10
Prediction of Fiber Orientation in Injection Molded Parts of Short-Fiber-Reinforced Thermoplastics* TAKAAKI MATSUOKA, JUN-ICHI TAKABATAKE, YOSHINORI INOUE. and HIDEROH TAKAHASHI Toyota Central Research & Development Laboratories, Znc. Nagakute, Aichi 480-1 1 Japan Fiber orientation induced by injection mold filling of short-fiber-reinforced thermoplastics (FRTP) causes anisotropy in material properties and warps molded parts. Predicting fiber orientation is important for part and mold design to produce sound molded parts. A numerical scheme is presented to predict fiber orientation in three-dimensional thin-walled molded parts of FRTP. Folgar and Tucker’s orientation equation is used to represent planar orientation behavior of rigid cylindrical fibers in concentrated suspensions. The equation is solved about a distribution function of fiber orientation by using a finite difference method with input of velocity data from a mold filling analysis. The mold filling is assumed to be nonisothermal Hele-Shaw flow of a non-Nexhonian fluid and analyzed by using a finite element method. To define a degree of fiber orientation, an orientation parameter is calculated from the distribution function against a typical orienta- tion angle. Computed orientation parameters were compared with measured thermal expansion coefficients for molded square plates of glass-fiber-reinforced polypropylene. A good correlation was found. INTRODUCTION n injection molding, computer simulation is emerg- I ing as an approach to improve productivity and quality of molded parts. The computational flow analysis and thermal analysis have been useful tools to optimize mold filling and mold cooling at the design stage (1 -3). However, they are not capable of predict- ing warpage of molded parts caused by the anisotropy of material properties such as the thermal expansion coefficient. For short-fiber-reinforced thermoplas- tics (FRTP), it is well known that material properties are anisotropic because of fiber orientation that is induced by mold filling. Therefore, the prediction of fiber orientation and anisotropic material properties is required for predicting warpage of FRTP molded parts. A qualitative prediction of fiber orientation was proposed by Lockett (4). The prediction is based on the flow field in the mold filling. The flow in practical complex cavity is considered to be combinations of three kinds of flow types, converging, diverging, and shearing flows. The fibers tend to align with flow direction in converging and shearing flows, but per- pendicular in diverging flow. Givler (5) developed a numerical simulation scheme of the fiber orientation by integration of Jeffery’s Thls paper was presented on the 37th annual meeting of the Soclety of Polymer Science, Japan. in 1988. orientation equation (6) along the streamline in dilute suspensions. Fibers are assumed to be free from other fibers in dilute suspensions. Numerical results were verified with experimental results for an end- gated bar with a molded-in hole (7). Hirai et al. (8) proposed a numerical model to predict fiber orienta- tion distribution by assuming that the behavior of fibers is completely governed by the flow state of matrix. These methods are used for quantitative pre- diction of fiber orientation in dilute suspensions. In commercially available FRTP, short glass fibers are usually filled in thermoplastics by 10 to 45 wt.%. The melt of the FRTP should be assumed to be con- centrated suspensions, in which there is the inter- action between fibers. Therefore, it is necessary to consider the interaction between fibers for simulat- ing the orientation behavior of fibers in the FRTP melt. Folgar and Tucker (9) proposed a model to describe the planar behavior of rigid fibers in concen- trated suspensions. Their model was applied to the prediction of fiber orientation in compression mold- ing (1 0). This paper presents a numerical scheme to predict planar fiber orientation in injection-molded parts of FRTP as concentrated suspensions by using Folgar and Tucker’s orientation model. The anisotropy of the thermal expansion coefficient in the molded parts was experimentally examined to validate numerical results by using an injection-molded square plate of glass-fiber-reinforced polypropylene (FRPP). POLYMER ENGINEERING AND SCIENCE, AUGUST 7990, Vol. 30, No. 76 957

Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

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Page 1: Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

Prediction of Fiber Orientation in Injection Molded Parts of Short-Fiber-Reinforced Thermoplastics*

TAKAAKI MATSUOKA, JUN-ICHI TAKABATAKE, YOSHINORI INOUE. and HIDEROH TAKAHASHI

Toyota Central Research & Development Laboratories, Znc. Nagakute, Aichi 480-1 1 Japan

Fiber orientation induced by injection mold filling of short-fiber-reinforced thermoplastics (FRTP) causes anisotropy in material properties and warps molded parts. Predicting fiber orientation is important for part and mold design to produce sound molded parts. A numerical scheme is presented to predict fiber orientation in three-dimensional thin-walled molded parts of FRTP. Folgar and Tucker’s orientation equation is used to represent planar orientation behavior of rigid cylindrical fibers in concentrated suspensions. The equation is solved about a distribution function of fiber orientation by using a finite difference method with input of velocity data from a mold filling analysis. The mold filling is assumed to be nonisothermal Hele-Shaw flow of a non-Nexhonian fluid and analyzed by using a finite element method. To define a degree of fiber orientation, an orientation parameter is calculated from the distribution function against a typical orienta- tion angle. Computed orientation parameters were compared with measured thermal expansion coefficients for molded square plates of glass-fiber-reinforced polypropylene. A good correlation was found.

