Fiber orientation in 3-D injection molded features: prediction and experiment

Transcription

1 Fiber orientation in 3-D injection molded features: prediction and experiment Brent E. VerWeyst, Charles L. Tucker III, Peter H. Foss, and John F. O Gara Department of Mechanical and Industrial Engineering, University of Illinois 1206 W. Green St., Urbana, IL Materials and Processes Lab, General Motors R&D and Planning Warren, MI Research and Development, Delphi Automotive Systems Warren, MI Submitted to International Polymer Processing June 18, 1999 Abstract We present a finite element method for predicting the fiber orientation patterns in 3-D injection molded features, and compare the predictions to experiments. The predictions solve the full balance equations of mass, momentum, and energy for a generalized Newtonian fluid. A second-order tensor is used to describe and calculate the local fiber orientation state. A standard Hele-Shaw molding filling simulation is used to provide inlet boundary conditions for the detailed finite element models, which are limited to the local geometry of each feature. The experiments use automated image analysis of polished cross-sections to determine fiber orientation as a function of position. Predictions compare well with experiments on a transverse rib, where the detailed calculation can be 2-D. Results of a 3-D calculation for a flow-direction rib also show generally good agreement with experiments. Some errors in this latter calculation are caused by not simulating the initial filling of the rib, due to computational limits. 1

2 1 Introduction When a short-fiber reinforced polymer is injection molded, the flow during mold filling creates patterns of preferential fiber orientation in the part. This leads to considerable point-to-point variation and anisotropy in the mechanical properties of the material. In order to anticipate the structural performance of the molded part (stiffness, strength, warpage, shrinkage), it is necessary to predict the fiber orientation pattern produced by the injection molding process. The capability of predicting fiber orientation has been available in many commercial injection molding software packages for several years [1, 2]. All of these programs use the generalized Hele-Shaw approximation [3] to calculate the velocity field in the mold cavity. These velocities are then used to determine the fiber orientation, using the scheme developed by Advani and Tucker [4] and Bay and Tucker [5]. A growing body of experimental studies show that, as long as the underlying assumptions of the Hele- Shaw approximation are satisfied, these fiber orientation predictions are quite accurate [6 10]. However, this approach also has severe limitations. The Hele-Shaw approximation assumes that the mold is locally a thin, flat cavity. Therefore, it does not provide accurate velocity fields in 3-D features, such as corners, ribs, gates, and bosses. For these and other similar geometries, a complete solution of the momentum equations is required. In this paper we present a model and numerical methods for predicting fiber orientation in 2-D and 3-D features in injection-molded parts, and we compare our predictions with experiments on parts molded from a typical fiber-reinforced molding compound. The main new feature of our model is that we solve the full equations for mass, momentum, and energy balance in their general form, using the actual geometry of the feature, rather than solving the Hele-Shaw equations on a simplified geometry. Similar calculations reported in the literature include steady state solutions for flow in an axisymmetric tube extrusion die [11] and in a planar contraction [12]. Transient calculations that include the motion of the free surface during mold filling have been performed by Ko and Youn [13] for a flat cavity geometry, and Vincent, Devilers, and Agassant [14] for a cylindrical runner joining to a thin disk-shaped cavity. Both of these calculations use a remeshing or mesh generation scheme to track the flow front, and so may be limited in the geometries they can treat. The only fully three-dimensional solutions for flow and fiber orientation to date were performed by Kabanemi et al. [15]. They used the Doi-Doraiswamy-Metzner model, which differs in detail from the fiber orientation models typically used in commercial software. The calculations reported here use the same underlying model for fiber orientation that has proved successful in simple injection-molded geometries. We use a fixed mesh with a volume-of-fluid (VOF) method to treat the moving flow front, which allows considerable geometric flexibility, and we use the same type of rheological models that are common in commercial injection molding simulations. The flow and heat transfer problems are coupled through a temperature-dependent viscosity and viscous dissipation. Our code can treat 2-D, axisymmetric, or 3-D geometries. Details of our numerical solution method are discussed elsewhere [16, 17], so here we summarize the main points of the model, and focus on comparison with experiments. It is seldom possible to simulate an entire injection-molded part using a full 3-D model, because of computational demands. However, it is feasible to simulate small segments of a part, i.e., features, in this way. Thus, we use a two-level approach, in which a standard Hele-Shaw simulation is used to model the entire part, and the results of this analysis provide the boundary conditions for more detailed 2-D and 3-D models of selected features. Our goal is to have a practical tool that accurately predicts fiber orientation patterns in molded features, and to use that information to determine the structural performance of the molded part. 2

