Fiber orientation and mechanical properties of short-fiber-reinforced injection-molded composites:...

Fiber Orientation and Mechanical Properties of Short-Fiber-Reinforced Injection-Molded

Composites: Simulated and Ex per i m ent al Res u I t s

MAHESH GUPTA and K. K. WANG

Sibley School of Mechanical and Aerospace Engineering Cornell University

Ithaca, New York 14853

A numerical simulation is presented that combines the flow simulation during injection molding with an efficient algorithm for predicting the orientation of short fibers in thin composite parts. Fiber-orientation state is represented in terms of a second-order orientation tensor. Fiber-fiber interactions are modeled by means of an isotropic rotary diffusion. The simulation predicts flow-aligned fiber orientation (shell region) near the surface with transversely aligned (core region) fibers in the vicinity of the mid-plane. The effects of part thickness and injection speed on fiber orientation are analyzed. Experimental measurements of fiber orientation in plaque-shaped parts for three different combinations of cavity thickness and injection speed are reported. It is found that gapwise-converging flow due to the growing layer of solidified polymer near the walls tends to flow-align the fibers near the entrance, whereas near the melt front, gapwise-diverging flow due to the diminishing solid layer tends to align the fibers transverse to the flow. The effect of this gapwise-converging-diverging flow is found to be especially significant for thin parts molded at slower injection speeds, which have a proportionately thicker layer of solidified polymer during the filling process. If the fiber orientation is known, predictions of the anisotropic tensile moduli and thermal-expansion coefficients of the composite are obtained by using the equations for unidirectional composites and taking an orientation average. These predictions are found to agree reasonably well with corresponding experimental measurements.

INTRODUCTION

ecause of its ability to produce net-shaped B components with high dimensional accuracy in a relatively short cycle time, injection molding is one of the most widely used polymer-processing methods. Injection molding of short-fiber-reinforced thermoplastics combines these inherent processing advantages of injection molding with the superior performance characteristics of composite materials (high strength-to-weight ratio, high stiffness, and possibly controlled anisotropy). Flow of a fiber suspension during injection molding results in a pref- erential orientation of short fibers. This flow-induced fiber orientation can vary significantly across the thickness of injection-molded parts. Typically, fibers are aligned transverse to the flow near the mid-plane of the part, but are aligned along the flow near the surface. As the composite solidifies, this flow-induced orientation of short fibers is frozen into the matrix

and becomes a key feature of the finished product. Even though the strength of short-fiber composites is lower and less anisotropic than that of long fiber composites, prediction of fiber orientation in short- fiber-reinforced injection-molded (SFRIM) parts is still important for good structural design, to account for anisotropy in such properties as the elastic moduli and thermal-expansion coefficients of such products.

The injection molding process starts with filling the cavity with hot polymer melt, which is followed by cooling until the part is sufficiently rigid to be ejected. During this solidification process, not only does the molded part shrink, but as a result of nonuniform cooling, asymmetric residual stresses also develop in complex parts. These asymmetric residual stresses can cause the part to warp after it is ejected from the mold. Anisotropy of the tensile moduli and thermal- expansion coefficients can thus significantly affect the shrinkage and warpage pattern of SFFUM parts. Therefore, an accurate prediction of the anisotropy

POLYMER COMPOSITES, OCTOBER 1993, Vol. 14, No. 5 367

Mahesh Gupta and K. K. Wang

in these properties (which, in turn, requires an accurate prediction of fiber orientation) in SFRIM parts is especially important.

With known injection-molding processing conditions, the objective of the present study is to predict the fiber orientation and mechanical and thermal properties of SFRIM parts. Such a code for predicting fiber orientation and the properties of SFRIM parts has been developed. The flow field has been decoupled from the fiber orientation, neglecting the effect of anisotropic fiber orientation on the flow field. This approach is valid for thin injection-molded parts provided the fiber orientation is sufficiently flat to make no significant contribution to the gapwise shear stress. Even though this paper is focused on estab- lishing the validity of the simulation by making detailed comparison with experimental results for the fiber orientation and anisotropic properties of simple SFRIM parts, the present computer program can also be applied to thin parts of complex three-dimensional geometry ( 1).

During injection molding, because of continuous cooling of the mold walls, a thin layer of solidified polymer develops near the walls. The thickness of this cold layer increases near the entrance and diminishes near the advancing fluid front. This growing-diminishing thermal boundary layer results in a gapwise-converging-diverging flow even for a uni- formly thick cavity. It will be seen later in this paper that this gapwise-converging-diverging flow can significantly affect the fiber orientation in thin injection- molded parts.

FIBER ORIENTATION

The orientation of a single fiber can be specified by a unit vector p directed along its axis or by the angles (0 , 4 ) shown in Fig. 1 . Jeffery (2) determined the equation of motion of a single ellipsoidal particle in Stokes flow of a Newtonian fluid. Jeffery’s equation has been shown to describe the motion of rigid cylindrical particles in Stokes flow, provided that an equivalent aspect ratio is used (3-5). To describe the motion of cylindrical fibers with large aspect ratio ( r p = I/d > 10, where 1 and d are the length and diameter, respectively, of the cylindrical fiber), Folgar and Tucker (6) used Jeffery’s equation with rp + m.

At higher fiber concentrations, fiber-fiber interactions result in a deviation in the fiber orientation from the orientation predicted by Jeffery’s equation. The nature and importance of the inter-fiber interactions depend upon the volume fraction, +,, of fibers in the suspension. In the dilute regime (df<< ( d/1>2), interactions between fibers are rare and the motion of fibers is completely defined by Jeffery’s equation. In the semi-dilute regime ((d/l)’ +Je d / l ) , interactions between fibers are quite frequent. However, fiber-fiber interactions are purely hydrodynamic and the change in orientation due to mechanical contact between fibers is not significant (7). In the semi- concentrated regime ( d / 1 << dJ << 1). which is of

x3

t

Fig. 1 . Dejnition of$ber orientation.

the most commercial importance, mechanical contact between fibers is frequent.*

In the semi-dilute regime, an effective diffusivity has been used in the literature (8) to describe the effects of hydrodynamic interactions on fiber orientation. In this approach, small changes in the orientation of a fiber due to the presence of its neighbors are modeled by means of a rotary diffusion. If the flow field and orientation distribution are anisotropic, the rotary diffusivity for fiber-fiber interactions is expected to be anisotropic. In the most general case, the rotary diffusivity is a second-order tensor (9, 10). However, if the probability of change in orientation due to fiber-fiber interactions is the same in all directions, then the rotary diffusivity is a scalar.

