Journal of Thermal Stresses, 25:523±538, 2002
Copyright # 2002 Taylor & Francis
0149-5739 /02 $12.00+.0 0
DOI: 10.108 0=01495730290074270
APPROXIMATE MODEL OF THERMAL RESIDUAL STRESS
IN AN INJECTION MOLDED PART
X. Zhang, X. Cheng, and K. A. Stelson
Department of Mechanical Engineering
University of Minnesota
Minneapolis, Minnesota, USA
M. Bhattacharya and A. Sen
Department of Biosystems and Agricultural Engineering
University of Minnesota
Minneapolis, Minnesota, USA
V. R. Voller
Department of Civil Engineering
University of Minnesota
Minneapolis, Minnesota, USA
A description and model of the thermally induced residual stress that forms in a ``platelike’’ injected molded polymer part is outlined. A linear heat balance integral (LHBI)
temperature pro®le approximation and the concept of a virtual adjunct mold (VAM) is
used to arrive at an approximate algebraic model that, upon providing material properties and processing conditions, gives an immediate evaluation of the residual stress
through the plate thickness. Results obtained with this model are in very close agreement with a full numerical solution and agree closely with experimental measurements
made on a range of starch-based biodegradabl e polymer samples.
Injection molding is used in the fabrication of a wide range of plastic products. In the
injection-molding process, molten polymer is injected into a mold and then is packed
under pressure, solidi®ed, and then cooled to a uniform temperature. Figure 1
schematically illustrates the solidi®cation stage of a ``platelike’’ part.
Injection molding introduces residual stresses, a stress state that exists in the
bulk of a material without application of any external load, in the ®nal product [1, 2].
It is important to know the residual stress state of the part to predict its performance
under load. Depending on the situation, residual stresses can be either detrimental or
Received 10 July 2001; accepted 10 November 2001.
This work was supported by the NSF Division of Design, Manufacturing and Industrial Innovation,
under grant DMI 97-00126 .
Address correspondence to Professor Vaughan Voller, Department of Civil Engineering, University
of Minnesota, 500 Pillsbury Drive S.E., Minneapolis, MN 55455. E-mail: volle001@umn.edu
523
524
X. ZHANG ET AL.
Figure 1. Schematic of solidi®cation of an injected molded plate.
bene®cial. Compressive residual stress on the surface can prevent the opening up of
cracks, thereby increasing the fatigue life. Conversely, when the stresses due to external loads are added to the residual stress, plastic yielding will begin at a lower
load. An asymmetric mold wall temperature distribution will cause asymmetric residual stresses; and if the part is not sti€ enough, it will warp after its ejection from
the mold [3].
The aim of this article is the development of a simple approximate model for
predicting the residual stresses in injected-molded parts fabricated from starch-based
biodegradable polymers. In a general case the residual stress in an injected molded
part could include a number of contributions , including
i.
Flow-induced residual stress, generated as a result of shear, normal, and extensional ¯ows during processing [4, 5];
ii. Packing stress, resulting from the high pressures imposed during packing;
iii. Thermal stress, formed during solidi®cation and cooling [6±8].
As a general rule of thumb, for most polymers of interest, the thermal residual
stresses dominate the ¯ow-induced stresses. Further, packing pressure and packing
time may not have a signi®cant e€ ect; for example, in starch-based polymer it has
been shown that the packing pressures account for approximately 5% of the observed residual stress [6]. Hence, the work in this article will only consider thermalinduced residual stresses that form in injection-molded polymer parts.
STRESS IN AN INJECTION MOLDED PART
525
In the ®rst instance, a description of the mechanisms that lead to a thermally
induced residual stress is provided and the governing equations of a standard model
are outlined. In a complete treatment, the standard model requires the coupled solution of a solidi®cation heat transfer and stress equilibrium equation, a set of calculations that can be relatively complex. In this work, a simple thermal treatment
that approximates the evolving thermal ®eld by a linear pro®le is proposed. This
simple treatment leads to an explicit expression for the residual stress distribution
through a platelike part. This approximate model is the key result of the article. The
article concludes with a demonstration of the proposed approximate model via the
comparison with full numerical treatments and experimental measurements.