3 ' 2 Governing equations 2.1 Fiber orientation The fiber orientation state at each point in the part is represented by a second-order orientation tensor [4], whose components are (1) Here, with Cartesian components, is a unit vector pointing along the axis of a single fiber, is the probability density function for fiber orientation, and the integral is taken over all directions. The orientation tensor can represent any type of orientation state, from perfectly aligned to random, and also describes the relative orientation between the fibers and the problem axes. We will also need the fourth-order orientation tensor, which is defined as The second-order orientation tensor is symmetric, and its trace equals unity. As a result, has only five independent components in a general 3-D problem. This greatly reduces the complexity of the calculation. The fibers are convected with the fluid, and they re-orient in response to the deformation and rotation of the fluid. We model these orientation dynamics using Here summation is implied on repeated indices, and! and " are the strain rate and vorticity tensors, (4) Equation (3) reflects the fact that the fibers orient according to the equation of Jeffery [18] for dilute suspensions, with an added rotary diffusivity # $ to model the randomizing effects of fiber-fiber interactions. is a constant that depends on fiber aspect ratio, and we use which is the limit for slender fibers. Following Folgar and Tucker [19] we set # $, where is the scalar magnitude of! and is a material constant called the interaction coefficient. is often determined by matching numerical predictions of fiber orientation to experiments for a simple shear flow [e.g., 9, 10]. 2.2 Closure approximation The presence of in Eqn. (3) represents a problem in obtaining a closed-form solution, since it contains unknown higher-order information. The standard approach is to approximate the fourth-order tensor in terms of the known second-order tensor using a closure approximation. The best available closures are the family of orthotropic closures [20] and the natural closure [21, 22]. The natural closure is based on an exact % relationship for the special case of &, while any orthotropic closure is fitted to a set of data pairs, which might be found by solving the detailed equation for. (2) (3) 3

4 A A ij DFC ORT ORF NAT ORL 0.0 A Gt Figure 1: Tensor components in simple shear flow for & ( & &. Data from Cintra and Tucker [20], except for the ORT curve. The ORT and NAT curves are indistinguishable in this figure. Comparisons by Cintra and Tucker [20] show that their fitted orthotropic closures are slightly more accurate than the natural closure for the typical range of & ( & to & ( & &. However, the orthotropic closures given by Cintra and Tucker tended to oscillate in simple shear flow for very small interaction coefficients, say ) & ( & &. To fit experimental data we typically need to use interaction coefficients in this range, so in this study we use a new orthotropic closure that alleviates this problem. This new closure was generated by Wetzel [23], using several improvements over the Cintra and Tucker procedures. In particular, the fitted functions were constrained to give exact results for important limiting cases of fiber orientation, such as fully aligned fibers or planar orientation states. The closure was then fitted to detailed data for &. Consequently, this closure should yield results which are very similar to those of the natural closure. Full details of this closure and its testing are given by VerWeyst [16]. An example of the performance of the new closure is shown in Fig. 1. Here the tensor components * * and * + *, + for selected closures are given as a function of time in a simple shear flow ( and + & - ), for & ( & &. The new closure (ORT) compares well to the reference results calculated by solving the detailed equation for, which are labelled DFC in the figure. The new closure is also almost identical to the natural closure, NAT. The remaining curves in the figure are the two orthotropic closures of Cintra and Tucker (ORF and ORL). The ORT closure is not quite as accurate as ORL in this example. However, it is better suited for numerical simulations because the ORL closure has a slight oscillation at large shear rates, which may interfere with the convergence of numerical solutions. 2.3 Balance equations In this work we use the general balance equations for mass, momentum, and energy. We assume that the fluid has constant density, specific heat, and conductivity, and we neglect body forces. With these assumptions the balance equations reduce to & (5) / 0 (6) (7) 4

