an analysis of thermal warpage in injection molded flat parts due to unbalanced cooling

An Analysis of Thermal Warpage in Injection Molded Flat Parts Due to Unbalanced Cooling

MICHAEL ST. JACQUES

Eastmun Kodak Company Rochester, New York 14650

To illustrate the potential effect of unbalanced cooling on warpage of flat parts, a simplified two-part analysis is presented. First a computational model for amorphous polymers cooling in an injection molding cavity is presented. The simulation is a finite difference solution of the one-dimensional, transient heat conduction equation with constant material properties. Plastic and mold temperature profiles are calculated through the cooling cycle and the transients from cycle to cycle are included. Temperatures are predicted for both sides of the mold allowing asymmetrical cooling to be analyzed. The model is verified analytically and is in agreement with pub- lished data. Secondly, a simplified method of predicting the thermal warpage of a flat part from calculated temperature profiles is discussed and illustrated. The relative effects on calculated part warpage of asymmetric mold geometry and cooling fluid temperature are predicted with this analysis method. The sensitivity of warpage to these design factors is illustrated for an example part.

INTRODUCTION or successful fabrication of plastic parts by the injec- F tion molding process, cooling and solidification

must be accomplished in the shortest possible time with part warpage held within specified limits. If a plastic part is cooled in too short aperiod of time it may deform or flow upon release from the mold cavity. On the other hand, if the part is cooled for an excessive period of time, the production cost becomes prohibitive and warpage may still occur.

Many of the injection molded parts manufactured today are either thin and flat, as illustrated in Fig. 1, or can be developed from thin sections. For simplicity only flat parts made of amorphous materials are considered in this paper. For a part of the type shown in Fig. 1, most of the cooling takes place through the largest surface area and the most severe temperature gradient is established through the thickness of the relatively low thermal conductivity plastic.

A number of mechanisms have been identified as causes of part warpage, including differences in cavity pressure, differential orientation, differential crys- tallinity, inhomogeneous thermal stresses caused by irregular geometry, and stresses frozen during packing. The one cause of warpage considered in this study is the thermal stress established by temperature gradients while the plastic is solidifying. In the case of flat parts the thermal stress results in bending if the temperature profile is asymmetric.

In this paper, unbalanced cooling is considered by studying two design factors which skew the cooling

temperature profiles. These are differences in the cooling water temperature on each side of the cavity and differences in the thickness of the mold from the surface of the cavity to the cooling channel.

The purpose of this investigation was to develop a method of estimating the relative effect of these two design variables and their combinations on part warpage (i. e. bending due to asymmetrical temperature distribution). Calculation of warpage for flat parts is basi- cally a two-fold procedure. First, the thermal history of the part must be predicted. Secondly, the calculated temperature profiles must be converted into thermal strains and subsequently related to the final deformed shape of the part after its release from the mold.

In order to predict the history of temperature profiles through the thickness of the part a computational heat

MOLD CAVITY

COOLING CHANNELS

Fig. 1 . Mold base and caoity (one side) w i th cooling channels.

POLYMER ENGINEERING AND SCIENCE, MARCH, 1982, Vol. 22, No. 4 24 1

Michael S t . Jacques

transfer model was developed. An investigation of the literature showed that a number of models have been written since the original work of Ballman and Shusman (1). The model due to Host and Osmers (2) is complete from the process standpoint but spatial temperature gradients required for this investigation were not included.

Variable material properties were extensively considered by Dietz (3) whose cooling model shows promis- ing correlation for a large cylindrical part. However, the cooling was symmetrical. Rigdahl (4) presented a finite element method of combined thermal and elastic stress analysis. The approach illustrates the potential usefulness of the finite element technique for situations requiring two-dimensional analysis. However, the model was not used in this study because of suspected aspect ratio problems with flat parts (i.e. the thickness being significantly smaller than the length or width) and the probable costs of implementation and running. The extensive mold filling analysis of Lord and Williams (5) accounts for the important flow effects such as viscous heating and development of a solidified layer of plastic. Their method can be used to predict transient temperature profiles in the mold cavity as well as in the melt. However, symmetry is assumed and transients between cycles appear to be excluded.