Folgar and Tucker (6) modeled the change in orientation caused by fiber-fiber interaction in the semi- concentrated regime by means of an isotropic rotary diffusion. Based upon their observations of fiber behavior in semi-concentrated suspensions, they suggested that the rotary diffusivity be represented as CI+, where y is the magnitude of the strain-rate tensor and C,, the interaction coefficient, is assumed to be an intrinsic property of the fiber suspension. An expression for C, in terms of the root-mean-square angle change caused by fiber-fiber interactions was also given in Ref. 6. However, in the absence of any knowledge of the angle change caused by mechanical contact between fibers, Folgar and Tucker (6) deter mined C, empirically.

Since the orientation of an individual fiber in the semi-dilute or semi-concentrated regimes is not defined deterministically by the flow field, a probability density function, +(O, 41, called the orientation distribution function, is generally used to describe the fiber-orientation state. The general equation gov- erning the time variation of the orientation distri-

‘In the literature some authors have used slightly different terminology referring to the above three regimes as dilute, semi-concentrated, and concentrated regimes. respectively.

368 POLYMER COMPOSITES, OCTOBER 1993, Vol. 14, No. 5

Fiber Orientation and Mechanical Properties of SFRTM Composites

bution function in the presence of isotropic rotary diffusivity ( Dr), as derived by Burgers ( 1 11, is given by

where w(0 ,$ ) is the angular velocity described by Jeffery’s equation. If the processing conditions during injection molding are known, Eq 1 could be solved to determine the orientation distribution function. How- ever, this distribution function depends not only upon the angles 0 and $ but also upon spatial position and time in the case of unsteady flow. The resulting approach would be highly computationally intensive with large storage requirements, making it somewhat inappropriate for numerical simulation. For a more efficient method of numerically simulating the orientation state of fibers, Advani and Tucker (12) used orientation tensors, which were originally introduced by Hand (13). Such tensors are defined as the dyadic products of the unit vector p averaged over all possible directions. with 4 as the weighting function. The definitions of second (a,) and fourth (a,)-order tensors, respectively, are

and

Since the coefficients of a generalized Fourier series of in terms of a set of spherical basis functions can be determined from the orientation tensors (141, use of orientation tensors to describe the orientation state of fibers is equivalent to approximating the orientation distribution function by a finite number of terms in a Fourier series. The evolution equation for the second-order tensor, as obtained by Hinch and Leal ( 15). is as follows:

-= Da, 0,a,+a,.LnT+h(+.a,+a,.+T-2a4: +) Dt

+ 2 Dr(I - 3a,) (4)

where a= ~ ( O U - Our) is the vorticity tensor, + = f (Vu + Our) the strain-rate tensor, I the identity tensor, and h depends upon the shape of the fiber. For an ellipsoidal particle, A = ( r z - 1/r,“ + 1) where re is the aspect ratio of the ellipsoid. For long cylindrical fibers, A = 1.

As seen in Eq 4, the evolution equation for the second-order orientation tensor contains a term with a fourth-order orientation tensor. In general, the evolution equation for any order orientation tensor contains the next higher-order orientation tensor. To obtain a closed set of equations, Hinch and Leal (1 5) approximated the fourth-order orientation tensor in terms of the second-order orientation tensor. For an accurate approximation of the fourth-order orientation tensor, Advani and Tucker (12, 16) suggested

replacing aijkl by a hybrid closure, namely

where S is the kronecker delta. The hybrid closure is a weighted average of

the so-called linear ( f= 0) and quadratic (f= 1) clos- ures. In particular, the linear and quadratic closure approximations are exact for completely random and perfectly aligned orientation distributions, respectively. Two possible expressions for the weighting function f (such that f = 1 for perfectly aligned fibers and f= 0 for completely random orientation distribution) were also suggested by Advani and Tucker (12, 16) as follows:

(6)

and f = 1 - 27det( aij) (7)

For planar flow, f and f are identical. However, for three-dimensional flows, Advani (17) found that J’ performed better than f when the hybrid closure was used in Eq 4. This has been confirmed in the present study also.

Tucker ( 18) examined the flow of fiber suspensions in narrow gaps (Fig. 21, which is typically the case during injection and compression molding of fiber- filled polymers. Since fiber orientation in narrow gaps is nearly planar ( p3 <( d m ) for flow of a fiber suspension in a narrow gap, the in-plane compe nents (all, aI2. and q,) are the dominant components. However, as shown in Ref. 18, the small out-of-plane components (al3, q,, and a33) play an important role in determining fiber orientation near the surface. Depending upon the particle number and the out-of-plane fiber orientation, Tucker (18)

Fig. 2. Narrow gap geometry. r f L is the characteristic length in the 1-2 plane, then the characteristic parameter E = b/ L e 1.


Mahesh Gupta and K . K . Wang

divided the flow of fiber suspensions into four regimes. He concluded that the flow is decoupled from the fiber orientation if the flow channel is narrow enough such that the fiber orientation is sufficiently flat to make no significant contribution to the gapwise shear stress (Regime I).

In the injection molding of thin parts, the flow near the walls tends to be shear dominated, whereas near the mid-plane, the flow is extensional-flow dominated since the shear vanishes along the mid- plane in the symmetric case. This extensional flow can be especially significant near a point gate, such as for a center-gated disk. Owing to this shearing/stretching nature of the flow in injection molding, the fibers tend to be aligned transverse to the flow near the mid-plane (core region) but aligned with the flow near the surface (shell region).