5 Here is velocity,. is density, 1 2 is specific heat, 4 is conductivity, / is pressure, and 5 is extra stress. In principle the thermal conductivity could be anisotropic and depend on the fiber orientation state. This could be accommodated in the present numerical framework, but here we neglect this effect. 2.4 Stress constitutive equation For fibers suspended in a Newtonian fluid, the extra stress 5 depends on both the deformation rate! and the fiber orientation state [24, 25]. This leads to a two-way coupling between the solutions for velocity and orientation. Our numerical scheme can easily handle this situation, but our calculations suggest that this coupling has little effect on the flow pattern for most geometries typical of mold filling [16, 17]. In fact, the contraction flow studied by Lipscomb et al. [25] and its reverse (an expansion flow) are the only geometries we studied where the coupling had a large effect on the flow pattern. Accordingly, the present calculations neglect the effect of fibers on the stress, and use a generalized Newtonian fluid, as is typical of Hele-Shaw mold filling simulations. Specifically, we use 0 6 (8) where 6 6 ' 3. We choose a Cross-WLF model for the viscosity [26, 27], a model that is widely used for injection molding simulation: > 8 9?: < = * A B Here 0, C, and 6 are empirical constants. Temperature dependence is introduced through the function which takes the WLF form D E F 1 * 3 3 G G (10) In the above equation, 6, 1 *, 1 +, and 3 G are empirical constants. 2.5 Boundary conditions The detailed mesh for a typical feature has an inlet boundary along a surface in the interior of the mold cavity. On this surface one must specify the velocity, temperature, and fiber orientation as a function of position. Note that a full velocity profile must be specified, in contrast to a Hele-Shaw simulation where only the inlet pressure is needed. Alternately one can specify the traction, instead of the velocity, on this surface. On the walls of the mold we set the velocity to zero (no slip) and fix the temperature. The fiber orientation equation, Eqn. (3), is hyperbolic, and only requires a boundary condition on the inlet. At the flow front we assume that surface tension in negligible and impose zero surface traction. Heat conduction through this surface is neglected (zero heat flux), and again no boundary condition is needed for fiber orientation. (9) 3 Numerical solution methods 3.1 Velocity, pressure, and temperature We use the Galerkin finite element method to solve for velocity, pressure, and temperature, as well as for fiber orientation. The software is constructed on top of FIDAP [28], and uses the built-in equations 5

6 H H Q H S S I I I I - * - I I W S I I U I I W I W for velocity, pressure, and temperature. The continuity, momentum, and energy equations are discretized by standard procedures, using a mixed formulation in which pressure is interpolated one order lower than velocity and temperature. Details of this is well-known formulation can be found in Engelman [28] or Reddy [29], and will not be repeated here. 3.2 Fiber orientation To facilitate the formulation of the discrete fiber orientation equations, the components of the second-order orientation tensor are written as a column vector, such that * * * + The vector H contains only the five independent components of. A matrix I of fiber orientation basis functions is defined, such that J K L M N (12) ' P for each O. That is, K L M * N is the vector of nodal values of * *, K L M + N contains nodal values of + +, and so forth. We use the same basis functions for orientation as for velocity and temperature. It follows that the gradient of H is Q J K L M N (13) Next, a weighted residual for the fiber orientation equation (Eqn. (3)) is defined by applying Galerkin s method. After making the necessary substitutions and rearranging the result, the residual is K L R N K L M N S T K V M N Q K L M N X & U (14) where T is a vector of weighting functions, and W is a source term which represents all of the terms on the right-hand side of Eqn. (3). Also K V M N is the vector of nodal values for the O component of velocity, and U is the matrix of basis functions for velocity. Rearranging Eqn. (14) and choosing T I yields the discrete form of the fiber orientation equation. K L M N X K V M N Q X K L M N S X (15) A fully implicit time integration scheme is used for all analyses in this work. Discretizing the temporal derivative in Eqn. (15) using the backward Euler scheme yields K V M N Q X K L M N Y I I I U S K L A * M N X Y I I $ (16) (11) 6

