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252 Y .C . Chen, C- H. Chen,S. C . Chen

Kubat6 measured the residual stress in plastic parts bya stress relaxation method. Only the average value ofthe residual stress over the whole of the part could beobtained. Siegmann et aL7 and Broutman8 applied alayer-removal technique to the solidified part. Theyemployed a high-speed milling machine to remove

uniform layers from the part surface and plotted theresulting curvatures as a function of the depth of theremoved layer. The unknown longitudinal residualstress was then derived by measuring the relative curva-tures along longitudinal and transverse dimensions ofthe plate. This was also a destructive measurement. Toprevent the part from being destroyed, a photoelasticitymethod can be utilized. From the fringe pattern pro-duced by light interference, one can obtain the birefrin-gence value for the part from the optical-stressrelationship and photoelastic theory. Although residualstress analysis using the photoelastic technique andfringe pattern information is tedious and time-consuming, the measured birefringence can be con-sidered as the relative residual stress distribution in themolded parts. Of all the methods of experimental stressa n a l y ~ i s , ~ - ~he photoelastic stress analysis method hasthe advantage of determining the complete stress dis-tribution at any location in the art.^-'^ On the otherhand, classic photoelasticity measurement demandsmaterials with high birefringence, leading to an exten-sive use of plastics as a model mate~ial.'~-'~ ince thebehavior of these materials is often different from thatof the prototype materials, their use distorts the simi-larity relationship. In many contemporary problemsthis distortion is unacceptable. A new method forwhole-field stress analysis called 'half-fringe photoelasti-city' (HFP), was developed to overcome these problems,integrating modern digital imaging technology andclassic photoelasticity. Small differences in birefringencecan be resolved; this permits materials and loads to bechosen so that no more than one-half a fringe orderappears in the area of interest. In addition, with the helpof a modern digital imaging technique, it is possible toread-out the fringe position and the fringe order auto-matically, without the need of a compensator to allowfor the optical path difference at any location in thepart. 8-zo

Although HFP can provide an efficient way ofexamining birefringence and residual stress distributionwithin the molded parts, the method can only beapplied to transparent materials. Therefore, it will be ofgreat help if a simulation tool can be used to predict thebirefringence in parts and the residual stress develop-ment during all stages of the injection molding process.Using these computer simulations for birefringence andresidual stress, the physical origin of the residual stressformation can be understood better on a scientific basis.Numerical simulation of the frozen-in birefringence andthe associated residual stresses in the molded partsusing a viscoelastic fluid model, however, is a difficult

task. Recent reviews have been The majordifficulty arises because of the viscoelastic nature of apolymer fluid subjected to large elastic deformationduring the molding process. Among different visco-elastic models employed for the simulation of injectionmolding, the Leonov model seems to be more flexible

and has promising capability for describing the visco-elastic nature of the polymer melt.zz-z8 In the first partof this paperz1 analysis of the injection molding process,based on the Leonov viscoelastic fluid model, has beendeveloped to study the effects of processing conditions,including melt temperature, mold temperature, fillingspeed and packing pressure, on the residual stress andbirefringence development in injection molded partsduring the entire molding process. The simulated resultsmay be verified experimentally. If the simulated resultsshow good consistency with measured values, then thesimulation can be applied to analyze the residual stressformation in molded parts which are not transparent.

In the present work, the birefringence of injectionmolded plates was measured using a half-fringe photo-elasticity system and the digital image analysis tech-nique. The effects of various processing conditions, suchas melt temperature, mold temperature, filling speedand packing pressure on the birefringence developmentin the parts were investigated. All experimental resultswere compared with simulated predictions calculated bythe Leonov viscoelastic fluid model, details of which arereported elsewhere.z1