A DESCRIPTION OF RESIDUAL STRESS FORMATION
IN A PLATE
Consider a thin plate part cooled from its edges; see Figure 2. As a ¯uid layer solidi®es, a so-called ``¯ow-strain’’ is initiated [7]. This strain occurs, under hydrostatic
stress, at the point of solidi®cation, where the semisolid layer deforms to attach to
the existing solid. The interaction between the development of the strain, due to
thermal shrinkage, and the initiation of the ¯ow-strain leads to a nonzero throughthickness stress distribution when the sheet is cooled to a uniform temperature. A
simple description based on Figure 2, which models the part as a set of lamella, can
throw more light on the key mechanisms. At the early stages of cooling, a solid layer
forms next to the cooling surface and it is reasonable to assume that the strain in this
solid layer, a decreasing function of time, is uniform through the thickness. As a new
¯uid lamella solidi®es and joins the solid, it needs to undergo a strain from its
Figure 2. Schematic of thermal residual stress formation.
526
X. ZHANG ET AL.
undeformed initial state. This strain, referred to as the ¯ow strain occurs under
negligible stress, takes a value of zero at the cooled surface and is a monotonically
increasing function of solid thickness alone. Since the ¯ow strain will be larger in the
center of the part, the additional e€ ect of thermal shrinkage will lead to a situation
where, if taken in isolation, the deformation of a lamella from the center of the part
will be larger than that of a lamella taken from the edge of the part. Hence, the net
result, when the lamella are integrated into the part and the condition of a uniform
solid strain is imposed, is to place lamella at the part center in tension and those at
the part edge in compression, behavior that matches the reported observations [7, 8].
A THERMAL-ELASTIC RESIDUAL STRESS MODEL
The thermal-elastic residual stress model presented here closely follows the work of
Osswald and Menges [7]. Since injection-molded parts are generally thin, we simplify
the analysis to a thin plate of thickness 2b, which is cooled from both sides. If
symmetry is assumed, the coupled mechanical and thermal analysis can be carried
out using a one-dimensional model in the half thickness, 0 µ z µ b; see Figure 1.
At some instant of the cooling process, three regions in Figure 1 can be identi®ed:
i. A liquid region that is strainfree and stressfree;
ii. A thin layer undergoing solidi®cation at temperature Ts. This layer undergoes
volume contraction and ¯ow resulting in a ``¯ow strain,’’ ev, which can also be
referred to as a ``viscous strain.’’ It is assumed that this strain occurs under
hydrostatic stress and is frozen in to the solid layer and, hence, is a function of
space alone.
iii. A full solid region. As the full solid region cools, a thermal strain,
eth = b(T(z; t) – Ts ), is formed, where b is the thermal expansion coe cient.
Under the assumption that the total strain, e(t), is, at a given instant in time, uniform
across the plate thickness these observations lead to the expression for the total
strain as a function of time
e(t) = ee (z; t) ‡ b(T(z; t) – Ts ) ‡ ev (z)
(1)
where ee is the elastic strain. In order to make forward progress from Eq. (1) a
method for evaluation of the ¯ow strain, ev (z), is required. From Hooke’s Law,
assuming that Poisson’s ratio and Young’s modulus are constants, the stress in the
solid layer, 0 µ z µ zs , is given by
s(z; t) =
E
(e(t) – b(T(z; t) – Ts ) – ev (z))
1– v
Further, equilibrium requires that the integral
Z zs
s(z; t) dz = 0
0
(2)
(3)
STRESS IN AN INJECTION MOLDED PART
527
Combining Eqs. (2) and (3) gives
Z
zs
[e(t) – b(T(z; t) – Ts ) – ev (z)] dz = 0
(4)
0
which upon manipulation and integration of the total strain gives
1
e(t) =
zs
Z
zs
[b(T(z; t) – Ts ) ‡ ev (z)] dz
0
(5)
At the solidi®cation front, z = zs , the thermal and elastic strains are zero; and by
Eq. (1), e(t) = ev (z),which means that Eq. (5) can be written as
ev (zs ) =
1
zs
Z
zs
0
[b(T(z; t) – Ts ) ‡ ev (z)] dz
(6)
Assuming no relaxation, this ¯ow strain is frozen in place and will not change.