7 where Y is the time increment, and K L A * M N $ is the known solution at the previous time step. Note that the source term W is evaluated from K L M N, the solution at the new time. The fiber orientation equation is hyperbolic, so the classical Galerkin solution is likely to exhibit nonphysical oscillations. We control these by adding streamline upwinding. Further details are given by Ver- Weyst [16] and Engelman [28]. We implement the orientation equation in FIDAP, taking advantage of the similarity between the fiber orientation equation and the mass transport equation, which is already present in the software. Setting the density to unity and the diffusivity to zero reduces this equation to 1 B 1 B R B (17) where 1 B is the species concentration and R B is the reaction rate of that species. Each independent component of the orientation tensor in Eqn. (11) is treated as a separate species, and the source terms are programmed to represent the right-hand side of Eqn. (3). The solution to the orientation equations at each time step is found using a full Newton-Raphson iteration. The different components of are strongly coupled through their source terms, so the equations for all components are formed into a single finite element equation set and iterated simultaneously. 3.3 Moving boundary The moving flow front is simulated using the volume of fluid (VOF) method [30]. A fill factor % is associated with each element, with % equal to the volume of fluid within the element divided by the volume of the element. % & indicates an empty element, and % a full element. The flow front is usually taken to be at the interpolated position corresponding to % & ( P. At each time step the fill factors are updated based on the flow rates crossing the element sides. This method imposes a limit on the time step, which essentially requires that the flow front move less than one element length per time step. For unfilled polymers, Michaeli et al. [31] have shown good agreement between experimental flow fronts and FIDAP VOF calculations during mold filling. Because Hele-Shaw simulations suppress 3-D details of the velocity field, they do not explicitly model the fountain flow in the vicinity of the flow front. In more sophisticated Hele-Shaw simulations, special treatments have been developed in an attempt to correctly predict the temperature and orientation fields in the fountain flow region [5, 32]. However, our calculations represent the entire 2-D or 3-D flow field explicitly, including the shape and position of the flow front. As a result, no other special treatment of the fountain flow region is required. 4 Experiments 4.1 Part geometry and molding conditions The part considered here is a rectangular plaque with eight ribs on its upper surface, shown in Fig. 2. The filling flow is from left to right. The 1, 2, and 3 directions will refer to the flow, crossflow, and gapwise directions, respectively. The structure on the left edge of the rib plaque is a runner and film gate. Downstream of the gate are five ribs of varying thickness oriented transverse to the flow, followed by three identical ribs oriented parallel to the flow. The base of the plaque is 254 mm long (excluding the gate), 76 mm wide, and 2.5 mm thick. The parts were molded from a polycarbonate resin (GE Plastics Lexan LS2) with thirty percent by weight glass fiber. Note that LS2 is sold as an unreinforced resin, and this material was specially com- 7

8 Figure 2: Photograph of the rib plaque. The two ribs indicated by X s were modeled. Figure 3: Finite element mesh for the transverse rib. pounded to include the glass fibers. The process conditions were: injection temperature 3 B P Z [ K, mold-wall temperature 3 \ ] ] ( [ Z K, and fill time ^ ( ] _ seconds. Even this simple part is too large to simulate in its entirety, given the available computational resources. Therefore, we chose to focus on one transverse rib and one flow-direction rib. The two ribs studied here are indicated in Fig. 2. These ribs are both 4 mm tall, measured from the top surface of the plaque, and 2 mm thick. The rib plaque is wide enough so that the side walls do not affect the behavior of the flow along the centerline. Hence, flow in the transverse rib can be treated as a two-dimensional problem in the 1-3 plane. The finite element mesh used to analyze the transverse rib is shown in Fig. 3. This mesh contains 1,903 nodes, and consists of four-noded quadrilateral elements. The mesh is 25.4 mm long in the flow direction, and begins 12.7 mm upstream of the centerline of the rib. The geometry of the flow-direction rib is truly three-dimensional, and a simplification to two dimensions would be inaccurate for this feature. Due to computational limitations, only the first third of the flowdirection rib was modeled. The finite element mesh for the flow-direction rib is shown in Fig. 4, and contains 3,841 nodes. Eight-noded brick elements were used. This mesh is 37.6 mm long in the flow direction and 12.7 mm wide. The inlet face, at the back left in Fig. 4, is 12.2 mm upstream of the front of the rib, leaving 15.4 mm of rib length that is simulated. We did not simulate the moving flow front in this rib; see the discussion of boundary conditions below. 4.2 Fiber orientation measurement Fiber orientation in the molded samples was measured by reflected light microscopy of polished sections, using an image analysis apparatus and algorithms developed by Clarke et al. [33] and Hine et al. [34]. The method is fully automated, and determines the three-dimensional orientation distribution function within finite-size areas. On the polished cross-section, each fiber appears as an ellipse. The apparatus relates the geometry of this 8