Besides this core-shell structure of the orientation distribution, a thin layer of randomly oriented fibers is sometimes found near the surface of molded parts. As explained by Bay and Tucker (19, 20), this randomly oriented region (skin layer) is caused by fountain flow near the melt front. In particular, fibers from the core region near the melt front move out- ward to the walls by passing through the fountain- flow region. At slow injection speed, these fibers near the wall may not have sufficient time to align with the flow before they are frozen in the polymer as a result of a relatively thick cold layer. Bay and Tucker (19, 20) detected a skin layer in a center-gated disk, which was molded at a Graetz number, Gr = heat-conduc- tion-time/mold-filling-time, of 22, whereas a film- gated strip molded at Gr = 272 had no skin layer. Bay (19) showed that a skin layer is likely if Gr < 50, whereas a skin layer will not form for Gr> 100. By integrating the orientation along path lines, Bay and Tucker (19, 20) determined the change in orientation of a fiber resulting from fountain flow, which also depends upon the gapwise location at which the fiber enters the fountain-flow region.

Numerical predictions for a,, and a33 obtained in Ref. 20 for both a film-gated strip and a center-gated disc were found to match closely with the experimental results (subscript 1 and 3 here denote the flow- and gapwise directions, respectively), although the predictions for a,, were quite different from the experimental results. This discrepancy was attributed (20) to the inaccuracy of the closure approximation. However, even though the experimentally obtained values for aI3 (20) were always much smaller than the predicted values, both results had a similar variation in the gapwise direction. In particular, a], is antisymmetric in the gapwise direction such that it vanishes at the mid-plane, but changes sharply to a fairly uniform value (of opposite sign) on either side of the mid-plane. On the other hand, the %3 compe nent of the orientation tensor in injection-molded parts is generally much smaller than a13.

Stover and Cohen (21). in their isothermal experimental study of motion of short fibers in narrow gaps, observed a “pole vaulting” motion of fibers in

the vicinity of wall. This “pole vaulting” motion, resulting from fiber-wall interaction, could give rise to a skin layer near the wall even at higher Graetz number, where the cold layer is relatively thin. In the present simulation, however, only the core/shell structure is addressed.

MECHANICAL PROPERTIES

Calculating the effective properties of a fiber,’ polymer composite as a homogeneous continuum involves averaging the properties of the two phases. If the fiber distribution is anisotropic, the averaging process must be weighted according to the orientation distribution. Such averages, called orientation averages (121, for the stiffness and thermal-expansion coefficients, which are fourth- and second-order tensors, respectively, are given by

and

(8 )

(9)

where K U k l and ai, are the stiffness and thermal- expansion coefficients for a unidirectional composite. In Eqs 8 and 9, the averaging volume is impli- citly assumed to be sufficiently large to contain many fibers but still small relative to the scale over which the macroscopic flow quantities can vary significantly.

Mechanical properties of unidirectional composites, which depend upon the properties of the two components, the volume fraction of fibers, and also the size and shape of the fibers, have been exten- sively investigated (22-30). Most of the analytical methods developed [self-consistent models (22-241, variational models (25). exact methods (2611 are not only complicated but are also limited to a small portion of the overall range of fiber orientation and packing geometry of interest. By approximating these complicated micromechanical results, Halpin and Tsai (27, 28) developed simple empirical expressions for composite moduli. For moderate volume fraction (4, < 0.5). the Halpin-Tsai equations have been found to be quite accurate (29). Even though more sophisti- cated equations are now available (30). the present work employs the Halpin-Tsai equations. because of their relative simplicity, to predict various elastic moduli. In particular, the general form of the Halpin- Tsai equations is given by

where p can be the two tensile moduli (El EZ2) or the two shear moduli (G12 or G,,), with subscript 1 representing the fiber direction. In Eq 10,


Fiber Orientation and Mechanical Properties of SFRIM Composites

where subscripts f and m denote fiber and matrix, respectively, and 5 depends upon the shape of the fibers and the packing geometry. For discontinuous, oriented fiber composites,

5E,, = 2(l /d)

6EZz =

lc,, = 1

Km/Grn lGZ3 = K,/G, + 2

where K , and G, are the bulk modulus and shear modulus, respectively, of the matrix.

On the other hand, the rule of mixture has been used in this work to model Poisson's ratio, namely

(1 1)

Concerning the longitudinal and transverse thermal-expansion coefficients for unidirectional composites, we employ the results of Schapery (31). That is,

v 1 2 = +fVf+ (1 - 4 , > v ,

E ~ ~ J + J + Ernam(1- +f) E f 4 f + E m ( 1 - +j)

a1 = (12)

a 2 = ( 1 + ~j)ar+f+ (1 + v,,,)a,(l - + f ) - a 1 ~ ~ 1 , (13)

where E is Young's modulus. Since unidirectional composites are transversely

isotropic, the orientation average of the stiffness and thermal-expansion coefficients must have the following form ( 12):

( K ) y k ~ = B,( a,jkL)

+ B,(a ,J6k[+ ak ,6 , J )

+ B 3 ( a i k 6 ~ l + a i16~k+ aj l s ik+ a jk6 i I )

+ B*(6,6kl)

+B5(s ik6j l+ 6t16jk) (14)

( = A, a , + A, (15)

where €3,. . . B5 and A,, A, can be easily determined by knowing the stiffness and thermal-expansion coefficients for the unidirectional composite.