INTRODUCTION

n injection molding, computer simulation is emerg- I ing as an approach to improve productivity and quality of molded parts. The computational flow analysis and thermal analysis have been useful tools to optimize mold filling and mold cooling at the design stage (1 -3). However, they are not capable of predict- ing warpage of molded parts caused by the anisotropy of material properties such as the thermal expansion coefficient. For short-fiber-reinforced thermoplas- tics (FRTP), it is well known that material properties are anisotropic because of fiber orientation that is induced by mold filling. Therefore, the prediction of fiber orientation and anisotropic material properties is required for predicting warpage of FRTP molded parts.

A qualitative prediction of fiber orientation was proposed by Lockett (4). The prediction is based on the flow field in the mold filling. The flow in practical complex cavity is considered to be combinations of three kinds of flow types, converging, diverging, and shearing flows. The fibers tend to align with flow direction in converging and shearing flows, but per- pendicular in diverging flow.

Givler (5) developed a numerical simulation scheme of the fiber orientation by integration of Jeffery’s

Thls paper was presented on the 37th annual meeting of the Soclety of Polymer Science, Japan. in 1988.

orientation equation (6) along the streamline in dilute suspensions. Fibers are assumed to be free from other fibers in dilute suspensions. Numerical results were verified with experimental results for an end- gated bar with a molded-in hole (7). Hirai et al. (8) proposed a numerical model to predict fiber orienta- tion distribution by assuming that the behavior of fibers is completely governed by the flow state of matrix. These methods are used for quantitative pre- diction of fiber orientation in dilute suspensions.

In commercially available FRTP, short glass fibers are usually filled in thermoplastics by 10 to 45 wt.%. The melt of the FRTP should be assumed to be con- centrated suspensions, in which there is the inter- action between fibers. Therefore, it is necessary to consider the interaction between fibers for simulat- ing the orientation behavior of fibers in the FRTP melt. Folgar and Tucker (9) proposed a model to describe the planar behavior of rigid fibers in concen- trated suspensions. Their model was applied to the prediction of fiber orientation in compression mold- ing (1 0).

This paper presents a numerical scheme to predict planar fiber orientation in injection-molded parts of FRTP as concentrated suspensions by using Folgar and Tucker’s orientation model. The anisotropy of the thermal expansion coefficient in the molded parts was experimentally examined to validate numerical results by using an injection-molded square plate of glass-fiber-reinforced polypropylene (FRPP).

POLYMER ENGINEERING AND SCIENCE, AUGUST 7990, Vol. 30, No. 76 957

Page 2: Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

Takaaki Matsuoka, Jun-ichi Takabatake, Yoshinori Inoue, and Hideroh Takahashi

THEORY

A numerical scheme has been developed to simu- late fiber orientation in three-dimensional thin- walled injection-molded parts. The fiber orientation and the polymer melt flow during mold filling are assumed to be planar. Figure 1 shows a schematic representation of the flow region in a thin cavity of molded parts and definition of the coordinates used. A local Cartesian coordinate system, x-y-z, is defined on the plate, which is located in the global coordinate system, X-Y-2 . The local x-y plane is parallel with the plate and the z axis is the thickness direction of the plate.

Polymer melt flow in the packing stage affects fibers in their orientation after filling also. As addi- tional polymer is forced into the cavity without empty space by holding pressure, the flow rate during pack- ing corresponds to the amount of the compression and shrinkage of polymer and is considered to be much smaller than the flow rate during filling. There- fore, the fiber orientation is mainly induced during filling and slightly changed during packing. Then, it is assumed that the influence of the flow during packing is negligible against its influence during fill- ing.

It is known that an injection-molded part of FRTP has the layered structure of fiber orientation (1 1, 12). The structure mainly consists of three layers: skin layer near the surface, core layer close to the center, and middle layer between skin and core layers. The structure of fiber orientation is different among them. Kenig (13) has described the mechanism of fiber orientation for each layer. There are four sources, diverging flow, converging flow, elonga- tional flow, and shear flow. Fibers align in the flow direction by these sources except diverging flow. The diverging flow causes fibers to orient perpendicular to the flow direction.