9 inlet wall symmetry plane wall symmetry plane outlet Figure 4: Finite element mesh for the flow-direction rib. The flow is from left to right. ellipse to the orientation of the fiber with respect to the sample axes. The surface of the sample is divided into a series of rectangular areas, or bins, and data for all of the fibers in a bin is averaged and reported as the fiber orientation (components of ) at the centroid of the bin. The averages must be computed using an orientation-dependent weighting factor, to correct for the fact that fibers of some orientations are more likely to appear on the section than others, but this correction is well understood [35]. Sampling error presents a more difficult problem, and care must be taken to ensure that each bin contains a sufficient number of fibers to obtain a representative average. Discretizing the surface into too many bins makes the measurements noisy, while using too few bins can average out important spatial variations of the orientation. When measuring the orientation in a plate [e.g., 8, 9], one can extend the sample areas in the flow direction to get more fibers in the sample, since orientation varies slowly in that direction. That approach is not available here, because we expect the orientation to vary with both directions over the cross-sections. As a result, our experimental data contain a fair amount of noise. For the transverse rib, a single cross-section in the 1-3 plane, taken through the center of the plaque, was analyzed. For the flow-direction rib two cross-sections transverse to the flow (in the 2-3 plane) were & ( analyzed, one ` a mm downstream of the front of the rib, and the other P a mm downstream. This latter position is beyond the end of the finite element mesh, but we compare the experiments here with the calculations 13 mm downstream of the front of the rib, because by this point the orientation changes very little in the flow direction. 4.3 Material properties and boundary conditions The Cross-WLF parameters for the polycarbonate molding compound appear in Table 1, along with the other physical properties of this material. For fiber orientation modeling we used an interaction coefficient of & ( & & P. This value was chosen to match experimental data for the * * tensor component in the shell region of a strip, molded from the same glass-filled polycarbonate resin, using the ORT closure for the calculations. Note that the value of obtained this way will vary according to the closure approximation that is used. A delicate numerical issue in this type of simulation is the calculation of viscosity at low temperatures. The polymer is treated as a fluid, which solidifies as it cools by acquiring a very large viscosity. The WLF equation, Eqn. (10), provides this behavior. However, as polymer approaches its glass transition temperature, the zero shear rate viscosity given by the WLF equation becomes very large, and the numerical calculations overflow. For most calculations, a variation in viscosity of four orders of magnitude is sufficient to distinguish between molten and frozen regions. Hence, in practice one must choose a no-flow temperature, below which the viscosity is very large, but independent of temperature. This temperature should be high enough to avoid numerical difficulties, but low enough to avoid altering the solution. For the transverse rib calculation, a no-flow temperature of 451 K was chosen, which gives a viscosity five orders of magnitude 9

10 P 6. 4 ` Table 1: Material properties and viscosity model parameters for polycarbonate with 30% by weight glass fiber (GE Plastics Lexan LS2). Parameter Value mb kg 1840 J kgc K 0.24 W mc K 0 ] ( [ Z d & e C * + ( & d & 1 * G K K Pa Pa J s higher than at the injection temperature. For the flow-direction rib calculations, 462 K was chosen, which allows the viscosity to change by four orders of magnitude. These no-flow temperatures were the minimum values for which the solutions converged. To determine the velocity and temperature boundary conditions, a mold filling analysis was performed on the entire rib plaque using C-MOLD, a commercial injection molding software package based on the Hele-Shaw approximation. The inlet locations for the two meshes were chosen halfway between each rib and its upstream neighbor. This places the inlet in a narrow rectangular channel, a geometry for which the Hele-Shaw approximation gives excellent results. The C-MOLD velocity and temperature solutions for each location were then averaged in time, at each node across the thickness, to obtain boundary conditions for the detailed feature calculations. Time averaging began when the flow front reached the given location ( ( [ ` s for the transverse rib and ( & s for the flow rib), and continued until the plaque was completely filled. Full details of the averaged boundary conditions are given by VerWeyst [16]. The inlet boundary conditions for fiber orientation could also have been obtained from the C-MOLD results, and this would be the normal approach in practice. However, we chose here to use experimentally measured orientations. When comparing the calculations to the experiments, this eliminates the possibility of disagreement due to inaccuracies in the Hele-Shaw orientation results. The inlet orientation values were obtained using the measurement method described above, on cross-sections in the 1-3 plane at the actual inlet locations. Figure 5 shows selected components of the orientation boundary condition at the inlet to the transverse rib. The orientation at the inlet to the flow-direction rib is similar. The fiber orientation at both locations shows a clear shell-core structure, which is typical of injection-molded parts. Note that the data in Fig. 5 does not seem to be affected by the ribs upstream of that position. This supports our approach of analyzing features separately, connected to the global geometry only through the Hele-Shaw simulation results. For the transverse rib we performed a complete filling analysis, starting from a nearly empty mesh. In the real part the flow in this region continues after the meshed region has filled. To model this part of the flow the mesh is first filled, then the boundary condition at the exit is changed to allow fluid to exit there, 10