The computational heat transfer model, calculation of warpage and illustration of the relative effect on part warpage will now be described. Since relative effects, as opposed to absolute values were the desired goal of the investigation, constant material properties were used and compressibility as well as viscoelastic effects were assumed negligible. However, the software was written so that inclusion of variable material properties and other effects could readily be included in future efforts.

COMPUTATIONAL MODEL OF COOLING Experimental studies demonstrate that temperature

profiles are very uniform on a mold cavity surface with a water cooling system similar to that shown in Fig. 1 . These observations imply that heat transfer primarily occurs through the thickness of the plastic as described by the one dimensional, transient heat conduction equation (Carslaw and Jaeger [6]).

It is possible to solve this equation analytically for constant material properties if the metal mold cavity adjacent to the plastic is held at a constant temperature. However, in a real molding situation the mold cavity temperature changes with time.

The basic cooling model is shown schematically in Fig. 2. The plastic is divided into 11 layers and each mold base is divided into 10 layers. The thickness of the mold base layers is varied to optimize the time con- stants. A transient energy balance is made for each layer by the finite difference method resulting in a set of 31 simultaneous differential equations. These are solved by Euler’s method (see Scarborough [7]) which was

COOLING FLUID

TEMPERATURES ARE

THIS BOUNDARY

I 4 I

MOLD, SIDE 1 (10 layers)

SIDE 1

Fig. 2. Computational model of cooling.

found to b e as accurate as t h e modified E u l e r predictor-corrector method and is much faster (also see Lord and Williams [5]). Instability was avoided by choosing a sufficiently small time step (3 milliseconds for the example parts to be considered). The model was written in FORTRAN and run on a Hewlett Packard 3000 computer. Computation time is about twice real- time so that a 10-second cycle takes approximately 20 seconds to compute. This is considered reasonable for our purposes and could be improved substantially, if necessary, by running the program on a faster computer.

Convective heat transfer occurs at the boundary between the metal mold base and the cooling fluid, and an empirical correlation for turbulent flow from Holman (8) was used to calculate the appropriate heat transfer coefficient. Adequate cooling channel spacing is assumed so that all of the thermal energy transferred from the plastic is absorbed by the mold sides and cooling fluid (see Fig. 2 ) . Also, the analysis deals with amorphous materials which vitrify without loss of latent heat at a specified glass transition temperature. Inti- mate contact is assumed at the plastic-metal interface and the series thermal conduction coefficients are combined in the usual manner. Polystyrene was selected for the calculations.

Initially, the plastic and metal layers are set to prede- termined starting temperatures. Typically, all 11 plastic layers are set to the melt temperature and the metal layers are set to their appropriate cooling fluid temperature. It might be reasonable to initialize the plastic

242 POLYMER ENGINEERING A N D SCIENCE, MARCH, 1982, Vol. 22, N o . 4


layers to other values. For example, the existence of a frozen layer created while the cavity is filling or viscous heating effects would dictate an initial temperature gradient which could be determined from an appropriate flow analysis (e.g. Lord and Williams [5] or Wang et a2. [Q]).

The cooling model calculates temperatures by solution of the above mentioned equations. A flow chart describing the computational procedure is shown in Fig. 3 and Table 1 shows all input and output data. A method was devised to simulate real operating conditions when the mold opens and the part is removed at the end of the specified cooling time. During this period, the plastic temperature nodes are replaced with a natural air convection boundary condition for a verti- cal plate obtained from McAdams (10). The cooling fluid boundary condition remains unchanged. The 20 remaining differential equations for the two mold bases are solved until the end of the specified cycle time.

To simulate start-up conditions, metal temperatures at the end of one cycle become the initial temperatures for the next cycle, and plastic temperatures are re- initialized to their original values. The entire computational procedure is repeated until the specified number of cycles is completed.

An example of a time-temperature history for one cycle is shown inFig. 4 . The plastic layers closest to the metal wall cool the fastest and the center layer is the last to cool. This thermal history is reasonable and expected. Also, the metal layers adjacent to the cooling fluid remain essentially unchanged. After the specified

R W I W W T .

