Abstract. 1 Introduction

Transcription

1 Computational simulation methods for molding of composite materials K.A. Olivero, C.F. Khoo, M.C. Altan School ofaerospace and Mechanical Engineering, University of Oklahoma, Norman, OK 73019, USA Abstract Currently used methods for the numerical simulation of the molding of shortfiber-reinforced composite materials are presented. Model formulations for both dilute and non-dilutefibersuspensions are discussed. Previous studies show that for dilute fiber suspensions, the orientation formation in a geometrically complex, three-dimensional, and time dependent flow can be obtained efficiently and without approximations in the governing equations. For such systems, it is possible to decouple the principal orientation directions and the degree of alignments with respect to the principal orientation directions. Thus, one can solve for principal orientation directions without obtaining degree of alignments. The flow induced orientation field in the vicinity of an inlet gate placed near a mold wall is obtained by using dilute suspension models. Orientation results are presented at different layers through the thickness, depicting the skin-core orientation structure and mold wall effects. 1 Introduction Injection molding offiber-reinforcedpolymers is one of the commonly used fabrication methods for thin-walled, geometrically complex, small to medium sized parts. Molded parts have high strength and stiffness properties and can be manufactured at high production rates. During injection mold filling, fibers, suspended in a high viscosity polymer, are translated and rotated, thus forming a so-called flow induced orientation structure. In order to predict mechanical and physical properties of the

2 68 Computer Methods in Composite Materials molded part, it is essential to fully characterize this flow induced orientation structure. In the last two decades, a variety of numerical methods have been developed to model orientation formation in complex mold cavities. All models start from Jeffery's theory [1] for the rotation of a single ellipsoidal particle immersed in Stokes flow. The assumptions involved in Jeffery's theory are listed in a number of articles [2,3]. One of the limitations of Jeffery's theory is the absence of hydrodynamic interactions between fibers. Such hydrodynamic interactions may become important in molding processes where fiber content usually ranges from 10 to 25%. To account for fiber to fiber interactions, Jeffery's theory has been modified to include a rotational diffusion term similar to the one used for microscopic Brownian particles. This method was initially proposed by Folgar and Tucker [4] based on their experimental observations. Subsequent theoretical studies attempted to model fiber to fiber interactions in non-dilute suspensions by including hydrodynamic and mechanical contact forces using slender body theory [5,6]. The orientation of a single fiber is represented by a unit vector aligned along its logitudinal axis. Jeffery's theory can be used to obtain time dependent rotation of this vector in an externally imposed velocity gradient field. In molding simulations, the velocity gradient is spatially non-uniform; hence, the fibers are subjected to varying velocity gradient along their path. To predict macroscopic anisotropic properties of molded parts, statistical distribution of orientation angles needs to be quantified throughout the mold cavity. This can be achieved by the use of an orientation probability density function or its moments, which are referred to as orientation tensors. The utility of orientation tensors is demonstrated by Advani and Tucker [7]. Currently, orientation tensors are used to characterize orientation microstructure in most injection molding simulations of composite materials [8-20]. All of these methods involve the use of closure approximations for higher order orientation tensors, except those published by Altan and coworkers [16-18]. Rao and Altan obtained closed form solution for the orientation field in a centergated disk [16] where very good agreement is obtained with the experimentally measured orientation tensor components [21] in a molded disk. The orientation field obtained by Rao and Altan was for dilute systems without any hydrodynamic interaction among fibers. The results of Altan and Rao demonstrate the advantages of solving governing orientation equations for dilute systems without closure approximations. Subsequently, Rao [22] developed a solution method

3 Computer Methods in Composite Materials 69 for orientation structure for dilute suspension flows applicable to complex mold geometries. In this study, we briefly discuss this method and present orientation results obtained around an inlet gate placed near a mold wall. 2 Solution method for flow-inducedorientation The rotation of a single ellipsoidal particle immersed in a visocus fluid is described by Jeffery's theory [1]. For a homogeneous flow where the velocity gradient tensor, du^ I ckj, is spatially uniform, the governing equation for the rotation of a spheroidal particle (i.e., ellipsoid of revolution) is given by, In eqn. (1), />,. is the time rate of change of the unit orientation vector, p., Qy, and Dy are the vorticity and strain rate tensors, respectively; and A is a parameter dependent on the particle aspect ratio, a^ as ^4-1. (2) For fiber suspensions containing numerous particles, the orientation probability density function (OPDF), Y(p, f), can be used to characterize the mechanical and rheological properties of the suspension. For a given velocity field, u(x, f), the velocity gradient tensor and the particle aspect ratio are sufficient to construct the OPDF. where A,-, is the inverse of the particle rotation tensor, Ey, defined as, de,. (4) By using eqns. (3) and (4), it can be shown that the inverse of the particle rotation tensor A,y needs to satisfy,,a',, = 0, (5) A',, = 0, (6) A',, = 0, (7) in the primed coordinates specified by base vectors of the maximum orientation directions [22]. The solution of eqns. (5-7) represents the principal orientation directions at a particular point within the mold cavity. In the principal orientation coordinates, the second-order moment, Sy, of the OPDF takes the following form,