11 - f A ij 0.4 A 11 A 33 A ^ x 3 Figure 5: Experimental data for the variation of selected fiber orientation components across the thickness, 12.7 mm upstream of the center of the transverse rib. g - h i, where i is the half-thickness of the plaque. and the calculation is restarted, and allowed to continue for an additional one second. For the flow-direction rib, insufficient computing resources were available to solve the complete filling analysis. Instead, we first determined a steady-state solution for the non-isothermal, non-newtonian flow, with the mesh completely filled. Note that this solution is independent of the fiber orientation results, since we assumed that rheology is independent of fiber orientation. Then the velocities from this solution were used to calculate a steady-state fiber orientation solution. This required integrating the fiber orientation calculation in time, starting from a 3-D random orientation everywhere, until a steady state was reached. This is not the best way to model the rib, and we chose this approach solely because of computational limitations. In general one would expect the orientation results computed this way to be accurate near the center of the channel wherever the velocity is significant. However, the results will probably be inaccurate near the mold surfaces, where the material arrives through the fountain flow, and in stagnant regions, which experience most of their deformation during the filling stage. 4.4 Results: transverse rib The calculated location of the flow front at selected times during the filling of the transverse rib is shown in Fig. 6. At & the rib is empty, except for a small amount of fluid (2.5 mm ) required to initiate the VOF calculation. The shape of the flow front as it advances towards the rib is approximately semicircular. Figures 6 (e) and 6 (f) show that the rib does not begin to fill until after the flow front reaches the downstream side of the rib. The slight irregularities in the flow fronts are due to discretization errors in the VOF method. The flow front within the rib advances slightly faster along the downstream face. The main channel continues to fill, even as the rib is filling. After the rib has filled the flow front advances more quickly in the main channel. The geometry is completely filled at & ( ` seconds. Figure 7 shows the fluid streamlines at & ( ` seconds. Most of the flow occurs near the midplane of the main channel, because of the frozen layers near the walls. The streamlines show a slight hump just below the rib. Figure 7 also indicates that fluid motion within the rib is negligible after the rib has filled. Temperature contours in the transverse rib at & ( ` seconds are shown in Fig. 8. This time corresponds to Fig. 6(l), when the finite element mesh for the rib is completely filled. At this time the interior of both the rib and the main plaque are still quite hot, with thin layers of cool fluid coating the wall. Note that the temperature contour is stretched downstream, and is beginning to cover the entrance to the rib. This is polymer that cools by being close to the upper mold wall, and is then advected downstream, across the entrance to the rib. This phenomenon persists as mold filling continues, and eventually a layer of cool 11

12 (a) j k l seconds. (b) j k l m l n o seconds. (c) j k l m l p q seconds. (d) j k l m l q r seconds. (e) j k l m l s q seconds. (f) j k l m l t q seconds. (g) j k l m u l n seconds. (h) j k l m u n l seconds. (i) j k l m u o o seconds. (j) j k l m u p t seconds. (k) j k l m u q o seconds. (l) j k l m u s o seconds. Figure 6: Movement of the flow front during the filling of the transverse rib. 12

13 polymer covers the rib entrance, even though the central part of the rib is still quite hot. This phenomenon has not been reported before, and may have interesting implications in terms of rib shrinkage and sink mark formation. Figure 9 shows an area scan of the fiber orientation in the transverse rib, as acquired by the image analyzer. Each fiber appears as an ellipse in this image, with nearly circular ellipses being fibers that are normal to the image plane, and highly elongated ellipses being fibers that lie nearly in the plane of the image. A narrow core of random orientation is present in the main channel beneath the rib, surrounded by shell layers where the fibers are aligned with the flow. This is a typical orientation for a flat plate, and is consistent with Fig. 5. Beneath the rib, the core is displaced slightly upward. Figure 9 also shows that the fibers are aligned parallel to the walls, at the corners leading into and out of the rib. In addition, the fibers are more strongly aligned in the 3 direction along the downstream (right) face of the rib than on the upstream face. Figure 10 shows the predicted fiber orientation. In this figure, the magnitude and direction of each vector is determined by the largest eigenvalue of and its associated eigenvector. That is, each line in Fig. 10 represents the main direction of orientation at that position, with long lines indicating highly aligned fibers and short lines indicating nearly random fibers. The same features are present in this prediction as were found in Fig. 9. Quantitative fiber orientation data for the * * and - - tensor components are shown in Figs. 11 and 13, respectively. Note that * * measures the degree of alignment with the 1 direction, while - - measures alignment in the 3 direction. These plots are colored using linear interpolations of the bin data. The purple areas in the experimental figures are outside of the part. In Fig. 11, the fibers are highly aligned in the 1 direction in the main part of the plaque. A very narrow core is also present. The core appears narrower than in Fig. 9, because it is spanned by only one bin. Figure 11 also indicates that the fibers at the top and bottom of the rib are aligned in the 1 direction. Figure 13 shows that the fibers are highly aligned in the 3 direction along both sides of the rib. Also note the narrow region at the walls, on either side of the rib, where the fibers are not aligned in the 3 direction. Finally, the fiber orientation is asymmetric about the centerline of the rib. Figures 12 and 14 show the predictions for the * * and - - tensor components, respectively. In general the predictions are a good match to the experiments. The prediction for * * in Fig. 12 shows an upward shift in the core directly beneath the rib. Figure 12 also indicates an aligned region at the top of the rib, though not to the same extent as in the experimental data. This is probably due to small errors in the fountain flow introduced by the VOF method, since the fiber orientation at the top of the rib is determined by the fountain flow as the rib is filled. The model predicts that the fibers align vertically (large value of - - ) on both faces of the rib (Fig. 14). The prediction shows the same vertically aligned regions as in Fig. 13. The model also predicts a narrow band of fibers not oriented in the 3 direction near the wall, which also appears in the experimental data. In addition, the model predicts a slightly larger aligned region on the upstream side of the rib than the experimental data. 4.5 Results: flow-direction rib Fiber orientation results near the beginning of the flow-direction rib are shown in Figs Note that these figures display cross sections in the 2-3 plane, which is perpendicular to the main direction of flow. The finite element results show only one half of the rib, and the right-hand edge of theses plots should be matched with the centerlines of the experimental plots. The experiments in Fig. 15 show that the fibers in the main part of the plaque are mostly aligned in the 1 direction, as are fibers in the tip of the rib. The calculations in Fig. 16 show the same behavior in the main plaque, but the prediction misses the alignment in the rib tip. In Fig. 17 we see that the fibers in the rib have 13