In the present work, the average fiber orientation for the composite part has been determined at the centroid of the triangular finite elements by averaging the in-plane components of a, at eleven equally spaced locations across the half-gap thickness of the part. Knowing the average fiber orientation and the properties for unidirectional composites ( E q s 10-13). the tensile moduli and thermal- expansion coefficients for SFRIM parts have been determined by using Eqs 14 and 15.

and

IMPLEMENTATION DETAILS

As mentioned, the flow field in this work has been decoupled from the fiber orientation with the result-

ing velocity field then used in calculating the fiber orientation. Following Hieber and Shen (32), the flow of polymer in a thin cavity in injection molding has been modeled as generalized Hele-Shaw flow for an incompressible, inelastic, and non-Newtonian fluid under nonisothermal conditions. Since the fiber orientation should be only weakly dependent upon the post-filling stage of injection molding because of a decreased flow field, only the mold-filling stage has been analyzed here. In addition, a uniform mold-wall temperature has been assumed for the upper and lower cavity walls, resulting in a symmetric gapwise distribution of temperature and velocity about the mid-plane of the cavity. As is typical, the effects of streamwise thermal convection as well as viscous heating have been included in the energy equation. The coupled flow and energy equations have been solved together by using a hybrid finite-element/ finite-difference scheme (32) together with a control- volume approach (33, 34) for handling automatic melt-front advancement.

As noted, even though the in-plane components of the second-order orientation tensor are much larger (because of the predominantly planar orientation distribution in SFRIM parts) compared to the out-of- plane components, these small out-of-plane components of the orientation tensor are critical for determining the fiber-orientation near the cavity walls. For an efficient simulation, the out-of-plane components of orientation tensor in this work have been assumed to be constant with their values being determined empirically. Note that even though a], changes sharply near the mid-plane, the terms involving the out-of-plane components of a2 in the evolution equation for the in-plane components ( E q 4 ) are identically zero at the mid-plane such that the variation of a13 near the mid-plane has little effect on the predicted fiber orientation in that vicinity. The simplification of determining the out-of-plane components of a2 empirically is further justified, since the interaction coefficient C, in Eq 4 is also determined empirically. Even though C, is constant in Refs. 6, 16, and 20 and in this work, it may in fact depend upon the orientation distribution and type of flow. Once a systematic way of determining the interaction coefficient is established, it would be appropriate to solve the evolution equations for the out-of-plane components of the orientation tensor as well. Pending such developments, the predictions presented in a later part of this paper are based upon assumed values of a13 = -0.01. %, = 0.0, and a,, = 0.01. These values, which are in reasonable agreement with the values determined in Refs. 9 and 10 for simple shear flow, have been determined by numerical experimentation with the present program as applied to the current data.

The same finite-element/finite-difference mesh has been used for the flow analysis and the orientation prediction, In particular, the evolution equation for the in-plane components of the orientation tensor ( E q 4 ) has been solved at the centroid of each trian-



gular element with an upwind scheme being used to caiculate the material derivative of the orientation tensor in Eq 4. To simulate the Hele-Shaw flow, constant-velocity triangular elements have been used in the flow simulation (33, 34). To calculate the in-plane velocity gradients at the centroid, we have determined the velocity at each of the three nodes of the triangular element by averaging the velocity over the elements containing each such node: the in-plane velocity gradients have then been determined by assuming a linear variation in velocity over the element. Since the in-piane viscous diffusion is neglected in the Hele-Shaw approximation, the resulting flow does not satisfjr the no-slip condition along the lateral boundaries. However, in calculating the in-plane velocity gradients in Eq 4, the velocity at all nodes on such lateral boundaries has been set to zero.

Since fiber orientation changes significantly across the thickness of the SFRIM parts, the in-plane components of the orientation tensor at the centroid of each triangular element have been determined at eleven equally spaced locations (same as for energy equation) across the half gap. The effect of fountain flow on fiber orientation, which sometimes (19, 20) gives rise to a skin layer, has been neglected in this work. In determining ( d a , / d t ) in Eq 4 for elements that become filled during the current time step, the orientation tensor has been initialized by averaging its value at the same gapwise location over the neigh- boring elements, which were previously filled.

MOLD GEOMETRY

Our experimental mold consists of a plaque of uniform thickness with planar geometry indicated in Fig. 3. Three different combinations of cavity thickness (2b) and filling time ($1 were examined, namely case i : b = 0.079 cm ( 1 /32" 1, tf = 0.33 s; case ii : b= 0.079 cm (1/32"), tLr= 1.76 s; case iii : b =

0.159 cm (1/16), t f = 0.43 s. The polymer was injected into the cavity from a circular runner with 0.95 cm (3/8") diameter. In all cases, the melt inlet temperature and mold-wall temperature were 3 1 0 and 1 l O T , respectively. To examine the fiber orientation, 1.27 X 1.27 cm samples were taken from four different locations (shown in Fig. 3) on the composite part. These samples were mounted in epoxy and pol- ished metallurgically. The microstructure at the surface of the samples was photographed at a 1 : 5 0 enlargement. To examine the fiber orientation in the core region, the portion of the sample consisting of the shell region was removed and the samples were photographed again at the same enlargement. These photographs were glued together to form montages of the microstructure of the samples. By picking up the end-points of the fibers in the micrographs, using an image processing digitizer, their orientation in the plane of the part was determined. The total number of fibers discretized in a montage ranged from 2000 to 3000. From this data on fiber orientation, the frequency of fiber occurrence

Gate

a 1.91 / I

I ' 5.72 1

- 1.27

12.7 I k- 7.62 -4

Fig. 3. Shape of experimental mold cavity a n d locations of the 1.27 X 1.27 cm samples taken to examine the$ber oriert tation. In this paper, the four sample locations have bcen referred by the letters in parentheses. (All the dimensions are in cm.)

in eighteen intervals of ten degrees each, between 0 to 180", was determined. Assuming a planar orientation, the orientation distribution function (q+) was approximated by the histograms presented in the next section, which have been obtained by normalizing the frequency distributions. That is,

where N, is the number of fibers in the histogram interval centered on angle

Tensile moduli and thermal-expansion coefficients of the composite part in the axial and transverse directions have been determined by using 2.54 X 1.27 cm samples from the locations shown in Fg. 4.

The injection-molding experiments have been conducted with a polyester/glass fiber suspension, namely Rynite 530 from Du Pont, which has - 19% fibers by volume (30% fibers by weight). Rheologi- cal properties of Rynite 530 as well as the mechari- ical properties and thermal-expansion coefficients of the glass fibers and polyester components (which are required to predict the properties of the composite parts) are given in the Appendix. For all predictions reported in the next section, an empirically determined value for the interaction coefficient C, of 0.00 1 has been used for the Rynite 530.

and A 4 = T / 18.