B IR EFRlN G EN CE M EASU R E MENT SYSTEM

Birefringencean d stress-optical relationship

When the refractive index for a material in one direc-tion of polarization is different from that in anotherdirection, the material is said to be birefringent, ordouble refractive. If stress is applied to an isotropicmaterial, the stress produces an index ellipsoid with arefractive index change as follows:

n, = nj = aa, + b(aj + ak ) for i , j , k = 1, 2, 3 (1)where a', r r z and a3 are the principal stresses, that is the

stresses in the direction of the principal axes, and n, isthe refractive index experienced by a light wave pol-arized in the direction of principal axis i. By taking thedifference between the two axes i an dj, one obtains

n, - j = (a - )@ , - o j ) = C(a i- r j ) (2)

where C is the stress-optic coefficient. This is alsoknown as the 'stress-optical law'. It can be seen that thechange in the index of refraction is proportional to thedifference in the principal stress. The stress-optic coeffi-cient is usually measured in Brewsters (Br):

1 Br = 10-lz(N/mZ)-l = lo1' Pa (3)

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Birefrigence development in injection molded parts. 11 253

Photoelastic theory the relationship

(1 1)= BIY

where B is a proportionality constant and y is the slopeof the CCD camera sensitivity curve. To calibrate thesystem for photoelasticity, a reference intensity I , is

introduced; eqn (6) is rewritten to incorporate this refer-ence intensity for a particular model polariscope com-bination. As a result, eqn (11) becomes

Z = BIIo sin(N~)]~ (12)

To obtain the relationship between fringe order N andgray level Z, the procedure reported previously was fol-lowed. From eqn (11) the maximum response of thesystem is

zm, = BIkm (13)Therefore

For a two-dimensional problem, the stress-optic law oflinear elastic stress birefringence is

where A is the relative retardation for the light atnormal incidence to the plane of the sample, al and a2are in-plane principal stresses, d is the thickness in thepropagation direction of the light, c is the relativestress-optic coefficient in Brewsters, and C s the stress-optic coefficient. Equation (4) can be rewritten as

( 5 )

where N (= A/2n) is the relative retardation in terms ofcomplete cycles of retardation, f, (=IZ/c) in units ofnumber/fringe-my when m is the materials fringe valueor the photoelastic fringe constant.

In a dark field circular polariscope set-up, the intens-ity of the transmitted light emerging from the analyzeris

(6)A2

I = I , sin - = I , sin(Nn)

where I , is a constant. Hence, extinction occurs whenI = 0, i.e. when

(7)A- = Nn for N = 0, 1, 2, 3, ..2

The resulting dark lines are called isochromatic fringesand the corresponding numbers are called full fringeorders. For these values, N is always an integer.

In the light field set-up for a circular polariscope, theintensity is given by

A2

I = I , COS -

Hence, extinction occurs when I = 0, i.e. when

n for N = 0, 1, 2, 3, ... (9)A 2 N + 12 2

-

or

These are often called half-order fringes. In the presentfringe measurement, the dark field circular polariscopeset-up was utilized.

Calibrationof the im age analysis system

In the image analysis system, the digital output value,or the gray level, Z, is related to the light intensity, I , by

For a selected half-fringe interval, N = 1/2 andsin(N,, n) = 1 ; qn (14) becomes

L--7 - sin(Nn)lZy

or

A set of values of N and Z can be obtained by employ-

ing the Tardy compensation method* on an unloadedspecimen. The least squares method is then used toestablish the parametric relationship between N and Z.An ifitercept term, A, will normally be introduced toaccount for the least squares fitting process. The relation-ship between fringe order N and gray level Z can bewritten as

If the digital photoelasticity method is applied to findthe fringe order N at a specific part location, the corre-sponding birefringence is then given by

NIZAn =-

d

PROCESS FORMULATION AND NUMERICALS IM U LATlO N

The mathematical modeling and the assumptions usedfor the simulation of the injection molding process arebasically similar to those of previous Therelevant components of the governing equations forcontinuity, momentum and energy describing the injec-tion molding of a simple plate in terms of the Hele-

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254

TABLE 1. Materi al properties of polystyrene

Y. C. Chen, C-H. Chen, S. C. Chen

Shaw flow model are

1. Thermal properties

2. Rheological properties (Leonov viscoelastic model) at 463 Kp = 940 kg/m3; C, = 21 00 J / (kg K) ; K , = 0.1 5 W/(m K)

N = 2; s = 0.09; 8, = 0.8s; 8 = 0.027 s; qo = 7000 Pas; /I, = 5440 Pas;q2 = 1500 Pas; T, = 373 K; Tre f= 407 K; C, = 20.378: C2 = 101.6 K