In the skin layer, fibers tend to align in the flow direction because of the fountain flow at the melt front, where the elongational flow is important. The thickness of the skin layer is very thin in general. In the middle layer, shear flow corresponding to velocity

Z z I f

FS Plate x FS Plate x

Fig. 1 . Schematic representation of a thin molded plate and coordinate systems used for the fiber orientation and mold filling analysis (X-Y-2: global Cartesian coordinate, x-y-z: local Cartesian coordinate).

gradient in the thickness direction causes fibers to orient to the flow direction. The maximum degree of orientation is expected at some distance from the part surface for nonisothermal flow as the maximum shear rate is attained at the boundary between flow and solid layers. On the other hand, if the flow is diverging, transverse fiber orientation is induced in the middle layer because the diverging flow is more effective than the shear flow regarding fiber orien- tation.

In the core layer, fiber orientation is perpendicular to the flow direction for the diverging flow and par- allel for the converging flow. As a practical molded part usually has a thin-walled complex geometry, almost all flow regions seem to be occupied by the diverging and converging flow. It is assumed that fiber orientation of the middle layer is almost the same as one of the core layer. Neglecting the skin layer because of its small thickness, it is expected that the material properties of molded parts mainly depend on fiber orientation in the core layer. The fiber orientation analysis explained in this paper considers the fiber orientation mechanism in the core layer.

Fiber Orientation

The orientation model developed by Folgar and Tucker (9) was used to predict the fiber orientation during mold filling. The model is for planar orienta- tion in concentrated suspensions of fibers, which are rigid cylinders with large aspect ratios (length/diam- eter, l/d >lo), in consideration of interactions be- tween fibers. In the local x-y coordinate system as shown in Fig. 2, the equation of the model is

a+ a+ a+ - + ux - + u, - at ax ay

a2+ a = Cli.,, - - -

ad2 ad

Y

t

Fig. 2. Planar fiber orientation defined with an orienta- tion angle 4 on the local x-y plane.

958 POLYMER ENGlNEERlNG A N D SCIENCE, AUGUST 1990, Vol. 30, N o . 16

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Prediction of Fiber Orientation

where t is the time, ux is the velocity in the x direction, u, is the velocity in the y direction, +xy is the magni- tude of the strain rate tensor, 4 is the orientation angle from the x axis, C, is the interaction coefficient determined experimentally, and $ is the orientation distribution function.

The function rl/ is a function of the angle 4. The integral of the function ic/ from a n angle d1 to 4z means the probability that any fiber will have a n orientation between angles d1 and d2. The function + must satisfy the following conditions:

To evaluate the degree of the fiber orientation, a n orientation parameterfp is defined from the function * as

fp = 2(COS24) - 1 (4)

(5) (C0S2d) = iz +(4)cos2(d - 4 W J .

The orientation parameter varies between -1 and 1 as shown in Fig. 3. The fibers will perfectly align parallel with the angle 4i forfp = 1 and perpendicu- larly forfp = - 1. Forfp = 0, they will be at random.

Polymer Melt Flow in the Mold Filling

The polymer melt flow in the mold filling is as- sumed to be nonisothermal Hele-Shaw flow of non- Newtonian fluid between narrow-gapped plates as shown in Fig. 1. The flow is described by the equa- tions of continuity, motion, and energy. For Hele- Shaw flow, the motion equation is integrated in the gapwise direction, the z-direction (l), and combined with the continuity equation to give the governing equation of the flow:

where P is the pressure and H is the thickness. ,G is the equivalent Newtonian viscosity (14) and is

f i = - (7) 3 IT':+ dr

-*d L + d I + d 1

f p = - 1 fp= 0 f p = 1 Perpendicular Random Paral le 1

Fig. 3. Schematic representation of fiber orientation de- scribed by using orientation parameter.

where T is the shear stress, T,, is the shear stress at the wall, i. is the shear rate.

The energy equation for nonsteady one-dimen- sional thermal conductivity problem with shear heat generation is

where T is the temperature, t is the time, s is the flow direction, and w is the velocity in the s direction. k is the thermal conductivity, Cp is the specific heat, p is the density, and 'I is the viscosity of the polymer melt. For crystalline polymer, the latent heat is lumped into the specific heat in neighborhood of the solidification temperature.

Since the polymer melt is non-Newtonian fluid, the viscosity depends not only on temperature but also on shear rate and is approximately described by the following power-law and Arrhenius-type model:

log(?,) = CI + Cz.lOg(;/o) + cs/(T + 273) (9)

where cI, c2, and c3 are experimental viscosity coef- ficients of polymer melt: ?u is the apparent viscosity: and +a is the apparent shear rate.

As the flow is nonisothermal, the solidification of polymer melt induced by cooling is taken into consid- eration in the calculation of the flow during filling. The gapwise field is classified into two layers, flow layer in which polymer is able to flow and solid layer in which there is no flow due to increase of viscosity and solidification of polymer. The thickness of the flow layer is determined with gapwise temperature profile obtained by solving the energy equation and no-flow temperature, under which polymer melt loses the ability to flow. Under nonisothermal condition, the thickness between two plates, H , is replaced with the thickness of the flow layer in Eq 6.The wall shear stress is defined at the boundary between flow and solid layers, and the integration is done only for the flow layer in Eq 7.

