first and glued to the machine table by a double side
tape. The first three cutting layers were 0.02 mm, and
the rest of layers were 0.06 mm until near the middle
plane. Totally, 10 layers were removed in the test. In
order to minimize the cutting stress and relaxation effects, different samples were used for the cutting of difTable 1
Mechanical data and parameters for time–temperature shift
function
Material data
Symbol
Value
YoungÕs modulus
PoissonÕs ratio
Reference temperature
Thermal expansion coefficient
(liquid state)
Thermal expansion coefficient
(glassy state)
E
m
Tr
a1
2.312 GPa
0.38
373 K
9.4 · 105 K1
a2
9.4 · 105 K1
c1
c2
14.22
47.01
The bending curvature can be estimated by the following
equations:
jx ¼
1
q
2
q¼
ð7Þ
Table 2
Rheological data: cross-WLF viscosity
2
L þ 4/
8/
ð8Þ
The definitions of L and / are shown in Fig. 3.
3. Experimental
A simple strip part with dimensions 168 · 30 · 1 mm3
was used in this study. Fig. 4 shows the finite element
Symbol
Value
n
s*
D1
D2
D3
A1
A2
0.2995
1.054 · 105 Pa
2.655 · 105 Pa s
373 K
0
23.74
373 K
2514
T.-H. Wang, W.-B. Young / European Polymer Journal 41 (2005) 2511–2517
where T is the temperature, c1 and c2 are material constants, and Tr is the reference temperature. The relaxation modulus G1 and G2 are described by the following
models:
Table 3
pvT data: two-domain Tait model
Symbol
Value
b1m
b2m
b3m
b4m
b1s
b2s
b3s
b4s
b5
b6
0.0009999 m3/kg
5.7487 · 107 m3/kg K
1.9213 · 108 Pa
0.0056107 K1
0.00099971 m3/kg
1.2130 · 107 m3/kg K
2.0579 · 108 Pa
0.00011384 K1
373.8 K
1.0678 · 107 K/Pa
ð13Þ
G2 ðtÞ ¼ 3juðtÞ
ð14Þ
and
uðtÞ ¼
m
X
r¼1
ferent depths. Three samples were used for each depth.
The layer removal was performed by a milling machine
with the conditions of a 12 mm diameter cutter, a feed
rate 25 mm/min, and a spindle speed 1000 rpm. In order
to understand the cutting stress introduced to the sample
during the machining. Some samples were annealed
under 90 C for 8 h. After cutting the annealed samples
to different depths, the measured deformation and curvature were found to be relatively small in our setup.
Therefore, the cutting stress was considered negligible
in our system.
4. Numerical simulation of the residual stress
t
gr exp hr
ð15Þ
where gr, l, and j are material constants, and hr are
relaxation times. Due to the condition of the solidification, the analysis of residual stress can be separated into
the following three cases:
1. The core region is above the no-flow temperature.
Since the core region is still under the packing pressure, rzz will equal to the local melt pressure, P(t),
and Drnzz is equivalent to the variation of local melt
pressure at time tn, DPn. The change of in-plane stress
can be written as [13]
Drnxx ¼ Drnyy
¼ ðuðDnn Þ 1Þrnxx þ a½DP n ðuðDnn Þ 1ÞP n þ CðDnn Þða 1Þð3jÞDenth
ð16Þ
and
For a thin part, the in-plane dimension is larger than
the thickness. The shear stresses are neglected in the
analysis of the in-mold residual stresses. The stresses
to be determined are the three normal stresses. The normal stress, rz, is assumed to be constant in the thickness
direction. The polymer is assumed to behave as an isotropic thermorheologically simple solid, in such a manner that stress components are related to histories of
strain components and temperature through appropriate
relaxation functions. The equations for the deviatoric
stresses and bulk stress are [13]