VARIULES. MAKE PRELIIINIRV

REDUCE SIDE 1 I m R F L C E

COEFFICIENT ID I C C W m l FOR SEPARATION

I +- N U Y E I I U U Y SOLVE

LOUITIONS FOI

IN YOLD. INCREASE TIME

b

at DIFFEIENTILL

m s n c cooLmG

. o O BEENREACHED?

STORE U L C U U T E O

FOR UTER

I N O 0 BLEHREICHEOI

IYITIILIZE MOLD TEYPERINRES TO CURRENTVLLUES

END

Fig. 3. Flow chart of computational model.

Table 1. Summary of Input Data and Output for Heat Transfer Model

Input

Plastic melt temperature ("F) Cooling water temperature on each side ( O F ) Thickness of plastic part (inches) Thickness of metal from face of mold cavity to cooling channel,

each side (inches) Cycle time (seconds) Cooling time (seconds) Separation time (seconds) Total number of cycles (dimensionless) Physical properties of plastic and mold material including:

Density (Lbm./(inch)') Heat Capacity (BTU/(Lbm., O F ) Thermal Conductivity (BTU/(sec., inch, "F)

output

Temperature of plastic and metal layers at each time up to the

Plots of temperature vs. time and plastic temperature profiles at completion time.

various times.

cooling time is reached (7 seconds, in Fig. 4 ) , the plastic is removed and only the metal temperatures are shown. As anticipated, cooling of the metal in air is relatively slow.

The results described above can be illustrated as temperature profiles through the thickness of the plastic at various times into the cooling cycle. Such a set of profiles is shown in Fig. 5 . It is seen that initially the profiles are severe but begin to level out after a few seconds.

As time progresses, the calculated temperatures begin to repeat themselves from one cycle to the next, similar to a production molding operation. Figure 6 illustrates the buildup of the maximum plastic temperature and the mold cavity surface temperature in the first 20 cycles of a molding operation as predicted by the model. Molding conditions are also shown in Fig. 6 and, in this case, cycle to cycle repetition occurs in about 12 cycles.

A comparison of the analytical solution (Carslaw and Jaeger [5]) to model predictions of cooling time (the time required to cool the central layer of plastic to the glass transition temperature) is given in Fig. 7 . Two

0. 1 1 , / / / 1

0. 2. 4. 6. 0. ' 10. ' TIME INTO CYCLE (seconds)

Fig. 4 . Temperature us. time for one cycle as predicted by computational model.

POLYMER ENGINEERING AND SCIENCE, MARCH, 1982, Vol. 22, No. 4 243


20,.

D, y.

500

400.

G e W a 3

300. d B W n

I-

200,

100.

0 PREDICTIONS OF MODEL

PIaslk I n W h Tmp.ohlre =4mo F ~ m i i n p water @ 800 F Born 8mea Mdd T h k k m u 4 . P b Born SMw lnllhl Mold Tamp.rNum = 110' F.

I I I I s.C' -1

I I I 1 .01 .02 .03 .04 .05

DISTANCE THROUGH THICKNESS OF PLASTIC (inches)

Fig. 5 . Plastic temperature profiles at uarious times as predicted by computational model.

I I 't

MAXIMUM PLASTIC TEMPERATURE AT EXIT TIME (7 seconds)

1 l O . t . ' \

I * E e

(_I 90.

* . . . . . . . . . . .

'MOLD CAVITY SURFACE TEMPERATURE AT END OF CYCLE (11 seconds)

80.

5 10 15 20 CYCLE NUMBER

0 Plastic Injection Temperature =490° F 0 Cooling Water @ 80 O F Both Sides 0 Part Thickness = .050 in 0 Mold Thickness = 1.0 in Both Sides

Initial Mold Temperature = 80°F.

Fig. 6 . Cyclic temperature buildup.

THICKNESS OF PART (INCHES)

Fig. 7. Comparison of analytical solution with model predictions of cooling time vs. part thickness.

analytical solution curves are plotted. The upper curve represents a mold surface temperature of 110°F which was chosen because it is close to the surface temperature predicted by the model. The lower curve is for a mold surface temperature of 80°F which is the cooling fluid temperature used in the model. Model predictions are for three part thicknesses and the calculations are for steady operating conditions which are reached after about 20 molding cycles. Time cycles were selected to resemble realistic operating values. The two analytical solutions bracket the model predictions which are close to the upper curve for each part thickness. Also, model predictions are close to the experimental results of Ballman and Shusman (1) when their lower glass transition temperature (186°F) is used.