NUMERICAL METHOD

The numerical method for solving the fiber orien- tation equation, Eq I , is a n implicit Crank-Nicolson scheme of the finite difference method. The fiber orientation function is calculated numerically from velocity data under boundary conditions of E q s 2 and 3. The velocity data are composed of velocities and velocity gradients in the x- and y-directions during mold filling. For the flow analysis of the mold filling, the finite element method for the flow equation, Eq 6, is combined with the finite difference method for the energy equation, Eq 8. The unknown variables are the pressure and temperature every filling time. The melt front is advanced by using the same method as the FAN (Flow Analysis Network) method (1).

Figure 4 shows the computational procedure. First, the flow analysis is executed with input of mold data, polymer data, and condition data. The mold data are a set of triangular plate elements generated by mesh- ing a geometry of molded parts. The polymer data are

POLYMER ENGlNEERlNG AND SCIENCE, AUGUST 1990, Vol. 30, No. 16 959

Page 4: Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

Takaaki Matsuoka. Jun-ichi Takabatake, Yoshinori Inoue, and Hideroh Takahashi

Condition Mold Polymer da ta da ta d a t a

8 B a I 1 /

Mold filling analysis

1 8 Cr 8 Velocity da ta I 1

Fiber or ientat ion analysis

1 a Orientation parameter

Fig. 4. Flow chart of the ftberorientation and mold filling analysis.

a set of thermal properties and viscosity coefficients of polymer melts. The condition data are molding conditions, which are composed of the maximum injection pressure, polymer temperature, mold tem- perature, and filling time (or flow rate). The elemental velocity and flow direction are computed and output- ted to a data file at a constant time interval during mold filling by the flow analysis. Second, velocity gradients are calculated from the velocity distribu- tions and stored in a temporary data file. Third, the fiber orientation function is calculated every filling time for each element by using the velocity data and a suitable interaction coefficient. At the end of filling, the orientation parameter against the typical angle, at which the value of the orientation function is maximum, is calculated from the orientation func- tion.

EXPERIMENTS

The in-plane anisotropy of thermal expansion coef- ficient in injection-molded parts was investigated to evaluate the fiber orientation because the fiber ori- entation has significant effects on the thermal ex- pansion coefficient. A square plate with sides 100 mm and thickness 3 mm was injection molded by using the mold that had a rectangular gate with sides 2 by 4 mm and length 2 mm at a center of a side of the plate. The material was glass-fiber-reinforced polypropylene (FRPP). The glass fractions of FRPP were 10, 20, and 30 percent by weight. To measure the thermal expansion coefficient, 3 mm by 3 mm by 17 mm bar specimens were machined from molded plates at 15 positions in the x and y directions as shown in Fig. 5. The measurement of the thermal expansion coefficient was made twice for each sam- ple from 25°C to 150°C at a constant temperature increment of 1°C per min. There were effects not only of the fiber orientation but also of residual strains induced by molding and machining on ther- mal expansion in the initial specimen. The first

measurements were made to eliminate the residual strains from the specimen. The thermal expansion coefficient was obtained from the results of the sec- ond measurements at temperatures of 50, 75, and 100°C.

The fiber orientation in the molded square plate was also investigated by X-ray observation. Metal fibers with a length of 2.5 mm and a diameter of 0.15 mm were filled in FRPP at a fraction of 1 % by weight as a tracer. Metal fibers and FRPP were thrown into the hopper of the injection machine and mixed to- gether between the cylinder and screw in the melting stage of injection molding.

CALCULATIONS

The geometry of the square plate was meshed into 200 elements and 121 nodes with the triangular element as shown in Fig. 6. The meshed model was commonly used for both the flow analysis and the fiber orientation analysis. Molding conditions and

Gate

Y

k 3 m m

L 100 Fig. 5. Sampling positions of bar specimens for measur- ing thermal expansion coefficients in an injection-molded square plate.

Y A

Ga

Fig. 6. Meshed model of the square plate with triangular elements.

960 POLYMER fNG/NffR/NG AND SCIENCE, AUGUST 1990, Yo/. 30, No. 16

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Prediction of Fiber Orientation

material properties used for the flow analysis are summarized in Table 1. The molding conditions were measured at the experiments. The fiber orientation analysis was made for six interaction coefficients of 0.2, 0.1, 0.05, 0.01. 0.005 and 0.001 to find a suita- ble value. The initial fiber orientations were assumed to be random at all elements. The orientation param- eter was calculated against the typical angle a t all elements and was also calculated against the x and y directions at all nodal points.