P = 186 M P a;3. Constants in Spencer-Gilmore type PV T equation of state

= 1220kg/m3; f i = 80 J/(kg K)

where t is the time, p the density, C, the specific heat, Kthe thermal conductivity, P the pressure, T the tem-perature, z is the gapwise direction, z the shear stressin the x direction and U the gapwise injection speed inthe x direction. The term $his the dissipation functiondefined el ~e wh er e. ~~

The constitutive equations used for the analysis com-prise the Leonov viscoelastic model described by

1

Nq k

' l o = c -= l ( l - ,

where qo is the zero-shear-rate viscosity, q k is the shearviscosity of the kth mode, e k is the relaxation time of thekth mode, s is a rheological constant (0 < s -= l), z is thestress tensor and Ck is the elastic strain tensor of the kthmode. For self-consistency there is a constraint on c k

such that

det[Ck] = 1 (25)

The temperature dependence of qo , k and e k is basedon the Williams-Landel-Ferry shift factor aT , namelyek(T) = ek(TO)aT/aTo and qk(T) = ?k(TO)a?'/aTo where

Tef is the reference temperature,sition temperature, and C, and C2 are constants.

is the glass tran-

Since the polymer melt is considered to be compress-ible, the change of melt density should follow a PVT(pressure-volume-temperature) relationship. In thisstudy, the Spencer-Gilmore type P V T equation is usedand is given by

where p, 9 and ff are constants. Once the pressure, thevelocity, the shear rate, the elastic strain tensor and thestress tensor have been calculated, the first normalstress difference NiF , the shear stress z , and the corre-sponding birefringence ANF of flow-induced origin inthe x-z plane are computed according to the followingequations :

ANF = CAT = C[NfF + 4~: , ]"~ (30)where C is the stress-optical coefficient taken16 as avalue of 4.8 x m3/N. The thermal-induced stressesare formulated under a plane strain condition, i.e.assuming that E, = 0, E,, = 0 and E,, = 0. As a result,the stress-strain relation can be described by

cfi

z x x = [( l - )E , + VE, - j - ]1 - - v2E

1 - - v2[YE, + (1 - V)E,- T ]T,, =

E

zxz = _ _ _2(1 + v ) (33)

where v is Poisson's ratio, E is Young's modulus, CI isthe thermal expansion coefficient and AT the tem-perature difference. Stress zy y also exists. However, itdoes not contribute to the birefringence formation ofthermal origin in the x-z plane. The stresses in eqns(31)-(34) are solved using the principle of virtual work."The first normal stress difference N IT and the corre-sponding birefringence ANT, resulting from thermal

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Birefrigence development in injection molded parts.II 255

*

uni t : mm

Fig. 1. Geometry of the injection molded plate.

stresses, are therefore expressed by

N I T = (Txx - zz)T

ANT = CAT = C[NfT + 4~:,]"~(35)

(36)

In the present case, ANT is about one to two orders ofmagnitude lower than ANF under even cooling condi-

Light So

-1 Analyzer2 Quarter-Wave Plate3 Polarizer

0 VMonitorLJI

0

Fig. 2. Schematic diagram of the half-fringe photoelasticsystem used for birefringence measurements.

tions for both sides of the mold cavity walls. Therefore,the measured birefringence is assumed to be of flow-induced origin. In order to compare the simulatedresults with measured birefringence, gapwise averagedvalues of ANF along the flow direction were computed.Detailed mathematical procedures for this have beenr e p ~ r t e d . ~ ' . ~ ~ll the material constants required forthe simulations are listed in Table 1.