Fiber Orientation and Mechanical Properties of SFRlM Composites

5.08 I

++ 2.54

R g . 4. Locations and sues of samples used to determine the gapwise-averaged tensile moduli and thermal-expansion coefficients in the axial and transverse directions. Samples labeled ( A. D.) and (T. D.) have been used to determine the properties in the axial and transverse directions, respectively. (All the dimensions are in cm.)

RESULTS AND DISCUSSION

To qualitatively depict the fiber orientation at the four sample locations, the micrographs, which originally showed fibers at 1 : 50 enlargement, were photc- copied with a 60% reduction (i.e., net enlargement of 1 : 20). Portions of these photocopies are shown in Figs 5 through 7, in which the core/shell structure of fiber orientation in the injection-molded parts is quite evident. As mentioned in the last section, the fiber-orientation micrographs have been discretized to determine the orientation distribution function. The orientation distributions obtained for the three molding cases are tabulated in Tables 1 through 3 and shown graphically for case i in Figs. 8 through 1 1 . At sample locations €3 and C (Figs. 9 and 10). the fibers are highly flow-aligned near the surface and strongly aligned transverse to the flow at the mid- plane. Similar behavior is seen at location A (Fig. 8) although, being closer to the gate, the degree of alignment is relatively low. At location D (Fig. 111 , the shearing nature (in the 1-2 plane) of the flow due to the no-slip condition at the lateral boundary causes the fibers to align along the flow not only at the surface but at the mid-plane also. However, as expected, the degree of alignment along the flow is higher at the surface.

Figures 12 through 14 show the predicted fiber orientation for the three molding cases at four different gapwise locations. The lines in Figs. 12 through

14 are drawn along the most probable direction of fiber orientation, with the length of these lines being proportional to the degree of alignment.* As expected, in all three cases the fibers at the mid-plane tend to be aligned transverse to the flow direction, whereas near the surface the fibers are highly flow aligned. In case ii, however, which was molded at a slower injection speed, the mid-plane prediction is seen to be more flow-aligned than in the other two cases. In turn, this can be attributed to a gapwise-converging flow caused by a relatively thicker solidified cold layer near the wall, which grows with distance from the gate (at least in the upper half of the domain in Fig. 1 3 ~ ) . To understand the effect of this thermal boundary layer on fiber orientation, the flow for each of the three molding cases has been simulated again under isothermal conditions. This has been done by omitting the viscous-heating term in the energy equation and replacing the cold-wall temperature with the injection temperature, 3 10°C. Under isothermal conditions, identical orientation distributions were obtained in molding cases i and ii, corresponding to different flow rates. On the other hand, for molding case iii under isothermal conditions, the orientation prediction at the mid-plane was the same as that in cases i and ii but different away from the mid-plane. By normalizing the evolution equation for the in-plane components of a2 ( E q 4). it can be easily confirmed that under isothermal conditions, a change in injection speed has no effect on fiber orientation (35). The normalized equations also indicate that a change in part thickness will affect the fiber orientation away from the mid-plane, although, under isothermal conditions, the orientation distribution of fibers at the mid-plane will remain the same (35).

As shown in Fig. 15, under nonisothermal conditions the temperature field during injection molding is characterized by thin cold boundary layers (36-40) near the walls and a hot core region away from the walls. The effect of the cold layer upon the stream-wise velocity profile has also been indicated in Fig. 15. In particular, the thermal boundary layer consists of a steady-state layer near the entrance of order ( ( Y X E f./ ) l I3 in thickness and a transient layer behind the advancing front with thickness of order (a i) I/’, where x is the distance from the gate, 01 is the thermal diffusivity, and i = t - x t / / L . At t,, the two layers overlap in the region where x / L - .$;/’. with tL = 01 t / b2, where both layers have thickness of order ( 0 1 t ~ ) ’ X . In effect, the polymer in most of the cold layer has solidified, whereas the material in the hot core is replenished continuously. Since the thickness of the cold boundary layer near the entrance increases with x , this gives a gapwise-converging flow. Accordingly, the mid-plane fiber orientation near the entrance is determined by two opposing factors. In the 1-2 plane, the diverging nature of the flow tries

’The lines are in fact drawn along thr direction 0 1 - the eigen-vrctor (rot-re- sponding to the larger of the two eigen-values) of the planar orientalion tcnsor with t h r length being proportional to the differrnce betwccn the two eigcn-values.



(a) (bl Fg. 5. fiber orientation micrographs for case i at mid-plane ( a ) and surface ( b). Note: these micrographs are shown enlarged 20 times.

to align the fibers transverse to the flow, whereas the gapwise-converging flow due to the cold boundary layer tries to align the fibers along the flow. At slower injection speeds, the cold layer is thicker and can result in a flow-aligned fiber orientation at the mid- plane of the cavity as indicated in Fig. 13a. The gapwise-converging nature of the flow due to the cold layer further promotes the flow-aligned fiber orientation because of the shear-dominant nature of the flow near the surface. Near the advancing front, owing to the diminishing thickness of the cold layer (as shown schematically in Fg. 15). the flow diverges in the gapwise direction. In turn, this gapwise-diverging nature of the flow tends to align the fibers transverse to the flow. To confirm this dependence of the fiber orientation upon a variable effective cavity thickness, additional isothermal calculations have been performed in which the cavity thickness is now pre- scribed to have a linearly increasing or decreasing variation with streamwise location. In particular, the fiber orientation at the entrance was initialized to a random distribution in the plane of the part and the polymer was allowed to slip at the lateral boundaries. Resulting predictions with a gapwise- converging (diverging) flow have confirmed an

increased tendency toward fiber orientation parallel (transverse) to the flow direction (35).