A simple model to estimate thermal expansion coef- ficient as a function of volume fraction and fiber orientation was presented by Fischer and Eyerer (1 5). When fibers are perpendicular to the direction of measuring thermal expansion, the thermal expan- sion coefficient of the composite, aCl is described by the mixing rule:

a,-] = a , - V,(a, - a,) (10)

where o ( ~ and a, are thermal expansion coefficients of matrix and fiber respectively. V , is the volume fraction of fibers. This case corresponds to a n ori- entation parameter of - 1. For parallel alignment of fibers with the measuring direction ( f p = 1). as if fibers had infinite length, aC2 is

where E,, and E, are elastic modulus of matrix and fiber. Using orientation parameter as the ratio of two models, f = ( f p - 1)/2, the combination model of both is described for any fiber orientation as follows:

(12) a c = a c l - f (ac* - a c 2 )

= + acz) - fp(ac1 - ac2)1/2.

They experimentally confirmed that this model al- lows the approximate calculation of thermal expan- sion as a function of fiber orientation by comparing measured thermal expansion coefficient with fiber orientation measured by image analysis.

As mentioned above, a linear relation is expected between orientation parameter and thermal expan- sion coefficient. Therefore, correlation between them is investigated. Nodal orientation parameters were compared with measured thermal expansion coeffi-

cients at same positions respectively to examine cor- relation between them and determine a suitable in- teraction coefficient. The correlation factor, r , was defined as

n

c ( fPt -fP)(at - 4 (13)

I= 1

,=1 i"' where

i n

The correlation was described as the following linear equation by the least squares method:

a = a. fp + b (16)

where a and b are experimental coefficients. Com- paring Eq 12 with Eq 16, a and b are

a = -(acl - ac2)/2 17)

Since acl is greater than ac2, negative correlation is expected.

RESULTS

Experimental thermal expansion coefficients are shown in Fig. 7 at a temperature of 100°C for fiber fractions of 10,20, and 30 wt.%. The thermal expan- sion coefficients were different by position and direc- tion for any fiber fraction. The difference between thermal expansion coefficients in the x and y direc- tions at the same position appeared as anisotropy induced by the fiber orientation. For example, the thermal expansion coefficient in the x direction was smaller than one in the y direction at the position near the side wall. It was considered that fibers aligned strongly parallel with the wall because the thermal expansion coefficient of the glass fiber is smaller than that of the polymer matrix. At increas- ing fiber fraction, the anisotropy of thermal expan-

Table 1. Molding Conditions and Material Properties Used for the Orientation and Mold Filling Analysis.

Glass Fiber Fraction (wt. %) 10 20 30

Injection temperature ("C) Mold temperature ("C) Filling time (s) Thermal conductivity (W/(m. K)) Density (kg/m3) Specific heat (J/(kg. K)) No-flow temperature ("C) Solidification temp. ("C) Latent heat (J/kg) Viscosity coefficients (Pa. s)

c1 c2 c3

223 41 3 0.1 70

960 2,759

155 124

99,230

2.081 -0.647

866

t

c t

0.1 79 1,030 2,600

156 126

89,600

2.901 -0.686

533

t

t

t

0.193 1,120 2,403

157 125

77,870

2.491 -6.970

773

POLYMER ENGINEERlNG AND SCIENCE, AUGUST 1990, Yo/. 30, No. 16 96 1

Page 6: Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

Takaaki Matsuoka, Jun-ichi Takabatake, Yoshinori Inoue, and Hideroh Takahashi

GF30w t % - F R P P Y 1x10-4 w c )

t + + + + +

+ + + + +

G F l O w t % - F R P P - 1 ~ 10-4 ( 1 m

e., k-t

+++++I

40

+++++/ t t t t t l

It f I + I

GF20w t % - F R P P w 1 4 0 - 4 wt) + + + + + + + + + + t t t t t ]

Fig. 7. Experimental thermal expansion coefficients at a temperature of I00"C for three kinds of fiber-reinforced polypropy1eneIA = fiberfraction of 10 wt.%; B = 20 wt.%; C = 30 wt.%).

sion coefficients was decreased. This would be due to the increase of the interaction between fibers.

In FRPP, orientation and crystallization of the ma- trix have a n effect on properties of molded parts (1 6). Thermal expansion coefficients were also measured for natural polypropylene. Significant anisotropy was not recognized in the measurement.

A result of X-ray observation of metal fiber distri- bution in the square plate is shown in Photo 1. The metal fibers were filled in FRPP as a tracer. Orien- tation of metal fibers was clearly observed. It was found that metal fibers tended to align in circumfer- ential direction around the gate, parallel with the side near the walls and in the flow direction at the final filling positions. The observed fiber orientation corresponded to the degree of the anisotropy of the measured thermal expansion coefficients at the same position of the plate.