Z t
oeij
sij ¼
G1 ðn n0 Þ 0 dt0
ð9Þ
ot
0
Z t
oðe eth Þ 0
s ¼
G2 ðn n0 Þ
dt
ð10Þ
ot0
0
where sij and eij are the deviatoric stress and strain, s is
the bulk stress, e is the average normal strain, and eth
is the thermal strain. The modified time scale n is related
to the shift function as
Z t
U½T ðxi ; t0 Þdt0
ð11Þ
nðxi ; tÞ ¼
0
ð 2l þ jÞ
a ¼ 4l3 þj
3
ð17Þ
2. The entire layers are below the no-flow temperature.
As all the layers are below the no-flow temperature
and the material does not detach from the mold wall
(rzz < 0), the boundary condition of no displacement
in the thickness direction is employed. One has [13]
Pnl
Dzi
ðuðDnn Þ 1Þrnzz 3jCðDnn ÞDenth
i¼1 CðDnn Þ
n
Drzz ¼
Pnl Dzi
i¼1 CðDnn Þ
ð18Þ
Drnxx
¼
Drnyy
¼ ðuðDnn Þ 1Þrnxx þ a Drnzz ðuðDnn Þ 1Þrnzz
ð19Þ
þ CðDnn Þða 1Þð3jÞDenth
3. The material detached from the mold wall. As the
material cools further, the compressive stress in the
thickness direction may drop to zero, and the material may detach from the mold wall. At this time,
the conditions of rzz = Drzz = 0 is employed. The variation of in-plane stress becomes [13]
Drnxx ¼ Drnyy
and
log U ¼
G1 ðtÞ ¼ 2luðtÞ
c1 ðT T r Þ
ðc2 þ T T r Þ
ð12Þ
¼ ðuðDnn Þ 1Þrnxx þ CðDnn Þð3jÞða 1ÞDenth
ð20Þ
The calculation of the residual stresses requires the simulation of the complete injection molding cycle. This
involves a set of mass and energy balance equations
together with constitutive relations. Detailed information regarding to the solution of the injection molding
cycle and the above residual stress equations can be
found in the literature [13,14].
Residual Stress (MPa)
T.-H. Wang, W.-B. Young / European Polymer Journal 41 (2005) 2511–2517
30
20
10
0
-10
-20
-30
-40
-50
-60
-70
-80
Melt temp. 200°C
Melt temp. 210°C
Melt temp. 220°C
-1
Pressure (MPa)
Measurement
Simulation
0.5
1.0
-0.6
-0.4
1.5
2.0
Time (sec.)
Fig. 5. Pressure trace at entrance.
2.5
3.0
Residual Stress (MPa)
30
20
10
0
-10
-20
-30
-40
-50
-60
-70
-80
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
0.6
0.8
1
Packing pressure 136MPa
Packing pressure 153MPa
Packing pressure 168MPa
-1
30
20
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-1
c
-0.2
Normalized Thickness
-0.8
-0.6
-0.4
b
Residual Stress (MPa)
Simulations of the developed residual stresses during
the molding process were performed using a code developed in the literature [13] for the viscoelastic model. The
commercial code of C-MOLD was used for calculations
of the residual stresses using an elastic model. Fig. 5
shows the comparison of the pressure traces from the
simulation prediction and experimental measurement
at the position of the gate. The melt pressure rises sharply during the filling stage and keeps about the same at
the packing stage. As the melt at the gate starts to freeze,
the pressure drops at the same time. The predicted pressure was in the same trend as the experimental measured
value.
For numerical simulation results, the residual stresses
of the molding part presented at the following is located
at the node position shown in Fig. 4. Fig. 6(a) shows the
simulated residual stresses in the thickness direction of
samples molded under different melt temperatures using
an elastic model. The thickness in the plot was normalized to from 1 to 1. A thin skin layer is under tensile
stress while a compression stress in the sublayer. There
were not many variations of the residual stresses for different melt temperatures in the range from 200 to
220 C. The effect of different packing pressures from
136 to 168 MPa on the residual stresses is shown in
Fig. 6(b). No obvious change was found in this case.
The effect of packing pressure on residual stress is usually found in the low packing pressure range. At this
range, the packing pressure drops to zero prior to the
180
160
140
120
100
80
60
40
20
0
0.0
-0.8
a
5. Results and discussion
2515
-0.2
0
0.2
0.4
Normalized Thickness
Mold temp. 50°C
Mold temp. 60°C
Mold temp. 70°C
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Normalized Thickness
Fig. 6. In-plane residual stress distributions for different: (a)
melt temperatures, (b) packing pressures, and (c) mold
temperatures calculated by an elastic model.
complete glass transition. In the high packing pressure,
the pressure distributions for different packing pressures
prior to entire glass transition only differ by a constant,
resulting in almost the same distribution of residual
stresses. However, the average cavity pressure for different packing pressure will be different and cause different
shrinkages. Fig. 6(c) shows the effects of different mold
temperatures from 50 to 70 C on the through thickness
residual stresses. Higher mold temperature tends to increase the tensile stress in the surface, but without obvious effect on the rest of the part. A slightly larger core
region was noticed for higher mold temperature. In
numerical simulation of the residual stresses of the injection part using the elastic model tends to overestimate
the stress level because of the ignorance of the stress
relaxation effect. As the melt freezes during the cooling
process, the stress relaxation effect is quite fast near
the transition point. A large portion of the frozen stress
T.-H. Wang, W.-B. Young / European Polymer Journal 41 (2005) 2511–2517
can be relaxed at this moment. Most of the residual
stress was built up at the following cooling process with
temperature far lower than the transition point.