Including the mold sides and transients from cycle to cycle in the model gives the mold designer a good quantitative idea of actual mold surface operating temperatures and the effects of selected production time cycles on cooling time and operatingconditions. Also, if design factors are such that asymmetric cooling results, that is significant because it creates a bending stress and subsequently warpage.

WARPAGE DUE TO THERMAL STRESS Upon cooling a flat part, the center layers remain at a

higher temperature than the sides adjacent to the mold walls and have a greater potential to shrink than the sides. As the part cools and the sides are bearing on the mold cavity walls, tensile strain is set up in the center layers compressing the outer layers as illustrated in Fig. 8. End effects tend to diminish the stress some- what. When the part is released from the mold cavity it can react in two ways. If cooling is symmetrical, every- thing is balanced about the center, and little or no deformation takes place. For very thin parts, the resul- tant forces can reach critical column buckling values. However, since the forces are internal, the ultimate result is unknown. If the cooling is asymmetrical, the stress distribution at ambient temperature (see F i g . 8) bends the part when it is released from the mold.

244 POLYMER ENGINEERING AND SCl€NC€, MARCH, 1982, Vol. 22, No. 4


/ I L--Yo*

Fig. 8. Temperature and stress distribution in plastic resulting from cooling.

The final deformation of the part or the warpage can be calculated in a simplified way if it is assumed that the elastic modulus of the material takes on a constant average value at the specified glass transition temperature and that viscoelastic effects are negligible. That is:

E = 0 , T > T ,

E = E , T I T ,

where E is the elastic modulus, T is the temperature, To is the glass transition temperature and E is the average value of the elastic modulus.

When the maximum plastic temperature reaches the glass transition point, and the part is restrained from bending by the cavity, the temperature profile through the thickness of the part (predicted by the cooling model) gives rise to a stress distribution which is evaluated in the following way (see Gatewood [9]):

(2)

u, = aE(T, - T ) (3) where u, is the normal stress at the nth layer after all eleven temperatures have reached ambient temperature, T, is the temperature of that layer at the time when the maximum plastic temperature reaches the glass transition point, T is the average temperature of all eleven layers at the same time and (Y is the average thermal expansion coefficient (assumed constant). A ,positive value of U, indicates tension.

The stress distribution is then converted to an equiva- lent bending moment by:

11

M = 2umAnSn (4) n = 1

where M is the bending moment, A, is the cross sectional area of the nth layer and 6 , is the distance of the nth layer from the center of the part. For the present model (see Fig. 2 ) ,

6 , , > 0 , l ~ n ~ 5

S,, = 0, n = 6 (5)

S,, < 0,

bending theory (see Timoshenko [12]):

7 5 n 5 11

Finally, the deformation is obtained using pure

d = MZ2(1 - pz))/(8E1) (6) where d is the maximum deflection of the plate or

warpage, 2 is its length, p is poisson's ratio, and 1 is the moment of inertia in bending.

Equation 6 is strictly applicable only for pure bending with the neutral axis located at the mid-plane of the part. In order to check the applicability of this approxi- mation technique to a plate with skewed stress distribution it was compared to a finite element model. The finite element analysis was performed using ANSYS (ANalysis SYStem, see DeSalvo and Swanson [13]), a well-known, commercially available program. Two dimensional isoparametric solid elements were used, and an asymmetric temperature distribution (for one of the cases to be discussed) was applied. The results of this analysis are shown in Fig. 9 where the warpage is seen to be 0.034 inches for a 4 x 4 inch flat polystyrene part of 0.050 inch thickness. The proposed analytical method (Eqs 2-6) gives the same result within the required accuracy. Therefore, the analytical method was em- ployed for this study because it is easier to use and is less costly than the finite element model for the part shape under consideration.

It must be noted that ignoring viscoelastic effects results in a more severe deformation than actual. Also, the values of the elastic modulus and the thermal expansion coefficient used for this analysis (Table 2 ) were averages selected to yield slightly greater predicted warpage than probably occurs. The predictions to be presented are intended to indicate the relative impor- tance or sensitivity of warpage to asymmetric mold geometry and cooling fluid temperature.