In addition, the effect of mold-wall temperature and thermal conductivity of the polymer upon predicted fiber orientation in a film-gated strip has also been studied. As the thermal conductivity of the polymer is increased, thereby increasing the thickness of the cold boundary layer, fibers at the mid-plane become more flow-aligned and the thickness of the core region decreases. Similar changes in fiber orientation are obtained when the mold-wall temperature is decreased, which also thickens the cold layer. Accordingly, in molding case ii, which has a thicker thermal boundary layer because of a slower injection speed, the fibers are more flow-aligned at the mid- plane and the core region is smaller. On the other hand, in the region closer to the melt front, which corresponds to gapwise-diverging flow (see schematic in Fig. 151, fibers in the mid-plane are transversely aligned even in case ii. As a further illustration, Fig. 16 shows the predicted fiber orientation for molding case ii when the cavity is approximately half filled. Since the thermal boundary layer is thinner at this time, fibers in the mid-plane are transversely aligned even for case ii.



(a) (b) Rg. 6. Rber orientation micrographs for case ii at mid-plane ( a ) and surface ( b). Note: these micrographs are shown enlarged 20 times.

With regard to case iii, it is noted that irrespective of the cavity thickness, the thickness of the cold boundary layer at tf is of order (a$)'/ ' . Hence, for the same tf, a thicker cavity will have a relatively smaller fraction occupied by the cold layer. Accord- ingly, a thicker core region is expected in case iii, as is seen in Fig. 14.

It might be noted further that the initial fiber orientation at the gate in Figs. 12 through 14 and 16 has been assumed to be aligned with the flow in the runner. Another initial condition that has been con- sidered in simulating the three molding cases is that the fibers are flow-aligned at the surface and transversely aligned at the mid-plane with a linear variation between. For these two different initial conditions, Figs 17 through 19 show the a,, component of the orientation tensor for each of the three molding cases, as well as the corresponding experimental results. The different initial conditions are seen to affect the predicted fiber orientation in a small region near the gate, beyond which the initial conditions do not have a significant effect on the predicted fiber orientation.

Using the definition of a, , ( Eq 2). the experimental values for a, at sample locations A, B, and C (Fig. 3)

for the three molding cases have been calculated from the orientation distributions indicated in Tables 1 through 3. At the faster injection speeds (molding cases i and iii), within a small distance ( - 4 cm) from the gate, the fibers get aligned along the flow at the surface and transverse to the flow at the mid-plane. This flow/transversely aligned orientation structure is maintained for most of the distance along the axis. Because of the gapwise-diverging nature of the flow in the region closer to the melt-front position, the value of a, , decreases slightly. At the slower injection speed (Fig. 181, fibers along the axis of the mid-plane have a very small degree of alignment; however, near the final melt-front position, because of the gapwise- diverging nature of the flow, fibers are transversely aligned as in cases i and iii.

To predict the mechanical properties, the average orientation of the fibers for the complete thickness of the composite part has been determined by averaging the orientation tensor at the eleven gapwise locations. Knowing the average orientation of the fibers for the composite part, mechanical properties are predicted by using the method described above. Even though the degree of alignment of fibers in injection-molded parts is generally quite high.



(a) (b) Fig. 7. Fiber orientation micrographs for case iii at mid-plane ( a ) and surface ( b). Note: these micrographs are shown enlarged 20 times.

Table 1. Orientation Distribution for Molding Case i. Location A Location B Location C Location D Angle

(") mid-plane surface mid-plane surface mid-plane surface mid-plane surface

5.0 15.0 25.0 35.0 45.0 55.0 65.0 75.0 85.0 95.0

105.0 115.0 125.0 135.0 145.0 155.0 165.0 175.0

0.144 0.1 95 0.147 0.122 0.1 27 0.099 0.096 0.091 0.122 0.096 0.134 0.296 0.248 0.261 0.248 0.1 72 0.1 70 0.096

0.254 0.296 0.292 0.222 0.1 58 0.129 0.101 0.065 0.074 0.084 0.076 0.068 0.100 0.137 0.164 0.166 0.21 1 0.267

0.051 0.045 0.077 0.045 0.106 0.055 0.090 0.119 0.318 0.360 0.575 0.414 0.180 0.148 0.106 0.067 0.067 0.042

0.546 0.385 0.254 0.128 0.096 0.067 0.049 0.042 0.062 0.023 0.025 0.039 0.072 0.086 0.126 0.198 0.328 0.338

0.01 5 0.036 0.054 0.067 0.076 0.091 0.142 0.163 0.236 0.503 0.527 0.370 0.221 0.142 0.082 0.061 0.058 0.021

0.436 0.31 2 0.143 0.124 0.056 0.049 0.044 0.025 0.027 0.027 0.043 0.043 0.075 0.097 0.191 0.320 0.482 0.370

0.234 0.187 0.104 0.101 0.065 0.062 0.062 0.047 0.053 0.062 0.083 0.068 0.181 0.21 6 0.246 0.365 0.365 0.362

0.807 0.497 0.263 0.099 0.073 0.019 0.019 0.01 4 0.017 0.01 7 0.019 0.025 0.021 0.037 0.095 0.133 0.277 0.431

376

~ ~ ~

POLYMER COMPOSITES, OCTOBER 1993, Vol. 14, No. 5

Fiber Orientation a n d Mechanical Properties of SFRIM Composites

Table 2. Orientation Distribution for Molding Case ii.