4 70 Computer Methods in Composite Materials 0 o o 0 0^33 As a result of having all nondiagonal 6\ components zero, it is possible to obtain closed-form expressions of 6\\, S^> &nd S'^ in this coordinate system. After all diagonal s'y values are calculated, it is straightfoward to transform s'y to the fixed, global coordinates*,^, andz. (8) 3 Numerical results In most molded composite materials, a complex three-dimensional orientation structure is observed near the inlet gate. Inlet gates are designed to have a relatively small entry diameter, thus leading to a planar decelerating flow (i.e., on the x-y plane). In addition to the planar velocity gradients, considerable shear rates through the cavity thickness (in the z-direction) try to align fibers in the local flow direction. The detailed orientation field around an injection gate located near a mold wall has not been studied. Depending on the relative magnitude of gap-thickness and the distance between the wall and the inlet gate, a complex three-dimensional microstructure is formed. The flow field for this case is obtained using a steady Hele-Shaw flow formulation. The planar flow kinematics is calculated by using a source placed at a unit distance from the mold wall with a mold thickness of The presence of the wall is modeled by the superposition of two sources. Thus, analytical expressions for the stream function, velocity components, u and v, and velocity gradient tensor, du^ I faj, are obtained. The streamlines near the inlet gate are shown in fig. 1. The variation of this velocity field through the cavity thickness is obtained by assuming a parabolic profile through the thickness. The velocity components u and v obtained from the Hele-Shaw formulation are dependent on all three coordinates, resulting in six non-zero velocity gradient tensor components. Consequently, it is possible to investigate the threedimensional orientation structure at different layers through the cavity thickness. After the flow paths are determined, the three-dimensional orientation field is solved along the flow pathlines as described in section 2. The components of second-order orientation tensor, Sy, are calculated and graphically depicted by using orientation ellipsoids. The planar projections of these three-dimensional ellipsoids are used to represent

5 Computer Methods in Composite Materials 71 planar orientation at different layers. A perfect circle indicates random orientation distribution (i.e. isotropic orientation), whereas a line segment indicates full alignment. Due to flow symmetry, only half of the mold is shown in the figures depicting orientation field. Figures 2-4 illustrate planar orientations at different layers through the thickness represented by z=0 (midplane), 0.4, and 0.8. In each figure, orientation ellipses are shown along seven streamlines originating from the inlet gate. An isotropic orientation is assumed at the inlet specified by Sn=S22=S^=l/3. Other inlet conditions can be specified at the inlet gate to investigate dependence of flowinduced orientation on the gate design. Olivero et al. [23] conducted a similar study to determine the effect of inlet orientation conditions in a planar mold containing a three to one sudden contraction. Inlet orientation conditions are found to have considerable effects only within the immediate neighborhood of the inlet gate. At large distances, induced orientation remained unaffected by the entry orientation. On the midplane, shear rates through the thickness vanish due to flow symmetry. Thus, the only nonzero velocity gradients are primarily tangentially extensional (i.e., radially decelerating). The radial deceleration causes fibers to align perpendicular to the flow, as seen in fig. 2. Higher shear rates through the thickness at z=0.4 and 0.8 induce alignment closer to flow direction as depicted in figs. 3 and 4, respectively. Those fibers approaching the mold wall on the symmetry line are found to have very strong alignment perpendicular to the flow at all layers through the thickness due to the very high radial deceleration in this small region. Fibers moving away from the symmetry line are found to have strong alignment perpendicular to the local flow at the midplane. At z=0.4 and 0.8, a preferred orientation near the local flow direction is observed at large distances from the inlet gate. The deviation of the preferred orientation from the local flow at these layers is induced by the effect of directionally varying planar shear along the pathlines combined with shear through the thickness. Figs. 2-4 indicate fibers at all layers form a steady orientation structure away from the gate which remains nonuniform through the thickness. 4 Concluding remarks A numerical method to calculate the three-dimensional orientation structure in a complex mold geometry is presented for the flow around an inlet gate located near a mold wall. Planar orientation results are shown at three layers through the mold thickness. The flow field for this