Hence, if information on the evolving transient thermal ®eld, T(z; t), and location of
the solidi®cation front, zs (t), are provided, Eq. (6) can be integrated at given values
of zs to determine the ¯ow strain pro®le, ev (z). This calculation needs to be continued
up to when the midplane, z = b, becomes solid, at which point the ¯ow strain pro®le
will be fully determined. From the calculated ¯ow strain pro®le, ev (z), the total
strain, and the residual stress distribution, when the plate has cooled to room
temperature, Tf , can be found from the expressions
etot
=
1
b
Z
b
0
[b(Tf – Ts ) ‡ ev (z)] dz
(7)
and
s(z) =
E
[etot – b(Tf – Ts ) – ev (z)]
1– v
(8)
respectively.
Numerical Solution
The transient temperature ®eld, T(z; t), and location of the solidi®cation front, zs,
needed in Eq. (6) can be determined numerically. For semicrystalline polymers a
model can be based on an enthalpy formulation [9, 10]
r
@H
@2 T
= k 2
@t
@z
(9)
where the enthalpy H = cp T ‡ gL, L is the latent heat of the polymer. Cp is the
speci®c heat, k the thermal conductivity, r the density, and g is the liquid fraction
528
X. ZHANG ET AL.
taking a value of zero in the solid and one in the liquid. Upon introducing the
dimensionless variables
z¤ =
z
b
t¤ =
T – Ts
Ts – Tf
cp (Ts – Tf )
hb
St =
Bi =
L
k
kSt t
T¤ =
r cp b2
H ¤ = StT ¤ ‡ g
(10)
where h is the convective heat transfer coe cient at the surface z = 0, the governing
equation becomes
@H ¤ @ 2 T ¤
=
@t¤
@z¤
subject to the boundary conditions
­
@T ¤ ­
­ = – Bi(T ¤ ‡ 1)
@z¤ ­
z= 0
and
(11)
­
@T ¤ ­
­= 0
@z¤ ­
z= b
(12)
The key dimensionless groups in the preceding equation are the Stefan number, St;
the ratio of sensible to latent heat, which controls the phase change; and the Biot
number, Bi, the ratio of convective heat removal to heat conduction, which controls
the cooling.
Note that the value of the heat transfer coe cient h is not strictly a constant as
assumed here, but a function of time, controlled by interface conditions between the
mold and part. The choice of a constant h is justi®ed upon noting that (i) the inclusion of a time-dependent heat transfer coe cient, assuming that reliable information is available, would greatly complicate the analysis in any modeling e€ ort;
and (ii) comparisons with experiments (see results that follow) indicate that reasonable residual stress distributions can be predicted using a constant representative
value of h.
Equations (11) and (12) can be readily solved using, for example, an explicit
enthalpy scheme [10]
HInew = HI ‡
Dt
[TI– 1 – 2TI ‡ TI‡ 1 ]
D z2
(13)
where I is a node counter on a space grid with uniform step D z; D t is a time step; the
superscript new refers to values at the new time level; and, for presentation convenience, the ¤ superscript has been dropped. In a given time step the solution of
Eq. (13) will provide an updated nodal enthalpy ®eld. The corresponding updated
temperature ®eld follows on setting
8
H– 1
>
< St
T=
0
>
:H
St
H>1
(14)
H<0
STRESS IN AN INJECTION MOLDED PART
529
The time at which the solidi®cation front reaches node I, that is, zs = ID z, is given
when the nodal enthalpy HI = 0.5. At this point the current nodal temperature
distribution can be used in a numerical integration of Eq. (6) to obtain a corresponding nodal distribution for ev(zs). These nodal values can subsequently be used
to calculate the residual stress pro®le through the part thickness.
RESULTS FROM NUMERICAL SOLUTION
The aim of this article is to arrive at a closed-form approximate model for prediction
of the residual stress across the part thickness. This is seen as an alternative to the
numerical solution presented previously. Intermediate results from the numerical
solution, however, are needed to guide the development of the approximate model.