Angle Location A Location B Location C Location D (“1 mid-plane surface mid-plane surface mid-plane surface mid-plane surface

5.0 0.137 0.409 0.1 57 0.796 0.063 0.551 0.384 0.925 15.0 0.140 0.489 0.1 33 0.313 0.046 0.364 0.279 0.341 25.0 0.155 0.390 0.118 0.191 0.076 0.194 0.140 0.089 35.0 0.133 0.205 0.1 14 0.112 0.046 0.1 02 0.089 0.069 45.0 0.148 0.140 0.137 0.075 0.055 0.051 0.062 0.032 55.0 0.159 0.1 06 0.1 37 0.01 7 0.055 0.031 0.078 0.036 65.0 0.177 0.049 0.239 0.01 7 0.1 09 0.034 0.027 0.01 2 75.0 0.174 0.042 0.169 0.037 0.235 0.027 0.031 0.024 85.0 0.251 0.057 0.283 0.020 0.445 0.065 0.043 0.032 95.0 0.155 0.038 0.204 0.007 0.483 0.031 0.050 0.024

105.0 0.21 8 0.034 0.251 0.01 0 0.51 2 0.020 0.074 0.020 115.0 0.229 0.057 0.1 81 0.037 0.281 0.037 0.147 0.053 125.0 0.199 0.072 0.173 0.037 0.139 0.054 0.283 0.045 135.0 0.166 0.110 0.1 73 0.075 0.067 0.102 0.275 0.036 145.0 0.092 0.110 0.094 0.071 0.109 0.112 0.275 0.057 155.0 0.129 0.152 0.1 02 0.160 0.034 0.197 0.1 98 0.21 1 165.0 0.100 0.197 0.079 0.347 0.059 0.405 0.151 0.300 175.0 0.103 0.208 0.1 22 0.541 0.050 0.486 0.279 0.556

C .- I

i

Orientation angle (degree?)

(b) Fig. 8. Fiber orienlation distributions a t sample IocationA f o r case i at mid-plane ( a) a n d surface (b).

because of the transversely aligned and flow-aligned fibers in the core and shell regions, respectively, the gapwise-averaged fiber orientation in the plane of the part is closer to a random distribution. This average fiber orientation for SFRIM parts will of course

Orientation angle (degrees)

Ib) Fig. 9. Fiber orientation distributions at sample location R.for cme i a t mid-plane ( a ) a n d surface ( b).

depend upon the relative extent of the core and shell regions, which strongly depends upon the processing conditions, material properties, and part geometry. For gapwise-averaged fiber orientation close to a random distcbution in the 1-2 plane, the weighting function f ( Eq 6) has been found to perform better



Table 3. Orientation Distribution for Molding Case iii.

- Location A Location B Location C Location D Angle (“1 mid-plane surface mid-plane surface mid-plane surface mid-plane surface

5.0 0.059 0.147 0.026 0.447 0.012 0.225 0.372 0.875 15.0 0.062 0.189 0.000 0.308 0.020 0.174 0.072 0.331 25.0 0.070 0.279 0.004 0.173 0.047 0.168 0.036 0.162 35.0 0.051 0.285 0.039 0.100 0.051 0.106 0.028 0.104 45.0 0.070 0.186 0.060 0.065 0.071 0.147 0.036 0.062 55.0 0.048 0.114 0.099 0.059 0.146 0.092 0.040 0.068 65.0 0.093 0.138 0.172 0.029 0.1 70 0.076 0.048 0.026 75.0 0.070 0.135 0.237 0.035 0.217 0.065 0.020 0.023 85.0 0.138 0.099 0.422 0.041 0.375 0.087 0.040 0.01 3 95.0 0.135 0.063 0.500 0.029 0.565 0.071 0.032 0.01 0

105.0 0.247 0.090 0.543 0.029 0.470 0.087 0.072 0.01 9 11 5.0 0.309 0.114 0.444 0.065 0.292 0.073 0.068 0.052 125.0 0.393 0.102 0.203 0.053 0.182 0.106 0.140 0.052 135.0 0.379 0.132 0.065 0.073 0.099 0.160 0.176 0.052 145.0 0.31 5 0.189 0.026 0.150 0.059 0.258 0.320 0.084 155.0 0.199 0.225 0.01 7 0.282 0.040 0.367 0.468 0.165 165.0 0.152 0.213 0.004 0.447 0.028 0.350 0.568 0.347 175.0 0.076 0.168 0.004 0.479 0.020 0.252 0.328 0.421

,- .- 0

2 ”

C .- 0 E L

5 25 45 65 85 105 125 145 165


b l Fg. 10. Fiber orientation distributions at sample location C for case i at mid-plane ( a ) and surface ( b).

than f’ ( Eq 7 ) when the hybrid closure approximation is used in Eq 14. For such a n orientation distribution, we have found that the resulting predicted elastic moduli often lie outside the upper and lower bounds when the hybrid closure based on f is used

5 25 45 65 85 105 125 145 165


(b) Rg. 11. Fiber orientation distributions at sample location D for case i at mid-plane (a ) and surface (b).

in Eq 14. For a highly aligned average fiber orienta-A tion distribution, both weighting functions f and f have been found to give good results. Accordingly, even though the weighting function f’ has been used for approximating a4 in Eq 4 , the weighting func-


Fiber Orientation and Mechanical Properties of SFHM Composites

(C) (d) Rg. 12. PredictedJber orientation for case i at x3 / b = 0 ( a ) , 0.3 ( b ) . 0.6 ( c ) and ( d ) .

(C) (d) Fig. 13. Predicted jber orientation for cuse ii at x3 / b = 0 (a) , 0.3 (b) , 0.6 ( c ) and 0.9 (d).

tion _t? has been used to approximate a4 in Eq 14. Figures 20 and 21 show the tensile moduli ( E ) and the thermal-expansion coefficients (a), respectively, for the three molding cases. Experimental values of E and c1 in the axial and transverse directions have been determined by using the samples shown in Fig. 4. Since the shell region in all three cases is thicker than the core region, the tensile modulus in the axial direction is larger than the tensile modulus in the transverse direction (as expected, an opposite trend is noted for the thermal-expansion coefficient). In particular, the predicted values for E and a in Figs. 20 and 21 follow the trends observed in

the experimental results. In comparison to the tensile moduli, the predicted values of the thermal- expansion coefficients are in better agreement with the experimental results. The additional error in tensile moduli prediction may be due to the closure approximation in Eq 14.

CONCLUSIONS

Short fibers in thin injection-molded parts predominantly lie in the plane of the part. Near the surface, shear-dominated flow tends to align the fibers with the flow, whereas in-plane stretching flow causes the fibers at the mid-plane to align transverse



(d (d) Fig. 14. Predicted Jber orientation for case iii at x, / b = 0 ( a ) , 0.3 ( b ) , 0.6 (c) and 0.9 (d).