For the numerical simulation using a viscoelastic
model, Fig. 7(a) shows the residual stress through the
thickness for different melt temperatures. The calculations included both the shear and bulk relaxation. Table
4 lists the relaxation data from literature [3] for ABS
plastics. In the core region, the packing pressure relaxes
near the glass transition temperature, resulting in lower
residual stress. On the other hand, the skin-layer freezeoff at the filling stage under a low pressure. During the
packing stage, packing pressure produces an elastic
strain in the skin layer, which releases as the packing
pressure drops off. The resulting stress is the tensile
stress induced by the thermal contraction. The residual
stress level shown in Fig. 7(a) is quite low compared
those from an elastic model due to the large stress relax-
Residual Stress (MPa)
10.0
melt temp. 200°C
melt temp. 210°C
melt temp. 220°C
2.5
gi
1
2
3
4
5
6
4.706e9
4.410e6
2.082e3
6.198e1
3.305e6
2.749e8
0.0605
0.1037
0.3467
0.4884
0.0004
0.0003
6
5
4
3
2
1
0
-1
-0.8
-0.6
-0.4 -0.2 0.0
0.2
0.4
Normalized Thickness
0.6
0.8
1.0
Fig. 8. Measured in-plane residual stress distribution.
-2.5
-0.8
-0.6
-0.4
a
-0.2 0.0
0.2
0.4
Normalized Thickness
0.6
0.8
1.0
0.6
0.8
1.0
10.0
7.5
Residual Stress (MPa)
hi
0.0
-5.0
-1.0
packing pressure 136MPa
packing pressure 153MPa
packing pressure 168MPa
5.0
2.5
0.0
-2.5
-5.0
-1.0
b
-0.8
-0.6
-0.4
-0.2 0.0
0.2
0.4
Normalized Thickness
30.0
Residual Stress (MPa)
i
-2
-1.0
7.5
5.0
Table 4
Relaxation data for thermal stress calculation [3]
Residual Stress (MPa)
2516
25.0
mold temp. 50°C
mold temp. 60°C
mold temp. 70°C
20.0
15.0
10.0
5.0
ation near and above the glass transition point. Fig. 7(b)
shows the effect of the packing pressure on the residual
stress. As the case in the elastic model, no major effect
can be found in the studied packing pressure range.
The effect of mold temperature is shown in Fig. 7(c).
The mold temperature has major effect at the stress level
at the skin layer since the material at this location is
quenched to the mold temperature at the filling stage.
As the mold temperature increases, the core region is
larger, resulting in more stress relaxation during the
packing and less residual stresses.
From the experiments, the deformations and curvatures of the samples can be measured. The resulting
through thickness residual stresses can thus be calculated using the measured data and Eq. (6). Experimental
result (symbols) of the residual stress is shown in Fig. 8
together with the simulation result (solid line). The predicted stress level and trend are close to the experimental
measurement. In the core region, the predicted value is
quite low as compared to the measured value. The possible reason might be that the used relaxation data overestimates the actual case, leading to the under-estimate
of the stress level in the core region.
0.0
-5.0
-1.0
c
-0.8
-0.6
-0.4
-0.2 0.0
0.2
0.4
Normalized Thickness
0.6
0.8
1.0
Fig. 7. In-plane residual stress distributions for different: (a)
melt temperatures, (b) packing pressures, and (c) mold
temperatures calculated by viscoelastic model.
6. Conclusions
The layer removal method was used to measure the
residual stresses at a flat thin-walled test sample. Moldings under different conditions were investigated to
T.-H. Wang, W.-B. Young / European Polymer Journal 41 (2005) 2511–2517
study the effects of the process conditions on the residual
stresses of a thin-walled product. The mold temperature
was found to affect the size of the core region and residual stress on the surface layer of a thin-walled part in
our studied range. The packing pressure was found to
be insensitive to the residual stresses in the studied
high-pressure range. The predicted stress level and trend
are close to the experimental measurement in viscoelastic model. The layer removal method was also found to
be adequate for a thin-walled part.
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