These results are presented in this section.
Transient residual stress pro®les, predicted with the numerical solution, at different times, corresponding to the property data given in Table 1, are shown in
Figure 3; these pro®les have a good agreement with the results presented by Eduljee
et al. [11].
Figure 4 shows three numerical predictions of the residual stress at room
temperature. Between each prediction, the value of the Stefan number is changed by
a factor of about 10 by increasing the latent heat value. It is noted that this change
does not have a signi®cant impact on the predicted level of stress; a factor-of-sixty
change in the value of the latent heat produces a maximum stress change less than
one half. Thus it can be concluded that residual stress predictions with the thermalelastic model outlined previously are relatively insensitive to changes in the latent
heat. This is a very worthwhile observation since the value of the latent heat in a
semicrystalline polymer is di cult to determine [7].
Table 1 Material data for the starch-based polymer
Material Data
Symbol
Value
Part half thickness
Young’s modulus=(1-Poisson’s ratio)
70% starch, 50% starch,
30% starch
Thermal expansion coe cient
(average)
Thermal conductivity
(average value)
Final temperature
Solidi®cation temperature
Density (average value)
Speci®c heat
Heat transfer coe cient
(average value)
b
E=(1– v)
5 mm
2163 MPa, 1435 MPa, 837 MPa
b
4.87 £ 10 – 5¯ C– 1
K
0.25 w=m=¯ C
Tf
Ts
r
Cp
h
20¯ C
115 ¯ C
1273 kg=m3
1674 J=kg=¯ C
1000 W=m2=¯ C
530
X. ZHANG ET AL.
Figure 3. Residual stress distribution at three di€ erent times (70% starch).
In interpreting the stress distribution in Figure 4, and later results of this article
(e.g., Figures 5 and 9), it is important to note that, due to the requirement that
stresses be in equilibrium, the area under a residual stress curve needs to be zero. A
given curve represents a balance between compressive stresses at the part surface and
Figure 4. Residual stress predictions at di€ erent Stefan number (changed by L).
STRESS IN AN INJECTION MOLDED PART
531
Figure 5. Residual stress predictions at di€ erent Biot numbers (70% starch).
tensile stresses at the part center. Since the residual stress curves are typically
parabolic in shape, this results in curves, representing di€ erent conditions, crossing
over each other at about the same point.
Unlike the latent heat value in the Stefan number, the value of the Biot number
a€ ects the residual stress ®eld. With a large Biot number, heat is rapidly removed
from the surface, resulting in high residual stresses. As the Biot number decreases,
the rate of heat removal slows and the residual stress is reduced [6]. Figure 5 shows
the stress ®elds resulting from di€ erent choices of Biot number. The di€ erence in
stress levels is very clear. Figure 6 shows, for two di€ erent Biot numbers, the temperature pro®les through the part thickness for various locations of the solidi®cation
front. Note the relative straight nature of the pro®le in the solid region.
Figure 6. Temperature pro®les at di€ erent solidi®cation thickness at two Biot numbers.
532
X. ZHANG ET AL.
THE LHBI-VAM APPROXIMATE SOLUTION
Observations of results obtained with the full numerical solution indicate that:
i.
ii.
The value of the residual stress does not depend on the rate at which the phasechange front zs (t) moves but rather on the shape of the temperature pro®le in
the solid at each zs position;
The shape of the temperature pro®le in the solid approaches a linear form when
the value of the Stefan number, St, decreases (which a limit analysis of the wellknown Neumann solution of the Stefan melting problem will readily show [10]).