Computed velocity distribution at the end of mold filling is shown in Fig. 8 for the glass fraction 20% FRPP. This is a n output result of the flow analysis. The arrow indicates the average velocity across the gap and the flow direction. The flow diverges radially from the gate decreasing the velocity. At the final filling positions, the flow converges to two corners on the right side of the figure increasing the velocity.

Figure 9 shows computed gapwise velocity, shear rate, and temperature profiles in a n element near the center of the plate at the end of filling. The temper- ature is rapidly changed near the wall and flat at the center because of flow and low thermal conductivity of polymer. There are solid layers on both walls, in which the temperature is less than the no-flow tem- perature of 156°C and velocity is zero. The thickness

Gate -+

Photo 1 . X-ray observation of metal fibers filled in FRPP as a tracer.

Gate

20cm/s

PI

- - . . . \ + . - - - - ,

F i n a l f i l l i n g p o s i t i o n Fig. 8. Computed uelocity distributions at the end of mold filling for 20 wt.% glass-fber-reinforced polypropylene.

201 h

rn \ 4 v

-&-

l o

v) \ E U v

2 100 5 L t:,

O L 0

40

E-'

c W

v

40 kj + (d L a, a E a, c

Fig. 9. Cornputedgapwiseprofiles of velocity, shear rate, and temperature near the center of the square plate at the end of filling for 20 wt.% glass-fiber-reinforced poly- prop ylene.

962 POLYMER ENGlNEERlNG AND SCIENCE, AUGUST 1990, Vol. 30, No. 16

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Prediction of Fiber Orientation

of the solid layer is about 0.4 mm on one side and the flow layer is about 2.2 mm. If all regions of the solid layers correspond to the skin layer, the thick- ness of the core and middle layer is about three quarters of the part thickness. Plug flow is clearly seen in the flow layer from velocity and shear rate profiles because the flow is a nonisothermal one of non-Newtonian fluid. The maximum value of shear rate appears slightly inside the flow-solid boundary because of high viscosity induced by low tempera- ture. For the parallel flow, fibers are aligned to the flow direction in the neighborhood of these locations by the shear flow. However, there is almost the di- verging flow or the converging flow in the square plate as is evident from the velocity distribution shown in Fig. 8. Because the shear flow is not effec- tive for the diverging flow, the fiber orientation in the middle layer is perpendicular to the flow direction as in the core layer. The fiber orientation in the core layer mainly depends on the diverging flow and the converging flow. Therefore, it is reasonably consid- ered that the thermal expansion coefficient mainly depends on fiber orientation like the core layer.

Figure 10 shows the predicted planar fiber orien- tation at the end of filling by using a interaction coefficient of 0.1. The interaction coefficient will be discussed later in detail. The direction of the segment means the typical orientation angle. The length of the segment means the orientation parameter against the typical angle. Since the orientation pa- rameters are calculated from the gapwise average velocity, the predicted fiber orientation seems to be similar to the core layer’s orientation. It is predicted that the fibers tend to align perpendicularly against flow direction around the gate, parallel with the side near the walls, and in the flow direction at the final filling positions. The flow can be classified to the diverging flow around the gate, the shearing flow along the wall, and the converging flow at the final filling position. The predicted results agreed with the idea of Lockett (4) qualitatively. The orientation pa- rameter is defined to evaluate the degree of the fiber orientation. Long segments mean that many fibers

Gate

f p = l . 0 - Fig. 10. Computed distribution oforientation parameters with a n interaction coefficient of 0.1 for 20 wt.% glass- f iber-reinforced polypropylene.

will align in the direction of the segment at each position. On the other hand, short segments mean that fibers will be near random. The results obtained agreed qualitatively with the X-ray photograph by eye.

There is at present no model to predict the inter- action coefficient, CI. Folgar and Tucker (9) experi- mentally determined interaction coefficients for sev- eral suspensions of nylon and polyester monofila- ment fibers in silicon oil by using a concentric cylinder apparatus, which provides a simple shearing flow. I t was found that the interaction coefficient depends on aspect ratio and volume fraction of fibers, and C I = 0.0081 for the nylon fiber suspension with a fiber volume fraction of 8% and nominal aspect ratio of 16. However, Jackson, et al. ( 1 0) measured a value of C I = 0.035 in the compression molding of the same suspension as used in Folgar’s experi- ments. Jackson, et al. discuss that the difference is due to the highly constrained fiber orientation in the x-y plane in the compression molding and that ori- entation behavior in constrained planar orientation requires a larger interaction coefficient than in un- constrained three-dimensional cases like Folgar’s ex- periments.