E X P E R I M E N TA L

A one line gated plate mold giving a specimen of 2 mmuniform thickness, as shown in Fig. 1, was built to

Fig. 3. (a) Fringe pattern of part molded without applying packing pressure. (b) Fringe pattern of part molded with 18MPapacking pressure. (c) Fringe pattern of part molded with 40MPa packing pressure. (d) Fringe pattern of part molded with

61.5 MPa packing pressure.

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Birefrigence development in injection molded parts.II 257

0

9 3.0 A I-Fill l i m e = l . 4 1 secPaekin; Pressure = 39.7 MPaWall Temperature = 25 ' C- e l t = 2 l 0 *C- ~ e l t = 2 3 0- melt=ZBO '

0.0 j I0 I00'g 0 20 40 6005 Distance from gate(mm)

Fig. 6. Distribution of the gapwise-averaged birefringence Analong the flow direction for different melt temperatures.

corresponding to filling times of 0-48, 0.71 and 1-41 s.As the filling time increases, the birefringence also

increases. The smaller the filling time, the higher theaveraged melt temperature at the end of the fillingprocess. As a result, the birefringence value is alsolower. In the filling process, hot melt enters the moldcavity. Once the melt contacts the cold cavity wall, the

.y

LL.0 40 80 80 100

t o 204 Distance from gate(mm)0

Fig. 7. Distribution of the gapwise-averaged birefringence Analong the flow direction for different mold temperatures.

/ - aeking kz:E:?8.0ressure=3S.7-

acking Preasure=61.5

Filling Time = 1.41 sec

01.y 0.0 .........,-. < . . , . . X I , . . . , . . , . .

20 40 80 80 t 00

Distance from gate(mm)L o

0

Fig. 8. Distribution of the gapwise-averaged birefringence Analong the flow direction for various packing pressures.

0. 00. 0 0. 2 0.4 0.6 0.8 1 O

,

(b)

Filling Time = 1.41 secTrnell = 230 *C

U

P l.0.0

0.0 0. 2 0.4 0. 6 0.8 0

z / b

Fig. 9. (a) Simulated distribution of birefringence An alongthe gapwise direction at the end of the filling, packing andcooling stages at location 3 for the case of 210C melt tem-perature. (b) Simulated distribution of birefringence An alongthe gapwise direction at the end of the filling, packing andcooling stages at location 3 for the case of 230C melt tem-

perature.

melt starts to cool down. The solidified thicknessincreases as the contact time with the mold wallincreases. Due to the fountain flow effect at the meltfront and the small flow length of the present experi-mental part, the averaged melt temperature in the

0D Distance from gate(mm)

Fig. 10. Comparison of the simulated and measured distribu-tions of the gapwise-averaged birefringence An along the flow

direction for different injection speeds.



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258 Y .C . Chen, C- H. Chen,S . C . Chen

1 ip.!? 0.0L O 20 40 KO 80 100a Distance from gate(mm)

0

Fig. 11. Comparison of th e simulated and measured distribu-tions of the gapwise-averaged birefringence An along the flow

direction for different melt temperatures.

gapwise direction varies only slightly along the flow

direction. Near the gate area (position 2), the thicknessof the solidified layer is larger than that away from gatearea (position 6 ) [see Fig. 11. Therefore, the velocity

-0 1.2

X

Y

--:2 .8

.-L 1

z

2

Pw 0.4

2

I

2 0.0

&V

Distance from gate(mm)

Fig. 12. Comparison of the simulated and measured distribu-tions of the gapwise-averagedbirefringence An along th e flow

direction for differentmold temperatures.

a

X

2 2.5-e 2.0EPL. 1.5

L

.I

z1.0

PW

k 0.5m

m

W.z 0.0F

CIk2

S i l a t d PaddngPrerrure 0 MPaS i ' P w P r e r r u r e l 4 MP.Mcasurcd:P.ckioePrc~urr 0 MPaMarud.packjngPnuureI4 YPS

Melt temperature=

210'W m I I temperature = 25 '

Fill ing Time = 1 .4 1 sec

m

0 20 40 60 80

Distance from gate(mm)

Fig. 13. Comparison of the simulated and measured distribu-tions of the gapwise-averaged birefringence An along th e flow

direction for various packing pressures.