6 72 Computer Methods in Composite Materials case is calculated using the superposition of two point sources and the Hele-Shaw flow model. Fibers are found to align perpendicular to the local flow at the mold midplane, and near the local flow direction at other layers. At large distances from the inlet gate, a steady orientation structure is formed which is nonuniform through the mold thickness. References [1] Jeffery, G.B., The motion of ellipsoidal particles immersed in a viscousfluid,proc. Royal Soc., A102, pp , [2] Altan, M.C., A review offiber-reinforcedinjection molding: Flow kinematics and particle orientation, J. Thermoplastic Composite Mails., 3,pp , [3] He, J., Olivero, K.A. & Altan, M.C., Orientation formation in planar mold filling: Theory and numerical predictions, MD-79, ASME, eds. H.P. Wang, L.-S. Turng, & J.-M. Marchal, Dallas, pp , [4] Folgar, F. & Tucker, C.L, Orientation behavior of fibers in concentrated suspensions, J. Rein. Plastic Comp., 3, pp , [5] Mani, R. & Koch, D.L., The effect of hydrodynamic interactions on the orientation distribution in a fiber suspension subject to simple shear flow, Phys. Fluids,7(3), pp , [6] Sundararajakumar, R.R & Koch, D.L, Structure and properties of sheared fiber suspensions with mechanical contacts, J. Non- Newtonian Fluid Mech., 73, pp , [7] Advani, S.G. & Tucker, C.L., The use of tensors to describe and predict fiber orientation in short fiber composites, J. Rheol, 31, pp , [8] Altan, M.C., Subbiah, S., Guferi, S.I. & Pipes, R.B., Numerical prediction of three-dimensional fiber orientation in Hele-Shaw flows, Polym. Eng. ScL, 30,pp , [9] defrahan, H.H., Verleye, V., Dupret, F. & Crochet, M.J., Numerical prediction of fiber orientation in injection molding, Polymer Eng. Sci., 32, pp , [10] Bay, R.S. & Tucker, C.L., Fiber orientation in simple injection moldings. Part I: Theory and numerical methods, Polym. Comp., 13, pp , [11] Gupta, M. & Wang, K.K., Fiber orientation and mechanical properties of short-fiber-reinforced injection-molded composites:

7 Computer Methods in Composite Materials 73 Simulated and experimental results, Polym. Comp., 14, pp , [12] Friedrichs, B., Gugeri, S.I., Subbiah, S. & Altan, M.C., A numerical approach to short fiber reinforced reaction injection molding (RRIM), J. Mails. Processing & Manufacturing ScL, 1, pp , [13] Matsuoka, T., Takabatake, J., Inque, Y. & Takahashis, H., Prediction of fiber orientation in injection molded parts of shortfiber-reinforced thermoplastics, Polym. Eng. Sci., 30, pp , [14] Chung, S.T. & Kwon, T.H., Coupled analysis of injection molding filling and fiber orientation, including in-plane velocity gradient effect, Polym. Comp., 17(6), pp , [15] Chung, S.T. & Kwon, T.H., Numerical simulation of fiber orientation in injection molding of short-fiber-reinforced thermoplastics, Polym. Eng. ScL, 35(7),pp , [16] Rao, B.N. & Altan, M.C., Closed-form solution for the orientation field in a center-gated disk, J. Rheol, 39(3), pp , [17] Rao, B.N. & Altan, M.C., Three-dimensional orientation structure in a radially diverging suspension flow between parallel plates, FED- 231; MD-66, ASME, eds. D.A. Siginer, & H.P. Wang, pp , [18] Olivero, K.A., He, J. & Altan, M.C., Orientation formation in planar mold filling: Experimental results, MD-79, ASME, eds. H.P. Wang, L.-S. Turng, & J.-M. Marchal, Dallas, pp , [19] Dupret, F., Verley, V. & Languillier, B., Numerical prediction of the molding of composite parts, FED-243; MD-78, ASME, eds. S.G. Advani & D.A. Siginer, pp , [20] Vincent, M., Devilers, E. & Agassant, J.-F., Fiber orientation calculation in injection moulding of reinforced thermoplastics, J. Non-Newtonian Fluid Mech., 73,pp , [21] Bay, R.S. & Tucker, C.L., Fiber orientation in simple injection moldings. Part II: Experimental results, Polym. Comp., 13, pp , [22] Rao, B.N., Rheology and flow of fiber suspensions in composite materials manufacturing, Ph.D. Thesis, University of Oklahoma, Norman, OK, [23] Olivero, K.A, Khoo, C.F. & Altan, M.C, Numerical prediction of flow induced orientation structure in injection molding, to be published in the Proc. of the 14* Annual Meeting of the Polymer Processing Society, Yokohama, Japan, 1998.

8 74 Computer Methods in Composite Materials Figure 1: Streamlines of a source next to a mold wall. Figure 2: Orientation structure at the midplane (i.e., z=0) formed by the inlet gate near a mold wall.

9 Computer Methods in Composite Materials 75 Figure 3: Orientation structure at 40% above the midplane (i.e., z=0.4) formed by the inlet gate near a mold wall. Figure 4: Orientation structure at 80% above the midplane (i.e., z=0.8) formed by the inlet gate near a mold wall.