Since fibers are highly constrained in the x-y plane in injection molding as in compression molding, the interaction coefficient must be determined by injec- tion-molding experiments. A number of techniques have been developed for measuring fiber orientation distribution in injection-molded parts (1 6, 17). Re- cently, image analysis has been applied to rapid measurement (15, 18). However, even when image analysis is used, much effort and time are needed to measure fiber orientation distribution. The chief ob- jective of fiber orientation analysis is to estimate anisotropic material properties of molded parts for prediction of warpage. Since there is a linear relation between the orientation parameter and the thermal expansion coefficient as described previously, deter- mination of the interaction coefficient was tried by comparing computed orientation parameter with measured thermal expansion coefficient directly. If correlation is obtained successfully, it can be used immediately to predict the thermal expansion coef- ficient by fiber orientation analysis with the deter- mined interaction coefficient.

Correlation factors between computed orientation parameters and measured thermal expansion coef- ficients are shown in Table 2 for interaction coeffi- cients C I varied from 0.001 to 0.2 at each tempera- ture. The glass fiber fraction is 20%. There was negative correlation between them for any interac- tion coefficient. When the fiber orientation becomes stronger, the thermal expansion coefficient becomes smaller. Increasing and decreasing C, from 0.1, the correlation factor decreased at every temperature. I t was found that the maximum correlation factor was obtained for C I = 0.1 independent of temperature. The correlation factor is significant a t the 0.1 % level for 30 samples. The decrease of correlation factor

POLYMER ENGINEERING AND SCIENCE, AUGUST 1990, Vol. 30, No. 16 963

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Takaaki Matsuoka, Jun-ichi Takabatake, Yoshinori Inoue, and Hideroh Takahashi

a,

Id -

was not so large by changing the interaction coeffi- cient from the suitable within the limits of this ex- amination. It is thought that the effect of fluid dy- namics is greater than the interactive effect since the injection-molding condition is under high shear rate. The fiber orientation in injection-molding is considered to be mainly governed by fluid dynamics and additionally by interaction.

For other glass fiber fractions, correlation factors are summarized in Table 3 at a temperature of 100°C. For obtaining better correlation, the interaction coef- ficient is small as the glass fiber fraction decreases, because of the reduction of the interaction between fibers. The aspect ratio of fiber has an effect on the interaction coefficient (9). Fiber lengths and diame- ters were measured in molded plates because it is reported that there is breakage of fibers during filling in injection molding (19). Fiber lengths were from about 0.1 to 1 .O mm for any fiber fraction. The means of fiber lengths were 0.42.0.37, and 0.51 mm for 10, 20, and 30 wt.% FRPP respectively. Since the fiber diameter was 14 pm, the mean aspect ratios were 30, 26, and 36 in order. It was not recognized that there was a significant difference among them due to broad distribution of fiber lengths. Therefore, it is thought that the effect of the aspect ratio is the same on the interaction coefficient for any fiber fraction in this experiment. The suitable interaction coefficients were 0.01 for 10% FRPP and 0.1 for 20 and 30% FRPP. I t was found that the correlation is good enough to validate the computed results for all fiber fractions.

Correlation plots are shown in Figs. 11, 12, and 13. The fractions of the glass fibers are 10, 20, and 30 wt.% respectively. The temperature is 100°C and

:x d i r e c t i o n 0 : y d i r e c t i o n

Table 2. Correlation Factors Between Computed Orientation Parameters and Experimental Thermal Expansion Coefficients

for 20 wt. Oh Glass-fiber-reinforced Polypropylene.

Temperature ("C)

Cl 50 75 100

0.001 -0.789 -0.785 -0.791 0.005 -0.790 -0.787 -0.794 0.01 -0.808 -0.809 -0.817 0.05 -0.824 -0.822 -0.826 0.1 -0.833 -0.834 -0.836 0.2 -0.821 -0.820 -0.826

Table 3. Correlation Factors Between Computed Orientation Parameters and Experimental Thermal Expansion Coefficients for Various Glass-fiber-reinforced Polypropylene Samples at a

Temperature of 100°C.

Glass Fiber Fraction (wt. %)

c1 10 20 30

0.001 -0.859 -0.791 -0.61 7 0.005 -0.877 -0.794 -0.638 0.01 -0.884 -0.817 -0.669 0.05 -0.872 -0.826 -0.690 0.1 -0.873 -0.836 -0.692 0.2 -0.878 -0.826 -0.668

1.5

1.0-

0.5 .

Fig. 1 1 , Correlation between computed orientation pa- rameters and experimental thermal expansion coeff i- cients for 10 wt.% glasslfiber-reinforced polypropylene at 100°C. The interaction coefficient is 0.01.