Borrowing some of the concepts from the well-known heat balance integral
(HBI) method [12], these observation s suggest that the temperature in the solid could
be modeled by an approximating pro®le that evolves with the movement of zs. If well
chosen, this approximating pro®le could be used to arrive at an analytical integration
of Eq. (6) and, in turn, an explicit analytical expression for the residual stress ®eld in
the part. A ®rst cut approach is to assume perfect thermal contact between the mold
and the part, that is, the Biot number approaches in®nity, and employ the linear
approximation
T(z; t) =
Ts – Tf
z ‡ Tf
zs (t)
(15)
Using Eq. (15) in Eqs. (6)±(8) does indeed lead to an analytical expression for the
stress ®eld (see derivation in the appendix)
³
´
E
b
¢ b(Tf – Ts ) ¢ ln – 1
s(z) =
2(1 – v)
z
(16)
As might be expected, however, there is a singularity at z = 0 (the surface), where,
when the ®rst solid skin forms, there is an in®nite temperature gradient. Nevertheless, when compared with a full numerical solution of a problem characterized by
a large Biot number (a large h), the approximate analytical and full numerical solutions for the residual stress are very close across the majority of the part thickness;
see Figure 7. With reference to Figure 5, however, it is clear that the performance of
the model in Eq. (16) will begin to deviate when smaller, more realistic, Biot numbers
are used; in this case, the assumption of a ®xed surface temperature of Tf is not valid.
The proposed approximate treatment can be remedied to account for the e€ ect
of the Biot number by using the ideas in the virtual adjunct mold (VAM) method
devised to account for ®nite heat transfer coe cients in the HBI modeling of steel
solidi®cation [13, 14]. The idea is to add a virtual extension to the polymer domain
with surface at z = – D ( D > 0) ®xed at temperature Tf. The thickness of the added
domain will depend on the value of the Biot number (thin if the Biot number is large,
thick if the Biot number is small). The VAM concept is illustrated schematically in
Figure 8. With the VAM the approximating linear pro®le takes the form
STRESS IN AN INJECTION MOLDED PART
Figure 7. Comparison numerical solution and initial HBIM approximation (70% starch).
Figure 8. The virtual adjunct concept.
533
534
X. ZHANG ET AL.
T(z; t) =
Ts – Tf
[z ‡ D ] ‡ Tf
zs (t) ‡ D
0 µ z µ zs (t)
(17)
When Eq. (17) is used in the residual stress model, Eqs. (6)±(8), an analytical
treatment can be realized, resulting in an explicit expression for the residual stress
distribution, an expression that does not have a singularity in the real domain
s(z) =
» µ
¶
¼
E
b‡ D
z
b(Tf – Ts ) ln
–
2(1 – v)
z‡ D
z‡ D
(18)
where a curve-®tting exercise indicates that the virtual adjunct is a function of the
Biot number Bi
D
b
= 0:04 ‡
1:37
Bi
(19)
Equations (18) and (19), referred to as the LHBI-VAM model, to signify the coupling of a linear HBI with a VAM, is the key result in this article. This model allows
for the immediate calculation of the residual stress that will result from a given injection molding set-up. For completeness, a detailed derivation of Eq. (18) is given in
the appendix.
RESULTS
Comparison Between Numerical Solution and Approximation
Figure 9 compares LHBI±VAM predictions of Eqs. (18) and (19) with a full numerical solution across a wide range of Biot numbers. These results indicate a very
high accuracy for the approximate LHBI±VAM solution.
Comparison Between Approximation and Experimental Results
A comparison of residual stresses from experimental measurements [6] on starchbased polymers (see Table 1) and the approximate LHBI±VAM solution is shown in
Figure 10. Agreement between the experiment and approximate model is good. The
most noticeable deviation occurs at the part surface where the LHBI±VAM predicts
higher compressive stresses than those measured. It should be noted, however, that:
i.
The majority of the part thicknesses of the LHBI±VAM model are in very close
agreement with the experimental measurements.
ii. No attempt has been made to improve the ®t of the LHBI±VAM model; the
data given in Table 1 [6] was obtained independently of the model.
iii. The experimental techniques were made via the well-known layer removal
technique [15]. This required a ®tting to determine the part curvature. It is
reasonable to expect that this ®tting may be subject to the most error at the
surface where large compressive stresses are formed.
STRESS IN AN INJECTION MOLDED PART
535
Figure 9. Comparison of numerical and LHBI-VAM predictions (70% starch).