/ / / / /

x19 x2

Fig. 15. Schematic of cold thermai boundary layer ( dashed) in frow during injection molding. Effect of cold layer upon stream-wise velocity proJle has also been indicated.

I 1 1

(C) (d) Fig. 16. PredictedPber orientation a t t = t , / Z . f o r case ii at x , / b = 0 ( a ) , 0.3 ( b ) , 0.6 (c) and 0.9 (d).

to the flow. At the mid-plane, fiber orientation in injection-molded parts can be determined by a two- dimensional analysis: however, away from the mid- plane, small out-of-plane fiber orientation plays an important role in determining the fiber orientation, which necessitates a three-dimensional analysis of fiber orientation. Fiber-fiber interactions can be rnod- eled as a rotary diffusion, which tries to randomize the fiber orientation.

Even for a cavity of uniform thickness, a growing layer of solidified polymer gives rise to an effective gapwise-converging flow near the entrance, which competes with the in-plane stretching flow and tries to align the fibers with the flow. Near the melt front,



1 .o

0.8

0.6 - - ?-

- Core (predicted, r.a.i.c.) Shell (predicted, r.a.i.c.)

- Core (predicted, i.t.a.i.c.) - Shell (predicted, f.t.a.i.c.)

Core (experimental) Shell (experimental)

---------_..___._ - - ---- -____ /

\

0.0 1 I I I I I

2 4 6 8 1 0 1 2 1 4 Axial distance (cm)

Fig. 17. Variation oj- the a , , component of the orientation tensor along the axis oftheplaquejor case i. (r.a.i.c.: runner- aligned orientation as initial condition; Jt.a.i.c.: pow-aligned orientation at the sucface, transversely aligned orientation at the mid-plane, with linear gap-wise variation between).

1 .o

0 .8

0.6 - -

5

0.4

0.2

0.0

Shell (predicted, r.a.1.c.) . Core (predicted, f.1.a.i.c.) - Shell (predicted, f.t.a.1.c.)

Core (experimental) Shell (expermental)

t------.-.---------

I I I I I I 2 4 6 8 1 0 1 2 1 4

Axial &stance (cm)

Fig. 18. Variation of the a , , component of the orientation tensor along the axis of the plaquefor case ii.

a diminishing solid layer gives rise to a gapwise- diverging flow, tending to cause the fibers near the melt front to align transverse to the flow. The relative extent of the solid layer is larger for thinner cavities and increases as the injection speed is reduced. For thin parts molded at slower injection speeds, the gapwise-converging flow near the entrance may domi- nate over the in-plane stretching flow to give a flow- aligned orientation even to the fibers at the mid-plane of the cavity, whereas, near the melt front, the gapwise-diverging flow gives a transversely aligned orientation to the fibers in the mid-plane of such parts.

In the present study, Halpin-Tsai equations for the elastic moduli and results of Schapery for the thermal-expansion coefficients of unidirectional composites have been combined with the predicted orientation distribution across the thickness of the injection-molded parts to calculate the gapwise-

-Core (predicted, r a.1.c.) Shell (predicted, r a.1.c.)

- Core (predicted, f.t.a.i.c.) - Shell (predicted. f.t.a.i.c.)

1 .o Shell (experimental)

---_ _..__.____.-_-________ - - _ - - - - 0 .8 , - - - , '

0.6 L< * 1

0 .0 ' I I I I I I 2 4 6 a 1 0 1 2 1 4

Axial distance (cm)

Eg. 19. Variation of the a,, component OJ the orientation tensor along the axis of the plaque for case iii.

0.8

2 0.7

0.6

-

v

w

0

0 .5 0

0.4 . P I I I I I i

(1) (1 1) (1 I l i

Case number

F'g. 20. Gapwise-averaged tensile moduli in the axial and transverse directionsfor the three molding cases.

0 ad [experimental)

atd (experimental)

7.0 T-- 0 ead (predicted)

6 .0 1 .

2.0 1 I I t I I

(1) (1 i) (I i I)

Case numbei

Fig. 21. Gapwise-averaged thermal-expansion coeficients in the axial and transverse directions for the three molding cases.



n 7 *

averaged values of these properties in the axial and transverse directions. Experimentally measured fiber orientation, tensile moduli, and thermal-expansion coefficients are also reported in this paper, with reasonable comparison being obtained with the predicted values.

APPENDIX

Material properties of Rynite 530 (41, 42):

Density ( p ) 950 kg/m3 Specific heat (C p )

Thermal conductivity (K) 0.22W/(m.K) Shear viscosity (7) (Pa.4

2.0 X lo3 J/(Kg.K)

770 17=

qo = Bexp(T,/T) 1 + (qOj/7*)1-n

0.6006

B

For matrix (polyester): Young's modulus ( E m ) Poisson's ratio ( v,) Thermal-expansion

coefficient ( (Y ,) For fibers (glass): Young's modulus ( Ef) Poisson's ratio ( vJ) Thermal-expansion

coefficient ( af) Diameter ( d ) Aspect ratio (Z/d) Volume fraction ( 4,-)

T b

8.665 Pa (Note: Such a low value of T* means that, in effect, the viscosity follows the power- law model.) 2.317 x Kg/(m.s) 1.3026 X lo4 OK

2.79 X lo9 N/m2 0.42

7.24 x 10" N/m2 0.22

0.56 x ~ o - ~ / o K ] O X m 25 0.19

ACKNOWLEDGMENTS

We would like to thank E. I. du Pont de Nemours & Co., particularly Dr. A. E. Hirsch, for the micrographs of the fiber orientation and Mr. Swapnil Shah (Cornell) for his painstaking work of digitizing the micrographs. We are also grateful to Dr. C. A. Hieber (Cornell) and R& D group of A. C. Technology (Ithaca) for their helpful discussions throughout this work. The investigation was partially supported by A. C. Technology and has been camed out under the Cornell Injection Molding Program (CIMP) which is supported by the CIMP Industrial Consortium.

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