14 Figure 7: Streamlines for the transverse rib at & ( ` seconds. The flow is from left to right. A E-01 B E+00 C E+00 D E+00 E E+00 F E+00 G E+00 H E+00 B D F H G E C A Figure 8: Finite element solution for the temperature in the transverse rib at & ( ` seconds. The temperature variable in this plot has been non-dimensionalized: a value of 1 indicates the injection temperature and a value of 0 indicates the wall temperature. 14

15 Figure 9: Scanned image of transverse rib. Figure 10: Fiber orientation vectors for the transverse rib at ( ` seconds. The magnitude and direction of each vector is determined by the largest eigenvalue of and its associated eigenvector. Compare to Fig

16 v wv v wx v wy v wz v w{ Figure 11: Measured * * tensor component for the transverse rib. The 1 direction is horizontal and the 3 direction is vertical..836e e e e e e e E-01 * * ( ` Figure 12: Finite element solution for the tensor component in the transverse rib at seconds. Compare to experiments in Fig

17 v wv v wx v wy v wz v w{ Figure 13: Measured - - tensor component for the transverse rib. The 1 direction is horizontal and the 3 direction is vertical..892e e e e e e e e ( ` Figure 14: Finite element solution for the tensor component in the transverse rib at seconds. Compare to experiments in Fig