RESULTS A series of computer analyses were made to examine

the relative effect of asymmetric geometry and cooling fluid temperature on thermal history and warpage. Each of these analyses were made over a period of 20 cycles so that steady operating conditions were reached. Cooling and cycle times were selected to be realistic production values and three part thicknesses were considered (up to 0.150 inches). The plastic injection temperature was 490°F for all cases. A summary of operating conditions and material properties is shown in Table 2 and the warpage calculations were made for a 4 in. x 4 in. part.

The effect of cooling the mold with fluids of different temperatures on each side is illustrated in Fig. 10. As the difference in cooling fluid temperatures between mold halves increases, the predicted warpage increases linearly. The thinnest part warps the greatest amount, as expected, because of its relatively small moment of inertia in bending. The different slopes of the three lines illustrate the sensitivity of warpage to part thickness.

Asymmetry in the plastic temperature profiles, established by having a different thickness of metal on each

Fig. 9. Results of Fnite element analysis.

POLYMER ENGINEERING AND SCIENCE, MARCH, 1982. VoI. 22, No. 4 245


30.

20.

10.

Table 2. Summary of Operating Conditions and Material Properties for Computer Analyses

A

-

-

-

COOLING FLUID TEMP. AS SHOWN

Operating Conditions

Plastic Injection Temperature, 490°F Cooling Fluid Temperatures (2sides), varied from 60"Fto 100°F Thickness of Metal Mold Base from Cavity Face to Cooling

Cycle times were selected for each of three part thicknesses as

Part Thickness Cooling Time Total Cycle Time

Channel (2 sides), varied from 1.0 in. to 2.5 in.

follows:

(sec.) (sec.) 0.050 in. 7 11 0.100 in. 13 17 0.150 in. 30 35

where the Cooling Time is the time during which the plastic remains in the closed mold cavity and theTotal CycleTime is the time between mold fills.

0.

Polystyrene Properties

CURVES C & D FOR DIFFERENCES IN -METAL THICKNESS AS SHOWN

I I I I 7

Heat Capacity, 0.32 Btu/(lb. "F) Average Elastic Modulus, 3.75 . lo5 psi Thermal Conductivity, 1.85 . Glass Transition Temperature, 220. "F Coefficient of Thermal Expansion, 4.86 . Density, 0.039 Ib. / i t~.~

mold side, is illustrated in Fig. 11. The warpage increases as the difference in metal thickness between mold sides becomes greater. The curve for the thickest part is linear while the result for the thinnest part is non-linear. Also, warpage for the thinnest part is greatest again because of its relatively low bending stiffness. The sensitivity to part thickness is not as pronounced as that shown in Fig. 10.

The sensitivity of warpage to part thickness is further illustrated in Fig. 12 , A comparison of the curves shows that the sensitivity is greater for the design condition of asymmetric cooling fluid temperatures than for the case of different metal thicknesses.

Btu/(sec. in. OF)

in./(in. "F)

- 0. 5. 10. 15. 20.

DIFFERENCE IN COOLING FLUID TEMPERATURES BETWEEEN MOLD SIDES F)

Fig. 10. Sensitivity of part warpage to difference in coolingfluid temperatures between mold sides for part thicknesses shown as predicted by model.

246

I i 1 I I i

Plastic lnjecllon Temperature =4W0 F Cooling Water @ 80" F Both Sides

30. t 3 z 2

2 g 10.

20.

Y 0

a

0.

- PART THICKNESS

-

P

1

DIFFERENCE IN METAL THICKNESS BETWEEN MOLD SIDES (IN.)

Fig. 11. Sensitivity of part warpage to difference in metal thickness between mold sides for part thicknesses shown as predicted by model.

I I I I I

40* t

Computations for cases where the two causes of warpage occur simultaneously show that the effects are approximately additive. Therefore, part warpages from Figs. 10 and 11 can be added (or subtracted) yielding a reasonable estimate of final part deformation.

POLYMER ENGINEERING AND SCIENCE, MARCH, 1982, Vol. 22, No. 4


SUMMARY The computational heat transfer model presented in

this paper appears to be a reasonable simulation of flat plastic parts cooling in an injection molding cavity. Analytical verification exists for certain conditions and some experimental evidence is available. All of the results presented are physically plausible.