1 . 5 7

0.5 1 :x d i r e c t i o n

o :y d i r e c t i o n 0.0 -1.0 0

F- O r i e n t a t i o n p a r a m e t e r f p Fig. 12. Correlation between computed orientation pa- rameters and experimental thermal expansion coeffi- cients f o r 20 wt.% glassgiber-reinforced polypropylene at 100°C. The interaction coefficient is 0.1.

c x 0 - .3

rd a X a,

: x d i r e c t i o n 0 : y d i r e c t i o n 0.0

- 1.0 0.0 1.0 O r i e n t a t i o n parameter f p c

Fig. 13. Correlation between computed orientation pa- rameters and experimental thermal expansion coeffi- cients for 30 wt.% glassgiber-reinforced polypropylene at 100°C. The interaction coefficient is 0.1.

964 POLYMER ENGINEERING AND SCIENCE, AUGUST 1990, Vol. 30, No. 16

Page 9: Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

Prediction of Fiber Orienta t ion

I 0

7 6 -

P 4 -

v

the interaction coefficient is the suitable value for each fraction described above. The open circle means data in the y-direction and the closed circle in the x- direction. The linear relationship between the orien- tation parameter, fp, and the thermal expansion coefficient, a , can be expressed as an equation of a = aLfp + b. The coefficients a and b calculated by the least squares method are summarized in Table 4. Values of a and b depend on the fiber fraction and temperature. With increasing temperature, a and b increase because of a large thermal expansion of PP at high temperature.

Absolute a and b are plotted in Fig. I 4 versus fiber weight fraction at a temperature of 100°C. The ex- perimental results as in Table 4 obtained in the way described above are compared with values calculated using the simple model, E q s 17 and 18 . The slope of the linear relation, a, which means an influence of fiber orientation on anisotropy of the thermal expan- sion coefficient, depends on the fiber fraction. Al- though it is expected that the influence is bigger as fiber fraction increases, the experimental value of a is found to be maximum at 20 wt.% fiber fraction. This tendency is also recognized in the solid curve calculated by the model, in which the maximum value of a appears at 21 wt.% and seems to be due to the decrease of b, which indicates the thermal expan- sion coefficient for random orientation. Experimen-

,

. . . . A .

a /-** 0

a b - '0 10 20 30 40 50

Fiber weight fraction ( % )

Fig. 14. Absolute coefficients a and b versusfiber weight fraction at a temperature of 100°C.

tal and calculated results are in good agreement at low fiber fraction but not at the higher fraction. The experimental values of a and b are smaller than the calculated ones at high fraction. The reason is thought to be that a n effect of fiber interaction on thermal expansion behavior is not considered in the model. The correlation equation can be useful to estimate anisotropy of thermal expansion coefficient from fiber orientation as a convenient method for warpage prediction of molded parts. It is desirable that a theoretical model that includes fiber interac- tion and layered structure be developed to predict thermal expansion coefficients for concentrated FRTP.

CONCLUSIONS

A numerical scheme has been developed to simu- late fiber orientation in the injection-molded thin- walled parts of short-fiber-reinforced thermoplastics as concentrated fiber suspensions by using Folgar and Tucker's model. Two experiments were per- formed to validate the numerical scheme for a square plate molded by glass-fiber-reinforced polypropyl- ene. Computed results were compared with (1) X-ray observation results of the distribution of metal fibers, which were filled in glass-fiber-reinforced polypro- pylene as a tracer, and (2) the experimental thermal expansion coefficients in the square molded parts, which is one of the anisotropic material properties induced by fiber orientation. There were good agree- ments between computed and experimental results for both comparisons. The validity of the scheme was confirmed experimentally. Furthermore, the linear relationship was obtained between computed orien- tation parameters and measured thermal expansion coefficients. It should be noted that this relationship can be used to estimate the thermal expansion coef- ficient of molded parts from the simulation of fiber orientation for predicting warpage of molded parts practically.

ACKNOWLEDGMENT

The authors are grateful to Dr. 0. Kamigaito and Dr. T. Kurauchi of Toyota Central Research and De- velopment Laboratories, Inc., for many helpful sug- gestions throughout the work.

Table 4. Coefficients of Linear Relationship Between Computed Orientation Parameters and Experimental Thermal Expansion Coefficients.

Glass Fiber Temperature Fraction (wt. YO) CI ("C) a b r

10 0.01 50 -3.71 x 1 0 - ~ 7.81 x 10-5 -0.846 75 -4.73 8.27 -0.887

100 -5.38 9.01 -0.884 20 0.1 50 -4.44 6.26 -0.833

75 -5.62 6.40 -0.834 100 -6.12 6.89 -0.836

30 0.1 50 -3.56 4.80 -0.700 75 -3.86 4.81 -0.680

100 -4.32 5.19 -0.692

POLYMER ENGINEERING AND SCIENCE, AUGUST 1990, Vol. 30, No. 76 965

Page 10: Prediction of fiber orientation in injection molded parts of short-fiber-reinforced thermoplastics

Takaaki Matsuoka, Jun-ichi Takabatake, Yoshinori inoue, and Hideroh Takahashi

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