profile at position 2 becomes more sharply delineatedwith an increase in the maximum value at the gapcenter. Thus, the associated shear stress and shear rateas well as the birefringence values also increase. As aresult, birefringence of the part decreases with increas-ing distance from the gate. Figure 6 shows the gapwise-averaged birefringence distribution for parts moldedunder different melt temperatures. The higher the melttemperature, the lower the birefringence value. Thesame situation occurs for variation of mold tem-perature, as seen in Fig. 7. Figure 8 shows the effect ofthe applied packing pressure on the birefringence dis-tribution. As the packing pressure increases, birefrin-gence of the part also increases. However, the effect ismore significant around the gate area, as seen by theincrease of the fringe pattern strips in Figs 3a-3d. Thisis because there exists a pressure drop in the flow direc-tion within the mold cavity. As a result, a location nearthe gate area is subjected to a larger molding pressurethan that away from the gate. In addition, the partlocated near the gate area becomes solidified later thanthat away from the gate. Therefore, flow-inducedresidual stress is more concentrated around the gatearea. Generally speaking, temperature and pressure arethe two dominant factors that determine the birefrin-gence development in the part during the moldingprocess.

Figures 9a and 9b show two typical examples of thesimulated birefringence distribution in the gapwise direc-tion at part location 3. The gapwise birefringence devel-opment at the end of the filling, packing and coolingstages of the injection molding process are illustrated. Adetailed picture resulting from the simulations of thegapwise birefringence development at different partlocations and at different molding stages was describedin the first part of this paper." Figure 10 shows a com-parison between the simulated and the measured dis-tributions of the gapwise-averaged birefringence forparts molded at different injection speeds. Simulationshows a similar effect of the injection speed on thebirefringence distribution. Simulation also shows asimilar dependence of birefringence on the variation ofmelt temperature, mold temperature and packing pres-sure, as illustrated in Figs 11-13. All the simulatedresults show fairly good agreement with the measuredbirefringence values from the engineering applicationviewpoint. The only slightly large difference occurs forthe case of molding under a very large packing pressure.

CONCLUSIONS

Birefringence in injection molded parts was measuredusing a polariscope and a digital image analysis system.The effects of various processing conditions includingmelt temperature, mold temperature, filling time andpacking pressure on the birefringence development in



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Bireffigence development in injection molded parts.I1 259

the molded parts were investigated. The experimentalresults were compared with numerical predictionsobtained from simulations. The following results were

found. 1989 Chapter 2.

REFER ENC ES

1 Isayev, A. I. 8~ Upadhyay, R. K., in Injection and CompressionMolding Fundamentals,ed. A. I. Isayev. Marcel Deker, New York,

The half-fringe photoelasticity method and the

digital image analysis technique provide an efficientway for the birefringence measurement of parts. Thepresent method permits materials and loads to bechosen so that no more than one-half a fringe orderappears in the area of interest. It also requires nocompensator to allow for the optical path differenceat any location in the part.It was found that temperature and pressure are thetwo dominant factors that determine the birefrin-gence development in the part. Frozen-in birefrin-gence of the molded part decreases with increasingmelt temperature, mold temperature and injectionspeed. Birefringence in parts also increases with

increasing packing pressure, especially around thegate area.Gapwise-averaged birefringence decreases withincreasing distance from the gate. Around, the gatearea the hot melt is in contact with the cold cavitywall longer than away from the gate in the fillingstage. Thus, the solidified melt is thicker near thegate, resulting in a larger flow stress and a corre-sponding beirefringence value. In the post-fillingprocess, there exists a pressure drop in the flow direc-tion within the mold cavity. As a result, a locationnear the gate area is subjected to a larger moldingpressure than one away from the gate. In addition, a

part located near the gate area becomes solidifiedlater than one away from the gate. Therefore, flow-induced residual stress is more concentrated aroundthe gate area.Simulation of the injection molding process based onthe Leonov viscoelastic fluid model was performed topredict the birefringence development of the moldedparts throughout the entire stages of the moldingprocess. Numerical simulations showed a similardependence of birefringence of parts on processingconditions. Simulation results were also consistentwith measured values.

A C K N O W L E D G E M E N T

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This work was supported by a grant, NSC 81-0405-E033-04, from the National Science Council of theRepublic of China.

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effect of proc cond on birefringence devl im

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