The model is recommended for estimating cooling time for flat parts. Parts which include ribs or other geometric complexities require more extensive two or three dimensional analysis. Because the model in- cludes the mold sides and transients from cycle to cycle, it should be very useful to mold designers in determining the effects of real operating conditions on thermal profiles.

The simplifying assumptions made in the analysis of warpage appear to be reasonable for evaluating the sensitivity of warpage to the two design factors considered. Experimental verification is required before the accuracy of the model can be conclusively evaluated. A more accurate approach for the prediction of warpage would be to use an elastic modulus and thermal expansion coefficient variable with temperature and to make continuous determination of the stress. Inclusion of viscoelastic effects would also im- prove the model.

The reported sensitivity of warpage to mold design factors illustrates the distinct possibility of troublesome operation from slight design imbalance. Hopefully, the analysis will help in overcoming such difficulties.

ACKNOWLEDGMENT The author gratefully acknowledges the efforts of

Ms. G. Wildeman for preparation of the figures appear- ing in this paper. Thanks are also expressed to Mr. R. Curtin for conducting the computerized literature search used in this work and to Mr. D. Onesti for consultation on the finite element stress analysis method. Finally, the author’s appreciation is extended to Mr. W. Kirk for his advice on the heat transfer analysis and to Dr. J. Greener and Dr. J. Wheeler for reviewing this paper and offering helpful comments and criticism.

NOMENCLATURE A,, = cross sectional area of the nth layer, in.’ C , = heat capacity, Btu/(lb. OF) d = maximum deflection of a plate (warpage), in. E = elastic modulus of the polymer, psi E = average elastic modulus of the polymer, psi 1 = moment of inertia in bending of a plate, in.4 k = thermal conductivity, Btu/(Sec. in. O F )

I = length of a plate, in. M = bending moment, in lb. n = subscript denoting the number of the plastic

t = time, sec. - T = temperature, O F

T = average temperature, OF T, x = Cartesian coordinate a

6,

p = density, lb./in.3 u, = normal stress on the nth layer of the plastic, psi

layer

= glass transition temperature of the polymer, O F

= thermal expansion coefficient of the polymer,

= distance of the nth layer from the center of the in./(in. OF)

plastic, in.

El.

1. 2.

3. 4. 5. 6.

7.

8.

9.

10.

11.

12.

13.

= Poisson’s ratio of the plastic

REFERENCES R. L. Ballman and T. Shusman, Mod. Plast., 37,126 (1959). R. F. Host and H. R. Osmers, SPE ANTEC Tech. Papers, 24, 77 (1978). W. Dietz, Polym. Eng. Sci., 18, 1030 (1978). M. Rigdahl, Inst. ofPoZym. Maters., 5, 43 (1976). H. A. Lord and G. Williams, Polyrn. Eng. Sci., 15,569 (1975). H. S. Carslaw and J. C. Jaeger, “Conduction of Heat in Solids,” Ch. 3, Oxford Clarendon, London (1959). J. B. Scarborough, “Numerical Mathematical Analysis,” 6th Edition, p. 310, The Johns Hopkins Press, Baltimore. J. P. Holman, “Heat Transfer,” p. 160, McGraw-Hill, New York (1968). K. K. Wang et al., “Computer Aided Injection Molding Sys- tem,” Progress Reports nos. 1 to 5, National Science Foun- dation Grant no. APR74-11490. W. H. McAdams, “Heat Transmission,” p. 241, McCraw- Hill, New York (1942). B. E. Gatewood, “Thermal Stresses,” p. 9, McGraw-Hill, New York (1957). S. Timoshenko, “Strength of Materials,” Part 11, p. 76, D. Van Nostrand Company, New York (1958). G. J. DeSalvo and J. A. Swanson, “ANSYS, Engineering Analysis System Users’ Manual,” Swanson Analysis Sys- tems Inc., Houston, PA. (1978).

POLYMER ENGINEERING AND SCIENCE, MARCH, 1982, Vol. 22, No. 4 247

an analysis of thermal warpage in injection molded flat parts due to unbalanced cooling

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