CONCLUSIONS
The ability to relate the residual stress in an injected molded part to processing
conditions and material properties is an important step toward process optimization.
Experimental measurements on the injection modeling of rectangular parts are
su cient to characterize the levels of stress that might be encountered when fabricating a given polymer material under given processing conditions [6].
Although many e€ ects contribute to residual stresses during injection molding
of polymers (e.g., packing pressure, viscoelastic relaxation, nonconstant material
properties, degree of crystallinity), its formation may be adequately modeled using a
one-dimensional thermoelastic treatment that assumes constant material properties.
This model requires the coupled solution of a thermal solidi®cation problem and
evaluation of the integral of stress through the part thickness. In a general setting,
a numerical solution is needed, requiring a discretization in both space and time, to
resolve the model fully. In this article it has been shown that, upon assuming (i) a
linear temperature pro®le for the solid and (ii) a VAM to account for convective
cooling, the solution of the residual stress model can be carried out analytically to
arrive at an explicit relationship for the residual stress (see Eqs. (18) and (19)). This
simple model allows for a direct assessment of the level of residual stress that will
occur for a given polymer material under given injection molding conditions. Across
a very wide range of conditions, the closed-form approximation, Eq. (18), provides
residual stress predictions that are very close to those obtained with a full numerical
solution. Predictions with the simple model also agree well with a range of experimental measurements made on starch-based biodegradable polymers.
536
X. ZHANG ET AL.
Figure 10. Comparison of LHBI-VAM predictions with experimental measurements.
The applicability of the proposed approach rests on assuming that the phase
change can be modeled as a sharp front moving from the part surface to the part
center. In a strict sense this should restrict the use of the approximation to crystalline
or semicrystalline polymers, but since the predictions from the model are relatively
STRESS IN AN INJECTION MOLDED PART
537
insensitive to the value of the Stefan number (see Figure 4) it is expected that the
approximation could also be used with some con®dence to obtain residual stress
pro®les for amorphous polymers. The method may not be accurate, however, for
situations with large packing pressures. Recent research [16] has shown that any
manipulation of the packing pressure can have an e€ ect on the residual stress pro®le.
Current work is aimed at addressing this de®ciency.
Taken as a whole the results in this article have shown that:
i.
The LHBI±VAM treatment provides an accurate solution of the one-dimensional thermal-elastic residual stress model.
ii. The formation of residual stresses is adequately modeled by a thermal-elastic
treatment alone.
iii. The LHBI±VAM model provides an accurate description of how the residual
stress in an injection molding process is a€ ected by process conditions and
material properties.
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538
X. ZHANG ET AL.
APPENDIX: DERIVATION OF THE LHBI-VAM MODEL, EQ. (18)
In – D µ z µ zs the approximating temperature pro®le is
T(z; t) =
Ts – Tf
T f zs ‡ T s D
z‡
zs ‡ D
zs ‡ D
and
b(T – Ts ) = b
Ts – Tf
¢ (z – zs )
zs ‡ D
Substituting into Eq. (6) with some rearrangement and simple integration gives
b Ts – Tf 2
zs ¢ ev (zs ) = – ¢
¢z ‡
2 zs ‡ D s
Z
zs
ev dz
0
Taking the derivative
dev
1
zs
= 2A ¢
– A¢
dz s
zs ‡ D
(zs ‡ D ) 2
where the constant A = b(Tf – Ts )=2. Since the ¯ow strain is a function of space
alone
dev
1
z
= 2A¢
– A¢
dz
z‡ D
(z ‡ D) 2
which has the general solution
ev = A ¢ ln(z ‡ D ) – A ¢
D
‡C
z‡ D
Substitution into Eq. (7) gives
etot = 2 A ‡ C ‡ A[ln(b ‡ D ) – 1]
From which, through Eq. (8), the LHBI±VAM expression for the stress at ®nal
cooling, Eq. (18) in the main text, results in
» µ
¶
¼
E
b‡ D
z
s(z) =
b(Tf – Ts ) ln
–
2(1 – v)
z‡ D
z‡ D
In the limit, D ! 0, the perfect thermal contact expression for the residual stress, Eq.
(16), follows.