18 considerable alignment in the 3 direction at this station, but the calculation in Fig. 18 does not completely reproduce this. These differences may be better understood by looking at Fig. 19(a), which shows the streamlines on the central symmetry plane of the rib (the front face in Fig. 4). Here we see that the top front corner of the rib is stagnant once the rib has been filled, similar to the tip of the transverse rib. We expect that the material in this area experiences most of its deformation during the filling process, and sees very little deformation during the steady-state flow used in this analysis. Thus, the orientation in the front of the rib is not accurately modeled by our calculation. The left side of Fig. 16 shows that the plaque to the side of the rib retains its classical core/shell structure, but the core disappears directly underneath the rib. The experiments in Fig. 15 contain a hint of this same effect, just to the right of center, but are too noisy to be definitive. The mechanism by which the core disappears underneath the rib is apparent in Fig. 19(b), which shows * * on the central symmetry plane. Comparing this figure to the streamlines, we see that upstream of the rib the core lies on the midplane of the plaque, but underneath the rib the fibers in the core are carried upwards away from the midplane, and gradually become oriented by shearing, while the midplane becomes occupied by fibers aligned in the flow direction, which were formerly in the lower shell layer in the plaque. Figure 19(b) suggests that along most of the rib the fibers are strongly oriented in the flow direction, and this is confirmed in the cross-sections in Figs Figure 20 shows clearly that the core underneath the rib has disappeared by this point, which matches the calculation in Fig. 21. However, in Fig. 22 we see some 3-direction orientation close to the side walls of the rib, which is not present in the calculations in Fig. 23. This is probably a fountain flow effect. As the rib fills there is a fountain flow that rolls material from the center of the rib to the surfaces. Since we did not do a filling calculation for this rib, our calculation misses this effect. However, the orientation away from the walls is predicted quite well. 5 Conclusions The predictions for fiber orientation compare quite well with the experiments, and suggest that the model is accurate. Both calculations and experiments display effects that are not represented in Hele-Shaw models of injection molding. Clearly, detailed 2-D and 3-D calculations of the type presented here can be used to represent the flow, heat transfer, and fiber orientation near 3-D molded features. The main simplifications of the model, that viscoelasticity and the coupling of fiber orientation to rheology are neglected, do not appear to affect the results for fiber orientation. Our general approach of analyzing the mold globally using a Hele-Shaw model, and then analyzing features locally using 2-D or 3-D models appears to work well. However, it is essential to model the transient filling of the feature, since there may be little or no flow in parts of the feature after it has been filled. It is also advisable to model the flow past the feature after it has been filled, since there may be significant changes in the temperature and orientation during this period. Measuring the complex 3-D fiber orientation patterns in and around a molded feature is not easy. The automated image analysis method used here works well, but the comparison of experiments to predictions suffers from noise due to sampling errors. The best way to alleviate this problem would be to mold many parts, polish the same cross-section on each part, analyze each section, and then combine the data for all the sections. Since the mathematical model of the process appears to be sound, it is worth considering issues of numerical efficiency. The numerical methods used in this work were chosen for accuracy and stability, and this limited the size of the problem we could attack. To make the treatment of 3-D features with transient filling a practical task, the solution needs to be made more efficient. Some possibilities are to use the 18

19 v wv v wx v wy v wz v w{ Figure 15: Measured * * tensor component in the 2-3 plane at & ( & ( ` a mm from the leading edge of the flow-direction rib. The 2 direction is horizontal and the 3 direction is vertical..852e e e e e e E E+00 Figure 16: Finite element solution for the * * tensor component in the 2-3 plane at 0.6 mm from the leading edge of the flow-direction rib. Compare to experiments in Fig

20 v wv v wx v wy v wz v w{ Figure 17: Measured - - tensor component in the 2-3 plane at & ( & ( ` a mm from the leading edge of the flow-direction rib. The 2 direction is horizontal and the 3 direction is vertical..889e e e e e e e e+00 Figure 18: Finite element solution for the - - tensor component in the 2-3 plane at 0.6 mm from the leading edge of the flow-direction rib. Compare to experiments in Fig

21 (a) Streamlines.852E E E E E E E E+00 (b) Contours of } } Figure 19: Streamlines and flow-direction orientation along the symmetry plane of the flow-direction rib. The 1 direction is horizontal and the 3 direction is vertical. discontinuous Galerkin method to discretize the fiber orientation equations, and to parallelize the solution process. This deserves attention in future work. Acknowledgements Financial support to the University of Illinois was provided by General Electric Company and General Motors Corporation. The work of Peter Foss and John O Gara was sponsored by General Motors Corporation as part of the Thermoplastic Engineering Design (TED) Venture with General Electric. The TED Venture is a Department of Commerce Advanced Technical Program administered by the National Institute of Standards and Technology. The FIDAP finite element software was made available by Fluent, Incorporated. References [1] E. Henry, S. Kjeldsen, and P. Kennedy. Fiber orientation and the mechanical properties of SFRP parts. SPE Tech. Papers, 40: , [2] C. S. Randall and H. H. Chiang. Applications of fiber orientation analysis in injection molding of fiber-filled composites. SPE Tech. Papers, 40: , [3] C. A. Hieber and S. F. Shen. A finite-element/finite-difference simulation of the injection-molding filling process. J. Non-Newtonian Fluid Mech., 7:1 32,

22 ~ ~ ~ ~ ~ ~ ƒ Figure 20: Measured * * tensor component in the 2-3 plane at P a mm from the leading edge of the flow-direction rib. The 2 direction is horizontal and the 3 direction is vertical..852e e e e e e E E+00 Figure 21: Finite element solution for the * * tensor component in the 2-3 plane at 13 mm from the leading edge of the flow-direction rib. Compare to experiments in Fig

23 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Figure 22: Measured - - tensor component in the 2-3 plane at P a mm from the leading edge of the flow-direction rib. The 2 direction is horizontal and the 3 direction is vertical..889e e e e e e e e+00 Figure 23: Finite element solution for the - - tensor component in the 2-3 plane at 13 mm from the leading edge of the flow-direction rib. Compare to experiments in Fig

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